principia mathematica - whitehead & bertrand russell. pag 1-20

8/2/2019 Principia Mathematica - Whitehead & Bertrand Russell. Pag 1-20

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PRINCIPIA MATHEMATICA


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CAMBRIDGE UNIVERSITY PRESSILonl1on: FETTER LANE, E.C.

C. F. CLAY, MANAGER

(!Fllinburgb: 100, PRINCES STREET

~trlin: A. ASHER AND CO.

lLeiMig: F. A. BROCKHAUS

j)lrw W~rk: G. P. PUTNAM'S SONS

J5ombal! anll ([:alcutta: MACMILLAN AND CO., LTD.

i,.


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l:>RINCIPIA MATHEMA TICA

BY

ALFRED NORTH WHITEHEAD, Sc.D., F.R.S.

Fellow and late Lecturer of Trinity College, Cambridge

AND

BERTRAND RUSSELL, M.A., F.R.S.

Lecturer and late Fellow of Trinity College, Cambridge

VOLUME I

Cambridgeat the University Press

19 10


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Q ra mb rilrg e :

PRINTED BY JOHN CLAY, M.A.

AT THE UNIVERSITY PRESS


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CONTENTS OF VOLUME I

PREFACE

INTRODUCTION

CHAPTER 1. PRELIMINARY EXPLANATIONS OF IDEAS AND NOTATIONS.

CHAPTER II. THE THEORY OF LOGICAL TYPES

CHAPTER III. INCOMPLETE SYMBOLS

PART I. MATHEMATICAL LOGIC.

Summary of Part I .

SECTION A. THE THEORY OJt' DEDDC'l'ION

*1. Primitive Ideas and Propositions

*2. Immediate Consequences of the Primitive Propositions*3. The Logical Product of two Propositions*4. Equivalence and Formal Rules*5. Miscellaneous Propositions

SECTION B. THEORY OF ApPARENT VARIABLES

*9. Extension of the Theory of Deduction from Lower to HigherTypes of Propositions.

dO. Theory of Propositions containing one Apparent Variable*11. Theory of two Apparent Variables.d2. The Hierarchy of Types and the Axiom of Reducibility

d3. Identity .d4. Descriptions

SECTION C. CLASSES AND RELATIONS

*20. General Theory of Classes*21. General Theory of Relations.*22. Calculus of Classes.*23. Calculus of Relations*24. The Universal Class, the Null-Class, and the Existence of Classes.*25. The Universal Relation, the Null Relation, and the Existence of

Relations

PAGE

V

1

4

3 9

6 9

9 1 ~

9 4

9 5

1 0 2

1 1 4

1 2 0

1 2 8

1 3 2

1 3 2

1 4 3

1 5 7

1 6 8

1 7 6

1 8 1

1 9 6

1 9 6

2 1 1

2 1 7

2 2 6

2 2 9 I

\2 4 1


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xu CONTENTS

SECTION D. LOGIC OF RELATIONS .PAGE

244

~. Descriptive Functions 245*31. Converses of Relations 251*32. Referents and Relata of a given Term with respect to a given

Relation 255*33. Domains, Converse Domains, and Fields of Relations . 260*34. The Relative Product of two Relations 269*35. Relations with Limited Domains and Converse Domains 278*36. Relations with Limited Fields 291*37. Plural Descriptive Functions . 293*38. Relations and Classes derived from a Double Descriptive Function 311

Note to Section D . 314

SECTION E. PRODUCTS AND SUMS OF CLASSES 317

~O. Products and Sums of Classes of Classes*41. The Product and Sum of a Class of Relations~2. Miscellaneous Propositions*43. The Relations of a Relative Product to its Factors

319331

336340

PART II. PROLEGOMENA TO CARDINAL ARITHMETIC.

Summary of Part II 345

SECTION A. U NIT CLASSES AND COUPLES

*50. Identity and Diversity as Relations*51. Unit Classes*52. The Cardinal Numbel' 1*53. Miscellaneous Propositions involving U nit Classes*54. Cardinal Couples*55. Ordinal Couples*56. The Ordinal Number s..

347

349356363368376383395

SECTION B. SUB-CLASSES, SUB-RELATIONS, AND RELATIVE TYPER

*60. The Sub-Classes of a given Class.~1. The Sub-Relations of a given Relation .*62. The Relation of Membership of a Class

*63. Relative Types of Classes~4. Relative Types of Relations .*65. On the Typical Definition of Ambiguous Symbols

404

406412414

419429434

SECTION C. ONE-MANY, MANy-ONE, AND ONE-ONE RELATIONS 437

*70. Relations whose Classes of Referents and of Relata belong to givenClasses. 439

*71. One-Many, Many-One, and One-One Relations 446*72. Miscellaneous Propositions concerning One-Many,Many-Oue, and

One-One Relations 462*73. Similarity of Classes 476*74. On One-Many and Many-One Relations with Limited Fields 490


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CONTENTS Xlll

SECTION D. SELECTIONS.

*80. Elementary Properties of Selections

~l. Selections from Many-One Relations*82. Selections from Relative Products.*83. Selections from Classes of Classes.*84. Classes of Mutually Exclusive Classes*85. Miscellaneous Propositions~. Conditions for the Existence of Selections

PAGE

500

505

5 1 95245 3 1

5405495 6 1

SECTION E. INDUCTIVE RELATIONS

*90.*91.*92.*93.

*94.*95.*96.*97.

On the Ancestral RelationOn Powers of a RelationPowers of One-Many and Many-One RelationsInductive Analysis of the Field of a Relation

On Powers of Relative ProductsOn the Equi-factor Relation .On the Posterity of a Term .Analysis of the Field of a Relation into Families

5 6 9

5 7 6

5856 0 1

6 0 7

6 1 7

6 2 6

6 3 7

654


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ALPHABETICAL LIST OF PROPOSITIONSREFERRED TO BY NAMES.

Name Number

Abs *2'01. I - :p:> C-' p.:> . '" PAdd *1'3. I-:q.:>.pvqAss *3'35. I-:p.p:>q.:>.qAssoc *1'5. I- :p V (q V 1 ") :> q V (p V '1')

Comm *2'04. I - : .p . ") . q") '1'::> :q . :> p") rComp *3'43. I- :.p - : Jq p") r , ") :P ") q. '1'Exp *3'3. I- : .p q . ") r :- : J :p . ") . q") '1'Fact *3'45. I- : .p"J q . :> :P > r , " ) . q. rI d *2'08. I-.p)p

Imp *3'31. I - : .P "J q") r : ) :p . q ") . r

Perm *1'4. I-:pvq.").qvpSimp *2'02. I-:q.").p")q

" *3'26. I-:p.q.-:J.p

" *3'27. l-:p.q.J.qSum *1'6. I- :. q " J r , ") : p V q . J. P V rSyll *2'05. I- : . q "J r . "): p ") q ") .P J r

" *2'06.I - : .p"J q. J : q ")1". ") P J r

" *3"33.I - :p "Jq . q ")1" J P J '1'

" *3'34. l-:qJ'1',pJq.J.p")rTaut *1'2. I-:pvp.").p \ .Transp *2'03. I-:p:> "-' q."). q") ,....,p i

" *2'15. 1-:"'p")q.")."-'q"Jp

" *2'16. l-:pJq."). "fJjJ"'p

" *2'17. 1-:"'q"J"'p.").p")q

" *3'37.I - : .p . q . ").1": ") :p. rv '1'. "J. "-'q

"~1. l-:pJq.= . "'qJ "'P

II *4"11. r i p e q ,: ,


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ERRATA.

p. 14, line 2, jor "states" read .. allows us to infer."p. 14, line 7, after "*3'03" insert "*1'7, *1'71, and *1'72."p. 15, last line but one, jor "function of c p x " read" function cpx . "p. 34, line Hi, jor "x" read" R."n, 6 ~ .. li . : : , . : :2 1 ,for ""l!!'! '!;e"" : ",gu, ti classes of classes."p. 86, line 2, after "must" insert " neither be nor."p. 91, line 8, delete "and in *3'03:'p. 103, line 7, jor "assumption" read" assertion."p. 103, line 25, at end of line, jor "q" read "r."

p. 218, last line but one, jor "A" read" 1\" [owing to brittleness of the type, thesame error is liable to occur elsewhere].

p. 382, last line but one, delete "in the theory of selections (*83'92) and."p. 487, line 13, for "*95" read "*94."p. 503, line 14, jor "*88'38" read "*88'36."


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PREFACE

T HE mathematical treatment of the principles of mathematics, which isthe subject of the present work, has arisen from the conjunction of twodifferent studies, both in the main very modern. On the one hand we havethe work of analysts and geometers, in the way of formulating and systematisingtheir axioms, and the work of Cantor and others on such matters as the theoryof aggregates. On the other hand we have symbolic logic, which, after anecessary period of growth, has now, thanks to Peano and his followers,acquired the technical adaptability and the logical comprehensiveness that areessential to a mathematical instrument for dealing with what have hithertobeen the beginnings of mathematics. From the combination of these twostudies two results emerge, namely (1) that what were formerly taken, tacitlyor explicitly, as axioms, are either unnecessary or demonstrable; (2) that thesame methods by which supposed axioms are demonstrated will give valuable

results in regions, such as infinite number, which had formerly been regardedas inaccessible to human knowledge. Hence the scope of mathematics isenlarged both by the addition of new subjects and by a backward extensioninto provinces hitherto abandoned to philosophy.

The present work was originally intended by us to be comprised ina second volume of The Principles of Mathematics. With that object inview, the writing of it was begun in 1900. But as we advanced, it becameincreasingly evident that the subject is a very much larger one than we hadsupposed; moreover on many fundamental questions which had been leftobscure and doubtful in the former work, we have now arrived at what webelieve to be satisfactory solutions. It therefore became necessary to makeour book independent of The Principles of Mathematics. We have, however,avoided both controversy and general philosophy, and made our statementsdogmatic in form. The justification for this is that the chief reason in favourof any theory on the principles of mathematics must always be inductive,i.e. it must lie in the fact that the theory in question enables us to deduceordinary mathematics. In mathematics, the greatest degree of self-evidenceis usually not to be found quite at the beginning, but at some later point;hence the early deductions, until they reach this point, give reasons rather

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V I PREFACE

for believing the premisses because true consequences follow from them, thanfor believing the consequences because they follow from the premisses.

In constructing a deductive system such as that contained in the present

work, there are two opposite tasks which have to be concurrently performed.On the one hand, we have to analyse existing mathematics, with a viewto discovering what premisses are employed, whether these premisses aremutually consistent, and whether they are capable of reduction to morefundamental premisses. On the other hand, when we have decided uponour premisses, we have to build up again as much as may seem necessaryof the data previously analysed, and as many other consequences of ourpremisses as are of sufficient general interest to deserve statement. Thepreliminary labour of analysis does not appear in the final presentation,which merely sets forth the outcome of the analysis in certain undefined

ideas and undemonstrated propositions. It is not claimed that the analysiscould not have been carried farther: we have no reason to suppose that it isimpossible to find simpler ideas and axioms by means of which those withwhich we start could be defined and demonstrated. All that is affirmed isthat the ideas and axioms with which we start are sufficient, not that theyare necessary.

In making deductions from our premisses, we have considered it essentialto carry them up to the point where we have proved as much as is true inwhatever would ordinarily be taken for granted. But we have not thoughtit desirable to limit ourselves too strictly to this task. It is customary toconsider only particular cases, even when, with our apparatus, it is just aseasy to deal with the general case. For example, cardinal arithmetic isusually conceived in connection with finite numbers, but its general lawshold equally for infinite numbers, and are most easily proved without anymention of the distinction between finite and infinite. Again, many of theproperties commonly associated with series hold of arrangements which arenot strictly serial, but have only some of the distinguishing properties ofserial arrangements. In such cases, it is a defect in logical style to provefor a particular class of arrangements what might just as well have beenproved more generally. An analogous process of generalization is involved,to a greater or less degree, in all our work. We have sought always themost general reasonably simple hypothesis from which any given conclusioncould be reached. For this reason, especially in the later parts of the book,the importance of a proposition usually lies in its hypothesis. The conclusionwill often be something which, in a certain class of cases, is familiar, but thehypothesis will, whenever possible, be wide enough to admit many casesbesides those in which the conclusion is familiar.

We have found it necessary to give very full proofs, because otherwiseit is scarcely possible to see what hypotheses are really required, or whether


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PREFACE. .

Vll

our results follow from our explicit premisses. (It must be remembered thatwe are not affirming merely that such and such propositions are true, but alsothat the axioms stated by us are sufficient to prove them.) At the same time,though full proofs are necessary for the avoidance of errors, and for convincing

those who may feel doubtful as to our correctness, yet the proofs of propositionsmay usually be omitted by a reader who is not specially interested in thatpart of the subject concerned, and who feels no doubt of our substantialaccuracy on the matter in hand. The reader who is specially interested insome particular portion of the book will probably find it sufficient, as regardsearlier portions, to read the summaries of previous parts, sections, and numbers,since these give explanations of the ideas involved and statements of theprincipal propositions proved. The proofs in Part I, Section A, however,are necessary, since in the course of them the manner of stating proofs isexplained. The proofs of the earliest propositions are given without the

omission of any step, but as the work proceeds the proofs are graduallycompressed, retaining however sufficient detail to enable the reader by thehelp of the references to reconstruct proofs in which no step is omitted.

The order adopted is to some extent optional. For example, we havetreated cardinal arithmetic and relation-arithmetic before series, but wemight have treated series first. To a great extent, however, the order isdetermined by logical necessities.

A very large part of the labour involved in wntmg the present work1as been expended on the contradictions and paradoxes which have infected

logic and the theory of aggregates. We have examined a great number ofhypotheses for dealing with these contradictions; many such hypotheseshave been advanced by others, and about as many have been invented byourselves. Sometimes it has cost us several months' work to convinceourselves that a hypothesis was untenable. In the course of such aprolonged study, we have been led, as was to be expected, to modify ourviews from time to time; but it gradually became evident to us that someform of the doctrine of types must be adopted if the contradictions were tobe avoided. The particular form of the doctrine of types advocated in thepresent work is not logically indispensable, and there are various other forms

equally compatible with the truth of our deductions. We have particularized,both because the form of the doctrine which we advocate appears to us themost probable, and because it was necessary to give at least one perfectlydefinite theory which avoids the contradictions. But hardly anything in ourbook would be changed by the adoption of a different form of the doctrineof types. In fact, we may go farther, and say that, supposing some otherway of avoiding the contradictions to exist, not very much of our book,except what explicitly deals with types, is dependent upon the adoption ofthe doctrine of types in any form, so soon as it has been shown (as we claim


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Vlll PREFACE

that we have shown) that it is possible to construct a mathematical logicwhich does not lead to contradictions. It should be observed that the wholeeffect of the doctrine of types is negative: it forbids certain inferences whichwould otherwise be valid, but does not permit any which would otherwise beinvalid. Hence we may reasonably expect that the inferences which thedoctrine of types permits would remain valid even if the doctrine shouldbe found to be invalid.

Our logical system is wholly contained in the numbered propositions,which are independent of the Introduction and the Summaries. TheIntroduction and the Summaries are wholly explanatory, and form no partof the chain of deductions. The explanation of the hierarchy of types inthe Introduction differs slightly from that given in *12 of the body of thework. The later explanation is stricter and is that which is assumedthroughout the rest of the book.

The symbolic form of the work has been forced upon us by necessity:without its help we should have been unable to perform the requisitereasoning. It has been developed as the result of actual practice, and is notan excrescence introduced for the mere purpose of exposition. The generalmethod which guides our handling of logical symbols is due to Peano. Hisgreat merit consists not so much in his definite logical discoveries nor in thedetails of his notations (excellent as both are), as in the fact that he firstshowed how symbolic logic was to be freed from its undue obsession with theforms of ordinary algebra, and thereby made it a suitable instrument for

research. Guided by our study of his methods, we have used great freedomin constructing, or reconstructing, a symbolism which shall be adequate todeal with all parts of the subject. No symbol has been introduced excepton the ground of its practical utility for the immediate purposes of ourreasomng.

A certain number of forward references will be found in the notes andexplanations. Although we have taken every reasonable precaution to securethe accuracy of these forward references, we cannot of course guarantee theiraccuracy with the same confidence as is possible in the case of backwardreferences.

Detailed acknowledgments of obligations to prevIOUS writers have notvery often been possible, as we have had to transform whatever we haveborrowed, in order to adapt it to our system and our notation. Our chiefobligations will be obvious to every reader who is familiar with the literatureof the su bject. In the matter of notation, we have as far as possible followedPeano, supplementing his notation, when necessary, by that of Frege or bythat of Schroder. A great deal of the symbolism, however, has had to be

new, not so much through dissatisfaction with the symbolism of others, asthrough the fact that we deal with ideas not previously symbolised. In all

principia mathematica - whitehead & bertrand russell. pag 1-20

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