Identity and Modality (Mind Association Occasional Series)

Identity and Modality MIND ASSOCIATION OCCASIONAL SERIES This series consists of occasional volumes of original paper...

Author: Fraser MacBride

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Identity and Modality

MIND ASSOCIATION OCCASIONAL SERIES This series consists of occasional volumes of original papers on predeﬁned themes. The Mind Association nominates an editor or editors for each collection, and may cooperate with other bodies in promoting conferences or other scholarly activities in connection with the preparation of particular volumes. Publications Ofﬁcer: M. A. Stewart Secretary: R. D. Hopkins

Identity and Modality Fraser MacBride

CLARENDON PRESS · OXFORD

1

Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With ofﬁces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © the several contributors 2006 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Identity and modality / [edited by] Fraser MacBride. p. cm.—(Mind Association occasional series) Includes bibliographical references and index. ISBN-13: 978–0–19–928574–7 (alk. paper) ISBN-10: 0–19–928574–8 (alk. paper) 1. Identity (Philosophical concept) 2. Modality (Theory of knowledge) I. MacBride, Fraser. II. Series. BD236.I4155 2006 111 .82—dc22 2006009859 Typeset by Laserwords Private Limited, Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, King’s Lynn, Norfolk ISBN 0–19–928574–8 978–0–19–928574–7 1 3 5 7 9 10 8 6 4 2

CONTENTS

Acknowledgements

vii

Notes on contributors

ix

Introduction Fraser MacBride Part I: Modality

1

11

1. The Limits of Contingency Gideon Rosen

13

2. Modal Infallibilism and Basic Truth Scott Sturgeon

40

3. The Modal Fictionalist Predicament John Divers and Jason Hagen

57

4. On Realism about Chance Philip Percival

74

Part II: Identity and Individuation 5. Structure and Identity Stewart Shapiro 6. The Identity Problem for Realist Structuralism II: A Reply to Shapiro Jukka Ker¨anen 7. The Governance of Identity Stewart Shapiro

107 109

146 164

vi / Contents 8. The Julius Caesar Objection: More Problematic than Ever Fraser MacBride

174

9. Sortals and the Binding Problem John Campbell

203

Part III: Personal Identity

219

10. Vagueness and Personal Identity Keith Hossack

221

11. Is There a Bodily Criterion of Personal Identity? Eric T. Olson

242

Index

261

ACKNOWLEDGEMENTS

This volume grew out of a conference held at the University of St Andrews in July 2000, the ﬁrst of an ongoing series of conferences held under the auspices of Arch´e (now the AHRC Research Centre for the Philosophy of Logic, Language, Mathematics and the Mind). The conference was funded by the British Academy, the British Society for the Philosophy of Science, the John Wright Trust, the Mind Association, the Royal Society of Edinburgh, the Scots Philosophical Club, and the School of Philosophical and Anthropological Studies at the University of St Andrews. The Analysis Trust and the Stirling–St Andrews Graduate Programme performed the invaluable service of subsidising graduate attendance at the conference. I gratefully acknowledge the support of the Arts and Humanities Research Council who funded a period of leave during which the volume was prepared for publication. Fraser MacBride Birkbeck College London

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NOTES ON CONTRIBUTORS

John Campbell isWillisS. andMarion Slusser Professor of Philosophy, University of California at Berkeley. He is the author of Past, Space and Self (1994) and Reference and Consciousness (2002). John Divers is a Professor of Philosophy at Shefﬁeld University. He is the author of Possible Worlds (Routledge 2002). Jason Hagen is a graduate student in the Philosophy Department at Purdue University. Keith Hossack is Lecturer in Philosophy at King’s College London. He is the author of The Metaphysics of Knowledge (Oxford University Press, forthcoming). Jukka Ker¨anen is a visiting Professor in the Philosophy Department at North western University. Fraser MacBride is Reader in Philosophy at Birkbeck College London. He is the author of several papers in metaphysics and the philosophy of mathematics. Eric T. Olson is Reader in Philosophy at Shefﬁeld University. He is the author of The Human Animal: Personal Identity Without Psychology (Oxford University Press, 1997). Philip Percival is Reader in Philosophy at Glasgow University. His publications include papers on metaphysics and philosophy of science. Gideon Rosen is Professor of Philosophy at Princeton University. He is the author (with John P. Burgess) of A Subject With No Object (Oxford University Press, 1997). Stewart Shapiro is O’Donnell Professor of Philosophy at the Ohio State University and Professorial Fellow in Arch´e: the AHRC Research Centre for the Philosophy of Logic, Language, Mathematics & Mind at the University of St Andrews. His is the author of Foundations Without Foundationalism (Oxford

x / Notes on Contributors University Press, 1991), Philosophy of Mathematics: Structure and Ontology (Oxford University Press, 1997), and Thinking about Mathematics: Philosophy of Mathematics (Oxford University Press, 2000). Scott Sturgeon is Reader in Philosophy at Birkbeck College London. He is the author of Matters of Mind (Routledge, 2000).

Introduction Fraser MacBride

The papers in this volume constellate about fundamental philosophical issues concerning modality and identity: How are we to understand the concepts of metaphysical necessity and possibility? Is chance a basic ingredient of reality? How are we to make sense of claims about personal identity? Do numbers require distinctive identity criteria? Does the capacity to identify an object presuppose an ability to bring it under a sortal concept? In order to provide a guide to the reader I will provide a brief overview of the content of the papers collected here and some of the interrelations that obtain between them.

Part I: Modality In ‘The Limits of Contingency’ Gideon Rosen sets out to examine the modal status of metaphysical and mathematical propositions. Typically such propositions—that, for example, universals or aggregates or sets exist—are claimed to be metaphysically necessary. But such claims of metaphysical necessity, Rosen maintains, are inherently deﬁcient. This is because the kinds of elucidation philosophers typically offer of the concept of metaphysical necessity fail to pin down a unique concept of necessity; in fact no conception exactly ﬁts the elucidations given, and at least two conceptions—which Rosen dubs ‘Standard’ and ‘Non-Standard’—ﬁt the elucidations equally well. According to the Standard Conception, the synthetic apriori truths of basic

2 / Introduction ontology are always necessary. By contrast, according to the Non-Standard Conception, such truths are sometimes contingent. Consider, for example, Armstrong’s claim that qualitative similarity between particulars is secured by the recurrence of immanent universals. By the lights of the Standard Conception this claim, if it is true, is metaphysically necessary. For whilst it is not a logical or a conceptual necessity—there is no reason think it’s denial self-contradictory or otherwise inconceivable—it is not aposteriori either. But, by the lights of the Non-Standard Conception, Armstrong’s claim is contingent. For other metaphysical accounts that eschew universals—in favour, for example, of duplicate tropes—are also compatible with the nature of the similarity relation. So, if it is true, Armstrong’s claim tells us only about how similarity happens to be secured in the actual world; in other possible worlds similarity is secured differently. Since philosophical elucidations of the concept of metaphysical necessity favour neither the Standard nor the Non-Standard Conception Rosen concludes that philosophical discourse about metaphysical necessity is shot through with ambiguity, an ambiguity that we ignore at our peril. In ‘Modal Infallibilism and Basic Truth’ Scott Sturgeon investigates further the relationship between metaphysical possibility and intelligibility. Most philosophers agree that apriori reﬂection provides at best a fallible guide to genuine possibility. The schema (L) that says: if a proposition is intelligible then it is genuinely possible, is generally recognized not to be valid. Nevertheless, Sturgeon argues, philosophers have frequently failed to practise what they preach. They have been led by (L) to advance contradictory claims about the fundamental structure of reality. Sturgeon provides as a representative example of the capacity of (L) to mislead, a battery of six basic metaphysical claims about change and identity that Lewis has advanced but together generate contradiction. They generate contradiction because, Sturgeon maintains, Lewis accepts at least one instance of (L), inferring from the intelligibility of objects that endure identically through time—Sturgeon calls these ‘enduring runabouts’—that such objects are genuinely possible. The contradiction Sturgeon uncovers suggests that (L), if it is true at all, must be signiﬁcantly qualiﬁed. But, Sturgeon argues, there is no restricted reading of (L) that is valid either. The ﬁrst restriction Sturgeon considers qualiﬁes (L) to accommodate Kripke’s insight that there are intelligible propositions that fail to mark genuine possibility because the negations of these propositions are aposteriori necessary. Sturgeon rejects (L) so qualiﬁed because there is at

Fraser MacBride / 3 least one intelligible proposition P whose equally intelligible negation ¬P fails to be a posteriori necessary but nevertheless it cannot be the case that P and ¬P are both genuinely possible. Sturgeon provides as an example of such a P the Lewisian proposition that concrete possible worlds are the truth-makers for claims of genuine possibility. After considering yet further unsatisfactory qualiﬁcations to (L) Sturgeon concludes that philosophers have been mislead by the ‘ep-&-met tendency’, the human tendency to fuse epistemic and metaphysical matters; what is required is to recognize where this tendency misleads us whilst—and this is where the task becomes almost insuperably difﬁcult—continuing to respect the fact that it is a cornerstone of our modal practice that intelligibility defeasibly marks genuine possibility. In ‘The Modal Fictionalist Predicament’, John Divers and Jason Hagen turn to consider the metaphysics of modality itself. According to ‘genuine modal realism’, the metaphysical status of modal statements is rendered perspicuous by translating claims about what is possible into (counterparttheoretic) claims about possible worlds. But the doctrine that there really are such outlandish entities as possible worlds encounters familiar metaphysical and epistemological difﬁculties. However the adherents of ‘modal ﬁctionalism’ maintain that the beneﬁts of possible worlds discourse may be secured without these associated costs. They attempt to achieve this by conceiving of possible worlds discourse as itself just an immensely useful ﬁction that does not commit us to the existence of possible worlds. Part of what makes modal ﬁctionalism plausible is what Divers has called a ‘safety result’: the result that translating our ordinary modal claims in and out of the ﬁctional discourse of possible worlds will never lead us astray. However Divers and Hagen question whether the modal ﬁctionalist is in a position to take advantage of this result. Two objections to modal ﬁctionalism have arisen over the decade since the doctrine was ﬁrst advanced. According to the ﬁrst objection, modal ﬁctionalism, despite surface appearances, is committed to the existence of a plurality of possible worlds. According to the second objection modal ﬁctionalism is not even consistent; its acceptance results in modal collapse, so that for any modal claim X, both X and ¬X are true. Divers and Hagen argue that each objection may be avoided by deft handling of the doctrine. But what, they maintain, modal ﬁctionalists cannot do is to avoid one or other of these objections whilst maintaining a right to the safety result that makes modal ﬁctionalism plausible in the ﬁrst place. Divers and Hagen conclude that

4 / Introduction modal ﬁctionalism is in a serious predicament. Modal ﬁctionalism must be rescued from this predicament if it is to be considered a genuine competitor to genuine modal realism. Philip Percival’s ‘On Realism about Chance’ considers the metaphysical status of another modal notion, namely chance. Chance, as Percival conceives it, is a single-case (applying to individual events), temporally relative (liable to change over time), objective probability (existing independently of what anyone thinks about it) towards which our cognitive attitudes are normatively constrained. Percival construes the question of whether chance exists as the question of whether there are objectively true statements of the form ‘the chance at time t of event E is r’. Famously, Lewis has advanced realism about chance but Percival takes issue with this assessment, arguing for scepticism about the kinds of reason one might give for realism about chance. One common reason for afﬁrming realism about chance is that chance may be used toexplain statistical phenomena or the warrantednessof certain credences. But, Percival argues, the notion of chance cannot perform this kind of explanatory role. Consequently, an inference to the best explanation of (e.g.) statistical phenomena cannot be employed to ground realism about chance. Another reason commonly offered for afﬁrming realism about chance is that chance may be analysed in terms of non-chance. If chance is analysable then either chance supervenes (relatively) locally upon non-chance or chance supervenes globally upon non-chance. Buthowever chance supervenes, Percival argues, no extant analysis—including Lewis’s ‘best-system’ analysis—succeeds. Percival concludes upon the sceptical reﬂection that there is little prospect of a correct analysis of chance being forthcoming in the future that vindicates realism about chance.

Part II: Identity and Individuation The next three papers reﬂect upon the identity and individuation of mathematical objects. In his ‘Structure and Identity’ Stewart Shapiro reﬂects upon the doctrine (advanced in his Philosophy of Mathematics: Structure and Ontology (Oxford: OUP, 1997)) that mathematical objects are places in structures where the latter are conceived as ante rem universals. This doctrine—that Shapiro dubs ‘ante rem structuralism’—suggests that there is no more to a mathematical object than the (structural) relations it bears to the other objects within

Fraser MacBride / 5 the structure to which it belongs. However, as Shapiro recognizes, when conceived in this way ante rem structuralism is open to a variety of criticisms. This is because there appears to be more to a mathematical object than the relations it bears to other objects within its parent structure. Mathematical objects enjoy relations to (i) items outside the mathematical realm (e.g. the concrete objects they are used to measure or count) and (ii) objects that belong to other structures inside the mathematical realm. Moreover, (iii) there are mathematical objects (e.g. points in a Euclidean plane) that are indiscernible with respect to their (structural) relations but nevertheless distinct. This makes it appear that ante rem structuralism is committed to the absurdity of identifying these objects. Shapiro seeks to overcome these difﬁculties by a series of interlocking manoeuvres. First, he seeks to overturn the metaphysical tradition about numbers, suggesting that it may be contingent whether a given mathematical object is abstract or concrete. Second, Shapiro questions whether mathematical discourse is semantically determinate. Finally, Shapiro rejects the requirement that ante rem structuralism provide for the non-trivial individuation of mathematical objects. In ‘The Identity Problem for Realist Stucturalism II: A Reply to Shapiro’ Jukka Ker¨anen argues that Shapiro nevertheless fails to provide an adequate account of the identity of numbers conceived as places in structures. According to Ker¨anen, it is an adequacy constraint upon the introduction of a type of object that some account be given of the kinds of fact that metaphysically underwrite the sameness and difference of objects of this type. More speciﬁcally, Ker¨anen favours the view that facts about the sameness and difference of objects must be underwritten by facts about the properties they possess or relations they stand in. He holds up set theory as an exemplar of a theory that meets this adequacy constraint, grounding the identity of sets—via the Axiom of Extensionality—in facts about their members. Ker¨anen doubts, however, whether ante rem structuralism can meet this adequacy constraint because there are no structural properties or relations that can be used to distinguish between (e.g.) the structurally indiscernible points in a Euclidean plane. Of course, the structuralist can meet the constraint by force majeure, positing a supply of haecceitistic properties to distinguish between structurally indiscernible objects. But, as Ker¨anen reﬂects, the positing of haecceities opens up the possibility of indiscernible structures that differ only haecceitistically. Since mathematical discourse lacks the descriptive resources to distinguish between these structures, this manoeuvre on the part of the structuralist

6 / Introduction threatens to render reference to mathematical objects deeply inscrutable. Ker¨anen concludes that the particular difﬁculties encountered by ante rem structuralism in particular reﬂect deep difﬁculties for ontological realism in general. ‘The Governance of Identity’ is Shapiro’s response to Ker¨anen. Shapiro ﬁrst concedes, for the sake of argument, the adequacy constraint on the introduction of a type of object Ker¨anen imposes. Shapiro then argues that indiscernible objects within a structure S may be distinguished by embedding S within a larger structure S∗ whose positions are discernible. Later, lifting the concession, Shapiro questions whether it is necessary to supply non-trivial identity conditions for a type of object introduced. He concludes rather that identity must be taken as primitive. In ‘The Julius Caesar Objection: More Problematic than Ever’ Fraser MacBride further explores issues surrounding the identity and individuation of numbers from a Fregean point of view. According to Frege it is a requirement upon the introduction of a range of objects into discourse that identity criteria are supplied for them—criteria that determine whether it is appropriate to label and then relabel an object on a different occasion as the same again. In order to introduce cardinal numbers into discourse Frege therefore proposed the following principle—Hume’s Principle that speciﬁes necessary and sufﬁcient conditions for the identity of cardinal numbers: the number of Fs = the number of G’s iff there is a 1-1 correspondence between the Fs and the Gs. Famously, however, Frege became dissatisﬁed with Hume’s Principle as a criterion of identity, maintaining that it failed even to settle whether (e.g.) the number two was identical or distinct to an object of an apparently quite different sort (e.g.) the man Caesar. MacBride subjects this difﬁculty—the so-called ‘Julius Caesar Objection’—to critical examination, arguing that beneath the superﬁcial simplicity of the problem that bedevilled Frege there lies a welter of distinct difﬁculties. These may be arranged along three different dimensions. (A) Epistemology: does the identity criterion supplied for introducing numbers into discourse provide warrant for the familiar piece of common-sense knowledge that numbers are distinct from persons? (B) Metaphysics: doesthe identity criterion given determine whether the things that are numbers might also be such objects as Caesar? (C) Meaning: does the identity criterion supplied bestow upon the expressions that purport to denote numbers the distinctive signiﬁcance of singular terms? It is because, MacBride argues, these different problems and the interrelations between them often

Fraser MacBride / 7 fail to be disentangled that (in part) the different (purported) solutions to the Julius Caesar—neo-Fregean and supervaluationist solutions—fail. MacBride concludes by suggesting that Frege may have been too strict in imposing the requirement that objects introduced into discourse have identity criteria, noting that not even sets have identity criteria in the strict sense Frege required. John Campbell’s ‘Sortals and the Binding Problem’ sets out to question the related doctrine that singular reference to an object depends upon a knowledge of the sort of object (whether a number or a man) to which one is referring. Part of what makes this doctrine plausible is the fact that, as Quine emphasized, our pointing to something remains ambiguous until the sort of thing that we are pointing is made evident. For example, I can point towards the river and variously be taken to refer to the river itself which continues downstream, a temporal part of the river that exists contemporaneously with my pointing gesture, the collection of water molecules that occupies the river when I point, and so on. But if I specify the sort of object to which I wish to draw your attention then it becomes determinate what I am pointing to. These kinds of consideration have led philosophers to adopt what Campbell calls ‘The Delineation Thesis’: Conscious attention to an object has to be focused by the use of a sortal concept that delineates the boundaries of the object to which you are attending. Campbell argues however that the delineating thesis is false. Instead, Campbell proposes, attention to an object arises from the way in which the visual system binds together the information it receives in various processing streams. Roughly speaking, the visual system does so by exploiting the location of an object together with the Gestalt organization of characteristics found at that location. Since this integration may be achieved without the use of a semantic classiﬁcation of an object as of a certain sort it appears that we can single out an object without the use of a sortal concept. Philosophers have nevertheless been mislead into supposing the Delineation Thesis because, Campbell maintains, of the typical use that is made of sortal concepts in demonstrative constructions (‘that mountain’) and our readiness to withdraw these constructions when it transpires that these sortal concepts are misapplied (when, for example, it turns out that our attention is being drawn to what is merely a hill). Campbell argues nonetheless that sortal concepts employed in demonstrative construction serve merely to orientate our attention to an object without necessarily contributing to the content of what is said by the use of these constructions.

8 / Introduction

Part III: Personal Identity The remaining two papers of the collection turn to a consideration of issues related to the identity of persons. In ‘Vagueness and Personal Identity’ Keith Hossack considers the inﬂuential ‘Bafﬂement Argument’ put forward by Bernard Williams, an argument that threatens to undermine the materialistic conception of the self. The well-known thought experiments about personal identity suggest that there are possible situations—where, for example, a subject undergoes ﬁssion—in which it is indeﬁnite whether the subject survives. If materialism is true it appears that this indeﬁniteness must be objective. For it appears that there are no sharp boundaries to the biological processes or physical mechanisms that sustain human life. By contrast, if dualism is true it appears that this indeﬁniteness can only be a matter of ignorance. For the kinds of issues in ethics and philosophy of religion that give rise to dualism suggest that the boundaries between souls must be sharp. Williams’s Bafﬂement Argument suggests however that we cannot make sense of objectively indeﬁnite identity in the case of persons, and so materialism must be abandoned. This is because we cannot make sense—we are bafﬂed by—the suggestion that it is objectively indeﬁnite whether I (or you) will continue to exist tomorrow. Hossack seeks to defend materialism by showing that the Bafﬂement Argument owes its persuasive force to a skewed conception of the self that fails to recognize that the correct way to understand the ‘I’ concept is as the intersection of subjective and objective ways of thinking about the self. What is wrong with the Cartesian conception of the self is that it fails to give due weight to the location of the self in the objective worldly order. But what is wrong with the bodily conception of the self—a conception advanced, for example, by Strawson and Evans—is that by identifying the self with the body it fails to sufﬁciently stress the subjective aspect of the ‘I’ concept. The mistake that underlies the Bafﬂement Argument, Hossack maintains, is a misguided solipsistic conception of the self that arises from focusing exclusively upon the subjective aspect of the ‘I’ concept. Once this mistake is corrected by giving proper weight to the place of persons in the objective order—without falling over into the corresponding failings of the bodily conception of the self—the Bafﬂement Argument need no longer pose a threat to materialism. Eric T. Olson’s ‘Is There a Bodily Criterion of Personal Identity?’ continues the theme of questioning how we relate to our bodies. One of the perennial debates about personal identity concerns whether we should adopt

Fraser MacBride / 9 a bodily criterion of personal identity as opposed, say, to a psychologistic criterion. But this debate only makes sense if there is such a thing as a bodily criterion of personal identity; about the existence of such a criterion Olson expresses scepticism. The bodily criterion is supposed to offer an account according to which we are our bodies or, at least, that our identity over time consists in the identity of our bodies. So the bodily criterion is supposed to be a non-trivial thesis about our bodies and how we are related to them that determines that we go where our bodies go. But, Olson argues, we cannot specify the bodily criterion in such a way as to ensure that it does what it is supposed to do. Olson’s argument for this conclusion proceeds by elimination, considering in turn a variety of different purported speciﬁcations of the bodily criterion. Either these criteria imply too little or they imply too much: either (1) they say nothing about, or leave it open that we may survive, the destruction of our bodies or (2) they imply that you could never be a foetus or a corpse. It may be suggested that the difﬁculties identiﬁed are a consequence of the surreptitious assumption of a Cartesian account of body ownership. But Olson dismisses this suggestion, arguing that the accounts of body ownership proposed by Shoemaker and Tye imply that the bodily criterion is not the substantial thesis debate assumes but a trivial consequence of materialism. How did such a depth of misunderstanding arise? Olson ventures a diagnosis. We are misled by the superﬁcial grammar of such expressions as ‘Wilma’s body’; in this case, an expression that appears to be the name of an object with which Wilma enjoys an especially intimate relationship. But, Olson argues, we should no more believe that ‘Wilma’s body’ names a special object than we should believe that the expression ‘Wilma’s mind’ names another object with which she enjoys a different, but not less intimate, relationship.

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Part I Modality

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1 The Limits of Contingency Gideon Rosen

1. What is Metaphysical Necessity? There are two ways to understand the question. We might imagine it asked by an up-to-date philosopher who grasps the concept well enough but wants to know more about what it is for a proposition to hold of metaphysical necessity. Alternatively, we might imagine it asked by a neophyte who’s never heard the phrase before and simply wants to know what philosophers have in mind by it. My main interest in this paper is the ﬁrst sort of question. But for several reasons it will help to begin with the second. Suppose it were your job to explain the concept of metaphysical necessity to a beginner. What might you say? The task is not straightforward. The concepts of metaphysical possibility and necessity are technical concepts of philosophy. Not only is the phrase ‘metaphysical necessity’ a bit of jargon. No ordinary word or phrase means exactly what the technical phrase is supposed to mean. So you cannot say, ‘Ah, it’s really very simple: What we call ‘‘metaphysical necessity’’ is what you call. . .’. If you’re going to explain the technical idiom to the neophyte you’re going to have to introduce him to a novel concept. You’re going to have to teach him how to make distinctions that he does not already know how to An earlier version of this paper was presented at the Arch´e conference in St Andrews in June 2000. I am grateful to Scott Sturgeon for his exemplary comments on that occasion and for extensive correspondence. I have been unable to take many of his important suggestions into account.

14 / The Limits of Contingency make. And so the question comes down to this: How is this novel concept to be explained? The best sort of explanation would be an informative deﬁnition: an explicit speciﬁcation, framed in ordinary terms, of what it means to say that P is metaphysically necessary. Unfortunately, no such deﬁnition is readily available. But we know in advance that it must be possible to get along without one. For the fact is that no one comes to master the concept of metaphysical necessity in this way. To the contrary, the project of deﬁnition and analysis in modal metaphysics invariably presupposes aprior grasp of the technical modal concepts. For up-to-date philosophers it always works like this: First, we learn what it means to say that P is metaphysically necessary. Then we look for an account of what this comes to in other terms. We may or may not ﬁnd one. It would not be surprising if the basic concepts of modal metaphysics were absolutely fundamental.1 But even if we do, our capacity to recognize it as correct will depend on prior grasp of the metaphysical modal idiom. And it is this prior grasp that we are attempting to inculcate in our neophyte. If we do not begin with a deﬁnition, we must offer some sort of informal elucidation. We all know roughly how this works in other parts of philosophy. The neophyte is presented with a battery of paradigms and foils, ordinary language paraphrases (with commentary), and bits and pieces of the inferential role of the target notion, and then somehow as a result of this barrage he cottons on. To ask how the concept of metaphysical necessity might be explained to the neophyte is to ask how this informal elucidation ought to go in the modal case. No doubt, there is more than one way to proceed. But here is one possibility.

2. An Informal Elucidation The ﬁrst thing to say is that metaphysical necessity is a kind of necessity. To say that P is metaphysically necessary is to say that P must be the case, that it has to be the case, that it could not fail to be the case, and so on. If the ordinary modal idioms were univocal, this would be enough. But it clearly isn’t enough. When I drop an apple there is a sense in which it cannot fail to fall. When I promise to meet you there is a sense in which I have to keep the date. But these claims involve two very different modal notions, and neither is a claim 1

For an argument to this effect, see Kit Fine (2002).

Gideon Rosen / 15 of metaphysical necessity. So granted, the metaphysical ‘must’ is a kind of ‘must’. The challenge remains to distinguish it from the many others. Some distinctions are easy. Thus, unlike the various practical and ethical ‘musts’, the metaphysical ‘must’ is alethic. If P is metaphysically necessary, then it’s true. And unlike the various epistemic and doxastic musts (e.g. the ‘must’ in ‘She must be home by now. She left an hour ago.’) claims of metaphysical necessity are not in general claims about what is known or believed. Other distinctions are less straightforward. Thus, some philosophers believe in something called logical necessity, and some believe in something called analytic or conceptual necessity, where a truth is logically necessary when some sentence that expresses it is true in all models of the language (or some such thing) and conceptually necessary when it is true in virtue of the concepts it contains (or some such thing). It is controversial whether these are genuine species of necessity. After all, it is one thing to say that P is necessary in some generic sense because it is a truth of logic. It is something else to say that P therefore enjoys a special sort of necessity. But if there is a distinctively logical or conceptual species of necessity, then it is (presumably) both alethic and non-epistemic, and in that case we must say something to distinguish metaphysical necessity from such notions. At this point the usual procedure is to invoke an epistemological distinction along with certain crucial paradigms. One says: ‘Unlike the various logical and semantic species of necessity, metaphysically necessary propositions are sometimes synthetic and aposteriori. To a ﬁrst approximation, the logicoconceptual necessities are accessible to ‘Humean reﬂection’. To suppose the falsity of a logical or a conceptual truth is to involve oneself in the sort of selfcontradiction or incoherence that a sufﬁciently reﬂective thinker might detect in the armchair simply through the exercise of his logical and semantic capacities. By contrast, some metaphysically necessary truths can be rejected without such incoherence. The most famous examples are the Kripkean necessities: true identities ﬂanked by rigid terms; truths about the essences of individuals, kinds, and stuffs. But one might also mention the claims of pure mathematics in this connection. Mathematical truths are among the paradigms of metaphysical necessity. But logicism not withstanding, it is not self-contradictory to reject mathematical objects across the board, or to deny selected existential principles such as the axiom of inﬁnity. So if substantive truths of these sorts can be necessary in the metaphysical sense, metaphysical necessity differs from logical or conceptual necessity. Indeed, the natural thing

16 / The Limits of Contingency to say is that metaphysical necessity is a strictly weaker notion, in the sense that some metaphysically necessities are neither logical nor conceptual necessities, but not vice versa. Let’s call any modality that is alethic, non-epistemic, and sometimes substantive or synthetic a real modality. So far we have it that metaphysical modality is a real modality. But this is still not enough to pin the notion down. For the same might be said of the various causal or nomic modalities: physical necessity, historical inevitability, technical impossibility (as in: ‘It’s impossible to fabricate an artiﬁcial liver’), and so on. And here the natural thing to say is that among the real modalities, the metaphysical modalities are absolute or unrestricted. Metaphysical necessity is the strictest real necessity and metaphysical possibility is the least restrictive sort of real possibility in the following sense: If P is metaphysically necessary, it is necessary in every real sense: If P is really possible in any sense, then it’s possible in the metaphysical sense. So if you can’t square the circle because it’s metaphysically impossible to square the circle, then it’s certainly not physically or biologically possible for you to do so. But if you can’t move faster than the speed of light because to do so would be to violate a law of nature, then it does not follow that superluminal velocities are metaphysically impossible. One has the palpable sense—though philosophy might correct it—that some of the laws of nature might have been otherwise. To say this is not just to say that these laws are not logical or conceptual truths. That is too obvious to be worth saying. And it’s certainly not to say that the laws of nature amount to physical contingencies. No, to entertain the philosophical suggestion that the laws might have been otherwise is to presuppose that there exists a genuine species of contingency ‘intermediate’ between physical contingency on the one hand and conceptual contingency on the other. Focus on this sense of contingency, it might be said, and you are well on your way to knowing what ‘metaphysical’ modality is supposed to be. It is the sort of modality relative to which it is an interesting question whether the laws of nature are necessary or contingent.

3. A Question about the Informal Elucidation Informal explanations of this sort are the indispensable starting point for modal metaphysics. In the end we may hope for more: an account of what it is for a proposition to be metaphysically necessary; an account of what in reality makes it the case that P

Gideon Rosen / 17 is metaphysically necessary when it is. But before we can ask these profound questions, we must identify our topic. We must distinguish metaphysical possibility and necessity from the various other species of possibility and necessity. And it is natural to suppose that for this restricted purpose something like the informal explanation sketched above should be sufﬁcient. In fact, this is almost universally supposed. A small handful of philosophers reject the notions of metaphysical possibility and necessity altogether.2 But among those who accept them, it is universally assumed that a question about the metaphysical modal status of any given proposition is clear and unambiguous, at least as regards the predicate. We may not be able to say what metaphysical necessity really is in its inner most nature. But thanks to the informal elucidation sketched above or something like it, we know enough about it to ask unambiguous questions about its nature and its extension. Our questions may be hard to answer. In some cases we may not even know where to begin. But even so, it is perfectly clear what is being asked when we ask whether P holds as a matter of a metaphysical necessity. One of my aims in this paper is to reconsider this supposition. I shall suggest that the informal explanation sketched above is consistent with two distinct conceptions of necessity and possibility; or better, since no single conception is fully consistent with the sketch, that two relatively natural conceptions ﬁt the elucidation equally well. If this is right then our working conception of metaphysical necessity is confused in the sense in which the Newtonian conception of mass is supposed to have been confused.3 Questions about metaphysical necessity are ambiguous, and where divergent resolutions of the ambiguity yield different answers, the modal question as we normally understand it has no answer. Indeed, I shall suggest that this is just what we should think about an interesting (though largely neglected) class of questions.

4. The Standard Conception and the Differential Class If there are two conceptions of metaphysical necessity, they must overlap considerably in extension. The informal explanation functions as a constraint 2

Dummett (1993: 453) calls it ‘misbegotten’, though he is elsewhere moderately sympathetic to a closely related notion of ‘ontic necessity’ (cf. Dummett 1973: 117; 1981: 30). Field (1989: esp. 235 ff.) expresses general skepticism about the notion. 3 Field (1974).

18 / The Limits of Contingency on both, and it includes a list of paradigms, a signiﬁcant number of which must count as ‘metaphysically necessary’ on any modal conception worth the name. On the account I propose to consider, the (non-indexical) logical truths and the conceptual truths more generally will count as necessary on both conceptions, as will the uncontroversial Kripkean necessities: (propositions expressed by) true identities ﬂanked by rigid terms and essential predications: propositions of the form Fa, where a is essentially F. The two conceptions will diverge in application to certain claims of fundamental ontology, which do not slot easily into any of these categories. For an example of the sort of claim I have in mind, consider the axioms of standard set theory. At least one is plausibly analytic—the axiom of extensionality, according to which sets are identical if and only if they have just the same members. Insofar as this axiom is uncontroversial, it does not entail that sets exist. It says that if sets exist, they are extensional collections. And since this is presumably part of what it is to be a set—or if you prefer, part of what the word ‘set’ means—the axiom of extensionality will count as metaphysically necessary on any reasonable conception of the notion.4 The same cannot be said for the remaining axioms. Consider the simplest: the pair set axiom: (Pairing) For any things x and y, there exists a set containing just x and y. In conjunction with Extensionality, Pairing entails that given a single non-set, inﬁnitely many sets exist. The truth of Pairing is not guaranteed by what it is to be a set, or by what the word ‘set’ means. It may lie in the nature of the sets to satisfy the principle, but only in the sense that if there are any sets, then it lies in their collective nature to conform to pairing. (That is part of what makes them the sets, it might be said.) It may be that no relation deserves the name ‘∈’ unless it satisﬁes the pairing axiom, just as nothing deserves the name ‘bachelor’ unless it is male. But it is not in the nature of bachelorhood to be instantiated; and likewise, it is not in the nature of the epsilon relation that something should bear it to something else. You can know full well what set membership is supposed to be—what it is to be a set, what the word ‘set’ means—without knowing whether any sets exist, and hence without knowing whether Pairing is true. 4 But see Frankel, Bar Hillel, and Levy (1973: 27–8). For discussion of the analyticity of extensionality, see Maddy (1997: 39).

Gideon Rosen / 19 What is the modal status of the Pairing axiom? Suppose it’s true. Is it a metaphysically necessary truth or a contingent one? As we have said, it is traditional to regard the truths of pure mathematics as paradigms of metaphysical necessity. On this view, while there may be room for dispute about whether sets exist, and if so, which principles they satisfy, there is no room for dispute about the modal status of those principles. If sets exist and satisfy Pairing, then Pairing holds of necessity. If sets do not exist (or if they do exist and somehow fail to satisfy the principle) then not only is Pairing false; it could not possibly have been true. What I call the Standard Conception of metaphysical necessity extends this familiar thought to a range of synthetic claims in metaphysics. As another example, consider classical mereology. Once again, some of the axioms are plausibly analytic. ‘A is part of A’; ‘If A is part of B and B of C, then A is part of C’. But the ‘analytic core’ of the theory does not entail that composite things exist, or that they must exist given the existence of at least objects. It says (in effect) that a relation counts as the mereological part-whole relation only if it is transitive and reﬂexive. But it does not say whether two things ever manage to stand in this relation. By way of contrast, consider the axiom that gives the theory its teeth. UMC: Whenever there are some things, there is something that they compose (where the Fs compose X iff every F is part of X and every part of X overlaps an F ). UMC is not a conceptual truth. Given anodyne input it delivers an entity composed of my head and your body, Cleopatra’s arms and Nixon’s legs. And whatever one thinks of such scattered monstrosities, it is not a sign of logicolinguistic confusion to reject them.5 Nor is it true in virtue of the nature of the part–whole relation. Once again, a conditional version of the principle might be accorded such a status, viz.: If there are mereological aggregates, then whenever there are some things, there is something they compose. But you can know perfectly well what a mereological aggregate is supposed to be (as the opponents of classical mereology clearly do) without being in a position to assert the unconditional version of UMC. UMC and pairing have at least this much in common. (a) They are substantive principles. They can be rejected without self-contradiction or 5

For more on the epistemological status of UMC, see Dorr and Rosen (2001).

20 / The Limits of Contingency absurdity. (b) They entail the existence of a distinctive sort of object (perhaps conditionally on the existence of things of some other, more basic, sort). (c) Their epistemological status is uncertain, but they are palpably more apriori than aposteriori. If they are empirical truths they are empirical truths of a peculiar sort, since it is hard to imagine a course of experience that would bear differentially upon their acceptability. (d) They concern matters of basic ontology. Unlike the principles of ornithology, for example, they are not concerned with what exists hereabouts. To put the point somewhat grandly, they concern the structure of the world, not just its inventory. (e) They are standardly regarded as metaphysically non-contingent. Philosophers have questioned the existence of sets and mereological aggregates. But hardly anyone has suggested that the basic principles governing such things might have been other than they are. These are some central features of what I shall call the Differential Class: the class of claims with respect to which the two conceptions of metaphysical necessity will diverge. On the Standard Conception, the synthetic apriori truths of basic ontology are always necessary. On the Non-Standard Conception, as I shall call it, they are sometimes contingent. This characterization of the Differential Class leaves much to be desired. Matters will improve somewhat as we proceed. But for now it may help to list some further examples. Existence claims elsewhere in mathematics, e.g. the existential principles of arithmetic and analysis. Neo-Fregean abstraction principles of the form ‘F(a) = F(b) iff a and b are equivalent in some respect’, e.g. ‘The temperature of a = the temperature of b iff a and b are in thermal equilibrium.’ Meinongian abstraction principles to the effect that for any (suitably restricted) class of properties, there exists an abstract entity (arbitrary object, subsistent entity) that possesses just those properties. Accounts of the ontological underpinnings of genuine similarity; e.g. the neo-Aristotelian claim that whenever a and b are genuinely similar, they have an immanent universal part in common. Accounts of the ontological underpinnings of persistence through time, e.g. the claim that whenever a persisting object exists at a time it has a momentary part that exists wholly at that time.

Gideon Rosen / 21 In each of these domains we are concerned with synthetic, seemingly nonempirical facts of metaphysics. The Standard Conception does not say which claims are true in these areas. But it does say that the truth, whatever it is, could not be otherwise. If Peter van Inwagen (1990) is right that a plurality of material things constitute a single thing only whether their activity constitutes the life of an organism, then the Standard Conception says this is so of necessity. If Hartry Field (1989) is right that abstract objects do not exist, then according to the Standard Conception, this sort of nominalism is a necessary truth.6

5. The Non-Standard Conception The Standard Conception is familiar. Insofar as you have any use for the concept of metaphysical necessity, it is probably your conception. The Differential Class is a class of metaphysical principles par excellence, and we normally take it for grantedthatmetaphysicshasa metaphysically non-contingentsubjectmatter.7 That’s what we think. But consider the Others: a tribe of outwardly competent philosophers whose contact with the mainstream has been intermittent over the past (say) thirty years. The Others share our tradition and they are concerned with many of the same problems. In particular, they take themselves to have absorbed the main lessons of the modal revolution of the 1960s. Metaphysical modality is the modality that mainly interests them, and they do not confuse it with analyticity and the other semantico-epistemological modalities. When they introduce the notion to their students their informal gloss is much like ours. In particular, they agree that the Kripkean ‘aposteriori’ necessities are paradigm cases of metaphysical necessity, along with the truths of logic and the analytic truths more generally. You’ve been looking in on the Others, reading their journals, attending their conferences; and so far as you can tell they might as well be some of 6

See also Field (1993). As noted above, Field himself rejects the notion of metaphysical necessity. For him, modal questions about abstract objects can only be questions about the conceptual modalities. Since Field himself regards both nominalism and its negation as non-self-contradictory in the relevant sense, he regards the existence of mathematical objects as a contingent matter. 7 With some exceptions. It is widely acknowledged, for example, that the debate over materialism (or physicalism) concerns a contingent proposition. (See Lewis 1983.) The suggestion to follow is that much of what passes under the name ‘ontology’ might be understood in a similar spirit.

22 / The Limits of Contingency Us. But now you see something that makes you wonder. In the philosophy of mathematics seminar, Professors P and N disagree about whether sets exist. According to P, the utility of set-theoretic mathematics gives us reason to believe the standard axioms. N agrees that set theory is useful, but points out that it is just as useful in worlds without sets as it is in worlds that have them, and so maintains, on grounds of economy, that set theory is best regarded as a useful ﬁction. You’ve heard most of this before, but you are struck by the suggestion that sets might exist in some worlds but not in others.8 You know that we sometimes indulge in loose talk of this sort amongst ourselves. But you want to know whether N takes the idea seriously. So you ask, and she answers: ‘I meant exactly what I said. Platonism may be proﬂigate, but it is not incoherent or self-contradictory. I can conceive a world in which sets exist. I can conceive a world in which they don’t. Each view thus corresponds to a metaphysical possibility. There might have been sets, but then again, there might not have been. The only question is which sort of world we inhabit. That’s what my colleague and I disagree about.’ You are ﬂabbergasted—not simply by the suggestion that the truths of mathematics might be contingent, but by the blithe transition from a claim of conceivability to a claim of metaphysical possibility. You point out that we’ve known for years that the inference from conceivability to possibility is no good. ‘There is no incoherence in the supposition that water is an element,’ you say. ‘But even so, we know that water could not possibly have been an element. You agree about this. So how can you be so blas´e about the corresponding inference in the case of sets?’ ‘Ah, but the cases are very different,’ says N. ‘The ancients could see no incoherence in the supposition that water is an element. Indeed, insofar as they had reason to believe that water was an element, they had reason to believe that there was no such incoherence. Perhaps this gives a sense in which it was conceivable for the ancients that water should have been an element. And if so, we agree: that sort of conceivability does not entail possibility. But when I say that a world containing sets is conceivable, I have in mind a somewhat different sort of conceivability. I’m talking about what we call informed or correct conceivability. Here’s the idea:’ ‘If the ancients could conceive a world in which water is an element, this is only because they were ignorant of certain facts about the natures of things. 8

See van Fraassen (1977).

Gideon Rosen / 23 In particular, it is because they did not know what it is to be water. They did not know that to be water just is to be a certain compound of hydrogen and oxygen—that to be a sample of water just is to be a quantity of matter predominantly composed of molecules of H2 O. This is not to say that they did not understand their word for water. But it’s one thing to understand a word, another to know the nature of its referent. The ancients could see no contradiction in the supposition that water is an element because they did not know that water is a compound by its very nature. But we know this; and given that we do, we can see that to suppose a world in which water is an element is to suppose a world in which a substance that is by nature a compound is not a compound. And that’s absurd.’ ‘In one sense of the phrase, P is conceivable for X if and only if that X can see no absurdity or incoherence in the supposition of a world in which P is true. Correct conceivability begins life as an idealization of this notion of relative conceivability. To a ﬁrst approximation, P is correctly conceivable iff it would be conceivable for a logically omniscient being who was fully informed about the natures of the things. The mind boggles at this sort of counterfactual, to be sure. But once we see what it amounts to, we can see that it is merely heuristic. If it’s true that an ideally informed conceiver would see no absurdity in the supposition of a P-world, this is because there is no such absurdity to be seen. The ideally informed conceiver is simply an infallible detector of latent absurdity. And once we see this we can drop the reference to the ideal conceiver altogether.’ ‘As we understand the notion, metaphysical possibility is, as it were, the default status for propositions. When the question arises, ‘‘Is P metaphysically possible?’’ the ﬁrst question we ask is ‘‘Why shouldn’t it be possible?’’ According to us, P is metaphysically possible unless there is some reason why it should not be—unless there is, as we say, some sort of obstacle to its possibility. Moreover, the only such obstacle we recognize is latent absurdity or contradiction.9 If the 9

What is an ‘absurdity’ in the relevant sense? For present purposes, it will sufﬁce to take an absurdity to be a formal contradiction: a proposition of the form P & not-P, or a = a. [This assumes that propositions, as distinct from the sentences that express them, may be said to have a ‘form’.] A complication arises from the fact that not everyone agrees that contradictions are absurd in the relevant sense. Dialethists maintain that some contradictions are not manifestly absurd. Nearly everyone else disagrees. This proponent of the Non-Standard Conception may remain neutral on this point. His fundamental contention is that a proposition is metaphysically impossible when it entails a manifest absurdity or impossibility. For the purposes of exposition, I assume that this

24 / The Limits of Contingency question arises, ‘‘Why shouldn’t there by a world at which P is true?’’ the only answer cogent response is a demonstration that the supposition that there is such a world involves a contradiction or some other manifest absurdity. (This is tantamount to a principle of plenitude. It has the effect that the space of possible worlds is as large as it can coherently be said to be.) Now, whether P harbors an absurdity is not in general an apriori matter. To say that it does is to say that P logically entails an absurdity given a full speciﬁcation of the natures of the items it concerns. And since these natures are often available only aposteriori, it is often an aposteriori matter whether P is correctly conceivable. That’s why it is often an aposteriori matter whether P is metaphysically possible.’10 The Others have adopted the Non-Standard Conception of the metaphysical modalities. According to this conception, correct conceivability—logical consistency with propositions that express the natures of things—is both necessary and sufﬁcient for metaphysical possibility. This need not be construed as a reductive analysis. It may be that the full account of correct conceivability must make use of metaphysical modal notions.11 But even if the equivalence is not reductive, it may nonetheless be true. And if it is then it would appear to yield a series of deviant verdict about the Differential truths. Consider the Pairing axiom once again. The axiom and its denial are both logically consistent. Moreover, it is plausible that both are correctly conceivable. To be sure, we have no adequate conception of what it is to be a set. But even in the absence of a fully explicit such conception, we can amounts to entailing a contradiction. But this is not strictly speaking a commitment of the view. A complete account of the Non-Standard Conception would involve an account of the more fundamental notion. 10

The Non-Standard Conception presented here is inspired by some remarks of Kit Fine. Fine (1994) deﬁnes metaphysical necessity as truth in virtue of the natures of things. However, Fine would not agree that the account is revisionary in the ways I have suggested. In particular, he would not agree that the account entails that the existential truths of mathematics and metaphysics are uniformly contingent. The question would seem to come down to whether natures are to be construed as ‘conditional’ or ‘Anti-Anselmian’ (see below): whether it can lie in the nature of some thing that it exist, or whether it can lie in the nature of some kind that it have instances whenever some more basic kind has instances. I am grateful to Fine for conversation on these questions and for his eye-opening seminar at Princeton in 1999. But he would certainly resist my abuse of his ideas in the present context. 11 The account of correct conceivability involves three ingredients: the notion of a proposition, the notion of logical entailment among propositions, and the notion of an absurdity or contradiction. It may well be that a correct account of some or all of these notions presupposes the notion of metaphysical necessity.

Gideon Rosen / 25 consider the alternatives. It is hardly plausible that it lies in the nature of the set-membership relation to violate the Pairing axiom.12 So either the nature of the relation is silent on whether Pairing is true, or it lies in the nature of the relation to satisfy the principle. In the former case it is automatic that both the axiom and its negation are correctly conceivable. So the relevant case is the latter. Here is what the Others have to say about it. ‘If it lies in the nature of the sets (or the relation of set-membership) to conform to Pairing, then it is indeed incoherent to suppose a world in which sets exist and Pairing is false. But that is not the negation of Pairing. The negation of the principle amounts to the claim that either there are no sets, or sets exist and some things X and Y lack a pair set. Our claim that the negation of the axiom is correctly conceivable depends on the thought that no contradiction follows from the supposition that sets do not exist. Because we can see no such absurdity, and we can’t see how more information about the natures of the items in question could make a difference, we conclude that the Pairing axiom and its negation both correspond to genuine possibilities.’ This little speech brings out an important feature of the Others’ talk of natures and essences. For the Others, all natures are conditional or Kantian or perhaps Anti-Anselmian. To say that it lies in the nature of the Fs to be G is to articulate a condition that a thing must satisfy if it is be an F. It is to give a (partial) account of that in virtue of which the Fs are F. It is not obviously incompatible with this interpretation that existence (or existence given the existence of things of some more basic kind) should be part of the nature of a thing or kind. But even when it is, it will not be incoherent to deny the existence of that thing or kind (or to deny it when the alleged condition has been satisﬁed). As Kant says in a somewhat different context: If, in an identical proposition, I reject the predicate while retaining the subject, contradiction results . . . But if we reject subject and predicate alike, there is no contradiction; for nothing is then left that can be contradicted. To posit a triangle and yet to reject its three angles is contradictory; but there is no contradiction in rejecting the triangle together with its three angles. (Critique of Pure Reason, A 595/B623)

12

Properly formulated. If it lies in the nature of set-membership that if sets exist then von Neumann-style proper classes exist as well, then the unrestricted version of Pairing given in the text will be ruled out by the nature of the membership relation.

26 / The Limits of Contingency In a similar spirit the Others say: If it lies in the nature of God to exist (or to exist necessarily), then to posit God and yet to reject his (necessary) existence is absurd. But there is no contradiction in rejecting God altogether. And similarly, if it lies in the nature of the sets to satisfy Pairing, then to posit a system of sets and yet to reject Pairing is absurd; but there is no contradiction in rejecting the sets along with Pairing. So even if Pairing is somehow constitutive of what it is to be a set, its negation is nonetheless correctly conceivable and therefore possible. Let’s consider one more application of the Non-Standard Conception, this time to a thesis about the constitution of ordinary particulars. D. M. Armstrong has long maintained that whenever two particulars resemble one another, this is because they share an immanent universal as a common part (Armstrong 1978). Let us grant the coherence of the very idea of an immanent universal, wholly located in distinct particulars. In fact, let us grant that in the actual world similarity works as Armstrong says it does. The question will then be whether it is absurd to suppose a world in which qualitative similarity is secured by some other mechanism: e.g. a world in which similar particulars are similar because they contain exactly resembling tropes, or because they instantiate one or another primitive similarity relation. For the sake of argument, we may suppose that these alternative theories are not conceptually confused or self-contradictory.13 On the Non-Standard Conception, the suggestion that they are nonetheless impossible must then amount to the claim that they are incompatible with the nature of qualitative similarity or some other item. But is that plausible? We have assumed that in the actual world qualitative similarity works as Armstrong says it does. And in light of this, someone might say, ‘So that is what qualitative similarity turns out to be. This is not an analytic matter; and it is not exactly an empirical matter either. But it is nonetheless the case that for two particulars to be similar just is for them to share an 13 Once again, it is hard to know whether this is the case. The alternatives have not been developed in sufﬁcient detail. However, the arguments typically brought against these and other proposals do not purport to show that the accounts are straightforwardly contradictory or incoherent. They purport to show that they are uneconomical, or implausible, or less explanatory than the alternatives, and so on. It is just barely possible that in the theory of universals there is in the end exactly one coherent (non-self-contradictory) position. If so, then the Standard Conception and the Non-Standard Conception will concur in calling it necessary. If not, then the two conceptions will diverge. The true account will be necessary in the Standard sense but contingent in the Non-Standard sense.

Gideon Rosen / 27 immanent universal as a common part.’ If this were correct, then in this case the Non-Standard Conception would support the orthodox verdict that the correct metaphysical account of similarity in the actual world amounts to a metaphysically necessary truth. We cannot rule this out without further investigation. But it is implausible on its face. Note that nothing in the story rules out worlds in which something like the trope theory or the primitive resemblance theory is correct: worlds in which there are no immanent universals wholly present in their instances, but in which particulars stand in relations of (let us say) quasi-similarity by virtue of satisfying one of these alternative theories. These quasi-similar particulars may look (quasi-)similar to observers. They may behave in (quasi-)similar ways in response to stimuli. They may be subject to (quasi-)similar laws. The proposal under consideration nonetheless entails that they are not really similar: that quasi-similarity stands to genuine similarity as fool’s gold stands to gold, or as Putnam’s XYZ stands to water. But on reﬂection this seems preposterous. If it walks like similarity and quacks like similarity then it is (a form of) similarity. If you were deposited in such a world (or if you could view it through your Julesvernoscope) and were fully informed both about its structure and about the structure of the actual world, would you be at all tempted to conclude that over there nothing resembles anything else? Surely not. Suppose that’s right. Then the various metaphysical accounts are all compatible with the nature of the similarity relation. The true theory (namely, Armstrong’s) tells us how similarity happens to be grounded. It describes the mechanism by which similarity is secured in the actual world, much as the atomic theory of ﬂuids describes how ﬂuidity happens to be realized in this world. But it goes well beyond a speciﬁcation of the underlying nature of similarity. And if that’s right—if the nature of similarity is in this sense thin—then the alternatives may be correctly conceivable, in which case they represent genuine possibilities according to the Others. We should pause to note a peculiar consequence of the Non-Standard Conception. The view suggests that many of the synthetic propositions of fundamental metaphysics are metaphysically contingent. But it does not say that these propositions are unknowable, or that they can only be known empirically. To the contrary, nothing in the view is incompatible with the thought that the powerful methods of analytic metaphysics supply an altogether reasonable canon for ﬁxing opinion on such matters. Now analytic methodology is for the most part an apriori matter. If the doctrine of immanent

28 / The Limits of Contingency universals is to be preferred as an account of qualitative similarity, this is because it is elegant, intrinsically plausible, philosophically fruitful, immune to compelling counterexample, and so on. All of these features are presumably available to apriori philosophical reﬂection insofar as they are available at all. The view therefore yields a new species of the so-called ‘contingent apriori’. One need not appeal to claims involving indexicals (‘I am here now’) or stipulative reference ﬁxing (‘Julius invented the zip’). According to the Others, the claims of basic ontology (including the existential claims of mathematics), are both contingent and apriori (insofar as they are knowable); but in this case the mechanism has nothing to do with indexicality.14

6. The Two Conceptions and the Informal Explanation Let us suppose—just for a moment—that the Non-Standard Conception is tolerably clear, in the sense that there might a coherent practice in which propositions are classiﬁed as ‘necessary’ or not depending on whether their negations are correctly conceivable. One might object that a notion of this sort, however interesting, does not deserve the name ‘metaphysical necessity’. After all, the main controls on this notion are supplied by the informal elucidation with which we began. A modal notion deserves to be called ‘metaphysical’ only to the extentthatitconformstothisaccount. Andthe Non-Standard Conception falls short in one obvious respect. We explain what we mean by ‘metaphysical necessity’ in part by holding up the truths of mathematics and fundamental ontology and saying, ‘You want to know what metaphysical necessity is supposed to be? It’s the sort of necessity that attaches to claims like that.’ Since the Non-Standard Conception threatens to classify many of these paradigms as ‘contingent’, this counts against regarding it as a conception of metaphysical necessity. 14 Note that if these mathematical and metaphysical truths are indeed both apriori and contingent, then the warrant for them (whatever it comes to) will presumably be available even in worlds where they are false. Apriori warrant is therefore fallible: an interesting result, but not a problem. Compare the force of considerations of simplicity in the empirical case. We are supposed to have reason to believe the simplest theory simply in virtue of its simplicity; but there are deceptive worlds in which the simplest empirically adequate theory is wildly false. This does not show that simplicity is not a reason for empirical belief; it just shows that in deceptive worlds a belief can be both false and justiﬁed. The present picture supports a similar conception of (one sort of) apriori warrant. Thanks to a referee for Oxford University Press on this point.

Gideon Rosen / 29 The charge is one of terminological impropriety, and as such it is ultimately inconsequential. But it seems to me that the Others have a telling response nonetheless. They may say, ‘Tu quoque. Our notion may not ﬁt your informal explanation to the letter. But neither does yours. We think we know what you mean by ‘‘metaphysical necessity’’. At any rate, we can construct a modal notion much like yours, relative to which the Differential truths are clearly necessary. But it is a restricted necessity, on a par with physical necessity. As we normally think, the laws of physics are metaphysically contingent: true in some genuinely possible worlds, false in others. But they are also necessary in a sense: true in each of a distinguished subclass of worlds. By our lights, what you call ‘‘metaphysical necessity’’ has a similar status. It does not amount to truth in every genuinely possible world, but rather to truth in each of a distinguished subclass of worlds: the worlds compatible with the basic facts—or perhaps one should say laws—of metaphysics: the most fundamental facts about ‘‘what there is and how it hangs together’’. This hypothesis squares brilliantly with your taxonomic practice. But it is at odds with the idea that the metaphysical modalities differ from the physical modalities in being unrestricted.’ Theinformalelucidationincludestheclaimthatthemetaphysicalmodalities are absolute among the real modalities. The Non-Standard Conception appears to satisfy the condition. It is certainly less restrictive than the Standard Conception, and it is hard to think of a natural modal conception of the relevant sort that is less restrictive.15 So if the Non-Standard Conception sins against the informal elucidation by reclassifying some of the paradigms, the Standard Conception sins against the absoluteness clause. This is the basis for my suggestion that while neither conception ﬁts the informal explanation to the letter, both conceptions ﬁt it well enough, and so bear roughly equal title to the name ‘metaphysical modality’.

7. Is the Non-Standard Conception Coherent? All of this assumes, of course, that the two conception are genuinely tenable. There are questions on both sides. Let’s begin with objections to the NonStandard Conception. 15

It is easy to construct gerrymandered up real modalities that are less restrictive.

30 / The Limits of Contingency The Others claim that apart from its heterodox classiﬁcation of claims in the Differential Class, the Non-Standard Conception amounts to a recognizable conception of metaphysical modality. But there are reasons to doubt this, some of which are quite familiar. Consider the following exemplary challenge. Let God be Anselm’s God—a necessarily existing perfect spirit—and consider the proposition that God exists. It is not incoherent to suppose there is a God; and pace Anselm, it is not incoherent to suppose there is not. The Non-Standard Conception therefore entails that Anselm’s God is a contingent being. But that’s absurd. If Anselm’s God exists at some world, He exists at all worlds by His very nature. So the Non-Standard Conception is incoherent. It entails that God’s existence is both necessary and contingent. There are several ways to approach the problem, some of which would require substantial modiﬁcation in the Non-Standard Conception. These modiﬁcations may be independently motivated. But it seems to me that the view has the resources to evade this particular problem as it stands. Let’s begin with a question. Anselm’s God is supposed to be a necessary being. But necessary in what sense? If he is supposed to be necessary in the Standard sense, there is no problem. It might well be a contingent matter in the Non-Standard sense whether the basic laws of metaphysics require the existence of a perfect spirit, just as it may be metaphysically contingent in the Standard sense whether the laws of physics require the existence of (say) gravitons. But it’s not very Anselmian to suppose that God’s perfection involves only Standard necessary existence. Surely, ’tis greater to exist in every genuinely possible world than merely to exist in every world that resembles actuality in basic respects. So if we admit the Non-Standard Conception, it will be natural to suppose that God’s existence is supposed to be Non-standardly necessary. But in that case we can afford to be less ecumenical. What would a necessary being in the Non-Standard sense have to be like? It would have to be a being whose non-existence is not correctly conceivable, which is to say: a being whose non-existence together with a complete speciﬁcation of the (conditional, Kantian, anti-Anselmian) natures of things logically entails a contradiction or some similar absurdity. But upon reﬂection it seems clear that there can be no

Gideon Rosen / 31 such thing. The Anti-Anselmian natures of things are given by formulae of the form: To be an F is to be . . . To be A is to be . . . But it seems clear that no collection of such formulae can yield a contradiction when conjoined with a negative existential proposition of the form There are no Fs, or. A does not exist The proposition that a Non-Standard necessary being sense exists is thus incoherent; it is not correctly conceivable. The proponent of the Non-Standard Conception may therefore resist the objection. The same response applies to non-theological versions of the objection. It is sometimes said, for example, that the idea of Number includes the idea of necessary existence, so that nothing counts as a number unless it exists necessarily. (Of course, the textbook deﬁnitions tend to omit this condition, just as they omit to mention that numbers do not exist in space and time. But still it might be said that our ‘full conception’ of the natural numbers entails that numbers exist necessarily if they exist at all.)16 The worry is that the Non-Standard theorist will be forced to concede that is coherent to suppose that numbers so-conceived exist, and also that it is coherent to deny their existence, in which case it will follow, absurdly, that numbers are both necessary and contingent. The response is to distinguish two senses in which numbers might be said to be necessary. If the claim is that numbers, if they exist, must be necessary in the Standard sense, then once again there is no problem. It might be contingent in the Non-Standard sense whether some Standardly Necessary Being exists. On the other hand, if the claim is that numbers must be necessary in the Non-Standard sense, then we may conclude straight away that numbers so-conceived are impossible, since it is not correctly conceivable to suppose that they exist. As a ﬁnal example, consider the claim that there exists an actual golden mountain. Since there is no golden mountain in the actual world, we know that this proposition is not possibly true. But is the proponent of the Non-Standard Conception entitled to this verdict? Is the supposition of a world in which 16

Balaguer (1998).

32 / The Limits of Contingency there exists an actually existing golden mountain logically incompatible with the natures of things? Couldn’t you know all there was to know about what it is to be gold, what it is to be a mountain, and what it is to be actual without being in a position to rule out the existence of an actual golden mountain? No. For there to be an actual golden mountain is for there to be a golden mountain in the actual world. And in the relevant sense, the actual world has its complete intrinsic nature essentially. To be the actual world is to be a world such that P, Q, . . . where these are all the contingently true propositions. Propositions of the form ‘Actually P’ are singular propositions about this world and will thus be true (or false) in virtue of the nature of the actual world. It follows that for propositions of this sort, the NonStandard Conception agrees with the Standard one. All such propositions are metaphysically non-contingent.

8. Objections to the Standard Conception There is much more to say about whether the Non-Standard Conception represents a tenable conception of the metaphysical modalities.17 But if we suppose that it does, then our critical focus naturally shifts to the Standard Conception. For once we have the Non-Standard Conception clearly in focus, it is no longer obvious that the Standard Conception represents a genuine alternative. A skeptic might suggest that it was just thoughtless acquiescence in tradition that led us to regard the substantive principles of fundamental ontology as metaphysically necessary according to our usual understanding of the notion. After all, if there really is no obstacle to the possibility of a world in which (say) mereological aggregates do not exist, is it really so obvious that such worlds should be deemed impossible? Presumably, we have never faced 17 In his very useful comments on an earlier version of this paper, Scott Sturgeon objected to the Non-Standard Conception on the ground that David Lewis’s theory of possibility—his version of modal realism—and its negation are both correctly conceivable, whereas it is absurd to suppose that a modal account of this sort might be a contingent truth. In response, I am inclined to say that Lewis’s metaphysics of many worlds, shorn of its modal gloss, is indeed contingent in the Non-Standard sense, and that no contradiction follows from this concession. On the other hand, Lewis’s package includes account of what it is for a truth to be necessary, and that account is either compatible with the nature of necessity (in which case the negation of Lewis’s theory is an impossibility) or incompatible with it (in which case Lewis’s theory itself is an impossibility).

Gideon Rosen / 33 the question directly. And it is tempting to suppose that when we do, our reaction should be not to reafﬁrm the Standard verdict, but rather to conclude that what I have been calling the Non-Standard conception really is our own conception and that we have been systematically misapplying it in such cases. Tobe sure, even given the tenability of the Non-StandardConception, we still know how to classify truths as necessary or contingent in the Standard sense. We still know how to identify the truths (or putative truths) of fundamental ontology, along with the uncontroversial metaphysical necessities. That is, we know how to apply the Standard Conception in practice. So never mind what we would say if we were to confront the question sketched above. Is there any reason to doubt that the Standard Conception as I have described it tracks a perfectly genuine modal distinction (even if it is not the only such distinction in the neighborhood?) Let’s not deny that it tracks a distinction. The question is whether that distinction amounts to a distinction in modal status. Let me explain. As we have seen, from the standpoint of the Non-Standard Conception, Standard metaphysical necessity is best seen as a restricted modality. To be necessary in the Standard sense is to hold, not in every genuinely possible world, but rather in every world that meets certain conditions. Now it is sometimes supposed that restricted modalities are cheap. After all, given any proposition, φ we can always introduce a ‘restricted necessity operator’ by means of a formula of the form φ (P) =df (φ → P). And in that case, there can be no objection to the Standard Conception. The trouble is that most such ‘restricted necessity operators’ do not correspond to genuine species of necessity. Let NJ be the complete intrinsic truth about the State of New Jersey, and say that P is NJ-necessary just in case NJ strictly implies P. It will then be NJ-necessary that Rosen is in Princeton, but NJ-contingent that Blair is in London. But of course we know full well that there is no sense whatsoever in which I have my location of necessity while Blair has his only contingently. So NJ-necessity is not a species of necessity. The moral is that one cannot in general infer, from the fact that a certain consequence (φ → P) holds of necessity, that there is any sense in which the consequent (P) holds of necessity. (If there were then every proposition would be necessary in a sense, even the contradictions.)

34 / The Limits of Contingency Now, metaphysical necessity on the Standard Conception is supposed to be a restriction of Non-Standard metaphysical necessity for which the restricting proposition φ is the conjunction of what we have been calling the ‘laws’ of metaphysics. The challenge is thus to show that Standard necessity so conceived amounts to a genuine species of necessity—that it is more like physical necessity than it is like NJ-necessity. It is unclear what it would take to meet this challenge. There is some temptation to say that φ-necessity amounts to a genuine species of necessity only when the restricting proposition φ has independent modal force—only when there is already some sense in which it must be true. But what could this mean? Consider the Mill–Ramsey–Lewis (MRL) account of the laws of nature, according to which a generalization L is a law just in case L is a theorem of every true account of the actual world that achieves the best overall balance of simplicity and strength (Lewis 1973). Let us grant that this standard picks out a tolerably well-deﬁned class of truths. Still, one might ask, ‘Why should propositions incompatible with the laws so conceived be called impossible?’ Consider a related class of truths: those propositions that would ﬁgure in every true account of the State of New Jersey that achieves the best overall balance of simplicity and strength. If the Encyclopedia Britannica is any guide, one such truth is the proposition that New Jersey is a haven for organized crime. But one needs a dark view of things to suppose that this proposition is in some sense necessary. It certainly doesn’t follow from the fact that it is important enough to be worth mentioning in a brief account of New Jersey that it enjoys a distinctive modal status. So why is it than when the MRL-theory in question is a theory about the entire world, we are inclined to credit its general theorems with some sort of necessity? One way with this sort of question is a sort of nominalism. There no objective constraints on which restricted necessities we recognize. We take an interest in some but not in others. We hold their associated restricting propositions ﬁxed in counterfactual reasoning for certain purposes. And in these cases we dignify the operator in question with a modal name. But our purposes might have been otherwise, and if they had been then we might have singled out a different set of operators. On this sort of view there can be no principled objection to the Standard Conception. The worst one can say is that the restricted necessity upon which it fastens is not particularly interesting or useful. But one cannot say that it fails to mark a genuine modal distinction,

Gideon Rosen / 35 for on the view in question any modal distinction we see ﬁt to mark as such is ipso facto genuine. If we set this sort of nominalism to one side, then one natural thing to say is that a putative restricted necessity counts as genuine only when the boundary it draws between the necessary and the contingent is non-arbitrary or nonad hoc from a metaphysical point of view. (Note that this is at best a necessary condition.) The truths about NJ are not a natural class from the standpoint of general metaphysics; nor are the Mill–Ramsey–Lewis generalizations about New Jersey. On the other hand, the most important general facts about nature as a whole may well be thought to constitute a metaphysically signiﬁcant class of facts. And if so, there would be no objection on this score to the idea that physical necessity deﬁned in Lewis’s way amounts to a genuine species of necessity. The Standard Conception of metaphysical necessity conditionalizes upon what we have been calling the basic laws or facts of fundamental ontology. Just as the Mill–Ramsey–Lewis laws of nature are supposed to represent the goal of one sort of natural science, the metaphysical laws are supposed to represent the goal of one sort of metaphysics: nuts and bolts systematic ontology. Clearly, there is no worry that these truths might constitute an arbitrary class from the standpoint of metaphysics. But it might still be wondered whether anything substantial can be said about what uniﬁes them, and in particular, about what ﬁts them to serve in the speciﬁcation of a restricted modality. I have a conjecture (and some rhetoric) to offer on this point. Consider the true propositions in the Differential class: the truths in the theory of universals and the metaphysics of material constitution; the truths about how abstract entities of various sorts are ‘generated’ from concrete things and from one another. To know these truths would not be to know which particulars there are or how they happen to be disposed in space and time. But it would be to know what might be called the form of the world: the principles governing how objects in general are put together. If the world is a text then these principles constitute its syntax. They specify the categories of basic constituents and the rules for their combination. They determine how non-basic entities are generated from or ‘grounded in’ the basic array. Worlds that agree with the actual world in these respects, though they may differ widely in their ‘matter’, are nonetheless palpably of a piece. They are constructed according to the same rules, albeit in different ways, and perhaps even from different ultimate ingredients. In this sense, they are like sentences in a single language. The metaphysically necessary truths on the Standard Conception may not be

36 / The Limits of Contingency absolutely necessary. But they hold in any world that shares the form of the actual world in this sense. Combinatorial theories of possibility typically take it for granted that the combinatorial principles characterize absolutely every possibility: that possible worlds in general share a syntax, as it were, differing only in the constituents from which they are generated or in the particular manner or their arrangement. The Non-StandardConception isnot strictly combinatorial in this sense, since it allows that the fundamental principles of composition—the syntax—may vary massively from world to world. The actual grammar is not privileged. Any coherent grammar will do. But the Standard Conception carves out an inner sphere within this larger domain: the sphere of worlds that share the combinatorial essence of actuality. As I have stressed, it is unclear what it takes to show that a class of truths is sufﬁciently distinguished to count as a legitimate basis for a restricted modality. Nonetheless, the foregoing may be taken to suggest that if any restricted modality is to be reckoned genuine, the restricted modality marked out by Standard Conception should be so reckoned.

9. Physical Necessity Reconsidered This way of thinking raises a question about the boundary between physical necessity and Standard metaphysical necessity. Some physical necessities will presumably be Standardly contingent. Suppose the laws of nature involve particular numerical constants that determine the strengths of the fundamental forces or the charges or masses of the fundamental particles. It will then be natural to suppose that the precise values of these constants are not aspects of the general combinatorial structure of the world and that they are therefore contingent in the Standard sense. But other claims that might feature in the Mill–Ramsey–Lewis theory of the natural world might be candidates for metaphysical necessity in the standard sense: that the laws of nature all assume a certain mathematical form (e.g. that they are quantum mechanical); that the space–time manifold has certain geometrical features, e.g.: that it has only one ‘time’ dimension; that the ultimate particles are excitation states of one-dimensional strings; and so on. It is not inconceivable that such physical features should be sufﬁciently basic to count as aspects of the underlying form or structure of the world: that any world in which such

Gideon Rosen / 37 physical features failed to be manifest, would fail to share a syntax with the actual world. And insofar as this is so, these physical truths should be reckoned metaphysically necessary on the Standard conception for the same reason that the facts of fundamental ontology are to be reckoned necessary on that conception. The point I wish to stress, however, is that on the present conception it is to be expected that the border between Standard metaphysical necessity and physical necessity should be vague—not simply because the notion of physical necessity (or a law of nature) is vague, but also because it is vague when a truth is ‘fundamental’ or ‘structural’ enough to count as part of the combinatorial essence of the world. This is not the prevailing view on this matter. Most writers take it for granted that the question whether a certain law of nature is also metaphysically necessary is a well-deﬁned question whose answer is in no way up for stipulation. On the present conception, that is unlikely to be the case. If the question is whether some given law of nature is a Non-Standard necessity, then indeed, for all we have said, it may be sharp. However hard it may be to ﬁnd the answer, the question then is whether the negation of the law is ruled out by the natures of the properties and relations it concerns, and we have seen no reason to believe that this question is a vague one. (There may be such reasons, but we have not seen them.)18 On the other hand, if the question is whether the law is a Standard metaphysical necessity, then we should expect that in some cases it will have no answer, since the boundary between structural or formal truths and mere ‘material’ truths has only been vaguely speciﬁed.

10. Conclusion We have distinguished two conceptions of metaphysical necessity, both of which cohere well enough with the usual informal explications to deserve the name. According to the Non-Standard Conception, P is metaphysically necessary when its negation is logically incompatible with the natures of things. According to the Standard Conception, P is metaphysically necessary when 18

For example, it might turn out to be a vague matter whether P holds in virtue of the nature of things. This is immensely plausible when P is a proposition about a particular organism or a biological species.

38 / The Limits of Contingency it holds in every (Non-Standard) possible world in which the actual laws of metaphysics also hold, where the basic laws of metaphysics are the truths about the form or structure of the actual world. Neither conception has received a fully adequate explanation. But if both are tenable, then our discourse about necessity is shot through with ambiguity. The ambiguity only matters when we are discussing the modal status of metaphysical propositions—or perhaps the modal status of certain laws of nature. But when it does matter, we ignore it at our peril. We are inclined to believe that questions about the modal status of the claims of mathematics and metaphysics are unambiguous. But if I’m right, that is not so. In particular, it may be metaphysically necessary in one sense that sets or universals or mereological aggregates exist, while in another sense existence is always a contingent matter.

References Armstrong, D. M. (1978), Universals and Scientiﬁc Realism, 2 vols. (Cambridge: CUP). Balaguer, Mark (1998), Platonism and Anti-Platonism in Mathematics (Oxford: OUP) Dorr, Cian, and Rosen, Gideon (2001), ‘Composition as a Fiction’, in R. Gale (ed.), Blackwell’s Guide to Metaphysics (Oxford: Blackwell), 151–74. Dummett, Michael (1973), Frege: Philosophy of Language (2nd edn., Cambridge, Mass.: Harvard University Press). (1981), The Interpretation of Frege’s Philosophy (Cambridge, Mass.: Harvard University Press). (1993), ‘Wittgenstein on Necessity: Some Reﬂections’, repr. in his The Seas of Language (Oxford: OUP), 446–61. Field, Hartry (1974), ‘Quine and the Correspondence Theory’, Philosophical Review 83: 200–28. (1989), Realism, Mathematics and Modality (Oxford: Blackwell). (1993), ‘The Conceptual Contingency of Mathematical Objects’, Mind, 102: 285–99. Fine, Kit (1994), ‘Essence and Modality’, in J. E. Tomberlin (ed.), Philosophical Perspectives 8: Logic and Language (Atascadero, Calif.: Ridgeview), 1–16. (2002), ‘The Varieties of Necessity’, in J. Hawthorne and T. Gendler (eds.), Conceivability and Possibility (Oxford: OUP), 253–81. Frankel, Abraham, Bar Hillel Yehoshua, and Levy, Azriel (1973), Foundations of Set Theory (2nd edn., Amsterdam: North Holland). Lewis, David (1973), Counterfactuals (Cambridge, Mass.: Harvard University Press).

Gideon Rosen / 39 (1983), ‘New Work for a Theory of Universals’, Australasian Journal of Philosophy, 61: 343–77. Maddy, Penelope (1997), Naturalism in Mathematics (Oxford: OUP). van Fraassen, Bas (1977), ‘Platonism’s Pyrrhic Victory’, in Marcus et al. (eds.), The Logical Enterprise (New Haven: Yale University Press), 39–50. van Inwagen, Peter (1990), Material Beings (Ithaca, NY: Cornell University Press).

2 Modal Infallibilism and Basic Truth Scott Sturgeon

1. Introduction This paper concerns intelligibility and possibility. It argues no linking principle of the form (L) Int(ø) ⇒ ø is valid. The notion of intelligibility will vary throughout the discussion. Details will be given as needed. The notion of possibility will remain constant. Let me explain it now. Modal operators will stand for genuine modality: ø will mean ø is genuinely possible; ø will mean ø is genuinely necessary. This will be our analogue of metaphysical modality. The latter is oft equated with logical modality, when logic is ‘broadly construed’. But that suggests there is something intrinsically apriori about metaphysical modality. I must blanche the discussion of just that bias. So I use new terms. They leave open whether genuine modality This paper is dedicated to David Lewis. It grew from a conference response to Gideon Rosen’s ‘The Limits of Contingency’. I thank Fraser MacBride for inviting that response, Rosen for his thought-provoking paper, and MacBride again for inviting the expansion of my thoughts on the day. My views of apriority and modality have evolved considerably since the conference. They can be found in Epistemic Norms (Oxford: OUP, forthcoming). Finally: thanks to Dorothy Edgington, Dominic Gregory, John Hawthorne, Jen Hornsby, Barry Lee, Stephan Leuenberger, Jonathan Lowe, Fraser MacBride, Mike Martin, Gideon Rosen, Nick Shea, Maja Spener, Alan Weir, Tim Williamson, Crispin Wright, and Dean Zimmerman for helpful feedback.

Scott Sturgeon / 41 plays a role in anything canonically apriori. By such modality I mean simply this: the most absolute realistic modal space. The idea, of course, is twofold. Its components can be glossed via possibility or necessity. Consider each in turn. When ø is genuinely possible, it is a mind- and language-independent fact that ø can happen.1 That fact does not spring from how we think or talk (even in the rational ideal). Genuine possibility is like genuine actuality. It does not depend on us for its existence. It does not depend on us for its nature. It is a realistic domain. Realism is a component of genuine possibility. And so is weakness: whenever there is a realistic sense in which ø can happen, ø is genuinely possible. Such possibility is the weakest kind of realistic possibility. It is entailed by every kind of realistic possibility. The ‘diamond face’ of genuine modality is a two-part affair. It is the most inclusive realistic space of possibility. Similarly: when ø is genuinely necessary, it is a mind- and languageindependent fact that ø must happen. That fact does not spring from how we think or talk (even in the rational ideal). Genuine necessity is like genuine actuality. It is a realistic domain. Realism is a component of genuine necessity. And so is strength: when ø is genuinely necessary, it is necessary in any realistic sense. Genuine necessity is the strongest kind of realistic necessity. It genuinely entails every kind of realistic necessity. The ‘box face’ of genuine modality is a two-part affair. It is the least inclusive realistic space of necessity. Our topic is whether intelligibility marks genuine possibility. We will work it in several stages. By way of motivation, Section 2 presents six claims of Basic Metaphysics. The very last is got by detachment from an instance of (L). Section 3 derives contradictions from the six. That prompts scrutiny of (L). Section 4 considers and rejects several readings. Section 5 diagnoses their defects. That leads to the best reading of (L). Section 6 contains its critique. Section 7 closes with remarks on reason and modality.

2. Six Claims of Basic Metaphysics (i) Instantaneous things last for an instant. Persisting things last longer. And notionally at least, they might do so three ways. They might last solely by composition from instantaneous things. They might last by no such composition. They might combine parts drawn from each category. The ﬁrst 1

Unless ø concerns mind or language, of course. I leave the point tacit.

42 / Modal Infallibilism and Basic Truth method of persistence is perdurance. The second is endurance. Mixed persistence has no common name. Suppose O1 , . . . , On manifest condition C. Suppose their so doing has nought to do with time or other things. It has just to do with them. In the event, C is an intrinsic condition. It is an intrinsic property when n equals 1. It is an intrinsic relation when n is larger. Our ﬁrst claim of Basic Metaphysics concerns intrinsics. Speciﬁcally, it concerns their shift. The key idea is simple. Since intrinsics have only to do with what satisﬁes them, they cannot be shifted. That would involve satisfying yet not satisfying a condition (which cannot be). Intrinsics are thus for life. They cannot be temporary. Appearances to the contrary mislead. Yet appearances are to the contrary, as we will see. Intrinsics do seem to shift. So how is the magic done? Through perdurance: seeming shift in intrinsics is none but intrinsically varying temporal parts. Such parts do not shift their intrinsics. But persisters into which they build may seem to; and they may do so because their temporal parts intrinsically vary. Intrinsic change is thing-level sleight of hand. It is done by change of intrinsically varying temporal parts. This is a matter of Basic Metaphysics. Our ﬁrst claim is thus (1) Necessarily, intrinsic change comes via perdurance. (ii) When intrinsic change is discussed, shape is the pat example. Lewis founds orthodoxy thus: Persisting things change their intrinsic properties. For instance shape: when I sit, I have a bent shape; when I stand, I have a straightened shape. Both shapes are temporary intrinsic properties; I have them only some of the time.2

And he polishes it thus: When I change my shape, that is not a matter of my changing relationship to other things, or my relationship to other changing things. I do the changing, all by myself.3

Shape is intrinsic. This is a matter of Basic Metaphysics. Our second claim is thus (2) Necessarily, shape is intrinsic. (iii) Intrinsics come in greater-than-one adicity. As Lewis notes when extending orthodoxy: 2

On the Plurality of Worlds (Oxford: Blackwell, 1986), 204–5. ‘Rearrangement of Particles: Reply to Lowe’, in his Papers in Epistemology and Metaphysics (Cambridge: CUP, 1999), 187. 3

Scott Sturgeon / 43 Exactly as some properties are just a matter of how the thing itself is, without regard to any relationship to any second thing, so some relations are just a matter of how two things are vis-`a-vis one another, without regard to any relationship to any third thing. The relation is intrinsic to the pair of relata. The ever-changing distances of particles from one another seem to be temporary intrinsic relations.4

Distance between things is intrinsic. Our third claim of Basic Metaphysics is thus (3) Necessarily, distance is intrinsic. (iv) Some things are none but the sum of their parts. And any collection seems to bring one of them with it: namely, the sum of its members. The sum is the least inclusive thing with each member as part. It is none but the sum of its parts. The view that sums perforce spring from collections is mereological universalism. And it too is a matter of Basic Metaphysics. Our fourth claim is thus (4) Necessarily, for any O1 , . . . , On there is a sum = [O1 + · · · + On ]. (v) If 1 and 2 have strictly identical parts, they are but one sum. After all, sums are none but the sum of their parts. Strict identity of part ensures that of sum. Our ﬁfth claim of Basic Metaphysics is thus (5) Necessarily, if 1 and 2 have strictly identical parts, 1 = 2 . (vi) So far so good. (1)–(5) are popular claims. They yield conﬂict, though, when married to this line from Lewis: There might be things that endure identically through time or space, and trace out loci that cut across all lines of [Humean] continuity. It is not, alas, unintelligible that there might be suchlike rubbish. Some worlds have it.5

These are enduring runabouts. Lewis infers their possibility from their intelligibility. And by that, of course, he means more than their understandability. Impossibilities may be understandable enough. By intelligibility Lewis means something like ‘free of conceptual difﬁculty after serious reﬂection’. We will have more to say about that from Section 4 onward. 4

Ibid. 192–3. Philosophical Papers (Oxford: OUP, 1986), vol. ii, p. x; see also ‘Rearrangement of Particles’, 195. 5

44 / Modal Infallibilism and Basic Truth The possibility of enduring runabouts is a matter of Basic Metaphysics. Our sixth claim is thus (6) Enduring runabouts are possible.

3. Two Problems Now we are in trouble. (6) interacts badly with the combination of (1), (2), (4), and (5); and it interacts badly with that of (1) and (3). In the ﬁrst case we get the problem of shapeshifting. In the second we get that of placeshifting. Consider each in turn. Shapeshifting: (6) says endurers can run about. (4) guarantees that at each moment they do so there is a sum composed just of them existing at that moment. (5) guarantees it is one sum. Hence the sum endures. Yet its parts runabout. The sum thereby changes shape. (1) and (2) thus ensure it perdures. No good! Nothing can perdure and endure. The former is done solely by composition from instantaneous things. The latter is done without such composition. The combination of (1), (2), (4), (5), and (6) conﬂict. They cannot all be true. This is the puzzle of shapeshifting. One might deny, of course, that enduring runabouts can shapeshift their sum. One might say they can only move so as to preserve sum shape. That would dissolve the puzzle. But it would do so by ﬁat. It would be too ad hoc for systematic philosophy. We should insist: if endurers can be at all, they can run about so as to shapeshift their sum. To reinforce the worry, let be a sum of enduring runabouts. Suppose it changes shape. Now echo Lewis: When changes its shape, that is not a matter of its changing relationship to other things, or its relationship to other changing things. does the changing, all by itself. At ﬁrst blush this looks incredible. After all, ’s changing shape is a matter of its runabout parts. They change its shape. The shift is precisely a matter of ’s relationship to changing things. It is a matter of ’s relationship to runabout parts. Yet none of those parts is identical to . There is a good sense, then, in which ’s shapeshift is due to its relation to other changing things. At second blush one can make out Lewis’s line. For as he puts it: ‘the ‘‘are’’ of composition is just the plural of the ‘‘is’’ of identity.’6 So in a good sense 6

‘Rearrangement of Particles’, 195.

Scott Sturgeon / 45 parts are not otherly relative to their sum. They are their sum. Their sum is them. ’s shapeshift is just the running about of its parts. It is an intrinsic change in an enduring thing (which can not be). Hence we face a genuine puzzle. (1), (2), (4), and (5) look good. They cannot all be true. The possibility of enduring runabouts yields conﬂict in Basic Metaphysics. That possibility was got from intelligibility. The conﬂict grows, at least in part, from the view that intelligibility yields genuine possibility. It grows from an instance of (L) Int(ø) ⇒ ø. That motivates questioning the validity of (L). Placeshifting: (6) says endurers can run about. Distance between them can shift. (3) guarantees that is intrinsic change. (1) then ensures they perdure. No good! Nothing can perdure and endure. The former is done solely by composition from instantaneous things. The latter is done without such composition. (1), (3), and (6) conﬂict. They cannot all be true. This is the puzzle of placeshifting. The genuine possibility of enduring runabouts yields conﬂict in Basic Metaphysics. That possibility is got from intelligibility. Here too conﬂict grows, at least in part, from the view that intelligibility yields genuine possibility. Here too it grows from an instance of (L).7 The schema’s validity is our main topic. It is one of some urgency. For as we have just seen, free movement from intelligibility to possibility breeds conﬂict in Basic Metaphysics. So we must ask: what kind of intelligibility, if any, marks genuine possibility? Are there readings of ‘Int(—)’ which make (L) valid?

4. Links Let us say ø is prima facie apriori coherent iff it is coherent after a bit of apriori reﬂection. And let us write pfac(ø) to express that. The naive apriori infallibilist says this is valid 7 It might be thought the shapeshifting and placeshifting puzzles show enduring runabouts are not intelligible. After all, the puzzles clearly show enduring runabouts are not free of conceptual difﬁculty after serious reﬂection (to borrow a phrase used earlier). But it is important to remember this: the difﬁculty attaches, in the ﬁrst instance, to the conjunction of (1) thru (6). To blame (6) for the bother ignores the fact that apriori warrant, like its aposteriori cousin, is holistic. Echoing Quine: apriori intelligibility faces the rub of contradiction as a corporate body.

46 / Modal Infallibilism and Basic Truth (p) pfac(ø) ⇒ ø.8 She says belief in ø based on pfac(ø) is infallible. But she is obviously wrong. After all, apriori reﬂection can self-correct. When we see that not all clear concepts yield extension sets, or that not all inﬁnite sets are equinumerous, for instance, apriori reﬂection self-corrects. (p) is invalid: genuine impossibilities can be prima facie apriori coherent. Inferring ø from pfac(ø) is at best a fallible affair. So let us idealize. Let us say ø is limit coherent iff it is coherent at the limit of apriori reﬂection. The idea, roughly, is that ø is coherent even when faced with all that is rational after ideal apriori reﬂection. Let us write lim(ø) to express that. The less-naive apriori infallibilist says this is valid (I) lim(ø) ⇒ ø. She says belief in ø based on lim(ø) is infallible. It is not obvious she is wrong. After all, refuting her was a key move in Kripke’s war to segregate apriority and modality. Yet win the war he did. For Kripke showed limit-case apriori reﬂection does not mark genuine possibility. He showed non-modal fact can rub out limit coherence. Something coherent at the limit of apriori reﬂection can be genuinely impossible; and non-modal fact can show it to be so. For instance: let be the proposition that David Lewis is Bruce LeCatt. Both and ¬ are limit coherent. No amount of reﬂection can show whether Lewis is LeCatt. The matter is not apriori. In fact they are one person. But you cannot tell by reﬂection, not even ideal reﬂection. You have to look. Since they are one, however, they cannot help but be so. There is no way to pull them apart. Not only is true, it is genuinely necessary. Hence ¬ is genuinely impossible. Despite its limit coherence, ¬ fails to be genuinely possible. It is a counter-instance to (l). That schema too is invalid. In refuting (l), Kripke’s strategy is clear. He locates non-modal claims such that (a) both they and their negation are limit coherent; and (b) it is apriori obliged, in the limit, to think the matter they raise is non-contingent. 8

I shall understand the validity of schemata to consist in their lack of counter-instance which is logically simple or the negation of such. That keeps discussion where it belongs, on base-case bother. Everything to follow could be recast, mutatis mutandis, in more general terms.

Scott Sturgeon / 47 So it is with , the proposition that Lewis is LeCatt. Not only is it aposteriori whether is true, it is apriori the matter is non-contingent. It is apriori that if is true, then it is necessary; and it is apriori that if ¬ is true, then it is necessary. However things turn out it could not have been otherwise. That is the view one should have in the limit of apriori reﬂection. Since the ﬁrst-order matter is not apriori, though, limit-case apriori reﬂection does not mark genuine possibility. That was Kripke’s insight. How should a less-naive apriori infallibilist react? Good question. Let us say Kripke claims are non-modal claims like and ¬. And let us say true Kripke claims are Kripke truths. Such truths are non-modal. They and their negations are limit coherent. But it is apriori the matter they raise is non-contingent. Then we deﬁne: ø is Kripke coherent iff it is limit coherent in light of Kripke truths. And we write kc(ø) to express that. We set kc(ø) = lim(ø/K), with K the set of Kripke truths. A conservative reaction to Kripke admits (l) is invalid but insists Kripke coherence marks genuine possibility. It replaces (l) with (k) kc(ø) ⇒ ø. This is mild aposteriori infallibilism. It says belief in ø based on kc(ø) is infallible. The view is at work, I believe, in a good deal of post-Naming-andNecessity philosophy. It retains the naive and less-naive apriorist’s infallibilism. It gently resists their apriorism. The view makes use of aposteriori matters. But it does so only by appeal to Kripke truths. What should we make of it? In particular, should we believe Kripke coherence marks genuine possibility? No. There are Kripke coherent claims that are not genuinely possible. There are counter-instances to (k). For example, consider the view that Lewisian worlds truthmake claims of genuine modality. The idea, basically, is that points in the space of genuine possibility are Lewisian worlds. Such worlds are maximal sums of spatiotemporally related parts. Lewis equates these sums with maximal genuine possibilities. So his idea is twofold: it postulates maximal sums of a certain sort; and it identiﬁes them with maximal genuine possibilities. Call the view ‘L’. Both L and ¬L are prima facie coherent. And they look limit coherent too. So a question is presently sharp: are there non-modal truths to rub out their apriori coherence? Are there truths expressible in non-modal idioms that render L or ¬L incoherent?

48 / Modal Infallibilism and Basic Truth I doubt it. Both L and ¬L look Kripke coherent: (7) kc(L) (8) kc (¬L). In English: it is Kripke coherent that genuine possibilities are Lewisian; and it is Kripke coherent they are not. Take apriori reﬂection to the limit, throw in Kripke truth, both L and ¬L stay coherent. 9 This spells trouble for (k). After all, (7), (8), and (k) jointly yield (9) L and (10) ¬L. But consider the view that Lewisian modal metaphysics is true but genuinely might not have been: (11) L & ¬L. This is genuinely impossible. Its ﬁrst conjunct entails its second is true only if there is a Lewisian truthmaker at which there is no such truthmaker. (11)’s conjuncts cannot both be true. So we have (12) (L ⊃ ¬¬L). Yet this and (9) yield (13) L, which conﬂicts with (10). So (9) rules out (10). Similarly, consider the view that Lewisian modal metaphysics is false but genuinely might have been true: (14) ¬ L & L. This too is genuinely impossible. For its possibility implies that of its right-hand side. Yet that leads to (9). As we have just seen, though, (9) leads via (12) to (13); and (13) conﬂicts with (14)’s left-hand side.10 The moral is clear: whichever 9

We know at least one Lewis world exists: the maximal spatiotemporal sum of which we are parts. Suppose no other Lewis world exists. Does this show L is false? No. To get that result one must also suppose that things could genuinely be other than they are. Yet that is a modal claim par excellence. By stipulation neither it nor its negation is a Kripke truth. 10 This line of thought leans on S5 for genuine modality. That is contentious, of course. See Graham Forbes, The Metaphysics of Modality (Oxford: OUP, 1985); Nathan Salmon, ‘The Logic of What Might Have Been’, Philosophical Review, 98 (1989), 3–34 and references therein. But notice: L is an identity claim. The conclusion of the argument can be got by appeal to the necessity of identity

Scott Sturgeon / 49 of L and ¬L turns out true, it is genuinely necessary as well. (9) and (10) cannot both be true. At least one Kripke coherence is genuinely impossible. (k) is invalid. Not surprisingly, the schema breaks down on the metaphysics of genuine modality. There are Kripke coherent claims about that metaphysics which genuinely cannot be true. Mild aposteriori infallibilism breeds conﬂict in modal metaphysics. It might be replied that L or ¬L is a Kripke truth. After all, the idea behind L is that claims like (l) (Donkeys talk) iff there is a Lewisian world at which donkeys talk reduce their modal left-hand to their non-modal right-hand sides. And if that is right, of course, those claims and their right-hand sides are in K. They are both Kripke truths. This would be good and bad news for (k). It would reveal putative counter-instances to be no such thing. But it would render the schema trivial. For if (l) and its ilk count as non-modal, that notion covers not only claims about actuality expressed in non-modal terms, but those about non-actuality so expressed. (k) is trivially valid. Genuine possibility trivial follows from non-modal facts so understood. On this view, Kripke coherence does imply genuine possibility. But that is not because bona ﬁde intelligibility has been shown to mark genuine modality. Nor is it because a mild non-modal weakening of that notion has been shown to do so. It is because ‘intelligibility’ has been stretched by ﬁat to cover genuine modality.

5. Diagnosis Consider and L, the propositions that Lewis is LeCatt and that genuine possibilities are Lewisian worlds. They bear the marks of Kripke truth: (a) both they and their negations are limit coherent; and (b) it is apriori obliged, in the limit, to think the matters they raise are non-contingent.11 and distinctness. That appeal entails nothing about the logic of genuine modality. I present the S5-argument because I think it best ediﬁes L’s modal status. I accept its underlying logic. But that logic is inessential to the point at hand. 11 Note satisﬁes (a) because it is aposteriori. It is to be gleaned from experience. L is not like that. It satisﬁes (a) because apriori concerns underdetermine its truth-value. That is why Lewis is right

50 / Modal Infallibilism and Basic Truth When a claim satisﬁes these principles, it is an apriori red ﬂag. It cannot be ruled out apriori. It cannot be ruled in apriori. The claim’s subject matter can be seen apriori to be non-contingent. And the same holds of its negation. Hence we can see, on purely apriori grounds, that apriori reﬂection must be insensitive to impossibility somewhere. Either a given red ﬂag or its negation is the rub. Either the ﬁrst cannot be true despite remaining in play at the limit of apriori reﬂection, or its negation cannot be true despite remaining in play at that limit. Impossibility slips through the net of apriori reﬂection.12 This prompts a strategy. Let us say ø is apriori open—or open for short—iff two things are true: (i) both ø and ¬ø are limit coherent; but (ii) it is not apriori obliged the matter they raise is non-contingent. And let us write o(ø) to express that. When ø is open, both ø and ¬ø are coherent at the limit of apriori reﬂection; but it is not settled there that ø is non-contingent. The thought is to replace (k) with (o) o(ø) ⇒ ø. This is seasoned apriori infallibilism. It says openness marks genuine possibility. If a claim and its negation are limit coherent, and it is not apriori they are non-contingent, then they are genuinely possible. What should we make of the idea? Well: neither Kripke-style claims like , nor Lewis-style claims like L, are counter-instances to (o). For none are open. None satisfy the schema’s antecedent. It is apriori the matter they raise is non-contingent. That is why they can play spoiler in apriori discussion of (l) and (k). But this fact about them precludes their cutting against (o). Further, we can see apriori that there can be no base-case apriori counterinstance to (o). For such a claim would be a logically simple (or negated to insist L-commitment should rest on L’s role in systematic philosophy. See On the Plurality of Worlds, 3–5. 12 This means there will be further systematic slip-ups by apriority. For instance, let R be any red ﬂag. Neither R nor its negation can be ruled out apriori. But it is apriori that R is necessary or impossible. So consider any contingent claim C which is apriori independent of R and its negation. If R is necessary, then (C&¬R) is limit coherent yet impossible. If R is impossible, then (C&R) is limit coherent yet impossible. False red ﬂags logically embed to generate systematic apriori blindspots. I ignore this in what follows, focusing on the root source of bother.

Scott Sturgeon / 51 logically simple) open apriori impossibility. Yet no such claim can be open and apriori impossible. The latter precludes the former. If such a claim is apriori impossible, it is apriori the matter it raises is non-contingent. But if it is apriori the matter it raises is non-contingent, it is not open. Hence no base-case claim can be open and apriori impossible. None can be an apriori counter-instance to (o). This looks good. (o) side-steps problems which infect (p), (l), and (k). And it does so in an explanatory way. Base-case apriori counter-instances to the latter two schemata have two things in common: they satisfy (a) and (b). They and their negation are limit coherent; and it is apriori the matter they raise is non-contingent. Openness rules out the last feature. For this reason, no base claim can apriori satisfy (o)’s antecedent yet fail to satisfy it is consequent. Having said that, there is an obvious shortcoming with (o). The principle is too restricted. The deﬁnition of openness guarantees it is closed under negation. ø is open iff ¬ø is open. (o) is valid, therefore, only if openness marks contingency. (o)’s validity guarantees that of (o)* o(ø) ⇒ ø is contingent. Yet there are non-contingent possibilities marked by apriori reﬂection. The fact that everything is itself, for instance. Surely we mark this possibility with apriori reﬂection. (o) misses it altogether. The schema is incomplete. It is suited to contingent possibilities (if any). It skips noncontingent ones. To plug the gap we need a condition designed for apriori necessities. So let us say ø is apriori forced—or forced for short—iff three things are true: (iii) ø is limit coherent, (iv) ¬ø is not limit coherent, but (v) it is apriori obliged that the matter they raise is non-contingent. And let us write f(ø) to express that. Then we can say ø is apriori apt—or apt for short—iff ø is open or forced. And we can write apt(ø) to express that. Then we swap (o) for (a) apt(ø) ⇒ ø. This is best-shot apriori infallibilism. It amounts to endorsing (o) o(ø) ⇒ ø

52 / Modal Infallibilism and Basic Truth and (f) f(ø) ⇒ ø. Best-shot apriori infallibilism says aptness marks genuine possibility. It builds its mark out of two conditions. One is designed for apriori necessity. The other is designed for such contingency. What should we make of it? Well, we have seen no base claim can be apriori open and impossible. There is such an apriori counter-instance to (a), therefore, only if a base claim can be apriori forced and impossible. Yet these too are incompatible. If you show something apriori to be impossible, it is limit coherent it is false. That means it is not forced. It fails condition (iv). Just as there can be no basic apriori counter-instance to (o), there can be none to (a). It too side-steps direct refutation. In my view, (a) grounds the best modal infallibilism. Just as (l) is motivated by weakness in (p), and (k) is motivated by weakness in (l), (a) is motivated by cracks in all three. It is designed to avoid their pitfalls; and it does so demonstrably. If there is a valid base-case link from apriority to genuine possibility, (a) schematizes it.

6. Problems I reject (a). I have no direct refutation, of course. The schema does not permit one. But I do have two indirect worries. The ﬁrst generalizes a point made with force by Gideon Rosen in ‘The Limits of Contingency’ (Chapter 1 in this volume). The second springs from my take on the ﬁt between epistemology and metaphysics. [A] Many claims of Basic Metaphysics are open. Apriori reﬂection does not determine a truth-value for them, nor does it say they are non-contingent. Stripped of its modal operator, for instance, (1) is like that: (1)− Intrinsic change comes via perdurance. This claim is Basic. When true it is not made so by anything else. Yet (1)− is open. For recall: intrinsic change is strictly impossible. Whenever it seems to occur, sleight of hand is in play. (1)− says it comes by perdurance. That claim is limit coherent. But so is the view that the trick comes by enduring things shifting properties which look to be, but are not, intrinsic. Were the adicity of

Scott Sturgeon / 53 them properly subtle—for instance, were they to involve a difﬁcult-to-spot relation to time—intrinsic change would look to happen; but it would come by endurance rather than perdurance. It would come by property- rather than thing-level sleight of hand. This too is limit coherent. Yet limit-case reﬂection does not secure the matter is non-contingent. It leaves it unsettled. Hence (1)− is open. It is an open Basic claim. As Rosen notes, though, Joe Metaphysician assumes Basic truths are noncontingent. He embraces the necessity of (1)− after deciding it is the best explanation of intrinsic change. When looking to explain such change, he seeks the best amongst coherent options. He concludes the best is necessary. By letting b(ø) mean ø is Basic, we can schematize his practice with (b) [b(ø) & ø] ⇒ ø. Joe assumes Basic truths are necessary. He accepts (b)* b(ø) ⇒ ø is non-contingent. And that makes for trouble. After all: (a) guarantees openness marks contingency, and (b) guarantees Basicness marks non-contingency. No claim can be contingent and non-contingent. Hence none can be Basic and open. We have just seen, though, that Basic claims can be open. So there is potential conﬂict in practice. Either (a) or (b) is invalid. Either aptness does not ensure possibility, or Basic truths need not be necessary. Joe must inﬂate his space of possibilities or reject openness as one of its marks. The ﬁrst option pays for (a) with (b). The second does the reverse. One of them must go. Joe cannot have it both ways. If aptness marks genuine possibility, open Basics of Metaphysics are contingent. If such Basics are not contingent, aptness does not mark genuine possibility. Rosen’s point spreads, of course, to any area of inquiry driven by (b)*. It covers all traditional areas of philosophy (ethics, logic, mind, and so forth). They all harbour debate about Basics. They all presume Basic truths are necessary. Yet the timeless nature of debate within them strongly suggests that incompatible claims about Basics are apt. In turn that means one of two things: either such Basics are contingent, or aptness does not mark necessity. [B]. Humans tend to fuse epistemic and metaphysical matters: credence with chance, certainty with truth, certainty with causal determination. I take it we see this in our students and ourselves. Call it the ep-&-met tendency. It explains, I think, why it is so natural to say apriori reﬂection infallibly depicts genuine

54 / Modal Infallibilism and Basic Truth possibility. But here as elsewhere nature should be resisted. For genuine modality is mind-and language-independent. It is a non-epistemic domain. This should ensure apriori reﬂection is at best a fallible guide to genuine possibility. That is how good epistemology and metaphysics ﬁt together. When doing the former on mind- and language-independent fact, the result is humble pie. It is epistemology cleansed of error-free capacities to interrogate reality. It is magic-free epistemology. It is fallibilism. Most philosophers agree. When it comes to modality, though, they do not apply the lesson. They write as if they accept an under-revised version of the infallibilist model induced by the ep-&-met tendency. They write as if they replace (l) with something like (k) Apriori reﬂection infallibly depicts genuine possibility unless corrected by Kripke truth. But (k) looks like magic. After all, the space of genuine possibility is fully realistic. Neither its contours nor contents spring from apriori practice. Nor do they spring from such practice tutored by Kripke truth. By my lights, then, (k) is no better than Credence about the future infallibly depicts its chance unless corrected by present categorical fact; or Certainty about the past infallibly depicts it unless corrected by present fact. These views only tempt when blinded by the ep-&-met tendency. Once it is thrown off they do not. So it should be with (k). The schema trades in magic epistemology. To think otherwise is to fall prey to the ep-&-met tendency. Or so it seems to me. We have seen directly, moreover, that (k) is invalid. We have found Kripke coherent impossibilities. Reﬂection on them led to (a) Apriori reﬂection infallibly depicts genuine possibility unless when admitting a claim and its negation it also admits their non-contingency. But that looks even more magical. After all, the idea is that apriori reﬂection infallibly marks genuine possibility unless it self-corrects. Yet the space of such possibility is fully realistic. It would be little short of miraculous, then, if our

Scott Sturgeon / 55 apriori practice had in-built corrections to hand whenever needed. Would that it were so! By my lights, (a) is no better than Non-degenerate credence about future outcome infallibly depicts its chance unless it admits the chance of such outcome is degenerate; or Perceptual belief about one’s surroundings is infallible unless one knows one suffers perceptual delusion. These are crazy views. They only tempt when blinded by the ep-&-met tendency. Once it is thrown off they do not. And so it should be with (a). The schema trades in magic epistemology. To think otherwise is to fall prey to the ep-&-met tendency. Or so it seems to me.

7. Coda Our conclusion is simple. No reading of (L) Int(ø) ⇒ ø is valid. But we should ask: what difference does that make? Well, some discussion of physicalism requires (L) to have a valid reading. It maintains one can work out apriori whether physicalism is true simply on the basis of facts aside from whether physicalism is true. It says one need only work out whether fundamental physicals imply the rest. If they do, physicalism is true. If they do not, it is false. If (L) has no valid reading, however, the question is moot. Physicalism does not require everything to follow apriori from fundamental physicals.13 Set aside such discussion of physicalism. Ask yourself this: does anything else turn on (L)’s validity? I cannot see that it does. After all, intelligibility is obviously some kind of guide to genuine possibility. The question is what kind. (L) says it is an infallible one. But the truth is more modest. Intelligibility is a fallible guide to genuine possibility. 13

See David Chalmers’s The Conscious Mind (Oxford: OUP, 1996) and Frank Jackson’s From Metaphysics to Ethics (Oxford: OUP, 1998). For discussion see my Matters of Mind (London: Routledge, 2001).

56 / Modal Infallibilism and Basic Truth Let Int(ø) ø mean ø’s intelligibility is defeasible reason for its genuine possibility. We have several good readings of (L)* Int(ø) ø. Indeed, we have six ready to hand: (p)* pfac(ø) ø, (l)* lim(ø) ø, (k)* kc(ø) ø, (o)* o(ø) ø, (f)* f(ø) ø, (a)* apt(ø) ø. Prima facie, limit and Kripke coherence are reason to think a claim is genuinely possible. So are openness, forcedness, and aptness. In each case an apriori (or quasi-apriori)14 condition points to genuine possibility. And in each case it does so defeasibly. The range of defeaters will vary from case to case. But the moral will not: intelligibility defeasibly marks genuine possibility. It is a good question why that is so. It is not a good question whether it is. There is no question but that intelligibility defeasibly marks genuine possibility. That fact is a cornerstone of our modal practice. Without it no practice remains.15 14

Kripke coherence is idealized apriority tutored by Kripke truth. Such truth is aposteriori. Kripke coherence is thus quasi-apriori. 15 For discussion of the link between sensuous imagination and genuine modality, see chapter 5 of Matters of Mind and references therein. And for a perspective at odds with the ﬁnal section of this effort, see my Epistemic Norms.

3 The Modal Fictionalist Predicament John Divers and Jason Hagen

1. Introduction A central aim of genuine modal realism (GMR) is to provide a translation schema: (R) A iff A* in which the instances of ‘A’ are modal claims and the instances of ‘A*’ are their possible-world (counterpart-theoretic) translations. Correspondingly, the modal ﬁctionalist (locus classicus, Rosen 1990) proposes that we translate modal claims via the weaker schema: (F) A iff According to GMR, A*. Thus, for instance, with regard to: (1) It is possible that there are talking donkeys the realist holds: (R1) It is possible that there are talking donkeys iff there is a world at which there are talking donkeys whilst the ﬁctionalist holds: Thanks to the following for various helpful comments and suggestions: Rod Bertolet, Jan Cover, Andrew Evenson, Fraser MacBride, Joseph Melia, Daniel Nolan, Scott Shalkowski, Richard Woodward and an anonymous referee.

58 / The Modal Fictionalist Predicament (F1) It is possible that there are talking donkeys iff According to GMR, there is a world at which there are talking donkeys. One who afﬁrms (F1) and its left-side is committed to afﬁrming its right-side but, it is natural to think, one is not thereby committed to the existence of any world other than our own. Further, the ﬁctionalist will interpret unqualiﬁed possible world claims—e.g. ‘There is a world at which there are talking donkeys’ or ‘There is a world at which there are blue swans’—as being elliptical for the qualiﬁed thoughts expressed by right-side instances of (F). Broadly construed, then, the ﬁctionalist proposal is that we can use realistic sounding possible-world talk, retain at least some of the philosophical beneﬁts of doing so, and yet avoid commitment to the existence of a plurality of worlds. The modal ﬁctionalist proposal has attracted a great deal of discussion in the decade since its inception. In this paper we will argue that the upshot is that modal ﬁctionalism is ensnared in a serious predicament. In Section 2, we summarize the two potentially lethal objections to ﬁctionalism that have emerged and the only result which suggests that modal ﬁctionalism is capable of delivering any of the speciﬁc theoretical beneﬁts associated with realistic quantiﬁcation over possible worlds. The ﬁctionalist predicament, we argue, is that she cannot avoid both objections while maintaining her claim on the result. We proceed by considering two different strategies that the ﬁctionalist may adopt in order to translate modal claims about the plurality of worlds. In relation to the ﬁrst strategy we argue (in Section 3) that the Brock–Rosen objection is avoided but the Hale objection is avoided only by resorting to measures that are desperate (Hale 1995b) but which also deprive the ﬁctionalist of her result. In Section 4 we turn to the second strategy and consider two relevant sub-strategies. On the direct sub-strategy, we show that the Brock–Rosen objection is avoided if and only if the Hale objection is not avoided. On the indirect sub-strategy, we show that the Brock–Rosen objection is not avoided and that absurdity follows from the ﬁctionalist schema.

2. Two Objections and One Beneﬁt We start with an objection developed independently by both Stuart Brock (1993) and Gideon Rosen (1993). Brock and Rosen argue that the ﬁctionalist is

John Divers and Jason Hagen / 59 covertly committed to the proposition: (P) There is a plurality of worlds. The basic form of their argument may be represented as follows: 1 2

According to GMR, at every world, P Hyp Necessarily P iff According to GMR, at Inst of (F) every world, P 1,2 (B3) Necessarily P 1,2 iff elim 1,2 (B4) P 3T Clearly, if this ontological commitment can be pinned on the ﬁctionalist, then ﬁctionalism fails in its primary aim. For regardless of whatever beneﬁts ﬁctionalism may deliver, they come at the cost of a commitment to possible worlds.1 The second objection is a dilemma developed by Bob Hale (1995a). Hale argues that the ﬁctionalist who holds P false is in trouble if she maintains that the falsehood of P is contingent or if she maintains that it is necessary. If the ﬁctionalist holds that P is contingently false, the argument proceeds, then the instance of (F) that deals with the modal claim ‘Possibly P’—i.e. it is possible that there is a plurality of worlds—cannot adequately capture the content of that claim. We believe that the contingency horn of Hale’s dilemma can be blunted (see Divers 1999a). So all that the ﬁctionalist must do in order to resist Hale’s dilemma is establish her right to assert that P is contingently false. That can be done by defending the view that P, though false, is possible. The ﬁctionalist’s right to assert that P is possible is established, we presume, if she can plausibly deny that P is impossible. Since we confront the latter issue at the ﬁrst premise of the impossibility horn of Hale’s dilemma, we will only develop that horn of the dilemma. The objection associated with the impossibility horn is that the impossibility of P commits the ﬁctionalist to the truth of any arbitrary modal claim ‘X’. The argument to this effect runs as follows: 1

(B1) (B2)

An extended version of the argument, as developed by Rosen (1993) in particular, produces (B1) as a consequence of prior premises. Starting with the premise that a certain modal claim is true—e.g. the claim that it is necessarily contingent that there are kangaroos—it deduces (B1) from the truth of the ﬁctionalist translation of that claim. In this way, the right to reject (B1) will depend on the truth of this starting premise. We will qualify our main points as required to take this more extensive variant of the argument into account, but the variant need not, and will not, ﬁgure prominently in our argument.

60 / The Modal Fictionalist Predicament 1 2 1,2

(H1) (H2) (H3)

Not (Possibly P) GMR strictly implies P Not (Possibly GMR)

1,2

(H4)

GMR strictly implies (X*)

5

(H5)

1,2,5

(H6)

If GMR strictly implies (X*) then (According to GMR, X*) According to GMR, X*

Hyp Hyp 1,2 strict implication 3 Paradox strict implication Hyp

4,5 modus ponens 7 (H7) X iff (According to GMR, X*) Inst of (F) 1,2,5,7 (H8) X 6,7 iff elim If the above objection stands, then modal ﬁctionalism is not so much as consistent since it delivers the conclusion that, for any modal claim X, both X and its negation (also a modal claim) are true.2 The beneﬁt claimed on behalf of the ﬁctionalist by Divers (1999b) is that she has the resources to prove a certain safety result, thereby providing justiﬁcation for the practice of doing modal logic by proxy in ﬁrst-order logic. The practice of doing modal logic by proxy is the familiar one in which we set about evaluating a modal inference as follows: start with the modal premises; ‘translate’ modal premises into ﬁrst-order claims about worlds; use ﬁrst-order logic to draw a worldly conclusion from the worldly premises; move back from the worldly conclusion to its modal translation; and ﬁnally, conclude that the modal conclusion follows from the modal premises. The safety result that has to be established is that, in the following sense, this practice of doing modal logic by proxy never leads us astray: 2 The Hale (1995a and b) objection and Rosen (1995) reply do focus on a counterfactual treatment of the ﬁctionalist preﬁx. But the counterfactual conditional functions in that context as a representative of non-material conditionals in general Thus consider Hale’s comment on his own dilemma : ‘We know two things about [the ﬁctionalist preﬁx]—(1) it has to be non-factive and (2) if my dilemma is to be avoided it cannot be phrased as a strong (strict or counterfactual) conditional or entailment’ (Hale 1995b: 75–6, our emphasis). What is important for our purposes is: (a) that the ﬁctionalist result discussed below depends on treating the preﬁx in terms of strict implication and (b) that the Hale objection applies when the preﬁx is so understood (which is not in doubt).

John Divers and Jason Hagen / 61 (SR) Necessarily, if worldly claim B* is a logical consequence of worldly claim A*, then modal claim B is a logical consequence of modal claim A.3 The ﬁctionalist can demonstrate this result by invoking the ﬁctionalist schema (F), if she proceeds as follows: (Step 1) Incorporate a modal characterization of logical consequence to give a modal version of the safety result as follows: (MSR) (A* strictly implies B*) strictly implies (A strictly implies B).4 (Step 2) Assume that the ﬁctional operator, ‘According to GMR’ is closed under strict implication, thus: (CSI) ( (According to GMR, X) and (X strictly implies Y) ) strictly implies (According to GMR, Y). (Step 3) Derive (in classical S4 modal propositional logic) the modal version of the safety result (MSR) from the necessitated ﬁctionalist schema (NF): (NF) Necessarily, (A iff According to GMR, A*). 3

The very natural thought that the ﬁctionalist cannot deliver this result by means of a semantic theory that involves quantiﬁcation over possible worlds is defended in Divers (1999b). 4 An anonymous referee makes a good point in suggesting that perhaps what the ﬁctionalist should be aiming to show in this regard is not (MSR) but a result about logical consequence characterized other than in modal terms. We make two points in reply. First, the move from (SR) to (MSR) is made on behalf of the ﬁctionalist in order to secure the safety result (SR). If the move is wrong or unjustiﬁed, the effect is to enhance our present anti-ﬁctionalist dialectic. If a proof of (MSR) ought not to be construed as a justiﬁcation of (SR), then so much the better for our present case and so much the worse for the ﬁctionalist. For the position would then be that the ﬁctionalist has no proper claim on the safety result (SR) and—a fortiori—no ground for claiming that she can maintain the safety result while meeting the objections. Secondly, perhaps the kind of result that the ﬁctionalist ought to seek to establish is: (SR*) Necessarily ( (If B* is derivable from A*) then Necessarily (If A then B) ). Fortunately, the revised result is immediately obtained from the original strict implication result (MSR): (MSR) Necessarily (If Necessarily (If A* then B*) then Necessarily (If A then B) ) given the necessity of the (ﬁrst-order) derivability relation: (ND) Necessarily ( (If B* is derivable from A*) then Necessarily (If A* then B*) ). All we need is the transitivity of strict implication. Moreover the same point holds for the related result concerning ﬁrst-order semantic consequence in place of ﬁrst-order derivability. So there is a sense in which (MSR) may be a stronger safety result than (SR).

62 / The Modal Fictionalist Predicament (Step 4) Show that the modal version of the safety result follows from the weaker schema (F) by establishing (again in classical S4 modal propositional logic) the lemma: (Lem) (F) strictly implies (NF). Here we should note two important points. First, the capacity to underwrite the practice of doing modal logic by proxy is the only substantial beneﬁt for which the ﬁctionalist has a supporting argument. It has been argued that ﬁctionalism cannot deliver a possible worlds semantics for modal languages (Divers 1995) and no one has argued that ﬁctionalism gives—say—an account of the truthmakers for modal claims or that it articulates an analysis of modal concepts in nonmodal terms. This is what we mean when we claim that the result is the only result which suggests that modal ﬁctionalism is capable of delivering any of the speciﬁc theoretical beneﬁts associated with realistic quantiﬁcation over possible worlds.5 Second, it may be possible to establish the safety result by some other means, but in the absence of any alternative demonstration, the strategy outlined above presently offers the ﬁctionalist her one and only guarantee of the result. Moreover, if our argument is correct, then, in abandoning the principles that she must abandon in order to meet the objections, the ﬁctionalist must weaken signiﬁcantly the logic that she has at her disposal for showing what follows from the ﬁctionalist schema. In light of these considerations, we think it crucial to the credibility of ﬁctionalism that the ﬁctionalist be able to show that she is entitled to implement the strategy outlined. Overall, then, our challenge to the ﬁctionalist is to develop her original proposal in a way that proves capable of implementing the above strategy for securing the safety result and of avoiding both the Brock–Rosen and 5 Thus, in our terms it is not a ‘beneﬁt’ of modal ﬁctionalism, as may be alleged, that it offers an ontologically economical or naturalistic account of modality. Our concern is with the content of the intimated ‘account’. Which kinds of substantial explanations—semantic, metaphysical, conceptual—if any, can ﬁctionalism deliver? If modal ﬁctionalism can deliver any beneﬁts then, perhaps it may have the advantage over rival theories that deliver similar beneﬁts (genuine realism, actualist realism) that it does so naturalistically and economically. But that is a secondary matter, and ﬁctionalism must deliver the explanatory goods (‘beneﬁts’) in order to earn attention for its ‘theoretical virtues’. It ought to be noted in this regard that Rosen (1990) does suggest that ﬁctionalism, unlike realism, delivers the epistemological beneﬁt of accounting for the role of imagination in the formation of our modal beliefs. While we cannot engage fully here with this claim on behalf of ﬁctionalism, we register the objection that the claim will be justiﬁed only if the ﬁctionalist is prepared to endorse the view that the truthmaking facts for modal claims are facts about a story (GMR).

John Divers and Jason Hagen / 63 Hale objections. We will argue that neither of the two main developments of ﬁctionalism can meet this challenge.

3. The First Development Following the observations of Harold Noonan (1994), Rosen (1995) responds to the Brock–Rosen objection by specifying a version of the realist theory GMR that incorporates the Lewis (1968) principles for translating the formulas of quantiﬁed modal logic into formulas of counterpart theory. On those principles, modal claims are translated into claims about the parts of worlds: what is necessarily the case is a matter of what is true of the parts of all worlds, what is possibly the case is a matter of what is true of the parts of some worlds, and what is non-modally the case is a matter of what is true of the parts of the world of which we are parts. So, for instance, the initial hypothesis of the Brock–Rosen argument: (B1) According to GMR, at every world, P is false. For within the resources of Lewis (1968), the claim that (2) At every world, P can only mean (R2) ∀x(Wx → ∃y∃z(Wy & Wz & ∼(y=z) & Iyx & Izx) ).6 But (R2)—the claim that every world has a plurality of worlds as parts —does not hold according to GMR. It is inconsistent with postulate (P2) of Lewis (1968: 27). However, by accepting this solution to the BrockRosen objection, Rosen knows that the ﬁctionalist commits herself to grasping the ‘impossibility’ horn of Hale’s dilemma. On the Lewis (1968) approach, (3) It is possible that P is translated as (R3) ∃x(Wx & ∃y∃z(Wy & Wz & ∼(y=z) & Iyx & Izx) ). 6

Here we use ‘Ixy’ to stand for the part-of relation, while making it explicit when the relata are worlds. This diverges from how Lewis (1968) originally interpreted the symbol as a primitive in-a-world relation, but still consistent with the relation that he says he had ‘foremost in mind’ in that 1968 article (1983: 39).

64 / The Modal Fictionalist Predicament (R3)—the claim that some world has a plurality of worlds as parts—does not hold according to GMR. Indeed, to underscore the point, the Lewis (1968) schema translates its negation, ‘Not Possibly P’, as a claim which is true according to GMR—i.e. that no world has a plurality of worlds as parts. Thus, given (F), the ﬁctionalist is committed to denying (3) and to afﬁrming the initial hypothesis of the Hale argument: (H1) Not (Possibly P). Rosen (1995) then pinpoints three aspects of Hale’s argument that might be resisted in order to ‘ﬁx’ ﬁctionalism. First, one might reject the hypothesis, (H5), which partly articulates the thesis that the ﬁctive preﬁx can be adequately ‘glossed’ by some strict conditional.7 Second, there is the move from (H3) to (H4) which depends on the standard ‘paradoxical’ treatment of strict implication on which any proposition whatever is strictly implied by any impossibility. Third, the ﬁctionalist might seek to avoid commitment to (H1) by adducing (non ad hoc) grounds on which to claim that (H1) lacks a truthvalue. Hale (1995b) characterizes all of these attempts to protect the ﬁctionalist proposal as ‘desperate ﬁxes’, and there is certainly scope for regarding each as unsatisfactory in its own right. But here, we will only point out the implications that each move has for the preservation of the safety result. Concerning the ﬁrst ﬁx, it is provable that (H5) is a consequence of the closure of the ﬁctional operator under strict implication (CSI) given the further premise that According to GMR, GMR: 1 (1) ( (According to GMR, GMR) & Hyp (Inst CSI) GMR strictly implies X*) ) strictly implies (According to GMR, X*) 2 (2) According to GMR, GMR Hyp 3 (3) GMR strictly implies X* Hyp 2,3 (4) (According to GMR, GMR) & 2,3 & Intro GMR strictly implies X* 1,2,3 (5) According to GMR, X* 1,4 strict imp 1,2 (6) If GMR strictly implies X*, then 3,5 CP According to GMR, X* 7

It is worth noting that (H5) is far weaker than any claim of meaning equivalence between the ﬁctional preﬁx construct, ‘According to GMR, A’ and the strict conditional ‘GMR strictly implies A’. (H5) is only a material conditional, and that is all the Hale argument requires.

John Divers and Jason Hagen / 65 We assume that ‘According to’ must be understood as reﬂexive so that there is no question of rejecting premise (2). Thus, rejection of the conclusion (H5) compels rejection of the closure principle (CSI). Since the assumption of (CSI) constitutes Step 2 in the argument for the safety result, rejection of (H5) leaves the ﬁctionalist with no legitimate claim on the result.8 Concerning the second ﬁx, we note that rejection of the principle (H4)—that anything is strictly implied by an impossibility—marks a retreat to a modal propositional logic that is much weaker than S4. But the proof of the safety result relies on the soundness of S4 at Step 3 and Step 4. The thought behind the third ﬁx is that one can avoid commitment to (H1) by claiming that the sentence ‘Not (Possibly P)’ lacks a truth-value. More fully, if we consider a version of GMR, say GMR*, in which the deﬁnition of ‘world’ proceeds via the unanalysable, ﬁctive, and theoretical term ‘worldmate’, then we treat ‘world’, in effect, as a ﬁctional term—one like ‘hobbit’, ‘gimbles’, etc.—that earns its meaning solely by virtue of the role that it plays in the ﬁction. We might then choose to treat such terms in a broadly Fregean manner so that some of the declarative sentences in which they occur, and notably those that do not fall within the syntactic scope of an appropriate ﬁctional operator, are deemed to lack a truth-value. So the sentence that expresses P—viz. ‘There is a plurality of worlds’—would be taken to lack a truth-value and so, consequently, would modal complexes such as ‘Not Possibly P’ and ‘Possibly P’ in which it occurs. There are many points at which this sort of rationale might be questioned, but we will only consider the impact that accepting such truth-value gaps has on the argument for the safety result. One such gap would appear, for example, at Step 1 of the argument with the statement of the modal version of the safety result: (MSR) (A* strictly implies B*) strictly implies (A strictly implies B) (MSR) will lack a truth-value since its antecedent lacks one; the instances of ‘A*’ and ‘B*’ will be unpreﬁxed formulas of counterpart-theory which feature talk of ‘worlds’. So while lack of truth-value for (H1) would be enough to 8

Reﬂecting on the point made in n. 4, above, perhaps the ﬁctionalist should not endorse the thesis that her preﬁx is closed under strict implication, but only that it is closed under logical consequence otherwise understood, e.g. as derivability or as semantic consequence. That being so, perhaps the present objection would fall. But the result would be placed in jeopardy since we have no guarantee that it follows from the alternative, weaker, closure principles.

66 / The Modal Fictionalist Predicament render the Hale argument unsound, the lack of truth-value of (MSR) would also be enough to render unsound the proof of the ﬁctionalist result. We conclude that none of Rosen’s ﬁxes will sustain the only available argument for the only beneﬁcial result that has been claimed on behalf of ﬁctionalism. Moreover, any of these ﬁxes will signiﬁcantly restrict the logic of ﬁctionalism, so that the capacity to deduce theoretically beneﬁcial consequences from the ﬁctionalist translation scheme is, in general, diminished. But before leaving the version of ﬁctionalism that incorporates the Lewis (1968) translation principles, it is important to emphasize a further and broader difﬁculty that ﬁctionalism inherits from them. The Lewis (1968) translation principles fail to provide truth-preserving translations for an important range of modal claims. The claims in question, which might be termed ‘advanced’ modal claims (Divers 1999a), involve modalizing about entities, or collections of entities, that are not (all) parts of any single spatiotemporally uniﬁed whole. Such claims include modal claims about collections of Lewisian worlds, understood as spatiotemporally disjoined individuals. However, amongst such claims we also ﬁnd, arguably—and certainly according to Lewis’s genuine modal realism—modal claims about properties, numbers, propositions, sets, and states of affairs. The Lewis (1968) translation principles for quantiﬁed modal logic are intended to deal only with modal formulas that express modal claims about ordinary, spatiotemporally uniﬁed individuals. Accordingly, advanced modal claims such as (4) It is possible that there are natural properties (5) For any proposition, it is contingent whether it is thought (6) 4 is necessarily even are not intended by Lewis to be within the expressive range of the formulas of (ﬁrst-order) quantiﬁed modal logic; counterpart-theoretic translations are not intended to apply to these cases. This is, at least, the obvious and charitable interpretation. The application of Lewis (1968) principles to such claims simply distorts truth-values as when, for example, (4) comes to express: (R4) ∃x(Wx & ∃y(Iyx & Ny) ). According to GMR, it is not the case that some world has a natural property—for Lewis, a transworld set—as a part.9 So the ﬁctionalist would be 9 For more on why the 1968 principles fail to properly account for advanced modal claims, and on why other variations on the 1968 approach are inadequate, see Divers (1999a).

John Divers and Jason Hagen / 67 committed to the falsehood of (4) and to other such errors. Incorporating the Lewis (1968) translation principles into the reading of (F), we get: (F4) It is possible that there are natural properties iff According to GMR, ∃x(Wx & ∃y(Iyx & Ny) ) whereby the falsehood of the right-side—according to GMR, some world has a natural property as a part—visits falsehood on the left. How signiﬁcant is all of this? Well, if the best version of genuine modal realism on offer is one that must restrict its translations to ordinary modal claims, that is certainly a signiﬁcant boundary on the realist programme. But the ﬁctionalist, who aims only to keep up with the realist, is no worse off than that. Of course, if a version of genuine realism is available which is capable of dealing just as adequately with the translation of advanced modal claims as it does with the translation of ordinary modal claims, then the ﬁctionalist is obliged to keep up with that version of realism. This last observation paves the way for a second development of the original ﬁctionalist proposal that the ﬁctionalist might explore in the effort to avoid the objections and secure the result.

4. The Second Development Divers (1999b) argued that the ﬁctionalist can solve all of her problems by incorporating the genuine modal realist approach to advanced modal claims presented in (Divers 1999a). This approach unfolds as follows.10 The genuine realistregardsour non-modal claimsaboutordinary individuals such as donkeys, swans, planets, etc., as implicitly world-restricted claims. For example: (7) There are donkeys is to be interpreted—by default, in ordinary contexts of use—as the (true) claim (R7) ∃x(Ixα & Dx) 10 The need for the genuine modal realist to consider a special treatment for modal claims about the plurality of worlds has been widely recognized in the modal ﬁctionalist literature (e.g. Menzies and Pettit 1994; Noonan 1994). The suggestions in those papers and in this as to the form that the treatment should take are developments of a hint given by Lewis (1986: 6) in remarks on restrictive modiﬁers.

68 / The Modal Fictionalist Predicament —that the actual world (α) has donkeys as parts. However, the genuine realist cannot regard true non-modal existential claims about properties, numbers, propositions—e.g. (P) There is a plurality of worlds (8) Natural properties exist —as claims in which the quantiﬁers are world-restricted. If read that way, they express falsehoods from the genuine realist’s standpoint—viz.: (RP) ∃x∃y(Ixα & Iyα & Wx & Wy & ∼ (y = x) ) (R8) ∃x(Ixα & Nx). If (P) and (8) are to express truths, as the genuine realist requires, the quantiﬁers they contain should be interpreted as unrestricted, ranging over all of logical space and not just the actual part of it—thus: (RP*) ∃x ∃y(Wx & Wy & −(y = x) ) (R8*) ∃x(Nx). Ordinary modal claims are interpreted by the realist as existential and universal quantiﬁcations into places held in corresponding non-modal claims by world-restricting terms. So we have the translations: (R9) There are donkeys iff ∃x(Ixα & Dx) (R10) It is possible that there are donkeys iff ∃y(Wy & ∃x(Iyx & Dx) ) (R11) It is necessary that there are donkeys iff ∀y(Wy → ∃x(Ixy & Dx) ). But sentences taken to express true advanced modal claims cannot be regarded as expressing generalizations of previously world-restricted contents. In sentences taken to express true advanced modal claims—e.g. (3) It is possible that P (4) It is possible that there are natural properties —the non-modal sentences that the modal items modify—viz. (P) There is a plurality of worlds (8) Natural properties exist never expressed world-restricted contents in the ﬁrst place. (Thus, see (RP*) and (R8*) above.) There is no world-restricting element afoot in the content of (P) or (8) to sustain subsequent existential or universal generalization.

John Divers and Jason Hagen / 69 As an alternative, the genuine realist can say that in advanced modal claims, the modal modiﬁers are redundant. When A is read as such a claim, the genuine realist should impose the advanced possibility translation schema: (RP-A) It is possible that A iff A. With regard to the typical advanced possibility claims that we have considered, this schema yields: (RP-A3) It is possible that there is a plurality of worlds iff ∃x ∃y(Wx & Wy & ∼(y=x) ) (RP-A4) It is possible that natural properties exist iff ∃x (Nx). By invoking the classical deﬁnitions of necessity and contingency in terms of possibility, the realist can derive from (RP-A) further schemas to deal with advanced necessity and contingency claims respectively: (RN-A) It is necessary that A iff A (RC-A) It is contingent that A iff A and not-A.11 Intuitively, what holds of logical space unrestrictedly is not a contingent matter, and in non-contingent matters what is possible is also necessary. Such intuitions are captured by these realist schemas. In formulating translation schemas for advanced modal claims, there are two ways that the ﬁctionalist might proceed. The most direct approach is to apply the general ﬁctionalist schema (F) to the genuine realist translations of advanced modal claims that are provided by (RP-A), (RN-A), and (RC-A). This approach produces the following: (FP-A) It is possible that A iff According to GMR, A (FN-A) It is necessary that A iff According to GMR, A (FC-A) It is contingent that A iff According to GMR, A & not-A. An indirect approach is to accept the directly obtained ﬁctionalist translation schema for one type of advanced modal claim (e.g. advanced possibility claims), and then use the classical deﬁnitions of the other modal operators in terms of that basic case to derive schemas for the other types of advanced modal claims. The two approaches—direct and indirect—are not equivalent. Using 11

In (RC-A), this is contingency in the factive sense: it is contingent that A iff A and possibly not-A. Hence, (RC-A) from (RP-A). Contrast contingency in the non-factive sense: it is contingent whether A iff it is not necessary that A and it is not impossible that A.

70 / The Modal Fictionalist Predicament the schema for advanced possibility as the basic case, the approaches provide the same schema for advanced necessity but different ones for advanced contingency. Beginning with the target sentence, (9) It is necessary that A, translation into its classical possibility equivalent gives us: (10) It is not the case that it is possible that not A. Applying (FP-A) to 10, the sentence obtained is equivalent to the right-side of (FN-A): (11) It is not the case that (According to GMR, not A). If it is not the case that not A is true according to GMR, then A is true according to GMR. Thus, the derived schema for advanced necessity would be equivalent to (FN-A): (DFN-A) It is necessary that A iff According to GMR, A. Turning to contingency, we begin with: (12) It is contingent that A, and translate it into its classical possibility equivalent: (13) A and possibly not A. Applying (FP-A) to (13), the derived advanced contingency schema is: (DFC-A) It is contingent that A iff (A and According to GMR, not A). The major difference between this derived schema and (FC-A) is that, given the consistency of GMR, (FC-A) renders all advanced contingency claims false whilst on the indirect approach some advanced contingency claims must be, from the ﬁctionalist’s perspective, true. For example, the ﬁctionalist must afﬁrm the crucial claim that it is contingent that not-P when that claim is translated as a right-side instance of (DFC-A): (14) It is not the case that P and According to GMR, P.12 However, we will now show that it does not matter whether the ﬁctionalist opts for the direct or indirect approach to the translation of advanced modal 12

Divers (1999b) had tried to earn for the ﬁctionalist the right to assert the contingent falsehood of P by appealing to (14), but we will now show why that move is unsatisfactory.

John Divers and Jason Hagen / 71 claims. Either way, she cannot escape her predicament. If the direct approach is to be available to the ﬁctionalist, then she must accept all three schemas (FP-A), (FN-A), and (FC-A). But given the acceptance of (FP-A) and (FN-A), the ﬁctionalist will be committed, as the realist is, to afﬁrming the equivalence of advanced possibility and necessity: (15) It is possible that A iff it is necessary that A. This is disastrous. For with respect to the instance (16) It is possible that P iff it is necessary that P, the ﬁctionalist needs to assert the left-side in order to avoid Hale’s objection but needs to deny the right-side in order to avoid a commitment (via the Brock–Rosen argument) to P. So the direct approach will not work. If the indirect approach is adopted, then the ﬁctionalist has to accept at least one directly parasitic schema from which the others are derived. But problems arise in connection with accepting any of them. The ﬁctionalist must reject the contingency schema (FC-A) in order to avoid Hale’s objection; one who accepts (FC-A) is constrained, given the consistency of GMR, to deny the contingent falsehood of P. The ﬁctionalist must reject the necessity schema (FN-A) in order to avoid what might be considered a generalized version of the Brock–Rosen objection. Where ‘Trans (Necessarily P)’ stands for whatever the appropriate ﬁctionalist translation of ‘Necessarily P’ is, a generalized form of the Brock–Rosen argument is this: 1 (1) Trans (Necessarily P) Hyp 2 (2) (Necessarily P) iff Trans (Necessarily P) Hyp 1,2 (3) Necessarily P 1,2 iff 1,2 (4) P 3, T Given (FN-A), the ﬁctionalist is committed to the soundness of an instance of this argument-form since she will be committed to the truth of its ﬁrst two premises: (1*) According to GMR, P (2*) Necessarily P iff According to GMR, P. Finally, the ﬁctionalist must reject the possibility schema since it commits her to an absurd consequence—that according to GMR there is no plurality of worlds—in conjunction with her claim that P is false:

72 / The Modal Fictionalist Predicament 1 1 3

(1) (2) (3)

1,3

(4)

Not-P Possibly (Not-P) Possibly (Not-P) iff According to GMR (Not-P) According to GMR, not-P

Hyp Possibility Introduction Instance of (FP-A) 2,3 iff13

So the indirect approach is not an option either.14 The second strategic development of ﬁctionalism, like the ﬁrst, fails to vindicate ﬁctionalism.

5. Conclusion The challenge that we put to the ﬁctionalist remains: show that ﬁctionalism can be developed in a way that justiﬁes doing modal logic by proxy (or delivers some other beneﬁt) whilst avoiding the disjunction of modal collapse and commitment to possible worlds. Since no development of the original proposal has thus far proved capable of meeting that challenge, we conclude that ﬁctionalism is in a serious predicament from which it must be rescued if it is to remain a viable option in modal philosophy. 13

Of course, the ﬁctionalist could block this argument against (FP-A) by rejecting the unrestricted validity of possibility introduction just as one could block the (generalized) Brock–Rosen argument by rejecting the validity of necessity elimination. But we take the validity of these rules to be so obviously constitutive of our alethic modal concepts that to abandon them would be to change the subject of translation. 14 There is another reason, besides the one given by the above argument by elimination, for rejecting the indirect approach to translating advanced modal claims. Once the ﬁctionalist rejects the direct approach, she must retreat to a restricted version of the ﬁctionalist schema (F) since she no longer endorses all instances of (F), certain advanced contingency claims being the exceptions. Yet the proof of the safety result depends on the necessitated ﬁctionalist schema (NF) at Step 3 and the equivalence of that schema to (F) at Step 4. Given that only a restricted version of the schema is now available to the ﬁctionalist, the safety result, and so her justiﬁcation of doing modal logic by proxy, does not cover all modal claims. The result may still be signiﬁcant even if it covers inferences involving only ordinary modal claims and it is plausible that that was all that the ﬁctionalist originally envisaged, but the restriction promises to put the ﬁctionalist at a disadvantage to the realist. As for the realist, there is no obvious reason why she should not be able to extend the ﬁrst-order, proxy treatment of modal inferences to cases involving advanced modal claims.

John Divers and Jason Hagen / 73

References Brock, Stuart (1993), ‘Modal Fictionalism: A Reply to Rosen’, Mind, 102: 147–50. Divers, John (1995), ‘Modal Fictionalism Cannot Deliver Possible World Semantics’, Analysis, 55: 81–9. (1999a), ‘A Genuine Modal Realist Theory of Advanced Modalizing’, Mind, 108: 217–39. (1999b), ‘A Modal Fictionalist Result’, Noˆus, 33: 317–46. Hale, Bob (1995a), ‘Modal Fictionalism—A Simple Dilemma’, Analysis, 55: 63–7. (1995b), ‘A Desperate Fix’, Analysis, 55: 74–81. Lewis, David (1968), ‘Counterpart Theory and Quantiﬁed Modal Logic’, Journal of Philosophy, 65: 113–26. (1983), ‘Postscript to ‘‘Counterpart Theory and Quantiﬁed Modal Logic’’ ’, Philosophical Papers, (Oxford: OUP), 39–46. (1986), On the Plurality of Worlds (Oxford: Basil Blackwell). Menzies, Peter, and Pettit, Philip (1994), ‘In Defence of Fictionalism about Possible Worlds’, Analysis, 54: 27–36. Noonan, Harold (1994), ‘In Defence of the Letter of Fictionalism’, Analysis, 54: 133–9. Rosen, Gideon (1990), ‘Modal Fictionalism’, Mind, 99: 327–54. (1993), ‘A Problem for Fictionalism about Possible Worlds’, Analysis, 53: 71–81. (1995), ‘Modal Fictionalism Fixed’, Analysis, 55: 67–73.

4 On Realism about Chance Philip Percival

‘Chance’ is a single-case, temporally relative, objective probability for which a normative principle of direct inference holds. It is single case in so far as it applies to particular events, such as the explosion of the Challenger Space Shuttle (or, better, to propositions asserting the occurrence of such events); it is temporally relative in so far as it changes over time, as when, seemingly, the chance of the Challenger exploding increased as the temperature dropped; it is objective in so far as claims of the form ‘the chance at t of E is r’ purport to be true independently of what anyone thinks; and a normative principle of direct inference holds for it in so far as the chance of an event, or at any rate a cognitive attitude, such as belief, or credence (i.e. partial belief), towards that chance, normatively constrains cognitive attitudes towards the event itself, in the manner, e.g. of a requirement that an agent who believes there to be a one half chance of heads on a certain coin-toss ought, in the absence of direct evidence regarding the outcome, to distribute his credence in heads and tails equally. These stipulations are not arbitrary: at worst, they constitute well-motivated sharpenings of one use of the word ‘chance’. But although features of a concept I am grateful to Jim Edwards, Barry Loewer, and Peter Milne for helpful conversations, as I am to the Arts and Humanities Research Board for funding the research leave during which the ﬁrst draft of this paper was written. I am unusually and especially heavily indebted to Christopher Hitchcock, who corrected several misunderstandings on my part. Without his careful and patient explanations of certain aspects of non-standard analysis and their bearing on the problem of ﬁt, my discussion of this problem in Section 2.3 and the Appendix would have contained several errors it now avoids. Responsibility for any errors that remain is entirely mine.

Philip Percival / 75 can be stipulated, the existence of something falling under the concept cannot be. It remains to be settled whether chance exists. To say that chance exists is to say that some statement of the form ‘the chance at t of E is r’ is objectively true. I shall focus on the question as to whether ‘non-trivial’ chance—that is, chance having values other than 0 or 1—exists. I call the view that it does ‘realism’ about chance.1 Thus deﬁned, realism about chance is neutral regarding whether chance is a (natural) property, tendency, disposition, or ‘display’ of any such.2 It is also neutral as to whether chance is analysable, Humean supervenient, or supervenient on non-chance.3 And it is neutral as to whether chance is constrained by our capacities to ascertain its values. It might not be neutral with respect to determinism: many authors maintain, albeit controversially, that the existence of non-trivial chance requires the falsity of determinism.4 But I shall not address the objection that realism about chance is false because determinism is true. To sideline it, I shall assume—as quantum mechanics leads many to suspect—that determinism is false.5

1 There is no established terminology in this area. Black’s (1998) use of ‘realism’ about chance lacks the neutrality of mine. 2 See Giere (1973), Mellor (1969, 1971), Popper (1959), and Sapire (1991). 3 Non-chance at a world w comprises everything at w other than chance and what presupposes chance. It includes all of w’s particular events, like deaths and the breaking of chemical bonds, and the instantiations at it of properties and relations like negative charge and being the father of. But other features of non-chance are controversial. For example, whether it includes causes at indeterministic worlds depends on controversial features of indeterministic causation. McDermott (1999: sect. 2.6) argues that even if realism about chance is correct, counterfactuality does not supervene on non-modal matters (including chance). Since he offers a counterfactual analysis of causation, in my terms this commits him to holding that non-chance includes indeterministic causes. But indeterministic causes are excluded from non-chance by those such as Lewis (1986: 175–84) and Noordhof (1999) who advance analyses of indeterministic causation in terms of chance. Either way, the thesis that chance supervenes on non-chance is neither entailed by, nor entails, the thesis that chance is ‘Humean supervenient’ in Lewis’s (1986: pp. ix–xvii) sense, since (i) it does not impose stringent requirements (such as localization to spacetime points) on the ‘non-chance’ properties in the subvenient base; (ii) it is not committed to the assumption that all combinatorially possible distributions of properties in the base are metaphysically possible, and (iii) it is not consistent with the supposition that chance is an unanalysable natural property localized at spacetime points and subject to the combinatorial hijinks described in (ii). (Cf. Sturgeon (1998).) 4 See Mellor (1971), Giere (1973: 475), and Lewis (1986: 117–21). Mellor (1982, 1995) subsequently retracts his earlier view, and argues that non-trivial chance is compatible with determinism. 5 Cf. Lewis (1986: 58–9).

76 / On Realism about Chance In recent years, the terms of the debate over realism about chance have been set by a series of studies in which David Lewis (1980, 1986, 1994) develops a stunning exposition and defence of (one version of ) the doctrine. Ironically, however, his ambivalence as to whether chance supervenes on non-chance embodies a dilemma that threatens realism’s coherence. At the outset, Lewis (1980, 1986) felt forced to eschew the supervenience of chance on account of so-called ‘undermining’: if chance were supervenient, the truth of the proposition that captures the way the subvenient base actually is would determine chance in such a way as to give some chance of that proposition being false, and, hence, of the chances not being what they in fact are.6 By the end, however, Lewis (1994: 484–5) embraces with evident relief a ‘best-systems’ analysis that ensures chance’s supervenience: if chance were not supervenient, rational credence would not be something it must be—namely, supervenient on symmetries and relative frequencies. Of course, this volte face does not entangle Lewis (1994) in double-think: he reckons the undermining difﬁculties he had raised earlier to have been resolved. Nevertheless, his unwavering advocacy of realism about chance notwithstanding, the dilemma still has bite: amongst realists about chance, each horn continues to attract passionate advocates.7 I shall not explore the possibility of developing the dilemma into a full-blown argument for the incoherence of the concept of chance, however. Although I shall argue that the best-systems analysis Lewis came ﬁnally to espouse is untenable, I 6

Undermining in this sense is barely credible in itself, but Lewis (1980: 109–13; 1986: pp. 121–31, xiv–xvii) is more concerned with its apparent inconsistency with the ‘Principal Principle’—a normative principle connecting credence about a proposition’s chance with credence in the proposition itself—that he takes to be constitutive of the notion of chance. (This principle is explained in n. 24, below.) Lewis’s worry has met with several responses. See Vranas (2002) for the retort that the apparent inconsistency rests on the assumption that the supervenience of chance on non-chance is necessary (an assumption questioned by Lewis’s claim that Humean supervenience is only contingently true). And see Thau (1994) and Hall (1994) for the retort that the Principal Principle is false, but not constitutive of the notion of chance. Lewis (1994) himself favours a third response. He holds that although the Principal Principle is constitutive of the notion of chance but only approximately true, this much is consistent with the existence of chance: he argues that chance is that feature of the world that best approximates the principles that are constitutive of chance. (See too Strevens (1995). Hall (2004) has further references.) 7 Regarding the ﬁrst horn, Bigelow, Collins, and Pargetter (1993) argue that quite apart from its impact on the Principal Principle, undermining is absurd in itself. (See n. 6, above.) Regarding the second horn, compare Hall’s (2004: 94) remark that ‘it is widely assumed that . . . it is unacceptably mysterious’ why, if chance is not Humean supervenient, rational credence should be constrained by chance in the way that it is.

Philip Percival / 77 shall not do so with a view to evaluating the dilemma. In particular, I shall have nothing to say about the undermining difﬁculties raised for this analysis in its ﬁrst horn. My aim is merely to question whether, on the assumption that realism about chance is coherent, there is any reason to believe it. Some take an afﬁrmative answer to this question to be relatively straightforward: contemporary science seems to attribute chances to various events, and, following Quine, this fact alone, they say, warrants belief in the existence of chance.8 I think this line of thought superﬁcial, however. That science appears to employ chance in describing the world, and formulates laws concerning it, is no doubt a prima facie reason for realism about chance. But the instrumentalist, constructivist, and anti-realist traditions in the philosophy of science have enough going for them for there to be no guarantee that this appearance will withstand critical scrutiny. Science is good, but it is not that good. More substantial groundings might be attempted for belief in chance, but I shall argue that they too are inadequate. One way to demonstrate the existence of chance would be to show that currently accepted facts about non-chance necessitate it, and some analyses of chance might purport to do just that. The least worse of those that are currently available—Lewis’s—fails, however, on account of a defect that has passed unnoticed in the furore over undermining (Section 2). In the absence of a successful analysis, the realist must search for an inductive reason for believing in chance. The most promising course would be to try to identify an explanatory role chance best fulﬁls, and then appeal to inference to the best explanation. There are two candidates for such a role, but neither gives the realist what he needs (Section 1).

1. Chance and Inference to the Best Explanation Chance has been invoked with a view to explaining statistical phenomena,9 and, more idiosyncratically, the warrantedness of certain credences.10 I shall argue that neither consideration supports an inference to the best explanation from non-chance to chance: chance is explanatorily vacuous with respect 8

Cf. Lewis’s (1986: 178) remark: ‘You may not like single-case chances—I don’t either—but I cannot see how to make sense of certain well-established scientiﬁc theories without them.’ 9 See Mellor (2000: 20) and Hall (2004: 110). Hall seems to maintain that it is only if chance does not supervene on non-chance that it can explain indeterministic statistical phenomena. 10 See Mellor (1969, 1971, and 1982).

78 / On Realism about Chance to indeterministic statistical phenomena (Section I.1), while attempts to introduce it as a ground of warrant beg the question (Section I.2).

1.1 Statistical Phenomena Let ‘statistical phenomena’ include both statistical regularities concerning the relative frequencies that possible outcomes of certain processes tend to in large-scale trials, together with these regularities’ instances and the individual events that make them up. A statistical regularity takes some such form as: for all large-scale trials on a set-up S, the relative frequency with which outcome O is obtained tends to (approximately) r. An instance of the regularity is then some such trial in which the relative frequency of the outcome O tends to (approximately) r. For example, a statistical regularity involving a fair coin might consist in the fact that there are three-hundred large-scale trials in which the coin is tossed, in each of which the relative frequency of heads tends to (approximately) 1/2. The fact that the relative frequency of heads tended to 1/2 on the seventeenth of these trials is an instance of the regularity. The individual events that comprise this instance are its successive tosses and their outcomes. Indeterminism must be defended if chances are to be advocated on the grounds that they play a vital role in the best explanation of statistical phenomena. This is a point regarding explanation that is independent of the metaphysical issue alluded to earlier regarding whether or not the existence of chances other than 0 and 1 is compatible with determinism: it remains valid even if non-trivial chance and determinism are compatible.11 If determinism is true, all actual phenomena not involving chance are inevitable at times prior to their occurrence: propositions describing the events that constitute them are entailed by propositions that describe prior (‘initial’) conditions and physical laws without mention of chance. Since entailments of this kind are characteristic of paradigmatic ‘deductive nomological’ explanations, both deterministic statistical regularities, and events comprising their instances, are explicable without invoking chance.12 No room remains for a claim of the 11

See p. 75, above. One might object that this attitude to the explanation of deterministic statistical regularities is insufﬁciently sensitive to the distinction between an instance of a statistical regularity and the regularity itself, and that though the fact that e.g. 100 successive deterministic tosses of a coin on some occasion yielded 47 heads might be explained in the manner envisaged, the fact 12

Philip Percival / 79 form: such events and regularities are best explained by invoking chance. If determinism is true, there are ideal explanations of them involving no appeal to chance. Although we might not know these explanations individually, or, even, have any means of knowing them, belief that determinism is true obliges belief that they exist. Nevertheless, I am happy to concede that belief in indeterminism is reasonable: quantum mechanics might well provide grounds for it. But what exactly is the crucial explanatory role chance is supposed to play with respect to indeterministic statistical phenomena? It is natural to focus ﬁrst on statistical regularities. Certainly, it is no easier to accept ‘brute’ indeterministic regularities of this kind than it is to accept the brute non-statistical regularities that have been the primary focus of the post-Humean controversy over fundamental deterministic laws. Indeed, whereas the latter have prompted some anti-Humeans to hypothesize a contingent relation of necessitation between universals in an attempt to reduce their apparent arbitrariness,13 the attempt to explain statistical regularities via chances seems more compelling. A natural explanation of why this should be so is suggested by claimed analogies between the theoretical roles played by chance and, e.g., mass and force.14 It views the difference between appeals to a necessitation relation, and appeals to chance, as the difference between pseudo-science and the real thing. Appealing to a relation of necessitation between universals so as to explain fundamental non-statistical regularities is pseudo-science: so doing tries to get something on the cheap when science stops. In contrast, appealing to chance to explain indeterministic statistical regularities is just science: talk of chance is integral to physics. One should be wary both of this explanation, and of the apparent asymmetry for which it purports to account, however. Even if talk of chance is integral to physics, it does not follow that science gives chance an explanatory role to play with respect to statistical (or any other) phenomena. Talk of numbers is integral to physics too, but it can hardly be supposed that physics exploits the explanatory power, e.g. of the number 49. Moreover, the simple model that underlies na¨ıve enthusiasm for the explanatory pretensions of chance is quite that it regularly happens that when the coin is tossed deterministically a large number of times approximately 50% heads results cannot be. The most this consideration could show, however, is that ideal explanations of deterministic statistical regularities must advert to statistical regularities amongst initial conditions. 13 14 See Dretske (1977) and Armstrong (1983). Giere (1973: 473–4, 486).

80 / On Realism about Chance untenable. It takes chance to play a direct role in the explanation of statistical regularities as follows: Explanans

Each G has chance r of being F.

Explanandum

The relative frequency of F’s in each ‘trial’ comprising large numbers of G’s tends to (approximately) r. That explanations of this kind have an immediate appeal is undeniable.15 But their appeal is illusory. It stems from an entailment that acts as an intermediary between the explanans and the explanandum, and once this entailment is made explicit, the explanation is exposed as a sham. The intermediary entailment is the proposition that the chance of the relative frequency of F’s among large numbers of G’s tending to (approximately) r is high. When this is made explicit, the model splits into two: Explanans1

Each G has chance r of being F.

Explanandum1 / Explanans2

The chance of the relative frequency of F’s among large numbers of G’s tending to r is high.

Explanandum2

The relative frequency of F’s in trials comprising large numbers of G’s tends to (approximately) r.16 Theﬁrstsub-explanationisimpeccable,sinceexplanans1 entailsexplanandum1 . But it is this fact alone that accounts for the original explanation’s appeal. For the second sub-explanation is spurious. Even if—and, perhaps, especially if—the chances involved do not supervene on non-chance, how is the fact that a proposition had a ‘chance’ that was ‘high’ supposed to explain its truth? Why, as must be the case if belief in the existence of chance is to be grounded in this simple model, is the explanation of an indeterministic event by its chance the better the higher that chance? Since there are neither immediate nor mediate deductive connections between the proposition asserting of an event that it has a high chance, and the proposition asserting the event’s occurrence, what is the connection between these two propositions such that the one explains the other? The only speciﬁable

Mellor (2000: 20) writes that ‘[T]he chance ch(H) = p of heads on a coin toss explains the fact (when it is fact) that the frequency f(H) of heads on many such tosses is close to p.’ 16 The reason Mellor (2000: 20) gives for his claim that the chance of heads on each toss of a coin explains the fact that a relative frequency of heads in the long run which approximates that chance results is that ‘(given the laws of large numbers) the chance of this fact is close to 1’. 15

Philip Percival / 81 connection is epistemic: the truth of the former warrants high credence in the latter. But in that case a dilemma arises. If the ‘explains’ relation is never epistemic, chance is impotent: the high chance of an event cannot explain it. If, however, the ‘explains’ relation is sometimes epistemic, chance is superﬂuous: we already possess epistemic relations in abundance—namely, evidential relations—in terms of which to explain indeterministic events without invoking chance.17 The force of the ﬁrst horn is clear enough, but let us consider an example that serves to illustrate the force of the second. Suppose that the relative frequency of heads when some coin is tossed a large number of times tends to 2/3. Realists themselves often emphasize that postulating a chance of heads on each toss equal to 2/3 to explain this fact only has bite in so far as this chance can be linked in a law-like way to other properties of the coin and the tossing device.18 In the case at hand, call these properties, ‘bias’ and ‘fairness’ respectively. Ex hypothesi, bias and fairness together comprise evidence, of certain strengths, that the outcome of a given toss will be heads, and that the relative frequency of heads in large-scale trials will tend to 2/3. So if epistemic relations can sufﬁce for explanatoriness, chance is an idle intermediary: bias and fairness possess the requisite explanatory power already. In part because of the failure of the simple model, some authors maintain that even if chance exists, indeterministic matters—indeterministic statistical regularities, their instances, and the events that constitute them—are inexplicable, and that those who maintain the contrary confuse the explanation of the chance of an event with an explanation of the event itself.19 On the other hand, their opponents take scepticism of this ilk to wrongly attribute the defects of the simple model to what is modelled, and turn to models of indeterministic explanation that are more reﬁned. Though these reﬁnements are to be applauded in some respects, they too fail to underpin realism about chance. This much is obvious in the case 17

Although Mellor (1976: 235–6) argues that an explanandum is better explained the higher its chances according to the explanans, even he does not insist that only chance provides probabilistic explanations. On his account, explanans and explanandum must be linked by a high epistemic probability. But while Mellor holds that chance can underpin this probability, he declines to rule out the possibility of explanations in which the epistemic probability is not so underpinned. He writes ‘Perhaps all explanatory probabilities are chances, perhaps not.’ 18 19 Mellor (1969: 26). Woodward (1989: sect. 3).

82 / On Realism about Chance of Railton’s (1978) deductive nomological model. Let E be an indeterministic event. According to Railton, the core of an explanation of E comprises a deductive argument from chance laws and initial conditions to the conclusion that, immediately prior to its occurrence, E had a chance r of occurring. The explanation is completed by conjoining this argument with the statement that E in fact occurred. The salient feature of this model is that how good the explanation is does not depend on the chance accorded the event to be explained: provided the chance is ascribed correctly, the explanation is equally good whatever the chance ascribed E. Clearly, belief in the existence of chance cannot possibly be grounded by an inference to the best explanation if the model is correct. For the model presupposes chance. There is no merit in the thought that it is reasonable to believe that chance exists because certain explanations that conform to Railton’s model are best. For the realist’s purposes, alternative models that exploit the thought that indeterministic events can have causal histories are more promising. Once this thought is granted, indeterministic events can be given both ‘plain’ explanations, and, pace Lewis (1986: 229–31), ‘contrastive’ ones too: for one can explain why an indeterministic event occurred by citing one of its causes, and one can explain why it occurred rather than some other event that might have occurred instead by citing a cause that meets certain conditions.20 Models of this kind suggest a less direct strategy for introducing chance via its role in the explanation of indeterministic events. In full generality, the strategy is this. What makes something an explanans of an explanandum is the existence of a certain kind of link between them. This link must be objective. In the deterministic case, logic itself provides the link. But logic cannot be the link in the indeterministic case, and whatever one calls it the link can only arise from chance. On this strategy, chances must be invoked not because they themselves explain indeterministic events, but because they make the explanation of one indeterministic event by another possible (in a way Railton’s model fails to capture). An obvious, and not unattractive instance of this strategy, is to take the link between explanans and explanandum to be causation, the idea being that indeterministic events are only explicable if 20 A ‘plain’ explanation of (the occurrence of ) an event E answers a ‘plain’ why-question of the form ‘Why did E occur?’ A ‘contrastive’ explanation of (the occurrence of ) an event E answers a ‘contrastive’ why-question of the form ‘Why did E occur rather than E*?’ See Hitchcock (1999) and Percival (2000) for defences of the claim that contrastive explanations of indeterministic events are possible.

Philip Percival / 83 they have causes, and that chances must be postulated in order to account for indeterministic causation. I doubt that this strategy can succeed, however, for two reasons. First, whether indeterministic causation presupposes chance is controversial. Consider the following example. Suppose R-atoms form a large population, and that 0.001% of them decay within the hour when left to their own devices, whereas 10% of them decay immediately upon being struck by a q-particle. Suppose decay is indeterministic. Consider an R-atom that is struck by a q-particle and then decays. Did the q-particle cause the atom’s decay? For some, an afﬁrmative answer turns on issues about chance—for example, on such matters as what the chance of the atom’s decay was when it was struck, what the chance of decay would have been had it not been struck, and whether the ﬁrst of these chances is greater than the second, etc. Much of the rationale for this diagnosis, however, stems from the conviction that indeterminism prevents counterfactuals like ‘had the q-particle not struck the atom, the atom would not have decayed when it did’ from being true. But if, as others maintain, this conviction is misplaced, the way is clear for a counterfactual analysis of causation to be retained in the shift from determinism to indeterminism. On this approach, the q-particle’s impact can be supposed to have caused the atom’s decay without having to assume that chance exists.21 Admittedly, on this approach, counterfactual truth can’t supervene on actuality: on the assumption of indeterminism, nothing actual—not even the entire course of history—sufﬁces to make it true that if the atom hadn’t been struck by the qparticle, it wouldn’t have decayed. To my mind, however, some realists about chance—namely, those who deny chance supervenes on non-chance—are in no position to dismiss those who deny that counterfactuality supervenes on actuality. A second objection to the strategy of arguing for the existence of chance indirectly, via the claim that only the existence of chance renders any explanation of indeterministic events possible, addresses a wider audience. It too focuses upon certain counterfactuals. Unlike the ﬁrst, however, it does not invoke non-supervenient counterfactuals. It concedes that indeterminism prevents counterfactuals like ‘had the atom not been struck by the q-particle, it would not have decayed when it did’ from being true. But it exploits 21

See McDermott (1999).

84 / On Realism about Chance counterfactuals about rational credence of the form ‘had the atom not been struck by the q-particle, rational credence in the atom’s decaying when it did would have been r’. Many counterfactuals of this ilk remain true under indeterminism. Of course, as Mellor is fond of observing, credences, rational or otherwise, cannot themselves explain the worldly events at which they are directed. This in itself does not prevent sufﬁcient conditions for the explanation of an indeterministic event being given in terms of rational credence, however. Suppose process P is indeterministic, that it had outcome E but might have had outcome E*, and that in large scale trials that token P repeatedly, the relative frequency of E tends to some r close to zero, while that of E* tends to some r* close to one. Let C be an event which ‘interferes’ with P, and that amongst tokens of P attended by C, the relative frequencies of E and E* are reversed. Relative to statistical evidence, then, rational credence in outcome E is close to one unless P is attended by C, in which case it is close to zero. Now consider an agent who, believing a token of P not to have been attended by C, is surprised when its outcome is E. His question ‘Why E rather than E*?’ might surely be answered to his subsequent satisfaction if he is told ‘Because appearances to the contrary, C attended P’. Under indeterminism, counterfactual dependence of rational credence can still provide the objective link between explanans and explanandum that explanation requires.

1.2 Temporally Relative Warrant Mellor (1971: 58–62) writes: I propose to account for chance in terms of a feature of the world, ascertainable by the methods of science, that warrants adopting some partial beliefs rather than others . . . On the present analysis, ascribing chances to single trials expresses the fact that their function is to warrant certain partial beliefs on the possible outcomes of such a trial.

Mellor’s basic idea seems to be that, primarily, the explanatory role of chance is to ground the warrantedness of certain credences. This idea generates an inference to the best explanation as follows: (1) There are times t and events E such that, at t, one’s credence in E ought to be r (0
Philip Percival / 85 So (3) There is a feature of the world—‘chance’ (of E to degree r)—that grounds these credences’ warrant. This inference reverses the customary normative transition from chance to credence: normatively, the direction is from (credence about) chance to credence, whereas the inference extracts the existence of chance from normatively constrained credence. In this, Mellor’s strategy bears some similarities to Lewis’s (1980).22 But it is different in two important respects. First, Lewis seeks merely to read off the formal properties of chance from those of credence: in contrast, in trying to establish the existence of chance via (warranted) credence, Mellor attempts, so to speak, for the matter of chance what Lewis attempts for its form.23 Secondly, whereas the normative principle governing chance to which Lewis appeals—the Principal Principle—is internalist in so far as it constrains rational credence in an event by the agent’s credence in the event’s chance,24 Mellor relies on a principle that is externalist: he takes a certain credence at t in an event to be warranted by the chance at t of the event itself. Mellor’s externalism about the normative power of chance is embodied in premise (2), and constitutes one respect in which this premise is controversial: some internalists will object that the idea of a feature of the world having the power to demand a certain credence is no less absurd than the moral realist’s idea of a feature of the world—moral wrongness—having the power to demand a certain action.25 But (2) is controversial in a second respect that is even more fundamental. For it presupposes that epistemic norms enjoy an 22

Lewis (1980: acknowledgement) writes ‘I am also much indebted to Mellor (1971), which presents a view very close to mine; exactly how close I am not prepared to say.’ 23 In particular, Lewis tries to show that chance is a probability, that it is temporally relative, and that the past has a chance that is equal to 1. (In contrast, I have accorded chance the ﬁrst two of these features by deﬁnition.) 24 Let X be the proposition that the chance at t of p is x, and E be any proposition compatible with X that is ‘admissible’ at t. Then the Principal Principle states that an initial credence function is reasonable only if credence in p conditional on X&E, is equal to x. (Here, admissible propositions ‘are the sort of information whose impact on credence about outcomes comes entirely by way of credence about the chances of those outcomes’. See Lewis (1980: 86–96).) 25 Cf. Black’s (1998: 384) remark that ‘credence makers [that are not Humean-supervenient] look metaphysically ‘‘queer’’ in rather the way in which John Mackie thought objective values queer—they build a strange bridge between what is ‘‘out there’’ in the world . . . and something from a different conceptual realm (reasonableness of conﬁdence in an occurrence)’.

86 / On Realism about Chance objectivity that is often disputed. After all, if there is no fact of the matter as to whether the credences in question are warranted—as there isn’t if sentences of the form ‘S’s credence to degree r in p at t is warranted’ have meanings of such a kind that their proper ‘assertoric’ use is restricted to the expression of certain pro-attitudes, rather than beliefs that are objectively true or false—then there is nothing relevant to be explained. Fortunately, there is no need to become embroiled in these controversies. The inference can be seen to fail even if (2) is conceded. Let us call the warranted credences adverted to in (1)–(3) ‘M-warranted’. Jointly, (1) and (2) yield: (M) For all r, p and t: credence r in p at t is M-warranted iff the chance of p at t is r. Given (M), M-warrant must share chance’s temporally relativity (and objectivity). It differs markedly, therefore, from the concepts of warrant that are employed in everyday epistemic appraisals. Centrally, these concepts invoke available evidence, and cognitive abilities. As such, they pertain to an agent’s state of information, and his perceptual and intellectual faculties, but not to time. With respect to them, Smith’s credence r at t in p might be rational, or warranted, even though Jones’s credence r at t in p is irrational, or unwarranted: for Smith might possess evidence for p, or a faculty like vision, that Jones lacks. In contrast, M-warrant is relative to time but not to anything else: if any agent’s credence r at t in p is M-warranted, so too is every agent’s. It follows that (1) is very far from being a datum. On the contrary, it begs the question as to whether there is any reason to believe that some credences enjoy M-warrant. Of course, an independent reason for believing in chance might well provide, via chance’s normativity, an indirect reason for believing in M-warrantedness. But this is irrelevant: only a direct reason will do, since the whole point of the inference under discussion is to introduce chance via M-warrant. Upon reﬂection, it is clear that there cannot be a direct reason for believing that some credences have M-warrant. The problem is not M-warrant’s temporal relativization as such, since temporally relativized notions of warrant are readily deﬁnable in terms of evidential warrant. Let Tw be the proposition that captures a possible world w’s course of history up to t. Then a temporally relativized notion of T-warrant is deﬁned by:

Philip Percival / 87 (T) For all w, r, p, and t: at w, credence r in p at t is T-warranted iff total evidence Tw uniquely evidentially warrants credence r in p.26 T-warrant is not M-warrant, however. At t, credence r in p is M-warranted iff at t, r is the chance of p. But the credences that are T-warranted at t do not have the same values as the chances at t. If t is very shortly after the Big Bang, the course of history up until t won’t constitute enough evidence to ensure that for all p, rational credence in p based on this evidence alone coincides with the chance of p at t. Other temporally relative notions of warrant are deﬁnable in terms of evidential warrant, however. To help motivate their deﬁnitions, and to see why none of them can be identiﬁed with a notion of M-warrant that serves the realist strategy under discussion, consider a paradigm case of empirical judgement regarding the chance of an event. The judgement is retrospective, and it is grounded in statistical information about the relative frequency of an event of that type in an appropriate reference class. For example, imagine a newly constructed downward-sloping pin table. There are eight slots in which a ball, ﬁrst projected by a rigid lever that has just one setting, and then winding its way down through the net of pins, might settle. Suppose one knows beforehand that, at unit-intervals, the table is to be tested a thousand times, beginning at t + 1. Let pm,n be the proposition that on the mth trial, the ball will settle in slot n. An observer S who at t distributes his credence over the p1,n is only rational if he distributes it in accordance with evidence ES,t he possesses at t, i.e. so that cS,t (p1,n ) = c(p1,n |ES,t ), where c is a credence function that properly respects evidence. Different rational observers will distribute their credences differently, in accordance with their possession of different evidence. If any of these credences are the same as the chances, that is just coincidence. Initially, then, an observer’s credences are appraised according to whether they respect his evidence. The appraisal is not in terms of M-warrant. Now consider the situation at t + 1001. The relative frequency of slot n after m trials is seen to be fm,n and, as m grows, fm,n is seen to appear to stabilize around rn . In the light of this new information, the various observers’ initial credences in the result of the ﬁrst trial can be reappraised. It can now be seen that, irrespective of the evidence then available to the agent, credences 26

Of course, such a deﬁnition idealizes customary evidential norms. At best, these norms constrain credences by evidence. Unlike (T), they do not require a unique distribution of credence conditional upon evidence.

88 / On Realism about Chance ct (p1,n ) = rn have certain virtues that went unrecognized initially.27 Let these virtues be acknowledged, retrospectively, by calling them ‘Rm -warranted’. The argument just given for the claim that an agent’s credences are subject to a notion of Rm -warrant that is independent of the evidence available to the agent, is the closest one can get to a direct argument for M-warrantedness. It falls short of what is required, however. Rm -warrant is not the requisite notion of M-warrant, even if it is temporally relative, in the sense that for all agents S and S*, cS,t (p1,n ) = r is Rm -warranted iff cS∗,t (p1,n ) = r is Rm -warranted. It is only the case that for some m, Rm -warrant is M-warrant, if there is a guarantee that the values of the credences entertained at t that are Rm -warranted are the chances at t. But there can be such a guarantee only if the chances at t are deﬁnable in terms of the statistical evidence in terms of which Rm -warrant is deﬁned. So a dilemma arises that is fatal to any attempt to introduce chance on the back of an independently introduced notion of Rm -warrant. If chance is not deﬁnable in terms of the statistical evidence in terms of which Rm -warrant is deﬁned, the attempt fails. But if chance is so deﬁnable, the attempt is superﬂuous. Unless there is some m such that chance is deﬁnable in terms of Rm -warrant, for no m is Rm -warrant the same as M-warrant. The moral is that Mellor’s strategy fails. There can be no direct, prior evidential ground for introducing M-warrant. To have reason to believe that some credences are M-warranted, one must ﬁrst have reason to believe that chance exists.

2. Analyses of Chance Attempts to ground realism about chance by an inference to the best explanation of some feature of non-chance do not succeed. The realist might seek to circumvent the need for such an inference, however, by trying to analyse chance in terms of non-chance.28 I shall argue that this strategy is to no avail. If chance can be analysed in terms of non-chance, it supervenes on non-chance. Analyses of chance in terms of non-chance can be classiﬁed according to the kind of supervenience thesis they 27 Exactly what these virtues are, and how highly they should be valued, is debatable, but irrelevant. 28 In saying this, I do not presume that chance lacks explanatory powers if it is analysable. The point is just that even if analysable chance explains some part of non-chance, the gap between non-chance and chance can be negotiated without inference to the best explanation.

Philip Percival / 89 entail: ‘molecular’ analyses entail a (relatively) local supervenience of chance on non-chance, while ‘holistic’ analyses entail one that is (relatively) global.29 I shall argue that extent molecular (Section 2.1) and holistic (Section 2.2–2.4) analyses of chance fail.

2.1 Frequentist Analyses Frequentist analyses of chance purport to deﬁne the chance of an event as the relative frequency with which a certain outcome is obtained in a reference class of ‘trials’.30 The naive manifestation of this hope deﬁnes the chance an event E has at a time t as the relative frequency of an event of the same kind as E in a reference class comprising actual instances of some process that would have to take place were E to occur. Often, however, as in the case of a coin that is tossed once and then destroyed, the reference class that seems most relevant is too small for the relative frequencies of outcomes in it to be plausibly thought a strictly reliable guide to chance. More sophisticated proposals attempt a deﬁnition of chance in terms of relative frequencies of outcomes in reference classes that are merely hypothetical. For example, irrespective of whether a coin is ever actually tossed, one might try deﬁning the chance of its landing heads if it is tossed on some speciﬁc occasion as the relative frequency (or limiting relative frequency) of heads that would obtain were it subjected to some ideal sequence of tosses. Unfortunately, although these proposals are in some respects an improvement on the naive one, they succumb to similar difﬁculties. Whether there are facts about the relative frequencies with which the possible outcomes of an indeterministic process would be obtained were it tokened repeatedly is controversial: since a fair but never tossed coin might have landed heads on one-thousand successive occasions had it been tossed a thousand times, can it be a fact that it would have landed otherwise?31 To be sure, McDermott’s (1999) 29

The supervenience of chance on non-chance is strictly ‘global’ iff for no proposition p is there a proper part of a world w on which the chance of p at t at w supervenes. If the supervenience of chance on non-chance is not strictly global, it is less global (equivalently, more ‘local’) the smaller are the various proper parts of non-chance at w on which the chances at w supervene. 30 Frequentist analyses of chance are not species of ‘frequentism’ in the classical sense. Classical frequentists, such as von Mises (1957: 17–18), simply deny that chance in my sense exists: they hold that although objective probability exists, it is not single-case. 31 Cf. Lewis (1986: 90, 176). Lewis (1986: 58–65) does maintain that {¬A, A →C, A → p(¬C) > 0} is consistent, but he holds that these propositions can only all be true together in the special case in which, had A occurred, the occurrence of ¬C would have been ‘quasi-miraculous’.

90 / On Realism about Chance ‘realism’ about counterfactuals paves the way for commonplace facts about what the outcome of an indeterministic process would have been had the process been instigated. But even if there is a fact about the relative frequency with which a possible outcome would have obtained had some indeterministic process been tokened n times, identifying this relative frequency with the chance (beforehand) of obtaining an outcome of that kind on the one occasion the process is actually tokened does violence to the concept of chance: there is always a positive chance that the two differ. Nor does resorting to the inﬁnite case resolve these difﬁculties. It is even more controversial to suppose that there are facts about what the limiting frequency of an outcome would have been if the relevant process had been tokened inﬁnitely often. But even if there are such facts, it is hard to see how such a sequence of tosses could avoid physical changes in the trial set-up that affect the chance distributions over outcomes on individual trials. And even if such changes were somehow avoided in a fact of the form ‘were an inﬁnite series of trials conducted on the trial set-up T, the limiting relative frequency with which outcome O would be obtained is r,’ the concept of chance still leaves room for a gap between the chance of O on any of those trials and the limiting relative frequency of its occurrence over all of them. Pace Mellor (2000: 20), such a gap remains a metaphysical possibility: there is no inconsistency, for example, in the idea of a fair coin being fairly tossed inﬁnitely often without ever landing heads.32 In any case, all frequentist analyses of chance face the problem of the reference class. Processes fall under not one ‘kind’ but many. The naive suggestion that, e.g., the chance of heads with a certain coin is the frequency of heads in an actual reference class invites the question: ‘Which reference class?’. 32

Cf. Giere (1973: 477–8) and Lewis (1986: 90). Following Mellor (1995: ch. 3), call the idea that events with chance 1 cannot but occur the ‘necessity condition’ on chance, and the idea that, necessarily, an inﬁnite series of independent trials each having a chance r of outcome O has a limiting relative frequency r of O, the ‘frequency condition’ on chance. By the Bernoulli theorem, the sequence of chances, on n independent trials on each of which the chance of O is r, of the relative frequency of O diverging by a small amount from r, tends to the limit zero as n increases. To reject the frequency condition on chance, as I have done, one must therefore take one of three (non-exclusive) options: (1) reject the necessity condition on chance; (2) deny that chance is deﬁned over an inﬁnite sequence of independent trials; or (3) opt for a non-standard probability calculus in which probabilities can take inﬁnitesimal values. In the hope of retaining the necessity condition on chance while keeping the domain over which chance is deﬁned broad, Lewis (1986: 61, 89–90, 333) embraces (3).

Philip Percival / 91 If the coin is a 1998 penny tossed left handed to a maximum height of three metres on to a stone ﬂoor, should the reference class comprise tosses with these features? If so, how ﬁne-grained should the characterization of the toss be? Resorting to hypothetical relative frequencies does not avoid the problem: what would count as repeated tokening of a process is as dependent on the selection of one of the many kinds under which the process falls as what does count as a tokening of it. Attempts at frequentist analyses of chance fail. But perhaps they fail merely because they are molecular.33 Let us consider whether a holistic analysis fares any better.

2.2 Lewis’s ‘Best System’ Analysis of Chance Lewis’s (1994) fundamental idea is that a logical link between chance and non-chance arises provided (local) matters of particular fact are taken in their entirety. On his analysis, the chance of an event at a world is determined by the entire course of history at that world. The chance (beforehand), e.g., of getting heads with this coin on that toss is determined not directly, by the local patterns of heads and tails on tosses of this coin, or on tosses of those kinds of coins of that type, or, even, on all tosses of all coins, but indirectly, by the pattern, literally, of all events. The problems facing attempts to analyse the chance of an event in terms of relative frequencies of that kind of event are removed at one stroke. There is no danger of there being many equally deserving reference classes, or of the preferred reference class being empty: so to speak, with respect to every chance event the ‘reference class’ comprises all events. The analysis is therefore holistic, and the supervenience thesis it entails global. Thus characterized, however, Lewis’s ‘analysis’ is merely structural, or programmatic. The totality of events at a possible world is not literally a reference class. It is all very well to deem molecularism to be at fault. But the devil is in the detail. How is a more holistic analysis of chance going to work? Lewis’s answer to this question builds upon his ‘best system’ analysis of nonprobabilistic laws of nature under the supposition of determinism. Regularity 33

The supervenience theses entailed by analyses in terms of actual relative frequencies are clearly local. But so too are those entailed by an analyses that rely on counterfactuals about what the outcomes of indeterministic processes would have been had they been instigated. Because such counterfactuals don’t supervene on actuality (as a condition of the analysis being successful), they constitute a small part of non-chance.

92 / On Realism about Chance theorists take laws of nature to be regularities. Since some regularities are clearly not laws—such as the fact (assuming it to be one) that all lumps of solid gold have a mass of less than a billion kilos—those that are laws must be hived off. Lewis (1973: 73–5; 1986: 121–4) effects the separation as follows. He holds that, for all worlds w, the laws at w are whatever regularities are expressed by theorems or axioms of whatever deductive system S best systematizes w’s entire course of history in the following sense: S does best with respect to (an appropriate weighting of) the competing criteria of strength and simplicity in a competition between deductive systems that are true at w and which are concerned only with ‘natural’ properties.34 Some regularities will not make it into the best system, and on the analysis these are distinguished from laws and classiﬁed as merely accidental. If chances are taken to be primitive natural properties, extending this analysis to the indeterministic case, and hence to chance laws, is almost trivial.35 If there are chance laws, and chance is a natural property, the deductive systems that do best in competition with one another will claim, truly, that there are certain regularities involving chance (just as, under the supposition of determinism, they make true claims concerning non-chancy matters involving mass and charge). Lewis suggests but one emendation. Regularities that have merely chanced to come about should not be accounted laws: they too are merely accidental. So deductive systems that entail regularities of this ilk must be excluded from the competition. In Lewis’s (1986: 126) own words, the result is as follows: Previously, we held a competition between all true systems. Instead, let us admit to the competition only those systems that are true not by chance; that is, those that not only are true, but also have never had any chance of being false. The ﬁeld of eligible competitors is thus cut down. But then the competition works as before. The best system is the one that achieves as much simplicity as is possible without excessive loss of strength, and as much strength as is possible without excessive loss of simplicity. A law is a regularity that is included, as an axiom or as a theorem, in the best system.

This proposal is an analysis of chance law, not an analysis of chance itself: it adverts to competing deductive systems the axioms and theorems of which 34

This account is unfaithful in one respect, since Lewis tends to identify laws not with regularities expressed by axioms or theorems of the best system, but with those axioms or theorems themselves. There is a familiar issue here, but it is not one which is relevant to the main concerns of this paper. 35 Lewis holds that determinism is incompatible with the existence of (non-trivial) chance. (See n. 4, above.)

Philip Percival / 93 embody single-case probability functions that are interpreted uncritically as chance functions. In the modiﬁcation by means of which Lewis aspires to analyse chance, such functions are left uninterpreted. So the competition is held between deductive systems that are only partially interpreted, and the notion of the best such system invoked with a view to deﬁning not just chance ‘laws’, but the uninterpreted probability functions too, and, hence, ‘chance’ itself. That is, what the chances of particular events are (at a world), and hence what regularities involving chances there are (at that world), and hence what the chance laws are (at that world), are matters to be deﬁned en masse in terms of which of the various deductive systems that articulate regularities involving as yet uninterpreted single-case probabilities does best (at that world) in the competition. Two conditions must be fulﬁlled for this strategy to have any hope of success. Firstly, since chance is temporally relative—the chance of an event happening during some temporal interval changing over time depending (at least in part) on which events chance to happen beforehand—for all worlds w, the deductive system that does best with respect to w must determine, in combination with (non-chance at) w, for each time t, a probability function Pt deﬁned over all the events that, at w, are chancy at t.36 Secondly, the criteria of ‘strength’ and ‘simplicity’, that sufﬁced to pick out the best system in the analysis of law alone, must be supplemented. When the project was merely to deﬁne ‘law’ (at a possible world), the best system analysis needed to look at the world in question to ascertain which deductive systems are true at that world. It could then select from among the deductive systems that are true the one that does best with respect to the further criteria of simplicity and strength. But once the more ambitious project is pursued of deﬁning ‘chance’ too, the requirement of truth cannot be employed to select the candidate systems on which the criteria of simplicity and strength are to operate: the probability ‘claims’ the systems embody are as yet uninterpreted, and so cannot be true or false. Without some further criterion to do duty for the previous requirement of truth, maximal strength and simplicity would be all too easily achievable independently of the world in question, and the laws of chance would turn out to be necessary (and absurd). To avoid this consequence, Lewis supplements the criteria of strength and simplicity with 36

In his earlier treatments, Lewis (1980: 91; 1986: 131–2) explicitly avoids the assumption that for all times, chance at a time is deﬁned over all propositions. He subsequently makes the assumption implicitly (Lewis 1994: 483). As Hoefer (1997) observes, much turns on which propositions chance at a time is deﬁned over.

94 / On Realism about Chance an additional criterion of ‘ﬁt’ between a deductive system and a world. In effect, this criterion does for the best system analysis of ‘chance’ (and ‘law’) what the requirement of truth does for the best system analysis of ‘law’ alone. It is not enough for one of the competing deductive systems to be maximally simple and strong. The uninterpreted ‘claims’ it makes must also ﬁt the world in question. More accurately, in Lewis’s (1994: 480) own words: The virtues of simplicity, strength, and ﬁt trade off. The best system is the system that gets the best balance of all three. As before, the laws are those regularities that are theorems of the best system. But now some of the laws are probabilistic. So now we can analyse chance: the chances are what the probabilistic laws of the best system say they are.37

Although this analysis has attracted much critical attention, most of it concerns the ‘undermining’ difﬁculties Lewis (1980: 111–12) once deemed fatal.38 None of it focuses on the difﬁculty I shall raise.39 Because this difﬁculty is fundamental, it besets modiﬁcations and restrictions of the analysis that have been proposed as resolutions of the undermining difﬁculties.40 I call it ‘the problem of ﬁt’.

2.3 The Problem of Fit Lewis’s elucidatory remarks concerning ‘ﬁt’ are disconcertingly sketchy. In the paper in which he eventually defends the best system analysis, he merely says: [S]ome [systems] will ﬁt the actual course of history better than others. That is, the chance of that course of history will be higher according to some systems than according to others. (Though it may well turn out that no otherwise satisfactory system makes the chance of the actual course of history very high; for this chance will come out as a product of chances for astronomically many chance events.)41 37 This statement is misleading. It invites a reading on which ‘the chances’ means ‘the (temporal) chances of particular events’. But of course the laws of chance no more state the chance (at some earlier time) of, e.g., the Challenger space shuttle exploding than do deterministic laws state when, e.g., a solar eclipse will occur. In either case, such matters of particular fact depend not just on laws, but on earlier matters of particular fact. 38 See n. 6, above. 39 This statement was true until the ﬁnal drafting. I only came to know of Elga’s (2004) identiﬁcation and treatment of the difﬁculty at the last minute. 40 See the proposals in Hoefer (1997) and Halpin (1998). 41 Lewis (1994: 480). This statement is also misleading. It appears to deﬁne the measure of a deductive system’s ‘ﬁt’ as what that system takes to be ‘the chance of [the actual] course of history’,

Philip Percival / 95 There are both conceptual and formal difﬁculties with this gloss. The supposition that candidate deductive systems will specify their own measures of ﬁt by specifying the chance of entire courses of history is in tension with the fact that chance is temporal. The chances of an event of a certain kind occurring in a temporal interval vary beforehand according, at least in part, to what chances to happen earlier. For example, although the chance of one hundred successive heads with a fair coin would have been very tiny at the outset, it would have reached one-half just before the hundredth toss had a run of 99 heads occurred in the meantime. Similarly, if, as Lewis supposes, the chances at t are deﬁned over entire futures Ft , chance at t will be deﬁned over entire courses of history Ht Ft too. (‘Ht ’ is an entire history up to an including t; ‘Ft ’ is an entire future after t. Since there is no chance of the past not occurring, at t the chance of Ht is 1.) Somehow, then, the best deductive systems must combine with facts about the non-chances at w to give each nomologically possible course of history Ht Ft different chances at different times—say, by entailing conditionals Ht → [Pt (Ht Ft ) = r]. Yet nothing in this requires that such systems specify non-temporal chances for the various Ht Ft , and there seems much in the concept of chance to prevent them from so doing. ‘Propensity’ theories of the ontology of chance, whereby chance is a measure of the ‘tendency’ of a cause to produce an effect, or the display of a dispositional property,42 should deem the notion of non-temporal chance confused. In presupposing that for all worlds w at which chance exists, the deductive system that does best at w does best, in effect, as a result of specifying the non-temporal chance (probability) at w of w’s course of history, Lewis seems merely to assume, counterintuitively and without warrant, that, at w, there is some such chance to be speciﬁed. Still, the conceptual difﬁculty of this assumption does not refute the spirit of his approach. Perhaps he might concede that chance is essentially temporal by merely declining to interpret whereas the best systems analysis has been seen to deﬁne chance in terms of the ﬁt of deductive systems employing an uninterpreted probability function. However, the threat of circularity is mere appearance. It is dissipated by the fact that both interpreted and partially interpreted deductive systems can be said to ﬁt a world. If a deductive system D employs an interpreted probability function P that accords a chance r to world w’s course of history, this is because the truth-condition of P’s claims is speciﬁable in terms of which among the deductive systems that employ an uninterpreted probability function best ﬁts w. These systems will include a deductive system that is just like D save for the fact that it employs an uninterpreted probability function formally similar to D’s chance function P. 42 Respectively, as in Giere (1973) and Mellor (1971).

96 / On Realism about Chance the non-temporal probability of a world’s course of history according to the deductive system which does best at that world as the world’s ‘chance’.43 So doing would invite the objection that if there are no non-temporal chances of courses of history at worlds, the competition between deductive systems ought to include systems that, like theories consisting in a set of history-to-chance conditionals of the form Ht → [Pt (A) = r], say nothing about non-temporal probabilities, the upshot being that some measure of a system’s ﬁt to a world other than a non-temporal probability it ascribes the world’s course of history must be found. But perhaps this objection would beg the question by refusing to take talk of ‘system’ in Lewis’s analysis seriously: perhaps the measure of ﬁt need not apply to systems that don’t ascribe non-temporal probabilities to worlds because such systems would do so badly by the criterion of simplicity that they can be excluded from serious consideration. Indeed, the best system analysis might try to turn the tables at this juncture by using this consideration against the intuition that chance is inherently temporal: one’s pre-theoretical intuitions notwithstanding, if the laws of chance are determined by which system does best, and any aspiring system that eschews a non-temporal probability function deﬁned over entire courses of history will be hopeless with respect to the criterion of simplicity, chance must be non-temporal in part after all.44 Be this as it may, let us turn to the formal difﬁculty in Lewis’s gloss on ﬁt, for it is this that I wish to emphasize. While Lewis (1994: 480) says that the chance of a world according to a competing deductive system will be a ‘product of chances for astronomically many chance events’, he does not say what the factors in this product are. For illumination, we must turn to a hint Lewis (1986: 128) gave earlier: [A] system ﬁts a world to the extent that the history of that world is a comparatively probable history according to that system . . . If the histories permitted by a system formed a tree with ﬁnitely many branch points and ﬁnitely many alternatives at each 43

Cf. Hoefer (1997). Hoefer argues that if chance supervenes on non-chance in a Humean way, there are independent reasons for denying that chance is deﬁned over propositions so ‘big’ as to be undermining. As he in effect observes (e.g. 1997: 328), supposing chance restricted to ‘small’ propositions is compatible with holding it to be expressed by a restriction of a probability function that is deﬁned over ‘big’ propositions too. 44 There is another alternative. Perhaps Lewis assumes that for every set S comprising, for all t, and A, a conditional Ht → [Pt (A) = r], S will entail a proposition that speciﬁes a non-temporal probability for each course of history.

Philip Percival / 97 point, and the system speciﬁed chances for each alternative at each branch point, then the ﬁt between the system and a branch would be the product of these chances along that branch; and likewise, somehow, for the general inﬁnite case.

Lewis (1986) says nothing more about the notion of ﬁt,45 and his use of the expression ‘somehow’ is well justiﬁed. There is no tenable inﬁnite analogue of this ‘product’ method for determining the probability of a world according to a system (and hence the system’s measure of ﬁt). Moreover, serious problems arise for attempts to construct an alternative. To see this, let us ﬁrst elaborate the ﬁnite case Lewis considers. Each deductive system in the competition holds ﬁnitely many courses of history nomologically possible. These courses of history can be represented by the branches of a ﬁnite tree. The nodes of the tree are ordered into levels doing duty for times. At each level t, each node ti at that level represents a nomologically possible momentary state of the world at time t. The branchings from ti represent the possible futures that the system says have some probability of coming about at t if (a) the momentary state of the world at t is as the node ti represents it as being, and (b) the momentary states of the world prior to t are as the nodes on the branch beneath ti represent them as being. Importantly, time is discrete: each node has immediate successor nodes that represent possible states that have some probability of coming about next. Each node ti is associated with a probability function Pti , that speciﬁes, for each of ti ’s immediate successor nodes ti+1,j , a probability at ti of ti+1,j . (So the values for the various j’s of Pti (ti+1,j ) sum to 1.) The probability (chance) of a branch of the tree is the product of the probabilities (chances), at each of the nodes on the branch, of the immediate successor nodes. Of course, this ﬁnite case is unrealistic: chance does not determine the topology of time, and it is more realistic to suppose that time is dense or continuous. On this supposition, however, no time has an immediate successor, and a strict analogue of the ﬁnite case is impossible: there can be no probabilities at the nodes on a branch of the next node on the branch, and so 45

It is unsurprising that in this earlier discussion Lewis (1986: 128) treats the notion of ﬁt offhandedly, since there he rejects the best system analysis of chance on account of the undermining difﬁculties. It is perfectly legitimate for him to bring his discussion to an end by remarking ‘never mind the details, if, as I think, the [best system analysis] won’t work anyway’. In contrast, when Lewis (1994) eventually takes the undermining difﬁculties to have been resolved, he ought perhaps to have reﬂected further on the trickiness of deﬁning ﬁt before embracing the analysis.

98 / On Realism about Chance no product of them. Let us consider, therefore, the prospects for an inexact analogue. Perhaps a node Bt on a branch B might contribute some other probability value to a product that is B’s overall probability. In particular, for all branches B and nodes Bt , if a probability B Pt is deﬁned at Bt over each higher branch-segment Bt St+k that is directly above Bt , the contribution Bt makes to a product equal to the probability of the branch B might be the limit of the sequence of probabilities B Pt (Bt St+k ) as k tends to zero.46 Unfortunately, however, thisproposal solvesone problem only toencounter another: although we now have probabilities for each of the nodes on a branch that can be multiplied together so as to get the probability of the whole branch, we have too many. Standardly, the product of these values would be useless as a measure of the probability of the branch even in the case where time is dense: for a denumerable product of values in the interval [0,1) is zero. Resorting to non-standard methods on which the value of such a product can be inﬁnitesimal might sidestep this problem.47 Yet so doing would be futile if time is not just dense but continuous. Standardly, continuous time is constituted by continuum many times. But if selecting at each node on a branch some unique probability value generates continuum many values, the product method of obtaining a value for the whole branch remains untenable: non-denumerable products are deﬁned neither standardly nor non-standardly. The only way left by which to try to uphold such a method would be to embrace the ideas of non-standard analysis exclusively. From within the conceptual scheme these ideas provide, entire courses of history taking place over continuous time will be represented by branches constituted not by continuum many, but by hyperﬁnitely many nodes.48 Continuous time would then no longer be a bar to the method, since products of hyperﬁnitely many reals or inﬁnitesimals are deﬁned in non-standard analysis. This manoeuvre is desperate, however. If the cardinality of continuous time is only contingently hyperﬁnite, adopting the non-standard conceptual scheme exclusively will leave the competing deductive systems ‘blind’ to some temporally continuous worlds at which chance exists (if it exists at any worlds). Biting the bullet and maintaining that chance does not exist at worlds at which there are continuum many times 46

A branch segment Bt St+k directly above Bt is the segment between Bt and a higher node Bt+k . It includes Bt+k , but not Bt itself. 47 Lewis advocates a non-standard calculus on which chances can take inﬁnitesimal values on independent grounds. (See n. 32, above.) 48 In non-standard analysis, hyperﬁnite numbers succeed the positive integers.

Philip Percival / 99 would be too hard a course. Maintaining that the cardinality of continuous time is non-contingently hyperﬁnite would be harder still. I conclude that Lewis is wrong to suggest that a measure of ﬁt in realistic cases can be modelled on a product method that is appropriate to the ﬁnite case. While Lewis’s own suggestion as to how to elucidate a measure of ﬁt between competing systems and worlds does not work, the question arises as to whether some alternative will. In particular, the question arises as to whether the classical treatment of stochastic processes might serve the best system analysis’s needs. I do not think it can. On this treatment, the various deductive systems in the competition are taken to specify an atemporal probability function P, deﬁned over entire courses of history, that is basic. Temporal probabilities Pt () are then deﬁned as conditionalizations P(-|Ht ) of this function on the course of history up to and including t. Various conceptual and formal difﬁculties beset this proposal, however, even if recourse is made to non-standard probabilities. Each version of it turns out to be at least as unattractive as the version that is committed to the following: (i) Lewis’s favoured necessity condition on chance is false—nomologically possible events can have zero chance—even though (ii) it is apriori that there is no possible world at which an event occurs that had zero chance of occurring; moreover (iii) it is apriori that chance at a time cannot depend on matters of particular fact at later times and, hence, on some views (including Lewis’s) (iv) it is apriori that backwards causation cannot occur.49

2.4 ‘Fit’ as a Primitive? At this point, a proponent of the best system analysis might think to take the degree to which a deductive system ﬁts a possible world to be a primitive notion. To my mind, however, this option is untenable. To adopt an analogy in the philosophy of mind, it should come as no surprise that there is no published 49 I say more about the difﬁculties that beset attempts to glean a measure of ﬁt from the classical treatment of stochastic processes in the Appendix. Elga (2004) agrees that allowing the competing deductive systems to employ non-standard probability functions won’t solve the problem of ﬁt. His grounds differ from mine, however. They are that with respect to simple possible ‘worlds’, amongst the ‘constructable’ non-standard probability functions, those that comply with the necessity condition on chance, and that in a certain sense approximate standard probability functions, give counterintuitive results: there are worlds such that a non-standard probability function that approximates the standard probability function that, intuitively, ﬁts the world best, has worse ﬁt than does a non-standard probability function that approximates some standard probability function that, intuitively, ﬁts the world badly.

100 / On Realism about Chance attempt to solve the difﬁculties confronting classical analytic behaviourism by resorting to a holistic doctrine according to which, for each mental state M, a subject S is in M iff S’s being in M best ﬁts S’s behaviour, where the notion of a mental state’s ‘ﬁtting’ a pattern of behaviour is left primitive. Of course, the term ‘ﬁt’ does invoke an epistemological notion on which we have an independent grip: there is a well-established practice of basing judgements of chance on, e.g., statistical evidence. But it would be quite wrong to protest that best system analyses merely adapt to logical and semantical ends an epistemological notion to which Lewis’s opponents—in particular, realists about chance who deny chance supervenes on non-chance—are committed. Denying that chance supervenes on non-chance does not commit one to a notion of ﬁt having the formal properties best system analyses of chance require. That realism about non-supervenient chance must offer an epistemology detailing ways in which evidence can support hypotheses about chances falls short of saying that it presupposes facts about how a course of history in its entirety bears evidentially on ‘complete’ chance hypotheses, i.e. regarding all matters that are chancy.

3. Conclusion My conclusion is no less modest than the aim with which I began. I have given grounds for scepticism regarding the strength of reasons one might give for realism about chance. Chance has no explanatory role with respect to non-chance that would ground realism about it via an inference to the best explanation, and while an analysis of chance in terms of non-chance might excuse the realist about chance of a need to make any such inference, extent analyses of chance fail. I would like to end, however, by speculating that there is little prospect that, at some time in the future, a correct analysis of chance will yield a warrant for belief that chance exists. If chance is analysable, Lewis’s best-systems analysis is surely along the right lines: only global facts about non-chance could possibly determine an event’s chance. But weakening Lewis’s presumption that the system that best ﬁts a world is the system that gives the world’s entire course of history highest probability (chance) won’t yield a warrant for believing in chance, even if so doing results in an analysis of chance that is correct. For a fundamental obstacle remains: even if the measure of a system’s ﬁt is

Philip Percival / 101 determined by the probability it gives some much smaller part of non-chance, there will no reason to believe that any system has best ﬁt independent of a prior reason for believing that chance exists. Elga’s (2004: 72) proposed account of ﬁt serves to illustrate the point.50 Elga suggests that one deductive system X ﬁts a world better than another system Y iff ‘the chances X assigns to the test propositions are predominantly greater than the corresponding chances that Y assigns’. It is clear that even if this criterion can be employed in a version of the best system analysis that is correct,51 no warrant for belief in the existence of chance would result. In the absence of an independent reason for realism about chance, there is simply no reason whatsoever to believe that one system assigns chances to the ‘test’ propositions that are predominantly greater than those that are assigned by all the other systems.

Appendix In effect, Lewis’s product method constructs the non-temporal probability a deductive system accords an entire course of history out of certain temporal probabilities that, given events prior to time t, the system says hold at t.52 That this method cannot be extended from discrete to continuous time need not undermine his more basic idea that the measure of a system’s ﬁt to a world is a (non-temporal) probability it accords the world: some other relationship might hold between this probability and the values the system 50

As I remarked in n. 39, above, I only became aware of Elga’s paper just before going to press. Elga leaves what the ‘test’ propositions are implicit in all but the case of the very simplest of idealized ‘worlds’. He also leaves the meaning of ‘predominantly greater’ open, and it is difﬁcult to see how the argument he gives for the claim that in this case the criterion yields the correct results might be extended to the case of possible worlds proper. Moreover, in objecting to the suggestion that a system’s ﬁt to a world should be identiﬁed with the value for the world’s course of history as given by a non-standard probability function the system employs, he presupposes the necessity condition on chance. Yet his own criterion ﬂaunts this condition. These all too hasty remarks notwithstanding, Elga’s treatment of the problem of ﬁt is a signiﬁcant advance. 52 Actually, this isn’t quite true even in the case Lewis considers of an indeterministic ﬁnite world with a ﬁrst moment and discrete time. The product of probabilities along a branch Lewis calls ‘the’ probability (chance) the system accords the entire branch is really temporal: it is a probability (chance) at the ﬁrst moment. Moreover, it is not the probability (chance) of an entire course of history either. It is the probability (chance) at the ﬁrst moment of the rest of history, excluding the ‘initial conditions’ that obtain at that moment and which determine that probability (chance). 51

102 / On Realism about Chance gives to temporal probabilities (chances) in the light of the world’s course of history. In particular, the classical treatment of stochastic processes suggests that temporal probabilities might be viewed as conditionalizations of a nontemporal probability function P. A deductive system in the competition for ‘best’ would specify a non-temporal probability function of this kind deﬁned over each course of history Ht Ft the system deems nomologically possible. The temporal probabilities Pt () that the system determines at a world w in combination with w’s course of history (or, more generally, w’s non-chances) are just the conditional probabilities P(-| Ht,w ). From this perspective, the complete theory of chance at a world is given by a single probability function deﬁned over entire courses of history (and hence over sets of them). The chances at a world at a time are given by conditionalizing this function on that world’s course of history up to and including that time. This method also encounters formal and conceptual problems, however. Chance no more precludes chance processes the outcome spaces of which are continuous than it precludes continuous time. Looked at standardly, then, the set of courses of history over which the probability function characterized by a competing deductive system is deﬁned will have (at least) the cardinality of the continuum. Standardly, only a denumerable subset of these courses of history can have a positive value: the rest must have probability zero. So the necessity condition on chance would be violated if this unconditional probability expresses non-temporal chance.53 Moreover, even if it doesn’t express this, the necessity condition on chance will still have to go: if a course of history has zero unconditional probability, it will have zero probability conditional on any history Ht , while ex hypothesi the conditional probability P(-| Ht ) is a temporal probability Pt () that expresses temporal chance. So we must choose between the standard perspective and the necessity condition on chance. Suppose then that we try to keep the necessity condition on chance by going non-standard, and giving each nomologically possible course of history at least inﬁnitesimal probability. Giving only inﬁnitesimal values to continuum many courses of history would bring disaster: it would be consistent to raise these values by a ﬁnite factor, and, in effect, no system would ﬁt better than any other. This result might be averted by adopting non-standard ideas exclusively, so that the cardinality of the courses of history over which a 53

According to the necessity condition on chance, an event which has zero chance is not nomologically possible. See n. 32, above.

Philip Percival / 103 system’s probability function is deﬁned is hyperﬁnite. The inﬁnitesimal values of the function will not then be ﬂexible in the same way, since raising them by any ﬁnite factor will result in their no longer summing to one. Yet exclusively adopting a non-standard perspective faces the same dilemma that arose when so doing was proposed as the saviour of Lewis’s product method. If the cardinality of the set of nomologically possible courses of history is contingently hyperﬁnite, competing deductive systems that are wedded to the non-standard conceptual scheme will be blind to some possible worlds. But biting the bullet and supposing that chance does not exist at worlds at which, e.g., continuum many courses of history are nomologically possible is too hard a course, while it is harder still to suppose that the cardinality of the set of nomologically possible courses of history at worlds at which chance exists is non-contingently hyperﬁnite. So the non-standard perspective is more trouble than its worth. Alternatively, then, we should try to avoid the disastrous consequences of assuming that systems give only inﬁnitesimal probability to each course of history they deem nomologically possible, not by requiring the set of these courses of history to have a hyperﬁnite cardinality, but by allowing some systems to give ﬁnite probabilities to some entire courses of events they deem nomologically possible. Yet this manoeuvre proves too slight. While the possibility is now admitted that some system will do best at some world in virtue of giving that world a ﬁnite probability, it will remain impossible for any system to do best at a world by giving that world a higher inﬁnitesimal probability than does any other similarly strong and simple system. Yet ruling out this much builds too much into the concept of chance. If the laws of chance at a world determine non-temporal chances at that world for entire courses of history, and some of those chances are inﬁnitesimal, why shouldn’t there be a world at which the chance of its own course of history is inﬁnitesimal? Whereas inﬁnitesimals were brought in to accommodate the intuition that some events have a positive chance that must be smaller than any positive real, we now ﬁnd ourselves asked to accept a deﬁnition of chance on which there is an a priori guarantee that events with inﬁnitesimal chance don’t happen. It remains to consider the possibility that these problems all stem from burdening the ‘conditional probability’ method of handling the interplay between non-temporal and temporal chance with the necessity condition on chance. The attractions of this condition notwithstanding, giving it up might

104 / On Realism about Chance be an acceptable price to pay for an otherwise satisfactory analysis. Yet even the retreat to zero chances would leave some of the problems already discussed intact. In particular, the guarantee that no course of history, and hence no event, with inﬁnitesimal chance ever occurs, will simply go over into the guarantee that no course of history, and hence no event, will transpire at a world at which it has zero chance. Finally, there is a further conceptual difﬁculty that sacriﬁcing the necessity condition on chance won’t solve. Adopting this method would rule out the temporally backwards dependence of chances at a time on events that chance to happen after that time (and, hence, on some analyses of causation, backwards causation too). Of course, backwards causation has been held to be conceptually incoherent. But Lewis himself doesn’t hold it so, and it is in any case unsatisfactory that it should be precluded by the exigencies of a solution to a formal problem that is generated by a purported analysis of chance. There is no way to modify the conditional probability method for establishing the desired interplay between temporal and non-temporal chance so as to accommodate backwards dependence of chance on later events, however. Only a temporal reference in the proposition conditioned on can make the conditional probability P(−|A) a temporal probability Pt (). But if there is a world at which a temporal probability Pt () is backwards dependent, conditionally, not just on w’s history up until t but on a later event A, so that the temporal reference in A is some t* later than t, there is no way back from t* so as to allow the conditional probability P(−| At∗ ) to be equated with the temporal probability Pt ().

References Armstrong, D. (1983), What is a Law of Nature? (Cambridge: CUP). Bigelow, J., Collins, J. and Pargetter, R. (1993), ‘The Big Bad Bug: What are the Humean’s Chances?’, British Journal of the Philosophy of Science, 44: 443–62. Black, R. (1998), ‘Chance, Credence, and the Principal Principle’, British Journal of the Philosophy of Science, 49: 371–85. Dretske, F. (1977), ‘Laws of Nature’, Philosophy of Science, 44: 248–68. Elga, A. (2004), ‘Inﬁnitesimal Chances and the Laws of Nature’, Australasian Journal of Philosophy, 82: 67–76.

Philip Percival / 105 Giere, R. (1973), ‘Objective Single Case Probabilities and the Foundations of Statistics’, in P. Suppes et al. (eds.), Logic, Methodology and the Philosophy of Science IV (Amsterdam: North Holland), 467–83. Hall, N. (1994), ‘Correcting the Guide to Objective Chance’, Mind, 103: 505–17. (2004), ‘Two Mistakes about Credence and Chance’, Australasian Journal of Philosophy, 82: 93–111. Halpin, J. (1998), ‘Lewis, Thau, and Hall on Chance and the Best-System Account of Law’, Philosophy of Science, 65: 349–60. Hitchcock, C. (1999), ‘Contrastive Explanation and the Demons of Determinism’, British Journal of the Philosophy of Science, 50: 585–612. Hoefer, C. (1997), ‘On Lewis’s Objective Chance: ‘‘Humean Supervenience Debugged’’ ’, Mind, 106: 321–34. Lewis, D. (1973), Counterfactuals (Oxford: Blackwell). (1980), ‘A Subjectivist’s Guide to Objective Chance’, in R. C. Jeffrey (ed.), Studies in Inductive Logic and Probability, ii (Berkeley and Los Angeles: University of California Press). (Page references are to the reprinting as ch. 19 of Lewis 1986.) (1986), Philosophical Papers, ii (Oxford: OUP). (1994), ‘Humean Supervenience Debugged’, Mind, 103: 473–90. McDermott, M. (1999), ‘Counterfactuals and Access Points’, Mind, 108: 291–334. Mellor, D. H. (1969), ‘Chance’, Proceedings of the Aristotelian Society, suppl. vol. 43: 11–36. (1971), The Matter of Chance (Cambridge: CUP). (1976), ‘Probable Explanation’, Australasian Journal of Philosophy, 54: 231–41. (1982), ‘Chance and Degrees of Belief ’, in Mellor (1991: ch. 14). (1991), Matters of Metaphysics (Cambridge: CUP). (1995), The Facts of Causation (London: Routledge). (2000), ‘Possibility, Chance and Necessity’, Australasian Journal of Philosophy, 78: 16–27. Noordhof, P. (1999), ‘Probabilistic Causation, Preemption and Counterfactuals’, Mind, 108: 95–125. Percival, P. (2000), ‘Lewis’s Dilemma of Explanation under Indeterminism Exposed and Resolved’, Mind, 109: 39–66. Popper, K. (1959), ‘The Propensity Interpretation of Probability’, British Journal for the Philosophy of Science, 10: 25–42. Railton, P. (1978), ‘A Deductive-Nomological Model of Probabilistic Explanation’, Philosophy of Science, 45: 206–26. Sapire, D. (1991), ‘General Causation’, Synthese, 86: 321–47. Strevens, M. (1995), ‘A Closer Look at the ‘‘New’’ Principle’, British Journal of the Philosophy of Science, 46: 545–61.

106 / On Realism about Chance Sturgeon, S. (1998), ‘Humean Chance: Five Questions for David Lewis’, Erkenntnis, 49: 321–35. Thau, M. (1994), ‘Undermining and Admissibility’, Mind, 103: 491–503. von Mises, R. (1957), Probability, Statistics and Truth (New York: Dover). Vranas, P. (2002), ‘Who’s Afraid of Undermining?’, Erkenntnis, 57: 321–35. Woodward, J. (1989), ‘The Causal Mechanical Model of Explanation’, in P. Kitcher and W. Salmon (eds.), Scientiﬁc Explanation (Minneapolis: University of Minnesota Press), 357–83.

Part II Identity and Individuation

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5 Structure and Identity Stewart Shapiro

The purpose of this paper is to further articulate my preferred version of mathematical structuralism in light of some criticisms of my Philosophy of Mathematics: Structure and Ontology (1997) and Michael Resnik’s Mathematics as a Science of Patterns (1997). Some of these criticisms are shortcomings in the original presentation, and so I take this opportunity to correct aspects of the view. The view in question is what I call ante rem structuralism, the thesis that mathematical structures exist prior to, and independent of, any exempliﬁcations they may have in the non-mathematical world. The contrast is with an approach that either adopts an Aristotelian in re view that a given structure exists only in the systems that exemplify it or the more common eliminative thesis that structures do not exist at all—talk of structures is to be paraphrased away. In his classic ‘What Numbers Could Not Be’, Paul Benacerraf (1965) proposed an eliminative view, and an intriguing variation on that theme is articulated in Geoffrey Hellman’s Mathematics Without Numbers (1989). I call Hellman’s view ‘modal eliminative structuralism’, since it avoids commitment to the (actual) existence of models of the various mathematical theories by careful placement of modal operators. Instead of speaking of systems of objects, Hellman speaks of possible systems. The ofﬁcial argument in my Structure and Ontology is that when I am indebted to Jonathan Kastin and Jukka Ker¨anen, as well as Geoffrey Hellman, Michael Resnik, Crispin Wright, Katherine Hawley, and especially Fraser MacBride for extensive and fruitful exchanges.

110 / Structure and Identity fully articulated, the various versions of structuralism roughly share their virtues and shortcomings, but I thought—and think—that the realist, ante rem approach is the most perspicuous of the lot. Before going further, I would like to acknowledge an orientation. As I see it, the goal of philosophy of mathematics is to interpret mathematics, and articulate its place in the overall intellectual enterprise. One desideratum is to have an interpretation that takes as much as possible of what mathematicians say about their subject as literally true, understood at or near face value. Call this the faithfulness constraint. According to ante rem structuralism, natural numbers are places in structures, and places in structures are bona ﬁde objects. This accords with faithfulness. Grammatically, numerals seem to function as singular terms, and according to ante rem structuralism, numerals are singular terms. Eliminative structuralism, on the other hand, interprets statements about mathematical objects as implicit generalizations over all systems of a certain sort, or all possible systems of a certain sort. In other words, on the eliminative program, numerals are understood to be bound variables, perhaps under a modal operator. This does not accord with faithfulness. A second, and weaker, desideratum is to develop an interpretation that does not go too much beyond what mathematicians say about their subject. Surely the philosopher is going to say some things that the mathematician does not say. Mathematicians, as such, do not usually address philosophical issues about their subject. For example, they do not say much about what the natural numbers are, nor how we obtain mathematical knowledge, nor how mathematics applies to the physical world. Presumably, philosophical questions about mathematics are not to be answered solely in mathematical terms. The second desideratum is to not attribute mathematical properties to mathematical objects unless those attributions are explicit or at least implicit in mathematics itself. Call this the minimalism constraint. Richard Dedekind’s philosophical methodology was to deﬁne a system of objects, and then abstract the structure of the system (e.g. Dedekind 1872, 1888). For example, he constructed the system of cuts in rationals, and then abstracted the real numbers from the cuts. According to Dedekind, the abstracted items—the real numbers—are not part of the system abstracted from, but are instead something ‘new’ which the mind freely ‘creates’ (see Shapiro 1997: ch. 5, §4). Dedekind’s friend Heinrich Weber suggested instead that real numbers be identiﬁed with cuts. Dedekind replied that there are

Stewart Shapiro / 111 many properties that cuts have which would sound very odd if applied to the corresponding real numbers (Dedekind 1932: iii. 489–90). For example, cuts have members. Do real numbers have members? Dedekind’s Benacerraf-type point is consonant with the minimalism constraint. Weber’s proposal assigns mathematical properties to real numbers that the mathematician does not. In the preface to his later treatise on the natural numbers, Dedekind (1888) agreed that Cantor and Weierstrass both gave ‘perfectly rigorous’, and so presumably acceptable, accounts of the real numbers, even though they were different from his own and from each other. But Dedekind wrote that his approach is ‘somewhat simpler, I might say quieter’ than the others. I presume he meant that his own account of the real numbers has less excess baggage than the others. Again, the minimalism constraint. Neither the faithfulness constraint nor the minimalism constraint are absolute. They are defeasible criteria, among others. Hellman, for example, competently argues that his modal reinterpretation of mathematics avoids some sticky philosophical problems that literalists like Resnik and myself must face (see also Hellman 2001). This, he argues, justiﬁes a non-literal reading of mathematical language. In a relatively recent note, W. V. O. Quine (1992) advocated a breathtaking extension of structuralism beyond mathematics to the entire web of belief. But he stopped short of adopting a realist view of structures (either ante rem or in re), even for mathematics: My global structuralism should not . . . be seen as a structuralist ontology. To see it thus would be to rise above naturalism and revert to the sin of transcendental metaphysics. My tentative ontology continues to consist of quarks and their compounds, also classes of such things, classes of such classes, and so on . . .

Quine’s argument here seems to be that since scientists (and mathematicians) do not themselves say that ante rem structures exist, and they do say that classes, quarks, and natural numbers exist, then we should accept classes, quarks, and numbers, but not ante rem structures. I submit that here Quine takes the minimalism constraint too far, or is misled by it. The proposal in my book, and presumably in Resnik’s too, is to interpret the mathematician’s own numbers as places in a structure. This does not strike me as pernicious metaphysics. The faithfulness constraint is to avoid contradicting standard mathematics and the minimalism constraint is to avoid attributing more mathematical properties to mathematical objects than

112 / Structure and Identity standard mathematics does. I submit that ante rem structuralism does neither of those. I have discussed, and defended, the faithfulness constraint in Shapiro (1994) and Chapter 1 of Shapiro (1997). I do not have any arguments in favor of the minimalism constraint, nor do I know how to (or even care to) articulate it further, to determine its limits for example. The minimalism constraint is more a matter of taste, in the sense that, other things equal, I would ﬁnd a ‘quieter’ philosophy more pleasing. Of course, my personal preferences carry no weight in a sober philosophical discussion. Philosophy must go beyond statements of personal taste. I mention the criteria here to provide some perspective, and to note that while I am still attracted to the minimalism constraint, I concede that it led me astray (see below). The ﬁrst issue to be dealt with here concerns the very characterization of ante rem structuralism. Its slogans either seem to be incoherent or point to outright falsehoods. In one way or another, the other issues presented here concern the identity relation on mathematical objects, i.e. the identity relation on places in structures. One issue concerns whether places from different structures can be, or should be, identiﬁed. Is the natural number 2 the same or different from the second von Neumann ordinal? Is the natural number 2 identical to the real number 2? Are there determinate facts answering these questions? Another issue concerns the identity relation within the same structure. What are we to make of structures that have different places with the same structural, or relational properties? This comes up whenever a structure has a non-trivial automorphism. One reading of the slogans of ante rem structuralism suggests that I am committed to identifying such places as the complex numbers i and −i since they have the same relational properties—despite the easy theorem that i and −i are distinct. Indeed, every point of Euclidean space has the same relations to the rest of space as every other point. Space is completely homogeneous. Do I have to say—absurdly—that there is only one point? Much of this paper concerns some criticisms of ante rem structuralism that appear in various notes and reviews, and I have encountered versions of them in conversation and correspondence with friends, colleagues, supporters, and opponents. The most articulate versions of the arguments are in a pair of papers, one by Jonathan Kastin, and the other by Jukka Ker¨anen (presented at the Paciﬁc Division of the American Philosophical Association, 1998, and his subsequent (2001) ). I will reformulate the criticisms myself, and take responsibility if I do not come up with the most compelling versions of them.

Stewart Shapiro / 113

1. What Is (Ante Rem) Structuralism? The main thesis of ante rem structuralism is that a natural number, say, is a place in the natural number structure, and the natural number structure exists independent of any concrete system that may exemplify it. So there are two ontological categories: structures and places. Shapiro (1997) focused on the structures, and perhaps did not say enough, and did not say the right things, about places. What is a natural number? Or, in other words, what is a place in the natural number structure? The problems turn on the slogans associated with this version of structuralism. Benacerraf (1965) himself initiated some of the rhetoric: [I]n giving the properties . . . of numbers you merely characterize an abstract structure . . . [T]he ‘elements’ of the structure have no properties other than those relating them to other ‘elements’ of the same structure . . . To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, to be followed by 4, 5, and so forth . . . What is peculiar to 3 is that it deﬁnes that role . . . by representing the relation that any third member of a progression bears to the rest of the progression. (Benacerraf 1965: 291)

Famously, Benacerraf argues from this that numbers are not objects. If numbers are sets, then they must be particular sets, and so natural numbers would have membership relations with other sets. Thus numbers will have set-theoretic properties which go beyond the relations of natural numbers to other natural numbers, contradicting the slogan. Benacerraf ’s point here is consonant with the above minimalism constraint. It invites the simple retort that natural numbers are not sets, but the argument is general. The underlying idea seems to be that any bona ﬁde object has some properties that do not concern its relations to natural numbers. So natural numbers are not bona ﬁde objects. Russell (1903: §242) makes a related point against Dedekind’s abstraction (see Hellman 2001: 194 n): . . . it is impossible that the ordinals should be, as Dedekind suggests, nothing but the terms of such relations as constitute a progression. If they are to be anything at all, they must be intrinsically something; they must differ from other entities as points from instants, or colours from sounds. What Dedekind intended to indicate was probably a deﬁnition by means of the principle of abstraction . . . But a deﬁnition so made always indicates some class of entities having . . . a genuine nature of their own . . .

114 / Structure and Identity One burden of my book was to dispute this: ‘the issue raised by Benacerraf . . . concerns the extent to which a place in a structure is an object’ (1997: 3). This is the matter before us now. Dummett (1991: ch. 23) dubs views like mine ‘mystical’ structuralism. According to that view, according to Dummett, ‘mathematics relates to abstract structures, distinguished by the fact that their elements have no non-structural properties’. Thus, for example, the zero place of the natural number structure ‘has no other properties than those which follow from its being the zero’ of that structure. Resnik’s early (1981) has similar passages: In mathematics, I claim, we do not have objects with an ‘internal’ composition arranged in structures, we have only structures. The objects of mathematics, that is, the entities which our mathematical constants and quantiﬁers denote, are structureless points or positions in structures. As positions in structures, they have no identity or features outside a structure.

Kastin’s ‘Shapiro’s mathematical structuralism’ identiﬁes passages in my book where I endorse the Benacerraf–Dummett–Resnik slogans about mathematical objects. In the introduction (1997: 5–6), I wrote that there ‘is no more to being the natural number 2 than being the successor of the successor of 0, the predecessor of 3, the ﬁrst prime, and so on’. Later in the book, I am only a little more careful: The essence of a natural number is its relations to other natural numbers . . . The number 2, for example, is no more and no less than the second position in the natural number structure; 6 is the sixth position . . . The essence of 2 is to be the successor of the successor of 0, the ﬁrst prime, and so on. (1997: 72)

The extra care here, if one can call it that, is the mention of essence. Kastin follows Benacerraf in pointing out that slogans like this seem to conﬂict with the thesis that numbers exist as objects. The slogans are incoherent, or misleading at best (see also Hellman 1999: 924). Start with Dummett’s ‘mystical’ version that mathematical objects ‘have no non-structural properties’. This is obviously false, on any realist reading. The number 3 has the property of being the number of my children (pictures on request), and 17 has the property of being Seymore’s favorite number. Clearly, ‘being the number of Stewart Shapiro’s children’ and ‘being Seymore’s favorite number’ are non-structural properties, no matter how liberally we manage to construe the notion of a structural property.

Stewart Shapiro / 115 These examples undermine the slogan by invoking contingent properties. One response would be to reformulate it to disallow these counterexamples: individual natural numbers have no non-structural necessary properties; the number 0 has no other necessary properties than those which follow from its being the zero of the natural number structure. In other words, for any property P and natural number n, if Pn then P is a property concerning how the natural numbers relate to each other. This does not work either. Traditional Judaism has it that, necessarily, the number 1 has the property of being the number of deities. Some forms of dualism have it that, necessarily, 2 is the number of parts of each human. Being the number of deities and being the number of parts of a human are not structural properties. Of course, I am not saying that traditional Judaism and this form of dualism are correct, but we don’t want ante rem structuralism to rule them out. Next retort: individual natural numbers have no non-structural mathematical properties; the number 0 has no other mathematical properties than those which follow from its being the zero of the natural number structure. For any mathematical property P and natural number n, if Pn then P is a property concerning how the natural numbers relate to each other. To pursue this suggestion, we would have to say what makes a property mathematical (as opposed to physical, theological, metaphysical). We can neatly beg the question by deﬁning a mathematical property of a number n to be a property that concerns how n relates to other numbers. But this ﬁat badly fails to capture the intuitive sense of ‘mathematical’. The number 0 has the property of being the number of real square roots of −1, the number 4 has the property of being the maximum number of colors needed for a map. These properties are mathematical, par excellence, and yet both go beyond the relations of the natural numbers to each other, and neither of the facts follows from the characterization of the natural number structure alone. Other mathematical objects and structures are involved. As noted above, the ‘ofﬁcial’ view in my book is that individual natural numbers have no non-structural essential properties. Shamefully, I did not say anything about what an essential property is. Let us see if we can do any better here. Simon Blackburn’s Oxford Dictionary of Philosophy (1994) deﬁnes essence to be the ‘basic or primary element in the being of a thing; the thing’s nature, or that without which it could not be what it is’. Blackburn adds that a ‘thing cannot lose its essence without ceasing to exist’.

116 / Structure and Identity In these terms, the supposedly careful version of my stated thesis is that the basic or primary element in the being of the natural number 2—the nature of the natural number 2—consists in its relations to other natural numbers. The number 2 could not be what it is without its relations to other natural numbers. This sounds all right, as far as it goes, but I admit it does not go very far. The last clause in Blackburn’s deﬁnition is particularly useless here, since presumably there is no sense to a number losing its relations to other numbers, and thereby ceasing to exist. It seems that the notion of ‘essential property’ has a more natural home with contingent objects. Let us suppose that this is resolved, and that we know what we are talking about with the essence/accident distinction concerning mathematical objects. Kastin and Hellman point out that, even so, numbers seem to have some nonstructural essential properties. For example, the number 2 has the property of being an abstract object, the property of being non-spatio-temporal, and the property of not entering into causal relations with physical objects. At least intuitively, these seem essential to the natural numbers, if the natural numbers exist. To paraphrase the Blackburn deﬁnition, being abstract, and the like, seem to be among the basic, or primary elements in the being of the natural numbers. If they were not abstract, they would not exist. Abstractness is certainly not an accidental property of a number—or is it? We will come back to this shortly. To continue the sequence of Lakatos-style monster-barring, perhaps we can try a pair of theses: (i) a given natural number, like 2, is uniquely characterized by its relations to other natural numbers, and (ii) other essential properties of the number 2 ﬂow from, or are consequences of, this characterization. When we say that there is no more to being the number 2 than being the successor of 1, the predecessor of 3, the ﬁrst prime, etc., we thereby also say that 2 is abstract, causally inert, etc. Anything which is no more than the successor of 1, the predecessor of 3, etc. is abstract, causally inert, etc. I do not know how to make this way out more illuminating and less problematic. Starting with (ii), it is not at all clear what notion of ‘consequence’ is in play. It is not a logical consequence of ‘2 is the successor of 1’, ‘2 is the predecessor of 3’, and the like, that 2 is abstract. Indeed, the abstractness of 2 is not a logical consequence of the sum total of the mathematical properties that 2 enjoys. Abstractness is not itself a mathematical property at all. Perhaps we can say that the abstractness of 2 follows from its characterization as a place in the natural number structure plus some conceptual and metaphysical

Stewart Shapiro / 117 truths about structures, but these metaphysical principles would have to be articulated and defended somehow. One might say instead that the essential properties of 2 supervene on the fact that 2 is a particular place in the natural number structure, but this requires a developed notion of supervenience. Supervenience is itself a modal notion, and it is not clear how to wield it in a non-trivial way when dealing with objects that supposedly exist of necessity (if they exist at all). I suppose we can say that in any possible world in which the natural number structure exists, its second place is abstract, causally inert, etc. But how does this help? Clause (i) of the retrenched thesis is that a given natural number is uniquely characterized by its relations to other natural numbers. The problem is that this does not manage to distinguish structuralism from any other view that takes numbers to be objects. Surely any realist and, for that matter, any intuitionist will agree that 2 is uniquely characterized by being the predecessor of 3, without even mentioning any of its other structural properties, like being the successor of 1 or being the ﬁrst odd prime. According to any view that accepts the existence of numbers, 2 is the predecessor of 3, and nothing else is. Moreover, on any reasonable characterization of supervenience, the Platonist will hold that all of the necessary properties of 2 supervene on this characterization. In the end, I think that ante rem structuralism is a version of traditional Platonism, or what I call realism in ontology, just because it takes numbers to exist as bona ﬁde objects. The problem is that we have now retrenched the slogan to the point where it does not say anything distinctively structuralist. Putting the monster-barring aside, there are at least two further ways out of thisdilemma. The ﬁrstrunsagainstacceptedwisdom concerning mathematical objects. The idea is to hold that abstractness, non-spatio-temporality, and the like, are indeed accidental properties of (some) mathematical structures and objects. Hear me out. Recall that on my view, structures are like traditional universals, in that they can have instances—systems of objects exemplify structures. Suppose that one takes an Aristotelian in re view of structures which are exempliﬁed by systems of concrete objects, and a more Platonic view of structures which have no concrete exempliﬁcations. On such a view, an exempliﬁed structure is located where the objects exemplifying it are located, and it inherits the causal efﬁcacy of those objects. Suppose that an announcer says: ‘the runner was thrown out by the shortstop.’ The natural, intended reading of this would be that the runner was thrown out by the person playing the of role of shortstop. This is what I call the ‘places are ofﬁces perspective’

118 / Structure and Identity (1997: 10). On the present proposal, we can also say that the position threw out the runner, since the position has the causal efﬁcacy of its occupant that day. The proposal is a structuralist version of Penelope Maddy’s (1990) account of sets of physical objects. For Maddy, sets of physical objects (and sets of those sets, etc.) are located where the physical objects are located and enjoy the physical properties of those objects. So for Maddy the singleton of the person playing shortstop threw out the runner (and the shortstop also threw out the singleton of the runner, but fortunately for the offense, only one out was recorded). For Maddy, pure sets, such as the empty set and ω, are not located anywhere. Here, physically unexempliﬁed structures are not located anywhere. The only place I know of where a similar view is broached is in David Armstrong (1989: 76), which argues against a corresponding ‘mixed’ orientation toward universals: Once you have uninstantiated universals you need somewhere special to put them, a ‘Platonic heaven’, as philosophers often say. They are not to be found in the ordinary world of space and time. And since it seems that any instantiated universal might have been uninstantiated—for example, there might have been nothing past, present, or future that had that property—then if uninstantiated universals are in a Platonic heaven, it will be natural to place all universals in that heaven.

Strictly speaking, abstracta are not ‘located’ anywhere. That is part of what it is to be abstract. So on the view in question here, an unexempliﬁed structure is not located anywhere. Once instantiated, the structure is located wherever the system is located. I have no objection to a metaphorical extension of the notion of ‘location’ to include uninstantiated universals, and if we want, we can use the colorful phrase ‘Platonic heaven’ for these ‘locations’. It seems to me, however, that Armstrong’s argument turns on taking this metaphor too seriously. Speaking strictly and literally, we have no need to postulate ‘somewhere special to put’ the unexempliﬁed structures (see also MacBride (1998: esp. 211) ), and so there is no point to ‘locating’ all structures in this special ‘place’. If we take the language less literally, perhaps Armstrong’s point is that since at least some universals—the uninstantiated ones—are abstract, then we might as well declare that all universals are abstract. Other things equal, this declaration does make for a simpler theory in one respect, but in the case of ante rem structuralism, perhaps other things are not equal. The proposed view of structures and the abstract-concrete dichotomy takes some of the ante out of ante rem structure. Perhaps. The view does have it that

Stewart Shapiro / 119 a structure exists independent of any instances it has. It is still ‘before the things’, to translate the Latin. But once a structure is exempliﬁed, it takes on the physical and causal properties of its instantiations. For what it is worth, the proposal goes well with the vague minimalism constraint broached above. Speaking strictly, the thesis that mathematical structures are abstract, acausal, etc. is not part of mathematics. In a sense, mathematics leaves this bit of metaphysics open. To be sure, mathematicians do not attribute causal or other physical properties to mathematical objects—to places in structures—but on the view in question, one would not expect them to. A given mathematical structure might not have any physical properties, and even if it does have physical properties, a mathematician as such does not care about them. As noted in my and Resnik’s books, the mathematician studies a structure independent of any physical exempliﬁcations it may have. On the present view, whether a given structure is abstract depends on how the physical world is, and pure mathematics does not concern itself with such matters. In Physics B, Aristotle delimits what he takes to be a distinctive feature of mathematical methodology: The next point to consider is how the mathematician differs from the physicist. Obviously physical bodies contain surfaces, volumes, lines, and points, and these are the subject matter of mathematics . . . Now the mathematician, though he too treats of these things (viz. surfaces, volumes, lengths, and points), does not treat them as (qua) the limits of a physical body; nor does he consider the attributes indicated as the attributes of such bodies. This is why he separates them, for in thought they are separable from motion, and it makes no difference nor does any falsity result if they are separated . . . [G]eometry investigates physical lengths, but not as physical . . . (193b–194a)

See also Metaphysics M (1077b–1078a). Aristotle’sidea seemstobe thatmathematicsconcernsitself with physical objects, butthe mathematician isnotconcerned with their physical properties. In thought, the geometer separates surfaces, lines, and points from the physical objects that contain them, and focuses on their geometric properties. Here, we have the mathematician separating structures from any systems they may be structures of. The physical exempliﬁcations are of no concern to the mathematician, qua mathematician. But with Aristotle, it does not follow that mathematical structures have no physical properties. According to the present proposal, some do and some don’t. For Aristotle, Plato’s mistake was to conclude that geometrical objects are metaphysically separate from their physical instantiations, just because

120 / Structure and Identity mathematicians manage to ignore certain physical aspects of their subject matter. But unlike Plato, Aristotle left no room for mathematical objects that have no physical instantiations. Here we leave it genuinely open whether a given structure has a physical exempliﬁcation. On the view in question, I would retract my offhand comment that ‘mathematical objects—places in structures—are abstract and causally inert’ (1997: 112). The closest I can come to this traditional wisdom is the Aristotelian comment that the practice of (pure) mathematics is independent of any physical properties a given structure may have. The context of my offhand comment was to note that structuralism is not compatible with crude versions of the causal theory of knowledge. That much is still correct. Since a given structure might not be physical, the epistemology of mathematics had better not be entirely causal. We may obtain knowledge of small structures via causal contact with their instances, as outlined in my book (1997: 112–16), but we do not learn about larger structures that way. These larger structures are not instantiated at all (in the physical realm). Strange or not, this is one way out. To summarize, on the resolution in question, we maintain that the essential properties of a natural number consist of its relations to other natural numbers, and then claim that the purported non-mathematical counterexamples, like abstractness, are not essential to the natural numbers. Adopting this proposal does not leave us completely home and dry. What of the property of being possibly abstract? Surely it is essential to the number 512, or the natural number structure itself, or the set-theoretic hierarchy, that it might be abstract, in the sense that there don’t have to be physical instantiations of this structure. Clearly, the property of being possibly abstract is not a structural property of that number or those structures. So we seem to have a counterexample to the proposal. To pursue the strategy, we would have to deny that modal locutions like ‘being possibly abstract’ correspond to properties, or else insist that the essence-accident distinction is out of place with them, or else we can restrict the slogan to non-modal properties, if this last act of ﬁat does not look too desperate. The other resolution of the problem concerning the slogans is perhaps disappointing, since it does not invoke an interesting metaphysical position. The ante rem structuralist can simply give up the slogan that all essential properties of natural numbers concern the relations of such numbers to other natural numbers. I would take back statements like there ‘is no more to

Stewart Shapiro / 121 being the natural number 2 than being the successor of the successor of 0, the predecessor of 3, the ﬁrst prime, and so on’ (1997: 5–6) and the ‘essence of a natural number is its relations to other natural numbers’ (1997: 72). Recall that according to the view that Dummett (1991: ch. 23) calls ‘mystical’ structuralism ‘mathematics relates to abstract structures, distinguished by the fact that their elements have no non-structural properties’. And Resnik’s ‘[t]he objects of mathematics . . . are structureless points or positions in structures. As positions in structures, they have no identity or features outside a structure.’ The present option is to concede the rejection of this ‘mystical structuralism’, since there are no ‘abstract structures’ in this sense. The concession does not affect the underlying philosophy of ante rem structuralism. The revisions are limited to which slogans we can use and take seriously, in order to postpone a detailed articulation of the view. The ofﬁcial stance is (still) that a natural number is a place in the natural number structure. The latter is construed after the fashion of an ante rem universal in that it exists independent of any exempliﬁcations it may have in the non-mathematical realm. Then we rely on the account of ante rem structures articulated in my book, or if you like it better, the account of patterns in Resnik’s book. The underlying view that numbers and other mathematical objects, so construed, are bona ﬁde objects is unaffected. The number 2, so construed, has lots of properties. Some of these properties, like being the number of doors in my dining room, are contingent. Some of the properties of the number 2, like being the number of parents of a normal human, may be biologically necessary. Other properties of the number 2 are metaphysical and still others are mathematical. Of the mathematical properties, some, like being the ﬁrst prime, are internal to the natural number structure. Others, like being the number of complex square roots of −1, are external to the natural number structure. It does not matter. On the view in question, every property that 2 enjoys comes in virtue of its being that place in the natural number structure. This is because that is what 2 is. It is only the slogan that must be given up.

2. Cross-Structural Identity We have it that each mathematical object is a place in a structure. Now we turn to the identity relation on mathematical objects. The next section focuses

122 / Structure and Identity on the identity relation on places within the same structure; here our topic is places from different structures. As with much philosophy of mathematics, the issue begins with the Benacerraf (1965). According to von Neumann’s interpretation of arithmetic, each natural number is the corresponding ﬁnite ordinal, so that 2 is {φ, {φ} }; according to Zermelo’s interpretation, 2 is { {φ} }. So which is 2? The structuralist answers to this question are that both are and that neither is. When we say that the ordinal {φ, {φ} } is 2, the ‘is’ is not identity. Rather, we say that the ordinal occupies a certain place in a given system that exempliﬁes the natural number structure—the system of ﬁnite ordinals. The ‘is’ in ‘{φ, {φ} } is 2’ is something like the ‘is’ of predication—I call it the ‘is’ of ofﬁce-occupancy (1997: 83). The same goes for the statement that { {φ} } is two. We could say instead that { {φ} } is a 2, or, more fully, that { {φ} } occupies the two-role in the Zermelo system. It is similar to saying that ‘Rabin was Prime Minister’ and ‘Begin was Prime Minister’, and then wondering which of them was the real Prime Minister. Both were and neither was. Both occupied the ofﬁce; neither is the ofﬁce. So far so good, but what of the ‘is’ of identity in the context of structuralism? What are we to make of statements of identity or difference between places in different structures? This is a situation where the faithfulness constraint and the minimalism constraint led me astray. Resnik says that, in such cases, there is no fact of the matter, or at least there is no fact unless the two structures are taken to be part of a larger one (Evans (1978) notwithstanding). I more or less agreed, but added a bizarre addendum. I take the liberty of quoting myself at length. I began by endorsing indeterminacy, suggesting a notion of relative identity (1997: 79–81): I would think that a good philosophy of mathematics need not answer questions like ‘Is Julius Caesar = 2?’ and [‘Is {φ, {φ}} = 2?’]. Rather, a philosophy of mathematics should show why these questions need no answers . . . It is not that we just do not care about the answers; we want to see why there is no answer to be discovered—even for a realist in ontology . . . One can form coherent and determinate statements about the identity of two numbers: 1 = 1 and 1 = 4 . . . But it makes no sense to pursue the identity between a place in the natural number structure and some other object, expecting there to be a fact of the matter. Identity between natural numbers is determinate; identity between numbers and other sorts of objects is not, and neither is identity between numbers and the positions of other structures . . . The Frege–Benacerraf questions do not have determinate answers, and they do not need them . . .

Stewart Shapiro / 123 We point toward a relativity of ontology, at least in mathematics. Roughly, mathematical objects are tied to the structures that constitute them. Benacerraf [1965, § III.A] himself espoused a related view, at least temporarily: . . . ‘Identity statements make sense only in contexts where there exist possible individuating conditions . . . [Q]uestions of identity contain the presupposition that the ‘entities’ inquired about both belong to some general category’ . . . In mathematics, at least, the notions of ‘object’ and ‘identity’ are unequivocal but thoroughly relative . . .

But the faithfulness constraint led me to balk at the clean relativity (1997: 81–2): I do not wish to go as far as Benacerraf in holding that identifying positions in different structures (or positions in a structure with other objects) [is] always meaningless. On the contrary, mathematicians sometimes ﬁnd it convenient, and even compelling, to identify the positions of different structures. This occurs, for example, when set theorists settle on the ﬁnite von Neumann ordinals as the natural numbers. They stipulate that 2 is {φ, {φ} } and so it follows that 2 = { {φ} }. For a more straightforward example, it is surely wise to identify the positions in the natural number structure with their counterparts in the integer, rational, real, and complex number structures. Accordingly, the natural number 2 is identical to the integer 2, the rational number 2, the real number 2, and the complex number 2 (i.e. 2 + 0i). Hardly anything could be more straightforward. For an intermediate case, mathematicians occasionally look for the ‘natural settings’ in which a structure is best studied. An example is the embedding of the complex numbers in the Euclidean plane, which illuminates both structures. It is not an exaggeration to state that some structures grow and thrive in certain environments . . . The point here is that cross-identiﬁcations like these are matters of decision, based on convenience, not matters of discovery.

Charles Parsons (1990: 334) also questions the extreme relativity broached just above: . . . one should be cautious in making such assertions as that identity statements involving objects of different structures are meaningless or indeterminate. There is an obvious sense in which identity of natural numbers and sets is indeterminate, in that different interpretations of number theory and set theory are possible which give different answers about the truth of identities of numbers and sets. In a lot of ordinary, mathematical discourse, where different structures are involved, the question of identity or non-identity of elements of one with elements of another just does not arise (even to be rejected). But of course some discourse about numbers and sets makes identity statements between them meaningful, and some of that . . . makes commitments as to the truth value of such identities. Thus it would be quite

124 / Structure and Identity out of order to say (without reference to context) that identities of numbers and sets are meaningless or that they lack truth-values.

My stated view at the time was that there are no cross-structural-identity facts that we start with, but identiﬁcation-decisions create such facts. Vann McGee (1997: 39) allows a role for this sort of stipulation in mathematics: What we do when we ‘identify’ the natural numbers with the von Neumann numbers is to adopt a convention assigning truth values to sentences like ‘5 ∈ 7’, to which [previous] usage, together with the mathematical facts, assigned no values. We are free to do this, in the same way that we are permitted, it if suits our purposes, to assign truth values to borderline attributions of vague terms by adopting conventions making the vague terms more precise. Identifying the numbers with the sets is not a matter of pruning our ontology, but rather a matter of reﬁning our usage. What there is is not a question of notational convenience, but how we use our symbols is, in large measure, a matter of convenience.

This passage comes on the heels of McGee’s use of Quine–Putnam considerations to reject ‘referential realism’, the view that mathematical language has a determinate reference relation. The purpose of McGee (1997) is to defend what I call ‘realism in truth-value’, the view that properly declarative mathematical sentences have objective, determinate truth-values, without this referential realism. In contrast, Shapiro (1997) proposes a referential realism. Accordingly, the singular terms of arithmetic have determinate reference, to places in the natural number structure. The singular terms of set theory also have determinate reference, to places in the set-theoretic hierarchy structure. The question here is whether some of the set-theoretic terms and some of the arithmetic terms refer to the same objects. Kastin correctly pointed out that if mathematical objects exist, as objects not of our making, then identities between them cannot be stipulated. Among existing objects, identities are discovered, not made. So the allowed stipulations are inconsistent with the realism in ontology (see also Hellman 1999: 924; 2001: 192). Nevertheless, some cross-structural identities are at least prima facie natural, and perhaps even inevitable. Suppose that during a lecture, a mathematician writes a closed integral on a blackboard, and asks the class to determine whether the number it denotes is a natural number. A philosophy student, fresh from a study of structuralism—having read part of Resnik’s book and/or

Stewart Shapiro / 125 part of mine—quickly gets the ﬂoor while the other students are busy trying to resolve the integral, and proudly announces the solution: there is no fact of the matter. He says that the integral denotes a place in the real number structure, and so it is indeterminate whether places from the real number structure are the same as or distinct from places in the natural number structure. Clearly, something has gone wrong; the philosophy student is confused. The mathematician asked a sensible, substantive mathematical question. The job of the students was to answer it, and our job as philosophers is to interpret it. Following the faithfulness constraint, I wanted the mathematician’s language to be understood literally, at face value. My solution, if you want to call it that, was that the mathematical community had stipulated that the non-negative whole numbers of the real number structure are indeed the natural numbers. This makes the mathematician’s question a proper mathematical one. Recall that Frege (1884: e.g. § 66) dropped his ﬁrst attempt to characterize the natural numbers since it provided no way to determine the truth-value of a statement of identity between a number and an object not given as a number. This became known as the ‘Caesar problem’. As above, I suggested that a clean resolution of the Caesar problem is a motivation for structuralism. The issue at hand here suggests that in the end, the ante rem structuralist does have a version of Frege’s Caesar problem, and has not resolved it. Resnik suggested that statements of identity between items from different structures are determinate only in the context of a theory, or structure, that encompasses both. If the structures are kept separate, there is no fact of the matter concerning whether their places are identical or distinct from each other. Let us use the word ‘rubric’ for a theory or structure that encompasses ranges of mathematical objects. Let us return to the imaginary mathematician, introduced above, who asked whether a given closed integral denotes a natural number. According to Resnik, since this mathematician is speaking of the natural numbers and the real numbers under a single rubric, there would be a determinate identity relation on the whole of her domain of discourse in that context. Obviously, this is what she intended. I am not sure that Resnik’s position is stable, however, since it invites a third-man type of regress. Suppose that a mathematician M1 is studying the natural numbers, and nothing else. So he is limited to a single structure. Mathematician M2 is the mathematician introduced above, who is talking about the natural numbers and the real numbers at the same time. Suppose we ask if the natural numbers of M1 are the same or different from

126 / Structure and Identity the natural numbers of M2 . Resnik tells us that in such cases, there is no fact of the matter. We have identity-facts only if there is a combined rubric for the natural numbers of M1 and the natural numbers of M2 , but by hypothesis, there isn’t. Our character M1 studies the natural numbers and nothing else. So we envision a third mathematician M3 who studies the natural numbers of M1 and those of M2 . Are her natural numbers the same as those of M1 and/or M2 ? No fact of the matter unless there is a rubric combining the natural numbers of M1 , M2 , and M3 . And on it goes. Recall that M2 studies the natural numbers and the real numbers. Suppose that by using the resources of the full theory, she proves something about the natural numbers. That is, M2 establishes a theorem in the language of arithmetic, but the proof invokes her extra resources, the real numbers. This is a common situation in mathematics, pretty close to the norm. Although the proof is not available to M1 , is the sentence nevertheless true of the natural numbers of M1 —or the natural numbers of any mathematician whether or not he looks beyond the natural number structure? On pain of massive revisions of mathematical practice (cf. the faithfulness constraint), we had better say is true of the natural number structure. Our ﬁrst mathematician M1 may not have the resources to know (until he meets M2 ), but is true in that context nonetheless. On Resnik’s view, however, we would be forced to say that there is no fact of the matter whether holds of the natural numbers of M1 . A straightforward reply might go like this: (*) Of course holds of the natural numbers of M1 . No matter how we settle the nice question of identity, it is clear that the natural numbers of M1 are isomorphic to those of M2 , and isomorphic structures are equivalent. So from the fact that holds of the natural numbers of M2 , it follows that it holds of the natural numbers of M1 . Q.E.D. The problem isthatin order tosaythis, andspeakof an isomorphism between the natural numbers of M1 and those of M2 , we have to bring the natural numbers of M1 under the same rubric as the natural numbers of M2 . Isomorphisms relate structures to each other. We can speak of an isomorphism between S and T only if we are speaking of S and T at the same time—under the same rubric. Suppose that we ourselves here and now utter (*). Are we talking about, or including, the natural numbers of M1 ? According to Resnik, there is no fact of the matter. And so we still do not know that holds of the natural numbers

Stewart Shapiro / 127 of M1 . In short, the dilemma is that in order to invoke the isomorphism, we have to bring the natural numbers of M1 under the same rubric as those of someone else. On Resnik’s view, as soon as we do that, there is no fact of the matter as to whether we are talking about the natural numbers of M1 . Perhaps instead we avoid talk of isomorphism and try the following: (**) There is a meta-theorem that the sentence holds of all ωsequences. Since the natural numbers of M1 constitute an ωsequence, we have that holds of those natural numbers. This is no better. The problem now is that in order to say (**), we must envision a rubric that consists of all ω-sequences, which, of course, would include the natural number structure. But does this encompassing rubric include the natural numbers of M1 , the mathematician who is just studying the natural numbers and not a universe containing all ω-sequences? One would think the new rubric includes the domain of M1 , since it speaks of all ω-sequences. But Resnik’s view does not sanction this manifest intuition. The rubrics are different and so there is no fact of the matter concerning identities between them. Is there then a fact of the matter as to whether the original natural numbers of M1 are the same as their counterparts in this new rubric, the one that speaks of all ω-sequences? On Resnik’s view, it seems, there is no such fact of the matter, and so we cannot apply this meta-theorem. Let’s review some options on the issue of identity between the places of different structures. One might insist on indeterminacy, following Resnik and my former self (minus the stipulation stuff). On this view, there is no fact of the matter whether the natural number 2 is identical to the real number 2 or the set {φ, {φ}}. On the option in question here, the locutions ‘the natural number 2’ and ‘the real number 2’ each have determinate reference—to a place in the natural number structure and a place in the real number structure respectively. The view is that there is no fact of the matter concerning whether these referents are identical or distinct. The indeterminacy is charged to the (mathematical) world, not to mathematical language. It is an ontological or metaphysical indeterminacy. This option runs against the Quinean dictum ‘no entity without identity’. Quine’s thesis is that within a given theory, language, or framework, there should be a deﬁnite criteria identity relation on its objects. Against this, the present option answers these Caesar-type questions of identity or distinctness with: ‘there is no fact of the matter.’

128 / Structure and Identity Of course, the Quinean dictum is not set in stone, and the ante rem structuralist is free to demur from it, but let us explore some further proposals that meet it. A second option would be to hold that there is a determinant identity relation on the places of different structures, but that in at least some cases, the identities are inscrutable. For example, there is indeed a fact of the matter whether the natural number 2 is identical with or distinct from the real number 2 and the set {φ, {φ} }, but we may have no way of knowing this. It is an epistemic indeterminacy. Given that we have no clue as to how at least some of these identities can be discovered, we may be left with some unknowable mathematical facts. So in this case, the indeterminacy is charged to our paltry epistemic situation. A third option, the one I propose here, is to insist that places from different structures are distinct. This ﬁts with at least one of my repeated remarks/slogans: ‘mathematical objects are tied to the structures that constitute them’. On the this third option, the natural number 2, the real number 2, and the set {φ, {φ} } are tied to different structures and so they are distinct objects. Also, as indicated in the long quote above, the present option would ﬁt in well with the slogan, discussed in the previous section, that the essence of a mathematical object is its relations to other objects in the same structure. The natural number 2 and the real number 2 and the set {φ, {φ} } enjoy different relations to different objects. It is part of the essence of the real number 2 that it has a square root, it is greater than this square root, and it is less than π . The natural number 2 does not have those properties. Similarly, it is part of the essence of the real number 3 that there is a number that is exactly half of it. The natural number 3 does not have that property. To quote myself again: ‘Julius Caesar and {φ, {φ} } have essential properties other than those relating to other places in the natural number structure’ (Shapiro 1997: 80). At the time, I said that this misses the point, but I guess it does not. On this view, the lingering indeterminacy is charged to the language. The phrase ‘the number 2’ is systematically ambiguous, denoting a place in the natural number structure, a place in the integer structure, a place in the real number structure, a place in the ordinal structure, etc. Even the phrase ‘the natural numbers’ is systematically ambiguous, denoting both the natural number structure itself and various substructures of other structures. For example, ‘the natural numbers’ denotes a substructure of the real numbers and another substructure of the complex numbers. So the phrase ‘the natural

Stewart Shapiro / 129 number 2’ is also ambiguous. In addition to denoting a place in the natural number structure, the term denotes the corresponding place in the real number structure. Our third option, then, is a sort of semantic indeterminacy. Damage control: I propose that even with this sharp and (admittedly) crude resolution to the ‘Caesar’ issue, we can still come reasonably close to the faithfulness and minimalism constraints. The crucial mathematical fact here is that isomorphic systems are equivalent. If B is isomorphic to B , then any sentence in the relevant language true of one is true of the other. This fact is what drives McGee’s (1997) aforementioned defense of realism in truth value without referential realism (and what I call ‘realism in ontology’). Here, the point is that it simply does not matter whether a mathematician is thinking of a given structure or a system that exempliﬁes it—perhaps a system constructed from the places of another structure. So the mathematician can go back and forth between the structure itself and any of the systems that exemplify it. So when I said in the book that the mathematical community stipulates that the ﬁnite ordinals are to be the natural numbers or that natural numbers are also real numbers, I should have said instead that they (permanently or temporarily) agreed to study a particular system that exempliﬁes the natural number structure. In doing so, they invoke what I call the ‘places-are-ofﬁces’ perspective. But they still shed light on the natural number structure, since the given system exempliﬁes that structure. This sort of stipulation does not undermine the ontological realism, or even the referential realism. The mathematicians’ decision to study the Von Neumann ﬁnite ordinals is not to say that these ordinals are the natural numbers. A study of these ordinals sheds light on the natural numbers structure because the ordinals exemplify the structure. The fact that the ﬁnite ordinals exemplify the natural number structure, of course, is not a matter of stipulation. Mathematicians make such agreements/stipulations for a variety of reasons. In the case of the von Neumann ordinals, the move is made for purposes of uniformity. The set-theorist needs ‘numbers’ for studying cardinality and it is convenient to avoid new sorts of variables. Since the given ω-sequence of sets will serve as natural numbers, he uses them for that purpose. The ﬁnite ordinals serve as natural numbers because they have the right structure. Other situations, like the embedding of the natural numbers in the reals, or the complex numbers in the Euclidean plane, have powerful ramiﬁcations within mathematics. That is, the decision to focus on a particular system sheds

130 / Structure and Identity new light on the underlying structure. It sometimes happens that we only grasp a given structure when it is embedded in a richer one. In present terms, we learn more about a given structure by studying a system that exempliﬁes the structure, this system consisting of places from another, richer structure. To quote Kreisel (1967: 166): . . . very often the mathematical properties of a domain D become only graspable when one embeds D in a larger domain D . Examples: (1) D integers, D complex plane; use of analytic number theory. (2) D integers, D p-adic numbers; use of p-adic analysis. (3) D surface of a sphere, D three-dimensional space; use of three-dimensional geometry. Non-standard analysis [also applies] here . . .

Some of the embeddings are extremely natural, and have come to be part of the framework for mathematics—to the point where names are shared. Once again, cases in point are the mutual embeddings of the natural numbers, the integers, the real numbers, etc. So, to repeat the resolution, the numeral ‘2’ denotes a natural number, a real number, a complex number, a ﬁnite ordinal, etc., and the phrase ‘natural numbers’ itself ambiguously refers to the natural number structure itself, a particular substructure of the integers, a particular substructure of the real numbers, etc. Similarly, ‘π ’ denotes both a real and a complex number. One result of this interpretation is that it is often left indeterminate just which structure and place we are talking about. However, I suggest that in the practice of mathematics—whether pure or applied—this indeterminacy does not interfere with communication or understanding. In virtually all cases, even if it is not clear from the context which structure, and thus which place, we are talking about, it does not matter since all of the relevant systems are isomorphic to each other. So we can talk about the structure and the systems exemplifying it at once. A shift in reference can take place silently and smoothly, and on the ﬂy. One can use a numeral to refer to a natural number and immediately discuss the properties of the corresponding real number, and transfer some of that information back to the natural number. So let us return to the aforementioned imaginary mathematician who wrote a closed integral and asked whether it denotes a natural number. On the interpretation suggested here, we can take his question literally, as referring to the natural number substructure of the real number structure. If one does not like this resolution of the ambiguity, we can render the query like this: ‘does

Stewart Shapiro / 131 this term denote the image of a natural number in the obvious and natural embedding?’ Not quite literal, but pretty close.

3. Identity and Indiscernibility This ﬁnal section concerns an issue that was raised in conversation by a few prominent philosophers and logicians. It is broached in Burgess (1999: 287–8) and Hellman (2001: 192–6), and is articulated in some detail in Ker¨anen’s ‘The Individuation Problem: Realist Structuralism is Dead’, and a follow up ‘The Identity Problem for Realist Structuralism’ (2001) (written in response to an ancestor of this paper). Ker¨anen’s ‘The Identity Problem for Realist Structuralism II’, Chapter 6 in this volume, is a reply to the present paper, and my brief ‘The Governance of Identity’, Chapter 7 in this volume is a brief reply to that. The problem concerns structures that have what we may call ‘indiscernible’ places. Let me illustrate with a few examples. In my chapter 4 (1997: 115–16), devoted to epistemology, I introduced the ﬁnite ordinal structures. The ordinalthree-structure, for example, has three places with a linear order. It is exempliﬁed by the system consisting of my children, taken in birth order. In the structure, each place can be uniquely characterized. One place comes ﬁrst, and it alone satisﬁes the formula ∀y(¬y < x). One place comes second, satisfying ∃y∃z(y < x & x < z); and one place comes third, satisfying ∀y(¬x < y). I also consider ﬁnite cardinal structures, which have no relations. They are about as simple as structures can get. The three-structure is exempliﬁed by my children, the ﬁve-structure by the starting players on a basketball team, etc. Each ﬁnite cardinal structure is completely homogeneous in the sense that there is no formula in the relevant language that distinguishes the places from each other. Indeed, since the structures have no relations, then the relevant language has no non-logical terminology. Any formula, with one free variable, that holds of any place of the three-structure, holds of the other two places. The Euclidean plane is another homogenous structure. Any formula (x), with only x free, in the language of Euclidean geometry, that holds of a given point holds of every point. The complex numbers, of course, are not homogeneous, but they provide a more subtle example of the phenomenon in question. Any formula (x), with only x free, in the language of complex analysis, that holds of the complex number i also holds of −i. Generally,

132 / Structure and Identity any formula in the language of complex analysis that holds of the complex number a + bi also holds of a − bi. So within the language of complex analysis, a + bi is indiscernible from a − bi. The situation under discussion arises whenever a structure has a non-trivial automorphism, a one-to-one function from the domain to itself (other than the identity mapping) that preserves the relations of the structure. Any permutation of a ﬁnite cardinal structure is an automorphism, for the simple reason that there are no relations to preserve. Any rigid translation or rotation of the Euclidean plane is an automorphism, and with the complex numbers, the function that takes a + bi to a − bi is an automorphism. Recall that isomorphic systems are equivalent. So if f is an automorphism of a given structure and fa = b, then for any formula (x), with only x free, we have (a) iff (b). So a is indiscernible from b, even if a = b. Any structure with a non-trivial automorphism violates the identity of indiscernibles—if the only ‘discernings’ we allow are formulas, with only one free variable, in the language of the structure. Is this a problem for ante rem structuralism? Ker¨anen argues that it is. His bold conclusion is that the very idea of extracting the structure of a system with a non-trivial automorphism, and treating that structure as an entity in its own right, is incoherent. The problem, he says, is that the ante rem structuralist is committed to the identity of indiscernibles, and this forces the structuralist to identify all of the places of each ﬁnite cardinal structure with each other, so that each structure has only one place after all. The ante rem structuralist must also identify the complex numbers i and −i, which contradicts the theorem that each number (other than 0) has two distinct square roots. So why is the structuralist committed to the identity of indiscernibles? Ker¨anen proposes a general metaphysical thesis that anyone who proposes a scientiﬁc or philosophical theory of a type of object must provide an account of how those objects are to be ‘individuated’. Let L be a linguistic practice and let a be an object in the purview of L. As Ker¨anen puts it, an ‘account of individuation’ for L would specify ‘the fact of the matter that makes a the object it is, distinct from any other object’ in the purview of L, by ‘providing a unique characterization thereof ’. Ker¨anen suggests that the burden on any advocate of any theory of objects is to ﬁll in the blank in the following: (IND) ∀x(x = a ≡

).

Stewart Shapiro / 133 He states that ‘the ontology of a practice must, by that theory’s lights, consist of items that can be referred to without ambiguity—that is, of properly individuated items.’ There is a question of what resources one is allowed to use in order to discharge this ‘individuation task’. Some philosophers have posited that each object has an ‘haecceity’, which is a property enjoyed by it and it alone. For those philosophers, the individuation task is trivially discharged, by ﬁlling in the blank in (IND) with a term for the haecceity for a: ∀x(x = a ≡ Ha (x) ), or perhaps just ∀x(x = a ≡ x = a). Ker¨anen concedes that if one does accept haecceities, then this unilluminating trick does the individuation job. However, philosophers who eschew haecceities—or philosophers who must eschew haecceities—have to resolve the individuation task some other way. Ker¨anen argues that when it comes to structures with non-trivial automorphisms, the ante rem structuralist leaves himself no room to resolve the individuation task. According to the slogans discussed in Section 1, above, the essential properties of each place of an ante rem structure consist of the relations between that place and other places in the same structure. So one would think that the sum total of the intra-structural relations would have to resolve the individuation task—if anything can. Unfortunately, I said as much myself. In deﬁning the notion of a freestanding structure, I wrote: ‘Every ofﬁce is characterized completely in terms of how its occupant relates to the occupants of the other ofﬁces in the structure’ (1997: 100). This, it seems, was a mistake, if ‘characterize completely’ entails ‘can be identiﬁed uniquely’, as seems plausible. In Ker¨anen’s terminology, it follows that intra-structural relations resolve the identity task. Presumably, a structuralist cannot accept haecceities for places, since a haecceity seems to be a non-structural property. Ker¨anen argues that we ante rem structuralists are committed to the following thesis: for any structure S and any place a of S, (STR) ∀x(x = a ≡ ∀( ∈ Sa ≡ (x) )), where the leftmost quantiﬁer ranges over the places of S, and Sa is the set of ‘relational predicates’ in the language of the structure S that satisfy the given place.

134 / Structure and Identity In this requirement, the ‘relational predicates’ in Sa are those that specify relations of a place in the structure to other places in the structure. For example, in arithmetic, ‘being the fourteenth prime’ is a relational predicate. According to Ker¨anen, the main formal criterion is that Sa contains no predicate whose description essentially involves a singular term denoting either a place of the structure or an object in a system exemplifying the structure. Ker¨anen then shows that (STR) entails the identity of indiscernibles, in the strong form indicated above: every object is uniquely discerned by a formula with one free variable. This, together with the aforementioned facts about isomorphism, entails that no ante rem structure can have non-trivial automorphisms, and so a structuralist account of Euclidean geometry and complex analysis is refuted. Is the individuation task a reasonable demand on a philosophical or scientiﬁc theory? This raises the age old problem of the identity of indiscernibles. I do not see how there can be a non-trivial resolution of the individuation task for any view that holds that the usual array of mathematical objects exist. The reason is that there are too many objects and not enough formulas. Recall the form for resolving the individuation task: (IND) ∀x(x = a ≡

).

For the task to be accomplished in a non-trivial way, the blank would be ﬁlled with a formula that does not have a singular term denoting a. Consider, ﬁrst, real analysis. There are uncountably-many real numbers, but any language used by the philosopher to characterize the real numbers has only countably many formulas. And so only countably many reals can be ‘individuated’ by formulas in the form (IND). Most real numbers cannot be so individuated. To be sure, for any pair a,b of distinct real numbers, there is a formula that applies to a and not to b. Suppose, without loss of generality, that a < b. Then there is a rational number r such that a < r < b. Since the rational number r is a quotient of integers, it can be identiﬁed uniquely, and so there is a formula equivalent to x < r. This formula is true of a but not true of b. However, the individuation task is not to merely distinguish any pair of distinct objects from each other, but to individuate each object. As Ker¨anen puts it, the job is to specify for each object a, ‘the fact of the matter that makes a the object it is, distinct from any other object’, by ‘providing a unique characterization thereof ’.

Stewart Shapiro / 135 We can solve the individuation problem for real analysis if we are allowed to ﬁll the blank in (IND) with a set of formulas. Let a be a real number and let be the collection of all formulas equivalent to x < s, where s is a rational number greater than a, together with the collection of all formulas equivalent to t < x, where t is a rational number less than a. Then, a satisﬁes every member of and, by the completeness property, no other real number does. The scope of this trick is limited. A language with a ﬁnite or countable alphabet has only continuum-many sets of formulas, and so at most continuum-many objects can be individuated by sets of formulas in that language. If a mathematical theory countenances more than continuum-many objects, then the individuation task cannot be resolved in this way. Most of the objects of functional analysis and set theory are still left unindividuated. Against this, one might think that the principle of extensionality resolves the individuation task for set theory. Two sets are identical if they have the same members. So for any given set a, the blank in (IND) would be ﬁlled with a statement that a has the members it does. But ﬁrst we need to individuate the members of a, so that we can refer to them uniquely. How is this accomplished? Starting with the empty set, each hereditarily ﬁnite set can be individuated with a single formula in the language of set theory. Sets of hereditarily ﬁnite sets can then be individuated with sets of formulas, as above. So by the foregoing trick, we can individuate the members of Vω+1 . But this is as far as it goes. A bit after that, we run out of sets of formulas. I suppose that the philosopher has the option of introducing a language with an uncountable alphabet, and then using formulas (or sets of formulas) from that language to ﬁll the blank in (IND). For example, we might include a name for each real number, or a name for each real-valued function, or a name for each set. Then we can satisfy (IND) in a trivial manner. But this just raises a question about how the characters in the language get individuated. In any case, someone who opts for realism in ontology and accepts the individuation task must either introduce haecceities or adopt some equally trivial resolution. Presumably, this route is not available to the ante rem structuralist, due to the slogans about how places-in-structures are characterized. Recall that an in re realist still speaks of the natural number structure, the complex number structure, etc. This Aristotelian philosopher thinks of the structures as existing, but only in the systems that exemplify them. I

136 / Structure and Identity submit that if the ante rem structuralist has an unsurmountable individuation problem, then so does this in re realist. By hypothesis, any system exemplifying the complex number structure will consist of properly individuated items. In the complex plane, for example, the two imaginary unit places are occupied by the ordered pairs 0, 1 , 0, −1 . Presumably, these two pairs of real numbers are themselves nicely individuated. Is there still a fact of the matter as to which place is occupied by which pair? Is it determinate that 0, 1 occupies the i place and not the −i place? We cannot stipulate that 0, 1 occupies the i place until we know which place that is. So, if we take the individuation task seriously, one cannot adopt an Aristotelian view of structures either. The only option left is to reject the existence of structures altogether, a thoroughgoing eliminative view, like those of Benacerraf and Hellman. So what of the eliminative structuralist? Ker¨anen seems to assume that such a philosopher does not have an unresolvable individuation problem. He says that for the eliminativist, the objects that ﬁll the places in structures are themselves individuated independently (of structuralism, presumably). For example, any system exemplifying the complex number structure will have two different objects in the i and −i places. Those objects are individuated somehow in the background theory. To assess whether this is sound, we need to enquire after the background theory. As I noted in the book several times, the background ontology for an eliminative program must be quite large, as large as the ontology of mathematics. So by the foregoing considerations, only a trivial resolution of the individuation task is available. Ker¨anen’s point, conﬁrmed in his contribution to this volume in Chapter 6, is that the eliminative structuralist is free to invoke haecceities, and thus a trivial resolution, while the ante rem structuralist, like Resnik or myself, does not have this option. We are saddled with the impossible task of a non-trivial resolution. For what it is worth, I have never been moved by the identity of indiscernibles. There are two ways that this thesis about identity can be understood, and both are contentious (at best). First, the identity of indiscernibles might be a metaphysical doctrine, stating that the universe of objects has a certain feature, namely that each object—each grain of sand, each piece of dandruff, each electron, each space–time point, each set—can be uniquely characterized, either by a formula of some language or by a property, propositional function, etc. If the individuation is supposed to be done with a formula, then the thesis is that our linguistic or semantic resources are somehow up to this task

Stewart Shapiro / 137 of individuation. But why think this? Why think that our human languages are capable of uniquely picking out each and every object in the universe, including the abstract objects? If the individuation is to be done with properties or propositional functions, regarded as objective and independent of language, then why think that the intensional realm of properties and propositional functions matches up to the realm of objects in the relevant way? What reason is there to think that the realm of properties and propositional functions is up to the task of individuating each and every object? Unless, of course, there are haecceities, in which case the identity of indiscernibles is trivially true, and not very interesting. If there are not enough properties to individuate each object, then there will be distinct objects that share all of their properties. These approaches to the identity of indiscernibles seem to presuppose that ‘object’ is some sort of metaphysical primitive. Our philosophy aims at determining the facts about these objects, and may succeed or fail in this task. The other orientation toward the identity of indiscernibles comes from a more Quinean approach to ontology (see Kraut 1980). It is an inversion of sorts. The idea is that the universe does not come pre-packaged into objects. Which objects there are is a function of our overall conceptual scheme. If, by using the totality of our conceptual resources, we cannot distinguish a from b—if everything true of a is true of b—then we should identify a and b. The proposal is that if a and b are truly indistinguishable, then we have a single object before us, not two. This seems to be a principle for restructuring the ship of Neurath, the primary job left to philosophy from this perspective. This Quine–Kraut proposal is of a piece with the criteria of simplicity and Occam’s razor. Notice that this requirement, as stated, is not as strong as the identity of indiscernibles. It does not say that (in a properly regimented language) we should be able to uniquely characterize each object. The Quine–Kraut proposal only says that we can distinguish any pair of distinct objects. As above, this much can be accomplished for the real numbers. However, we would still have to reject the places in the ﬁnite cardinal structures, since we have left ourselves no resources to distinguish the places in this structure. Perhaps this is an acceptable loss. Any work that those structures do can be performed by the corresponding ﬁnite ordinal structures, and, as above, the places of ﬁnite ordinal structures are nicely individuated internally. When talking about cardinality, we would just ignore the ordering in the ordinal structures.

138 / Structure and Identity Nevertheless, I submit that the Quine–Kraut requirement is not warranted for mathematics. The fact is that we have powerful and eminently useful theories which violate the distinguishability requirement. Euclidean space, complex analysis, and set theory are cases in point. Consider geometry. As we have seen, there is no non-trivial way to individuate or distinguish the points, and yet it is a theorem of Euclidean geometry that there are lots of different points. Since it is not an option to identify every point in Euclidean space—so that there is just one point after all—the only way to satisfy the Quine–Kraut requirement would be to add resources to Euclidean geometry to distinguish the points (by breaking the non-trivial automorphisms). We could, for example, drop the theory altogether and replace it with analytic geometry, deﬁned over n-tuples of real numbers. The talk of synthetic Euclidean geometry would consist of simply ignoring the analytic parts of the structure. Similarly, we could do without the complex number structure altogether, and make do with the system of ordered pairs of real numbers instead. Notice, however, that this maneuver to salvage the Quine–Kraut requirement demands revisions in established mathematical and scientiﬁc theories, a rather unQuinean conclusion. As we have seen, the case of set theory is even worse. Although there are no non-trivial automorphisms of the set-theoretic hierarchy, cardinality considerations show that the relevant distinctions cannot be made, let alone the relevant individuations. To say the least, invoking haecceities would also be a very unQuinean move. We might allow distinctions involving resources other than formulas with one free variable. With complex analysis, we can distinguish i from −i as follows: the pair i, −i satisﬁes the formula x + y = 0, and the pair i, i does not. If i were identical to −i, then the ordered pair i, −i would be the same as the pair i, i , and so these pairs would satisfy the same formulas. Of course, we still have no way of telling, among i and −i, which is which, but maybe we do not need that ability. In Euclidean geometry, if a and b are distinct points, then the pair a, b satisﬁes a formula that a and b determine only one line, while the pair a, a does not satisfy this formula. We can apply the technique generally, if we can countenance minor revisions. If S is any system of objects, let S be the same system with another binary relation ≺, which is a linear order (or a well-order). Then if a and b are distinct, then the pair a, b satisﬁes the formula (x ≺ y ∨ y ≺ x) and the pair a, a does not. We add the new relation symbol just to discharge our task of distinguishing the items in our ontology.

Stewart Shapiro / 139 Ker¨anen raises the possibility of solving the individuation problem using formulas with more than one free variable, and he shows that this does not work (due to the fact that isomorphic systems are equivalent). Here we use formulas with two free variables to distinguish two objects a, b, by showing that a, a is distinct from a, b . The isomorphism theorem does not undermine this. If the identity sign is a primitive of the language, and we have singular terms denoting each object, then we can trivially individuate each object. In complex analysis, i satisﬁes x = i and nothing else does. If this trivial resolution is allowed, then the Quine–Kraut requirement is no requirement at all. Every theory meets it. In set theory, if a and b are distinct sets, then we have that ∃x( (x ∈ a & x ∈ / b) ∨ (x ∈ / b & x ∈ a) ). Does this manage to distinguish a and b? Alternately, we can let c be a name of such an x, and note that the sets a, b are distinguished by (c ∈ a & c ∈ / b) ∨ (c ∈ / b & c ∈ a). If the ﬁrst disjunct holds, then a satisﬁes c ∈ x and b does not; if the second disjunct holds then a satisﬁes c ∈ / a and b does not. This last assumes that we can use names for sets in making the distinctions. The rejection (or trivialization) of the identity of indiscernibles is quite general. The foregoing arguments are independent of the theses concerning structuralism. Anything I said that suggests that I am committed to a non-trivial resolution of the individuation task must be withdrawn. As noted above, Ker¨anen correctly derives the identity of indiscernibles from the above thesis (STR). Thus I agree that (STR) entails that there is a non-trivial resolution of the individuation task. However, if ante rem structuralism is to be a full philosophy of mathematics, then there should be ante rem structures of Euclidean space, the complex numbers, and the set-theoretic hierarchy. So for us ante rem structuralists, there can be no non-trivial resolution of the individuation task. Thus, we must reject (STR). But why think that Resnik and I are committed to it? One way around (STR) would be to introduce haecceities for places in structures. So the two square roots of −1 have different haecceities, and each point of Euclidean space has its own haecceity. Notice that this move does not resolve the issue of the previous section. On the present plan, there is a haecceity for the natural number 2 and there is a haecceity for the real number 2. But this, by itself, does not say (one way or another) whether those two haecceities, and thus those two places, are identical or distinct from each other.

140 / Structure and Identity Once again, however, haecceities run against the slogans like ‘the essence of each mathematical object consists of its relations to other places in the same structure’. If a given mathematical object has a haecceity, that haecceity is essential to that object, and said haecceity does not concern its relations to other objects in the same structure. In Section 1, I broached the possibility of dropping the slogans, in favor of a more detailed articulation of the theses of ante rem structuralism. But let us maintain the slogans for the sake of this argument, and drop haecceities. Let a be a complex number b + ci, where b and c are real numbers and c = 0. It would follow that the set Sa of relational properties enjoyed by a includes all of its essential properties. Let a be b − ci. Then a satisﬁes every member of Sa . Indeed, according to the slogans, the essential/relational properties of a are the same as those of a. Yet we know that a is distinct from a. So it seems that if we maintain the slogan about essences, we have two distinct objects having all of their essential properties in common. This does not strike me as problematic. If we do not invoke haecceities, then why should we think that distinct objects always have distinct essences? Why think that two distinct objects cannot have all of their essential properties in common? Thus, I do not see how the ante rem structuralist is committed to a crucial premise for the identity of indiscernibles, and for Ker¨anen’s principle (STR). There is, however, at least one embarrassing remark in my book on this matter: ‘Quine’s thesis is that within a given theory, language, or framework, there should be deﬁnite criteria for identity among its objects. There is no reason for structuralism to be the single exception to this’ (1997: 92). The context of that remark was Resnik’s claim that there is sometimes no fact of the matter concerning when two structures are the same or different. What I meant—or think I meant, or should have meant—was that if we are to develop a theory of structures, then there must be a determinate identity relation between structures. Given Quine’s slogan, which I endorsed, there is no room for a view that takes structures seriously as objects and leaves the identity relation between structures indeterminate. Surely the same goes for places within a given structure (independently of the issues from the previous section). When it comes to mathematical objects—places within a given structure—identity must be determinate. Given two places in the same structure, it is determinate whether they are identical or distinct. Either a = b or a = b. However, I regret the use of the phrase ‘criteria for identity’ in the remark, since it seems to suggest a commitment to a non-trivial resolution of

Stewart Shapiro / 141 Ker¨anen’s individuation task, and with the slogans of ante rem structuralism, to (STR) and the problems that engenders. If I indeed meant that structuralism requires a criterion for individuation, then I hereby take the remark back. Hellman (2001: 193–4) raises an extension of the foregoing complaint against ante rem structuralism. He suggests that the places of an ante rem structure must be determined solely by the relations of the structure, and this, he suggests, is incoherent: . . . on Shapiro’s view . . . numerals denote the ‘places’ in a unique, archetypical structure answering to what all progressions have in common . . . But the ‘places’ of the ante rem archetype (call it N, ϕ, 1 ) are entirely determined by the successor function ϕ, and derivative from it in the sense of being identiﬁed merely as the terms of the ordering induced by ϕ. Now in the case of an in re structure, we understand a particular successor relation in the ordinary way as arising from the given relata, reﬂecting, e.g. an arrangement of some sort . . . But if the relata are not already given but depend for their very identity upon a given ordering, what content is there to talk of ‘the ordering’? What can ‘succession’ mean, if we are abstracting from all in re cases and if we can’t even speak of relata without making reference to the relation intended? . . . This, I submit, is a vicious circularity: in a nutshell, to understand the relata, we must be given the relation, but to understand the relation, we must already have access to the relata.

Hellman paraphrases the objection as follows: ‘a purely structuralist view of objects serving as relata cannot be sustained without departing substantially from a structuralist view of structural relations, in effect, of structures themselves’ (2001: 195 n). I do not know what to make of the metaphors of the successor relation of the natural number structure being ‘given’, and our ‘having access’ to the places of the structure. Presumably, Hellman does not mean to saddle the ante rem structuralist with a Platonist epistemology that posits some sort of direct access to structures and their places and relations. From the more literal parts of the passage, Hellman means that the ante rem structuralist is committed to a thesis that the places of a structure S are somehow ‘determined’ or ‘identiﬁed’ by the relations of S. Metaphysically, the relations are prior to the places, and the relations ﬁx the places. This cannot be correct. Consider, once again, the ﬁnite cardinal structures, which are degenerate cases of ante rem structures. The four places of the cardinal four structure, for example, cannot be ‘determined’ by the relations of the structure, for the simple reason that the structure has no relations

142 / Structure and Identity (other than identity). I did not mean to give the impression that the relations of a structure are prior to its places. To be frank, I had not considered this bit of metaphysics before, but the view must be that a structure is determined by its places and its relations. Neither is prior to the other. In the introduction, I did write that ‘Structures are prior to places in the same sense that any organization is prior to the ofﬁces that constitute it. The natural number structure is prior to ‘‘6’’, just as ‘‘baseball defense’’ is prior to ‘‘shortstop’’ or ‘‘U.S. Government’’ is prior to ‘‘Vice President’’ ’ (1997: 9). Later in the book, I used the word ‘objects’ instead of ‘places’: ‘The structure is prior to the mathematical objects it contains, just as any organization is prior to the ofﬁces that constitute it’ (1997: 78). With these passages, I said (or meant to say) that a given structure is ontologically or metaphysically prior to its places. This is the central ontological thesis of ante rem structuralism. I did not intend the passage to imply that the relations of each structure are prior to its places (nor that its places are prior to its relations). After broaching this objection, Hellman (2001: 195) asks ‘In what sense can ante rem structuralism claim to be distinguishable from straightforward objects platonism? Such platonism, after all, can readily speak of structures and isomorphisms all it likes, once given its abstract archetypes.’ As above, ante rem structuralism is an instance of the view I call ‘realism-in-ontology’. Admittedly, it might be a variant of traditional platonism, at least as far as ontology is concerned (see Shapiro 1997: 72–4). Perhaps it is just a difference of emphasis. Hellman (2001: 195–6) argues that even if the structuralist manages to introduce a ‘privileged successor-type relation on an ante rem structure’, he is then faced with a Benacerraf-type problem of multiple instantiations: For if we suppose, for the sake of argument, that an ante rem progression, N, ϕ, 1 is somehow attained, we immediately see that indeﬁnitely many others, explicitly deﬁnable in terms of this one, qualify equally well as candidates to serve as the referents of our numerals. We need only permute . . . the ﬁrst two elements [of N], deﬁning ϕ ∗ from ϕ in the obvious way . . . [T]he structure N, ϕ ∗ , 1∗ is . . . able to serve as ante rem archetypical progression every bit as well as N, ϕ, 1 [n.b., 1∗ is 2]. Indeed, on what conceivable ground are we able to say that one and not the other is ‘the result of Dedekind abstraction’? It makes no sense to say that 1 is ‘really ﬁrst’ for ‘ﬁrst’ has no meaning except relative to a successor-type function. Relative to ϕ ∗ , 1∗ [i.e. 2], not 1, is ‘ﬁrst’ . . . Shapiro . . . does not avoid the multiplicity leading to Benacerraf ’s argument, now directed at ante rem structures. That any number of such structures

Stewart Shapiro / 143 are equally well qualiﬁed in all respects to count as the result of Dedekind abstraction seems to undermine that very notion.

I do not know what to make of this objection. In the terminology of my book, N, ϕ ∗ , 1∗ is not a structure. Rather, it is a system consisting of the places of the natural number structure N, ϕ, 1 under the deﬁned relations. This system exempliﬁes the natural number structure, as do countless other systems. I cannot give a knock-down argument against the claim that the numerals of ordinary arithmetic refer to the system N, ϕ ∗ , 1∗ , rather than the (original) structure. Indeed, I have no argument that the numerals of ordinary arithmetic do not refer to the Von Neumann ﬁnite ordinals or the Zermelo numerals. The underlying theme of the project (and indeed of Hellman’s own modal eliminative structuralism) is the now commonplace observation that the language of arithmetic only determines its ontology up to isomorphism. My proposal is that we interpret the ﬁrst-order variables of arithmetic as ranging over the places of the ante rem structure. In Hellman’s terms, I propose that we take the ante rem structure as the result of Dedekind abstraction. Given that isomorphic systems are equivalent, I cannot rule out alternate interpretations. I also cannot rule out the hypothesis that there is more than one ante rem structure isomorphic to the natural numbers (i.e., N, ϕ, 1 ). Hellman is correct that an ontological realist cannot simply stipulate that there is at most one structure for each isomorphism type. I also cannot rule out the possibility that different structures share their elements (contra the proposal toward the end of the previous section). So perhaps the realm of ante rem structures contains both the postulated N, ϕ, 1 and N, ϕ ∗ , 1∗ . Unless simplicity or Ockham’s razor can be invoked, but I do not want to broach that issue right now. It is best to remain modest. In responding to the above objections, I do not claim to have made plain, once and for all, the notion of an ante rem structure. Parsons (1990: 335) wrote that the . . . absence of notions whose non-formal properties really matter . . . makes mathematical objects on the structuralist view continue to seem elusive, and encourages the belief that there is some scandal to human reason in the idea that there are such objects. My claim is that something close to the conception of objects of this kind, already encouraged by the modern development of arithmetic, geometry, and algebra, is forced on us by higher set theory.

144 / Structure and Identity I only hope that I have made ante rem structuralism sufﬁciently clear and interesting for the discussion to continue.

References Armstrong, D. M. (1989), Universals: An Opinionated Introduction (Boulder, Colo.: Westview Press). Benacerraf, P. (1965), ‘What Numbers Could Not Be’, Philosophical Review, 74: 47–73. Blackburn, S. (1994), The Oxford Dictionary of Philosophy (Oxford: OUP). Burgess, J. (1999), Review of Shapiro (1997), Notre Dame Journal of Formal Logic, 40: 283–91. Dedekind, R. (1872), Stetigkeit und irrationale Zahlen, (Brunswick, Vieweg); trans. as Continuity and Irrational Numbers, in Essays on the Theory of Numbers, ed. W. W. Beman (New York: Dover Press, 1963), 1–27. (1888), Was Sind und was Sollen die Zahlen? (Brunswick, Vieweg); trans. as The Nature and Meaning of Numbers, in Essays on the Theory of Numbers, ed. W. W. Beman (New York, Dover Press, 1963), 31–115. (1932), Gesammelte Mathematische Werke, iii, ed. R. Fricke, E. Noether, and O. Ore (Brunswick: Vieweg). Dummett, M. (1991), Frege: Philosophy of Mathematics (Cambridge, Mass.: Harvard University Press). Evans, G. (1978), ‘Can there be vague objects’, Analysis 38, 208. Frege, G. (1884), Die Grundlagen der Arithmetik (Breslau: Koebner); trans. J. Austin as The Foundations of Arithmetic (2nd edn., New York: Harper, 1960). Hellman, G. (1989), Mathematics Without Numbers (Oxford: OUP). (1999), Review of Shapiro (1997), Journal of Symbolic Logic, 64: 923–6. (2001), ‘Three Varieties of Mathematical Structuralism’, Philosophia Mathematica, 9/3: 184–211. Kastin, J. (1998), ‘Shapiro’s Mathematical Structuralism’ (manuscript). Ker a¨ nen, J. (1999), ‘The Identity Problem: Realist Structuralism is Dead’, read at the Paciﬁc Meeting of the American Philosophical Association, Mar. 1999. (2001), ‘The Identity Problem for Realist Structuralism’, Philosophia Mathematica, 9/3: 308–30. Kraut, R. (1980), ‘Indiscernibility and Ontology’, Synthese, 44: 113–35. Kreisel, G. (1967), ‘Informal Rigour and Completeness Proofs’, Problems in the Philosophy of Mathematics, ed. I. Lakatos (Amsterdam: North Holland), 138–86. MacBride, F. (1998), ‘Where are Particulars and Individuals?’, Dialectica, 52: 203–27. McGee, Vann (1997), ‘How We Learn Mathematical Language’, Philosophical Review, 106: 35–68.

Stewart Shapiro / 145 Maddy, P. (1990), Realism in Mathematics (Oxford: OUP). Parsons, C. (1990), ‘The Structuralist View of Mathematical Objects’, Synthese, 84: 303–46. Quine, W. V. O. (1992), ‘Structure and Nature’, Journal of Philosophy, 89: 5–9. Resnik, M. (1981), ‘Mathematics as a Science of Patterns: Ontology and Reference’, Noˆus, 15: 529–50. (1997), Mathematics as a Science of Patterns (Oxford: OUP). Russell, B. (1903), The Principles of Mathematics (London: Allen & Unwin). Shapiro, S. (1994), ‘Mathematics and Philosophy of Mathematics’, Philosophia Mathematica, 2/3: 148–60. (1997), Philosophy of Mathematics: Structure and Ontology (New York: OUP).

6 The Identity Problem for Realist Structuralism II: A Reply to Shapiro Jukka Ker¨anen

In his Philosophy of Mathematics: Structure and Ontology, Stewart Shapiro gave what is perhaps the most comprehensive presentation of realist structuralism to date. In my ‘The Identity Problem for Realist Structuralism’,1 I argued that any theory with the same general shape as that of Shapiro’s will either fail to present a tenable account of mathematical ontology or else must betray the very motivations from which it stems. While the argument in that paper was intended to criticize realist structuralism in general, Shapiro is right to identify his theory as a primary target and in ‘Structure and Identity’, Chapter 5 in this volume, he comes to its defense. Shapiro’s paper has two parts, the ﬁrst being a reply to certain criticisms by Jonathan Kastin, the second being a reply to me. In this paper, I will give a detailed reaction to the latter. I will argue that Shapiro has not yet provided a satisfactory solution to the ‘identity problem’ for places in structures. In fact, I will conclude by suggesting that any form of realism is either open to the identity problem or else faces the well-known Benacerraﬁan problem of ‘multiple reductions’ in a new guise. Special thanks are due to Ken Manders and John McDowell. 1 Ker¨anen (2001).

Jukka Ker¨anen / 147

1. Identity and Individuation The argument in ‘The Identity Problem’ had two main components. On the one hand, I showed that mathematical structures typically contain places that are indiscernible if individuated solely by the relations they have to one another. On the other hand, I argued that any account of place-identity available to the realist structuralist entails that relationally indiscernible places are identical. Since she maintains that mathematical singular terms denote places in structures, the realist structuralist is therefore committed to saying that, for example, 1 = −1 in the group of integers (Z, +). I called this predicament the identity problem. The claim that constituted the ﬁrst component of my argument is, I should think, generally agreed to be true; it is the second component that is now under dispute. The core of the second component consisted of establishing two claims. First, I argued that any theory of ontology must be able to furnish an account of identity for the objects of which it is a theory. I proposed that the following is a natural way of furnishing such an account. Given a language L and given two quantiﬁers ‘∀x’ and ‘∀y’ ranging over the domain of discourse of L, just complete the blank in the following identity schema: (IS) ∀x∀y(x = y ⇔

).

When suitably completed, the result is an account that speciﬁes the circumstances under which two singular terms of L denote the same object and, more generally, speciﬁes what governs the identity of objects in the domain of discourse of L. The second claim of the second component, perhaps the most contentious aspect of my argument overall, was that the realist structuralist can only employ intra-structural relational properties in completing the blank in (IS). Let φ(x) indicate that x has the (possibly complex) property φ. Given any system S, and given two quantiﬁers ‘∀x’ and ‘∀y’ ranging over the places of the structure S of S, I argued that the realist structuralist should complete the blank in (IS) as follows: (STR) ∀x∀y(x = y ⇔ ∀φ(φ ∈ ⇒ (φ(x) ⇔ φ(y) ) ) ), where ‘∀φ’ is taken to be ranging over the relational properties in S. I further argued that the set should only contain relations that can be speciﬁed without making explicit reference to particular elements in S. In this way, each

148 / The Identity Problem: Reply to Shapiro place of S is individuated by the ‘purely structural’ relational properties its occupants have to the occupants of the other places of S.2 Thus, whenever two distinct elements in S that have the same intra-systemic relational properties, according to (STR) they occupy the same place of S. While this conclusion should seem prima facie plausible, in many cases it leads into absurdities. It entails, for example, that in the additive group of integers (Z, +), the 1-place is the −1-place. Since the realist structuralist takes mathematical singular terms to denote places in structures, she would now have to say that 1 = −1. The same problem arises for any structure whose instantiating systems have non-trivial automorphisms. I should note that there are two minor differences between the published version of my argument outlined above and the earlier version Shapiro is addressing in his ‘Structure and Identity’. First, in the earlier version, I began by proposing that an account of identity could be furnished by completing the blank in the following schema: (IS*) ∀x(x = a ⇔

).

I derived the crucial principle (STR) from this schema and pointed out that it is a form of the ‘Leibniz Principle’ of the Identity of Indiscernibles for places in structures. In the later, published version, I began by proposing that an account of identity could be furnished by completing the blank in the schema (IS) above. I showed that (STR) is simply a particular way of completing this schema and, again, pointed out that it is a form of the Leibniz Principle. I should emphasize that in either case, the Leibniz Principle itself does not carry any direct argumentative weight. I will return to this issue in Section 3, below. Let us brieﬂy clarify the relationship between (IS) and (IS*). Given two quantiﬁers ‘∀x’ and ‘∀y’ ranging over the domain of discourse of L, it is easy to show that ∀x∀y(x = y ⇔ ∀φ(φ ∈ ⇒ (φ(x) ⇔ φ(y) ) ) ), and For all singular terms ‘a’ in L, ∀x(x = a ⇔ ∀φ(φ ∈ ⇒ (φ(x) ⇔ φ(a) ) ) ), 2

I framed my argument in this way in order to avoid the question of the individuation of the relations themselves. In any case, the outcome resembles very closely what Shapiro tells us about places and their occupants, and as long as satisﬁes my restriction, there is no problem in viewing the individuating relations as relations in S also.

Jukka Ker¨anen / 149 are interderivable in ordinary ﬁrst-order logic, provided every element in the domain of discourse of L has a singular term denoting it. However, most mathematical languages do not satisfy this proviso and hence, I ended up choosing (IS) as my starting point. After all, we would like an account of identity for L to apply to all objects in the domain of L, not just to the ones with singular terms denoting them. Second, in the earlier version, I spoke of predicates individuating objects. I am indebted to Shapiro for pointing out the dubiousness of this approach; accordingly, the later, published version of the argument is formulated in terms of properties. Still, as we shall presently see, his concerns here are different from mine. I will return to this issue in Section 2, below. Shapiro structures his response to my argument around the distinction between what he calls a ‘trivial’ and ‘non-trivial’ way of completing the blank in the identity schema.3 The trivial way of completing the blank amounts to saying that each object is individuated by its ‘primitive thisness’, or haecceity, and that there is simply nothing further to say. The non-trivial way of completing the blank requires us to specify which general properties4 (or predicates) individuate objects. With this distinction in view, Shapiro’s response to my argument has four principal components. First, he argues that to furnish a non-trivial way of completing the blank is too much to ask of any theory of (mathematical) ontology: our (mathematical) languages simply do not have the resources for uniquely characterizing every object in every domain of (mathematical) objects. I take it that this is his main argument; henceforth, I shall call it the ‘trivializing’ objection. Second, Shapiro points out that if realist structuralism falls prey to the identity problem, then so does its chief rival, eliminative structuralism. Third, he distinguishes two ways of construing the Leibniz Principle and argues that, ultimately, neither one can serve my argumentative purposes. Finally, he attempts to ‘analyze away’ the apparent commitment he made in Philosophy of Mathematics to furnishing a non-trivial account of identity for places in structures. I will address each of these components in turn. 3

In the earlier version of my argument, I called the challenge of completing the blank in (IS) the ‘individuation task’; Shapiro follows me in doing so. I shall avoid using this phrase here since, as I shall explain below, there is a sense in which it is misleading. 4 A general property is one that can be had by several distinct objects, as opposed to a property that can be had by only one object. For example, being yellow is a general property, whereas being identical to a is not.

150 / The Identity Problem: Reply to Shapiro

2. The Trivializing Objection Shapiro tells us that he does not see ‘how there can be a non-trivial resolution of the individuation task for any view that holds that the usual array of mathematical objects exists’. The reason for this inability is that ‘there are too many objects and not enough formulas’. For example, ‘there are uncountablymany real numbers, but any language used by the philosopher to characterize the real numbers has only countably many formulas’. There is a certain ‘trick’ that would allow us to individuate the real numbers but, as Shapiro notes, ‘the scope of this trick is limited’. In the end, ‘most of the objects of functional analysis and set theory are left unindividuated’ (p. 135, above). Shapiro concludes that my challenge for realist structuralism is one that no theory of mathematical ontology could meet except trivially, by adopting Haecceitism. It seems to me that Shapiro’s argument conﬂates two distinct theoretical tasks. In order to see this, we must get clear on what an ‘account of identity’ is, and what it is not supposed to accomplish. In ‘The Identity Problem’, I said that: There are two basic requirements. First, given what we have just said, an account of identity for the domain of discourse of L should specify the circumstances under which ‘a = b’ is true in L by specifying what sort of facts govern the identity of the objects referred to in L. In other words, an account of identity should specify the circumstances under which ‘a’ and ‘b’ denote the same object. Second, although the ﬁrst requirement is phrased in terms of singular terms, an account of identity for the domain of discourse of L should obviously apply to all objects in that domain, not just to the ones with singular terms denoting them. (2001: 312)

Ofﬁcially, then, I did not want to claim that in order to furnish an account of identity for a given domain one must be able to explicitly specify a unique characterization of each and every object in it. All I wanted to claim was that one must complete the identity schema in a way that indicates schematically what governs the identity of objects in that domain. For example, according to Zermelo–Frankel set theory, the identity of any set is governed by which sets are its members. According to Haecceitism, the identity of any object is governed by its haecceity. Introducing a bit of terminology may help make the point clearer. First, an account of identity for a language L is simply a universally quantiﬁed formula that indicates what governs the identity of objects in the domain of discourse of L. Second, a condition of identity for a given object a in that domain is an instance

Jukka Ker¨anen / 151 of some account of identity for L. Finally, sometimes we can explicitly display what the condition of identity for a given object a amounts to. For example, ﬁrst, the account of identity for the language of Zermelo–Frankel set theory is simply the Axiom of Extensionality: ∀x∀y(x = y ⇔ ∀z(z ∈ x ⇔ z ∈ y) ). Second, the condition of identity for the set {∅} in Zermelo–Frankel set theory is ∀x(x = {∅} ⇔ ∀z(z ∈ x ⇔ z ∈ {∅}) ). Finally, third, for small sets such as {∅} we can explicitly display what its condition of identity amounts to: ∀x(x = {∅} ⇔ (∅ ∈ x ∧ ∀z(z = ∅ ⇒ z ∈ / x) ) ). This should make it clear that in order to furnish an account of identity for L, it is certainly not required that we be able to explicitly display what the condition of identity amounts to for every object in its domain. Neither is it required that we be able to write down the condition of identity for every object. Note, indeed, that if this was required, we could not furnish even the trivial account of identity for the real numbers, contra Shapiro. As Shapiro himself emphasizes, our mathematical languages do not have uncountably many singular terms and hence, there are not enough instances of the formula Hx , ‘the haecceity of x’, for all real numbers. Shapiro’s prose suggests that he might now raise the following objection. Our inability to write down the condition of identity for an object means that that object is somehow ‘left unindividuated’ (p. 135, above). However, I am unable to make sense of this phrase in the present context. For the purposes of my argument in ‘The Individuation Problem’, I assumed that realism about mathematical ontology is true. Given this assumption, there seems to be little sense in saying that some objects in the domain of discourse of L are ‘not individuated’. Such objects exist quite independently of our cognitive activities; therefore, of course they are individuated regardless of our cognitive activities. Each one is the object it is, distinct from any other, and (so I argued) there is some property that makes it so, whether or not we are able to specify what that property is. This is the reason why I now feel that the phrase ‘individuation task’ is misleading in the realist context. After all, it is not as if we are doing the ‘individuating’ for the objects in a given domain generally, let alone for each one individually. That we are unable to specify explicitly the property that individuates a given object does not mean that that object is somehow ‘left unindividuated’.

152 / The Identity Problem: Reply to Shapiro I should note that the foregoing line of thought relies on a conception of ontology that is of a piece with what Shapiro calls the ‘metaphysical’ way of construing the Leibniz Principle. The idea is that the universe comes ‘prepackaged’ into objects, all of which are uniquely individuated by some (possibly complex) property. This conception of ontology is to be contrasted with what Shapiro calls the ‘Quinean’ conception according to which the universe does not come pre-packaged into objects. ‘Which objects there are is a function of our overall conceptual scheme.’ Thus, ‘if, by using the totality of our conceptual resources, we cannot distinguish a from b [. . .] then we should identify a and b’ (p. 137, above). It is not entirely clear to me which conception of ontology Shapiro favors, although his tentative admission that the identity of places might be governed by haecceities would suggest he is not a Quinean in the sense relevant here. Whatever the case, it seems to me that Shapiro’s ‘trivializing’ objection is equally faulty under the Quinean conception. For, surely even the Quinean will agree that the fact that we do not have enough singular terms to pick out each real number individually is compatible with the idea that real numbers are properly ‘individuated’.5 Even if Shapiro were to go Quinean, he would still have to recognize that in order to have a domain of properly individuated objects in view, it is not necessary to be able to write down the condition of identity for each one. Having an account of identity will sufﬁce. I submit, then, that my argument is not open to Shapiro’s ‘trivializing’ objection. It is perfectly compatible with saying, for example, that the identity of objects in Zermelo–Frankel set theory is governed by the Axiom of Extensionality and thus, that the Axiom of Extensionality constitutes a nontrivial account of identity. Even though we cannot write down the condition of identity for each and every set, the Axiom of Extensionality in no way commits us to denying that sets we ordinarily take to be distinct really are distinct. In contrast, the realist structuralist’s account of identity (STR) commits her to saying that some objects we (suppose we) know to be distinct are in fact identical. Again, this has nothing to do with our inability to write down the condition of identity for every place in every structure. Rather, (STR) entails that in some cases the conditions of identity, were we to write them down, would be identical even though we (suppose we) know that the objects in question are distinct. The realist structuralist is in a peculiarly uncomfortable situation. 5 Indeed, as Shapiro notes, otherwise the Quinean conception of ontology is actually refuted. See p. 138, above.

Jukka Ker¨anen / 153

3. The Leibniz Principle I will now make six brief remarks concerning the skepticism Shapiro expresses about the Leibniz Principle. As I noted above, Shapiro distinguishes two ways of construing this principle, the ‘metaphysical’ and the ‘Quinean’. I will consider each one in turn. (1) Against the metaphysical conception, Shapiro considers the status of the Leibniz principle relative to two possibilities. The ﬁrst possibility is that it is our ‘linguistic or semantic resources’ that govern the identity of objects. But, ‘why think that our human languages are capable of uniquely picking out each and every object in the universe, including abstract objects?’ (p. 137, above). Two remarks. First, again, this query seems to conﬂate the task of furnishing and account of identity for a domain on the one hand, and the task of writing down a condition of identity for each object in a domain on the other. Second, I do not see how our ‘linguistic or semantic resources’ could govern the identity of objects under the metaphysical conception of ontology since according to that conception, the universe comes ‘pre-packaged’ into objects. The second possibility is that it is ‘properties or propositional functions, regarded as objective and independent of language’ that govern the identity of objects. But, ‘why think that the intensional realm of properties and propositional functions matches up to the realm of objects in the relevant way? What reason is there to think that the realm of properties and propositional functions is up to the task of individuating each and every object?’ (p. 137, above). Three remarks. First, although I do not know how to argue the point in general, it seems to me that in the case of pure sets it is obvious that there are enough properties to individuate them. Given any set s at any level in the hierarchy, there is a well-deﬁned collection of sets that belong to s, and only to s, by the Axiom of Extensionality. It seems entirely natural to say that for any set r, there is a unique property r belongs to that is enjoyed by all and only those sets to which r belongs. Hence, given any set s, there is a well-deﬁned collection of properties that s, and only s, has. This shows that there are enough properties to individuate every pure set and hence, enough properties to individuate enough objects to serve as the domain of any mathematical theory. If there is a problem here, it is not one of cardinality. Second, no matter what one makes of the strength of the foregoing argument, Shapiro would still have to produce some reason against thinking that the

154 / The Identity Problem: Reply to Shapiro two realms match up in the relevant way. After all, the majority of traditional metaphysics supposes that the identity of objects is governed by properties. Third, as Shapiro acknowledges, there does not seem to be any quick, prima facie objection to the ‘trivial’ possibility that each object at least has the unique property being identical to itself. Immediately after the query quoted above, he adds: ‘Unless, of course, one believes in haecceities, in which case the identity of indiscernibles is trivially true, and not very interesting’ (p. 137, above). All I really need to say here is that as far as the argument in ‘The Identity Problem’ is concerned, it is beside the point whether the Leibniz principle is interesting; the only relevant issue is whether it is true. According to the haecceity account, it is. In sum, there are two possibilities here. One, if the ‘two realms’ match up in the ‘non-trivial’ way, I can run my argument in ‘The Identity Problem’ unchanged. Two, if the ‘two realms’ match up only in the ‘trivial’ way, each object being individuated by its haecceity, I will argue in Section 5 that the realist structuralism theory of ontology is confronted with a new problem. (2) Against the Quinean construal of the Leibniz Principle, Shapiro points out that the ontologies of many mathematical theories ﬂy in its face. For example, the domain of functional analysis has many more objects than we could ever hope to individuate, or even pair-wise distinguish, by using our conceptual resources. One remark. Again, in order to furnish an account of identity for a given domain, it is not required that we be able to ‘uniquely pick out’ each object in that domain. If the Quinean takes it, as I think she should, that an account of identity is all we really need here,6 this easy refutation of the Quinean picture is not available. I submit, then, that Shapiro has not yet given any convincing reason to think that the Leibniz principle should be rejected in the case of mathematical objects. My own attitude about the Leibniz principle is this. I believe that given any domain of objects, there is some fact that metaphysically underwrites the distinctness of any two distinct objects in that domain.7 I will argue in Section 4, below, that that feature must be analyzable in terms of properties of individual objects. Hence, I believe that given any domain of objects, some formula of the form ∀x∀y(x = y ⇔ ∀φ(φ ∈ ⇒ (φ(x) ⇔ φ(y) ) ) ), 6 7

I think that Quine himself would ﬁnd this perfectly acceptable. See e.g. Quine (1992: 23 ff.). See Ker¨anen (2001: 312–13).

Jukka Ker¨anen / 155 is true of that domain—a form of the Leibniz principle. I would like to emphasize that to maintain this is not so metaphysically contentious as one might suppose. The point is that the old disputes over the Leibniz principle concern the status of robust forms thereof. It is indeed contentious whether there is always some true formula of this form if it is required that the set contains general properties. However, the ‘trivial’ accounts of identity do not require this. They merely require that each object has the unique property being identical to itself, and this much virtually any opponent of the Leibniz principle would allow—including, as we have seen, Shapiro himself. Since I am placing no apriori constraints on the contents of the set , my position should seem acceptable to them.

4. The ‘Trivial’ Accounts of Identity I will now comment on Shapiro’s efforts to ‘analyze away’ the apparent commitment he made in Philosophy of Mathematics to furnishing a non-trivial account of identity for places in structures. My overall strategy will be to argue, ﬁrst, that Shapiro has no choice but to adopt Haecceitism and, second, that doing so will vitiate realist structuralism. Shapiro’s principal motivation for wanting to analyze away the apparent commitment to furnishing a non-trivial account of identity is that he thinks such an account cannot be furnished for domains of mathematical theories. However, as I have tried to show in the foregoing two sections, Shapiro’s argumentrestson a conﬂation of twovery differenttheoretical tasks: furnishing an account of identity on the one hand, and specifying a condition of identity on the other hand. I maintain, accordingly, that his adoption of a trivial account of identity for places in structures has no serious motivation. Shapiro could nevertheless stand his ground and maintain that even if his ‘trivializing’ objection is set aside, he can avoid the Identity Problem by showing that the realist structuralist does not have to furnish a non-trivial account of identity. I will now consider the two ways in which Shapiro suggests this could be accomplished. The ﬁrst, of course, is the adoption of Haecceitism. The second is the idea that distinct objects can have all their essential properties in common. As to the ﬁrst, I have always conceded that there is indeed no logical obstacle to adopting Haecceitism about places in structures. I will argue in Section 5, below, however, that doing so would compromise one of the central

156 / The Identity Problem: Reply to Shapiro motivations of structuralism. If my argument there is accepted, Shapiro would then have to go with his second proposal. I will now argue that, in fact, the second proposal commits him to Haecceitism as well. Recall that essential properties of an object are the ones it has necessarily. Thus, if φE (x) indicates that x has the property φ essentially, Shapiro’s second proposal amounts to endorsing (a) ∀x∀y(x = y ⇒ ∀φ(φE (x) ⇔ φE (y) ) ) while rejecting (b) ∀x∀y(x = y ⇐ ∀φ(φE (x) ⇔ φE (y) ) ). Thus, Shapiro concludes, he ‘does not see how the ante rem structuralist is committed to [the] crucial premise for the identity of indiscernibles, and for Ker¨anen’s principle (STR)’ (p. 140, above). I maintain that, strictly speaking, there are no distinct but essentially indiscernible objects. Again, given any domain of objects, I believe that there must be some fact that metaphysically underwrites the distinctness of any two distinct objects in that domain. Suppose that you think that the objects a and b are essentially indiscernible and yet distinct; you must still think that there is something about the world that is responsible for the objects being two and not one. Suppose that you think that the non-essential properties of a and b are not up to the task;8 you must still think that there is something about the world that is. Suppose that you think that it is a primitive, ‘brute’ fact that a and b are two objects there rather than one; surely you must still think that there is something about each one that makes it the case that it is the object it is, and not the other. For, surely you want to be able to make sense of each object being identical to itself, and distinct from all others, no matter what other objects there are.9 Thus, you must think that each object at least has the property being identical to itself. Note that we cannot say that it is the very same property of ‘being identical to itself’ for then we would again have no metaphysical foundation for saying that the two objects really are two and not one. Thus, we must understand the description of this property ‘indexically’, 8

I take it to be obvious that individuating properties must be essential (although, perhaps, not vice versa). We do not want to say that, for example, the identity of number 1 is governed in part by 1 having the (non-essential) property of being the number of stars in our solar system. 9 I take it that a bona ﬁde structuralist is precisely committed to denying this. However, at this point I am assuming that Shapiro has rejected the idea that mathematical objects are individuated by their intra-structural relational properties.

Jukka Ker¨anen / 157 so that each object has a unique property being identical to itself. But, surely the property being identical to itself is an essential property of an object if anything is. It follows trivially that (b) ∀x∀y(x = y ⇐ ∀φ(φE (x) ⇔ φE (y) ) ). Thus, if you reject (b), you are committed to adopting (b), a contradiction. Of course, all the foregoing argument really shows is that if you reject (b) in general, you must still allow the following: if H is the set of properties of the form being identical to x, then ∀x∀y(x = y ⇔ ∀φ(φ ∈ H ⇒ (φ(x) ⇔ φ(y) ) ) is true of the domain in question—the most common construal of Haecceitism. Shapiro is free to maintain the possibility of essentially indiscernible objects as long as he excludes haecceities from the scope of this claim, admitting that the identity of such objects is governed by their haecceities. I submit, then, that Shapiro has no choice but adopt Haecceitism. Recall that in Philosophy of Mathematics, Shapiro set out to provide a theory of mathematical ontology. One of the central claims of ‘The Identity Problem’ was that in order to provide such a theory one must furnish some account of identity for the objects in the domains under consideration—a claim that Shapiro does not dispute. Simply announcing that there can be essentially indiscernible objects does not amount to discharging this task. But, as I have tried to show, once Shapiro adopts the possibility of essentially indiscernible objects, the only account of identity available to him is Haecceitism. It is my impression that Shapiro would not regard this possibility as repugnant to the motivations of structuralism; I will next argue that, alas, it is.

5. Realist Structuralism Reconsidered In Section 1 of ‘Structure and Identity’, Shapiro considers four possible ways of construing the fundamental structuralist thesis that ‘the essence’ of a mathematical object is its relations to other mathematical objects: (a) All necessary properties of mathematical objects are intra-structural; (b) All mathematical properties of mathematical objects are intrastructural; (c) All essential properties of mathematical objects are intra-structural;

158 / The Identity Problem: Reply to Shapiro (d) (1) Each mathematical object is uniquely characterized by its relations to other objects in the same structure, and (2) its other essential properties are consequences of this being the case. After discussing problems with each of these four possibilities, Shapiro regroups by returning to (c). The problem with (c) was that some non-structural properties of mathematical objects are very plausibly essential; being non-spatio-temporal might be an example (see p. 117, above). Shapiro proposes two ways in which the realist structuralist could neutralize this problem. First, she could insist that the putative non-structural essential properties are not essential after all (p. 120, above). In some cases, however, doing so would be rather implausible. Thus, second, the realist structuralist could simply reject the slogan that all essential properties of mathematical objects are intra-structural. In more detail, Shapiro’s ﬁnal proposal is this: (1) While some extra-structural properties are essential, (2) each property a mathematical object has ‘comes in virtue of’ its being the place it is in the structure to which it belongs. Shapiro ‘would like to take back statements like ‘‘the essence of a natural number is its relations to other natural numbers’’ ’ (p. 120, above). All this, however, is supposed ‘not [to] affect the underlying philosophy of structuralism’. Rather, ‘the revisions to ante rem structuralism are limited to which slogans we can use and take seriously’. ‘The ofﬁcial stance is (still) that a natural number is no more and no less than a place in the natural number structure’ (p. 121, above). I want ﬁrst to clarify the status of Shapiro’s ﬁnal proposal. To say that mathematical objects are places in structures does not amount to giving a theory of mathematical ontology. To genuinely illuminate the nature of mathematical ontology, the structuralist must also provide a theory of structures and places in structures. I applaud Shapiro for doing the former, but do not see that he has ever even attempted the latter. At the very least, he should endeavor to say what makes a given place the place it is. In other words, he should furnish an account of identity for places in a given structure.10 And, as I argued in 10

As far as ontology goes, the view Shapiro proposes is essentially classical Platonism. What remains structuralist is his emphasis on the insight that mathematical objects are arranged in structures and, presumably, that epistemology is to be built on this insight. The only genuinely structuralist component of Shapiro’s ﬁnal proposal is the claim that every property that a mathematical object enjoys ‘comes in virtue of’ its being the place it is in the structure to which it

Jukka Ker¨anen / 159 the preceding section, given his other commitments, Haecceitism is the only account of identity available to Shapiro. Suppose, then, that the realist structuralist adopts Haecceitism, maintaining that the identity of places in structures in governed by haecceities, their ‘primitive thisness’. The problem with this move is that the resulting theory of mathematical ontology will be unable to solve the ontological problem that motivated it in the ﬁrst place. Recall that the core motivation of the ontological project of (any kind of) structuralism is the observation that, given any mathematical theory, there are indeﬁnitely many systems of objects that could in principle serve as the domain of that theory. This is problematic to a serious realist because she would like to say that, given any categorical mathematical theory, any singular term in the language of that theory denotes a unique object. Since we have no means of singling out any one system as the domain of any given theory, the realist ambition is apparently defeated. Further, if one accepts Benacerraf ’s well-known argument, it follows that mathematical objects are not objects at all, given that we have no reason to take, say, the number 2 to be any particular one of the indeﬁnitely many candidates. The realist structuralist solves this problem, quite plausibly, by proposing that the domain of a given theory is simply the structure all the systems exemplifying it have in common. The kicker, then, is this. If the realist structuralist adopts Haecceitism, the foregoing problem re-emerges on the level of structures. For, if the identity of places in mathematical structures is governed by haecceities, there will be indeﬁnitely many distinct copies of the same structure. The point of realist structuralism was precisely that while there are many systems that exemplify any given structure, the structure itself is unique because the relations satisﬁed by the multiple occupants of each place individuate the place itself uniquely. But, if the identity of places is governed by haecceities, there is no reason to suppose that there is just one unique structure of any given isomorphism type. At the very least, the realist structuralist is forced to admit that there is no reason to suppose that each mathematical singular term has a unique referent. Finally, again, if one accepts Benacerraf’s argument,11 it follows that mathematical objects are not objects at all. belongs (p. 121, above). However, in discussing his proposal (d) a few pages earlier, Shapiro himself points out that it is very unclear how one should construe a phrase such as ‘comes in virtue of’. 11 As, I take it, the average realist structuralist does.

160 / The Identity Problem: Reply to Shapiro

6. Conclusion: The Identity Problem for Realism I shall now step away from realist structuralism and consider the prospect for a realist theory of mathematical ontology more generally. For, it seems to me that the foregoing argument against realist structuralism is easily recast as an argument against ontological realism in general.12 A realist about mathematical ontology maintains that mathematical objects exist independently of our cognitive activities. Accordingly, she maintains that each mathematical object is the object it is, identical only to itself and distinct from any other, independently of our cognitive activities. Moreover, she maintains that any categorical mathematical theory T has a unique domain of discourse, and that the singular terms of T denote objects in that domain and the quantiﬁers of T range over that domain. I maintain that given any such theory, the ontological realist should furnish an account of identity for the domain of discourse of the language L of T. There are four basic options. The identity of mathematical objects is governed by either: (a) (b) (c) (d)

intra-structural relational properties, extra-structural relational properties, non-relational properties, or haecceities.

It seems to me that adopting any one of these four options leaves the realist in a difﬁcult predicament. I will discuss each one in turn. (a) Suppose that the identity of mathematical objects is governed by their intra-structural relational properties. But, as I argued in ‘The Identity Problem’, this leads into absurdities. (b) Suppose that the identity of mathematical objects is governed by extrastructural relational properties. There are two basic options here. First, one could suppose that the identity of mathematical objects is governed by their relations to physical objects. This is extremely implausible, however, particularly in the case of very large domains. Second, one could suppose that the identity of mathematical objects is governed by their relations to non-physical objects outside their ‘native’ structure. The most plausible way of ﬂeshing out this proposal is to suppose that each mathematical object is individuated by 12 I will only consider what Shapiro would call ‘metaphysical’ realism, setting aside the ‘Quinean’ variant. I intend to examine the possibility of Quinean realism at length elsewhere.

Jukka Ker¨anen / 161 the relations in some rigid structure (one without automorphisms), avoiding the identity problem. For example, we could suppose that each object in the group of integers (a non-rigid structure) is individuated by the relations in the ring of integers (a rigid structure). The problem with this proposal is that it seems extremely unlikely that every non-rigid structure has a rigid structure associated to it. What, for example, would be the rigid structure associated with the Euclidean plane? Or, what of abstract groups in general?13 (c) Suppose that the identity of mathematical objects is governed by their non-relational properties. The problem with this supposition is that mathematical objects (famously) do not have any non-relational properties, save for the trivial one of being identical to themselves.14 (d) Suppose that the identity of mathematical objects is governed by their haecceities. The problem with this supposition is, again, that there is no reason to think that each mathematical singular term has a unique, well-deﬁned referent. Further, we must again confront the old Benacerraf problem of multiple reductions: mathematical objects are not objects at all, given that there is no reason to take, say, the number 2 to be any particular one of the indeﬁnitely many candidates. Finally, since places in structures are individuated by haecceities, the realist structuralist solution to the problem fails. The ontological realist can of course concede these points and adopt a form of eliminative structuralism. That is, she can concede that there is no speciﬁcally mathematical ontology: a mathematical theory is about any system of objects that satisﬁes its axioms. Mathematical statements are not to be taken at face value: apparent singular terms, such as numerals, are disguised bound variables ranging over elements in systems, and the identity of such elements is governed by haecceities. This is not the place to discuss the merits of such a view; I only want to emphasize that eliminative structuralism does not count as ontological realism proper to the extent that by its lights, (categorical) mathematical theories do not have unique domains of discourse.15 13

Recall that all abstract groups of order greater than 2 are non-rigid. Now, there is the obvious thought that perhaps mathematical objects are simply sets in the pure hierarchy. Recall, then, that it was precisely this thought that gave rise to the problem of multiple reductions and hence, structuralism in the ﬁrst place. 14 Perhaps mathematical objects have some general properties that can be considered nonrelational; being its own square root might be an example. It is clear, however, that there are not enough such properties to individuate mathematical objects. 15 I take this occasion to record my attitude about the status of eliminative structuralism vis-`a-vis the identity problem. As Shapiro surmises (p. 136, above), I think that the eliminative

162 / The Identity Problem: Reply to Shapiro Finally, I suppose the realist could insist that the identity of mathematical objects is primitive. However, as I explained in Section 4, above, I do not regard this position to be genuinely distinct from Haecceitism. In sum, I suggest here, and will argue at greater length elsewhere, that the ontological realist simply cannot furnish a plausible account of identity for mathematical objects. If we take the metaphysics of mathematical ontology seriously, we will have to take individuation seriously. And, if we take individuation seriously, we must take the identity problem and the Benacerraf problem seriously. And, ﬁnally, if we take these two problems seriously, we will see that there is something fundamentally wrong with ontological realism about mathematics. Given what I have said thus far, the reader might expect me to now come out in favor of nominalism about mathematical objects. For what it is worth, then, I will not.16 While it is not possible to discuss my own views at length here, I shall conclude this paper by giving a brief indication of the direction in which I think the resolution is to be found. I believe that there are mathematical objects, and I believe that (most) mathematical singular terms have unique referents. Thus, I believe that ontological realism about mathematics is true. However, I also believe that we must re-examine what we can and should mean when we say that there are mathematical objects. Indeed, we must re-examine what we mean when we say that there are physical objects. For, I believe that the problems with realism about mathematical objects result from transporting our naive realism about physical objects into our theory of mathematical objects. I believe with Carnap that ontological questions can only be raised and answered in the context of a speciﬁc conceptual enterprise. In particular, I propose that all object-employing conceptual practices contain two interrelated sets of norms that may be called a standard of identity and a standard of existence. These sets of norms govern the way in which singular terms, quantiﬁers, and predicate expressions function. My claim, then, is that all we can and need to mean by ‘there are mathematical objects’ is that our mathematical conceptual practices have consistent standards of identity and existence. That structuralist is in principle free to adopt Haecceitism about the elements in the various systems. By so doing, she can avoid both the identity problem, as well as the problem of multiple reductions (unlike her realist counterpart). Assessing whether eliminative structuralism is ultimately tenable is, of course, beyond the scope of the present paper. 16 For example, I feel that there is much to be said for Burgess’s call to resist (and perhaps even combat) revisionism about the surface grammar of mathematical language.

Jukka Ker¨anen / 163 is, the existence and identity claims that we commit ourselves to by adopting these standards do not lead into contradictions. The picture is that in the case of physical objects, the standards of existence and identity place requirements, among other things, on the physical world, whereas in the mathematical case, the requirements are ones pertaining only to consistency and, perhaps, the quality of understanding. Yet, I maintain, mathematical and physical objects are objects in the same sense: items that our singular terms denote, that are or are not identical, that have properties, and so on. What is metaphysically central about physical objects is not that their conditions of existence involve spatially localized and temporally persisting hunks of matter but rather, that we have consistent and unambiguous standards of existence and identity for them in the ﬁrst place. It is in this regard that non-physical objects are no different from physical objects. Given this framework of conceptual practices and norms, my proposal is simply this: it is our conceptual resources generally that govern the identity of mathematical objects. Consider, for example, the problematic case of nonrigid structures. Even though there will be objects in such structures that have the same intra-structural relational properties, we can simply decree how many distinct copies of each given place-type there are. This does not mean that what we can and cannot decree is arbitrary, depending as it does on our previous commitments. In particular, the standards of identity and existence impose various constraints on our ontological decisions, notably consistency. Beyond such complex conceptual interrelations, however, numbers are and can be whatever we want them to be.

References Kastin, J. (1998), ‘Shapiro’s Mathematical Structuralism’ (manuscript). Ker a¨ nen, J. (2001), ‘The Identity Problem for Realist Structuralism’, Philosophia Mathematica, 93: 308–30. Quine, W. V. (1992), Pursuit of Truth (Cambridge, Mass.: Harvard University Press). Shapiro, S. (1997), Philosophy of Mathematics: Structure and Ontology (New York: OUP).

7 The Governance of Identity Stewart Shapiro

My exchange with Jukka Ker¨anen exempliﬁes the long tradition of using the thoughtful objections from opponents as an opportunity to clarify, further articulate, and correct philosophical views. ‘Structure and Identity’, Chapter 5 in this volume, changes and extends aspects of the ante rem structuralism developed in my (1997). Ker¨anen’s ‘The Identity Problem for Realist Structuralism’ (2001) and ‘The Identity Problem for Realist Structuralism II: A Reply to Shapiro’ Chapter 6 in this volume, constitute an advance in the articulation of his metaphysical views from their public origin in his 1999 APA paper. At some point, he and I may realize that we are talking at cross-purposes, or we may end up with a clash of brute intuitions, and will agree to disagree. But we are not there yet. Let us review the basics. Ker¨anen claims that any theorist who proposes an ontology owes an ‘account of identity’. We discharge the obligation by ﬁlling in the blank in the following scheme: (IS) ∀x∀y(x = y ≡

).

Presumably, we are not allowed to use the identity sign in discharging the obligation. It won’t do to just put ‘x = y’ in the blank, resulting in the tautology ∀x∀y(x = y ≡ x = y). The account (IS) speciﬁes ‘what governs the identity of objects in the domain of discourse’. The crucial issue seems to be to determine what it is to be ‘governed’, and why identity needs to be governed by something.

Stewart Shapiro / 165 We are told that in set theory, the axiom of extensionality does the job: ∀x∀y(x = y ≡ ∀z(z ∈ x ≡ z ∈ y)). The extensionality principle shows how identity is governed by facts about membership. The issue Ker¨anen raises concerns the account of identity for places of structures. From the slogans in Shapiro (1997) and certain metaphysical principles, Ker¨anen argues that the ante rem structuralist is committed to (STR) ∀x∀y(x = y ≡ ∀φ(φ ∈ → (φ(x) ≡ φ(y)))). The variable φ ranges over the relational properties of a given structure S, and the set ‘should only contain relations that can be speciﬁed without making explicit reference to particular elements in’ the structure S. Notice that (STR) requires that indiscernible places be identiﬁed. So (STR) entails that ante rem structuralism fails for ‘non-rigid’ structures—those with at least one non-trivial automorphism. In the complex plane, for example, (STR) entails that i = −i, which is absurd. In ‘Structure and Identity’, I proposed a modus tollens at this point, rejecting (STR), and, in particular, rejecting the provisos on the set . To do this, I have to either take back or reformulate some of the slogans and other remarks in Shapiro (1997). Ker¨anen maintains that the ante rem structuralist is still committed to (STR), based on metaphysical principles concerning identity. I’d be inclined to reject those metaphysical views, which would just amount to a clash of intuitions. But I am not satisﬁed to leave it like that, at least not yet. Let us take a look at the metaphysics. In response to a point about our ability to make distinctions with formulas in a ﬁxed language, Ker¨anen states that the problem he raises against structuralism is not one of uniquely describing each object. It is not a matter of whether the structuralist, or anyone else, has the resources to pick out each object. This is not possible for any theory (structuralist or otherwise) with a sufﬁciently large ontology, assuming that the background language—the framework in which we give an account of identity—is restricted to some ﬁxed number of terms. We are told instead that the project is to give a criterion of identity. A theorist who proposes an ontology has to say, in general or schematic terms, what makes each object the object it is, distinct from any other. We are also told that the individuation should be done with properties (and relations), and not with the predicates available in this or that language.

166 / The Governance of Identity There is always the following, second-order logical truth: ∀x∀y(x=y

≡

∀X(Xx ≡ Xy) )

(see Shapiro (1991, 63)). Will this work? If so, we would wrap up the ‘account of identity’ for every second-order theory at once. I suppose that this depends on what the second-order variables range over. It would not be very deep or interesting to require that individuation be done with sets (which is how the above sentence is shown to be a logical truth). If the background theory is extensional, and allows the singleton construction, then there is no real problem with identity. Singletons are the extensional counterpart of haecceities, without being controversial. Clearly, Ker¨anen requires that individuation be done with intensional properties, and the issue turns on what properties, or what kinds of properties, the theorist is allowed to use when giving the criterion of identity. The gloss on (STR) in ‘Identity Problem II’ says that the set ‘should only contain relations that can be speciﬁed without making explicit reference to particular elements in’ the given structure. I presume that this is not to be taken literally. If the relations that individuate the places of a structure have to be speciﬁed in some manner or other, then it is, after all, a matter of what we can say in a given language—the language of the background metaphysics. And of course we cannot do very many speciﬁcations with a ﬁnite or countable language. The cardinality problem returns. Ker¨anen’s remarks are consonant with a tradition that takes properties and relations to have a structure analogous to formulas in language. There are conjunctive properties, disjunctive properties, etc. Ker¨anen speaks of the ‘complexity’ of properties. For example, he mentions ‘relational’ properties. These are monadic properties derived from relations. A typical example is the property of being a child, which derives from the parent–child relation. Let us say that the property of being a child invokes the parent–child relation. Similarly, some properties invoke individual objects. For example, the property of being Seymore’s child invokes Seymore. The property of being larger than the White House invokes the While House. So perhaps Ker¨anen means that the members of the set in (STR) should not invoke the places of the structure in question, in this sense. This replaces the earlier talk of the elements of not containing singular terms denoting places. But then what can the members of invoke? If it is just the relations speciﬁed in the deﬁnition of the structure, then cardinality grounds alone

Stewart Shapiro / 167 preclude (STR). Most structures have only a few relations, not enough to generate uncountably many individuations. Perhaps Ker¨anen means that we should be able to specify the members of schematically. We are required to specify the kinds of properties in by using the relations of the structure. Ker¨anen says that set theory has no identity problem, since it is rigid (i.e. has no non-trivial automorphisms). To paraphrase Ker¨anen, let r be a set. Then there is a property Mr of ‘having r as a member’: for any set t, the property Mr holds of t if and only if r ∈ t. Of course, if we cannot characterize r uniquely, then we cannot ‘specify’ the property Mr . But this is not relevant. Ker¨anen holds that for each set r, there is such a property. The axiom of extensionality entails that these membership properties sufﬁce to individuate each set t: it has all and only the properties Mr where r ∈ t. So the membership properties tell us what makes t the set it is, distinct from any other. I am not sure if Ker¨anen thinks that the set-theoretic hierarchy meets (STR). It depends on whether the properties Mr meet the provisos that he hoists on the structuralist for set . The properties Mr are clearly relational, since they invoke the membership relation, and this relation can surely be speciﬁed without making explicit reference to particular elements in the set-theoretic hierarchy. The relation is just ‘∈’. The property Mr is a set-theoretic analogue of the ‘son of Seymore’ property. However, it is also clear that Mr invokes the set r. I am not sure if the ‘r’ in Mr counts as an explicit ‘reference’ to a particular element of the set-theoretic hierarchy. It sure looks like it does. Consider the cardinal three structure. It has three places and no relations. It is exempliﬁed by my children. This is case of a non-rigid structure which supposedly fails its identity problem. Suppose that we postulate ‘distinction properties’: For each place a in the cardinal structure, let Da be the property of being distinct from a. So for any place x, Da x if and only if a = x. The distinction properties are analogous to the membership properties above. They are relational properties, assuming that distinctness is a bona ﬁde relation. If distinction properties are allowed in the set , then the cardinal three structure satisﬁes (STR). Presumably, the problem here is that distinction properties invoke the distinctness relation, which, in turn, invokes identity. We might as well include haecceities. Recall that a structure is rigid if it has no non-trivial automorphisms. Ker¨anen suggests that for the ante rem structuralist, any non-rigid rem structure has an unsolvable identity problem. I submit that this is mistaken, even by Ker¨anen’s own lights.

168 / The Governance of Identity Consider, for example, a structure S with a single relation, <. The axioms of S are that < is a linear order, there is no ﬁrst or last element, and for any two elements x, y, there are only ﬁnitely-many z, such that x < z < y. Every model of this theory is isomorphic to the integers, under ‘less-than’. So it is categorical. But the structure S is not rigid. Let n be any integer. The function that takes x to x + n is an automorphism. The only relation in the structure is <, and we have that x < y if and only if x + n < y + n. If Ker¨anen is correct that the principle of extensionality provides an account of identity for set theory, then the following (IL) ∀x∀y(x = y ≡ ∀z(z < x ≡ z < y) ) provides an account of identity for S. Indeed, (IL) holds in S and it has exactly the same form as extensionality. The <-relation governs identity in S in exactly the same way that membership supposedly governs identity in the set-theoretic hierarchy. Sets are individuated by their members; the places of S are individuated by the places that are ‘smaller’ in the order. We can also mimic Ker¨anen’s membership properties. For each place r in S, let Lr be the property of being larger than r in the given ordering. That is, for any place s of S, Lr s if and only if r < s. If membership properties are a legitimate means of individuating sets, then these ‘smallness properties’ sufﬁce to individuate the places of S. Each place t has all and only the smallness properties Lr , where r < s. So if the set-theoretic hierarchy satisﬁes (STR), then so does S, even though the latter is not rigid. What of Ker¨anen’s formal argument that non-rigid structures fail (STR)? This is explicitly tied to the fact that isomorphic systems are equivalent—any sentence in the language of the theory that is true of one is true of the other. Suppose that f is an automorphism, and s, t are distinct places in the structure such that fs = t. Let A(x) be a formula in the language of the structure, in which the variable x is free. Then A(s) if and only if A(t). So s is indiscernible from t. The problem here is that the isomorphism theorem only shows that s and t cannot be distinguished (nor individuated) with a formula in the language of the structure. However, as above, ‘Identity Problem II’ is explicit that the account of identity is not a matter of ﬁnding a formula that picks out each object uniquely. Ker¨anen says instead that the individuation is to be done with properties. The theorem that isomorphic structures are equivalent does not entail that if an automorphism f takes s to t, then for every property P, we have Ps if and only if Pt. The theorem only entails this for properties that are deﬁnable

Stewart Shapiro / 169 in the language. In particular, automorphisms do not preserve the smallness properties that individuate the places of S. Of course, this is not an argument that every structure satisﬁes (STR). To adjudicate this matter, we would need a fuller account (and justiﬁcation) of the properties that are allowed in the set . Let me concede, for the sake of argument, that complex analysis, Euclidean space, and the ﬁnite cardinal structures all fail (STR). Incidentally, another example is set theory with urelements. Any permutation of the urelements gives rise to an automorphism of the set-theoretic structure. As above, I remain unconvinced the ante rem structuralist is committed to (STR). But again, for the sake of argument, let us suppose for a while that (STR) is a genuine requirement. So there is no ante rem structure for complex analysis, Euclidean space, the ﬁnite cardinal structures, and set theory with urelements. Under these circumstances, the best option for the ante rem structuralist is to embed each of these structures in one that does meet (STR), following a suggestion in my other contribution to this volume. For example, the cardinal three structure would be replaced with the ordinal three structure. That structure has three places, and a linear order. Any work, or any application, of the cardinal structure can be done with the corresponding ordinal structure. We can always ignore the order. Similarly, the complex number structure would be replaced with R2 , with complex addition and multiplication deﬁned in the usual way. In effect, this allows us to use the linear order on the real numbers to distinguish i ( 0,1 ) from −i ( 0,−1 ): since the real number 1 can be distinguished from −1, the pair 0,1 is distinguished from 0,−1 . Ker¨anen wonders what rigid structure might be associated with the Euclidean plane. The structure of R2 works here too, if we include the operations of complex analysis. This is the insight behind analytic geometry. We could still pursue synthetic, Euclidean geometry by ignoring the analytic elements of the structure. Those are the theorems that are invariant under the choice of a reference frame and metric. Let T be any structure. Suppose that we add a linear order (or perhaps a well-order) to the places of T, or to be precise, we embed T into another structure that has a linear order (or a well-order). Then (STR) holds for the richer structure, following the trick we used with the structure S. Identity on the richer structure is governed by the order. I do not know if every structure the ante rem structuralist might want in his repertoire can be accommodated. That is, I do not know if every structure

170 / The Governance of Identity can be embedded in one that satisﬁes (STR). This depends on a version of global choice. But surely the usual structures—the ones actually studied in mathematics—can be so embedded. Let us now cancel the concessions to (STR), made for the sake of argument. Once again, in Ker¨anen (2001) and ‘Identity Problem II’ we are told that the principle of extensionality ∀x∀y(x = y ≡ ∀z(z ∈ x ≡ z ∈ y)) is in the form (IS) and gives an account of identity for set theory. The axiom shows how identity is governed by the facts about membership. If I wanted to be ornery, I might demand an ‘account of membership’ for set theory. This would be to ﬁll in the blank in the following scheme: (MS) ∀x∀y(x ∈ y ≡

).

To paraphrase Ker¨anen, this is to ask for the facts that govern the membership relation. If x ∈ y then there must be some fact that underwrites this connection. For any sets x, y, someone who thinks that x ∈ y must think that there is something about the world that is responsible for these objects to be related this way. My ornery counterpart demands an articulation of this fact. I demand an ‘account of membership’, just as Ker¨anen demands an account of identity. And, of course, we cannot use the membership relation in whatever ﬁlls the blank in (MS), just as we cannot use identity in (MS). If the set theorist takes the bait, and somehow ﬁlls in (MS), the mischief could continue. Someone might demand an ‘account’ of the items used in the blank of (MS). Where does it end? The reply, I presume, is that membership is a primitive of set theory. This means that we cannot—and certainly do not have to—give an ‘account’ of it. But if membership is such a primitive, then why isn’t identity another primitive? Why do we have to give an ‘account’ of identity? To be sure, there is an important asymmetry between identity and membership. One of the epistemological strategies of ante rem structuralism is that the axioms of a coherent, categorical theory constitute an implicit deﬁnition of the primitive terms of the theory. The axioms of Euclidean geometry, for example, implicitly deﬁne ‘point’, ‘line’, and ‘plane’ (see Shapiro 1997: 129–36; 1997: 157–70 for historical background). The axioms of second-order ZFC are not categorical, but they do serve as at least part of an implicit deﬁnition of membership. Although it is not in the form (IS), the axiomatization itself is (part of) the account of membership in the only sense that one can demand an ‘account’.

Stewart Shapiro / 171 In some cases, the axioms of a theory include the identity relation. But we do not think of them as giving an implicit deﬁnition of identity. On the contrary, identity is presupposed in giving the implicit deﬁnition. For example, the second-order axioms of Peano arithmetic provide an implicit deﬁnition of ‘natural number’, ‘successor’, ‘addition’, and ‘multiplication’. This is done in terms of identity. It would not be correct to say instead that the axioms of Peano arithmetic implicitly deﬁne ‘natural number’, ‘successor’, ‘addition’, ‘multiplication’, and ‘identity’ (and the other logical terminology ‘&’, ‘¬’, ‘∀’). On the contrary, identity (and the other logical terminology) is presupposed in giving the implicit deﬁnition of the non-logical terms. As noted above, one can give an explicit deﬁnition of identity in secondorder logic, but perhaps that begs the present questions. If one uses only ﬁrst-order resources (or Henkin semantics for second-order languages), it is not possible to give an implicit deﬁnition of identity. Let I be a binary relation symbol, and let be any ﬁrst-order axiomatization that is a purported implicit deﬁnition of I as identity. Of course, does not include identity as a logical symbol. Any model of has models in which I is not identity. The usual axioms for identity force I to be a congruence, but not the identity relation. For what it is worth, my view is that we understand identity (and the other logical terminology) well enough, and can use it (and them) to give implicit deﬁnitions of various mathematical structures. For example, we declare that the cardinal three structure has three distinct places. Nothing in the language of the theory distinguishes those places, but they are three, and not one, nevertheless. I remain unconvinced that a theory of a structure somehow needs an ‘account of identity’ in Ker¨anen’s sense. We know what identity is. If we think of identity as one of the relations of a structure, then haecceities are indeed relational properties of it. But I do not wish to engage this bit of metaphysics. Ker¨anen and I may be at cross-purposes after all. Toward the end of ‘Identity Problem II’, he points out that his metaphysical views undermine any ontological realism toward non-rigid theories. So the ante rem structuralist is not the only one who would be inclined to reject the metaphysical views in question. Anyone who holds that Euclidean space and complex analysis, for example, describe collections of objects that exist independent of the mind, must ﬁnd a way to reject the metaphysical principles broached in Ker¨anen’s papers.

172 / The Governance of Identity The very end of ‘Identity Problem II’ consists of a sketch of Ker¨anen’s own views on ontology. On the negative side, he writes that if we take the metaphysics of mathematical ontology seriously, then we have to take the problem of how they are individuated seriously, and if we do that we face the identity problem (and the Benacerraf problem of multiple realizations). This, he claims, undermines ‘ontological realism’, which I call ‘realism in ontology’. But Ker¨anen goes on to reject nominalism—the view that mathematical objects do not exist. He says that there are mathematical objects, and that most of the singular terms of mathematical theories have unique referents. Moreover, mathematical and physical objects are objects ‘in the same sense’. The conclusion that ontological realism is true seems to be a direct contradiction of the negative part. To resolve this, Ker¨anen suggests that we need a new account of mathematical existence. We are told that our conceptual resources ‘govern’ the identity of mathematical objects. Little more than consistency is required for us to speak of mathematical objects. So what about non-rigid structures, such as Euclidean geometry and complex analysis? Ker¨anen says that even though there are objects in these structures that have the same intra-structural properties, we can simply ‘decree’ how many distinct copies of each given place-type there are. Well, this is exactly what the ante rem structuralist does in deﬁning structures. So at least so far, Ker¨anen’s remarks are of a piece with my own characterization of ante rem structuralism—the very view he attacks in the original talk (1999), in (2002), and in ‘Identity Problem II’. Indeed, he seems to adopt my otherwise controversial principle that any coherent implicit deﬁnition characterizes at least one structure. To take a simple example, consider the cardinal three structure. As I deﬁne it, the structure has three places, and no relations (other than identity). It is about as non-rigid as structures get. Since there are no relations, any bijection of the places is an automorphism. The deﬁnition of this structure amounts to a ‘decree’ that there are three distinct ‘copies’ of each place-type, a decree that Ker¨anen explicitly allows. Ker¨anen admits that the brief remarks at the end of ‘Identity Problem II’ are only a start on proposals for ontology. For now, it looks like those proposals amount to a rejection of the metaphysical principles that underlie the identity problem. This may constitute a new metaphysics, which stands opposed to the principles invoked in the attack on ante rem structuralism. I eagerly look forward to the promised further developments.

Stewart Shapiro / 173

References Ker a¨ nen, J. (1999), ‘The Identity Problem: Realist Structuralism is Dead’, read at the Paciﬁc Meeting of the American Philosophical Association, Mar. 1999. (2001), ‘The Identity Problem for Realist Structuralism’, Philosophia Mathematica, 9/3: 308–30. Shapiro, S. (1991), Foundations without Foundationalism: A Case for Second-Order Logic (Oxford: OUP). (1997), Philosophy of Mathematics: Structure and Ontology (New York: OUP).

8 The Julius Caesar Objection: More Problematic than Ever Fraser MacBride

1. Introduction Is it possible that Julius Caesar was not only a person but also a number? Might the conqueror of Gaul—beneath his material guise—have been the bearer of numerical properties? Common sense appears to inform us otherwise. Caesar was a man and no doubt added up sums. But he could hardly have been the result of a subtraction. And what appears to be true of Caesar also appears to be true of other concrete items. They enjoy the beneﬁts of being spatially related and causally interacting with their neighbours. By contrast, numbers appear quite different kinds of thing. They stand in equations and ﬁgure in the ancestral of the successor relation to zero. It seems that there is a Moorean fact, basic to our ordinary ways of thinking, any adequate philosophy of mathematics must accommodate. The fact that must be respected is the fact that no concrete item is a number. A theory encounters the ‘Caesar’ problem when it fails to meet this adequacy constraint. ¨ Thanks to audiences at the Universities of Bristol, Dusseldorf, and St Andrews for their helpful comments. I would also like to thank Peter Clark, Bill Demopoulos, Katherine Hawley, Alex Oliver, Michael Potter, Graham Priest, Stephanie Schlitt, Stewart Shapiro, Crispin Wright, and an anonymous reader for Oxford University Press. I gratefully acknowledge the support of the Leverhulme Trust whose award of a Philip Leverhulme Prize made possible the writing of this paper.

Fraser MacBride / 175 These remarks serve only to introduce an aspect of the Caesar problem. The Caesar problem demands our attention alongside other fundamental issues that ask for elucidation and justiﬁcation of the basic structure of our conceptual scheme—for example, the problem of universals or the problem of change. And, like them, the Caesar problem does not engage a single subject matter easily isolated. It engages a variety of logical, semantical, and metaphysical questions and their interrelations. The failure to recognize the rich and varied nature of the Caesar problem has resulted in a failure to appreciate the sort of difﬁculties that confront the provision of a satisfactory resolution. It may also have resulted in the relative neglect of the Caesar problem, a problem that deserves a rightful place alongside other more venerable, indeed ancient, philosophical concerns.

2. What is the ‘Caesar Problem’? 2.1 Frege Frege brought the ‘Caesar problem’ to the attention of analytic philosophy in the course of his attempt to introduce abstract objects (directions, numbers) into ordinary discourse. Frege began his discussion by forging the now familiar connection between the notions of object and identity. Self-standing objects (persons, mountains) may be identiﬁed and then re-identiﬁed on different occasions and from different perspectives. So expressions that are used to refer to objects must have associated with them a class of statements (Frege calls them ‘recognition statements’) that settle identity criteria for these objects: If for us the symbol a is to denote an object, then we must have a criterion which determines in every case whether b is the same as a, even if it is not always in our power to apply this criterion. (Frege 1950: §62).

Recognition statements determine when it is appropriate to label and then relabel an object on a different occasion with the same expression. If abstract objects are to be introduced into ordinary discourse then a class of recognition statements must also be supplied for them. Frege attempted to meet this constraint by stipulating identity criteria for the abstract objects he proposed to introduce (1950: §§62–5). Numbers are stipulated to be identical just if their associated concepts are 1-1 correspondent: (HP) Nx:Fx = Nx:Gx iff F 1-1 G

176 / The Julius Caesar Objection Directions are stipulated to be identical just if their associated lines are parallel: (D =) Dir (a) = Dir (b) iff a is parallel to b. However, Frege soon came to doubt whether these deﬁnitions supply genuine recognition statements. He ﬁrst concentrated his attention upon the case of (D =): In the proposition ‘the direction of a is identical with the direction of b’ the direction of a plays the part of an object, and our deﬁnition affords us a means of recognizing this object as the same again, in case it should happen to crop up in some other guise, as the direction of b. But this means does not provide for all cases. It will not, for instance, decide for us whether England is the same as the direction of the Earth’s axis—if I may be forgiven an example that looks nonsensical. Naturally, no one is going to confuse England with the direction of the Earth’s axis—but that is no thanks to our deﬁnition of direction. (1950: §66)

Frege then went on to question (HP) ‘for the same reasons’ (1950: §68). Frege appears to have deliberated in the following manner. Genuine recognition statements distinguish an object of a given kind K from all other objects. This means that they must determine the truth-values of two sorts of identity statement—pure and impure. A pure identity statement says that an object of kind K is identical to another object also explicitly stated to be of kind K (‘Kx = Ky’). By contrast, an impure identity statement says that an object of kind K is identical to another object that may not be described in K terms (‘Kx = q’). (D =) and (HP) notably determine truth-values for pure identity statements concerning, respectively, directions and numbers. They each provide a rule for determining the truth-value of identity statements in which, respectively, dual occurrences of direction and number terms ﬂank the identity sign (‘Dir (a) = Dir (b)’, ‘Nx:Fx = Nx:Gx’) by appeal to the obtaining of a familiar equivalence relation (‘a is parallel to b’, ‘F 1-1 G’). But (D =) and (HP) fail to settle truth-values for impure identity statements (‘Dir (a) = the Earth’s axis’, ‘Nx:Fx = Caesar’) in which there are only single occurrences of direction and number terms. They simply say nothing about statements of this form. As a consequence (D =) and (HP) fail to supply identity criteria for the objects they are designed to introduce.

2.2 Dimensions of the Caesar Problem These rudimentary reﬂections do not, however, reveal the depth and breadth of issues the Caesar problem raises. Beneath the superﬁcial simplicity of Frege’s

Fraser MacBride / 177 reasoning lies a welter of distinct worries. These concerns may be arranged along three distinct dimensions: (A) Epistemology: do the identity criteria supplied for directions and numbers provide any creditable warrant for a familiar piece of knowledge, namely that directions and numbers are distinct from such objects as Caesar? (B) Metaphysics: dothe identity criteria suppliedfor directionsandnumbers determine whether the things that are directions or numbers might also be such objects as Caesar? (C) Meaning: do the identity criteria supplied for the novel expressions putatively denoting directions and numbers bestow upon them the distinctive signiﬁcance of singular terms? Whether these differentdimensionsof concern may ultimately be distinguished will depend upon a variety of prior theoretical choices. For example, in the context of a robust realism—say, aposteriori realism—that draws a clear separation between ontological and linguistic issues, the metaphysical and meaning dimensions are held apart. In the context of a minimalist approach to ontology, these different dimensions of the Caesar problem will, by contrast, be intimately related. In advance of settling anterior realist or anti-realist choices the Caesar problem is likely to resist canonical speciﬁcation. The Caesar problem also raises different issues in the cases Frege considers (number and direction). Here attention will be given primarily to the numerical case.1

2.3 The Epistemological Caesar Problem According to one epistemological version of the Caesar problem suggested by Frege’s remarks, we already know that Caesar is not a number. So adequate identity criteria for numbers should settle that Caesar is not a number. But (HP) leaves it open whether Caesar is or isn’t a number. Therefore, (HP) fails to supply adequate identity criteria for the entities they respectively purport to introduce. Whilst this epistemological concern possesses considerable prima facie force. a number of questionable assumptions are made that require (in this context) to be legitimated: (i) Common Sense: we ordinarily know that Caesar is not a number. 1

See Hale and Wright (2001: 352–66) for an extended discussion of the direction case.

178 / The Julius Caesar Objection (ii) Discrimination: adequate identity criteria for numbers should enable a subject who grasps them to discriminate numbers from all other kinds of object (however presented). First, let it be granted that we do ordinarily know that Caesar is not a number (Common Sense). According to the version of the Caesar problem delineated, putative identity criteria for numbers that fail to conﬁrm that ordinary item of knowledge are inadequate. But the question needs to be asked: in what sense inadequate? The guiding thought must be that identity criteria that fail to meet this constraint fail to isolate the subject matter—the numbers—with which ordinary discourse deals. In other words, this version of the Caesar problem assumes that if (HP) is to be used to talk about an intended range of familiar numbers these identity criteria must provide for all the discriminations that we would ordinarily make between these and other objects (Discrimination). However, Discrimination appears too strong an assumption to make. It may reasonably be supposed that the ability to refer to a given kind of object presupposes some capacity to recognize items of that kind. But it would also be unreasonable to claim that a novice who only possesses limited recognitional skills could never light upon the same subject matter as a thinker with more reﬁned discriminatory capacities. It therefore remains open that (HP) provides a rudimentary but nevertheless genuine basis for identifying and re-identifying numbers. They allow us to identify and distinguish between, respectively, numbers (that are given as such) even though they fail to underwrite the sort of discriminatory abilities required to identify and distinguish between numbers and others objects differently presented. These reﬂections invite a reﬁnement of the epistemological concern. It is certainly true that there is a distinction to be drawn between succeeding—on however a rudimentary basis—in directing thought and talk upon a given range of objects, and being able to draw sophisticated, ﬁne-grained distinctions amongst the objects in question. But the possibility of directing thought and talk upon a subject matter presupposes a basic grasp of the sort of objects under consideration. But someone who has only grasped (HP) has no grasp of the sorts of objects that are under consideration. They don’t even know whether persons are numbers. So they can hardly claim to be able to think or talk about numbers. However, this response is also contestable. Imagine a child otherwise ignorant of arithmetic who is taught (HP). For all that has been said so far this child may—within the arithmetical language game—go on to reliably

Fraser MacBride / 179 distinguish between different numbers (presented as such). According to the result called Frege’s Theorem, (HP) entails the fundamental truths of arithmetic (the Peano Postulates) (Wright 1983: 158–69; Boolos 1987). So, continuing the fantasy, we can also imagine that the child goes on to develop basic arithmetical skills (addition, multiplication, etc.). This means that the child may mingle at school with peers who are taught arithmetic in the ordinary fashion and come home with good test results in maths. Do we really want to say that this computationally competent child does not succeed in talking about numbers? Does the fact that he expresses incomprehension when asked whether Caesar is a number immediately settle that his test results are nothing more than a sham? Would it not be more appropriate to say that this child trains his thought upon numbers well enough but lacks an additional piece of metaphysical knowledge (that Caesar is not a number)? Another assumption made by the epistemological version of the Caesar problem also merits scrutiny. According to Common Sense we already know Caesar is not a number. But do we know this? If we do, it is certainly not in virtue of grasping Frege’s—let’s face it, pretty decent—guesses at identity criteria for numbers and directions. So how else might we know this? It may seem that there is no pressing need to answer this question. For there is a tendency in contemporary philosophy to assign to common sense an epistemologically and theoretically innocent nature. As a result the verdicts of common sense are simply taken for granted. However, it is important to bear in mind the possibility that common sense may itself be corrupt, nothing other than the consequence of signiﬁcant—albeit prolonged and low level—theoretical labour. As Russell once remarked upon the common sense understanding of such notions as ‘thing’ and ‘object’: the thing was invented by prehistoric metaphysicians to whom common-sense is due. (Russell 1911: 148)

It may indeed be that our common sense understanding delivers the verdict that Caesar is no object. But it remains open that the endorsement of Common Sense may incur signiﬁcant epistemic costs. There appear to be two possibilities concerning the putative knowledge that Caesar is no number. Either this knowledge is immediate or it is derived. It is important to realize that the former option is far from plausible. It is part of the beguiling nature of the Caesar problem that when we try to form a clear and distinct idea of Caesar we do not ﬁnd it explicitly represented there

180 / The Julius Caesar Objection that he cannot be a number. Nor when we try to form a clear and distinct idea of a number (say, zero) do we ﬁnd it explicitly represented that zero cannot be a person. Rather we encounter a modest silence on these matters. Caesar has personal properties. Zero has numerical properties. But it is neither explicitly ruled in nor ruled out that Caesar might be zero. So if we really do know that Caesar is not a number then there must be some argument implicit in our ordinary understanding that shows this to be the case. Since the metaphysical version of the Caesar problem concerns the availability of just such an argument let us turn our attention there.

2.4 The Metaphysical Caesar Problem According to this development of Frege’s reasoning, it is impossible for radically different kinds of object to overlap. (HP) fails to preclude the possibility that the same objects fall under radically different kinds (persons, numbers). This is because it leaves open whether a range of identity statements concerning objects drawn from disparate kinds are true or false (for example, ‘Nx:Fx = Caesar’). Therefore, (HP) fails to provide adequate identity criteria for the objects it is intended to introduce. This version of the Caesar problem rests upon the following apparently sane and sensible assumption (iii) Sortal Exclusion: such radically contrasting kinds of objects as numbers and persons cannot overlap. But part of what makes the metaphysical Caesar problem so problematic is that it is far from clear what legitimate grounds for Sortal Exclusion there might be. It is frequently asserted that it is simply ‘absurd’ to suppose otherwise (Parsons 1990: 308–9). However, brute intuition has proved a notoriously unreliable guide in theory construction. So an argument for Sortal Exclusion is wanted. One tempting strategy is to argue that numbers are abstract whereas persons are concrete and thereby obtain Sortal Exclusion as a conclusion. Waive the usual concerns about whether the abstract–concrete distinction is in good enough shape to distinguish between mutually exclusive classes of abstract and concrete items (Burgess and Rosen 1997: 12–25). Just suppose for current purposes that concrete objects are located and capable of entering into causal interaction whereas abstract objects are not. It still does not follow that the kinds in question cannot overlap unless it is also presupposed that numbers are abstract and persons are concrete.

Fraser MacBride / 181 Arguments for this last claim may appear readily forthcoming. It may be thought that there are more numbers than concrete objects. Perhaps there are inﬁnitely many of the former and only ﬁnitely many of the latter. So, it may be concluded, numbers cannot be concrete objects. Since persons are concrete it follows that persons aren’t numbers. But this argument is too quick. It neglects to rule out the possibility that some numbers (ﬁnitely many of them) are concrete objects. Another line of argument appeals to the necessary truth of a wide range of mathematical claims. Necessary truths require necessary existences to serve as their immutable subject matter. Since concrete objects are contingent it follows that the objects picked out by mathematical truths cannot be concrete. But this argument is also too quick. It assumes that the sentences that express necessary truths must refer to necessary existences. This assumption may be questioned. The necessary truth of a sentence may be sustained by virtue of its constituent terms picking out different objects at different possible worlds. In other words, the argument assumes that numerical terms are rigid designators. However, if numerical terms are non-rigid then—for all that has so far been established—they may at a given world pick out concrete objects (Caesar amongst them). A variation on this argument appeals to the role of numerical terms in contingent counterfactual claims of applied arithmetic. Consider a range of counterfactual circumstances in which the number of Fs remains the same even though concrete non-Fs pass either in or out of existence from one circumstance to the next. It follows that the expression ‘the number of Fs’ cannot refer to any concrete non-F. For example, the number of moons of Mars is two. We can entertain counterfactual circumstances in which the number of moons of Mars remains two even though Caesar had never existed. So the number of moons of Mars cannot be Caesar. (Of course, it may take further discussion to show that the numerical terms that ﬁgure in the statements of pure arithmetic cannot pick out concrete items either.) Here it is assumed that the numbers applied to concepts in counterfactual circumstances actually exist there. Then since the concrete objects at issue do not exist in those circumstances the desired consequence follows that the numbers in question are none of the concrete things. But this assumption is far from obligatory. The application of numbers to concepts in counterfactual circumstances may not rest upon an ability to identify the numbers that exist in those circumstances. Rather it may rest upon an ability to identify and count

182 / The Julius Caesar Objection with numbers in the actual world and then use these numbers to count the objects falling under concepts in counterfactual circumstances from here. It is therefore left open whether numbers exist or not in any given counterfactual circumstance. Alternatively, the relation expressed by ‘is the same number as’ may express an equivalence relation (a trans-world relation), weaker than identity, between the shifting referents of numerical terms. If so, the fact that the number of Fs remains the same across counterfactual circumstances fails to determine that the number of Fs is not concrete in some world. It may also be argued that concrete objects are contingent whereas numbers are necessary and so no person can be a number. These claims may appear beyond question. Surely it is ‘manifest’ that Caesar is no necessary existent (Hale and Wright 2001: 366)? Surely it is beyond doubt that numbers cannot be contingent? But these assumptions have been questioned. Field has argued that numbers are contingent existences; after all, he claims, there is no logical incoherence in the suggestion that numbers might fail at a given world to exist.2 As part of a defence of a simple form of quantiﬁed modal logic (including, crucially, the Barcan formula: ♦∃xφ → ∃x♦φ) Linsky and Zalta have argued that every object exists necessarily.3 However, the point is not only that these assumptions have been questioned. More signiﬁcantly, the point is that a failure to appreciate that these assumptions may be questioned constitutes a failure to appreciate the problematic character of the (metaphysical) Caesar problem itself. A common ﬁrst reaction to the Caesar problem is to take it as manifest that Caesar is no number. So Sortal Exclusion is simply taken for granted. But we are able to see our way past this initial response when it is appreciated that it is no straightforward matter to settle whether different kinds of objects—that are apparently as unlike as objects can be—are really distinct. The matter is difﬁcult to settle because (in part) it is not explicitly written into the nature of persons that they are not numbers (or vice versa). But nor is it explicitly written into the nature of persons or numbers that they are contingent or necessary. So to assume on manifest grounds that Caesar cannot be necessary or that numbers cannot be contingent is simply to ignore the problematic character of the Sortal Exclusion assumption. What is wanted here is just an 2

See Field (1993). Field’s proposal is contested by Hale and Wright (1992, 1994). Field responds in his (1993). For further discussion see MacBride (1999: 443–7). 3 See Linsky and Zalta (1994, 1996). Williamson defends a similar position in his (1998).

Fraser MacBride / 183 instance of what is wanted generally: a principled account of why objects of one kind cannot possess features (necessary existence, abstractness) usually associated with different kinds. In response it may be claimed that it is plainly constitutive of being a person to be contingent. After all, persons come to be and pass away. They begin and cease to exist. They might not have existed. They can hardly be necessary existents! Of course, this train of thought will hardly settle that Caesar is not a number unless it is also shown that numbers cannot be contingent. But, more signiﬁcantly, ask yourself the question: is that an accurate statement of what we know to be true of persons? Might it not be more accurate to say that persons take on and then throw off a material guise and it is left open—a matter upon which speculation may never cease—what, if anything, happened before, next or whether they might never have existed? I do not mean to suggest that persons continue to exist without bodies or as bare abstract entities. The point is rather that to legitimate the Sortal Exclusion assumption that underwrites the metaphysical version of the Caesar problem it must be demonstrated that no person is a number. And if the resolution is to be theoretically satisfying this fact must somehow be guaranteed by the underlying nature of persons and numbers. But since there is no immediate incompatibility between being a number and being a person the intriguing difﬁculty we have to confront is that we have apparently no idea of how Sortal Exclusion might be legitimated. In any case appeal to different principles of modal existence is far too coarse-grained a basis upon which to ground Sortal Exclusion. For suppose that one were to become convinced that persons exist necessarily. Would one then feel any more comfortable with the suggestion that Caesar is a number? Or, alternatively, suppose that one already believed God, or some other plausibly non-numerical item, exists necessarily. Would it then be legitimate to suggest God is a number? The intuitive response—not to mention the theological one—is likely to be that it is not. And until it is established that there cannot be different (non-overlapping) kinds of necessary existent this response cannot be rejected out of hand. So what gives rise to this intuitive response? It does not appear—as one might initially have thought—to arise from any overt incompatibility between the different kinds in question. Instead it appears to result from the fact that we have no intellectual stomach for irresolvable metaphysical inscrutability. For if Caesar is a number then this identity is simply brute. The different ranges of properties associated with being a person and being a number are

184 / The Julius Caesar Objection so distinct in kind that there is nothing that might be said to render this identity transparent to the understanding. It is entirely opaque how a single object could be the subject of such diverse properties. Consequently even an exhaustive investigation (at the limit of enquiry) of the personal properties of Caesar will not enable us to decide whether Caesar is a number, and if so, which one. Similarly, no amount of investigation of the numerical properties of 4 will determine whether it is also a person. The issues surrounding the Caesar problem encroach here upon traditional, metaphysical concerns about the nature of substance, about what makes an object a uniﬁed, integrated whole. Consider the following remarks from Leibniz: I also maintain that substances (material or immaterial) cannot be conceived in their bare essence, devoid of activity; that activity is of the essence of subject in general . . . it must be borne in mind above all that the modiﬁcations which can occur to a single subject naturally and without miracles must arise from limitations and variations in a real genus, i.e. of a constant and absolute inherent nature. . . . Whenever we ﬁnd some quality in a subject, we ought to believe that if we understood the nature of both the subject and the predicate we would conceive how the quality could arise from it. So within the order of nature (miracles apart) it is not at God’s arbitrary discretion to attach this or that quality haphazardly to substances. He will never give them any which are not natural to them, that is, which cannot arise from their nature as explicable modiﬁcations . . . what is natural must be such as could become distinctly conceivable by anyone admitted into the secrets of things. (Leibniz 1981: 65–6)

Two relevant thoughts may be distinguished here. First, it is claimed that substances cannot be conceived as bare particulars. Second, it is stated that the exhibition of properties by a substance must somehow be rendered intelligible by the underlying ‘real kind’ of the substance. Both thoughts plausibly militate against the identiﬁcation of Caesar with a number. For if Caesar is a number then the subject that underlies the relevant personal properties and the subject that underlies the relevant numerical properties can be no more than barely identical. Moreover, if it is Caesar’s nature to be human then he cannot also be a number. For the possession of numerical properties is rendered not one whit intelligible by an underlying human nature. Obviously these reﬂections present no decisive case. It is arguable that (in certain limit cases) bare identities may be properly admitted. Moreover, if one is willing to admit such conjunctive kinds as being a person and a number then Caesar’s underlying nature will render his possession of numerical qualities intelligible after all. Of

Fraser MacBride / 185 course, this raises the question of whether conjunctive kinds are themselves intelligible. More signiﬁcantly, Leibniz’s views on substance ﬂow from his endorsement of the principle of sufﬁcient reason—a principle that demands the intrinsic intelligibility of the universe. But since we have jettisoned the principle it is difﬁcult to see how it can be maintained that the world ought to be intrinsically intelligible. The demand that the world should conform to the patterns of our thoughts about it, that it should be transparent to even our idealized understanding, appears no more than conceit. Should we therefore continue to maintain Sortal Exclusion or should we be prepared to simply leave it open that—for all that we know—Caesar is a number?

2.5 The Meaning-Theoretic Caesar Problem This version of the Caesar problem questions whether (HP) succeeds in conferring contenton the natural number expressionsitpurportsto introduce. Frege initially sought to introduce numerical terms (‘Nx:Fx’) contextually by ﬁxing the content of identity sentences in which they occur. However, (HP) fails to settle the content of all the identity contexts—speciﬁcally contexts of the form ‘Nx:Fx = q’—in which the introduced expressions feature. Recall: (HP) simply fails to say anything about the signiﬁcance of contexts that feature a singleton occurrence of the numerical operator. Therefore, (HP) fails to bestow the signiﬁcance of singular terms upon the expression it introduces. In fact, this version of the Caesar problem is multiply ambiguous. It all depends upon what ‘content’ is taken to mean. If ‘content’ means sense then the complaint comes down to this. (HP) fails to determine whether identity contexts of the form ‘Nx:Fx = q’ have any sense. This calls into question whether (HP) ﬁxes a sense even for sentences of the superﬁcially tractable form ‘Nx:Fx = Nx:Gx’. When it is articulated in terms of sense the meaningtheoretic version of the Caesar problem may be developed in the following manner. Frege sought to introduce numerical terms by ﬁxing the truth-conditions of the identity sentences in which they occur. However, if this method of ﬁxing truth-conditions is to result in the introduction of genuine numerical terms—singular terms that purport to stand for objects—then the sentences of the form ‘Nx:Fx = Nx:Gx’ whose truth-conditions are ﬁxed must be genuinely logically complex. Sentences of this form must be understood as

186 / The Julius Caesar Objection saying of Nx:Fx that it satisﬁes the predicate ‘. . . = Nx:Gx’. Consequently, (HP) will only succeed in conferring individual signiﬁcance on the component numerical terms of the sentences whose truth conditions it ﬁxes if it also determines a meaning for such predicates as ‘. . . = Nx:Gx’ and ‘Nx:Fx = . . .’. To achieve this (HP) must also ﬁx truth-conditions for all the sentences in which these predicates occur. So (HP) must also ﬁx truth-conditions for sentences of the form ‘Nx:Fx = q’ (where ‘q’ is any singular term whatsoever). But (HP) only ﬁxes truth-conditions for sentences of the form ‘Nx:Fx = Nx:Gx’. It does not provide truth-conditions for sentences of any other form. Therefore, (HP) fails to provide a basis for supposing ‘Nx:Fx = Nx:Gx’ is logically complex and that the expressions it contains are genuine singular terms. However, if ‘content’ means reference then the complaint is quite different. The failure of (HP) to settle truth-values for the identities ‘Nx:Fx = q’ is interpreted as a failure to determine a deﬁnite referent for each of the numerical expressions introduced. It is then doubted whether expressions of the form ‘Nx:Fx’ are referential in the ﬁrst place. Benacerraf famously propounded a related argument (see his 1965 and Kitcher 1975). In this particular case set-theoretic terms are taken as values of ‘q’. According to Zermelo’s set-theoretic deﬁnition of natural number, 0 is the empty set and the successor function takes x to the unit set of x. Von Neumann deﬁned the natural numbers a different way: 0 is the empty set but the successor function takes x to the union of x and the singleton of x. Benacerraf argued that the use of arithmetical vocabulary fails to settle whether ordinary numerals refer to the Zermelo numbers or the von Neumann numbers (whether ‘2 = {{Ø}}’ or ‘2 = {Ø, {Ø}}’ is true). The problem is that each of these set-theoretic progressions serves as an equally effective model of the number theory embodied in ordinary usage. He concluded that the semantic function of ordinary arithmetical expressions must be other than referential. The problem that Benacerraf isolates for ordinary numerical terms is often assimilated to the Caesar problem itself (see, for example, Shapiro (1997: 78–81) ). However, this would be a mistake for several reasons. As we have seen, the Caesar problem has epistemological and metaphysical dimensions that Benacerraf’s argument fails to capture. But even if attention is focused solely upon the meaning-theoretic Caesar problem there are other reasons to resist the assimilation. To begin with, the Caesar problem Frege confronted concerned the signiﬁcance of expressions (‘Nx:x = x’, ‘Ny:[y = Nx:x = x]’ . . .)

Fraser MacBride / 187 artiﬁcially introduced by means of a deﬁnition (HP), whereas Benacerraf’s problem concerns the signiﬁcance of ordinary terms (‘0’, ‘1’ . . .) that may or may not have been introduced this way. Putting this issue aside, the ‘sense’ and ‘reference’ versions of the meaning-theoretic Caesar problem need to be kept separated. The latter species of argument does not deny the signiﬁcance of contexts of the form ‘Nx:Fx = q’ but moves from the existence of distinct eligible referents for the same numerical terms to the conclusion that the semantic function of these expressions (‘Nx:Fx’) cannot be referential. By contrast, the former sort of argument moves from the failure of (HP) to address the status of ‘Nx:Fx = q’ to doubt whether such contexts are signiﬁcant at all. Despite the important differences that obtain between these divergent versions of the meaning-theoretic Caesar problem they are subject to a generic doubt. The problematic character of the relevant class of Caesar problems is revealed in the high semantic threshold they each impose on the introduction of genuine singular terms: (iv) Semantic Threshold: in order to confer signiﬁcance on the terms and predicates purportedly introduced by (HP) their signiﬁcance should be everywhere ﬁxed. The obvious doubt to entertain here is whether these differing versions of the Caesar problem impose too high a threshold. More or less extreme forms of this doubt may be entertained. For example, it may be claimed that no context of the form ‘Nx:Fx = q’ requires to have its signiﬁcance ﬁxed in order to introduce singular numerical terms. A related view is evidenced in Carnap’s contention that such ‘mixed’ contexts that feature both mathematical and non-mathematical terms are actually nonsense: 2. ‘Caesar is a prime number’ . . . (2) is meaningless. ‘Prime number’ is a predicate of numbers; it can neither be afﬁrmed or denied of a person. . . . The fact that the rules of grammatical syntax are not violated easily seduces one at ﬁrst glance into the erroneous opinion that one has still to do with a statement, albeit a false one. But ‘a is a prime number’ is false iff a is divisible by a natural number different from a and from 1; evidently it is illicit to put here ‘Caesar’ for ‘a’. This example has been chosen that the nonsense is easily detectable. . . . (Carnap 1932: 67–8)

If such mixed contexts are nonsense then it can hardly be an adequacy constraint on the introduction of numerical terms that a meaning is ﬁxed

188 / The Julius Caesar Objection for these contexts.4 A less extreme doubt will discriminate between different sentences of the form ‘Nx:Fx = q’ where ‘q’ takes different sorts of values. It may be that there is no need for (HP) to ﬁx the signiﬁcance of some of these contexts in order to effect the introduction of genuine singular terms. It may also be that genuinely signiﬁcant sentences that take different values for ‘q’ generate different obstacles for the introduction of numerical singular terms. Examination of (an admittedly) provisional schedule of the different values ‘q’ may take reveals the range of distinct issues involved. First, ‘q’ may take values that in advance of a consideration of the Caesar problem we might have taken to denote paradigmatic extra-mathematical objects (‘Nx:Fx = Caesar’). A distinction may also be drawn between the extra-mathematical cases that feature reference to contingent as opposed to necessary non-mathematical entities (‘Nx:Fx = the Earth’s axis’, ‘Nx:Fx = the True’). Alternatively, ‘q’ may take values that denote objects characteristic of the mathematical domain. Some of these cases will feature reference to elements of progressions that plausibly might have been taken to be mathematical, but not distinctively numerical, progressions ( (‘Nx:Fx = {Ø}’) (Benacerraf’s examples may be located here). Others will involve reference to elements of numerical, but not distinctively arithmetical, series (‘Nx:Fx = 2real ’). Finally, there is the special case where the identities in question concern the relation between the objects denoted by the putatively arithmetical terms (HP) introduces and the objects denoted by the numerals of ordinary arithmetic (‘Nx:Fx = 2natural ’). The ability to settle the signiﬁcance of one of these different forms may not result in an ability to settle the signiﬁcance of another. For example, we may be able to determine that ‘Nx:Fx’ refers to a mathematical rather than a non-mathematical object. But then we may be unable to determine whether it refers to an item drawn from one rather than another mathematical progression. These are not the only cases a consideration of which may be expected to shed light upon the signiﬁcance of numerical terms. There are particular concerns about reference generated by the special character of series that exhibit non-trivial automorphisms (Brandom 1996). For example, in complex 4

Benacerraf makes the related suggestion that ‘identity statements make sense only in contexts where there exist possible individuating conditions’ (see his 1965: 285–9). Resnik offers a further alternative: that impure identities ‘should be banned from a proper language for science’ (Resnik 1997: 244).

Fraser MacBride / 189 number theory −1 has two square roots (i and −i). But there is no way to settle within the theory which square root our use of the signs ‘i’ or ‘−i’ denotes. It seems that we cannot settle the reference of these terms. This is because every predicate (not containing ‘i’ or ‘−i’) that is true of one of them is true of the other. There are also general concerns about reference that apply irrespective of the character of the series in question (Hodes 1984: 134–5). For the sake of argument, suppose that the series of natural numbers N have been singled out as the referents of the ordinary numerals. Then an alternative progression may be formed from N that serves just as well as a source of eligible referents for ordinary numerals. To see this we need merely permute a ﬁnite number of elements of N and make compensating adjustments to the successor function. For example, we might let ‘4’ designate 5 and ‘5’ designate 4 and employ ‘successor’ to stand for the function that differs from the successor function only in assigning 3 to 5, 5 to 4, and 4 to 6. Since there are indeﬁnitely many ways of so permuting the elements of N there is no telling which number is referred to by a given numeral. It is sometimes thought that all these different concerns are expressive of the same problem, the Caesar problem. This would be a mistake. The Caesar problem originally arose as a result of the inability of (HP) to settle the signiﬁcance of identity claims of the form ‘Nx:Fx = q’. However, the concerns that have just been raised about the reference of numerical terms are not occasioned by any doubt about whether some identity is meaningful or true. In the former case they turn upon the mathematical character of the complex number series. In the latter case, the difﬁculty raised does not concern agreement or disagreement about some object language sentence. So, for example, it is accepted that ‘4 = 4’ is true whereas ‘4 = 5’ is false. The difﬁculty confronted stems from another source. It stems—ﬁguratively speaking—from stepping back from our own language once the truth-values of all the sentences have been settled and then considering the myriad different ways in which it may be reinterpreted. It follows that a happy solution to the Caesar problem that generates truth conditions and values for sentences of the form ‘Nx:Fx = q’ cannot be expected to help (directly) in resolving more recherch´e concerns of this sort. Correlatively, the inability of a given solution of the Caesar problem to settle issues that arise once truth-conditions and values have been settled need cast no doubt upon the credentials of the resolution in question qua provider of truth-conditions and values.

190 / The Julius Caesar Objection

3. Two Solutions A ﬁrst encounter with the Caesar problem often occasions a denial, a denial that there is any signiﬁcant problem to be addressed. We are so convinced that there is something amiss with the identiﬁcation of Caesar and a number that it often takes a good deal of theoretical orientation before it is even appreciated that (HP) fails to secure the result that Caesar is no number. The preceding discussion sketched in a preliminary way some of the different issues that underlie the Caesar problem. It is to be hoped that sufﬁcient structure has been imposed to enable us to question the character of our pre-theoretic conviction and see that many distinct epistemological, metaphysical, and meaningtheoretic forces may be at work inducing the belief that Caesar is no number. A failure to appreciate or effectively treat of its many different dimensions undermines several proposed solutions to the Caesar problem. Two such solutions—supervaluationism and neo-Fregeanism—will be examined here.

3.1 The Supervaluationist Solution In its most familiar guise, supervaluationism provides a method for dealing with the semantic phenomenon of vague predicates (Fine 1975). There are (apparently) no sharp boundaries between the objects to which vague predicates apply and those objects to which they do not apply. Vague predicates have borderline cases where they neither clearly apply nor fail to apply. Nevertheless, these predicates may be ‘precisiﬁed’: a sharp boundary may be ﬁxed for their application. But there are many different ways of precisfying a vague predicate and it would be arbitrary to choose one to express the ‘real meaning’ of the predicate. So the supervaluationist attempts to account for the phenomenon of borderline cases by taking into account all the possible precisiﬁcations of a vague predicate. According to the supervaluationist account, a sentence is true iff it is true on all precisiﬁcations of its constituent vague expressions, false iff it is false on all precisiﬁcations, and neither true nor false iff it is true on some but not other precisiﬁcations. The supervaluationist interprets vagueness as a species of semantic indeterminacy. Predicates do not turn out to be vague because they apply to vague objects. They turn out to be vague because language users have not chosen between different possible precisiﬁcations of them. The inability of (HP) to settle whether Caesar is a number may likewise be interpreted as a consequence of semantic indeterminacy (cf. Field 1974;

Fraser MacBride / 191 McGee 1997; Shapiro: Chapters 5 and 7, in this volume). (HP) determines that the numerical terms it introduces refer to numbers (so presented) but fails to determine whether objects otherwise depicted are so picked out. This is because (HP) is a semantically indeterminate principle. There are many different possible precisiﬁcations of it and (HP) does not select between them. According to some precisiﬁcations of (HP), Caesar is a number; according to others, Caesar is not. Consequently, it is neither true nor false that Caesar is a number. Of course, it is open to us to constrain the admissible precisiﬁcations of (HP). We may choose to unite (HP) with the additional principle that no Nx:Fx is a person. Then there will be no precisiﬁcation upon which Caesar is picked out by a numerical term. It will be false that Caesar is a number. Viewed from the supervaluationist perspective Frege overreacted to the Caesar problem. Frege interpreted the inability of (HP) to settle whether Caesar was a number to be a symptom of an underlying malady, the failure of (HP) to introduce referring expressions. But really what was signalled by the inability of (HP) to settle a deﬁnite reference for number words was the indeterminacy of the terms it introduced. To use the terminology of the previous section, Frege simply set the semantic threshold for introducing referential expressions too high. He required that referring expressions must refer determinately and therefore failed to recognize that the terms introduced by (HP) referred indeterminately. In order to appreciate some of the problems that attend a supervaluationist account it is useful to consider a more simple and direct approach to the Caesar problem. When reﬂecting upon the inability of (HP) to determine a truth-value for the sentence ‘Julius Caesar is the number of planets’, Dummett once suggested that the difﬁculty could be swept aside with ease. He wrote: ‘it would be straightforward to provide by direct stipulation for the falsity of such sentences’ (see his 1967: 111).5 There are two relevant difﬁculties associated with this proposal. First, it is unclear how a direct stipulation that sufﬁced for the falsity of all such sentences might be constructed. Of course, it is true that no numerical thing is a non-numerical thing. But this doesn’t need a stipulation to make it 5 It should be noted that Dummett’s views have evolved since 1967. Nevertheless, he appears to remain at least willing to countenance the possibility that the Caesar problem is the consequence of semantic indeterminacy. In the course of a discussion of Benacerraf he writes: ‘the received senses of numerical terms do not impose any one speciﬁc identiﬁcation of the natural numbers’ (Dummett 1990: 179).

192 / The Julius Caesar Objection so. Moreover, it must not be forgotten that what is at issue is the reference of numerical terms and the application conditions of numerical predicates. So this truth—whose sentential expression makes play with the relevant class of problematic vocabulary the reference and application of which are in question—provides no guide to whether a given sentence is true or false as a consequence of it. This suggests that it might be better to proceed piecemeal, providing a range of stipulations to distinguish numbers from different sorts of non-numerical objects. But, as we have seen, it is no easy matter to settle whether numbers do or do not possess a feature—for example, contingency or concreteness—characteristic of a given sort (Section 2.4, above). This ﬁrst difﬁculty is a clue to the second. Suppose that Caesar leads a double life. Suppose that in addition to leading his material existence Caesar is also a number. In that case the stipulation that sentences that say Caesar is a number are all false cannot succeed. For some of these sentences will be true and true sentences cannot be stipulated to be false. So Dummett’s strategy of directly stipulating the falsity of the relevant range of sentences presupposes that Caesar is no number. Stipulation cannot sufﬁce as a basis for determining that Caesar is no number. A similar difﬁculty afﬂicts the more sophisticated supervaluationist strategy. According to the supervalutionist, the inability of (HP) to settle the truth-value of such sentences as ‘Julius Caesar is the number of planets’ is a consequence of semantic indeterminacy. In other words, (HP) may be precisiﬁed in a number of arbitrary ways and some of these precisiﬁcations make such sentences true, others make them false. But this assumes that it is legitimate to precisify the numerical vocabulary introduced in such a way as make it false that Caesar is a number. But suppose again that Caesar is a number. Suppose that facts about his identity and distinctness from every other number are determined by facts about 1-1 correspondences between concepts. Then there are no precisiﬁcations upon which Caesar is not a number. Any attempt to precisify (HP) in this way will conﬂict with underlying metaphysical facts. Consequently, it will be inappropriate to apply the semantic machinery of supervaluationism to explicate the apparent indeterminacy of the sentences in question. It will not be the case that there are some precisiﬁcations upon which these sentences are true and some precisiﬁcations upon which they are false. The same difﬁculty will attend the attempt to precisify (HP) by adjoining further stipulations (for example, that no number is contingent).

Fraser MacBride / 193 What this reveals is that both the direct stipulation strategy and supervaluationism rest upon a common assumption, the assumption that (HP) is semantically indeterminate. They are motivated by the idea that the vocabulary introduced by (HP) is semantically neutral in the sense that it enjoins no commitments that might conﬂict with antecedent facts. It is because, they presume, the introduced vocabulary is free of such commitments that it is possible to stipulate or precisify its use without there being any risk of offending against any antecedent fact. If, however, the vocabulary introduced fails to be neutral in this regard—if the use stipulated for it may conﬂict with antecedent facts—then the mechanisms employed to resolve the indeterminacy will misﬁre. It follows that we can have no assurance that direct stipulation or supervaluationism succeed in resolving the Caesar problem in the absence of an argument that (HP) is semantically indeterminate. But neither purported solution shows this. They suppose (HP) is semantically indeterminate and then seek to accommodate that happenstance. The point deserves emphasis. (HP) fails to explicitly address the signiﬁcance of identity contexts of the form ‘Nx:Fx = q’. The direct stipulation and supervaluationist strategies construe this silence to be a symptom of the semantic indeterminacy of the terms (HP) introduces. But the silence of (HP) may be interpreted differently. It may be taken to reﬂect the epistemological inscrutability of impure identities, the fact that we can just never know whether Caesar is a number. Alternatively—and these do not exhaust the alternatives—the silence may betoken the meaninglessness of impure identities, the failure of (HP) to ﬁx any signiﬁcance for identities of this form. In the former case, the terms introduced have a deﬁnite (albeit unknown) reference. In the latter case, the terms have no sense and do not refer—not even indeﬁnitely. In either case the imposition of a supervaluationist semantics upon contexts of the form ‘Nx:Fx = q’ will fail to remedy the underlying malady. One of the most signiﬁcant tasks facing any resolution of the Caesar problem is that of determining the character of the problem itself. Call this the circumscription problem, the problem of circumscribing the character of the problem that demands resolution. The direct stipulation and supervaluationist strategies fail to address this problem. They assume rather than show that the Caesar problem has a certain character (semantic indeterminacy). As a result the direct stipulation and supervaluationist strategies fail to resolve the Caesar problem.

194 / The Julius Caesar Objection

3.2 The Neo-Fregean Solution By contrast to the supervaluationist, the neo-Fregean proposes a solution to the Caesar problem that relies upon a distinctive ‘philosophical ontology’ (Hale and Wright 2001: 385–96).6 The solution is framed in the context of a theory of categories. A category is a collection of objects that share a common criterion of identity. Different categories are distinguished by the different criteria of identity associated with them. Now consider the possibility that an object belonging to one category is identical to an object drawn from another. The neo-Fregean claims that such trans-sortal identiﬁcations possess a distinctive epistemological feature: ‘there is simply no provision for or against such identities’; there are no encompassing identity criteria available that would allow us to settle whether objects drawn from distinct categories are the same or different (Hale and Wright 2001: 394). On the basis of the claim that trans-categorical are evidence transcendent in this way the neo-Fregean presents a dilemma. Either it is granted that there are true trans-categorical identities or it is not. If there are such identities then Frege may be convicted of overestimating the gravity of the Caesar problem. For this so-called ‘problem’ arises from the inability of (HP) to settle a statement concerning the identity of objects drawn from different categories (persons and numbers). But if it is a general truth that such identities cannot be settled then it can signal no defect in (HP) that it fails to settle a truth-value for impure identity claims. Alternatively, it may be denied that there are any true trans-categorical identities. But then Frege may be convicted of underestimating the capacity of (HP) to solve the Caesar problem. For the criteria of identity that (HP) stipulates for numbers are distinct from the characteristic criteria of personal identity (whatever package of psychological and bodily conditions that might be). Therefore persons and numbers belong to different categories and this fact sufﬁces for their numerical difference. So either the Caesar problem dissolves or it is solved. Either way, Frege failed to show that (HP) provided a defective mechanism for introducing numbers. 6

An earlier neo-Fregean treatment proposed to solve the Caesar problem by demonstrating that—despite Frege’s proclamation to the contrary—(HP) does sufﬁce for the falsity of impure identity contexts. This result was to be achieved by appealing to differences between the types of consideration that serve as canonical grounds for determining facts of personal and numerical identity (Wright 1983: 107–17). In response to criticisms by Rosen (1993: 171–4) and Sullivan and Potter (1997), the neo-Fregean adopted the position described above (Hale and Wright 2001: 367–85). See MacBride (2003: 128–35) for an overview and assessment of the debate.

Fraser MacBride / 195 Thedetailsoftheneo-Fregeanviewareclearlyopentoquestion.Consider,for example, the neo-Fregean claim that trans-categorical identities are evidence transcendent. This claim is motivated by the reﬂection that if it is legitimate to countenance the identity of Caesar with a number then it is equally legitimate to countenance the identity of, say, Frege with a Roman statue (an object drawn from another distinct category). They both constitute cases of bare, imponderable identities (Hale and Wright 2001: 394). But if persons and artefacts turn out to fall under a common category—the category of physical object—then it remains opens that the facts of identity and distinctness amongst persons and artefacts may be settled by veriﬁable considerations (for example, spatio-temporal duration and location). Clearly, the relevant notion of category requires greater development before the neo-Fregean proposal can be properly assessed. The claim that all trans-categorical identities share an evidential status may be questioned for another reason. Suppose Frege has mass m and the Roman statue mass n > m. Then it is natural to reason in the following way: nothing can have mass of both n and n > m values; so Frege and the statue cannot be identical. If we are to countenance the possibility that an arbitrary object (Frege) is really identical to an object located at another place (at the same time) with different intrinsic properties then this line of reasoning will have to be shown to be somehow at fault. I have described this elsewhere as the ‘problem of spatial intrinsics’ (by analogy with the more familiar problem of temporary intrinsics; see MacBride (1998: 223–7) for further details). To solve this problem we must give up some ordinary assumptions about intrinsic property possession. We will have to give up the assumption that intrinsic properties (mass, shape, etc.) are possessed simpliciter, that is, independently of spatial location. Instead we will have to think something of the following sort: intrinsic properties are possessed relative to spaces (and perhaps times too). And then there will be no incompatibility generated by Frege possessing mass and shape relative to the slice of space–time carved out by his life, and, another mass and shape relative to the duration and location of the Roman statue. What this suggests is a surprising result. Far less damage is done to our ordinary ways of thinking by countenancing the possibility of an identity between numbers and persons than by seriously entertaining the idea that different sorts of physical objects might be identical. By contrast to the latter, the identiﬁcation of numbers and persons does not force any revision or particular view of the way in which intrinsic properties are possessed. This reﬂects once

196 / The Julius Caesar Objection again a beguiling aspect of the Caesar problem noted earlier: the fact that there is no overt incompatibility between being a number and being a person. Of course, these are considerations of detail that may very well be addressed in the context of a fuller development of the neo-Fregean approach. Nevertheless, it is worthy of note that there are such details to be negotiated. For the neoFregean intends their philosophical ontology to be placed at the service of a logicist philosophy of mathematics. The greater the metaphysical depths the neo-Fregean must fathom to make good their claims the less likely it appears that the relevant theory of categories should draw on merely logical concepts and techniques for its expression. Independently of such considerations how well does the neo-Fregean solution fare with respect to the critical task of negotiating the various different aspects of the Caesar problem? How does the neo-Fregean solution fare with respect to the circumscription problem? Unlike the supervaluationist the neo-Fregeans do not assume that contexts of the form ‘Nx:Fx = q’ are semantically indeterminate. But they do make the contrasting assumption that such contexts are meaningful and determinate. Having made that assumption the neo-Fregean then sets about demonstrating that either the truth-values of impure contexts are imponderable or settled by categorytheoretic consideration. Recall, however, the version of the Caesar problem that denied (HP) supplied impure identities with any meaning whatsoever (Section 2.5, above). In that case, contrary to the neo-Fregean solution, impure identities cannot have any sort of truth-value, unfathomable or otherwise. Unfortunately, the neo-Fregean fails to address the circumscription problem. As a result the neo-Fregean fails to provide a solution to the Caesar problem. However, the neo-Fregean does offer two arguments to undermine the contention that one might rest content with the situation that Carnap was willing to tolerate—the situation where impure identities lack a truthcondition or value (Hale and Wright 2001: 340–5).7 One argument operates at the level of understanding and appeals to Evans’s Generality Constraint (Evans 1982: 100–5). Construed as a linguistic principle this constraint exercises a control on the understanding of sentences. To understand an expression is 7 It is Heck (1997), rather than Carnap, that provides the immediate target of Hale and Wright’s arguments. Heck argues that a Fregean-style proof on the inﬁnity of the number series may be carried out in the context of a many-sorted logic. He refrains, however, from endorsing the view that the kinds of Caesar problem under consideration here may be resolved by adopting a many-sorted logic.

Fraser MacBride / 197 to understand the contribution it makes to the meaning of all the signiﬁcant sentential contexts in which it occurs. So a subject may only grasp a particular sentence if he or she grasps the range of signiﬁcant sentences that result from the permutation of understood constituents. Consider the possibility currently at issue: that a subject may understand a range of pure numerical identities (‘Nx:Fx = Nx:Gx’) and pure personal identities (‘Caesar = Julius’) but fail to comprehend the signiﬁcance of mixed identities (‘Nx:Fx = Caesar’). The Generality Constraint appears to rule this possibility out. For if the subject ‘fully understands’ the pure identities then they must also understand the sentences that result from the permutation of the constituent terms ‘Nx:Fx’, ‘Nx:Gx’, ‘Caesar’, ‘Julius’, and ‘. . . = . . .’. But impure identities occur amongst the results of such a permutation. So it cannot be the case that (HP) serves to ﬁx the signiﬁcance of pure numerical contexts whilst neglecting entirely the signiﬁcance of impure contexts. This argument is open to question. First, maintaining the discussion at the level of understanding, it may be granted that having a full understanding of pure identities requires that a subject must be able to understand all the signiﬁcant permutations of them—impure identities included. But suppose that (HP) provides only a partial understanding of pure numerical identities. In that case there need be no conﬂict with the Generality Constraint. For a subject whose understanding of pure numerical identities is mediated by (HP) and yet fails to comprehend the signiﬁcance of impure identities need be a subject with only a partial understanding of the signiﬁcance of pure identities. Second, the argument stands in need of qualiﬁcation. The Generality Constraint does not require that a subject understand the range of grammatical sentences that result from the permutation of understood constituents. It requires only that a subject understand the resulting range of signiﬁcant sentences. Suppose impure identities even though grammatical are meaningless. It follows that the Generality Constraint fails to rule out the possibility of understanding pure but not impure identities. The neo-Fregean argues however that impure identities are meaningful and so this possibility does come into conﬂict with the Generality Constraint. Hale and Wright declare ‘the thought dies hard that identity is categorically appropriate simply to any object’ and offer two considerations in favour of the contended signiﬁcance of impure identities (2001: 344, 350–1). The ﬁrst consideration incorporates an appeal to the contrapositive of Leibniz’s Law (‘the diversity of the dissimilar’). Suppose that Caesar and Nx:x = x belong

198 / The Julius Caesar Objection to mutually exclusive categories (persons and numbers). Then Caesar is a person whereas Nx:x = x is not. Since Caesar has a property (being a person) that Nx:x = x lacks it follows that they are distinct. So the relevant impure identity (‘Caesar = Nx:x = x’) is false rather than meaningless. However, this argument presupposes that claims of trans-categorical distinctness are themselves meaningful and capable of receiving a truth-value. But if transcategorical identity statements are meaningless then so are trans-categorical distinctness statements. So the argument from the diversity of the dissimilar begs the question against the view that impure identities are meaningless. The neo-Fregean therefore offers a second consideration. Hale and Wright argue that it is ‘utterly unclear’ how a case may be made for the claim that impure identities lack a sense. An appeal to intuitions of signiﬁcance is likely to be unsatisfactory because our intuitions do not speak in unison. An appeal to stipulation is also out of order. They conclude: ‘what is needed is a principled reason for denying that this conﬁguration of individually signiﬁcant words adds up to an expression which, taken as a whole, expresses something true or false’ (Hale and Wright 2001: 351). But this argument also appears to beg the question. The meaning-theoretic version of the Caesar problem arises from the fact that (HP) fails to settle any signiﬁcance for impure identities—it leaves a semantic gap there. This provides a principled reasoning for doubting whether impure identities do express something. Of course, if the default assumption is made that identity contexts exhibit a free wheeling compositionality—so every grammatical permutation of them is meaningful—then the difﬁculties encountered in settling the signiﬁcance of impure identities provide no grounds for doubting that they have a meaning. They will have a meaning regardless. But since the Caesar problem raises the question of whether every grammatical identity has a sense the assumption of free wheeling compositionality can hardly be relied upon to bolster a solution to the problem itself. The neo-Fregean also supplies a metaphysical argument to undermine the contention that impure identities are meaningless. Appeal is made to Frege’s avowed ‘Platonism’: the contention that numbers belong to an inclusive domain of objects.8 The neo-Fregean reasons that if numbers are to belong to 8

A further argument the neo-Fregean might plausibly employ here appeals to the fact that we appear (at least) to be able to simultaneously count objects drawn from distinct categories. For example, it seems intelligible to ask: how many natural numbers less than ten and planets

Fraser MacBride / 199 such a domain then there must be a fact of the matter about which objects the numbers are. In other words, there must be a determinate truth about whether a number is identical or distinct to any other object (however presented). The neo-Fregean concludes that if Platonism is to be a legitimate position then it cannot be the case the impure identities are meaningless. But if there is any legitimacy to the concern that—for all (HP) settles—impure contexts lack a sense then this argument simply places a question mark over whether Platonism is a legitimate position. This in turn raises a doubt concerning the effectiveness of any purported solution to the Caesar problem—the neo-Fregean solution included—that presupposes Platonism.

4. Conclusion LetusreturntoFrege’soriginalformulationoftheCaesarproblem(Section 2.1). Frege linked the notions of object and identity and then claimed that if a symbol a is to be used to denote an object then we must have available a criterion that determines in every case whether b is the same as a. We have seen that considerable difﬁculties confront any attempt to supply such a criterion. This suggests that it may be time to reconsider whether Frege was right to tie together the notions of object and identity in the manner he proposed. It is true that objects may be identiﬁed and re-identiﬁed and seen from different perspectives. No doubt it is the capacity of objects to be identiﬁed and re-identiﬁed in this way that is responsible for our treating discourse about objects in a realist fashion. But there does not appear to be anything in our ordinary interaction with objects that determines objects must be capable of being identiﬁed and re-identiﬁed from every point of view. Rather it appears that we ‘track’ objects across a range of relevant situations and perspectives. It would be an imposition to suppose that the ‘tracking conditions’ with which we habitually operate are identity criteria that tacitly determine the presence or absence of an object across all situations. One may therefore wonder whether of the solar system are there? This question appears to presuppose that there are facts of the matter concerning the identity and distinctness of numbers and planets. But appearances may be deceptive. Perhaps the question should be interpreted as asking for the result of the addition of the number of numbers less than ten and the number of the planets. Then no facts of trans-categorical identity will be called upon to settle the answer to the question.

200 / The Julius Caesar Objection Frege made any legitimate demand when he required identity criteria for the objects he planned to introduce. The doctrine that the notions of object and identity cannot be separated has, however, become deeply entrenched. It is therefore unlikely that this suggestion will be readily received. But there is something perplexing about this persistent adherence to Fregean doctrine. For, despite the conviction that every object has identity criteria, proponents of the view have been beggared to provide any. This goes for all kinds of objects, all the way up from quantum particles to persons. Even sets—often presented as the paradigm of objects with clear and distinct identity criteria—fail to meet the prevailing standards. For the Axiom of Extensionality that purportedly provides criteria of identity for sets neglects to make any mention of times or possible worlds. As a consequence Extensionality fails to determine whether sets are identical or distinct at different times or different worlds. So even someone who submits that there are well-deﬁned identity criteria for sets must overcome a version of the Caesar problem. Serious engagement with the Caesar problem and its presuppositions may well assist in identifying and assessing the conﬂicting intellectual forces that give rise to this perplexing situation.

References Benacerraf, P. (1965), ‘What Numbers Could Not Be’, Philosophical Review, 74: 47–73; repr. in P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics: Selected Readings, (2nd edn., Cambridge: CUP, 1983), 272–94. Boolos, G. (1987), ‘The Consistency of Frege’s Foundations of Arithmetic’ in J. J. Thomson (ed.), On Being and Saying: Essays for Richard Cartwright (Cambridge, Mass.: MIT Press), 3–20. Brandom, R. (1996), ‘The Signiﬁcance of Complex Numbers for Frege’s Philosophy of Mathematics’, Proceedings of the Aristotelian Society, 96: 293–325. Burgess, J., and Rosen, G. (1997), A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics (Oxford: OUP). ¨ Carnap, R. (1932), ‘Uberwindung der Metaphysik durch logische Amalyse der Sprache’, Erkenntnis, 2: 219–41; repr. as ‘The Elimination of Metaphysics through Logical Analysis of Language’, in A. J. Ayer (ed.), Logical Positivism (London: Allen & Unwin, 1959), 60–81. Dummett, M. (1967), ‘Frege’s Philosophy’, in P. Edwards (ed.), The Encyclopaedia of Philosophy (New York: MacMillan), 225–37; repr. in Dummett, Truth & Other Enigmas (London: Duckworth, 1978), 87–115.

Fraser MacBride / 201 (1990), Frege: Philosophy of Mathematics (London: Duckworth). Evans, G. (1982), The Varieties of Reference (Oxford: OUP). Field, H. (1974), ‘Quine and the Correspondence Theory’, Philosophical Review, 83: 200–28. (1980), Science without Numbers (Oxford: Blackwell). (1993), ‘The Conceptual Contingency of Mathematical Objects’, Mind, 102: 285–99. Fine, K. (1975), ‘Vagueness, Truth and Logic’, Synthese, 30: 265–300. Frege, G. (1950), The Foundations of Arithmetic, trans. J. L. Austin (Oxford: Blackwell). (1964), The Basic Laws of Arithmetic, trans. and ed. M. Furth (Berkeley and Los Angeles: University of California Press). Hale, B., and Wright, C. (1992), ‘Nominalism and the Contingency of Abstract Objects’, Journal of Philosophy, 89: 111–35. (1994), ‘A reductio absurdum: Field on the Contingency of Mathematical Objects’, Mind, 103: 169–84. (2001), ‘To Bury Caesar . . . ’ in their The Reason’s Proper Study (Oxford: OUP, 2001), 335–96. Heck, R. (1997), ‘The Julius Caesar Objection’, in Heck (ed.), Language, Thought and Logic (Oxford: OUP, 1997), 273–308. Hodes, H. (1984), ‘Logicism and the Ontological Commitments of Arithmetic’, Journal of Philosophy, 81: 123–49. Kitcher, P. (1975), ‘The Plight of the Platonist’, Noˆus, 12: 119–36. Leibniz, G. W. (1981), New Essays on Human Understanding, trans. and ed. P. Remnant and J. Bennett (Cambridge: CUP). Linsky, B., and Zalta, E. (1994), ‘In Defence of the Simplest Quantiﬁed Logic’, Philosophical Perspectives, 8: 432–58. (1996), ‘In Defence of the Contingently Nonconcrete’, Philosophical Studies, 84: 283–94. MacBride, F. (1998), ‘Where are Particulars and Universals?’, Dialectica, 52: 203–27. (1999), ‘Listening to Fictions: A Study of Fieldian Nominalism’, British Journal for the Philosophy of Science, 50: 431–55. (2000), ‘On Finite Hume’, Philosophia Mathematica, 8/2: 150–9. (2002), ‘Could Nothing Matter?’, Analysis, 62: 125–35. (2003), ‘Speaking with Shadows: A Survey of Neo-Logicisim’, British Journal for the Philosophy of Science, 54: 103–63. (2005), ‘The Julio Cesar Problem’, Dialectica, 59: 223–36. McGee, V. (1997), ‘How We Learn Mathematical Language’, Philosophical Review, 106: 35–68.

202 / The Julius Caesar Objection Parsons, C. (1964), ‘Frege’s Theory of Number’, in M. Black (ed.), Philosophy in America (London: Allen & Unwin), 180–203; repr. in W. Demopoulos (ed.), Frege’s Philosophy of Mathematics (Cambridge, Mass.: Harvard University Press, 1995), 182–210. (1990), ‘The Structuralist View of Mathematical Objects’, Synthese, 84: 303–46. Resnik, M. (1997), Mathematics as a Science of Patterns (Oxford: OUP). Rosen, G. (1993), ‘The Refutation of Nominalism (?)’, Philosophical Topics, 21: 149–86. Russell, B. (1911), ‘On the Relation of Sense-Data to Physics’, in his Mysticism and Logic (New York: Longmans, Green & Co), 139–70. Shapiro, S. (1997), Philosophy of Mathematics: Structure and Ontology (Oxford: OUP). Sullivan, P., and Potter, M. (1997), ‘Hale on Caesar’, Philosophia Mathematica, 5: 135–53. Williamson, T. (1998), ‘Bare Possibilia’, Erkenntnis, 48: 257–73. Wright, C. (1983), Frege’s Conception of Numbers as Objects (Aberdeen: Aberdeen University Press).

9 Sortals and the Binding Problem John Campbell

1. The Delineation Thesis Philosophers have sometimes maintained very strong theses of sortal dependence; theses to the effect that singular reference depends on knowledge of the sort of the object to which one is referring. They have held that the ability to single out in experience a tree, or a mountain, or a lion depends on grasp of such sortal concepts as ‘tree’, ‘mountain’, or ‘lion’. Their basic point is that there can be different things of different kinds in the same place at the same time. Suppose we consider a bend in the river. What object is there, at that point? There is, of course, the river, which continues on downstream. There is also a stage of a river: that is, the river as it is at this particular moment, for example. A ‘river-stage’ so deﬁned cannot last longer than a moment, unlike the river. There is also a collection of water molecules at that point. This collection will have been diffused in the sea by tomorrow, unlike the river itself. There is also a stage of that collection of water molecules, at the same place: a momentary stage in the life of that collection of water molecules. Quine (1953) gave the basic argument here when he said: Thanks to Rae Langton for a most helpful reply when an earlier version of this paper was presented at the Arche meeting, and to the participants for discussion. Thanks also to David Wiggins, and to the participants at an Oriel group meeting at which I presented another early draft. I have also beneﬁted from remarks by an anonymous referee. This paper is excerpted from the discussion of these topics in my Reference and Consciousness (Oxford: OUP, 2002).

204 / Sortals and the Binding Problem Pointing is of itself ambiguous as to the temporal spread of the indicated object. Even given that the indicated object is to be a process with considerable temporal spread, and hence a summation of momentary objects, still pointing does not tell us which summation of momentary objects is intended, beyond the fact that the momentary object at hand is to be in the desired summation. Pointing . . . could be interpreted either as referring to the river . . . or as referring to the [collection of water-molecules] . . . , or as referring to any one of an unlimited number of less natural summations . . . such ambiguity is commonly resolved by accompanying the pointing with such words as ‘this river’, thus appealing to a prior concept of a river as one distinctive type of time-consuming process. (Quine 1953: 67)

The argument here is explicitly about pointing, which is a way of orienting attention—it is one of the control mechanisms of attention—rather than conscious attention itself. And you might argue—indeed, in this chapter, I will argue—that Quine’s argument should be read as an argument about the control of attention, rather than about what is involved in consciously singling out an object. But that is not the natural reading of the signiﬁcance of Quine’s comment. On the natural reading, Quine’s comments imply that if you are to single out an object in experience, you have to be employing a relevant sortal concept, to delineate the boundaries of the object singled out. The idea is that that sortal concepts play, as it were, a top-down role in making it possible to pick out objects in experience. We could put the idea in terms of what I will call the Delineation Thesis: The Delineation Thesis: Conscious attention to an object has to be focused by the use of a sortal concept which delineates the boundaries of the object to which you are attending. It is, indeed, very often assumed that if you are to attend to Fs, you must already have the concept of an F (cf., e.g., Fodor 1998). But straight off, that is hard to believe. Of course, you cannot be intentionally attending to Fs without having the concept of an F; but for all that, you can be attending to an F without knowing that it is an F and without having that concept. It is a fundamental point that demonstrative reference to an object can succeed even though you do not know what sort of thing you are dealing with. In a garden centre I exclaimed at the beauty of a particular plant, only to be told that it was plastic. There was no relevant ambiguity about which thing I was talking about. I had plainly referred to the plastic plant. But a plastic plant is not a plant, any more than a fake Rembrandt is a Rembrandt. You can succeed in singling out

John Campbell / 205 an object despite a mistaken belief about what sort of thing it is. A still more radical case is the case in which you succeed in singling out the object even though you have not the slightest idea what sort of thing it is. Suppose that by some chance, an ordinary teacup from today survives for a thousand years. And suppose that when it is discovered by our descendants then, they and their circumstances have changed so dramatically that they could have considerable conceptual difﬁculty in understanding what a teacup is. To grasp the concept they would need to have a lot of background ﬁlled in of which they know nothing as yet. But the discovered teacup prompts a lot of discussion. There are learned conferences speculating as to its nature. A strong body of opinion leans to the view that it ‘probably had some religious signiﬁcance’. It is kept in a glass case in a museum. Given the intense discussion it receives, it would be absurd to say that our descendants have not managed to ‘single it out’. They can certainly make demonstrative reference to it; their experience of the object is sufﬁcient to specify uniquely which thing is in question. In both these cases, the plant and the teacup, what is happening is that your visual system is managing to bind together the information from a single thing, and you are consequently able to attend consciously to it, even though you have not managed to apply the right sortal concept to it. Application of the correct sortal concept seems therefore to be a more sophisticated phenomenon than conscious attention to the object; the Delineation Thesis is simply false. In this essay I want to go further into the possibility that the principles used by the visual system to bind together all the information derived from a single object may, in effect, play something of the role traditionally given to sortal concepts. These principles achieve that effect, I will argue, by making it possible to attend visually to an object even in the absence of knowledge as to what sort of thing it is. It is the principles used by the visual system to bind together information as all true of a single object that allow the subject to delineate the object in experience. And it is your capacity for conscious visual attention to the object that provides you with knowledge of the reference of a demonstrative referring to that thing.

2. The Binding Problem According to Quine, the involvement of the sortal concept is needed for there to be a determinate answer to the question: ‘to which object is the subject

206 / Sortals and the Binding Problem consciously attending?’. If we do not appeal to the subject’s grasp of a sortal concept, how could we say what the difference is between attending to a river and attending to a collection of water molecules, for example? Different styles of conscious attention will be used in attending to different sorts of object. For example, if you are consciously attending to a person over a period of time, the way in which you keep track of that person will be quite different to the way in which you keep track of a valley to which you are attending over a period of time. In consciously attending to the person you will, for example, keep track of the movements of that person; but in the case of the valley you will make no such allowance for movement (certainly none for movement with respect to the ﬂanking mountains). I want to propose that what underpins these differences in style of visual attention is not a difference in which sortal concept the subject is using, but a difference in the style of binding that the visual system is using. The binding problem is the problem the visual system has of putting together different features as features of a single object. The visual system processes different characteristics of objects in different specialized processing streams. Shape, size, colour, and motion, for example, each have processing streams devoted to them. Since visual information-processing involves computation of the object’s various characteristics in separate processing streams, the visual system has the problem of putting together all the information, in various processing streams, that relates to one and the same object. Although different areas in the visual cortex are used to process different features of objects, such as motion and depth, colour and shape, or location, there is no brain area where the ﬁring of neurons corresponds to the objects we see. So how are different features put together as features of a single object? Here is one way it could happen. Suppose that in each processing stream, not only is there information relating to the kind of feature in question—shape or motion or whatever—but also, information about where the feature is located. If all the processing streams carry location information as well as information relating to the speciﬁc property being processed, this gives the beginning of a solution to the binding problem: features found at the same location can be put together as features of one and the same object. The evidence both from cognitive studies, and from physiology, is that something like this solution is indeed what our visual system uses (cf. Treisman 1996; Zeki 1993). Of course location could not be the only criterion used: if you think of a hand grasping a glass, for example, the parts of the hand are

John Campbell / 207 no closer to one another than they are to the glass, but the hand and the glass are visibly different things. This means that location cannot be the only criterion used, and the natural suggestion is that the appeal to location must be supplemented with an appeal to something like Gestalt grouping principles in putting together the features of the hand as features of a single object, and the features of the glass as features of a single object. A keen sortalist might say that the visual system has to use processes which we can as theorists characterize using sortal concepts: that the visual system uses location plus the classiﬁcation ‘hand’, or location plus the classiﬁcation ‘glass’, in solving the binding problem. But there are two problems with this. First, such empirical evidence as there is does seem to suggest that it is general Gestalt principles, rather than speciﬁc sortal classiﬁcations, that are used in binding (Palmer and Rock 1994; Prinzmetal 1995). Secondly, as we shall see in a moment, the evidence from neuropsychology is that sortal classiﬁcation is a more sophisticated phenomenon than visual binding; visual binding can take place without sortal classiﬁcation, but sortal classiﬁcation in the absence of binding seems impossible. In any case, an appeal to sortal concepts as part of an empirical hypothesis about the way in which the visual system works is not what philosophers have traditionally had in mind when they appealed to sortal concepts; the suggestion was rather that an explicit grasp of sortal concepts by the subject was essential to the delineation of perceived objects. In summary, we can think of visual binding as exploiting the location of the object together with something like the Gestalt organization of the characteristics found at that location. And, course, the visual system can bind together features over time, as when we keep track of a moving object, and across sensory modality, as when we assign heard speech to the person seen before us. This is a more primitive phenomenon, involving a more primitive level of information-processing content, than the application of the subject’s grasp of sortal concepts. I would argue, furthermore, that we should think of the complex binding parameter used in vision as playing a role also in the subject’s conscious attention to the object. When the subject consciously attends to an object with the aim of ﬁnding out more about it, information about the object has to be selected from the various specialized processing streams in the visual system; as I said, there seems to be no point at which all and only the information relating to a single object is put together in the visual system. The particular value of the binding parameters—location together with the relevant Gestalt

208 / Sortals and the Binding Problem principles—in effect provides a way of identifying the object which can be used in recruiting information about that object in various processing streams, or to single out the object for the purpose of acting on the object. So the singling-out of an object in experience need not involve the application of sortal concepts; only the mechanisms of binding. Whether you are consciously attending to a river or a mass of molecules, for example, will show up in how your visual system binds together information from the thing over time. If you have to keep moving downstream to keep track of the object of your attention, then you are attending to a collection of water molecules rather than a river. If, on the other hand, you are binding together information from any point in the course of the river, as all relating to a single object, then you are attending to the river itself. The distinction between consciously attending to a collection of water molecules, and consciously attending to a river, is not particularly hard to draw, even without appealing to any grasp of sortal concepts by the subject. These differences in style of attention amount to differences in what I called the complex binding parameter used by the visual system in putting together the information true of the object. The binding parameter for perception of a person will have to allow for the possibility of movement by the person; the binding parameter for a valley will not have to allow for any possibility of movement by the valley. As I said, the complex binding parameter in effect provides an address for the object, by which it can be identiﬁed, at the level of conscious attention, in a way that can be used in recruiting information from various processing streams to allow veriﬁcation of propositions about the object, and action on the object. So the style of conscious attention to the object that is appropriate will depend on what sort of object is in question. The use of one style of conscious attention rather than another—that is, the use of one type of complex binding parameter rather than another—is nonetheless a more primitive phenomenon than the ability to use sortal concepts to classify the objects to which you can attend. Animals other than humans plainly have a repertoire of binding strategies available to them: a cat keeping track of a mouse is plainly using different binding strategies than a cat keeping track of its home. It seems evident, also, that you could be using various styles of conscious attention in keeping track of various of the things around you in a new environment, for example, without yet having learned what sorts of things any of them are. Philosophers interested in the fact that we can make demonstrative reference to various quite different sorts of physical object have frequently proposed

John Campbell / 209 that our understanding of sortal concepts has a foundational role to play in making it possible for us to refer to objects of these various sorts. What I am proposing in this essay, though, is that grasp of sortal concepts is a more sophisticated matter than is the mere capacity for demonstrative reference. What is important, in our capacity for demonstrative reference to different sorts of object, is our capacity to engage in different styles of conscious attention. And indeed, that capacity to engage in different styles of conscious attention is what plays the roles that have so often been assigned to our grasp of sortal concepts. Another way to put the point is in terms of the classical distinction between ‘associative’ and ‘apperceptive’ visual agnosias (Lissauer 1890; for discussion see Farah 2000). On the one hand, according to the classical distinction, there are ‘associative’ agnosias in which the patient sees the object, but does not recognize which object he is seeing—so the patient may be able to give quite a full description of the volumetric properties of, for example, a glove, but be at a loss to say what such a thing might be used for (cf. Humphreys and Riddoch 1987 on ‘semantic access agnosia’). A patient of this type seems perfectly capable of using and understanding perceptual demonstratives. Such patients seems to be plain counterexamples to the Delineation Thesis; there is no question but that they have singled out the object in experience, though they are unable to give a semantic classiﬁcation of it. The patient is even, in this kind of case, able to ﬁnd what sort of thing it is that he is looking at; his understanding of the demonstrative means that he has the resources to understand a demonstration that the object is of this or that sort; it may be only the capacity for speciﬁcally visual recognition of the sort of the object that is impaired. What makes it so compelling that the patient is able to understand visual demonstratives is that his situation is, after all, not so very different from that of an ordinary subject given an unfamiliar view of an object of a familiar type, or a view of an object of a sort he has never seen before. In those cases vision alone does not allow you to classify the object. But it is nonetheless compelling that in those cases you are still able to identify the object demonstratively, and to formulate hypotheses about that thing, or plans to act on it thus and so. Consider, in contrast, the so-called ‘apperceptive’ agnosics. An agnosic of this type may be able to copy complex ﬁgures, even though he insists that he has no idea what he is drawing. The patient will have no idea which objects there are to be found in the scene he has successfully copied, even though that is evident to an ordinary subject looking at the copy. Humphreys and Riddoch (1987) suggest that what is wrong here is, in effect, a problem with binding (cf. their discussion of ‘integrative

210 / Sortals and the Binding Problem agnosia’). The patient is able to see the simple components of the scene and where they are; it is just that there is a difﬁculty with visual integration. When such an agnosic views a scene, is he in a position to understand demonstrative reference to the object before him? It seems evident that he is not. He cannot, on the basis of vision, verify propositions about the object he is seeing; he is in no position to act on the object, and most fundamentally, his experience does not provide him with knowledge of which object is in question. It thus seems that for experience of an object to provide you with knowledge of the reference of a demonstrative referring to that object, there must be sufﬁcient integration of the object in experience—the various features or parts of the object must provide experience of it as a coherent single thing—but this experience of the object as a coherent unity need not involve semantic classiﬁcation of the object as an object of this or that sort.

3. What Justiﬁes Binding? So far I have talked about the binding strategies that the visual system uses to put together the various properties of a single object. The question I want to address now is: what causes and justiﬁes our use of those strategies? Is there a role for sortal concepts in explaining how we come to use those strategies, or in explaining why those strategies are correct? Consider, for example, the use of location as a binding parameter. As I said earlier, it seems likely that the visual system uses location as a fundamental parameter in binding together different features as features of the same object: features at the same location are, as a ﬁrst approximation, assigned to the same object. Someone who thinks there is a foundational place for sortal concepts in explaining how there can be conscious attention to objects may press the question: Why is binding together features at the same location the right procedure for the visual system to use? What causes and justiﬁes the use of this binding procedure? The null hypothesis is that there is nothing which justiﬁes the use of one binding procedure rather than another. On this hypothesis, which binding procedures the visual system uses is more fundamental than either the question which objects there are in the environment, or the question which sortal concepts the subject employs. It is more fundamental than the question which objects there are in the environment, because, the argument runs,

John Campbell / 211 the very notion of the ‘environment’ is always relative to a particular type of creature; it depends on the ecological niche the creature inhabits. And, on this view, the way in which the perceptual system solves the binding problem is one of the things that determines what kind of world the creature inhabits. Moreover, on this view, which procedure the visual system uses to solve the binding problem is also more fundamental than the question which sortal concepts we use. For which sortal concepts we use depends partly on what our perceptual skills are; the most fundamental sortal concepts are observational concepts, which we can apply to objects on the basis of perception alone. But then the way in which our perceptual system solves the binding problem is one of the things which determines what kinds of sortal concepts we can have and use. So our way of solving the binding problem cannot be justiﬁed by reference to our use of one rather than another collection of sortal concepts. Rather, our grasp of sortal concepts simply has to work with whatever solution the visual system ﬁnds to the binding problem. In any area, there is always, I think, a presumption in favour of the null hypothesis; but it is a startling thought, that the choice of a particular procedure to solve the binding problem may be quite unconstrained by anything other than the internal requirements of the visual system. We can see what the null hypothesis is saying by supposing we consider for a moment what would have to be happening for the visual system to be solving the binding problem in a radically non-standard way. Quine and Goodman used to talk about spatiotemporally scattered objects, which they claimed were just as real as anything else. So a typical spatiotemporally scattered object might comprise the top of one chair plus the base of a lightbulb. Non-standard binding would involve putting together the perceived properties of the top of the chair together with the perceived properties of the base of the lightbulb as properties of a single, albeit spatiotemporally scattered, object. These features would be combined to give an ‘object token’, which could then be compared to stored representations to determine its characteristics. We could have a still more radically non-standard form of binding. The properties of the top of the chair would all be bound, and the properties of the base of the lightbulb would all be bound, on the picture I just gave; the odd part is just putting the two together. But we could in principle have binding in which no two features from the same location were put together. The redness at this location, the squareness at that location, and the uprightness from a still further location could all be put together to give a single ‘object token’, to be compared to stored

212 / Sortals and the Binding Problem representations. The upshot would be a kind of collection of spatiotemporally scattered tropes. According to the null hypothesis, there is no justiﬁcation to be given for the visual system operating in the ordinary way rather than in the ways I just described; there is no objective advantage in the standard approach. The visual system proceeds in whatever way it does; that is a primitive datum. The way in which the visual system proceeds will determine which object tokens are constructed, and that in turn will determine what stored object representations the subject has. This in turn will determine what the sorts of objects are in the subject’s environment; it is up to the subject to use one system of object representations rather than another to delineate which things in the surroundings he is thinking and talking about. It is natural to protest, as against the null hypothesis, that the way in which we actually bind objects has objective advantages over these bizarre alternatives. But the natural suspicion is that this protest is just a parochial, conservative reaction which elevates habit into a kind of transcendent superiority. What reason could be given for thinking that one way of binding features into objects is better than another? I think we can immediately say that it is not believable that there are no normative constraints on binding. There do appear occasionally individuals whose visual systems have problems with binding—see, for example, Friedman-Hill et al. (1995), or Robertson et al. (1997). These patients are quite seriously impaired. Though they may, for example, be above chance at saying which features are present in a scene displayed to them, they will be at chance when saying which combinations of features are present. Or, as in the case of the patient described by Humphreys and Riddoch (1987), they may be able to copy a drawing of a complex scene accurately, but be doing it without any identiﬁcation of the objects involved. It is difﬁcult to accept that since there are no norms of binding, these patients cannot be described as impaired. Rather, it seems that there must be normativity here, since these patients are so palpably impaired. However, you might have a view on which getting it right in your use of particular binding procedures is simply a matter of doing it the same way as everyone else. That is, you might have a view of binding which is like the kind of account sometimes given of what it is to be going right or wrong in English grammar. On this picture there is such a thing as getting it right or wrong in grammar; but ultimately, rightness and wrongness are just a question of whether you are in step with other people speaking the same language. Similarly, you might have, as it were, a ‘community view’ of binding

John Campbell / 213 procedures, on which rightness or wrongness is simply a matter of agreement or disagreement with others in your community. This view implies that the only problem with a non-standard binding strategy, such as putting together features at different locations, and the only problem with the impaired patients I just mentioned, is social. Their only problem is that they bind differently to other people. But that seems entirely inadequate as an analysis of the problem. The problem with these subjects is that they cannot see the objects around them. The sortalist proposal, at this point, contains two elements. One is a causal hypothesis: that our visual systems use the binding strategies they do because we have to sortal concepts that we do. The other element in the sortalist proposal is normative: that what deﬁnes the objective of the binding strategies being used is the need to keep faith with the system of sortal concepts used by the subject. It has to be said, though, that neither of these ideas seems compelling. Given the similarities between human vision and vision in other species, it seems somewhat unlikely that the use of particular binding strategies in humans evolved in response to our possession of particular sortal concepts. And it is difﬁcult anyway to see how our possession of particular sortal concepts could have come ﬁrst; our grasp of the concepts of particular sorts of observable objects depends on our abilities to perceive them, which in turn depends on having the relevant binding strategies in use already. Grasp of a system of sortal concepts thus seems to depend causally on possession of a relevant set of binding strategies, which makes it hard to see how there could also be a causal dependence in the other direction. The second element in the sortalist proposal is also difﬁcult to accept. According to the second element, it is our system of sortal concepts that deﬁnes the objectives of the binding strategies that we use. Since the system of sortal concepts that we have presumably developed later than the binding strategies that we use, the sortal concepts are in effect providing a kind of after the fact justiﬁcation for the use of those binding strategies, on this view. But if we have in fact developed a system of binding strategies in virtue of which we can see various sorts of objects, then we do not need the development of sortal concepts to provide a justiﬁcation for the use of those strategies. The objective is simply to see the relevant objects, and binding strategies achieve that. To develop a system of sortal concepts, consequently upon this achievement, and then maintain that the objectives of the binding strategies were after all being set by this system of sortal concepts, adds nothing. We already have it in place

214 / Sortals and the Binding Problem that the objective of the binding strategies is to let us see the various sorts of objects around us, and there is no place for a further level of justiﬁcation.

4. Sortals as Orienting Attention There are two quite simple points which together can make it seem that there must be something right about the Delineation Thesis. One is that we usually do use sortal concepts in demonstrative constructions; very often, in discussion, you would not know which thing was in question unless your interlocutor used a sortal. The second point is that if someone uses an incorrect sortal classiﬁcation in a demonstrative construction, then what they have said cannot be regarded as correct as it stands; once the facts are known, the statement has to be withdrawn and replaced by a use of the correct sortal classiﬁcation. These points together can make it seem that (a) sortal concepts are indeed essential to demonstrative reference, and (b) cases in which there seems to be singular reference but incorrect sortal classiﬁcation are somehow not cases in which the singular reference has after all been successful. But actually, these points need to be independently explained and they do not in the end offer any support to the Delineation Thesis. I will take them in turn. On the ﬁrst point, there is a role for sortal concepts in demonstrative identiﬁcation which is much less fundamental than the kind of role allotted to them by the Delineation Thesis, but which is commonplace and pervasive. When I commented on Quine’s discussion of pointing, I remarked on the distinction between conscious attention to an object, and the control, or causation, of conscious attention. We can distinguish between the factors in virtue of which you can be said to be consciously attending to one particular object rather than anything else, and the factors which caused you to be attending to that object rather than anything else. For example, in the case of vision, it is arguable that you attend to a speciﬁc object in virtue, in part, of the fact that you are attending to the location the object is at. In contrast, I might orient your attention to the object by pointing, or by physically turning your head towards it while waggling the object in front of you. You can see the distinction between the two cases. Suppose that attending to the object is partly constituted by attending to the place, while the turning of your head is merely a cause of your attending to the object. Then you could have attended to the object, in the very same way, as a result of some other cause.

John Campbell / 215 For example, I could simply have pointed to the thing, or you might have become interested in it spontaneously. The upshot could still have been that you consciously attended to it in the very same way. As I said, we do not in ordinary communication conﬁne ourselves to saying ‘this’ and ‘that’; we very often do use phrases of the form, ‘this F’, where F is indeed a sortal term. Does this show that the Delineation Thesis is after all correct? I think it does not. We can distinguish between two ways in which a descriptive component can ﬁgure in demonstrative reference. Suppose you and I are strolling through the park and I point my umbrella and say something about ‘that clock to the left of the fountain’. How does the descriptive component ‘to the left of the fountain’ relate to the demonstrative ‘that clock’ in this case? There are two possibilities. One is that it states a descriptive condition which has to be met by any object, if it is to be the reference of my term. So if there is no clock to the left of the fountain, I have not referred to anything. The other possibility is that the phrase ‘to the left of the fountain’ is just a way of orienting your attention, exactly on a par with the ﬂick of my umbrella, which aims to direct your attention onto an object I have already identiﬁed, prior to my use of the phrase or the umbrella. The identiﬁcation of the object came ﬁrst, and now I am trying, by hook or by crook, to direct your attention to that thing, and it does not really matter how I achieve that effect. So if the clock is not really to the left of the fountain, but I have still managed to direct your attention to it by means of my use of the phrase, then my term still refers and you understand what I have said. In these terms, we can say that one commonplace role for sortals in demonstrative identiﬁcation is to orient attention to one object rather than another. The sortal really does play a role in the singling out of the object: it is what causes the orientation of attention to one thing rather than another. If I say to you, ‘that is very old’, you will very often have no idea which thing I am talking about until I supply the sortal, even if I point in roughly the right direction. You use the sortal to orient your attention onto the right object. But since the sortal here is functioning merely to orient your attention onto the right object, it can play its role successfully even if the object is not actually of that sort. That is why the notion of a ‘plant’ can be playing a role in referring to an object which is not actually a plant; it is ‘good enough’ in the sense that anyone looking in roughly the right direction will have their attention directed onto the right object by the sortal, even though the thing is not in fact a plant.

216 / Sortals and the Binding Problem To say that the role of the sortal is merely to orient attention towards the right object, though, is also to say that the use of the sortal is dispensable. You could in principle have your attention oriented towards that object by some other cause. This is what happens in the case of the long-preserved teacup. Our descendants manage to orient their attention onto the thing without the use of any sortals at all. This place for the role of sortals does not give them the kind of constitutive role that the Delineation Thesis envisages in making it the case that you are consciously attending to one thing rather than another. It is merely an external causal factor in the act of conscious attention. But it does explain why typically you simply would not understand a demonstrative ‘this’ which was not, implicitly or explicitly, accompanied by a sortal. This point, about the role of sortals in demonstrative reference, should not be interpreted as a point about the semantic treatment of complex demonstratives in English. We can contrast simple demonstratives, such as ‘this’ and ‘that’, with complex demonstratives, such as ‘this river’ or ‘that mountain’. Suppose that the contribution of the nominal ‘F’ in a phrase of the form ‘that F’ were merely to orient attention to the demonstrated object. Then the truth of a statement of the form ‘that F is G’, would not require that the object referred to be F. But it does have to be acknowledged that when you use a demonstrative ‘that F’ to identify an object which is not in fact F, your statement cannot be regarded as correct. On the face of it, at any rate, if I point to a sundial and say, ‘that clock is one hour slow’, then what I say is not literally true; it implies that there is a clock which is one hour slow, and there may be no such clock. One analysis of the situation is that in addition to orienting attention, the nominal ‘F’ restricts what the demonstrative can refer to (cf. Kaplan 1989a, 1989b). On the view I am recommending, it would be surprising if ordinary English did work in this way. For demonstrative reference, on the view I am recommending, does not intrinsically require that you get it right about the sort of the object. But it would in principle be possible for English to impose an extrinsic requirement on reference: that when the simple demonstrative is coupled with a nominal, the nominal must apply to the object for the simple demonstrative to refer to it. It may be, though, that the nominal in a complex demonstrative contributes to the truth-conditions of a sentence containing it otherwise than by imposing a condition on the reference of the demonstrative. Strawson (1950) suggested that a sentence of the form ‘that F is G’ says the same thing as a sentence

John Campbell / 217 of the form, ‘that is the F which is G’. On this interpretation, the simple demonstrative refers without any conditions on reference being imposed by the nominal, though the nominal may indeed serve the pragmatic function of orienting attention. But the nominal does contribute to the truth-conditions of the sentence. The sentence cannot be true unless the nominal applies to the object to which the simple demonstrative refers. In a recent discussion, Lepore and Ludwig (2000) provide an analysis of the same general type, in which we have a self-standing use of a simple demonstrative and a quantiﬁcational analysis of the role of the nominal: The key to understanding demonstratives in complex demonstratives is to see the concatenation of a demonstrative with a nominal, as in ‘That F’, as itself a form of restricted quantiﬁcation, namely as equivalent to ‘[The x: x is that and x is F]’. (Lepore and Ludwig 2000: 229; cf. King 2001)

On this analysis, the simple demonstrative refers without the nominal playing any role in ﬁxing its reference. Such an analysis nonetheless acknowledges that a statement of the form ‘that F is G’ cannot be true unless the object referred to is F; this is the second point with which we had to deal. And this provides no comfort to a proponent of the Delineation Thesis, which we must abandon.

References Farah, Martha J. (2000), The Cognitive Neuroscience of Vision (Oxford: Blackwell). Fodor, J. A. (1998), ‘There and Back Again: A Review of Annette Karmiloff-Smith’s Beyond Modularity’, in In Critical Condition, 127–42 (Cambridge, Mass.: MIT Press). Friedman-Hill, S. R., Robertson, L. C., and Treisman, A. (1995), ‘Parietal Contributions to Visual Feature Binding: Evidence from a Patient with Bilateral Lesions’, Science, 269: 853–5. Humphreys, Glyn W, and Riddoch, M. Jane (1987), To See but Not to See: A Case Study of Visual Agnosia (London: Erlbaum). Kaplan, David (1989a), ‘Demonstratives’, in Joseph Almog, John Perry, and Howard Wettstein (eds.), Themes from Kaplan (Oxford: OUP), 481–563. (1989b), ‘Afterthoughts’, in Joseph Almog, John Perry, and Howard Wettstein (eds.), Themes from Kaplan (Oxford: OUP), 565–614. King, Jeffrey C. (2001), Complex Demonstratives: A Quantiﬁcational Account (Cambridge, Mass.: MIT Press).

218 / Sortals and the Binding Problem Lepore, Ernest, and Ludwig, Kirk (2000), ‘The Semantics and Pragmatics of Complex Demonstratives’, Mind, 109: 199–240. Lissauer, H. (1890), ‘Ein fall von seelenblindheit nebst einem Beitrage zur Theorie derselben’, Archiv fur Psychiatrie und Nervenkrankenheiten, 21: 222–70. Palmer, Stephen E., and Rock, Irvin (1994), ‘Rethinking Perceptual Organization: The Role of Uniform Connectedness’, Psychonomic Bulletin and Review, 1: 29–55. Prinzmetal, William (1995), ‘Visual Feature Integration in a World of Objects’, Current Directions in Psychological Science, 4: 90–4. Quine, W. V. O. (1953), ‘Identity, Ostension and Hypostasis’, in From a Logical Point of View (Cambridge, Mass.: Harvard University Press). (1960), Word and Object (Cambridge, Mass.: MIT Press). Robertson, L., Treisman, A., Friedman-Hill, S., and Grabowecky, M. (1997), ‘The Interaction of Spatial and Object Pathways: Evidence from Balint’s Syndrome’, Journal of Cognitive Neuroscience, 9: 254–76. Strawson, P. F. (1950), ‘On Referring’, Mind, 59: 320–44. Treisman, Anne (1996), ‘The Binding Problem’, Current Opinion in Neurobiology, 6: 171–8. Zeki, Semir (1993), A Vision of the Brain (Oxford: Blackwells).

Part III Personal Identity

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10 Vagueness and Personal Identity Keith Hossack

1. Unclarity of Personal Identity If someone journeys by matter transfer booth, or undergoes a brain ﬁssion, it can seem unclear whether he himself will survive the adventure. The wellknown thought-experiments about personal identity raise questions which we cannot see clearly how to answer. What is the explanation of irresoluble unclarity about personal identity? We might say that sometimes there is no precise fact of the matter about identity of persons; that would be to explain the unclarity as a case of objective indeﬁniteness. We might instead say that in every case there is always a precise fact, but that in some cases we do not know how to discover it; that would be to explain the unclarity as simply our own ignorance. If materialism is true then probably the explanation of the unclarity is objective indeﬁniteness. One version of materialism says that persons are biological beings. We already know enough about biology to know that probably nature draws no sharp boundaries in the biological realm; that means there can be borderline cases where there is no fact of the matter whether a particular animal has survived, or whether a particular biological organ such as a brain has survived. A different version of materialism identiﬁes a person’s life with a psychological process sustained by an underlying physical I am grateful for very helpful comments to David Bell, David Papineau, Mark Sainsbury, and an anonymous referee; and to audiences at Shefﬁeld, Lampeter, and Sussex.

222 / Vagueness and Personal Identity mechanism. Again, we already know enough about mechanisms to know that, whether natural or artiﬁcial, probably there are no sharp boundaries in the realm of mechanisms. So standard versions of materialism seem committed to the view that personal identity can be objectively indeﬁnite. Cartesian dualism, in contrast, standardly denies that there can be borderline cases of personal identity: in the realm of the soul, dualists think, the relevant boundaries are always sharp. Dualism explains unclarity about personal identity as due to our own ignorance, not objective indeﬁniteness.1 Historically, dualism has been motivated less by abstract metaphysics than by issues in ethics and the philosophy of religion. For example, Kant held that it is a requirement of pure practical reason that moral agents be able to hope that if they so act as to be worthy of happiness, then they will in fact attain happiness. But happiness and worthiness of happiness seem uncorrelated in this present life, so Kant argued that belief in a future life is practically necessary.2 But this seems to require that the person who is to enjoy happiness in the future life be deﬁnitely the one who had made himself worthy of happiness; it would be unsatisfactory if it were less than deﬁnite whether the right person was enjoying the happiness. Dualist resistance to indeﬁniteness of identity is the reason that Reid and others were led to the initially odd-seeming doctrine that a person is a monad, i.e. a simple thing that has no parts.3 The motivation is that if we replace enough parts of a complex thing like the ship of Theseus, it 1

The epistemic theory of vagueness of Williamson (1994) invokes ignorance to explain unclarity even in borderline cases. A dualist and a materialist who subscribes to the epistemic theory will therefore be in agreement that unclear cases of personal identity are cases of ignorance. Where they will differ is over whether the unclear cases are ever borderline cases of personal identity, for the dualist will deny it, but the materialist will afﬁrm it. According to Williamson, vagueness and borderline cases are due to a special kind of ignorance which arises because of limitations of our powers of conceptual discrimination (1994: 237). Williamson says that if a concept is vague, it is because we are unable to discriminate it from other concepts we might have used instead, which draw the line in a slightly different place from where we actually draw it. This inability of ours, itself epistemic, reﬂects something ontological, namely the absence of a natural division where the line would naturally fall: in the absence of a natural division, there is nothing to ‘stabilize’ the extension of the concept; that is why the concept is vague (ibid. 231.) Thus even on the epistemic theory, the materialist will say that personal identity can be objectively indeﬁnite, because of an absence of natural boundaries; whereas the dualist will deny this. 2 See Kant (1933: ‘The Canon of Pure Reason’, A820–31); (1993: ‘On Assent Arising from a Need of Reason’, pp. 149–53). 3 ‘A part of a person is a manifest absurdity. A person is something indivisible, and is what Leibniz calls a monad’ (Reid 1983: Essay III, ch. 4).

Keith Hossack / 223 can seem indeﬁnite whether it survives; whereas a simple thing is not subject to this kind of indeﬁniteness. The question whether personal identity is determinate therefore seems a central issue in the debate between materialism and dualism. But can we even make sense of objectively indeﬁnite identity in the case of persons? Bernard Williams has suggested that there is something bafﬂing in the very idea of it. Williams (1970) invites one to imagine one has fallen into the clutches of a mad surgeon. The surgeon tells one that he will bring it about that tomorrow there will be two human persons A and B, each with a plausible claim to be oneself. A will be bodily continuous with oneself; B will be psychologically continuous with oneself; and either A or B will be tortured tomorrow. One naturally wonders whether one will still be surviving tomorrow, and if so whether one will be in pain. In normal cases bodily and psychological criteria of personal identity go together, but the point of Williams’s example is to oblige us to consider a case where they come apart. It may seem tempting to say there is no fact of the matter whether one will be A or B; for example, on the grounds that our concepts are simply not designed to cover such situations. But Williams thinks this would encounter an ‘extraordinary difﬁculty’. He writes: there seems to be an obstinate bafﬂement in mirroring in my expectations a situation in which it is conceptually undecidable whether I occur.4

The reason is as follows: Central to the expectation of [situation] S is the thought of how it will be for me, the imaginative projection of myself as participant in S. Suppose it is for conceptual reasons undecidable whether it involves me or not. The subject has an incurable difﬁculty about how he may think about S. If he engages in projective imaginative thinking (about how it will be for him) he implicitly answers the unanswerable question; if he thinks that he cannot engage in such thinking, he answers it in the opposite direction.5

If the supposition of indeﬁnite personal identity really faces an ‘incurable difﬁculty’, we might think we could develop Williams’s point into an argument against materialism generally. Since there can be borderline cases of bodily continuity, and borderline cases of psychological continuity, it follows that standard versions of materialism imply that there can be borderline cases 4

Williams (1973: 61).

5

Ibid.

224 / Vagueness and Personal Identity of personal identity, and hence that there can be indeﬁniteness of personal identity. But if indeﬁnite personal identity is so bafﬂing as to be incoherent, a version of materialism that implies it must itself be incoherent. I shall call this the Bafﬂement Argument against materialism. Richard Swinburne states it as follows: An awkwardness for any such empiricist theory is that there will inevitably be borderline cases. But there seems no intermediate possibility between a certain future conscious being being that person, and not being that person.6

Geoffrey Madell also endorses it. He writes: What I fear about the future pain is simply its being felt by me. We, rightly, ﬁnd unintelligible the idea that there could be a pain in the future which is in part mine and in part not. I claim, then, that the empiricist view of personal identity is incoherent in itself.7

The Bafﬂement Argument is open to the reply that it is not at all bafﬂing to learn that a person vaguely identical with oneself will be tortured tomorrow; I know exactly what to expect, for I know how to describe it from the thirdperson perspective, and what I must expect is that. If it is insisted that bafﬂement remains because one cannot effect the imaginative projection of oneself into the situation, the reply is that this is just false. One can project oneself quite vividly into the pain that the person who may or may not be oneself will feel tomorrow. So one can imagine the torture tomorrow ‘from the inside’, and one can conceive of its being indeﬁnite whether that will happen to oneself. So why should it be insisted that we still cannot really make sense of indeﬁnite personal identity? Despite this reply, it seems to me that the Bafﬂement Argument retains a residual ability to puzzle us. What seems so hard to imagine is not just the situation tomorrow, but one’s whole course of experience between today and tomorrow. Consider a variant of the mad surgeon’s experiment, where he permits only A to be alive tomorrow. One is told today that it is objectively indeﬁnite whether one is identical with A, and that A will be tortured tomorrow. How is one to imagine one’s future course of experience? If one is deﬁnitely not identical with A, one will not survive until tomorrow, and so one will have a course of experience that will end before tomorrow. If one is 6

Swinburne (1974: 240).

7

Madell (1989: 32–3).

Keith Hossack / 225 deﬁnitely identical with A, one will have a course of experience that extends through today into torture tomorrow. Either of these courses of experience is imaginable, but it is not so easy to imagine ‘from the inside’ a course of experience that is somehow borderline between the two. To put it another way, what one would like to know is whether the light of one’s own particular consciousness—this consciousness—will be on or off tomorrow, and what seems so bafﬂing is the answer that it will not deﬁnitely be on, but it will not deﬁnitely be off either. In this paper I attempt a diagnosis of this bafﬂement. I argue that the concept of a person is a theoretical concept of a theory that combines two distinctive components, namely a subjective conception of conscious states, and an objective conception of psychological agents. Exclusive concentration on the objective component is misguided, but it is equally wrong to overemphasize the subjective component. I suggest that the latter mistake is what underlies the Bafﬂement Argument: by focusing excessively on the subjective component it induces a solipsistic perspective from which indeﬁnite personal identity is indeed bafﬂing. But although indeﬁnite identity may bafﬂe the solipsist, it need not bafﬂe those whose concept of person gives proper weight to the place of persons as part of the objective order.

2. ‘I’—the ‘Essential’ Indexical What is a person? I shall assume that x is a person only if x has a mind, and that x has a mind iff x has the capacity to know something. One way to know something, and therefore to have a mind, is to instantiate a conscious state; for I take it that a state S is conscious iff necessarily if x is in S then x is consciously aware of x’s being in S. To say that a state is conscious is to say that its subject is aware of it, and hence is aware of what is in fact a state of itself. In that sense any conscious mind is conscious of itself. It does not follow, however, that such a mind is self-conscious, for it may lack a concept of its Self. Because of the possibility of such conceptual deﬁciency, awareness of what is in fact a state of itself need not give a mind self-awareness. There is no contradiction in supposing that a complex event could have a simple mode of presentation: if event E is the instantiation of state S by entity x, and M is a mode of presentation of E, it need not follow that M has a part which presents S and a part which presents x. In

226 / Vagueness and Personal Identity particular, then, a mental event might have a simple mode of presentation, in which case it would fail to have parts which present either the instantiated mental state or the instantiating mental subject. The distinction between mere consciousness and full self-consciousness was relied on by Locke to distinguish between minds generally and the special kinds of minds that are persons. According to Locke, a person is: a thinking intelligent Being, which has reason and reﬂection, and can think of it self as it self, the same thinking being in different times and places.8

Here ‘it self ’ seems to be occurring as the English indirect reﬂexive, i.e. the pronoun used in indirect speech to report the content expressed by a speaker who refers to himself as ‘I’.9 Locke’s deﬁnition then amounts to the claim that a being with a mind is a person if it is able to think of itself by means speciﬁcally of the concept or mode of presentation that is expressed by the word ‘I’. A concept is individuated by its cognitive role, which is a matter for epistemology. So on Locke’s assumptions, the philosophy of the self is one of those rare cases where metaphysics is best done by epistemological methods; if a self is that which can grasp a self-concept, we study the Self best by studying the self-concept, and therefore by studying its cognitive role. The cognitive role of the self-concept expressed by ‘I’ has at least the three following distinctive features: (1) Warrant: My normal warrant for ‘I am in pain’ ceases to be a warrant if a co-referring expression is substituted for ‘I’. I know by immediate introspection that I am in pain, but even if I am A, I cannot know by introspection that A is in pain—unless I know that I am A. The same applies to other contents warranted by introspection. (2) Desire: ‘Thank goodness I am not to be tortured!’ This expression of relief will not be apt if a co-referring expression is substituted for ‘I’. It is not the same to say ‘Thank goodness A is not to be tortured!’, even if I am A. I am not relieved that B is to be tortured and A is not—what’s so good about that? Here is a characteristic kind of non-contrastive relief which can only be properly expressed in English by the ﬁrst-person and its cognates, thus: I am glad I will not be tortured myself (indirect reﬂexive). (3) Action: Substitution of another mode of presentation for the ﬁrstperson can destroy the rationalization of action. Suppose I desire that the 8

Locke (1975: Bk. ii, ch. xxvii, § 9).

9

Geach (1957).

Keith Hossack / 227 race should start, and believe that if I wave my arm the race will start. The desire and belief jointly rationalize my trying to wave my arm, but they do not rationalize my trying to wave A’s arm, unless I know that I am A. Perry and others give a psychologistic account of this ‘essential indexical’ role of ‘I’-thoughts, by appeal to the supposed causal powers of ﬁrst-personal ‘belief-states’.10 But we can give an account that is more Fregean in spirit if we regard conscious action as rationalized by a practical syllogism whose conclusion is the action. A practical syllogism is formally valid iff its action conclusion and its belief premiss formally entail its desire premiss, e.g.: (desire) the race starts (belief ) if I wave my arm then the race starts (action) I wave my arm A syllogism rationalizes an agent’s conscious action only if it describes it under that mode of presentation in which the action is given to the agent’s consciousness. Now consciousness of waving my arm warrants my thought that I am waving my arm, but it does not warrant the thought that A is waving his arm, even if I am A. So the following practical syllogism is not formally valid: (desire) the race starts (belief ) if A waves his arm then the race starts (action) I wave my arm Here the action and the belief do not formally entail the thing desired.11 The action would get me what I want, but I do not know that, and cannot rationally act on this belief and desire. Rationality can be restored here only by adding as an extra premiss the belief ‘I am A’.

3. How Do I Know I Exist? To investigate the self-concept, we must investigate its cognitive role. This will include an investigation of what warrants a subject’s judgements about himself. One judgement I am inclined to make about myself is that I exist: what warrants that? 10

Perry (1979).

11

cf. Campbell (1994: ch. 3).

228 / Vagueness and Personal Identity I cannot come to know of my existence by ordinary perception. I see my body, and thus have warrant for believing ‘This body exists’. Given some forms of materialism, I know of what is in fact myself that it exists. But I need not yet know that I exist, because the identity ‘I am that body’ would be informative for me. Nor can I do better by turning my mental gaze within. Even if there really were an inner gaze and, luckier than Hume, I managed to catch a glimpse of my cartesian Ego, still I would not know that I exist, for the judgement ‘I am that Ego’ would be informative for me. I cannot know of my own existence apriori. It is of course apriori that any token of ‘I exist’, including this one, is true. So I can know apriori of a token of ‘I exist’ which is in fact produced by me that it is true. But still I do not know apriori that I exist, for I do not know apriori that I produced that token. I cannot know of my existence by introspection. Following Reid I will use ‘consciousness’ to denote that immediate introspective knowledge one has of one’s own conscious states.12 Then consciousness will be part of my warrant for such knowledgeable judgements as ‘I am in pain’, or ‘I am thinking that snow is white’. But mere consciousness cannot warrant my judgement that I exist. For although consciousness informs me of the existence of a state of what is in fact myself, it cannot inform me of the existence of my self; as is shown by the existence of minds that are conscious but not self-conscious. Descartes missed this point in his famous paralogism Cogito, ergo sum. But his critics such as Hume and Kant did not miss it; they correctly insisted that mere consciousness of a conscious state does not on its own warrant a belief in the existence of the subject of the conscious states. As Lichtenberg put the point: To say cogito is already to say too much, as soon as we translate it I think. (1990: Notebook K, 18, p. 168)

According to Lichtenberg, all that is warranted on the basis of consciousness alone is a report of the conscious states in a subjectless style: We should say it thinks, just as we say it lightens. (ibid.)

Wittgenstein endorsed this manoeuvre. Discussing the word ‘I’ in Philosophical Remarks he says: It would be instructive to replace this way of speaking by another in which immediate experience would be represented without using the personal pronoun; We could 12 ‘Consciousness is that immediate knowledge we have of our present thoughts and purposes, and in general of all the present operations of our minds’ (Reid 1983: I. 7).

Keith Hossack / 229 adopt the following way of representing matters: if I, L.W. have toothache, then that is expressed by means of the proposition ‘There is toothache.’13

We can imagine constructing a Lichtenbergian language from our ordinary language. Whenever ordinary language contains a predicate which I am warranted in applying to myself by consciousness together with knowledge that the conscious state is a state of myself, we introduce into the Lichtenbergian language a corresponding subjectless sentence, for which the consciousness alone is warrant enough. Then we can replace ‘I am thinking’ by ‘It thinks’, ‘I am in pain’ by ‘It pains’, and so on. Each Lichtenbergian sentence expresses a content warrantable by consciousness alone; equivalently, it expresses an introspective mode of presentation of the relevant conscious state. A Lichtenbergian language is of course the Wittgensteinian ‘private language’, which ‘only I myself can understand’.14 In what follows, I side-step the question whether such a language is genuinely possible. All I require is the more modest assumption that there exist introspective modes of presentation of conscious states that are not modes of presentation of the Self. These are the modes of presentation which the Lichtenbergian ‘private language’ would express, supposing a private language were possible. A private language may be a ﬁction, but it is a useful one here, since it provides a convenient means of discussing these modes of presentation. A Lichtenbergian sentence is, by a convenient pun, subjectless in a double sense. First, it is subjectless because it is not a subject-predicate sentence. Instead it has the holophrastic logical form of a Quinean ‘observation sentence’; the sentence calculus is adequate for its formal representation. Secondly, a Lichtenbergian sentence is subjectless because it expresses a conscious mental state without explicit mention of the owner or subject of that state. Thus someone could speak Lichtenbergian even if he had no concept of a mental subject. He could state the occurrence of states of affairs which were in fact states of a subject, without having a concept of the subject. We can imagine someone following Descartes’s method of doubt, believing only what he ﬁnds indubitable. He no longer believes in outer things. He can even doubt his own existence, and pace Descartes the doubt is not selfrefuting—to say dubito ‘is already to say too much’. This skepticism is extreme indeed. It goes beyond even what Russell called ‘solipsism of the present 13

Wittgenstein (1975: 88).

14

Wittgenstein (1958b: §256).

230 / Vagueness and Personal Identity moment’, i.e. the belief that only I and my present experiences are real. Our skeptic goes further, for he lacks belief not only in the existence of other minds, but also in his own. Imagine our skeptic has been sunk in his skepticism so long that it is no longer right to say he doubts his own existence; for he has now entirely lost the very concept of himself. I shall call such a skeptic a ‘Lichtenbergian solipsist’. (I say ‘solipsist’ to accord with what I take to be Wittgenstein’s usage, but the terminology is of course a little misleading: the Lichtenbergian solipsist does not think that he is the only person who exists, for he lacks so much as the concept of his own or other minds.) Our solipsist follows Lichtenberg’s procedure, always saying ‘It thinks’ instead of ‘I think’, and so on. Is he hopelessly stuck in his solipsism? Or is there a way he can rationally come to believe again that he exists? If we can discern such a way, that would throw light on the tacit warrants we non-solipsists have for the belief ‘I exist’, and hence clarify the cognitive role of the self-concept.

4. Beyond Solipsism of the Present Moment A Lichtenbergian solipsist arrives at his epistemological predicament by doubting all his aposteriori sources of knowledge other than consciousness. He can attain knowledge of his own existence again only if he starts to draw on the other sources of knowledge that he has. If he is prepared to trust his memory, then he can have knowledge of the past. He can arrive at a conception of the future by remembering that in the past the present was not yet, i.e. was future. He can thus advance to a Lichtenbergian sentential language enriched with tense operators like in the past and always. A second source of knowledge he might come to trust is induction, i.e. the ability to project a remembered regularity into the future. That enables him to know Lichtenbergian psychological laws. For example: Always, if it pains, it is desired that it does not pain. Compare the subjectless meteorological law: Always, if it rains, it pours. Knowledge of a Lichtenbergian psychological law is knowledge of a genuine law of subjective psychology, and as I shall argue later, such knowledge plays a key role in our understanding of other minds. But no amount of such

Keith Hossack / 231 knowledge on its own will lead the solipsist to knowledge of the Self as one of the things that objectively exist, for the very conclusive reason that our solipsist may as yet completely lack the concept of a thing. The solipsist can arrive at this concept by trusting another source of knowledge—abduction, i.e. inference to the best explanation by means of a theory. To think of himself as ‘the same thinking thing at different times and places’ (Locke), our solipsist will need to have concepts of objective locations in time and space. Kant argued that one cannot have a conception of time and space without a conception of the physical objects that occupy time and space; this entails that Lockean self-consciousness requires concepts of physical objects. We arrive at the claim that (for human beings) there can be no conception of oneself without a conception of the physical world in which one is located. Is the converse true, that one have no conception of the physical world without a concept of oneself as located within it? Strawson and others have argued that it is true, on the grounds that one could not regard one’s experience as experience of independent objects without some rudimentary theory of perception which relates the content of one’s experience to one’s own location and the physical objects perceptible from that location.15 If so, the transition from Lichtenbergian solipsism to knowledge of the physical world would necessarily bring with it knowledge of one’s own existence. The claim would appear to be that if our solipsist is to have experiential knowledge of material objects in space and time, he will need to develop at least a rudimentary physics in which physical objects, places, and times are the theoretical entities. The solipsist’s evidence for the theory will be the course of his own conscious experience, so to link the theoretical entities with his consciousness, he will need to supplement the physics with ‘bridge laws’ or ‘correspondence rules’ that amount to a rudimentary psycho-physical theory of perception. This theory must link physical states of affairs with states accessible to consciousness, and the key point is that in the case of human beings it must do so by linking conscious experience to physical impingements on the theorist’s own body. Thus without something to play the role of his own body in the theory, there can be no prospect of a thinker having warrant for belief in physical objects. If I have a concept of what is my body in this sense, do I have a concept of myself? Must I take the adjective ‘my’ as the possessive case of the substantive ‘I?’ 15

Strawson (1966). For a careful discussion of this and the ensuing literature, see Cassam (1997).

232 / Vagueness and Personal Identity I think we must deny this if we wish to maintain a sharp division between mere mindedness and being a person. Example: it might not be deﬁnitely wrong to say that an animal can have the concept of its body, since it can acquire through perception a warranted belief that, e.g., there is food to its right; even so, it presumably remains deﬁnitely wrong to attribute to the animal a Lockean concept of itself. Our solipsist’s concept of his body will be embedded in a theory which posits physical objects as the ‘theoretical entities’, with the conscious experiential states of the theorist himself as the ‘observation sentences’. I see no reason why this theory could not be developed by a solipsist who conceived of his own conscious states only in the subjectless style of Lichtenberg. The solipsist would acquire a concept of what is in fact his own body, and would think of it under the special mode of presentation it has in virtue of its central role in his theory of perception. We may suppose that the solipsist employs a special name to express this special mode of presentation of his own body; perhaps he adopts for it the name ‘Centre’.16 Our solipsist has attained to a minimal physics, with an embedded theory of perception giving a special role to Centre. But in view of the possibility of a Lichtenbergian ‘observation language’, his doing all this does not entail that he has formed a concept of himself; he might remain a solipsist. In The Blue Book, Wittgenstein describes a solipsist who knows about the physical world, and is aware of his own conscious states, but lacks so much as the concept of his own or other minds. [T]he man whom we call a solipsist, and who says that only his own experiences are real . . . would say that it was inconceivable that experiences other than his own were real.17

Despite his extensive or even complete knowledge of the physical world and the psycho-physical laws relating his conscious states to the physical states of his body, this solipsist still lacks the concept expressed by the word ‘I’.

5. The Solipsist as Rational Agent Can this solipsist be a fully rational conscious agent, acting rationally upon his beliefs and desires, and learning rationally from his experience? We saw that 16

Wittgenstein (1975: 89).

17

Wittgenstein (1958a: 59).

Keith Hossack / 233 ‘I’-thoughts have an ‘essential indexical’ cognitive role, which is indispensable for our rationality. The solipsist has no self-concept, and therefore no ‘I’thoughts: but his Lichtenbergian sentences can play a corresponding ‘essential indexical’ role in his rational economy. (1) Warrant. Consciousness is the salient warrant for self-attributions like ‘I am in pain’. The solipsist has just the same warrant for the corresponding Lichtenbergian thought ‘It pains’. (2) Desire. There is a particular non-contrastive relief I feel when I learn I will not be tortured. The solipsist will feel the same non-contrastive relief when he learns that it is not Centre that will be badly injured, for from psycho-physical laws he knows he can infer ‘It will not pain’, and be relieved at that. His relief is non-contrastive, in just the required way: necessarily so, for he lacks the concepts of self and others, and so cannot draw the distinction that a contrastive relief would require. (3) Action. Action normally requires intentional movement of one’s body, so if a practical syllogism is to be formally valid its belief premiss must refer to the agent’s body under the mode of presentation whereby he is conscious of it in voluntary movement. But that is the mode of presentation under which the solipsist thinks of his body when he refers to it as ‘Centre’. So the solipsist’s action can be properly rationalized by a Lichtenbergian practical syllogism. For example, if he waves his arm in order to start the race, we have: (desire) the race starts (belief ) if Centre’s arm is waved, then the race starts (action) Centre’s arm is waved So even in the absence of a self-concept, our solipsist can think thoughts with an ‘essential indexical’ rational role, and so he can be a rational agent.

6. A Functionalist Conception of Human Beings At this stage about the only remaining difference between the Wittgensteinian solipsist and ourselves is the solipsist’s continuing lack of a self-concept. Because the solipsist has no concept of a self, he lacks the concept of a person, and cannot be aware of other human beings as persons. However, he knows of their bodies and can predict how their bodies will move by using physics.

234 / Vagueness and Personal Identity Moreover there seems nothing to prevent the solipsist from acquiring a theory of objective psychology, which will allow him more easily to predict how human bodies will move. An example of an objective psychology that is cognitively accessible to the solipsist is functionalism; for example, the functionalist ‘folk psychology’ that some say all normal human beings tacitly know. A functionalist theory says that human beings are functional agents with bodies; perhaps human beings are identical with their bodies, but this is a question functionalism need not settle. The psychological states of a psychological agent explain why his body moves as it does, in accordance with the functional role of each state. Such a functionalism would be fully cognitively accessible to a Wittgensteinian solipsist, but knowledge of the existence of psychological agents in the sense of functionalism is not yet knowledge of other minds. Functionalism treats pain as a psychological state, but it need not treat it as a conscious state. A fully developed functionalism may include functional analogues of introspection and qualitative character; for example, it may say that normally if x is in pain then x reliably believes that x is in pain and desires that x is not in pain, but this does not entail that the pain, the belief, or the desire are conscious states. There need be nothing in a deﬁnition of functional pain that requires that pain is a conscious state.18

7. Other Minds Theory Although our solipsist is by now an expert physicist and an expert objective psychologist, he still does not know that there are any other conscious beings in the world. He does not know this because he still lacks the very concept of a conscious being or a conscious state. He is aware of what are in fact his own conscious states, but thinks of them only in the subjectless Lichtenbergian way. To form a concept of the mental subject, the solipsist needs to conceive of his subjectless ‘It pains’ as being made true by a fact of subject-predicate form, 18 Here I am assuming that the functionalist theory in question is within the cognitive grasp of our solipsist. Might a more elaborate version of functionalism be able to link pain with consciousness? A conscious state is one that its subject is aware of, i.e. a state such that he knows he instantiates it. To build consciousness into ‘functionalism’ would therefore require a functionalist analysis of knowledge itself; it seems excessive to require our solipsist to come up with such an analysis, given that no philosopher has so far been able to do so.

Keith Hossack / 235 i.e. by the instantiation of some state by some entity. He must posit the existence of a class of conscious states corresponding to each of his Lichtenbergian sentences, and then he can posit a type of entity that can instantiate these conscious states. Only then will he be able so much as to grasp the hypothesis that conscious entities exist, and that he himself is one of them. How can the solipsist ﬁnd evidence for this hypothesis? He must link the hypothesis in a theoretically fruitful way to the rest of what he knows. He already knows that each normal human body is associated with an agent in the sense of functionalism who ‘owns’ that body; in particular, he knows there is an agent that ‘owns’ Centre. Suppose the solipsist calls this agent the ‘Ego’. The solipsist can now propose to identify the Ego with the entity that instantiates conscious pain when the Lichtenbergian sentence ‘It pains’ is true. Similarly, the solipsist knows there is a state that realizes the functional role of pain in human beings. He can propose to identify this state with the state instantiated by the Ego when ‘It pains’ is true. He will propose corresponding property identiﬁcations for other sentences of the Lichtenbergian language. A rich theory now emerges, in which the solipsist conceives of the Ego as both an agent in the sense of functionalism and as the conscious subject of the states of subjective consciousness expressed by Lichtenbergian sentences. Because the Lichtenbergian sentences are indexical, the solipsist’s concept of the Ego is an indexical one. Thus the solipsist has attained to a self-concept at last, and he may as well start using the indexical word ‘I’ to express it. By ‘I’ he expresses a theoretical concept of the following theory: I am the ‘owner’ of Centre; I exist; whenever it pains, I am in pain, whenever it thinks, I am thinking, etc. Such an I-concept will have the proper ‘essential indexical’ rational role, since it will straightforwardly inherit that role from the corresponding Lichtenbergian sentences. It seems plausible that this is in fact the very concept expressed by the English indexical ‘I’. Our solipsist has now arrived at a psychological theory that is more powerful than functionalism alone. For by attributing conscious states to other human beings, he enables the Lichtenbergian psychological laws he knows to generalize to other minds. So now he has a new method of understanding other people: he can put himself in the other person’s place by means of projective imaginative thinking. The Lichtenbergian psychological laws he knows from his own case can by this means be extended to apply to other people too, and thus he comes to understand other people better than he

236 / Vagueness and Personal Identity would if he were to rely on functionalism alone. The success of the method is evidence of the correctness of its assumptions; the solipsist is therefore warranted in believing in his own existence precisely because and to the extent that he is warranted in believing in the existence of other persons. This is the deep point of Wittgenstein’s discussion of the problem of other minds: skepticism about other minds is an unstable position, because anyone seriously skeptical about the existence of other people will undermine the evidence he has for his own existence. Ontological commitment to oneself is inseparable from ontological commitment to others.

8. Is Personal Identity Indeﬁnite? Wittgenstein suggests in The Blue Book that ‘as subject’ uses of the ﬁrst-person pronoun can be seen as immune to error through mis-identiﬁcation of the subject. There is no question of recognizing a person when I say I have toothache. To ask ‘are you sure it’s you who have pains?’ would be nonsensical. And now this way of stating our idea suggests itself: that it is as impossible that in making the statement ‘I have toothache’ I should have mistaken another person for myself, as it is to moan with pain by mistake, having mistaken someone else for me.19

As Wittgenstein perhaps intended, if there is indeed such a phenomenon as immunity to error through mis-identiﬁcation, it can be neatly explained in the light of the Lichtenbergian theory of the word ‘I’. For a person can be mistaken in the ‘as subject’ belief ‘I am in pain’ only if he can be mistaken in what warrants it. But the warrant is his awareness of pain under the introspective mode of presentation expressed by the Lichtenbergian sentence ‘It pains’. Since the Lichtenbergian sentence is subjectless, there seems no possibility of mistakenly believing it as a result of mis-identifying its subject; therefore ‘as subject’ uses of ‘I’ are immune from error arising from error about the identity of the subject. Keeping this point in mind, let us return to our consideration of the Bafﬂement Argument against materialism. The argument says it is hard to make sense of information that it is indeﬁnite whether a person who will 19

Wittgenstein (1958a: 67).

Keith Hossack / 237 be in pain tomorrow will be me. The suggestion is that, from the ﬁrstperson perspective, there is something incoherent in the materialist claim that personal identity can be indeﬁnite. There is an analogy here with immunity to error through mis-identiﬁcation, which we can bring out by putting the suggestion of the anti-materialists in the following way: ‘as subject’ uses of ‘I’ are immune from indeﬁniteness arising from indeﬁniteness about the identity of the subject. A neat Lichtenbergian explanation might again seem to be available. Consider again the situation where the mad surgeon is going to bring it about that the only candidate for being oneself tomorrow will be person A, who will be a borderline case of bodily and/or psychological continuity with oneself. One would like to know if one will still exist tomorrow. Since A will be tortured tomorrow, and one will exist tomorrow only if one is identical with A, then if one will exist tomorrow, one will be in pain tomorrow, and so one’s corresponding Lichtenbergian sentence ‘It will pain tomorrow’ is true now. So a way to express what one would like to know about one’s survival is this: if A is tortured tomorrow, will it pain tomorrow? Here I am expressing things from the Lichtenbergian perspective of a solipsist, but intuitively this does seem to be, at least in the present case, exactly ‘what matters in survival’.20 Now from a Lichtenbergian perspective, indeﬁniteness of identity and survival is indeed bafﬂing; for the sentence ‘If A is tortured tomorrow, it will pain tomorrow’ seems immune from indeﬁniteness about the subject, since it does not even employ a concept of the subject. Note that the Lichtenbergian sentence is not otherwise indeﬁnite, for even borderline torture is deﬁnite pain. When an opponent of materialism feels bafﬂement about how his personal identity can be indeﬁnite tomorrow, I suggest it is because he has slipped into thinking of things in this Lichtenbergian way. The subjectless sentence ‘It will pain tomorrow’ does indeed appear immune from indeﬁniteness about the subject. However, this appearance is misleading. For a vague object can make a sentence vague even if the sentence itself expresses no concept of the vague object. This can happen if the sentence concerned is indexical, for an indexical sentence may rely on the context to contribute an object that enters its truth-conditions, and if the context disobligingly contributes a vague object, then the indexical sentence itself will be vague. 20

Parﬁt (1971).

238 / Vagueness and Personal Identity Now non-solipsists know that Lichtenbergian sentences are indexical, so they understand that the subject does indeed enter the truth-condition of ‘It will pain tomorrow’: A token of ‘It will pain tomorrow’ by S at t is true iff S is in pain the day after t So non-solipsists understand that a token of ‘It will pain tomorrow’ will be indeﬁnite if it is indeﬁnite whether the person speaking at t is identical with the person who is tortured the day after t. Of course, the solipsist is bafﬂed to understand this, for he lacks the concept of a person. But the fact that the solipsist is bafﬂed does not mean that the rest of us ought to be.

9. Conclusion The Bafﬂement Argument says that from the ﬁrst-person perspective there is something incoherent in the materialist claim that personal identity can be indeﬁnite. My suggestion has been that the anti-materialist’s so-called ‘ﬁrst-person perspective’ is not the genuinely ﬁrst-person perspective of one who conceives of himself as part of the objective order. Rather it is the merely Lichtenbergian perspective of the solipsist, who does not conceive of himself at all. When an anti-materialist feels bafﬂement about how his personal identity can be indeﬁnite tomorrow, I suggest it is because he is thinking of things from this subjectless point of view. He can escape the bafﬂement easily enough by going beyond solipsism and seeing Lichtenbergian sentences as indexical with respect to the subject of experience. My account of the self-concept turned on the identiﬁcation of states of the conscious subject with the properties that realize the functional properties of objective psychology. If the account is correct, it will follow that in our present state of knowledge we cannot answer the question whether personal identity can be indeﬁnite. For our present concept of the self combines a functional with a subjective conception of conscious states, and neither of these can contribute any information on the question of the real essence of consciousness. For the time being at least, although we know what consciousness does, and although we know what it is like to be conscious, we know nothing of what consciousness is. It may turn out that a theory of the real essence of consciousness can be completed within biology, or within the theory of physical mechanisms; if

Keith Hossack / 239 so, then personal identity will have turned out to be indeﬁnite. But for all we presently know, the realm of consciousness may be one where nature in fact does draw sharp boundaries. If so, personal identity will not after all be objectively indeﬁnite, and the most plausible versions of materialism will have been refuted. But this is not something we can decide in advance apriori, either by the Bafﬂement Argument or indeed by any other species of knowledge argument for dualism.

Appendix: A Lichtenbergian Reconstruction of the Bafﬂement Argument Assume the scenario of Williams’s paper ‘The Self and the Future’. Subject S is told today that tomorrow there will be a person A, who will deﬁnitely be the only person to suffer pain tomorrow, and who is such that it is objectively indeﬁnite whether S = A. If materialism is true, what S is told is possibly true. But it is not possibly true, and so materialism is false. ‘Proof’: Use ‘’ as the ‘deﬁnitely’ operator. What S is told is (1) and (2) below: (1) ¬(S = A)& ¬¬(S = A) (2) (∀x)(x will be in pain tomorrow ↔ x = A) Assume you are deﬁnitely S. (3) (I am S) (4) (S = A ↔ I am A)

(from (3))

Now ‘imaginatively project’ yourself into A’s situation tomorrow: (5) (I am A ↔ will be in pain tomorrow)

(from (2))

In view of the connection between ‘I’ and your Lichtenbergian private language: (6) (I will be in pain tomorrow ↔ It will pain tomorrow) The pain in question is to be torture, i.e. deﬁnite pain. Since the Lichtenbergian sentence expresses no concept of the subject, it cannot be indeﬁnite whether it will pain tomorrow. So: (7) (It will pain tomorrow) v ¬ (It will pain tomorrow) We may assume that for the operator ‘’ the following rule of inference is valid: Deﬁnite Modus Ponens: p, (p → q) |- q

240 / Vagueness and Personal Identity Then we have from (4), (5), and (6) (8) (It will pain tomorrow) |- (S = A) (9) ¬ (It will pain tomorrow) |- ¬ (S = A) And hence by (7) we may conclude: (10) (S = A) v ¬ (S = A) (10) contradicts (1), so either what S was told is not possibly true, and so materialism is false, or else the method of projective imagining (premisses (5 & 6)) is not possible in this case. But it would be ‘bafﬂing’ to have a situation where projective imagining is not possible, since it is constitutive of the ‘I’-concept. So materialism is false.

References Campbell, J. (1994), Past, Space and Self (Cambridge, Mass.: MIT Press). Cassam, Q. (1997), Self and World (Oxford: OUP). Geach, P. (1957), ‘On Beliefs about Oneself ’, Analysis, 18; rep. in his Logic Matters (Oxford: Blackwell, 1972). Kant, I. (1933), Critique of Pure Reason, trans. Norman Kemp Smith (London: MacMillan). (1993), Critique of Practical Reason, trans. Lewis White Beck (Englewood Cliffs, NJ: Prentice-Hall). Lichtenberg, G. C. (1990), Aphorisms, trans. R. J. Hollingdale (Harmondsworth: Penguin Classics). Locke, J. (1975), An Essay concerning Human Understanding (Oxford: OUP). Madell, G. (1989), ‘Personal Identity and the Mind/Body Problem’, in J. R. Smithies and J. Beloff (eds.), The Case for Dualism (Charlottesville, Va.: University of Virginia Press). Parﬁt, D. (1971), ‘Personal Identity’, Philosophical Review, 80: 3–27. Perry, J. (1979), ‘The Problem of the Essential Indexical’, Noˆus, 13: 3–21. Reid, T. (1983), Essays on the Intellectual Powers of Man, excerpted in Inquiry and Essays, ed. R. Beanblossom and K. Lehrer (Indianopolis: Hackett Publishing Company). Strawson, P. (1966), The Bounds of Sense (London: Methuen). Swinburne, R. G. (1974), ‘Personal Identity’, Proceedings of the Aristotelian Society, 74: 231–47.

Keith Hossack / 241 Williams, B. (1973), ‘The Self and the Future’ (1st pub. 1970), repr. in his Problems of the Self (Cambridge: CUP). Williamson, T. (1994), Vagueness (London: Routledge). Wittgenstein, L. (1958a), The Blue and Brown Books (Oxford: Basil Blackwell). (1958b), Philosophical Investigations, trans. E. Anscombe (Oxford: Basil Blackwell). (1975), Philosophical Remarks, ed. R. Rhees, trans. R. Hargreaves and R. White (Oxford: Basil Blackwell).

11 Is There a Bodily Criterion of Personal Identity? Eric T. Olson

1. One of the main problems of personal identity is supposed to be how we relate to our bodies. A few philosophers endorse what is called a ‘bodily criterion of personal identity’: they say that we are our bodies, or that our identity over time consists in the identity of our bodies. Many more deny this—typically on the grounds that we can imagine ourselves coming apart from our bodies. But both sides agree that the bodily criterion is an important view, which anyone thinking about personal identity must consider. I have never been able to work out what the bodily criterion is supposed to be. Despite my best efforts, I have not found any clear position that plays the role in debates on personal identity that everyone takes the bodily criterion to play. What role is that? What is the bodily criterion supposed to be? Well, ﬁrst it is supposed to be a thesis about our bodies and how we relate to them. Second, it is supposed to be, or imply, an account of what it takes for us to persist through time. Speciﬁcally, it should imply that we go where our bodies go: it should rule out our having different bodies at different times, or surviving without a I thank Keith Allen, Andrew Deakin, Katherine Hawley, Jane Heal, Dan Hewitt, Diana Kendall, Hugh Mellor, Trenton Merricks, Gonzalo Rodriguez-Pereyra, an anonymous referee, and the audience at the 2000 St Andrews conference for comments on earlier drafts.

Eric T. Olson / 243 body. Third, the bodily criterion is supposed to be a substantive metaphysical claim that is neither trivially true nor trivially false. That we have bodies is uncontentious, or at least no more contentious than the existence of other physical objects; but there is supposed to be room for disagreement about whether we are our bodies. This is not just the old debate between dualism and materialism: it is supposed to be possible for a materialist—someone who thinks that we are made entirely of matter—to reject the bodily criterion by denying that we are identical with our bodies. Any thesis that deserves to be called the bodily criterion of personal identity ought to have at least these three features. It should also be compatible with the things that virtually all philosophers say about our bodies in other contexts. For instance, both those who accept the bodily criterion and those who deny it agree that a person’s body does not cease to exist simply because that person loses the ability to think. The bodily criterion should respect this; otherwise both sides of the debate would say, ‘that’s not what I meant by the bodily criterion’. Let us try to formulate a thesis that meets this modest standard.

2. The bodily criterion suggests two different thoughts: that a person’s identity consists in the identity of something called her body, and that we are identical with our bodies. These are not the same: Ayer, for instance, held the ﬁrst but not the second (1936: 194). But they are closely related, and it is no accident that they are commonly afﬁrmed or denied together. They are certainly supposed to be compatible. (This is another thesis about bodies that both friends and foes of the bodily criterion agree on.) Let us begin with the ﬁrst thought, that our identity is determined by the identity of our bodies. Here is a typical statement: Person A at time t1 is identical to person B at time t2 if and only if A and B have the same body. (Garrett 1998: 45) I take this to mean that one is, necessarily, that past or future person whose body then is the very thing that is one’s body now: Necessarily, for any person x existing at a time t and any person y existing at another time t∗ , x = y if and only if the thing that is x’s body at t is the thing that is y’s body at t∗ . Let us call this the Standard Bodily Criterion, or Standard Criterion for short.

244 / A Bodily Criterion of Personal Identity? Is this the bodily criterion? Well, suppose you were to lapse into an irreversible vegetative state: your upper brain is destroyed, while the lower parts that direct your ‘vegetative’ functions remain intact. The result is an animal that is fully alive in the biological sense—as alive as an oyster is alive—but with no mental features whatever. Most philosophers, I think, would say that your body still exists in this case. Of course, the Standard Criterion does not itself say this. It doesn’t say when something is the same body as an earlier one, and when not. This raises a problem: the bodily criterion will tell us nothing about our identity through time unless we have at least some idea of what it takes for a person’s body to persist; yet no one has ever produced a serious account of the identity conditions of human bodies. But never mind. In order to give the bodily criterion a run for its money, let us pretend that we know what it takes for a person’s body to persist. Speciﬁcally, let us accept that your body persists as a vegetable when you lapse into an irreversible vegetative state. Now what happens to you in this case? The bodily criterion ought to imply that you go where your body goes. If your body persists as a vegetable, then according to the bodily criterion you too ought to persist as a vegetable. But the Standard Criterion implies no such thing. That is because it applies only in cases where we have a person existing at one time and a person existing at another time; and there is no person left behind after your upper brain is destroyed. The usual view of personhood, anyway, is that a thing needs certain mental properties, or at least the capacity to acquire them, in order to count as a person. You might think that advocates of the bodily criterion should reject any account of what it is to be a person that involves mental properties. But there is no reason why they should. Such an account ought to be perfectly compatible with the bodily criterion. There is no evident contradiction in saying that beings with such-and-such mental properties are identical with their bodies, or that they persist if and only if their bodies do. The bodily criterion is not meant to be a deﬁnition of ‘person’, but rather an account of our identity through time. Moreover, if the bodily criterion is to be in competition with other accounts of personal identity, those accounts must all give identity conditions for people in the same sense of the word; otherwise they will simply be about different things. And what non-psychological account of personhood could advocates of the bodily criterion give? Now since a human vegetable has no mental properties and no capacity to acquire them, it is presumably not a person. So the Standard Criterion implies nothing about whether you could come to be a human vegetable—even

Eric T. Olson / 245 supposing that your body could come to be a vegetable. For the same reason, it implies nothing about what happens to you when you die. Your body may persist as a corpse when you die; but if a corpse is not a person, the Standard Criterion does not imply that you persist as a corpse. It simply does not apply here, for this is not a case in which we have ‘a person x existing at a time t and a person y existing at another time t∗ ’. The Standard Criterion even allows for you to have different bodies at different times, if at one of those times you are not a person. Suppose you get a bad case of senile dementia—so bad that you no longer count as a person. For all the Standard Criterion says, you may end up as a demented non-person with a different body from the one you have now. The Standard Criterion tells us far less than the bodily criterion was supposed to tell us. In fact the Standard Criterion is not really an account of what it takes for us to persist through time at all, but only an account of what it takes for us to persist as people—that is, what it takes for us to persist as long as we remain people. It tells us nothing about what would happen to us if we ceased to be people and became vegetables or corpses. Now many philosophers assume that every person is a person essentially, or at any rate that nothing could be a person at one time and a non-person at another. That entails that no person could persist without remaining a person. This assumption leads philosophers to ask what it takes for us to persist through time by asking what it takes for a person existing at one time to be identical with a person existing at another time; and it is this question that the Standard Criterion purports to answer. If being a person implies having mental properties, however, the assumption that people are people essentially is questionable. It entails the contentious metaphysical claim that whatever is a person at one time must necessarily cease to exist (not just cease to be a person) if it loses all of its mental properties. This is something that no advocate of the bodily criterion would accept. The bodily criterion is supposed to imply that we may end as unthinking vegetables or corpses, and hence (on the usual view of personhood) that we are people only contingently. If we are to take the bodily criterion seriously, then, we cannot assume at the outset that people are people essentially. We cannot ask about our identity over time in a way that presupposes something that is incompatible with the bodily criterion. Since the Standard Criterion does make this assumption, it is not an acceptable statement of the bodily criterion.

246 / A Bodily Criterion of Personal Identity?

3. We can avoid this problem by asking the right question about our identity over time (Olson 1997: 21–6): Under what conditions is a person existing at one time identical with anything, person or not, existing at another time? This suggests what we might call the Revised Bodily Criterion: Necessarily, any person x who exists at a time t is identical with something y that exists at another time t∗ (whether or not y is a person then) if and only if the thing that is x’s body at t is the thing that is y’s body at t∗ . That is, you are that past or future being whose body then is the thing that is your body now. If all people are essentially people, the standard and revised criteria are equivalent; but we cannot assume that here. But the Revised Criterion is not the bodily criterion either. Where the Standard Criterion implied too little, the Revised Criterion implies too much. The Standard Criterion implied nothing about whether you could one day be a vegetable or a corpse. The Revised Criterion implies that you deﬁnitely could not be; nor, for that matter, were you ever a foetus. And this is so even if our bodies were once foetuses and may one day be vegetables or corpses. Suppose you die peacefully and your body continues to exist as a corpse. Do you still exist? The Revised Criterion says that you do if and only if something that exists then has as its body the thing that is now your body: if the corpse has the same body then as you have now, the corpse is you. Alternatively, if the corpse is the very thing that is now your body, and if it is then the body of anything, you are the thing whose body it then is. Otherwise you no longer exist. (I assume that the corpse and the corpse’s body are the only candidates, after your death, for being the thing that is now your body.) What, then, is the corpse’s body? Or what thing is it the body of? That depends on what it is for something to be someone’s (or something’s) body. For any things x and y, under what conditions is x the body of y? This important question is almost universally ignored. Philosophers speak freely of ‘people’s bodies’ without saying a word about how they understand that phrase. If there are such things as our bodies, however, and if it makes sense to ask how we relate to them, this question must have an answer. It may be a vague or a messy or a complicated answer: for instance, there may be pairs of objects such that, for one reason or another, it is neither deﬁnitely true nor deﬁnitely false that one is the body of the other. But a vague or messy or complicated

Eric T. Olson / 247 answer is still an answer. There must be such an account if the bodily criterion of personal identity is to have any content at all. (Note that an account of what it is for someone to be embodied—to have a body—does not yet tell us which thing one’s body is. Nor does an account of what it is for a body to be someone’s body, until we know what counts as a ‘body’ in the relevant sense.) The most common account of what it is for something to be someone’s body is roughly this: your body is that material object by means of which you perceive and act in the physical world. More precisely, it is the largest such object: your left hand is not your body, even though you use it to perceive and act. You get knowledge of the physical world only by the way it impinges upon your body, via your sense organs. And you can affect the physical world only by affecting your body. To move your pencil, you have to move your body; but you can move your body directly, just by intending to, and without thereby moving anything else (save parts of it). I once called this the Cartesian Account of what makes something someone’s body—not because it implies Cartesian dualism, but because Descartes held something like it. The Cartesian Account is probably not right as it stands: for one thing, it appears to conﬂict with the conviction that limbs and organs that you can neither move nor feel may still be parts of your body (Olson 1997: 145–8). But let us ignore these worries and suppose that something like the Cartesian Account is correct. The Revised Criterion then amounts to this: Necessarily, any person x who exists at a time t is identical with something y that exists at a time t∗ if and only if the largest material object that x can move and feel directly at t is the largest material object that y can move and feel directly at t∗ . A moment’s reﬂection shows that this is not an acceptable statement of the bodily criterion. A corpse cannot learn about the physical world by any means whatever, let alone by virtue of changes in any particular material object. Nor can it move any material object just by intending to move it, for it cannot intend anything. Nothing is the corpse’s body on the Cartesian Account. For similar reasons, the corpse is not the body of anything. The Revised Criterion thus implies that you would cease to exist if you lost all your mental capacities, for without them you would have no body, and the Revised Criterion rules out your surviving without a body. You could never come to be a corpse, or a vegetable, even if the thing that is now your body came to be a corpse or a vegetable. Nor were you ever a foetus. During the

248 / A Bodily Criterion of Personal Identity? ﬁrst half or two-thirds of its gestation, anyway, a human foetus has no mental properties, and therefore no body. Nor was any foetus your body, for a foetus (on the Cartesian Account) is not the body of anything. So you did not exist when the foetus that produced you was developing in the womb, even if the thing that is now your body existed then. You don’t always go where your body goes. Your body—the thing that is now your body—exists at times when you don’t exist. It follows that you are not identical with your body. The Revised Criterion is actually incompatible with the idea that people are their bodies. This is clearly not what the bodily criterion was supposed to be.

4. We might try to avoid these troubles by moving to the second way of understanding the bodily criterion, namely the idea that people are their bodies. Call this Corporealism. As we have seen, the Standard Criterion does not imply Corporealism, since for all it says your body might outlive you; and the Revised Criterion is incompatible with it. There are at least two different versions of Corporealism. Weak Corporealism says that you are identical with that thing that is now your body. That is: Necessarily, for any person x existing at a time t, x is the thing that is x’s body at t. Though this does not explicitly say what it takes for us to persist through time, it implies that you are that past or future being that is identical with the thing that is now your body: Necessarily, for any person x existing at a time t and any y existing at another time, x = y if and only if y is the thing that is x’s body at t. So your identity is the identity of your body insofar as you and your body are one and the same. I doubt whether Weak Corporealism is what anyone means by the bodily criterion, for Weak Corporealism allows for a person to survive without a body. It even allows for people to exchange bodies. At any rate it does so on the Cartesian Account of what it is for something to be someone’s body. Suppose, as Weak Corporealism has it, that you are the thing that is now your body. That is, you yourself are the largest material object through which you now perceive the world, and which you can now move just by intending to move. Now imagine that surgeons cut the sensory and motor nerves that

Eric T. Olson / 249 connect your brain with the rest of you, but leave intact the connections that direct your life-sustaining functions. The resulting being would be alive but entirely blind, deaf, numb, and paralysed. (I take this to be physically possible, though it would no doubt be a delicate business in practice.) Presumably you could survive this: the resulting being would be you. At any rate the bodily criterion shouldn’t rule it out. On the Cartesian Account, however, you would thereby cease to have a body, for there would no longer be any material object that you could move or feel. Thus, you could survive without having any body at all, even if you are identical with your body. This is not because you are only contingently identical with your body, but because your body only contingently has the property of being your body. It is easy enough to alter the story so that you get a new body. Let the surgeons hook up the motor and sensory nerves leaving your brain to the appropriate nerves of my spinal cord (and my optic nerves, etc.) by wires or radio links, so that you can move my limbs and perceive via my sense organs in just such a way that the thing that was once my body comes to be your body. Adding appropriate links between my brain and your spinal cord would make the thing that was previously your body, my body. We could exchange bodies. You could even outlive your body: if we destroyed the thing that comes to be your body in this way, you would still exist. This is precisely the sort of thing that the bodily criterion was meant to rule out. Yet it is compatible with Weak Corporealism. We could avoid this by moving to Strong Corporealism, the view that each person must, at every time when she exists, be identical with the thing that is her body then: Necessarily, for any person x existing at any time t and any y existing at any time t∗ , x = y if and only if y is the thing that is x’s body at t and x is the thing that is y’s body at t∗ . That is, you are identical with that thing that is now your body and which is, at every other time when it exists, its own body. The persistence of your body as such is necessary and sufﬁcient for you to persist. This rules out your existing without a body or exchanging bodies. However, it also rules out your existing at a time when you have no mental capacities. For you would then be unable to act or perceive, and so nothing, not even yourself, would count as your body. Strong Corporealism therefore implies that you were never a foetus and could never come to be a vegetable or a corpse. In fact your body was never a foetus and could never come to be a

250 / A Bodily Criterion of Personal Identity? vegetable or a corpse, since you are your body and you couldn’t do so. For that matter, Strong Corporealism implies that both you and your body would perish if surgeons disconnected the sensory and motor nerves leaving your brain. No one sympathetic towards the bodily criterion would accept that. Strong Corporealism is not what anyone meant by the bodily criterion either.

5. We have not yet found a statement of the bodily criterion that does what that view is supposed to do. The Revised Criterion implied that you could never be a foetus or a vegetable or a corpse, even if your body is, and ruled out our being identical with our bodies. Weak Corporealism allowed for people to survive without a body and to change bodies. Strong Corporealism implied that neither people nor their bodies were ever foetuses, or could come to be vegetables or corpses, or could survive total paralysis. None of these views is the bodily criterion we learned about as students. Now many of these difﬁculties can be laid at the door of the Cartesian Account of body ownership. So we might try a different account of what makes something someone’s body. There are not many to choose from. The literature offers two main rivals to the Cartesian Account. Shoemaker counts a thing as your body if its relation to you is near enough to what the Cartesian Account speciﬁes.1 Someone who is completely paralysed may still have a body, he says, if there is something that she would be able to move just by intending to, were it not damaged or disabled. Roughly, x is y’s body if and only if y would be able to move and feel x in the ‘Cartesian’ way if x were intact. This would rule out your surviving without a body or exchanging bodies with someone else as a result of your sensory and motor nerves’ being cut or re-routed, making Weak Corporealism look more like what the bodily criterion was supposed to be. But Shoemaker’s account still appears to imply that a foetus, vegetable, or corpse neither has a body nor is the body of anything. A foetus is intact, yet until at least mid-gestation it cannot move or feel anything, or be moved or felt in the relevant way. There is nothing the foetus relates to in anything like the Cartesian way, and not just because a 1 (1976: 115). What he actually offers is an account of what it is to be embodied, not of what it is for something to be someone’s body; so I extrapolate.

Eric T. Olson / 251 few nerves are cut. Nor is there anything that a vegetable or a corpse relates to in anything like the way in which you relate to the thing you can now move and feel. Nothing there is in any way capable of doing any moving or feeling. If a foetus, vegetable, or corpse neither has a body nor is the body of anything, then Weak Corporealism still implies that you could exist without a body (assuming that the thing that is now your body was once a foetus or could one day be a vegetable or corpse). Even if Shoemaker’s account could be amended to avoid these problems, it faces a more serious obstacle: it implies all by itself that we are identical with our bodies. The implication is not merely that we are identical with our bodies if the bodily criterion is true. On Shoemaker’s account, the very idea of what it is for something to be someone’s body implies that we are our bodies. Or at least it does so given an assumption that most philosophers, whether friends or foes of the bodily criterion, would accept, namely that we are made of the same matter as our bodies. That is, we are things of ﬂesh and blood, and not immaterial souls, bundles of perceptions, or what have you; and we are no larger or smaller than our bodies. Call this Standard Materialism. It implies that you yourself are the largest thing that you can move and feel in the direct Cartesian way, or that you would be able to move and feel in that way if it were intact. (At any rate you are one such thing. I will return to this complication in the next section.) Hence, on Shoemaker’s account of what it is for something to be someone’s body, you qualify as your own body on Standard Materialism. But the bodily criterion is not supposed to follow from Standard Materialism. Many Standard Materialists, including Shoemaker himself, deny that we are our bodies and reject the bodily criterion root and branch. The view that we are made of the same matter as our bodies but are nonetheless numerically different from them is widely held. Shoemaker’s account of body ownership would make this view self-contradictory. I take this to be unacceptable. Tye (1980) proposes that a person’s body is the bearer of those physical and spatio-temporal properties truly predicable of her in ordinary language. The idea is this: we say truly, when we aren’t doing philosophy, that Margaret Thatcher is human (in the biological sense), is female, is over ﬁve feet tall, and so on. Even those who think that in the strict and philosophical sense Thatcher is an immaterial substance or a bundle of perceptions will agree that this naive description is in some sense accurate. What makes it accurate, Tye says, is that these things are strictly true of the material object that is Thatcher’s body. So that is what her body is.

252 / A Bodily Criterion of Personal Identity? This view is in some ways more promising than the Cartesian Account and its variants. But it too makes Standard Materialism imply that we are identical with our bodies; for in that case those physical and spatio-temporal properties truly predicable of you in ordinary language really are, in the strict philosophical sense, properties of you. So neither of these alternative accounts of body ownership is any help in formulating the bodily criterion.

6. The content of the bodily criterion depends on what it is for something to be someone’s body. We want an account of this relation that would enable the bodily criterion to play the role it is supposed to play in debates about personal identity. The account should not imply that most of those who claim to reject the bodily criterion are in fact committed to it; in particular, it shouldn’t make Standard Materialism a version of the bodily criterion. None of the accounts of body ownership currently on offer meets this standard. Perhaps we can do better by inventing one of our own. We seem to need something more stringent than those we have considered: not just anything that you would be able to move and feel in the Cartesian way if it were intact, or that bears those physical and spatio-temporal properties truly predicable of you in ordinary language, counts as your body. Here are three suggestions. 1. Your body is that body that bears those physical and spatio-temporal properties truly predicable of you in ordinary language, or that you would be able to move and feel in the Cartesian way if it were intact. The bodily criterion would then be the claim that you are that body. Standard Materialists may agree that you bear the relevant physical and spatio-temporal properties, but disagree about whether you are a body; so they can disagree about whether you are identical with your body, and hence about whether the bodily criterion is true. But what is a body? Not just any material object, or even any connected material object that contrasts with its surroundings, for according to Standard Materialism you are a body in that sense. A human body, then. (Set aside worries about non-human people.) But what is a human body? If it is just the body of a human person, then the proposed amendment—replacing ‘thing’ in Tye’s account with ‘body’—adds nothing. Perhaps a human body is a human organism. The bodily criterion would then be the claim that each of us is identical with a human animal: Animalism, as it is sometimes called.

Eric T. Olson / 253 Animalism does have many of the features that the bodily criterion is supposed to have. It is neither trivially true nor trivially false. It looks consistent with Standard Materialism, but does not follow from it. And it implies something about what it takes for us to persist through time, namely that our persistence conditions are those of animals. But Animalism is not the bodily criterion. It does not belong to the idea of one’s body that it must be an animal, let alone a human animal. At any rate, people often say that someone could have a partly or wholly inorganic body: our bodies might one day include artiﬁcial limbs, or even be entirely robotic. But no animal—no biological organism—could be partly or wholly inorganic. If you cut off an animal’s limb and replace it with something inorganic, the animal only gets smaller and has an inorganic prosthesis attached to it. So the view that people might one day have partly or wholly inorganic bodies allows for a person to be identical with his body without being an animal, or to be an animal without being identical with his body. Many philosophers even say that an animal’s body is always a different thing from the animal itself. They say that an animal stops existing when it dies, but unless its death is particularly violent its body persists as a corpse; or they say that an animal, by virtue of large-scale changes of parts, can have different bodies at different times. That would mean that no one could be both an animal and identical with his body. Animalism would be incompatible with the bodily criterion. I am in no position to say whether these claims about bodies are true. But they are not obviously false; and they don’t seem to conﬂict with the very idea of one’s body. So animalism is not a thesis about how we relate to our bodies. The idea of one’s body doesn’t come into it. One could of course stipulate: ‘By ‘‘x’s body’’ I shall mean the biological organism that bears those physical and spatio-temporal properties truly predicable of x in ordinary language.’ But it would be a stipulation, and no part of what anyone ordinarily means by ‘body’. The bodily criterion was supposed to be about bodies in some ordinary sense of the term. Thus, animalism is not the bodily criterion. 2. Some say that a person’s body by deﬁnition lacks any mental properties (Wiggins 1976: 152). I suppose it is absurd to suppose that Bertrand Russell’s body denied the existence of God. The reason, on this view, is that Russell’s body obviously did not have that belief. So we might say that your body is that bearer of those physical and spatio-temporal properties truly predicable of you in ordinary language (or that thing that you can move and feel in the relevant way) that has no mental properties. Standard Materialists could then deny that we are our bodies.

254 / A Bodily Criterion of Personal Identity? But this would make Corporealism trivially false. If part of what it is to be one’s body were to be something that lacks any mental properties, then it would be evident that we are not our bodies, for it is evident (eliminative materialism aside) that we have mental properties. Yet no one thinks that the bodily criterion is trivially false. 3. Perhaps your body is that bearer of those physical and spatio-temporal properties truly predicable of you in ordinary language that has brute physical (non-psychological) identity conditions.2 Many philosophers say that our identity through time consists in some sort of mental continuity. They will deny that we are identical with our bodies in this sense, even if they are Standard Materialists. The Standard Materialists who take us to have brute physical identity conditions are those who accept the bodily criterion. The bodily criterion would be the claim that you are identical with your body in this sense. This is the most promising attempt to state the bodily criterion that I know of. For all that, it faces two serious technical problems. The ﬁrst arises from a snag in Tye’s account of body ownership and others that I have ignored up to now. What thing is it that bears those physical and spatio-temporal properties truly predicable of you in ordinary language? Presumably a certain human organism does. But many philosophers believe that other things do too. Any Standard Materialist who denies that you are your body believes that the same matter can make up more than one material object at once. That suggests that a certain hunk of ﬂesh or mass of matter now stands to ‘your’ organism as your body stands to you, or as (according to a popular view) a hunk of clay stands to a clay statue made from it. Owing to metabolic turnover, an organism coincides with different masses of matter at different times. If, as many believe, you coincide both with a mass of matter and with an organism, then there are at least two things that bear those physical and spatio-temporal properties truly predicable of you in ordinary language and have brute physical identity conditions. This could mean either of two things. If your body must be the only thing that has the relevant features, then nothing is your body. Yet both those who accept the bodily criterion and those who deny it agree that we have bodies. Or we might say that anything with the relevant features is your body—one of your bodies, that is. So the animal and the mass of matter coinciding with you would both be bodies of yours: you would have at least two bodies. Worse, 2

I owe this inspired suggestion to Katherine Hawley.

Eric T. Olson / 255 those bodies would have different identity conditions. Your ‘animal body’ would continue to exist as long as it remains biologically alive (so I should argue, anyway). Your ‘matter body’ would remain your body only as long as that mass of matter remains in human form: a fraction of a second. The moment your metabolism assimilates or expels a particle, the mass would begin to disperse. In a few months’ time it would be widely scattered. (If a mass of matter must be connected, it exists only momentarily.) Which body would it be that, according to the bodily criterion, is you, or whose identity determines your identity? Of course, there may be no such things as masses of matter. There may be only one thing with brute physical identity conditions that bears those physical and spatio-temporal properties truly predicable of you in ordinary language. But the intelligibility of the bodily criterion ought not to turn on this deep metaphysical issue. This problem plagues other accounts of what makes something someone’s body as well: many philosophers believe that there is more than one largest thing that you would be able to move and feel in the Cartesian way if it were intact. Here is the second problem. Some philosophers say that we are human organisms, but deny that those organisms have any identity conditions, brute physical or otherwise (Merricks 1998). Our identity over time does not consist in anything other than itself. Nearly everyone would call this a version of the bodily criterion. But it would not be a version of the bodily criterion on the current proposal. Perhaps the bodily criterion is the view that we are those things that have the relevant physical and spatio-temporal properties and lack psychological identity conditions. But consider the view of Lowe (1996: 35 f., 41 ff.) and Baker (2000) that we have no identity conditions, yet coincide with human organisms numerically different from ourselves. Lowe and Baker deny that this is a version of the bodily criterion, and no one has ever disputed this description. Yet it would be a version of the bodily criterion on our revised proposal, implying that we ourselves are our bodies. I can see no good solution to either of these problems.

7. The accounts of body ownership that Descartes, Shoemaker, and Tye propose imply that the bodily criterion is not the controversial view that many materialist philosophers oppose, but rather something that follows trivially

256 / A Bodily Criterion of Personal Identity? from Standard Materialism. And our attempts to ﬁnd an account of body ownership that avoided this problem merely turned up more trouble. So we have not yet found the bodily criterion. You might conclude from this that there is no one bodily criterion, but rather a number of different bodily criteria that philosophers have failed to distinguish. The problem, you might think, is merely that those who ask whether we are our bodies, or whether our identity through time is determined by the identity of our bodies, have not said which notion of ‘body’ they mean. But none of these versions is acceptable as a statement of the bodily criterion. And if they really were versions of the bodily criterion, it ought to be possible to say explicitly what they are versions of, and why. There ought to be a more general thesis about how we relate to our bodies that the various versions entail. This would be the bodily criterion we seek. Yet we have been unable to ﬁnd such a thesis. What could it mean, then, to say that Animalism (for instance) is a version of the bodily criterion? If there is any unifying idea behind the various views that are called versions of the bodily criterion, it is that we are material things whose identity consists in some sort of brute physical continuity: Necessarily, any person x who exists at a time t is a material thing, and is identical with any y that exists at another time t* if and only if x relates at t in some brute physical way to y as it is at t*. Given an appropriate speciﬁcation of the ‘brute physical way’, this claim seems perfectly intelligible. It is an important position in debates on personal identity. And it seems to follow from most views that are called versions of the bodily criterion.3 Is this what we were after? I think not. The Anodyne Bodily Criterion, as we might call it, is the bodily criterion in name only. It doesn’t mention bodies. That notion doesn’t come into it. That was its virtue: if it were about our bodies and how we relate to them, it would presumably be equivalent to one of the unsatisfactory formulations we considered earlier. We needn’t talk about our bodies in order to say that we are material and that our identity consists in brute physical continuity. Introducing that notion only makes trouble. You can call brute physical continuity ‘bodily’ continuity if you like, but you will only come to grief if you try to understand it in terms of things called people’s bodies. If the 3

Merricks’s view, mentioned at the end of the previous section, is an exception.

Eric T. Olson / 257 bodily criterion is nothing more than the Anodyne Criterion, it is not a thesis about bodies at all. Stated carefully, it is not the view that we are identical with our bodies, or that our identity consists in the identity of our bodies. But then we might as well say that there is no bodily criterion of personal identity.

8. Let me now venture a diagnosis of the problem. I suspect that the reason we have been unable to ﬁnd the bodily criterion of personal identity is that there is none. Why did we all think there was? Presumably it was because we took phrases like ‘Wilma’s body’ to be names of objects. To say that Wilma has a muscular body, we supposed, is to state a relation between Wilma and a certain muscular object. Having a muscular body is like having a powerful car, except that one has one’s body in a more intimate sense than one has one’s car. Without some sort of assumption like this there can be no question of how we relate to our bodies, and no bodily criterion worthy of the name. Our inability to state the bodily criterion, if nothing else, suggests that this linguistic assumption is false. If this is hard to believe, consider a closely related term: mind. Does the phrase ‘Wilma’s mind’ purport to name an object? When we say that Wilma has a clever mind, are we stating a relation between Wilma and a certain clever object? Is having a clever mind like having a clean shirt, except that one has one’s mind in a more intimate sense than one has one’s shirt? I think most philosophers would say no. Otherwise many metaphysical questions would arise: What is this clever thing made of? How big is it? How clever is it? (As clever as Wilma?) How exactly does it relate to Wilma? (Is it identical with her? Or a part of her?) What is it about this mind that makes it Wilma’s, rather than someone else’s? Could it have been the mind of someone else? And so on. These are questions no one asks. But if minds were things that people in some sense own, they would have to have answers. This is not to deny that real questions are sometimes put in the language of people’s minds. We may ask, for instance, whether a person’s identity consists in the identity of her mind. But no one means this literally. It is only an informal way of asking something we should put more carefully in other terms: whether a person’s identity consists in some sort of psychological unity and continuity, for instance. When we speak carefully, apparent reference to the mind as an owned mental object disappears. To speak of someone’s

258 / A Bodily Criterion of Personal Identity? mind is presumably just to speak in a loose and colourful way of her mental properties, where the notion of a mental property can be explained without talking about things called minds. To say that Wilma has a clever mind is to just say that Wilma is clever (with the implication, perhaps, that cleverness is a mental property: though Wilma may be tall, we don’t say that her mind is tall). I suggest that talk of people’s bodies is like talk of people’s minds. Our bodies are no more physical things that we somehow have than our minds are mental things that we have. To speak of a person’s body is just to speak in a loose and colourful way about her ‘bodily’ properties, where the notion of a bodily property can be explained without talking about things called people’s bodies. To say that Wilma has a muscular body is just to say that Wilma is muscular. (With the implication, perhaps, that being muscular is a bodily property. What makes it absurd to say that Russell’s body denied the existence of God may be the absurd implication that denying the existence of God is a bodily property.) The human body is the subject matter of anatomy and physiology only in the notional sense that the mind or psyche is the subject matter of psychology. The true subjects of those sciences are the anatomical, physiological, and mental properties of human beings.4 When we speak carefully, reference to the body as a physical object whose relation to its owner is open to question disappears. That is why the only acceptable statement of the bodily criterion is not about bodies. We should no more expect there to be a bodily criterion of personal identity relating people to their bodies than we should expect there to be a ‘mental’ criterion of personal identity relating people to things called minds. If this is right, there is no meaningful question of how we relate to our bodies, any more than there is a real question of how we relate to our minds. There may be real questions that the phrase ‘how we relate to our bodies’ suggests: whether we are made entirely of matter, for instance, or whether our identity through time consists in some sort of brute physical continuity. But they are not really questions about things called bodies. There is no bodily criterion of personal identity strictly so called. More generally, there is no concept of a body as a thing that each of us has. This would have wide implications. The notion of one’s body as an owned object plays an essential role in a great deal of philosophical thought. The landscape would look rather different without it. 4 Versions of this view are defended in van Inwagen (1980), Rosenberg (1998: 71–3), and Olson (1997: 152 f.).

Eric T. Olson / 259 I am not conﬁdent that this diagnosis is correct. Perhaps I simply haven’t been clever enough to understand the bodily criterion. But there is clearly some sort of problem here. Those who believe that there is a bodily criterion of personal identity ought to tell us what they think it is.

References Ayer, A. J. (1936), Language, Truth, and Logic (London: Gollancz). Baker, L. R. (2000), Persons and Bodies (Cambridge: CUP). Garrett, B. (1998), Personal Identity and Self-Consciousness (London: Routledge). Lowe, E. J. (1996), Subjects of Experience (Cambridge: CUP). Merricks, T. (1998), ‘There are No Criteria of Identity over Time’, Noˆus, 32: 106–24. Olson, E. (1997), The Human Animal (New York: UP). Rosenberg, J. (1998), Thinking Clearly about Death (2nd edn., Indianapolis: Hackett). Shoemaker, S. (1976), ‘Embodiment and Behavior’, in A. Rorty (ed.), The Identities of Persons (Berkeley and Los Angeles: University of California Press). Tye, M. (1980), ‘In Defense of the Words ‘‘Human Body’’ ’, Philosophical Studies, 38: 177–82. van Inwagen, P. (1980), ‘Philosophers and the Words ‘‘human body’’ ’, in van Inwagen (ed.), Time and Cause (Dordrecht: Reidel). Wiggins, D. (1976), ‘Locke, Butler, and the Stream of Consciousness’, in A. Rorty (ed.), The Identities of Persons (Berkeley and Los Angeles: University of California Press).

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INDEX

abstract objects and the Caesar problem 175–6 numbers as 116, 117–19 and the Standard Conception of metaphysical necessity 21 action and the ‘I’ self-concept 226–7 and the solipsist as rational agent 233 advanced modal claims, and modal ﬁctionalism 66–71 agnosics, and sortal concepts 209–10 analytic necessity 15–16 analytical methodology, and the Non-Standard Conception of metaphysical necessity 27–8 Animalism, and bodily criterion 252–3, 256 Anodyne Bodily Criterion 256–7 Anselmian God, and the Non-Standard Conception of metaphysical necessity 30–1 ante rem structuralism 4–5, 109, 110, 111, 112, 113–21 and the Caesar problem 125 and eliminative structuralism 136, 143 and the governance of identity 164, 165, 169–72 and haecceities 140 Hellman’s objections to 141–3 and the identity of indiscernibles 5, 132, 133, 139 and the individuation problem 136 and Ker¨anen’s principle (STR) 169–70 Ker¨anen’s reply to Shapiro 5–6, 146–63 aposteriori infallibilism 47, 49 aposteriori realism, and the Caesar problem 177 appercetive agnosics, and sortal concepts 209–10

apriori infallibilism 45–9, 50–2 apriori necessities 51 apriori red ﬂags 50 apriori reﬂection 50, 52, 53–5 Aristotelianism, and mathematical structuralism 135–6 Aristotle, on mathematical methodology 119–20 Armstrong, D.M. 2, 26, 27, 118 axiom of extensionality 5, 151, 152, 165, 167 and the Caesar problem 200 Ayer, A.J. 243 backwards causation, and chance 104 Bafﬂement Argument against materialism, and personal identity 8, 224–5, 236–8, 239–40 Baker, L.R. 255 basic truth, and modal infallibilism 40–56 Benacerraf, Paul 136, 159, 161, 162 and the Caesar problem 186–7, 188 ‘What Numbers Could Not Be’ 109, 113, 114, 122 binding, and sortal concepts 7, 205–14 Blackburn, Simon, Oxford Dictionary of Philosophy deﬁnition of essence 115, 116 bodily criterion of personal identity 8–9, 242–59 ‘animal bodies’ and ‘matter bodies’ 254–5 and Animalism 252–3, 256 Anodyne Bodily Criterion 256–7 and Corporealism 248–50, 249–50, 254 and minds 9, 257–8 Revised Bodily Criterion 246–8, 250 Shoemaker’s account of 9, 250–1 Standard Bodily Criterion 243–5 and Standard Materialism 251, 252, 253, 256

262 / Index bodily criterion of personal identity (cont.) Tye’s account of 9, 251–2, 254 Brock, Stuart, objection to modal ﬁctionalism 58–63 Burgess, J. 131 Caesar problem 6–7, 174–202 dimensions of the problem 176–7 epistemological 6, 177–80 and mathematical structuralism 125, 127, 128, 129 meaning-theoretic 6–7, 177, 185–9, 198 metaphysical 6, 177, 180–5 neo-Fregean solution 194–9 and object and identity 199–200 supervaluationist solution 190–3 what the problem is 175–89 Campbell, John, ‘Sortals and the Binding Problem’ 7, 203–18 cardinal numbers, identity of 6 Carnap, R. 162, 196 Cartesian Account, of bodily criterion 247, 248, 249, 250–1, 252, 255–6 Cartesian dualism, and personal identity 8, 222–3 chance see realism about chance circumscription problem, and the Caesar problem 193, 196 classical mereology, and the Standard Conception of metaphysical necessity 19 common sense, and the epistemological Caesar problem 177, 178, 179 conceptual necessity 15 concrete objects, and the Caesar problem 181, 182 conditional probability, and chance 103–4 consciousness, and self-consciousness 226 contingency, limits of 1–2, 13–38 contingent apriori, and the Non-Standard Conception of metaphysical necessity 28 contingent properties and mathematical structuralism 115 of numbers 121 Corporealism, and bodily criterion 248–50, 249–50, 254

corpses, and bodily criterion 245, 246, 247, 251, 253 correct conceivability, and metaphysical necessity 22–3 credences, and realism about chance 84–8 cross-structural identity in mathematics 121–31 Dedekind, Richard 110–11, 113, 143 delineation thesis, and sortals 7, 203–5, 209, 214, 215, 216, 217 demonstratives, and sortal concepts 7, 215–17 Descartes, Ren´e 228, 229, 255 desire and the ‘I’ self-concept 226 and the solipsist as rational agent 233 determinism, and realism about chance 75, 78–9, 83, 91, 92 Differential Class, of metaphysical necessity 20–1, 30, 35–6 discrimination, and the epistemological Caesar problem 177, 178–9 distance, and basic metaphysics 43 Divers, John and Hagen, Jason, ‘The Modal Fictionalist Predicament’ 3–4, 57–73 dualism and bodily criterion 9, 243 and mathematical structuralism 115 and personal identity 8, 222–3 Dummett, M. 17 and the Caesar problem 191, 192 and ‘mystical’ structuralism 114, 121 Elga, A. 99, 101 eliminative structuralism 136, 149, 161 enduring runabouts 2, 43–4 and placeshifting 44, 45 and shapeshifting 44–5 epistemological Caesar problem 6, 177–80 epistemology and metaphysics (ep-&-met tendency) 3, 53–4 and personal identity 226 essential idexical role, of ‘I’ thoughts 225–7, 233 essential properties, and mathematical structuralism 115–17, 120–1

Index / 263 Euclidean geometry and the governance of identity 169, 170–1, 172 and mathematical structuralism 131–2, 138 Evans, G. 8 generality constraint and the Caesar problem 196–8 existence, and personal identity 227–30 faithfulness constraint, and mathematical structuralism 111–12, 123, 125, 129 ﬁctionalism see modal ﬁctionalism Field, Hartry 21 Fine, Kit 24 ﬁnite cardinal structures 131, 132, 141–2 and Ker¨anen’s principle (STR) 169 ﬁnite ordinals, and mathematical structuralism 122, 129, 131 ﬁt, and realism about chance 94–100, 100–3 foetuses, and bodily criterion 246, 247–8, 249–50, 250–1 Frege, G. 125 and the Caesar problem 6–7, 175–7, 179, 185, 191, 194–9, 200 frequentist analyses of chance 89–91 functionalist conception of human beings 233–4 Garrett, B. 243 generality constraint, and the Caesar problem 196–8 genuine modal realism (GMR), and modal ﬁctionalism 3, 57–8, 63–72 genuine modality 40–1, 54 genuine necessity 41 genuine possibility, and intelligibility 55–6 Gestalt principles, and sortal concepts 7, 207–8 golden mountain example, and the Non-Standard Conception of metaphysical necessity 31–2 Haecceitism and the identity of indiscernibles 139–40 and the Leibniz Principle 154

and ontological realism 161, 162 and realist structuralism 5–6, 149, 159 and the trivializing objection 150, 155–7 Hagen, Jason see Divers, John and Hagen, Jason Hale, B. and the Caesar problem 198 objection to modal ﬁctionalism 59, 60, 63, 64, 65–6, 71 happiness, and unclarity of personal identity 22 Hellman, Geoffrey 111, 116, 131, 136 Mathematics Without Numbers 109 modal eliminative structuralism 143 objection to ante rem structuralism 141–3 Hossack, Keith, ‘Vagueness and Personal Identity’ 7, 221–41 Hume, D. 228 Humean reﬂection, on metaphysical necessity 15 Hume’s Principle 6 Humphreys, G.W. 209–10, 212 ‘I’ self-concept, and personal identity 8, 225–7, 233 identity and the Caesar problem 6–7, 174–202 cross-structural identity in mathematics 121–31 governance of 164–74 and mathematical structuralism 131–44 problem for realist structuralism 5–6, 146–63 trivial accounts of 155–7 see also personal identity indeterminacy, and mathematical structuralism 122–3, 127, 128–9, 130 indeterministic statistical phenomena, and realism about chance 78–84 indiscernability, and mathematical structuralism 131–44 individuation and the Caesar problem 6–7, 174–202 and the governance of identity 165–6, 168–9

264 / Index individuation task and the identity of indiscernibles 134–7, 139, 141 and Ker¨anen’s reply to Shapiro 150–1 infallibilism, model infallibilism and basic truth 40–56 informed/correct conceivability, and metaphysical necessity 22–3 intelligibility, and metaphysical possibility 2–3, 40–56 intrinsic property possession, and the Caesar problem 195–6 intrinsics, and basic metaphysics 42–3 Inwagen, Peter van 21 isomorphism theorem, and mathematical structuralism 139 Julius Caesar objection see Caesar problem Kant, I. 25, 222, 228, 231 Kastin, Jonathan 112, 114, 116, 124, 146 Ker¨anen, Jukka 112 and eliminative structuralism 136 ‘Identity Problem for Realist Structuralism II’ 5–6, 131, 146–63, 164, 165, 171, 172 Ker¨anen’s principle (STR) 139, 140, 141, 148, 152, 165, 166, 167, 168–70 and mathematical structuralism 131, 132–4, 139 and Shapiro’s ‘The Governance of Identity’ 6, 131, 164–74 Kitcher, P. 186 Kreisel, G. 130 Kripke coherence 56 Kripke truth, and modal infallibilism 2–3, 46, 47, 48, 49, 50, 56 Kripkean necessities, and metaphysical necessity 15, 18, 21 laws of nature, and metaphysical necessity 34–5, 36, 37, 38 Leibniz, G.W. 184 Leibniz Principle, and realist structuralism 148, 149, 152, 153–5 Lepore, E. 217

Lewis, David 32, 40, 42–3, 44–5, 46–7, 49 and advanced modal claims 66–7 ‘best-system’ analysis of chance 4, 76–7, 85, 91–4, 100–1, 104 chance and the problem of ﬁt 94–7, 99, 100–1 Lichtenberg, G.C., and personal identity 228, 229, 230, 231, 232, 234, 235–6, 237–8, 239 location, as a binding parameter 210 Locke, John 226, 231 logical modality 40 logical necessity 15–16 Lowe, E.J. 255 Ludwig, K. 217 MacBride, Fraser 1–9, 40 ‘The Julius Caesar Objection: More Problematic than Ever’ 6–7, 174–202 McDermott, M. 89–90 McGee, Vann 124, 129 Maddy, Penelope 118 Madell, Geoffrey 224 materialism and bodily criterion 9, 243, 251, 252, 253 256 and personal identity 8, 221–2, 223–5, 236–8, 239–40 mathematical ontology, realist theory of 160–3 mathematical structuralism 4–5, 109–46 cross-structural identity 121–31 identity and indiscernability 131–44 and mathematical properties 115 model eliminative structuralism 109 see also ante rem structuralism mathematical truths, and metaphysical necessity 15 mathematicians and natural numbers 124–5, 125–7, 129, 130–1 and philosophers 110 meaning-theoretic Caesar problem 6–7, 177, 185–9, 198 Mellor, D.H., and realism about chance 80, 81, 84–5, 90 membership properties, and the governance of identity 168, 168–9, 170 mereological universalism 43

Index / 265 metaphysical Caesar problem 6, 177, 180–5 metaphysical conception, of the Leibniz Principle 153–4 metaphysical modality 16, 29 metaphysical necessity 1–2, 13–38 deﬁning 13–14 Differential Class of 20–1, 30, 35–6 informal elucidation of 14–17 Non-Standard Conception of 1, 2, 20, 21–8, 33, 36, 37–8 and physical necessity 36–7 Standard Conception of 1, 2, 17–32, 29, 32–6, 37–8 metaphysical possibility 17 and basic metaphysics 41–4, 53 and intelligibility 2–3, 40–56 metaphysics, and the governance of identity 165 Mill-Ramsey-Lewis (MRL) account of the laws of nature, and metaphysical necessity 34–5, 36 minds, and bodily criterion of personal identity 9, 257–8 minimalism constraint, and mathematical structuralism 111, 112, 113, 119, 129 modal eliminative structuralism 143 modal ﬁctionalism 3–4, 57–73 and advanced modal claims 66–71 Brock–Rosen objection to 58–63 and genuine modal realism (GMR) 3, 57–8, 63–72 modal logic, and modal ﬁctionalism 60–2, 72 modal realism 32 modal status, and the Standard Conception of metaphysical necessity 33 modality 1–4 chance 4, 74–106 and metaphysical necessity 1–2, 13–38 metaphysics of 3–4 modal ﬁctionalism 3–4, 57–73 modal infallibilism and basic truth 40–56 model eliminative structuralism 109 natural numbers, and mathematical structuralism 124–5, 125–31 neo-Fregean solution, to the Caesar problem 194–9

nominals, and sortal concepts 216–17 Non-Standard Conception of metaphysical necessity 1, 2, 20, 21–32, 37–8 coherence of 29–32 and the informal explanation 28–9 and objections to the Standard Conception 33, 36 and physical necessity 37 Noonan, Harold 63 object and identity, and the Caesar problem 199–200 objects see abstract objects; sortal concepts Olson, Eric T., ‘Is There a Bodily Criterion of Personal Identity?’ 8–9, 242–59 ontological realism, identity problem for 6, 160–3, 172 ontology and the governance of identity 164, 165 and metaphysical necessity 28, 35, 37 orienting attention, sortals as 214–17 other minds theory, and personal identity 234–6 The Others, and the Non-Standard Conception of metaphysical necessity 21–2, 24, 25, 26, 28, 29 Pairing axiom, and metaphysical necessity 18–19, 19–20, 24–5, 26 Parsons, Charles 123–4, 143 particulars, and the Non-Standard Conception of metaphysical necessity 26–7 Percival, Philip, ‘On Realism about Chance’ 4, 74–106 perdurance, and basic metaphysics 42 Perry, J. 227 persistence, and basic metaphysics 41–2 personal identity and the Bafﬂement Argument against materialism 8, 224–5, 236–8, 239–40 bodily criterion of 8–9, 242–59 and the functionalist conception of human beings 233–4 and the ‘I’ self-concept 8, 225–7, 233 as indeﬁnite 236–8 and knowledge of existence 227–30 and other minds theory 234–6

266 / Index personal identity (cont.) and the solipsistic conception of the self 8, 225, 230–6 unclarity of 8, 221–5 vagueness and 8, 221–41 physical necessity, and metaphysical necessity 36–7 physicalism, and intelligibility 55 physics and metaphysical necessity 29 and realism about chance 79–80 placeshifting, and enduring runabouts 44, 45 Platonism and the Caesar problem 198–9 and mathematical structuralism 117, 118, 119–20, 141, 142 and metaphysical necessity 22 pointing, and sortal concepts 7, 204, 214–15 practical syllogisms, and the ‘I’ self-concept 227 primitive resemblance theory, and the Non-Standard Conception of metaphysical necessity 27 Principal Principle, and realism about chance 85 ‘propensity’ theories of the ontology of chance 95 Putnam, H. 27 qualitative similarity, and the Non-Standard Conception of metaphysical necessity 26–7 Quine, W.V.O. and mathematical structuralism 111, 127–8, 137, 140 Quinean observation sentences 229 and sortal concepts 7, 203–4, 211, 214 Quine–Kraut requirement, and mathematical structuralism 137–8 Quinean conception, of the Leibniz Principle 152, 153, 154–5 Railton, P., deductive normological mode 82–3 rational agents, solipsists as 232–3 real modalities, and metaphysical necessity 16

realism and genuine modality 41 identity problem for ontological realism 160–3 realism about chance 4, 74–106 analyses of chance 88–100 and backwards causation 104 deﬁning 75 and determinism 75 and ﬁt 94–100, 100–3 frequentist analyses of 89–91 and indeterministic statistical phenomena 78–84 Lewis’s ‘best-system’ analysis of 4, 76–7, 85, 91–4, 100–1, 104 and temporally relative warrant 84–8 realist structuralism, identity problem for 5–6, 146–63 referential realism, and mathematical structuralism 124, 129 Reid, T. 222, 228 relational properties, and the governance of identity 166–8 relative conceivability, and metaphysical necessity 23 Resnik, Michael and the identity of indiscernibles 139 Mathematics as a Science of Patterns 109, 111, 114, 119, 121, 122 and natural numbers 125–6 restricted modality, and metaphysical necessity 36 Revised Bodily Criterion 246–8, 250 Riddoch, M.J. 209–10, 212 Rosen, Gideon objection to modal ﬁctionalism 58–63, 64 ‘The Limits of Contingency’ 1–2, 13–39, 40, 52, 53 Russell, B. 113, 179, 229–30 semantic access agnosia 209 semantic indeterminacy, and the Caesar problem 190–1 semantic threshold, and the Caesar problem 187 senile dementia, and bodily criterion 245

Index / 267 set theory and the axiom of extensionality 5, 151, 152, 165, 167 and the Caesar problem 200 and Ker¨anen’s principle (STR) 169, 170 and mathematical structuralism 135, 138 and metaphysical necessity 18, 22, 25, 38 shape, and basic metaphysics 42–3 shapeshifting, and enduring runabouts 44–5 Shapiro, Stewart ‘The Governance of Identity’ 6, 131, 164–74 Ker¨anen’s reply to 5–6, 131, 146–63 and the Leibniz Principle 148, 149, 152, 153–5 Philosophy of Mathematics: Structure and Ontology 109, 146, 149, 155, 157 and realist structuralism 157–9 ‘Structure and Identity’ 4–5, 109–46, 165 and trivial accounts of identity 155–7 trivializing objection 150–2 see also ante rem structuralism Shoemaker, S., account of body ownership 9, 250–1, 255–6 similarity, and the Non-Standard Conception of metaphysical necessity 27 solipsistic conception of the self 8, 225, 230–3 and functionalism 233–4 and other minds theory 234–6 sortal concepts 7, 203–18 and the binding problem 205–10 and the delineation thesis 7, 203–5, 209, 214, 215, 216, 217 justiﬁcation for binding 7, 210–14 as orienting attention 214–17 sortal exclusion, and the Caesar problem 180, 182–3 spatiotemporally scattered objects, binding of 211–12 Standard Bodily Criterion 243–5 Standard Conception of metaphysical necessity 1, 2, 17–21, 29, 37–8 objections to 32–6 and physical necessity 37 Standard Materialism, and bodily criterion 251, 252, 253, 256 standards of identity and existence 162–3

statistical phenomena, and realism about chance 78–84 Strawson, P.F. 8, 216–17, 231 Strong Corporealism, and bodily criterion 249–50 Sturgeon, Scott ‘Moral Infallibilism and Basic Truth’ 2–3, 40–56 and the Non-Standard Conception of metaphysical necessity 32 sufﬁcient reason principle, and the Caesar problem 185 supervaluationist solution, and the Caesar problem 190–3 supervenience, and mathematical structuralism 117 Swinburne, Richard 224 temporally relative warrant, and realism about chance 84–8 trope theory, and the Non-Standard Conception of metaphysical necessity 27 Tye, M., account of body ownership 9, 251–2, 254, 255–6 UMC, and the Standard Conception of metaphysical necessity 19–20 unclarity of personal identity 221–5 vagueness, and personal identity 8, 221–41 vegetative states, and standard bodily criterion 244–5, 246, 247, 249–50, 251 von Neumann numbers and the Caesar problem 186 and mathematical structuralism 122, 124, 129, 143 warrant and the ‘I’ self-concept 226 and the solipsist as rational agent 233 Weak Corporealism, and bodily criterion 248–9, 250, 251 Weber, Heinrich 110–11 Williams, Bernard on personal identity 8, 223–4 ‘The Self and the Future’ 239

268 / Index Williamson, T. 222 Wittgenstein, L. 228–9, 230 The Blue Book 232, 236 Wright, C. 198

Zermelo system and the Caesar problem 186 and mathematical structuralism 122 Zermelo-Frankel set theory 150–1, 152