PRINCIPIA MATHEMATICA
CAMBRIDGE UNIVERSITY PRESS
London: FETTER LANE, E. C.
C. F. CLAY, Manager
Edinburgh: 100, PRINCES STREET
Berlin: A. ASHER AND CO.
Leipzig: F. A. BROCKHAUS
New York: G. P. PUTNAM'S SONS
Bombay and Calcutta: MACMILLAN AND CO., Ltd.
All rights reserved
PRINCIPIA MATHEMATICA
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
Cambridge
at the University Press
1910
Cambridge:
PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS
CONTENTS OF VOLUME I
PAGE
PREFACE....................................................................................................................................................................................................................................................v
INTRODUCTION....................................................................................................................................................................................................................................................1
Chapter I. Preliminary Explanations of Ideas and Notations.................................................................................................................................................................................................................................................... 4
Chapter II. The Theory of Logical Types.................................................................................................................................................................................................................................................... 39
Chapter III. Incomplete Symbols.................................................................................................................................................................................................................................................... 69
PART I. MATHEMATICAL LOGIC.
Summary of Part I.................................................................................................................................................................................................................................................... 91
Section A. The Theory of Deduction.................................................................................................................................................................................................................................................... 94
*1. Primitive Ideas and Propositions.................................................................................................................................................................................................................................................... 95
*2. Immediate Consequences of the Primitive Propositions.................................................................................................................................................................................................................................................... 102
*3. The Logical Product of two Propositions.................................................................................................................................................................................................................................................... 114
*4. Equivalence and Formal Rules.................................................................................................................................................................................................................................................... 120
*5. Miscellaneous Propositions.................................................................................................................................................................................................................................................... 128
Section B. Theory of Apparent Variables.................................................................................................................................................................................................................................................... 132
*9. Extension of the Theory of Deduction from Lower to Higher Types of Propositions.................................................................................................................................................................................................................................................... 132
*10. Theory of Propositions containing one Apparent Variable.................................................................................................................................................................................................................................................... 143
*11. Theory of two Apparent Variables.................................................................................................................................................................................................................................................... 157
*12. The Hierarchy of Types and the Axiom of Reducibility.................................................................................................................................................................................................................................................... 168
*13. Identity.................................................................................................................................................................................................................................................... 176
*14. Descriptions.................................................................................................................................................................................................................................................... 181
Section C. Classes and Relations.................................................................................................................................................................................................................................................... 196
*20. General Theory of Classes.................................................................................................................................................................................................................................................... 196
*21. General Theory of Relations.................................................................................................................................................................................................................................................... 211
*22. Calculus of Classes.................................................................................................................................................................................................................................................... 217
*23. Calculus of Relations.................................................................................................................................................................................................................................................... 226
*24. The Universal Class, the Null-Class, and the Existence of Classes.................................................................................................................................................................................................................................................... 229
*25. The Universal Relation, the Null Relation, and the Existence of Relations.................................................................................................................................................................................................................................................... 241
Section D. Logic of Relations.................................................................................................................................................................................................................................................... 244
*30. 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
*40. Products and Sums of Classes.................................................................................................................................................................................................................................................... 319
*41. The Product and Sum of a Class of Relations.................................................................................................................................................................................................................................................... 331
*42. Miscellaneous Propositions.................................................................................................................................................................................................................................................... 336
*43. The Relations of a Relative Product to its Factors.................................................................................................................................................................................................................................................... 319
PART II. PROLEGOMENA TO CARDINAL ARITHMETIC.
Summary of Part II.................................................................................................................................................................................................................................................... 345
Section A. Unit Classes and Couples.................................................................................................................................................................................................................................................... 347
*50. Identity and Diversity as Relations.................................................................................................................................................................................................................................................... 349
*51. Unit Classes.................................................................................................................................................................................................................................................... 356
*52. The Cardinal Number 1.................................................................................................................................................................................................................................................... 363
*53. Miscellaneous Propositions involving Unit Classes.................................................................................................................................................................................................................................................... 368
*54. Cardinal Couples.................................................................................................................................................................................................................................................... 376
*55. Ordinal Couples.................................................................................................................................................................................................................................................... 383
*56. The Ordinal Number 2.................................................................................................................................................................................................................................................... 395
Section B. Sub-Classes, Sub-Relations, and Relative Types.................................................................................................................................................................................................................................................... 404
*60. The Sub-Classes of a given Class.................................................................................................................................................................................................................................................... 406
*61. The Sub-Relations of a given Realtion.................................................................................................................................................................................................................................................... 412
*62. The Relation of Membership of a Class.................................................................................................................................................................................................................................................... 414
*63. Relative Types of Classes.................................................................................................................................................................................................................................................... 419
*64. Relative Types of Relations.................................................................................................................................................................................................................................................... 429
*65. On the Typical Definition of Ambiguous Symbols.................................................................................................................................................................................................................................................... 434
Section C. One-Many, Many-One, and One-One Relations.................................................................................................................................................................................................................................................... 437
*70. Relations whose Classes of Referents and of Relata belong to given Classes.................................................................................................................................................................................................................................................... 439
*71. One-Many, Many-One, and One-One Relations.................................................................................................................................................................................................................................................... 446
*72. Miscellaneous Propositions concerning One-Many, Many-One, and One-One Relations.................................................................................................................................................................................................................................................... 462
*73. Similarity of Classes.................................................................................................................................................................................................................................................... 476
*74. On One-Many and Many-One Relations with Limited Fields.................................................................................................................................................................................................................................................... 490
Section D. Selections.................................................................................................................................................................................................................................................... 500
*80. Elementary Properties of Selections.................................................................................................................................................................................................................................................... 505
*81. Selections from Many-One Relations.................................................................................................................................................................................................................................................... 519
*82. Selections from Relative Products.................................................................................................................................................................................................................................................... 524
*83. Selections from Classes of Classes.................................................................................................................................................................................................................................................... 531
*84. Classes of Mutually Exclusive Classes.................................................................................................................................................................................................................................................... 540
*85. Miscellaneous Propositions.................................................................................................................................................................................................................................................... 549
*88. Conditions for the Existence of Selections.................................................................................................................................................................................................................................................... 561
Section E. Inductive Relations.................................................................................................................................................................................................................................................... 569
*90. On the Ancestral Relation.................................................................................................................................................................................................................................................... 576
*91. On Powers of a Relation.................................................................................................................................................................................................................................................... 585
*92. Powers of One-Many and Many-One Relations.................................................................................................................................................................................................................................................... 601
*93. Inductive Analysis of the Field of a Relation.................................................................................................................................................................................................................................................... 607
*94. On Powers of Relative Products.................................................................................................................................................................................................................................................... 617
*95. On the Equi-factor Relation.................................................................................................................................................................................................................................................... 626
*96. On the Posterity of a Term.................................................................................................................................................................................................................................................... 637
*97. Analysis of the Field of a Relation into Families.................................................................................................................................................................................................................................................... 654
ALPHABETICAL LIST OF PROPOSITIONS REFERRED TO BY NAMES.
| Name | | Number | | ||
| Abs | *2·01. | 
| |||
| Add | *1·3. | 
| |||
| Ass | *3·35. | 
| |||
| Assoc | *1·5. | 
| |||
| Comm | *2·04. | :.p.\supset .q\supset r:\supset :q.\supset .p\supset r}}
| |||
| Comp | *3·43. | :.p\supset q.p\supset r.\supset :p.\supset .q.r}}
| |||
| Exp | *3·3. | :.p.q.\supset .r:\supset :p.\supset .q\supset r}}
| |||
| Fact | *3·45. | :.p\supset q.\supset :p.r.\supset .q.r}}
| |||
| Id | *2·08. | 
| |||
| Imp | *3·31. | :.p.\supset .q\supset r:\supset :p.q.\supset .r}}
| |||
| Perm | *1·4. | 
| |||
| Simp | *2·02. | 
| |||
| " | *3·26. | 
| |||
| " | *3·27. | 
| |||
| Sum | *1·6. | :.q\supset r.\supset :p\lor q.\supset .p\lor r}}
| |||
| Syll | *2·05. | :.q\supset r.\supset :p\supset q.\supset .p\supset r}}
| |||
| " | *2·06. | :.p\supset q.\supset :q\supset r.\supset .p\supset r}}
| |||
| " | *3·33. | 
| |||
| " | *3·34. | 
| |||
| Taut | *1·2. | 
| |||
| Transp | *2·03. | 
| |||
| " | *2·15. | :\sim p\supset q.\supset .\sim q\supset p}}
| |||
| " | *2·16. | 
| |||
| " | *2·17. | :\sim q\supset \sim p.\supset .p\supset q}}
| |||
| " | *3·37. | :.p.q.\supset .r:\supset :p.\sim r.\supset .\sim q}}
| |||
| " | *4·1. | 
| |||
| " | *4·11. | 
| |||
ERRATA.
p. 14, line 2, for "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, for "function of 
" read "function ."
p. 34, line 15, for "x" read "R."
p. 68, line 20, for "classes" read "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, for "assumption" read "assertion."
p. 103, line 25, at end of line, for "q" read "r."
p. 218, last line but one, for "A" read "
" [owing to brittleness of the type, the same 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, for "*88·38" read "*88·36."