Collected Papers, vol. 4: Foundations of Logic 1903-05

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THE COLLECTED PAPERS OF BERTRAND RUSSELL

The McMaster University Edition

Volume I Cambridge Essays, r888-99 Volume 2 Philosophical Papers, r896-99 Volume 3 Toward the "Principles of Mathematics", r900-02 Volume 4 Foundations of Logic, r903-05 Volume 6 Logical and Philosophical Papers, r909-I3 Volume 7 Theory of Knowledge: The I9I3 Manuscript Volume 8 The Philosophy of Logical Atomism and Other Essays, r9r4-r9 Volume 9 Essays on Language, Mind and Matter, r9r9-26 Volume 12 Contemplation and Action, r902-r4 Volume 13 Prophecy and Dissent, r9r4-r6

,

BERTRAND RUSSELL

Foundations of Logic

1903-05

Edited by Alasdair Urquhart with the assistance of Albert C. Lewis

Bertrand Russell with his friend G. Lowes Dickinson (left) at Bagley Wood, probably 1905.

I~ London and New York

First published 1994 by Routledge 11 New Fetter Lane, London EC4P 4EE

THE COLLECTED PAPERS OF BERTRAND RUSSELL

Simultaneously published in the USA and Canada by Routledge a division of Routledge, Chapman and Hall Inc. 29 West 35th Street, New York, NY rnoo1 GENERAL EDITOR All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Bertrand Russell's unpublished letters, Papers 1-15, 21, 22, and Appendix 1 •j McMaster University 1993. Papers 16, 17-20, 24-34 and Appendices 11 and III The Bertrand Russell Estate 1903, 1904, 1905. Editorial matter and the translation printed as Paper 23 11 Alasdair Urquhart. Funds to edit this volume were provided by a major editorial grant from the Social Sciences and Humanities Research Council of Canada and by McMaster University.

British Library Cataloguing in Publication Data Russell, Bertrand Foundations of Logic, 1903-05. (Collected Papers of Bertrand Russell; V,

4)

I. Title II. Urquhart, Alasdair III. Lewis, Albert C. IV. Series 192 ISBN 0---415-09406-2 Library of Congress Cataloging in Publication Data Russell, Bertrand, 1872-1970. Foundations of logic: 1903-05 /Bertrand Russell; edited by Alasdair Urquhart with the assistance of Albert C. Lewis. p. cm. - (The collected papers of Bertrand Russell; v. 4) Includes bibliographical references and indexes. I. Logic, Symbolic and mathematical. 2. Mathematics-Philosophy. I. Urquhart, Alasdair. II. Lewis, Albert C. III. Title. IV. Series: Russell, Bertrand, 1872-1970. Selections. 1983; v. 4. B1649.R91 1983 Vol. 4 [BC135) 192 s-dc20 93-5603 [160) ISBN 0---415-09406-2

Typeset in l0/12 pt. Plantin by The Bertrand Russell Editorial Project, McMaster University and printed and bound in Great Britain at the University Press, Cambridge

John Passmore (Australian National University)

ADVISORY EDITORIAL BOARD

I. Grattan-Guinness (Middlesex University) Jock Gunn (Queen's University) Francess G. Halpenny (University of Toronto) Royden Harrison (University of Warwick) Leonard Linsky (University of Chicago) H. C. G. Matthew (St. Hugh's College, Oxford) D. F. Pears (Christ Church, Oxford) John M. Robson (University of Toronto) Alan Ryan (Princeton University) Katharine Tait

Contents

Abbreviations Introduction Acknowledgements Chronology

Xll

xiii xiv xlvii

PART I. EARLY FOUNDATIONAL WORK

1 Classes [ r 903] a Draft of *12 to *16 b *12·5 etc. c General Theory of Classes 2 Relations [1903] 3 Functions [1903] a Functions and Objects b Primitive Propositions for Functions c No Greatest Cardinal d Functional Complexes e Complexes and Functions

3 5 31 35

38 49 50 53 56 65

69

PART II. THE ZIG-ZAG THEORY

4 5 6 7 8 9 10

Outlines of Symbolic Logic [1904] On Functions, Classes and Relations [1904] On Functions [1904] Fundamental Notions [1904] On the Functionality of Denoting Complexes [1904] On the Nature of Functions [1904] On Classes and Relations [1905] PART III. THE THEORY OF DENOTING

11 12 13

On the Meaning and Denotation of Phrases [1903] Dependent Variables and Denotation [1903] Points about Denoting [1903] V11

77 85

96 Ill

260 264 273

CONTENTS

Vlll

14 15 16

On Meaning and Denotation [1903] On Fundamentals [1905] On Denoting [1905]

CONTENTS

314

ANNOTATION

359 414

TEXTUAL NOTES

673

BIBLIOGRAPHICAL INDEX

713

GENERAL INDEX

727

PART IV. PHILOSOPHY OF LOGIC AND MATHEMATICS

17 18 19 20 21 22 23

Meinong's Theory of Complexes and Assumptions [1904] The Axiom of Infinity [1904] Headnote to Replies to Hugh MacColl (19-20) Non-Euclidean Geometry [1904] The Existential Import of Propositions [1905) The Nature of Truth [1905] Necessity and Possibility [1905] On the Relation of Mathematics to Symbolic Logic [1905]

431 475 479 482 486 490 507 521

PART V. PHILOSOPHICAL REVIEWS

24 25 26

27 28 29 30 31 32 33 34

Headnote to Two Reviews of Work on Leibniz (24-25) Recent Work on the Philosophy of Leibniz [1903] Review of Couturat, Opuscules et fragments inedits de Leibniz [I 904] Review of Geissler, Die Grundsiitze und das Wesen des Unendlichen in der Mathematik und Philosophie [1903] Headnote to Two Reviews of Moore (27-28) Principia Ethica [1903] The Meaning of Good [1904] Review of Delaporte, Essai philosophique sur !es geometries non-euclidiennes [ l 904] Review of Hinton, The Fourth Dimension [1904] Review of Petronievics, Principien der Metaphysik [1905] Headnote to Two Reviews of Poincare (32-33) Science and Hypothesis [1905) Review of Poincare, Science and Hypothesis [1905] Review of Meinong and Others, Untersuchungen zur Gegenstandstheorie und Psychologie [1905]

535 537 562 564 567 568 571 576 578 581 585 586 589 595

APPENDICES

Frege on the Contradiction Comments on Definitions of Philosophical Terms III Sur la relation des mathematiques a la logistique II

MISSING AND UNPRINTED PAPERS

lX

607 620 622

Illustrations

frontispiece Bertrand Russell with his friend G. Lowes Dickinson (left) at Bagley Wood, probably 1905.

between pages 44 and 45 The three-dimensional projective geometry of fifteen points as constructed by Russell. See A584: u-13. II-III

The house that Russell had built in Bagley Wood, where he and his wife lived from 1905 to 1910. The photographs were published in 1924 in Country Life.

IV

Page 15 of Paper 2, showing Russell's method of changing Peanostyle notation to Fregean. See 44: 3-24.

v

Page

VI

Page 707 of Paper 7. Seep. 206 and A206: 15.

VII

Page 772 of Paper 7. See pp. 236-7.

VIII

The first page of Paper 15. See 360: 1-25.

l

of Paper 3a. See 50: 2-18.

Plates I and IV-VIII are photographs of documents in the Bertrand Russell Archives at McMaster University. Plate I is reduced to seventy percent of the original size. Plates IV-VIII are reduced from their original sizes, which are given at the head of each set of textual notes.

xi

Abbreviations

To GIVE THE reader an uncluttered text, abbreviations and symbols have been kept to a minimum. The few necessary to the referencing system are as follows. The papers printed in the volume are given a boldface number for easy reference. For example, "On Denoting" is Paper 16. Angle brackets in the text distinguish editorial insertions from Russell's use of square brackets. A dagger Ct) indicates that the text carries Russell's own commentary which will be found in the Annotation. (See Introduction xliii.) Bibliographical references are usually in the form of author, date and page, e.g. "Russell 1967, 20". Consultation of the Bibliographical Index shows that this reference is to The Autobiography of Bertrand Russell, Vol. l: 1872-1914 (London: George Allen and Unwin, 1967), p. 20. The location of archival documents cited in the edition is the Bertrand Russell Archives at McMaster University ("RA"), unless a different location is given. File numbers of documents in the Russell Archives are provided only when manuscripts of papers printed here are cited or when files are difficult to identify. Cross-references to annotations are preceded by "A" and followed by page and line numbers (as in "A5: 27"). Cross-references to textual notes are preceded by "T". Further abbreviations are used in the Textual Notes, but they are identified at the beginning of each set of notes.

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Introduction

I. LOGIC AND THE PARADOXES

Russell described his first encounter with mathematical logic as a honeymoon period of intellectual delight. After meeting Giuseppe Peano at the International Congress of Philosophy in Paris (July 1900), Russell studied and mastered the mathematical logic of Peano and his school, and wrote his earliest papers on logic. Of this period, he wrote: The time was one of intellectual intoxication. My sensations resembled those one has after climbing a mountain in a mist, when, on reaching the summit, the mist suddenly clears, and the country becomes visible for forty miles in every direction. For years I had been endeavouring to analyse the fundamental notions of mathematics, such as order and cardinal numbers. Suddenly, in the space of a few weeks, I discovered what appeared to be definitive answers to the problems which had baffled me for years .... Intellectually, the month of September 1900 was the highest point of my life. (Russell 1967, 145) It was not long before the mist descended again. Peano's work had inspired Russell to attempt the derivation of pure mathematics on a purely logical basis, a programme later known as "logicism". Russell enlisted the help of the mathematician Alfred North Whitehead in this large enterprise. All mathematical concepts were to be defined in purely logical terms, and all mathematical theorems derived from purely logical axioms. In 1901, Russell began writing the logical deduction of mathematics from logic.' This led him to examine Cantor's proof that there is no greatest cardmal number. This result conflicted with Russell's assumption that there was a universal class, having all objects as members, which ought to have the greatest cardinal number. Close analysis of the diagonal argument used in Cantor's proof (Cantor 1892) led to the discovery of the paradox of the class of all classes that are not members of themselves. On asking whether this class is or is not a member of itself Russell found that either answer implies its contradictory (Russell 1~67, 147). The xiii

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INTRODUCTION

emergence of the paradox, which Russell called "the Contradiction", can be studied in Volume 3 of the Collected Papers. Six years were to pass before Russell and Whitehead found a workable solution to the difficulties posed by the Contradiction and related paradoxes. Most of the logical papers in this volume turn around the central problem of the paradoxes, as one approach after another is adopted, then abandoned, in attempts to evade them. It is not until 1905, with the discovery of the Theory of Descriptions, that a glimmering of hope emerges from the wreckage of earlier theories. Russell's marriage broke down in 1902. Although in that year he had told his first wife Alys that he no longer loved her, they continued to live together, in spite of an almost complete estrangement. It is not surprising that Russell describes this period of his life as one in which he was assailed by "intellectual and emotional problems which plunged me into the darkest despair that I have ever known" (1967, 145). The logical paradoxes emerged at an awkward moment, when Russell had already written most of the penultimate draft of The Principles of Mathematics. Rather than hold up its publication indefinitely, he took the manuscript of his book to the printer in May 1902 before finding a solution (Blackwell 1984-85, 280-1). His initial reaction was that the Contradiction could be avoided by some simple modification of the primitive propositions of logic. It appears from Russell's correspondence that he embarked on the publication of the book in the hope that he could dispose of the Contradiction in an appendix after the printing of the main text. On 8 August 1902, in a letter to Gottlob Frege (Frege 1976, 226-7), he proposed a theory of types, similar to that elaborated in Russell 1903, 5238, as a solution. However, this attempt ended in failure. The theory of types of 1902 assigns types to classes, relations and functions, but not to propositions; as a result the theory contains a paradox generated by applying Cantor's diagonal construction to classes of propositions. Russell described this new paradox in a letter to Frege of 29 September 1902 (Frege 1976, 230-1), and again in Appendix B of the Principles. His disappointment in the failure of this attempt at patching his logical system emerges in a letter to the philosopher Louis Couturat written on the same day: "When they began printing my book, I thought I could avoid these contradictions, but I see now that I was mistaken, so that my book has lost a great deal of its value." Russell gives no solution to the paradox in the Principles, but exhorts students of logic to study it. The paradoxes threw in doubt the joint work in logic on which Whitehead and Russell had been engaged since late 1900. The collaboration between the two men began when they discovered that Russell's projected Volume II of the Principles and Whitehead's projected Volume II of his Universal Algebra had much in common. Russell enlisted Whitehead as a

collaborator in writing his own Volume n; Whitehead temporarily put aside work on his book (which he never completed). The scale of the project grew as they worked, until it eventually became a separate book. The earliest reference to a change in title is in a letter from Russell to Couturat dated 21 August 1906: "Sad to say, it will be a long time before our work on the second volume is finished. We are thinking of making it into an independent book, which we shall call Principia Mathematica." During the period of Volume 4, then, the collaborators still thought of the book as the second volume of the Principles. It was not written (as many of Russell's other books were) by a simple process of accumulation of material, followed by a more or less straightforward writing of a draft of the final work. Rather, Whitehead and Russell produced the book by continually rewriting earlier drafts. They based the first drafts on their early published papers on symbolic logic and cardinal arithmetic, and the lecture notes of their courses on symbolic logic. Russell's lecture notes are discussed below; the mathematician G. H. Hardy, in a paper on infinite cardinal numbers (1904, 94) mentions Whitehead's lectures during the winter of 1902-03 on "the application of symbolic logic to the theory of aggregates", but there seems to be no other remaining evidence about this course. The parts of their book that gave the most trouble were the foundational portions, which were largely Russell's responsibility. The difficulty lay in attempting to preserve as much as possible of the derivation of mathematics from the original (inconsistent) foundations, while basing it on reworked primitive propositions that should not lead to contradictions. Unfortunately, these two goals were often in conflict, forcing modification of later developments in the light of the changing primitive propositions. Whitehead and Russell seem to have done a good deal of preliminary work on Volume II during the latter half of 1902. Russell wrote to his friend Gilbert Murray from the Whiteheads on 16 September 1902:

XlV

I am very deep in proofs and Symbolic Logic: I created, with all an artist's passion for the perfect, a new treatment of Symbolic Logic, and to my joy Whitehead finds that it has all the beauty and perfection that I hoped. None of these early manuscripts on logic seems to have survived, except for some undated pieces by Whitehead that are discussed below. Systematic work on a draft of Volume II seems to have been begun early in 1903, and is recorded in letters from Russell to Alys written during his visits to the Whiteheads. On 20 January he wrote: "I got some more work done with Alfred, and we planned the arrangement of Volume n", and on 5 February: "Alfred has been doing a lot of work of an interesting

XVI

INTRODUCTION

kind, which I must get up for Volume II." On 26 May, he wrote: "Alfred has worked like a horse since he got my letter, and has done a lot of things that have to be considered: we have to adopt a joint policy before we can go on with the writing out of our book, and that demands discussion." Russell lectured on symbolic logic at Cambridge r9or-02. His notes for this course, the first of its kind in England, may be considered the earliest draft of Volume II of the Principles. Apparently there is only a single remaining leaf of these notes, headed "Lecture II. Logic of propositions" (1901b). The file of rough notes RA 230.03085 (described in the section on missing and unprinted papers) refers to "Lectures *39·6" and "Lectures *122·5". If the high proposition numbers are an indication, Russell must have carried the derivation of mathematics from logic quite far. It seems likely that the primitive ideas and propositions used in the lectures were essentially the same as the eight logical constants and twenty primitive propositions listed in the Principles (1903, II-26). This conjecture is confirmed by Russell's boast in a letter to Couturat of 23 March 1902: Did I tell you that in my lecture course at Cambridge I deduced all of pure mathematics, including geometry, from 8 undefined ideas and 20 unproved propositions? I give purely logical definitions of number, numbers and various types of space. I believe this work would have pleased Leibniz; it comes closer to his ideas than any other work I know. The logical foundations adopted in the Principles were soon to be replaced. In 1902 Russell produced a new version of the foundations of the logic of propositions and quantifiers; his draft was the object of severe criticism by Whitehead quoted below in the section on their collaboration. The citations of logical propositions in Paper la show that b.Y early 1903 Russell was using a new set of axioms and rules, which were essentially the same as those set down explicitly in Paper 4. The foundations of the logic of propositions and quantifiers remained essentially unchanged, except for some notational changes, from 1902 to mid-1905. The changes from the version of the Principles show the influence of Frege. In his earlier work, Russell had followed Peano in adopting two devices that fulfil the role of the quantifiers "for all" and "there exists''. The first is the notion of formal implication, indicated by a subscripted conditional sign; the expression '\f>(x). =>x. !/J(x)", for example, is to be read: "For all values of x, if (x) then !/J(x)". The second is the existence symbol 3, which applies to classes; the expression "go:" means "The class a is non-empty". When combined with set abstraction notation, this second notation served as a version of the existential quantifier.

INTRODUCTION

XVII

notations were discarded as primitive ideas, under the influence of Russell had studied Frege's work in the summer of 1902, and wrote a detailed account of it in Appendix A of the Principles. The resulting familiarity with Frege's higher standards of logical rigour and deeper conceptual analysis led to a succession of changes in Russell's work, of which the first was a change in the primitive logical notions. From Frege's point of view, Peano's logical primitives confuse ideas that should be analyzed separately. Thus the concept of formal implication mixes the ideas of the conditional with that of quantification, while the use of the class existence svmbol makes existential quantification depend on the theory of classes. Whitehead and Russell followed Frege's lead in adopting the material conditional and universal quantification as logical primitives, while retaining Russell's earlier definition of negation in terms of formal implication (1903, I 8). The elimination of the existence symbol as a logical primitive is particularly important because it was clearly undesirable to make the well-established theory of the existential quantifier depend on the theory of classes, thrown in doubt by the Contradiction. The new notation for quantifiers, like most of the notation in Principia lYiathematica, was due to Whitehead (Russell 1948). Three related manuscripts by Whitehead have survived, perhaps among the earliest to be written in the new notation. They may have been written in late 1902, or perhaps early 1903. The first of this group of manuscripts is a draft entitled "The Logic of Propositional Functions" (RA 710.057490). This contains a rough draft of the foundations of the logic of quantification, using the material conditional and universal quantification as primitive notions (negation and the existential quantifier are defined). Russell used essentially this version of quantification theory as a basis for Paper la; the propositions cited in that paper often correspond exactly to propositions demonstrated in Whitehead's manuscript, except that (for example) Whitehead's *9·36 has been renumbered as *10·36. The second of this group is entitled "Theory of Identity" (RA 710.057497). In this manuscript Whitehead employs Schroder's notation for identity and diversity ( l' and O') and develops their basic properties on the basis of the definition of identity employed, for example, in Paper 4. The last and in some ways most interesting of this group is entitled "The Logic of Ranges" (RA 710.057489). The word "range" is a translation of Frege's Werthverlauf; the manuscript itself shows the influence of Frege, although Whitehead uses Peano's notation for class abstraction. The main primitive proposition *1l·12 is a transcription of Basic Law v of Frege l 893 into Peano's symbolism: ( x) : x •

= . !/Jx : • = . x 3 ( x) l 'x 3 ( !/Jx).

INTRODUCTION

INTRODUCTION

Whitehead deduces the unrestricted comprehension axiom from it in the form

1902, and was able to add a note to the proofs of Appendix A of the Principles, calling attention to Frege's attempted solution of the paradox, and adding: "As it seems very likely that this is the true solution, the reader is strongly recommended to examine Frege's argument on the point" (1903, Russell's notes on Frege's solution are printed as Appendix I. Frege, to whom Russell had communicated his paradox in a letter of 16 June 1902, learnt that his system was inconsistent as the printing of Volume II of his Grundgesetze der Arithmetik was nearing completion. He reacted by adding a hastily written appendix, making an ad hoc modification to his system in the hope of avoiding Russell's contradiction. This last-minute patch is inadequate, as a contradiction can be deduced in the amended system on the assumption that there are at least two objects; this was proved by Lefaiewski in 1938 (see Quine 1955). Frege's later silence may be taken as evidence that he soon became aware of its insufficiency. Frege attempted to solve the paradox by modifying his Basic Law v, so that two propositional functions might determine equal classes, while not having the same value for all arguments (the original Basic Law v, which leads to the Russell paradox, is discussed below). Russell attempted to rework the theory of classes by using this idea of Frege; Paper la is a draft of the corresponding portions of Volume II of the Principles, while 2 is a draft of the theory of the relations, written about the same time (early in x903). The resulting theory, which retains some of the earlier terminology of the Principles, such as the term "quadratic" for propositional functions leading to contradictions, is formally consistent, but (like Frege's) inadequate, since an inconsistency results if some elementary existence statements are added. Russell's discovery of this fact is recorded in the rough notes lb, which show how the introduction of the empty class leads immediately to a contradiction. These notes also sketch another attack on the problem, again derived from a passage in the appendix of Frege 1903; it is developed further in le. No other manuscripts have survived that mention this abortive attack on the paradoxes; it was to be replaced shortly by a more radical approach. In March 1903, Russell succumbed to one of his recurrent fits of depression. On 21 March he wrote to Gilbert Murray: "Nothing stirs me, nothing seems worth doing or worth having done: the only thing that I strongly feel worth while would be to murder as many people as possible so as to diminish the amount of consciousness in the world. These times have to be lived through: there is nothing to be done with them." In early April, having made no progress with Volume II, but driven to accomplish something, he wrote a review article on Meinong (17). Whitehead, on the other hand, had been forging ahead. Russell wrote Alys from Cambridge on 21 April 1903: "The place looked lovely yesterday, but I hardly saw it, as Alfred pounced upon me at once, and kept me working all through the

XVlll

Whitehead must have seen shortly that this leads to the Contradiction, for he added a later marginal note "Not true", while to* 11·12 he added: "Not true when cf> and 1/1 are quadratic functions-the (x): x. =. tjfx :. ::::>. etc. always true." The last of these three Whitehead manuscripts also contains a comment on Russell's work: The impression I have from a careful study is of very elaborate definitions which are not used, some proofs very careful, others equally important carried out by common sense in the style of Euclid. The proofs are the only interest of this subject-no one will read this unless the proofs are careful and minute. Russell's work at this time shows the gradually increasing influence of Frege, but in his logical practice he retained many of the methods of Peano; the formal conditional and the existence symbol survive even in Principia Mathematica, as the definitions *10·02 and *24·03. This persistence of earlier notation and ideas is shown most clearly in Paper 2, a large part of which was written in terms of the class existence operator, later systematically replaced by the existential quantifier. The same paper also uses Peano's technique of conditional definitions, in which certain operations are not defined on the whole logical universe, but only for a restricted subclass of entities defined by the antecedent of the definition. Frege had argued eloquently against such definitions (1898 and 1903, 6978). Russell, however, in his work of 1903 continued Peano's practice, justifying himself on the grounds of practical convenience, though he expresses his theoretical agreement with Frege by condemning such definitions as "formally vicious" (2, p. 45). Russell and Whitehead seem to have abandoned them finally towards the middle of 1903. In a letter of 19 June 1903 to Russell quoted at greater length below (pp. xxxix-xl), Whitehead alludes to the fact that "everything is defined for all values of variables", something that would only hold true if definitions under hypotheses had been given up for good. Writing to the mathematician Philip Jourdain on 15 May 1904, Russell noted: "Definitions under a hypothesis are to my mind always objectionable, and both Whitehead and I now avoid them wholly" (Grattan-Guinness 1977, 30). The failure of his early version of the theory of types had for the moment left Russell with no way out of the paradoxes. He had received the second volume of Frege's Grundgesetze der Arithmetik around the end of

XlX

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INTRODUCTION

afternoon till nearly dinner-time. He has done a great deal of work in the Vac, and proposes serious changes in the part of Vol. II which was already by way of being done. That is not the way to bring things nearer to completion!"

basic distinction between functions and objects. In other words, Russell's first version of the functional theory is similar to the system of the Grundgesetze, but with the basic distinction between functions and objects erased; expressions for functions can stand in the place of names, as they cannot in Frege's system. Some formal details of the new system are to be found in rough notes that must have been written around the middle of 1903, printed here as Paper 3. The system sketched in these notes is remarkably economical in its choice of primitive notions, which are: material implication, universal quantification, function application, substitution of an argument in a function, and functional abstraction. A characteristic notational feature of the 1903 manuscripts from May onwards is the use of a vertical bar for function application, as in the expression lx; from April 1904 onwards, this was replaced by do duty for z). I treated as an entity. All went well till I came to consider the function W, where

z(

W(). This brought back the contradiction, and showed that I had gained nothing by rejecting classes. (Grattan-Guinness 1977, 78) Contrary to what one might expect from these later remarks, Russell continued to work on modified versions of the functional theory. His approach was to restrict the primitive propositions of the original theory so that they applied only to "functional complexes". Complexes such as ~ f(f) that led to paradoxical results were to be ruled out as non-functional; Russell hoped that in spite of such restrictions it would be possible to make the class of functional complexes wide enough to include all those needed in the derivation of mathematics. Some details of attempts along these lines are given in Papers 3c, 3d, and 3e. In 3e, Russell lists versions of the primitive propositions of 3b, restricted to functional complexes; he also enumerates "inadmissable functions" which lead to contradictions. During 1903 and 1904, Russell was essentially following a version of Frege's strategy in attempting to solve the paradoxes. His plan of attack can be broken down into three parts. First, analyze as carefully as possible

INTRODUCTION

INTRODUCTION

the conditions giving rise to the contradictions; this involves giving the most general versions possible of the arguments leading to the paradoxes. Second, isolate the formal properties of the propositional functions that lead to the paradoxes. Third, turn these formal properties into definitions by postulating that a propositional function determines a class unless it satisfies these formal properties. Paper la furnishes a good example of the approach. In the definitions* 12·5 and * 12·5 l of that paper, Russell defines two propositional functions F(x) and F'(x); they depend on a given function f which takes propositional functions as arguments. The succeeding propositions *12·52 to * 12·57 are based on generalized versions of the argument underlying the Contradiction (the function f generalizes the class abstraction operator x 3 (x)). Subsequently, Russell postulates that the "quadratic" functions are exactly those equivalent to functions having the form F(x) or F'(x); two propositional functions are to determine the same class provided neither are quadratic (*12·6). Although the proposal of la was a failure, Russell continued to follow essentially the same plan into I904; various attempts along these lines are found in Papers 5, 6 and 7. The strategy seems reasonable, since all the paradoxes known to Russell involved some version of Cantor's diagonal argument; thus one might hope to give a formal characterization of those propositional functions that result from a form of diagonalization. In the summer of 1903, Russell rewrote the purely logical parts of Volume II. He recorded his progress in a letter to Couturat of June 20:

My work goes badly: new difficulties come up as fast as the old ones are solved. The political events interest me immensely, and if I could get away from her, I could work at Free Trade as well as mathematics; but I am spending on not being cruel to her as much energy as would make a whole political campaign. (Russell r985, 25)

xxii

The part of my second volume dealing with symbolic logic is nearly finished; however, it is possible that I shall find things in it to correct, and that I shall have to redo it all. I think that at present the development of it is at once deeper and simpler than in the sketch I gave in the first volume. · Towards the end of October, Russell sent the manuscript of this part to Couturat. Couturat responded on 8 November with a lengthy critique, complaining of the obscurity of the ten primitive propositions adopted by Russell. The primitive propositions to which Couturat objected appear to have been essentially the same as those given in Paper 4. In his reply of November 12, after defending himself against Couturat's criticisms, Russell indicates the plan for Volume n: "The numbers *10-*19 are concerned with functions and classes; the numbers *20-*29 with relations. Then we proceed to cardinal arithmetic." In spite of the optimism of May l 903, the parts on functions and classes proved much harder to write than the purely logical ones. In the course of some uncharitable remarks about his wife Alys, he noted in his journal on 26 July:

XXlll

On 29 July he wrote to Alys's contemporary and friend Lucy Donnelly: "The last four months I have been working like a horse, and have achieved almost nothing. I discovered in succession seven brand-new difficulties, of which I solved the first six. When the seventh turned up, I became discouraged, and decided to take a holiday before going on." Several months later, little progress had been achieved in surmounting the new difficulties. Russell wrote to Couturat on 20 October about the problems caused by the Contradiction in the theory of functions and classes: "I have tried at least a hundred different theories, but sooner or later found myself at a dead end. However, I go on hoping." Towards the end of 1903, Russell abandoned the logical struggle and threw himself into a campaign in favour of free trade, giving many speeches on this topic in January and March 1904 (the details of this episode are in Russell r985). On 15 March he told Jourdain in a letter that he would be returning to work on logic in a fortnight and after a walking tour in the latter half of March with Desmond Maccarthy, he took up work on foundations again in April (Grattan-Guinness r977, 25). III. THE ZIG-ZAG THEORY

The logical work of l 904 followed the general lines of the functional theory of 1903. As before, the concept of function plays a central role, with the smooth-breathing abstraction operator an important notation; Russell's strategy is to develop formal criteria to separate those expressions leading to legitimate functions from those leading to paradox. It is possible to glean something of Russell's work in April 1904 from six letters from Whitehead written between 23 April and 30 April, which comment on and extend some manuscript material by Russell. Whitehead begins his letter of 23 April with the remark, "Your work is admirable. I think that you are in sight of land", then proceeds to detailed comments on notation and primitive propositions. Russell's manuscript contained a "Proposition" and "Primitive proposition", which Whitehead reformulates as:

=x. 'x f- :: ,.., '() :. (tJ.Y): tf/(x, y). =x. 'x :. :J: (tJ.): 1//(x, x). =x. 'x. f- :: (tJ.): t//(x, x). ="''x. 'x: :J :. ,.., '() :. (tJ.Y): tf/(x, y).

and

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INTRODUCTION

INTRODUCTION

The first of these propositions can be understood as a generalized version of the Contradiction; the consequent of the proposition follows from the antecedent by considering the propositional function ,.., 1//(x, x), then following the argument for Russell's paradox (where 1/; replaces the membership relation). At the end of the same letter Whitehead introduces the notation (1x). (p'x for definite descriptions. The new theory whose central assumption was the Primitive Proposition above was based on setting down conditions under which a propositional form containing two occurrences of a variable was equivalent to a form containing only one (recall that the construction leading to the Contradiction rests on forming the "diagonal" function ,.,,, (x £ x) from the two-place propositional function x Ey). Whitehead suggested the use of the notation cp!x, cp!(x, x) and so on for functional expressions in which the different occurrences of variables are explicitly marked. He also suggested ("for brevity") the basic definition:

The subsequent letters of April continue this general theme of attempting to set down primitive propositions for reducibility sufficient to ensure the existence of the classes needed in later mathematical developments. Whitehead's letter of 16 May 1904 to Russell suggests a somewhat different approach. Although the concept of reducibility still occupies a central position, the smooth-breathing abstraction operator is no longer primitive, being replaced by the primitive idea "u Kl .-(x)

Df [*12, *13] Df [*12, *13]

These two definitions are only to apply within numbers *12 and *13: temporary definitions of this sort will have the range of their application indicated as above, by square brackets after the letters Df. The F in these definitions is a function off; thus G(x) would have the same relation to g(x).

*12·52

*12·53

(1)

[f- :. F'{f(F')}. =: () :f(F') l'f(). :>. -{f(F')}: :> :f(F') l' f(F'). :> . ,.., F' {f(F' )}

(1)

I·. (1). *11·2. *5·6. :J f- Prop]

*12·56

f-: (3) · [f( ) l 'f(F') • {f(F' )} ] [f-:,.., F' {f(F' )} • = . (3 )[f( ) l 'f(F'). {f(F')}]

(1)

(1)

IO

f-. (1). *12·55. :> f- Prop] *12·57

l·F{f(F')} [f-. *12·55. *11 ·2. :> f-:,.., F' {f(F')} .f(F') l' f(F'): :> f- Prop]

The above six propositions should be compared with *12·42·43. They prove that, whatever function f( ) we may define as the class determined by , there will always be, for a suitable value of , another function ' determining the same class, but not equivalent to . That is to say, the two functions x E 'a and x E" a must be distinguished, whatever definition of a class we may adopt. Functions having the form of F and F' above are those that we call quadratic. The above peculiar results only occur when we are not only dealing with such functions, but also take as their arguments terms defined by means of quadratic functions. It seems allowable to assume, therefore, that, in other cases, any two functions determining the same class are equivalent. We shall put (retaining the above definitions of F and F' as functions off): *12·58

v. (3f). {(x): x. = .F'x} *12·581

20

Quad().= :(3f).{(x):x.=.Fx}. Quadratic( If!). = . (3f). [():If!().=. - {f()}]

*12·59

Simple(). = ...... Quad()

Df Df

Df

*12·6

f-::Simple(). :J :.(!f!):.{x3(x)} l'{x3(!/Jx)}. :>:(x):x.=.!f!x

f-. (3). [- {f(F)} .f(F) l 'f( )] [f-: F {f(F)}. =. (3). [- {f(F)} .f(F) l 'f( )] f-.(1).*12·52.:>f- Prop]

[*12·52.:>f-Prop]

The second of these definitions is only inserted for the sake of completeness; in regard to functions of the first order, only the first is relevant.

f-F{f(F)} [f- :. -F{f(F)}. = :() :f(F) l'f(). :>. {f(F)}: :> :f(F) l'f(F). :> .F{f(F)}: :> :F{f(F)} f-. (1). *5·6. :> f- Prop]

*12· 55

t--F'{f(F)} f-,.., F' {f(F' )}

Pp

i._e. any two functions which define the same class are equivalent provided either of them is not quadratic.

30

10

10

VOL. 4 FOUNDATIONS OF LOGIC, 1903-05

*12·60

I-·:. (x): .f{g( :./( I- Prop]

Thus where the above conditions are fulfilled, f{g(¢, t/I)} is a function of f( .-al'-b [I-:. Hp.:>: x £a. ='x. x Eb::>: x,..., Ea. =x. x,..., Eb:.:> I- Prop] n'k=x3(uEk.:>,,.xEu) Df

This is the definition of the class-product of k: it is read "product of k". (An apostrophe, or a single inverted comma, will always, following Peano, be used for "of"; but we shall define each separate use of it, and not give a general definition of "of".) If k is a class whose members are classes ~ 'k represents the common part of the members of k, i.e. the class of terms' 3o (1f any) which are members of every member of k. Thus e.g. if k were the class of London clubs, n'k would be the class of those persons (if such there be) who belong to every club in London. If k is not a class whose members are classes, n'k is a class which has no members, i.e. the nullclass; and the same is the case if k is a class of classes which have no common part, e.g. the nations of Europe considered as classes of citizens.

VOL. 4 FOUNDATIONS OF LOGIC, 1903-05

20

*14·71

u'k

=

X3{3u3(xi::u.ui::k)}

Df

I-. (3). *2·31 . :::>I-:: (p):: q. :::> : p Em.:::> . p :.

=:.pun.:J:q.::>.p (4). *10·23. :JI-;; (p) :. q. :J: p Em. :J. p :. =:(p):pi::m.::>.q:Jp 1-.(1).(2).(5).:::>I- Prop]

This is the definition of the class-sum of k: it is read "sum of k". If k is a class whose members are classes, u'k represents the class of all terms belonging to some k or other; thus in the above instance of the London clubs, u'k represents the class of all persons who have a club. If k is not a class of classes, u'k is a class having no members, i.e. the null-class. *14·72

10

A'm. = :pi::m.:Jp·P

v'm.

=

:.(s):.pi::m.:::>P.p:Js::J.s

I- : p Fm. :J . p :J v'm

*14·85

I- :.q:J A'm. == :pi::m. :JP .q:J P [1-::q:JA'm.==:.q.:J:(p):pi::m.:J.p I-. *10· 13. :J I-:: q. :J : (p): p Em. :J . p :. ==:.(p):.q.:J:pi::m.:J.p I-. Comm. :JI-:: q. :J :pi:: m. :J .p :. ==:.pi:: m. :J: q. :J .p

[I-. *2·3. :JI-:. p £ m. :JP. p :J s: :J: p Em. :J. p :J s I- . Imp . :J I- : . p i:: m . :J . p :J s : :J : p Em . p . :J . s I- . * 3· 3 . :J I- : : p Em . p . :::> : . p Em . p . :J . s : :J . s I- • (1). (2). (3). :J I- : : p Em. p . :J : . p Em. :JP. p :J s: :J . s 1-.(4).*2·31·4.:Jl-::pi::m.p. :J:.(s):.p0m.:JP.p:Js::J.s f-. (5). Exp. :JI- Prop] I- : . v'm :J q . == : p Em . :JP . p :J q

[l-.*2·3~.:Jl-::v'm.:J:.p0m.:Jp.p:Jq::J.q s

I-. (1). Comm. :JI- ::p Em. :JP. p :J q: :J. v'm :J q I- . *3·4. :J I- : . v'm :J q. :J : p :J v'm. :J . p :J q I- . *3·4. :J I- : : p :J v'm. :J . p :J q: :J : . p Em. :J . p :J v'm: :J : p Em . :J . p :J q I-. (3). (4). Syll. :JI- ::v'm :J q. :J :.p0m. :J .p :Jv'm: :J:p0m.:J.p:Jq I-. (5). *2·31·4. :JI-:: v'm :J q. :J :. (p) :.pi:: m. :J .p :J v'm:

Df

(1) (2) (3)

IO

(4) (5)

(1)

(2) (3)

(4)

20

(5)

:J:pi::m.:J.p:Jq (6) I- . * 10· 22 . :J I- : : ( p) : . p Em . :J . p :J v'm : :J : p Em . :J . p :J q : . :J : . p Em . :JP . p :J v'm : :J : p Em • :JP . p :J q (7) I-. (6). (7). Syll. :JI-:: v'm :J q. :J :.pi:: m. :JP .p :J v'm:

This defines the proposition-sum of m; it is read "sum of m". It is equivalent to "m is a class of which at least one member is a true proposition". It is also equivalent to the negation of "pi:: m. :JP.,..,, p" (see * 14·86). The definition of p v q in *4· l cannot be extended to a class of propositions, but the present definition is an extension of *4·23. v'm will be false if m is not a class, or is a class none of whose members are propositions, or is a class whose members, when they are propositions, are false. *14·82

(5)

This proposition is an extension of *3·81, and thence of Composition.

Df

This is the definition ofthe proposition-product of m; it is read, like* 14·7, "product of m". A'm is the function "every member of m is a true proposition". Thus A'm is a propositional function of m, which is true when m is a class of true propositions, but is not true in any other case. This definition extends, to any class of propositions, the logical product defined for two propositions in *3· l. A'm is much less important than n'k, and rarely occurs. We might define A', where is any function, as (p). :JP. p. Thus A' would mean: "All terms satisfying are true propositions." The function -. p :::> r

(1) (2) (3)

*14·86

30

22

f-. (1). (2). (3). *2·31 . :J f- : . (p):. p Em. :J . ,..., p: =:(r):pEm.:::>.p:::>r (4) f-.(4).*10·23.:::>f-:.pEm.:::>p.,..,p:=:(r):pEm.:::>P.p:::>r (5) f-.*14·85.:::>f-:.(q):.v'm:::>q.=:pEm.:::>P.p:::>q (6) f-. (6). *10·23. :::> f- :. (q): v'm :::> q: =: (q) :p Em. :::>P. p :::> q (7) f-.(5).(7).:::>f-:.pEm.:::>P.,..,p:=:(q):v'm:::>q (8) f-. *l0· 13. :::> f- :. (q): v'm :::> q: =: v'm. :::>. (q)q (9) f-. (8). (9). :::> f- :. p Em. =>p.,.., p: =.,..., v'm (10) f-. (10). Transp. :::> f- Prop] IO

20

1 CLASSES

VOL. 4 FOUNDATIONS OF LOGIC, 1903-05

*15. The Class-Calculus. There is a close analogy between many of the formulae in the theory of implication and correlated formulae in the theory of the inclusion of classes, i.e. of the function a Cb. If we replace, in any proposition of *3-*5, p. q by an b, pv q by au b, ,..., p by -a, and p :::> q in a subordinate implication by a Cb, while leaving the principal implication-sign unchanged, then, provided each side of the principal implication-sign is a function of implications only, i.e. does not contain any single variable as a factor, we shall as a rule obtain a true proposition. Thus p :::> p becomes a Ca;

*15·01

*15· l l

p:::>q.q:::>r.:::>.p:::>r aCb.bCc.:::>.aCc,

becomes

which is one form of the syllogism in Barbara; p:::>s.q:::>s. :::> .pvq:::>s a Cc. b Cc.:::>. au b Cc; pvq:::>r.:::>:p:::>r.v.q:::>r au b Cc.:::>: a Cc. v. b Cc,

becomes but becomes

30

which is false. 2 When we have such expressions asp:::> q. :::> . r, we cannot apply this method; for neither (a Cb) Cc nor a Cb. :::> . c gives a homogeneous formula such as we want. An expression such asp.:::>. q :::> r must be changed, before the method is applied, top. q. :::>. r, which becomes an b Cc. The analogues to propositions of *3-*5 are obtained by substituting x Ea, x Eb, x Ec for p, q, r, and then varying x in accordance with *2·31. We shall only give a few of the proofs, since all proceed in exactly the same way. *15·0 2

f-.a =a

On this point, see remarks following *15·37, below.

23

[f-. *3·690. :::> f-: x Ea.=. XE a (1) f-.(l).*2·31.:::>f-:(x):xEa.=.XEa (2) f-. *10· 14. :::> f- :. (x): x Ea.=. x Ea:=: (x): x Ea.:::>. x Ea: (x): x Ea.:::>. XE a (3) f- .(2).(3) .(*14·5). :::> f- .a Ca .ac a (4) f- . (4). (*14·5 l) . :::> f- Prop] f-:a==b.=:xEa.=x.xEb [( d 4· 5. 51) f- : . a = b • = : x Ea • :::> x . x Eb : x Eb . :::> x • x Ea ( 1) f-. (1) .'d0· 14. :::> f- Prop] f-.anbCa (1) [f-. *3·52. :::> f-: x Ea. x Eb.:::> . x Ea f- . *12 ·77 . (* 14·41) . :::> f- : XE a n b . :::> . XE a . XE b (2) f-. (1). (2). :::> f-: XE an b. :::>.XE a (3) f-. (3). *2·31. :::> f-: XE an b. =>x. XE a::::> f- Prop] f-:aCb.bCc.:::>.aCc [f-. Syll. :::> f- :. x Ea.:::>. x Eb: x Eb.:::>. x Ec: :J :XE a. :J .XEC (1) f-. (1). *2·31. :::> f- :. (x) :. XE a.:::>. XE b: x Eb.:::>. x EC: :J : XE a. :J . X EC (2) f-. (2). *10·22. :J f- ;; (x) ;, X Ea. :J. XE b: XE b, :J. X EC:. :::>: (x): x Ea.:::>. x EC (3) f- . * 10· 14. :::> f- : : (x):. x Ea. :::> . x Eb : x Eb. :::> . x EC:. =:.(x):XEa.:::>.xEb:.(x):.xEb.:::>.xEC (4) I-. (3). (4). :::> f- :: (x): x Ea.:::>. x Eh:. (x): x Eb.:::>. x EC:. :::> : (x): x Ea. :::> . x EC:: :::> f- Prop]

The above proposition affirms that if all a is b, and all bis c, then all a is c. This is one form of the syllogism in Barbara, and it was on account of this analogue that *2·7, *3·4 and *3·60 were called "Syllogism". The other form of Barbara is: "If xis an a, and all a is b, then xis ab." The distinction of the two forms is due to Peano. The above proposition * 15· ll will also be cited as "Syll". *15· l 2

f- : . b n c ECls . :::> : a C b • a C c • :::> . a C b n c [f-. Comp.:::> f- :. x Ea.:::>. x Eb: x Ea.:::>. x EC: :::>:xEa.:::>.x£b.xEC (1) f-. (1). *10·22. :::> f- :: (x) :. XE a.:::>. x Eb: x Ea.:::>. XE c :. :::>: (x): XE a.:::>. XE b. x EC (2)

IO

20

30

24

1 CLASSES

VOL. 4 FOUNDATIONS OF LOGIC, 1903-05

f-. (2). *10· 14. ::::> f- :. x Ea. =>x. x Eb: x Ea.::::> x. x Ec: :>:xEa.:>x.xEb.xEC f-. *12·7·75. ::::> f-:.bnu:Cls.:>:xEb.xEC.:>.xEbnc f- . (4) . *2·31·4 . ::::> f- : . b n c ECls • ::::> : XE b . x EC . ::::> x • XE b n c f-. (3). (5). *10·22. ::::> f- Prop]

[~

*15· 13

20

*15· 14

30

*15· 15

*15·16

. d2·78. ::::> f-: Hp.::::>. a, b ECls (1) f-. (1). *15· 15. ::::> f- :. Hp.::::>: Hp.=. XE an b :. ::::> f- :Hp.::::> .XEanb]

(3) (4) ( 5)

If b n c is not a simple class (Cls), while band care so, the inference from x Eb. x Ec to x Eb n c fails, and thus the proposition may not be true. But the possibility of this case is excluded by *12·81, as we shall now show. IO

b, c ECls • ::::> . b n c ECls [f-. * 12·75. ::::> f- :: Hp.::::> :. bl' x 3 ( x). cl' x 3 ( tf!x). ::::> : x Eb. =x. x: x Ec. =x. !/Ix (1) f- . ( 1) . *10·4. ::::> f- : : Hp. ::::> : • b l' x 3 ( x). c 1' x 3 ( tf!x). ::::> : x Eb • x EC . = x- x • tf!x (2) f-. *12·75. (*12·66). ::::> f- :. Hp.::::>: (3). {Simple(). bl' x 3 (x)}: (3!/I). {Simple(tf!). cl' x 3 (t/Jx)} (3) f-. *12·81·68. ::::> f-: Simple(). Simple(t/I). ::::>. Spl{x 3 (x. t/Jx)} (4) f-. (2). (4). *12· l. ::::> f- :. bl' x 3 (x). Simple(). cl' X3(!/Jx). Simple(t/I). ::::>. Spl{x 3 (xc b. x EC)}.::::>. Spl(b n c) (5) (6) f-. (3). (5). *10·32. ::::> f-: Hp.::::>. Spl(b nc) f-. (6). *12·75. ::::> f- Prop]

f-:XF:a.aCb.::::>.xEb

[f-. *2·3. ::::> f- :. a Cb.::::>: x Ea.::::>. x Eb f-. (1). Comm. Imp.::::> f- Prop]

*15· 1l.

*15·21

*15·22

f-:x£a.x£b.::::>.x£anb

f-:. a, b ECls . ::::> : a C c • ::::> . a n b C c [f-. *15· 15. ::::> f- :. Hp.::::>: x Ea n b. =x. x Ea. x Eb 1-. *2·3. ::::> f- :. a Cc.::::>: x Ea.::::>. x EC: :>:xEa.xEb.:>.xEc f- .(2). *2·31·4. ::::> f- :.acc.::::> :xEa .XEb. =>x.XEC I·. (1). (3). ::::> f- Prop] ~· : • a, c ECls . ::::> : a C b • ::::> . an c C b n c [f-. Fact.::::> f- :. x Ea.::::>. x Eb:::::>: x Ea. x EC.::::>. x Eb. x EC

IO

(1)

(2) (3)

(1)

~.(1).*15·16.:>f-:.XEa.:>.XEb:

*15·23

::::> :XEG.XEC. ::::> .xEbnc f-. (2). * 15· 15. ::::> f-:: a, c ECls. ::::> :. x Ea.::::> . x Eb: ::::> :xEanc. ::::> .xEbnc f-. (3). *2·31. *10·22. ::::> f- Prop] f- : . a, c ECls . ::::> : a C b • c C d • ::::> . a n c C b n d (f-. *3·72. ::::> f- :. XE a.::::>. XE b: X EC.::::>. XE d:

(2)

::::>:xEa.xEc.::::>.xEb.xEd

(1)

(1)

f-. (1). *2·31. *10·22. ::::> f- :. a Cb. c Cd.

(2) (3) (4) (5) (6)

I·. (2). *15· 15· 16. ::::> f- Prop]

::::>:xEa.xEc.::::>x.xEb.xEd

*15·24 *15·241

20

(3)

(2)

f-anal'x3(xEa) [f-. *3·67. ::::> f-: x Ea. x Ea.==. x Ea:::::> f- Prop] I· : a ECls . ::::> . a n a = a [*15·24. *12·77]

30

This proposition does not hold if a is a quadratic class. *15·25

(1)

(1)

Th'15 is the syllogism in Barbara with a particular subject for the minor pre m1·5 ,s · The traditional formal logic confounds this proposition with

f-:aCb.aCc.::::>.aCbnc

[f-. *12·78. ::::> f- :. Hp.=: x Ea. =>x. x Eb. b ECls: x Ea • ::::> x • x Ec • c ECls f-. (1). *2·3. ::::> f- :. Hp.::::>: x Ea.::::>. XE b. b ECls: x Ea • ::::> . x EC • c ECls f-. (2). *3·71. ::::> f- :. Hp.::::>: x Ea.::::>. x Eb. x EC. b, c ECls f-. (3). *15·13. ::::> f- :. Hp.::::> :xw. ::::>. x Eh .xu. b ncECls f-. *12·7·75. ::::> f- :.x Eb. XEC. b ncECls. ::::>.XE bnc f-. (4). (5). ::::> f- :. Hp.::::>: XE a.::::>. x Eb n c f-. (6). *2·31·4. ::::> f- Prop] f- : . a, b ECls • ::::> : x Ea n b • = x • x Ea • x Eb [f- . * 15· 13 . ::::> f- : Hp. ::::> . an b ECls f-. (1). *12·7. ::::> f- :. Hp.::::>: x Ea n b. =x. x Ea. x Eb]

25

f-anb l' bna

[f-. *3·66. ::::> f-: x Ea. x Eb.==. x Eb. x Ea 1-. (1). *2·31. *12· l. ::::> f- Prop]

(1)

Note that identity always implies equality, but the converse holds neither for individuals nor for quadratic classes.

*15·26

[f- . *4· 23 . :J f- : : XE a . V. XE b : =:.xEa.:J.s:xEb.:J.s::Js.s:. :J : . XE a. :J . XE U: XE b. :J . XE U: :Ju.XE U :.

[*3·68]

f-(anb)ncl'an(bnc)

:J :. XE a. :J x. XE U: XE b. :J x. XE U: :Ju.XE U :.

The inference from *3·68 is conducted as that from *3·66 was conducted in *15·25.

*15·27 *15·28

IO

*15·281 *15·282 *15-3 *15·30

anbnc = (anb)nc

:J:.aCu.bCu.:Ju.XEU f- . Sim p . :J f- : : . s . :J : : a C u . b C u . :Ju . x E u :

Df

b.b c.:J.a=c f- :. a= b. :J: a Cc.=. b Cc f- :. a= b. :J: c Ca.=. c Cb f-au b 1' bua

:J :. XE a. :J. S: XE b. :J. S: :J. S

f-(aua)l'{x3(xca)}

*15·301

f-. (1). *2·31. *12· 1. :J f- Prop] f- : a E Cls . :J . a u a 1' a

* 15·31

f- : a, b E Cls . :J . a u b E Cls

:J :. ,.., {x Ea. :J. x Ea: x Eb. :J. x Ea: :J. x Ea}:.

(1)

:J :. -{x Ea. :J. x Eu: x Eb. :J. x Eu: :J,,. x Eu}:. :J :. ,.., {x Ea. :J x. x Eu: x Eb. :J x. x Eu: :J,,. x Eu}:.

[* 15·30. * 12·77]

:J :. -{a Cu. b Cu. :J,,. x Eu}:.

:J:.aCu.bCu.:Ju.xcu::J.s f- . ( 3) . Comm . :J f- : : . ,.., s . :J : : a C u . b C u . :J,, . x E u :

[f-. *12·75. :J f-: Hp. :J. (34>). {Simple(¢). a 1' x 3 Cx)}. (31p). {Simple( I/I). b I' x 3 (!/Ix)}

(1)

f-. * 12·82·68. :J f-: Simple( x : • ¢x . :J : !/Jx • v . (3x) . {x :J !/J : x l' k' x} : . v :. x. :J. !/Jx :. !/Jx. :J: Ill If! [f- . * 14· 36 . ::J f- : . Hp . ::J : cf>x • = . x E{x 3 (cf>x)} .

:.f(!/J).:J.f(xEU.:Jx.lf!x: +u) 1(1/1°, f) · = :.f(!/1°) · :J .f(!/11) /(!/JI, f). = :.f(!/11) • :J .f(!/12) =

Io

Df.

ClsN(u).

f- :. u C x 3 (cf>x). ::Ju.,.., cf>(u): ::J. Min() [f-: *14·2: cf>(u). ::Ju. u E {x 3 (cf>x)}: ::J f- : • cf>( u) . ::J : u C x 3 ( cf>x) • v . u E{ x 3 ( cf>x)} f- :. Hp. ::J: cf>(u). ::J,, ....... (u C x 3 (cf>x))

:/(lndiv, f) .f(lndiv). :J1.f(cf>)

=

= . (g:c/>). {ClassN( cf>).

cf>(u)}

Df Df.

Then CJsN(u) asserts that all the members of u are of the same type, i.e. all individuals, all simple classes, all Cls2, all Cls3 , or etc. We shall have to suppose ClsN(u) always. To get two classes u, v of the same type, we want ClsN(w). u, v Ew.

20

We shall have ClsN :J Pure.

=.XE{x3(!/Jx)}.=.!/Jx] *14·38

/(lf!,f).

ClassN(cf>).

Df

Here Min stands for minimum; cf> is a minimum function in relation to its class. *14·36

Df Df

1

I

20

37

1 CLASSES

VOL. 4 FOUNDATIONS OF LOGIC, 1903-05

Also and

u l 'y 3 (cf>y): x Cu. :Jx.,..., cf>(x): :J : cf>x. =x. XE u x Cy 3 (cf>y). :Jx.,..., cf>(x): :J . Min cf>. Pure(y 3 cf>y).

Consider e.g. the case of numbers:

1

30

(1)

Nc'u = V3(vsimu).

Here we do not have

(2)

w C v 3 (vsimu). :J.,...,. (wsimu).

f- . (1). (2). ::J f- : . Hp. ::J : cf>(u). ::J 11 • u E{x 3 (cf>x)} : . ::J f- Prop] Consequently a function not satisfied by any sub-class of its own class is a minimum function.

Thus such a class is not pure, and the above devices serve no purpose. But Ne is a pure class: no class of numbers is a number. Put

Pure(cf>).

=

:xCy3(cf>y).:Jx • .-x

Df.

30

2 RELATIONS

2

Relations [1903]

THIS PAPER CONSISTS of manuscript notes on the theory of relations forming part of a draft of Volume II of the Principles. Five pages of undated comments by Whitehead entitled "Relations" (RA 710.057496) appear to refer to an early version of this theory. Whitehead begins: The first two pages of the 'logic of relations' are very muddled. The symbols used are (x, y) defined as the couple with sense (z)}. =: (z): "'"'3(x, y)3 {z l' (x-?y). cf>(x, y)}: =: (z): (x, y). -{z l' (x-?y). cf>(x, y)}

(2)

1-(2). *2·3. X-7Y. *11 ·2. :J

z

-7

(3)

l-:.-3z3{cf;(z)}.:J.-cf>(x,y) -7

1-(3). *2·31 . *2·4. :J I-:. - 3z 3 { 4> (z)}. :J : (x, y)."'"' cf>(x, y) -7

1-(4). Transp. :JI-:. 3(x, y)3 cf>(x, y). :J. 3z3 { 4> (z)} H3·53. Transp. :JI-:,.., cf>(x, y). :J. -{z l' (x-?y). cf>(x, y)} 1-(6). *2·31. *20·6. :JI-: (x, y).,..., cf>(x, y). :J. (x, y). -{z l' (x-?y). cf>(x, y)} 1-(7). *2·31·4. :JI-:. (x, y). -cf>(x, y). :J : (z): (x, y).,..., {z l' (x-?y). cf>(x, y)}

20

(4)

(5) (6) (7)

.(, :::._{~,,~\~~~ .••

'(";:~

(8)

-7

1-(2). (8). :JI-:. (x, y). -cf>(x, y). :J. -3z3 { c/>(z)} :. -7

:J I- :3z3{ cf>(z)}. :J .3(x, y)3{cf>(x, y)} 1-(5). (9). :J 1- Prop]

(9) 4

The propositions concerning ";/;(z) corresponding to *21·21·24 are similarly proved. They are .,.......

*21·25

I-:: (x, y). 4> (x-?y): =: (x, y). cf>(y, x) :.

.,.......

(x, y): 4> (x-?y). *21·26 l-:3z3 cf>(z). = .3(x,y)3{cf>(y, x)} 30 *21·3 DR= x3{3y 3(x/R/y)} .,.......

=. cf>(y, x) Df

*21·31

DR =y3{3x3(x/R/y)}

Df

*21'32

FR= DRuDR

Df

three-dimensional projective geometry of fifteen points as constructby Russell. See As84: II-13.

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