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BERTRAND RUSSELL
Toward "Principia
Mathematica"
1905-08
Edited by
Gregory H. Moore
London and New York
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Contents
Illustrations x
Abbreviations xii
Introduction xiii
Acknowledgements xciii
Chronology xcv
PART I. NO-CLASSES THEORIES AND
SUBSTITUTIONAL THEORIES, 1905-06
General Headnote to Part I 3
1 The Theory of Implication [1905-06] 14
2 On Some Difficulties in the Theory of Transfinite
Numbers and Order Types [1905-06] 62
3 EarlyWork on the SubstitutionalTheory [1905] 90
a A Letter to Hardy on Substitution 93
b On Substitution 97
4 Developing the Substitutional Theory [1906-07] 104
a Substitution 106
b A Theory of Determination 113
c *20ff. 119
d Verbal Definitions 124
e A Paradox of the Substitutional
Theory 1255 Two Drafts on Substitution [1906] 126
a On Substitution 129
b A Partial Draft 233
6 A PaperWithdrawn from Publication [1906] 236
a On the Substitutional Theory of Classes
and Relations: Abstract 241
b On the Substitutional Theory of Classes
and Relations 243
7 Logic in Which Propositions Are Not Entities [1906] 2628 On the Functional Theory of Propositions, Classes
and Relations [1906] 268
9 The Paradoxes of Logic [1906] 273
10 Multiplicative Axiom [1906] 297
vii
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viii CONTENTS
11 The Paradox ofthe Liar [1906] 3J7
a The Paradox of the Liar 320
b A Partial Draft 3^9
12 List ofPropositions [1906] 375
PART II. THEORIES OF TRUTH, 1906-08
General Headnote to Part II 4X7
13 Two Reviews of Joachim [1906] 419
a What IsTruth? 423
b Review of Joachim, The Nature of Truth 427
14 On the Nature of Truth [1907] 433
15 The Nature of Truth [1907] 455
16 WilliamJames's Conception of Truth [1908] 465
PART III. FROM SUBSTITUTIONAL THEORIES TO THE
RAMIFIED THEORY OF TYPES, 1906-08
General Headnote to Part III 489
17 Corrections Required in Present Work [1906] 492
18 Early Drafts on theTheory of Types [1906-08] 495
a Types 498
b On Types 515
c Notes on Types 520
d FourthTheory 525e Individuals 529
19 Fundamentals [1907] 53620 The Regressive Method of Discovering the Premises
of Mathematics [1907] 57121 "If" and "Imply", A Reply to Mr. MacColl [1908] 58122 Mathematical Logic as Based on the Theory of
Types [1908] 58523 Partial Drafts of Principia Mathematica [c.1908] 626
a * 10.Theory of One Apparent Variable 629b Deduction of Theory of Propositions of HigherType
from That of Those ofLower Type 632c *27. The Hierarchy of Types 635d *92. The Schroder-Bernstein Theorem 641e *130. Selections from Relations 645
PART IV. REVIEWS ON FOUNDATIONS OF MATHEMATICS
24 M. Poincare's Science et hypothese [1906] 65925 Two Reviews of MacColl [1906] 66X
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26
27
28
29
30
31
32
33
34
35
36
I
CONTENTS ix
a Review of MacColl, Symbolic Logic and
ItsApplications, in the Athenaeum 666
b Review of MacColl, Symbolic Logic and
ItsApplications, in Mind 670
Review of Pastore, Logicaformale dedotta dalla
considerazione di modelli meccanici [1906] 676
The Study of Logic [1906] 678
Two Reviews of Meinong [1906-07] 680
a Review of Meinong, Uber die Erfahrungsgrundlagen
unseres Wissens [1906] 685
b Review of Meinong, Uber die Stellung der Gegenstands-
theorie im System der Wissenschaften [1907] 689
Mr.Haldane on Infinity [1908] 693
PART V. OTHER PHILOSOPHICAL REVIEWS AND WRITINGS
Is Reason Irrational? [1906]
Metaphysics for the Man of Action [1907]
Spinoza's Moral Code [1907]
Newton's Philosophy [1908]
Determinism and Morals [1908]
Review ofEssays, Philosophical and
Psychological, in Honorof William James [1908]
A Reply to Dr. Schiller [1908]
APPENDICES
Les Paradoxes de la logique [1906] 743
Comments on Definitions ofPhilosophical Terms [1905-06] 762
Notes on Ward's Comments on ThePrinciples of
Mathematics
[1905] 765
Berry's Letters to Russell [1904-10] 770
MISSING AND UNPRINTED TEXTS 783
ANNOTATION 785
TEXTUAL NOTES 839
BIBLIOGRAPHICAL INDEX 883
SYMBOLS INDEX 907
GENERAL INDEX 923
701
705
713
716
719
729
735
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Illustrations
frontispiece
Bertrand Russell: Campaign photograph for the Wimbledon By-
Election, 1907.
between pages c and 1
Verso of folio 31 of Paper 19 (ra 230.030760). The notation $ | x used
in this verso shows that the verso dates from 1903 (see Russell 1994,
49j 53)3 almost four years before Paper 19 was written. The verso is
unusual indating so longbefore the text on the recto (January 1907).
Verso of folio
35 ofPaper 19. The notations
(f>'x and ~
'fi'x used in
this verso indicate that it dates from between February 1904 and July
1905; see pp. lxiv, 17. The marginal note to replace the notation
{(f>lxYy with ) shows the verso to have been commented on later
than July 1905. The last two formulas onthe verso, unlike the rest, are
in Whitehead's handwriting.
Part of Paper 4b (verso 16 of Paper 10, ra 230.030750), showing
some of the notation for the substitutional theory, in particular the
notation (f>Q ... ?).
Folio 6, from Paper 10, showing the quantifiedvariable directly under
the implication sign, rather than after it.
Folio 26, from Paper 10,illustrating four quantified variables directlyunder one implication sign on two occasions.
Third leaf of Paper 12, showing his use of the notations (x,y) respectively. The notations
(?) and 0(??) are found in Russell's letter of 22 November 1905 to
Couturat and in Whitehead's letter of 21 February 1906 to Russell.
x
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ILLUSTRATIONS xi
vii Verso of folio 3 of Paper 17 (ra 230.030740). It deals with an attempt
to rule out the existence of what are now called Dedekind cardinals
(i.e. the cardinal of a set which is, in Russell's terminology, not induc
tive and not reflexive; that is, in modern terminology, a set that is
infinite but Dedekind-finite). See p. 492. Russell's use
of Cantor'snotation K0 shows that this folio dates from 1906; see A772:12.
viii Verso (ra 230.031240-Fi) about Likeness and *151 of Principia. By
the time of Paper 22 (p. 622), "likeness" has been replaced by
"Smor" or "similar ordinally", and so this verso dates from before
that, but not long before. This verso also has 1>Cls, which is a rela
tively late notation found first at 521: 7.
ix First page of letter of 11 January 1908 from Russell to Whitehead on
types (ra 710.057503). See pp. lxxix-lxxx. Very few letters from
Russell to Whitehead survive since Whitehead had all letters to him
destroyed on his death.
x Verso of folio 39 of Paper 19. This was presumably part of a draft by
Whitehead of a section ofPrincipia. It is one of a very few extant parts
of such a draft in Whitehead's handwriting.