Introduction to Mathematical PhilosophyIntroduction to Mathematical Philosophy by Bertrand Russell Originally published by George Allen & Unwin, Ltd., London. May 1919. Online Corrected

Introduction toMathematicalPhilosophy

by

Bertrand Russell

Originally published byGeorge Allen & Unwin, Ltd., London.May . Online Corrected Edition

version . (February , ), based onthe “second edition” (second printing) of

April , incorporating additionalcorrections, marked in green.

iii

[Russell’s blurb from the orig-inal dustcover:]

This book is intended forthose who have no previousacquaintance with the top-ics of which it treats, and nomore knowledge of mathe-matics than can be acquiredat a primary school or even atEton. It sets forth in elemen-tary form the logical defini-tion of number, the analysis ofthe notion of order, the mod-ern doctrine of the infinite,

iv

and the theory of descriptionsand classes as symbolic fic-tions. The more controversialand uncertain aspects of thesubject are subordinated tothose which can by now beregarded as acquired scien-tific knowledge. These areexplained without the use ofsymbols, but in such a wayas to give readers a generalunderstanding of the methodsand purposes of mathematicallogic, which, it is hoped, willbe of interest not only to thosewho wish to proceed to a more

v

serious study of the subject,but also to that wider circlewho feel a desire to know thebearings of this importantmodern science.

CONTENTS

Contents . . . . . . . . . viPreface . . . . . . . . . . xEditor’s Note . . . . . . . xvii

I. The Series of Nat-ural Numbers . . .

II. Definition of Num-ber . . . . . . . . .

CONTENTS vii

III. Finitude and Math-ematical Induction

IV. The Definition ofOrder . . . . . . .

V. Kinds of Relations VI. Similarity of Re-

lations . . . . . . . VII. Rational, Real,

and Complex Num-bers . . . . . . . . .

VIII. Infinite CardinalNumbers . . . . . .

IX. Infinite Series andOrdinals . . . . . .

CONTENTS viii

X. Limits and Conti-nuity . . . . . . . .

XI. Limits and Conti-nuity of Functions

XII. Selections andthe MultiplicativeAxiom . . . . . . .

XIII. The Axiom of In-finity and LogicalTypes . . . . . . . .

XIV. Incompatibilityand the Theory ofDeduction . . . . .

XV. Propositional Func-tions . . . . . . . .

XVI. Descriptions . . . . XVII. Classes . . . . . . . XVIII.Mathematics and

Logic . . . . . . . . Index . . . . . . . . . . . Appendix: Changes to

Online Edition . .

PREFACE

This book is intended essen-tially as an “Introduction,”and does not aim at giving anexhaustive discussion of theproblems with which it deals.It seemed desirable to setforth certain results, hithertoonly available to those whohave mastered logical sym-

x (original page v)

bolism, in a form offering theminimum of difficulty to thebeginner. The utmost endeav-our has been made to avoiddogmatism on such questionsas are still open to seriousdoubt, and this endeavour hasto some extent dominated thechoice of topics considered.The beginnings of mathemat-ical logic are less definitelyknown than its later portions,but are of at least equal philo-sophical interest. Much ofwhat is set forth in the follow-ing chapters is not properlyxi (original page v)

to be called “philosophy,”though the matters concernedwere included in philosophyso long as no satisfactory sci-ence of them existed. Thenature of infinity and conti-nuity, for example, belongedin former days to philosophy,but belongs now to mathemat-ics. Mathematical philosophy,in the strict sense, cannot,perhaps, be held to includesuch definite scientific re-sults as have been obtainedin this region; the philoso-phy of mathematics will nat-xii (original page v)

urally be expected to dealwith questions on the fron-tier of knowledge, as to whichcomparative certainty is notyet attained. But speculationon such questions is hardlylikely to be fruitful unless themore scientific parts of theprinciples of mathematics areknown. A book dealing withthose parts may, therefore,claim to be an introductionto mathematical philosophy,though it can hardly claim, ex-cept where it steps outside itsprovince, to be actually deal-xiii (original page v)

ing with a part of philosophy.It does deal, | however, with abody of knowledge which, tothose who accept it, appearsto invalidate much traditionalphilosophy, and even a gooddeal of what is current in thepresent day. In this way, aswell as by its bearing on stillunsolved problems, mathe-matical logic is relevant tophilosophy. For this reason,as well as on account of theintrinsic importance of thesubject, some purpose maybe served by a succinct ac-xiv (original pages v–vi)

count of the main results ofmathematical logic in a formrequiring neither a knowl-edge of mathematics nor anaptitude for mathematicalsymbolism. Here, however,as elsewhere, the method ismore important than the re-sults, from the point of viewof further research; and themethod cannot well be ex-plained within the frameworkof such a book as the follow-ing. It is to be hoped thatsome readers may be suffi-ciently interested to advancexv (original page vi)

to a study of the method bywhich mathematical logic canbe made helpful in investigat-ing the traditional problemsof philosophy. But that is atopic with which the follow-ing pages have not attemptedto deal.

BERTRAND RUSSELL.

xvi (original page vi)

EDITOR’S NOTE

[The note below was writtenby J. H. Muirhead, LL.D., ed-itor of the Library of Philoso-phy series in which Introduc-tion to Mathematical Philoso-phy was originally published.]

Those who, relying on thedistinction between Mathe-

xvii (original page vii)

matical Philosophy and thePhilosophy of Mathematics,think that this book is out ofplace in the present Library,may be referred to what theauthor himself says on thishead in the Preface. It is notnecessary to agree with whathe there suggests as to thereadjustment of the field ofphilosophy by the transfer-ence from it to mathematicsof such problems as those ofclass, continuity, infinity, inorder to perceive the bearingof the definitions and discus-xviii (original page vii)

sions that follow on the workof “traditional philosophy.” Ifphilosophers cannot consentto relegate the criticism ofthese categories to any of thespecial sciences, it is essential,at any rate, that they shouldknow the precise meaningthat the science of mathemat-ics, in which these conceptsplay so large a part, assigns tothem. If, on the other hand,there be mathematicians towhom these definitions anddiscussions seem to be anelaboration and complicationxix (original page vii)

of the simple, it may be wellto remind them from the sideof philosophy that here, aselsewhere, apparent simplic-ity may conceal a complexitywhich it is the business ofsomebody, whether philoso-pher or mathematician, or,like the author of this volume,both in one, to unravel.

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CHAPTER ITHE SERIES OF

NATURAL NUMBERS

Mathematics is a study which,when we start from its mostfamiliar portions, may be pur-sued in either of two oppositedirections. The more famil-iar direction is constructive,towards gradually increas-ing complexity: from integersto fractions, real numbers,

(original page )

complex numbers; from ad-dition and multiplication todifferentiation and integra-tion, and on to higher math-ematics. The other direction,which is less familiar, pro-ceeds, by analysing, to greaterand greater abstractness andlogical simplicity; instead ofasking what can be definedand deduced from what is as-sumed to begin with, we askinstead what more generalideas and principles can befound, in terms of which whatwas our starting-point can be (original page )

defined or deduced. It is thefact of pursuing this oppositedirection that characterisesmathematical philosophy asopposed to ordinary mathe-matics. But it should be un-derstood that the distinctionis one, not in the subject mat-ter, but in the state of mind ofthe investigator. Early Greekgeometers, passing from theempirical rules of Egyptianland-surveying to the gen-eral propositions by whichthose rules were found to bejustifiable, and thence to Eu- (original page )

clid’s axioms and postulates,were engaged in mathematicalphilosophy, according to theabove definition; but whenonce the axioms and postu-lates had been reached, theirdeductive employment, as wefind it in Euclid, belonged tomathematics in the | ordinarysense. The distinction be-tween mathematics and math-ematical philosophy is onewhich depends upon the in-terest inspiring the research,and upon the stage whichthe research has reached; not (original pages –)

upon the propositions withwhich the research is con-cerned.

We may state the same dis-tinction in another way. Themost obvious and easy thingsin mathematics are not thosethat come logically at the be-ginning; they are things that,from the point of view oflogical deduction, come some-where in the middle. Just asthe easiest bodies to see arethose that are neither verynear nor very far, neither verysmall nor very great, so the (original page )

easiest conceptions to graspare those that are neithervery complex nor very simple(using “simple” in a logicalsense). And as we need twosorts of instruments, the tele-scope and the microscope, forthe enlargement of our visualpowers, so we need two sortsof instruments for the en-largement of our logical pow-ers, one to take us forward tothe higher mathematics, theother to take us backward tothe logical foundations of thethings that we are inclined (original page )

to take for granted in mathe-matics. We shall find that byanalysing our ordinary math-ematical notions we acquirefresh insight, new powers,and the means of reachingwhole new mathematical sub-jects by adopting fresh linesof advance after our backwardjourney. It is the purpose ofthis book to explain math-ematical philosophy simplyand untechnically, withoutenlarging upon those por-tions which are so doubtfulor difficult that an elementary (original page )

treatment is scarcely possible.A full treatment will be foundin Principia Mathematica; thetreatment in the present vol-ume is intended merely as anintroduction.

To the average educatedperson of the present day,the obvious starting-point ofmathematics would be theseries of whole numbers,Cambridge University Press, vol.

i., ; vol. ii., ; vol. iii., .By Whitehead and Russell.

(original page )

, , , , . . . etc. |

Probably only a person withsome mathematical knowl-edge would think of begin-ning with instead of with, but we will presume thisdegree of knowledge; we willtake as our starting-point theseries:

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

and it is this series that weshall mean when we speak of (original pages –)

the “series of natural num-bers.”

It is only at a high stageof civilisation that we couldtake this series as our starting-point. It must have requiredmany ages to discover thata brace of pheasants and acouple of days were both in-stances of the number : thedegree of abstraction involvedis far from easy. And the dis-covery that is a number musthave been difficult. As for , itis a very recent addition; theGreeks and Romans had no (original page )

such digit. If we had been em-barking upon mathematicalphilosophy in earlier days, weshould have had to start withsomething less abstract thanthe series of natural num-bers, which we should reachas a stage on our backwardjourney. When the logicalfoundations of mathematicshave grown more familiar, weshall be able to start furtherback, at what is now a latestage in our analysis. But forthe moment the natural num-bers seem to represent what (original page )

is easiest and most familiar inmathematics.

But though familiar, theyare not understood. Very fewpeople are prepared with adefinition of what is meant by“number,” or “,” or “.” It isnot very difficult to see that,starting from , any other ofthe natural numbers can bereached by repeated additionsof , but we shall have to de-fine what we mean by “adding,” and what we mean by“repeated.” These questionsare by no means easy. It was (original page )

believed until recently thatsome, at least, of these firstnotions of arithmetic must beaccepted as too simple andprimitive to be defined. Sinceall terms that are defined aredefined by means of otherterms, it is clear that humanknowledge must always becontent to accept some termsas intelligible without defi-nition, in order | to have astarting-point for its defi-nitions. It is not clear thatthere must be terms whichare incapable of definition: it (original pages –)

is possible that, however farback we go in defining, wealways might go further still.On the other hand, it is alsopossible that, when analysishas been pushed far enough,we can reach terms that re-ally are simple, and thereforelogically incapable of the sortof definition that consists inanalysing. This is a questionwhich it is not necessary forus to decide; for our purposesit is sufficient to observe that,since human powers are fi-nite, the definitions known to (original page )

us must always begin some-where, with terms undefinedfor the moment, though per-haps not permanently.

All traditional pure math-ematics, including analyticalgeometry, may be regarded asconsisting wholly of proposi-tions about the natural num-bers. That is to say, the termswhich occur can be defined bymeans of the natural numbers,and the propositions can bededuced from the propertiesof the natural numbers—withthe addition, in each case, of (original page )

the ideas and propositions ofpure logic.

That all traditional puremathematics can be derivedfrom the natural numbers is afairly recent discovery, thoughit had long been suspected.Pythagoras, who believed thatnot only mathematics, buteverything else could be de-duced from numbers, was thediscoverer of the most seriousobstacle in the way of what iscalled the “arithmetising” ofmathematics. It was Pythago-ras who discovered the exis- (original page )

tence of incommensurables,and, in particular, the incom-mensurability of the side ofa square and the diagonal. Ifthe length of the side is inch,the number of inches in thediagonal is the square root of, which appeared not to be anumber at all. The problemthus raised was solved onlyin our own day, and was onlysolved completely by the helpof the reduction of arithmeticto logic, which will be ex-plained in following chapters.For the present, we shall take (original page )

for granted the arithmetisa-tion of mathematics, thoughthis was a feat of the verygreatest importance. |

Having reduced all tradi-tional pure mathematics tothe theory of the natural num-bers, the next step in logicalanalysis was to reduce thistheory itself to the smallestset of premisses and unde-fined terms from which itcould be derived. This workwas accomplished by Peano.He showed that the entire the-ory of the natural numbers (original pages –)

could be derived from threeprimitive ideas and five prim-itive propositions in additionto those of pure logic. Thesethree ideas and five proposi-tions thus became, as it were,hostages for the whole of tra-ditional pure mathematics.If they could be defined andproved in terms of others, socould all pure mathematics.Their logical “weight,” if onemay use such an expression,is equal to that of the wholeseries of sciences that havebeen deduced from the theory (original page )

of the natural numbers; thetruth of this whole series isassured if the truth of the fiveprimitive propositions is guar-anteed, provided, of course,that there is nothing erro-neous in the purely logicalapparatus which is also in-volved. The work of analysingmathematics is extraordinar-ily facilitated by this work ofPeano’s.

The three primitive ideas inPeano’s arithmetic are:

, number, successor.

(original page )

By “successor” he means thenext number in the natural or-der. That is to say, the succes-sor of is , the successor of is , and so on. By “num-ber” he means, in this connec-tion, the class of the naturalnumbers. He is not assumingthat we know all the membersof this class, but only that weknow what we mean when wesay that this or that is a num-

We shall use “number” in thissense in the present chapter. After-wards the word will be used in a moregeneral sense.

(original page )

ber, just as we know what wemean when we say “Jones is aman,” though we do not knowall men individually.

The five primitive propo-sitions which Peano assumesare:

() 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.() Any property which be-

(original pages –)

longs to , and also to thesuccessor of every numberwhich has the property,belongs to all numbers.

The last of these is the prin-ciple of mathematical induc-tion. We shall have much tosay concerning mathemati-cal induction in the sequel;for the present, we are con-cerned with it only as it occursin Peano’s analysis of arith-metic.

Let us consider briefly thekind of way in which the the-

(original page )

ory of the natural numbersresults from these three ideasand five propositions. To be-gin with, we define as “thesuccessor of ,” as “the suc-cessor of ,” and so on. Wecan obviously go on as longas we like with these defini-tions, since, in virtue of (),every number that we reachwill have a successor, and, invirtue of (), this cannot beany of the numbers alreadydefined, because, if it were,two different numbers wouldhave the same successor; and (original page )

in virtue of () none of thenumbers we reach in the se-ries of successors can be .Thus the series of successorsgives us an endless series ofcontinually new numbers. Invirtue of () all numbers comein this series, which beginswith and travels on throughsuccessive successors: for (a) belongs to this series, and(b) if a number n belongs to it,so does its successor, whence,by mathematical induction,every number belongs to theseries. (original page )

Suppose we wish to definethe sum of two numbers. Tak-ing any number m, we definem+ asm, andm+(n+) as thesuccessor of m + n. In virtueof () this gives a definition ofthe sum of m and n, whatevernumber n may be. Similarlywe can define the product ofany two numbers. The readercan easily convince himselfthat any ordinary elementaryproposition of arithmetic canbe proved by means of ourfive premisses, and if he hasany difficulty he can find the (original page )

proof in Peano.It is time now to turn to the

considerations which make itnecessary to advance beyondthe standpoint of Peano, who| represents the last perfec-tion of the “arithmetisation”of mathematics, to that ofFrege, who first succeededin “logicising” mathematics,i.e. in reducing to logic thearithmetical notions whichhis predecessors had shownto be sufficient for mathe-matics. We shall not, in thischapter, actually give Frege’s (original pages –)

definition of number and ofparticular numbers, but weshall give some of the reasonswhy Peano’s treatment is lessfinal than it appears to be.

In the first place, Peano’sthree primitive ideas—namely,“,” “number,” and “suc-cessor”—are capable of aninfinite number of differentinterpretations, all of whichwill satisfy the five primitivepropositions. We will givesome examples.

() Let “” be taken to mean, and let “number” be (original page )

taken to mean the numbersfrom onward in the seriesof natural numbers. Then allour primitive propositions aresatisfied, even the fourth, for,though is the successor of, is not a “number” inthe sense which we are nowgiving to the word “number.”It is obvious that any numbermay be substituted for inthis example.

() Let “” have its usualmeaning, but let “number”mean what we usually call“even numbers,” and let the (original page )

“successor” of a number bewhat results from adding twoto it. Then “” will standfor the number two, “” willstand for the number four,and so on; the series of “num-bers” now will be

, two, four, six, eight . . .

All Peano’s five premisses aresatisfied still.

() Let “” mean the num-ber one, let “number” meanthe set

(original page )

, , , , , . . .

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

It is clear that such exam-ples might be multiplied in-definitely. In fact, given anyseries

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

which is endless, contains norepetitions, has a beginning,and has no terms that cannot (original pages –)

be reached from the begin-ning in a finite number ofsteps, we have a set of termsverifying Peano’s axioms. Thisis easily seen, though the for-mal proof is somewhat long.Let “” mean x, let “number”mean the whole set of terms,and let the “successor” of xnmean xn+. Then

() “ is a number,” i.e. x isa member of the set.

() “The successor of anynumber is a number,” i.e. tak-ing any term xn in the set, xn+is also in the set. (original page )

() “No two numbers havethe same successor,” i.e. if xmand xn are two different mem-bers of the set, xm+ and xn+are different; this results fromthe fact that (by hypothesis)there are no repetitions in theset.

() “ is not the successor ofany number,” i.e. no term inthe set comes before x.

() This becomes: Anyproperty which belongs tox, and belongs to xn+ pro-vided it belongs to xn, belongsto all the x’s. (original page )

This follows from the cor-responding property for num-bers.

A series of the form

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

in which there is a first term, asuccessor to each term (so thatthere is no last term), no repe-titions, and every term can bereached from the start in a fi-nite number of steps, is calleda progression. Progressionsare of great importance in theprinciples of mathematics.As we have just seen, every (original page )

progression verifies Peano’sfive axioms. It can be proved,conversely, that every serieswhich verifies Peano’s five ax-ioms is a progression. Hencethese five axioms may be usedto define the class of pro-gressions: “progressions” are“those series which verifythese five axioms.” Any pro-gression may be taken as thebasis of pure mathematics: wemay give the name “” to itsfirst term, the name “number”to the whole set of its terms,and the name “successor” to (original page )

the next in the progression.The progression need not becomposed of numbers: it maybe | composed of points inspace, or moments of time, orany other terms of which thereis an infinite supply. Each dif-ferent progression will giverise to a different interpreta-tion of all the propositions oftraditional pure mathematics;all these possible interpreta-tions will be equally true.

In Peano’s system there isnothing to enable us to distin-guish between these different (original pages –)

interpretations of his primi-tive ideas. It is assumed thatwe know what is meant by“,” and that we shall not sup-pose that this symbol means or Cleopatra’s Needle orany of the other things that itmight mean.

This point, that “” and“number” and “successor”cannot be defined by means ofPeano’s five axioms, but mustbe independently understood,is important. We want ournumbers not merely to verifymathematical formulæ, but (original page )

to apply in the right way tocommon objects. We want tohave ten fingers and two eyesand one nose. A system inwhich “” meant , and “”meant , and so on, mightbe all right for pure math-ematics, but would not suitdaily life. We want “” and“number” and “successor”to have meanings which willgive us the right allowance offingers and eyes and noses.We have already some knowl-edge (though not sufficientlyarticulate or analytic) of what (original page )

we mean by “” and “” andso on, and our use of numbersin arithmetic must conformto this knowledge. We cannotsecure that this shall be thecase by Peano’s method; allthat we can do, if we adopthis method, is to say “weknow what we mean by ‘’and ‘number’ and ‘successor,’though we cannot explainwhat we mean in terms ofother simpler concepts.” Itis quite legitimate to say thiswhen we must, and at somepoint we all must; but it is (original page )

the object of mathematicalphilosophy to put off sayingit as long as possible. By thelogical theory of arithmeticwe are able to put it off for avery long time.

It might be suggested that,instead of setting up “” and“number” and “successor” asterms of which we know themeaning although we can-not define them, we mightlet them | stand for any threeterms that verify Peano’s fiveaxioms. They will then nolonger be terms which have (original pages –)

a meaning that is definitethough undefined: they willbe “variables,” terms concern-ing which we make certain hy-potheses, namely, those statedin the five axioms, but whichare otherwise undetermined.If we adopt this plan, ourtheorems will not be provedconcerning an ascertained setof terms called “the naturalnumbers,” but concerning allsets of terms having certainproperties. Such a procedureis not fallacious; indeed forcertain purposes it represents (original page )

a valuable generalisation. Butfrom two points of view itfails to give an adequate basisfor arithmetic. In the firstplace, it does not enable usto know whether there areany sets of terms verifyingPeano’s axioms; it does noteven give the faintest sugges-tion of any way of discoveringwhether there are such sets.In the second place, as al-ready observed, we want ournumbers to be such as can beused for counting commonobjects, and this requires that (original page )

our numbers should have adefinite meaning, not merelythat they should have certainformal properties. This defi-nite meaning is defined by thelogical theory of arithmetic.

(original page )

CHAPTER IIDEFINITION OF

NUMBER

The question “What is a num-ber?” is one which has beenoften asked, but has only beencorrectly answered in ourown time. The answer wasgiven by Frege in , in hisGrundlagen der Arithmetik.

The same answer is given morefully and with more development in

(original page )

Although this book is quiteshort, not difficult, and of thevery highest importance, itattracted almost no attention,and the definition of numberwhich it contains remainedpractically unknown untilit was rediscovered by thepresent author in .

In seeking a definition ofnumber, the first thing to beclear about is what we maycall the grammar of our in-

his Grundgesetze der Arithmetik, vol.i., .

(original page )

quiry. Many philosophers,when attempting to definenumber, are really settingto work to define plurality,which is quite a differentthing. Number is what is char-acteristic of numbers, as manis what is characteristic ofmen. A plurality is not aninstance of number, but ofsome particular number. Atrio of men, for example, is aninstance of the number , andthe number is an instanceof number; but the trio is notan instance of number. This (original page )

point may seem elementaryand scarcely worth mention-ing; yet it has proved toosubtle for the philosophers,with few exceptions.

A particular number is notidentical with any collectionof terms having that number:the number is not identicalwith | the trio consisting ofBrown, Jones, and Robinson.The number is somethingwhich all trios have in com-mon, and which distinguishesthem from other collections.A number is something that (original pages –)

characterises certain collec-tions, namely, those that havethat number.

Instead of speaking of a“collection,” we shall as a rulespeak of a “class,” or some-times a “set.” Other wordsused in mathematics for thesame thing are “aggregate”and “manifold.” We shallhave much to say later onabout classes. For the present,we will say as little as pos-sible. But there are someremarks that must be madeimmediately. (original page )

A class or collection maybe defined in two ways thatat first sight seem quite dis-tinct. We may enumerateits members, as when wesay, “The collection I meanis Brown, Jones, and Robin-son.” Or we may mention adefining property, as whenwe speak of “mankind” or“the inhabitants of London.”The definition which enu-merates is called a definitionby “extension,” and the onewhich mentions a definingproperty is called a definition (original page )

by “intension.” Of these twokinds of definition, the oneby intension is logically morefundamental. This is shownby two considerations: ()that the extensional definitioncan always be reduced to anintensional one; () that theintensional one often cannoteven theoretically be reducedto the extensional one. Eachof these points needs a wordof explanation.

() Brown, Jones, and Robin-son all of them possess a cer-tain property which is pos- (original page )

sessed by nothing else inthe whole universe, namely,the property of being eitherBrown or Jones or Robinson.This property can be usedto give a definition by inten-sion of the class consisting ofBrown and Jones and Robin-son. Consider such a formulaas “x is Brown or x is Jones orx is Robinson.” This formulawill be true for just three x’s,namely, Brown and Jones andRobinson. In this respect itresembles a cubic equationwith its three roots. It may be (original page )

taken as assigning a propertycommon to the members ofthe class consisting of thesethree | men, and peculiar tothem. A similar treatment canobviously be applied to anyother class given in extension.

() It is obvious that in prac-tice we can often know a greatdeal about a class withoutbeing able to enumerate itsmembers. No one man couldactually enumerate all men,or even all the inhabitants ofLondon, yet a great deal isknown about each of these (original pages –)

classes. This is enough toshow that definition by exten-sion is not necessary to knowl-edge about a class. But whenwe come to consider infiniteclasses, we find that enumera-tion is not even theoreticallypossible for beings who onlylive for a finite time. We can-not enumerate all the naturalnumbers: they are , , ,, and so on. At some pointwe must content ourselveswith “and so on.” We cannotenumerate all fractions or allirrational numbers, or all of (original page )

any other infinite collection.Thus our knowledge in regardto all such collections can onlybe derived from a definitionby intension.

These remarks are relevant,when we are seeking the defi-nition of number, in three dif-ferent ways. In the first place,numbers themselves form aninfinite collection, and can-not therefore be defined byenumeration. In the secondplace, the collections having agiven number of terms them-selves presumably form an (original page )

infinite collection: it is to bepresumed, for example, thatthere are an infinite collec-tion of trios in the world, forif this were not the case thetotal number of things in theworld would be finite, which,though possible, seems un-likely. In the third place, wewish to define “number” insuch a way that infinite num-bers may be possible; thuswe must be able to speak ofthe number of terms in aninfinite collection, and such acollection must be defined by (original page )

intension, i.e. by a propertycommon to all its membersand peculiar to them.

For many purposes, a classand a defining characteris-tic of it are practically inter-changeable. The vital differ-ence between the two consistsin the fact that there is onlyone class having a given set ofmembers, whereas there arealways many different char-acteristics by which a givenclass may be defined. Men |may be defined as featherlessbipeds, or as rational animals, (original pages –)

or (more correctly) by thetraits by which Swift delin-eates the Yahoos. It is thisfact that a defining charac-teristic is never unique whichmakes classes useful; other-wise we could be content withthe properties common andpeculiar to their members.

Any one of these properties

As will be explained later, classesmay be regarded as logical fictions,manufactured out of defining char-acteristics. But for the present itwill simplify our exposition to treatclasses as if they were real.

(original page )

can be used in place of theclass whenever uniqueness isnot important.

Returning now to the def-inition of number, it is clearthat number is a way of bring-ing together certain collec-tions, namely, those that havea given number of terms. Wecan suppose all couples inone bundle, all trios in an-other, and so on. In this waywe obtain various bundles ofcollections, each bundle con-sisting of all the collectionsthat have a certain number of (original page )

terms. Each bundle is a classwhose members are collec-tions, i.e. classes; thus each isa class of classes. The bundleconsisting of all couples, forexample, is a class of classes:each couple is a class withtwo members, and the wholebundle of couples is a classwith an infinite number ofmembers, each of which is aclass of two members.

How shall we decide wheth-er two collections are to be-long to the same bundle? Theanswer that suggests itself is: (original page )

“Find out how many memberseach has, and put them inthe same bundle if they havethe same number of mem-bers.” But this presupposesthat we have defined num-bers, and that we know howto discover how many terms acollection has. We are so usedto the operation of countingthat such a presuppositionmight easily pass unnoticed.In fact, however, counting,though familiar, is logicallya very complex operation;moreover it is only available, (original page )

as a means of discoveringhow many terms a collec-tion has, when the collectionis finite. Our definition ofnumber must not assume inadvance that all numbers arefinite; and we cannot in anycase, without a vicious circle,| use counting to define num-bers, because numbers areused in counting. We need,therefore, some other methodof deciding when two collec-tions have the same numberof terms.

In actual fact, it is simpler (original pages –)

logically to find out whethertwo collections have the samenumber of terms than it isto define what that numberis. An illustration will makethis clear. If there were nopolygamy or polyandry any-where in the world, it is clearthat the number of husbandsliving at any moment wouldbe exactly the same as thenumber of wives. We do notneed a census to assure us ofthis, nor do we need to knowwhat is the actual number ofhusbands and of wives. We (original page )

know the number must bethe same in both collections,because each husband hasone wife and each wife hasone husband. The relation ofhusband and wife is what iscalled “one-one.”

A relation is said to be “one-one” when, if x has the rela-tion in question to y, no otherterm x′ has the same relationto y, and x does not have thesame relation to any term y′

other than y. When only thefirst of these two conditions isfulfilled, the relation is called (original page )

“one-many”; when only thesecond is fulfilled, it is called“many-one.” It should be ob-served that the number isnot used in these definitions.

In Christian countries, therelation of husband to wife isone-one; in Mahometan coun-tries it is one-many; in Tibet itis many-one. The relation offather to son is one-many; thatof son to father is many-one,but that of eldest son to fatheris one-one. If n is any number,the relation of n to n + isone-one; so is the relation of (original page )

n to n or to n. When weare considering only positivenumbers, the relation of n ton is one-one; but when neg-ative numbers are admitted,it becomes two-one, since nand −n have the same square.These instances should suf-fice to make clear the notionsof one-one, one-many, andmany-one relations, whichplay a great part in the princi-ples of mathematics, not onlyin relation to the definition ofnumbers, but in many otherconnections. (original page )

Two classes are said to be“similar” when there is a one-one | relation which correlatesthe terms of the one classeach with one term of theother class, in the same man-ner in which the relation ofmarriage correlates husbandswith wives. A few prelimi-nary definitions will help usto state this definition moreprecisely. The class of thoseterms that have a given re-lation to something or otheris called the domain of thatrelation: thus fathers are the (original pages –)

domain of the relation of fa-ther to child, husbands arethe domain of the relation ofhusband to wife, wives are thedomain of the relation of wifeto husband, and husbandsand wives together are thedomain of the relation of mar-riage. The relation of wife tohusband is called the converseof the relation of husband towife. Similarly less is the con-verse of greater, later is theconverse of earlier, and so on.Generally, the converse of agiven relation is that relation (original page )

which holds between y and xwhenever the given relationholds between x and y. Theconverse domain of a relationis the domain of its converse:thus the class of wives is theconverse domain of the rela-tion of husband to wife. Wemay now state our definitionof similarity as follows:—

One class is said to be “sim-ilar” to another when there isa one-one relation of which theone class is the domain, whilethe other is the converse do-main. (original page )

It is easy to prove () thatevery class is similar to itself,() that if a class α is similarto a class β, then β is similarto α, () that if α is similar toβ and β to γ , then α is similarto γ . A relation is said to bereflexive when it possesses thefirst of these properties, sym-metrical when it possesses thesecond, and transitive whenit possesses the third. It isobvious that a relation whichis symmetrical and transitivemust be reflexive throughoutits domain. Relations which (original page )

possess these properties arean important kind, and it isworth while to note that sim-ilarity is one of this kind ofrelations.

It is obvious to commonsense that two finite classeshave the same number ofterms if they are similar, butnot otherwise. The act ofcounting consists in estab-lishing a one-one correlation| between the set of objectscounted and the natural num-bers (excluding ) that areused up in the process. Ac- (original pages –)

cordingly common sense con-cludes that there are as manyobjects in the set to be countedas there are numbers up tothe last number used in thecounting. And we also knowthat, so long as we confineourselves to finite numbers,there are just n numbers from up to n. Hence it followsthat the last number used incounting a collection is thenumber of terms in the collec-tion, provided the collectionis finite. But this result, be-sides being only applicable (original page )

to finite collections, dependsupon and assumes the factthat two classes which aresimilar have the same numberof terms; for what we do whenwe count (say) objects isto show that the set of theseobjects is similar to the setof numbers to . The no-tion of similarity is logicallypresupposed in the operationof counting, and is logicallysimpler though less familiar.In counting, it is necessary totake the objects counted in acertain order, as first, second, (original page )

third, etc., but order is not ofthe essence of number: it isan irrelevant addition, an un-necessary complication fromthe logical point of view. Thenotion of similarity does notdemand an order: for exam-ple, we saw that the numberof husbands is the same asthe number of wives, withouthaving to establish an order ofprecedence among them. Thenotion of similarity also doesnot require that the classeswhich are similar should befinite. Take, for example, the (original page )

natural numbers (excluding )on the one hand, and the frac-tions which have for theirnumerator on the other hand:it is obvious that we can cor-relate with /, with /,and so on, thus proving thatthe two classes are similar.

We may thus use the notionof “similarity” to decide whentwo collections are to belongto the same bundle, in thesense in which we were ask-ing this question earlier in thischapter. We want to make onebundle containing the class (original page )

that has no members: this willbe for the number . Thenwe want a bundle of all theclasses that have one member:this will be for the number. Then, for the number ,we want a bundle consisting| of all couples; then one ofall trios; and so on. Givenany collection, we can definethe bundle it is to belong toas being the class of all thosecollections that are “similar”to it. It is very easy to see thatif (for example) a collectionhas three members, the class (original pages –)

of all those collections that aresimilar to it will be the classof trios. And whatever num-ber of terms a collection mayhave, those collections thatare “similar” to it will havethe same number of terms. Wemay take this as a definitionof “having the same numberof terms.” It is obvious that itgives results conformable tousage so long as we confineourselves to finite collections.

So far we have not sug-gested anything in the slight-est degree paradoxical. But (original page )

when we come to the actualdefinition of numbers we can-not avoid what must at firstsight seem a paradox, thoughthis impression will soon wearoff. We naturally think thatthe class of couples (for ex-ample) is something differentfrom the number . But thereis no doubt about the classof couples: it is indubitableand not difficult to define,whereas the number , in anyother sense, is a metaphysicalentity about which we cannever feel sure that it exists or (original page )

that we have tracked it down.It is therefore more prudentto content ourselves with theclass of couples, which weare sure of, than to hunt fora problematical number which must always remainelusive. Accordingly we setup the following definition:—

The number of a class is theclass of all those classes that aresimilar to it.

Thus the number of a cou-ple will be the class of allcouples. In fact, the class ofall couples will be the number (original page )

, according to our defini-tion. At the expense of a littleoddity, this definition securesdefiniteness and indubitable-ness; and it is not difficult toprove that numbers so definedhave all the properties that weexpect numbers to have.

We may now go on to definenumbers in general as any oneof the bundles into which sim-ilarity collects classes. A num-ber will be a set of classes suchas that any two are similar toeach | other, and none outsidethe set are similar to any in- (original pages –)

side the set. In other words,a number (in general) is anycollection which is the num-ber of one of its members; or,more simply still:

A number is anything whichis the number of some class.

Such a definition has a ver-bal appearance of being cir-cular, but in fact it is not.We define “the number of agiven class” without usingthe notion of number in gen-eral; therefore we may definenumber in general in terms of“the number of a given class” (original page )

without committing any logi-cal error.

Definitions of this sort arein fact very common. Theclass of fathers, for example,would have to be defined byfirst defining what it is to bethe father of somebody; thenthe class of fathers will be allthose who are somebody’s fa-ther. Similarly if we want todefine square numbers (say),we must first define whatwe mean by saying that onenumber is the square of an-other, and then define square (original page )

numbers as those that are thesquares of other numbers.This kind of procedure is verycommon, and it is importantto realise that it is legitimateand even often necessary.

We have now given a defi-nition of numbers which willserve for finite collections.It remains to be seen how itwill serve for infinite collec-tions. But first we must decidewhat we mean by “finite” and“infinite,” which cannot be

(original page )

done within the limits of thepresent chapter.

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CHAPTER IIIFINITUDE ANDMATHEMATICAL

INDUCTION

The series of natural num-bers, as we saw in ChapterI., can all be defined if weknow what we mean by thethree terms “,” “number,”and “successor.” But we maygo a step farther: we can de-fine all the natural numbers

(original page )

if we know what we meanby “” and “successor.” Itwill help us to understand thedifference between finite andinfinite to see how this can bedone, and why the method bywhich it is done cannot be ex-tended beyond the finite. Wewill not yet consider how “”and “successor” are to be de-fined: we will for the momentassume that we know whatthese terms mean, and showhow thence all other naturalnumbers can be obtained.

It is easy to see that we can (original page )

reach any assigned number,say ,. We first define “”as “the successor of ,” thenwe define “” as “the succes-sor of ,” and so on. In thecase of an assigned number,such as ,, the proof thatwe can reach it by proceedingstep by step in this fashionmay be made, if we have thepatience, by actual experi-ment: we can go on until weactually arrive at ,. Butalthough the method of ex-periment is available for eachparticular natural number, it (original page )

is not available for proving thegeneral proposition that allsuch numbers can be reachedin this way, i.e. by proceedingfrom step by step from eachnumber to its successor. Isthere any other way by whichthis can be proved?

Let us consider the ques-tion the other way round.What are the numbers thatcan be reached, given theterms “” and | “successor”?Is there any way by which wecan define the whole class ofsuch numbers? We reach , (original pages –)

as the successor of ; , asthe successor of ; , as thesuccessor of ; and so on. Itis this “and so on” that wewish to replace by somethingless vague and indefinite. Wemight be tempted to say that“and so on” means that theprocess of proceeding to thesuccessor may be repeatedany finite number of times; butthe problem upon which weare engaged is the problem ofdefining “finite number,” andtherefore we must not use thisnotion in our definition. Our (original page )

definition must not assumethat we know what a finitenumber is.

The key to our problem liesin mathematical induction. Itwill be remembered that, inChapter I., this was the fifthof the five primitive propo-sitions which we laid downabout the natural numbers.It stated that any propertywhich belongs to , and tothe successor of any numberwhich has the property, be-longs to all the natural num-bers. This was then presented (original page )

as a principle, but we shallnow adopt it as a definition. Itis not difficult to see that theterms obeying it are the sameas the numbers that can bereached from by successivesteps from next to next, but asthe point is important we willset forth the matter in somedetail.

We shall do well to beginwith some definitions, whichwill be useful in other connec-tions also.

A property is said to be“hereditary” in the natural- (original page )

number series if, whenever itbelongs to a number n, it alsobelongs to n+ , the successorof n. Similarly a class is said tobe “hereditary” if, whenever nis a member of the class, so isn+ . It is easy to see, thoughwe are not yet supposed toknow, that to say a propertyis hereditary is equivalent tosaying that it belongs to allthe natural numbers not lessthan some one of them, e.g. itmust belong to all that are notless than , or all that arenot less than , or it may (original page )

be that it belongs to all thatare not less than , i.e. to allwithout exception.

A property is said to be “in-ductive” when it is a heredi-tary | property which belongsto . Similarly a class is “in-ductive” when it is a heredi-tary class of which is a mem-ber.

Given a hereditary class ofwhich is a member, it fol-lows that is a member ofit, because a hereditary classcontains the successors of itsmembers, and is the suc- (original pages –)

cessor of . Similarly, givena hereditary class of which is a member, it follows that is a member of it; and soon. Thus we can prove by astep-by-step procedure thatany assigned natural number,say ,, is a member ofevery inductive class.

We will define the “poster-ity” of a given natural num-ber with respect to the rela-tion “immediate predecessor”(which is the converse of “suc-cessor”) as all those terms thatbelong to every hereditary (original page )

class to which the given num-ber belongs. It is again easyto see that the posterity of anatural number consists ofitself and all greater naturalnumbers; but this also we donot yet officially know.

By the above definitions,the posterity of will consistof those terms which belongto every inductive class.

It is now not difficult tomake it obvious that the pos-terity of is the same setas those terms that can bereached from by successive (original page )

steps from next to next. For,in the first place, belongs toboth these sets (in the sensein which we have defined ourterms); in the second place,if n belongs to both sets, sodoes n+ . It is to be observedthat we are dealing here withthe kind of matter that doesnot admit of precise proof,namely, the comparison of arelatively vague idea with arelatively precise one. The no-tion of “those terms that canbe reached from by succes-sive steps from next to next” (original page )

is vague, though it seems as ifit conveyed a definite mean-ing; on the other hand, “theposterity of ” is precise andexplicit just where the otheridea is hazy. It may be takenas giving what we meant tomean when we spoke of theterms that can be reachedfrom by successive steps.

We now lay down the fol-lowing definition:—

The “natural numbers” arethe posterity of with respect tothe | relation “immediate prede-cessor” (which is the converse (original pages –)

of “successor”).We have thus arrived at a

definition of one of Peano’sthree primitive ideas in termsof the other two. As a re-sult of this definition, two ofhis primitive propositions—namely, the one assertingthat is a number and theone asserting mathematicalinduction—become unneces-sary, since they result from thedefinition. The one assertingthat the successor of a naturalnumber is a natural number isonly needed in the weakened (original page )

form “every natural numberhas a successor.”

We can, of course, easilydefine “” and “successor”by means of the definitionof number in general whichwe arrived at in Chapter II.The number is the num-ber of terms in a class whichhas no members, i.e. in theclass which is called the “null-class.” By the general defini-tion of number, the numberof terms in the null-class isthe set of all classes similar tothe null-class, i.e. (as is easily (original page )

proved) the set consisting ofthe null-class all alone, i.e.the class whose only memberis the null-class. (This is notidentical with the null-class:it has one member, namely,the null-class, whereas thenull-class itself has no mem-bers. A class which has onemember is never identicalwith that one member, as weshall explain when we cometo the theory of classes.) Thuswe have the following purelylogical definition:— is the class whose only

(original page )

member is the null-class.It remains to define “suc-

cessor.” Given any number n,let α be a class which has nmembers, and let x be a termwhich is not a member of α.Then the class consisting of αwith x added on will have n+members. Thus we have thefollowing definition:—

The successor of the numberof terms in the class α is thenumber of terms in the classconsisting of α together with x,where x is any term not belong-ing to the class. (original page )

Certain niceties are re-quired to make this defini-tion perfect, but they need notconcern us. It will be remem-bered that we | have alreadygiven (in Chapter II.) a logi-cal definition of the numberof terms in a class, namely,we defined it as the set of allclasses that are similar to thegiven class.

We have thus reduced Pea-no’s three primitive ideas to

See Principia Mathematica, vol. ii.∗.

(original pages –)

ideas of logic: we have givendefinitions of them whichmake them definite, no longercapable of an infinity of dif-ferent meanings, as they werewhen they were only deter-minate to the extent of obey-ing Peano’s five axioms. Wehave removed them from thefundamental apparatus ofterms that must be merelyapprehended, and have thusincreased the deductive artic-ulation of mathematics.

As regards the five prim-itive propositions, we have (original page )

already succeeded in makingtwo of them demonstrableby our definition of “natu-ral number.” How stands itwith the remaining three? Itis very easy to prove that isnot the successor of any num-ber, and that the successor ofany number is a number. Butthere is a difficulty about theremaining primitive proposi-tion, namely, “no two num-bers have the same successor.”The difficulty does not ariseunless the total number ofindividuals in the universe is (original page )

finite; for given two numbersm and n, neither of which isthe total number of individu-als in the universe, it is easyto prove that we cannot havem + = n + unless we havem = n. But let us suppose thatthe total number of individu-als in the universe were (say); then there would be noclass of individuals, andthe number would be thenull-class. So would the num-ber . Thus we should have = ; therefore the succes-sor of would be the same as (original page )

the successor of , although would not be the sameas . Thus we should havetwo different numbers withthe same successor. This fail-ure of the third axiom cannotarise, however, if the numberof individuals in the world isnot finite. We shall return tothis topic at a later stage.

Assuming that the numberof individuals in the universeis not finite, we have now suc-ceeded not only in defining

See Chapter XIII.

(original page )

Peano’s | three primitive ideas,but in seeing how to prove hisfive primitive propositions,by means of primitive ideasand propositions belongingto logic. It follows that allpure mathematics, in so far asit is deducible from the the-ory of the natural numbers, isonly a prolongation of logic.The extension of this resultto those modern branches ofmathematics which are notdeducible from the theory ofthe natural numbers offers nodifficulty of principle, as we (original pages –)

have shown elsewhere.

The process of mathemat-ical induction, by means ofwhich we defined the naturalnumbers, is capable of gen-eralisation. We defined thenatural numbers as the “pos-terity” of with respect to therelation of a number to its im-mediate successor. If we callthis relation N, any number mwill have this relation tom+.For geometry, in so far as it is

not purely analytical, see Principlesof Mathematics, part vi.; for rationaldynamics, ibid., part vii.

(original page )

A property is “hereditary withrespect to N,” or simply “N-hereditary,” if, whenever theproperty belongs to a numberm, it also belongs to m+ , i.e.to the number to which m hasthe relation N. And a numbern will be said to belong tothe “posterity” of m with re-spect to the relation N if n hasevery N-hereditary propertybelonging to m. These defini-tions can all be applied to anyother relation just as well asto N. Thus if R is any relationwhatever, we can lay down (original page )

the following definitions:—A property is called “R-

hereditary” when, if it belongsto a term x, and x has the rela-tion R to y, then it belongs toy.

A class is R-hereditary whenits defining property is R-hereditary.

These definitions, and the gener-alised theory of induction, are due toFrege, and were published so long agoas in his Begriffsschrift. In spiteof the great value of this work, I was, Ibelieve, the first person who ever readit—more than twenty years after itspublication.

(original page )

A term x is said to be an“R-ancestor” of the term y ify has every R-hereditary prop-erty that x has, provided x isa term which has the relationR to something or to whichsomething has the relation R.(This is only to exclude trivialcases.) |

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

We have framed the abovedefinitions so that if a term isthe ancestor of anything it isits own ancestor and belongs (original pages –)

to its own posterity. This ismerely for convenience.

It will be observed that if wetake for R the relation “par-ent,” “ancestor” and “pos-terity” will have the usualmeanings, except that a per-son will be included amonghis own ancestors and poster-ity. It is, of course, obvious atonce that “ancestor” must becapable of definition in termsof “parent,” but until Fregedeveloped his generalised the-ory of induction, no one couldhave defined “ancestor” pre- (original page )

cisely in terms of “parent.”A brief consideration of thispoint will serve to show theimportance of the theory. Aperson confronted for thefirst time with the problem ofdefining “ancestor” in termsof “parent” would naturallysay that A is an ancestor of Zif, between A and Z, there area certain number of people, B,C, . . . , of whom B is a child ofA, each is a parent of the next,until the last, who is a parentof Z. But this definition is notadequate unless we add that (original page )

the number of intermediateterms is to be finite. Take, forexample, such a series as thefollowing:—

−, − , − , −

, . . .

, , , .

Here we have first a seriesof negative fractions with noend, and then a series of pos-itive fractions with no begin-ning. Shall we say that, in thisseries, −/ is an ancestor of/? It will be so according tothe beginner’s definition sug-gested above, but it will not be

(original page )

so according to any definitionwhich will give the kind ofidea that we wish to define.For this purpose, it is essentialthat the number of interme-diaries should be finite. But,as we saw, “finite” is to bedefined by means of mathe-matical induction, and it issimpler to define the ances-tral relation generally at oncethan to define it first only forthe case of the relation of nto n + , and then extend itto other cases. Here, as con-stantly elsewhere, generality (original page )

from the first, though it may| require more thought at thestart, will be found in the longrun to economise thought andincrease logical power.

The use of mathematicalinduction in demonstrationswas, in the past, something ofa mystery. There seemed noreasonable doubt that it wasa valid method of proof, butno one quite knew why it wasvalid. Some believed it to bereally a case of induction, inthe sense in which that word

(original pages –)

is used in logic. Poincare

considered it to be a principleof the utmost importance, bymeans of which an infinitenumber of syllogisms couldbe condensed into one argu-ment. We now know that allsuch views are mistaken, andthat mathematical inductionis a definition, not a principle.There are some numbers towhich it can be applied, andthere are others (as we shallsee in Chapter VIII.) to which

Science and Method, chap. iv.

(original page )

it cannot be applied. We de-fine the “natural numbers”as those to which proofs bymathematical induction canbe applied, i.e. as those thatpossess all inductive prop-erties. It follows that suchproofs can be applied to thenatural numbers, not in virtueof any mysterious intuitionor axiom or principle, butas a purely verbal proposi-tion. If “quadrupeds” aredefined as animals havingfour legs, it will follow thatanimals that have four legs (original page )

are quadrupeds; and the caseof numbers that obey math-ematical induction is exactlysimilar.

We shall use the phrase “in-ductive numbers” to meanthe same set as we have hith-erto spoken of as the “naturalnumbers.” The phrase “in-ductive numbers” is prefer-able as affording a reminderthat the definition of this setof numbers is obtained frommathematical induction.

Mathematical induction af-fords, more than anything (original page )

else, the essential characteris-tic by which the finite is dis-tinguished from the infinite.The principle of mathemati-cal induction might be statedpopularly in some such formas “what can be inferred fromnext to next can be inferredfrom first to last.” This is truewhen the number of inter-mediate steps between firstand last is finite, not other-wise. Anyone who has ever |watched a goods train begin-ning to move will have noticedhow the impulse is commu- (original pages –)

nicated with a jerk from eachtruck to the next, until at lasteven the hindmost truck isin motion. When the train isvery long, it is a very long timebefore the last truck moves.If the train were infinitelylong, there would be an infi-nite succession of jerks, andthe time would never comewhen the whole train wouldbe in motion. Nevertheless, ifthere were a series of trucksno longer than the series ofinductive numbers (which, aswe shall see, is an instance (original page )

of the smallest of infinites),every truck would begin tomove sooner or later if theengine persevered, thoughthere would always be othertrucks further back which hadnot yet begun to move. Thisimage will help to elucidatethe argument from next tonext, and its connection withfinitude. When we come toinfinite numbers, where ar-guments from mathematicalinduction will be no longervalid, the properties of suchnumbers will help to make (original page )

clear, by contrast, the almostunconscious use that is madeof mathematical inductionwhere finite numbers are con-cerned.

(original page )

CHAPTER IVTHE DEFINITION OF

ORDER

We have now carried our anal-ysis of the series of naturalnumbers to the point wherewe have obtained logical defi-nitions of the members of thisseries, of the whole class of itsmembers, and of the relationof a number to its immediatesuccessor. We must now con-

(original page )

sider the serial character of thenatural numbers in the order, , , , . . . We ordinar-ily think of the numbers as inthis order, and it is an essentialpart of the work of analysingour data to seek a definition of“order” or “series” in logicalterms.

The notion of order is onewhich has enormous impor-tance in mathematics. Notonly the integers, but also ra-tional fractions and all realnumbers have an order ofmagnitude, and this is essen- (original page )

tial to most of their mathe-matical properties. The orderof points on a line is essentialto geometry; so is the slightlymore complicated order oflines through a point in aplane, or of planes througha line. Dimensions, in ge-ometry, are a developmentof order. The conception ofa limit, which underlies allhigher mathematics, is a serialconception. There are partsof mathematics which do notdepend upon the notion oforder, but they are very few in (original page )

comparison with the parts inwhich this notion is involved.

In seeking a definition oforder, the first thing to realiseis that no set of terms has justone order to the exclusion ofothers. A set of terms hasall the orders of which it iscapable. Sometimes one or-der is so much more familiarand natural to our | thoughtsthat we are inclined to regardit as the order of that set ofterms; but this is a mistake.The natural numbers—or the“inductive” numbers, as we (original pages –)

shall also call them—occurto us most readily in orderof magnitude; but they arecapable of an infinite numberof other arrangements. Wemight, for example, considerfirst all the odd numbers andthen all the even numbers;or first , then all the evennumbers, then all the oddmultiples of , then all themultiples of but not of or, then all the multiples of but not of or or , and soon through the whole seriesof primes. When we say that (original page )

we “arrange” the numbers inthese various orders, that is aninaccurate expression: whatwe really do is to turn ourattention to certain relationsbetween the natural numbers,which themselves generatesuch-and-such an arrange-ment. We can no more “ar-range” the natural numbersthan we can the starry heav-ens; but just as we may noticeamong the fixed stars eithertheir order of brightness ortheir distribution in the sky,so there are various relations (original page )

among numbers which maybe observed, and which giverise to various different ordersamong numbers, all equallylegitimate. And what is trueof numbers is equally trueof points on a line or of themoments of time: one orderis more familiar, but othersare equally valid. We might,for example, take first, on aline, all the points that haveintegral co-ordinates, then allthose that have non-integralrational co-ordinates, then allthose that have algebraic non- (original page )

rational co-ordinates, and soon, through any set of com-plications we please. Theresulting order will be onewhich the points of the linecertainly have, whether wechoose to notice it or not; theonly thing that is arbitraryabout the various orders of aset of terms is our attention,for the terms themselves havealways all the orders of whichthey are capable.

One important result of thisconsideration is that we mustnot look for the definition of (original page )

order in the nature of the setof terms to be ordered, sinceone set of terms has many or-ders. The order lies, not in theclass of terms, but in a relationamong | the members of theclass, in respect of which someappear as earlier and some aslater. The fact that a class mayhave many orders is due to thefact that there can be manyrelations holding among themembers of one single class.What properties must a rela-tion have in order to give riseto an order? (original pages –)

The essential characteris-tics of a relation which is togive rise to order may be dis-covered by considering thatin respect of such a relationwe must be able to say, ofany two terms in the classwhich is to be ordered, thatone “precedes” and the other“follows.” Now, in order thatwe may be able to use thesewords in the way in which weshould naturally understandthem, we require that the or-dering relation should havethree properties:— (original page )

() If x precedes y, y mustnot also precede x. This is anobvious characteristic of thekind of relations that lead toseries. If x is less than y, y isnot also less than x. If x is ear-lier in time than y, y is not alsoearlier than x. If x is to theleft of y, y is not to the left ofx. On the other hand, rela-tions which do not give rise toseries often do not have thisproperty. If x is a brother orsister of y, y is a brother or sis-ter of x. If x is of the sameheight as y, y is of the same (original page )

height as x. If x is of a dif-ferent height from y, y is ofa different height from x. Inall these cases, when the re-lation holds between x and y,it also holds between y and x.But with serial relations sucha thing cannot happen. A rela-tion having this first propertyis called asymmetrical.

() If x precedes y and yprecedes z, x must precede z.This may be illustrated by thesame instances as before: less,earlier, left of. But as instancesof relations which do not have (original page )

this property only two of ourprevious three instances willserve. If x is brother or sisterof y, and y of z, x may not bebrother or sister of z, since xand z may be the same person.The same applies to differenceof height, but not to same-ness of height, which has oursecond property but not ourfirst. The relation “father,” onthe other hand, has our firstproperty but not | our second.A relation having our secondproperty is called transitive.

() Given any two terms (original pages –)

of the class which is to beordered, there must be onewhich precedes and the otherwhich follows. For example,of any two integers, or frac-tions, or real numbers, one issmaller and the other greater;but of any two complex num-bers this is not true. Of anytwo moments in time, onemust be earlier than the other;but of events, which may besimultaneous, this cannot besaid. Of two points on a line,one must be to the left of theother. A relation having this (original page )

third property is called con-nected.

When a relation possessesthese three properties, it is ofthe sort to give rise to an or-der among the terms betweenwhich it holds; and whereveran order exists, some relationhaving these three propertiescan be found generating it.

Before illustrating this the-sis, we will introduce a fewdefinitions.

() A relation is said to be

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an aliorelative, or to be con-tained in or imply diversity,if no term has this relationto itself. Thus, for example,“greater,” “different in size,”“brother,” “husband,” “fa-ther” are aliorelatives; but“equal,” “born of the sameparents,” “dear friend” arenot.

() The square of a relationis that relation which holdsbetween two terms x and zwhen there is an intermediateThis term is due to C. S. Peirce.

(original page )

term y such that the givenrelation holds between x andy and between y and z. Thus“paternal grandfather” is thesquare of “father,” “greater by” is the square of “greater by,” and so on.

() The domain of a relationconsists of all those terms thathave the relation to somethingor other, and the conversedomain consists of all thoseterms to which something orother has the relation. Thesewords have been already de-fined, but are recalled here (original page )

for the sake of the followingdefinition:—

() The field of a relationconsists of its domain andconverse domain together. |

() One relation is said tocontain or be implied by an-other if it holds whenever theother holds.

It will be seen that an asym-metrical relation is the samething as a relation whosesquare is an aliorelative. Itoften happens that a rela-tion is an aliorelative withoutbeing asymmetrical, though (original pages –)

an asymmetrical relation isalways an aliorelative. For ex-ample, “spouse” is an aliorela-tive, but is symmetrical, sinceif x is the spouse of y, y is thespouse of x. But among transi-tive relations, all aliorelativesare asymmetrical as well asvice versa.

From the definitions it willbe seen that a transitive rela-tion is one which is implied byits square, or, as we also say,“contains” its square. Thus“ancestor” is transitive, be-cause an ancestor’s ancestor (original page )

is an ancestor; but “father”is not transitive, because afather’s father is not a fa-ther. A transitive aliorela-tive is one which contains itssquare and is contained indiversity; or, what comes tothe same thing, one whosesquare implies both it anddiversity—because, when arelation is transitive, asymme-try is equivalent to being analiorelative.

A relation is connected when,given any two different termsof its field, the relation holds (original page )

between the first and the sec-ond or between the secondand the first (not excludingthe possibility that both mayhappen, though both cannothappen if the relation is asym-metrical).

It will be seen that the rela-tion “ancestor,” for example,is an aliorelative and transi-tive, but not connected; it isbecause it is not connectedthat it does not suffice to ar-range the human race in aseries.

The relation “less than or (original page )

equal to,” among numbers,is transitive and connected,but not asymmetrical or analiorelative.

The relation “greater orless” among numbers is analiorelative and is connected,but is not transitive, for if x isgreater or less than y, and y isgreater or less than z, it mayhappen that x and z are thesame number.

Thus the three properties ofbeing () an aliorelative, () |transitive, and () connected,are mutually independent, (original pages –)

since a relation may have anytwo without having the third.

We now lay down the fol-lowing definition:—

A relation is serial when it isan aliorelative, transitive, andconnected; or, what is equiva-lent, when it is asymmetrical,transitive, and connected.

A series is the same thing asa serial relation.

It might have been thoughtthat a series should be the fieldof a serial relation, not the se-rial relation itself. But thiswould be an error. For exam- (original page )

ple,

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

are six different series whichall have the same field. Ifthe field were the series, therecould only be one series witha given field. What distin-guishes the above six seriesis simply the different or-dering relations in the sixcases. Given the ordering re-lation, the field and the orderare both determinate. Thus

(original page )

the ordering relation may betaken to be the series, but thefield cannot be so taken.

Given any serial relation,say P, we shall say that, in re-spect of this relation, x “pre-cedes” y if x has the relationP to y, which we shall write“xPy” for short. The threecharacteristics which P musthave in order to be serial are:

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

() P must imply P, i.e. if x

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precedes y and y precedesz, x must precede z.

() If x and y are two differentterms in the field of P, weshall have xPy or yPx, i.e.one of the two must pre-cede the other.

The reader can easily convincehimself that, where thesethree properties are foundin an ordering relation, thecharacteristics we expect ofseries will also be found, andvice versa. We are thereforejustified in taking the above

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as a definition of order | orseries. And it will be observedthat the definition is effectedin purely logical terms.

Although a transitive asym-metrical connected relationalways exists wherever thereis a series, it is not alwaysthe relation which wouldmost naturally be regardedas generating the series. Thenatural-number series mayserve as an illustration. Therelation we assumed in con-sidering the natural numberswas the relation of immediate (original pages –)

succession, i.e. the relationbetween consecutive integers.This relation is asymmetri-cal, but not transitive or con-nected. We can, however,derive from it, by the methodof mathematical induction,the “ancestral” relation whichwe considered in the preced-ing chapter. This relation willbe the same as “less than orequal to” among inductiveintegers. For purposes of gen-erating the series of naturalnumbers, we want the relation“less than,” excluding “equal (original page )

to.” This is the relation of mto n when m is an ancestorof n but not identical with n,or (what comes to the samething) when the successor ofm is an ancestor of n in thesense in which a number isits own ancestor. That is tosay, we shall lay down thefollowing definition:—

An inductive number m issaid to be less than anothernumber n when n possessesevery hereditary propertypossessed by the successorof m. (original page )

It is easy to see, and not dif-ficult to prove, that the rela-tion “less than,” so defined, isasymmetrical, transitive, andconnected, and has the in-ductive numbers for its field.Thus by means of this rela-tion the inductive numbersacquire an order in the sensein which we defined the term“order,” and this order is theso-called “natural” order, ororder of magnitude.

The generation of series bymeans of relations more orless resembling that of n to (original page )

n + is very common. Theseries of the Kings of England,for example, is generated byrelations of each to his suc-cessor. This is probably theeasiest way, where it is ap-plicable, of conceiving thegeneration of a series. In thismethod we pass on from eachterm to the next, as long asthere | is a next, or back to theone before, as long as there isone before. This method al-ways requires the generalisedform of mathematical induc-tion in order to enable us to (original pages –)

define “earlier” and “later” ina series so generated. On theanalogy of “proper fractions,”let us give the name “properposterity of x with respect toR” to the class of those termsthat belong to the R-posterityof some term to which x hasthe relation R, in the sensewhich we gave before to “pos-terity,” which includes a termin its own posterity. Revertingto the fundamental defini-tions, we find that the “properposterity” may be defined asfollows:— (original page )

The “proper posterity” of xwith respect to R consists ofall terms that possess every R-hereditary property possessedby every term to which x hasthe relation R.

It is to be observed that thisdefinition has to be so framedas to be applicable not onlywhen there is only one termto which x has the relation R,but also in cases (as e.g. that offather and child) where theremay be many terms to whichx has the relation R. We definefurther: (original page )

A term x is a “proper ances-tor” of y with respect to R if ybelongs to the proper poster-ity of x with respect to R.

We shall speak for short of“R-posterity” and “R-ances-tors” when these terms seemmore convenient.

Reverting now to the gener-ation of series by the relationR between consecutive terms,we see that, if this methodis to be possible, the relation“proper R-ancestor” must bean aliorelative, transitive, andconnected. Under what cir- (original page )

cumstances will this occur?It will always be transitive:no matter what sort of rela-tion R may be, “R-ancestor”and “proper R-ancestor” arealways both transitive. Butit is only under certain cir-cumstances that it will bean aliorelative or connected.Consider, for example, therelation to one’s left-handneighbour at a round dinner-table at which there are twelvepeople. If we call this relationR, the proper R-posterity of aperson consists of all who can (original page )

be reached by going round thetable from right to left. Thisincludes everybody at thetable, including the personhimself, since | twelve stepsbring us back to our starting-point. Thus in such a case,though the relation “properR-ancestor” is connected, andthough R itself is an aliorel-ative, we do not get a seriesbecause “proper R-ancestor”is not an aliorelative. It is forthis reason that we cannot saythat one person comes beforeanother with respect to the (original pages –)

relation “right of” or to itsancestral derivative.

The above was an instancein which the ancestral rela-tion was connected but notcontained in diversity. An in-stance where it is containedin diversity but not connectedis derived from the ordinarysense of the word “ancestor.”If x is a proper ancestor of y,x and y cannot be the sameperson; but it is not true thatof any two persons one mustbe an ancestor of the other.

The question of the circum- (original page )

stances under which seriescan be generated by ances-tral relations derived fromrelations of consecutivenessis often important. Some ofthe most important cases arethe following: Let R be amany-one relation, and letus confine our attention tothe posterity of some term x.When so confined, the rela-tion “proper R-ancestor” mustbe connected; therefore allthat remains to ensure its be-ing serial is that it shall becontained in diversity. This (original page )

is a generalisation of the in-stance of the dinner-table.Another generalisation con-sists in taking R to be a one-one relation, and includingthe ancestry of x as well as theposterity. Here again, the onecondition required to securethe generation of a series isthat the relation “proper R-ancestor” shall be containedin diversity.

The generation of order bymeans of relations of consecu-tiveness, though important inits own sphere, is less general (original page )

than the method which usesa transitive relation to definethe order. It often happensin a series that there are aninfinite number of interme-diate terms between any twothat may be selected, how-ever near together these maybe. Take, for instance, frac-tions in order of magnitude.Between any two fractionsthere are others—for exam-ple, the arithmetic mean ofthe two. Consequently thereis no such thing as a pair ofconsecutive fractions. If we (original page )

depended | upon consecu-tiveness for defining order,we should not be able to de-fine the order of magnitudeamong fractions. But in factthe relations of greater andless among fractions do notdemand generation from rela-tions of consecutiveness, andthe relations of greater andless among fractions have thethree characteristics which weneed for defining serial rela-tions. In all such cases the or-der must be defined by meansof a transitive relation, since (original pages –)

only such a relation is ableto leap over an infinite num-ber of intermediate terms.The method of consecutive-ness, like that of counting fordiscovering the number ofa collection, is appropriateto the finite; it may even beextended to certain infiniteseries, namely, those in which,though the total number ofterms is infinite, the num-ber of terms between any twois always finite; but it mustnot be regarded as general.Not only so, but care must (original page )

be taken to eradicate fromthe imagination all habits ofthought resulting from sup-posing it general. If this is notdone, series in which thereare no consecutive terms willremain difficult and puzzling.And such series are of vitalimportance for the under-standing of continuity, space,time, and motion.

There are many ways inwhich series may be gener-ated, but all depend uponthe finding or constructionof an asymmetrical transitive (original page )

connected relation. Some ofthese ways have considerableimportance. We may take asillustrative the generation ofseries by means of a three-term relation which we maycall “between.” This methodis very useful in geometry,and may serve as an intro-duction to relations havingmore than two terms; it is bestintroduced in connection withelementary geometry.

Given any three points ona straight line in ordinaryspace, there must be one of (original page )

them which is between theother two. This will not bethe case with the points ona circle or any other closedcurve, because, given anythree points on a circle, wecan travel from any one to anyother without passing throughthe third. In fact, the notion“between” is characteristic ofopen series—or series in thestrict sense—as opposed towhat may be called | “cyclic”series, where, as with peopleat the dinner-table, a suffi-cient journey brings us back (original pages –)

to our starting-point. Thisnotion of “between” may bechosen as the fundamentalnotion of ordinary geometry;but for the present we willonly consider its applicationto a single straight line and tothe ordering of the points ona straight line. Taking anytwo points a, b, the line (ab)consists of three parts (besidesa and b themselves):

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

(original page )

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

tween x and b.() Points y such that b is be-

tween y and a.

Thus the line (ab) can be de-fined in terms of the relation“between.”

In order that this relation“between” may arrange thepoints of the line in an orderfrom left to right, we needcertain assumptions, namely,the following:—

() If anything is between a

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and b, a and b are not identi-cal.

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

() Anything between a andb is not identical with a (nor,consequently, with b, in virtueof ()).

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

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

() If x and y are betweena and b, then either x and y (original page )

are identical, or x is between aand y, or x is between y and b.

() If b is between a and xand also between a and y, theneither x and y are identical, orx is between b and y, or y isbetween b and x.

These seven properties areobviously verified in the caseof points on a straight line inordinary space. Any three-term relation which verifiesthem gives rise to series, asmay be seen from the fol-lowing definitions. For thesake of definiteness, let us (original page )

assume | that a is to the leftof b. Then the points of theline (ab) are () those betweenwhich and b, a lies—these wewill call to the left of a; ()a itself; () those between aand b; () b itself; () thosebetween which and a lies b—these we will call to the rightof b. We may now define gen-erally that of two points x, y,on the line (ab), we shall saythat x is “to the left of” y inany of the following cases:—

() When x and y are both to

(original pages –)

the left of a, and y is be-tween x and a;

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

() When x is a, and y is be-tween a and b or is b or isto the right of b;

() When x and y are both be-tween a and b, and y is be-tween x and b;

() When x is between a andb, and y is b or to the rightof b;

() When x is b and y is to theright of b;

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() When x and y are both tothe right of b and x is be-tween b and y.

It will be found that, fromthe seven properties which wehave assigned to the relation“between,” it can be deducedthat the relation “to the leftof,” as above defined, is aserial relation as we definedthat term. It is importantto notice that nothing in thedefinitions or the argumentdepends upon our meaningby “between” the actual re-

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lation of that name whichoccurs in empirical space: anythree-term relation having theabove seven purely formalproperties will serve the pur-pose of the argument equallywell.

Cyclic order, such as thatof the points on a circle, can-not be generated by meansof three-term relations of“between.” We need a re-lation of four terms, whichmay be called “separation ofcouples.” The point may beillustrated by considering a (original page )

journey round the world. Onemay go from England to NewZealand by way of Suez orby way of San Francisco; wecannot | say definitely thateither of these two places is“between” England and NewZealand. But if a man choosesthat route to go round theworld, whichever way roundhe goes, his times in Englandand New Zealand are sepa-rated from each other by histimes in Suez and San Fran-cisco, and conversely. Gen-eralising, if we take any four (original pages –)

points on a circle, we can sep-arate them into two couples,say a and b and x and y, suchthat, in order to get from ato b one must pass througheither x or y, and in order toget from x to y one must passthrough either a or b. Underthese circumstances we saythat the couple (a, b) are “sep-arated” by the couple (x, y).Out of this relation a cyclicorder can be generated, in away resembling that in whichwe generated an open orderfrom “between,” but some- (original page )

what more complicated.

The purpose of the latterhalf of this chapter has beento suggest the subject whichone may call “generation ofserial relations.” When suchrelations have been defined,the generation of them fromother relations possessingonly some of the propertiesrequired for series becomesvery important, especially inthe philosophy of geometryCf. Principles of Mathematics, p.

(§), and references theregiven.

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and physics. But we can-not, within the limits of thepresent volume, do more thanmake the reader aware thatsuch a subject exists.

(original page )

CHAPTER VKINDS OFRELATIONS

A great part of the philos-ophy of mathematics is con-cerned with relations, andmany different kinds of rela-tions have different kinds ofuses. It often happens thata property which belongs toall relations is only importantas regards relations of certain

(original page )

sorts; in these cases the readerwill not see the bearing ofthe proposition asserting sucha property unless he has inmind the sorts of relations forwhich it is useful. For reasonsof this description, as well asfrom the intrinsic interest ofthe subject, it is well to havein our minds a rough list ofthe more mathematically ser-viceable varieties of relations.

We dealt in the preced-ing chapter with a supremelyimportant class, namely, se-rial relations. Each of the (original page )

three properties which wecombined in defining series—namely, asymmetry, transitive-ness, and connexity—has itsown importance. We will be-gin by saying something oneach of these three.

Asymmetry, i.e. the propertyof being incompatible withthe converse, is a characteris-tic of the very greatest interestand importance. In order todevelop its functions, we willconsider various examples.The relation husband is asym-metrical, and so is the relation (original page )

wife; i.e. if a is husband of b, bcannot be husband of a, andsimilarly in the case of wife.On the other hand, the rela-tion “spouse” is symmetrical:if a is spouse of b, then b isspouse of a. Suppose now weare given the relation spouse,and we wish to derive the re-lation husband. Husband is thesame as male spouse or spouseof a female; thus the relationhusband can | be derived fromspouse either by limiting thedomain to males or by lim-iting the converse domain to (original pages –)

females. We see from thisinstance that, when a sym-metrical relation is given, it issometimes possible, withoutthe help of any further rela-tion, to separate it into twoasymmetrical relations. Butthe cases where this is possi-ble are rare and exceptional:they are cases where thereare two mutually exclusiveclasses, say α and β, such thatwhenever the relation holdsbetween two terms, one of theterms is a member of α andthe other is a member of β— (original page )

as, in the case of spouse, oneterm of the relation belongs tothe class of males and one tothe class of females. In sucha case, the relation with itsdomain confined to α will beasymmetrical, and so will therelation with its domain con-fined to β. But such cases arenot of the sort that occur whenwe are dealing with series ofmore than two terms; for ina series, all terms, except thefirst and last (if these exist),belong both to the domainand to the converse domain (original page )

of the generating relation, sothat a relation like husband,where the domain and con-verse domain do not overlap,is excluded.

The question how to con-struct relations having someuseful property by meansof operations upon relationswhich only have rudiments ofthe property is one of consid-erable importance. Transitive-ness and connexity are easilyconstructed in many caseswhere the originally given re-lation does not possess them: (original page )

for example, if R is any rela-tion whatever, the ancestralrelation derived from R bygeneralised induction is tran-sitive; and if R is a many-onerelation, the ancestral relationwill be connected if confinedto the posterity of a giventerm. But asymmetry is amuch more difficult propertyto secure by construction. Themethod by which we derivedhusband from spouse is, as wehave seen, not available in themost important cases, such asgreater, before, to the right of, (original page )

where domain and conversedomain overlap. In all thesecases, we can of course ob-tain a symmetrical relationby adding together the givenrelation and its converse, butwe cannot pass back fromthis symmetrical relation tothe original asymmetrical re-lation except by the help ofsome asymmetrical | relation.Take, for example, the re-lation greater: the relationgreater or less—i.e. unequal—is symmetrical, but there isnothing in this relation to (original pages –)

show that it is the sum of twoasymmetrical relations. Takesuch a relation as “differing inshape.” This is not the sum ofan asymmetrical relation andits converse, since shapes donot form a single series; butthere is nothing to show that itdiffers from “differing in mag-nitude” if we did not alreadyknow that magnitudes haverelations of greater and less.This illustrates the fundamen-tal character of asymmetry asa property of relations.

From the point of view of (original page )

the classification of relations,being asymmetrical is a muchmore important characteris-tic than implying diversity.Asymmetrical relations implydiversity, but the converse isnot the case. “Unequal,” forexample, implies diversity,but is symmetrical. Broadlyspeaking, we may say that,if we wished as far as possi-ble to dispense with relationalpropositions and replace themby such as ascribed predicatesto subjects, we could succeedin this so long as we confined (original page )

ourselves to symmetrical rela-tions: those that do not implydiversity, if they are transitive,may be regarded as assertinga common predicate, whilethose that do imply diversitymay be regarded as assertingincompatible predicates. Forexample, consider the relationof similarity between classes,by means of which we de-fined numbers. This relationis symmetrical and transitiveand does not imply diversity.It would be possible, thoughless simple than the proce- (original page )

dure we adopted, to regardthe number of a collection asa predicate of the collection:then two similar classes willbe two that have the samenumerical predicate, whiletwo that are not similar willbe two that have differentnumerical predicates. Sucha method of replacing rela-tions by predicates is formallypossible (though often veryinconvenient) so long as therelations concerned are sym-metrical; but it is formallyimpossible when the relations (original page )

are asymmetrical, becauseboth sameness and differenceof predicates are symmetri-cal. Asymmetrical relationsare, we may | say, the mostcharacteristically relational ofrelations, and the most impor-tant to the philosopher whowishes to study the ultimatelogical nature of relations.

Another class of relationsthat is of the greatest use isthe class of one-many rela-tions, i.e. relations which atmost one term can have toa given term. Such are fa- (original pages –)

ther, mother, husband (exceptin Tibet), square of, sine of,and so on. But parent, squareroot, and so on, are not one-many. It is possible, formally,to replace all relations by one-many relations by means ofa device. Take (say) the rela-tion less among the inductivenumbers. Given any numbern greater than , there will notbe only one number havingthe relation less to n, but wecan form the whole class ofnumbers that are less thann. This is one class, and its (original page )

relation to n is not shared byany other class. We may callthe class of numbers that areless than n the “proper an-cestry” of n, in the sense inwhich we spoke of ancestryand posterity in connectionwith mathematical induction.Then “proper ancestry” is aone-many relation (one-manywill always be used so as toinclude one-one), since eachnumber determines a singleclass of numbers as consti-tuting its proper ancestry.Thus the relation less than can (original page )

be replaced by being a mem-ber of the proper ancestry of.In this way a one-many re-lation in which the one is aclass, together with member-ship of this class, can alwaysformally replace a relationwhich is not one-many. Peano,who for some reason alwaysinstinctively conceives of arelation as one-many, deals inthis way with those that arenaturally not so. Reductionto one-many relations by thismethod, however, though pos-sible as a matter of form, does (original page )

not represent a technical sim-plification, and there is everyreason to think that it doesnot represent a philosophi-cal analysis, if only becauseclasses must be regarded as“logical fictions.” We shalltherefore continue to regardone-many relations as a spe-cial kind of relations.

One-many relations are in-volved in all phrases of theform “the so-and-so of such-and-such.” “The King of Eng-land,” | “the wife of Socrates,”“the father of John Stuart (original pages –)

Mill,” and so on, all describesome person by means of aone-many relation to a giventerm. A person cannot havemore than one father, there-fore “the father of John StuartMill” described some one per-son, even if we did not knowwhom. There is much to sayon the subject of descriptions,but for the present it is rela-tions that we are concernedwith, and descriptions areonly relevant as exemplifyingthe uses of one-many rela-tions. It should be observed (original page )

that all mathematical func-tions result from one-many re-lations: the logarithm of x, thecosine of x, etc., are, like thefather of x, terms described bymeans of a one-many relation(logarithm, cosine, etc.) to agiven term (x). The notionof function need not be con-fined to numbers, or to theuses to which mathematicianshave accustomed us; it canbe extended to all cases ofone-many relations, and “thefather of x” is just as legiti-mately a function of which x (original page )

is the argument as is “the log-arithm of x.” Functions in thissense are descriptive functions.As we shall see later, thereare functions of a still moregeneral and more fundamen-tal sort, namely, propositionalfunctions; but for the presentwe shall confine our attentionto descriptive functions, i.e.“the term having the relationR to x,” or, for short, “the R ofx,” where R is any one-manyrelation.

It will be observed that if“the R of x” is to describe (original page )

a definite term, x must be aterm to which something hasthe relation R, and there mustnot be more than one termhaving the relation R to x,since “the,” correctly used,must imply uniqueness. Thuswe may speak of “the fatherof x” if x is any human be-ing except Adam and Eve;but we cannot speak of “thefather of x” if x is a table ora chair or anything else thatdoes not have a father. Weshall say that the R of x “ex-ists” when there is just one (original page )

term, and no more, having therelation R to x. Thus if R isa one-many relation, the R ofx exists whenever x belongsto the converse domain of R,and not otherwise. Regarding“the R of x” as a function inthe mathematical | sense, wesay that x is the “argument”of the function, and if y is theterm which has the relation Rto x, i.e. if y is the R of x, theny is the “value” of the functionfor the argument x. If R is aone-many relation, the rangeof possible arguments to the (original pages –)

function is the converse do-main of R, and the range ofvalues is the domain. Thus therange of possible argumentsto the function “the fatherof x” is all who have fathers,i.e. the converse domain ofthe relation father, while therange of possible values forthe function is all fathers, i.e.the domain of the relation.

Many of the most impor-tant notions in the logic ofrelations are descriptive func-tions, for example: converse,domain, converse domain, field. (original page )

Other examples will occur aswe proceed.

Among one-many relations,one-one relations are a spe-cially important class. Wehave already had occasion tospeak of one-one relationsin connection with the def-inition of number, but it isnecessary to be familiar withthem, and not merely to knowtheir formal definition. Theirformal definition may be de-rived from that of one-manyrelations: they may be de-fined as one-many relations (original page )

which are also the conversesof one-many relations, i.e. asrelations which are both one-many and many-one. One-many relations may be de-fined as relations such that, ifx has the relation in questionto y, there is no other termx′ which also has the relationto y. Or, again, they may bedefined as follows: Given twoterms x and x′, the terms towhich x has the given relationand those to which x′ has ithave no member in common.Or, again, they may be de- (original page )

fined as relations such thatthe relative product of one ofthem and its converse impliesidentity, where the “relativeproduct” of two relations Rand S is that relation whichholds between x and z whenthere is an intermediate termy, such that x has the relationR to y and y has the relationS to z. Thus, for example, ifR is the relation of father toson, the relative product of Rand its converse will be therelation which holds betweenx and a man z when there is (original page )

a person y, such that x is thefather of y and y is the son ofz. It is obvious that x and zmust be | the same person. If,on the other hand, we take therelation of parent and child,which is not one-many, we canno longer argue that, if x is aparent of y and y is a child of z,x and z must be the same per-son, because one may be thefather of y and the other themother. This illustrates that itis characteristic of one-manyrelations when the relativeproduct of a relation and its (original pages –)

converse implies identity. Inthe case of one-one relationsthis happens, and also therelative product of the con-verse and the relation impliesidentity. Given a relation R,it is convenient, if x has therelation R to y, to think of yas being reached from x byan “R-step” or an “R-vector.”In the same case x will bereached from y by a “back-ward R-step.” Thus we maystate the characteristic of one-many relations with which wehave been dealing by saying (original page )

that an R-step followed by abackward R-step must bringus back to our starting-point.With other relations, this is byno means the case; for exam-ple, if R is the relation of childto parent, the relative productof R and its converse is therelation “self or brother orsister,” and if R is the relationof grandchild to grandparent,the relative product of R andits converse is “self or brotheror sister or first cousin.” Itwill be observed that the rela-tive product of two relations (original page )

is not in general commutative,i.e. the relative product of Rand S is not in general thesame relation as the relativeproduct of S and R. E.g. therelative product of parent andbrother is uncle, but the rel-ative product of brother andparent is parent.

One-one relations give acorrelation of two classes,term for term, so that eachterm in either class has itscorrelate in the other. Suchcorrelations are simplest tograsp when the two classes (original page )

have no members in common,like the class of husbands andthe class of wives; for in thatcase we know at once whethera term is to be considered asone from which the correlatingrelation R goes, or as one towhich it goes. It is convenientto use the word referent for theterm from which the relationgoes, and the term relatumfor the term to which it goes.Thus if x and y are husbandand wife, then, with respectto the relation | “husband,” xis referent and y relatum, but (original pages –)

with respect to the relation“wife,” y is referent and x re-latum. We say that a relationand its converse have opposite“senses”; thus the “sense” of arelation that goes from x to yis the opposite of that of thecorresponding relation fromy to x. The fact that a relationhas a “sense” is fundamental,and is part of the reason whyorder can be generated bysuitable relations. It will beobserved that the class of allpossible referents to a givenrelation is its domain, and the (original page )

class of all possible relata isits converse domain.

But it very often happensthat the domain and conversedomain of a one-one relationoverlap. Take, for example,the first ten integers (exclud-ing ), and add to each; thusinstead of the first ten integerswe now have the integers

, , , , , , , , , .

These are the same as those wehad before, except that hasbeen cut off at the beginningand has been joined on at (original page )

the end. There are still tenintegers: they are correlatedwith the previous ten by therelation of n to n+ , which isa one-one relation. Or, again,instead of adding to each ofour original ten integers, wecould have doubled each ofthem, thus obtaining the in-tegers

, , , , ,, , , , .

Here we still have five ofour previous set of integers,

(original page )

namely, , , , , . The cor-relating relation in this caseis the relation of a number toits double, which is again aone-one relation. Or we mighthave replaced each numberby its square, thus obtainingthe set

, , , , , , , ,, .

On this occasion only threeof our original set are left,namely, , , . Such pro-cesses of correlation may bevaried endlessly. (original page )

The most interesting caseof the above kind is the casewhere our one-one relationhas a converse domain whichis part, but | not the whole,of the domain. If, insteadof confining the domain tothe first ten integers, we hadconsidered the whole of theinductive numbers, the aboveinstances would have illus-trated this case. We may placethe numbers concerned in tworows, putting the correlatedirectly under the numberwhose correlate it is. Thus (original pages –)

when the correlator is the re-lation of n to n + , we havethe two rows:

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

When the correlator is the re-lation of a number to its dou-ble, we have the two rows:

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

When the correlator is therelation of a number to its

(original page )

square, the rows are:

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

In all these cases, all inductivenumbers occur in the top row,and only some in the bottomrow.

Cases of this sort, where theconverse domain is a “properpart” of the domain (i.e. a partnot the whole), will occupy usagain when we come to dealwith infinity. For the present,we wish only to note that they

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exist and demand considera-tion.

Another class of correla-tions which are often im-portant is the class called“permutations,” where thedomain and converse domainare identical. Consider, forexample, the six possible ar-rangements of three letters:

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

(original page )

c, b, a |

Each of these can be obtainedfrom any one of the others bymeans 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 is obvious thatthe combination of two per-mutations is again a permuta-tion, i.e. the permutations of agiven class form what is calleda “group.”

These various kinds of cor-relations have importance in (original pages –)

various connections, somefor one purpose, some foranother. The general notionof one-one correlations hasboundless importance in thephilosophy of mathematics, aswe have partly seen already,but shall see much more fullyas we proceed. One of its useswill occupy us in our nextchapter.

(original page )

CHAPTER VISIMILARITY OFRELATIONS

We saw in Chapter II. that twoclasses have the same numberof terms when they are “sim-ilar,” i.e. when there is a one-one relation whose domain isthe one class and whose con-verse domain is the other. Insuch a case we say that thereis a “one-one correlation” be-

(original page )

tween the two classes.In the present chapter we

have to define a relation be-tween relations, which willplay the same part for themthat similarity of classes playsfor classes. We will call thisrelation “similarity of rela-tions,” or “likeness” when itseems desirable to use a dif-ferent word from that whichwe use for classes. How islikeness to be defined?

We shall employ still thenotion of correlation: we shallassume that the domain of the (original page )

one relation can be correlatedwith the domain of the other,and the converse domain withthe converse domain; but thatis not enough for the sort ofresemblance which we desireto have between our two re-lations. What we desire isthat, whenever either relationholds between two terms, theother relation shall hold be-tween the correlates of thesetwo terms. The easiest ex-ample of the sort of thing wedesire is a map. When oneplace is north of another, the (original page )

place on the map correspond-ing to the one is above theplace on the map correspond-ing to the other; when oneplace is west of another, theplace on the map correspond-ing to the one is to the leftof the place on the map cor-responding to the other; andso on. The structure of themap corresponds with that of| the country of which it is amap. The space-relations inthe map have “likeness” to thespace-relations in the countrymapped. It is this kind of (original pages –)

connection between relationsthat we wish to define.

We may, in the first place,profitably introduce a certainrestriction. We will confineourselves, in defining like-ness, to such relations as have“fields,” i.e. to such as permitof the formation of a singleclass out of the domain andthe converse domain. Thisis not always the case. Take,for example, the relation “do-main,” i.e. the relation whichthe domain of a relation hasto the relation. This relation (original page )

has all classes for its domain,since every class is the domainof some relation; and it hasall relations for its conversedomain, since every relationhas a domain. But classes andrelations cannot be added to-gether to form a new singleclass, because they are of dif-ferent logical “types.” We donot need to enter upon thedifficult doctrine of types, butit is well to know when weare abstaining from enteringupon it. We may say, withoutentering upon the grounds for (original page )

the assertion, that a relationonly has a “field” when it iswhat we call “homogeneous,”i.e. when its domain and con-verse domain are of the samelogical type; and as a rough-and-ready indication of whatwe mean by a “type,” we maysay that individuals, classes ofindividuals, relations betweenindividuals, relations betweenclasses, relations of classes toindividuals, and so on, are dif-ferent types. Now the notionof likeness is not very usefulas applied to relations that are (original page )

not homogeneous; we shall,therefore, in defining like-ness, simplify our problem byspeaking of the “field” of oneof the relations concerned.This somewhat limits the gen-erality of our definition, butthe limitation is not of anypractical importance. Andhaving been stated, it need nolonger be remembered.

We may define two rela-tions P and Q as “similar,” oras having “likeness,” whenthere is a one-one relation Swhose domain is the field of (original page )

P and whose converse domainis the field of Q, and whichis such that, if one term hasthe relation P | to another, thecorrelate of the one has therelation Q to the correlate ofthe other, and vice versa. Afigure will make this clearer.

z w

x y

Q

P

S S

(original pages –)

Let x and y be two terms hav-ing the relation P. Then thereare to be two terms z, w, suchthat x has the relation S to z,y has the relation S to w, andz has the relation Q to w. Ifthis happens with every pairof terms such as x and y, andif the converse happens withevery pair of terms such as zand w, it is clear that for everyinstance in which the relationP holds there is a correspond-ing instance in which the re-lation Q holds, and vice versa;and this is what we desire to (original page )

secure by our definition. Wecan eliminate some redundan-cies in the above sketch of adefinition, by observing that,when the above conditions arerealised, the relation P is thesame as the relative productof S and Q and the converseof S, i.e. the P-step from x toy may be replaced by the suc-cession of the S-step from x toz, the Q-step from z to w, andthe backward S-step from wto y. Thus we may set up thefollowing definitions:—

A relation S is said to be (original page )

a “correlator” or an “ordinalcorrelator” of two relationsP and Q if S is one-one, hasthe field of Q for its conversedomain, and is such that P isthe relative product of S andQ and the converse of S.

Two relations P and Q aresaid to be “similar,” or to have“likeness,” when there is atleast one correlator of P andQ.

These definitions will befound to yield what we abovedecided to be necessary.

It will be found that, when (original page )

two relations are similar, theyshare all properties whichdo not depend upon the ac-tual terms in their fields. Forinstance, if one implies di-versity, so does the other; ifone is transitive, so is theother; if one is connected, sois the other. Hence if one isserial, so is the other. Again,if one is one-many or one-one, the other is one-many | orone-one; and so on, throughall the general properties ofrelations. Even statements in-volving the actual terms of the (original pages –)

field of a relation, though theymay not be true as they standwhen applied to a similar re-lation, will always be capableof translation into statementsthat are analogous. We areled by such considerationsto a problem which has, inmathematical philosophy, animportance by no means ad-equately recognised hitherto.Our problem may be stated asfollows:—

Given some statement in alanguage of which we knowthe grammar and the syn- (original page )

tax, but not the vocabulary,what are the possible mean-ings of such a statement, andwhat are the meanings of theunknown words that wouldmake it true?

The reason that this ques-tion is important is that itrepresents, much more nearlythan might be supposed, thestate of our knowledge ofnature. We know that cer-tain scientific propositions—which, in the most advancedsciences, are expressed inmathematical symbols—are (original page )

more or less true of the world,but we are very much at sea asto the interpretation to be putupon the terms which occur inthese propositions. We knowmuch more (to use, for a mo-ment, an old-fashioned pairof terms) about the form ofnature than about the matter.Accordingly, what we reallyknow when we enunciate alaw of nature is only that thereis probably some interpreta-tion of our terms which willmake the law approximatelytrue. Thus great importance (original page )

attaches to the question: Whatare the possible meanings ofa law expressed in terms ofwhich we do not know thesubstantive meaning, but onlythe grammar and syntax? Andthis question is the one sug-gested above.

For the present we will ig-nore the general question,which will occupy us againat a later stage; the subject oflikeness itself must first befurther investigated.

Owing to the fact that,when two relations are sim- (original page )

ilar, their properties are thesame except when they de-pend upon the fields beingcomposed of just the terms ofwhich they are composed, itis desirable to have a nomen-clature which collects | to-gether all the relations thatare similar to a given rela-tion. Just as we called theset of those classes that aresimilar to a given class the“number” of that class, so wemay call the set of all thoserelations that are similar to agiven relation the “number” (original pages –)

of that relation. But in or-der to avoid confusion withthe numbers appropriate toclasses, we will speak, in thiscase, of a “relation-number.”Thus we have the followingdefinitions:—

The “relation-number” of agiven relation is the class of allthose relations that are similarto the given relation.

“Relation-numbers” arethe set of all those classesof relations that are relation-numbers of various relations;or, what comes to the same (original page )

thing, a relation-number is aclass of relations consistingof all those relations that aresimilar to one member of theclass.

When it is necessary tospeak of the numbers of class-es in a way which makes itimpossible to confuse themwith relation-numbers, weshall call them “cardinal num-bers.” Thus cardinal numbersare the numbers appropri-ate to classes. These includethe ordinary integers of dailylife, and also certain infinite (original page )

numbers, of which we shallspeak later. When we speak of“numbers” without qualifica-tion, we are to be understoodas meaning cardinal numbers.The definition of a cardinalnumber, it will be remem-bered, is as follows:—

The “cardinal number” ofa given class is the set of allthose classes that are similarto the given class.

The most obvious appli-cation of relation-numbersis to series. Two series maybe regarded as equally long (original page )

when they have the samerelation-number. Two fi-nite series will have the samerelation-number when theirfields have the same cardi-nal number of terms, andonly then—i.e. a series of (say) terms will have the samerelation-number as any otherseries of fifteen terms, but willnot have the same relation-number as a series of or terms, nor, of course, thesame relation-number as arelation which is not serial.Thus, in the quite special (original page )

case of finite series, there isparallelism between cardinaland relation-numbers. Therelation-numbers applicableto series may be | called “se-rial numbers” (what are com-monly called “ordinal num-bers” are a sub-class of these);thus a finite serial number isdeterminate when we knowthe cardinal number of termsin the field of a series havingthe serial number in question.If n is a finite cardinal num-ber, the relation-number ofa series which has n terms is (original pages –)

called the “ordinal” number n.(There are also infinite ordi-nal numbers, but of them weshall speak in a later chapter.)When the cardinal number ofterms in the field of a series isinfinite, the relation-numberof the series is not determinedmerely by the cardinal num-ber, indeed an infinite numberof relation-numbers exist forone infinite cardinal num-ber, as we shall see when wecome to consider infinite se-ries. When a series is infinite,what we may call its “length,” (original page )

i.e. its relation-number, mayvary without change in thecardinal number; but whena series is finite, this cannothappen.

We can define addition andmultiplication for relation-numbers as well as for car-dinal numbers, and a wholearithmetic of relation-num-bers can be developed. Themanner in which this is to bedone is easily seen by con-sidering the case of series.Suppose, for example, thatwe wish to define the sum of (original page )

two non-overlapping series insuch a way that the relation-number of the sum shall be ca-pable of being defined as thesum of the relation-numbersof the two series. In the firstplace, it is clear that there isan order involved as betweenthe two series: one of themmust be placed before theother. Thus if P and Q arethe generating relations of thetwo series, in the series whichis their sum with P put beforeQ, every member of the fieldof P will precede every mem- (original page )

ber of the field of Q. Thus theserial relation which is to bedefined as the sum of P andQ is not “P or Q” simply, but“P or Q or the relation of anymember of the field of P toany member of the field ofQ.” Assuming that P and Q donot overlap, this relation is se-rial, but “P or Q” is not serial,being not connected, sinceit does not hold between amember of the field of P and amember of the field of Q. Thusthe sum of P and Q, as abovedefined, is what we need in (original page )

order | to define the sum oftwo relation-numbers. Simi-lar modifications are neededfor products and powers. Theresulting arithmetic does notobey the commutative law:the sum or product of tworelation-numbers generallydepends upon the order inwhich they are taken. But itobeys the associative law, oneform of the distributive law,and two of the formal laws forpowers, not only as applied toserial numbers, but as appliedto relation-numbers generally. (original pages –)

Relation-arithmetic, in fact,though recent, is a thoroughlyrespectable branch of mathe-matics.

It must not be supposed,merely because series affordthe most obvious applicationof the idea of likeness, thatthere are no other applicationsthat are important. We havealready mentioned maps, andwe might extend our thoughtsfrom this illustration to geom-etry generally. If the systemof relations by which a ge-ometry is applied to a certain (original page )

set of terms can be broughtfully into relations of like-ness with a system applyingto another set of terms, thenthe geometry of the two setsis indistinguishable from themathematical point of view,i.e. all the propositions arethe same, except for the factthat they are applied in onecase to one set of terms and inthe other to another. We mayillustrate this by the relationsof the sort that may be called“between,” which we consid-ered in Chapter IV. We there (original page )

saw that, provided a three-term relation has certain for-mal logical properties, it willgive rise to series, and may becalled a “between-relation.”Given any two points, we canuse the between-relation todefine the straight line deter-mined by those two points; itconsists of a and b togetherwith all points x, such thatthe between-relation holdsbetween the three points a,b, x in some order or other.It has been shown by O. Ve-blen that we may regard our (original page )

whole space as the field of athree-term between-relation,and define our geometry bythe properties we assign toour between-relation. Nowlikeness is just as easily | de-finable between three-termrelations as between two-termrelations. If B and B′ aretwo between-relations, so thatThis does not apply to elliptic

space, but only to spaces in which thestraight line is an open series. ModernMathematics, edited by J. W. A. Young,pp. – (monograph by O. Veblen on“The Foundations of Geometry”).

(original pages –)

“xB(y, z)” means “x is betweeny and z with respect to B,”we shall call S a correlator ofB and B′ if it has the field ofB′ for its converse domain,and is such that the relationB holds between three termswhen B′ holds between theirS-correlates, and only then.And we shall say that B is likeB′ when there is at least onecorrelator of B with B′. Thereader can easily convincehimself that, if B is like B′

in this sense, there can be nodifference between the geom- (original page )

etry generated by B and thatgenerated by B′ .

It follows from this thatthe mathematician need notconcern himself with the par-ticular being or intrinsic na-ture of his points, lines, andplanes, even when he is spec-ulating as an applied math-ematician. We may say thatthere is empirical evidenceof the approximate truth ofsuch parts of geometry as arenot matters of definition. Butthere is no empirical evidenceas to what a “point” is to be. (original page )

It has to be something that asnearly as possible satisfies ouraxioms, but it does not haveto be “very small” or “with-out parts.” Whether or notit is those things is a matterof indifference, so long as itsatisfies the axioms. If wecan, out of empirical material,construct a logical structure,no matter how complicated,which will satisfy our geomet-rical axioms, that structuremay legitimately be calleda “point.” We must not saythat there is nothing else that (original page )

could legitimately be calleda “point”; we must only say:“This object we have con-structed is sufficient for thegeometer; it may be one ofmany objects, any of whichwould be sufficient, but that isno concern of ours, since thisobject is enough to vindicatethe empirical truth of geom-etry, in so far as geometry isnot a matter of definition.”This is only an illustrationof the general principle thatwhat matters in mathematics,and to a very great extent in (original page )

physical science, is not theintrinsic nature of our terms,but the logical nature of theirinterrelations.

We may say, of two similarrelations, that they have thesame | “structure.” For mathe-matical purposes (though notfor those of pure philosophy)the only thing of importanceabout a relation is the cases inwhich it holds, not its intrinsicnature. Just as a class may bedefined by various differentbut co-extensive concepts—e.g. “man” and “featherless bi- (original pages –)

ped”—so two relations whichare conceptually different mayhold in the same set of in-stances. An “instance” inwhich a relation holds is tobe conceived as a couple ofterms, with an order, so thatone of the terms comes firstand the other second; the cou-ple is to be, of course, suchthat its first term has the rela-tion in question to its second.Take (say) the relation “fa-ther”: we can define what wemay call the “extension” ofthis relation as the class of all (original page )

ordered couples (x, y) whichare such that x is the fatherof y. From the mathematicalpoint of view, the only thingof importance about the rela-tion “father” is that it definesthis set of ordered couples.Speaking generally, we say:

The “extension” of a re-lation is the class of thoseordered couples (x, y) whichare such that x has the relationin question to y.

We can now go a step fur-ther in the process of abstrac-tion, and consider what we (original page )

mean by “structure.” Givenany relation, we can, if it is asufficiently simple one, con-struct a map of it. For the sakeof definiteness, let us take arelation of which the exten-sion is the following couples:ab, ac, ad, bc, ce, dc, de, wherea, b, c, d, e are five terms, nomatter what. We may make a“map” of this relation by tak-ing five points on a plane andconnecting them by arrows, asin the accompanying figure.What is revealed by the map

(original page )

d c

a b

e

is what we call the “structure”of the relation.

It is clear that the “struc-ture” of the relation does not (original page )

depend upon the particularterms that make up the fieldof the relation. The field maybe changed without chang-ing the structure, and thestructure may be changedwithout changing the field—for | example, if we wereto add the couple ae in theabove illustration we shouldalter the structure but not thefield. Two relations have thesame “structure,” we shallsay, when the same map willdo for both—or, what comesto the same thing, when ei- (original pages –)

ther can be a map for theother (since every relationcan be its own map). Andthat, as a moment’s reflec-tion shows, is the very samething as what we have called“likeness.” That is to say, tworelations have the same struc-ture when they have likeness,i.e. when they have the samerelation-number. Thus whatwe defined as the “relation-number” is the very samething as is obscurely intendedby the word “structure”—aword which, important as it (original page )

is, is never (so far as we know)defined in precise terms bythose who use it.

There has been a great dealof speculation in traditionalphilosophy which might havebeen avoided if the impor-tance of structure, and thedifficulty of getting behind it,had been realised. For exam-ple, it is often said that spaceand time are subjective, butthey have objective counter-parts; or that phenomena aresubjective, but are caused bythings in themselves, which (original page )

must have differences inter secorresponding with the dif-ferences in the phenomena towhich they give rise. Wheresuch hypotheses are made,it is generally supposed thatwe can know very little aboutthe objective counterparts. Inactual fact, however, if thehypotheses as stated werecorrect, the objective coun-terparts would form a worldhaving the same structure asthe phenomenal world, andallowing us to infer from phe-nomena the truth of all propo- (original page )

sitions that can be stated inabstract terms and are knownto be true of phenomena. Ifthe phenomenal world hasthree dimensions, so mustthe world behind phenomena;if the phenomenal world isEuclidean, so must the otherbe; and so on. In short, everyproposition having a commu-nicable significance must betrue of both worlds or of nei-ther: the only difference mustlie in just that essence of indi-viduality which always eludeswords and baffles descrip- (original page )

tion, but which, for that veryreason, is irrelevant to sci-ence. Now the only purposethat philosophers | have inview in condemning phenom-ena is in order to persuadethemselves and others thatthe real world is very differentfrom the world of appearance.We can all sympathise withtheir wish to prove such avery desirable proposition,but we cannot congratulatethem on their success. It istrue that many of them do notassert objective counterparts (original pages –)

to phenomena, and these es-cape from the above argu-ment. Those who do assertcounterparts are, as a rule,very reticent on the subject,probably because they feelinstinctively that, if pursued,it will bring about too muchof a rapprochement betweenthe real and the phenomenalworld. If they were to pursuethe topic, they could hardlyavoid the conclusions whichwe have been suggesting. Insuch ways, as well as in manyothers, the notion of structure (original page )

or relation-number is impor-tant.

(original page )

CHAPTER VIIRATIONAL, REAL,AND COMPLEX

NUMBERS

We have now seen how to de-fine cardinal numbers, andalso relation-numbers, ofwhich what are commonlycalled ordinal numbers area particular species. It willbe found that each of thesekinds of number may be in-

(original page )

finite just as well as finite.But neither is capable, as itstands, of the more famil-iar extensions of the idea ofnumber, namely, the exten-sions to negative, fractional,irrational, and complex num-bers. In the present chapterwe shall briefly supply logicaldefinitions of these variousextensions.

One of the mistakes thathave delayed the discoveryof correct definitions in thisregion is the common ideathat each extension of number (original page )

included the previous sorts asspecial cases. It was thoughtthat, in dealing with posi-tive and negative integers,the positive integers might beidentified with the originalsignless integers. Again it wasthought that a fraction whosedenominator is may be iden-tified with the natural numberwhich is its numerator. Andthe irrational numbers, suchas the square root of , weresupposed to find their placeamong rational fractions, asbeing greater than some of (original page )

them and less than the others,so that rational and irrationalnumbers could be taken to-gether as one class, called“real numbers.” And whenthe idea of number was fur-ther extended so as to include“complex” numbers, i.e. num-bers involving the square rootof −, it was thought that realnumbers could be regarded asthose among complex num-bers in which the imaginarypart (i.e. the part | which wasa multiple of the square rootof −) was zero. All these sup- (original pages –)

positions were erroneous, andmust be discarded, as we shallfind, if correct definitions areto be given.

Let us begin with positiveand negative integers. It isobvious on a moment’s con-sideration that + and −must both be relations, andin fact must be each other’sconverses. The obvious andsufficient definition is that+ is the relation of n + ton, and − is the relation ofn to n + . Generally, if m isany inductive number, +m (original page )

will be the relation of n + mto n (for any n), and −m willbe the relation of n to n +m.According to this definition,+m is a relation which is one-one so long as n is a cardinalnumber (finite or infinite) andm is an inductive cardinalnumber. But +m is under nocircumstances capable of be-ing identified with m, whichis not a relation, but a class ofclasses. Indeed, +m is everybit as distinct from m as −mis.

Fractions are more interest- (original page )

ing than positive or negativeintegers. We need fractionsfor many purposes, but per-haps most obviously for pur-poses of measurement. Myfriend and collaborator Dr A.N. Whitehead has developeda theory of fractions speciallyadapted for their applicationto measurement, which is setforth in Principia Mathemat-ica. But if all that is neededis to define objects having therequired purely mathemati-

Vol. iii. ∗ff., especially .

(original page )

cal properties, this purposecan be achieved by a simplermethod, which we shall hereadopt. We shall define thefraction m/n as being that re-lation which holds betweentwo inductive numbers x, ywhen xn = ym. This definitionenables us to prove thatm/n isa one-one relation, providedneither m nor n is zero. Andof course n/m is the converserelation to m/n.

From the above definition itis clear that the fraction m/is that relation between two (original page )

integers x and y which con-sists in the fact that x = my.This relation, like the relation+m, is by no means capableof being identified with theinductive cardinal numberm, because a relation and aclass of classes are objects |of utterly different kinds. It

Of course in practice we shallcontinue to speak of a fraction as (say)greater or less than , meaning greateror less than the ratio /. So long asit is understood that the ratio / andthe cardinal number are different, itis not necessary to be always pedantic

(original pages –)

will be seen that /n is alwaysthe same relation, whateverinductive number n may be;it is, in short, the relation of to any other inductive car-dinal. We may call this thezero of rational numbers; itis not, of course, identicalwith the cardinal number .Conversely, the relation m/is always the same, whateverinductive number m may be.There is not any inductive car-dinal to correspond to m/.

in emphasising the difference.

(original page )

We may call it “the infinityof rationals.” It is an instanceof the sort of infinite that istraditional in mathematics,and that is represented by“∞.” This is a totally differentsort from the true Cantorianinfinite, which we shall con-sider in our next chapter. Theinfinity of rationals does notdemand, for its definition oruse, any infinite classes orinfinite integers. It is not,in actual fact, a very impor-tant notion, and we coulddispense with it altogether if (original page )

there were any object in doingso. The Cantorian infinite, onthe other hand, is of the great-est and most fundamental im-portance; the understandingof it opens the way to wholenew realms of mathematicsand philosophy.

It will be observed that zeroand infinity, alone among ra-tios, are not one-one. Zerois one-many, and infinity ismany-one.

There is not any difficultyin defining greater and lessamong ratios (or fractions). (original page )

Given two ratios m/n and p/q,we shall say that m/n is lessthan p/q if mq is less thanpn. There is no difficulty inproving that the relation “lessthan,” so defined, is serial, sothat the ratios form a series inorder of magnitude. In this se-ries, zero is the smallest termand infinity is the largest.If we omit zero and infinityfrom our series, there is nolonger any smallest or largestratio; it is obvious that if m/nis any ratio other than zeroand infinity, m/n is smaller (original page )

and m/n is larger, thoughneither is zero or infinity, sothat m/n is neither the small-est | nor the largest ratio, andtherefore (when zero and in-finity are omitted) there isno smallest or largest, sincem/n was chosen arbitrarily.In like manner we can provethat however nearly equal twofractions may be, there are al-ways other fractions betweenthem. For, let m/n and p/qbe two fractions, of whichp/q is the greater. Then it iseasy to see (or to prove) that (original pages –)

(m+ p)/(n+ q) will be greaterthan m/n and less than p/q.Thus the series of ratios isone in which no two termsare consecutive, but there arealways other terms betweenany two. Since there are otherterms between these others,and so on ad infinitum, it isobvious that there are an infi-nite number of ratios betweenany two, however nearly equalthese two may be. A series

Strictly speaking, this statement,as well as those following to the end of

(original page )

having the property that thereare always other terms be-tween any two, so that notwo are consecutive, is called“compact.” Thus the ratiosin order of magnitude form a“compact” series. Such serieshave many important prop-erties, and it is important toobserve that ratios afford aninstance of a compact seriesgenerated purely logically,without any appeal to space

the paragraph, involves what is calledthe “axiom of infinity,” which will bediscussed in a later chapter.

(original page )

or time or any other empiricaldatum.

Positive and negative ratioscan be defined in a way anal-ogous to that in which we de-fined positive and negative in-tegers. Having first definedthe sum of two ratios m/n andp/q as (mq + pn)/nq, we define+p/q as the relation of m/n +p/q to m/n, where m/n is anyratio; and −p/q is of course theconverse of +p/q. This is notthe only possible way of defin-ing positive and negative ra-tios, but it is a way which, for (original page )

our purpose, has the merit ofbeing an obvious adaptationof the way we adopted in thecase of integers.

We come now to a moreinteresting extension of theidea of number, i.e. the exten-sion to what are called “real”numbers, which are the kindthat embrace irrationals. InChapter I. we had occasionto mention “incommensu-rables” and their | discoveryby Pythagoras. It was throughthem, i.e. through geometry,that irrational numbers were (original pages –)

first thought of. A square ofwhich the side is one inch longwill have a diagonal of whichthe length is the square rootof inches. But, as the an-cients discovered, there is nofraction of which the square is. This proposition is provedin the tenth book of Euclid,which is one of those booksthat schoolboys supposed tobe fortunately lost in the dayswhen Euclid was still usedas a text-book. The proof isextraordinarily simple. If pos-sible, let m/n be the square (original page )

root of , so that m/n = ,i.e. m = n. Thus m is aneven number, and thereforem must be an even number,because the square of an oddnumber is odd. Now if m iseven, m must divide by ,for if m = p, then m = p.Thus we shall have p = n,where p is half of m. Hencep = n, and therefore n/pwill also be the square root of. But then we can repeat theargument: if n = q, p/q willalso be the square root of ,and so on, through an unend- (original page )

ing series of numbers that areeach half of its predecessor.But this is impossible; if wedivide a number by , andthen halve the half, and so on,we must reach an odd numberafter a finite number of steps.Or we may put the argumenteven more simply by assum-ing that the m/n we start withis in its lowest terms; in thatcase, m and n cannot both beeven; yet we have seen that, ifm/n = , they must be. Thusthere cannot be any fractionm/n whose square is . (original page )

Thus no fraction will ex-press exactly the length of thediagonal of a square whoseside is one inch long. Thisseems like a challenge thrownout by nature to arithmetic.However the arithmeticianmay boast (as Pythagoras did)about the power of numbers,nature seems able to bafflehim by exhibiting lengthswhich no numbers can esti-mate in terms of the unit. Butthe problem did not remainin this geometrical form. Assoon as algebra was invented, (original page )

the same problem arose asregards the solution of equa-tions, though here it took ona wider form, since it alsoinvolved complex numbers.

It is clear that fractionscan be found which approachnearer | and nearer to hav-ing their square equal to .We can form an ascendingseries of fractions all of whichhave their squares less than ,but differing from in theirlater members by less thanany assigned amount. That isto say, suppose I assign some (original pages –)

small amount in advance, sayone-billionth, it will be foundthat all the terms of our se-ries after a certain one, saythe tenth, have squares thatdiffer from by less than thisamount. And if I had as-signed a still smaller amount,it might have been necessaryto go further along the series,but we should have reachedsooner or later a term in theseries, say the twentieth, afterwhich all terms would havehad squares differing from by less than this still smaller (original page )

amount. If we set to work toextract the square root of bythe usual arithmetical rule,we shall obtain an unendingdecimal which, taken to so-and-so many places, exactlyfulfils the above conditions.We can equally well form adescending series of fractionswhose squares are all greaterthan , but greater by con-tinually smaller amounts aswe come to later terms of theseries, and differing, sooneror later, by less than any as-signed amount. In this way (original page )

we seem to be drawing a cor-don round the square root of, and it may seem difficultto believe that it can perma-nently escape us. Neverthe-less, it is not by this methodthat we shall actually reachthe square root of .

If we divide all ratios intotwo classes, according as theirsquares are less than or not,we find that, among thosewhose squares are not lessthan , all have their squaresgreater than . There is nomaximum to the ratios whose (original page )

square is less than , andno minimum to those whosesquare is greater than . Thereis no lower limit short of zeroto the difference between thenumbers whose square is alittle less than and the num-bers whose square is a littlegreater than . We can, inshort, divide all ratios intotwo classes such that all theterms in one class are lessthan all in the other, there isno maximum to the one class,and there is no minimum tothe other. Between these two (original page )

classes, where√ ought to

be, there is nothing. Thusour | cordon, though we havedrawn it as tight as possible,has been drawn in the wrongplace, and has not caught

√.

The above method of divid-ing all the terms of a seriesinto two classes, of which theone wholly precedes the other,was brought into prominenceby Dedekind, and is there-fore called a “Dedekind cut.”Stetigkeit und irrationale Zahlen,

nd edition, Brunswick, .

(original pages –)

With respect to what happensat the point of section, thereare four possibilities: () theremay be a maximum to thelower section and a minimumto the upper section, () theremay be a maximum to the oneand no minimum to the other,() there may be no maximumto the one, but a minimumto the other, () there may beneither a maximum to the onenor a minimum to the other.Of these four cases, the firstis illustrated by any series inwhich there are consecutive (original page )

terms: in the series of integers,for instance, a lower sectionmust end with some numbern and the upper section mustthen begin with n + . Thesecond case will be illustratedin the series of ratios if wetake as our lower section allratios up to and including, and in our upper sectionall ratios greater than . Thethird case is illustrated if wetake for our lower section allratios less than , and for ourupper section all ratios from upward (including itself). (original page )

The fourth case, as we haveseen, is illustrated if we putin our lower section all ratioswhose square is less than ,and in our upper section allratios whose square is greaterthan .

We may neglect the first ofour four cases, since it onlyarises in series where thereare consecutive terms. In thesecond of our four cases, wesay that the maximum of thelower section is the lower limitof the upper section, or ofany set of terms chosen out (original page )

of the upper section in such away that no term of the uppersection is before all of them.In the third of our four cases,we say that the minimum ofthe upper section is the up-per limit of the lower section,or of any set of terms chosenout of the lower section insuch a way that no term ofthe lower section is after allof them. In the fourth case,we say that | there is a “gap”:neither the upper section northe lower has a limit or a lastterm. In this case, we may (original pages –)

also say that we have an “irra-tional section,” since sectionsof the series of ratios have“gaps” when they correspondto irrationals.

What delayed the true the-ory of irrationals was a mis-taken belief that there must be“limits” of series of ratios. Thenotion of “limit” is of the ut-most importance, and beforeproceeding further it will bewell to define it.

A term x is said to be an“upper limit” of a class α withrespect to a relation P if () (original page )

α has no maximum in P, ()every member of α whichbelongs to the field of P pre-cedes x, () every member ofthe field of P which precedesx precedes some member of α.(By “precedes” we mean “hasthe relation P to.”)

This presupposes the fol-lowing definition of a “maxi-mum”:—

A term x is said to be a“maximum” of a class α withrespect to a relation P if x is amember of α and of the fieldof P and does not have the re- (original page )

lation P to any other memberof α.

These definitions do not de-mand that the terms to whichthey are applied should bequantitative. For example,given a series of moments oftime arranged by earlier andlater, their “maximum” (ifany) will be the last of themoments; but if they are ar-ranged by later and earlier,their “maximum” (if any) willbe the first of the moments.

The “minimum” of a classwith respect to P is its maxi- (original page )

mum with respect to the con-verse of P; and the “lowerlimit” with respect to P is theupper limit with respect tothe converse of P.

The notions of limit andmaximum do not essentiallydemand that the relation inrespect to which they are de-fined should be serial, butthey have few important ap-plications except to caseswhen the relation is serial orquasi-serial. A notion which isoften important is the notion“upper limit or maximum,” to (original page )

which we may give the name“upper boundary.” Thus the“upper boundary” of a set ofterms chosen out of a series istheir last member if they haveone, but, if not, it is the firstterm after all of them, if thereis such a term. If there is nei-ther | a maximum nor a limit,there is no upper boundary.The “lower boundary” is thelower limit or minimum.

Reverting to the four kindsof Dedekind section, we seethat in the case of the firstthree kinds each section has (original pages –)

a boundary (upper or loweras the case may be), while inthe fourth kind neither hasa boundary. It is also clearthat, whenever the lower sec-tion has an upper boundary,the upper section has a lowerboundary. In the second andthird cases, the two bound-aries are identical; in the first,they are consecutive terms ofthe series.

A series is called “Dedekin-dian” when every section hasa boundary, upper or lower asthe case may be. (original page )

We have seen that the seriesof ratios in order of magnitudeis not Dedekindian.

From the habit of beinginfluenced by spatial imagi-nation, people have supposedthat series must have lim-its in cases where it seemsodd if they do not. Thus,perceiving that there was norational limit to the ratioswhose square is less than ,they allowed themselves to“postulate” an irrational limit,which was to fill the Dedekindgap. Dedekind, in the above- (original page )

mentioned work, set up theaxiom that the gap must al-ways be filled, i.e. that everysection must have a boundary.It is for this reason that serieswhere his axiom is verifiedare called “Dedekindian.” Butthere are an infinite numberof series for which it is notverified.

The method of “postulat-ing” what we want has manyadvantages; they are the sameas the advantages of theft overhonest toil. Let us leave themto others and proceed with (original page )

our honest toil.It is clear that an irrational

Dedekind cut in some way“represents” an irrational. Inorder to make use of this,which to begin with is nomore than a vague feeling,we must find some way ofeliciting from it a precise defi-nition; and in order to do this,we must disabuse our mindsof the notion that an irrationalmust be the limit of a set ofratios. Just as ratios whosedenominator is are not iden-tical with integers, so those (original page )

rational | numbers which canbe greater or less than irra-tionals, or can have irrationalsas their limits, must not beidentified with ratios. Wehave to define a new kind ofnumbers called “real num-bers,” of which some will berational and some irrational.Those that are rational “corre-spond” to ratios, in the samekind of way in which the ration/ corresponds to the integern; but they are not the same asratios. In order to decide whatthey are to be, let us observe (original pages –)

that an irrational is repre-sented by an irrational cut,and a cut is represented by itslower section. Let us confineourselves to cuts in which thelower section has no maxi-mum; in this case we will callthe lower section a “segment.”Then those segments that cor-respond to ratios are thosethat consist of all ratios lessthan the ratio they correspondto, which is their boundary;while those that represent ir-rationals are those that haveno boundary. Segments, both (original page )

those that have boundariesand those that do not, aresuch that, of any two pertain-ing to one series, one mustbe part of the other; hencethey can all be arranged in aseries by the relation of wholeand part. A series in whichthere are Dedekind gaps, i.e.in which there are segmentsthat have no boundary, willgive rise to more segmentsthan it has terms, since eachterm will define a segmenthaving that term for bound-ary, and then the segments (original page )

without boundaries will beextra.

We are now in a position todefine a real number and anirrational number.

A “real number” is a seg-ment of the series of ratios inorder of magnitude.

An “irrational number” is asegment of the series of ratioswhich has no boundary.

A “rational real number” isa segment of the series of ra-tios which has a boundary.

Thus a rational real num-ber consists of all ratios less (original page )

than a certain ratio, and it isthe rational real number cor-responding to that ratio. Thereal number , for instance, isthe class of proper fractions. |

In the cases in which wenaturally supposed that anirrational must be the limitof a set of ratios, the truthis that it is the limit of thecorresponding set of rationalreal numbers in the series ofsegments ordered by wholeand part. For example,

√ is

the upper limit of all thosesegments of the series of ra- (original pages –)

tios that correspond to ratioswhose square is less than .More simply still,

√ is the

segment consisting of all thoseratios whose square is lessthan .

It is easy to prove that theseries of segments of any se-ries is Dedekindian. For, givenany set of segments, theirboundary will be their logicalsum, i.e. the class of all thoseterms that belong to at leastone segment of the set.

For a fuller treatment of the sub-

(original page )

The above definition ofreal numbers is an exampleof “construction” as against“postulation,” of which wehad another example in thedefinition of cardinal num-bers. The great advantageof this method is that it re-quires no new assumptions,but enables us to proceed de-

ject of segments and Dedekindian re-lations, see Principia Mathematica, vol.ii. ∗–. For a fuller treatment ofreal numbers, see ibid., vol. iii. ∗ff.,and Principles of Mathematics, chaps.xxxiii. and xxxiv.

(original page )

ductively from the originalapparatus of logic.

There is no difficulty indefining addition and multi-plication for real numbers asabove defined. Given two realnumbers µ and ν, each be-ing a class of ratios, take anymember of µ and any memberof ν and add them togetheraccording to the rule for theaddition of ratios. Form theclass of all such sums obtain-able by varying the selectedmembers of µ and ν. Thisgives a new class of ratios, and (original page )

it is easy to prove that thisnew class is a segment of theseries of ratios. We define itas the sum of µ and ν. Wemay state the definition moreshortly as follows:—

The arithmetical sum of tworeal numbers is the class of thearithmetical sums of a mem-ber of the one and a memberof the other chosen in all pos-sible ways. |

We can define the arith-metical product of two realnumbers in exactly the sameway, by multiplying a member (original pages –)

of the one by a member of theother in all possible ways. Theclass of ratios thus generatedis defined as the product ofthe two real numbers. (In allsuch definitions, the seriesof ratios is to be defined asexcluding and infinity.)

There is no difficulty in ex-tending our definitions to pos-itive and negative real num-bers and their addition andmultiplication.

It remains to give the defi-nition of complex numbers.

Complex numbers, though (original page )

capable of a geometrical inter-pretation, are not demandedby geometry in the same im-perative way in which irra-tionals are demanded. A“complex” number means anumber involving the squareroot of a negative number,whether integral, fractional,or real. Since the square ofa negative number is posi-tive, a number whose squareis to be negative has to be anew sort of number. Usingthe letter i for the square rootof −, any number involving (original page )

the square root of a nega-tive number can be expressedin the form x + yi, where xand y are real. The part yi iscalled the “imaginary” partof this number, x being the“real” part. (The reason forthe phrase “real numbers”is that they are contrastedwith such as are “imaginary.”)Complex numbers have beenfor a long time habitually usedby mathematicians, in spiteof the absence of any precisedefinition. It has been simplyassumed that they would obey (original page )

the usual arithmetical rules,and on this assumption theiremployment has been foundprofitable. They are requiredless for geometry than for al-gebra and analysis. We desire,for example, to be able to saythat every quadratic equationhas two roots, and every cu-bic equation has three, andso on. But if we are con-fined to real numbers, such anequation as x + = has noroots, and such an equation asx − = has only one. Ev-ery generalisation of number (original page )

has first presented itself asneeded for some simple prob-lem: negative numbers wereneeded in order that subtrac-tion might be always possible,since otherwise a − b wouldbe meaningless if a were lessthan b; fractions were needed| in order that division mightbe always possible; and com-plex numbers are needed inorder that extraction of rootsand solution of equations maybe always possible. But ex-tensions of number are notcreated by the mere need for (original pages –)

them: they are created by thedefinition, and it is to the def-inition of complex numbersthat we must now turn ourattention.

A complex number may beregarded and defined as sim-ply an ordered couple of realnumbers. Here, as elsewhere,many definitions are possible.All that is necessary is thatthe definitions adopted shalllead to certain properties. Inthe case of complex numbers,if they are defined as orderedcouples of real numbers, we (original page )

secure at once some of theproperties required, namely,that two real numbers arerequired to determine a com-plex number, and that amongthese we can distinguish afirst and a second, and thattwo complex numbers areonly identical when the firstreal number involved in theone is equal to the first in-volved in the other, and thesecond to the second. What isneeded further can be securedby defining the rules of addi-tion and multiplication. We (original page )

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 couplesof real numbers, (x,y) and(x′, y′), their sum is to be thecouple (x+x′, y+y′), and theirproduct is to be the couple(xx′ − yy′, xy′ + x′y). By thesedefinitions we shall securethat our ordered couples shall (original page )

have the properties we desire.For example, take the productof the two couples (, y) and(, y′). This will, by the aboverule, be the couple (−yy′,).Thus the square of the cou-ple (,) will be the couple(−,). Now those couples inwhich the second term is are those which, according tothe usual nomenclature, havetheir imaginary part zero; inthe notation x + yi, they arex + i, which it is natural towrite simply x. Just as it isnatural (but erroneous) | to (original pages –)

identify ratios whose denom-inator is unity with integers,so it is natural (but erroneous)to identify complex numberswhose imaginary part is zerowith real numbers. Althoughthis is an error in theory, itis a convenience in practice;“x+ i” may be replaced sim-ply by “x” and “ + yi” by“yi,” provided we rememberthat the “x” is not really a realnumber, but a special case ofa complex number. And wheny is , “yi” may of course bereplaced by “i.” Thus the (original page )

couple (,) is representedby i, and the couple (−,) isrepresented by −. Now ourrules of multiplication makethe square of (,) equal to(−,), i.e. the square of i is−. This is what we desired tosecure. Thus our definitionsserve all necessary purposes.

It is easy to give a geometri-cal interpretation of complexnumbers in the geometry ofthe plane. This subject wasagreeably expounded by W. K.Clifford in his Common Senseof the Exact Sciences, a book of (original page )

great merit, but written beforethe importance of purely log-ical definitions had been re-alised.

Complex numbers of ahigher order, though muchless useful and importantthan those what we have beendefining, have certain usesthat are not without impor-tance in geometry, as maybe seen, for example, in DrWhitehead’s Universal Alge-bra. The definition of complexnumbers of order n is obtainedby an obvious extension of the (original page )

definition we have given. Wedefine a complex number oforder n as a one-many rela-tion whose domain consistsof certain real numbers andwhose converse domain con-sists of the integers from to n. This is what would or-dinarily be indicated by thenotation (x, x, x, . . . xn),where the suffixes denote cor-relation with the integers usedas suffixes, and the correlation

Cf. Principles of Mathematics,§, p. .

(original page )

is one-many, not necessarilyone-one, because xr and xsmay be equal when r and s arenot equal. The above defini-tion, with a suitable rule ofmultiplication, will serve allpurposes for which complexnumbers of higher orders areneeded.

We have now completed ourreview of those extensions ofnumber which do not involveinfinity. The application ofnumber to infinite collectionsmust be our next topic.

(original page )

CHAPTER VIIIINFINITE CARDINAL

NUMBERS

The definition of cardinalnumbers which we gave inChapter II. was applied inChapter III. to finite numbers,i.e. to the ordinary naturalnumbers. To these we gavethe name “inductive num-bers,” because we found thatthey are to be defined as num-

(original page )

bers which obey mathematicalinduction starting from . Butwe have not yet consideredcollections which do not havean inductive number of terms,nor have we inquired whethersuch collections can be saidto have a number at all. Thisis an ancient problem, whichhas been solved in our ownday, chiefly by Georg Can-tor. In the present chapterwe shall attempt to explainthe theory of transfinite orinfinite cardinal numbers as itresults from a combination of (original page )

his discoveries with those ofFrege on the logical theory ofnumbers.

It cannot be said to be cer-tain that there are in factany infinite collections in theworld. The assumption thatthere are is what we call the“axiom of infinity.” Althoughvarious ways suggest them-selves by which we mighthope to prove this axiom,there is reason to fear thatthey are all fallacious, andthat there is no conclusivelogical reason for believing it (original page )

to be true. At the same time,there is certainly no logicalreason against infinite collec-tions, and we are thereforejustified, in logic, in inves-tigating the hypothesis thatthere are such collections. Thepractical form of this hypoth-esis, for our present purposes,is the assumption that, if nis any inductive number, n isnot equal to n + . Varioussubtleties arise in identifyingthis form of our assumptionwith | the form that asserts theexistence of infinite collec- (original pages –)

tions; but we will leave theseout of account until, in a laterchapter, we come to considerthe axiom of infinity on itsown account. For the presentwe shall merely assume that,if n is an inductive number, nis not equal to n+. This is in-volved in Peano’s assumptionthat no two inductive num-bers have the same successor;for, if n = n+ , then n− andn have the same successor,namely n. Thus we are as-suming nothing that was notinvolved in Peano’s primitive (original page )

propositions.Let us now consider the

collection of the inductivenumbers themselves. This isa perfectly well-defined class.In the first place, a cardi-nal number is a set of classeswhich are all similar to eachother and are not similar toanything except each other.We then define as the “induc-tive numbers” those amongcardinals which belong to theposterity of with respectto the relation of n to n + ,i.e. those which possess every (original page )

property possessed by andby the successors of posses-sors, meaning by the “succes-sor” of n the number n + .Thus the class of “inductivenumbers” is perfectly definite.By our general definition ofcardinal numbers, the numberof terms in the class of induc-tive numbers is to be definedas “all those classes that aresimilar to the class of induc-tive numbers”—i.e. this set ofclasses is the number of theinductive numbers accordingto our definitions. (original page )

Now it is easy to see thatthis number is not one of theinductive numbers. If n isany inductive number, thenumber of numbers from to n (both included) is n + ;therefore the total number ofinductive numbers is greaterthan n, no matter which of theinductive numbers n may be.If we arrange the inductivenumbers in a series in orderof magnitude, this series hasno last term; but if n is aninductive number, every se-ries whose field has n terms (original page )

has a last term, as it is easy toprove. Such differences mightbe multiplied ad lib. Thus thenumber of inductive numbersis a new number, differentfrom all of them, not possess-ing all inductive properties.It may happen that has acertain | property, and that ifn has it so has n + , and yetthat this new number does nothave it. The difficulties thatso long delayed the theory ofinfinite numbers were largelydue to the fact that some, atleast, of the inductive prop- (original pages –)

erties were wrongly judgedto be such as must belong toall numbers; indeed it wasthought that they could not bedenied without contradiction.The first step in understand-ing infinite numbers consistsin realising the mistakennessof this view.

The most noteworthy andastonishing difference be-tween an inductive num-ber and this new numberis that this new number isunchanged by adding orsubtracting or doubling or (original page )

halving or any of a numberof other operations which wethink of as necessarily makinga number larger or smaller.The fact of being not alteredby the addition of is usedby Cantor for the definitionof what he calls “transfinite”cardinal numbers; but for var-ious reasons, some of whichwill appear as we proceed, itis better to define an infinitecardinal number as one whichdoes not possess all induc-tive properties, i.e. simply asone which is not an inductive (original page )

number. Nevertheless, theproperty of being unchangedby the addition of is a veryimportant one, and we mustdwell on it for a time.

To say that a class has anumber which is not alteredby the addition of is thesame thing as to say that, ifwe take a term x which doesnot belong to the class, we canfind a one-one relation whosedomain is the class and whoseconverse domain is obtainedby adding x to the class. For inthat case, the class is similar (original page )

to the sum of itself and theterm x, i.e. to a class havingone extra term; so that it hasthe same number as a classwith one extra term, so thatif n is this number, n = n+ .In this case, we shall also haven = n − , i.e. there will beone-one relations whose do-mains consist of the wholeclass and whose converse do-mains consist of just one termshort of the whole class. Itcan be shown that the casesin which this happens are thesame as the apparently more (original page )

general cases in which somepart (short of the whole) canbe put into one-one relationwith the whole. When thiscan be done, | the correlatorby which it is done may besaid to “reflect” the wholeclass into a part of itself; forthis reason, such classes willbe called “reflexive.” Thus:

A “reflexive” class is onewhich is similar to a properpart of itself. (A “proper part”is a part short of the whole.)

A “reflexive” cardinal num-ber is the cardinal number of (original pages –)

a reflexive class.We have now to consider

this property of reflexiveness.One of the most striking

instances of a “reflexion” isRoyce’s illustration of themap: he imagines it decidedto make a map of Englandupon a part of the surface ofEngland. A map, if it is ac-curate, has a perfect one-onecorrespondence with its orig-inal; thus our map, which ispart, is in one-one relationwith the whole, and mustcontain the same number of (original page )

points as the whole, whichmust therefore be a reflexivenumber. Royce is interestedin the fact that the map, if it iscorrect, must contain a map ofthe map, which must in turncontain a map of the map ofthe map, and so on ad infini-tum. This point is interesting,but need not occupy us at thismoment. In fact, we shall dowell to pass from picturesqueillustrations to such as aremore completely definite, andfor this purpose we cannotdo better than consider the (original page )

number-series itself.The relation of n to n + ,

confined to inductive num-bers, is one-one, has the wholeof the inductive numbers forits domain, and all except for its converse domain. Thusthe whole class of inductivenumbers is similar to what thesame class becomes when weomit . Consequently it is a“reflexive” class according tothe definition, and the num-ber of its terms is a “reflexive”number. Again, the relationof n to n, confined to in- (original page )

ductive numbers, is one-one,has the whole of the induc-tive numbers for its domain,and the even inductive num-bers alone for its conversedomain. Hence the total num-ber of inductive numbers isthe same as the number ofeven inductive numbers. Thisproperty was used by Leib-niz (and many others) as aproof that infinite numbersare impossible; it was thoughtself-contradictory that | “thepart should be equal to thewhole.” But this is one of (original pages –)

those phrases that dependfor their plausibility uponan unperceived vagueness:the word “equal” has manymeanings, but if it is takento mean what we have called“similar,” there is no con-tradiction, since an infinitecollection can perfectly wellhave parts similar to itself.Those who regard this as im-possible have, unconsciouslyas a rule, attributed to num-bers in general propertieswhich can only be proved bymathematical induction, and (original page )

which only their familiaritymakes us regard, mistakenly,as true beyond the region ofthe finite.

Whenever we can “reflect”a class into a part of itself, thesame relation will necessarilyreflect that part into a smallerpart, and so on ad infinitum.For example, we can reflect,as we have just seen, all theinductive numbers into theeven numbers; we can, by thesame relation (that of n ton) reflect the even numbersinto the multiples of , these (original page )

into the multiples of , andso on. This is an abstract ana-logue to Royce’s problem ofthe map. The even numbersare a “map” of all the induc-tive numbers; the multiples of are a map of the map; themultiples of are a map of themap of the map; and so on.If we had applied the sameprocess to the relation of n ton + , our “map” would haveconsisted of all the inductivenumbers except ; the mapof the map would have con-sisted of all from onward, (original page )

the map of the map of themap of all from onward;and so on. The chief use ofsuch illustrations is in orderto become familiar with theidea of reflexive classes, sothat apparently paradoxicalarithmetical propositions canbe readily translated into thelanguage of reflexions andclasses, in which the air ofparadox is much less.

It will be useful to givea definition of the numberwhich is that of the inductivecardinals. For this purpose (original page )

we will first define the kindof series exemplified by theinductive cardinals in orderof magnitude. The kind ofseries which is called a “pro-gression” has already beenconsidered in Chapter I. It is aseries which can be generatedby a relation of consecutive-ness: | every member of theseries is to have a successor,but there is to be just onewhich has no predecessor, andevery member of the series isto be in the posterity of thisterm with respect to the rela- (original pages –)

tion “immediate predecessor.”These characteristics may besummed up in the followingdefinition:—

A “progression” is a one-one relation such that thereis just one term belonging tothe domain but not to the con-verse domain, and the domainis identical with the posterityof this one term.

It is easy to see that a pro-gression, so defined, satisfies

Cf. Principia Mathematica, vol. ii.∗.

(original page )

Peano’s five axioms. The termbelonging to the domain butnot to the converse domainwill be what he calls “”; theterm to which a term has theone-one relation will be the“successor” of the term; andthe domain of the one-onerelation will be what he calls“number.” Taking his fiveaxioms in turn, we have thefollowing translations:—

() “ is a number” be-comes: “The member of thedomain which is not a mem-ber of the converse domain (original page )

is a member of the domain.”This is equivalent to the ex-istence of such a member,which is given in our defini-tion. We will call this member“the first term.”

() “The successor of anynumber is a number” be-comes: “The term to whicha given member of the do-main has the relation in ques-tion is again a member of thedomain.” This is proved asfollows: By the definition, ev-ery member of the domainis a member of the posterity (original page )

of the first term; hence thesuccessor of a member of thedomain must be a memberof the posterity of the firstterm (because the posterityof a term always contains itsown successors, by the gen-eral definition of posterity),and therefore a member ofthe domain, because by thedefinition the posterity of thefirst term is the same as thedomain.

() “No two numbers havethe same successor.” This isonly to say that the relation is (original page )

one-many, which it is by defi-nition (being one-one). |

() “ is not the successor ofany number” becomes: “Thefirst term is not a member ofthe converse domain,” whichis again an immediate resultof the definition.

() This is mathematical in-duction, and becomes: “Ev-ery member of the domain be-longs to the posterity of thefirst term,” which was part ofour definition.

Thus progressions as wehave defined them have the (original pages –)

five formal properties fromwhich Peano deduces arith-metic. It is easy to show thattwo progressions are “similar”in the sense defined for simi-larity of relations in ChapterVI. We can, of course, derivea relation which is serial fromthe one-one relation by whichwe define a progression: themethod used is that explainedin Chapter IV., and the re-lation is that of a term to amember of its proper poster-ity with respect to the originalone-one relation. (original page )

Two transitive asymmetri-cal relations which generateprogressions are similar, forthe same reasons for whichthe corresponding one-onerelations are similar. The classof all such transitive genera-tors of progressions is a “serialnumber” in the sense of Chap-ter VI.; it is in fact the smallestof infinite serial numbers, thenumber to which Cantor hasgiven the name ω, by whichhe has made it famous.

But we are concerned, forthe moment, with cardinal (original page )

numbers. Since two progres-sions are similar relations, itfollows that their domains(or their fields, which are thesame as their domains) aresimilar classes. The domainsof progressions form a cardi-nal number, since every classwhich is similar to the do-main of a progression is easilyshown to be itself the domainof a progression. This cardi-nal number is the smallest ofthe infinite cardinal numbers;it is the one to which Cantorhas appropriated the Hebrew (original page )

Aleph with the suffix , to dis-tinguish it from larger infinitecardinals, which have othersuffixes. Thus the name of thesmallest of infinite cardinalsis ℵ.

To say that a class has ℵterms is the same thing as tosay that it is a member of ℵ,and this is the same thing asto say | that the members ofthe class can be arranged in aprogression. It is obvious thatany progression remains aprogression if we omit a finitenumber of terms from it, or (original pages –)

every other term, or all exceptevery tenth term or every hun-dredth term. These methodsof thinning out a progressiondo not make it cease to be aprogression, and therefore donot diminish the number ofits terms, which remains ℵ.In fact, any selection from aprogression is a progressionif it has no last term, howeversparsely it may be distributed.Take (say) inductive numbersof the form nn, or nn

n. Such

numbers grow very rare in thehigher parts of the number (original page )

series, and yet there are justas many of them as there areinductive numbers altogether,namely, ℵ.

Conversely, we can addterms to the inductive num-bers without increasing theirnumber. Take, for example,ratios. One might be inclinedto think that there must bemany more ratios than inte-gers, since ratios whose de-nominator is correspond tothe integers, and seem to beonly an infinitesimal propor-tion of ratios. But in actual (original page )

fact the number of ratios (orfractions) is exactly the sameas the number of inductivenumbers, namely, ℵ. This iseasily seen by arranging ratiosin a series on the followingplan: If the sum of numera-tor and denominator in oneis less than in the other, putthe one before the other; if thesum is equal in the two, putfirst the one with the smallernumerator. This gives us theseries

, , , , ,

, , , ,

, . . .

(original page )

This series is a progression,and all ratios occur in it sooneror later. Hence we can arrangeall ratios in a progression, andtheir number is therefore ℵ.

It is not the case, however,that all infinite collectionshave ℵ terms. The numberof real numbers, for example,is greater than ℵ; it is, infact, ℵ , and it is not hard toprove that n is greater thann even when n is infinite. Theeasiest way of proving this isto prove, first, that if a classhas n members, it contains n

(original page )

sub-classes—in other words,that there are n ways | of se-lecting some of its members(including the extreme caseswhere we select all or none);and secondly, that the numberof sub-classes contained in aclass is always greater thanthe number of members of theclass. Of these two proposi-tions, the first is familiar inthe case of finite numbers,and is not hard to extend toinfinite numbers. The proofof the second is so simple andso instructive that we shall (original pages –)

give it:In the first place, it is clear

that the number of sub-classesof a given class (say α) is atleast as great as the number ofmembers, since each memberconstitutes a sub-class, andwe thus have a correlation ofall the members with someof the sub-classes. Hence itfollows that, if the numberof sub-classes is not equal tothe number of members, itmust be greater. Now it is easyto prove that the number isnot equal, by showing that, (original page )

given any one-one relationwhose domain is the membersand whose converse domainis contained among the setof sub-classes, there mustbe at least one sub-class notbelonging to the converse do-main. The proof is as follows:

When a one-one correlationR is established between allthe members of α and someThis proof is taken from Can-

tor, with some simplifications:see Jahresbericht der DeutschenMathematiker-Vereinigung, i. (),p. .

(original page )

of the sub-classes, it may hap-pen that a given member xis correlated with a sub-classof which it is a member; or,again, it may happen that x iscorrelated with a sub-class ofwhich it is not a member. Letus form the whole class, β say,of those members x which arecorrelated with sub-classesof which they are not mem-bers. This is a sub-class of α,and it is not correlated withany member of α. For, takingfirst the members of β, eachof them is (by the definition (original page )

of β) correlated with somesub-class of which it is not amember, and is therefore notcorrelated with β. Taking nextthe terms which are not mem-bers of β, each of them (by thedefinition of β) is correlatedwith some sub-class of whichit is a member, and thereforeagain is not correlated withβ. Thus no member of α iscorrelated with β. Since Rwas any one-one correlationof all members | with somesub-classes, it follows thatthere is no correlation of all (original pages –)

members with all sub-classes.It does not matter to the proofif β has no members: all thathappens in that case is thatthe sub-class which is shownto be omitted is the null-class.Hence in any case the numberof sub-classes is not equal tothe number of members, andtherefore, by what was saidearlier, it is greater. Combin-ing this with the propositionthat, if n is the number ofmembers, n is the number ofsub-classes, we have the theo-rem that n is always greater (original page )

than n, even when n is infi-nite.

It follows from this proposi-tion that there is no maximumto the infinite cardinal num-bers. However great an infi-nite number n may be, n willbe still greater. The arithmeticof infinite numbers is some-what surprising until one be-comes accustomed to it. Wehave, for example,

(original page )

ℵ + = ℵ,ℵ +n = ℵ, where n is any

inductive number,

ℵ = ℵ.

(This follows from the case ofthe ratios, for, since a ratio isdetermined by a pair of induc-tive numbers, it is easy to seethat the number of ratios is thesquare of the number of in-ductive numbers, i.e. it is ℵ;but we saw that it is also ℵ.)

ℵn = ℵ, where (original page )

n is any inductive number.(This follows from

ℵ = ℵby induction; for if

ℵn = ℵ,then

ℵn+ = ℵ = ℵ.)But ℵ > ℵ.In fact, as we shall see later,ℵ is a very important num-ber, namely, the number ofterms in a series which has“continuity” in the sense inwhich this word is used byCantor. Assuming space andtime to be continuous in this (original page )

sense (as we commonly doin analytical geometry andkinematics), this will be thenumber of points in space orof instants in time; it will alsobe the number of points inany finite portion of space,whether | line, area, or vol-ume. After ℵ, ℵ is the mostimportant and interesting ofinfinite cardinal numbers.

Although addition andmultiplication are always pos-sible with infinite cardinals,subtraction and division nolonger give definite results, (original pages –)

and cannot therefore be em-ployed as they are employedin elementary arithmetic.Take subtraction to beginwith: so long as the num-ber subtracted is finite, allgoes well; if the other numberis reflexive, it remains un-changed. Thus ℵ − n = ℵ, ifn is finite; so far, subtractiongives a perfectly definite re-sult. But it is otherwise whenwe subtract ℵ from itself;we may then get any result,from up to ℵ. This is easilyseen by examples. From the (original page )

inductive numbers, take awaythe following collections of ℵterms:—

() All the inductive num-bers—remainder, zero.

() All the inductive num-bers from n onwards—re-mainder, the numbers from to n−, numbering n terms inall.

() All the odd numbers—remainder, all the even num-bers, numbering ℵ terms.

All these are different waysof subtracting ℵ from ℵ,and all give different results. (original page )

As regards division, verysimilar results follow fromthe fact that ℵ is unchangedwhen multiplied by or orany finite number n or by ℵ.It follows that ℵ divided byℵ may have any value from up to ℵ.

From the ambiguity of sub-traction and division it resultsthat negative numbers andratios cannot be extendedto infinite numbers. Addi-tion, multiplication, and ex-ponentiation proceed quitesatisfactorily, but the inverse (original page )

operations—subtraction, divi-sion, and extraction of roots—are ambiguous, and the no-tions that depend upon themfail when infinite numbers areconcerned.

The characteristic by whichwe defined finitude was math-ematical induction, i.e. we de-fined a number as finite whenit obeys mathematical induc-tion starting from , and aclass as finite when its num-ber is finite. This definitionyields the sort of result thata definition ought to yield, (original page )

namely, that the finite | num-bers are those that occur inthe ordinary number-series ,, , , . . . But in the presentchapter, the infinite numberswe have discussed have notmerely been non-inductive:they have also been reflexive.Cantor used reflexiveness asthe definition of the infinite,and believes that it is equiv-alent to non-inductiveness;that is to say, he believes thatevery class and every cardinalis either inductive or reflex-ive. This may be true, and (original pages –)

may very possibly be capableof proof; but the proofs hith-erto offered by Cantor andothers (including the presentauthor in former days) are fal-lacious, for reasons which willbe explained when we cometo consider the “multiplica-tive axiom.” At present, it isnot known whether there areclasses and cardinals whichare neither reflexive nor in-ductive. If n were such acardinal, we should not haven = n+ , but n would not beone of the “natural numbers,” (original page )

and would be lacking in someof the inductive properties.All known infinite classes andcardinals are reflexive; butfor the present it is well topreserve an open mind as towhether there are instances,hitherto unknown, of classesand cardinals which are nei-ther reflexive nor inductive.Meanwhile, we adopt the fol-lowing definitions:—

A finite class or cardinal isone which is inductive.

An infinite class or cardinalis one which is not inductive. (original page )

All reflexive classes and cardi-nals are infinite; but it is notknown at present whether allinfinite classes and cardinalsare reflexive. We shall returnto this subject in Chapter XII.

(original page )

CHAPTER IXINFINITE SERIESAND ORDINALS

An “infinite series” may bedefined as a series of whichthe field is an infinite class.We have already had occa-sion to consider one kind ofinfinite series, namely, pro-gressions. In this chapterwe shall consider the subjectmore generally.

(original page )

The most noteworthy char-acteristic of an infinite se-ries is that its serial numbercan be altered by merely re-arranging its terms. In thisrespect there is a certain oppo-siteness between cardinal andserial numbers. It is possibleto keep the cardinal numberof a reflexive class unchangedin spite of adding terms to it;on the other hand, it is possi-ble to change the serial num-ber of a series without addingor taking away any terms, bymere re-arrangement. At the (original page )

same time, in the case of anyinfinite series it is also possi-ble, as with cardinals, to addterms without altering theserial number: everything de-pends upon the way in whichthey are added.

In order to make mattersclear, it will be best to be-gin with examples. Let usfirst consider various differ-ent kinds of series which canbe made out of the inductivenumbers arranged on variousplans. We start with the series

(original page )

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

which, as we have alreadyseen, represents the smallestof infinite serial numbers, thesort that Cantor calls ω. Letus proceed to thin out this se-ries by repeatedly performingthe | operation of removing tothe end the first even numberthat occurs. We thus obtain insuccession the various series:

(original pages –)

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

and so on. If we imagine thisprocess carried on as long aspossible, we finally reach theseries

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

in which we have first all theodd numbers and then all theeven numbers. (original page )

The serial numbers of thesevarious series are ω + , ω +, ω+ , . . . ω. Each of thesenumbers is “greater” than anyof its predecessors, in the fol-lowing sense:—

One serial number is saidto be “greater” than anotherif any series having the firstnumber contains a part hav-ing the second number, but noseries having the second num-ber contains a part having thefirst number.

If we compare the two series

(original page )

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

we see that the first is simi-lar to the part of the secondwhich omits the last term,namely, the number , butthe second is not similar toany part of the first. (This isobvious, but is easily demon-strated.) Thus the second se-ries has a greater serial num-ber than the first, accordingto the definition—i.e. ω + (original page )

is greater than ω. But if weadd a term at the beginningof a progression instead of theend, we still have a progres-sion. Thus + ω = ω. Thus+ω is not equal to ω+. Thisis characteristic of relation-arithmetic generally: if µ andν are two relation-numbers,the general rule is that µ + νis not equal to ν +µ. The caseof finite ordinals, in whichthere is equality, is quite ex-ceptional.

The series we finally reach-ed just now consisted of first (original page )

all the odd numbers and thenall the even numbers, and itsserial | number is ω. Thisnumber is greater than ω orω + n, where n is finite. It isto be observed that, in accor-dance with the general defi-nition of order, each of thesearrangements of integers is tobe regarded as resulting fromsome definite relation. E.g. theone which merely removes to the end will be defined bythe following relation: “x andy are finite integers, and ei-ther y is and x is not , or (original pages –)

neither is and x is less thany.” The one which puts firstall the odd numbers and thenall the even ones will be de-fined by: “x and y are finiteintegers, and either x is oddand y is even or x is less thany and both are odd or bothare even.” We shall not trou-ble, as a rule, to give these for-mulæ in future; but the factthat they could be given is es-sential.

The number which we havecalled ω, namely, the num-ber of a series consisting of (original page )

two progressions, is some-times called ω . . Multiplica-tion, like addition, dependsupon the order of the factors:a progression of couples givesa series such as

x, y, x, y, x, y, . . .

xn, yn, . . . ,

which is itself a progression;but a couple of progressionsgives a series which is twiceas long as a progression. It istherefore necessary to distin-guish between ω and ω . .

(original page )

Usage is variable; we shalluse ω for a couple of pro-gressions and ω . for a pro-gression of couples, and thisdecision of course governsour general interpretationof “α . β” when α and β arerelation-numbers: “α . β” willhave to stand for a suitablyconstructed sum of α relationseach having β terms.

We can proceed indefinitelywith the process of thinningout the inductive numbers.For example, we can placefirst the odd numbers, then (original page )

their doubles, then the dou-bles of these, and so on. Wethus obtain the series

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

of which the number is ω,since it is a progression ofprogressions. Any one of theprogressions in this new seriescan of course be | thinned outas we thinned out our originalprogression. We can proceedto ω, ω, . . . ωω, and so on;however far we have gone, we

(original pages –)

can always go further.The series of all the ordi-

nals that can be obtained inthis way, i.e. all that can beobtained by thinning out aprogression, is itself longerthan any series that can beobtained by re-arranging theterms of a progression. (Thisis not difficult to prove.) Thecardinal number of the classof such ordinals can be shownto be greater than ℵ; it is thenumber which Cantor callsℵ. The ordinal number of theseries of all ordinals that can (original page )

be made out of an ℵ, taken inorder of magnitude, is calledω. Thus a series whose ordi-nal number is ω has a fieldwhose cardinal number is ℵ.

We can proceed from ωand ℵ to ω and ℵ by a pro-cess exactly analogous to thatby which we advanced fromω and ℵ to ω and ℵ. Andthere is nothing to prevent usfrom advancing indefinitelyin this way to new cardinalsand new ordinals. It is notknown whether ℵ is equalto any of the cardinals in the (original page )

series of Alephs. It is not evenknown whether it is compara-ble with them in magnitude;for aught we know, it may beneither equal to nor greaternor less than any one of theAlephs. This question is con-nected with the multiplicativeaxiom, of which we shall treatlater.

All the series we have beenconsidering so far in thischapter have been what iscalled “well-ordered.” A well-ordered series is one whichhas a beginning, and has con- (original page )

secutive terms, and has a termnext after any selection of itsterms, provided there are anyterms after the selection. Thisexcludes, on the one hand,compact series, in which thereare terms between any two,and on the other hand serieswhich have no beginning, orin which there are subordi-nate parts having no begin-ning. The series of negativeintegers in order of magni-tude, having no beginning,but ending with −, is notwell-ordered; but taken in the (original page )

reverse order, beginning with−, it is well-ordered, beingin fact a progression. Thedefinition is: |

A “well-ordered” series isone in which every sub-class(except, of course, the null-class) has a first term.

An “ordinal” numbermeans the relation-numberof a well-ordered series. It isthus a species of serial num-ber.

Among well-ordered series,a generalised form of mathe-matical induction applies. A (original pages –)

property may be said to be“transfinitely hereditary” if,when it belongs to a certainselection of the terms in aseries, it belongs to their im-mediate successor providedthey have one. In a well-ordered series, a transfinitelyhereditary property belong-ing to the first term of theseries belongs to the wholeseries. This makes it possibleto prove many propositionsconcerning well-ordered se-ries which are not true of allseries. (original page )

It is easy to arrange theinductive numbers in serieswhich are not well-ordered,and even to arrange them incompact series. For example,we can adopt the followingplan: consider the decimalsfrom · (inclusive) to (ex-clusive), arranged in orderof magnitude. These form acompact series; between anytwo there are always an infi-nite number of others. Nowomit the dot at the beginningof each, and we have a com-pact series consisting of all (original page )

finite integers except such asdivide by . If we wish toinclude those that divide by, there is no difficulty; in-stead of starting with ·, wewill include all decimals lessthan , but when we removethe dot, we will transfer tothe right any ’s that occur atthe beginning of our decimal.Omitting these, and returningto the ones that have no ’sat the beginning, we can statethe rule for the arrangementof our integers as follows: Oftwo integers that do not begin (original page )

with the same digit, the onethat begins with the smallerdigit comes first. Of two thatdo begin with the same digit,but differ at the second digit,the one with the smaller sec-ond digit comes first, but firstof all the one with no seconddigit; and so on. Generally, iftwo integers agree as regardsthe first n digits, but not asregards the (n+ )th, that onecomes first which has eitherno (n+ )th digit or a smallerone than the other. This ruleof arrangement, | as the reader (original pages –)

can easily convince himself,gives rise to a compact seriescontaining all the integersnot divisible by ; and, aswe saw, there is no difficultyabout including those thatare divisible by . It followsfrom this example that it ispossible to construct compactseries having ℵ terms. Infact, we have already seen thatthere are ℵ ratios, and ratiosin order of magnitude form acompact series; thus we havehere another example. Weshall resume this topic in the (original page )

next chapter.Of the usual formal laws

of addition, multiplication,and exponentiation, all areobeyed by transfinite car-dinals, but only some areobeyed by transfinite ordinals,and those that are obeyedby them are obeyed by allrelation-numbers. By the“usual formal laws” we meanthe following:—

(original page )

I. The commutative law:α + β = β +α andα × 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 lawdoes not hold, the above formof the distributive law must bedistinguished from

(β +γ)α = βα +γα.

As we shall see immedi- (original page )

ately, one form may be trueand the other false.

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

All these laws hold for car-dinals, whether finite or in-finite, and for finite ordinals.But when we come to infi-nite ordinals, or indeed torelation-numbers in general,some hold and some do not. (original page )

The commutative law doesnot hold; the associative lawdoes hold; the distributive law(adopting the convention | wehave adopted above as regardsthe order of the factors in aproduct) 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

(original pages –)

still hold, but not the law

αγ . βγ = (αβ)γ ,

which is obviously connectedwith the commutative law formultiplication.

The definitions of multi-plication and exponentiationthat are assumed in the abovepropositions are somewhatcomplicated. The reader whowishes to know what theyare and how the above lawsare proved must consult thesecond volume of PrincipiaMathematica, ∗–. (original page )

Ordinal transfinite arith-metic was developed by Can-tor at an earlier stage than car-dinal transfinite arithmetic,because it has various techni-cal mathematical uses whichled him to it. But from thepoint of view of the philoso-phy of mathematics it is lessimportant and less fundamen-tal than the theory of transfi-nite cardinals. Cardinals areessentially simpler than ordi-nals, and it is a curious his-torical accident that they firstappeared as an abstraction (original page )

from the latter, and only grad-ually came to be studied ontheir own account. This doesnot apply to Frege’s work, inwhich cardinals, finite andtransfinite, were treated incomplete independence ofordinals; but it was Cantor’swork that made the worldaware of the subject, whileFrege’s remained almost un-known, probably in the mainon account of the difficultyof his symbolism. And math-ematicians, like other peo-ple, have more difficulty in (original page )

understanding and using no-tions which are comparatively“simple” in the logical sensethan in manipulating morecomplex notions which are |more akin to their ordinarypractice. For these reasons, itwas only gradually that thetrue importance of cardinalsin mathematical philosophywas recognised. The impor-tance of ordinals, though byno means small, is distinctlyless than that of cardinals,and is very largely merged in

(original pages –)

that of the more general con-ception of relation-numbers.

(original page )

CHAPTER XLIMITS ANDCONTINUITY

The conception of a “limit”is one of which the impor-tance in mathematics has beenfound continually greaterthan had been thought. Thewhole of the differential andintegral calculus, indeed prac-tically everything in highermathematics, depends upon

(original page )

limits. Formerly, it was sup-posed that infinitesimals wereinvolved in the foundationsof these subjects, but Weier-strass showed that this is anerror: wherever infinitesimalswere thought to occur, whatreally occurs is a set of fi-nite quantities having zero fortheir lower limit. It used tobe thought that “limit” wasan essentially quantitative no-tion, namely, the notion of aquantity to which others ap-proached nearer and nearer,so that among those others (original page )

there would be some differ-ing by less than any assignedquantity. But in fact the no-tion of “limit” is a purelyordinal notion, not involv-ing quantity at all (except byaccident when the series con-cerned happens to be quan-titative). A given point on aline may be the limit of a setof points on the line, withoutits being necessary to bring inco-ordinates or measurementor anything quantitative. Thecardinal number ℵ is thelimit (in the order of magni- (original page )

tude) of the cardinal numbers, , , . . . n, . . . , although thenumerical difference betweenℵ and a finite cardinal isconstant and infinite: from aquantitative point of view, fi-nite numbers get no nearer toℵ as they grow larger. Whatmakes ℵ the limit of the fi-nite numbers is the fact that,in the series, it comes imme-diately after them, which is anordinal fact, not a quantitativefact. |

There are various forms ofthe notion of “limit,” of in- (original pages –)

creasing complexity. The sim-plest and most fundamentalform, from which the rest arederived, has been already de-fined, but we will here repeatthe definitions which lead toit, in a general form in whichthey do not demand that therelation concerned shall beserial. The definitions are asfollows:—

The “minima” of a class αwith respect to a relation P arethose members of α and thefield of P (if any) to which nomember of α has the relation (original page )

P.The “maxima” with respect

to P are the minima with re-spect to the converse of P.

The “sequents” of a classα with respect to a relation Pare the minima of the “suc-cessors” of α, and the “succes-sors” of α are those membersof the field of P to which everymember of the common partof α and the field of P has therelation P.

The “precedents” with re-spect to P are the sequentswith respect to the converse (original page )

of P.The “upper limits” of α

with respect to P are the se-quents provided α has nomaximum; but if α has a max-imum, it has no upper limits.

The “lower limits” with re-spect to P are the upper limitswith respect to the converse ofP.

Whenever P has connex-ity, a class can have at mostone maximum, one minimum,one sequent, etc. Thus, in thecases we are concerned within practice, we can speak of (original page )

“the limit” (if any).When P is a serial relation,

we can greatly simplify theabove definition of a limit. Wecan, in that case, define firstthe “boundary” of a class α,i.e. its limit or maximum, andthen proceed to distinguishthe case where the bound-ary is the limit from the casewhere it is a maximum. Forthis purpose it is best to usethe notion of “segment.”

We will speak of the “seg-ment of P defined by a classα” as all those terms that have (original page )

the relation P to some oneor more of the members ofα. This will be a segment inthe sense defined | in ChapterVII.; indeed, every segment inthe sense there defined is thesegment defined by some classα. If P is serial, the segmentdefined by α consists of allthe terms that precede someterm or other of α. If α has amaximum, the segment willbe all the predecessors of themaximum. But if α has nomaximum, every member of αprecedes some other member (original pages –)

of α, and the whole of α istherefore included in the seg-ment defined by α. Take, forexample, the class consistingof the fractions

, , , , . . .,

i.e. of all fractions of the form − /n for different finitevalues of n. This series of frac-tions has no maximum, andit is clear that the segmentwhich it defines (in the wholeseries of fractions in order ofmagnitude) is the class of all

(original page )

proper fractions. Or, again,consider the prime numbers,considered as a selection fromthe cardinals (finite and infi-nite) in order of magnitude. Inthis case the segment definedconsists of all finite integers.

Assuming that P is serial,the “boundary” of a class αwill be the term x (if it ex-ists) whose predecessors arethe segment defined by α.

A “maximum” of α is aboundary which is a memberof α.

An “upper limit” of α is a (original page )

boundary which is not a mem-ber of α.

If a class has no boundary,it has neither maximum norlimit. This is the case of an“irrational” Dedekind cut, orof what is called a “gap.”

Thus the “upper limit” of aset of terms α with respect to aseries P is that term x (if it ex-ists) which comes after all theα’s, but is such that every ear-lier term comes before someof the α’s.

We may define all the “up-per limiting-points” of a set (original page )

of terms β as all those thatare the upper limits of setsof terms chosen out of β. Weshall, of course, have to distin-guish upper limiting-pointsfrom lower limiting-points. Ifwe consider, for example, theseries of ordinal numbers:

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

ω, . . . , |

the upper limiting-points ofthe field of this series are thosethat have no immediate pre- (original pages –)

decessors, i.e.

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

The upper limiting-points ofthe field of this new series willbe

, ω, ω, . . . ω, ω +ω . . .

On the other hand, the seriesof ordinals—and indeed everywell-ordered series—has nolower limiting-points, becausethere are no terms except thelast that have no immediate (original page )

successors. But if we considersuch a series as the seriesof ratios, every member ofthis series is both an upperand a lower limiting-pointfor suitably chosen sets. Ifwe consider the series of realnumbers, and select out of itthe rational real numbers, thisset (the rationals) will haveall the real numbers as up-per and lower limiting-points.The limiting-points of a setare called its “first derivative,”and the limiting-points of thefirst derivative are called the (original page )

second derivative, and so on.With regard to limits, we

may distinguish various gradesof what may be called “conti-nuity” in a series. The word“continuity” had been usedfor a long time, but had re-mained without any precisedefinition until the time ofDedekind and Cantor. Eachof these two men gave a pre-cise significance to the term,but Cantor’s definition is nar-rower than Dedekind’s: a se-ries which has Cantorian con-tinuity must have Dedekin- (original page )

dian continuity, but the con-verse does not hold.

The first definition thatwould naturally occur to aman seeking a precise mean-ing for the continuity of serieswould be to define it as con-sisting in what we have called“compactness,” i.e. in the factthat between any two termsof the series there are others.But this would be an inade-quate definition, because ofthe existence of “gaps” in se-ries such as the series of ratios.We saw in Chapter VII. that (original page )

there are innumerable waysin which the series of ratioscan be divided into two parts,of which one wholly precedesthe other, and of which thefirst has no last term, | whilethe second has no first term.Such a state of affairs seemscontrary to the vague feel-ing we have as to what shouldcharacterise “continuity,” and,what is more, it shows that theseries of ratios is not the sortof series that is needed formany mathematical purposes.Take geometry, for example: (original pages –)

we wish to be able to say thatwhen two straight lines crosseach other they have a pointin common, but if the series ofpoints on a line were similarto the series of ratios, the twolines might cross in a “gap”and have no point in common.This is a crude example, butmany others might be givento show that compactness isinadequate as a mathematicaldefinition of continuity.

It was the needs of geom-etry, as much as anything,that led to the definition of (original page )

“Dedekindian” continuity. Itwill be remembered that wedefined a series as Dedekin-dian when every sub-class ofthe field has a boundary. (It issufficient to assume that thereis always an upper bound-ary, or that there is alwaysa lower boundary. If one ofthese is assumed, the othercan be deduced.) That is tosay, a series is Dedekindianwhen there are no gaps. Theabsence of gaps may arise ei-ther through terms havingsuccessors, or through the ex- (original page )

istence of limits in the absenceof maxima. Thus a finite se-ries or a well-ordered seriesis Dedekindian, and so is theseries of real numbers. Theformer sort of Dedekindianseries is excluded by assumingthat our series is compact; inthat case our series must havea property which may, formany purposes, be fittinglycalled continuity. Thus we areled to the definition:

A series has “Dedekin-dian continuity” when it isDedekindian and compact. (original page )

But this definition is stilltoo wide for many purposes.Suppose, for example, thatwe desire to be able to assignsuch properties to geometricalspace as shall make it certainthat every point can be speci-fied by means of co-ordinateswhich are real numbers: thisis not insured by Dedekindiancontinuity alone. We want tobe sure that every point whichcannot be specified by rationalco-ordinates can be specifiedas the limit of a progressionof points | whose co-ordinates (original pages –)

are rational, and this is a fur-ther property which our def-inition does not enable us todeduce.

We are thus led to a closerinvestigation of series withrespect to limits. This inves-tigation was made by Cantorand formed the basis of hisdefinition of continuity, al-though, in its simplest form,this definition somewhat con-ceals the considerations whichhave given rise to it. We shall,therefore, first travel throughsome of Cantor’s conceptions (original page )

in this subject before givinghis definition of continuity.

Cantor defines a series as“perfect” when all its pointsare limiting-points and allits limiting-points belong toit. But this definition doesnot express quite accuratelywhat he means. There is nocorrection required so far asconcerns the property that allits points are to be limiting-points; this is a property be-longing to compact series, andto no others if all points areto be upper limiting- or all (original page )

lower limiting-points. But ifit is only assumed that theyare limiting-points one way,without specifying which,there will be other seriesthat will have the propertyin question—for example, theseries of decimals in which adecimal ending in a recurring is distinguished from thecorresponding terminatingdecimal and placed immedi-ately before it. Such a seriesis very nearly compact, buthas exceptional terms whichare consecutive, and of which (original page )

the first has no immediatepredecessor, while the secondhas no immediate successor.Apart from such series, theseries in which every point isa limiting-point are compactseries; and this holds withoutqualification if it is specifiedthat every point is to be anupper limiting-point (or thatevery point is to be a lowerlimiting-point).

Although Cantor does notexplicitly consider the matter,we must distinguish differentkinds of limiting-points ac- (original page )

cording to the nature of thesmallest sub-series by whichthey can be defined. Cantorassumes that they are to bedefined by progressions, orby regressions (which are theconverses of progressions).When every member of ourseries is the limit of a pro-gression or regression, Cantorcalls our series “condensed initself” (insichdicht). |

We come now to the secondproperty by which perfectionwas to be defined, namely,the property which Cantor (original pages –)

calls that of being “closed”(abgeschlossen). This, as wesaw, was first defined as con-sisting in the fact that all thelimiting-points of a seriesbelong to it. But this onlyhas any effective significanceif our series is given as con-tained in some other largerseries (as is the case, e.g., witha selection of real numbers),and limiting-points are takenin relation to the larger se-ries. Otherwise, if a series isconsidered simply on its ownaccount, it cannot fail to con- (original page )

tain its limiting-points. WhatCantor means is not exactlywhat he says; indeed, on otheroccasions he says somethingrather different, which is whathe means. What he reallymeans is that every subor-dinate series which is of thesort that might be expectedto have a limit does have alimit within the given series;i.e. every subordinate serieswhich has no maximum hasa limit, i.e. every subordinateseries has a boundary. ButCantor does not state this for (original page )

every subordinate series, butonly for progressions and re-gressions. (It is not clear howfar he recognises that this isa limitation.) Thus, finally,we find that the definition wewant is the following:—

A series is said to be “clos-ed” (abgeschlossen) when everyprogression or regression con-tained in the series has a limitin the series.

We then have the furtherdefinition:—

A series is “perfect” whenit is condensed in itself and (original page )

closed, i.e. when every term isthe limit of a progression orregression, and every progres-sion or regression containedin the series has a limit in theseries.

In seeking a definition ofcontinuity, what Cantor hasin mind is the search for adefinition which shall applyto the series of real numbersand to any series similar tothat, but to no others. For thispurpose we have to add a fur-ther property. Among the realnumbers some are rational, (original page )

some are irrational; althoughthe number of irrationals isgreater than the number of ra-tionals, yet there are rationalsbetween any two real num-bers, however | little the twomay differ. The number of ra-tionals, as we saw, is ℵ. Thisgives a further property whichsuffices to characterise conti-nuity completely, namely, theproperty of containing a classof ℵ members in such a waythat some of this class occurbetween any two terms of ourseries, however near together. (original pages –)

This property, added to per-fection, suffices to define aclass of series which are allsimilar and are in fact a serialnumber. This class Cantordefines as that of continuousseries.

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

A “median class” of a seriesis a sub-class of the field suchthat members of it are to befound between any two termsof the series.

Thus the rationals are a (original page )

median class in the series ofreal numbers. It is obviousthat there cannot be medianclasses except in compact se-ries.

We then find that Cantor’sdefinition is equivalent to thefollowing:—

A series is “continuous”when () it is Dedekindian,() it contains a median classhaving ℵ terms.

To avoid confusion, we shallspeak of this kind as “Can-torian continuity.” It will beseen that it implies Dedekin- (original page )

dian continuity, but the con-verse is not the case. All serieshaving Cantorian continuityare similar, but not all serieshaving Dedekindian continu-ity.

The notions of limit andcontinuity which we have beendefining must not be con-founded with the notions ofthe limit of a function for ap-proaches to a given argument,or the continuity of a func-tion in the neighbourhood ofa given argument. These aredifferent notions, very impor- (original page )

tant, but derivative from theabove and more complicated.The continuity of motion (ifmotion is continuous) is aninstance of the continuity of afunction; on the other hand,the continuity of space andtime (if they are continuous)is an instance of the continu-ity of series, or (to speak morecautiously) of a kind of conti-nuity which can, by sufficientmathematical |manipulation,be reduced to the continuityof series. In view of the funda-mental importance of motion (original pages –)

in applied mathematics, aswell as for other reasons, itwill be well to deal brieflywith the notions of limits andcontinuity as applied to func-tions; but this subject will bebest reserved for a separatechapter.

The definitions of conti-nuity which we have beenconsidering, namely, those ofDedekind and Cantor, do notcorrespond very closely to thevague idea which is associatedwith the word in the mind ofthe man in the street or the (original page )

philosopher. They conceivecontinuity rather as absenceof separateness, the sort ofgeneral obliteration of dis-tinctions which characterisesa thick fog. A fog gives animpression of vastness with-out definite multiplicity ordivision. It is this sort of thingthat a metaphysician meansby “continuity,” declaring it,very truly, to be characteristicof his mental life and of thatof children and animals.

The general idea vaguelyindicated by the word “con- (original page )

tinuity” when so employed,or by the word “flux,” is onewhich is certainly quite differ-ent from that which we havebeen defining. Take, for exam-ple, the series of real numbers.Each is what it is, quite defi-nitely and uncompromisingly;it does not pass over by imper-ceptible degrees into another;it is a hard, separate unit, andits distance from every otherunit is finite, though it canbe made less than any givenfinite amount assigned in ad-vance. The question of the (original page )

relation between the kind ofcontinuity existing among thereal numbers and the kindexhibited, e.g. by what we seeat a given time, is a difficultand intricate one. It is notto be maintained that the twokinds are simply identical, butit may, I think, be very wellmaintained that the mathe-matical conception which wehave been considering in thischapter gives the abstract log-ical scheme to which it mustbe possible to bring empiricalmaterial by suitable manip- (original page )

ulation, if that material is tobe called “continuous” in anyprecisely definable sense. Itwould be quite impossible | tojustify this thesis within thelimits of the present volume.The reader who is interestedmay read an attempt to justifyit as regards time in particularby the present author in theMonist for −, as well asin parts of Our Knowledge ofthe External World. With theseindications, we must leavethis problem, interesting as itis, in order to return to topics (original pages –)

more closely connected withmathematics.

(original page )

CHAPTER XILIMITS AND

CONTINUITY OFFUNCTIONS

In this chapter we shall beconcerned with the defini-tion of the limit of a function(if any) as the argument ap-proaches a given value, andalso with the definition ofwhat is meant by a “continu-ous function.” Both of these

(original page )

ideas are somewhat technical,and would hardly demandtreatment in a mere intro-duction to mathematical phi-losophy but for the fact that,especially through the so-called infinitesimal calculus,wrong views upon our presenttopics have become so firmlyembedded in the minds ofprofessional philosophers thata prolonged and considerableeffort is required for their up-rooting. It has been thoughtever since the time of Leib-niz that the differential and (original page )

integral calculus required in-finitesimal quantities. Math-ematicians (especially Weier-strass) proved that this is anerror; but errors incorporated,e.g. in what Hegel has to sayabout mathematics, die hard,and philosophers have tendedto ignore the work of suchmen as Weierstrass.

Limits and continuity offunctions, in works on ordi-nary mathematics, are definedin terms involving number.This is not essential, as Dr

(original page )

Whitehead has shown. Wewill, however, begin with thedefinitions in the text-books,and proceed afterwards toshow how these definitionscan be generalised so as toapply to series in general, andnot only to such as are numer-ical or numerically measur-able.

Let us consider any ordi-nary mathematical functionfx, where | x and fx are both

See Principia Mathematica, vol. ii.∗–.

(original pages –)

real numbers, and fx is one-valued—i.e. when x is given,there is only one value that fxcan have. We call x the “ar-gument,” and fx the “valuefor the argument x.” When afunction is what we call “con-tinuous,” the rough idea forwhich we are seeking a pre-cise definition is that smalldifferences in x shall corre-spond to small differences infx, and if we make the dif-ferences in x small enough,we can make the differencesin fx fall below any assigned (original page )

amount. We do not want, if afunction is to be continuous,that there shall be suddenjumps, so that, for some valueof x, any change, howeversmall, will make a change infx which exceeds some as-signed finite amount. Theordinary simple functions ofmathematics have this prop-erty: it belongs, for example,to x, x, . . . logx, sinx, andso on. But it is not at all dif-ficult to define discontinuousfunctions. Take, as a non-mathematical example, “the (original page )

place of birth of the youngestperson living at time t.” Thisis a function of t; its value isconstant from the time of oneperson’s birth to the time ofthe next birth, and then thevalue changes suddenly fromone birthplace to the other.An analogous mathematicalexample would be “the inte-ger next below x,” where x isa real number. This functionremains constant from oneinteger to the next, and thengives a sudden jump. Theactual fact is that, though con- (original page )

tinuous functions are morefamiliar, they are the excep-tions: there are infinitely morediscontinuous functions thancontinuous ones.

Many functions are discon-tinuous for one or several val-ues of the variable, but contin-uous for all other values. Takeas an example sin/x. Thefunction sin θ passes throughall values from − to everytime that θ passes from −π/to π/, or from π/ to π/,or generally from (n− )π/to (n+)π/, where n is any (original page )

integer. Now if we consider/x when x is very small, wesee that as x diminishes /xgrows faster and faster, sothat it passes more and morequickly through the cycle ofvalues from one multiple ofπ/ to another as x becomessmaller and smaller. Conse-quently sin/x passes moreand more quickly from − | to and back again, as x growssmaller. In fact, if we take anyinterval containing , say theinterval from −ε to +ε whereε is some very small num- (original pages –)

ber, sin/x will go throughan infinite number of oscilla-tions in this interval, and wecannot diminish the oscilla-tions by making the intervalsmaller. Thus round aboutthe argument the functionis discontinuous. It is easy tomanufacture functions whichare discontinuous in severalplaces, or in ℵ places, oreverywhere. Examples willbe found in any book on thetheory of functions of a realvariable.

Proceeding now to seek a (original page )

precise definition of whatis meant by saying that afunction is continuous fora given argument, when argu-ment and value are both realnumbers, let us first define a“neighbourhood” of a num-ber x as all the numbers fromx − ε to x+ ε, where ε is somenumber which, in importantcases, will be very small. It isclear that continuity at a givenpoint has to do with what hap-pens in any neighbourhood ofthat point, however small.

What we desire is this: If (original page )

a is the argument for whichwe wish our function to becontinuous, let us first definea neighbourhood (α say) con-taining the value fa which thefunction has for the argumenta; we desire that, if we take asufficiently small neighbour-hood containing a, all valuesfor arguments throughout thisneighbourhood shall be con-tained in the neighbourhoodα, no matter how small wemay have made α. That is tosay, if we decree that our func-tion is not to differ from fa

(original page )

by more than some very tinyamount, we can always find astretch of real numbers, hav-ing a in the middle of it, suchthat throughout this stretchfx will not differ from fa bymore than the prescribed tinyamount. And this is to remaintrue whatever tiny amount wemay select. Hence we are ledto the following definition:—

The function f (x) is saidto be “continuous” for theargument a if, for every posi-tive number σ , different from, but as small as we please, (original page )

there exists a positive num-ber ε, different from , suchthat, for all values of δ whichare numerically | less than ε,the difference f (a+ δ)− f (a) isnumerically less than σ .

In this definition, σ firstdefines a neighbourhood off (a), namely, the neighbour-hood from f (a)−σ to f (a) +σ .The definition then proceedsto say that we can (by meansof ε) define a neighbourhood,A number is said to be “numeri-

cally less” than ε when it lies between−ε and +ε.

(original pages –)

namely, that from a−ε to a+ε,such that, for all argumentswithin this neighbourhood,the value of the function lieswithin the neighbourhoodfrom f (a) − σ to f (a) + σ . Ifthis can be done, howeverσ may be chosen, the func-tion is “continuous” for theargument a.

So far we have not definedthe “limit” of a function fora given argument. If we haddone so, we could have de-fined the continuity of a func-tion differently: a function is (original page )

continuous at a point whereits value is the same as thelimit of its values for ap-proaches either from above orfrom below. But it is only theexceptionally “tame” functionthat has a definite limit as theargument approaches a givenpoint. The general rule is thata function oscillates, and that,given any neighbourhood ofa given argument, howeversmall, a whole stretch of val-ues will occur for argumentswithin this neighbourhood.As this is the general rule, let (original page )

us consider it first.Let us consider what may

happen as the argument ap-proaches some value a frombelow. That is to say, we wishto consider what happens forarguments contained in theinterval from a− ε to a, whereε is some number which, inimportant cases, will be verysmall.

The values of the functionfor arguments from a−ε to a (aexcluded) will be a set of realnumbers which will define acertain section of the set of (original page )

real numbers, namely, the sec-tion consisting of those num-bers that are not greater thanall the values for argumentsfrom a − ε to a. Given anynumber in this section, thereare values at least as great asthis number for argumentsbetween a − ε and a, i.e. forarguments that fall very littleshort | of a (if ε is very small).Let us take all possible ε’s andall possible correspondingsections. The common part ofall these sections we will callthe “ultimate section” as the (original pages –)

argument approaches a. Tosay that a number z belongs tothe ultimate section is to saythat, however small we maymake ε, there are argumentsbetween a− ε and a for whichthe value of the function isnot less than z.

We may apply exactly thesame process to upper sec-tions, i.e. to sections that gofrom some point up to thetop, instead of from the bot-tom up to some point. Herewe take those numbers thatare not less than all the val- (original page )

ues for arguments from a − εto a; this defines an uppersection which will vary as εvaries. Taking the commonpart of all such sections forall possible ε’s, we obtain the“ultimate upper section.” Tosay that a number z belongs tothe ultimate upper section isto say that, however small wemake ε, there are argumentsbetween a− ε and a for whichthe value of the function isnot greater than z.

If a term z belongs both tothe ultimate section and to (original page )

the ultimate upper section,we shall say that it belongsto the “ultimate oscillation.”We may illustrate the mat-ter by considering once morethe function sin/x as x ap-proaches the value . We shallassume, in order to fit in withthe above definitions, thatthis value is approached frombelow.

Let us begin with the “ul-timate section.” Between −εand , whatever ε may be,the function will assume thevalue for certain arguments, (original page )

but will never assume anygreater value. Hence the ul-timate section consists of allreal numbers, positive andnegative, up to and including; i.e. it consists of all nega-tive numbers together with, together with the positivenumbers up to and including.

Similarly the “ultimate up-per section” consists of allpositive numbers togetherwith , together with the neg-ative numbers down to andincluding −. (original page )

Thus the “ultimate oscil-lation” consists of all realnumbers from − to , bothincluded. |

We may say generally thatthe “ultimate oscillation” of afunction as the argument ap-proaches a from below con-sists of all those numbers xwhich are such that, howevernear we come to a, we shallstill find values as great as xand values as small as x.

The ultimate oscillationmay contain no terms, or oneterm, or many terms. In the (original pages –)

first two cases the functionhas a definite limit for ap-proaches from below. If theultimate oscillation has oneterm, this is fairly obvious. Itis equally true if it has none;for it is not difficult to provethat, if the ultimate oscillationis null, the boundary of theultimate section is the sameas that of the ultimate uppersection, and may be definedas the limit of the functionfor approaches from below.But if the ultimate oscillationhas many terms, there is no (original page )

definite limit to the functionfor approaches from below.In this case we can take thelower and upper boundariesof the ultimate oscillation (i.e.the lower boundary of theultimate upper section andthe upper boundary of theultimate section) as the lowerand upper limits of its “ulti-mate” values for approachesfrom below. Similarly we ob-tain lower and upper limitsof the “ultimate” values forapproaches from above. Thuswe have, in the general case, (original page )

four limits to a function forapproaches to a given argu-ment. The limit for a givenargument a only exists whenall these four are equal, andis then their common value.If it is also the value for theargument a, the function iscontinuous for this argument.This may be taken as definingcontinuity: it is equivalent toour former definition.

We can define the limit of afunction for a given argument(if it exists) without passingthrough the ultimate oscil- (original page )

lation and the four limits ofthe general case. The defi-nition proceeds, in that case,just as the earlier definitionof continuity proceeded. Letus define the limit for ap-proaches from below. If thereis to be a definite limit forapproaches to a from below,it is necessary and sufficientthat, given any small numberσ , two values for argumentssufficiently near to a (but bothless than a) will differ | by lessthan σ ; i.e. if ε is sufficientlysmall, and our arguments (original pages –)

both lie between a− ε and a (aexcluded), then the differencebetween the values for thesearguments will be less thanσ . This is to hold for any σ ,however small; in that casethe function has a limit forapproaches from below. Simi-larly we define the case whenthere is a limit for approachesfrom above. These two limits,even when both exist, neednot be identical; and if theyare identical, they still neednot be identical with the valuefor the argument a. It is only (original page )

in this last case that we callthe function continuous forthe argument a.

A function is called “contin-uous” (without qualification)when it is continuous for ev-ery argument.

Another slightly differentmethod of reaching the def-inition of continuity is thefollowing:—

Let us say that a function“ultimately converges into aclass α” if there is some realnumber such that, for thisargument and all arguments (original page )

greater than this, the valueof the function is a memberof the class α. Similarly weshall say that a function “con-verges into α as the argumentapproaches x from below” ifthere is some argument y lessthan x such that throughoutthe interval from y (included)to x (excluded) the functionhas values which are membersof α. We may now say that afunction is continuous for theargument a, for which it hasthe value fa, if it satisfies fourconditions, namely:— (original page )

() Given any real numberless than fa, the function con-verges into the successors ofthis number as the argumentapproaches a from below;

() Given any real numbergreater than fa, the functionconverges into the predeces-sors of this number as the ar-gument approaches a from be-low;

() and () Similar condi-tions for approaches to a fromabove.

The advantage of this formof definition is that it analyses (original page )

the conditions of continuityinto four, derived from con-sidering arguments and val-ues respectively greater or lessthan the argument and valuefor which continuity is to bedefined. |

We may now generalise ourdefinitions so as to apply toseries which are not numer-ical or known to be numer-ically measurable. The caseof motion is a convenient oneto bear in mind. There is astory by H. G. Wells whichwill illustrate, from the case (original pages –)

of motion, the difference be-tween the limit of a functionfor a given argument andits value for the same argu-ment. The hero of the story,who possessed, without hisknowledge, the power of re-alising his wishes, was beingattacked by a policeman, buton ejaculating “Go to——”he found that the policemandisappeared. If f (t) was thepoliceman’s position at timet, and t the moment of theejaculation, the limit of thepoliceman’s positions as t ap- (original page )

proached to t from belowwould be in contact with thehero, whereas the value for theargument t was —. But suchoccurrences are supposed tobe rare in the real world, andit is assumed, though withoutadequate evidence, that allmotions are continuous, i.e.that, given any body, if f (t) isits position at time t, f (t) is acontinuous function of t. It isthe meaning of “continuity”involved in such statementswhich we now wish to defineas simply as possible. (original page )

The definitions given forthe case of functions whereargument and value are realnumbers can readily be adapt-ed for more general use.

Let P and Q be two re-lations, which it is well toimagine serial, though it isnot necessary to our defini-tions that they should be so.Let R be a one-many relationwhose domain is contained inthe field of P, while its con-verse domain is contained inthe field of Q. Then R is (in ageneralised sense) a function, (original page )

whose arguments belong tothe field of Q, while its val-ues belong to the field of P.Suppose, for example, thatwe are dealing with a particlemoving on a line: let Q bethe time-series, P the series ofpoints on our line from leftto right, R the relation of theposition of our particle onthe line at time a to the timea, so that “the R of a” is itsposition at time a. This illus-tration may be borne in mindthroughout our definitions.

We shall say that the func- (original page )

tion R is continuous for the ar-gument | a if, given any inter-val α on the P-series contain-ing the value of the functionfor the argument a, there is aninterval on the Q-series con-taining a not as an end-pointand such that, throughout thisinterval, the function has val-ues which are members of α.(We mean by an “interval” allthe terms between any two;i.e. if x and y are two membersof the field of P, and x has therelation P to y, we shall meanby the “P-interval x to y” all (original pages –)

terms z such that x has the re-lation P to z and z has the re-lation P to y—together, whenso stated, with x or y them-selves.)

We can easily define the“ultimate section” and the“ultimate oscillation.” To de-fine the “ultimate section” forapproaches to the argumenta from below, take any ar-gument y which precedes a(i.e. has the relation Q to a),take the values of the func-tion for all arguments up toand including y, and form the (original page )

section of P defined by thesevalues, i.e. those members ofthe P-series which are earlierthan or identical with someof these values. Form all suchsections for all y’s that pre-cede a, and take their commonpart; this will be the ultimatesection. The ultimate uppersection and the ultimate oscil-lation are then defined exactlyas in the previous case.

The adaptation of the defi-nition of convergence and theresulting alternative defini-tion of continuity offers no (original page )

difficulty of any kind.We say that a function R is

“ultimately Q-convergent intoα” if there is a member y ofthe converse domain of R andthe field of Q such that thevalue of the function for theargument y and for any argu-ment to which y has the rela-tion Q is a member of α. Wesay that R “Q-converges intoα as the argument approachesa given argument a” if there isa term y having the relation Qto a and belonging to the con-verse domain of R and such (original page )

that the value of the functionfor any argument in the Q-interval from y (inclusive) to a(exclusive) belongs to α.

Of the four conditions thata function must fulfil in orderto be continuous for the argu-ment a, the first is, putting bfor the value for the argumenta: |

Given any term having therelation P to b, R Q-convergesinto the successors of b (withrespect to P) as the argumentapproaches a from below.

The second condition is ob- (original pages –)

tained by replacing P by itsconverse; the third and fourthare obtained from the first andsecond by replacing Q by itsconverse.

There is thus nothing, inthe notions of the limit of afunction or the continuity ofa function, that essentiallyinvolves number. Both can bedefined generally, and manypropositions about them canbe proved for any two se-ries (one being the argument-series and the other the value-series). It will be seen that (original page )

the definitions do not involveinfinitesimals. They involveinfinite classes of intervals,growing smaller without anylimit short of zero, but theydo not involve any intervalsthat are not finite. This isanalogous to the fact that if aline an inch long be halved,then halved again, and so onindefinitely, we never reachinfinitesimals in this way: af-ter n bisections, the length ofour bit is /n of an inch; andthis is finite whatever finitenumber n may be. The process (original page )

of successive bisection doesnot lead to divisions whose or-dinal number is infinite, sinceit is essentially a one-by-oneprocess. Thus infinitesimalsare not to be reached in thisway. Confusions on such top-ics have had much to do withthe difficulties which havebeen found in the discussionof infinity and continuity.

(original page )

CHAPTER XIISELECTIONS AND

THEMULTIPLICATIVE

AXIOM

In this chapter we have to con-sider an axiom which can beenunciated, but not proved,in terms of logic, and whichis convenient, though not in-dispensable, in certain por-tions of mathematics. It is

(original page )

convenient, in the sense thatmany interesting proposi-tions, which it seems natu-ral to suppose true, cannotbe proved without its help;but it is not indispensable,because even without thosepropositions the subjects inwhich they occur still exist,though in a somewhat muti-lated form.

Before enunciating the mul-tiplicative axiom, we mustfirst explain the theory of se-lections, and the definition ofmultiplication when the num- (original page )

ber of factors may be infinite.In defining the arithmetical

operations, the only correctprocedure is to construct anactual class (or relation, inthe case of relation-numbers)having the required numberof terms. This sometimes de-mands a certain amount ofingenuity, but it is essential inorder to prove the existenceof the number defined. Take,as the simplest example, thecase of addition. Suppose weare given a cardinal numberµ, and a class α which has µ (original page )

terms. How shall we defineµ + µ? For this purpose wemust have two classes havingµ terms, and they must notoverlap. We can constructsuch classes from α in variousways, of which the followingis perhaps the simplest: Formfirst all the ordered coupleswhose first term is a class con-sisting of a single member ofα, and whose second term isthe null-class; then, secondly,form all the ordered coupleswhose first term is | the null-class and whose second term (original pages –)

is a class consisting of a sin-gle member of α. These twoclasses of couples have nomember in common, and thelogical sum of the two classeswill have µ+µ terms. Exactlyanalogously we can defineµ+ ν, given that µ is the num-ber of some class α and ν isthe number of some class β.

Such definitions, as a rule,are merely a question of asuitable technical device. Butin the case of multiplication,where the number of factorsmay be infinite, important (original page )

problems arise out of the defi-nition.

Multiplication when thenumber of factors is finite of-fers no difficulty. Given twoclasses α and β, of which thefirst has µ terms and the sec-ond ν terms, we can defineµ×ν as the number of orderedcouples that can be formed bychoosing the first term out ofα and the second out of β. Itwill be seen that this defini-tion does not require that αand β should not overlap; iteven remains adequate when (original page )

α and β are identical. Forexample, let α be the classwhose members are x, x, x.Then the class which is usedto define the product µ × µ isthe class of couples:

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

This definition remains ap-plicable when µ or ν or bothare infinite, and it can be ex-tended step by step to three orfour or any finite number offactors. No difficulty arises as (original page )

regards this definition, exceptthat it cannot be extended toan infinite number of factors.

The problem of multipli-cation when the number offactors may be infinite arisesin this way: Suppose we havea class κ consisting of classes;suppose the number of termsin each of these classes isgiven. How shall we definethe product of all these num-bers? If we can frame ourdefinition generally, it will beapplicable whether κ is finiteor infinite. It is to be observed (original page )

that the problem is to be ableto deal with the case when κis infinite, not with the casewhen its members are. If |κ is not infinite, the methoddefined above is just as ap-plicable when its membersare infinite as when they arefinite. It is the case when κ isinfinite, even though its mem-bers may be finite, that wehave to find a way of dealingwith.

The following method ofdefining multiplication gener-ally is due to Dr Whitehead. (original pages –)

It is explained and treated atlength in Principia Mathemat-ica, vol. i. ∗ff., and vol. ii.∗.

Let us suppose to beginwith that κ is a class of classesno two of which overlap—say the constituencies in acountry where there is no plu-ral voting, each constituencybeing considered as a classof voters. Let us now set towork to choose one term outof each class to be its repre-sentative, as constituencies dowhen they elect members of (original page )

Parliament, assuming that bylaw each constituency has toelect a man who is a voter inthat constituency. We thusarrive at a class of representa-tives, who make up our Par-liament, one being selectedout of each constituency. Howmany different possible waysof choosing a Parliament arethere? Each constituency canselect any one of its voters,and therefore if there are µvoters in a constituency, it canmake µ choices. The choicesof the different constituencies (original page )

are independent; thus it isobvious that, when the totalnumber of constituencies isfinite, the number of possi-ble Parliaments is obtainedby multiplying together thenumbers of voters in the var-ious constituencies. Whenwe do not know whether thenumber of constituencies isfinite or infinite, we may takethe number of possible Parlia-ments as defining the productof the numbers of the sepa-rate constituencies. This isthe method by which infi- (original page )

nite products are defined. Wemust now drop our illustra-tion, and proceed to exactstatements.

Let κ be a class of classes,and let us assume to beginwith that no two members ofκ overlap, i.e. that if α and βare two different members ofκ, then no member of the oneis a member of the other. Weshall call a class a “selection”from κ when it consists ofjust one term from each mem-ber of κ; i.e. µ is a “selection”from κ if every member of µ (original page )

belongs to some member | ofκ, and if α be any member ofκ, µ and α have exactly oneterm in common. The classof all “selections” from κ weshall call the “multiplicativeclass” of κ. The number ofterms in the multiplicativeclass of κ, i.e. the number ofpossible selections from κ, isdefined as the product of thenumbers of the members ofκ. This definition is equallyapplicable whether κ is finiteor infinite.

Before we can be wholly sat- (original pages –)

isfied with these definitions,we must remove the restric-tion that no two membersof κ are to overlap. For thispurpose, instead of definingfirst a class called a “selec-tion,” we will define first arelation which we will call a“selector.” A relation R willbe called a “selector” from κif, from every member of κ,it picks out one term as therepresentative of that mem-ber, i.e. if, given any memberα of κ, there is just one termx which is a member of α and (original page )

has the relation R to α; andthis is to be all that R does.The formal definition is:

A “selector” from a class ofclasses κ is a one-many rela-tion, having κ for its conversedomain, and such that, if x hasthe relation to α, then x is amember of α.

If R is a selector from κ, andα is a member of κ, and x isthe term which has the rela-tion R to α, we call x the “rep-resentative” of α in respect ofthe relation R.

A “selection” from κ will (original page )

now be defined as the domainof a selector; and the multi-plicative class, as before, willbe the class of selections.

But when the members ofκ overlap, there may be moreselectors than selections, sincea term x which belongs to twoclasses α and β may be se-lected once to represent α andonce to represent β, givingrise to different selectors inthe two cases, but to the sameselection. For purposes ofdefining multiplication, it isthe selectors we require rather (original page )

than the selections. Thus wedefine:

“The product of the num-bers of the members of a classof classes κ” is the number ofselectors from κ.

We can define exponentia-tion by an adaptation of theabove | plan. We might, ofcourse, define µν as the num-ber of selectors from ν classes,each of which has µ terms.But there are objections tothis definition, derived fromthe fact that the multiplica-tive axiom (of which we shall (original pages –)

speak shortly) is unnecessarilyinvolved if it is adopted. Weadopt instead the followingconstruction:—

Let α be a class having µterms, and β a class having νterms. Let y be a member ofβ, and form the class of all or-dered couples that have y fortheir second term and a mem-ber of α for their first term.There will be µ such couplesfor a given y, since any mem-ber of α may be chosen forthe first term, and α has µmembers. If we now form all (original page )

the classes of this sort thatresult from varying y, we ob-tain altogether ν classes, sincey may be any member of β,and β has ν members. Theseν classes are each of thema class of couples, namely,all the couples that can beformed of a variable memberof α and a fixed member of β.We define µν as the number ofselectors from the class con-sisting of these ν classes. Orwe may equally well define µν

as the number of selections,for, since our classes of cou- (original page )

ples are mutually exclusive,the number of selectors is thesame as the number of selec-tions. A selection from ourclass of classes will be a setof ordered couples, of whichthere will be exactly one hav-ing any given member of β forits second term, and the firstterm may be any member ofα. Thus µν is defined by theselectors from a certain set ofν classes each having µ terms,but the set is one having acertain structure and a moremanageable composition than (original page )

is the case in general. Therelevance of this to the mul-tiplicative axiom will appearshortly.

What applies to exponen-tiation applies also to theproduct of two cardinals.We might define “µ × ν” asthe sum of the numbers of νclasses each having µ terms,but we prefer to define it asthe number of ordered cou-ples to be formed consistingof a member of α followed bya member of β, where α has µterms and β has ν terms. This (original page )

definition, also, is designed toevade the necessity of assum-ing the multiplicative axiom. |

With our definitions, wecan prove the usual formallaws of multiplication andexponentiation. But there isone thing we cannot prove:we cannot prove that a prod-uct is only zero when one ofits factors is zero. We canprove this when the numberof factors is finite, but notwhen it is infinite. In otherwords, we cannot prove that,given a class of classes none (original pages –)

of which is null, there mustbe selectors from them; orthat, given a class of mutuallyexclusive classes, there mustbe at least one class consist-ing of one term out of eachof the given classes. Thesethings cannot be proved; andalthough, at first sight, theyseem obviously true, yet re-flection brings gradually in-creasing doubt, until at lastwe become content to registerthe assumption and its con-sequences, as we register theaxiom of parallels, without (original page )

assuming that we can knowwhether it is true or false. Theassumption, loosely worded,is that selectors and selec-tions exist when we shouldexpect them. There are manyequivalent ways of stating itprecisely. We may begin withthe following:—

“Given any class of mutu-ally exclusive classes, of whichnone is null, there is at leastone class which has exactlyone term in common witheach of the given classes.”

This proposition we will (original page )

call the “multiplicative ax-iom.” We will first give var-ious equivalent forms of theproposition, and then con-sider certain ways in whichits truth or falsehood is ofinterest to mathematics.

The multiplicative axiom isequivalent to the propositionthat a product is only zerowhen at least one of its factorsis zero; i.e. that, if any num-ber of cardinal numbers beSee Principia Mathematica, vol. i.

∗. Also vol. iii. ∗–.

(original page )

multiplied together, the resultcannot be unless one of thenumbers concerned is .

The multiplicative axiom isequivalent to the propositionthat, if R be any relation, andκ any class contained in theconverse domain of R, thenthere is at least one one-manyrelation implying R and hav-ing κ for its converse domain.

The multiplicative axiom isequivalent to the assumptionthat if α be any class, and κ allthe sub-classes of α with theexception | of the null-class, (original pages –)

then there is at least one selec-tor from κ. This is the formin which the axiom was firstbrought to the notice of thelearned world by Zermelo, inhis “Beweis, dass jede Mengewohlgeordnet werden kann.”

Zermelo regards the axiom asan unquestionable truth. Itmust be confessed that, un-til he made it explicit, mathe-maticians had used it withouta qualm; but it would seemMathematische Annalen, vol. lix.

pp. –. In this form we shall speakof it as Zermelo’s axiom.

(original page )

that they had done so uncon-sciously. And the credit dueto Zermelo for having madeit explicit is entirely indepen-dent of the question whetherit is true or false.

The multiplicative axiomhas been shown by Zermelo,in the above-mentioned proof,to be equivalent to the propo-sition that every class can bewell-ordered, i.e. can be ar-ranged in a series in whichevery sub-class has a firstterm (except, of course, thenull-class). The full proof of (original page )

this proposition is difficult,but it is not difficult to seethe general principle uponwhich it proceeds. It uses theform which we call “Zermelo’saxiom,” i.e. it assumes that,given any class α, there is atleast one one-many relation Rwhose converse domain con-sists of all existent sub-classesof α and which is such that, ifx has the relation R to ξ, thenx is a member of ξ. Such arelation picks out a “represen-tative” from each sub-class; ofcourse, it will often happen (original page )

that two sub-classes have thesame representative. WhatZermelo does, in effect, is tocount off the members of α,one by one, by means of R andtransfinite induction. We putfirst the representative of α;call it x. Then take the rep-resentative of the class con-sisting of all of α except x;call it x. It must be differentfrom x, because every repre-sentative is a member of itsclass, and x is shut out fromthis class. Proceed similarly totake away x, and let x be the (original page )

representative of what is left.In this way we first obtain aprogression x, x, . . . xn, . . .,assuming that α is not finite.We then take away the wholeprogression; let xω be the rep-resentative of what is left of α.In this way we can go on untilnothing is left. The succes-sive representatives will forma | well-ordered series con-taining all the members of α.(The above is, of course, onlya hint of the general lines ofthe proof.) This proposition iscalled “Zermelo’s theorem.” (original pages –)

The multiplicative axiom isalso equivalent to the assump-tion that of any two cardinalswhich are not equal, one mustbe the greater. If the axiom isfalse, there will be cardinalsµ and ν such that µ is neitherless than, equal to, nor greaterthan ν. We have seen that ℵand ℵ possibly form an in-stance of such a pair.

Many other forms of the ax-iom might be given, but theabove are the most importantof the forms known at present.As to the truth or falsehood of (original page )

the axiom in any of its forms,nothing is known at present.

The propositions that de-pend upon the axiom, withoutbeing known to be equivalentto it, are numerous and impor-tant. Take first the connectionof addition and multiplica-tion. We naturally think thatthe sum of ν mutually exclu-sive classes, each having µterms, must have µ× ν terms.When ν is finite, this can beproved. But when ν is infinite,it cannot be proved withoutthe multiplicative axiom, ex- (original page )

cept where, owing to somespecial circumstance, the ex-istence of certain selectorscan be proved. The way themultiplicative axiom entersin is as follows: Suppose wehave two sets of ν mutuallyexclusive classes, each havingµ terms, and we wish to provethat the sum of one set has asmany terms as the sum of theother. In order to prove this,we must establish a one-onerelation. Now, since there arein each case ν classes, thereis some one-one relation be- (original page )

tween the two sets of classes;but what we want is a one-onerelation between their terms.Let us consider some one-onerelation S between the classes.Then if κ and λ are the twosets of classes, and α is somemember of κ, there will be amember β of λ which will bethe correlate of α with respectto S. Now α and β each have µterms, and are therefore sim-ilar. There are, accordingly,one-one correlations of α andβ. The trouble is that there areso many. In order to obtain (original page )

a one-one correlation of thesum of κwith the sum of λ, wehave to pick out one correlatorof α with β, and similarly forevery other pair. This requiresa selection from a set of classes| of correlators, one class ofthe set being all the one-onecorrelators of α with β. If κand λ are infinite, we cannotin general know that such aselection exists, unless we canknow that the multiplicativeaxiom is true. Hence we can-not establish the usual kind ofconnection between addition (original pages –)

and multiplication.This fact has various curi-

ous consequences. To beginwith, we know that ℵ =ℵ ×ℵ = ℵ. It is commonlyinferred from this that thesum of ℵ classes each havingℵ members must itself haveℵ members, but this infer-ence is fallacious, since we donot know that the number ofterms in such a sum is ℵ×ℵ,nor consequently that it is ℵ.This has a bearing upon thetheory of transfinite ordinals.It is easy to prove that an ordi- (original page )

nal which has ℵ predecessorsmust be one of what Cantorcalls the “second class,” i.e.such that a series having thisordinal number will have ℵterms in its field. It is alsoeasy to see that, if we take anyprogression of ordinals of thesecond class, the predecessorsof their limit form at most thesum of ℵ classes each hav-ing ℵ terms. It is inferredthence—fallaciously, unlessthe multiplicative axiom istrue—that the predecessorsof the limit are ℵ in number, (original page )

and therefore that the limitis a number of the “secondclass.” That is to say, it is sup-posed to be proved that anyprogression of ordinals of thesecond class has a limit whichis again an ordinal of the sec-ond class. This proposition,with the corollary that ω (thesmallest ordinal of the thirdclass) is not the limit of anyprogression, is involved inmost of the recognised theoryof ordinals of the second class.In view of the way in whichthe multiplicative axiom is in- (original page )

volved, the proposition and itscorollary cannot be regardedas proved. They may be true,or they may not. All that canbe said at present is that wedo not know. Thus the greaterpart of the theory of ordinalsof the second class must beregarded as unproved.

Another illustration mayhelp to make the point clearer.We know that × ℵ = ℵ.Hence we might suppose thatthe sum of ℵ pairs must haveℵ terms. But this, though wecan prove that it is sometimes (original page )

the case, cannot be provedto happen always | unless weassume the multiplicative ax-iom. This is illustrated bythe millionaire who boughta pair of socks whenever hebought a pair of boots, andnever at any other time, andwho had such a passion forbuying both that at last hehad ℵ pairs of boots and ℵpairs of socks. The problemis: How many boots had he,and how many socks? Onewould naturally suppose thathe had twice as many boots (original pages –)

and twice as many socks ashe had pairs of each, and thattherefore he had ℵ of each,since that number is not in-creased by doubling. But thisis an instance of the difficulty,already noted, of connect-ing the sum of ν classes eachhaving µ terms with µ × ν.Sometimes this can be done,sometimes it cannot. In ourcase it can be done with theboots, but not with the socks,except by some very artifi-cial device. The reason forthe difference is this: Among (original page )

boots we can distinguish rightand left, and therefore we canmake a selection of one outof each pair, namely, we canchoose all the right boots or allthe left boots; but with socksno such principle of selectionsuggests itself, and we can-not be sure, unless we assumethe multiplicative axiom, thatthere is any class consistingof one sock out of each pair.Hence the problem.

We may put the matter inanother way. To prove that aclass has ℵ terms, it is nec- (original page )

essary and sufficient to findsome way of arranging itsterms in a progression. Thereis no difficulty in doing thiswith the boots. The pairs aregiven as forming an ℵ, andtherefore as the field of a pro-gression. Within each pair,take the left boot first andthe right second, keeping theorder of the pair unchanged;in this way we obtain a pro-gression of all the boots. Butwith the socks we shall have tochoose arbitrarily, with eachpair, which to put first; and an (original page )

infinite number of arbitrarychoices is an impossibility.Unless we can find a rule forselecting, i.e. a relation whichis a selector, we do not knowthat a selection is even theo-retically possible. Of course,in the case of objects in space,like socks, we always can findsome principle of selection.For example, take the centresof mass of the socks: therewill be points p in space suchthat, with any | pair, the cen-tres of mass of the two socksare not both at exactly the (original pages –)

same distance from p; thus wecan choose, from each pair,that sock which has its centreof mass nearer to p. But thereis no theoretical reason whya method of selection such asthis should always be possi-ble, and the case of the socks,with a little goodwill on thepart of the reader, may serveto show how a selection mightbe impossible.

It is to be observed that, if itwere impossible to select oneout of each pair of socks, itwould follow that the socks (original page )

could not be arranged in a pro-gression, and therefore thatthere were not ℵ of them.This case illustrates that, ifµ is an infinite number, oneset of µ pairs may not containthe same number of termsas another set of µ pairs; for,given ℵ pairs of boots, thereare certainly ℵ boots, but wecannot be sure of this in thecase of the socks unless we as-sume the multiplicative axiomor fall back upon some fortu-itous geometrical method ofselection such as the above. (original page )

Another important probleminvolving the multiplicativeaxiom is the relation of reflex-iveness to non-inductiveness.It will be remembered that inChapter VIII. we pointed outthat a reflexive number mustbe non-inductive, but that theconverse (so far as is known atpresent) can only be proved ifwe assume the multiplicativeaxiom. The way in which thiscomes about is as follows:—

It is easy to prove that areflexive class is one whichcontains sub-classes having (original page )

ℵ terms. (The class may, ofcourse, itself have ℵ terms.)Thus we have to prove, ifwe can, that, given any non-inductive class, it is possibleto choose a progression outof its terms. Now there is nodifficulty in showing that anon-inductive class must con-tain more terms than any in-ductive class, or, what comesto the same thing, that if αis a non-inductive class andν is any inductive number,there are sub-classes of α thathave ν terms. Thus we can (original page )

form sets of finite sub-classesof α: First one class havingno terms, then classes having term (as many as there aremembers of α), then classeshaving | terms, and so on.We thus get a progression ofsets of sub-classes, each setconsisting of all those thathave a certain given finitenumber of terms. So far wehave not used the multiplica-tive axiom, but we have onlyproved that the number ofcollections of sub-classes of αis a reflexive number, i.e. that, (original pages –)

if µ is the number of membersof α, so that µ is the numberof sub-classes of α and

µis

the number of collections ofsub-classes, then, provided µis not inductive,

µmust be

reflexive. But this is a longway from what we set out toprove.

In order to advance be-yond this point, we must em-ploy the multiplicative axiom.From each set of sub-classeslet us choose out one, omittingthe sub-class consisting of thenull-class alone. That is to say, (original page )

we select one sub-class con-taining one term, α, say; onecontaining two terms, α, say;one containing three, α, say;and so on. (We can do this ifthe multiplicative axiom is as-sumed; otherwise, we do notknow whether we can alwaysdo it or not.) We have now aprogression α, α, α, . . . ofsub-classes of α, instead of aprogression of collections ofsub-classes; thus we are onestep nearer to our goal. Wenow know that, assuming themultiplicative axiom, if µ is (original page )

a non-inductive number, µ

must be a reflexive number.The next step is to notice

that, although we cannot besure that new members ofα come in at any one speci-fied stage in the progressionα, α, α, . . . we can be surethat new members keep oncoming in from time to time.Let us illustrate. The class α,which consists of one term,is a new beginning; let theone term be x. The classα, consisting of two terms,may or may not contain x; (original page )

if it does, it introduces onenew term; and if it does not,it must introduce two newterms, say x, x. In this caseit is possible that α consistsof x, x, x, and so intro-duces no new terms, but inthat case α must introduce anew term. The first ν classesα, α, α, . . . αν contain, atthe very most, +++ . . .+νterms, i.e. ν(ν + )/ terms;thus it would be possible, ifthere were no repetitions inthe first ν classes, to go onwith only repetitions from (original page )

the (ν + )th | class to theν(ν +)/th class. But by thattime the old terms would nolonger be sufficiently numer-ous to form a next class withthe right number of members,i.e. ν(ν + )/ + , thereforenew terms must come in atthis point if not sooner. Itfollows that, if we omit fromour progression α, α, α, . . .all those classes that are com-posed entirely of membersthat have occurred in pre-vious classes, we shall stillhave a progression. Let our (original pages –)

new progression be calledβ, β, β . . . (We shall haveα = β and α = β, becauseα and α must introduce newterms. We may or may nothave α = β, but, speakinggenerally, βµ will be αν , whereν is some number greater thanµ; i.e. the β’s are some of theα’s.) Now these β’s are suchthat any one of them, say βµ,contains members which havenot occurred in any of the pre-vious β’s. Let γµ be the partof βµ which consists of newmembers. Thus we get a new (original page )

progression γ, γ, γ, . . .(Again γ will be identicalwith β and with α; if α doesnot contain the one member ofα, we shall have γ = β = α,but if α does contain this onemember, γ will consist of theother member of α.) Thisnew progression of γ’s con-sists of mutually exclusiveclasses. Hence a selectionfrom them will be a progres-sion; i.e. if x is the memberof γ, x is a member of γ,x is a member of γ, and soon; then x, x, x, . . . is a pro- (original page )

gression, and is a sub-class ofα. Assuming the multiplica-tive axiom, such a selectioncan be made. Thus by twiceusing this axiom we can provethat, if the axiom is true, everynon-inductive cardinal mustbe reflexive. This could alsobe deduced from Zermelo’stheorem, that, if the axiom istrue, every class can be well-ordered; for a well-orderedseries must have either a finiteor a reflexive number of termsin its field.

There is one advantage in (original page )

the above direct argument, asagainst deduction from Zer-melo’s theorem, that the aboveargument does not demandthe universal truth of the mul-tiplicative axiom, but only itstruth as applied to a set ofℵ classes. It may happenthat the axiom holds for ℵclasses, though not for largernumbers of classes. For thisreason it is better, when | it ispossible, to content ourselveswith the more restricted as-sumption. The assumptionmade in the above direct ar- (original pages –)

gument is that a product ofℵ factors is never zero unlessone of the factors is zero. Wemay state this assumption inthe form: “ℵ is a multipliablenumber,” where a number νis defined as “multipliable”when a product of ν factorsis never zero unless one ofthe factors is zero. We canprove that a finite number isalways multipliable, but wecannot prove that any infinitenumber is so. The multiplica-tive axiom is equivalent to theassumption that all cardinal (original page )

numbers are multipliable. Butin order to identify the reflex-ive with the non-inductive,or to deal with the problemof the boots and socks, or toshow that any progression ofnumbers of the second classis of the second class, we onlyneed the very much smallerassumption that ℵ is multi-pliable.

It is not improbable thatthere is much to be discoveredin regard to the topics dis-cussed in the present chapter.Cases may be found where (original page )

propositions which seem toinvolve the multiplicative ax-iom can be proved withoutit. It is conceivable that themultiplicative axiom in itsgeneral form may be shownto be false. From this pointof view, Zermelo’s theoremoffers the best hope: the con-tinuum or some still moredense series might be provedto be incapable of having itsterms well-ordered, whichwould prove the multiplica-tive axiom false, in virtue ofZermelo’s theorem. But so (original page )

far, no method of obtainingsuch results has been discov-ered, and the subject remainswrapped in obscurity.

(original page )

CHAPTER XIIITHE AXIOMOFINFINITY ANDLOGICAL TYPES

The axiom of infinity is an as-sumption which may be enun-ciated as follows:—

“If n be any inductive car-dinal number, there is at leastone class of individuals hav-ing n terms.”

If this is true, it follows, of

(original page )

course, that there are manyclasses of individuals having nterms, and that the total num-ber of individuals in the worldis not an inductive number.For, by the axiom, there is atleast one class having n + terms, from which it followsthat there are many classes ofn terms and that n is not thenumber of individuals in theworld. Since n is any induc-tive number, it follows thatthe number of individuals inthe world must (if our axiombe true) exceed any inductive (original page )

number. In view of what wefound in the preceding chap-ter, about the possibility ofcardinals which are neitherinductive nor reflexive, wecannot infer from our axiomthat there are at least ℵ indi-viduals, unless we assume themultiplicative axiom. But wedo know that there are at leastℵ classes of classes, sincethe inductive cardinals areclasses of classes, and forma progression if our axiom istrue.

The way in which the need (original page )

for this axiom arises may beexplained as follows. One ofPeano’s assumptions is that notwo inductive cardinals havethe same successor, i.e. thatwe shall not have m+ = n+unless m = n, if m and n areinductive cardinals. In Chap-ter VIII. we had occasion touse what is virtually the sameas the above assumption ofPeano’s, namely, that, if n isan inductive cardinal, | n isnot equal to n + . It mightbe thought that this could beproved. We can prove that, (original pages –)

if α is an inductive class, andn is the number of membersof α, then n is not equal ton+ . This proposition is eas-ily proved by induction, andmight be thought to imply theother. But in fact it does not,since there might be no suchclass as α. What it does implyis this: If n is an inductive car-dinal such that there is at leastone class having n members,then n is not equal to n + .The axiom of infinity assuresus (whether truly or falsely)that there are classes having (original page )

n members, and thus enablesus to assert that n is not equalto n+ . But without this ax-iom we should be left withthe possibility that n and n+might both be the null-class.

Let us illustrate this possi-bility by an example: Supposethere were exactly nine indi-viduals in the world. (As towhat is meant by the word“individual,” I must ask thereader to be patient.) Then theinductive cardinals from upto would be such as we ex-pect, but (defined as + ) (original page )

would be the null-class. It willbe remembered that n+maybe defined as follows: n + is the collection of all thoseclasses which have a term xsuch that, when x is takenaway, there remains a class ofn terms. Now applying thisdefinition, we see that, in thecase supposed, + is a classconsisting of no classes, i.e.it is the null-class. The samewill be true of + , or gener-ally of + n, unless n is zero.Thus and all subsequentinductive cardinals will all be (original page )

identical, since they will all bethe null-class. In such a casethe inductive cardinals willnot form a progression, norwill it be true that no two havethe same successor, for and will both be succeeded bythe null-class ( being itselfthe null-class). It is in orderto prevent such arithmeticalcatastrophes that we requirethe axiom of infinity.

As a matter of fact, so longas we are content with thearithmetic of finite integers,and do not introduce either (original page )

infinite integers or infiniteclasses or series of finite in-tegers or ratios, it is possibleto obtain all desired resultswithout the axiom of infinity.That is to say, we can deal withthe addition, |multiplication,and exponentiation of finiteintegers and of ratios, butwe cannot deal with infiniteintegers or with irrationals.Thus the theory of the trans-finite and the theory of realnumbers fails us. How thesevarious results come aboutmust now be explained. (original pages –)

Assuming that the numberof individuals in the worldis n, the number of classesof individuals will be n.This is in virtue of the gen-eral proposition mentioned inChapter VIII. that the num-ber of classes contained in aclass which has n members isn. Now n is always greaterthan n. Hence the number ofclasses in the world is greaterthan the number of individ-uals. If, now, we suppose thenumber of individuals to be ,as we did just now, the num- (original page )

ber of classes will be , i.e.. Thus if we take our num-bers as being applied to thecounting of classes instead ofto the counting of individuals,our arithmetic will be nor-mal until we reach : thefirst number to be null willbe . And if we advanceto classes of classes we shalldo still better: the number ofthem will be , a numberwhich is so large as to stag-ger imagination, since it hasabout digits. And if weadvance to classes of classes (original page )

of classes, we shall obtaina number represented by raised to a power which hasabout digits; the numberof digits in this number willbe about three times . Ina time of paper shortage it isundesirable to write out thisnumber, and if we want largerones we can obtain them bytravelling further along thelogical hierarchy. In this wayany assigned inductive car-dinal can be made to find itsplace among numbers whichare not null, merely by travel- (original page )

ling along the hierarchy for asufficient distance.

As regards ratios, we havea very similar state of affairs.If a ratio µ/ν is to have the ex-pected properties, there mustbe enough objects of whateversort is being counted to insurethat the null-class does notsuddenly obtrude itself. Butthis can be insured, for anygiven ratio µ/ν, without the

On this subject see PrincipiaMathematica, vol. ii. ∗ff. On thecorresponding problems as regardsratio, see ibid., vol. iii. ∗ff.

(original page )

axiom of | infinity, by merelytravelling up the hierarchya sufficient distance. If wecannot succeed by countingindividuals, we can try count-ing classes of individuals; ifwe still do not succeed, wecan try classes of classes, andso on. Ultimately, howeverfew individuals there may bein the world, we shall reacha stage where there are manymore than µ objects, what-ever inductive number µ maybe. Even if there were no in-dividuals at all, this would (original pages –)

still be true, for there wouldthen be one class, namely,the null-class, classes ofclasses (namely, the null-classof classes and the class whoseonly member is the null-classof individuals), classes ofclasses of classes, at thenext stage, , at the nextstage, and so on. Thus no suchassumption as the axiom ofinfinity is required in order toreach any given ratio or anygiven inductive cardinal.

It is when we wish to dealwith the whole class or series (original page )

of inductive cardinals or of ra-tios that the axiom is required.We need the whole class of in-ductive cardinals in order toestablish the existence of ℵ,and the whole series in orderto establish the existence ofprogressions: for these results,it is necessary that we shouldbe able to make a single classor series in which no induc-tive cardinal is null. We needthe whole series of ratios inorder of magnitude in orderto define real numbers as seg-ments: this definition will not (original page )

give the desired result unlessthe series of ratios is compact,which it cannot be if the totalnumber of ratios, at the stageconcerned, is finite.

It would be natural to sup-pose—as I supposed myself informer days—that, by meansof constructions such as wehave been considering, theaxiom of infinity could beproved. It may be said: Letus assume that the numberof individuals is n, where nmay be without spoiling ourargument; then if we form the (original page )

complete set of individuals,classes, classes of classes, etc.,all taken together, the numberof terms in our whole set willbe

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

which is ℵ. Thus taking allkinds of objects together, andnot | confining ourselves toobjects of any one type, weshall certainly obtain an infi-nite class, and shall thereforenot need the axiom of infinity.So it might be said.

(original pages –)

Now, before going into thisargument, the first thing toobserve is that there is anair of hocus-pocus about it:something reminds one of theconjurer who brings thingsout of the hat. The man whohas lent his hat is quite surethere wasn’t a live rabbit in itbefore, but he is at a loss tosay how the rabbit got there.So the reader, if he has a ro-bust sense of reality, will feelconvinced that it is impossi-ble to manufacture an infinitecollection out of a finite col- (original page )

lection of individuals, thoughhe may be unable to say wherethe flaw is in the above con-struction. It would be a mis-take to lay too much stress onsuch feelings of hocus-pocus;like other emotions, they mayeasily lead us astray. But theyafford a prima facie groundfor scrutinising very closelyany argument which arousesthem. And when the above ar-gument is scrutinised it will,in my opinion, be found tobe fallacious, though the fal-lacy is a subtle one and by no (original page )

means easy to avoid consis-tently.

The fallacy involved is thefallacy which may be called“confusion of types.” To ex-plain the subject of “types”fully would require a wholevolume; moreover, it is thepurpose of this book to avoidthose parts of the subjectswhich are still obscure andcontroversial, isolating, forthe convenience of beginners,those parts which can be ac-cepted as embodying mathe-matically ascertained truths. (original page )

Now the theory of types em-phatically does not belongto the finished and certainpart of our subject: much ofthis theory is still inchoate,confused, and obscure. Butthe need of some doctrine oftypes is less doubtful thanthe precise form the doctrineshould take; and in connec-tion with the axiom of infinityit is particularly easy to seethe necessity of some suchdoctrine.

This necessity results, forexample, from the “contra- (original page )

diction of the greatest cardi-nal.” We saw in Chapter VIII.that the number of classescontained in a given class isalways greater than the | num-ber of members of the class,and we inferred that there isno greatest cardinal number.But if we could, as we sug-gested a moment ago, add to-gether into one class the indi-viduals, classes of individuals,classes of classes of individ-uals, etc., we should obtaina class of which its own sub-classes would be members. (original pages –)

The class consisting of all ob-jects that can be counted, ofwhatever sort, must, if therebe such a class, have a car-dinal number which is thegreatest possible. Since all itssub-classes will be membersof it, there cannot be moreof them than there are mem-bers. Hence we arrive at acontradiction.

When I first came uponthis contradiction, in the year, I attempted to discoversome flaw in Cantor’s proofthat there is no greatest cardi- (original page )

nal, which we gave in ChapterVIII. Applying this proof tothe supposed class of all imag-inable objects, I was led to anew and simpler contradic-tion, namely, the following:—

The comprehensive classwe are considering, which isto embrace everything, mustembrace itself as one of itsmembers. In other words, ifthere is such a thing as “ev-erything,” then “everything”is something, and is a mem-ber of the class “everything.”But normally a class is not a (original page )

member of itself. Mankind,for example, is not a man.Form now the assemblageof all classes which are notmembers of themselves. Thisis a class: is it a member ofitself or not? If it is, it isone of those classes that arenot members of themselves,i.e. it is not a member of it-self. If it is not, it is not oneof those classes that are notmembers of themselves, i.e. itis a member of itself. Thus ofthe two hypotheses—that itis, and that it is not, a mem- (original page )

ber of itself—each impliesits contradictory. This is acontradiction.

There is no difficulty inmanufacturing similar con-tradictions ad lib. The solutionof such contradictions by thetheory of types is set forthfully in Principia Mathemat-ica, and also, more briefly, inarticles by the present authorin the American Journal | ofMathematics and in the RevueVol. i., Introduction, chap. ii., ∗

and ∗; vol. ii., Prefatory Statement.“Mathematical Logic as based on

(original pages –)

de Metaphysique et de Morale.

For the present an outline ofthe solution must suffice.

The fallacy consists in theformation of what we maycall “impure” classes, i.e.classes which are not pureas to “type.” As we shall see ina later chapter, classes are log-ical fictions, and a statementwhich appears to be about aclass will only be significant if

the Theory of Types,” vol. xxx., ,pp. –.“Les paradoxes de la logique,”

, pp. –.

(original page )

it is capable of translation intoa form in which no mention ismade of the class. This placesa limitation upon the waysin which what are nominally,though not really, names forclasses can occur significantly:a sentence or set of symbolsin which such pseudo-namesoccur in wrong ways is notfalse, but strictly devoid ofmeaning. The suppositionthat a class is, or that it is not,a member of itself is mean-ingless in just this way. Andmore generally, to suppose (original page )

that one class of individu-als is a member, or is not amember, of another class ofindividuals will be to supposenonsense; and to constructsymbolically any class whosemembers are not all of thesame grade in the logical hi-erarchy is to use symbols ina way which makes them nolonger symbolise anything.

Thus if there are n indi-viduals in the world, and n

classes of individuals, we can-not form a new class, con-sisting of both individuals (original page )

and classes and having n+ n

members. In this way theattempt to escape from theneed for the axiom of infinitybreaks down. I do not pre-tend to have explained thedoctrine of types, or donemore than indicate, in roughoutline, why there is needof such a doctrine. I haveaimed only at saying just somuch as was required in or-der to show that we cannotprove the existence of infinitenumbers and classes by suchconjurer’s methods as we have (original page )

been examining. There re-main, however, certain otherpossible methods which mustbe considered.

Various arguments profess-ing to prove the existence ofinfinite classes are given inthe Principles of Mathematics,§ (p. ). | In so far asthese arguments assume that,if n is an inductive cardinal,n is not equal to n + , theyhave been already dealt with.There is an argument, sug-gested by a passage in Plato’sParmenides, to the effect that, (original pages –)

if there is such a number as, then has being; but isnot identical with being, andtherefore and being are two,and therefore there is such anumber as , and togetherwith and being gives a classof three terms, and so on.This argument is fallacious,partly because “being” is nota term having any definitemeaning, and still more be-cause, if a definite meaningwere invented for it, it wouldbe found that numbers donot have being—they are, in (original page )

fact, what are called “logi-cal fictions,” as we shall seewhen we come to consider thedefinition of classes.

The argument that thenumber of numbers from to n (both inclusive) is n + depends upon the assump-tion that up to and includingn no number is equal to itssuccessor, which, as we haveseen, will not be always true ifthe axiom of infinity is false.It must be understood thatthe equation n = n+ , whichmight be true for a finite n if (original page )

n exceeded the total numberof individuals in the world,is quite different from thesame equation as applied toa reflexive number. As ap-plied to a reflexive number, itmeans that, given a class of nterms, this class is “similar”to that obtained by addinganother term. But as appliedto a number which is too greatfor the actual world, it merelymeans that there is no class ofn individuals, and no class ofn + individuals; it does notmean that, if we mount the (original page )

hierarchy of types sufficientlyfar to secure the existence ofa class of n terms, we shallthen find this class “similar”to one of n+ terms, for if n isinductive this will not be thecase, quite independently ofthe truth or falsehood of theaxiom of infinity.

There is an argument em-ployed by both Bolzano andDedekind to prove the exis-

Bolzano, Paradoxien des Un-endlichen, .Dedekind, Was sind und was sollen

die Zahlen? No. .

(original page )

tence of reflexive classes. Theargument, in brief, is this: Anobject is not identical withthe idea of the | object, butthere is (at least in the realmof being) an idea of any object.The relation of an object to theidea of it is one-one, and ideasare only some among objects.Hence the relation “idea of”constitutes a reflexion of thewhole class of objects into apart of itself, namely, into thatpart which consists of ideas.Accordingly, the class of ob-jects and the class of ideas are (original pages –)

both infinite. This argumentis interesting, not only onits own account, but becausethe mistakes in it (or what Ijudge to be mistakes) are of akind which it is instructive tonote. The main error consistsin assuming that there is anidea of every object. It is, ofcourse, exceedingly difficultto decide what is meant byan “idea”; but let us assumethat we know. We are thento suppose that, starting (say)with Socrates, there is the ideaof Socrates, and then the idea (original page )

of the idea of Socrates, andso on ad inf. Now it is plainthat this is not the case in thesense that all these ideas haveactual empirical existence inpeople’s minds. Beyond thethird or fourth stage theybecome mythical. If the ar-gument is to be upheld, the“ideas” intended must be Pla-tonic ideas laid up in heaven,for certainly they are not onearth. But then it at once be-comes doubtful whether thereare such ideas. If we are toknow that there are, it must (original page )

be on the basis of some logi-cal theory, proving that it isnecessary to a thing that thereshould be an idea of it. Wecertainly cannot obtain thisresult empirically, or apply it,as Dedekind does, to “meineGedankenwelt”—the world ofmy thoughts.

If we were concerned toexamine fully the relation ofidea and object, we shouldhave to enter upon a numberof psychological and logicalinquiries, which are not rele-vant to our main purpose. But (original page )

a few further points shouldbe noted. If “idea” is to beunderstood logically, it maybe identical with the object, orit may stand for a description(in the sense to be explainedin a subsequent chapter). Inthe former case the argumentfails, because it was essentialto the proof of reflexivenessthat object and idea shouldbe distinct. In the secondcase the argument also fails,because the relation of ob-ject and description is not |one-one: there are innumer- (original pages –)

able correct descriptions ofany given object. Socrates(e.g.) may be described as “themaster of Plato,” or as “thephilosopher who drank thehemlock,” or as “the husbandof Xantippe.” If—to take upthe remaining hypothesis—“idea” is to be interpretedpsychologically, it must bemaintained that there is notany one definite psychologicalentity which could be calledthe idea of the object: there areinnumerable beliefs and atti-tudes, each of which could be (original page )

called an idea of the object inthe sense in which we mightsay “my idea of Socrates isquite different from yours,”but there is not any centralentity (except Socrates him-self) to bind together various“ideas of Socrates,” and thusthere is not any such one-onerelation of idea and object asthe argument supposes. Nor,of course, as we have alreadynoted, is it true psychologi-cally that there are ideas (inhowever extended a sense) ofmore than a tiny proportion of (original page )

the things in the world. For allthese reasons, the above argu-ment in favour of the logicalexistence of reflexive classesmust be rejected.

It might be thought that,whatever may be said of log-ical arguments, the empiricalarguments derivable fromspace and time, the diver-sity of colours, etc., are quitesufficient to prove the ac-tual existence of an infinitenumber of particulars. I donot believe this. We have noreason except prejudice for (original page )

believing in the infinite ex-tent of space and time, at anyrate in the sense in whichspace and time are physicalfacts, not mathematical fic-tions. We naturally regardspace and time as continuous,or, at least, as compact; butthis again is mainly preju-dice. The theory of “quanta”in physics, whether true orfalse, illustrates the fact thatphysics can never afford proofof continuity, though it mightquite possibly afford disproof.The senses are not sufficiently (original page )

exact to distinguish betweencontinuous motion and rapiddiscrete succession, as any-one may discover in a cinema.A world in which all motionconsisted of a series of smallfinite jerks would be empiri-cally indistinguishable fromone in which motion was con-tinuous. It would take uptoo much space to | defendthese theses adequately; forthe present I am merely sug-gesting them for the reader’sconsideration. If they arevalid, it follows that there is (original pages –)

no empirical reason for believ-ing the number of particularsin the world to be infinite,and that there never can be;also that there is at presentno empirical reason to be-lieve the number to be finite,though it is theoretically con-ceivable that some day theremight be evidence pointing,though not conclusively, inthat direction.

From the fact that the infi-nite is not self-contradictory,but is also not demonstrablelogically, we must conclude (original page )

that nothing can be known apriori as to whether the num-ber of things in the world isfinite or infinite. The conclu-sion is, therefore, to adopt aLeibnizian phraseology, thatsome of the possible worldsare finite, some infinite, andwe have no means of knowingto which of these two kindsour actual world belongs. Theaxiom of infinity will be truein some possible worlds andfalse in others; whether it istrue or false in this world, wecannot tell. (original page )

Throughout this chapterthe synonyms “individual”and “particular” have beenused without explanation. Itwould be impossible to ex-plain them adequately with-out a longer disquisition onthe theory of types than wouldbe appropriate to the presentwork, but a few words beforewe leave this topic may dosomething to diminish theobscurity which would other-wise envelop the meaning ofthese words.

In an ordinary statement (original page )

we can distinguish a verb,expressing an attribute or re-lation, from the substantiveswhich express the subject ofthe attribute or the terms ofthe relation. “Cæsar lived”ascribes an attribute to Cæsar;“Brutus killed Cæsar” ex-presses a relation betweenBrutus and Cæsar. Using theword “subject” in a gener-alised sense, we may call bothBrutus and Cæsar subjectsof this proposition: the factthat Brutus is grammaticallysubject and Cæsar object is (original page )

logically irrelevant, since thesame occurrence may be ex-pressed in the words “Cæsarwas killed by Brutus,” whereCæsar is the grammatical sub-ject. | Thus in the simpler sortof proposition we shall havean attribute or relation hold-ing of or between one, twoor more “subjects” in the ex-tended sense. (A relation mayhave more than two terms: e.g.“A gives B to C” is a relationof three terms.) Now it of-ten happens that, on a closerscrutiny, the apparent sub- (original pages –)

jects are found to be not reallysubjects, but to be capable ofanalysis; the only result ofthis, however, is that new sub-jects take their places. It alsohappens that the verb maygrammatically be made sub-ject: e.g. we may say, “Killingis a relation which holds be-tween Brutus and Cæsar.” Butin such cases the grammar ismisleading, and in a straight-forward statement, followingthe rules that should guidephilosophical grammar, Bru-tus and Cæsar will appear as (original page )

the subjects and killing as theverb.

We are thus led to the con-ception of terms which, whenthey occur in propositions,can only occur as subjects, andnever in any other way. This ispart of the old scholastic def-inition of substance; but per-sistence through time, whichbelonged to that notion, formsno part of the notion withwhich we are concerned. Weshall define “proper names”as those terms which can onlyoccur as subjects in propo- (original page )

sitions (using “subject” inthe extended sense just ex-plained). We shall further de-fine “individuals” or “partic-ulars” as the objects that canbe named by proper names.(It would be better to definethem directly, rather than bymeans of the kind of symbolsby which they are symbolised;but in order to do that weshould have to plunge deeperinto metaphysics than is de-sirable here.) It is, of course,possible that there is an end-less regress: that whatever (original page )

appears as a particular is re-ally, on closer scrutiny, a classor some kind of complex. Ifthis be the case, the axiomof infinity must of course betrue. But if it be not the case, itmust be theoretically possiblefor analysis to reach ultimatesubjects, and it is these thatgive the meaning of “particu-lars” or “individuals.” It is tothe number of these that theaxiom of infinity is assumedto apply. If it is true of them, itis true | of classes of them, andclasses of classes of them, and (original pages –)

so on; similarly if it is false ofthem, it is false throughoutthis hierarchy. Hence it is nat-ural to enunciate the axiomconcerning them rather thanconcerning any other stage inthe hierarchy. But whether theaxiom is true or false, thereseems no known method ofdiscovering.

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CHAPTER XIVINCOMPATIBILITY

AND THE THEORY OFDEDUCTION

We have now explored, some-what hastily it is true, thatpart of the philosophy ofmathematics which does notdemand a critical examina-tion of the idea of class. Inthe preceding chapter, how-ever, we found ourselves con-

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fronted by problems whichmake such an examinationimperative. Before we canundertake it, we must con-sider certain other parts ofthe philosophy of mathemat-ics, which we have hithertoignored. In a synthetic treat-ment, the parts which we shallnow be concerned with comefirst: they are more funda-mental than anything thatwe have discussed hitherto.Three topics will concern usbefore we reach the theory ofclasses, namely: () the theory (original page )

of deduction, () proposi-tional functions, () descrip-tions. Of these, the third isnot logically presupposed inthe theory of classes, but itis a simpler example of thekind of theory that is neededin dealing with classes. It isthe first topic, the theory ofdeduction, that will concernus in the present chapter.

Mathematics is a deductivescience: starting from cer-tain premisses, it arrives, bya strict process of deduction,at the various theorems which (original page )

constitute it. It is true that,in the past, mathematical de-ductions were often greatlylacking in rigour; it is truealso that perfect rigour is ascarcely attainable ideal. Nev-ertheless, in so far as rigouris lacking in a mathematicalproof, the proof is defective; itis no defence to urge that com-mon sense shows the resultto be correct, for if we wereto rely upon that, it would bebetter to dispense with argu-ment altogether, | rather thanbring fallacy to the rescue of (original pages –)

common sense. No appeal tocommon sense, or “intuition,”or anything except strict de-ductive logic, ought to beneeded in mathematics afterthe premisses have been laiddown.

Kant, having observed thatthe geometers of his day couldnot prove their theorems byunaided argument, but re-quired an appeal to the figure,invented a theory of math-ematical reasoning accord-ing to which the inference isnever strictly logical, but al- (original page )

ways requires the support ofwhat is called “intuition.” Thewhole trend of modern math-ematics, with its increasedpursuit of rigour, has beenagainst this Kantian theory.The things in the mathematicsof Kant’s day which cannot beproved, cannot be known—forexample, the axiom of par-allels. What can be known,in mathematics and by math-ematical methods, is whatcan be deduced from purelogic. What else is to belongto human knowledge must (original page )

be ascertained otherwise—empirically, through the sensesor through experience in someform, but not a priori. Thepositive grounds for this the-sis are to be found in PrincipiaMathematica, passim; a contro-versial defence of it is givenin the Principles of Mathemat-ics. We cannot here do morethan refer the reader to thoseworks, since the subject istoo vast for hasty treatment.Meanwhile, we shall assumethat all mathematics is deduc-tive, and proceed to inquire as (original page )

to what is involved in deduc-tion.

In deduction, we have oneor more propositions calledpremisses, from which we in-fer a proposition called theconclusion. For our purposes,it will be convenient, whenthere are originally severalpremisses, to amalgamatethem into a single proposi-tion, so as to be able to speakof the premiss as well as ofthe conclusion. Thus we mayregard deduction as a processby which we pass from knowl- (original page )

edge of a certain proposition,the premiss, to knowledge ofa certain other proposition,the conclusion. But we shallnot regard such a process aslogical deduction unless it iscorrect, i.e. unless there is sucha relation between premissand conclusion that we have aright to believe the conclusion| if we know the premiss tobe true. It is this relation thatis chiefly of interest in thelogical theory of deduction.

In order to be able validlyto infer the truth of a proposi- (original pages –)

tion, we must know that someother proposition is true, andthat there is between the twoa relation of the sort called“implication,” i.e. that (as wesay) the premiss “implies” theconclusion. (We shall definethis relation shortly.) Or wemay know that a certain otherproposition is false, and thatthere is a relation between thetwo of the sort called “disjunc-tion,” expressed by “p or q,”

We shall use the letters p, q, r, s, tto denote variable propositions.

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so that the knowledge that theone is false allows us to inferthat the other is true. Again,what we wish to infer may bethe falsehood of some propo-sition, not its truth. This maybe inferred from the truthof another proposition, pro-vided we know that the twoare “incompatible,” i.e. thatif one is true, the other isfalse. It may also be inferredfrom the falsehood of anotherproposition, in just the samecircumstances in which thetruth of the other might have (original page )

been inferred from the truthof the one; i.e. from the false-hood of p we may infer thefalsehood of q, when q impliesp. All these four are cases ofinference. When our mindsare fixed upon inference, itseems natural to take “im-plication” as the primitivefundamental relation, sincethis is the relation which musthold between p and q if we areto be able to infer the truthof q from the truth of p. Butfor technical reasons this isnot the best primitive idea (original page )

to choose. Before proceedingto primitive ideas and defi-nitions, let us consider fur-ther the various functions ofpropositions suggested by theabove-mentioned relations ofpropositions.

The simplest of such func-tions is the negative, “not-p.” This is that function ofp which is true when p isfalse, and false when p is true.It is convenient to speak ofthe truth of a proposition, orits falsehood, as its “truth-

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value”; i.e. truth is the “truth-value” of a true proposition,and falsehood of a false one.Thus not-p has the oppositetruth-value to p. |

We may take next disjunc-tion, “p or q.” This is a func-tion whose truth-value istruth when p is true and alsowhen q is true, but is false-hood when both p and q arefalse.

Next we may take conjunc-tion, “p and q.” This has truth

This term is due to Frege.

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for its truth-value when p andq are both true; otherwise ithas falsehood for its truth-value.

Take next incompatibility,i.e. “p and q are not both true.”This is the negation of con-junction; it is also the disjunc-tion of the negations of p andq, i.e. it is “not-p or not-q.” Itstruth-value is truth when p isfalse and likewise when q isfalse; its truth-value is false-hood when p and q are bothtrue.

Last take implication, i.e. “p (original page )

implies q,” or “if p, then q.”This is to be understood in thewidest sense that will allowus to infer the truth of q ifwe know the truth of p. Thuswe interpret it as meaning:“Unless p is false, q is true,”or “either p is false or q istrue.” (The fact that “implies”is capable of other meaningsdoes not concern us; this isthe meaning which is conve-nient for us.) That is to say, “pimplies q” is to mean “not-por q”: its truth-value is to betruth if p is false, likewise if q (original page )

is true, and is to be falsehoodif p is true and q is false.

We have thus five func-tions: negation, disjunction,conjunction, incompatibility,and implication. We mighthave added others, for exam-ple, joint falsehood, “not-pand not-q,” but the above fivewill suffice. Negation differsfrom the other four in beinga function of one proposition,whereas the others are func-tions of two. But all five agreein this, that their truth-valuedepends only upon that of the (original page )

propositions which are theirarguments. Given the truthor falsehood of p, or of p andq (as the case may be), we aregiven the truth or falsehoodof the negation, disjunction,conjunction, incompatibility,or implication. A function ofpropositions which has thisproperty is called a “truth-function.”

The whole meaning of atruth-function is exhaustedby the statement of the cir-cumstances under which itis true or false. “Not-p,” for (original page )

example, is simply that func-tion of p which is true whenp is false, and false when pis true: there is no further |meaning to be assigned to it.The same applies to “p or q”and the rest. It follows thattwo truth-functions whichhave the same truth-value forall values of the argument areindistinguishable. For exam-ple, “p and q” is the negationof “not-p or not-q” and viceversa; thus either of these maybe defined as the negation ofthe other. There is no further (original pages –)

meaning in a truth-functionover and above the conditionsunder which it is true or false.

It is clear that the abovefive truth-functions are notall independent. We can de-fine some of them in terms ofothers. There is no great diffi-culty in reducing the numberto two; the two chosen in Prin-cipia Mathematica are negationand disjunction. Implicationis then defined as “not-p orq”; incompatibility as “not-por not-q”; conjunction as thenegation of incompatibility. (original page )

But it has been shown by Shef-fer that we can be contentwith one primitive idea for allfive, and by Nicod that thisenables us to reduce the prim-itive propositions requiredin the theory of deductionto two non-formal principlesand one formal one. For thispurpose, we may take as ourone indefinable either incom-patibility or joint falsehood.

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

January .

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We will choose the former.Our primitive idea, now, is

a certain truth-function called“incompatibility,” which wewill denote by p/q. Nega-tion can be at once definedas the incompatibility of aproposition with itself, i.e.“not-p” is defined as “p/p.”Disjunction is the incompat-ibility of not-p and not-q, i.e.it is (p/p) | (q/q). Implica-tion is the incompatibilityof p and not-q, i.e. p | (q/q).Conjunction is the negationof incompatibility, i.e. it is (original page )

(p/q) | (p/q). Thus all our fourother functions are defined interms of incompatibility.

It is obvious that there isno limit to the manufactureof truth-functions, either byintroducing more argumentsor by repeating arguments.What we are concerned withis the connection of this sub-ject with inference. |

If we know that p is trueand that p implies q, we canproceed to assert q. There isalways unavoidably somethingpsychological about infer- (original pages –)

ence: inference is a methodby which we arrive at newknowledge, and what is notpsychological about it is therelation which allows us toinfer correctly; but the actualpassage from the assertionof p to the assertion of q is apsychological process, and wemust not seek to represent itin purely logical terms.

In mathematical practice,when we infer, we have alwayssome expression containingvariable propositions, say pand q, which is known, in (original page )

virtue of its form, to be truefor all values of p and q; wehave also some other expres-sion, part of the former, whichis also known to be true forall values of p and q; and invirtue of the principles of in-ference, we are able to dropthis part of our original ex-pression, and assert what isleft. This somewhat abstractaccount may be made clearerby a few examples.

Let us assume that we knowthe five formal principlesof deduction enumerated in (original page )

Principia Mathematica. (M.Nicod has reduced these toone, but as it is a complicatedproposition, we will beginwith the five.) These fivepropositions are as follows:—

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

() q implies “p or q”—i.e.the disjunction “p or q” is truewhen one of its alternatives istrue.

() “p or q” implies “q orp.” This would not be re-quired if we had a theoreti- (original page )

cally more perfect notation,since in the conception ofdisjunction there is no orderinvolved, so that “p or q” and“q or p” should be identi-cal. But since our symbols,in any convenient form, in-evitably introduce an order,we need suitable assumptionsfor showing that the order isirrelevant.

() If either p is true or “q orr” is true, then either q is trueor “p or r” is true. (The twistin this proposition serves toincrease its deductive power.) | (original page )

() If q implies r, then “p orq” implies “p or r.”

These are the formal prin-ciples of deduction employedin Principia Mathematica. Aformal principle of deductionhas a double use, and it is inorder to make this clear thatwe have cited the above fivepropositions. It has a use asthe premiss of an inference,and a use as establishing thefact that the premiss impliesthe conclusion. In the schemaof an inference we have aproposition p, and a proposi- (original page )

tion “p implies q,” from whichwe infer q. Now when we areconcerned with the principlesof deduction, our apparatusof primitive propositions hasto yield both the p and the “pimplies q” of our inferences.That is to say, our rules ofdeduction are to be used, notonly as rules, which is theiruse for establishing “p impliesq,” but also as substantivepremisses, i.e. as the p of ourschema. Suppose, for exam-ple, we wish to prove that ifp implies q, then if q implies (original page )

r it follows that p implies r.We have here a relation ofthree propositions which stateimplications. Put

p = p implies q, p =

q implies r, p = p implies r.

Then we have to prove thatp implies that p impliesp. Now take the fifth of ourabove principles, substitutenot-p for p, and rememberthat “not-p or q” is by defi-nition the same as “p impliesq.” Thus our fifth principle

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yields:

“If q implies r, then ‘p im-plies q’ implies ‘p impliesr,’” i.e. “p implies thatp implies p.” Call thisproposition A.

But the fourth of our princi-ples, when we substitute not-p, not-q, for p and q, and re-member the definition of im-plication, becomes:

“If p implies that q implies r,then q implies that p im-plies r.”

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

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

Now we proved by means ofour fifth principle that

“p implies that p impliesp,” which was what wecalled A.

Thus we have here an instanceof the schema of inference,

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since A represents the p of ourscheme, and B represents the“p implies q.” Hence we arriveat q, namely,

“p implies that pimplies p,”

which was the proposition tobe proved. In this proof, theadaptation of our fifth princi-ple, which yields A, occurs asa substantive premiss; whilethe adaptation of our fourthprinciple, which yields B, isused to give the form of the

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inference. The formal andmaterial employments of pre-misses in the theory of deduc-tion are closely intertwined,and it is not very importantto keep them separated, pro-vided we realise that they arein theory distinct.

The earliest method of ar-riving at new results froma premiss is one which isillustrated in the above de-duction, but which itself canhardly be called deduction.The primitive propositions,whatever they may be, are (original page )

to be regarded as assertedfor all possible values of thevariable propositions p, q, rwhich occur in them. We maytherefore substitute for (say)p any expression whose valueis always a proposition, e.g.not-p, “s implies t,” and soon. By means of such substi-tutions we really obtain setsof special cases of our orig-inal proposition, but from apractical point of view we ob-tain what are virtually newpropositions. The legitimacyof substitutions of this kind (original page )

has to be insured by meansof a non-formal principle ofinference.

We may now state the oneformal principle of inferenceto which M. Nicod has re-duced the five given above.For this purpose we will firstshow how certain truth-functionscan be defined in terms of in-compatibility. We saw alreadythatNo such principle is enunciated

in Principia Mathematica or in M.Nicod’s article mentioned above. Butthis would seem to be an omission.

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p | (q/q) means “p implies q.” |We now observe that

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

For this expression means “pis incompatible with the in-compatibility of q and r,” i.e.“p implies that q and r are notincompatible,” i.e. “p impliesthat q and r are both true”—for, as we saw, the conjunctionof q and r is the negation oftheir incompatibility.

Observe next that t | (t /t)means “t implies itself.” This (original pages –)

is a particular case of p |(q/q).

Let us write p for the nega-tion of p; thus p/s will meanthe negation of p/s, i.e. it willmean the conjunction of p ands. It follows that

(s/q) | p/s

expresses the incompatibilityof s/q with the conjunctionof p and s; in other words, itstates that if p and s are bothtrue, s/q is false, i.e. s and qare both true; in still simpler

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words, it states that p and sjointly 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 formalprinciple of deduction is

P | π/Q,

in other words, P implies bothπ and Q.

He employs in addition onenon-formal principle belong-ing to the theory of types (original page )

(which need not concern us),and one corresponding to theprinciple that, given p, andgiven that p implies q, we canassert q. This principle is:

“If p | (r /q) is true, and p istrue, then q is true.” From thisapparatus the whole theoryof deduction follows, exceptin so far as we are concernedwith deduction from or tothe existence or the univer-sal truth of “propositionalfunctions,” which we shallconsider in the next chapter.

There is, if I am not mis- (original page )

taken, a certain confusionin the | minds of some au-thors as to the relation, be-tween propositions, in virtueof which an inference is valid.In order that it may be validto infer q from p, it is onlynecessary that p should betrue and that the proposition“not-p or q” should be true.Whenever this is the case, it isclear that q must be true. Butinference will only in fact takeplace when the proposition“not-p or q” is known other-wise than through knowledge (original pages –)

of not-p or knowledge of q.Whenever p is false, “not-por q” is true, but is uselessfor inference, which requiresthat p should be true. When-ever q is already known to betrue, “not-p or q” is of coursealso known to be true, but isagain useless for inference,since q is already known, andtherefore does not need to beinferred. In fact, inferenceonly arises when “not-p or q”can be known without ourknowing already which ofthe two alternatives it is that (original page )

makes the disjunction true.Now, the circumstances underwhich this occurs are thosein which certain relations ofform exist between p and q.For example, we know thatif r implies the negation of s,then s implies the negation ofr. Between “r implies not-s”and “s implies not-r” there isa formal relation which en-ables us to know that the firstimplies the second, withouthaving first to know that thefirst is false or to know thatthe second is true. It is un- (original page )

der such circumstances thatthe relation of implication ispractically useful for drawinginferences.

But this formal relation isonly required in order thatwe may be able to know thateither the premiss is false orthe conclusion is true. It is thetruth of “not-p or q” that isrequired for the validity of theinference; what is requiredfurther is only required forthe practical feasibility ofthe inference. Professor C. I.

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Lewis has especially studiedthe narrower, formal relationwhich we may call “formal de-ducibility.” He urges that thewider relation, that expressedby “not-p or q,” should not becalled “implication.” That is,however, a matter of words.| Provided our use of wordsis consistent, it matters lit-tle how we define them. Theessential point of differencebetween the theory which ISee Mind, vol. xxi., , pp. –

; and vol. xxiii., , pp. –.

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advocate and the theory ad-vocated by Professor Lewisis this: He maintains that,when one proposition q is“formally deducible” fromanother p, the relation whichwe perceive between them isone which he calls “strict im-plication,” which is not therelation expressed by “not-por q” but a narrower relation,holding only when there arecertain formal connectionsbetween p and q. I maintainthat, whether or not there besuch a relation as he speaks (original page )

of, it is in any case one thatmathematics does not need,and therefore one that, ongeneral grounds of economy,ought not to be admitted intoour apparatus of fundamen-tal notions; that, wheneverthe relation of “formal de-ducibility” holds between twopropositions, it is the case thatwe can see that either the firstis false or the second true, andthat nothing beyond this factis necessary to be admittedinto our premisses; and that,finally, the reasons of detail (original page )

which Professor Lewis ad-duces against the view whichI advocate can all be met indetail, and depend for theirplausibility upon a covert andunconscious assumption ofthe point of view which I re-ject. I conclude, therefore,that there is no need to admitas a fundamental notion anyform of implication not ex-pressible as a truth-function.

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CHAPTER XVPROPOSITIONAL

FUNCTIONS

When, in the preceding chap-ter, we were discussing propo-sitions, we did not attempt togive a definition of the word“proposition.” But althoughthe word cannot be formallydefined, it is necessary to saysomething as to its meaning,in order to avoid the very com-

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mon confusion with “propo-sitional functions,” which areto be the topic of the presentchapter.

We mean by a “proposition”primarily a form of wordswhich expresses what is eithertrue or false. I say “primar-ily,” because I do not wishto exclude other than ver-bal symbols, or even merethoughts if they have a sym-bolic character. But I think theword “proposition” should belimited to what may, in somesense, be called “symbols,” (original page )

and further to such symbolsas give expression to truthand falsehood. Thus “two andtwo are four” and “two andtwo are five” will be proposi-tions, and so will “Socrates isa man” and “Socrates is not aman.” The statement: “What-ever numbers a and b may be,(a + b) = a + ab + b” is aproposition; but the bare for-mula “(a+ b) = a + ab+ b”alone is not, since it assertsnothing definite unless we arefurther told, or led to sup-pose, that a and b are to have (original page )

all possible values, or are tohave such-and-such values.The former of these is tac-itly assumed, as a rule, in theenunciation of mathematicalformulæ, which thus becomepropositions; but if no suchassumption were made, theywould be “propositional func-tions.” A “propositional func-tion,” in fact, is an expressioncontaining one or more unde-termined constituents, | suchthat, when values are assignedto these constituents, the ex-pression becomes a propo- (original pages –)

sition. In other words, it isa function whose values arepropositions. But this latterdefinition must be used withcaution. A descriptive func-tion, e.g. “the hardest propo-sition in A’s mathematicaltreatise,” will not be a propo-sitional function, although itsvalues are propositions. Butin such a case the proposi-tions are only described: ina propositional function, thevalues must actually enunciatepropositions.

Examples of propositional (original page )

functions are easy to give:“x is human” is a proposi-tional function; so long asx remains undetermined, itis neither true nor false, butwhen a value is assigned tox it becomes a true or falseproposition. Any mathemati-cal equation is a propositionalfunction. So long as the vari-ables have no definite value,the equation is merely an ex-pression awaiting determina-tion in order to become a trueor false proposition. If it is anequation containing one vari- (original page )

able, it becomes true whenthe variable is made equal toa root of the equation, other-wise it becomes false; but if itis an “identity” it will be truewhen the variable is any num-ber. The equation to a curvein a plane or to a surface inspace is a propositional func-tion, true for values of the co-ordinates belonging to pointson the curve or surface, falsefor other values. Expressionsof traditional logic such as“all A is B” are propositionalfunctions: A and B have to be (original page )

determined as definite classesbefore such expressions be-come true or false.

The notion of “cases” or“instances” depends uponpropositional functions. Con-sider, for example, the kind ofprocess suggested by what iscalled “generalisation,” andlet us take some very primi-tive example, say, “lightningis followed by thunder.” Wehave a number of “instances”of this, i.e. a number of propo-sitions such as: “this is a flashof lightning and is followed (original page )

by thunder.” What are theseoccurrences “instances” of?They are instances of thepropositional function: “If xis a flash of lightning, x is fol-lowed by thunder.” The pro-cess of generalisation (withwhose validity we are | fortu-nately not concerned) consistsin passing from a number ofsuch instances to the univer-sal truth of the propositionalfunction: “If x is a flash oflightning, x is followed bythunder.” It will be foundthat, in an analogous way, (original pages –)

propositional functions arealways involved whenever wetalk of instances or cases orexamples.

We do not need to ask, or at-tempt to answer, the question:“What is a propositional func-tion?” A propositional func-tion standing all alone may betaken to be a mere schema, amere shell, an empty recep-tacle for meaning, not some-thing already significant. Weare concerned with propo-sitional functions, broadlyspeaking, in two ways: first, (original page )

as involved in the notions“true in all cases” and “true insome cases”; secondly, as in-volved in the theory of classesand relations. The second ofthese topics we will postponeto a later chapter; the firstmust occupy us now.

When we say that some-thing is “always true” or “truein all cases,” it is clear that the“something” involved cannotbe a proposition. A proposi-tion is just true or false, andthere is an end of the mat-ter. There are no instances or (original page )

cases of “Socrates is a man”or “Napoleon died at St He-lena.” These are propositions,and it would be meaninglessto speak of their being true“in all cases.” This phraseis only applicable to propo-sitional functions. Take, forexample, the sort of thing thatis often said when causation isbeing discussed. (We are notconcerned with the truth orfalsehood of what is said, butonly with its logical analysis.)We are told that A is, in everyinstance, followed by B. Now (original page )

if there are “instances” of A, Amust be some general conceptof which it is significant to say“x is A,” “x is A,” “x is A,”and so on, where x, x, xare particulars which are notidentical one with another.This applies, e.g., to our previ-ous case of lightning. We saythat lightning (A) is followedby thunder (B). But the sep-arate flashes are particulars,not identical, but sharing thecommon property of beinglightning. The only way of ex-pressing a | common property (original pages –)

generally is to say that a com-mon property of a numberof objects is a propositionalfunction which becomes truewhen any one of these ob-jects is taken as the value ofthe variable. In this case allthe objects are “instances” ofthe truth of the propositionalfunction—for a propositionalfunction, though it cannot it-self be true or false, is true incertain instances and false incertain others, unless it is “al-ways true” or “always false.”When, to return to our exam- (original page )

ple, we say that A is in everyinstance followed by B, wemean that, whatever x may be,if x is an A, it is followed bya B; that is, we are assertingthat a certain propositionalfunction is “always true.”

Sentences involving suchwords as “all,” “every,” “a,”“the,” “some” require propo-sitional functions for theirinterpretation. The way inwhich propositional functionsoccur can be explained bymeans of two of the abovewords, namely, “all” and (original page )

“some.”There are, in the last anal-

ysis, only two things that canbe done with a propositionalfunction: one is to assert thatit is true in all cases, the otherto assert that it is true in atleast one case, or in some cases(as we shall say, assuming thatthere is to be no necessaryimplication of a plurality ofcases). All the other uses ofpropositional functions can bereduced to these two. Whenwe say that a propositionalfunction is true “in all cases,” (original page )

or “always” (as we shall alsosay, without any temporalsuggestion), we mean that allits values are true. If “φx”is the function, and a is theright sort of object to be anargument to “φx,” then φais to be true, however a mayhave been chosen. For ex-ample, “if a is human, a ismortal” is true whether a ishuman or not; in fact, everyproposition of this form istrue. Thus the propositionalfunction “if x is human, x ismortal” is “always true,” or (original page )

“true in all cases.” Or, again,the statement “there are nounicorns” is the same as thestatement “the propositionalfunction ‘x is not a unicorn’ istrue in all cases.” The asser-tions in the preceding chapterabout propositions, e.g. “‘p orq’ implies ‘q or p,’” are reallyassertions | that certain propo-sitional functions are true inall cases. We do not assert theabove principle, for example,as being true only of this orthat particular p or q, but asbeing true of any p or q con- (original pages –)

cerning which it can be madesignificantly. The conditionthat a function is to be signif-icant for a given argument isthe same as the condition thatit shall have a value for thatargument, either true or false.The study of the conditionsof significance belongs to thedoctrine of types, which weshall not pursue beyond thesketch given in the precedingchapter.

Not only the principles ofdeduction, but all the prim-itive propositions of logic, (original page )

consist of assertions that cer-tain propositional functionsare always true. If this werenot the case, they would haveto mention particular thingsor concepts—Socrates, or red-ness, or east and west, or whatnot—and clearly it is not theprovince of logic to makeassertions which are true con-cerning one such thing orconcept but not concerninganother. It is part of the def-inition of logic (but not thewhole of its definition) thatall its propositions are com- (original page )

pletely general, i.e. they allconsist of the assertion thatsome propositional functioncontaining no constant termsis always true. We shall re-turn in our final chapter tothe discussion of proposi-tional functions containingno constant terms. For thepresent we will proceed to theother thing that is to be donewith a propositional function,namely, the assertion that it is“sometimes true,” i.e. true inat least one instance.

When we say “there are (original page )

men,” that means that thepropositional function “x isa man” is sometimes true.When we say “some menare Greeks,” that means thatthe propositional function“x is a man and a Greek” issometimes true. When wesay “cannibals still exist inAfrica,” that means that thepropositional function “x isa cannibal now in Africa” issometimes true, i.e. is true forsome values of x. To say “thereare at least n individuals inthe world” is to say that the (original page )

propositional function “α isa class of individuals and amember of the cardinal num-ber n” is sometimes true, or,as we may say, is true for cer-tain | values of α. This formof expression is more conve-nient when it is necessary toindicate which is the variableconstituent which we are tak-ing as the argument to ourpropositional function. Forexample, the above propo-sitional function, which wemay shorten to “α is a class ofn individuals,” contains two (original pages –)

variables, α and n. The axiomof infinity, in the language ofpropositional functions, is:“The propositional function‘if n is an inductive number,it is true for some values ofα that α is a class of n indi-viduals’ is true for all possiblevalues of n.” Here there is asubordinate function, “α is aclass of n individuals,” whichis said to be, in respect of α,sometimes true; and the asser-tion that this happens if n isan inductive number is saidto be, in respect of n, always (original page )

true.The statement that a func-

tion φx is always true is thenegation of the statement thatnot-φx is sometimes true, andthe statement that φx is some-times true is the negation ofthe statement that not-φx isalways true. Thus the state-ment “all men are mortals”is the negation of the state-ment that the function “x isan immortal man” is some-times true. And the state-ment “there are unicorns” isthe negation of the statement (original page )

that the function “x is not aunicorn” is always true. Wesay that φx is “never true”or “always false” if not-φxis always true. We can, ifwe choose, take one of thepair “always,” “sometimes”as a primitive idea, and de-fine the other by means ofthe one and negation. Thus

For linguistic reasons, to avoidsuggesting either the plural or the sin-gular, it is often convenient to say“φx is not always false” rather than“φx sometimes” or “φx is sometimestrue.”

(original page )

if we choose “sometimes” asour primitive idea, we candefine: “‘φx is always true’ isto mean ‘it is false that not-φxis sometimes true.’” But forreasons connected with thetheory of types it seems morecorrect to take both “always”and “sometimes” as primi-tive ideas, and define by theirmeans the negation of propo-sitions in which they occur.That is to say, assuming thatwe have already | defined (oradopted as a primitive idea)the negation of propositions (original pages –)

of the type to which φx be-longs, we define: “The nega-tion of ‘φx always’ is ‘not-φxsometimes’; and the negationof ‘φx sometimes’ is ‘not-φxalways.’” In like manner wecan re-define disjunction andthe other truth-functions, asapplied to propositions con-taining apparent variables, interms of the definitions andprimitive ideas for proposi-tions containing no apparentvariables. Propositions con-taining no apparent variablesare called “elementary propo- (original page )

sitions.” From these we canmount up step by step, us-ing such methods as have justbeen indicated, to the theoryof truth-functions as appliedto propositions containingone, two, three . . . variables,or any number up to n, wheren is any assigned finite num-ber.

The forms which are takenas simplest in traditional for-mal logic are really far from

The method of deduction is givenin Principia Mathematica, vol. i. ∗.

(original page )

being so, and all involve theassertion of all values or somevalues of a compound propo-sitional function. Take, tobegin with, “all S is P.” Wewill take it that S is definedby a propositional functionφx, and P by a propositionalfunction ψx. E.g., if S is men,φx will be “x is human”; if Pis mortals, ψx will be “there isa time at which x dies.” Then“all S is P” means: “‘φx im-plies ψx’ is always true.” Itis to be observed that “all Sis P” does not apply only to (original page )

those terms that actually areS’s; it says something equallyabout terms which are not S’s.Suppose we come across anx of which we do not knowwhether it is an S or not; still,our statement “all S is P” tellsus something about x, namely,that if x is an S, then x is a P.And this is every bit as truewhen x is not an S as when x isan S. If it were not equally truein both cases, the reductio adabsurdum would not be a validmethod; for the essence of thismethod consists in using im- (original page )

plications in cases where (asit afterwards turns out) thehypothesis is false. We mayput the matter another way.In order to understand “all Sis P,” it is not necessary to beable to enumerate what termsare S’s; provided we knowwhat is meant by being an Sand what by being a P, we canunderstand completely whatis actually affirmed | by “all Sis P,” however little we mayknow of actual instances of ei-ther. This shows that it is notmerely the actual terms that (original pages –)

are S’s that are relevant in thestatement “all S is P,” but allthe terms concerning whichthe supposition that they areS’s is significant, i.e. all theterms that are S’s, togetherwith all the terms that are notS’s—i.e. the whole of the ap-propriate logical “type.” Whatapplies to statements aboutall applies also to statementsabout some. “There are men,”e.g., means that “x is human”is true for some values of x.Here all values of x (i.e. allvalues for which “x is human” (original page )

is significant, whether true orfalse) are relevant, and notonly those that in fact are hu-man. (This becomes obviousif we consider how we couldprove such a statement to befalse.) Every assertion about“all” or “some” thus involvesnot only the arguments thatmake a certain function true,but all that make it signifi-cant, i.e. all for which it has avalue at all, whether true orfalse.

We may now proceed withour interpretation of the tra- (original page )

ditional forms of the old-fashioned formal logic. Weassume that S is those terms xfor which φx is true, and P isthose for which ψx is true. (Aswe shall see in a later chap-ter, all classes are derived inthis way from propositionalfunctions.) Then:

“All S is P” means “‘φx im-plies ψx’ is always true.”

“Some S is P” means “‘φxandψx’ is sometimes true.”

“No S is P” means “‘φx im-plies not-ψx’ is always

(original page )

true.”“Some S is not P” means “‘φx

and not-ψx’ is sometimestrue.”

It will be observed that thepropositional functions whichare here asserted for all orsome values are not φx andψx themselves, but truth-functions of φx and ψx forthe same argument x. Theeasiest way to conceive of thesort of thing that is intendedis to start not from φx and ψxin general, but from φa and

(original page )

ψa, where a is some constant.Suppose we are considering“all men are mortal”: we willbegin with

“If Socrates is human,Socrates is mortal,” |

and then we will regard “Soc-rates” as replaced by a vari-able x wherever “Socrates” oc-curs. The object to be securedis that, although x remains avariable, without any definitevalue, yet it is to have the samevalue in “φx” as in “ψx” whenwe are asserting that “φx im- (original pages –)

plies ψx” is always true. Thisrequires that we shall startwith a function whose valuesare such as “φa implies ψa,”rather than with two separatefunctions φx and ψx; for ifwe start with two separatefunctions we can never securethat the x, while remainingundetermined, shall have thesame value in both.

For brevity we say “φx al-ways implies ψx” when wemean that “φx implies ψx” isalways true. Propositions ofthe form “φx always implies (original page )

ψx” are called “formal impli-cations”; this name is givenequally if there are severalvariables.

The above definitions showhow far removed from thesimplest forms are such propo-sitions as “all S is P,” withwhich traditional logic be-gins. It is typical of the lack ofanalysis involved that tradi-tional logic treats “all S is P”as a proposition of the sameform as “x is P”—e.g., it treats“all men are mortal” as ofthe same form as “Socrates is (original page )

mortal.” As we have just seen,the first is of the form “φxalways implies ψx,” while thesecond is of the form “ψx.”The emphatic separation ofthese two forms, which waseffected by Peano and Frege,was a very vital advance insymbolic logic.

It will be seen that “all Sis P” and “no S is P” do notreally differ in form, except bythe substitution of not-ψx forψx, and that the same appliesto “some S is P” and “someS is not P.” It should also be (original page )

observed that the traditionalrules of conversion are faulty,if we adopt the view, which isthe only technically tolerableone, that such propositionsas “all S is P” do not involvethe “existence” of S’s, i.e. donot require that there shouldbe terms which are S’s. Theabove definitions lead to theresult that, if φx is alwaysfalse, i.e. if there are no S’s,then “all S is P” and “no S is P”will both be true, | whateverP may be. For, according tothe definition in the last chap- (original pages –)

ter, “φx implies ψx” means“not-φx or ψx,” which is al-ways true if not-φx is alwaystrue. At the first moment, thisresult might lead the readerto desire different definitions,but a little practical expe-rience soon shows that anydifferent definitions would beinconvenient and would con-ceal the important ideas. Theproposition “φx always im-plies ψx, and φx is sometimestrue” is essentially composite,and it would be very awkwardto give this as the definition of (original page )

“all S is P,” for then we shouldhave no language left for “φxalways implies ψx,” which isneeded a hundred times foronce that the other is needed.But, with our definitions, “allS is P” does not imply “someS is P,” since the first allowsthe non-existence of S and thesecond does not; thus con-version per accidens becomesinvalid, and some moods ofthe syllogism are fallacious,e.g. Darapti: “All M is S, allM is P, therefore some S is P,”which fails if there is no M. (original page )

The notion of “existence”has several forms, one ofwhich will occupy us in thenext chapter; but the funda-mental form is that which isderived immediately from thenotion of “sometimes true.”We say that an argument a“satisfies” a function φx ifφa is true; this is the samesense in which the roots ofan equation are said to sat-isfy the equation. Now if φxis sometimes true, we maysay there are x’s for which itis true, or we may say “ar- (original page )

guments satisfying φx ex-ist.” This is the fundamentalmeaning of the word “exis-tence.” Other meanings areeither derived from this, orembody mere confusion ofthought. We may correctlysay “men exist,” meaningthat “x is a man” is some-times true. But if we make apseudo-syllogism: “Men exist,Socrates is a man, thereforeSocrates exists,” we are talk-ing nonsense, since “Socrates”is not, like “men,” merely anundetermined argument to a (original page )

given propositional function.The fallacy is closely analo-gous to that of the argument:“Men are numerous, Socratesis a man, therefore Socratesis numerous.” In this case itis obvious that the conclu-sion is nonsensical, but | inthe case of existence it is notobvious, for reasons whichwill appear more fully in thenext chapter. For the presentlet us merely note the factthat, though it is correct to say“men exist,” it is incorrect, orrather meaningless, to ascribe (original pages –)

existence to a given particularx who happens to be a man.Generally, “terms satisfyingφx exist” means “φx is some-times true”; but “a exists”(where a is a term satisfyingφx) is a mere noise or shape,devoid of significance. It willbe found that by bearing inmind this simple fallacy wecan solve many ancient philo-sophical puzzles concerningthe meaning of existence.

Another set of notions as towhich philosophy has alloweditself to fall into hopeless (original page )

confusions through not suf-ficiently separating proposi-tions and propositional func-tions are the notions of “modal-ity”: necessary, possible, andimpossible. (Sometimes con-tingent or assertoric is usedinstead of possible.) The tradi-tional view was that, amongtrue propositions, some werenecessary, while others weremerely contingent or asser-toric; while among false propo-sitions some were impossible,namely, those whose con-tradictories were necessary, (original page )

while others merely happenednot to be true. In fact, how-ever, there was never any clearaccount of what was added totruth by the conception of ne-cessity. In the case of proposi-tional functions, the threefolddivision is obvious. If “φx”is an undetermined value ofa certain propositional func-tion, it will be necessary if thefunction is always true, pos-sible if it is sometimes true,and impossible if it is nevertrue. This sort of situationarises in regard to probability, (original page )

for example. Suppose a ballx is drawn from a bag whichcontains a number of balls: ifall the balls are white, “x iswhite” is necessary; if someare white, it is possible; ifnone, it is impossible. Here allthat is known about x is thatit satisfies a certain propo-sitional function, namely, “xwas a ball in the bag.” Thisis a situation which is generalin probability problems andnot uncommon in practicallife—e.g. when a person callsof whom we know nothing (original page )

except that he brings a let-ter of introduction from ourfriend so-and-so. In all such |cases, as in regard to modalityin general, the propositionalfunction is relevant. For clearthinking, in many very di-verse directions, the habit ofkeeping propositional func-tions sharply separated frompropositions is of the utmostimportance, and the failure todo so in the past has been adisgrace to philosophy.

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CHAPTER XVIDESCRIPTIONS

We dealt in the precedingchapter with the words alland some; in this chapter weshall consider the word the inthe singular, and in the nextchapter we shall consider theword the in the plural. It maybe thought excessive to devotetwo chapters to one word, butto the philosophical mathe-matician it is a word of very (original page )

great importance: like Brown-ing’s Grammarian with theenclitic δε, I would give thedoctrine of this word if I were“dead from the waist down”and not merely in a prison.

We have already had occa-sion to mention “descriptivefunctions,” i.e. such expres-sions as “the father of x” or“the sine of x.” These are tobe defined by first defining“descriptions.”

A “description” may be oftwo sorts, definite and indefi-nite (or ambiguous). An indef- (original page )

inite description is a phraseof the form “a so-and-so,”and a definite description isa phrase of the form “the so-and-so” (in the singular). Letus begin with the former.

“Who did you meet?” “Imet a man.” “That is a veryindefinite description.” Weare therefore not departingfrom usage in our terminol-ogy. Our question is: Whatdo I really assert when I as-sert “I met a man”? Let usassume, for the moment, thatmy assertion is true, and that (original page )

in fact I met Jones. It is clearthat what I assert is not “I metJones.” I may say “I met aman, but it was not Jones”; inthat case, though I lie, I do notcontradict myself, as I shoulddo if when I say I met a |manI really mean that I met Jones.It is clear also that the personto whom I am speaking canunderstand what I say, even ifhe is a foreigner and has neverheard of Jones.

But we may go further: notonly Jones, but no actual man,enters into my statement. This (original pages –)

becomes obvious when thestatement is false, since thenthere is no more reason whyJones should be supposed toenter into the propositionthan why anyone else should.Indeed the statement wouldremain significant, though itcould not possibly be true,even if there were no man atall. “I met a unicorn” or “I meta sea-serpent” is a perfectlysignificant assertion, if weknow what it would be to bea unicorn or a sea-serpent, i.e.what is the definition of these (original page )

fabulous monsters. Thus itis only what we may call theconcept that enters into theproposition. In the case of“unicorn,” for example, thereis only the concept: there isnot also, somewhere amongthe shades, something unrealwhich may be called “a uni-corn.” Therefore, since it issignificant (though false) tosay “I met a unicorn,” it isclear that this proposition,rightly analysed, does notcontain a constituent “a uni-corn,” though it does contain (original page )

the concept “unicorn.”The question of “unreal-

ity,” which confronts us atthis point, is a very importantone. Misled by grammar, thegreat majority of those logi-cians who have dealt with thisquestion have dealt with iton mistaken lines. They haveregarded grammatical form asa surer guide in analysis than,in fact, it is. And they havenot known what differencesin grammatical form are im-portant. “I met Jones” and “Imet a man” would count tra- (original page )

ditionally as propositions ofthe same form, but in actualfact they are of quite differ-ent forms: the first names anactual person, Jones; whilethe second involves a proposi-tional function, and becomes,when made explicit: “Thefunction ‘I met x and x is hu-man’ is sometimes true.” (Itwill be remembered that weadopted the convention ofusing “sometimes” as not im-plying more than once.) Thisproposition is obviously notof the form “I met x,” which (original page )

accounts | for the existenceof the proposition “I met aunicorn” in spite of the factthat there is no such thing as“a unicorn.”

For want of the apparatusof propositional functions,many logicians have beendriven to the conclusion thatthere are unreal objects. Itis argued, e.g. by Meinong,

that we can speak about “thegolden mountain,” “the round

Untersuchungen zur Gegenstands-theorie und Psychologie, .

(original pages –)

square,” and so on; we canmake true propositions ofwhich these are the subjects;hence they must have somekind of logical being, sinceotherwise the propositions inwhich they occur would bemeaningless. In such theo-ries, it seems to me, there isa failure of that feeling forreality which ought to be pre-served even in the most ab-stract studies. Logic, I shouldmaintain, must no more admita unicorn than zoology can;for logic is concerned with (original page )

the real world just as truly aszoology, though with its moreabstract and general features.To say that unicorns have anexistence in heraldry, or in lit-erature, or in imagination, isa most pitiful and paltry eva-sion. What exists in heraldryis not an animal, made offlesh and blood, moving andbreathing of its own initiative.What exists is a picture, or adescription in words. Simi-larly, to maintain that Hamlet,for example, exists in his ownworld, namely, in the world (original page )

of Shakespeare’s imagination,just as truly as (say) Napoleonexisted in the ordinary world,is to say something deliber-ately confusing, or else con-fused to a degree which isscarcely credible. There isonly one world, the “real”world: Shakespeare’s imag-ination is part of it, and thethoughts that he had in writ-ing Hamlet are real. So arethe thoughts that we have inreading the play. But it is ofthe very essence of fiction thatonly the thoughts, feelings, (original page )

etc., in Shakespeare and hisreaders are real, and that thereis not, in addition to them, anobjective Hamlet. When youhave taken account of all thefeelings roused by Napoleonin writers and readers of his-tory, you have not touched theactual man; but in the case ofHamlet you have come to theend of him. If no one thoughtabout Hamlet, there would benothing | left of him; if no onehad thought about Napoleon,he would have soon seen to itthat some one did. The sense (original pages –)

of reality is vital in logic, andwhoever juggles with it bypretending that Hamlet hasanother kind of reality is do-ing a disservice to thought. Arobust sense of reality is verynecessary in framing a correctanalysis of propositions aboutunicorns, golden mountains,round squares, and other suchpseudo-objects.

In obedience to the feelingof reality, we shall insist that,in the analysis of proposi-tions, nothing “unreal” is tobe admitted. But, after all, if (original page )

there is nothing unreal, how, itmay be asked, could we admitanything unreal? The reply isthat, in dealing with propo-sitions, we are dealing in thefirst instance with symbols,and if we attribute signifi-cance to groups of symbolswhich have no significance,we shall fall into the error ofadmitting unrealities, in theonly sense in which this ispossible, namely, as objectsdescribed. In the proposition“I met a unicorn,” the wholefour words together make a (original page )

significant proposition, andthe word “unicorn” by itselfis significant, in just the samesense as the word “man.” Butthe two words “a unicorn”do not form a subordinategroup having a meaning ofits own. Thus if we falselyattribute meaning to thesetwo words, we find ourselvessaddled with “a unicorn,” andwith the problem how therecan be such a thing in a worldwhere there are no unicorns.“A unicorn” is an indefinitedescription which describes (original page )

nothing. It is not an indefinitedescription which describessomething unreal. Such aproposition as “x is unreal”only has meaning when “x”is a description, definite orindefinite; in that case theproposition will be true if“x” is a description which de-scribes nothing. But whetherthe description “x” describessomething or describes noth-ing, it is in any case not aconstituent of the proposi-tion in which it occurs; like“a unicorn” just now, it is not (original page )

a subordinate group havinga meaning of its own. Allthis results from the fact that,when “x” is a description, “xis unreal” or “x does not ex-ist” is not nonsense, but isalways significant and some-times true. |

We may now proceed todefine generally the meaningof propositions which con-tain ambiguous descriptions.Suppose we wish to makesome statement about “a so-and-so,” where “so-and-so’s”are those objects that have a (original pages –)

certain property φ, i.e. thoseobjects x for which the propo-sitional function φx is true.(E.g. if we take “a man” as ourinstance of “a so-and-so,” φxwill be “x is human.”) Let usnow wish to assert the prop-erty ψ of “a so-and-so,” i.e. wewish to assert that “a so-and-so” has that property which xhas when ψx is true. (E.g. inthe case of “I met a man,” ψxwill be “I met x.”) Now theproposition that “a so-and-so”has the property ψ is not aproposition of the form “ψx.” (original page )

If it were, “a so-and-so” wouldhave to be identical with x fora suitable x; and although (ina sense) this may be true insome cases, it is certainly nottrue in such a case as “a uni-corn.” It is just this fact, thatthe statement that a so-and-sohas the property ψ is not ofthe form ψx, which makesit possible for “a so-and-so”to be, in a certain clearly de-finable sense, “unreal.” Thedefinition is as follows:—

The statement that “an object

(original page )

having the property φ hasthe property ψ”

means:

“The joint assertion of φxandψx is not always false.”

So far as logic goes, this isthe same proposition as mightbe expressed by “some φ’s areψ’s”; but rhetorically there isa difference, because in theone case there is a sugges-tion of singularity, and in theother case of plurality. This,however, is not the important (original page )

point. The important point isthat, when rightly analysed,propositions verbally about“a so-and-so” are found tocontain no constituent repre-sented by this phrase. Andthat is why such propositionscan be significant even whenthere is no such thing as aso-and-so.

The definition of existence,as applied to ambiguous de-scriptions, results from whatwas said at the end of the pre-ceding chapter. We say that“men exist” or “a man exists” (original page )

if the | propositional function“x is human” is sometimestrue; and generally “a so-and-so” exists if “x is so-and-so”is sometimes true. We mayput this in other language.The proposition “Socrates is aman” is no doubt equivalent to“Socrates is human,” but it isnot the very same proposition.The is of “Socrates is human”expresses the relation of sub-ject and predicate; the is of“Socrates is a man” expressesidentity. It is a disgrace to thehuman race that it has chosen (original pages –)

to employ the same word “is”for these two entirely differ-ent ideas—a disgrace which asymbolic logical language ofcourse remedies. The iden-tity in “Socrates is a man” isidentity between an objectnamed (accepting “Socrates”as a name, subject to qualifica-tions explained later) and anobject ambiguously described.An object ambiguously de-scribed will “exist” when atleast one such proposition istrue, i.e. when there is at leastone true proposition of the (original page )

form “x is a so-and-so,” where“x” is a name. It is characteris-tic of ambiguous (as opposedto definite) descriptions thatthere may be any number oftrue propositions of the aboveform—Socrates is a man, Platois a man, etc. Thus “a man ex-ists” follows from Socrates, orPlato, or anyone else. Withdefinite descriptions, on theother hand, the correspondingform of proposition, namely,“x is the so-and-so” (where“x” is a name), can only betrue for one value of x at most. (original page )

This brings us to the subject ofdefinite descriptions, whichare to be defined in a wayanalogous to that employedfor ambiguous descriptions,but rather more complicated.

We come now to the mainsubject of the present chap-ter, namely, the definition ofthe word the (in the singular).One very important pointabout the definition of “a so-and-so” applies equally to“the so-and-so”; the definitionto be sought is a definitionof propositions in which this (original page )

phrase occurs, not a defini-tion of the phrase itself inisolation. In the case of “aso-and-so,” this is fairly ob-vious: no one could supposethat “a man” was a definiteobject, which could be de-fined by itself. | Socrates is aman, Plato is a man, Aristotleis a man, but we cannot inferthat “a man” means the sameas “Socrates” means and alsothe same as “Plato” meansand also the same as “Aristo-tle” means, since these threenames have different mean- (original pages –)

ings. Nevertheless, when wehave enumerated all the menin the world, there is noth-ing left of which we can say,“This is a man, and not onlyso, but it is the ‘a man,’ thequintessential entity that isjust an indefinite man withoutbeing anybody in particular.”It is of course quite clear thatwhatever there is in the worldis definite: if it is a man itis one definite man and notany other. Thus there cannotbe such an entity as “a man”to be found in the world, as (original page )

opposed to specific men. Andaccordingly it is natural thatwe do not define “a man” it-self, but only the propositionsin which it occurs.

In the case of “the so-and-so” this is equally true, thoughat first sight less obvious. Wemay demonstrate that thismust be the case, by a con-sideration of the differencebetween a name and a definitedescription. Take the propo-sition, “Scott is the author ofWaverley.” We have here aname, “Scott,” and a descrip- (original page )

tion, “the author of Waverley,”which are asserted to applyto the same person. The dis-tinction between a name andall other symbols may be ex-plained as follows:—

A name is a simple symbolwhose meaning is somethingthat can only occur as subject,i.e. something of the kind that,in Chapter XIII., we defined asan “individual” or a “particu-lar.” And a “simple” symbolis one which has no parts thatare symbols. Thus “Scott”is a simple symbol, because, (original page )

though it has parts (namely,separate letters), these partsare not symbols. On the otherhand, “the author of Waver-ley” is not a simple symbol,because the separate wordsthat compose the phrase areparts which are symbols. If,as may be the case, whateverseems to be an “individual”is really capable of furtheranalysis, we shall have to con-tent ourselves with what maybe called “relative individ-uals,” which will be termsthat, throughout the context (original page )

in question, are never anal-ysed and never occur | other-wise than as subjects. And inthat case we shall have cor-respondingly to content our-selves with “relative names.”From the standpoint of ourpresent problem, namely,the definition of descriptions,this problem, whether theseare absolute names or onlyrelative names, may be ig-nored, since it concerns dif-ferent stages in the hierarchyof “types,” whereas we haveto compare such couples as (original pages –)

“Scott” and “the author ofWaverley,” which both applyto the same object, and do notraise the problem of types.We may, therefore, for themoment, treat names as capa-ble of being absolute; nothingthat we shall have to say willdepend upon this assump-tion, but the wording may bea little shortened by it.

We have, then, two things tocompare: () a name, which isa simple symbol, directly des-ignating an individual whichis its meaning, and having (original page )

this meaning in its own right,independently of the mean-ings of all other words; () adescription, which consists ofseveral words, whose mean-ings are already fixed, andfrom which results whateveris to be taken as the “mean-ing” of the description.

A proposition containinga description is not identicalwith what that propositionbecomes when a name is sub-stituted, even if the namenames the same object as thedescription describes. “Scott (original page )

is the author of Waverley” isobviously a different propo-sition from “Scott is Scott”:the first is a fact in literaryhistory, the second a trivialtruism. And if we put anyoneother than Scott in place of“the author of Waverley,” ourproposition would becomefalse, and would therefore cer-tainly no longer be the sameproposition. But, it may besaid, our proposition is essen-tially of the same form as (say)“Scott is Sir Walter,” in whichtwo names are said to apply (original page )

to the same person. The replyis that, if “Scott is Sir Wal-ter” really means “the personnamed ‘Scott’ is the personnamed ‘Sir Walter,’” then thenames are being used as de-scriptions: i.e. the individual,instead of being named, isbeing described as the per-son having that name. Thisis a way in which names arefrequently used | in practice,and there will, as a rule, benothing in the phraseology toshow whether they are beingused in this way or as names. (original pages –)

When a name is used directly,merely to indicate what weare speaking about, it is nopart of the fact asserted, orof the falsehood if our asser-tion happens to be false: itis merely part of the symbol-ism by which we express ourthought. What we want toexpress is something whichmight (for example) be trans-lated into a foreign language;it is something for which theactual words are a vehicle,but of which they are no part.On the other hand, when we (original page )

make a proposition about “theperson called ‘Scott,’” the ac-tual name “Scott” enters intowhat we are asserting, andnot merely into the languageused in making the assertion.Our proposition will now bea different one if we substi-tute “the person called ‘SirWalter.’” But so long as weare using names as names,whether we say “Scott” orwhether we say “Sir Walter”is as irrelevant to what weare asserting as whether wespeak English or French. Thus (original page )

so long as names are used asnames, “Scott is Sir Walter” isthe same trivial propositionas “Scott is Scott.” This com-pletes the proof that “Scottis the author of Waverley” isnot the same proposition asresults from substituting aname for “the author of Wa-verley,” no matter what namemay be substituted.

When we use a variable,and speak of a propositionalfunction, φx say, the pro-cess of applying general state-ments about φx to particular (original page )

cases will consist in substi-tuting a name for the letter“x,” assuming that φ is a func-tion which has individuals forits arguments. Suppose, forexample, that φx is “alwaystrue”; let it be, say, the “lawof identity,” x = x. Then wemay substitute for “x” anyname we choose, and we shallobtain a true proposition. As-suming for the moment that“Socrates,” “Plato,” and “Aris-totle” are names (a very rashassumption), we can inferfrom the law of identity that (original page )

Socrates is Socrates, Plato isPlato, and Aristotle is Aris-totle. But we shall commit afallacy if we attempt to infer,without further premisses,that the author of Waverley isthe author of Waverley. Thisresults | from what we havejust proved, that, if we sub-stitute a name for “the authorof Waverley” in a proposition,the proposition we obtain isa different one. That is tosay, applying the result toour present case: If “x” is aname, “x = x” is not the same (original pages –)

proposition as “the author ofWaverley is the author of Wa-verley,” no matter what name“x” may be. Thus from thefact that all propositions ofthe form “x = x” are true wecannot infer, without moreado, that the author of Waver-ley is the author of Waverley.In fact, propositions of theform “the so-and-so is the so-and-so” are not always true: itis necessary that the so-and-soshould exist (a term whichwill be explained shortly). Itis false that the present King (original page )

of France is the present Kingof France, or that the roundsquare is the round square.When we substitute a descrip-tion for a name, propositionalfunctions which are “alwaystrue” may become false, if thedescription describes nothing.There is no mystery in thisas soon as we realise (whatwas proved in the precedingparagraph) that when we sub-stitute a description the resultis not a value of the proposi-tional function in question.

We are now in a position to (original page )

define propositions in whicha definite description occurs.The only thing that distin-guishes “the so-and-so” from“a so-and-so” is the impli-cation of uniqueness. Wecannot speak of “the inhab-itant of London,” becauseinhabiting London is an at-tribute which is not unique.We cannot speak about “thepresent King of France,” be-cause there is none; but we canspeak about “the present Kingof England.” Thus proposi-tions about “the so-and-so” (original page )

always imply the correspond-ing propositions about “a so-and-so,” with the addendumthat there is not more thanone so-and-so. Such a propo-sition as “Scott is the author ofWaverley” could not be true ifWaverley had never been writ-ten, or if several people hadwritten it; and no more couldany other proposition result-ing from a propositional func-tion φx by the substitution of“the author of Waverley” for“x.” We may say that “the au-thor of Waverley” means “the (original page )

value of x for which ‘x wrote| Waverley’ is true.” Thus theproposition “the author ofWaverley was Scotch,” for ex-ample, involves:

() “x wrote Waverley” is notalways false;

() “if x and y wrote Waver-ley, x and y are identical”is always true;

() “if x wrote Waverley, xwas Scotch” is always true.

These three propositions,translated into ordinary lan-guage, state: (original pages –)

() at least one person wroteWaverley;

() at most one person wroteWaverley;

() whoever wrote Waverleywas Scotch.

All these three are implied by“the author of Waverley wasScotch.” Conversely, the threetogether (but no two of them)imply that the author of Wa-verley was Scotch. Hence thethree together may be takenas defining what is meant bythe proposition “the author of

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Waverley was Scotch.”We may somewhat simplify

these three propositions. Thefirst and second together areequivalent to: “There is a termc such that ‘x wrote Waverley’is true when x is c and is falsewhen x is not c.” In otherwords, “There is a term c suchthat ‘x wrote Waverley’ is al-ways equivalent to ‘x is c.’”(Two propositions are “equiv-alent” when both are true orboth are false.) We have here,to begin with, two functionsof x, “x wrote Waverley” and (original page )

“x is c,” and we form a func-tion of c by considering theequivalence of these two func-tions of x for all values of x;we then proceed to assert thatthe resulting function of c is“sometimes true,” i.e. that itis true for at least one valueof c. (It obviously cannot betrue for more than one valueof c.) These two conditionstogether are defined as givingthe meaning of “the author ofWaverley exists.”

We may now define “theterm satisfying the function (original page )

φx exists.” This is the generalform of which the above is aparticular case. “The authorof Waverley” is “the term sat-isfying the function ‘x wroteWaverley.’” And “the so-and-so” will | always involve ref-erence to some propositionalfunction, namely, that whichdefines the property thatmakes a thing a so-and-so.Our definition is as follows:—

“The term satisfying thefunction φx exists” means:

“There is a term c such thatφx is always equivalent to ‘x (original pages –)

is c.’”In order to define “the au-

thor of Waverley was Scotch,”we have still to take account ofthe third of our three propo-sitions, namely, “Whoeverwrote Waverley was Scotch.”This will be satisfied by merelyadding that the c in questionis to be Scotch. Thus “the au-thor of Waverley was Scotch”is:

“There is a term c such that() ‘x wrote Waverley’ is al-ways equivalent to ‘x is c,’

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() c is Scotch.”

And generally: “the term sat-isfying φx satisfies ψx” is de-fined as meaning:

“There is a term c such that() φx is always equivalentto ‘x is c,’ () ψc is true.”

This is the definition of propo-sitions in which descriptionsoccur.

It is possible to have muchknowledge concerning a termdescribed, i.e. to know manypropositions concerning “the (original page )

so-and-so,” without actuallyknowing what the so-and-sois, i.e. without knowing anyproposition of the form “x isthe so-and-so,” where “x” isa name. In a detective storypropositions about “the manwho did the deed” are ac-cumulated, in the hope thatultimately they will sufficeto demonstrate that it was Awho did the deed. We mayeven go so far as to say that,in all such knowledge as canbe expressed in words—withthe exception of “this” and (original page )

“that” and a few other wordsof which the meaning varieson different occasions—nonames, in the strict sense,occur, but what seem likenames are really descrip-tions. We may inquire sig-nificantly whether Homer ex-isted, which we could not doif “Homer” were a name. Theproposition “the so-and-soexists” is significant, whethertrue or false; but if a is theso-and-so (where “a” is aname), the words “a exists”are meaningless. It is only (original page )

of descriptions |—definite orindefinite—that existence canbe significantly asserted; for,if “a” is a name, it must namesomething: what does notname anything is not a name,and therefore, if intended tobe a name, is a symbol de-void of meaning, whereas adescription, like “the presentKing of France,” does not be-come incapable of occurringsignificantly merely on theground that it describes noth-ing, the reason being that itis a complex symbol, of which (original pages –)

the meaning is derived fromthat of its constituent sym-bols. And so, when we askwhether Homer existed, weare using the word “Homer”as an abbreviated description:we may replace it by (say)“the author of the Iliad andthe Odyssey.” The same con-siderations apply to almost alluses of what look like propernames.

When descriptions occur inpropositions, it is necessaryto distinguish what may becalled “primary” and “sec- (original page )

ondary” occurrences. Theabstract distinction is as fol-lows. A description has a“primary” occurrence whenthe proposition in which itoccurs results from substi-tuting the description for “x”in some propositional func-tion φx; a description has a“secondary” occurrence whenthe result of substituting thedescription for x in φx givesonly part of the propositionconcerned. An instance willmake this clearer. Consider“the present King of France is (original page )

bald.” Here “the present Kingof France” has a primary oc-currence, and the propositionis false. Every proposition inwhich a description which de-scribes nothing has a primaryoccurrence is false. But nowconsider “the present Kingof France is not bald.” Thisis ambiguous. If we are firstto take “x is bald,” then sub-stitute “the present King ofFrance” for “x,” and then denythe result, the occurrence of“the present King of France”is secondary and our propo- (original page )

sition is true; but if we areto take “x is not bald” andsubstitute “the present Kingof France” for “x,” then “thepresent King of France” has aprimary occurrence and theproposition is false. Confu-sion of primary and secondaryoccurrences is a ready sourceof fallacies where descriptionsare concerned. |

Descriptions occur in math-ematics chiefly in the form ofdescriptive functions, i.e. “theterm having the relation R toy,” or “the R of y” as we may (original pages –)

say, on the analogy of “the fa-ther of y” and similar phrases.To say “the father of y is rich,”for example, is to say that thefollowing propositional func-tion of c: “c is rich, and ‘x be-gat y’ is always equivalent to‘x is c,’” is “sometimes true,”i.e. is true for at least one valueof c. It obviously cannot betrue for more than one value.

The theory of descriptions,briefly outlined in the presentchapter, is of the utmost im-portance both in logic andin theory of knowledge. But (original page )

for purposes of mathematics,the more philosophical partsof the theory are not essen-tial, and have therefore beenomitted in the above account,which has confined itself tothe barest mathematical re-quisites.

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CHAPTER XVIICLASSES

In the present chapter weshall be concerned with the inthe plural: the inhabitants ofLondon, the sons of rich men,and so on. In other words,we shall be concerned withclasses. We saw in Chapter II.that a cardinal number is tobe defined as a class of classes,and in Chapter III. that thenumber is to be defined as (original page )

the class of all unit classes, i.e.of all that have just one mem-ber, as we should say but forthe vicious circle. Of course,when the number is definedas the class of all unit classes,“unit classes” must be definedso as not to assume that weknow what is meant by “one”;in fact, they are defined ina way closely analogous tothat used for descriptions,namely: A class α is said to bea “unit” class if the proposi-tional function “‘x is an α’ isalways equivalent to ‘x is c’” (original page )

(regarded as a function of c) isnot always false, i.e., in moreordinary language, if there isa term c such that x will be amember of α when x is c butnot otherwise. This gives us adefinition of a unit class if wealready know what a class isin general. Hitherto we have,in dealing with arithmetic,treated “class” as a primitiveidea. But, for the reasons setforth in Chapter XIII., if forno others, we cannot accept“class” as a primitive idea.We must seek a definition on (original page )

the same lines as the defi-nition of descriptions, i.e. adefinition which will assigna meaning to propositions inwhose verbal or symbolic ex-pression words or symbols ap-parently representing classesoccur, but which will assign ameaning that altogether elim-inates all mention of classesfrom a right analysis | of suchpropositions. We shall thenbe able to say that the sym-bols for classes are mere con-veniences, not representingobjects called “classes,” and (original pages –)

that classes are in fact, likedescriptions, logical fictions,or (as we say) “incompletesymbols.”

The theory of classes is lesscomplete than the theory ofdescriptions, and there arereasons (which we shall givein outline) for regarding thedefinition of classes that willbe suggested as not finally sat-isfactory. Some further sub-tlety appears to be required;but the reasons for regard-ing the definition which willbe offered as being approx- (original page )

imately correct and on theright lines are overwhelming.

The first thing is to re-alise why classes cannot beregarded as part of the ulti-mate furniture of the world. Itis difficult to explain preciselywhat one means by this state-ment, but one consequencewhich it implies may be usedto elucidate its meaning. Ifwe had a complete symboliclanguage, with a definition foreverything definable, and anundefined symbol for every-thing indefinable, the unde- (original page )

fined symbols in this languagewould represent symbolicallywhat I mean by “the ultimatefurniture of the world.” I ammaintaining that no symbolseither for “class” in general orfor particular classes wouldbe included in this apparatusof undefined symbols. On theother hand, all the particularthings there are in the worldwould have to have nameswhich would be includedamong undefined symbols.We might try to avoid thisconclusion by the use of de- (original page )

scriptions. Take (say) “the lastthing Cæsar saw before hedied.” This is a descriptionof some particular; we mightuse it as (in one perfectly le-gitimate sense) a definition ofthat particular. But if “a” is aname for the same particular,a proposition in which “a”occurs is not (as we saw in thepreceding chapter) identicalwith what this propositionbecomes when for “a” we sub-stitute “the last thing Cæsarsaw before he died.” If ourlanguage does not contain (original page )

the name “a,” or some othername for the same particu-lar, we shall have no meansof expressing the proposi-tion which we expressed bymeans of “a” as opposed tothe one that | we expressedby means of the description.Thus descriptions would notenable a perfect language todispense with names for allparticulars. In this respect,we are maintaining, classesdiffer from particulars, andneed not be represented byundefined symbols. Our first (original pages –)

business is to give the reasonsfor this opinion.

We have already seen thatclasses cannot be regarded asa species of individuals, onaccount of the contradictionabout classes which are notmembers of themselves (ex-plained in Chapter XIII.), andbecause we can prove that thenumber of classes is greaterthan the number of individu-als.

We cannot take classes inthe pure extensional way assimply heaps or conglomera- (original page )

tions. If we were to attemptto do that, we should find itimpossible to understand howthere can be such a class asthe null-class, which has nomembers at all and cannotbe regarded as a “heap”; weshould also find it very hardto understand how it comesabout that a class which hasonly one member is not iden-tical with that one member.I do not mean to assert, or todeny, that there are such en-tities as “heaps.” As a math-ematical logician, I am not (original page )

called upon to have an opin-ion on this point. All thatI am maintaining is that, ifthere are such things as heaps,we cannot identify them withthe classes composed of theirconstituents.

We shall come much nearerto a satisfactory theory if wetry to identify classes withpropositional functions. Ev-ery class, as we explainedin Chapter II., is defined bysome propositional functionwhich is true of the membersof the class and false of other (original page )

things. But if a class can bedefined by one propositionalfunction, it can equally wellbe defined by any other whichis true whenever the first istrue and false whenever thefirst is false. For this reasonthe class cannot be identifiedwith any one such proposi-tional function rather thanwith any other—and given apropositional function, thereare always many others whichare true when it is true andfalse when it is false. We saythat two propositional func- (original page )

tions are “formally equiva-lent” when this happens. Twopropositions are | “equivalent”when both are true or bothfalse; two propositional func-tions φx, ψx are “formallyequivalent” when φx is al-ways equivalent to ψx. It isthe fact that there are otherfunctions formally equivalentto a given function that makesit impossible to identify aclass with a function; for wewish classes to be such thatno two distinct classes haveexactly the same members, (original pages –)

and therefore two formallyequivalent functions will haveto determine the same class.

When we have decided thatclasses cannot be things ofthe same sort as their mem-bers, that they cannot be justheaps or aggregates, and alsothat they cannot be identifiedwith propositional functions,it becomes very difficult to seewhat they can be, if they areto be more than symbolic fic-tions. And if we can find anyway of dealing with them assymbolic fictions, we increase (original page )

the logical security of our po-sition, since we avoid the needof assuming that there areclasses without being com-pelled to make the oppositeassumption that there are noclasses. We merely abstainfrom both assumptions. Thisis an example of Occam’s ra-zor, namely, “entities are notto be multiplied without ne-cessity.” But when we refuseto assert that there are classes,we must not be supposed tobe asserting dogmatically thatthere are none. We are merely (original page )

agnostic as regards them: likeLaplace, we can say, “je n’aipas besoin de cette hypothese.”

Let us set forth the condi-tions that a symbol must ful-fil if it is to serve as a class. Ithink the following conditionswill be found necessary andsufficient:—

() Every propositionalfunction must determine aclass, consisting of those argu-ments for which the functionis true. Given any proposi-tion (true or false), say aboutSocrates, we can imagine (original page )

Socrates replaced by Platoor Aristotle or a gorilla or theman in the moon or any otherindividual in the world. Ingeneral, some of these substi-tutions will give a true propo-sition and some a false one.The class determined willconsist of all those substitu-tions that give a true one. Ofcourse, we have still to decidewhat we mean by “all thosewhich, etc.” All that | we areobserving at present is that aclass is rendered determinateby a propositional function, (original pages –)

and that every propositionalfunction determines an ap-propriate class.

() Two formally equivalentpropositional functions mustdetermine the same class, andtwo which are not formallyequivalent must determinedifferent classes. That is,a class is determined by itsmembership, and no two dif-ferent classes can have thesame membership. (If a classis determined by a functionφx, we say that a is a “mem-ber” of the class if φa is true.) (original page )

() We must find some wayof defining not only classes,but classes of classes. We sawin Chapter II. that cardinalnumbers are to be definedas classes of classes. The or-dinary phrase of elementarymathematics, “The combina-tions of n things m at a time”represents a class of classes,namely, the class of all classesof m terms that can be se-lected out of a given class ofn terms. Without some sym-bolic method of dealing withclasses of classes, mathemati- (original page )

cal logic would break down.() It must under all cir-

cumstances be meaningless(not false) to suppose a classa member of itself or not amember of itself. This resultsfrom the contradiction whichwe discussed in Chapter XIII.

() Lastly—and this is thecondition which is most dif-ficult of fulfilment—it mustbe possible to make proposi-tions about all the classes thatare composed of individuals,or about all the classes thatare composed of objects of (original page )

any one logical “type.” If thiswere not the case, many usesof classes would go astray—for example, mathematicalinduction. In defining theposterity of a given term, weneed to be able to say that amember of the posterity be-longs to all hereditary classesto which the given term be-longs, and this requires thesort of totality that is in ques-tion. The reason there is adifficulty about this conditionis that it can be proved to beimpossible to speak of all the (original page )

propositional functions thatcan have arguments of a giventype.

We will, to begin with, ig-nore this last condition andthe problems which it raises.The first two conditions maybe | taken together. They statethat there is to be one class,no more and no less, for eachgroup of formally equivalentpropositional functions; e.g.the class of men is to be thesame as that of featherlessbipeds or rational animalsor Yahoos or whatever other (original pages –)

characteristic may be pre-ferred for defining a humanbeing. Now, when we saythat two formally equivalentpropositional functions maybe not identical, althoughthey define the same class, wemay prove the truth of theassertion by pointing out thata statement may be true ofthe one function and false ofthe other; e.g. “I believe thatall men are mortal” may betrue, while “I believe that allrational animals are mortal”may be false, since I may be- (original page )

lieve falsely that the Phœnix isan immortal rational animal.Thus we are led to considerstatements about functions, or(more correctly) functions offunctions.

Some of the things that maybe said about a function maybe regarded as said about theclass defined by the function,whereas others cannot. Thestatement “all men are mor-tal” involves the functions “xis human” and “x is mortal”;or, if we choose, we can saythat it involves the classes (original page )

men and mortals. We can in-terpret the statement in eitherway, because its truth-valueis unchanged if we substitutefor “x is human” or for “x ismortal” any formally equiv-alent function. But, as wehave just seen, the statement“I believe that all men aremortal” cannot be regardedas being about the class de-termined by either function,because its truth-value maybe changed by the substitu-tion of a formally equivalentfunction (which leaves the (original page )

class unchanged). We willcall a statement involving afunction φx an “extensional”function of the function φx, ifit is like “all men are mortal,”i.e. if its truth-value is un-changed by the substitution ofany formally equivalent func-tion; and when a function ofa function is not extensional,we will call it “intensional,”so that “I believe that all menare mortal” is an intensionalfunction of “x is human” or “xis mortal.” Thus extensionalfunctions of a function φx

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may, for practical | purposes,be regarded as functions ofthe class determined by φx,while intensional functionscannot be so regarded.

It is to be observed that allthe specific functions of func-tions that we have occasionto introduce in mathematicallogic are extensional. Thus,for example, the two funda-mental functions of functionsare: “φx is always true” and“φx is sometimes true.” Eachof these has its truth-valueunchanged if any formally (original pages –)

equivalent function is sub-stituted for φx. In the lan-guage of classes, if α is theclass determined by φx, “φxis always true” is equivalentto “everything is a memberof α,” and “φx is sometimestrue” is equivalent to “α hasmembers” or (better) “α hasat least one member.” Take,again, the condition, dealtwith in the preceding chapter,for the existence of “the termsatisfying φx.” The conditionis that there is a term c suchthat φx is always equivalent (original page )

to “x is c.” This is obviouslyextensional. It is equivalentto the assertion that the classdefined by the function φx isa unit class, i.e. a class havingone member; in other words, aclass which is a member of .

Given a function of a func-tion which may or may notbe extensional, we can alwaysderive from it a connected andcertainly extensional functionof the same function, by thefollowing plan: Let our origi-nal function of a function beone which attributes to φx

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the property f ; then considerthe assertion “there is a func-tion having the property f andformally equivalent to φx.”This is an extensional func-tion of φx; it is true when ouroriginal statement is true, andit is formally equivalent tothe original function of φx ifthis original function is exten-sional; but when the originalfunction is intensional, thenew one is more often truethan the old one. For exam-ple, consider again “I believethat all men are mortal,” re- (original page )

garded as a function of “x ishuman.” The derived exten-sional function is: “There is afunction formally equivalentto ‘x is human’ and such that Ibelieve that whatever satisfiesit is mortal.” This remainstrue when we substitute “xis a rational animal” | for “xis human,” even if I believefalsely that the Phœnix is ra-tional and immortal.

We give the name of “de-rived extensional function” tothe function constructed asabove, namely, to the func- (original pages –)

tion: “There is a functionhaving the property f andformally equivalent to φx,”where the original functionwas “the function φx has theproperty f.”

We may regard the derivedextensional function as havingfor its argument the class de-termined by the function φx,and as asserting f of this class.This may be taken as the def-inition of a proposition abouta class. I.e. we may define:

To assert that “the class de-termined by the function φx

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has the property f” is to as-sert thatφx satisfies the exten-sional function derived fromf.

This gives a meaning toany statement about a classwhich can be made signifi-cantly about a function; and itwill be found that technicallyit yields the results which arerequired in order to make atheory symbolically satisfac-tory.

See Principia Mathematica, vol. i.pp. – and ∗.

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What we have said just nowas regards the definition ofclasses is sufficient to sat-isfy our first four conditions.The way in which it securesthe third and fourth, namely,the possibility of classes ofclasses, and the impossibilityof a class being or not beinga member of itself, is some-what technical; it is explainedin Principia Mathematica, butmay be taken for granted here.It results that, but for our fifthcondition, we might regardour task as completed. But (original page )

this condition—at once themost important and the mostdifficult—is not fulfilled invirtue of anything we havesaid as yet. The difficulty isconnected with the theory oftypes, and must be brieflydiscussed.

We saw in Chapter XIII.that there is a hierarchy oflogical types, and that it isa fallacy to allow an object

The reader who desires a fullerdiscussion should consult PrincipiaMathematica, Introduction, chap. ii.;also ∗.

(original page )

belonging to one of these tobe substituted for an objectbelonging to another. | Nowit is not difficult to show thatthe various functions whichcan take a given object a asargument are not all of onetype. Let us call them alla-functions. We may takefirst those among them whichdo not involve reference toany collection of functions;these we will call “predica-tive a-functions.” If we nowproceed to functions involv-ing reference to the totality (original pages –)

of predicative a-functions, weshall incur a fallacy if we re-gard these as of the same typeas the predicative a-functions.Take such an every-day state-ment as “a is a typical French-man.” How shall we definea “typical Frenchman”? Wemay define him as one “pos-sessing all qualities that arepossessed by most French-men.” But unless we confine“all qualities” to such as donot involve a reference to anytotality of qualities, we shallhave to observe that most (original page )

Frenchmen are not typical inthe above sense, and thereforethe definition shows that tobe not typical is essential toa typical Frenchman. This isnot a logical contradiction,since there is no reason whythere should be any typicalFrenchmen; but it illustratesthe need for separating offqualities that involve refer-ence to a totality of qualitiesfrom those that do not.

Whenever, by statementsabout “all” or “some” of thevalues that a variable can sig- (original page )

nificantly take, we generatea new object, this new ob-ject must not be among thevalues which our previousvariable could take, since, ifit were, the totality of val-ues over which the variablecould range would only bedefinable in terms of itself,and we should be involvedin a vicious circle. For exam-ple, if I say “Napoleon hadall the qualities that make agreat general,” I must define“qualities” in such a way thatit will not include what I am (original page )

now saying, i.e. “having allthe qualities that make a greatgeneral” must not be itself aquality in the sense supposed.This is fairly obvious, and isthe principle which leads tothe theory of types by whichvicious-circle paradoxes areavoided. As applied to a-functions, we may supposethat “qualities” is to mean“predicative functions.” Thenwhen I say “Napoleon had allthe qualities, etc.,” I mean |“Napoleon satisfied all thepredicative functions, etc.” (original pages –)

This statement attributes aproperty to Napoleon, but nota predicative property; thuswe escape the vicious circle.But wherever “all functionswhich” occurs, the functionsin question must be limited toone type if a vicious circle is tobe avoided; and, as Napoleonand the typical Frenchmanhave shown, the type is notrendered determinate by thatof the argument. It would re-quire a much fuller discussionto set forth this point fully,but what has been said may (original page )

suffice to make it clear thatthe functions which can takea given argument are of aninfinite series of types. Wecould, by various technicaldevices, construct a variablewhich would run through thefirst n of these types, where nis finite, but we cannot con-struct a variable which willrun through them all, and,if we could, that mere factwould at once generate a newtype of function with the samearguments, and would set thewhole process going again. (original page )

We call predicative a-func-tions the first type of a-func-tions; a-functions involvingreference to the totality of thefirst type we call the secondtype; and so on. No variablea-function can run through allthese different types: it muststop short at some definiteone.

These considerations arerelevant to our definition ofthe derived extensional func-tion. We there spoke of “afunction formally equiva-lent to φx.” It is necessary (original page )

to decide upon the type ofour function. Any decisionwill do, but some decisionis unavoidable. Let us callthe supposed formally equiv-alent function ψ. Then ψappears as a variable, andmust be of some determinatetype. All that we know nec-essarily about the type of φis that it takes arguments ofa given type—that it is (say)an a-function. But this, aswe have just seen, does notdetermine its type. If we areto be able (as our fifth requi- (original page )

site demands) to deal withall classes whose membersare of the same type as a,we must be able to define allsuch classes by means of func-tions of some one type; thatis to say, there must be sometype of a-function, say the nth,such that any a-function isformally | equivalent to somea-function of the nth type.If this is the case, then anyextensional function whichholds of all a-functions ofthe nth type will hold of anya-function whatever. It is (original pages –)

chiefly as a technical meansof embodying an assump-tion leading to this result thatclasses are useful. The as-sumption is called the “axiomof reducibility,” and may bestated as follows:—

“There is a type (τ say) ofa-functions such that, givenany a-function, it is formallyequivalent to some function ofthe type in question.”

If this axiom is assumed,we use functions of this typein defining our associatedextensional function. State- (original page )

ments about all a-classes(i.e. all classes defined by a-functions) can be reducedto statements about all a-functions of the type τ . Solong as only extensional func-tions of functions are in-volved, this gives us in prac-tice results which would oth-erwise have required the im-possible notion of “all a-functions.” One particularregion where this is vital ismathematical induction.

The axiom of reducibilityinvolves all that is really es- (original page )

sential in the theory of classes.It is therefore worth while toask whether there is any rea-son to suppose it true.

This axiom, like the multi-plicative axiom and the axiomof infinity, is necessary forcertain results, but not forthe bare existence of deduc-tive reasoning. The theoryof deduction, as explained inChapter XIV., and the lawsfor propositions involving“all” and “some,” are of thevery texture of mathemati-cal reasoning: without them, (original page )

or something like them, weshould not merely not ob-tain the same results, but weshould not obtain any resultsat all. We cannot use them ashypotheses, and deduce hy-pothetical consequences, forthey are rules of deductionas well as premisses. Theymust be absolutely true, orelse what we deduce accord-ing to them does not evenfollow from the premisses.On the other hand, the ax-iom of reducibility, like ourtwo previous mathematical (original page )

axioms, could perfectly wellbe stated as an hypothesiswhenever it is used, insteadof being assumed to be actu-ally true. We can deduce | itsconsequences hypothetically;we can also deduce the conse-quences of supposing it false.It is therefore only convenient,not necessary. And in view ofthe complication of the theoryof types, and of the uncer-tainty of all except its mostgeneral principles, it is im-possible as yet to say whetherthere may not be some way of (original pages –)

dispensing with the axiom ofreducibility altogether. How-ever, assuming the correctnessof the theory outlined above,what can we say as to the truthor falsehood of the axiom?

The axiom, we may ob-serve, is a generalised formof Leibniz’s identity of indis-cernibles. Leibniz assumed,as a logical principle, that twodifferent subjects must differas to predicates. Now pred-icates are only some amongwhat we called “predicativefunctions,” which will in- (original page )

clude also relations to giventerms, and various proper-ties not to be reckoned aspredicates. Thus Leibniz’s as-sumption is a much stricterand narrower one than ours.(Not, of course, according tohis logic, which regarded allpropositions as reducible tothe subject-predicate form.)But there is no good rea-son for believing his form,so far as I can see. Theremight quite well, as a mat-ter of abstract logical possi-bility, be two things which (original page )

had exactly the same pred-icates, in the narrow sensein which we have been usingthe word “predicate.” Howdoes our axiom look when wepass beyond predicates in thisnarrow sense? In the actualworld there seems no way ofdoubting its empirical truthas regards particulars, owingto spatio-temporal differen-tiation: no two particularshave exactly the same spatialand temporal relations to allother particulars. But thisis, as it were, an accident, a (original page )

fact about the world in whichwe happen to find ourselves.Pure logic, and pure math-ematics (which is the samething), aims at being true, inLeibnizian phraseology, in allpossible worlds, not only inthis higgledy-piggledy job-lotof a world in which chancehas imprisoned us. There isa certain lordliness which thelogician should preserve: hemust not condescend to derivearguments from the things hesees about him. |

Viewed from this strictly (original pages –)

logical point of view, I donot see any reason to believethat the axiom of reducibilityis logically necessary, whichis what would be meant bysaying that it is true in all pos-sible worlds. The admissionof this axiom into a systemof logic is therefore a defect,even if the axiom is empiri-cally true. It is for this reasonthat the theory of classes can-not be regarded as being ascomplete as the theory of de-scriptions. There is need offurther work on the theory (original page )

of types, in the hope of ar-riving at a doctrine of classeswhich does not require sucha dubious assumption. Butit is reasonable to regard thetheory outlined in the presentchapter as right in its mainlines, i.e. in its reduction ofpropositions nominally aboutclasses to propositions abouttheir defining functions. Theavoidance of classes as en-tities by this method must,it would seem, be sound inprinciple, however the detailmay still require adjustment. (original page )

It is because this seems indu-bitable that we have includedthe theory of classes, in spiteof our desire to exclude, as faras possible, whatever seemedopen to serious doubt.

The theory of classes, asabove outlined, reduces it-self to one axiom and onedefinition. For the sake of def-initeness, we will here repeatthem. The axiom is:

There is a type τ such that ifφ is a function which can take agiven object a as argument, thenthere is a function ψ of the type (original page )

τ which is formally equivalentto φ.

The definition is:If φ is a function which can

take a given object a as argu-ment, and τ the type mentionedin the above axiom, then to saythat the class determined by φhas the property f is to say thatthere is a function of type τ , for-mally equivalent to φ, and hav-ing the property f.

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CHAPTER XVIIIMATHEMATICS AND

LOGIC

Mathematics and logic, his-torically speaking, have beenentirely distinct studies. Math-ematics has been connectedwith science, logic with Greek.But both have developedin modern times: logic hasbecome more mathematicaland mathematics has become

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more logical. The conse-quence is that it has nowbecome wholly impossibleto draw a line between thetwo; in fact, the two are one.They differ as boy and man:logic is the youth of mathe-matics and mathematics is themanhood of logic. This viewis resented by logicians who,having spent their time in thestudy of classical texts, areincapable of following a pieceof symbolic reasoning, andby mathematicians who havelearnt a technique without (original page )

troubling to inquire into itsmeaning or justification. Bothtypes are now fortunatelygrowing rarer. So much ofmodern mathematical work isobviously on the border-lineof logic, so much of modernlogic is symbolic and formal,that the very close relation-ship of logic and mathematicshas become obvious to everyinstructed student. The proofof their identity is, of course,a matter of detail: startingwith premisses which wouldbe universally admitted to be- (original page )

long to logic, and arriving bydeduction at results which asobviously belong to mathe-matics, we find that there isno point at which a sharp linecan be drawn, with logic to theleft and mathematics to theright. If there are still thosewho do not admit the iden-tity of logic and mathematics,we may challenge them to in-dicate at what point, in thesuccessive definitions and |deductions of Principia Math-ematica, they consider thatlogic ends and mathematics (original pages –)

begins. It will then be obviousthat any answer must be quitearbitrary.

In the earlier chapters ofthis book, starting from thenatural numbers, we have firstdefined “cardinal number”and shown how to generalisethe conception of number,and have then analysed theconceptions involved in thedefinition, until we foundourselves dealing with thefundamentals of logic. In asynthetic, deductive treat-ment these fundamentals (original page )

come first, and the naturalnumbers are only reachedafter a long journey. Suchtreatment, though formallymore correct than that whichwe have adopted, is more dif-ficult for the reader, becausethe ultimate logical conceptsand propositions with whichit starts are remote and unfa-miliar as compared with thenatural numbers. Also theyrepresent the present frontierof knowledge, beyond whichis the still unknown; and thedominion of knowledge over (original page )

them is not as yet very secure.It used to be said that math-

ematics is the science of “quan-tity.” “Quantity” is a vagueword, but for the sake of ar-gument we may replace itby the word “number.” Thestatement that mathemat-ics is the science of numberwould be untrue in two dif-ferent ways. On the one hand,there are recognised branchesof mathematics which havenothing to do with number—all geometry that does notuse co-ordinates or measure- (original page )

ment, for example: projec-tive and descriptive geometry,down to the point at whichco-ordinates are introduced,does not have to do with num-ber, or even with quantity inthe sense of greater and less.On the other hand, throughthe definition of cardinals,through the theory of induc-tion and ancestral relations,through the general theoryof series, and through thedefinitions of the arithmeti-cal operations, it has becomepossible to generalise much (original page )

that used to be proved onlyin connection with numbers.The result is that what wasformerly the single study ofArithmetic has now becomedivided into a number of sep-arate studies, no one of whichis specially concerned withnumbers. The most | elemen-tary properties of numbersare concerned with one-onerelations, and similarity be-tween classes. Addition isconcerned with the construc-tion of mutually exclusiveclasses respectively similar to (original pages –)

a set of classes which are notknown to be mutually exclu-sive. Multiplication is mergedin the theory of “selections,”i.e. of a certain kind of one-many relations. Finitude ismerged in the general studyof ancestral relations, whichyields the whole theory ofmathematical induction. Theordinal properties of the var-ious kinds of number-series,and the elements of the the-ory of continuity of functionsand the limits of functions,can be generalised so as no (original page )

longer to involve any essentialreference to numbers. It is aprinciple, in all formal rea-soning, to generalise to the ut-most, since we thereby securethat a given process of deduc-tion shall have more widelyapplicable results; we are,therefore, in thus generalisingthe reasoning of arithmetic,merely following a preceptwhich is universally admittedin mathematics. And in thusgeneralising we have, in effect,created a set of new deductivesystems, in which traditional (original page )

arithmetic is at once dissolvedand enlarged; but whetherany one of these new deduc-tive systems—for example,the theory of selections—is tobe said to belong to logic orto arithmetic is entirely arbi-trary, and incapable of beingdecided rationally.

We are thus brought face toface with the question: Whatis this subject, which maybe called indifferently eithermathematics or logic? Is thereany way in which we can de-fine it? (original page )

Certain characteristics ofthe subject are clear. To be-gin with, we do not, in thissubject, deal with particularthings or particular proper-ties: we deal formally withwhat can be said about anything or any property. Weare prepared to say that oneand one are two, but not thatSocrates and Plato are two,because, in our capacity oflogicians or pure mathemati-cians, we have never heard ofSocrates and Plato. A worldin which there were no such (original page )

individuals would still be aworld in which one and oneare two. It is not open to us,as pure mathematicians or lo-gicians, to mention anythingat all, because, if we do so,| we introduce something ir-relevant and not formal. Wemay make this clear by ap-plying it to the case of thesyllogism. Traditional logicsays: “All men are mortal,Socrates is a man, thereforeSocrates is mortal.” Now itis clear that what we mean toassert, to begin with, is only (original pages –)

that the premisses imply theconclusion, not that premissesand conclusion are actuallytrue; even the most traditionallogic points out that the ac-tual truth of the premisses isirrelevant to logic. Thus thefirst change to be made in theabove traditional syllogism isto state it in the form: “If allmen are mortal and Socratesis a man, then Socrates ismortal.” We may now ob-serve that it is intended toconvey that this argumentis valid in virtue of its form, (original page )

not in virtue of the particu-lar terms occurring in it. Ifwe had omitted “Socrates is aman” from our premisses, weshould have had a non-formalargument, only admissible be-cause Socrates is in fact a man;in that case we could not havegeneralised the argument. Butwhen, as above, the argumentis formal, nothing dependsupon the terms that occur init. Thus we may substitute αfor men, β for mortals, and xfor Socrates, where α and βare any classes whatever, and (original page )

x is any individual. We thenarrive at the statement: “Nomatter what possible values xand α and β may have, if allα’s are β’s and x is an α, thenx is a β”; in other words, “thepropositional function ‘if allα’s are β’s and x is an α, then xis a β’ is always true.” Here atlast we have a proposition oflogic—the one which is onlysuggested by the traditionalstatement about Socrates andmen and mortals.

It is clear that, if formalreasoning is what we are aim- (original page )

ing at, we shall always ar-rive ultimately at statementslike the above, in which noactual things or propertiesare mentioned; this will hap-pen through the mere de-sire not to waste our timeproving in a particular casewhat can be proved gener-ally. It would be ridiculous togo through a long argumentabout Socrates, and then gothrough precisely the sameargument again about Plato.If our argument is one (say)which holds of all men, we (original page )

shall prove it concerning “x,”with the hypothesis “if x is aman.” With | this hypothe-sis, the argument will retainits hypothetical validity evenwhen x is not a man. But nowwe shall find that our argu-ment would still be valid if,instead of supposing x to bea man, we were to supposehim to be a monkey or a gooseor a Prime Minister. We shalltherefore not waste our timetaking as our premiss “x is aman” but shall take “x is anα,” where α is any class of (original pages –)

individuals, or “φx” where φis any propositional functionof some assigned type. Thusthe absence of all mention ofparticular things or propertiesin logic or pure mathematicsis a necessary result of the factthat this study is, as we say,“purely formal.”

At this point we find our-selves faced with a problemwhich is easier to state thanto solve. The problem is:“What are the constituents ofa logical proposition?” I donot know the answer, but I (original page )

propose to explain how theproblem arises.

Take (say) the proposition“Socrates was before Aristo-tle.” Here it seems obviousthat we have a relation be-tween two terms, and thatthe constituents of the propo-sition (as well as of the cor-responding fact) are simplythe two terms and the rela-tion, i.e. Socrates, Aristotle,and before. (I ignore the factthat Socrates and Aristotleare not simple; also the factthat what appear to be their (original page )

names are really truncateddescriptions. Neither of thesefacts is relevant to the presentissue.) We may represent thegeneral form of such propo-sitions by “xRy,” which maybe read “x has the relation Rto y.” This general form mayoccur in logical propositions,but no particular instance ofit can occur. Are we to inferthat the general form itself isa constituent of such logicalpropositions?

Given a proposition, such as“Socrates is before Aristotle,” (original page )

we have certain constituentsand also a certain form. Butthe form is not itself a newconstituent; if it were, weshould need a new form toembrace both it and the otherconstituents. We can, in fact,turn all the constituents ofa proposition into variables,while keeping the form un-changed. This is what we dowhen we use such a schemaas “xRy,” which stands forany | one of a certain class ofpropositions, namely, thoseasserting relations between (original pages –)

two terms. We can proceedto general assertions, such as“xRy is sometimes true”—i.e.there are cases where dualrelations hold. This assertionwill belong to logic (or mathe-matics) in the sense in whichwe are using the word. Butin this assertion we do notmention any particular thingsor particular relations; no par-ticular things or relations canever enter into a propositionof pure logic. We are left withpure forms as the only pos-sible constituents of logical (original page )

propositions.I do not wish to assert pos-

itively that pure forms—e.g.the form “xRy”—do actuallyenter into propositions of thekind we are considering. Thequestion of the analysis ofsuch propositions is a dif-ficult one, with conflictingconsiderations on the one sideand on the other. We cannotembark upon this questionnow, but we may accept, as afirst approximation, the viewthat forms are what enter intological propositions as their (original page )

constituents. And we mayexplain (though not formallydefine) what we mean by the“form” of a proposition asfollows:—

The “form” of a proposi-tion is that, in it, that remainsunchanged when every con-stituent of the proposition isreplaced by another.

Thus “Socrates is earlierthan Aristotle” has the sameform as “Napoleon is greaterthan Wellington,” though ev-ery constituent of the twopropositions is different. (original page )

We may thus lay down, asa necessary (though not suffi-cient) characteristic of logicalor mathematical propositions,that they are to be such as canbe obtained from a proposi-tion containing no variables(i.e. no such words as all, some,a, the, etc.) by turning everyconstituent into a variable andasserting that the result is al-ways true or sometimes true,or that it is always true in re-spect of some of the variablesthat the result is sometimestrue in respect of the oth- (original page )

ers, or any variant of theseforms. And another way ofstating the same thing is tosay that logic (or mathemat-ics) is concerned only withforms, and is concerned withthem only in the way of stat-ing that they are always or |sometimes true—with all thepermutations of “always” and“sometimes” that may occur.

There are in every languagesome words whose sole func-tion is to indicate form. Thesewords, broadly speaking, arecommonest in languages hav- (original pages –)

ing fewest inflections. Take“Socrates is human.” Here“is” is not a constituent of theproposition, but merely in-dicates the subject-predicateform. Similarly in “Socra-tes is earlier than Aristotle,”“is” and “than” merely in-dicate form; the propositionis the same as “Socrates pre-cedes Aristotle,” in whichthese words have disappearedand the form is otherwise in-dicated. Form, as a rule, canbe indicated otherwise thanby specific words: the order (original page )

of the words can do most ofwhat is wanted. But this prin-ciple must not be pressed. Forexample, it is difficult to seehow we could convenientlyexpress molecular forms ofpropositions (i.e. what we call“truth-functions”) withoutany word at all. We saw inChapter XIV. that one wordor symbol is enough for thispurpose, namely, a word orsymbol expressing incompat-ibility. But without even onewe should find ourselves indifficulties. This, however, is (original page )

not the point that is importantfor our present purpose. Whatis important for us is to ob-serve that form may be the oneconcern of a general propo-sition, even when no wordor symbol in that propositiondesignates the form. If wewish to speak about the formitself, we must have a wordfor it; but if, as in mathemat-ics, we wish to speak aboutall propositions that have theform, a word for the form willusually be found not indis-pensable; probably in theory (original page )

it is never indispensable.Assuming—as I think we

may—that the forms of propo-sitions can be represented bythe forms of the propositionsin which they are expressedwithout any special wordsfor forms, we should arriveat a language in which ev-erything formal belonged tosyntax and not to vocabulary.In such a language we couldexpress all the propositions ofmathematics even if we didnot know one single word ofthe language. The language (original page )

of | mathematical logic, if itwere perfected, would be sucha language. We should havesymbols for variables, such as“x” and “R” and “y,” arrangedin various ways; and the wayof arrangement would indi-cate that something was beingsaid to be true of all valuesor some values of the vari-ables. We should not needto know any words, becausethey would only be needed forgiving values to the variables,which is the business of theapplied mathematician, not (original pages –)

of the pure mathematician orlogician. It is one of the marksof a proposition of logic that,given a suitable language,such a proposition can be as-serted in such a language bya person who knows the syn-tax without knowing a singleword of the vocabulary.

But, after all, there arewords that express form, suchas “is” and “than.” And inevery symbolism hitherto in-vented for mathematical logicthere are symbols having con-stant formal meanings. We (original page )

may take as an example thesymbol for incompatibilitywhich is employed in build-ing up truth-functions. Suchwords or symbols may occurin logic. The question is: Howare we to define them?

Such words or symbols ex-press what are called “logi-cal constants.” Logical con-stants may be defined ex-actly as we defined forms; infact, they are in essence thesame thing. A fundamentallogical constant will be thatwhich is in common among a (original page )

number of propositions, anyone of which can result fromany other by substitution ofterms one for another. Forexample, “Napoleon is greaterthan Wellington” results from“Socrates is earlier than Aris-totle” by the substitution of“Napoleon” for “Socrates,”“Wellington” for “Aristotle,”and “greater” for “earlier.”Some propositions can be ob-tained in this way from theprototype “Socrates is earlierthan Aristotle” and some can-not; those that can are those (original page )

that are of the form “xRy,”i.e. express dual relations. Wecannot obtain from the aboveprototype by term-for-termsubstitution such proposi-tions as “Socrates is human”or “the Athenians gave thehemlock to Socrates,” becausethe first is of the subject- |predicate form and the sec-ond expresses a three-termrelation. If we are to haveany words in our pure logicallanguage, they must be suchas express “logical constants,”and “logical constants” will (original pages –)

always either be, or be de-rived from, what is in com-mon among a group of propo-sitions derivable from eachother, in the above manner,by term-for-term substitution.And this which is in commonis what we call “form.”

In this sense all the “con-stants” that occur in puremathematics are logical con-stants. The number , forexample, is derivative frompropositions of the form:“There is a term c such thatφxis true when, and only when, (original page )

x is c.” This is a function of φ,and various different propo-sitions result from giving dif-ferent values to φ. We may(with a little omission of inter-mediate steps not relevant toour present purpose) take theabove function of φ as whatis meant by “the class deter-mined by φ is a unit class”or “the class determined byφ is a member of ” ( be-ing a class of classes). In thisway, propositions in which occurs acquire a meaningwhich is derived from a cer- (original page )

tain constant logical form.And the same will be found tobe the case with all mathemat-ical constants: all are logicalconstants, or symbolic abbre-viations whose full use in aproper context is defined bymeans of logical constants.

But although all logical (ormathematical) propositionscan be expressed wholly interms of logical constants to-gether with variables, it isnot the case that, conversely,all propositions that can beexpressed in this way are log- (original page )

ical. We have found so fara necessary but not a suffi-cient criterion of mathemat-ical propositions. We havesufficiently defined the char-acter of the primitive ideas interms of which all the ideasof mathematics can be de-fined, but not of the primitivepropositions from which all thepropositions of mathematicscan be deduced. This is a moredifficult matter, as to which itis not yet known what the fullanswer is.

We may take the axiom of (original page )

infinity as an example of aproposition which, thoughit can be enunciated in log-ical terms, | cannot be as-serted by logic to be true.All the propositions of logichave a characteristic whichused to be expressed by say-ing that they were analytic,or that their contradictorieswere self-contradictory. Thismode of statement, however,is not satisfactory. The law ofcontradiction is merely oneamong logical propositions; ithas no special pre-eminence; (original pages –)

and the proof that the contra-dictory of some propositionis self-contradictory is likelyto require other principles ofdeduction besides the law ofcontradiction. Nevertheless,the characteristic of logicalpropositions that we are insearch of is the one which wasfelt, and intended to be de-fined, by those who said thatit consisted in deducibilityfrom the law of contradiction.This characteristic, which,for the moment, we may calltautology, obviously does not (original page )

belong to the assertion thatthe number of individualsin the universe is n, what-ever number n may be. Butfor the diversity of types, itwould be possible to provelogically that there are classesof n terms, where n is any fi-nite integer; or even that thereare classes of ℵ terms. But,owing to types, such proofs,as we saw in Chapter XIII.,are fallacious. We are leftto empirical observation todetermine whether there areas many as n individuals in (original page )

the world. Among “possi-ble” worlds, in the Leibniziansense, there will be worldshaving one, two, three, . . .individuals. There does noteven seem any logical neces-sity why there should be evenone individual—why, in fact,there should be any world atall. The ontological proof ofthe existence of God, if it wereThe primitive propositions in

Principia Mathematica are such as toallow the inference that at least oneindividual exists. But I now view thisas a defect in logical purity.

(original page )

valid, would establish the log-ical necessity of at least oneindividual. But it is generallyrecognised as invalid, and infact rests upon a mistakenview of existence—i.e. it failsto realise that existence canonly be asserted of somethingdescribed, not of somethingnamed, so that it is meaning-less to argue from “this is theso-and-so” and “the so-and-soexists” to “this exists.” If wereject the ontological | argu-ment, we seem driven to con-clude that the existence of a (original pages –)

world is an accident—i.e. it isnot logically necessary. If thatbe so, no principle of logiccan assert “existence” exceptunder a hypothesis, i.e. nonecan be of the form “the propo-sitional function so-and-so issometimes true.” Propositionsof this form, when they occurin logic, will have to occur ashypotheses or consequencesof hypotheses, not as com-plete asserted propositions.The complete asserted propo-sitions of logic will all be suchas affirm that some proposi- (original page )

tional function is always true.For example, it is always truethat if p implies q and q im-plies r then p implies r, orthat, if all α’s are β’s and x isan α then x is a β. Such propo-sitions may occur in logic, andtheir truth is independent ofthe existence of the universe.We may lay it down that, ifthere were no universe, allgeneral propositions wouldbe true; for the contradictoryof a general proposition (aswe saw in Chapter XV.) is aproposition asserting exis- (original page )

tence, and would thereforealways be false if no universeexisted.

Logical propositions aresuch as can be known a priori,without study of the actualworld. We only know froma study of empirical factsthat Socrates is a man, butwe know the correctness ofthe syllogism in its abstractform (i.e. when it is stated interms of variables) withoutneeding any appeal to experi-ence. This is a characteristic,not of logical propositions in (original page )

themselves, but of the way inwhich we know them. It has,however, a bearing upon thequestion what their naturemay be, since there are somekinds of propositions which itwould be very difficult to sup-pose we could know withoutexperience.

It is clear that the defini-tion of “logic” or “mathemat-ics” must be sought by tryingto give a new definition ofthe old notion of “analytic”propositions. Although wecan no longer be satisfied to (original page )

define logical propositionsas those that follow from thelaw of contradiction, we canand must still admit that theyare a wholly different class ofpropositions from those thatwe come to know empirically.They all have the character-istic which, a moment ago,we agreed to call “tautology.”This, | combined with the factthat they can be expressedwholly in terms of variablesand logical constants (a logi-cal constant being somethingwhich remains constant in a (original pages –)

proposition even when all itsconstituents are changed)—will give the definition of logicor pure mathematics. For themoment, I do not know how todefine “tautology.” It wouldbe easy to offer a definitionwhich might seem satisfac-tory for a while; but I know

The importance of “tautology”for a definition of mathematics waspointed out to me by my former pupilLudwig Wittgenstein, who was work-ing on the problem. I do not knowwhether he has solved it, or evenwhether he is alive or dead.

(original page )

of none that I feel to be sat-isfactory, in spite of feelingthoroughly familiar with thecharacteristic of which a defi-nition is wanted. At this point,therefore, for the moment, wereach the frontier of knowl-edge on our backward journeyinto the logical foundations ofmathematics.

We have now come to anend of our somewhat sum-mary introduction to mathe-matical philosophy. It is im-possible to convey adequatelythe ideas that are concerned (original page )

in this subject so long as weabstain from the use of logi-cal symbols. Since ordinarylanguage has no words thatnaturally express exactly whatwe wish to express, it is nec-essary, so long as we adhere toordinary language, to strainwords into unusual mean-ings; and the reader is sure,after a time if not at first, tolapse into attaching the usualmeanings to words, thus ar-riving at wrong notions as towhat is intended to be said.Moreover, ordinary grammar (original page )

and syntax is extraordinarilymisleading. This is the case,e.g., as regards numbers; “tenmen” is grammatically thesame form as “white men,” sothat might be thought to bean adjective qualifying “men.”It is the case, again, whereverpropositional functions areinvolved, and in particular asregards existence and descrip-tions. Because language ismisleading, as well as becauseit is diffuse and inexact whenapplied to logic (for whichit was never intended), log- (original page )

ical symbolism is absolutelynecessary to any exact or thor-ough treatment of our subject.Those readers, | therefore,who wish to acquire a masteryof the principles of mathe-matics, will, it is to be hoped,not shrink from the labourof mastering the symbols—alabour which is, in fact, muchless than might be thought. Asthe above hasty survey musthave made evident, there areinnumerable unsolved prob-lems in the subject, and muchwork needs to be done. If any (original pages –)

student is led into a seriousstudy of mathematical logicby this little book, it will haveserved the chief purpose forwhich it has been written.

(original page )

INDEX

[Online edition note: This is ahyperlinked recreation of theoriginal index. The page num-bers listed are for the originaledition.]

Aggregates, .Alephs, , , , .Aliorelatives, .

(original page )

All, ff.Analysis, .Ancestors, , .Argument of a function, ,.

Arithmetising ofmathematics, .

Associative law, , .Axioms, .

Between, ff., .Bolzano, n.Boots and socks, .Boundary, , , .

Cantor, Georg, , , n., (original page )

, , , , .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.

(original page )

Contradictions, ff.Convergence, .Converse, , , .Correlators, .Counterparts, objective, .Counting, , .

Dedekind, , , n.Deduction, ff.Definition, ; extensional and

intensional, .Derivatives, .Descriptions, , ,ff.

Dimensions, .Disjunction, . (original page )

Distributive law, , .Diversity, .Domain, , , .

Equivalence, .Euclid, .Existence, , , .Exponentiation, , .Extension of a relation, .

Fictions, logical, n., ,.

Field of a relation, , .Finite, .Flux, .Form, . (original page )

Fractions, , .Frege, , [], n., , ,n.

Functions, ; descriptive,, ; intensional andextensional, ;predicative, ;propositional, , ,ff.

Gap, Dedekindian, ff., .Generalisation, .Geometry, , , , ,, ; analytical, , .

Greater and less, , .

(original page )

Hegel, .Hereditary properties, .

Implication, , ;formal, .

Incommensurables, , .Incompatibility, ff., .Incomplete symbols, .Indiscernibles, .Individuals, , , .Induction, mathematical,ff., , , .

Inductive properties, .Inference, ff.Infinite, ; of rationals, ;

Cantorian, ; of cardinals, (original page )

ff.; and series andordinals, ff.

Infinity, axiom of, n., ,ff., .

Instances, .Integers, positive and

negative, .Intervals, .Intuition, .Irrationals, , . |

Kant, .

Leibniz, , , .Lewis, C. I., , .Likeness, . (original pages –)

Limit, , ff., ff.; offunctions, ff.

Limiting points, .Logic, , , ff.;

mathematical, v, , .Logicising of mathematics,.

Maps, , ff., .Mathematics, ff.Maximum, , .Median class, .Meinong, .Method, vi.Minimum, , .Modality, . (original page )

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,

(original page )

ff., ; non-inductive, ,; real, , , ;reflexive, , ; relation,, ; serial, .

Occam, .Occurrences, primary and

secondary, .Ontological proof, .Order, ff.; cyclic, .Oscillation, ultimate, .

Parmenides, .Particulars, ff., .Peano, ff., , , , ,, .

(original page )

Peirce, n.Permutations, .Philosophy, mathematical, v,.

Plato, .Plurality, .Poincare, .Points, .Posterity, ff.; proper, .Postulates, , .Precedent, .Premisses of arithmetic, .Primitive ideas and

propositions, , .Progressions, , ff.

(original page )

Propositions, ; analytic,; elementary, .

Pythagoras, , .

Quantity, , .

Ratios, , , , .Reducibility, axiom of, .Referent, .Relation-numbers, ff.Relations, asymmetrical, ,; connected, ;many-one, ; one-many,, ; one-one, , , ;reflexive, ; serial, ;similar, ff.; squares of,

(original page )

; symmetrical, , ;transitive, , .

Relatum, .Representatives, .Rigour, .Royce, .

Section, Dedekindian, ff.;ultimate, .

Segments, , .Selections, ff.Sequent, .Series, ff.; closed, ;

compact, , , ;condensed in itself, ;Dedekindian, , , ;

(original page )

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, .

(original page )

Tautology, , .The, , ff.Time, , , .Truth-function, .Truth-value, .Types, logical, , ff., ,.

Unreality, .

Value of a function, , .Variables, , , .Veblen, .Verbs, .

Weierstrass, , . (original page )

Wells, H. G., .Whitehead, , , , .Wittgenstein, n.

Zermelo, , .Zero, .

(original page )

CHANGES TOONLINE EDITION

This Online Corrected Edi-tion was created by Kevin C.Klement; this is version .(February , ). It is basedon the April so-called“second edition” publishedby Allen & Unwin, which,

by contemporary standards,was simply a second printingof the original editionbut incorporating various,mostly minor, fixes. This edi-tion incorporates fixes fromlater printings as well, andsome new fixes, mentionedbelow. The pagination ofthe Allen & Unwin editionis given in the footer, withpage breaks marked with thesign “|”. These are in red, asare other additions to the textnot penned by Russell.

Thanks to members of the

Russell-l and HEAPS-l mail-ing lists for help in checkingand proofreading the ver-sion, including Adam Killian,Pierre Grenon, David Blitz,Brandon Young, RosalindCarey, and, especially, JohnOngley. A tremendous debt ofthanks is owed to KennethBlackwell of the BertrandRussell Archives/ResearchCentre, McMaster Univer-sity, for proofreading the bulkof the edition, checking itagainst Russell’s handwrittenmanuscript, and providing

other valuable advice andassistance. Another largedebt of gratitude is owed toChristof Graber who com-pared this version to the printversions and showed remark-able aptitude in spotting dis-crepancies. I take full re-sponsibility for any remain-ing errors. If you discoverany, please email me at [email protected].

The online edition differsfrom the Allen & Un-win edition, and reprintingsthereof, in certain respects.

Some are mere stylistic differ-ences. Others represent cor-rections based on discrepan-cies between Russell’s manu-script and the print edition, orfix small grammatical or typo-graphical errors. The stylisticdifferences are these:

• In the original, footnotenumbering begins anewwith each page. Sincethis version uses differ-ent pagination, it wasnecessary to numberfootnotes sequentially

through each chapter.Thus, for example, thefootnote listed as note on page of this edi-tion was listed as note on page of the original.

• With some exceptions,the Allen & Unwin edi-tion uses linear fractionsof the style “x/y” mid-paragraph, but verticalfractions of the form “ xy ”in displays. Contraryto this usual practice,those in the display on

page of the original(page of this edi-tion) were linear, buthave been convertedto vertical fractions inthis edition. Similarly,the mid-paragraph frac-tions on pages , , and of the original(pages , , and here) were printedvertically in the original,but here are horizontal.

The following more signifi-cant changes and revisions are

marked in green in this edi-tion. Most of these result fromKen Blackwell’s comparisonwith Russell’s manuscript. Afew were originally noted inan early review of the book byG. A. Pfeiffer (Bulletin of theAmerican Mathematical Society: (), pp. –).

. (page n. / originalpage n.) Russell wrotethe wrong publicationdate () for the sec-ond volume of PrincipiaMathematica; this has

been fixed to .

. (page / original page) “. . . or all that areless than . . . ” ischanged to “. . . or allthat are not less than . . . ” to match Rus-sell’s manuscript andthe obviously intendedmeaning of the passage.This error was notedby Pfeiffer in butunfixed in Russell’s life-time.

. (page / original

page ) “. . . either bylimiting the domain tomales or by limiting theconverse to females” ischanged to “. . . eitherby limiting the domainto males or by limitingthe converse domainto females”, which ishow it read in Russell’smanuscript, and seemsbetter to fit the context.

. (page / originalpage ) “. . . providedneither m or n is zero.”

is fixed to “. . . providedneither m nor n is zero.”Thanks to John Ongleyfor spotting this error,which exists even inRussell’s manuscript.

. (page n. / originalpage n.) The word“deutschen” in the orig-inal’s (and the manu-script’s) “Jahresberichtder deutschen Mathe-matiker-Vereinigung” hasbeen capitalized.

. (page / original

page ) “. . . of a classα, i.e. its limits or max-imum, and then . . . ”is changed to “. . . ofa class α, i.e. its limitor maximum, and then. . . ” to match Russell’smanuscript, and the ap-parent meaning of thepassage.

. (page / originalpage ) “. . . the limitof its value for approach-es either from . . . ” ischanged to “. . . the limit

of its values for ap-proaches either from. . . ”, which matchesRussell’s manuscript,and is more appropriatefor the meaning of thepassage.

. (page / originalpage ) The ungram-matical “. . . advantagesof this form of defini-tion is that it analyses. . . ” is changed to “. . .advantage of this formof definition is that it

analyses . . . ” to matchRussell’s manuscript.

. (page / originalpage ) “. . . all termsz such that x has therelation P to x and zhas the relation P to y. . . ” is fixed to “. . . allterms z such that x hasthe relation P to z andz has the relation P toy . . . ” Russell himselfhand-corrected this inhis manuscript, but notin a clear way, and at his

request, it was changedin the printing.

. (page / originalpage ) The words“correlator of α with β,and similarly for everyother pair. This requiresa”, which constitute ex-actly one line of Rus-sell’s manuscript, wereomitted, thereby amal-gamating two sentencesinto one. The missingwords are now restored.

. (page / original

page ) The passage“. . . if x is the memberof y, x is a member ofy, x is a member of y,and so on; then . . . ” ischanged to “. . . if x isthe member of γ, x isa member of γ, x is amember of γ, and soon; then . . . ” to matchRussell’s manuscript,and the obviously in-tended meaning of thepassage.

. (page / original

page ) The words“and then the idea ofthe idea of Socrates” al-though present in Rus-sell’s manuscript, wereleft out of previous printeditions. Note that Rus-sell mentions “all theseideas” in the next sen-tence.

. (pages – / orig-inal page ) The twofootnotes on this pagewere misplaced. Thesecond, the reference

to Principia Mathemat-ica ∗, was attached inprevious versions to thesentence that now refersto the first footnote inthe chapter. That foot-note was placed threesentences below. Thefootnote references havebeen returned to wherethey had been placed inRussell’s manuscript.

. (page / originalpage ) “. . . the nega-tion of propositions of

the type to which x be-longs . . . ” is changedto “. . . the negation ofpropositions of the typeto which φx belongs. . . ” to match Russell’smanuscript. This is an-other error noted byPfeiffer.

. (page / originalpage ) “Suppose weare considering all “menare mortal”: we will. . . ” is changed to “Sup-pose we are considering

“all men are mortal”:we will . . . ” to matchthe obviously intendedmeaning of the passage,and the placement of theopening quotation markin Russell’s manuscript(although he here usedsingle quotation marks,as he did sporadicallythroughout). Thanksto Christof Graber forspotting this error.

. (page / originalpage ) “. . . as op-

posed to specific man.”is fixed to “. . . as op-posed to specific men.”Russell sent this changeto Unwin in , andit was made in the printing.

. (page / originalpage ) The “φ” in“. . . the process of ap-plying general state-ments about φx to par-ticular cases . . . ”, presentin Russell’s manuscript,was excluded from the

Allen & Unwin print-ings, and has been re-stored.

. (page / originalpage ) The “φ” in“. . . resulting from apropositional functionφx by the substitutionof . . . ” was excludedfrom previous pub-lished versions, thoughit does appear in Rus-sell’s manuscript, andseems necessary for thepassage to make sense.

Thanks to John Ongleyfor spotting this error,which had also beennoted by Pfeiffer.

. (page / originalpages –) The twooccurrences of “φ” in“. . . extensional func-tions of a function φxmay, for practical pur-poses, be regarded asfunctions of the class de-termined by φx, whileintensional functionscannot . . . ” were omit-

ted from previous pub-lished versions, butdo appear in Russell’smanuscript. Againthanks to John Ongley.

. (page / originalpage ) The Allen& Unwin printings havethe sentence as “Howshall we define a “typi-cal” Frenchman?” Here,the closing quotationmark has been moved tomake it “How shall wedefine a “typical French-

man”?” Although Rus-sell’s manuscript is notentirely clear here, itappears the latter wasintended, and it alsoseems to make moresense in context.

. (page / originalpage ) “There is atype (r say) . . . ” hasbeen changed to “Thereis a type (τ say) . . . ” tomatch Russell’s manu-script, and conventionsfollowed elsewhere in

the chapter.

. (page / originalpage ) “. . . dividedinto numbers of sep-arate studies . . . ” hasbeen changed to “. . .divided into a numberof separate studies . . . ”Russell’s manuscriptjust had “number”, inthe singular, withoutthe indefinite article.Some emendation wasnecessary to make thepassage grammatical,

but the fix adopted hereseems more likely whatwas meant.

. (page / originalpage ) The passage“the propositional func-tion ‘if all α’s are β andx is an α, then x is aβ’ is always true” hasbeen changed to “thepropositional function‘if all α’s are β’s and xis an α, then x is a β’ isalways true” to matchRussell’s manuscript, as

well as to make it con-sistent with the otherparaphrase given earlierin the sentence. Thanksto Christof Graber fornoticing this error.

. (page / originalpage ) “. . . with-out any special wordfor forms . . . ” has beenchanged to “. . . with-out any special wordsfor forms . . . ”, whichmatches Russell’s man-uscript and seems to fit

better in the context.

. (page / originalpage ) The originalindex listed a referenceto Frege on page , butin fact, the discussionof Frege occurs on page. Here, “” is crossedout, and “[]” inserted.

Some very minor corrections topunctuation have been madeto the Allen & Unwin printing, but not marked ingreen.

a) Ellipses have been reg-ularized to three closeddots throughout.

b) (page / originalpage ) “We may de-fine two relations . . . ”did not start a new para-graph in previous edi-tions, but does in Rus-sell’s manuscript, and ischanged to do so.

c) (page / originalpage ) What appearsin the and laterprintings as “. . . is thefield of Q. and which is

. . . ” is changed to “. . . isthe field of Q, and whichis . . . ”

d) (page / originalpage ) “. . . a rela-tion number is a class of. . . ” is changed to “. . .a relation-number is aclass of . . . ” to matchthe hyphenation in therest of the book (and inRussell’s manuscript). Asimilar change is madein the index.

e) (page / originalpage ) “. . . and “feath-

erless biped,”—so two. . . ” is changed to “. . .and “featherless biped”—so two . . . ”

f) (pages – / orig-inal pages –) Onemisprint of “proges-sion” for “progression”,and one misprint of“progessions” for “pro-gressions”, have beencorrected. (Thanks toChristof Graber for notic-ing these errors in theoriginal.)

g) (page / original

page ) In the Allen& Unwin printing, the“s” in “y’s” in what ap-pears here as “Formall such sections forall y’s . . . ” was itali-cized along with the “y”.Nothing in Russell’smanuscript suggestsit should be italicized,however. (Again thanksto Christof Graber.)

h) (page / originalpage ) In the Allen& Unwin printing, “Lety be a member of β . . . ”

begins a new paragraph,but it does not in Rus-sell’s manuscript, andclearly should not.

i) (page / originalpages –) Thephrase “well ordered”has twice been changedto “well-ordered” tomatch Russell’s man-uscript (in the first case)and the rest of the book(in the second).

j) (page / originalpage ) “The wayin which the need for

this axiom arises may beexplained as follows:—One of Peano’s . . . ” ischanged to “The wayin which the need forthis axiom arises maybe explained as follows.One of Peano’s . . . ” andhas been made to start anew paragraph, as it didin Russell’s manuscript.

k) (page / originalpage ) The accenton “Metaphysique”, in-cluded in Russell’s man-uscript but left off in

print, has been restored.l) (page / original

page ) “. . . or whatnot,—and clearly . . . ” ischanged to “. . . or whatnot—and clearly . . . ”

m) (page / originalpage ) Italics havebeen added to one oc-currence of “Waverley”to make it consistentwith the others.

n) (page / originalpage ) “. . . most dif-ficult of fulfilment,—itmust . . . ” is changed

to “. . . most difficult offulfilment—it must . . . ”

o) (page / originalpage ) In the Allen &Unwin printings, “Soc-rates” was not itali-cized in “. . . we maysubstitute α for men, βfor mortals, and x forSocrates, where . . . ”Russell had markedit for italicizing in themanuscript, and it seemsnatural to do so for thesake of consistency, so ithas been italicized.

p) (page / originalpage ) The word“seem” was not itali-cized in “. . . a defini-tion which might seemsatisfactory for a while. . . ” in the Allen & Un-win editions, but wasmarked to be in Rus-sell’s manuscript; it isitalicized here.

q) (page / originalpage ) Under “Re-lations” in the index,“similar, ff;” has beenchanged to “similar,

ff.;” to match thepunctuation elsewhere.

There are, however, a num-ber of other places where theprevious print editions differfrom Russell’s manuscript inminor ways that were left un-changed in this edition. Fora detailed examination of thedifferences between Russell’smanuscript and the print edi-tions, and between the variousprintings themselves (includ-ing the changes from the to the printings not doc-

umented here), see KennethBlackwell, “Variants, Mis-prints and a BibliographicalIndex for Introduction to Math-ematical Philosophy”, Russelln.s. (): –.

p Bertrand Russell’s Intro-duction to Mathematical Philos-ophy is in the Public Domain.http://creativecommons.

org/licenses/publicdomain

cba This typeset-

ting (including LATEX code)and list of changes are li-censed under a Creative Com-mons Attribution—ShareAlike .United StatesLicense.http://creativecommons.

org/licenses/by-sa/./us/