Parts of Classes David Lewis
With an appendix by John P. Burgess, A. P. Hazen, .
and David Lewis
Copyright David Lewis 1991
First published 1991
Basil Blackwell Ltd 108 Cowley Road, Oxford. OX4 1JF, UK
Basil Blackwell, Inc. 3 Cambridge Center
Cambridge, Massachusetts 02142. USA
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic. mechanical, photocopying, recording or
otherwise, without the prior permission of the publisher.
Except in the United States of America. this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and
without a similar condition including this condition being imposed on the subsequent purchaser.
British Library Cataloguing in.Publication Data A CIP catalogue record for this book is available from the British
Library of Congress Cataloging in Publication Data
Lewis, David K. Parts of classes/David Lewis.
p. cm. Includes bibliographical references.
ISBN Q..631-17655-1 - ISBN Q..631-17656-X (pbk.) 1. Set theory. 2. Whole and parts (philosophy) L Title.
QA248.L48 1991 511.3'22 - dc20 89--49771
Typeset in 12 on 14pt Bembo by Graphicraft Typesetters Ltd., Hong Kong
Printed in Great Britain by Billings & Sons Ltd., Worcester
QA 24B .lAB 1991
1 T~king Classes Apart 1 1.1 Fusions and Classes 1 1.2 Classes and their Subclasses 3 1.3 Are there any other Parts of Classes? 6 1.4 The Nun Set 10 1.5 Consequences of the Main Thesis 15 1.6 More Redefinitions 17 1.7 Sethood and Proper Classes 18 1.8 A Map of Reality 19 1.9 Nominalistic Set Theory Revisited 21
2 The Trouble with Classes 29 2.1 Mysterious Singletons 29 2.2 Van Inwagen's Tu Quoque 35 2.3 On Relations that Generate 38 2.4 Quine on Urelements and Singletons 41 2.5 The Lasso Hypothesis 42
2.6 Ramsifying out the Singleton Function 45 2.7 Metaphysics to the Rescue? 54 2.8 Credo 55
A Framework for Set Theory 3.1 Desiderata for a Framework 3.2 Plural Quantification 3.3 Choice 3.4 Mereology 3.5 Strife over Mereology 3.6 Composition as Identity 3.7 Distinctions of Size
Set Theory for Mereologists 4.1 The Size of Reality 4.2 Mereologized Arithmetic 4.3 Four Theses Regained 4.4 Set Theory Regained 4.5 Ordinary Arithmetic and Mereologized
Arithmetic 4.6 What's in a Name? 4.7 Intermediate Systems 4.8 Completeness of Me reologi zed
61 61 62 71 72 75 81 88
93 93 95 98
107 109 112
113 Appendix on Pairing 121
by John P. Burgess, A. P. Hazen, and David Lewis
There is more mereology in set theory than we usually think. The parts of a class are exactly the subclasses (except that, for this purpose, the null set should not count as a class). The notion of a singleton, or unit set, can serve as the distinctive primitive of set theory. The rest is mereology:a class is the fusion of its singleton subclasses, something is a member of a class iff its singleton is part of that class. If we axiomatize set theory with singleton as primitive (added to an ontologically innocent framework of plural quantification and mereo-logy), our axioms for 'singleton' closely resemble the Peano axioms for 'successor'. From these axioms, we can regain standard iterative set theory.
Alas, the notion of a singleton was never properly ex-plained: talk of collecting many into one does not apply to one-membered sets, and in fact introduces us only to the mereology in set theory. I wonder how it is possible for us to understand the primitive notion of singleton, if indeed we really do.
In March 1989, after this book was mostly written, I learned belatedly that its main thesis had been anticipated in the
Preface 'ensemble theory' of Harry C. Bunt. 1 He says, as I do, that the theory of part and whole applies to classes; that subclasses are the parts of classes, and hence that singletons - unit classes - are the minimal parts of classes; and that, given the theory of part and whole, the member-singleton relation may replace membership generally as the primitive notion of set theory. There is some significant difference from the start, since Bunt accepts a null individual and rejects individual atoms. But the more important differences come later: in my discussion of the philosophical consequences of the main the-sis, and in our quite different ways of formulating set theory within a mereological framework. I think there is difference enough to justify retelling the story.
Later in 1989 I had another surprise. I learned how we can, in effect, quantify over relations without benefit of the re-sources of set theory. (All we need is the framework: plural quantification and mereology.) John P. Burgess discovered one way to do it; A. P. Hazen discovered a different way. It was not quite too late to add an appendix, jointly written with Burgess and Hazen, in which these methods are pre-sented and applied. They open the way to a new 'structuralist' interpretation of set theory, on which the primitive notion of singleton vanishes and my worries about whether it is well understood vanish too.
Good news; but not a full answer to my worries. The set theory of the future may go structuralist, if it likes. But I can't very well say that set theory was implicitly structuralist all along, even before the discoveries were made that opened
1 Harry C. Bunt, Mass Terms and Model- Theoretic Semantics (Cambridge University Press, 1985), pp. 53 -72, 233 - 301.
Preface the way. The set theory of the present which means the bulk of present-day mathematics rests on the primitive notion of singleton. Is it rotten in its foundation? I dare not make that accusation lightly! Philosophers who repudiate all that they cannot understand have very often gone astray. Maybe my worries are misguided; maybe somehow, I know not how, we have understood the member-singleton relation (and with it membership generally) well enough all along. If so, structuralism is a real solution to an unreal problem, and we might as well go on as before.
The book reflects my indecision. I whinge at length about the primitive notion of singleton (section 2.1), but then I mock those philosophers who refuse to take mathematics as they find it (section 2.8). The book mostly proceeds under the working assumption that singleton is a legitimate primi-tive notion. But also it presents the structuralist alternative, in section 2.6 and in the appendix. (What's said taking singleton as primitive will not in any case go to waste. All but section 4.6 admits of structuralist reinterpretation.) Structuralism is at least a very welcome fallback. But I would much prefer a good answer to my worries about primitive singleton, so that I could in good conscience take mathematics as I find it.
I thank all those who have helped by discussion of this material, especially D. M. Armstrong, Donald Baxter, Paul Benacerraf, George Boolos, Phillip Bricker,John P. Burgess, M. J. Cresswell, Jennifer Davoren, Hartry Field, A. P. Hazen, Daniel Isaacson, Mark Johnston, Graham Oppy, Hilary Putnam, W. V. Quine, Denis Robinson, Barry Taylor, and Peter van Inwagen. I thank Nancy Etchemendy for preparing the figures in the appendix. I am indebted to the Melbourne Semantics Group, where this material was first presented in 1984; to Stanford University, where
Preface a more developed version was presented as the 1988 Kant Lectures; to Harvard University, for research support under a Santayana Fellowship in 1988; and to the Boyce Gibson Memorial Library.
David Lewis Princeton, January 1990
1 Taking Classes Apart
1.1 Fusions and Classes Mereology is the theory of the relation of part to whole, and kindred notions. 1 One of these kindred notions is that of a mereological fusion, or sum: the whole composed of some given parts.
The fusion of all cats is that large, scattered chunk of cat-stuff which is composed of all the cats there are, and nothing else. It has all cats as parts. There are other things that have all cats as parts. But the cat-fusion is the least such thing: it is included as a part in any other one.
It does have other parts too: all cat-parts are parts of it, for instance cat-whiskers, cat-cells, cat-quarks. For parthood is transitive; whatever is part of a cat is thereby part of a part of the cat-fusion, and so must itself be part of the cat-fusion.
The cat-fusion has still other parts. We count it as a part of itself: an improper part, a part identical to the whole. But also
1 See sections 3.4 and 3.5 for further discussion of mereology.
Taking Classes Apart it has plenty of proper parts - parts not identical to the whole - besides the cats and cat-parts already mentioned. Lesser fusions of cats, for instance the fusion of my two cats Magpie and Possum, are proper parts of the grand fusion of all cats. Fusions of cat-parts are parts of it too, for instance the fusion of Possum's paws plus Mag~ie's whiskers, or the fusion of all cat-tails wherever they be. Fusions of several cats plus several cat-parts are parts of it. And yet the cat-fusion is made of nothing but cats, in this sense: it has no part that is entirely distinct from each an