Russell & Whitehead's Principia Mathematica

by Bertrand Russell, Alfred North Whitehead

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.	$\scriptstyle {\vdash :p\supset \sim p.\supset .\sim p}$
Add	✱1·3.	$\scriptstyle {\vdash :q.\supset .p\lor q}$
Ass	✱3·35.	$\scriptstyle {\vdash :p.p\supset q.\supset .q}$
Assoc	✱1·5.	$\scriptstyle {\vdash :p\lor (q\lor r).\supset .q\lor (p\lor r)}$
Comm	✱2·04.	$\scriptstyle {\vdash :.p.\supset .q\supset r:\supset :q.\supset .p\supset r}$
Comp	✱3·43.	$\scriptstyle {\vdash :.p\supset q.p\supset r.\supset :p.\supset .q.r}$
Exp	✱3·3.	$\scriptstyle {\vdash :.p.q.\supset .r:\supset :p.\supset .q\supset r}$
Fact	✱3·45.	$\scriptstyle {\vdash :.p\supset q.\supset :p.r.\supset .q.r}$
Id	✱2·08.	$\scriptstyle {\vdash .p\supset p}$
Imp	✱3·31.	$\scriptstyle {\vdash :.p.\supset .q\supset r:\supset :p.q.\supset .r}$
Perm	✱1·4.	$\scriptstyle {\vdash :p\lor q.\supset .q\lor p}$
Simp	✱2·02.	$\scriptstyle {\vdash :q.\supset .p\supset q}$
Simp„	✱3·26.	$\scriptstyle {\vdash :p.q.\supset .p}$
Simp„	✱3·27.	$\scriptstyle {\vdash :p.q.\supset .q}$
Sum	✱1·6.	$\scriptstyle {\vdash :.q\supset r.\supset :p\lor q.\supset .p\lor r}$
Syll	✱2·05.	$\scriptstyle {\vdash :.q\supset r.\supset :p\supset q.\supset .p\supset r}$
Syll„	✱2·06.	$\scriptstyle {\vdash :.p\supset q.\supset :q\supset r.\supset .p\supset r}$
Syll„	✱3·33.	$\scriptstyle {\vdash :p\supset q.q\supset r.\supset .p\supset r}$
Syll„	✱3·34.	$\scriptstyle {\vdash :q\supset r.p\supset q.\supset .p\supset r}$
Taut	✱1·2.	$\scriptstyle {\vdash :p\lor p.\supset .p}$
Transp	✱2·03.	$\scriptstyle {\vdash :p\supset \sim q.\supset .q\supset \sim p}$
Transp„	✱2·15.	$\scriptstyle {\vdash :\sim p\supset q.\supset .\sim q\supset p}$
Transp„	✱2·16.	$\scriptstyle {\vdash :p\supset q.\supset .\sim .q\supset \sim p}$
Transp„	✱2·17.	$\scriptstyle {\vdash :\sim q\supset \sim p.\supset .p\supset q}$
Transp„	✱3·37.	$\scriptstyle {\vdash :.p.q.\supset .r:\supset :p.\sim r.\supset .\sim q}$
Transp„	✱4·1.	$\scriptstyle {\vdash :p\supset q.\equiv .\sim q\supset \sim p}$
Transp„	✱4·11.	$\scriptstyle {\vdash :p\equiv q.\equiv .\sim p\equiv \sim q}$

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 $\scriptstyle {\phi {\hat {x}}}$" read "function $\scriptstyle {\phi {\hat {x}}}$."

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 "$\scriptstyle {\dot {\text{A}}}$" [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."