Oxford Studies in Metaphysics [8] 9780199682904

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OXFORD STUDIES IN METAPHYSICS

OXFORD STUDIES IN METAPHYSICS Editorial Advisory Board: David Chalmers (Australasian National University and New York University) Andrew Cortens (Boise State University) Tamar Szabó Gendler (Yale University) Sally Haslanger (MIT) John Hawthorne (Oxford University) Mark Heller (Syracuse University) Hud Hudson (Western Washington University) Kathrin Koslicki (University of Colorado, Boulder) E. J. Lowe (University of Durham) Kris McDaniel (Syracuse University) Brian McLaughlin (Rutgers University) Trenton Merricks (University of Virginia) Kevin Mulligan (Université de Genève) Theodore Sider (Cornell University) Timothy Williamson (Oxford University)

Managing Editor Marco Dees (Rutgers University)

OXFORD STUDIES IN METAPHYSICS Volume 8

Edited by Karen Bennett and Dean W. Zimmerman

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom 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. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors 2013 The moral rights of the authors have been asserted First Edition published in 2013 Impression: 1 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, by licence 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 work in any other form and you must impose this same condition on any acquirer

Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available ISBN 978–0–19–968290–4 (Hbk.) 978–0–19–968291–1 (Pbk.) As printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

PREFACE Oxford Studies in Metaphysics is dedicated to the timely publication of new work in metaphysics, broadly construed. The subject is taken to include not only perennially central topics (e.g. modality, ontology, and mereology) but also metaphysical questions that emerge within other subfields (e.g. philosophy of mind, philosophy of science, and philosophy of religion). Each volume also contains an essay by the winner of the Oxford Studies in Metaphysics Younger Scholar Prize, an annual award described within. K. B. & D. W. Z. Ithaca, NY & New Brunswick, NJ

CONTENTS The Oxford Studies in Metaphysics: Younger Scholar Prize

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FINDING THE FUNDAMENTAL 1 Naturalness Cian Dorr and John Hawthorne 2 Fundamental Properties of Fundamental Properties Maya Eddon 3 Absolutism vs Comparativism about Quantity Shamik Dasgupta

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ONTOLOGICAL COMMITMENTS: WORDS AND SLOTS 4 Modal Quantification without Worlds Billy Dunaway

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5 Slots in Universals Cody Gilmore

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MEREOLOGY 6 Against Parthood Theodore Sider

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7 Composition as General Identity Aaron J. Cotnoir

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8 Cut the Pie Any Way You Like? Cotnoir on General Identity Katherine Hawley

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THE A-THEORY OF TIME 9 Living on the Brink, or Welcome Back, Growing Block! Fabrice Correia and Sven Rosenkranz 10 Fighting the Zombie of the Growing Salami David Braddon-Mitchell

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viii | Contents 11 Changing Truthmakers: Reply to Tallant and Ingram Ross P. Cameron

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Author Index

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THE OXFORD STUDIES IN METAPHYSICS YOUNGER SCHOLAR PRIZE Sponsored by the Marc Sanders Foundation* and administered by the editorial board of Oxford Studies in Metaphysics, this annual essay competition is open to scholars who are within ten years of receiving a Ph.D. or students who are currently enrolled in a graduate program. (Independent scholars should enquire of the editor to determine eligibility.) The award is $8,000. Winning essays will appear in Oxford Studies in Metaphysics, so submissions must not be under review elsewhere. Essays should generally be no longer than 10,000 words; longer essays may be considered, but authors must seek prior approval by providing the editor with an abstract and word count by 1 November. To be eligible for next year’s prize, submissions must be electronically submitted by 31 January (paper submissions are no longer accepted). Refereeing will be blind; authors should omit remarks and references that might disclose their identities. Receipt of submissions will be acknowledged by e-mail. The winner is determined by a committee of members of the editorial board of Oxford Studies in Metaphysics, and will be announced in early March. At the author’s request, the board will simultaneously consider entries in the prize competition as submissions for Oxford Studies in Metaphysics, independently of the prize. Previous winners of the Younger Scholar Prize are: Thomas Hofweber, “Inexpressible Properties and Propositions”, Vol. 2; Matthew McGrath, “Four-Dimensionalism and the Puzzles of Coincidence”, Vol. 3;

* The Marc Sanders Foundation is a non-profit organization dedicated to the revival of systematic philosophy and traditional metaphysics. Information about the Foundation’s other initiatives may be found at .

x | Younger Scholar Prize Cody Gilmore, “Time Travel, Coinciding Objects, and Persistence”, Vol. 3; Stephan Leuenberger, “Ceteris Absentibus Physicalism”, Vol. 4; Jeffrey Sanford Russell, “The Structure of Gunk: Adventures in the Ontology of Space”, Vol. 4; Bradford Skow, “Extrinsic Temporal Metrics”, Vol. 5; Jason Turner, “Ontological Nihilism”, Vol. 6; Rachael Briggs and Graeme A. Forbes, “The Real Truth About the Unreal Future”, Vol. 7; Shamik Dasgupta, “Absolutism vs. Comparativism about Quantities”, Vol. 8. Nicholas K. Jones, “Multiple Constitution”, forthcoming in Vol. 9. Enquiries should be addressed to:

FINDING THE FUNDAMENTAL

1. Naturalness Cian Dorr and John Hawthorne 1. INTRODUCTION In the wake of David Lewis’s seminal paper ‘New Work for a Theory of Universals’ (Lewis 1983b), a certain use of the word ‘natural’ has become widespread in metaphysics and beyond. In this usage, properties can be classified as more or less natural, with perfectly natural properties as a limiting case. For example, Lewis would claim that being negatively charged is much more natural than being either negatively charged or part of a spoon, and may even be perfectly natural.1 Some philosophers have enthusiastically taken up this way of talking, perhaps with extensions and modifications. Others regard it as marking a grave turn for the worse in contemporary metaphysics. Many others prefer to avoid it, motivated not by any settled conviction that it is a bad thing, but by the sense that if they were to employ it, they would be tying their philosophical fortunes to a piece of controversial metaphysical speculation. What is at stake in the debate between the enthusiasts and the sceptics? Frustratingly, the differences are often articulated in terms of differing attitudes. The sceptics are said to ‘reject’ the distinction between natural and unnatural properties, while the enthusiasts are said to ‘accept’ or ‘countenance’ it, and perhaps even to ‘take it as primitive’. But it is far from clear what it means to have any of these attitudes to a distinction; and in any case, autobiographical claims of the form ‘I reject/accept/take as primitive this distinction’ are not the sorts of things around which we should be structuring philosophical debates. Meanwhile, when enthusiasm and scepticism 1 Following Lewis, we will use ‘property’ in such a way as to include relations; we will use ‘monadic property’ when we want to talk about properties in the usual sense.

4 | Cian Dorr and John Hawthorne are given propositional content, there is great variation as regards how the contents are characterized. In many of the works of naturalness enthusiasts, the only vision of the sceptical alternative that comes into view seems to involve wild claims such as that it is never the case that one thing is more similar to a second thing than to a third thing, or notoriously obscure claims to the effect that facts of this or that sort fail to be ‘objective’.2 On the other hand, discussions of the role naturalness plays in Lewis’s thought often present the idea as a bold and idiosyncratic ‘metaphysical posit’, analogous in its justificatory status to Lewis’s modal realism––the sort of thing whose final justification would require a comparative assessment of various grand philosophical systems. Our aim in this paper is not to take sides in the debate between naturalness enthusiasts and naturalness sceptics, but to bring some structure to the terrain, replacing displays of contrasting nebulous attitudes with a range of relatively precise and independently debatable questions. Our main strategy is familiar from Lewis’s own treatment of novel theoretical terms (Lewis 1970). According to the model presented in that paper, any theory expressed using a newly introduced predicate ‘F’ is analytically equivalent to its expanded postulate—the claim that there is a unique property that does all the things that F-ness does according to the original theory. (A theory’s expanded postulate is a close relative of its Ramsey sentence, which omits the uniqueness claim.) And assuming the original theory logically entails ‘Something is F’, it is also analytically equivalent to that claim: for ‘Something is F’ to be true, ‘F-ness’ has to refer, which it can only do if the expanded postulate is true. If we prefer to avoid the use of the new vocabulary, we can thus do so without losing anything of cognitive significance by replacing both the debate about whether the original theory is true, and the apparently quite different debate about whether anything at all is F, with the debate about whether the expanded postulate is true. If we apply this treatment to Lewis’s theory of naturalness, we will take the question whether some properties are more natural than others to be equivalent to the question whether Lewis’s entire theory of naturalness is true, and we will take both of these questions to be 2 The sceptical view is often associated with Goodman (1978). For the problems with the obvious ways of interpreting denials of objectivity, see Rosen (1994).

Naturalness | 5 equivalent to the question whether there is a unique ranking of properties that plays all the roles that the naturalness ranking plays according to Lewis’s theory.3 As Lewis recognized, this theory of novel terms is too rigid. Sometimes, a non-empty predicate is introduced into the language as part of a theory that uses it to make many false claims. The most obvious way this can happen is for the theorist to explicitly indicate that one of the sentences of the theory is intended to have the status of a definition of the new predicate. But in many other cases, it can be far from obvious what kind of semantic profile we should think of the novel vocabulary as having, even if we know exactly which portions of the overall role defined by the theory are satisfied (and which are uniquely satisfied).4 The fact that the expanded postulate includes a uniqueness claim also poses problems, in many cases, for the claim that it is analytically entailed by the original theory.5 Moreover, other aspects of Lewis’s metasemantics which we will discuss later suggest that there may be cases where a vocabulary-introducing theory is false although its expanded postulate is true.6 However, one can agree that Lewis’s theory of theoretical terms is flawed in all these ways while accepting its central methodological moral: namely, that the focus of the debate between enthusiasts and sceptics about some new piece of vocabulary should be on the question how close the relevant theoretical role comes to being

3 Note that Lewis’s talk of relative naturalness is not just about an ordering: he wants to be able to ask questions like ‘Is F-ness much more natural than G-ness, or only a little bit more natural?’ When we speak of ‘rankings’ we mean not just orderings, but items with a rich enough structure to interpret such questions. 4 Lewis (1970) suggests that the role the original model assigns to the expanded postulate should properly be played by the claim that the relevant theoretical role comes near enough to being realized, and has a unique nearest realizer. 5 Carnap (1947) proposes a theory like Lewis’s except that the role of the expanded postulate is played by the theory’s ‘Ramsey sentence’, which omits the uniqueness claim. Lewis (1997: 347) suggests a more tolerant view that allows a term-introducing theory to be true even when its theoretical roles are multiply realized, provided that the many realizers are ‘sufficiently alike’, with reference failure occurring only when the many realizers are ‘sufficiently different’; in the former case, it will be a vague matter what the new terms apply to. 6 We are thinking of cases where some property that isn’t too far from playing the relevant role is sufficiently more natural than the unique property that plays the role perfectly that the new predicate ends up expressing it.

6 | Cian Dorr and John Hawthorne satisfied (or satisfied uniquely). Wholehearted enthusiasts will want to claim that the entire role is uniquely satisfied, while thoroughgoing sceptics will not only claim that the entire role is unsatisfied, but say the same about various interesting fragments and variants of the role. And of course all sorts of intermediate positions will be available, which take different fragments and variants of the role to be satisfied. The idea that this richly structured landscape of possible views should be the focus for the debate between enthusiasts and sceptics about a new vocabulary item does not require us to think that answers to questions couched in terms of that vocabulary (including the question ‘Are there any F things at all?’) can be straightforwardly read off an answer to the question which portions of the relevant theoretical role are satisfied. There will be plenty of scope for further disagreement here as well. But typically, when the parties to the debate disagree as regards how to map questions expressed using the new vocabulary onto role-related questions, it will be a bad idea for them to spend much of their time debating the former questions. There is a strong danger that such debates will be infected with the pathology characteristic of ‘merely verbal disputes’, whatever the nature of that pathology might be. More specifically, the problem is that one’s policy for using the new vocabulary will depend in part on one’s answers to very detailed and localized questions about the semantics of theoretical terms, which are unlikely to be of much relevance to the subject matter to which the term-introducing theory was supposed to be a contribution. Thus for example, the question ‘Are there any F things at all?’ will be answered negatively both by those who think that some property comes very close to doing all the things that F-ness does according to the original theory but hold a draconian view of theoretical terms on which even this is not good enough to prevent ‘F’ from being empty, and by those who have a much more tolerant view of what it takes to introduce a non-empty predicate but think that relevant theoretical roles are so far from being satisfied that ‘F’ fails to meet even this low standard. The best policy is first to get as clear as we can on the answers to the questions we can state without using the new predicate. For those who don’t care about tricky puzzle cases in metasemantics, this might be enough; those who do care can conduct a parallel debate about what we should think about the exten-

Naturalness | 7 sion of the new predicate, conditional on various answers to those questions. These morals apply whenever new vocabulary is introduced as part of a controversial theory, whether in science or in philosophy. In particular, they apply to ‘natural’. We propose, then, that the debate between naturalness enthusiasts and naturalness sceptics should be conducted in a way that gives a central role to the question how much of the theoretical role defined by the use of ‘natural’ by Lewis and his followers is satisfied by some ranking of properties.7 For many pieces of philosophical jargon, this advice would be hard to follow. All too often, such terminology comes to us as part of a large system of interrelated terminology which we would need to Ramsify out simultaneously in order to make dialectical progress, but which is so pervasive in the relevant theory that the result of Ramsification risks triviality. In these cases, the debate between enthusiasts and sceptics will have to be approached in some other way. Fortunately, Lewis’s theory of naturalness is exemplary in this regard. Lewis propounds a broad array of claims about naturalness, which connect it with a wide range of other subject matters, and thereby provide a richly articulated structure for the debate about the extent to which the role is satisfied. Section 2 of the present paper will set out the role, while section 3 will consider some arguments that bear on the question how much of it is satisfied. We should emphasize that we are not suggesting that ‘natural’ is analytically, or even extensionally, equivalent to anything of the form ‘has the property of properties that plays such-and-such role’.8 Naturalness enthusiasts will surely think that there are important psychological and epistemological differences between belief in their theory of naturalness and belief in its Ramsey sentence (or its

7 We are thus in agreement with Sider (2011: 10), whose central positive claim on behalf of the notion of ‘structure’ (a close cousin of naturalness) is that its associated ‘inferential role’ is occupied. 8 Still less are we proposing this as a ‘reduction’ of naturalness. Whatever it means to give a reduction of something, one is not supposed to give reductions that go in circles. Thus reducing ‘natural’ to ‘having a property of properties that does suchand-such’, where doing such-and-such is partly specified in terms of ‘similar’, would prevent one from reducing ‘similar’ to anything specified in terms of ‘natural’. For reasons we will discuss in section 5, we think it is dangerous to treat the notion of ‘reduction’ as unproblematic common ground in the debate about naturalness.

8 | Cian Dorr and John Hawthorne expanded postulate).9 Some will want to draw a sharper contrast in this case than they would draw between, say, belief in Maxwell’s theory of electromagnetism and belief in its Ramsey sentence.10 The only status we are claiming for the Ramsey sentence, and its weakenings and variants, is that of being a good thing to focus on if one is looking for an articulate, argument-driven debate. Is there really nothing more to Lewis’s enthusiasm about naturalness than the claim that a unique property of properties plays the relevant role? You might think that this debate completely misses out on the central point at issue. What about the question whether there are objective joints in reality? Whether all properties are ‘on a par’? Whether the structure of the realm of properties is ‘elitist’ or ‘egalitarian’? The problem with these questions as foci for debate is they seem to be nothing more than variants of the question ‘Are there any natural properties?’, or ‘Are some properties more natural than others?’ For example, it is uncontroversial that there are some respects in which properties fail to be ‘on a par’; and the obvious answer to the question ‘How do you mean, on a par?’ is ‘With respect to naturalness’. If this is right, the negative moral of our general discussion of theoretical terms comes into play, namely that it is unhelpful for the debate between enthusiasts and sceptics about some novel vocabulary item to focus on questions expressed using that item. The answers to such questions will unhelpfully depend on the details of one’s approach to the metasemantics of theoretical jargon. For example, some who say that no property is more natural than any other will think that the Lewisian role comes very close to being satisfied, while accepting a draconian metasemantics on which even this is not good enough to prevent ‘natural’ from being defective. Meanwhile, some who accept that some properties are more natural than others will think the trend towards giving ‘natural’ a central role in metaphysics is completely lamentable, but endorse a forgiving metasemantics according to which the manifold errors made by Lewis and his followers 9 This is certainly true of Sider (2011), who says that ‘if the entire theory of this book were replaced with its Ramsey sentence, omitting all mention of fundamentality, something would seem to be lost’ (11). 10 For example, Chalmers (2012, chapter 7) is sympathetic to the thought that while the concept of fundamentality is ‘conceptually primitive’, the concept of negative charge is not.

Naturalness | 9 do not prevent ‘natural’ from acquiring a non-trivial extension, any more than the errors of astrology prevent ‘being a Gemini’ from having a non-trivial extension. Indeed, this reason for not spending much time on questions like ‘Are all properties on a par?’ applies even if we refuse to treat them as tantamount to ‘Are some properties more natural than others?’ (as we might if we think of expressions like ‘on a par’ as less tightly tied to Lewis’s particular theoretical commitments than ‘natural’ itself). The same problem arises, namely that people’s answers will depend on a complex mixture of their metasemantical views about the conditions for the relevant expressions to be non-empty, together with views about the extent to which certain associated roles (specified without using any such vocabulary) are satisfied. While formulae like ‘Properties are not all on a par’ are useful devices for initially conveying the flavour of one’s view, the idiosyncratic interpretative questions they raise make them poorly suited to serve as the central focus of any argument-driven debate.11 This is not to say there is nothing more going on in the debate between enthusiasts and sceptics about naturalness than the question how much of the Lewisian role is satisfied. In sections 4, 5, and 6 we will consider some further questions that might be thought central to the debate. A number of these turn out to be red herrings. However, we do identify one other fruitful topic for debate, namely the question whether and to what extent expressions like ‘natural’, ‘more natural than’, and ‘perfectly natural’ are vague. Some naturalness sceptics will want to claim that all these expressions are massively vague; some naturalness enthusiasts will want to claim that at least one of them is perfectly precise. Since these questions about vagueness are more or less orthogonal to the questions about role satisfaction that we will be discussing in the next two sections, the upshot will be that there are two good axes along which the debate about naturalness can be structured.

11 One might gloss ‘All properties are not on a par’ as something like ‘There is a metaphysically interesting ranking of properties’ or ‘There is a metaphysically interesting property that is had by some but not all properties’. But ‘interesting’ is prima facie much too vague for the kind of debate we are trying to foster, and ‘metaphysically’ only makes things worse, since few questions are less interesting than the question how metaphysics is to be demarcated from other branches of philosophy.

10 | Cian Dorr and John Hawthorne 2. THE NATURALNESS ROLE The aim of the present section is to list Lewis’s central theoretical claims involving the word ‘natural’, taking ‘New Work’ as our main text. We should stress again that we are not trying to suggest that any of the principles on our list should be accorded any kind of definitional status. (Given the important role paradigm cases play in introducing people to the concept of naturalness, this is an especially unpromising territory for sustaining claims of analyticity.) We don’t even claim that the rejection of any one of these principles amounts to a departure from full-blooded enthusiasm about naturalness—certainly, many of them have been explicitly rejected by philosophers who think of themselves as fully in agreement with Lewis about the importance of naturalness in metaphysics. Our aim is just to survey interesting questions in the general vicinity of the debate between sceptics and enthusiasts about naturalness. This does not require isolating any claims as singly or jointly analytic of naturalness. Now to the list. 1. Supervenience: Everything supervenes on the perfectly natural properties. There are several relevant ways of making Supervenience precise. Setting aside glosses that presuppose modal realism, the most obvious interpretation of Supervenience is that whenever two possible worlds differ as regards the truth value of any proposition, they differ as regards the truth value of at least one proposition predicating a perfectly natural monadic property of a particular object, or predicating a perfectly natural relation of a sequence of particular objects. A second gloss on Supervenience, more in keeping with Lewis’s ‘anti-haecceitism’, still treats it as a claim of propositional supervenience, but restricts the domain of supervenient propositions to qualitative ones (e.g. that there are at least seven blue chairs), while restricting the supervenience basis to propositions about the pattern of perfectly natural properties (e.g. perhaps, that there are at least 1070 negatively charged items).12 The third possible gloss is a claim about qualitative indiscernibility as a relation between individuals, 12

What does it mean for a proposition to be ‘about the pattern of perfectly natural properties’? One possible definition uses possible worlds: it is for the proposition not to divide any pair of worlds w1 and w2 for which there is a bijection from the domain

Naturalness | 11 as opposed to worlds: necessarily, if there is a permutation of the domain of all objects that maps x to x¢ and preserves all perfectly natural properties and their negations, x is qualitatively indiscernible from x¢.13 (‘Qualitatively indiscernible’ here expresses a relation that holds between distinct objects only in perfectly symmetric worlds, such as worlds of two-way eternal recurrence.) The fourth gloss extends this to a notion of cross-world qualitative indiscernibility: if there is a bijection π from the domain of w to the domain of w¢ that maps x to x¢, such that for any perfectly natural property F and objects y1, . . . , yn, y1, . . . , yn instantiate F at w iff π(y1), . . . , π(yn) instantiate F at w’, then x as it is at w is qualitatively indiscernible from x¢ as it is at w¢. The fourth gloss entails the third, since we can take w = w¢; it also entails the second, given that it cannot be true that x at it is at w is qualitatively indiscernible from y as it is at w¢ unless the same qualitative propositions are true at w and w¢.14,15

of w1 to that of w2 which preserves perfectly natural properties. Another is more linguistic: it is for the proposition to be expressible in some language (perhaps infinitary) whose non-logical vocabulary is limited to predicates expressing perfectly natural properties. A third is algebraic: it is for the proposition to be contained in the smallest algebra of propositions and properties that contains all perfectly natural properties and is closed under a certain range of logical operations. 13 Perhaps this should be strengthened to read: if there is a permutation of the domain of all objects that maps x1 to x1¢ and . . . and maps xn to xn¢ and preserves all perfectly natural properties and their negations, then for any qualitative relation R, Rx1 . . . xn iff Rx1¢ . . . xn¢. 14 One could also try to cash out Supervenience using the standard definitions of strong and weak individual supervenience (Kim 1984), but the resulting claims are too strong, and too implausible by Lewis’s lights, to be usefully thought of as part of the naturalness role. Given how Lewis is thinking, it would not be at all surprising to suppose that a certain chair and a certain table instantiate exactly the same monadic perfectly natural properties. Indeed, it might well be that neither the chair nor the table instantiates any monadic perfectly natural properties—Lewis takes seriously the hypothesis that only point-sized objects do so. If so, the property being a chair does not even weakly supervene on the monadic perfectly natural properties, according to the standard definition. And since weak and strong supervenience as standardly defined are relations between sets of monadic properties, it is not clear what it would even mean to ask whether being a chair weakly or strongly supervenes on the set of all perfectly natural properties and relations. 15 In stating these versions of Supervenience, we have helped ourselves to quantification over possible worlds and over objects existing in arbitrary possible worlds. Further issues arise if one attempts to cash them out in a way that is consistent with the widely believed ‘contingentist’ view that some things are such that they could fail to be identical to anything. This project is relatively straightforward for the first, second, and third glosses, but the attempt to extend it to the fourth gloss plunges us into the extremely difficult question what sense, if any, contingentists can make of quantification over sets of incompossible objects (see Williamson 2013: chapter 7).

12 | Cian Dorr and John Hawthorne The various glosses on Supervenience come apart in several interesting ways. Note, first, that the test provided by the first gloss is, under plausible assumptions, consistent with such hypotheses as that existence, or truth, or instantiation, is the one and only perfectly natural property. For example, if one believes in facts, it may be plausible to think that all truths about the world supervene on truths about which facts exist; but in that case, the propositions attributing existence to particular facts will together constitute a supervenience base for everything. Certain views on which material objects are extremely abundant may generate the same result, for example by entailing that every material object coincides with a world-bound material object. Similarly, given an abundant ontology of propositions, all propositions will supervene on the propositions about which propositions are true, and given an abundant ontology of properties, all propositions will supervene on propositions about what instantiates what. Perhaps the first gloss can be refined so as to rule out these deeply un-Lewisian suggestions— most obviously, we might impose some restriction on the entities whose perfectly natural properties can figure in the supervenience base.16 The second, third, and fourth glosses on Supervenience, by contrast, already prohibit these super-minimalistic proposals about what the perfectly natural properties are. At least, they do so on the assumption that we have some independent grip on the notion of qualitativeness in terms of which they are stated. (Some speculations put pressure on standard judgements about qualitativeness— for example, it is standard to suppose that the property of having a certain mass is qualitative while the property of being located in a particular place is not, but this is disrupted by the speculation (see Arntzenius and Dorr 2012) that one’s mass is a matter of occupying a point in a ‘mass space’ whose ontological status is similar to that of ordinary space.) Another notable divide between the glosses on Supervenience is this: the first and second are consistent with the hypothesis that familiar everyday objects (tables, trees, people . . .) neither instantiate any perfectly natural properties, nor stand in any perfectly natu16 If, unlike Lewis, we had a notion of perfect naturalness applicable to objects, the restriction could be to the perfectly natural objects. We will discuss the prospects for such a distinction further towards the end of section 2.

Naturalness | 13 ral relations to anything, while still being qualitatively discernible from one another. By contrast, the third and fourth glosses require at least one of any two qualitatively discernible objects to instantiate at least one perfectly natural property, or stand in at least one perfectly natural relation.17 2. Independence: The perfectly natural properties are mutually independent. Lewis entertained several different claims that can be regarded as precisifications of Independence. In ‘New Work’, the main focus is on a claim of Non-supervenience: no perfectly natural property is such that the facts about it supervene on the facts about all the other perfectly natural properties. In conjunction with Supervenience, this is equivalent to the claim that the perfectly natural properties constitute a minimal supervenience base for everything, where the relevant sense of ‘supervenience base for everything’ could be spelled out in any of the ways considered earlier.18 Another kind of independence claim is the principle of Recombination discussed in Lewis 1986. The basic idea here is that for any two parts of worlds, there is a single world containing a duplicate of each.19 Given the connection between perfect naturalness and duplication (to be discussed later), this entails that, for example, no two perfectly natural properties are such that it is impossible for them to be instantiated in the same world.20 The basic idea can be

17 So, for example, the first and second glosses, unlike the third and fourth, are consistent with the proposal that while there are many things, not all qualitatively indiscernible, there is only one thing (the Absolute?) that instantiates any perfectly natural properties or relations. 18 The following stronger claim in the same direction is also worth considering: it never happens that the complete description of a world in terms of some subset of the perfectly natural properties entails the complete description of that world in terms of the rest. 19 Lewis’s version of Recombination includes the proviso ‘size and shape permitting’, whose intended interpretation is not exactly clear. While he mostly applies the proviso in connection with the cardinality-based worries discussed later, the mention of shape as well as size might suggest that there would be exceptions even to the basic, two-object version of Recombination. But we will not worry about this: it seems plausible that even infinitely extended objects can be duplicated together in a world with higher dimensions. 20 Note that Recombination is consistent with the claim that some perfectly natural properties supervene on others.

14 | Cian Dorr and John Hawthorne strengthened along a few dimensions: (i) We could generalize from pairs to pluralities, although as Lewis points out, paradox will threaten if we impose no cardinality requirement whatsoever on the pluralities. (ii) We could strengthen the principle to allow any number of duplicates of each of the items (again subject to cardinality constraints). (iii) We could claim not only that some world contains duplicates of the items perhaps along with other things, but also that some world is entirely composed of (is a fusion of) duplicates of the items. (iv) We could try somehow to capture the idea that the duplicates can be ‘in any arrangement’: the thought is that the intrinsic natures of things do not much constrain the perfectly natural relations they bear to one another, although it is not clear how to articulate this precisely.21 Lewis (2009) entertains an even stronger independence principle, ‘Combinatorialism’, according to which the distinct ‘parts of reality’ which can be freely recombined ‘include not only spatiotemporal parts, but also abstract parts—specifically, the fundamental [perfectly natural] properties’ (209). This means, for example, that no perfectly natural property is entailed by any other. The general idea might be spelled out as follows: in an appropriate language in which all predicates express perfectly natural properties, the only sentences that express metaphysically necessary propositions are the logical truths. This can be fine-tuned in several ways, depending on how we specify the ‘appropriate language’ and the notion of logical truth. (i) We can make the principle stronger by allowing the language to contain infinitary operators, infinitary blocks of quantifiers, and/or higher-order quantifiers.22 (ii) We could adopt the standard conception of logical truth, on which ‘∃x∃y y≠x’ does not 21 Lewis’s version speaks only of spatiotemporal relations, but it is not clear exactly which spatiotemporal relations he has in mind: he would probably not want to be committed to the existence of a possible world in which a duplicate of a large doughnut fits inside the hole of a duplicate of a much smaller doughnut. 22 If we use the infinitary language L∞,∞, in which we can take conjunctions and disjunctions of arbitrary sets of formulae, and quantify arbitrary sets of variables simultaneously, then so long as we do not think that the perfectly natural properties are too numerous to form a set, we can fully specify any set-sized model for a language with predicates corresponding to all perfectly natural properties. Given the cardinality restrictions that need to be built into Recombination to avoid paradox, it is plausible that the infinitary version of Combinatorialism entails all reasonable interpretations of Recombination.

Naturalness | 15 count as logically true, or the alternative conception (defended in Williamson 1999) according to which all truths involving only logical vocabulary count as logical truths. (iii) We could think of the quantifiers in the ‘appropriate language’ as restricted somehow— e.g. to concrete objects, or to some unspecified collection of objects— or as unrestricted. In the latter case, if we also adopt the standard conception of logical truth, we will be committed to the metaphysical possibility of there being very few objects.23 (iv) We could allow the ‘appropriate language’ to contain names for some or all objects, as well as predicates, thereby ruling out a wide range of putative de re necessities involving the given objects.24 We should be clear that there is no chance that a version of the naturalness role containing just Supervenience and Independence will single out the set of perfectly natural properties uniquely, even if we adopt the strongest interpretations of those principles. If any set of properties satisfies this fragment of the role, so do many other sets of properties. For example, if a set of properties satisfies Supervenience and Combinatorialism, so does the set of their negations, and so does a set which replaces two properties F and G with F-iff-G and 23

If one takes the quantifiers in the ‘appropriate language’ to be distinct from those of ordinary language (see Dorr 2007), one might combine this with the claim that ‘Necessarily, there are infinitely many sets’ is true when interpreted in the ordinary way. If one wanted to make such a distinction, one would naturally hope for some helpful way of singling out the intended interpretation of the quantifiers. A salient option here is to say that the relevant quantifier-meanings are the most natural ones (cf. Sider 2011: chapter 9) having a certain basic logical profile. Given that standard semantic theories take quantifiers to express properties of properties, or relations between properties, or relations between properties and propositions, there is nothing especially surprising in the idea that they can be assessed as more or less natural. However, once we start talking about properties of properties, the need to decide what we are going to do about the property-theoretic paradoxes becomes urgent; we will discuss one possible response to this later. Note that one could say that there is a unique most natural property with the relevant logical profile without saying that any such property is perfectly natural; indeed, if we extend Combinatorialism to properties of properties in the obvious way, it entails that if there are any perfectly natural properties of properties, their instantiation by different properties is independent in a way that is not true for any property with the logical profile required to be an interpretation of ‘∃’. 24 Note that we would need a version of Combinatorialism that allows certain names into the appropriate language if we want it to entail the version of Non-supervenience according to which the propositions about which particular things instantiate a given perfectly natural property never supervene on the propositions about which particular things instantiate all the others.

16 | Cian Dorr and John Hawthorne F-iff-not-G. Other techniques will generate a very large proliferation of families satisfying Supervenience and Combinatorialism on the assumption that there is at least one such family. Given a set of properties S, say that w1 and w2 are S-opposites iff there is a bijection π from the domain of w1 to that of w2 such that whenever F∈S, Fx1 . . . xn at w1 iff it is not the case that Fπ(x1) . . . π(xn) at w2, and say that a proposition P is S-invariant if it never distinguishes between two worlds that are S-opposites. (For example, propositions about the cardinality of the universe are automatically S-invariant.) Suppose we have some set S that satisfies Supervenience (gloss 4) and Combinatorialism; then for any S-invariant proposition P, the set SP = {F-iff-P: F ∈ S} will also satisfy Supervenience and Combinatorialism.25 3. Duplication: If some bijection from the parts of x to the parts of y maps x to y and preserves all perfectly natural properties, x and y are duplicates. 4. Non-duplication: If no bijection from the parts of x to the parts of y that maps x to y preserves all perfectly natural properties, x and y are not duplicates.

25 To prove that SP satisfies Supervenience, suppose that π is an SP-preserving bijection from the domain of w to that of w¢ that maps x to x¢. It cannot be that w is a P-world and w¢ is not, since in that case it would have to be the case that for each F∈S, Fx1 . . . xn at w iff not Fπ(x1) . . . π(xn) at w¢, in which case w and w¢ are S-opposites, which is ruled out by the S-invariance of P. But if w1 and w2 are both P-worlds, or are both not-P worlds, π must also be an S-preserving bijection, so that x at w is qualitatively isomorphic to x¢ at w¢. To prove that SP satisfies Combinatorialism, consider a logically consistent sentence φ in a language whose atomic predicates stand in one-to-one correspondence with members of S. Let Q1, Q2, and Q3 be the propositions expressed by φ under, respectively, an interpretation on which the atomic predicates express the corresponding members of S; an interpretation on which they express the negations of the corresponding members of S; and an interpretation on which they express the corresponding members of SP. Since S satisfies Combinatorialism, Q1 is metaphysically possible. So is Q2, since the result of negating every atom in a logically consistent sentence is always logically consistent. We need to show that Q3 is also metaphysically possible. Since Q1 ∧ P is equivalent to Q3 ∧ P, while Q2 ∧ ¬P is equivalent to Q3 ∧ ¬P, it suffices to show that at least either Q1 ∧ P or Q2 ∧ ¬P is metaphysically possible. But this follows from the S-invariance of P: given that every Q2-world is the S-opposite of a Q1-world, so if no Q1-worlds are P-worlds, no Q2-worlds can be P-worlds either; since we know there are some Q2-worlds, we can conclude that in that case there must be some (Q2 ∧ P)-worlds.

Naturalness | 17 The concept of duplication is supposed to be intuitive: it is the relation that would hold between the copies produced by an ideal copying machine (Lewis 1983b: 355).26 For Lewis, the concept of duplication is tightly connected to that of an intrinsic property: an intrinsic property is one that never divides duplicates within or across worlds; duplicates are things which share all their intrinsic properties. However, others have found this connection more problematic (Francescotti 1999; Eddon 2011). For one thing, it entails that anything necessarily equivalent to an intrinsic property is itself intrinsic—a claim that might give you pause, if you take seriously the suggestion that the property of being identical to Prince Charles is distinct from, but necessarily equivalent to, the property of being descended from such-and-such sperm and egg, or that the property being a cube is distinct from, but necessarily equivalent to, the property being a cube and either five metres from a sphere or not five metres from a sphere.27 To avoid distraction by these issues, we will focus on duplication rather than intrinsicness. 26 Duplication and Non-duplication are endorsed in Lewis (1986: 61). Two other ideas about the connection between duplication and naturalness are also to be found in Lewis’s work. Lewis (1983b) gives a simpler account on which duplication is simply the sharing of all perfectly natural properties. However, getting that account to work requires a very rich supply of perfectly natural properties. For example, chairs would have to have many perfectly natural properties if any two non-duplicate chairs are distinguished by some perfectly natural property. Since such an abundance of perfectly natural properties fits poorly with many other components of the role (such as Independence), we suspect that it is a slip, and will concentrate on the 1986 account. Langton and Lewis (1998) and Lewis (2001) explore a different account of duplication in terms of comparative rather than perfect naturalness. Lewis accepted this account as well as Duplication/Non-duplication: for him, the interest of the Langton-Lewis account was that it could be addressed to ‘philosophers more risk-averse than Lewis’, who doubt that it ‘makes sense to single out a class of perfectly natural properties’. The Langton-Lewis account has proved much more controversial than Duplication/Non-duplication, even among naturalness enthusiasts: for some criticism, see Marshall and Parsons (2001) and Hawthorne (2001). 27 Another source of concern about Lewis’s account of ‘intrinsic’ in terms of ‘duplicate’ involves the need to make sense of cross-world duplication. It is by no means obvious that philosophers who do not endorse Lewis’s modal realism should even regard claims like ‘x at w1 is a duplicate of y at w2’ as intelligible. After all, not just any two-place relation among objects corresponds in any interesting way to a four-place relation among two objects and two worlds—for example, it is hard to see what nontrivial sense could be made of ‘x at w1 kicks y at w2’. However, those who endorse Duplication and Non-duplication have some natural options for making non-trivial sense of ‘x at w1 is a duplicate of y at w2’. The most obvious strategy is to take it as equivalent to ‘there is a bijection f from things that are part of x at w1 to things that are

18 | Cian Dorr and John Hawthorne 5. Empiricism: The right method for identifying actuallyinstantiated perfectly natural properties is empirical. For Lewis, the relevant empirical method is one that involves paying close attention to developments in physics. The claim is not, of course, that every word that physicists use is to be counted as expressing a perfectly natural property: Lewis would not be sympathetic to the suggestion that being a Nobel prize winner is perfectly natural. Even if we only looked at the words the physicists use when stating what they call ‘laws’, we will be apt to find our list of perfectly natural properties contaminated by properties like being a measurement, being an experiment, and being an observer, whose presence on the list would disturb many of the other roles. As we are understanding Empiricism, it does not even require the thought that the single words that physicists use ever express perfectly natural properties—for example, it is compatible with Empiricism to maintain that the relation ‘the mass of x is between the masses of y and z’ is perfectly natural, even though physicists prefer to encode mass using numerical mass values.28 Nor does endorsement of Empiricism, as we are construing it, require agreement with Lewis about the special role of physics. A view that treats all the sciences as equally good guides to perfect naturalness (e.g. Schaffer 2004) will still count as conforming with Empiricism (although clearly such a view will fit less well with Independence). Those with dualistic leanings might even wish to add something like introspection as another

part of y at w2, such that f(x) = y, and for every perfectly natural n-ary relation R, R(z1, . . . , zn) at w1 iff R(f(z1), . . . , f(zn)) at w2. This definition is, however, problematic if the facts about what there is are contingent—in deciding whether ‘x at w1 is a duplicate of y at w2’ is true, we do not want to be limited to considering the properties at w1 and w2 of actually existing parts of x and y. It is not clear whether there is a way for contingentists to simulate quantification over ‘non-actual objects’, and over set-theoretic constructions out of objects existing at different possible worlds, which would allow them to avoid this problem (see Williamson 2013: chapter 7). 28

There are other ways in which physics could be a useful guide to (some of) the instantiated perfectly natural properties without any such properties being expressed by the predicates of physics. According to Chalmers (1996: 154), for example, ‘mass is an extrinsic property that can be “realized” by different intrinsic properties in different worlds’. While Chalmers never mentions naturalness, the picture suggested might be one where, even though the extrinsic properties expressed by physical predicates are not perfectly natural, each of them stands in the ‘realization’ relation to a unique perfectly natural property.

Naturalness | 19 relevant empirical method. The kinds of views we want Empiricism to rule out are those on which the task of determining whether a property is perfectly natural is primarily a matter of a priori reflection. One example is the suggestion that existence is the one and only perfectly natural property, which we considered in connection with Supervenience earlier. We will consider more views of this sort in section 3.5.29 6. Simplicity: One property is more natural than another iff the former has a definition in terms of perfectly natural properties that is simpler than any definition of the latter in terms of perfectly natural properties. ‘Definitions’ of a property here are simply expressions which provide necessary and sufficient conditions. A definition ‘in terms of perfectly natural properties’ will be an expression in a language in which all syntactically simple non-logical vocabulary expresses perfectly natural properties, and in which only certain standard connectives figure as ways of building complex expressions.30 We don’t think it is in the spirit of Lewis’s thinking to be too legalistic about symbol-counting as a measure of the simplicity of an expression. For example, it would not go against the spirit of Simplicity to claim that disjunctions detract more from simplicity than conjunctions. Nor would it go against the spirit of Simplicity to rank the

29 Lewis may allow that a few relations can be revealed to be perfectly natural by a priori methods, for example identity and parthood (Lewis 1986: 67 n. 47). Whether these should count as perfectly natural is a vexed issue: they don’t fit so well with Independence, but do fit quite well with many of the other roles. 30 Should we allow the non-logical vocabulary of the language to contain names alongside predicates for perfectly natural properties? If we do not, the risk is that Simplicity will be completely silent about the relative naturalness of haecceitistic properties like living in Oxford: only on the widely rejected view that such properties supervene on the qualitative will they have any definitions in the canonical language. If we do, the risk is that all properties will count as very natural. For example, if we have names for properties and a predicate ‘instantiates’, every property will have a definition of the form ‘instantiates p’; even if we only allow names for particulars, we will be in trouble if our ontology of particulars is an abundant one in which, e.g. there is a particular that is at each world composed of all and only the grue things at that world. If we had a notion of perfect naturalness that applied to objects, we could allow the canonical language to contain names for only the perfectly natural objects. We will come back to the question what it might mean for an object to be natural towards the end of this section.

20 | Cian Dorr and John Hawthorne simplicity of an expression by counting the number of states in the smallest Turing machine that outputs that expression, even though this will assign high simplicity scores to some quite long, but regular, expressions.31 7. Laws: The conjunction of all the laws of nature can be expressed simply in terms of perfectly natural properties. For Lewis, of course, the status of Laws is intimately bound up with a Humean analysis of lawhood under which, necessarily, the laws are whichever generalizations follow from the system of propositions that achieves an optimal combination of simplicity and ‘strength’. (Lewis says little about how one should go about measuring strength. Given that there are infinitely many possible worlds, presumably what is needed is a measure on the space of possible worlds, where the strength of a proposition is given by some monotonically decreasing function of the measure of the set of worlds where it is true. Considerations of naturalness might have a role to play in specifying the relevant measure, perhaps by way of a metric of resemblance among worlds.) Lewis’s analysis does not obviously entail our Laws—perhaps there are worlds where considerations of strength lead to a notvery-simple best system—but Lewis was clearly optimistic that the actual world is not one of these. For our purposes, even if Laws is contingent, it is more useful to focus on it than on Lewis’s final analysis of lawhood, given that we are trying to articulate some

31 In a hyperintensionalist account of properties, one would expect there to be some notion of definition more demanding than simply that of necessary and sufficient conditions. However, many hyperintensionalists (e.g. Soames 2002) would also want to posit a rich supply of ‘unstructured’ properties that lack non-trivial definitions, in the demanding sense. If we cashed out Simplicity using the demanding notion, it seems we will have to count all of these unstructured properties as perfectly natural. If there are a lot of them—if, for example, every property is necessarily equivalent to some unstructured property—this will fit very badly with the rest of the naturalness role. On the other hand, hyperintensionalists will also be uncomfortable with the version of Simplicity on which the relevant notion of definition is just that of giving necessary and sufficient conditions, since this requires necessarily equivalent properties to be equally natural. Perhaps some hybrid story would allow the hyperintensionalist to use something like Simplicity to rank both structured and unstructured properties, using an initial scale for the unstructured properties plus further length-of-definition penalties for the structured ones.

Naturalness | 21 naturalness-related debates that aren’t just repackagings of familiar debates about the Humean programme.32 Given Simplicity, Laws is more or less equivalent to the following claim: Laws*: The conjunction of the laws of nature is very natural. Making sense of Laws* requires extending the notion of naturalness from properties and relations to propositions, but this is no great conceptual departure if we think of propositions as the 0-ary analogues of properties and relations, or as properties of worlds. And note that if one doesn’t like Simplicity, one might have reasons for resisting Laws that would not extend to Laws*. Note that even if we knew exactly which propositions were laws, given an intensional conception of propositions there is no hope that Laws (or Laws*) could be used all by itself to determine which properties are perfectly natural—at best, one could rule out certain candidate lists of perfectly natural properties. Some authors discuss a principle relating naturalness to lawhood which looks as if it could be used to establish the perfect naturalness of certain properties: namely, that the natural properties are those that ‘figure in’ the laws (cf. Sider 2011: 15). However, for the notion of ‘figuring in’ to do this kind of work, we would need to use a notion of lawhood that applies to structured propositions, and that can thereby apply to some but not all members of a family of necessarily equivalent propositions. Unless one had some independent grip on which properties are perfectly natural, it is very hard to see how one could be confident in elevating one member of such a family to the status of lawhood. 8. Similarity: The more natural a property is, the more it makes for similarity among things that share it. 9. Dissimilarity: The more natural a property is, the more it makes for dissimilarity among things that are divided by it.

32 The notion of lawhood employed by Laws had better be understood quite strictly. We shall not consider how naturalness might relate to more relaxed notions of lawhood that encompass generalizations with a high objective chance of being true, or which have a merely ceteris paribus status, and so on.

22 | Cian Dorr and John Hawthorne Lewis dwells heavily on these aspects of the role when he is introducing readers to the concept of naturalness. This is in part because of his interest in the continuity between the theory of naturalness and the traditional doctrine of universals, the central arguments for which turned on premises about similarity, or ‘having something in common’. Note that while claims in the vicinity of Similarity are more common in the literature, it is Dissimilarity that most directly captures the metaphor according to which natural properties ‘carve nature at its joints’. In this metaphor, the naturalness of a property turns on the amount of discontinuity at the boundary it draws between the things that have it and those that lack it. This suggests a relatively straightforward modal gloss on Dissimilarity: a property’s degree of naturalness is given by (some monotonically increasing function of) the minimum possible degree of dissimilarity between an instance of the property and a non-instance. However, this gloss has some surprising consequences. It entails that a property and its negation are always equally natural. It also entails that the conjunction or disjunction of some properties is never less natural than all of those properties, since any two things divided by the conjunction or disjunction of some properties must be divided by at least one of them. This does not fit well with other components of the role: Simplicity and Magnetism (to be considered shortly) both suggest that the conjunction and disjunction of some properties is often less natural than any of them (especially when the properties are numerous); while the Non-supervenience version of Independence tells us that negations, conjunctions, and disjunctions of perfectly natural properties are never perfectly natural.33 One could also give Similarity a modal gloss, according to which a property’s degree of naturalness is given by (some monotonically decreasing function of) the maximum degree of dissimilarity that

33

The modal gloss on Dissimilarity produces further unexpected results if we limit ourselves to intra-world dissimilarity. For example, the property being extremely unlike anything else (in one’s world) will have to be counted as highly natural. Things go more smoothly if one is willing to take a cross-world perspective on the relevant dissimilarity claims. However, it is not obvious how to think of cross-world similarity from a non-modal-realist point of view. We could speak of ‘the degree of dissimilarity between x1 at w1 and x2 at w2’; or we could use a notion of dissimilarity between complete qualitative profiles as a surrogate for the modal realist’s ontology.

Naturalness | 23 could obtain between two instances of the property. But the consequences of this interpretation of Similarity are even more unexpected than those of the modal interpretation of Dissimilarity. For example, being both negatively charged and grue will be at least as natural as being negatively charged. And on plausible assumptions, the most natural properties will all have to be complete qualitative profiles (properties that entail every qualitative property with which they are consistent). A ranking that works like this fits poorly with all the other components of the role, with the exception of Supervenience. One way to deal with these issues would be to keep the basic idea of the modal glosses on Similarity and/or Dissimilarity while throwing in a ‘ceteris paribus’ clause, which allows for some slippage between the naturalness ranking and the ‘maximum dissimilarity of sharers’/ ‘minimum dissimilarity of dividees’ rankings. The obvious worry about such a move is that it will make questions about the co-satisfiability of parts of the naturalness role that include Similarity or Dissimilarity too vague for fruitful debate to be possible. It would certainly help a lot if we could say something more articulate about the nature of the further factors that make for divergence between the rankings.34 A very different way of cashing out both Similarity and Dissimilarity would picture the degree of similarity between two objects as arising from some kind of comparison of two scores, one derived by ‘adding up’ the degrees of naturalness of all the properties they share, and another derived by ‘adding up’ the degrees of naturalness of all the properties that divide them.35 The higher the former number, and the lower the latter number, the more similar the 34 Lewis (2011) employs a conception of the naturalness–similarity link under which two separate similarity-theoretic factors can ‘detract from’ the naturalness of a property: spread (maximum dissimilarity distance between instances) matters, but so does scatter (‘the way non-instances are interspersed with instances’) (Lewis 2001: 391). A property whose set of possible instances was convex under the dissimilarity metric would have low scatter. (Cf. ‘Criterion P’ in Gärdenfors 2000, which defines a natural property as ‘a convex region of a domain in a conceptual space’.) However, since complete qualitative profiles have minimal scatter as well as minimal spread, scatter does not help to explain why they should not be counted as maximally natural. 35 Given the way Similarity and Dissimilarity have been formulated, they do not clearly rule out a view where the degree of similarity also depends on some further factors having nothing to do with sharing or being divided by natural properties.

24 | Cian Dorr and John Hawthorne objects are.36 Making rigorous sense of this approach will of course require somehow controlling for the fact that any two things share uncountably many properties, and are divided by uncountably many properties, a task we will not try to undertake here.37 Note that whereas the modal glosses on Similarity and Dissimilarity suggest that the facts about each property’s degree of naturalness can be read off the totality of facts about the degrees of similarity among actual and possible objects, the ‘additive’ glosses plausibly will leave us quite a lot of freedom in the assignment of naturalness scores, even when all the similarity facts are held fixed. As an exercise, imagine there are only n consistent complete qualitative profiles. To complete the total package of facts about qualitative similarity, we need just n(n−1)/2 numbers, one for each unordered pair of qualitative profiles. Our task is to recover these numbers from an assignment of naturalness scores to each of the 2n qualitative properties. Even without knowing anything about the function which yields the degrees of similarity as functions of the naturalness scores, we can see that it would have to work in quite bizarre ways for there to be only one assignment of naturalness scores which generates the given degrees of similarity. Another difficult interpretative question raised by Similarity and Dissimiliarity concerns the manifest vagueness and context-sensitivity of ‘similar’ (and ‘more similar’). In one context, our answer to the question ‘Which of these two people is more similar to this third person?’ might be driven by facts about the relevant people’s appearances; in another, we will ignore the appearances and focus only on facts about their personalities. Even among contexts where both appearance and personality are relevant, there are differences 36 We could implement this by ranking the degree of similarity as the difference between the two numbers, or their ratio, or some other function that is increasing in its first argument and decreasing in its second. Given that the two scores seem closely related, we might also consider computing degrees of similarity based on only one of them. 37 Some will be comfortable glossing ‘making for similarity’ using an ideology of grounding or in virtue of. But we note that Lewis was not comfortable giving ideology of this sort any important role in his theorizing. And even those who are comfortable need to be careful here. Given that similarity facts are non-fundamental, and that all non-fundamental facts are supposed to be grounded in fundamental facts, it is a challenge to articulate a distinctive grounding-theoretic connection between similarity and naturalness. (Thanks here to Ted Sider.)

Naturalness | 25 as regards their relative importance. And in any ordinary context, there are lots of borderline cases of ‘more similar’—quadruples of objects of which ‘. . . is more similar to . . . than . . . is to . . .’ is neither definitely true nor definitely false. This raises two related worries about Similarity and Dissimilarity. One worry is that context-sensitivity makes them toothless: the effect of endorsing Similarity or Dissimilarity is simply to establish an esoteric local context in which the use of ‘similar’ is forced to fit in the relevant ways with the use of ‘natural’. The other worry is that vagueness makes them untenable: one might suppose that ‘similar’ is so very vague (in the relevant philosophical contexts) that there is nothing we could say using it that would be both interesting and definitely true. On one very controversial picture, there is a certain range of uses of ‘more similar’ by metaphysicians, within which its context-sensitivity is resolved in exactly the same way, and which are all perfectly precise. The proponent of this picture might liken the context-sensitivity of ‘more similar’ to the context-sensitivity involved in the fixing of quantifier domains, as this is understood by fans of absolutely unrestricted quantification. Just as speeches like ‘Let us quantify unrestrictedly’ arguably force a particular, precise resolution of the context-sensitivity of quantifiers, so it might be thought that we can force a particular, precise resolution of the context-sensitivity of ‘more similar’ by making a speech like ‘Let all respects of similarity matter, and let us stipulate nothing about their relative importance’. However, this view looks very implausible, and we know of no evidence that Lewis endorsed it. It is just too arbitrary to suppose that the relevant speech manages to impose a particular definite answer to a question like ‘Is Bill Clinton more similar to Albert Einstein than to Fred Astaire?’ And once this is conceded, even the claim that the relevant philosophical uses all involve exactly the same resolution of the context-sensitivity of ‘more similar’ looks problematic. On many accounts of vagueness, including Lewis’s, vagueness always involves a kind of context-sensitivity, since we are always free to sharpen up a vague expression by settling some of its borderline cases. If vagueness and context-sensitivity are closely related in this way, it is hard to see how any reasonably broad class of uses of some expression could involve exactly the same resolution of its context-sensitivity unless they also involve rendering it perfectly precise.

26 | Cian Dorr and John Hawthorne Of course, this controversial picture need not be endorsed by those who want to use ‘more similar’ in theorizing about naturalness. In philosophy, we often get by perfectly well using language that is quite vague and context-sensitive. Local practices can spring up in which the vagueness and context-sensitivity are kept within limits; this can happen even when participants have no helpful way of explaining which interpretations of the relevant expressions are the intended ones. But while this might be how things stand as regards the relevant theoretical uses of ‘more similar’, it would be dialectically inappropriate simply to presuppose that it is the case in a debate with ‘naturalness sceptics’. In that context, we will perhaps do best to find ways of casting the claims in such a way as to avoid using the vocabulary which is suspected of being too vague or context-sensitive to be theoretically useful. The cause of clarity will best be served by semantic ascent. Instead of cashing out the connection between naturalness and similarity using the objectlevel Similarity and Dissimilarity, we could replace them with claims involving some kind of quantification over contexts, or over the relations which are admissible interpretations of ‘more similar’ in some contexts. Here are some thoughts one might try out in this connection: • Similarity/Dissimilarity is true in some not-too-unusual context. • Similarity/Dissimilarity is true in every not-too-unusual context.38 • The more natural a property P is, the more unusual a context needs to be for ‘Things that have P can be very similar to things that lack P’/ ‘Things that have P can be very dissimilar from one another’ to be definitely true/not definitely false at it. • The less the total naturalness of the properties that divide x and y, and the greater the total naturalness of the properties they share, the more unusual a context needs to be for ‘x and y are very dissimilar’ to be definitely true at it.

38 Note that if we cash out Similarity/Dissimilarity as involving existential quantification over functions, it will be possible for them both to be true across a wide range of contexts which interpret ‘similar’ differently but agree on the interpretation of ‘natural’.

Naturalness | 27 • The more the total naturalness of the properties that divide x1 and x2 exceeds the total naturalness of the properties that divide y1 and y2, the more unusual a context needs to be for ‘x1 and x2 are more similar than y1 and y2’ to be definitely true at it. Each of these claims requires us to make sense, in some rough-andready way, of the degree to which a context is unusual (as regards the interpretation of ‘similar’ and related vocabulary). We could try cashing this out either in some quasi-statistical way, or by means of questions like ‘How much special priming does it take to get into this context?’39 We will not attempt to single out any one principle as the right one to focus on in debates about how much of the naturalness role is satisfied. But we are hopeful that fully-fledged enthusiasts for naturalness will be able to find something that they can accept in this general vicinity. The connection they want to make between naturalness and similarity is surely not just supposed to apply to the relation expressed by ‘similar’ in philosophical contexts—for example, it is surely part of the vision that even in the contexts where we are primarily concerned with resemblance in people’s characters, more natural character-related properties will count for more than less natural ones. 10. Magnetism: The more natural a property is, the easier it is to refer to, ceteris paribus. In Lewis’s thought, this aspect of the role of naturalness is presented in the form of a certain proto-theory about how semantic facts supervene on certain non-semantic facts. The most widely discussed version of this theory is an account of linguistic interpretation according to which it is necessary and sufficient for an interpretation to be correct that it does the best job of simultaneously balancing two factors—‘use’ (interpreting people as disposed to speak the truth) and ‘eligibility’ (assignment of natural meanings).

39 If we go for a statistical construal, we will probably find it beneficial to consider not just contexts in which ‘similar’ is used by English-speakers in the actual world, but some broader range of contexts encompassing many different possible worlds, and expressions in different languages that play a role like that of ‘similar’. See the discussion of Magnetism in what follows for some ways in which one might make sense of the required measure over possible worlds.

28 | Cian Dorr and John Hawthorne But for Lewis, this was just a toy theory. In his considered view, the primary role for naturalness is in the theory of mental content, although naturalness does also play a subsidiary role in the story about how semantic facts supervene on mental ones (see Lewis 1992). (What, exactly, is the role of naturalness in Lewis’s final theory of mental content? ‘New Work’ presents the following simplified story: for C and V to be, respectively, the credence and value function of a certain agent a is for them to achieve an optimal balance of fit—being such that the options to which C assigns the highest expected V-value are those a actually takes—and certain desiderata of humanity, among which is that of eligibility—that ‘the properties the subject supposedly believes or desires or intends himself to have’ not be too severely unnatural (1992: 375).40 As it stands, this view looks inconsistent with certain other considered commitments of Lewis’s. For one thing, Lewis’s philosophy of mind is functionalist as opposed to behaviourist, but the view just sketched is a form of behaviourism: facts about the agent’s internal structure are relevant only in so far as they make a difference to the agent’s dispositions to act. Also, the view is hard to square with Lewis’s claim, that, at least ideally, one has high credence in all the propositions (no matter how unnatural) that are entailed by other propositions in which one has high credence. The second tension could be remedied by taking eligibility as the desideratum that C and V themselves should be as natural as possible. (This requires making sense of naturalness for relations between properties and numbers, but it is hard to see why this should be regarded as more problematic than naturalness for any other relations.) Figuring out how to remove the first tension requires perusing some of Lewis’s other works in the philosophy of mind, such as Lewis (1980). Here is one way a non-behaviouristic analysis might go. Step one: analyse ‘a has credence function C and value function V’ as ‘the three-place relation that plays the credence-value role for a’s species holds between a, C and V’. Step two: analyse ‘R plays the credence-value role for species s’ as a matter of R’s achieving an optimal balance of several desiderata, one of which is that it should not be too common 40 Recall that according to Lewis (1979), the basic objects of credence and value are properties rather than propositions.

Naturalness | 29 for members of s to perform actions that are not optimal according to the C and V to which they are mapped by R, and another of which is that R itself should not be too unnatural. On this way of doing things, the generalization that people tend not to have very unnatural credence and value functions stems from the generalization that natural Rs tend not to map people onto such functions. It is fine to interpret a person as having some rather weird and arbitrary C and V, so long as we have reason to think that their internal structure is weird and arbitrary in some corresponding way, so that a reasonably natural R can map them onto that C and V.41) We won’t be concerned here with the fortunes of any particular account of the supervenience of content-theoretic facts on facts of other kinds. Our purposes are better served by the bare-bones formulation of Magnetism, which could be integrated in many ways into a larger and more ambitious theory of content, and which does not even presume that a reductive theory of content is available. What could ‘easy to refer to’ mean, taken apart from any particular reductive programme? Here is the basic thought. Sometimes, our referring to a given property with a word depends on lots of detailed facts about our use of that word: the property is hard to refer to in the same sense in which the bullseye of a target is hard to hit. On other occasions, the fact that we refer to a given property with a word is much less sensitive to the exact details of use. In these cases, referring to this property rather than any other is a lesser achievement, like hitting some much larger region on the surface of a target. As a very crude first pass, the degree to which a property is easy to refer to might be measured by the number of worlds in which it is referred to. But since the relevant sets of worlds are infinite, simply counting the worlds is no good. What we need 41 The idea that the naturalness desideratum applies in the first instance to relations between people and contents, rather than directly to contents, is reminiscent of Sider’s claim that the generalization that referents tend to be natural is to be explained by the fact that ‘the reference relation must be a joint carving one’ (Sider 2011: section 3.2; and cf. Williams 2007 and Hawthorne 2007). But note that the relation R that plays the credence-value role for a given species is distinct from the relation being an a, C, V such that a’s credence function is C and a’s value function is V. Thus the fact that R is natural is not directly relevant to the question how natural the latter relation is. Indeed, Lewis often treats such functional properties as highly unnatural: for example, the property of having some property that plays the pain role is much less natural than the property that actually plays the role (Lewis 1983b: 349).

30 | Cian Dorr and John Hawthorne to make sense of this thought is something like a measure on possible worlds. One shouldn’t be excessively sceptical here: measures over certain sets of nomically possible worlds are quite integral to the practice of physics and other sciences. This suggests that one can at the very least make sense of the notion of ‘easiness’ required by Magnetism by appealing to some such measure. Given that Magnetism incorporates a ceteris paribus caveat, the claim that a given ordering satisfies this version of the Magnetism role will be tricky to evaluate. But since the spaces of nomically possible worlds on which these measures are defined will usually contain a vast variety of possible language-users and thinkers, it is unlikely that the ceteris paribus clause will need to be interpreted so liberally as to deprive Magnetism of any bite whatsoever. Of course, this is not the only possible way of making sense of the notion of ‘easiness’ in Magnetism. Other interpretations can be derived from other measures over possible worlds, for example an epistemologically-based measure of ‘a priori plausibility’, or a measure extracted from the dissimilarity-distances between worlds, or a measure defined directly in terms of naturalness. Plausibly, some of these more global measures will allow one to place less reliance on the ceteris paribus clause than is required by interpretations of Magnetism in which the relevant measure is one derived from physics. However, physics-based interpretations of ‘easy to refer to’ are likely to be especially useful in the context of the debate between naturalness enthusiasts and naturalness sceptics, since the sceptics are less likely to have qualms about their intelligibility.42 This idea of ‘reference’ to a property could be cashed out in different ways. One choice point concerns speaker versus semantic reference. Another concerns expression by a simple word versus by a complex predicate. And further issues arise in connection with vagueness. Some supervaluationists work in a framework where reference is thought of as being (or as needing to be replaced by) a one-many relation of ‘candidate reference’ or ‘partial denotation’ (cf. Field 1973). This suggests two measures of magnetism for a property: how easily can it be a candidate referent, and how easily can it be a determinate referent (i.e. the one and only candidate 42

See Dorr and Hawthorne MS for more on the relevant physics-based measures.

Naturalness | 31 referent)? The former, candidacy-based notion may behave quite strangely vis-à-vis Lewis’s original vision. If it is fairly commonplace for communities to have a word like ‘bald’ that is vague across a wide range of hair-distribution properties, whereas the flourishing of physics is a modally rare event, then many hair-distribution properties might prove easier to have as candidate referents than the property of being an electron. By contrast, if determinate reference is what counts, concerns about the rarity of physics might not be at all disruptive, if the study of physics is the most common route to determinate reference in the realm of the concrete. Other theorists of vagueness are happy to work with a predicate ‘refers simpliciter’, conceived of as unique but often very vague. One could use this to gloss Magnetism, at the risk that its extensive vagueness will generate an awful lot of vagueness in ‘easy to refer to simpliciter’— for example, if it is not definitely false that ‘bald’ refers simpliciter to exactly the same hair-distribution property at all not-too-distant worlds, no precisification of ‘bald’ will definitely fail to be easy to refer to. We will generally work with a gloss on ‘easy to refer to’ as ‘easy to determinately refer to’, since this seems to fit more unproblematically with the rest of the role.43 Finally, we should make explicit a claim about naturalness that Lewis presupposes in much of his discussion: 11. Necessity: Facts about a property’s degree of naturalness are non-contingent. The denial of Necessity would be quite alien to Lewis’s thought, and would require rethinking many of the other components of the role.44 Recall that Supervenience requires whatever properties are in fact perfectly natural to be such that, whenever any two possible 43 One might, like Williamson (1994), have the view that despite the fact that it is vague what ‘bald’ refers to simpliciter at each world in our modal neighbourhood, it is still definitely true that its reference simpliciter varies from world to world in a very fine-grained way. In that case it will also be definitely true that neither baldness itself, nor any of the other hair-distribution properties which ‘bald’ does not definitely fail to express, is easy to refer to (at least in our modal neighbourhood), so that glossing Magnetism in terms of reference simpliciter is more promising. See Dorr and Hawthorne MS for further discussion of the considerations for and against such extreme ‘semantic plasticity’. 44 See Lewis (1986: 61 n. 44). The view that the naturalness facts are contingent is favourably entertained by Cameron (2010).

32 | Cian Dorr and John Hawthorne worlds differ (or differ qualitatively), they differ in the distribution of those properties. This generates some pressure to think that there are properties that are actually perfectly natural but uninstantiated. Are there other possible worlds that differ from the actual world just as regards the relative naturalness of these alien properties? It is hard to believe that there are; but it is also hard to think of any reasonable story about how the contingent facts about the naturalness of these alien properties could supervene on any other facts at the worlds where they are uninstantiated. Likewise, Similarity and Dissimilarity relate truths about the naturalness ordering to modal facts (about possible levels of similarity and so on); at least on an S5-friendly conception of metaphysical modality, it is obscure how one could coherently combine this with the thought that the naturalness ordering is contingent.45 Of course, those who reject Necessity might be able to formulate surrogates for the other Lewisian principles that preserve some of their spirit. For example, Supervenience might be replaced by a claim to the effect that no two possible worlds at which exactly the same properties are perfectly natural differ qualitatively without differing in the pattern of instantiation of those properties. Magnetism might be replaced by a claim to the effect that for any given property, it is hard to refer to that property in a world where it is not very natural. And so on. The denier of Necessity faces a range of difficult and delicate questions about the possible distributions of perfect and relative naturalness. Among the properties which are not in fact perfectly natural, which ones could be perfectly natural? One possible view is that every property whatsoever could have been perfectly natural.46 While it seem like an attraction of this view that it enables us to dodge the need for making a distinction between the possibly perfectly natural properties and the rest, plenty of other awkward questions remain. For example, assuming that it is still necessary that no perfectly natural property supervenes on all the rest, there 45 With weaker modal logics, a whole range of decision points open up, including an S4-rejecting view that keeps Necessity while denying that the naturalness ordering is necessarily necessary. 46 One might want to make an exception for necessary and impossible properties, and perhaps also for certain cardinality-related properties which could never be part of any minimal supervenience basis over any set of worlds.

Naturalness | 33 must be limits on which sets of properties can be perfectly natural together; and it is hard to think of a good way to answer questions like ‘Which properties could be perfectly natural in a world where being a spoon was perfectly natural?’ Also, it is unclear whether there is any prospect of making headway with the question, concerning a given list of properties, which among the possible patterns of distribution are consistent with all of them being perfectly natural, and which are consistent with none of them being perfectly natural. For example, could the properties that are in fact perfectly natural have been distributed as they actually are while the property of being perfectly natural had a different distribution?47 One possible retreat that is still rather plenitudinous in spirit is to say that while not every property could be perfectly natural, it is still true that necessarily, every property is coextensive with at least one possibly perfectly natural property. Of course, there are also much more restrictive views available; for example, one might think that it is only qualitative properties, or intrinsic properties, or properties featuring in some interesting way in the special sciences, that are coextensive with possibly perfectly natural properties. Finally, the most conservative way to deny Necessity involves saying that the only properties that are not perfectly natural but could be are uninstantiated ones. However, this last view threatens to collapse into a notational variant on Lewisian orthodoxy, since it suggests the generalization that being possibly perfectly natural and instantiated is necessary and sufficient for being perfectly natural. Since we don’t have much sense of how to steer a disciplined path through this garden of decision points, we will assume Necessity for the remainder of our discussion.48 * * * 47 If perfect naturalness is itself perfectly natural, the answer is obviously no, but a question remains about the extent to which the other perfectly natural properties could have a matching distribution compossibly with a different list of perfectly natural properties. 48 One motivation for denying Necessity is the thought that there are possible worlds where, unlike the actual world, electrons are made of smaller particles, and where electronhood fails to be perfectly natural in the same way that being a hydrogen atom fails to be perfectly natural at the actual world. If one were gripped by this thought, one might be tempted to say too that there is a world where hydrogen atoms are simple, and where being a hydrogen atom is perfectly natural. But this conflicts with the compelling thought that to be a hydrogen atom is to be an atom containing exactly one proton—about as compelling an instance of the necessary a posteriori as one

34 | Cian Dorr and John Hawthorne The eleven claims on our list give us enough to go on for the purposes of this paper, although there are certainly other roles for naturalness which can with some plausibility be extracted from Lewis’s discussion, and which have been taken up to varying degrees in subsequent work. For example, it is clear that Lewis wants to make some connection between the fact that greenness is more natural than grueness and the epistemological fact that a disposition to infer that all emeralds are green from the evidence that all observed emeralds have been green is more reasonable than a disposition to infer that all emeralds are grue from the evidence that all observed emeralds have been grue. However, Lewis does not provide us with many clues about the form this connection should take. Given Lewis’s other commitments, one would hope to be able to characterize it in a Bayesian framework, where facts about the rationality of inductive inferences boil down to facts about which prior credence functions are reasonable. If one could make sense of comparisons of naturalness for probability functions—and there is no obvious obstacle to doing so, given that such functions can be thought of as relations between propositions and numbers—one could propose a view where the reasonableness of having a certain probability function play the role of one’s priors is tied directly to the naturalness of that function.49 But this can’t be right, since some probability functions which other parts of the role suggest are quite natural would make horribly unreasonable priors. Consider for

could hope to find. Perhaps, however, the proponent of the view will say that while this account of what it is to be a hydrogen atom is correct, there is a property which is coextensive with being a hydrogen atom at worlds where the list of perfectly natural properties is what it actually is, but which is instantiated by simples, and perfectly natural, at certain other worlds. Our own view is that the claim that electrons are simple is just as good a candidate to be necessary if true as the claim that hydrogen atoms aren’t. 49 One issue that we would need to resolve in formulating an appropriate measure of naturalness for probability functions comes from the fine-grained considerations that seem to play a role in individuating propositions (conceived of as the objects of credence). If we think of propositions as including Fregean ‘guises’, our measure will somehow need to take into account the naturalness of the guises as well as, or instead of, the naturalness of the properties of which they are guises. For example, the projectibility of greenness looks rather less when the relevant guise is given by the description ‘Fred’s actual favourite property’. These issues also complicate the attempt to connect naturalness to epistemic value.

Naturalness | 35 example a probability function that treats all coin tosses as fair and independent: having this play the role of one’s prior credence function would mean being disposed to assign credence 1/2 to the proposition that a coin will land Heads the next time it is tossed irrespective of facts about the coin’s track record, even when the track record strongly suggests that the coin is biased. However, there are more attractive ways in which we might connect naturalness to the reasonableness of priors. We could say, for example, that a reasonable prior credence function can be constructed by taking a weighted average of many probability functions, in such a way that more natural probability functions are weighted more heavily, and roughly equally natural probability functions are weighted roughly equally. (Note that if the connection between naturalness and inductive rationality turns on the naturalness of probability functions, there will be no straightforward route from a claim about the naturalness of a property of objects to any epistemological claim. There is no obvious need to employ any notion of the ‘degree of projectibility’ of a property; and if we do manage to devise a measure of projectibility that makes sense in a Bayesian framework, there is no obvious reason to expect that only natural properties will achieve high projectibility scores.) A second normative concept which one might want to connect to naturalness is that of intrinsic epistemic value. While Lewis says little about this, it plays a starring role in Sider (2011). One attractive idea relates naturalness to the fact that some true beliefs (or items of knowledge) are more valuable than others. If we care about the truth in the way we should, we will prefer the opportunity to find out whether neutrinos have mass to the opportunity to memorize the contents of a telephone directory. So long as we can make sense of naturalness for propositions as well as properties, it is tempting to explain this by claiming that the more natural a true proposition is, the more epistemic value one achieves in believing it (or knowing it, or having high credence in it), at least ceteris paribus. If we accept this claim, how should we generalize it to false propositions? Is believing a natural false proposition worse or better than believing an unnatural one? Sider’s discussion, in which naturalness and truth are treated as two independent ‘aims of belief’,

36 | Cian Dorr and John Hawthorne might be read as suggesting that it is better: a natural false belief is good in one way and bad in another, while an unnatural false belief is bad in both ways (see Sider 2011: 62). But this strikes us as odd. If forced to choose, shouldn’t one prefer a pill that will inculcate millions of false beliefs about strangers’ phone numbers to one that will inculcate a mistaken belief as regards whether neutrinos have mass? Moreover, if one believes a false natural proposition P (e.g. that electrons are more massive than protons), one will thereby be in a position to infer a vast array of less natural consequences of that proposition (P or at least one zebra wears pyjamas; P or at least two zebras wear pyjamas; . . .). A good theory of epistemic value should not entail that there is great disvalue attached to actually carrying out these inferences. (Sider also endorses a different, and according to him stronger, version of the thesis about epistemic value, according to which it is ‘better to think and speak in joint-carving terms’, and ‘worse to employ non-joint-carving concepts’ (2011: 61). This strikes us as much more problematic than the ‘weak version’ of the thesis, in so far as it is supposed to be a claim about intrinsic value. It is easy to see how speaking or thinking in a language whose syntactically simple expressions have unnatural semantic values could have various kinds of instrumental disvalue. It will mean one has to work much harder even to entertain the true and natural propositions belief in which brings epistemic value according to the weak version of the thesis. Indeed, on some views, it may make it impossible to entertain these propositions—if one thinks of propositions as structured entities, one might suppose that no natural propositions can be expressed in a language whose syntactically simple predicates express unnatural properties. Moreover, having a badly engineered language of thought may also tempt one to form false beliefs about naturalness and similarity. But how could anything of intrinsic value turn on the nature of one’s system of representation?) The putative normative components of the naturalness role have an important role to play in the debate between naturalness enthusiasts and naturalness sceptics; they are especially strongly evoked by labels like ‘elitism’ and ‘egalitarianism’. But we are not going to have much more to say about them, in part because they don’t figure much in Lewis, and in part because we haven’t yet come up

Naturalness | 37 with many interesting things to say about their relation to other aspects of the role.50 Some will be thinking that our list has left out an absolutely crucial component of the naturalness role. What about the claim that the perfectly natural properties are exactly the fundamental properties, or that the more natural a property is, the closer it is to being fundamental?51 What about the claim that all facts obtain in virtue of, or reduce to, the facts about the instantiation of the perfectly natural properties? Or what about the claim that the perfectly natural properties are the ones that give reality its structure? Our reason for not putting this sort of thing into the role is that the dispute we are trying to illuminate is one in which one side—that of the ‘naturalness sceptics’—is sceptical not just about ‘natural property’ but about expressions like ‘fundamental property’, ‘irreducible property’, etc. For this dispute, our methodological advice to focus on questions that can be asked without using the contentious vocabulary requires ‘fundamental’, ‘irreducible’, and the rest to be Ramsified out along with ‘natural’.52 * * * 50 Weatherson (forthcoming) suggests that there might be no ranking that satisfies both the role of ‘M-naturalness’ (our Supervenience, Duplication, Non-duplication, Laws, Similarity, and Dissimilarity) and that of ‘E-naturalness’ (the connection with inductive rationality). He suggests two arguments: (i) greenness is highly projectible but not very M-natural; (ii) electronhood is very M-natural but not very projectible (the claim that observed electrons do such-and-such does not by itself support the claim that all electrons do such-and-such, unless one also has positive reason to believe that electronhood is M-natural). These arguments depend on what seems to us an overly simple way of cashing out E-naturalness, as essentially equivalent to ‘projectibility’. The second argument also raises subtle issues connected with the role of ‘guises’ in the epistemic part of the naturalness role—the problem might be attributed to a lack of naturalness in the guises under which we think of electronhood rather than to a lack of naturalness in electronhood itself. 51 When Lewis uses ‘fundamental’ as a term of metaphysical art, he treats it as interchangeable with ‘perfectly natural’ (e.g. in Lewis 2009). 52 Of course, we can also imagine a debate about naturalness in which both sides take it for granted that, say, some properties are more fundamental than others; this debate will involve new questions about which of the other naturalness roles are satisfied by the fundamentality ranking. Many of the arguments about this question will recapitulate arguments about the extent to which our naturalness roles are co-satisfiable. For example, if one assumes that the fundamental properties form a supervenience base, one can argue that the fundamental properties are not all easy to refer to from the premise that the properties that are easy to refer to do not form a supervenience base. For the introduction of fundamentality into the role to give rise to new and interesting arguments, the disputants will need to find ways of supporting claims about fundamentality that do not depend on the assumption that fundamentality satisfies one of the other naturalness roles.

38 | Cian Dorr and John Hawthorne Throughout the discussion so far, we have followed Lewis in using ‘natural’ as a predicate of properties, conceived of as ‘abundant’. In treating properties as abundant, we have implicitly been helping ourselves to the instances of the schema ‘The property of being F is, necessarily, instantiated by all and only the things that are F’. While this is all very convenient, anyone who wants to talk like this had better face up eventually to the fact that the schema is inconsistent: it entails that the property of not instantiating oneself instantiates itself iff it doesn’t instantiate itself.53 This is not the place for a survey of all the candidates for a consistent replacement for the inconsistent practice. Let us just mention the best known, namely to officially replace all talk about properties with higher-order quantification, i.e. quantification into predicate position (see Williamson 2003). In this way of talking, ‘natural’ would no longer appear as a predicate of the same syntactic type as ‘cheerful’: rather, it would become a higher-order predicate—something that takes a predicate and makes a sentence. Or rather, it would have to give way to an infinite family of higher-order predicates: ‘natural((i))’ (taking a one-place first-order predicate as argument), ‘natural((i,i))’ (taking a two-place first-order predicate as argument), etc.54 Similarly, we would have many different versions of ‘more natural’, ‘perfectly natural’, etc. Indeed, this may only be the tip of the iceberg. If we had been wanting to talk about the naturalness of properties of properties, we will also need versions of ‘natural’ of higher types: ‘natural(((i)))’, ‘natural((((i))))’, ‘natural(((((i)))))’, . . . as well as ‘natural(((i),(i)))’, ‘natural((i,(i),((i))))’, and so on. The question whether we have any need for versions of ‘natural’ taking higher-order predicates as arguments will depend on how much interesting work we were originally hoping to do by talking about the naturalness of properties of properties. In so far as the focus is on the theory of reference, there seems to be no relevant difference between the orders: for example, we can discuss how easy it is to have an expression whose meaning is this or that generalized quantifier. On the other hand, similarity for properties is a far 53 This is inconsistent in classical logic; new possibilities open up for those who regard ‘P iff not-P’ as consistent (see Field 2008). 54 The subscripts indicate type, in the usual notation: an expression of type i is a singular term, and an expression of type (t1 . . . , tn) is something that yields a sentence when combined with an expression of type t1, . . ., and an expression of type tn.

Naturalness | 39 more rarefied concern than similarity for objects, while duplication for properties is deeply obscure. If one has versions of ‘natural’ of higher types, one might also wonder whether there would be any use in a predicate ‘natural(i)’ that makes a sentence when combined with a term referring to an object. What work could be done by a notion of naturalness that allows us to make discriminations among objects? Here one can’t appeal to contingent features of objects such as how sharply they stand out from their surroundings, at least assuming that Necessity is not to be sacrificed. Nor do Similarity, Dissimilarity, Duplication, or Non-duplication give such a predicate much life. Magnetism, on the other hand, seems well suited to making discriminations between objects, especially if we have an abundant ontology with objects corresponding to all sorts of arbitrary modal profiles. If one wants to speak of perfectly natural objects, the obvious strategy is to use them to refine the first of our glosses on Supervenience, so that the universal supervenience base consists of propositions of the form Fα1 . . . αn, where α1 . . . αn as well as F are perfectly natural. Independence could then be taken as requiring certain forms of independence among those special propositions—on the Combinatorialist version, the upshot would be that the perfectly natural objects are modally completely interchangeable. As at the level of properties, these ideas about the role played by object naturalness may push in different directions. For example, the suggested revision of Supervenience encourages such views as that the perfectly natural objects include all fundamental particles, or all spacetime points, whereas Magnetism suggests a starkly different assessment of the naturalness of such objects relative to, e.g. the number two.55 We will not have anything more to say about natural objects in the present paper. One consequence of moving to a higher-order approach is that certain apparently intelligible thoughts can no longer be encapsulated in a finite sentence. For example, no finite sentence corresponds to ‘There are no perfectly natural relations of arity greater than three’. The problem is that every variable has to have a particular syntactic type, and only finitely many types of variables can 55 Given the problem of the many, the judgement that people are easier to refer to than spacetime points is not straightforward.

40 | Cian Dorr and John Hawthorne occur bound in any finite sentence. For similar reasons, the higherorder approach will not give us any finite sentence that could serve as a replacement for ‘All facts supervene on the totality of facts about which things have which perfectly natural properties’. If we can understand what these claims are getting at, our ability to do so seems to rest on our ability to grasp what would be meant by certain sentences in an infinitary higher-order language. This is controversial, but those who are sceptical about our ability to reach into the infinite in this way should probably not be amenable to the higher-order approach in the first place.56 The idea that it is best to theorize in this area using an expression that can make a sentence by combining with expressions other than singular terms, rather than an ordinary predicate applying to properties or entities of any other sort, is a key theme in Sider (2011). However, it is worth noticing some features of Sider’s regimentation that distinguish it from a higher-order approach of the sort gestured at earlier. (i) Rather than many different versions of ‘natural’, Sider suggests that one could legitimately use a single, promiscuous expression S that can univocally combine with expressions of myriad types—including its own promiscuous type. (In section 4 we will discuss Sider’s argument that ‘SS ’ is not only meaningful but true.) (ii) Another aspect of the promiscuity of S is that it can combine with all manner of expressions, many of which would not typically be thought of as having ‘semantic values’. He includes variable-binders (e.g. ‘λ’), and it is not clear whether he has any grounds for denying the well-formedness of ‘S un’ (an S followed by the morpheme ‘un’), ‘S ,’ (an S followed by a comma), or ‘S (’ (an S followed by a left parenthesis).57 (iii) Given that Sider’s S can combine with expressions where it is obscure what it would

56 In so far as the expressibility problems merely involve the absence of any higherorder sentence corresponding to a quantification over relations of arbitrary arity, we might remedy it by using monadic predicates applying to ordered tuples as surrogates for polyadic predicates. In interpreting ‘natural’ applied to such predicates, we would need to be careful not to confuse the original sense of ‘natural’ with the new sense that enables us to use ordered pairs as surrogates. Those who are very concerned about expressibility will likely doubt that we could understand the new sense. 57 Note that if we could make sense of questions like these, they would not be questions about the relevant bits of language, any more than ‘S red’ is about the word ‘red’.

Naturalness | 41 mean to quantify into the relevant position, or to flank an identitylike symbol with such expressions, his notation leaves it obscure what it would mean to claim exhaustiveness on behalf of a particular list of S -claims. Even for predicates, it is not clear whether any of the extant ways of making sense of identity would fit with Sider’s purposes, given the extremely fine-grained distinctions he wants to make.

3. HOW MUCH OF THE ROLE IS SATISFIED? Our aim in this section is to draw attention to a range of subsets of the roles listed in section 2 for which one might reasonably doubt their joint satisfiability. In some cases we will simply point to a set of claims whose joint satisfiability is not obvious; in others, we will present distinctive reasons for doubting the relevant co-satisfiability claim. (We make no claim of exhaustiveness.) While this will give the section a somewhat negative cast, we will not be endorsing any of the reasons for doubt; nor do we want to get sucked into a debate about the proper location of burdens of proof. Our main aim is to show how, once the debate about naturalness is structured as we think it should be, the ‘naturalness sceptic’ becomes a more interesting and dialectically formidable character than the Goodmanian crypto-idealists that we tend to encounter in the works of naturalness enthusiasts. In general, the fragments of the role for which doubts about satisfiability arise comprise more than one of the principles. For example, given a general framework in which we don’t worry about the existence of properties and facts, and conceive of properties as necessary existents, it is unproblematic that there is a collection of properties upon which all facts supervene, namely the collection of all properties. Thus, barring a challenge to this background framework (one that is not normally the focus for disputes about naturalness), there is no worry about the satisfiability of Supervenience taken by itself. We could make it true simply by interpreting ‘more natural’ in such a way that no property is more natural than any other, so that all properties whatsoever count as perfectly natural. Similarly, if one interprets Magnetism in the bare-bones way sketched earlier, rather than treating it as a placeholder for Lewis’s entire

42 | Cian Dorr and John Hawthorne theory of linguistic and mental content, doubts about its satisfiability look far-fetched: surely some properties are easier to refer to than others. But as soon as one starts to combine the roles, space for reasonable doubt opens up.58

3.1. Supervenience + Magnetism Whatever exactly we mean by ‘easy to refer to’, it looks unlikely that the properties that are tied for first place will be numerous enough to provide a supervenience base for everything. Of course, Magnetism as we stated it contains a ceteris paribus clause, which could allow for some slippage here. Nevertheless, one could consistently maintain that all supervenience bases involve properties that are so hard to refer to that there is no reasonable interpretation of the ‘ceteris paribus’ clause which would allow us to sustain the combination of Magnetism and Supervenience.

3.2. Supervenience + Magnetism + (Non-duplication or Independence) Even if there are interpretations of ‘perfectly natural’ that satisfy Supervenience and Magnetism, new worries may arise when we include a third principle. For example, if every supervenience base whose members are reasonably easy to refer to contains at least one property that divides duplicates, Non-duplication will not be jointly satisfiable with Supervenience and Magnetism. In thinking about the joint satisfiability of Supervenience, Magnetism, and Independence, we need to distinguish between the different versions of Independence. Non-supervenience should not be hard to satisfy: given the ceteris paribus clause in Magnetism, we do not have to say that every property that is maximally easy to refer to is perfectly natural, so we need not be concerned even if it turns out that 58 We will not be considering any tensions there might be between the principles on our list and intuitive judgements about certain properties’ degrees of naturalness. These aren’t the kind of arguments that are appropriate in debates about naturalness scepticism. Even for those who are completely comfortable using ‘natural’, there is much to be gained—at least by way of dialectical clarity—by trying to settle questions about which portions of the role are satisfied at all before going on to ask which of them are satisfied by naturalness.

Naturalness | 43 some maximally easy-to-refer-to properties supervene on others. However, there is more reason to worry about stronger versions of Independence, which require a richer denial of necessary connections between the perfectly natural properties beyond a failure of supervenience. One might reasonably suppose that every set of reasonably easy-to-refer-to properties is either too small to be a supervenience base or else full of rich modal connections which would violate the stronger construals of Independence.

3.3. Independence + ((Duplication and Non-duplication) or Supervenience) Suppose there were only two perfectly natural monadic properties, F and G, and one perfectly natural relation, R. Then by Duplication, there could not be more than eight equivalence classes of duplicate atoms. For given any nine atoms, there must be at least two atoms x and y such that Fx↔Fy, Gx↔Gy, and Rxx↔Ryy; Duplication tells us that any such atoms x and y are duplicates. Moreover, Combinatorialism and Non-duplication jointly entail that it is possible that there are at least eight equivalence classes of duplicate atoms: Combinatorialism entails that there could be eight atoms no two of which satisfy the above biconditionals, and Non-duplication tells us that any duplicate atoms would have to satisfy the biconditionals. And the point obviously generalizes: if there is some finite number n of perfectly natural properties, the maximum possible number of equivalence classes of duplicate atoms is exactly 2n. Thus, anyone who wants to remain non-committal as regards whether the maximum number of such equivalence classes is a power of two has reason to doubt whether there is any interpretation of ‘perfectly natural’ that satisfies Combinatorialism, Duplication, and Non-duplication.59 59 Note that if we appeal to the infinitary-language version of Combinatorialism, the result extends to the case where the number of perfectly natural properties is infinite. Thus, even those who are confident that there could be infinitely many atoms no two of which are duplicates do not automatically escape the worry, since not all infinite cardinalities are powers of two. There are also interesting issues here for the view that there are at least as many perfectly natural properties as sets. A natural generalization of Combinatorialism to such a view suggests that in that case, every plurality of perfectly natural properties is such that possibly, there is an atom that instantiates just those properties. Given

44 | Cian Dorr and John Hawthorne Similar cardinality-based considerations can be used to put pressure on the combination of Combinatorialism with Supervenience. If F, G, and R are the only perfectly natural properties, then the second gloss on Supervenience tells us that there are at most eight qualitatively discernible worlds containing exactly one object, while Combinatoralism tells us that all eight combinations are metaphysically possible. In general, the number of equivalence classes of qualitatively indiscernible one-object worlds is guaranteed to be a power of two. Here again, those who wish to be non-committal will have reason to be hesitant about the combination of Supervenience and Combinatorialism.60 However, these kinds of worries do not arise in any obvious way for the other interpretations of Independence—for example, given that there are supervenience bases at all, there must be some minimal ones.61

3.4. Duplication + (Independence or Dissimilarity or Laws or Magnetism) Consider the following prima facie plausible claims: (i) it is impossible that there should be duplicate objects with different masses, and (ii) any positive real number could be the mass in grams of some mereological atom. Given Duplication, these commitments impose a severe constraint on the extension of ‘perfectly natural’. Lewisian modal realism, it will then follow that there are more atoms than perfectly natural properties. We would thus require three more-than-set-sized cardinalities: that of the perfectly natural properties, that of the atoms, and that of the fusions of atoms, whereas for Lewis 1986, any things that are fewer than the atoms are few enough to form a set. 60 One can also, with more work, use Combinatorialism and Supervenience to calculate the number of equivalence classes of worlds with domains of any given size, given a specification of the number of perfectly natural relations of each arity. As in the one-object case, there are plenty of numbers that cannot be generated in this way. 61 For example, take the set of all properties of the form being in a Q world, where Q is a complete qualitative world-description, and throw away one of its members; the result is guaranteed to be a minimal supervenience base in the sense corresponding to the second gloss on Supervenience (any qualitatively discernible worlds differ in the distribution of perfectly natural properties). The hypothesis that every supervenience base has a proper subset that is also a supervenience base is not quite so easy to rule out on the interpretations of ‘supervenience base’, but it does not look very promising.

Naturalness | 45 For one thing, it must be infinite, since any finite set will (as noted in section 3.3) allow for only finitely many equivalence classes of duplicate atoms.62 Which properties should we throw into the set so as to give us the necessary plenitude of mass-difference makers? The most obvious strategy (and one that Lewis seems sometimes to endorse) is to say that all the maximally determinate mass properties, such as having a mass of 4.183748 grams, are perfectly natural. However, this move makes for severe tension with several other parts of the package.63 • There is a worry about some versions of Independence. The determinate mass properties seem to be pairwise incompatible, whereas according to Combinatorialism, perfectly natural properties never are.64 • There is a worry about Dissimilarity. It is quite intuitive to think that every degree of similarity short of the maximum possible degree is consistent with difference in mass. For it would be strange if the function that takes a real number x and returns the maximum degree of similarity that could obtain between two objects whose mass differed by x grams failed to be continuous at 0. On this picture, being divided by a determinate mass property like being exactly three grams would seem to do practically nothing to ‘make for dissimilarity’. (Contrast the property having some mass or other, which seems a much more potent dissimilarity-maker.) • There is a worry about Laws. Propositions which express functional relationships between mass and other quantities will score terribly for simplicity if we have to express them as giant disjunctions of claims about all the determinate mass properties.

62 Note that if (ii) were weakened to the claim that any positive real number could be the mass of some object (perhaps non-atomic), it would have no such implication. 63 Many of the points that follow are also discussed in chapter 11 of Hawthorne (2006). 64 Also, if there are necessary truths about how the masses of composite objects depend on the facts about the masses of their parts, the version of Independence mentioned in footnote 18, according to which the complete description of a world in terms of only some of the perfectly natural properties never entails its complete description in terms of the rest, is threatened. In a world with two point particles of masses 2 and 3 grams whose fusion has mass 5 grams, each of the three mass properties is redundant.

46 | Cian Dorr and John Hawthorne We could try to avoid this problem by including further masstheoretic relations on our list of perfectly natural properties alongside the determinate masses, but in that case we will have to give up even the Non-supervenience version of Independence.65, 66 • There is a worry about Magnetism, since determinate mass properties seem to be very hard to refer to.67 (Perhaps it is easy given how things actually are for us to refer to various integer multiples of one gram—e.g. because a certain platinum bar whose mass happens to be one kilogram is crisply demarcated from its surroundings. But this is irrelevant to Magnetism as we are understanding it, given that at almost all close worlds the bar will have a marginally different mass, so that people do not use predicates like ‘three grams’ to express the same properties they actually express.) In response to this worry, one could of course lean on the ceteris paribus clause we built into Magnetism. If we take the use-plus-eligibility theory as our model, it is easy to see how ceteris could fail in this case to be paribus. For on that account, even a perfectly natural property could be very hard to refer to, if it belongs to a large family of ‘competitor’ perfectly natural properties which will typically score roughly equally well as regards ‘use’. On this picture, if we want to predict how easy it is to refer to a property, its absolute level of naturalness is less useful than the degree to which it is more natural than its competitors. This way of sharpening up Magnetism is not inevitable, but is clearly in the spirit of Lewis’s account. 65 Another option is to replace Laws with Laws*, give up Simplicity, and say that the supervenient relations required for a simple statement of the laws are very natural, but not perfectly natural. 66 Eddon (2013) explores a related route from the claim that determinate mass properties are perfectly natural to the denial of Non-supervenience. She argues (appealing to Laws, Similarity, and Dissimilarity) that metric relations among determinate mass properties (like being the sum of) are very natural, and (implicitly appealing to something like Simplicity) that some such relations must be perfectly natural. Since the facts about the holding of these relations are non-contingent, they trivially supervene on everything. 67 Recall that we are glossing Magnetism in terms of ease of determinate reference, for reasons explained earlier.

Naturalness | 47 Counting the determinate masses as perfectly natural is not the only way to prevent atoms that differ in mass from ever sharing all their perfectly natural properties. Another approach would include all the properties corresponding to intervals on the mass scale, e.g. being between three and four grams in mass. Or, more subtly, we could use properties of the form having a one in the nth place of the binary expansion of one’s mass in grams.68 We could also achieve the required discriminations by including families of hybrid properties like being square while having mass three grams, or even being square if and only if one’s mass is three grams. Some of these approaches can plausibly escape concerns arising from Independence. However, it is hard to see any escape from the worries about Magnetism and Dissimilarity. Suppose there is a possible world in which each determinate mass is instantiated by a mereological atom, where these atoms are otherwise as similar as can be. Given (i) (the claim that duplicates can never differ in mass) and Duplication, whatever properties we choose to count as perfectly natural will have to divide all these things. But it seems to be possible that some of the atoms in the world in question attain any degree of similarity consistent with lack of duplication; and, relatedly, that there are no easy-to-refer-to properties which are instantiated by only some of the atoms.69 One could respond to these concerns by discarding Duplication or by rejecting (i). These moves open up many new candidate lists of mass-relevant perfectly natural properties. One might, for example, use relations of mass-betweenness and mass-congruence holding among massive objects, as in Field (1980). Or one might posit a ‘mass space’ conceived in a substantivalist way, with a perfectly natural ‘occupation’ relation between massive objects and points in

68

Cf. Moss (2012). At worlds where the total mass of all particles is finite, we can make non-trivial divisions using easy-to-refer-to properties like being less than half the total mass of the universe; but these divisions are not available in the world we were imagining because of its infinite mass. Moreover, counting properties like that as perfectly natural would not sit well with Non-duplication, so long as we are comfortable with the ideology of transworld duplication: the claim that if x is less than half the total mass of the universe at w and y is more than half of the total mass of the universe at w seems to be consistent with the claim that x at w is a duplicate of y at w. 69

48 | Cian Dorr and John Hawthorne the space, together with suitable relations to structure mass space (see Arntzenius and Dorr 2012: section 8.3).70, 71

3.5. Empiricism + Independence According to ‘dispositional essentialists’ about the theoretical vocabulary of physics, predicates like ‘negatively charged’ express properties that necessarily involve the disposition to behave in certain ways under certain circumstances. One some versions of the idea, an upshot is that many interesting sentences entirely couched in the proprietary terminology of fundamental physics (and hence not involving words like ‘cause’, ‘disposed’, etc.) that are normally taken to express merely nomically necessary truths are in fact metaphysically necessary. For example, perhaps it is necessarily not the case that there are two particles, one negatively charged and the other positively charged, which remain relatively at rest throughout eternity in a world with no other massive objects. If this kind of dispositional essentialism is correct, those who want to count the properties expressed by such theoretical predicates as perfectly natural

70 There is precedent for the thought that there are certain entities which, although not among an object’s parts, are still such that relational properties involving them are ‘intrinsic’ and hence as relevant to questions of duplication. For example, some believers in universals might think that, although universals are not parts of the objects that instantiate them, an object’s instantiation relations to universals should play the same kind of role in our theory of duplication as its relations to its parts. 71 A question for those who reject the combination of (i) and Duplication is whether to keep the following weakening: if there is some permutation of the domain of all objects which maps x to y and preserves all perfectly natural properties and their negations, then x and y are alike in mass. (Here ‘the domain of all objects’ includes not only points and regions of spacetime but also points and regions of mass space, if there are such things.) There are apparent possibilities which put even this thought under pressure. Consider a static infinite plenum with uniform, positive mass-density, and choose an arbitrary point o. Let our permutation π map each point x onto the point twice as far from o in the same direction as x, and extend this to regions in the natural way. (If we are substantivalists about mass space, let π also map each point of mass space onto the point that is eight times further than it from the zero point.) While this π obviously fails to preserve mass, it will plausibly preserve all the properties and relations that are perfectly natural according to the theorists we are considering. If they admit that the case is possible and accept the third gloss on Supervenience, they will have to say that determinate masses are haecceitistic rather than qualitative properties.

Naturalness | 49 will have to give up some versions of Independence—at least Combinatorialism, and probably also Recombination. Other dispositional essentialists may not want to go along with the suggestion that the proprietary vocabulary of actual-world physics yields interesting metaphysical necessities. For example, they may think that it is possible that a negatively charged and positively charged particle should remain relatively at rest without the presence of any other massive objects, provided that there are some instances of alien properties whose dispositional effects mask the disposition to attract that is conferred by the facts of opposite charge. Perhaps the right sorts of alien maskers could sustain pretty much any logically consistent pattern of physical properties. But such a view still threatens some versions of Independence. For to preserve Supervenience, some of the relevant alien maskers must be perfectly natural, so that the fact that oppositely charged particles cannot remain at rest in a world without any of the alien maskers will still constitute a counterexample to the infinitary-language version of Combinatorialism.72

3.6. (Empiricism or Duplication) + (Dissimilarity or Magnetism) Suppose that (like many dispositional essentialists) you think that it is impossible that electrons should behave in such a way as to render the actual laws of nature radically, pervasively false. You might still be willing to countenance the possibility that electrons should exist in a world where there are some small, localized exceptions to the actual laws. Such tolerance for small exceptions is plausible in a wide range of cases where a predicate is associated with a theory in such a way that it could not apply to anything in a world where the theory was radically and pervasively false; it has considerable attractions even in the realm of particle physics. This combination of views—call it ‘tolerant physical necessitarianism’— strongly suggests that predicates like ‘electron’ are vague. For we can construct a Sorites sequence of sentences expressed using such 72 The view in question will also be inconsistent with the version of Recombination that entails that the world could consist entirely of duplicates of any given objects. For given Non-duplication, a world composed entirely of duplicates of actual-world objects cannot contain any instances of alien natural properties.

50 | Cian Dorr and John Hawthorne predicates, starting with a sentence which entails the actual ‘electron’-laws, and ending with a sentence which entails that these laws are radically and pervasively false; for familiar reasons, it will be hard to maintain that there is a definitely correct answer to the question which is the first sentence to express a metaphysical impossibility. As we discussed under Magnetism, the fact that our predicate ‘F’ is vague strongly supports the claim that F-ness is not in the relevant sense easy to refer to. Thus, in so far as tolerant physical necessitarianism is supposed to apply to all predicates introduced in the course of scientific theorizing, it makes the combination of Empiricism and Magnetism hard to sustain. Tolerant physical necessitarianism also generates a tension between Empiricism and Dissimilarity, if we take the latter to concern a ‘cross-world’ notion of dissimilarity. Take a possible world w near the outer limits of the set of worlds at which there are electrons: at w, the exceptions to the actual laws are about as widespread as it is possible for them to be, consistent with there being any electrons at all. It would be tempting for the tolerant physical necessitarian to suppose that there is another world w¢ that is very similar to w, but that just crosses the line, so that there are no electrons at w¢. This claim of similarity among worlds suggests that the following claim about cross-world similarity of objects will be true: ‘for some x and y, x is an electron in w and and y is not an electron in w¢ and x as it is in w is very similar to y as it is in w¢’. This seems inconsistent with the claim that electronhood satisfies the condition that highly natural properties have to satisfy according to Dissimilarity, namely making for a high degree of dissimilarity among the objects divided by it. It is interesting to consider whether views like tolerant physical necessitarianism and dispositional essentialism provide reason to doubt the co-satisfiability of subsets of the naturalness role that do not include Empiricism. To investigate this question, we will need to consider what the perfectly natural properties might be, if they are not the kinds of properties liable to be expressed by theoretical predicates of physics or other empirical sciences. One possibility is to say that the instantiated perfectly natural properties are not expressed by any of our predicates; but this makes for prima facie difficulties with Magnetism. A more promising avenue is to look to metaphysics to provide the relevant predicates. Here are some possible candidates:

Naturalness | 51 ‘identical to’ ‘part of’ ‘member of’ ‘is a duplicate of’ ‘. . . resembles . . . at least as much as . . . resembles . . .’ (cf. Price 1953) ‘. . . is the degree to which . . . resembles . . .’ (cf. Rodriguez-Pereyra 2002) ‘is a natural class’73 ‘are, collectively, natural’ (cf. Dorr 2007) ‘instantiates’ (cf. Armstrong 1978)74 ‘are compresent’ (cf. Russell 1940; Williams 1953)

Let us use call predicates like these, and the properties they express, ‘structural’.75 (For our purposes it is not important to specify the exact boundaries of this category.) If we drop Empiricism, might all or most of the remainder of the naturalness role be played by some ranking in which all of the top-rated properties are structural in character? The answer is not obvious. Let us consider some components of the role for which doubts are especially likely to arise. • Supervenience: Humeans will likely deny that the facts about which things instantiate which structural properties constitute a supervenience base for everything. For example, they may think that there is a world where positive and negative charge have switched places, in which there are exactly the same things standing in exactly the same pattern of structural relations. However, those sympathetic to dispositional essentialism may be unmoved by such putative counterexamples. Questions which dispositional essentialists like to press, such as ‘What could make the difference between the actual world and one where positive and negative charge have switched places?’,

73 It is sometimes taken for granted that the ‘natural classes’ are just the extensions of the (perfectly?) natural properties. But in the present setting, the interest of the idea that the property of being a natural class is perfectly natural depends on rejecting this assumption. We would want to be able to say, e.g. that the class of electrons is natural, even though neither the property of being an electron nor any property coextensive with it is perfectly natural. 74 We are thinking here especially of instantiation taken as a relation between objects and universals, where these are conceived of as entities of a completely different sort from the abundant ‘properties’ with which we have been concerned up to now. 75 This is of course a quite different use of ‘structural’ from that in Sider (2011), where ‘structural’ is roughly synonymous with our ‘perfectly natural’.

52 | Cian Dorr and John Hawthorne seem to suggest a picture where the facts about (e.g.) negative charge do supervene on the structural facts.76 • Independence: Most of the items on our list of structural predicates figure in sentences (e.g. ‘Whenever x is a duplicate of y and y is a duplicate of z, x is a duplicate of z’) which are not logically valid but seem to express metaphysically necessary truths. Counting those properties as perfectly natural would thus violate Combinatorialism. However, it is not obvious that this is true of every predicate on the list: Dorr (2007) claims that it is not true of ‘are collectively natural’, for example. • Duplication: It is hard to believe that the mere fact that the parts of two objects (e.g. a hydrogen atom and its antimatter counterpart) stand in the same pattern of structural relations suffices for those objects to be duplicates. Moreover, most of the structural relations on our list seem to be either necessarily reflexive or necessarily irreflexive: if our list of perfectly natural properties was composed of such relations, Duplication would require us to accept the outlandish claim that necessarily, all mereological atoms are duplicates of one another. • Magnetism suggests that the perfectly natural properties should be properties we are capable of expressing precisely. The obvious vagueness of our actual use of expressions like ‘. . . resembles . . . at least as much as . . . resembles . . .’ thus poses an obstacle to the view that comparative resemblance satisfies a version of the perfect-naturalness role that includes Magnetism. But this is not a decisive obstacle, since one might hold that some distinctive theoretical way of using the relevant resemblance

76 Suppose we think there is a possible world where the universe is mirror-symmetric except that the mirror image of each particle is an anti-particle. In such a world, the permutation that maps each object to its mirror image will preserve all structural properties. Thus, if we want to say that all perfectly natural properties are structural and preserve the third gloss on Supervenience, we can admit such a possibility only if we are prepared to classify being an electron as a non-qualitative, or haecceitistic property—one which can only be defined in terms of perfectly natural properties if we allow ourselves to mention certain particular objects in the definition. For example, ‘is an electron’ might be equivalent to ‘belongs to a natural class which plays suchand-such nomological role in relation to the other natural classes, and instances of which are much more common on Earth than instances of the other class that plays that nomological role’. For more on the options for defining predicates like ‘electron’ in structural terms, see the discussion of ‘structural nominalism’ in Dorr (2007).

Naturalness | 53 predicate suffices to eliminate its vagueness. The same objection may be made, with varying degrees of plausibility, to the other structural predicates on our list.

3.7. Similarity + (Simplicity or Dissimilarity or Magnetism) Similarity suggests that the degrees of naturalness of a property and its negation are often quite far apart. For on a natural reading of Similarity, it entails that no non-trivial property with at least a moderate degree of naturalness could have instances that are extremely dissimilar from one another. But in so far as we are allowed to rely on our intuitive judgements about similarity, it seems that it could very easily happen that there are three objects each of which is extremely dissimilar from the other two. If there were three such objects, it would have to be the case that for every property, either it or its negation has two extremely dissimilar instances. Thus Similarity seems to require us to say that it never happens that a property and its negation are both even moderately natural.77 By contrast, Simplicity, Magnetism, and (on some interpretations) Dissimilarity all encourage the view that a property’s negation is at most slightly less natural than it. This follows from a crude implementation of Simplicity in terms of symbol counting, and it is unclear how this could be affected by a more liberal interpretation. Since reference to the negation of a property is easily achieved once one is in a position to refer to that property, Magnetism will not draw any sharp contrast between properties and their negations. And as we have already pointed out, the obvious modal gloss on Dissimilarity entails that there is never any difference in naturalness between a property and its negation, since the objects divided by a property are also divided by its negation. 77 A different Similarity-based argument that the negation of a property is often much less natural than it can be extracted from the claim that determinate masses are very natural: given that the determinate masses are pairwise inconsistent, it is necessary that any two objects share the negations of continuum many of them. The claim that determinate masses are very natural could be motivated by Duplication in conjunction with certain ancillary premises as in section 3.4, or directly by Similarity together with the claim that objects having exactly the same determinate mass are always very similar. However, as emerged in the discussion in section 3.4, this claim also risks trouble with Independence, Magnetism, Laws, and Dissimilarity.

54 | Cian Dorr and John Hawthorne A parallel tension arises in the case of disjunction. Similarity suggests that the disjunction of two very natural properties can be extremely unnatural. By contrast, Simplicity and Magnetism suggest that the disjunction of two very natural properties will still be fairly natural. And, as we have seen, the modal gloss of Dissimilarity entails that the disjunction of two properties is never less natural than both of them.78 Perhaps, however, these apparent tensions can be made to disappear by adopting the ‘additive’ interpretation of Similarity and Dissimilarity, according to which the degree of similarity between two objects depends on the total naturalness of all the properties they share and/or the total naturalness of all the properties that divide them. If we embrace the picture where the negation of a property is at most a little less natural than it and the disjunction of two very natural properties is still quite natural, then the former value will have a quite high lower bound. For given any two very natural properties F and G, F ∨ G, ¬F ∨ G, F ∨ ¬G, and ¬F ∨ ¬G will all be pretty natural, and any two objects whatsoever must inevitably share at least one of these properties. But the additive interpretation 78

We could also consider the question whether there is a systematic difference between conjunction and disjunction. Similarity is naturally read as entailing that there is. By contrast, Magnetism suggests that there isn’t. Of course, on a non-hyperintensional picture of properties, it is easy to contrive examples where the conjunction of two properties is much easier or harder to refer to than their disjunction. For example, the disjunction of being negatively charged and not extremely happy and being negatively charged and not extremely unhappy is being negatively charged, which is quite easy to refer to; their conjunction, being negatively charged and neither extremely happy nor extremely unhappy, is not easy to refer to. But such examples can be constructed in both directions, and suggest no systematic reference-theoretic difference between the operations of conjunction and disjunction. (Meanwhile, in a hyperintensional theory of properties, one might take it that properties have a structure and can only be expressed by predicates that share this structure; if so, then assuming that creatures are not much more or less likely to have a word for and than to have a word for or, there will never be a big difference in ease of reference between the conjunction of two properties and their disjunction.) Similarly, the most straightforward implementations of Simplicity will treat conjunction and disjunction symmetrically; and while we could of course introduce various asymmetries by hand (e.g. by tweaking the symbol-counting measure to give extra weight to disjunctions), it is a tall order to set things up in such a way as to fit with the judgements elicited by Similarity. For the spirit of Simplicity surely requires the disjunction of seventeen perfectly natural properties to be much less natural than the disjunction of two of them; whereas Similarity suggests that in many cases, the disjunction of two is already about as unnatural as any property.

Naturalness | 55 is consistent with the thought that even when objects are as dissimilar as they could possibly be, they share many quite natural properties. For two objects to be extremely dissimilar, the total naturalness of the properties they share need not be close to zero—it is enough if it is about as low as it could possibly be.

3.8. (Similarity or Dissimilarity or Magnetism) + (Independence or Empiricism) + Simplicity According to one kind of moral realism (as developed, e.g. in Wedgwood 2007), certain moral properties, such as the property of being a morally permissible action, are very easy to refer to. Provided that a linguistic community has some word which they use to regulate their conduct in a certain characteristic way, that word will express the property of being morally permissible, even if their pattern of judgements using the relevant word are very foreign to our own. It would be unsurprising if proponents of this view also took moral permissibility to play an important role in making for similarity and dissimilarity. One possibility for those who take moral permissibility to be very easy to refer to and/or important in making for similarity and dissimilarity is to count moral permissibility as a perfectly natural property. However, this conflicts with the spirit of Empiricism, since natural science does not draw our attention in the relevant way to normative properties such as this. It also raises a worry for various versions of Independence, in so far as moral permissibility seems to be bound up in rich patterns of necessary connections with other similarity-making, dissimilarity-making, and easy-to-refer-to properties (including the properties with which physics is concerned).79

79 Wedgwood himself denies that the permissibility facts supervene on the physical facts in the sense that any two possible worlds with the same physical facts have the same permissibility facts. However, by denying the S4 and S5 principles for metaphysical necessity, he manages to combine the rejection of supervenience in this form with the acceptance of another kind of supervenience claim: necessarily, the totality of physical facts entails the totality of permissibility facts. The key is that whenever two physically indiscernible but morally discernible worlds are possible relative to the actual world, they are not possible relative to one another (Wedgwood 2007: chapter 9).

56 | Cian Dorr and John Hawthorne The alternative is to say that moral permissibility is very natural but not perfectly natural. But this leads to a problem with Simplicity, since plausibly any properties in terms of which moral permissibility can be given a simple definition (e.g. morally good, or maximizing happiness, or not prohibited by God’s commands) will also lead to the problems with Empiricism and Independence noted in the previous paragraph.80 This kind of issue also arises in other domains. For example, there are views where consciousness or personhood are very easy to refer to, or of great similarity-theoretic importance, while being of no concern to physics; these will generate an analogous dilemma.81 One might worry, indeed, that the Magnetism-based version of the problem will arise all over the place—aren’t properties like being a table, being red, and being a battle very easy to refer to?82 However, in these latter cases, one feels more comfortable answering no by appealing to the vagueness of words like ‘table’, ‘red’, and ‘battle’, which precludes any property from being their determinate referent.83, 84 80 As we formulated Empiricism, it says nothing at all about what the perfectly natural properties uninstantiated at the actual world are like. But given Empiricism, it is very implausible that there is a simple expression in the language of physics that provides necessary and sufficient conditions for moral permissibility even within the ‘inner sphere’ of worlds where no alien perfectly natural properties are instantiated; if there is no way to do this, there is also no simple metaphysically necessary and sufficient condition for moral permissibility in terms of the instantiated perfectly natural properties together with any properties not instantiated in the inner sphere. 81 For the case of personhood, see Hawthorne (2006: chapter 9). 82 The version of the problem based on Dissimilarity does not seem to arise for being red and so on. A very purplish red thing can be extremely similar to a very reddish purple thing; a very peaceful battle can be extremely similar to a very turbulent altercation. Similarity might be thought to generate problems, but it is hard to see how to get these going without assuming a modal gloss on Similarity, which as we have already noted leads to many other highly unexpected judgements about naturalness. 83 One can also make a vagueness-based case against the magnetism of being a battle, etc., even if one glosses ‘easy to refer to’ as ‘easy to refer to simpliciter’, provided that one holds the ‘extreme plasticity’ view of reference simpliciter mentioned in footnote 42. 84 Of course, one can construct Sorites-like arguments for ‘morally permissible action’. (Van Fraassen 1980 attributes one to Sextus Empiricus that begins, ‘Touching your mother’s big toe with your little finger is morally permissible’.) Do these count against the claim that moral permissibility is easy to refer to? Proponents of the Wedgwood-style view will perhaps bite the bullet and think that there is a non-vague line in such series. The hardest cases for such bullet-biting involve Sorites series where what varies is not the moral quality of an action, but the claims of some event to count as an action at all (cf. the literature on actions as things ‘caused in the right way’ by certain mental states).

Naturalness | 57 3.9. Simplicity + (Dissimilarity or Magnetism) As already noted, the modal gloss on Dissimilarity entails that for any set of properties, no matter how large and miscellaneous, the conjunction and disjunction of the properties in the set never does worse than all of them as regards ‘making for dissimilarity’. This is true even when all of the properties concerned ‘make for dissimilarity’ to a degree far less than the maximum possible degree. A ranking based on this interpretation of Dissimilarity thus fails to behave in anything like the way predicted by Simplicity. One can also easily imagine metasemantic views in which the ranking of properties in terms of how easy it is to refer to them behaves in ways that fail to fit with Simplicity, so that there are big differences in magnetism among properties which are far from the top of the magnetism scale, without significant differences in the complexity of their definitions in terms of items higher up the scale.

3.10. (Similarity or Dissimilarity) + Magnetism Can we make sense of the thought that there are some properties which, although very easy to refer to, do little or nothing to ‘make for’ similarity or dissimilarity? If there were such properties, there would be a severe tension between Magnetism and Similarity or Dissimilarity. However, in so far as a property is easy to refer to, it is hard to see how it could be all that hard for speakers to get themselves into a context where the sharing of that property by two objects counts in favour of applying some cognate of ‘similar’ to them, while division by that property counts against it. While there certainly will be other contexts where ‘similar’ expresses a relation to which the property in question is irrelevant, that is true of any property. Thus, so long as we think of Similarity and Dissimilarity as generalizations about the relations that get expressed by ‘similar’ and its cognates at a range of actual and hypothetical contexts, the idea that a magnetic property could be systematically irrelevant to similarity or dissimilarity becomes hard to sustain. One rather common kind of context, at least in philosophy, is one where ‘similar’ expresses an internal qualitative relation—a relation which holds between x and y whenever it holds between any

58 | Cian Dorr and John Hawthorne duplicate of x and any duplicate of y. It is natural to suppose that in these contexts, extrinsic properties don’t ‘make for similarity’ at all—or at least, that this is true of purely extrinsic properties, those that are consistent with every consistent intrinsic profile. But it is clear that if we can make sense of ‘easy to refer to’ at all, there are many extrinsic properties which are quite easy to refer to, and some of these can be argued to be purely extrinsic (e.g. being between two massive objects). If you insist on construing Similarity and Dissimilarity as uttered from such intrinsicness-favouring contexts, their tension with Magnetism will thus be quite stark. However, since there are plenty of contexts in which the relations expressed by ‘similar’ are not internal, this tension might not show up if Similarity and Dissimilarity are construed as generalizations over many contexts. The other direction in which a radical disconnect might arise would involve a property that scores highly by the lights of Similarity or Dissimilarity, while being very hard to refer to. As we have already pointed out, this kind of disconnect is inevitable on the modal glosses of Similarity and Dissimilarity. Conjunctions of (compossible) similarity-conducive properties will be even more similarity-conducive; conjunctions and disjunctions of dissimilarityconducive properties will never be less dissimilarity-conducive. Since this holds no matter how numerous and miscellaneous the properties in question might be, the resulting rankings will clearly clash with Magnetism in a way that cannot plausibly be cushioned by the ceteris paribus clause. However, it is much less clear how the ‘additive’ construal of Similarity and Dissimilarity could force us to classify a hard-to-refer-to property as highly natural, since as we have pointed out, the additive approach leaves one plenty of plenty of latitude in assigning naturalness scores even when all the facts about degrees of similarity are held fixed. (Still, it is far from obvious how one could set up a Magnetismfriendly naturalness ordering in such a way as to recover the apparently obvious fact that objects whose masses are close are more similar to one another than objects one of which is much more massive than the others.85 Magnetism suggests that all properties of 85 Some theorists will simply deny this fact. For example, if one thought of mass facts as involving location in mass space, one might be led to think in the relevant sense of ‘similar’, difference in mass has no more bearing on similarity than spatial distance.

Naturalness | 59 the form being more than/less than x grams in mass are on a par as regards naturalness, as are all conjunctions or disjunctions of two such properties. But any two objects with different masses will share continuum many properties of these kinds, and be divided by continuum many others. Sustaining the additive construals of Similarity and Dissimilarity thus requires finding some principled way of ‘adding’ these infinities in such a way as to get a higher total for items that are close in mass. Here it is hard to say anything very precise in the absence of a positive programme for dealing with the infinity-related problems for the additive construals.)

4. HOW VAGUE IS ‘NATURAL’? Lewis’s discussion of relative naturalness suggests that he would have been happy to accept that ‘more natural than’ is rather vague. From the standpoint of someone who thought that there was a precise fact of the matter as regards the relative naturalness of any two properties, Simplicity in particular seems a very poor basis for a research programme: it is hard to imagine that the questions that arise when one is trying to figure out how to measure simplicity all have definitely correct answers. On the other hand, Lewis often gives the impression of thinking that ‘perfectly natural’ is not at all vague. ‘Perfectly natural’ is used to state various speculative hypotheses—for example, the hypothesis of Humean Supervenience, according to which the only actually-instantiated perfectly natural properties are spatiotemporal relations and monadic properties of point-sized things and spatiotemporal relations—and nowhere does Lewis consider the deflating thought that it is just vague whether these hypotheses are true or not. Metaphysicians often aspire to set themselves apart from ordinary discourse by formulating certain central questions in a precise language, and in the tradition inaugurated by Lewis, it is envisaged that ‘perfectly natural’ will play a central role in making good on this aspiration. The question whether and to what extent ‘perfectly natural’ is vague can thus serve as a second axis for debate between ‘naturalness enthusiasts’ and ‘naturalness sceptics’. It’s not just that the claim that ‘perfectly natural’ is vague goes against something Lewis

60 | Cian Dorr and John Hawthorne happened to think. The more general point is that often, when philosophers introduce theoretical jargon with which one is unhappy, the appropriate complaint to make is not that the jargon is literally meaningless or empty, but that it is hopelessly vague, to such an extent that it provides no scope for epistemic advance—for pretty much any interesting argument in which the jargon plays an essential role, it will simply be indefinite whether the conclusion is true.86, 87 This question is largely orthogonal to the questions discussed in the previous section, about the extent to which the naturalness role is satisfied. It is coherent to suppose that ‘perfectly natural’ is vague even though the entire role is satisfied. On one possible view, the entire role is satisfied by several different classes of properties, and several of these classes are candidate extensions for ‘perfectly natural’.88 On another possible view, while it is definitely true that the role is uniquely satisfied and that the perfectly natural properties satisfy it, there is no set of properties that definitely satisfy the role and hence no set of properties that definitely are all and only the perfectly natural ones.89 Meanwhile, there are also coherent views on which ‘perfectly natural’ manages to be precise in spite of the fact that the Lewisian role is quite far from being satisfied (let alone uniquely satisfied). 86 Of course, this talk of ‘interesting’ questions and ‘epistemic advances’ is itself quite vague, and the task of precisifying it raises delicate questions. For example, even when a word is ‘horribly’ vague, we should probably allow that there might be interesting arguments of a metasemantic character which settle some definite truths about its extension. 87 Any list of examples where such a treatment is appropriate will inevitably be tendentious. Each of the following terms is on at least one of our lists: ‘analytic’, ‘acquaintance’, ‘justified belief’, ‘dispositional property’, ‘autonomous action’, ‘free will’. 88 Note that because the classes in question satisfy Supervenience and Independence, the union of two of them will not satisfy the role. 89 Sustaining the claim that there is no unique satisfier requires exploiting the ceteris paribus clause in Magnetism. Whatever ‘easy to refer to’ means, it is inconsistent to suppose that there are two distinct classes of properties each of which contains properties that are easier to refer to than any property outside the class. The vagueness of ‘easy to refer’ does not affect this point: at most, it prevents there from being any class of which it is definite that it contains all and only the properties that are maximally easy to refer to. By contrast, the package that combines unique realization and vagueness need not place any special weight on the ceteris paribus clause. (An analogous point holds for Similarity and Dissimilarity.)

Naturalness | 61 There are any number of episodes in the history of science that illustrate the fact that even theories that are full of errors can sometimes be used to introduce useful, precise theoretical vocabulary. We think that this is one of the clear insights behind the idea of reference magnetism; it can certainly be embraced even by those who deny that the whole naturalness role is even close to being satisfied. Perhaps, then, ‘perfectly natural’ determinately expresses a certain property of properties which satisfies only a small portion of the role, just as, arguably, Descartes’s term ‘quantity of motion’ determinately expresses momentum despite the fact that Descartes’s laws of motion are a complete disaster. For instance, ‘perfectly natural’ might definitely pick out a particular minimal supervenience base whose members are not at all easy to refer to. The question whether ‘perfectly natural’ is vague is intimately connected with the question how easy to refer to (‘magnetic’) the property of being a perfectly natural property is. Given the connection between vagueness and magnetism, those who claim that ‘perfectly natural’ is precise should think that perfect naturalness is fairly magnetic—it is not plausible that our success in determinately referring to it is some kind of modal freak. Conversely, those who think that ‘perfectly natural’ is vague will have some reason to say that perfect naturalness is not very magnetic. In Williamson’s theory of vagueness, for example, ‘perfectly natural’ will be vague only if there are close worlds where its reference is different; but if perfect naturalness is magnetic, most worlds where people refer to it lack close neighbours of that sort. In a use-and-eligibility framework, the idea would be that it is improbable that the use facts should be almost, but not quite, enough to overcome the pull of eligibility. So long as we preserve Magnetism as part of the naturalness role, there is thus pressure on those who hold ‘perfectly natural’ to be precise to claim that perfect naturalness is at least fairly natural, and on those who hold that ‘perfectly natural’ is vague to deny this.90 However, there is no obvious vagueness-related reason to want to say that perfect naturalness is perfectly natural.

90 Those who reject Magnetism, on the other hand, could consistently hold that perfect naturalness is very magnetic but not very natural, and indeed that no perfectly natural property is very magnetic. The magnetism of a property of properties is one thing, that of its instances another.

62 | Cian Dorr and John Hawthorne Once questions about whether properties instantiate themselves are in view, we are pushing up against the property-theoretic paradoxes. If our resolution of those paradoxes embraces a type-theoretic hierarchy of the sort described in section 2, the question ‘how natural is perfect naturalness?’ is illegitimate. On the obvious reconstruction, it will give way to an infinite series of questions, using an infinite hierarchy of naturalness predicates: How natural(((i))) (perfectlynatural((i)))? How natural(((i,i)))(perfectly-natural((i,i)))? . . . How natural((((i)))) (perfectly-natural(((i))))? . . . It is not immediately obvious that these would need to be answered in the same way. For example, one might think that it is very easy to have an expression that definitely means(i,((i))) perfectly-natural((i)), and on that account conclude that very-natural(((i)))(perfectly-natural((i))), while rejecting the analogous argument for the claim that very-natural((((i))))(natural(((i)))).) Replacing the linguistic question ‘How vague is “perfectly natural”?’ with non-linguistic questions like ‘How magnetic is perfect naturalness?’ and ‘How natural is perfect naturalness?’ might look to be a step forward for the debate. However, one also loses something important in setting the linguistic questions aside in this way. Someone might hold that ‘perfectly natural’ ended up very vague, and expressing on each precisification a very unmagnetic, very unnatural property of properties, while also holding that if only Lewis had left out some especially problematic part of his paper he would have succeeded in giving ‘perfectly natural’ a different semantic profile, on which it would have been perfectly precise. On this view, Lewisian materials provide a path to precision. There is a property of properties in the vicinity which even creatures without superhuman abilities are capable of expressing precisely. But achieving this requires us to jettison some tendencies in our use which, as things stand, make for vagueness.91 Someone who holds this seems 91 The idea that deleting some of the role would achieve precision might, but need not, be based on the thought that the whole role is unsatisfied while the role obtained by the relevant deletion is satisfied. But the thought could also be that the fragment in question is close enough to being satisfied by a magnetic property of properties which is much further from satisfying the entire role. And of course, the envisaged ‘path to precision’ might involve some change in use other than simply deleting part of the role. In principle, given that minute differences in use can sometimes make the difference between vagueness and precision—consider a Sorites sequence from a world where a word is vague to one where it is precise—even some apparently trivial variation in our practice with ‘perfectly natural’ might be thought capable of rendering it precise.

Naturalness | 63 to agree about something important with those who hold that ‘perfectly natural’ is already precise. If we want to use the concept of vagueness to help articulate the disagreement between naturalness enthusiasts and naturalness sceptics, we need to consider such paths to precision alongside the questions about the vagueness of ‘perfectly natural’ as we actually use it. The idea that ‘perfectly natural’ is at least somewhat vague has considerable appeal. For there are cases where it seems tempting to think that there are some perfectly natural properties within a certain set, but where the question exactly which of them are perfectly natural seems quite intractable. In such cases it is often a great relief to be able to sidestep the challenging question by denying that it has a definitely right answer. (Of course, different theories of vagueness explain the nature of this relief in quite different ways.) For example, while the most commonly encountered axiomatization of topology takes ‘open region’ as the key undefined predicate, everything goes just as smoothly if one treats ‘closed region’ as undefined, defining ‘open region’ as ‘complement of a closed region’. It is quite tempting to think that at least one property from the topological family is perfectly natural; but one might well be embarrassed to think that there is a definitely right answer to the question whether it is openness or closedness that is perfectly natural. Similar issues emerge whenever we have a set of interdefinable properties none of which does strikingly better than the others on the score of Similarity, Dissimilarity, or Magnetism. Consider the various predicates treated as primitive in rigorous axiomatizations of Euclidean geometry: while the best-known system (Tarski 1959) uses a three-place ‘between’ and four-place ‘congruent’, many other approaches have been explored—for example, Pieri (1908) uses ‘x and y are equally far from z’ as its sole primitive predicate. Realworld non-Euclidean geometry generates a similar array of choice points.92 It is tempting to think that some properties from the geometrical realm are perfectly natural. And while there may be metaphysical considerations which rule out some of the packages, it is optimistic to think that these will leave no one with no residual

92

See Arntzenius and Dorr (2012).

64 | Cian Dorr and John Hawthorne awkward questions.93 Analogous issues arise in classical extensional mereology, different axiomatizations of which variously treat parthood, proper parthood, overlap, and disjointness as primitive; and in the theory of location, where the candidate perfectly natural relations include weak location, entire location (insideness), pervasive location, and exact location (see Parsons 2007).94 In each of these cases, another way to dodge the need to choose is by claiming that all of the properties in question are perfectly natural. In so far as the proffered ‘interdefinitions’ really are modally adequate, this will mean sacrificing Non-supervenience (‘No perfectly natural property supervenes on all the rest’). Even if one were willing to sacrifice this aspect of the role, one would not thereby avoid awkward questions: once one has started extending the list of perfectly natural properties, it will be hard to know where to stop. (In topology, is connectedness perfectly natural? Path-connectedness? Hausdorffness?. . . .) 93 Duplication provides a potentially powerful tool for rejecting certain candidate lists of perfectly natural geometrical properties, for those who are willing to rely on pre-theoretic judgements about the geometric conditions that duplicates need to satisfy. For example, if betweenness and congruence are the only perfectly natural geometric properties, Duplication yields the surprising result that any two regions which contain the same number of points are duplicates, so long as none of the points are between any others or exactly as far apart as any others (Skow 2007). Similarly, one might use Duplication to argue that neither topological openness nor topological closedness is perfectly natural, since any region, whether open, closed, or neither, can have a duplicate that is both open and closed by virtue of not being topologically connected to anything else. (Maudlin 2010 raises some related worries about the standard approach to topology.) This mode of argument thus rules out many candidate lists of perfectly natural properties which might otherwise seem attractive. But it would be too much to hope that it will never leave us with awkward dilemmas which it would be nice to be able to dodge by appeal to vagueness: it is not hard to think of competing lists which generate exactly the same conditions on duplication. (Example: x and y are equally long and x is longer than y, considered as relations on line segments.) Another kind of metaphysical argument that might be relevant turns on the claim that it is possible for there to be gunky space, which might tell against approaches where the perfectly natural geometrical properties are properties that can only be instantiated by points. Also, if one thought that there was a difference between a world with just one point—a zero-dimensional manifold—and a world with just one object that is not a point, this might also provide a filter. 94 The rejecters of Empiricism considered in section 3.6 will think that none of the properties considered so far are perfectly natural, since according to them the only perfectly natural properties are ‘structural’ ones, like duplication or resemblance. But awkward dilemmas might well arise even within the structural realm.

Naturalness | 65 On the other hand, if we are willing to respond to these dilemmas by invoking vagueness in ‘perfectly natural’, we can if we wish embrace all versions of Independence. So for example, we might say that it is definite that just one of openness and closedness is perfectly natural, but indefinite which one it is.95 Considerations of the sort we have just been discussing do nothing to support the claim that ‘perfectly natural’ is horribly vague in the sense given earlier—roughly, that no interesting argument one could formulate in terms of it has a definitely true conclusion. Moreover, they are consistent with a perspective on which the vast majority of properties are definitely not perfectly natural.96 However, there is an interesting line of thought which might be thought to destabilize the view that ‘perfectly natural’ is only mildly vague. The central moves in this line of thought are inspired by Sider (2011: section 7.13), although the connection to vagueness is not one that directly concerns Sider. Sider’s target is ‘Melianism’, the view that ‘Structure is not structural’. In a nutshell, Sider’s argument is that ‘If structure is not perfectly structural then it is disjunctive and therefore highly nonstructural’, and therefore incapable of being genuinely explanatory. The line of thought we are interested in takes the extreme non-naturalness of perfect naturalness to pose a distinct (though related) threat, namely that ‘perfectly natural’ is horribly vague. The argument can be spelled out as follows: 1. If ‘perfectly natural’ is vague, then perfect naturalness is not perfectly natural. 2. If so, the simplest definitions of perfect naturalness in terms of the perfectly natural properties are long disjunctions of the form ‘is identical to P1, or is identical to P2, or . . .’. 3. If so, perfect naturalness is extremely unnatural. 4. If so, ‘perfectly natural’ is horribly vague. We see several ways of resisting this argument. 95

We will need to posit a measure of higher-order vagueness too, if we want to avoid a definite cut-off between the topological properties that are definitely not perfectly natural and those that are candidates to be perfectly natural. 96 They might, on the other hand, support the claim that no property is definitely perfectly natural. But neither the question ‘How many things definitely fall under the predicate?’ nor the distinct question ‘How many things definitely fail to fall under the predicate?’ provides a very good diagnostic for the kind of vagueness that makes for theoretical fruitlessness.

66 | Cian Dorr and John Hawthorne Step 1 we have discussed already. If we endorse Magnetism, there is a decent prima facie case that any predicate of ours that manages to express a perfectly natural property does so precisely. It is hard to see how to argue that ‘perfectly natural’ is one of the exceptions to this generalization. The interesting ways of resisting Step 1 thus seem to require giving up on Magnetism.97 One kind of theorist who will have no trouble resisting Step 2 is one who thinks that there just aren’t very many perfectly natural properties, so that the definition ‘property identical to P1 or identical to P2 or . . . or identical to Pn’ will not be long. There are various views of this kind that we think deserve to be taken seriously. One such view takes actual-world physics, reconstructed in such a way as to rely on a small family of relations (e.g. mass-betweenness and so on) rather than infinite families of determinates, as the sole guide to the realm of perfectly natural properties; another such view rejects Empiricism in favour of a short list of ‘structural’ perfectly natural properties (e.g. a single comparative resemblance relation). Other ways of rejecting Step 2 will appeal to non-list-like definitions, expressed in terms of putatively perfectly natural properties of properties other than perfect naturalness itself. For example, Mundy (1987) suggests a picture where there are infinite families of perfectly natural properties of objects (the determinate masses, the determinate distances . . .) and just a handful of perfectly natural properties of properties (e.g. addition and betweenness relations), which only the perfectly natural properties of objects instantiate. On such a view, ‘perfectly natural’ might have a short definition along the lines of ‘property that is either identical to addition, or identical to betweenness, or such as to bear addition to some other properties, or such as to bear betweenness to some other properties’. Step 3 will certainly not seem very compelling to any theorist who rejects Simplicity. But even those who accept Simplicity can resist this step so long as they are not wedded to symbol counting as a measure of the complexity of a definition. As we pointed out in 97 One might try to reconcile Magnetism with the rejection of Step 1 by appealing to a non-linguistic theory of vagueness, according to which there are properties such that it is a vague matter what instantiates them. In this setting, if one thought that the property of perfect naturalness was a vague one, ‘perfectly natural’ could definitely express this property while still being vague.

Naturalness | 67 our initial discussion of Simplicity, some long disjunctions are intuitively much less complex than others—for example, long disjunctions which are regular enough to be printed out by a small Turing machine can strike us as quite simple. If we were trying to devise a measure of complexity for formulae in a formal language with many atomic predicates, it would not be unreasonable to assign a low complexity-score to a long disjunction with exactly one disjunct for each atomic predicate, where the disjuncts are all short and all identical except for the substitution of one atomic predicate for another. Thus it is open to us to claim that the disjunction ‘identical to P1 or identical to P2 or . . .’, even if it is very long, counts as rather simple in the sense at issue. Step 4 also looks eminently resistible. While it is plausible that we do not determinately refer to extremely unnatural properties, it is hard to see why predicates whose candidate referents were extremely unnatural would have to be horribly vague. There are many vague predicates—including vague predicates that figure in explanatorily successful theories in the special sciences—each of whose admissible precisifications might plausibly be classified as extremely unnatural. For example, given the vast array of independent arbitrary decisions that one would have to make in order to precisely specify any of the precisifications of ‘alive’, or ‘bird’, or ‘parliamentary democracy’, or ‘demand shock’, anyone who has sympathies for anything in the direction of Simplicity must surely think that the precisifications are each very unnatural. If so, the fact that all the precisifications of a predicate are extremely unnatural does not block the relevant kind of theoretical fruitfulness.98 (Sider’s argument differs from the one we have just discussed in two ways. First, as already noted, his conclusion is that if perfect naturalness is not perfectly natural, it fails to be ‘explanatory’. Second, his emphasis seems not so much to be on the length of the definition as on the mere fact of its being a disjunction. We are suspicious of the idea that no property whose simplest definition in terms of 98 Note that there may be some other positive naturalness-theoretic status that predicates can enjoy even when their precisifications are very unnatural. For example, a natural second-order property might unify a group of unnatural precisifications. Thus the present point does not straightforwardly block all connections between theoretical fruitfulness and naturalness.

68 | Cian Dorr and John Hawthorne the perfectly natural properties is disjunctive is explanatory.99 Suppose betweenness is perfectly natural; then it is plausible that the simplest definition of ‘x, y, and z are collinear’ is ‘x is between y and z, or y is between x and z, or z is between x and y’. But this hypothesis does not seem to vitiate geometrical explanations involving collinearity. Moreover, it seems obvious that disjunctive facts do sometimes figure quite legitimately in humdrum explanations: for example, if I have decided to put everything in the room that is either a jellyfish or a spoon into a certain box, the fact that something is either a jellyfish or a spoon seems quite relevant to explaining its ending up in that box.100, 101)

5. ‘IS NATURALNESS PRIMITIVE?’ Some discussions of naturalness focus on a question which might seem to be orthogonal to the questions we have discussed so far: whether naturalness is ‘primitive’ (‘basic’, ‘irreducible’). The idea that there is an important question here may be encouraged by Lewis’s discussion (1986: 63–9) of a certain range of options for theorizing about naturalness between which he claims to be undecided. Two of these options involve explicit conceptual analyses of ‘natural’ (in terms of universals or tropes), while the third consists in taking the distinction between natural and unnatural properties as primitive. However, it is somewhat mysterious what theoretical

99

Nor are we clear what the relevant notion of disjunctiveness would be. Obviously any disjunction can be eliminated in favour of negation and conjunction, but this surely marks no difference in explanatory power. Also, given set-theoretic or property-theoretic resources, subtler ways of eliminating disjunction in favour of quantification become possible; for example, we could replace ‘x is F or x is G’ with ‘x belongs to every set that contains both the set of F things and the set of G things’. Sider seems to be confident that ordinary special-science predicates are not disjunctive, but we find it hard to see what the basis for this confidence would be. The usual functional analyses of such predicates do not settle this question, since they involve expressions like ‘tends to cause’ that are not perfectly natural according to Sider. 100 Disjunctive facts also seem often to be useful in explaining other disjunctive facts. Since it is plausible that if ‘perfectly natural’ is disjunctive, ‘refers’, ‘similar’, and many other such expressions are disjunctive in a similar way, this makes it hard to raise a disjunctiveness-related worry for the naturalness-based explanations that Sider focuses on, e.g. ‘Why are these objects similar?’ or ‘Why is this property easy to refer to?’ 101 Thanks to Ted Sider for discussion of the material in this section.

Naturalness | 69 posture this ‘naturalness primitivism’ amounts to. Does being a naturalness primitivist just mean using the word ‘natural’ without endorsing any particular analysis of it? Naturalness primitivism on this gloss is not a claim but a course of action, and indecision about it, if it comes to anything, will just be a kind of practical indecision about what to do. Moreover, on this gloss Lewis just is a naturalness primitivist, as opposed to being ‘undecided’, given that he does not endorse the universal-based or trope-based analyses. If the indecision is to be theoretical and not practical, then ‘naturalness primitivism’ needs to be some kind of claim. But what claim is it to be? Given that the alternatives are presented as involving conceptual analyses, the obvious candidate is the claim that there is no true conceptual analysis of ‘natural’, or of the concept natural. But what does that mean? There is a picture of the enterprise of conceptual analysis as part of cognitive science—the claim that a predicate has a conceptual analysis is something like the claim that we understand it by translating it into a syntactically complex expression in Mentalese. But this is (a) surely not what Lewis had in mind (see Lewis 1995); (b) of dubious philosophical interest beyond the philosophy of mind; (c) unlikely to make any interesting distinction between ‘natural’ and any other moderately technical term in philosophy; and (d) not legitimately thought of as equivalent to any claim about naturalness itself as opposed to our particular vehicles for thinking about it, since there is no obvious reason why a single property should not be expressed by both simple and complex Mentalese expressions. Is the claim just that there is no complex predicate in our language, not involving ‘natural’, that is necessarily equivalent to ‘natural’? Even if that claim were true—and it is quite problematic when one takes context-sensitive and rigidifying vocabulary into account—it seems much too strongly tied to the contingent facts about the array of words available in English. We could introduce single words ‘F’ and ‘G’ in such a way as to make ‘a property is F iff it is natural or mentioned in the Bible’ and ‘a property is G iff it is natural or not mentioned in the Bible’ express necessary truths; then ‘a property is natural iff it is F and G’ will express a necessary truth.102 Examples like this suggest that the 102 The same problem arises if we replace ‘necessary’ by ‘a priori’ or ‘necessary and a priori’.

70 | Cian Dorr and John Hawthorne pertinent way in which a predicate might be primitive is to be understood in naturalness-theoretic terms: φ is primitive iff it is syntactically simple, and not equivalent to any complex formula in any language whose simple predicates express properties more natural than the property expressed by φ. If we hold fixed Supervenience and allow the relevant formulae to be infinitary and to contain names for any objects we wish, the upshot is that a predicate is primitive iff it expresses a perfectly natural property. This makes ‘primitive’ a predicate of predicates; perhaps we can talk of properties as ‘primitive’ in an extended sense, but the ‘primitive’ properties will then either just be the perfectly natural ones, or those that are both perfectly natural and expressed by some simple predicate. So on this approach, the debate over ‘naturalness primitivism’ adds nothing to the debate about whether naturalness is perfectly natural, a debate which we have already discussed in section 4.103 Some metaphysicians are comfortable with a use of ‘primitive’ which does not take a detour through a predicate of predicates or concepts. For example, Sider (1996) considers (and argues against) a view about naturalness which he calls ‘Primitivism’: ‘the belief that naturalness is ontologically basic, incapable of reductive analysis’. A pressing question for those who want to theorize using both ‘primitive’ and ‘perfectly natural’ as predicates of properties is whether these should be taken as equivalent, or whether they can somehow be given a separate life. Are we supposed to leave it open that some properties might be primitive but highly unnatural? Or that some perfectly natural properties are distinguished from others by their failure to be primitive? Prima facie, the remarks theorists make in introducing us to the relevant uses of ‘primitive’ and ‘natural’ aren’t different enough to give us much of a grip on such suggestions. 103 Two-dimensionalists (e.g. Chalmers 2012) will presumably want to replace claims about some predicate’s ‘expressing a property’ in this discussion with claims about the primary intensions of predicates, since conceptual analyses are supposed to concern the a priori realm. Thus the question whether ‘natural’ is primitive will turn on the question whether its primary intension is maximally natural—as natural as the primary intension of any expression could ever be, remembering that primary intensions are always functions from centred worlds to extensions. It is not obvious that two-dimensionalists’ answer to this question should be the same as their answer to the question whether naturalness (the secondary intension of ‘natural’) is perfectly natural.

Naturalness | 71 It is relatively easy to get a sense for a view on which ‘perfectly natural’ and ‘metaphysically primitive’ fail to be coextensive—or at least fail to be definitely coextensive—because one or the other is rather vague. One thought is that ‘perfectly natural’ ended up vague because nothing satisfied the full role associated with it, or because of the presence in the role of vague words like ‘similar’ and ‘refers’, whereas because of the thinner role associated with ‘primitive’, there was nothing to resist the magnetic pull of its actual meaning. On this view, many, perhaps all, of the precisifications of ‘perfectly natural’ will be neither primitive nor very natural, while primitiveness would seem to be at least pretty natural (on all the precisifications of ‘pretty natural’); a key further question is whether primitiveness is primitive. (It might alternatively be suggested that ‘perfectly natural’ is precise while ‘primitive’ is vague, but this view is a bit harder to get a grip on: if perfect naturalness is magnetic enough to be precisely expressed by ‘perfectly natural’, what could there be in our use of ‘primitive’ to prevent it from picking up on the same magnetic meaning?) But either way, it does not seem that having both ‘metaphysically primitive’ and ‘perfectly natural’ available in the lexicon of metaphysics will open up a range of new questions for fruitful debate. It is of course conceivable that there are two or more non-coextensive, fairly magnetic properties of properties in the general vicinity of ‘naturalness’ talk, although we are not sure what would motivate such a hypothesis. Those who hold this will of course think that there is real progress to be made by introducing two or more terms associated with the different pathways to precision (though it may do more harm than good to have one of them be the old term ‘natural’). In this setting the old question ‘Is naturalness natural?’ will give way to multiple successors: we can (bracketing type-theoretic worries) ask whether each of the new properties of properties has itself, and whether each has the other. These remarks about ‘metaphysically primitive’ apply, mutatis mutandis, to many other bits of metaphysical jargon which might be used to raise questions about naturalness. Is naturalness fundamental? Is it true in reality that any properties are natural (Fine 2001)? Do facts about naturalness obtain in virtue of other facts, such as facts about similarity and reference, or do facts of those kinds obtain in virtue of facts about naturalness? The theoretical work that is

72 | Cian Dorr and John Hawthorne supposed to be done by all of these locutions has much in common with the work that is supposed to be done by ‘natural’: it is far from clear what point there would be in distinguishing the question whether the property of being F is perfectly natural from the question whether F-ness is fundamental, or whether it is (or could be) true in reality that things are F, or whether things that are F are F in virtue of nothing. If there is no useful distinction to be made, these questions will again collapse into the question ‘Is naturalness natural?’; which will be intimately associated with questions like ‘Is fundamentality fundamental?’; ‘Is it ever true in reality that it is true in reality that P?’; ‘Does the fact that a fact obtains in virtue of nothing obtain in virtue of anything?’ Even if there is a distinction to be made, the most likely scenario is one where only one of the relevant bits of vocabulary is precise, in which case questions articulated using both will not be especially helpful. Only on the surprising and tendentious hypothesis that there are two different pathways to precision in the vicinity will a multiplication of questions be any advance.104 One might also think that there are interesting further questions concerning the explanatory power of naturalness, for example ‘Do facts about naturalness explain other facts, such as facts about similarity and reference, or do facts of those kinds explain facts about naturalness?’ One job ‘explains’ could be doing here is standing for some metaphysically heavyweight relation such as ‘grounding’; if so, then the earlier remarks about ‘in virtue of’ will also carry over here. But most ordinary uses of ‘explain’ are much more lightweight 104 Some metaphysicians who are comfortable using ‘fundamental’ understand it in a way that makes it fairly easy to get a grip on the suggestion that the ranking of properties in terms of how fundamental they are comes apart from the naturalness ranking (as argued in Bennett MS, chapter 4). In this use, ‘fundamental’ is a predicate applicable to all sorts of entities, not just properties; the non-fundamental entities are the ‘ontologically second-rate’ ones—holes, wrinkles, smirks. . . . (Cf. Sider 2011: sections 8.6 and 8.7.) For those who talk like this, it would not be considered outré to suppose that the only fundamental entities are subatomic particles, or that there is only one fundamental entity (Schaffer 2009). The view that properties (even perfectly natural properties) are not among the fundamental entities is thus not at all surprising. Nor is there any glaringly obvious reason to want to say that some properties are more fundamental than others in this ontological sense. Sets of fundamental things will probably be said to be all equally fundamental—all one step away from the ground floor—and some might think that properties of fundamental things are like sets in this respect.

Naturalness | 73 in character. Moreover, ordinary uses of ‘explain’ are also famously context-sensitive and undemanding: an enormous variety of achievements in unifying and connecting various of one’s beliefs can, in the right context, license a ‘because’ speech. In view of this, assuming that there are truths about naturalness at all, it would be absurd to deny that there are contexts where those claims can figure in true explanations of various kinds.

6. HAVE WE MISSED THE HEART OF THE MATTER? Consider a theorist who counts as an out-and-out naturalness enthusiast as far as our two questions are concerned. She believes that there is a ranking of properties that plays all of the roles of Lewisian naturalness, and that ‘perfectly natural’ precisely expresses the property of coming first in this ranking. We imagine that some naturalness enthusiasts will have a lingering feeling that despite all this, our theorist could, deep down, still be in the grip of an ‘egalitarian’ metaphysics, in a way that our tests have failed to diagnose. To bring out this feeling, imagine that our theorist makes the following speech: Yes, there is a naturalness ranking: it is a ranking of properties whose top level constitutes an independent supervenience base for everything, and which relates in certain distinctive ways to duplication, laws, similarity, reference, and rationality. Moreover, perfect naturalness is very natural (perhaps even perfectly natural), and thus easy to refer to; and we do, in fact, refer to it precisely. But there are also many other rankings out there. For example, there is the naturalness* ranking: it is a very different ranking of properties, whose top level also constitutes an independent supervenience base for everything. It does not relate in the specified ways to duplication, laws, similarity, reference, or rationality. However, there are properties of duplication*, lawhood*, similarity*, reference*, and rationality* which relate to naturalness* exactly as duplication, lawhood, similarity, reference, and rationality relate to naturalness. Perfect naturalness* is not very natural, and not very easy to refer to. But it is very natural* (perhaps even perfectly natural*), and thus easy to refer* to; and we do, in fact, refer* to it precisely.

You might feel that in making this speech, our theorist has unmasked herself as a closet Goodmanian. Her picture seems to be one in which the whole system of concepts including ‘natural’, ‘similar’, and so on is just spinning in the void, rather than being anchored in

74 | Cian Dorr and John Hawthorne the world in the way true naturalness enthusiasts suppose. This might prompt a search for some elusive further piece of doctrine which could allow the true enthusiasts to distinguish their view from the ‘deflationary’ view towards which the theorist’s speech seemed to be gesturing. But we can see no intelligible contrast in this vicinity. No one who accepts the first paragraph of the speech should reject the second paragraph: once you Ramsify enough words out of a theory, it is inevitable that the resulting roles will be multiply satisfied if they are satisfied at all. And the speech does not say that naturalness and naturalness* are in any sense ‘on a par’—it explicitly states that perfect naturalness is very natural, whereas perfect naturalness* is not. One might be tempted to think that there is some other sense of ‘on a par’ in play, such that our theorist, unlike the true naturalness enthusiasts, thinks that naturalness and naturalness* are ‘on a par’ in that sense. But this temptation should be resisted, since it is completely obscure what the other sense might be and how it is supposed to be different from ‘equally natural’. * * * In the last two sections, we have surveyed a range of questions that might be thought to add further dimensions to the debate between naturalness enthusiasts and naturalness sceptics. But in so far as the questions we have found make any sense, they do not seem to add much to the questions investigated earlier in the paper. We suggest that the issues of role-satisfaction and vagueness provide the most fertile territory for future debate in this area.105 University of Oxford REFERENCES Armstrong, David M. (1978) A Theory of Universals, vol. 2 of Universals and Scientific Realism. Cambridge: Cambridge University Press. Arntzenius, Frank and Cian Dorr (2012) ‘Calculus as Geometry’, in Frank Arntzenius, Space, Time, and Stuff: 213–78. Oxford: Oxford University Press. 105 We are grateful to to Frank Arntzenius, Susanne Bobzien, and Jeremy Goodman, and to audiences at NYU, Geneva, Oxford, Princeton, St. Andrews, and at the 2012 Metaphysical Mayhem in Rutgers. Special thanks to Ted Sider, for many helpful discussions of these issues that began many years ago, and to Karen Bennett and Dean Zimmerman, for their helpful comments and their patience.

Naturalness | 75 Bennett, Karen (MS) Making Things Up. Cameron, Ross (2010) ‘Vagueness and Naturalness’, Erkenntnis 72: 281–93. Carnap, Rudolf (1947) Meaning and Necessity: A Study in Semantics and Modal Logic. Chicago: University of Chicago Press. Chalmers, David (1996) The Conscious Mind. Oxford: Oxford University Press. —— (2012) Constructing the World. Oxford: Oxford University Press. Dorr, Cian (2007) ‘There Are No Abstract Objects’, in Theodore Sider, John Hawthorne, and Dean Zimmerman (eds), Contemporary Debates in Metaphysics: 32–64. Malden, MA: Wiley-Blackwell. —— and John Hawthorne (MS) ‘Semantic Plasticity and Speech Reports’. Eddon, Maya (2011) ‘Intrinsicality and Hyperintensionality’, Philosophy and Phenomenological Research 82: 314–36. —— (2013) ‘Fundamental Properties of Fundamental Properties’, this volume. Field, Hartry (1973) ‘Theory Change and the Indeterminacy of Reference’, Journal of Philosophy 70: 462–81. —— (1980) Science Without Numbers. Oxford: Blackwell. —— (2008) Saving Truth from Paradox. Oxford: Oxford University Press. Fine, Kit (2001) ‘The Question of Realism’, Philosopher’s Imprint 1. Francescotti, Robert (1999) ‘How to Define Intrinsic Properties’, Noûs 33: 590–609. Gärdenfors, Peter (2000) Conceptual Spaces: The Geometry of Thought. Cambridge, MA: MIT Press. Goodman, Nelson (1978) Ways of Worldmaking. Indianapolis: Hackett. Hawthorne, John (2001) ‘Intrinsic Properties and Natural Relations’, Australasian Journal of Philosophy 63: 399–403. —— (2006) Metaphysical Essays. Oxford: Oxford University Press. —— (2007) ‘Craziness and Metasemantics’, Philosophical Review 116: 427–40. Kim, Jaegwon (1984) ‘Concepts of Supervenience’, Philosophy and Phenomenological Research 45: 153–76. Langton, Rae and David Lewis (1998) ‘Defining “Intrinsic” ’, Philosophy and Phenomenological Research 58: 333–45. Reprinted in Lewis 1999: pp. 116–32. Lewis, David (1970) ‘How to Define Theoretical Terms’, Journal of Philosophy 67: 427–46. Reprinted in Lewis 1983a: 78–96. —— (1979) ‘Attitudes De Dicto and De Se’, Philosophical Review 88: 513–43. Reprinted with postscripts in Lewis 1983a: 133–60. —— (1980) ‘Mad Pain and Martian Pain’, in Ned Block (ed.), Readings in Philosophy of Psychology, vol. 1: 216–32. Cambridge, MA: Harvard University Press. Reprinted with postscript in Lewis 1983a: 122–32.

76 | Cian Dorr and John Hawthorne Lewis, David (1983a) Philosophical Papers, vol. 1. Oxford: Oxford University Press. —— (1983b) ‘New Work for a Theory of Universals’, Australasian Journal of Philosophy 61: 343–77. Reprinted in Lewis 1999: 8–55. —— (1986) On the Plurality of Worlds. Oxford: Blackwell. —— (1992) ‘Meaning Without Use: Reply to Hawthorne’, Australasian Journal of Philosophy 70: 106–10. —— (1995) ‘Reduction of Mind’, in Samuel Guttenplan (ed.), The Blackwell Companion to the Philosophy of Mind. Oxford: Blackwell, pp. 412–31. Reprinted in Lewis 1999: 291–324. —— (1997) ‘Naming the Colours’, Australasian Journal of Philosophy 75: 325–42. Reprinted in Lewis 1999: 332–58. —— (1999) Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press. —— (2001) ‘Redefining “Intrinsic” ’, Philosophy and Phenomenological Research 63: 381–98. —— (2009) ‘Ramseyan Humility’, in David Braddon-Mitchell and Robert Nola (eds), Conceptual Analysis and Philosophical Naturalism: 203–22. Cambridge, MA: MIT Press. Marshall, Dan and Josh Parsons (2001) ‘Langton and Lewis on “Intrinsic” ’, Philosophy and Phenomenological Research 63: 347–51. Maudlin, Tim (2010) ‘Time, Topology and Physical Geometry’, Aristotelian Society supplementary volume 84: 63–78. Moss, Sarah (2012) ‘Solving the Color Incompatibility Problem’, Journal of Philosophical Logic 41: 841–51. Mundy, Brent (1987) ‘The Metaphysics of Quantity’, Philosophical Studies 51: 29–54. Parsons, Josh (2007) ‘Theories of Location’, in Dean Zimmerman (ed.), Oxford Studies in Metaphysics, vol. 3. Oxford: Oxford University Press, pp. 201–31. Pieri, Mario (1908) ‘La geometria elementare istituita sulle nozioni di “punto” e “sfera” ’, Memorie di matematica e di fisica della Società Italiana delle Scienze (series 3) 15: 345–450. Translated as chapter 3 of Elena Anne Marchisotto and James T. Smith (2007), The Legacy of Mario Pieri in Geometry and Arithmetic. Boston: Birkhäuser. Price, H. H. (1953) Thinking and Experience, second edition (1969). London: Hutchinson. Rodriguez-Pereyra, Gonzalo (2002) Resemblance Nominalism: A Solution to the Problem of Universals. Oxford: Oxford University Press. Rosen, Gideon (1994) ‘Objectivity and Modern Idealism: What Is the Question?’, in Michaelis Michael and John (O’Leary-) Hawthorne (eds),

Naturalness | 77 Philosophy in Mind: The Place of Philosophy in the Study of Mind: 277–319. Dordrecht: Kluwer. Russell, Bertrand (1940) An Inquiry into Meaning and Truth. London: Allen & Unwin. Schaffer, Jonathan (2004) ‘Two Conceptions of Sparse Properties’, Pacific Philosophical Quarterly 85: 92–102. —— (2009) ‘On What Grounds What’, in David Chalmers, David Manley, and Ryan Wasserman (eds), Metametaphysics: 347–83. Oxford: Oxford University Press. Sider, Theodore (1996) ‘Naturalness and Arbitrariness’, Philosophical Studies 81: 283–301. —— (2011) Writing the Book of the World. Oxford: Oxford University Press. Skow, Bradford (2007) ‘Are Shapes Intrinsic?’, Philosophical Studies 133: 111–30. Soames, Scott (2002) Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford: Oxford University Press. Tarski, Alfred (1959) ‘What Is Elementary Geometry?’, in Leon Henkin, Patrick Suppes, and Alfred Tarski (eds), The Axiomatic Method: With Special Reference to Geometry and Physics: 16–29. Amsterdam: NorthHolland. van Fraassen, Bas C. (1980) The Scientific Image. Oxford: Clarendon Press. Weatherson, Brian (forthcoming) ‘The Role of Naturalness in Lewis’s Theory of Meaning’, forthcoming in Journal for the History of Analytic Philosophy. Wedgwood, Ralph (2007) The Nature of Normativity. Oxford: Oxford University Press. Williams, D. C. (1953) ‘The Elements of Being’, Review of Metaphysics 7: 3–18, 171–92. Williams, Robert (2007) ‘Eligibility and Inscrutability’, Philosophical Review 116: 361–99. Williamson, Timothy (1994) Vagueness. London: Routledge. —— (1999) ‘Existence and Contingency’, Proceedings of the Aristotelian Society supplementary volume 73: 181–203. Reprinted with corrections in Proceedings of the Aristotelian Society 100 (2000): 321–43. —— (2003) ‘Everything’, in John Hawthorne and Dean Zimmerman (eds), Philosophical Perspectives 17: Language and Philosophical Linguistics: 415–65. Oxford: Blackwell. —— (2013) Modal Logic as Metaphysics. Oxford: Oxford University Press.

2. Fundamental Properties of Fundamental Properties M. Eddon 1. INTRODUCTION Two grams mass and 3 coulombs charge are examples of quantitative properties. Such properties have certain structural features that other sorts of properties lack. How should we account for the distinctive structure of quantity? The answer to this question will depend, in large part, on one’s other metaphysical commitments. In this paper I focus on the metaphysical framework offered by David Lewis. I shall argue that, given the Lewisian framework, the most satisfying theory of quantity employs second-order relations. I then argue that the properties and relations invoked by a theory of quantity must be what Lewis calls perfectly natural, or fundamental. Together, these claims entail that there are perfectly natural secondorder relations. The thesis that there are perfectly natural second-order relations has an interesting consequence. The perfectly natural properties are generally taken to provide a minimal supervenience base for the qualitative facts. If we adopt perfectly natural second-order relations, however, then this is a mistake. The perfectly natural properties do indeed comprise a supervenience base, but this supervenience base is not minimal. This paper proceeds as follows. In sections 2 and 3, I lay out some background assumptions, and sketch some of the structural features of quantity. In section 4, I assess several accounts of quantity, and argue that the one best suited to a Lewisian framework posits perfectly natural second-order relations. In section 5, I address worries that an account of the structural features of quantity, in terms of the perfectly natural, is not required. If such an account is not provided, I argue, then many accounts that make use of perfectly natural properties and relations are untenable. In section 6, I use the results of the previous sections to argue that the

Fundamental Properties of Fundamental Properties | 79 perfectly natural properties and relations do not comprise a minimal supervenience base.

2. BACKGROUND I will assume a broadly Lewisian framework. I assume that we can quantify over possibilia, and I assume that possible individuals are world-bound. I assume that properties are abundant.1 For any set of possibilia, there is at least one property that applies to all and only the members of that set. Likewise, for any set of ordered n-tuples, there is at least one n-ary relation that applies to all and only the n-tuples in that set.2 Note that abundance extends to higher-order properties and relations as well. For every set of properties, there is a property that applies to all and only the members of that set.3 Every n-place function is associated with a set of ordered n + 1-tuples, where the first n elements are the arguments of the function, and the last element is the function’s output given those arguments.4 Since every function is associated with a set of ordered n-tuples, and every set of ordered n-tuples is associated with a relation, every function is associated with a corresponding relation.5 I assume that in addition to the abundant properties, there is a privileged set of perfectly natural properties that carve nature at the joints. The perfectly natural properties are, intuitively, those in virtue of which all else obtains; once we fix the distribution of perfectly natural properties, we fix everything else.

1

See Lewis ([1983] 1999: 9). I take no stand on whether properties and relations should be identified with sets of their instances. (See Eddon (2011) for reasons to think they should not.) Also, I ignore issues about whether properties should be identified (or associated) with sets as opposed to classes. 3 See Lewis (1986b: 50): “I do not want to restrict myself to properties of individuals alone; properties themselves have properties. Properties must therefore be sets so that they may be members of other sets.” 4 I.e. a function f(x1, . . ., xn) is associated with the set {, , . . .}. 5 See Bigelow and Pargetter (1990: 45) and Lewis (1991: 50–2). I take no stand on whether functions should be identified (as opposed to just associated) with relations. 2

80 | M. Eddon Perfectly natural properties figure prominently in a number of Lewis’s analyses. For instance, two objects are duplicates iff they share all their perfectly natural properties, and their parts can be put into correspondence in such a way that corresponding parts have the same perfectly natural properties and stand in the same perfectly natural relations.6 Given duplication, we can analyze intrinsicality: a property P is intrinsic iff for all duplicates, either both have P or both lack it.7 And, in somewhat more involved ways, perfectly natural properties play key roles in Lewis’s analysis of laws (see section 5.2), counterfactuals (see section 5.3), and causation (see section 5.4).

3. QUANTITIES Among the perfectly natural properties are the quantitative properties posited by an ideal physics.8 These include properties such as 2 grams mass, 3 coulombs charge, −1/2 spin, and so on.9 (For simplicity, I restrict my attention to scalar quantities as opposed to, for instance, vector quantities.) Quantitative properties divide into kinds or “families”; e.g. all mass properties belong to the mass family, and all charge properties belong to the charge family, but no mass property belongs to the charge family and vice versa. Quantitative properties like these have distinctive features that qualitative properties—such as being a shoe—do not. In particular, there are two characteristic features of quantitative properties: Ordering. Distance.

6

Quantitative properties of the same family can be ordered; e.g. 2 grams mass is less than 3 grams mass. Quantitative properties of the same family stand in distance relations to one another; e.g. 2 grams mass and 3 grams mass are 1 gram mass apart.10

See Lewis ([1983] 1999: 27) and Lewis (1986b: 61). See Lewis ([1983] 1999: 26) and Lewis (1986b: 62). 8 See Lewis (1986b: 60). Physics need not provide a complete inventory of the perfectly natural properties, but it plausibly provides at least a partial inventory. 9 Denby (2001) argues that determinate properties such as these are fundamental rather than perfectly natural. In this paper, however, I use the terms interchangeably. 10 For some discussion of these features of quantities, see Ellis (1966: 24–38), Mundy (1987), Swoyer (1987), Bigelow and Pargetter (1988), and (1989), inter alia. 7

Fundamental Properties of Fundamental Properties | 81 These features of quantitative properties are generally taken to be necessary. So, claims like the following are taken to be either necessarily true or necessarily false: 1. 2 grams mass is less than 3 grams mass 2. the distance between 2 grams mass and 3 grams mass is 1 gram Call this the Necessity Assumption.

4. THEORIES OF QUANTITY Recall that properties are abundant: there is a property for every set of possibilia. Given this assumption, all rival accounts of quantity agree on which properties and relations exist. What they disagree on is which of these properties and relations play a role in grounding the structure of quantity. More precisely: they disagree on which belong to the elite set of perfectly natural properties and relations. In this section, I explore several accounts of quantity that have been offered in the literature.11 I argue that the account of quantity best suited to the Lewisian framework employs second-order relations. A presupposition of this section is that some account must be provided. In the following section, I address those who reject this presupposition.

4.1. Numerical relations Consider my pen, which has 10 grams mass. One might take this at face value, and say that the number 10 literally plays a role in grounding this fact. So, one might say, the fact that my pen has 10 grams mass obtains in virtue of there being a perfectly natural relation—the mass-in-grams relation—that holds between my pen and the number 10. We now have a naive account of quantity: quantities are grounded in perfectly natural relations between objects and numbers. Call this the numerical relations theory. Suppose object a has 2g mass and 11 The authors discussed do not always have an abundant conception of properties in mind (Armstrong, for instance, believes that properties are sparse), but nothing hangs on this.

82 | M. Eddon object b has 3g mass. According to the numerical relations theory, there are no perfectly natural intrinsic mass properties that a and b instantiate. Rather, a bears the perfectly natural mass-in-grams relation to the number 2, and b bears the perfectly natural mass-in-grams relation to the number 3. And because the distance between the numbers 3 and 2 is 1, the distance in mass between a and b is 1 gram.12 (A notational variant of the numerical relations theory has been proposed by Weatherson (2006). Weatherson argues that in place of perfectly natural determinate quantitative properties (like 2 grams mass), there are perfectly natural functions from objects to numbers, where a perfectly natural function is “such that there is some perfectly natural quantity such that for any x, f(x) is the value that quantity takes with respect to x” (2006: 488). Since a function is (identical to, or associated with) a set of ordered pairs, and a set of ordered pairs is (identical to, or associated with) a relation, this amounts to the view that there are perfectly natural relations between objects and numbers.) Numerical relations theories such as these appear to give a straightforward explanation of the distinctive structure of quantities: quantitative properties simply inherit the ordering and distance structure standardly associated with numbers. But such theories have significant drawbacks. First, employing relations to numbers does not, by itself, yield a complete account of the structure of quantitative properties. For now, let us assume that numbers exist (later we shall discuss dropping this assumption). The standard distance function, or metric, defined over numbers is one according to which the distance between any two numbers is the absolute value of their difference (i.e. d(x, y) = ∣ y – x ∣).13 But this is not the only metric that may be 12 A more sophisticated variant of the numerical relations theory takes there to be a three-place perfectly natural relation holding between pairs of objects and numbers. For instance, if a has 2 grams mass and b has 3 grams mass, then there is a perfectly natural mass-ratio relation holding among a, b, and the number 2/3. See Mundy (1988) for such a view. While this view is immune from the worry that choice of units is a matter of convention, the other worries for numerical relations theories raised in this section still apply. 13 A metric over a set S is a function d: S × S → R+ such that, for all x, y, z ∈ S:

(1) d(x, y) = d(y, x) (symmetry) (2) d(x, y) = 0 iff x = y (identity) (3) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Fundamental Properties of Fundamental Properties | 83 defined over them. Another metric is the p-adic metric, according to which the distance between any two numbers is the p-adic norm of their difference.14 This metric yields the result that when the difference between two numbers is divisible by a large power of a prime number, the distance between them is small, and when the difference between them is divisible by a small power of a prime, the distance between them is large.15 Consider, for example, the 3-adic metric, which yields the following: d(3, 5) = 1 d(3, 9) = 1/3 d(3, 84) = 1/81 Given the 3-adic metric, the distance between 3 and 5 is much greater than the distance between 3 and 84. Which of these metrics correctly captures the quantitative structure of properties like mass? Presumably, a proponent of this position will claim that it is the standard metric. But then we must say why the standard metric is the correct one to employ, and a p-adic metric is not. One explanation is that the numbers themselves stand in ordering and distance relations to one another, some of which are more fundamental than others. The most fundamental capture the structure of the real numbers, and these are the ones that yield the standard metric over numbers. Other explanations may be explored as well. But in any case, numerical relations theories do not, by themselves, provide complete accounts of the structural features of quantity. A second worry about numerical relations theories is that features that seem merely conventional may be construed as part of the fundamental structure of the world. Consider an object c with 10g mass. According to the numerical relations theory, c bears the mass-in-grams relation to the number 10. But we could equally well have said that c has 0.01 kilograms or 0.353 ounces. So in addition to bearing mass-in-grams to the number 10, c bears mass-in-kilograms to the number 0.01 and mass-in-ounces to the number 0.353. Indeed, 14 P-adic norm: For any p-adic number a/b·pn, where p is a prime number that does not divide either a or b, ∣a/b·pn∣p = 1/pn. P-adic metric: d(x, y) = |y − x|p. 15 See Holly (2001), inter alia.

84 | M. Eddon for any positive real number n, there is some scale of measurement according to which c has n units of mass—and thus for any n, c bears some mass relation to n. Are all of these mass relations perfectly natural, or just one of them? If we choose one of these relations over the rest, we are left with the unhappy result that a certain scale of measurement is metaphysically privileged. This seems absurd—the units of measurement we use are purely a matter of convention.16 But if all of them are perfectly natural, then we are left with the unhappy result that there is massive redundancy in the perfectly natural properties. Moreover, we have no explanation for why so many of them march in lockstep. These worries highlight an underlying methodological concern with the strategy of using numerical relations to account for the structural features of quantities like mass. We can represent the mass of an object using nearly any scale whatsoever—so choice of scale, and thus the number used to represent any given magnitude, should not enter into an account of the fundamental structure of quantity. To say that the fundamental facts about mass consist in relations to numbers is to confuse our representation of the fundamental facts with the facts themselves.17 A third reason to avoid numerical relations theories is a certain kind of methodological conservatism. The existence and structure of numbers are matters of considerable controversy, and it seems that a theory of quantity should not stand or fall depending on how those matters are resolved. For whether or not numbers exist, quantitative properties have structure that needs to be accounted for. There are several reasons why numerical relations theories may initially seem promising. One reason is simplicity: such theories appear to give straightforward accounts of the structure of quantity. 16 For more discussion of conventions in measurement scales, see Ellis (1966: 74–86), Field (1980: 45), Milne (1986), Mundy (1989), and Hawthorne (2006). 17 Ellis makes a similar observation when he warns against making inferences from the features of the representation of a property to the features of the property itself: “From the fact that the set of real positive numbers is everywhere dense, it does not follow that the sequence of heat states is everywhere dense. From the fact that this set of numbers is bounded below, but unbounded above, it does not follow that the set of heat states is bounded below and unbounded above. This is an important point, and we need to be constantly reminded of it, not only in connection with temperature measurement, but in connection with other kinds of measurement as well” (1966: 99).

Fundamental Properties of Fundamental Properties | 85 But as we’ve seen, they do not provide complete accounts of the structure of quantity. And complete extensions of them may be more complicated. Another reason is a desire to avoid unexplained necessary connections among properties. But if we hold that, say, the mass-in-grams and mass-in-ounces relations are both fundamental, then we still have unexplained necessary connections between these two relations. And if we say that only one of them is fundamental, then we avoid unexplained necessary connections at the cost of claiming that the fundamental structure of reality discriminates between grams and ounces. So while numerical relations theories may have some prima facie appeal, upon reflection these theories have little to recommend them.

4.2. Intrinsic structure of quantitative properties There is some intuitive pull to the thought that the ordering and distance relations the hold among quantitative properties are, in some sense, fixed by the intrinsic structure of the quantities themselves. David M. Armstrong has proposed an account of quantity that develops this intuitive idea. According to Armstrong, quantitative properties are structural universals. A structural universal is a universal that has other universals as constituents, where constituency is a primitive relation analogous to mereological parthood.18 On Armstrong’s account, every quantitative universal has an infinite number of “nested” constituent universals. Intuitively, the more constituents two quantitative universals share, the “closer” they are. For example, the 5 grams mass universal shares many constituents with the 4 grams mass universal, hence the property of 5 grams mass is close to the property of 4 grams mass.19 This structure succeeds in organizing quantities into families: mass universals have only other mass universals as constituents, charge universals have only other charge universals as constituents, etc. It also provides an ordering of quantities within families: the 5 18 Is the constituency relation between universals itself perfectly natural? If so, then this theory is technically a second-order theory, and therefore has the same surprising consequences as other second-order theories. For a defense of second-order theories of quantity (though not this theory in particular) see sections 4.4, 5, and 6. 19 See Armstrong (1978: 116–31), (1988), and (1989: 102–7).

86 | M. Eddon grams mass universal has the 4 grams mass universal as a constituent, but not vice versa. But structural universals cannot accommodate the distance structure of quantitative properties.20 To see why, let’s consider two different scenarios. Let’s first suppose there is a possible world w at which just a finite number of mass quantities are instantiated.21 Suppose there are only three objects with mass at w. Since Armstrong denies that there exist uninstantiated universals, it follows that there are only three mass universals—a, b, and c—at w. Now suppose these universals have the following structure: a has no constituents, a is the only constituent of b, and a and b are the only constituents of c. On Armstrong’s account, b is equidistant from a and c, no matter what mass values these universals correspond to. But if a is the 2 grams mass universal, b is the 10 grams mass universal, and c is the 11 grams mass universal, then b is not equidistant from a and c. The trouble is this: we’ve no guarantee that the gaps between the constituents of a universal are all the same “size.” We’ve no guarantee that the distance between 3 grams mass and 4 grams mass is the same as the distance between 4 grams mass and 5 grams mass, or that the distance between 2 grams mass and 10 grams mass is not the same as the distance between 10 grams mass and 11 grams mass. In order to ensure that the distance between 2 grams mass and 10 grams mass is greater than that between 10 grams mass and 11 grams mass, we must insist that the 10 grams mass universal has other constituents besides the 2 grams mass universal—it also has 9 grams mass, 8 grams mass, etc. So let’s do that—let’s now suppose that it does have those mass universals as constituents. Indeed, let’s suppose that there is a mass universal for every possible value of mass. (We can do this either by requiring the world to be such that, for every mass quantity, there is some thing at the world that instantiates that quantity, or by adopting a platonist conception of properties. Let us put aside worries about whether either of these constraints is plausible.) Now we have a new problem. Since there are uncountably many possible values of mass, there are uncountably many mass quantities. 20 Armstrong’s account raises a number of other worries as well. See Eddon (2007) for a more detailed examination of this theory and its limitations. 21 Armstrong would probably deny the possibility of such a world (see Armstrong 1988: 315).

Fundamental Properties of Fundamental Properties | 87 So, every mass universal has an infinite number of constituents. But then every mass universal has the same number of constituents as every other mass universal—an infinite number. The 4 grams mass universal has as many universals in common with the 3 grams mass universal as it does with the 100 grams mass universal. Since the extent to which universals share constituents determines the distance between values, the result is that the 4 grams mass universal is equidistant from 3 grams mass and 100 grams mass. But that is the wrong result. In sum, the intrinsic structure of quantitative properties does not, by itself, fix all the structural features of quantity; we are still missing the relations of distance between quantitative properties.22 So let us try a different tack. The theories of quantity explored in the following sections have one feature in common: each invokes fundamental relations of some sort in order to account for the structure of quantities.

4.3. First-order relations: Betweenness and congruence One sort of relational account of quantity appeals to fundamental first-order relations. A nominalist version of such an account has been proposed by Hartry Field (1980). Since we are working within a realist framework, let us borrow Field’s apparatus and sketch a realist counterpart of the theory. On this account, there is a pair of perfectly natural, first-order relations corresponding to every family of quantitative properties. The first is a three-place betweenness relation, a Bet bc, which can be intuitively understood as “a is between b and c.” The second is a 22 One might try to resuscitate Armstrong’s account by replacing constituent universals with amphibians—entities capable of repeated occurrence as well as duplication (see Lewis [1986a] 1999: 98). Suppose there is a “smallest” unit of mass—say, 1 gram. Then, on this account, the universal 2 grams mass has two 1 gram mass amphibians as parts, and the universal 3 grams mass has three 1 gram mass amphibians as parts. The distance between 2 grams mass and 3 grams mass is determined by “counting up” the number of amphibians (see Wilson 2000 for an account of arithmetic that assimilates the counting (or natural) numbers with tally marks). But this attempt to rescue Armstrong’s account is not successful. First, it’s not clear that the notion of an amphibian is coherent. Second, putting aside the dubious status of amphibians, this account gets off the ground only if we assume that properties like mass are not continuous, which is false.

88 | M. Eddon four-place congruence relation, ab Cong cd, which can be intuitively understood as “the distance between a and b is congruent to the distance between c and d.” For example, suppose object a has 2 grams mass, b has 3 grams mass, c has 6 grams mass, and d has 7 grams mass. Then b is mass-between a and c, or a Mass-Bet bc; and a and b are mass-congruent to c and d, or ab Mass-Cong cd. There are three worries one might have with this sort of account. First, the representation and uniqueness theorems Field employs require certain assumptions. For instance, they require that at any world with massive objects, one of these objects must have zero mass.23 And they require that the masses of the objects at any world be, intuitively, “evenly spaced.” (If there is a 1g-mass object, a 2g-mass object, and a 5g-mass object, there must also be a 3g-mass object and a 4g-mass object.)24 But these assumptions are implausible—indeed, it’s not obvious that the actual world satisfies them.25 Second, adopting this account results in the conflation of intuitively distinct metaphysical possibilities. Consider a world w like ours, but where all the quantities are doubled in value. (For example, an object with 2 grams mass in our world has 4 grams mass in w.) Because the ratios between quantities remain exactly the same, every claim about the betweenness and congruence relations that is true at our world is also true at w, and vice versa. But then what is the difference between w and the actual world? If all we have are intra-world betweenness and congruence relations, then we cannot accommodate the intuition that w is qualitatively distinct from the actual world.26 Third, if we adopt this sort of account, then we lose much of the utility of perfectly natural properties. Consider duplication: two things are duplicates iff they have the same perfectly natural properties, and their parts bear the same perfectly natural relations to one another. When combined with Field’s account of quantity, this definition yields some unwelcome results. Suppose all my perfectly natural properties and relations are quantitative, so that all of my intrinsic 23

See Krantz et al. (1971: 148–50). See Krantz et al. (1971: 82–3 and 172–3). 25 See Mundy (1987: 32 and 1989), Melia (1998), Liggins (2003), Hawthorne (2006), and Meyer (2009) for some discussion of this and related objections. 26 Though see Dasgupta (2013) for an attempt at accommodating this intuition without appealing to cross-world betweenness and congruence relations. 24

Fundamental Properties of Fundamental Properties | 89 properties supervene on the betweenness and congruence relations holding between my parts. Then any possible individual that preserves these betweenness and congruence relations counts as my duplicate, even if it is, intuitively, twice as massive or half the size. Now, one might deny that this intuition should hold much weight, and thus deny that any duplicate of me really is twice as massive or half the size. But in many cases this tactic is implausible. Consider a world at which one of these possible individuals, call it Duplicate1, coexists with another, call it Duplicate2, where Duplicate1 is half the size and twice as massive as Duplicate2. Given Field’s account, each is a duplicate of the other. And this is so even though the Fieldian will agree that Duplicate1 is half the size of Duplicate2, and Duplicate1 is twice as massive as Duplicate2. These implausible implications will filter into any account that makes use of a notion of duplication—including intrinsicality, supervenience, determinism, and so on. Of the three worries raised, the first two may be mitigated by adopting an ontology of possible worlds. The betweenness and congruence relations may then hold among possibilia in different possible worlds, guaranteeing that we have the objects needed to ground the intuitively correct numerical representation of quantitative properties. But this move only exacerbates the third worry— that we lose much of the theoretical utility of the perfectly natural properties. We’ve already seen that difficulties arise for duplication. And when duplication goes awry, so does much else. Consider the thesis of determinism: on Lewis’s characterization, laws of nature L are deterministic iff among the worlds where L holds, any two worlds that have duplicate initial temporal segments are duplicates simpliciter.27 Now consider two worlds, w1 and w2, alike in their laws of nature. World w1 contains just three point-sized objects, all with the same mass. World w2 is, intuitively, exactly the same as w1 until time t. At t, the masses of the objects in both worlds increase, but the masses of the objects at w2 increase twice as much as the masses of the corresponding objects at w1. Worlds w1 and w2 are exactly alike with respect to their intra-world patterns of betweenness and congruence. By definition, they are duplicates. Therefore, w1 and w2 do 27

See Lewis ([1983] 1999: 31–2).

90 | M. Eddon not constitute a counterexample to the claim that the laws of nature at w1 and w2 are deterministic. But this seems like the wrong result. On Field’s original account, where betweenness and congruence relations hold only between individuals located at the same world, one could resist this result by denying that worlds w1 and w2 are distinct. But on the account under consideration, the Fieldian does recognize w1 and w2 as distinct possibilities, because these worlds differ in their inter-world relations. And so this move is not available. Theses formulated in terms of supervenience also cause trouble. Consider the following supervenience claim: the distribution of mental properties supervenes on the distribution of perfectly natural physical properties. Now consider a possible world w that is a physical duplicate of our world, but where all the corresponding masses are doubled and all the corresponding charges are halved. Suppose that nothing is conscious at w. It seems that the existence of such a world does not constitute a counterexample to physicalism, because the fundamental physical structure of w is very different from the fundamental physical structure of the actual world. On Field’s account, however, w is a physical duplicate of our world, and thus w is a counterexample to physicalism. That seems like the wrong result. Again, on Field’s original account, one could resist this result by denying that w is distinct from the actual world (thereby denying that nothing is conscious in w). But on the account under consideration, the Fieldian does recognize w as a possibility distinct from the actual world, since it differs from the actual world with respect to its inter-world relations. And similar worries arise with other supervenience claims as well. Another example: the distribution of perfectly natural properties and relations are supposed to comprise a basis for characterizing a world exhaustively. Consider two worlds, w1 and w2, with the same distribution of perfectly natural properties and relations. The interworld mass relations are such that parts of w1 are twice as massive as corresponding parts of w2. On the account under consideration, the proposition that inhabitants of w1 are twice as massive as those of w2 is true. So which quantitative facts obtain at a world is partly a matter of the relations their inhabitants bears to other-worldly things. In that case, it seems that the perfectly natural properties and relations do not suffice to characterize a world exhaustively,

Fundamental Properties of Fundamental Properties | 91 since they do not suffice to characterize a world’s quantitative structure. Again, the original Fieldian could deny that w1 and w2 are distinct worlds, but the Fieldian who adopts cross-world betweenness and congruence relations cannot. Given these unpalatable consequences of allowing cross-world betweenness and congruence relations, one might retreat to the original Fieldian view and claim that betweenness and congruence relations hold only among objects located at the same world. But given the broadly Lewisian framework we’re assuming, it’s not clear this is even an option. The axioms governing Field’s system entail that betweenness is connected: for any x, y, and z over which betweenness is defined, either x Bet yz or y Bet xz or z Bet xy.28 So consider three massive objects, a, b, and c, each located at a different world. The requirement that betweenness is connected entails that there is some mass-betweenness relation that holds among a, b, and c—and so it entails that there are cross-world mass-betweenness relations. Moreover, this result isn’t confined to mass; connectedness requires cross-world spatiotemporal-betweenness relations as well. And this result conflicts with Lewis’s requirement that possible worlds are spatiotemporally isolated from one another.29 Anyone who wishes to retain the utility of perfectly natural properties in these and other areas will not welcome these results. For those who would like to situate an account of quantity within the Lewisian framework, a Fieldian account of quantity holds little appeal.

4.4. Second-order relations: Less than or equal to and sum of Finally, let’s turn to an account of quantity offered by Brent Mundy (1987). Mundy assumes a platonist conception of properties, according to which properties exist necessarily even if nothing instantiates them. Included among the first-order properties are quantities such as 2 grams mass. In addition, there are two perfectly natural secondorder relations holding between the quantitative properties: a twoplace relation less than or equal to (≾) and a three-place relation sum of (*). The less than or equal to relation generates the ordering structure of quantity; for example, 2 grams mass ≾ 3 grams mass ≾ 4 grams 28

See Krantz et al. (1971: 172).

29

See Lewis (1986b: 69–81).

92 | M. Eddon mass, and so on. The sum of relation generates the distance structure of quantity; for example, the sum of 2 grams mass and 3 grams mass is 5 grams mass (i.e. *(2 grams mass, 3 grams mass, 5 grams mass)).30 This theory avoids the objections raised against the accounts of quantity considered earlier. It does not quantify over numbers, and so steers clear of the worries associated with numerical relations theories. It does not rely solely on the intrinsic quantitative properties, and so avoids the problems that beset purely non-relational accounts, such as Armstrong’s. Finally, because it adopts a platonist conception of properties, it avoids many of the counterintuitive implications of first-order relational theories, such as Field’s.31 For these reasons, Mundy’s account has significant advantages over its rivals. This gives us reason to prefer it. And since Mundy’s account makes use of perfectly natural second-order relations, this gives us reason to admit perfectly natural second-order relations.32 (There is a first-order version of this view, which might be more agreeable to those squeamish about higher-order properties and relations. Call this the Squashed Mundy View. On the Squashed Mundy View, the second-order relations ≾ and * are replaced with family-specific first-order relations. So, for instance, there is a perfectly natural first-order relation mass less than to equal to 30

See Mundy (1987: 37–8) for the formal presentation. See Mundy (1987: 32) for comparisons of first- and second-order theories. 32 Another second-order account has been proposed by Bigelow and Pargetter (1988 and 1990). In place of first-order quantitative properties, they posit first-order relations like twice as massive as and half as long as. And in place of the second-order relations ≾ and *, they posit second-order “relations of proportion.” These secondorder relations are intended to “impose an ordering” on the first-order relations, “explain[ing] how one thing can be closer to a second in, say, mass than it is to a third” (1990: 59). For example, it seems we want to say that the relation twice as massive as is “bigger” than the relation half as massive as; indeed, twice as massive as is necessarily “four times bigger” than half as massive as. And this is what the secondorder relations of proportion are supposed to capture. The trouble with this account is that it is not clear how the second-order relations of proportion that Bigelow and Pargetter posit could capture the ordering and distance structure of quantity. Very roughly, the worry is that these second-order relations do not impose any axiomatic constraints on the instantiation pattern of first-order relations, and so do nothing to help capture the structural features of quantity (see Forge 1995 and Eddon 2013). In any case, even if one is unpersuaded by this criticism, the central moral of the paper applies: Bigelow and Pargetter’s theory employs second-order necessary relations that must be construed as perfectly natural (as I argue in section 5), which in turn entails that the perfectly natural properties do not comprise a minimal supervenience base (as I argue in section 6). 31

Fundamental Properties of Fundamental Properties | 93 (≾m), and a perfectly natural first-order relation mass sum of (*m). Now, suppose object a instantiates 2 grams mass, b instantiates 3 grams mass, and c instantiates 5 grams mass. On the Squashed Mundy View, a ≾m b ≾m c, and c is the mass sum of a and b (i.e.*m(a, b, c)). While this view avoids positing perfectly natural secondorder relations, it requires that objects in different worlds stand in perfectly natural relations to one another.33 And any objection against the original Mundy view on the grounds that the relations it posits are redundant, and hence not perfectly natural (see section 5), applies equally well to the Squashed Mundy View. Thus a Lewisian has little reason to prefer the Squashed Mundy View to the original.)

5. PERFECTLY NATURAL SECOND-ORDER RELATIONS Let’s take stock. I have argued that the account of quantity best suited to a Lewisian framework appeals to perfectly natural second-order relations. Still, there are a few reasons one may doubt that perfectly natural relations of this sort are really required. One might argue that because these second-order relations supervene on the distribution of first-order quantitative properties, we get them “for free.” Since these relations are necessary,34 they trivially supervene on any property whatsoever (a fortiori they supervene on the first-order quantitative properties), and so they contribute only redundantly to any characterization of reality. Therefore, they do not belong to the elite set of perfectly natural properties and relations. Once we have the perfectly natural intrinsic quantitative properties, nothing more needs to be said about the ordering and distance relations that hold between them. In a related vein, one might be reticent to posit an objective distinction between natural second-order relations and gerrymandered ones without evidence that such a distinction is required 33

Thanks to Boris Kment here. The relevant notion of a “necessary relation” coincides with Lewis’s definition of an internal relation: R(x, y) is internal iff for any possible duplicate of x, x', and any possible duplicate of y, y', R(x, y) iff R(x', y') (Lewis (1986b: 62). I assume that properties exist at every world, although they may not have instances at every world. 34

94 | M. Eddon beyond an account of the structure of quantity. After all, a more conservative explanation for why we’re inclined to posit such a distinction is that some second-order relations are interesting or useful to us, and nothing more. If the distinction between perfectly natural quantitative second-order relations and gerrymandered ones is not required to do any philosophical work, then it seems unduly extravagant to claim that certain second-order relations are metaphysically privileged. In essence, both of these challenges deny that an account of the structural features of quantity is needed. In the following sections, I address these two skeptical challenges. I argue that unless the ordering and distance structure of quantity is taken to be perfectly natural, many accounts that make use of perfectly natural first-order properties are untenable. The same considerations weigh in favor of taking some second-order relations to carve reality at the joints, and so to contribute—non-redundantly—to a complete characterization of reality. In section 6, I discuss a consequence of the claim that these second-order relations are perfectly natural: the perfectly natural properties do not comprise a minimal supervenience base. To what extent do these worries, and my replies, hang on the details of Mundy’s account? The first worry applies to any account of quantity that employs necessary relations, while the second applies to any account of quantity whatsoever. My responses address both worries. For simplicity, I assume that Mundy’s account is correct. But my defense of perfectly natural metric relations can be tailored to fit other accounts of quantity as well.

5.1. Resemblance Because properties are abundant, there is always some gerrymandered property that any two things will share, similar or not. Consider two electrons, e1 and e2. These electrons share the property being e1 or being e2. They also share the property of having 1.6 × 10−19 coulombs charge. But only the latter property captures an objective similarity between the two. So the sharing of some properties makes for genuine resemblance, and the sharing of others does not. How can we distinguish between them? Enter natural properties: when two things share a perfectly

Fundamental Properties of Fundamental Properties | 95 natural property, they genuinely resemble one another in some respect.35 Quantitative properties cause trouble for this analysis. Suppose we have three objects—a, b, and c. Each instantiates a different amount of mass, and none instantiates any other perfectly natural property. It seems that the degree to which they objectively resemble each other is given by the distances between their mass quantities. Suppose a instantiates 0.001g mass, b instantiates 0.002g mass, and c instantiates 5000 metric tons of mass. Which of these is least like the others? The answer seems to be c.36 Yet the characterization of resemblance as the sharing of natural properties will not help us here. Objects a, b, and c instantiate different quantities of mass, and so they do not share any perfectly natural properties. We might posit another fundamental property—the determinable mass—which a, b, and c all instantiate, and use this to capture the fact that they resemble one another. But this won’t help us capture degrees of resemblance—for instance, that b resembles a more closely than c. Nor will appealing to non-fundamental second-order relations help us. For among the first-order properties, there are myriad second-order distance relations, each of which delivers a different judgment as to which objects are more similar. For instance, there are second-order relations corresponding to “discrete metrics,” such that every mass quantity is equidistant from every other one (and zero distance from itself ). There are second-order relations corresponding to “doubling metrics,” such that the distance between, say, 3 grams mass and 4 grams mass is double the distance between 2 grams mass and 3 grams mass, which is double the distance between 1 gram mass and 2 grams mass. There are secondorder relations corresponding to “switchy metrics” that effectively switch the “position” of some of the quantities, so that, say, 3 grams mass is closer to 500 grams mass than to 2 grams mass. And so on. If we want to capture the intuition that objects a and b resemble each other more than either resembles c, we need to say that one of these relations is privileged, or perfectly natural.

35 36

See Lewis ([1983] 1999: 13 and 1986b: 60). See also Denby (2001: 299).

96 | M. Eddon At this stage, one might wonder whether these intuitions about the resemblance of quantities should be taken so seriously. Even if 2 grams mass is closer to 1 gram mass than to 100 grams mass, it is far from obvious that there is a fact of the matter about whether 2 grams mass objectively resembles 1 gram mass more than 100 grams mass. But, as we’ll see, this is just the tip of the iceberg.

5.2. Laws Let the best system of a world be the set of true sentences that provides, on balance, the simplest and most informative description of the distribution of the perfectly natural properties and relations at that world. On Lewis’s analysis of laws, the laws at a world are the regularities entailed by the best system of the world. But how shall we judge the simplicity and informativeness of candidate systems? We need to place some restrictions on the language they are stated in. Otherwise, a system may be maximally strong and simple, but only artificially so. Consider a language with the predicate F, where F applies to all and only the things at the world in question. Then the system containing only the sentence “for all x, x is an F” will be maximally simple and maximally informative. This is not the result we want.37 Enter natural properties. If we require the best system to be expressed in a language whose predicates refer only to the perfectly natural properties and relations, then this language won’t include predicates like F and we avoid the worry just stated. So, the laws of nature are the regularities entailed by the system that, when stated in the language of natural properties, best balances simplicity and informativeness.38, 39 37

See Lewis ([1983] 1999: 42). See Lewis ([1983] 1999: 39–43 and [1994] 1999: 231–2). 39 One objection to the Lewisian analysis of laws is that the predicates appearing in standard formulations of the laws of physics do not express perfectly natural properties (see Hawthorne (2006: 236–7)). For instance, Newton’s second law of motion F = ma does not state that the determinable property force is literally identical to the determinable property resulting from “multiplying” mass by acceleration. “Equals,” “multiplication”, and other mathematical operations apply to numerals, not to properties. Rather, given the appropriate numerical representations of force, mass, and acceleration properties, F = ma expresses a relationship between these values that reflects the nomic relationship between the determinate quantities. Exactly how the 38

Fundamental Properties of Fundamental Properties | 97 Let’s look more carefully at the criterion of simplicity. Consider a chaotic world where the trajectories of particles have few discernible patterns or regularities. Suppose that there are only three instances at which a (non-zero) force is applied to an object. In the first instance, an 8N force is applied to a 1kg particle, and the particle accelerates at a rate of 8m/s2. In the second instance, a 10N force is applied to a 3kg particle, and the particle accelerates at a rate of 0.1m/s2. And in the third instance, a force of 2N is a applied to a 6kg particle, and the particle accelerates at a rate of 4/(3π) m/s2. Now consider a candidate system of laws for this world that consists of only the following force law:

(L) F = ma for all m ≤ 2, and F = (m + 7)2 a for all 2 < m < 5, and F = (mπ / 4) a for all m ≥ 5 where we’re employing the standard (intuitively correct) distance relations between mass quantities, and using a numerical representation of these relations that yields values in kilograms. With respect to simplicity, a system consisting only of (L) does not appear to score very well. But now consider a metric function d' that differs from the standard one in the following way: for every quantity m that the standard representation assigns a number ≤ 2, d' assigns m, and for every quantity m which the standard representation assigns a number between 2 and 5, d' assigns (m + 7)2, and for every quantity m which the standard representation assigns a number ≥ 5, d' assigns mπ/4. Given d', we can reformulate the law like this:

(L') F = ma Now the system looks much simpler. When comparing the simplicity of the laws of this system to those of other candidate systems, which formulation should we use? Both are stated in the language of natural properties, and both are equally informative. But they are not equally simple. perfectly natural predicates enter into this representation is not a straightforward matter, and I will not address it here (see Denby (2001) for a proposal). I note only that however the matter is resolved, the problem raised in this section remains. For the standard formulations of the laws—the formulations used in assessing simplicity and informativeness—presuppose a numerical representation of quantitative properties that respects the perfectly natural second-order relations. If there are no perfectly natural second-order relations to respect, then there’s no reason to use one formulation rather than another when assessing simplicity.

98 | M. Eddon This is problematic. Given gerrymandered second-order relations, even the simplest system can be made to look extraordinarily complex. And systems of arbitrarily complex laws can be made simple if these relations are chosen judiciously. Which should we use when evaluating candidate systems? One option is to use whichever ones yield the simplest formulation of the laws of that system. But since virtually any system can be made to look simple, this move threatens to render the “simplicity” constraint vacuous. To avoid this result, there need to be constraints on which formulations of the laws are permissible. The permissible formulations are the ones that respect the genuine ordering and distance relations among quantities—i.e. the perfectly natural ones. And it’s these formulations that should be used when assessing the simplicity of a candidate system.

5.3. Counterfactuals According to Lewis’s analysis of counterfactuals, a counterfactual is true at w iff some world at which both the antecedent and the consequent are true is closer to w than any world at which the antecedent is true but the consequent is not (see Lewis (1973b)). Closeness of worlds is a matter of objective resemblance—with the relevant aspects of objective resemblance determined by context—and objective resemblance is grounded in the perfectly natural properties and relations. Consider the following simple counterfactual: if I had let go of the pen, it would have fallen. This claim is true just in case a world at which I let go of the pen and it falls is closer to the actual world than any world at which I let go of the pen and it does not fall. In most contexts, the closest worlds will be ones where the laws of nature are the same (or very similar to) the actual laws, where the history up until the time I drop the pen matches (or very closely matches) the actual history, and so on. So cross-world comparisons are critical in evaluating counterfactual claims, and these crossworld comparisons consist in assessing the degrees of resemblance in various respects among worlds. Now consider a different counterfactual. Suppose Bob is a very good basketball player. But Bob is small—he is only five feet tall, with a mass of 60kgs—and so is not tall enough to play college-level basketball. It seems the following counterfactual is true: “If Bob had

Fundamental Properties of Fundamental Properties | 99 been a foot taller, he would have been on the college basketball team.” This counterfactual is true just in case a world where Bob is a foot taller and plays college basketball is closer to the actual world than any world where Bob is a foot taller but is not on the basketball team. Now consider two possible candidates for being the closest world, w1 and w2. At both w1 and w2, Bob is a foot taller. But at w1, Bob has a mass of 500kgs, and so is too heavy to be on the college basketball team. At w2, on the other hand, Bob is athletic and his mass is a bit more than his actual mass—say, 75kgs. Which of these worlds is closer to the actual world? It depends on which second-order relations are relevant when comparing Bob’s mass at the actual world to his mass at w1 and at w2. After all, there are some relations according to which 60kgs is closer to 500kgs than to 75kgs, and so w1 is closer to the actual world than w2. But this is the wrong result. Of course, worlds w1 and w2 differ from the actual world in many ways—not just with respect to Bob’s mass. For example, since Bob weighs 500kgs at w1, he probably eats a lot more in w1 than he does in w2 or the actual world. And all of these differences (or at least those deemed relevant by context) must be taken into account when assessing the relative closeness of w1 and w2. But appeal to these differences does not abrogate the need for perfectly natural secondorder relations. In general, the worlds under consideration will differ from the actual world with respect to their quantitative values in lots of ways. If we want to get the right answers when we assess counterfactuals, the closest worlds need to be ones where these quantitative differences are small. So we need to make sure that we assess similarity with respect to the right second-order relations—the perfectly natural ones.

5.4. Causation Here is a rough characterization of Lewis’s counterfactual analysis of causation: C causes E iff both C and E occur, and if C had not occurred then E would not have occurred.40 The latter condition is analyzed thus: a world where neither C nor E occurs is closer to the 40

See Lewis ([1973a] 1986c: 159–213 and [1983] 1999: 43–5).

100 | M. Eddon actual world than any world where C does not occur and E does occur. Suppose an enormous asteroid, A, is headed straight for the Earth, and we are powerless to stop it. The asteroid is so massive that if it hits any part of the planet—even if it barely grazes the surface—the impact will be disastrous, and the Earth will be destroyed. Unbelievably, just before A is about to enter the Earth’s atmosphere, it collides with another asteroid, B, and the collision deflects A’s path just enough so that the Earth is saved from obliteration. Had there been any small variation in circumstances—if A’s trajectory had been slightly different, if A had been travelling at a slightly different speed, if A had begun its journey just a few moments earlier or later—then A would not have serendipitously collided with B, and would have gone on to hit the Earth. News anchors across the globe proclaim: “It’s a miracle! If A had not collided with B, then A would have hit the Earth. The Earth is saved because of an extremely unlikely event!” (Fox News adds: “Proof of God!”) Consider the following causal claim: (C) A’s hitting B caused the Earth to be saved from total annihilation. Given a Lewisian analysis of causation, (C) is true iff a world at which A does not hit B and the Earth is annihilated by A’s impact is closer than any world at which A does not hit B and the Earth is not annihilated. In the relevant context of utterance, (C) seems true. The closest world at which A does not collide with B is one where A hits the Earth. Now consider two other worlds, w1 and w2. Suppose that A’s trajectory at w1 differs from its trajectory at the actual world by just a small amount—small enough so that it hurtles right past B instead of colliding with it. So at w1, A does not collide with B, and A hits the Earth. At w2, on the other hand, A’s trajectory is radically different from its trajectory at the actual world. At w2, A travels in an entirely different direction. So A not only avoids colliding with B at w2, A also avoids colliding with the Earth. Furthermore, suppose that w1 and w2 are relevantly similar to the actual world in other respects. Is w1 or w2 closer to the actual world? It depends on which second-order relations are relevant to comparing A’s trajectory at the

Fundamental Properties of Fundamental Properties | 101 actual world to its trajectory at w1 and w2. There will be some distance relations according to which A’s trajectory at w2 is closer to A’s actual trajectory than A’s trajectory at w1. If we use one of these to evaluate objective resemblance, w2 will be closer to the actual world than w1. So (C) may well be false. But that is the wrong result. If we want to get the right results when it comes to assessing claims of causal dependence, we need to assess similarity with respect to the right second-order relations—the perfectly natural ones.

6. MINIMALITY According to Lewis, the perfectly natural properties and relations do not just characterize the world exhaustively, they also characterize it minimally: “there are only just enough of them to characterize things completely and without redundancy” (Lewis 1986b: 60).41 The intuitive idea that the perfectly natural properties are sparse and selective is generally cashed out in terms of supervenience:42, 43 Minimality:

The set of perfectly natural properties does not supervene on any proper subset of them.

But if some second-order relations are perfectly natural, as I have argued, then Minimality conflicts with the Necessity Assumption, according to which facts regarding the structural features of quantity are necessary (see section 2). Given an account that employs second-order relations among quantitative properties, the Necessity Assumption entails that if, for instance, 2 grams mass bears the less than or equal to relation to 3 grams mass, then this relation holds necessarily; there is no possible world where 2 grams mass is greater than 3 grams mass. Likewise, if the sum of relations holds among 2 grams mass, 3 grams mass, and 5 grams mass, then this relation holds

41

See also Lewis ([1983] 1999: 12): “The world’s universals should comprise a minimal basis for characterising the world completely.” 42 There are various ways to formulate the Minimality constraint. But whichever formulation one chooses, the result is: if we adopt perfectly natural second-order relations of the sort I’ve been considering, the perfectly natural properties fail to comprise a minimal supervenience base. 43 Note that it is a mistake to construe Minimality as part of the definition of perfect naturalness, for reasons given in Sider (1996).

102 | M. Eddon necessarily; there is no possible world where 2 grams mass and 3 grams mass sum to anything other than 5 grams mass. Since these second-order relations are necessary, they supervene on anything whatsoever.44 A fortiori, P supervenes on any of the perfectly natural properties and relations. So if P is necessary, and the set of perfectly natural properties and relations includes P, then the set violates Minimality. So which must go, Minimality or the Necessity Assumption? If we drop the Necessity Assumption, we effectively give up the game. For if these relations are not necessary, then it is hard to see how they can play the required roles in accounts of resemblance, duplication, laws, counterfactuals, causation, etc. An example: consider three objects— a, b, and c—located in worlds w1, w2, and w3, respectively. Object a has 2 grams mass, b has 3 grams mass, and c has 100 grams mass. Clearly, b is closer in mass to a than to c. (And, if the objects are alike in all other respects, b resembles a more than c.) But if the distances between mass quantities may vary from world to world, then there are no non-world-relative facts about the mass distances between a, b, and c. And there are no non-world-relative facts about whether b is closer to a or to c. Gone is any analysis that employs some notion of closeness between worlds that is tied to the distances between quantities.45 A better option is to keep the Necessity Assumption and drop Minimality. One might try to replace Minimality with another claim along similar lines—perhaps one might say that the perfectly natural properties and relations comprise a minimal in virtue of base. But 44 Roughly, P supervenes on Q iff there is no change in P without a corresponding change in Q. If there is never any change in P (because P obtains necessarily), then P supervenes on anything, including Q. 45 One might argue that another way to save Minimality is to claim that some second-order relations are more natural than any others, but none are perfectly natural. (Thanks to Cody Gilmore who first raised this objection to me.) I think this is difficult to sustain. First, the heirarchy of naturalness is supposed to be determined, somehow, by the perfectly natural. But there is nothing about the perfectly natural first-order quantitative properties that “makes” some second-order relations more natural than others. For if there were, an account of quantity need not appeal to second-order relations at all—whether perfectly natural or only somewhat natural—since the “right” second-order relations would be entailed by the first-order quantitative properties alone. Second, the original motivation for adopting a primitive distinction between perfectly natural properties and all the rest was the indispensable role perfectly natural properties play in a swath of metaphysical projects. The second-order relations play those very same roles. So to deny them the status of perfectly natural is to give up on many of the reasons for positing perfectly natural properties in the first place.

Fundamental Properties of Fundamental Properties | 103 this is hardly compulsory. Dropping Minimality does not mean that the perfectly natural properties are thereby abundant and undiscriminating.46 They are still sparse and selective, they are still the ones that carve nature at the joints, and they are still the ones that make for objective similarity and difference.47 University of Massachusetts, Amherst REFERENCES Armstrong, David M. (1978) Universals and Scientific Realism, Volume II: A Theory of Universals. Cambridge: Cambridge University Press. —— (1988) “Are Quantities Relations? A Reply to Bigelow and Pargetter,” Philosophical Studies 54: 305–16. —— (1989) Universals: An Opinionated Introduction. Colorado: Westview Press. Bigelow, John and Robert Pargetter (1988) “Quantities,” Philosophical Studies 54: 287–304. —— (1990) Science and Necessity. Cambridge: Cambridge University Press. Dasgupta, Shamik (2013) “Absolutism vs Comparativism About Quantity,” in Karen Bennett and Dean Zimmerman (eds), Oxford Studies in Metaphysics. Oxford: Oxford University Press. Denby, David (2001) “Determinable Nominalism,” Philosophical Studies 102: 297–327. Eddon, M. (2007) “Armstrong on Quantities and Resemblance,” Philosophical Studies 136: 385–404. —— (2011) “Intrinsicality and Hyperintensionality,” Philosophy and Phenomenological Research 82: 314–36. —— (2013) “Quantitative Properties,” Philosophy Compass. Ellis, Brian (1966) Basic Concepts of Measurement. Cambridge: Cambridge University Press. Field, Hartry (1980) Science Without Numbers. Princeton, NJ: Princeton University Press. Forge, John (1995) “Bigelow and Pargetter on Quantities,” Australasian Journal of Philosophy 73: 594–605. 46

Nor does allowing necessary relations of this sort violate the Humean proscription of necessary connections—at least not as interpreted by Lewis as “the principle that anything can coexist with anything else, at least provided they occupy distinct spatiotemporal positions” (Lewis 1986b: 88). 47 Thanks to Cody Gilmore, Boris Kment, Raul Saucedo, Ted Sider, Brad Skow, Kelly Trogdon, and Dean Zimmerman. Special thanks to Chris Meacham for extensive comments and discussion.

104 | M. Eddon Hawthorne, John (2006) “Quantity in Lewisian Metaphysics,” in John Hawthorne, Metaphysical Essays. Oxford: Oxford University Press. Holly, Jan E. (2001) “Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields,” American Mathematical Monthly 108: 721–8. Krantz, D., Luce, R., Suppes, P., and Tversky, A. (1971) Foundations of Measurement, Volume 1. New York: Academic Press. Lewis, David (1973a) “Causation,” Journal of Philosophy 70: 556–67, reprinted with postscripts in Lewis 1986c. —— (1973b) Counterfactuals. Oxford: Blackwell. —— (1983) “New Work for a Theory of Universals,” Australasian Journal of Philosophy 61: 343–77, reprinted in Lewis 1999. —— (1986a) “Against Structural Universals,” Australasian Journal of Philosophy 64: 25–46, reprinted in Lewis 1999. —— (1986b) On the Plurality of Worlds. Oxford: Blackwell. —— (1986c) Philosophical Papers, Vol. II. Oxford: Oxford University Press. —— (1991) Parts of Classes. Oxford: Blackwell. —— (1994) “Humean Supervenience Debugged,” Mind 103: 473–90, reprinted in Lewis 1999. —— (1999) Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press. Liggins, David (2003) “On Being Twice as Heavy,” Philosophia Mathematica 11: 203–7. Melia, Joseph (1998) “Field’s programme: Some interference,” Analysis 58: 63–71. Meyer, Glen (2009) “Extending Hartry Field’s Instrumental Account of Applied Mathematics to Statistical Mechanics,” Philosophia Mathematica 17: 273–312. Milne, Peter (1986) “Hartry Field on Measurement and Intrinsic Explanation,” British Journal for the Philosophy of Science 37: 340–6. Mundy, Brent (1987) “The Metaphysics of Quantity,” Philosophical Studies 51: 29–54. —— (1988) “Extensive Measurement and Ratio Functions,” Synthese 75: 1–23. —— (1989) “On Quantitative Relationist Theories,” Philosophy of Science 56: 582–600. Sider, Theodore (1996) “Intrinsic Properties,” Philosophical Studies 83: 1–27. Swoyer, Chris (1987) “Metaphysics of Measurement,” in John Forge (ed.), Measurement, Realism and Objectivity. Dordrecht: D. Reidel Publishing. Weatherson, Brian (2006) “Asymmetric Magnets Problem,” Philosophical Perspectives 20: 479–92. Wilson, Jessica (2000) “Could Experience Disconfirm the Propositions of Arithmetic?” Canadian Journal of Philosophy 30: 55–84.

3. Absolutism vs Comparativism about Quantity Shamik Dasgupta We naturally think that material bodies have weights, sizes, masses, densities, volumes, and charges; that there are spatial distances between them, and temporal durations between events involving them. These are all features that fall under the category of quantity. In this paper I discuss a question that arises for all quantities but which is best illustrated by the case of mass. The property of having mass is a determinable that appears to have two kinds of determinates. On the one hand, we naturally think that something with mass has a determinate intrinsic property, a property it has independently of its relationships with other material bodies. But we also think that things with mass stand in various determinate mass relationships with one another, such as x being more massive than y or x being twice as massive as y. My question is: of the intrinsic masses and the mass relationships, which are fundamental? According to a view I will call absolutism, the intrinsic masses are fundamental. Loosely speaking, the view is that the most fundamental facts about material bodies vis-à-vis their mass include facts about which intrinsic mass they possess. The absolutist does not deny that things with mass stand in determinate mass relationships, and she might even agree that those relationships are fundamental too. But more likely she will think that those relationships hold in virtue of the intrinsic masses: if my laptop is twice as massive as my cup, the idea is that this is (at least partly) in virtue of the particular intrinsic mass that they each possess.1 In contrast, comparativism is the view that the most fundamental facts about material bodies

1 I say “at least partly” because an absolutist might say that it is in virtue of their intrinsic masses and various higher-order relations between those intrinsic masses (see for example Mundy (1987)). But this issue will not concern us here.

106 | Shamik Dasgupta vis-à-vis their mass just concern how they are related in mass, and all other facts about their mass hold in virtue of those relationships.2 I will describe both these views more precisely in section 1, and as we will see an analogous issue arises for all quantities. Given the central role that quantities play in our understanding of the natural world, the question of absolutism vs comparativism is central to any inquiry into what the natural world most fundamentally consists in. Moreover, there is more at stake in the issue than just our understanding of quantities, for comparativism brings with it a commitment to fundamental external relations and these have historically been treated with some suspicion. It is therefore surprising that the contemporary metaphysics literature contains relatively little discussion of the issue. In this paper I motivate and defend comparativism. But I am less interested in pressing that particular view as I am in formulating the issue and broadly surveying what I take to be the more important lines of argument for each position. The main consideration in favor of comparativism is developed at the end of the paper, in section 8, and is based on the idea that we can only ever observe mass relationships. More fully, the idea is that even if material bodies possessed the intrinsic masses posited by the absolutist, those intrinsic masses would be undetectable in a very strong sense: the structure of the physical laws governing our world would guarantee that they could never have an effect on our senses. As is well known, absolute velocity and absolute simultaneity are undetectable in this very same sense, and for this reason most contemporary metaphysicians and physicists call them “redundant” or “superfluous” and dispense with them on Occamist grounds. My thought is that the same reasons should compel us to dispense with the intrinsic masses posited by the absolutist. More precisely, the Occamist principle I appeal to is that positing undetectable structure is a vice, in the sense that if one theory of the material world posits undetectable structure that another does not then all else being equal we should prefer the latter. All else being equal, then, we should prefer comparativism. Now, whether all else is indeed equal depends on whether there are stronger countervailing 2 Along, perhaps, with higher-order facts about those mass relationships, though again this issue does not concern us here. Thanks to David Baker for helpful suggestions about how to formulate these views.

Absolutism vs Comparativism about Quantity | 107 reasons to reject comparativism. So in sections 2–7, the bulk of the paper, I consider a number of potential objections to comparativism and argue that none are convincing; hence my preference for comparativism. The discussion will mostly be limited to the case of mass, and I will make the simplifying assumption that the correct physical theory of mass is a classical, Newtonian theory. The assumption is of course false, but much of the paper discusses rather general, philosophical considerations that do not depend on the physical details, and in those discussions this simplifying assumption is harmless. Considerations from physics are discussed towards the end of the paper, and there the assumption may be less harmless. But I continue to assume it because the considerations from physics turn out to be conceptually rather delicate and so it is worth clarifying them in the context of this simplifying assumption before asking whether they generalize to more realistic physical theories. I think the considerations stand a good chance of generalizing to at least some other quantities and physical theories, but whether this is so is a question for another time.

1. MORE ON ABSOLUTISM AND COMPARATIVISM The absolutism vs comparativism issue per se has not received much discussion. To be sure, views that count as absolutist have been defended by Armstrong (1978) and (1988), Mundy (1987), and Lewis (1986b); and views that count as comparativist have been defended by Ellis (1966), Bigelow and Pargetter (1988) and (1990), and Field (1980) and (1985). But those discussions are often intertwined with other issues about quantities, so it will help to clarify the issue I have in mind before considering arguments either way. I understand both views in terms of the idiom of one kind of fact “holding in virtue of” another. As I use the phrase, this is an explanatory idiom: to say that a fact holds in virtue of another is to say that the latter explains the former in a distinctively metaphysical sense. To illustrate, imagine asking what explains Europe’s being at war in 1939. A causal explanation might describe events during the preceding 50 years that led, say, Chamberlain to declare war. But there is another kind of explanation that would try to say what goings on in Europe at the time made it count as a continent at war in the

108 | Shamik Dasgupta first place. Regardless of what caused the conflict, someone in search of this second explanation recognizes that war is not a sui generis state and that there must therefore have been something about the continent that made it count as being a continent at war rather than (say) one at peace. A plausible answer is that it was at war in virtue of how its citizens were acting, for example that some were committing politically motivated acts of violence against others. As I use the phrase, an explanation of this second kind is a statement of that in virtue of which Europe was at war. I take this kind of explanation to be reasonably intuitive: regardless of the truth of this explanation of why Europe was at war, we seem to understand the claim reasonably well.3 Earlier I said that the absolutist thinks that the most fundamental facts about material bodies vis-à-vis their mass include facts about which intrinsic mass they possess. We can put this more precisely in terms of the idiom just described: the view is that there are facts about which intrinsic mass a given material body possesses that are not explained (in the metaphysical sense just described) in other terms—in particular, that are not explained in terms of its mass relationships with other bodies.4 As I said earlier, the absolutist need not deny that things with mass stand in various mass relationships, but she will likely think that when they do this is at least partly explained (in this metaphysical sense) by the particular intrinsic mass that each body has.5 In contrast, the comparativist thinks that all facts about material bodies vis-à-vis their mass are explained (in this metaphysical sense) in terms of the mass relationships between bodies.6 This leaves open what kinds of mass

3 The term “ground” has been used by Kit Fine and Gideon Rosen to describe this sense of metaphysical explanation (see Fine (2001) and (2012), and Rosen (2010)). 4 This might be because there are facts about which intrinsic mass a material body possesses that are unexplained. Or, if there are infinite descending chains of explanations, it might be because any such fact is explained by another fact of the same kind; that is, another fact about which intrinsic mass something has. 5 Though this is not enforced: the absolutist might conceivably think that there are also unexplained facts about mass relationships along with unexplained facts about intrinsic masses. But I do not know of anyone who holds this view or of any reasons in its favor so I will largely ignore it in what follows. 6 Along, perhaps, with higher-order facts about the relations themselves; see footnote 2.

Absolutism vs Comparativism about Quantity | 109 relations are most explanatorily basic: they might be mass ratios, such as an object being twice as massive as another; orderings, such as an object being more massive than another; or even just linear structures, such as an object lying between two others in mass. But this in-house dispute will not matter for our purposes. For simplicity I will often assume that the comparativist under discussion is of the first type, but nothing will hang on this.7 As stated, both views are claims about what actually holds in virtue of what and so are neutral on the relation between intrinsic mass and mass relationships in other possible worlds. One might of course argue that comparativism is necessarily true if true at all (mutatis mutandis for absolutism), but I will not take a stand on this issue here.8 I have not yet mentioned facts about mass in a particular scale, such as my laptop’s being 2 kgs, and one might consider this omission strange since it is this sort of fact that we most often express when talking about mass. But it is not immediately obvious whether this is ultimately a fact that holds in virtue of my laptop’s intrinsic nature or in virtue of its mass relationships, so to avoid begging questions it is best to state the absolutism/comparativism issue without mentioning these facts and leave their status as a further question (I discuss this question in section 5). I have so far assumed that “holding in virtue of” is a relation between facts and I will continue to do so for ease of prose. But those wary of facts may express claims about what holds in virtue of what with the sentential operator “because.” Thus, when I previously said that Europe’s being at war holds in virtue of facts about the actions of its citizens at that time, one could restate this without reference to facts as Europe was at war in 1939 because a battalion of troops marched to battle and . . . 9 7 Baker (manuscript a) discusses which kinds of relations the comparativist should and should not think are explanatorily basic. And Baker (manuscript b) discusses the idea that the comparativist might introduce what he calls ‘mixed relations’. These are inter-quantity relations, for example my mass being twice your length. I will not discuss these mixed relations here. 8 Both views, I should say, are neutral as to the status of facts about the nomic relation between mass and other quantities. The issue just concerns the status of “catergorical” facts concerning the masses of things. 9 This way of expressing claims about what holds in virtue of what is suggested by Fine (see his (2001) and (2012)). Strictly speaking, the right-hand side will consist of a list of sentences rather than a conjunction, but this complication does not matter to us here.

110 | Shamik Dasgupta so long as “because” is understood in the metaphysical rather than causal sense. On this way of talking, the absolutist will assert things like My laptop is more massive than my cup because my laptop has the intrinsic mass M and my cup has the intrinsic mass M*.

whereas the comparativist will deny such a claim. Moreover, I have so far made free reference to such things as intrinsic masses and mass relations. For example, the natural interpretation of the displayed sentence takes “M” to be a term referring to a property and the expression “has the instrinsic mass” to be a relational predicate holding of my laptop and the property M. But nominalists wary of properties and relations can also make sense of the absolutism vs comparativism issue by reading the displayed sentence in another way. For example, they might take “has the intrinsic mass M” to be a primitive monadic predicate containing no referential devices at all. Still, in what follows I will continue to refer to properties and relations for ease of prose.10 Although I have focused on the case of mass, it should be clear that the same issue arises for other quantities too. For example, consider the case of spatial distance. When material bodies X and Y stand in the determinable relation of being spatially related, there are two kinds of determinate relationships that they enter into. On the one hand, it is natural to think that X stands in a determinate distance relation to Y. But it is also natural to think they stand in various comparative spatial relations to other bodies, for example the relation of X being twice as far from Y as Z is from W, or even just of X being further from Y than Z is from W. But of the former fact about the distance between X and Y and the latter fact about the comparative distance between X, Y, Z, and W, which is fundamental? The absolutist says that the most fundamental facts about distance include facts of the former kind, while the comparativist says

10 This is not to say that the absolutism/comparativism issue is entirely independent of the issue of realism about properties. One might argue, for example, that the nominalist has a hard time being an absolutist since her vocabulary would then be required to include an infinite number of primitive predicates, one for each determinate intrinsic mass. Still, logically speaking the issues are orthogonal and in what follows I will not be concerned with considerations that depend on a resolution to the question of realism about properties.

Absolutism vs Comparativism about Quantity | 111 that all facts about distance are ultimately explained (in the metaphysical sense) in terms of facts of the latter kind.11 This is why I used the terminology “absolutism” and “comparativism” in the case of mass rather than “intrinsicalism” and “relationalism”: when we generalize to other quantities such as spatial distance, the facts that the absolutist takes to be most fundamental are themselves facts about relationships. As I said, my main reason for endorsing comparativism (developed in section 8) is that the intrinsic masses posited by the absolutist would be undetectable in a very strong sense. The Occamist principle described earlier deems this to be a significant mark against absolutism. However, it is not a decisive mark against the view and if there were stronger countervailing considerations against comparativism, then absolutism would remain our allthings-considered best theory. So in sections 2–7, I consider six kinds of objection to comparativism—objections from intuition, from modality, from semantics, from kilograms, from Humeanism, and from physics—and argue that none are compelling. All things considered, then, I believe that comparativism is the preferable view.

2. OBJECTIONS FROM INTUITION Let us start with objections to comparativism from intuition. When first introduced to the issue, absolutism strikes many as being the more intuitive and plausible view. If my laptop is more massive than my cup, it initially seems that this is (at least in part) because of their intrinsic masses. Moreover, comparativism conflicts with an intuitive “locality” principle concerning mass, namely that given a connected region of spacetime R composed of two sub-regions R1 and R2, the fundamental facts about mass within R1 and the fundamental facts about mass within R2 together determine the fundamental facts about mass within R.12 This principle strikes us as 11 The absolutism/comparativism issue about distance should not be confused with the substantivalism/relationalism issue. The latter issue concerns the relata of spatial relations and asks whether they are, fundamentally speaking, material bodies or regions of space. Whichever way that dispute is resolved, we may then raise the absolutism/comparativism debate by asking whether, at the fundamental level, those relata stand in two-place absolute distance relations or four-place comparative relations. 12 Thanks to Eliot Michaelson for elucidating this locality principle for me.

112 | Shamik Dasgupta intuitively correct, but the comparativist must deny it since the mass relationships within R1 and those within R2 do not determine the mass relationships between a body in the one region and a body in the other. So let us agree that absolutism is initially the more plausible view. One might then argue that this is a reason to think that it is true. But the last sentence conflates a number of arguments. One unconvincing argument is that just as we are endowed with a reliable faculty of perception, we are also endowed with a reliable faculty of intuition which delivers the verdict that absolutism is true. To this we might well object that there is good reason to doubt that we have a faculty of this sort (for one thing, anatomists and neurologists have yet to find anything corresponding to it). Here we need not deny that we have a faculty of intuition that is a reliable guide to math and logic; all we need insist is that we do not have one that delivers reliable verdicts about what the fundamental physical properties and relations are, for it is this that absolutists and comparativists disagree about. The denial of such a faculty should therefore be uncontroversial. Still, one might argue that the intuitive plausibility of absolutism carries epistemic weight without appealing to a faculty of intuition. For example, one might say that absolutism is a “Moorean truth,” a proposition in whose truth we are more certain than any premise used in an argument to the contrary. Or one might point out that absolutism is our starting point in the inquiry, and then argue for a principle of epistemic conservatism according to which our starting point is (defeasibly) justified merely by virtue of being our starting point. Either way, the upshot would be that our initial absolutist inclinations are epistemically significant. In response, I do not object to the principle of epistemic conservativism or to Moorean approaches to some questions in philosophy. In particular, I concede that absolutism’s initial plausibility is at least some reason to believe it. But any such reason is defeated by my Occamist argument to the contrary. To see this, consider the case of absolute simultaneity. While it is initially plausible that there is such a thing as simultaneity, most would agree that considerations from special relativity are enough to defeat any consideration from Mooreanism or epistemic conservativeness in simultaneity’s favor.13 13 To the Moorean, this shows that our belief in simultaneity is not, after all, more certain than the premises of any argument against it. To the epistemic conservative,

Absolutism vs Comparativism about Quantity | 113 Now, I take the initial plausibility of simultaneity to be at least as strong as the initial plausibility of absolutism. Therefore, since my reason to reject absolutism is the same as our reason to dispense with simultaneity (I leave it until section 8 to make good on this claim), it will be strong enough to defeat considerations from Mooreanism or epistemic conservativeness in absolutism’s favor.

3. OBJECTIONS FROM MODALITY Other objections to comparativism appeal to modal considerations. The idea behind all these objections is that the comparativist cannot make sense of certain situations that seem, at least on the face of it, to be possible. I believe that those with absolutist inclinations are often moved by these kinds of considerations, so I will consider four such objections in some detail. The first objection can be put aside reasonably quickly. The objection is that while it seems possible for everything’s mass to double tonight at midnight, the comparativist cannot make sense of this since the mass relationships between things tomorrow would be exactly the same as they were today and so by the comparativist’s own lights there would have been no change. But this objection is not compelling, for the comparativist may claim that some of the fundamental facts about mass concern how something at one time is related in mass to something at another time. If so, she can make sense of the possibility after all. A second, more compelling objection along these lines appeals to the intuition that it seems possible for everything’s mass to always have been double what it actually is. The possibility of “uniform doubling” under consideration here is not one in which everything’s mass doubles at a particular time, but rather one in which the entire history of the universe is just like ours with the one exception that at any given time, everything’s mass is double what it actually is at that time. The objection is then that while this intuitively seems to be possible, the comparativist can make no sense of it.14 it shows that while our belief in simultaneity may have been epistemically privileged by virtue of being our starting point, its privilege was not enough to ward off arguments to the contrary. 14 Eddon (2013) objects to comparativism in this way. Hawthorne (2006) outlines the objection and seems to take it to have some force, but he does not explicitly endorse it.

114 | Shamik Dasgupta But why think that the comparativist cannot make sense of the possibility of uniform doubling? At this point the objector needs to choose whether to express her objection with modal operators or in the framework of possible worlds. In the latter framework, the possibility of uniform doubling is represented by a possible world just like ours but with the one exception that everything’s mass is double what it actually is. To argue that the comparativist can make no sense of such a world, the objector would appeal to a plausible “necessitation” principle: that if a fact Y holds in virtue of some facts, the Xs, then every world in which the Xs obtain is also a world in which Y obtains.15 This necessitation principle is standard in the literature on the in virtue of relation and is extremely plausible. For example, if the fact that Europe was at war in 1939 holds in virtue of facts about the actions of its citizens, then those actions are what accounts for and makes it the case that Europe was in a state of war rather than (say) a state of peace. Consequently, it seems that any world in which the citizens of Europe act in that way must also be a world in which the continent is at war, just as the principle implies. Now, along with the necessitation principle, comparativism implies that worlds agreeing on all the mass relationships agree on all facts about mass. But the so-called “doubled” world agrees with the actual world on all mass relationships; hence according to the comparativist it is not a world that differs from the actual world regarding any fact about mass and is therefore not a world in which everything’s mass is doubled. That is the objection stated in the framework of possible worlds. It can also be stated using modal operators by formulating the necessitation principle in terms of modal operators instead. In what follows I will only discuss the possible worlds version of the objection just outlined. The details of the modal operator version differ subtly, but there is no space to discuss those details and in any case the main morals of our discussion in terms of possible worlds will carry over. 15 I am simplifying here. As I describe in section 5, I believe that the in virtue of relation is irreducibly plural in the sense that some facts, the Ys, can hold in virtue of some facts, the Xs, even though no Y taken on its own holds in virtue of anything. If that is right, the necessitation principle should be rephrased with plural variables as follows: if the Ys hold in virtue of the Xs, then every world in which the Xs all obtain is also a world in which the Ys all obtain. But this complication is not relevant to the current objection, so I will ignore it here.

Absolutism vs Comparativism about Quantity | 115 How might the comparativist respond to the objection? One response is to become a modal realist in Lewis’s sense and say that the fundamental facts of the world are really facts concerning a plurality of concrete worlds. The comparativist may then think that the fundamental facts concerning mass relationships include how objects in different worlds relate to one another in mass. A comparativist of this sort will point out that the actual world and the doubled world disagree on their inter-world mass relations, and therefore the necessitation principle does not imply that they agree on all facts about mass. But I do not wish to rest my case on this response. Putting aside the unpopularity of modal realism, a more important worry is whether the modal realist can legitimately allow fundamental relationships between objects in different worlds. For example, the generalization of this approach in the case of spatial distance is that fundamental spatial relations hold between bodies in different possible worlds, and one might argue that this conflicts with Lewis’s account of a possible world as the mereological sum of spatiotemporally related things. So it is worth asking whether the comparativist can respond to the objection without appealing to fundamental inter-world mass relationships. I believe she can. In the first place, she may argue that her failure to make sense of the possibility of uniform doubling is no real vice; and in the second place she may argue that she can, perhaps surprisingly, make sense of the possibility without fundamental inter-world mass relations after all. Both responses seem reasonable to me, so I will discuss each one. Start with the first. The objection rests on the intuition that a uniform doubling of mass really is possible, but is this right? I find that my inclinations here depend on my theoretical convictions: when absolutism strikes me as attractive it seems possible, but when I am in the grip of comparativism I feel that the possibility is a silly philosophical mistake. This should not be surprising, since the absolutist and the comparativist are both likely to agree that if absolutism is true, then uniform doubling in mass is possible. Now, if our intuition that doubled worlds are possible rests on a prior belief that absolutism is true, the current argument would at best collapse into the argument from intuition just considered or, at worst, beg the question. So the question is whether we have an inclination to think that uniform doubling is possible that is independent of any prior

116 | Shamik Dasgupta belief in absolutism, and if so how strong that inclination is. I am not sure how one might go about answering this question, but an answer is crucial to the current argument. For now, then, it seems reasonable to take the issue of uniform doubling to be a case of “spoils to the victor.” The second response is that the comparativist can, perhaps surprisingly, make sense of the possibility of uniformly doubling without appealing to fundamental inter-world mass relationships. To see how let us turn to the third modal objection to comparativism, for our response to it will provide the materials needed to make sense of uniform doublings. So, putting aside for a moment the possibility of everything’s mass being doubled, note that it is surely possible for just my laptop to have been twice as massive as it actually is. The third modal objection is that the comparativist cannot even make sense of this. On the face of it, this objection is far more powerful than the last. For while we might reasonably deny that it is possible for everything’s mass to have been doubled, we must surely agree that my laptop could have been more massive than it is. If the comparativist cannot even make sense of this latter possibility, that is a vice indeed. I am therefore surprised not to have seen or heard this argument expressed by those with absolutist inclinations. But why think that the comparativist can make no sense of the possibility? Again, the objection can be developed in the framework of possible worlds or with modal operators, but let us stick to the former. The idea, then, is that she can perfectly well make sense of a world W just like ours except that the mass-ratio between my laptop and all other things is double what it actually is. But without interworld mass relationships, there is no fact of the matter as to whether W is a world in which my laptop is twice as massive as it actually is, or one in which my laptop is the same mass and everything else is half as massive as they actually are. Notice that if the actual bodies and those in W had intrinsic masses, then the problem would not arise, for those intrinsic masses would determine mass relationships between bodies in the one world and bodies in the other. But without fundamental inter-world mass relationships, the comparativist has no resources to make a similar inter-world comparison.16 16 The situation is not improved by noting that comparativism is a contingent claim and allowing the material bodies in W to have intrinsic masses. For so long as

Absolutism vs Comparativism about Quantity | 117 How might the comparativist respond? I believe she can accuse the objection of resting on an incorrect model of how a possible world represents my laptop’s mass and introduce a better model that allows her to make sense of the possibility in question. To see this, it will help to work with a specific model of how a possible world represents something de re of my laptop in the first place. I will work with Lewis’s famous proposal that it does so not by containing my laptop but by containing one of its counterparts, though nothing hangs on this choice. Given this assumption, the world W introduced above can be re-described as a world containing a counterpart of every actual material body such that if my laptop is r times as massive as another body x, my laptop’s counterpart in W is 2r times as massive as x’s counterpart in W. Now, consider an object other than my laptop, such as my cup. Notice that the mass relationships it enters into are extremely similar to those entered into by its counterpart in W. The only difference is that while my cup is (say) half as massive as my laptop, my cup’s counterpart in W is one fourth as massive as my laptop’s counterpart in W. Other than that, my cup’s mass relationships to all other bodies are exactly the same as its counterpart’s relationships to theirs. But this is not so with my laptop: the mass relationships it enters into differ systematically from those that its counterpart in W enters into by a factor of 2. So the comparativist may say that it is in virtue of this asymmetry that W represents my laptop as being twice as massive as it actually is and everything else as being the same mass as they actually are. In effect, the comparativist just introduced a “mass-counterpart” relation in addition to the ordinary, Lewisian counterpart relation. Since my cup and its counterpart in W resemble one another with respect to their mass role, we call them mass-counterparts. And because my cup’s counterpart is also its mass-counterpart, W represents my cup as being the same mass as it actually is. Here the mass-counterpart relation is doing analogous work to Lewis’s counterpart relation: just as the latter is not identity but stands in for it when determining what a world represents de re, the masscounterpart relation is not the same-mass-as relation but stands in for it when determining what a world represents about mass. the actual material bodies lack intrinsic masses, there remains no fact of the matter as to whether W is a world in which my laptop is twice as massive as it actually is.

118 | Shamik Dasgupta Indeed, the comparativist can introduce a slew of mass-counterpart relations, one for each real number. My cup’s mass role resembles its counterpart’s mass role, so we call them mass1-counterparts. My laptop’s mass role does not resemble its counterpart’s mass role in the same way, but since the mass ratios my laptop stands in are uniformly half those of its counterpart, their mass roles resemble each other modulo a factor of 2. As a result, the comparativist can call them mass2-counterparts. More generally, she can say that x and y are massr-counterparts just in case x’s mass role resembles y’s mass role modulo a factor of r. And she may then propose the general principle that, relative to a counterpart relation and a set of mass-counterpart relations, W represents an actual object x as being r times as massive as it actually is just in case x has a counterpart in W that is also x’s massr-counterpart. Relative to the mass-counterpart relations just described, this delivers the desired result that W represents my laptop as being twice as massive as it actually is. The ordinary counterpart relation is context-sensitive, in the sense that the features of individuals relevant to determining whether they resemble one another, and therefore whether they are counterparts, are sensitive to the context in which the modal claim is made. Similarly, the comparativist can allow that those aspects of a body’s mass role relevant to determining what its massr-counterparts are depend on the conversational context. For example, a context in which mass relationships to my laptop are particularly salient might be one in which my laptop’s counterpart in W is also my laptop’s mass1-counterpart (since the latter agrees with my laptop on all mass relationships to itself). Relative to this masscounterpart relation, W represents my laptop as being the same mass as it actually is and everything else as being half as massive as they actually are! Although the discussion so far assumed Lewis’s own theory of de re modality, the mass-counterpart theory just introduced is consistent with many other theories, including ersatz ones. There is of course much more to say about it, but much of the ensuing discussion will resemble the literature on Lewis’s own counterpart theory, so instead let me return to the second modal argument we left earlier and explain how the comparativist can use the mass-counterpart theory just introduced to make sense of the

Absolutism vs Comparativism about Quantity | 119 possibility of uniform doubling. The problem, remember, was that a “uniformly doubled” world agrees with our world on all mass relationships and so, according to comparativism and the necessitation principle outlined earlier, also agrees with our world on all facts about mass whatsoever and is therefore not a doubled world after all. But with mass-counterpart theory in hand, I believe the comparativist can accuse the objection of ignoring the distinction between worlds and possibilities. That is, she can respect the necessitation principle and concede that she can make no sense of a uniformly doubled world, but insist that she can nonetheless make perfectly good sense of the possibility of uniform doubling. To see how, note that it is a familiar fact that worlds and possibilities come apart in ordinary counterpart theory. To use Lewis’s example, I might have been either one of a pair of twins: I might have been the first born, and I might have been the second born.17 Here we have one possible world with twins but two possibilities, one in which my counterpart is the first born and the other in which my counterpart is the second born. Indeed, we have already seen an analogous distinction between worlds and possibilities in masscounterpart theory: The world W discussed earlier represented two possibilities depending on which mass-counterpart relation we focused on, one in which my laptop is twice as massive as it is and one in which everything else is half as massive as they are. In his discussion of worlds and possibilities, Lewis also says that our counterparts need not always be in other worlds. When I consider the unhappy possibility of being my neighbor Fred, Fred himself (my worldmate) is acting as my counterpart and represents me as having all his properties.18 In this case the actual world, along with a certain counterpart relation, is representing a non-actual possibility for me. With mass-counterpart theory, the comparativist can model the possibility of uniform doubling analogously to how Lewis models the possibility of my being Fred, namely by using the actual world along with a suitable mass-counterpart relation. To see how, note that the possibility of uniform doubling is a possibility for all the

17

See Lewis (1986a), p. 231.

18

See Lewis (1986a), p. 232.

120 | Shamik Dasgupta material bodies taken together. If we let S be an ordered set of all those bodies, then according to Lewisian counterpart theory a world represents a possibility for those bodies by containing a counterpart of S. Well, surely S can be its own counterpart in normal contexts. Moreover, S can also be its own mass2-counterpart in normal contexts. This is because S’s mass role—the pattern of mass relations entered into by the members of S—resembles its mass role perfectly modulo a factor of 2: after all, the pattern of mass relations are exactly as they would be were everything doubled in mass! And relative to these counterpart and mass-counterpart relations, our mass-counterpart theory—suitably generalized to apply to ordered sets—implies that the actual world itself represents the possibility of uniform doubling. So in this sense, uniform doublings of mass are possible even for the comparativist. This is not to deny that there is another sense in which uniform doublings of mass are impossible for the comparativist. What we just did, effectively, is distinguish between two modal notions. One is the notion that quantifies over what I called “possible worlds” earlier and might be called “strict” possibility. This is the notion on which the earlier necessitation principle is true, that if a fact Y holds in virtue of some facts, the Xs, then every possible world in which the Xs obtain is also a possible world in which Y obtains. The comparativist should concede that she can make no sense of the strict possibility of uniform doublings—indeed, as the third modal objection rightly points out, the comparativist cannot even make sense of the strict possibility of my laptop being more massive than it actually is! But there is another modal notion that quantifies over what I called “possibilities” earlier, where a possibility can be thought of as a triple of a possible world W, an assignment of objects to a counterpart in W, and an assignment of objects to a massr-counterpart in W. We might call this “loose” possibility.19 As argued earlier, the comparativist can then make perfectly good sense of the loose possibility of my laptop’s being more massive than it actually is, and indeed the loose possibility of uniform doublings. 19 If one is attached to the idea that it is possible that P iff there is a world in which P is true, one should feel free to call the possibilities “possible worlds” instead when using the loose notion. The nomenclature is not what matters here.

Absolutism vs Comparativism about Quantity | 121 What the modal objections to comparativism need to establish, then, is not just that the envisaged situations (uniform doublings of mass, or just my laptop being more massive than it actually is) are possible in some generic sense but rather that they are strictly possible, for it is only this latter claim that the comparativist must deny. I have seen no attempt by absolutists to establish that they are strictly possible and I am skeptical that they can do so. At the very least, the burden is now on the absolutist to establish the required premise. For this reason I remain unmoved by these modal objections to comparativism. There is a more general lesson here, namely that when engaging in modal reasoning (as these objections do) it is crucial to take great care to clarify the modal notion in use. For there may be many notions of possibility related to but distinct from strict possibility. If so, then when faced with the objection that she cannot make sense of this or that possibility, the comparativist can always explore the response of saying that the envisaged situation is possible in one of these other senses even if it is impossible in the strict sense. Indeed, I have no particular allegiance to the counterpart-theoretic response developed earlier and am open to resting my response on another alternative to strict possibility, if one can be developed. The counterpart-theoretic response is dialectically useful, though, since it nicely illustrates the existence of at least one of these other modal notions.20 It is worth mentioning a fourth kind of modal argument against comparativism, namely that the comparativist cannot make sense of a possible world containing just one massive body.21 In particular, one argument is that she cannot make sense of it having the determinable property of having mass, while a second argument is that she cannot make sense of it having any particular determinate mass (say, the mass of an electron). But in response to the first argument, the 20 David Chalmers distinguishes two notions of metaphysical possibility that he calls prior and posterior possibility (he mentions the idea briefly in footnote 5, p. 449 of his (2012) and is developing the idea in other work). His notion of prior possibility is, I think, the same as the notion of strict possibility in use here. It is less clear to me how his notion of posterior possibility is related to the notion of loose possibility discussed in the text, though his notion of posterior possibility is certainly more general, since the notion of loose possibility is (as defined earlier) restricted to the case of mass. 21 Thanks to Michaela McSweeney for helping me to appreciate the force of this argument. Similar arguments are given by Eddon (2013).

122 | Shamik Dasgupta comparativist can say that something has the determinable property of having mass if it stands in a determinate mass relation, and then point out that the lone particle stands in such a relation to itself, namely the same-mass-as relation. And in response to the second objection, the comparativist can appeal to mass-counterpart theory and say that whether the particle counts as having the mass of an electron depends on the mass-counterpart relations allowed by the conversation in which the world is being discussed. I conclude, then, that objections to comparativism based on modal considerations are not convincing.

4. THE OBJECTION FROM SEMANTICS One might instead try to refute comparativism on semantic grounds. To see how, recall Kripke’s famous claim that we use the term “1 meter” with the stipulation that it is to refer to the length of the standard meter in Paris.22 The analogous view in the case of mass is that we use “1 kilogram” with the stipulation that it is to refer to the mass of that lump of platinum-iridium alloy in Paris that serves as our standard of measurement, known as the International Prototype Kilogram (IPK). But the entity that the Kripkean theory takes the referent of “1 kilogram” to be, namely the mass of IPK, sounds suspiciously like an intrinsic property of IPK. After all, if the fundamental facts about mass were just facts about mass relationships, it is difficult to see what “the mass” of IPK could possibly be. So, the argument goes, if comparativism were true, then “1 kilogram” would fail to refer and sentences like “My laptop is 2 kilograms” would fail to be true. Of course, the comparativist might bite the bullet and concede that kilogram sentences are not true. According to this “error theory” response, the term “kilogram” is similar to “phlogiston”: both were used with the stipulation that they are to refer to whatever entity satisfies some description, but since nothing answers to the description, they both fail to refer. The main difference between the two cases is that there is a pragmatic reason to continue using the term “kilogram” that is lacking in case of “phlogiston.” For as long as our use of “kilograms” is governed by the inference rule 22

See Kripke (1972).

Absolutism vs Comparativism about Quantity | 123 a is r kilograms b is s kilograms Therefore, a is r/s times as massive as b we can use “kilogram” as a convenient way of storing and communicating information about mass ratios even if sentences containing it are not true.23 If this error theory sounds radical and unwarranted, compare it to the case of absolute simultaneity. If semantic investigation revealed that the truth of our ordinary talk requires there to be such a thing as absolute simultaneity but it subsequently turned out for reasons of physics and metaphysics that there is no such thing, we would have no qualms concluding that our ordinary talk is in error. Similarly, if the semantic argument presented a moment ago showed that the truth of “kilogram” sentences requires the truth of absolutism but it subsequently turned out for reasons of physics and metaphysics that absolutism is false, we should have no qualms accepting the resulting error theory. But while this error theory is defendable, there is no need for the comparativist to adopt it since the semantic objection fails to establish that the truth of “kilogram” sentences requires the truth of absolutism in the first place. In fact, it fails for two reasons. First, the Kripkean theory of reference-fixing it presupposes is false. To see this, imagine reading in The Times that the French have been subjecting us to an illusion that makes IPK appear twice as massive as it actually is. Imagine that the article explains that the illusion has been systematic, so that whenever we used IPK to calibrate our measuring instruments, the calibration succeeded even though we were misled about the properties of the lump. So, if we were to put IPK on one of the many calibrated measuring instruments around the world, it would read “500 grams” rather than “1 kilogram.” How would we report this discovery? Intuitively, by saying that we discovered the surprising fact that IPK is 500 grams! But the Kripkean theory predicts otherwise. Since the theory is that “1 kilogram” is stipulated to refer to the mass of IPK whatever that mass is, it implies that the article should report instead that while the standard object is (of course) still 1 kilogram, all other material bodies are actually half the mass in kilograms that we previously 23

I discuss the role of this form of inference in more detail in Dasgupta (manuscript).

124 | Shamik Dasgupta thought they were. And this, I claim, is not how we would intuitively report it. But there is a second and perhaps more decisive reason why the semantic objection fails to establish that the truth of “kilogram” sentences requires absolutism. Even if we granted the Kripkean theory of reference, the argument is supposed to be that the entity to which “1 kilogram” is stipulated to refer, namely the mass of IPK, is not identical to IPK’s mass relationships, and it concludes that the comparativist must say that there is no such thing. But this last step is a non sequitur. All the comparativist claims is that the most fundamental facts about mass are facts about mass relationships; it is perfectly consistent with this that there is such a thing as the mass of IPK which is not identical to any mass relationships, so long as any fact of the matter concerning it holds in virtue of facts about IPK’s mass relationships. So the comparativist is free to agree that there is such a thing as the mass of IPK to which the term “1 kilogram” refers after all. The non sequitur exhibited by the semantic objection is vividly exemplified in the following case. Consider a physicalist who claims that all facts hold in virtue of facts concerning physical entities, and imagine an objector who says “The term ‘stock market’ refers to the stock market, but the stock market is not a physical entity; therefore your physicalism is false.” In response, our physicalist will surely point out that the argument misses its mark entirely: Her view was never that everything is a physical entity but rather that all facts about the world hold in virtue of facts concerning physical entities. The comparativist can say exactly the same about the semantic objection to comparativism. In sum, I do not believe that there is semantic evidence that the truth of “kilogram” sentences requires absolutism to be true. But even if there were, I believe that the appropriate response for the comparativist would be to adopt an error theory about “kilograms.”

5. THE OBJECTION FROM KILOGRAMS Since comparativism holds that all facts about mass hold in virtue of mass relationships, one might naturally try to refute the view by finding a counterexample, i.e. a fact about mass that does not hold in virtue of mass relationships. For example, consider the fact that

Absolutism vs Comparativism about Quantity | 125 my laptop is 2 kgs. If one can argue that there are no mass relationships in virtue of which this obtains, one would naturally take oneself to have refuted comparativism. Absolutists should find this strategy promising, for I believe that there are good arguments to the effect that my laptop’s being 2 kgs does not hold in virtue of any mass relationships. Unfortunately there is no room to discuss these arguments in full detail, but let me say something to motivate the idea.24 First, note than an absolutist has no problem accounting for my laptop’s being 2 kgs: she can say that it is either identical to, or else holds in virtue of, my laptop’s having a certain intrinsic mass. If absolutism were true, this would be an extremely plausible view. For if material bodies really did have intrinsic masses, it would be natural to think that terms of the form “r kilograms” would refer to those properties (even if the Kripkean view about what fixes the referents of the terms is incorrect). If so, then it is almost irresistible to say that my laptop’s being 2 kgs is either identical to, or else holds in virtue of, its having a certain intrinsic mass; namely the one that is the referent of “2 kgs.” But the fact that this account of my laptop’s being 2 kgs is so natural and satisfying shows that, at least intuitively, the mass relationships that it enters into are entirely irrelevant when it comes to explaining what makes it 2 kgs. My laptop stands in all sorts of mass relationships to standard objects in Paris and measuring instruments in Paraguay and electrons on Pluto, but the fact that the absolutist’s explanation is so satisfying shows that, intuitively, all these relationships are irrelevant to an explanation of its being 2 kgs. Therefore, the argument goes, whichever mass relationships the comparativist picks in order to explain its being 2 kgs, she will violate our intuitions as to what is relevant to explaining that fact. Of course, the comparativist might concede this and reply that revising our opinions about what is explanatorily relevant is a natural consequence of theoretical inquiry. To an extent, this reply is well taken. But all hands should agree that this would be a significant revision of pre-theoretic belief and therefore counts as at least a point against her view. As I said, there is no space to develop this kind of argument in detail. But let us give the absolutist the benefit of the doubt and 24

These arguments are developed in more detail in Dasgupta (manuscript).

126 | Shamik Dasgupta suppose that there are no mass relationships in virtue of which my laptop is 2 kgs. Where does this leave the comparativist? Of course, one option would be to adopt the error theory described earlier, according to which there are no facts about mass-in-kilograms in the first place. But I believe that there is a better option. The key is to recognize that the in virtue of relation is irreducibly plural, in the sense that a plurality of facts Y can sometimes hold in virtue of another plurality of facts X even though no Y when taken on its own holds in virtue of anything. Given this “pluralistic” conception of the in virtue of relation, the comparativist may take the set K of all kilogram facts and the set R of all facts about mass relationships, and propose that the members of K (plurally) hold in virtue of the members of R even though no kilogram fact taken on its own holds in virtue of anything. This view neatly sidesteps the problem of relevance discussed earlier, for R does not contain irrelevant information when it comes to explaining the members of K. To be sure, R contains irrelevant information when explaining my laptop’s being 2 kgs, such as information about its mass relationships to electrons on Pluto, but since K contains facts about how massive those electrons are in kilograms the relationships between them seem perfectly relevant when explaining K’s members all together. By adopting this position, the comparativist can then agree that there is a fact of my laptop’s being 2 kgs (contra error theory), concede that there are no mass relationships in virtue of which it obtains, and yet nonetheless insist that this is perfectly consistent with comparativism since it remains the case that all the facts about mass in kilograms when taken together as a plurality hold in virtue of the underlying mass relationships. There are many virtues of this view. One is that the comparativist can respect the intuition described earlier that facts about my laptop’s mass relationships are not part of what explains its being 2 kgs. They are not part of the explanation of this fact, on this view, because the fact on its own has no explanation in the first place! Another virtue is that it neatly explains why absolutism is initially the more intuitive and attractive view. For as we have seen, we have a strong intuition that my laptop’s being 2 kgs does not hold in virtue of its mass relationships to other bodies. According to the current approach, the absolutist’s mistake is just to take this to imply that its being 2 kgs must be explained in terms of its intrinsic nature, when instead the correct conclusion is that we

Absolutism vs Comparativism about Quantity | 127 can only explain facts about mass in kilograms when they are taken all together as a plurality. The absolutist’s mistake is therefore understandable, but a mistake nonetheless. Now, I have not argued that this pluralist explanation of kilogram facts is satisfactory, and unfortunately there is no space to do so here.25 But at the very least, it is clear that if the in virtue of relation is irreducibly plural, then refuting comparativism is significantly more difficult than one might have thought. For it is then not enough for the absolutist to argue that my laptop’s being 2 kgs fails to hold in virtue of its mass relationships. In addition, she would need to argue that the plurality of kilogram facts taken together do not hold in virtue of the totality of mass relationships. Until she shows this, comparativism remains a live option. 6. THE OBJECTION FROM HUMEANISM Humean Supervenience (HS), says Lewis, is the view that all there is to the world is a vast mosaic of local matters of particular fact, just one little thing and then another . . . We have geometry: a system of external relations of spatio-temporal distances between points . . . And at those points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated . . . And that is all. There is no difference without a difference in the arrangement of qualities. All else supervenes on that.26

So stated, HS is inconsistent with comparativism, since HS asserts that the fundamental physical quantities like mass are intrinsic and are instantiated at single points of spacetime. But HS is supported in the literature by a wide range of arguments. If it is inconsistent with comparativism, the argument would be, so much the worse for comparativism. However, most arguments for HS are perfectly consistent with comparativism. To see this, note that HS is the conjunction of two theses: one stating that everything supervenes on the categorical nature of the physical world, and a second describing what the categorical nature of the physical world is like. The quote from Lewis focuses on the second and says that the categorical nature of the world consists in the distribution of intrinsic properties across 25

I motivate and defend this pluralist explanation at some length in Dasgupta (manuscript). 26 Lewis (1986b), p. ix.

128 | Shamik Dasgupta spacetime. And that second thesis is indeed inconsistent with comparativism. But much of the literature on HS focuses on the first thesis, the view that everything else—including chances, causes, counterfactuals, minds, morals, etc.—supervenes on the world’s categorical nature. And a brief glance at that literature reveals that none of the arguments depend on whether those underlying categorical facts consist in the instantiation of intrinsic properties (as Lewis says) or in the instantiation of comparative relations (as the comparativist says). Even if one is moved by those arguments, one may still adopt comparativism. To be sure, the second thesis does play a role in Lewis’s metaphysics. For example, he famously analyzes de re modals in terms of counterparts, and he says that objects are counterparts insofar as they resemble each other, and he says that resemblance is ultimately a matter of sharing intrinsic properties. But one can easily restate his view in terms friendly to the comparativist by allowing resemblance to ultimately be a matter of participating in the same pattern of relations instead. Indeed, Lewis only thought that resemblance was ultimately a matter of sharing intrinsic properties because he thought that it was a matter of sharing perfectly natural features, and the second thesis of HS states that those perfectly natural features are all intrinsic (save for geometric relations). Thus, if one gives up that second thesis in favor of comparativism and allows that some perfectly natural features are relations, it will follow from the rest of his system that resemblance is sometimes a matter of participating in the same pattern of relations after all. As a result, a large chunk of Lewis’s system remains essentially unchanged even if we endorse comparativism. This is not to say that all of Lewis’s views are easily recast in comparativist terms. Still, once one sees how many of them can be, the quote from Lewis stating that the categorical world consists in the distribution of intrinsic properties sounds less like an essential part of his view and more like a convenient working assumption. 7. OBJECTIONS FROM PHYSICS The objections to comparativism considered so far have been broadly a priori, but is there empirical evidence against the view? If

Absolutism vs Comparativism about Quantity | 129 we could see the intrinsic mass had by a given material body or detect it with the help of mechanical devices, that would presumably count as empirical evidence for absolutism. But I will argue in section 8 that if material bodies had the intrinsic masses posited by the absolutist, those intrinsic masses would be invisible to the naked eye and undetectable by any physically possible device. Still, if the absolutist could show that intrinsic mass is indispensible to our best confirmed scientific theories, one might then think that empirical evidence confirming those theories would thereby count as empirical evidence that each material body has an intrinsic mass even if we cannot tell which particular one it is. To see how this idea might be developed, consider one simple law governing mass, f = ma, and let us pretend for simplicity that our best confirmed physical theory states that it is the only law governing the motions of material bodies. Now, consider a world W exactly like ours with the one exception that everything’s mass is double what it actually is. One might argue that if the equation f = ma actually obtains, then it does not obtain in W since doubling everything’s mass while leaving their forces and accelerations unchanged would break the equality.27 Since W is just like our world in all mass relational respects, the argument would be that the truth of f = ma depends not just on the mass relations between things but also on which intrinsic masses they have. Therefore, empirical evidence confirming f = ma is ipso facto empirical evidence confirming absolutism. But the argument does not convince, for it depends on a controversial interpretation of the equation “f = ma.” Taken at face value,

27 This is where it helps to ignore any other laws specifying the force acting on each particle and pretend that f = ma is the only law governing our world. For we thereby sidestep the complication that with the identification of inertial mass and gravitational mass, mass plays a unique role in classical mechanics: not only is it a “brake” on acceleration as described in f = ma, it is also a determiner of the gravitational force between things as described in the inverse-square gravitational force law. The fact that it plays this dual role might tempt one to think that doubling everything’s mass would preserve the truth of the classical mechanical laws, since the increase in gravitational forces would be counter-balanced by the increased “brake” effect experienced by each body. But even if this line of reasoning were sound, it would not generalize to other quantities. I consider f = ma in isolation from whatever force laws it might couple with precisely because we are looking for general considerations.

130 | Shamik Dasgupta it states a mathematical relationship between the numbers and vectors that represent force, mass, and acceleration in a particular scale. But what does it state about the quantities themselves? There are at least two interpretations, one on which it states something about absolute quantities and another on which it states something about comparative quantities. On the first interpretation, it states that the absolute masses, accelerations, and forces all line up in a specific way. For example, part of the content of the equation on this interpretation is that anything with the determinate intrinsic mass M which is accelerating at a determinate rate A will have a particular determinate force F acting on it, and so on for other determinate masses, accelerations, and forces. And, in the other direction, part of the content of the equation will be that anything with the particular determinate force F acting on it will either have the determinate mass M and be accelerating at the determinate rate A, or else will have the determinate mass 2M and be accelerating at the determinate rate A/2 (where 2M is the intrinsic mass that is twice M and A/2 is the determinate rate of acceleration that is half A), and so on for other pairs of determinate masses and accelerations. More generally, we might express this interpretation as follows: (L1) For any material thing x, (a) For any reals r1 and r2, if x has mass r1M and acceleration r2A, then x has force r1 r2F acting on it. (b) For any real r3, if x has force r3F acting on it, then there are reals r4 and r5 whose product is r3, such that x has mass r4M and acceleration r5A.28 By contrast, the second interpretation of the equation takes it to state a connection between mass relationships, force relationships, and acceleration relationships. For example, part of the content of the equation on this interpretation is that if a body x is twice as massive as another body y and an equal force is applied to both, then y will accelerate at twice the rate as x. More generally: (L2) For any material things x and y, (a) For any reals r1 and r2, if x is r1 times as massive as y and is accelerating at r2 times the rate of y, then x has r1r2 times as much force acting on it than y. 28 I am bracketing for simplicity details that arise from the fact that A and F are vector quantities.

Absolutism vs Comparativism about Quantity | 131 (b) For any real r3, if x has r3 times as much force acting on it than y, then there are reals r4 and r5 whose product is r3, and such that x is r4 times as massive as y and is accelerating r5 times the rate of y.29 The argument under consideration assumed that the actual physical laws do not obtain in the world W in which all masses are doubled. We can now see that whether this is true depends on whether the fundamental law governing the quantities is (L1) or (L2). The assumption is true if the fundamental law is (L1), since in W the absolute masses, forces, and accelerations line up differently than they actually do. But the assumption is false if the fundamental law is (L2), for (L2) only talks of mass relationships and those are the same in W as they are in the actual world. Indeed, it is clear that if the fundamental law is (L2), then whether it obtains does not depend on material bodies having the intrinsic masses posited by the absolutist, contra the argument under consideration. So in order to run the indispensability argument, the absolutist must argue that the empirical evidence confirming f = ma is evidence that confirms (L1) and disconfirms (L2). Only then would the intrinsic masses posited by the absolutist be indispensable to what is confirmed by the empirical evidence. But what evidence would favor (L1) over (L2)? The difference between the laws is this: (L1) implies that if the state of the world at present differed only in that everything’s mass were double what it actually is, things would proceed to accelerate at half their actual rate.30 (L2) does not imply this, because the mass ratios would be exactly the same in the doubled state and that is all that the law makes reference to. So the two laws issue these different predictions, but how could we test which prediction is correct? The obvious idea is to construct two isolated laboratories that are exactly alike at an initial time except for the fact that one is a doubled-mass version of the other. One might think

29 This is a very simplistic example of a comparativist rendering of a physical law. For a more realistic attempt to render physical theories in comparativist-friendly terms, see Field (1980). I will discuss (L2) here because it is particularly easy to grasp, but the discussion is intended to generalize to more realistic comparativist-friendly laws such as those described by Field. 30 Again, I am bracketing the effect that doubling everything’s mass might have on the forces acting on things. See footnote 27.

132 | Shamik Dasgupta that if the bodies in the doubled-mass laboratory proceed to accelerate at half the rate as the bodies in the other, this would confirm (L1) and disconfirm (L2). But on further reflection this experimental outcome is predicted by (L2) and would therefore not disconfirm it. After all, the bodies in the doubled-mass laboratory are twice as massive as their counterparts in the other and are subjected to the same forces as those counterparts, and the experimental outcome is that they accelerate at half the rate as their counterparts. But this is exactly what (L2) would predict! The trouble is that the two laws make different predictions about what would happen if the entire world were doubled in mass, but when attempting to test which prediction is correct we can do no better than to compare different parts of the world (our two laboratories) and the laws make exactly the same prediction about what would then occur. How else might we obtain evidence that confirms (L1) and disconfirms (L2)? Suppose we could see or detect which particular intrinsic mass, force, and acceleration each material body has, and suppose it turned out that they were lined up in the way stated by (L1).31 Would this confirm (L1) and disconfirm (L2)? No, because that observation is consistent with the hypothesis that material bodies have absolute quantities but are governed by (L2). For if that were the case, then the absolute quantities are bound to line up in one way or another; it would just be a matter of accident (or perhaps, more specifically, a matter of initial conditions) that they line up as they do whereas according to (L1) they would line up like that as a matter of law. It therefore appears that there is no possible evidence that would confirm (L1) and disconfirm (L2). Under the pretense that f = ma is the only law of our best confirmed physics, then, the intrinsic masses posited by the absolutist are not indispensable to our best physics after all.32 The pretense is of course false, but it helps to illustrate the problems that face this kind of “indispensability 31 I will argue in section 8 that we could never see or detect such a thing. But I do not want to rest my argument here on the results of that section. So here I am supposing, per impossible, that we can make such observations just for the sake of argument. 32 Baker (manuscript a) argues that the idea that the fundamental laws of our world are comparativist-friendly laws like (L2) (or, more realistically, the kind of laws that Field expresses in his (1980)) has serious problems. There is no space here to discuss his arguments in the detail they deserve but I hope to do so in further work. Very

Absolutism vs Comparativism about Quantity | 133 argument” for absolutism. As I said in the introduction, whether this line of reasoning generalizes to other theories and quantities is a question I leave for another time.33

8. IN FAVOR OF COMPARATIVISM 8.1. The Occamist argument Having surveyed a number of objections to comparativism, I find none convincing. But is there any positive reason to be a comparativist? I believe there is. I will argue that if material bodies really did have the intrinsic masses posited by the absolutist, those intrinsic masses would be undetectable. Our Occamist principle says that it is a mark against a theory if it posits undetectable structure, so this is reason to prefer comparativism. This “epistemic” argument is not the only possible argument for comparativism. Some might favor comparativism purely on grounds of ontological parsimony, the idea being that while everyone recognizes mass relations, it is only the absolutist who goes further and posits extra intrinsic masses that are not explicable in terms of those mass relationships. Others might favor comparativism because it leads to a simpler axiomatization of a theory of mass.34 But my epistemic argument is particularly important. For if we could see or otherwise detect which intrinsic mass a given body has—if the intrinsic masses posited by the absolutist were really part of our “data”—then a comparativist theory that dispensed with those intrinsic masses would not be empirically adequate. And briefly, the problems he describes concern modal properties of the laws, such as whether they are deterministic. Now I argued in section 3 that when discussing comparativism, it is important to think carefully about how modal claims about quantities are to be evaluated. There I argued in favor of a “mass-counterpart” interpretation of modal claims. I believe that if the comparativist endorses this mass-counterpart theory, she has all the resources needed to respond to Baker’s objections. But I hope to defend this idea in future work. 33 Field (1980) has famously made a good start at expressing a portion of physics in purely comparativist terms. However, the current point does not depend on the success of Field’s project, which was to express physics without reference to numbers, sets, or other abstracta. For example, (L2) serves the comparativist’s purposes but it freely quantifies over real numbers. 34 This is an idea that Field emphasizes in his (1985). Thanks to a referee for encouraging me to think about this consideration.

134 | Shamik Dasgupta in that case comparativism should obviously be rejected outright purely on grounds of empirical inadequacy, and the proposed benefits of ontological parsimony and theoretical simplicity would then be neither here nor there. So a hidden assumption in these other arguments for comparativism must be that the intrinsic masses posited by the absolutist are not detectable after all, which is precisely what my epistemic argument tries to establish. Much depends on what I mean by “undetectable.” If I used the term to include anything that we cannot see with the naked eye, our Occamist principle would recommend that we become radical scientific anti-realists and dispense with so-called “theoretical” entities such as electrons. But that is not how I use the term here. Instead, something is undetectable in my sense of the term if, roughly speaking, it follows from the structure of the laws of motion governing our world that it is physically impossible for it to have an impact upon our senses. Electrons are therefore detectable in this sense because there are physically possible processes, such as those that occur in particle accelerators, by which the presence of an electron can be made to have an impact on our senses via its impact on (say) the movement of a dial or an image produced on a computer screen. In contrast, features like absolute velocity and absolute simultaneity are undetectable in my sense: even if they were real, it turns out that the laws of motion governing our world are set up in such a way as to guarantee that it would be impossible for them to ever have an impact on our senses. That is why they are considered to be “redundant” or “superfluous” in modern physics, and most physicists and metaphysicians therefore believe on Occamist grounds that those features are not real after all. I will argue here that the same goes for intrinsic mass. The Occamist principle I use here is not verificationist: there is no claim that talk of these undetectable features is meaningless. It is just a principle of theory choice, a principle stating that (all else being equal) we should favor those theories without undetectable features. Nor does our Occamist principle say that we should always dispense with undetectable features. It just says that undetectable features are undesirable, so that all else being equal—or at least near enough equal—we should prefer theories like comparativism that dispense with them. But in the previous sections I have been arguing that there are no overwhelming reasons to reject comparativism and therefore that all is indeed near enough equal after all.

Absolutism vs Comparativism about Quantity | 135 Our crucial premise, then, is that if material bodies really possessed the kind of intrinsic mass posited by the absolutist, those intrinsic masses would be undetectable in our sense of the term. How should we argue for this? There is a reasonably well-known argument for the analogous claim in the case of absolute velocity, so let us rehearse it before applying it to the case of intrinsic mass.

8.2. The case of velocity What is absolute velocity? We often talk of a material body’s velocity relative to another body: a car might have a velocity of 65 mph in a particular direction relative to the highway and 10 mph in the same direction relative to the train traveling alongside it. But how fast is it really going, independent of any material reference point? If there is an answer to this question, that is a statement of its absolute velocity. Now if there were such a thing as absolute velocity, why would it be undetectable? At least naively, one might think that the speedometer found in an ordinary car is a device that detects the vehicle’s absolute velocity. But we can argue that such a device at best measures relative velocity and that absolute velocity is undetectable after all.35 For in order to detect absolute velocity, there would need to be some physically possible process that, when initiated at t0 to measure the absolute velocity of a given body, will generate a reading— an image on a computer screen, say, or the position of a needle—that indicates what that body’s velocity was at t0. Moreover, the outcome that would be produced if the body were traveling at one velocity must be discernibly different from the outcome that would be produced if it had a different velocity, on pain of our not being able to tell what velocity a given outcome indicates.36 So, if we simply wanted to measure whether a given body was in a state of absolute

35 This argument that follows has received perhaps its clearest written expression in Roberts (2008). I heard similar arguments orally in seminars given by Tim Maudlin at Rutgers and David Albert at Columbia. However, all these theorists run the argument in different ways. My presentation here overlaps with my presentation in Dasgupta (2009) and (2011). 36 At least, that is the ideal: in practice, we do not mind if the outcomes that would be produced by velocities differing only by some tiny amount are indiscernible. More accurately, then, what we require is that the outcomes would be discernible when the

136 | Shamik Dasgupta rest or absolute motion, the process would need to produce one outcome if the body was at rest at t0—for example an inscription of “At rest”—and a discernibly different outcome if the body was moving at t0—an inscription of “Moving,” say. Finally, since the process is a physical process, the outcome produced will depend on the physical laws governing our world. Putting this all together, we can therefore say that absolute velocity is detectable only if there is a physically possible device which at a given time t0 has two properties: first that, according to the laws, it will display “At rest” on a computer screen at a later time t1 iff it was presented with a body at rest at t0; and second that, according to the laws, it will display “Moving” on a computer screen at t1 iff it was presented with a body that was moving at t0.37 But according to most of our best confirmed physical theories, it is physically impossible for a device to have both properties. For suppose I take a device with the first property and present it with a body at rest at t0, and it therefore displays “At Rest” at t1. We can show that it does not have the second property by considering a world W just like ours with the one exception that at all times the absolute velocity of all bodies is five mph greater in a certain specified direction. Now, W is a world in which the device is presented with a moving body at t0, and yet—since the relative positions of all bodies at all times are (by construction) the same in W as they actually are—the device still displays “At rest” at t1. But it turns out that according to our best physics, the laws of motion governing W are the same as those governing our world. Therefore, the behavior of the device in W represents how it behaves according to our laws of motion; hence it does not have the second property listed earlier. QED. To be clear, the argument here does not assume that absolute velocity can only be detected by a device with these two properties. The assumption is rather that if absolute velocity is detectable, then it is detectable with a device with these two properties (even if it is

velocities differ by more than some amount x, in which case we say that the process measures absolute velocity up to an accuracy of x. 37 We use biconditionals here because we not only want each initial velocity to issue in a readable outcome; we also want each outcome to be uniquely associated with that initial velocity so that we know what the outcome indicates.

Absolutism vs Comparativism about Quantity | 137 also detectable in other ways too). The argument then is that it is not detectable with a device of this kind, therefore it is not detectable. I will discuss the assumption further in section 8.5.38 This is not to say that speedometers in cars are useless, for the argument here is consistent with the thesis that they detect the car’s velocity relative to a given body such as the road. All the argument shows is that they do not detect the car’s absolute velocity. And of course if absolute velocity is undetectable, our Occamist principle then recommends (all else being equal) endorsing a theory of spacetime according to which there is no such thing as absolute velocity.39

8.3. Undetectable by us Returning to the case of intrinsic mass, let us suppose for reductio that material bodies have the intrinsic masses posited by the absolutist. For similar reasons to those discussed in the case of velocity, I claim that which particular intrinsic mass each body has would be undetectable.40 38

Thanks to a referee for urging me to clarify the assumption used here. Readers may recognize this as one line of reasoning that can be used to extract metaphysical conclusions (in this case about the structure of spacetime) from facts about the symmetry or invariance groups of the fundamental laws of motion (in this case the relevant symmetry group being uniform velocity boosts). The general idea of using symmetries to motivate various metaphysical views is not new. See Earman (1989) chapter 3, and North (2009) for discussions of this idea with respect to the metaphysics of spacetime. It is also used (explicitly and implicitly) in contemporary discussions of diffeomorphism invariance in General Relativity and permutation invariance in Quantum Mechanics to motivate various “structuralist” metaphysical conclusions (see Ladyman and Ross 2007 for a discussion of both these issues, and see Dasgupta (2011) for a discussion of the former). Where I may differ from others is in my belief that the best such arguments go by way of intermediary conclusions about what is undetectable. That is, the best arguments in my view first use facts about symmetries to draw an intermediary conclusion to the effect that some putative feature is undetectable, and only then use the Occamist principle to draw the metaphysical conclusion that those undetectable features are not real. The question of whether symmetry arguments must take this “epistemic” form is vexed and not one I can discuss here, though I hope to do so in future work. In any event, there is no need to settle the matter here for we can evaluate the argument in the text on its own merits regardless of whether there are other ways of drawing metaphysical conclusions from symmetries. Thanks to a referee for encouraging me to point out the connections between the discussion in the text and these other debates. 40 I should stress that the argument concerns the kind of intrinsic mass as posited by the absolutist; that is, the kind of intrinsic mass the possession of which is not 39

138 | Shamik Dasgupta The claim may initially sound implausible. After all, my laptop and my cup feel different in mass when I pick them up, so am I not thereby detecting the intrinsic mass of each item? Similarly, one might naturally think that an ordinary bathroom scale is a device that allows us to detect the intrinsic mass of the object placed on it by displaying its mass in the position of a needle. But it turns out that this is a mistake: just as speedometers at best allow us to detect the velocity of a car relative to the highway, so too picking things up or putting them on bathroom scales at best allows us to detect the mass relationships between things and not their intrinsic masses. The argument is a little more complex than in the case of velocity, but let us start by running an analogous argument and modify it as required. For convenience, let us suppose that the terms “1 kg” and “2 kgs” label particular intrinsic masses. Following the discussion in the previous section, we can then say that intrinsic mass is detectable only if there is a physically possible device which at a given time t0 has two properties: first, that (according to the laws) it will display “1 kg” on a computer screen at a later time t1 iff it was presented with a 1 kg object at t0; and second, that (according to the laws) it will display “2 kgs” on a computer screen at t1 iff it was presented with a 2 kg object at t0. We now argue that it is physically impossible for a device to have both properties. To this end, suppose I take a device with the first property and present it with a 1 kg object at t0 and it therefore displays “1 kg” on a screen at t1. We show that it does not have the second property by considering a possible world W just like ours with the one exception that everything’s mass is double what it actually is; that is, a world in which our device is presented with a 2 kg object at t0 but in which it nonetheless displays “1 kg” at t0. And we now make a key assumption, namely that the laws of motion governing W are the same as those governing our world. It then follows that the behavior of the device in W represents how it behaves according to our laws of motion; hence it does not have the second property listed earlier. QED.

explicable in terms of mass relationships. As I argued in section 5, the comparativist can agree that there are facts about my laptop’s mass in kilograms such as its being 2 kgs, and perhaps this is a fact about its intrinsic mass in some sense of the term “intrinsic.” But she insists that these facts hold in virtue of facts about mass relationships, so my argument is not directed at these kinds of intrinsic masses.

Absolutism vs Comparativism about Quantity | 139 But is this key assumption correct? As before, let us suppose for simplicity that our best confirmed physical theory states that there is only one law of motion governing our world: f = ma. In section 7 we saw that this equation might express one of two things: a law governing absolute quantities like (L1), or a law governing comparative quantities like (L2). We also saw that if the actual law turns out to be of the second kind, then it would obtain in W too, while if the actual law turns out to be of the former kind, then it would not obtain in W (in the latter case, W does not represent how our device behaves according our laws of motion and it remains open that it has the two properties after all). So the assumption that the laws of W are the same as the actual laws amounts to the assumption that the laws are like (L2) and govern comparative quantities. Can we assume this without begging questions? The assumption does not presuppose the falsity of absolutism, since the hypothesis that the laws govern comparative quantities is consistent with the idea that material bodies also have intrinsic quantities in virtue of which those comparative quantities obtain. But is it reasonable for the comparativist to believe that the assumption is true? Recall that in section 7 we argued that there is no possible evidence that would confirm the hypothesis that laws govern absolute quantities but disconfirm the hypothesis that they govern comparative quantities. Importantly, the argument did not assume that intrinsic mass is undetectable, so we can appeal to the result of that argument without begging questions. So, if the evidence does not settle the matter either way, what is it reasonable to believe? This is a delicate issue in epistemology, concerning what it is reasonable to believe given certain evidence. A “permissive” view of rationality would say that if the evidence equally supports P and not-P, it is reasonable to believe either. On this view, the comparativist may reasonably believe that the actual laws govern comparative quantities, in which case she can run the earlier argument as written. But one might worry that the argument would then be dialectically weak, since on this permissive view it would also be reasonable for the absolutist to believe that the laws govern absolute quantities in which case she will remain unmoved by the argument. Moreover, other views in epistemology would insist that if the evidence equally supports two views, we should remain agnostic between the two rather than believe either.

140 | Shamik Dasgupta Luckily there is no need to settle the issue here. For what the argument two pages earlier shows is that whether or not a device with the two properties listed there is physically possible depends on which kind of laws govern our world. More specifically, the argument shows that a given device can have those two properties only if our laws govern absolute rather than comparative quantities. Therefore, my evidence that a given device has those two properties can be no stronger than my evidence that the laws govern absolute quantities rather than comparative quantities. But the conclusion of section 7 was that there is no possible evidence that the actual laws govern absolute quantities and not comparative quantities. Therefore, regardless of the delicate epistemic question of what it is reasonable to believe given certain evidence, all hands should agree that I can have no evidence that a given device has the two properties listed earlier. And this conclusion is strong enough for our purposes. For even if a given device does in fact have the two properties and gives a reading of “1 kg” when I present it with a material body, the fact remains that if I have no evidence that the device has those two properties, then the reading gives me no evidence as to what the body’s intrinsic mass is. So our conclusion is that even if absolute mass is in some sense detectable by the device, it is not detectable by us, which is the point we were trying to establish all along. The argument here trades on the familiar point that the outcome of a measurement depends on three things: the value of the feature being measured, the initial state of the device being used to measure the feature, and the laws that govern the interaction between the feature and the device. If we know enough about the last two factors, we can use the outcome of the measurement to infer what the value of the feature was. But if we do not know enough about what laws govern our world, then we may not be in a position to make the inference. The argument here is that if we lack evidence as to whether our laws govern absolute or comparative quantities, then we have exactly this kind of handicap when trying to detect which particular intrinsic mass a given body has. 8.4. A second argument Interestingly, one can argue for the same conclusion without relying on the premise that there is no evidence that the actual law governs

Absolutism vs Comparativism about Quantity | 141 absolute quantities rather than comparative quantities. Suppose, perhaps per impossible, that we were to acquire evidence that the actual law governs absolute quantities and that devices with the two properties listed earlier are therefore physically possible. One can argue that there would still be some evidential uncertainty as to the details of what that law is in such a way that we could never acquire evidence that a given device has the two properties. To see this, suppose we are given a device and are asked to determine whether it has the two properties listed earlier. One of those properties was that it will behave in a certain way according to the laws governing it, namely it will register “1 kg” on a computer screen at t1 iff it is presented with a 1 kg object at t0. Now suppose that, in fact, the device in front of us will behave like that according to (L1). And suppose that (L1) is the actual law, so the device really does have the property. Now (L1) does not obtain in W but something closely related does, namely the result of replacing all occurrences of “M” with “2M” (where this latter term refers to the mass that is double that of M). Call this law (L1*). Then what W shows is that if everything’s mass were uniformly doubled, then, according to (L1*), the device will not behave in the same way. Rather, it will register “1 kg”’ on a computer screen at a later time t1 iff it is presented with a 2 kg object at t0, rather than with a 1 kg object. So whether or not the device in front of us has the required property depends on which of two hypotheses is true: hypothesis H, which attributes to the device its actual mass and states that (L1) obtains, or H*, which attributes to the device the mass it has in W and states that (L1*) obtains. But one can argue that no evidence could possibly favor either hypothesis over the other. Let us assume that W is indiscernible from the actual world, in the sense that everything in W looks and feels and smells and tastes just like it does in the actual world (I will discuss this assumption in a moment). Then it follows that no empirical evidence would falsify either hypothesis. And (L1) and (L1*) are of exactly the same form: both are equally simple, elegant, unifying, and explanatory. So neither hypothesis trumps the other on any theoretical virtue we use to choose between hypotheses that agree on the empirical data. But the device in front of us has the required property only if hypothesis H is true. Therefore, since there can be no evidence that would favor H over H*, there can be no evidence that the device has the required property. And as we just

142 | Shamik Dasgupta saw when discussing the second strategy, this means that even if a given device does in fact have that property, the readings it delivers will give us no evidence as to what a body’s intrinsic mass is and therefore intrinsic mass will remain undetectable by us.41 Admittedly, this line of argument is rather more involved than the one I initially outlined. But still, some comparativists might be interested in developing it.

8.5. Indiscernibility We have assumed that if intrinsic mass is detectable, it is detectable by processes that indicate mass with inscriptions of “1 kg” and “2 kgs.”42 This is not to deny that it might be detectable in other ways too; our assumption was just that if it is detectable at all, it must at least be detectable in this way. Our strategy was then to argue that intrinsic mass is not detectable in this way; hence it is not detectable at all. How plausible is the assumption? Borrowing an idea of Albert’s, the comparativist might argue that it should be extremely plausible. For given anything D that counts as an intrinsic mass detector, it is presumably possible for us to decide in advance to record the result of the measurement produced by D by writing “1 kg” or “2 kgs” on a piece of paper depending on what the result is. If so, then the result of coupling D with our decision to record the outcome of D in that way constitutes a composite device that detects mass with the inscriptions “1 kg” and “2 kg,” and our assumption is vindicated.43 41 This argument appeals to the assumption that the actual world and W are indiscernible, and one might worry that this begs the question by assuming that absolute mass is undetectable. But the worry is misplaced. At most, the indiscernibility of the two worlds just shows that absolute mass is invisible to the naked eye, but it remains open that it is possible to build measuring devices that might reveal them to us. The fact that the worlds have such similar laws is then being used to show that no such device is possible. 42 In full, remember, we assumed that intrinsic mass is detectable only if there is a physically possible device which at a given time t0 has two properties: first, that (according to the laws) it will display “1 kg” on a computer screen at a later time t1 iff it was presented with a 1 kg object at t0; and second, that (according to the laws) it will display “2 kgs” on a computer screen at t1 iff it was presented with a 2 kg object at t0. 43 See Albert (manuscript) for more on this way of thinking about detectability.

Absolutism vs Comparativism about Quantity | 143 Still, one might try to resist the assumption. For one might try to argue that there are physically possible processes by which something’s intrinsic mass can have a discernible effect on the qualitative character of our experience, and that we can therefore detect which intrinsic mass a given body has by noticing what kind of qualitative experience we enjoy at the end of the measurement process. According to this objection, my argument so far only shows is that it would then be physically impossible for us to record the result of the measurement by writing “1 kg” or “2 kgs” on a piece of paper depending on what the result is. In response, the comparativist might emphasize that the scenario being envisaged is extremely implausible. For it seems compelling that whenever I am able to enjoy two discernibly different qualitative states, I am also able to produce some bodily movement— inscribing “1 kg,” say, or putting my arm in the air—in response to the one experience and not the other. The objection must therefore deny what we take to be an obvious fact about our mental life. But there are two further responses that the comparativist might want to explore. One is to deny that differences in intrinsic mass have any effect on the qualitative character of our experience. To argue this, the comparativist might argue that the doubled world W would be indiscernible from the actual world, in the sense that everything would look and feel and taste and smell exactly the same as it actually does. But is this true? The question is subtle. In the case of velocity, we have empirical evidence that a boosted world would be indiscernible from the actual world, based on our experiences in trains. For we have experienced reasonably small environments that are to some extent isolated from external interference and have noticed that they look the same while in motion as they do at rest in the station. Now this evidence is not conclusive, since it remains open that if everything were put in smooth motion, there would be some discernible difference in the qualitative nature of our experiences (perhaps our visual field would be tinged with yellow). But it seems that our best theory of what determines the character of our conscious states implies that the subject’s state of absolute motion is not a relevant factor. Now in the case of mass we lack the same kind of empirical evidence that the doubled world W is indiscernible from the actual world, since we have had no experience of reasonably isolated

144 | Shamik Dasgupta environments that differ only in a doubling of mass. Nonetheless, there is some reason to believe that W would be indiscernible, for our best theory of what determines the character of our conscious states seems to imply that intrinsic masses are not a relevant factor. Insofar as one’s conscious life is determined by physical facts at all, it seems to be determined by the positions of particles composing one’s brain (and perhaps the local environment). When a headache pill cures your pain, that is because it altered the positions of various particles in your brain, not because it made any of those particles more massive. Since W agrees with the actual world in all facts about particle positions, there is some reason to think it is indiscernible after all. A full discussion of this issue would take us too far into the philosophy of mind. But in any case the issue is not crucial, for a third response to the initial objection is to argue that even if intrinsic mass has an effect on the character of our experience, it would still be undetectable! To see this, suppose that the effect that intrinsic mass has on our experience means that our visual field would be tinged with yellow in W. Does this mean that we can infer, on the basis of the character of our visual field, which intrinsic mass each thing has? It does not, for to make that inference we would need to appeal to a hypothesis stating which intrinsic masses give rise to which sort of experience. But the hypothesis that one set of intrinsic masses (i.e. those instantiated in the actual world) give rise to my actual experiences and the hypothesis that another set of intrinsic masses (i.e. those instantiated in W) give rise to my actual experiences are both left open by my actual experience, and are both equally simple, elegant, explanatory, and so on; and so the fact that I am enjoying my actual experiences is not evidence for either hypothesis over the other! As a result, my enjoying these experiences does not put me in a position to infer which particular mass any given body has. So here are three responses available to the comparativist. I will not try to assess which response is most plausible, but I should emphasize that one’s choice of response is likely to affect the extent to which the argument will generalize to other quantities. For example, if the comparativist makes the second response, resting her case on the fact that W is indiscernible, then the argument will only generalize to a given quantity if uniform transformations of its absolute

Absolutism vs Comparativism about Quantity | 145 values while leaving the comparative values fixed result in an indiscernible world. And if the comparativist makes the third response, resting her case on the fact that the experiential difference in W leaves open which intrinsic masses are instantiated, then the argument will only generalize to a given quantity if the experiential difference in those transformed worlds still leaves open which absolute values are instantiated. 8.6. Inexpressible ignorance This, then, is my reason for preferring comparativism, at least in the case of mass. If material bodies had the intrinsic masses posited by the absolutist, those masses would be undetectable. Our Occamist principle states that all else being near enough equal, this is a mark against absolutism and in favor of comparativism. I argued in sections 2–7 that there are no decisive objections to comparatativism, so all else appears near enough equal. This Occamist argument rests on the premise that the intrinsic masses posited by the absolutist would be undetectable, but there is a sense in which the state of ignorance we would be in with regards to which intrinsic mass each body has would be inexpressible. For how could we express it? If I said that I do not know whether everything is twice as massive as it actually is, I would be mistaken since I know very well they are not! The trouble is that I just described the non-actual possibility W in such a way that I can infer from my description that it is non-actual. To remedy this, I could try giving each absolute mass a name and describing the non-actual possibility in those terms. For example, suppose that Kripke was right that we use terms of the form “n kgs” with the reference-fixing stipulation that it is to refer to the absolute mass that is n times that had by the standard kilogram in Paris. Since W differs in the mass in kilograms of my laptop, I could try saying that I do not know whether my laptop is 2 kgs or 4 kgs. But this is not clearly right either. By discovering that my laptop is twice as massive as the standard kilogram, and then appealing to the reference-fixing stipulation, I can infer that the sentence “My laptop is 2 kgs” is true. And while knowing that a sentence is true does not imply knowing the proposition it expresses, many theorists would say that in this

146 | Shamik Dasgupta case I would know that my laptop is 2 kgs. So this is the sense in which my ignorance is inexpressible: there is a sense in which I do not know which absolute mass my laptop has, but there is no sentence s that expresses its absolute mass for which I can truly say “I do not know that s.” Those who identify “knowing which” with “knowing that” might now conclude that I am not ignorant of which intrinsic mass my laptop has after all, but this would be to let theoretical opinion obscure the phenomena. For earlier I described a clear and vivid sense in which the particular intrinsic mass had by my laptop lies beyond our epistemic grasp, and we can naturally express this by saying that we do not “know which” intrinsic mass it has in one sense of that term. If this state of ignorance cannot be analyzed in terms of “knowing that,” so be it: we would be guilty of theoretical prejudice if we concluded that there is no such state at all. This is not to deny that there is an important question as to what the state consists in, but that is a question for another time.44

9. CONCLUSION The question of absolutism vs comparativism has received relatively little discussion, and I consider this a significant lacuna in our understanding of what the natural world fundamentally consists in. In this paper, I have tried to clarify what the issue amounts to and describe where I see the major battle lines as lying. I believe that comparativism is probably the correct view for mass, but if I

44 In this respect the case of absolute mass diverges from the case of absolute velocity, for our ignorance in the latter case is expressible: I can truly say “I do not know whether I am at rest.” Indeed, the case of absolute mass is more akin to the case of absolute location in space. For if there were such a thing as absolute space, then worlds that differ only in a uniform shift of all matter three feet to the right would look and feel and smell the same, and as in the case of velocity we can argue that no physically possible measuring device could reveal which particular region of space we are in. For this reason, our position in absolute space is undetectable. Still, as Maudlin (1993) points out, there is nothing I can say to express what I am ignorant of, for it is clearly false to say that I cannot not know whether I am here or three feet to the right of here! In this regard I agree with Maudlin entirely. But he went on to argue that there is no sense at all in which I am ignorant of my location in space, and here I believe that he made the mistake described in the last paragraph of allowing theoretical prejudice to obscure the phenomena. I give some reasons to reject Maudlin’s view in my (2011).

Absolutism vs Comparativism about Quantity | 147 have not convinced you of that I hope to have shown that the issue is important and that there is interesting further work to do in the area.45 Princeton University REFERENCES Albert, D. Manuscript. “The Technique of Significables.” Armstrong, D. 1978. A Theory of Universals: Volume 2. Cambridge: Cambridge University Press. Armstrong, D. 1988. “Are Quantities Relations? A Reply to Bigelow and Pargetter.” Philosophical Studies 54: 305–316. Baker, D. Manuscript a. “Some Consequences of Physics for the Comparative Metaphysics of Quantity.” Baker, D. Manuscript b. “Comparativism with Mixed Relations”. Bigelow, J. and R. Pargetter 1988. “Quantities.” Philosophical Studies 54: 287–316. Bigelow, J. and R. Pargetter 1990. Science and Necessity. Cambridge: Cambridge University Press. Chalmers, D. J. 2012. Constructing the World. Oxford: Oxford University Press. Dasgupta, S. 2009. “Individuals: An Essay in Revisionary Metaphysics.” Philosophical Studies 145: 35–67. Dasgupta, S. 2011. “The Bare Necessities.” Philosophical Perspectives 25: 115–60. Dasgupta, S. Manuscript. “On the Plurality of Grounds.” Earman, J. 1989. World Enough and Space-Time. Cambridge, MA: MIT Press. Eddon, M. 2013. “Fundamental Properties of Fundamental Properties.” This volume, pp. 78–104. Ellis, B. 1966. Basic Concepts of Measurement. Cambridge: Cambridge University Press. Field, H. 1980. Science without Numbers. Princeton, NJ: Princeton University Press.

45 Parts of this material were presented in November 2010 at California State University Los Angeles, and in January 2011 at Leeds University and the University of Oxford. Thanks to members of these audiences for all their helpful feedback. Thanks also to David Baker, Alexi Burgess, Kenny Easwaran, Maya Eddon, Kit Fine, Dustin Locke, Michaela McSweeney, Eliot Michaelson, David Plunkett, and three referees for extremely helpful comments on earlier drafts of this paper.

148 | Shamik Dasgupta Field, H. 1985. “Can We Dispense With Space-Time?” PSA 2: 33–90. Fine, K. 2001. “The Question of Realism.” Philosopher’s Imprint 1 (1): 1–30. Fine, K. 2012. “Guide to Ground.” In F. Correia and B. Schnieder (eds), Metaphysical Grounding: Understanding the Structure of Reality. Cambridge: Cambridge University Press. Hawthorne, J. 2006. “Quantity in Lewisian Metaphysics.” In John Hawthorne, Metaphysical Essays. Oxford: Clarendon, pp. 229–38. Kripke, S. 1972. Naming and Necessity. Oxford: Blackwell. Ladyman, J. and D. Ross. 2007. Every Thing Must Go: Metaphysics Naturalized. Oxford: Oxford University Press. Lewis, D. 1986a. On the Plurality of Worlds. Oxford: Blackwell. Lewis, D. 1986b. Philosophical Papers: Volume 2. Oxford: Oxford University Press. Maudlin, T. 1993. “Buckets of Water and Waves of Space: Why Spacetime Is Probably a Substance.” Philosophy of Science 60 (2): 183–203. Mundy, B. 1987. “The Metaphysics of Quantity.” Philosophical Studies 51: 29–54. North, J. 2009. “The ‘Structure’ of Physics: A Case Study.” Journal of Philosophy 106: 57–88. Roberts, J. 2008. “A Puzzle about Laws, Symmetries and Measurable Quantities.” British Journal for the Philosophy of Science 59: 143–68. Rosen, G. 2010. “Metaphysical Dependence: Grounding and Reduction.” In R. Hale and A. Hoffman (eds), Modality: Metaphysics, Logic, and Epistemology. Oxford: Oxford University Press, pp. 109–36.

ONTOLOGICAL COMMITMENTS: WORDS AND SLOTS

4. Modal Quantification without Worlds Billy Dunaway This paper is about avoiding commitment to an ontology of possible worlds with two primitives: a hyperintensional connective like ‘in virtue of,’ and primitive quantification into predicate position. I argue that these tools, which some believe can be independently motivated, render dispensable the ontology of possible worlds needed by traditional analyses of modality. They also shed new light on the notion of truth-at-a-world.

1. PRELIMINARIES: REDUCTIVE ANALYSES OF MODALITY AND QUANTIFICATION David Lewis’s familiar “modal realism” gives an account of the modal operators ◻ and ◇ (understood as expressing metaphysical necessity and possibility, respectively) that takes on additional ontological commitments to achieve a reduction in ideological complexity. It involves, as Lewis says, a trade-off: “[i]t offers an improvement in what Quine calls ideology, paid for in the coin of ontology.”1 For Lewis, the extra ontology consists in a multitude of maximal isolated regions of spacetime,2 which he calls “worlds.” These feature in the following quantificational analysis of ‘◇p ’ (where the quantifier ∃w ranges over the domain of Lewisian worlds):

1 Lewis (1986: 4). Lewis explicitly offers this quote as a description of the justification for the set-theoretic axioms and their accompanying ontology. (In this context, he goes on: “It’s an offer you can’t refuse. The price is right; the benefits in theoretical unity and economy are well worth the entities.”) But his point in describing the settheoretic axioms in this way is that the justification for an ontology of possible worlds is similar, though presumably Lewis doesn’t think that it is as obvious that in the modal case the trade-off is worthwhile. 2 Lewis (1986: 70–2).

152 | Billy Dunaway 1.

∃w: p is true at w.

The advantages of such an analysis go beyond the elimination of ◻ and ◇ from our primitive ideology; with an ontology of worlds in place, Lewis goes on to assemble an impressive array of further explanatory tasks for them.3 Many philosophers, however, have been unwilling to agree with Lewis that the ontological cost is justified by the promised elimination of modal idioms. They prefer to complicate their ideology slightly by holding that worlds are not maximal isolated regions of spacetime as Lewis describes them, but are rather entities that have primitive modal features. ‘◇p’ is still analyzed in terms of (1), but now with the notion of a possible world left either as primitive, or analyzed partly in modal terms.4 Many of the explanatory benefits of a possible worlds analysis depend not on the Lewisian reductive project, but rather on its quantificational character. Formulations of global supervenience and analyses of properties, semantic content, and counterfactuals can all be carried out with non-Lewisian worlds in place—we don’t lose all of the explanatory benefits of an ontology of possible worlds simply by retaining ‘possible’ in our primitive ideology.5 Many of Lewis’s explanatory insights, in other words, rely on the existence of a certain kind of entity to quantify over, and do not rely on the further reductive project of giving an account of these

3

These are found in Lewis (1986: ch. 1). Examples include Adams (1974), Stalnaker (1976), Plantinga (1978), and Soames (2007). Even though these views are not fully reductive, there are differences between them, primarily in what they take worlds to be. I discuss some of these differences in section 3, on truth-at-a-world. 5 There are exceptions: since Lewis can allow that (unrestricted) objectual quantifiers range over possible objects, there are analyses that quantify over these objects (and not merely worlds, Lewisian or otherwise) which are not available to the nonreductivist. Eli Hirsch (1997) points out that the “imperfect communities” from Goodman (1951) fail to qualify as counterexamples to the thesis that the natural properties are the resemblance-conferring properties, as the resemblance-conferring properties are those properties P such that: 4

For any x and y, if x has P and y lacks P, there is a z such that z has P and, for any w, if w has P and w is at least as similar to y as z is, then w is more similar to x than y is. (Hirsch (1997: 51))

Those who are not Lewisian realists cannot use mere quantification over worlds to get the same result—the definition essentially relies on the possibility of cross-world comparison. See also Lewis (1986: 13–14) for more on cross-world comparison.

Modal Quantification without Worlds | 153 entities in wholly non-modal terms. All that matters is that we don’t leave the modal operators unanalyzed.6 This brings us to the starting point of the present paper. The quantificational analyses we have mentioned so far all employ quantifiers that bind variables in nominal position; i.e. the bound variables occur in a syntactic position which can be occupied by names, or other referential expressions.7 The quantifiers are interpreted in the familiar way, “ranging over” a domain of objects (in this case, worlds). But there is growing interest in a proposal from Arthur Prior (1971) which holds, to a rough approximation, that we can include second-order quantifiers among our primitive ideology. These devices are second-order simply in the sense that they bind variables in predicate-position, as ∃F does in (2): 2.

∃F: a is F.

It is worth stressing that there may be other senses of ‘secondorder’ that I am not appealing to here—one might think, for instance, that the term designates quantifiers ranging over sets of objects, or quantifiers interpreted as plural quantifiers. These are different meanings for the term ‘second-order’ than the one I am interested in here, since they settle by definition questions about the proper interpretation of quantification into predicate position—questions that I wish to leave open to substantive argumentation. The designation ‘second-order,’ as I will use it, is a purely syntactic one, referring only to the position of bound variables. This non-semantic characterization is crucial, because in what follows I will take seriously the substantive thesis that second-order quantifiers can be taken as primitive. Such an understanding requires rejecting the idea that (2) is really shorthand for something like (3) or (4): 3. ∃x: x is a property and a instantiates x; 4. ∃s: s is a set and a is a member of s. 6 For a development of this latter kind of view, see Forbes (1989: 78), who holds that “modal operators provide the fundamental means of expression of modal facts.” See Melia (1992) for more discussion of this view. 7 I.e. they have the following syntactic feature: given a well-formed quantified sentence of the form ‘∃x: . . . x . . . ,’ replacing the bound occurrence of the variable with a proper name like ‘Sally’ and removing the quantified expression results in the wellformed sentence ‘. . . Sally . . . .’

154 | Billy Dunaway Indeed, according to the primitive understanding, any interpretation of second-order quantifiers that interprets them as shorthand for quantifiers that bind variables in nominal position (such as (3) or (4)) is unnecessary.8 They are primitive because they require no further analysis. I will postpone until section 2 consideration of reasons for thinking that (2) does not require analysis in terms of nominal quantification. But here is a preview of what second-order quantification thus understood might buy for us in the metaphysics of modality. Primitive second-order quantificational expressions are not, I will suggest, ontologically committing: (2), taken as primitive, is committed to the existence of the thing a names, and nothing else. This suggests the following possibility. We could replace the quantifier in (1) with a primitive second-order quantifier; this amounts (very roughly) to analyzing ‘◇p’ in terms of the second-order equivalent of a sentence which says that there is some possible way for things to be, W, and things’ being W entails p. Since encoding this kind of analysis in a sentence with a genuine second-order quantifier is not ontologically committing, we would appear to be in a position to give an analysis of ◻ and ◇ that is free of ontological commitments to worlds, but still has the benefits of familiar quantificational analyses. Working out the details, however, will show that things are not so simple—we will need to help ourselves to further resources that analyses which quantify into nominal position only are not committed to. This is because the notion of truth-at-a-world appears in (1), and most of the ordinary accounts which accept that there are possible worlds (of some kind or other) have the resources to say what truth-at-a-world is. But since second-order quantifiers bind variables in different syntactic positions, we are faced with the task of understanding constructions of the form “is true at F,” where 8 The “plural” analysis of second-order quantifiers in Boolos (1984) also quantifies into a nominal position (with a plural quantifier). Boolos suggests that

∃X: Xa is to be analyzed as There are some things such that a is one of them. The variable (‘them’) bound by the quantifier in the analysis is in nominal position, as it can be replaced with a (plural) noun, like ‘the Italians’ or ‘the hoarders.’ This kind of analysis is then among those we rule out when we say that second-order quantifiers are primitive. For more discussion, see Rayo and Yablo (2001).

Modal Quantification without Worlds | 155 grammatical instances are obtained by replacing F with a predicate. I will show how we can understand such constructions in a way that helps the second-order analysis. But it will require allowing something like a primitive ‘in virtue of’ in our ideology, which is another controversial resource, although one many metaphysicians think we need anyway. Thus, the main goal in what follows will be to show how those who have already complicated their ideology with primitive second-order quantification and a hyperintensional ‘in virtue of’-like connective can avoid further ontological commitment in their analysis of ◻ and ◇, while still reaping all the benefits of a quantificational analysis. As I introduce these additional resources in greater detail, I will give some indication of why we might think that they are legitimate and perhaps necessary. Arguments for these positions have been presented in greater detail elsewhere, however; my primary aim is to show what we can do in the theory of modality once these resources are in place.

2. PRIMITIVE SECOND-ORDER QUANTIFICATION As we noted in the previous section, second-order quantifiers, which bind variables in predicate position, are primitive when they are not analyzed in terms of sentences containing nominal quantification only. Presumably Quine had a nominal analysis in mind when he accused second-order quantification of being “set theory in sheep’s clothing”: he thought 2.

∃F: a is F

must quantify, in the final analysis, over sets.9 Quantification over sets in 4.

∃s: s is a set and a is a member of s

evidently quantifies into a nominal position, as grammatical instances of (4) result from dropping the quantifier ∃s and substituting referring expressions like ‘Sally’ or ‘the set of all red things’ for the variable s. And substituting a predicate like ‘is red’ for s does not produce a grammatical instance. By taking sentences like (2) instead as primitive, we forgo the possibility of explaining what the 9

Quine (1986).

156 | Billy Dunaway sentence means by invoking nominal quantification in this way. Not only is the Quinean analysis unavailable; any analysis that proceeds by specifying a domain of objects over which the quantifier ranges, plus a relation the referent of a bears to some members of that domain, is inadmissible as an explanation of (2). Some will think that, given this gloss on what primitive secondorder quantification is, it cannot even be intelligible. One reason, which we should confront at the outset, is that this is due to constraints on what can count as quantification. This line of thought might be expanded by imagining someone who makes the following series of claims: “if primitive second-order quantifiers don’t range over any objects, they can’t be genuine quantifiers. A construction can’t be quantificational unless it quantifies over some things. Primitive second-order quantifiers by definition don’t range over anything. So there is something incoherent in the notion of primitive second-order quantification.” The response to this type of complaint is to give the objector the word ‘quantification.’ What is important to the view is that second-order expressions like ∃F in (2) don’t require nominal interpretations. What is not essential is that, in addition, these expressions are called the same thing as the familiar constructions which bind variables in nominal positions. We can replace the conventional quantifier expressions ∃ and ∀ with alternative notation, and call them something else.10 What does matter is simply that there are intelligible constructions that obey quantifier-like introduction and elimination rules and serve the quantifier-like purpose of allowing us to speak more generally. Just as nominal quantifiers allow us to speak more generally than our practice of naming and referring 10 Here is one suggestion for a new name. Predicates plausibly don’t introduce new ontological commitments (see the end of this section for more discussion), but they can still be said to qualify how things are—the sentence ‘a and b are both red’ constrains how things are, even if it introduces no entities beyond the referents of a and b. It carries a new metaphysical commitment, though not a new commitment in ontology (see Sider (2012: 96) for related discussion). Sentences involving primitive second-order quantifiers impose similar qualifications of how things are, albeit in a more general way. ‘∃F: a and b are both F’ doesn’t require that a and b are both red, but it does impose some restrictions on how a and b are qualified. We could then label the constructions that bind predicate-position variables in (2) as ‘qualifiers’ and name the operation they perform ‘qualification’ to distinguish it from quantification, understood to be restricted to nominal quantification. Qualifiers, like predicates, would carry metaphysical commitments, but introduce no new ontology. Thanks to David Manley for the terminological suggestion.

Modal Quantification without Worlds | 157 typically allows, the “quantifiers” binding variables in predicateposition are similarly tools for speaking more generally (though in this instance the devices allow us to speak more generally than we typically speak by using predicates). Of course, there are legitimate questions about whether we really can understand primitive second-order quantifiers, about whether they are ontologically committing, and whether we need to complicate our ideology with them. I will address these questions in turn.

2.1. Second-order home languages Peter van Inwagen (2004) is one philosopher who thinks that second-order quantifiers, taken as primitive, are unintelligible. He says: “[q]uantification into non-nominal positions is meaningless unless (a) the non-nominal quantifiers are understood substitutionally . . . ; or, (b) it is understood as a kind of shorthand for nominal quantification over properties, taken together with a two-place predicate (corresponding to the ‘ε’ of set-theory) along the lines of ‘x has y’ or ‘x exemplifies y.’”11 The worry, in short, is that we can’t understand second-order expressions unless they are analyzed in nominal terms. But Rayo and Yablo (2001) suggest that another means to understanding these sentences shouldn’t be very hard to come by, as English contains non-nominal quantificational expressions, and these English idioms can serve in translations of a formal language with second-order quantification. Take a sentence with quantification into predicate-position, as in (5): 5.

∃X: John is X.

Rayo and Yablo point out that there is a reading of (6), a sentence of ordinary English, on which it is equivalent to (5) and yet does not contain a nominal quantifier: 6.

John is something.

On the relevant (and most natural) reading, ‘something’ in (6) occurs in predicate position, as replacing it with ‘kind’ results in an instance that is grammatical in English, while replacing it with a name like ‘Steve’ is ungrammatical. (Thus ‘is’ in (6) on this reading 11

van Inwagen (2004: 124).

158 | Billy Dunaway is the ‘is’ of predication.) This is perhaps more apparent when ‘something’ is fronted to obtain the sentence ‘there is something John is.’ It seems much more natural to say ‘there is something John is—kind’ than to say ‘there is something John is—Steve.’ So (6) can naturally be read as an English-language translation of (5) that doesn’t resort to nominal quantification. The point isn’t limited to simple sentences like (5): more complex formal sentences with second-order quantification can be rendered in ordinary English by using ‘that’ and ‘so’ in predicate-position in a manner resembling bound variables. The sentences 7. ∃X: John is X and Sally is X; 8. ∃R: San Francisco Rs Los Angeles and Boston Rs New York. have the following English translations: 9. John is something and Sally is that too; 10. San Francisco is somehow related to Los Angeles and Boston is so related to New York.12 There is then a good case that English—a language with which we are already competent—provides the tools to understand sentences with second-order quantification, and to do so without translating them into sentences that contain only quantification into nominal positions. (Thus, even though this approach provides a translation of second-order sentences, they are still primitive in an important sense, as the translations themselves contain second-order expressions.) This doesn’t show that the Rayo and Yablo approach yields a complete understanding of second-order quantifiers by providing translations of occurrences in arbitrarily complex constructions.13 But it isn’t clear that we need translations for every such sentence: in the event that translation into ordinary English is an inadequate tool for acquiring complete understanding, we might treat natural language as a means to acquiring a beginner’s grasp of sentences with quantification into predicate-position. We can

12 Rayo and Yablo (2001: 80–5). A compositional translation scheme into regimented English is presented on p. 84. 13 Manley (2009: 401–2) suggests that ‘∀X∃Y(Xa→Yb)’ is one sentence which has no translation into the second-order idioms of English.

Modal Quantification without Worlds | 159 then move on to the “direct method,” as Timothy Williamson explains: We may have to learn second-order languages by the direct method, not by translating them into a language with which we are already familiar. After all, that may well be how we come to understand other symbols in contemporary logic, such as ⊃ and ◇: we can approximate them by ‘if’ and ‘possibly’, but for familiar reasons they may fall short of perfect synonymy, and we certainly do not employ ⊃ and ◇ as synonyms for the complex discourses in which we explain how they differ subtly in meaning from ‘if’ and ‘possibly’. At some point, we learn to understand the symbols directly; why not use the same method for ∀F? We must learn to use higher-order languages as our home language.14

With Rayo and Yablo’s natural language translations in hand, plus Williamson’s direct method, we should be optimistic that we can understand primitive second-order constructions. Van Inwagen’s claim that we can’t understand them without translations into nominal quantification begins to look unpromising.

2.2. Ontological commitment The next issue concerns the ontology required by second-order quantificational constructions taken as primitive. Those who claim that they can understand what these expressions mean typically hold that they carry no ontological commitments to additional objects. I will canvass some arguments in the literature for this conclusion, but let us first note how this is not at all surprising once we take the contrast between the semantics for ordinary nominal quantifiers and primitive second-order quantifiers seriously. When second-order quantifiers are taken as primitive, in saying what sentences like ‘∃F: a is F’ mean, we use another second-order quantifier. (Whether it is an English construction like “somehow” or ‘something’ or the homophonic formal expression ∃F will depend on whether we are following a Rayo and Yablo-like translation manual or Williamson’s direct method.) Thus at no point in our gloss of sentences with second-order quantifiers do we refer to or quantify over entities in virtue of the presence of the second-order

14

Williamson (2003: 459).

160 | Billy Dunaway construction; any semantic unpacking will not reveal commitments that were hidden in the surface form.15 Ordinary quantification into the nominal position by sentences like ‘∃x: x is G,’ by contrast, does require the existence of something in virtue of the semantics for ∃x. This is revealed in the standard semantics for the quantifier, where an entity which satisfies G must be a member of the domain ∃x ranges over in order for the sentence to be true. In short, the existence of things cannot be read off from the meaning of primitive second-order quantifiers in the way it can be read off from the standard semantics for nominal quantifiers. The viability of this position depends in large part on a related position concerning the ontological commitments of ordinary predicates. Many philosophers, including Quine and others who do not count as friends of primitive second-order quantification, hold that predication by itself introduces no new ontological commitments. Quine, for instance, says [T]he word ‘red’ . . . is true of each of sundry individual entities which are red houses, red roses, red sunsets; but there is not, in addition, any entity whatever, individual or otherwise, which is named by the word ‘redness’.16

Against this background, Van Cleve (1994) notes: It would be extremely surprising if it were the need to speak generally that first ushered in universals. Could one hold that the specific predication 11.

Tom is tall

makes no commitment to universals, but that as soon as we are forced to generalize and say 12.

∃F Tom is F

we do recognize the existence of universals? That seems highly unlikely. If the existentially quantified formula (12) is legitimate at all, it follows

15

As an example of “semantic unpacking” that does reveal hidden commitments, consider the approaches to ‘∃F: a is F’ that offer nominal analyses in terms of quantification over (for example) sets. Since the superficially second-order sentence claims on this analysis that there is a set that a is a member of, it carries hidden ontological commitments, namely a commitment to the existence of sets. This commitmentinducing analysis disappears when the sentence is taken as primitive; on this view, the only answer to the question “What does ‘∃F: a is F’ mean?” is “That ∃F: a is F.” 16 Quine (1953a: 10). See also van Inwagen (2004).

Modal Quantification without Worlds | 161 from (11), and cannot reveal any ontological commitment not already inherent in (11).17

The Quinean position on the ontological commitments of ordinary predication, which Van Cleve takes as a starting point, is not universally accepted. A certain kind of realist about properties holds that they are needed for explaining predication; that is, predications of the form ‘a is G’ hold because some more basic fact holds— namely, the fact that a instantiates G-ness. This is denied by “Ostrich Nominalists” who, like Quine, hold that there is no need to explain predication at all. That something is red is basic; the predication stands in no need of further explanation.18 The friend of primitive second-order quantification adopts a position that is very much at home given Ostrich Nominalism, but is difficult to reconcile with its denial. In particular, the Ostrich Nominalist holds that a surface-level predication is fine as it is, unanalyzed, while her opponent insists that what it is to predicate G of an entity needs to be explained in terms of a relation (instantiation, say) that entity bears to abstracta of some kind (e.g. the property G-ness). The question of whether second-order quantification is to be taken as primitive is similarly the question of whether ‘∃F: a is F‘ is fine as it is, unanalyzed, or if it needs to be unpacked in terms of a sentence of the form 13.

∃x: a Rs x,

where R in (13) expresses a relation like instantiation and the quantifier ranges over properties.

17 Van Cleve (1994: 587), my numbering. See also Rayo and Yablo (2001: 79–80) for a similar point, as well as Wright (2007): “[S]tatements resulting from quantification into places occupied by expressions of a certain determinate syntactic type need not and should not be conceived as introducing a type of ontological commitment not already involved in the truth of statements configuring expressions of that type” (Wright (2007: 159)). 18 Armstrong (1978) advocates for the realist view. Van Cleve (1994) explores the Ostrich Nominalist view (including the possibility of invoking primitive secondorder quantifiers to handle difficulties) which is also defended in Devitt (1980). It is worth noting in this context that Ostrich Nominalists need not be nominalists, full stop; they just need to hold that ontological commitment to properties doesn’t derive from the need to explain predication (since they think there is no such need). They can still hold that there are other needs which introduce the commitment.

162 | Billy Dunaway As the quote from Van Cleve suggests, it is very difficult to see why quantification into predicate-position should be ontologically committing in the way (13) is, given Ostrich Nominalism—if the simple predication doesn’t need to be analyzed in terms of properties, then second-order generalizations likewise can be free of such an analysis. If, on the other hand, predication by itself does bring in commitments to properties, then it is very natural to think that second-order quantification would be ontologically committing, ranging over a domain of properties that explain the truth of predications. We haven’t said anything to settle the debate in favor of Ostrich Nominalism here, but it is a familiar position, and one which makes it very natural to accept that primitive second-order quantifiers carry no ontological commitments.

2.3. Expressive needs The previous two subsections detail reasons for thinking that primitive second-order quantifiers are intelligible and ontologically noncommitting. But, as we noted in section 1, using them in metaphysical theorizing would require a complication of our ideology, and it is worth asking whether there is any reason to do this. There are arguments in the literature that such reasons do exist. According to these arguments, devices of primitive second-order quantification provide the means for expressing important facts in some domain or other: they are well suited, we can say, to meet our expressive needs. Rayo and Yablo (2001: 82) point out that there are important differences between the following two sentences, as (14) is demonstrably false, while (15) is “on one reading quite true”: 14.

Take any objects you like, there’s an object containing them and nothing else; 15. Take any objects you like, they are something that the rest of the objects are not. Of course, a nominal analysis of ‘something’ in (15) on which it quantifies over sets would render it false for the same reasons that (14) is false. But by treating the ‘something’ in (15) as an unanalyzed (and hence non-committing) quantifier into the predicate position, it is consistent to give (14) and (15) different truth values.

Modal Quantification without Worlds | 163 Williamson (2003) draws a similar conclusion in the face of Russell-like paradoxes that result from defining validity and logical consequence in terms of truth under all interpretations, and allowing interpretations to be among the objects that our quantifiers range over.19 His preferred solution is to go second-order: The underlying assumption is that generalizing always amounts to generalizing into name position, that all quantification in the end reduces to firstorder quantification. But that Quinean assumption is not forced on us . . . It is therefore more natural . . . to think of subscript position in [“is trueI” and “is trueJ”] as predicate position rather than name position . . . In defining logical consequence, we generalize into predicate position in a second- (or higher-)order meta-language. We reject the question ‘What are we to generalize over?’ because inserting a predicate in the blank in ‘We are to generalize over . . .’ produces an ill-formed string.

Primitive second-order quantifiers, then, have a good claim to be able to meet our expressive needs with respect to articulating intuitive claims about sets and defining logical consequence without inconsistency. Much more deserves to be said about these issues. Of more interest to us here, however, is what we can buy in our theorizing about modality with primitive second-order quantifiers that are (i) intelligible in the absence of further (nominal) analysis, (ii) not ontologically committing, and (iii) promising as a tool for meeting our expressive needs.

3. COMPLICATIONS: TRUTH-AT-A-WORLD The forgoing discussion of second-order quantification has, in certain respects, a very straightforward application to modality. In particular, the move from nominal to second-order quantification in modal analyses can be quite natural given views on the proper analysis of ◻ and ◇ like those of Robert Stalnaker (1976: 70) and Scott Soames (2007: 251). These views emphasize that, strictly speaking, saying that (1) quantifies over worlds is misleading: 1.

∃w: p is true at w.

Instead, we should speak of the quantifier in an analysis of ‘◇p’ as ranging over ways the world might have been. What go by the name 19

Williamson (2003: 426).

164 | Billy Dunaway “possible worlds” are really, on this view, maximal properties that might have been instantiated. Letting S range over these possibly instantiated maximal properties, the analysis of ‘◇p’ is more perspicuously written as 16. ∃S: p is true at S. Given this innocuous amendment to (1), it seems that we can apply the forgoing discussion of second-order quantification in a straightforward manner. For most purposes, sentences which are ontologically committed to properties can be replaced by non-committing sentences containing a well-chosen predicate. In a simple case, the sentence ‘John is tall’ gives the same information about John and his height as ‘John has the property being tall’ does; the only difference is that the latter does so by making reference to a property.20 Once primitive second-order quantification is in the picture, there is an analogous way to find sentences that have the same import as general statements about properties, but without the same ontological commitments. What the sentence ‘there is some property John has’ says is captured by ‘∃F: John is F,’ or ‘John is something.’ The only difference is that the latter sentences say something about John without invoking properties, just as the analogy to the pair ‘John is tall’ and ‘John has the property being tall’ would predict. By insisting that worlds-based analyses of ◻ and ◇ should be read as quantifying over maximal properties, the views of Stalnaker and Soames make it extremely natural to move to an analysis where the quantification is second-order, binding variables in predicate-position, and leaving properties out of the picture. There is, however, a major hitch in this seemingly innocuous move; we cannot reap the ontological benefits of a shift to secondorder quantifiers while leaving the rest of the theory intact. By going second-order, we alter the grammatical form of sentences containing the expression ‘is true at.’ On the ordinary analyses, ‘is true at’ takes a formula and a world-name, or a variable ranging over worlds, as arguments. (Here I lump worlds construed as objects, and worlds construed as giant properties in the manner of Soames and Stalnaker, under the single heading of “worlds.” These views 20 Here and throughout I assume the Quinean/Ostrich Nominalist view of predication discussed in section 2.

Modal Quantification without Worlds | 165 agree that it is a nominal expression which designates the things at which claims are true or false. They differ over the nature of these entities; I will return to this difference later.) But by revising the analysis to incorporate second-order quantifiers, ‘is true at’ no longer takes nominals as arguments: by replacing a nominal quantifier by a quantifier that binds predicate-position variables and leaving everything else in tact, we arrive at the following “analysis”: 17.

∃F: p is true at F.

But now the second argument of ‘true at’ is filled by a second-order variable, and so is of a type that generates grammatical instances when the variable is replaced by expressions like ‘is red.’ A change in grammatical or syntactic type is not problematic per se, but in this case it isn’t clear what sentences like (17) can even mean with the second-order variable in place. The difficulties extend further. Many accounts that take ‘is true at’ in (1) to be a relation to a world are in a position to give a further analysis of the notion. Plantinga (1978: ch. 4) and Adams (1974: 255) claim that worlds are set-theoretic entities: either sets of states of affairs (for Plantinga) or propositions (for Adams). Worlds, on these views, have a kind of structure. Since p in (1) can be associated with a proposition or state of affairs, it is then very natural to say that for p to be true at a world is just for the relevant proposition or state of affairs to be a member of that world.21 I don’t say that these theorists necessarily accept these analyses; Plantinga (1978: 49) seems to acknowledge the possibility of giving the set-theoretic definition, but officially embraces a definition in counterfactual terms like the one I discuss later.22 The point here is that such an analysis is available on this style of view. Consider also Stalnaker: propositions, on his view, are just sets of possible worlds. The analysis of truth-at-aworld can then go the other way and hold that p is true at w just in case w is a member of the proposition expressed by p.

21 This at least applies to atomic p. If worlds don’t contain complex states of affairs or propositions (those expressed by sentences containing negation and other logical operations, as well as modal operators), then truth-at-a-world for the propositions or states of affairs associated with complex sentences must be defined in terms of truthat-a-world for the atomics, in the usual way. 22 Thanks to Dean Zimmerman for pointing this out to me.

166 | Billy Dunaway Once we give up on an account that quantifies over worlds, we lose the possibility of giving an analysis of truth-at-a-world in terms of set-membership or other structural relations. We can’t say that worlds are things which can be constituents of sets or propositions, and we can’t say that they are things with a set-theoretic structure. This is because there is nothing over which we quantify that has the kind of structure that can be useful in analysis. But this is not unique to the second-order view; even some accounts that do quantify over worlds do not posit enough structure to allow for a set-theoretic analysis. These views also have to say something else about truthat-a-world. Consider a view on which worlds are giant properties in the style of Stalnaker and Soames, where these properties have no further structure—they are not “structured properties” composed out of further constituents—and where propositions are not sets of possible worlds.23 A view of this kind is found in Soames (2007), and it must use alternative resources to explicate truth-at-a-world. Soames (2007, 2011) suggests that the counterfactual locution is suited to the task of explaining truth-at-a-world under these conditions. In particular, we can say that for p to be true at w is for the following to hold: 18.

If w were to be instantiated, then p would be true.24

It is worth noting that, if this is a genuine analysis of the notion of truth-at-a-world, then the counterfactual cannot (as some theorists hold) have an analysis in terms of truth at a possible world. Lewis (1973) and Stalnaker (1968) propose that the truth of a counterfactual depends on what is true at nearby worlds where its antecedent is true. (18) would on these views be true, then, just in case p is true at the nearest world(s) where w is instantiated. This appeals to the notion of truth-at-a-world, though, and so the resulting analysis would be circular. 23 Propositions might still be set-theoretic entities of another kind on this view— perhaps Russellian propositions with ordinary objects and properties for constituents. All that matters is that a proposition is not the kind of thing that is guaranteed to have all the worlds with respect to which it is true as constituents; for then a Stalnaker-style analysis on which p is true at w when w is a member of the proposition expressed by p is unavailable. 24 See, in particular, Soames (2007: 267) and Soames (2011: 126).

Modal Quantification without Worlds | 167 The point here is not that a Soames-style account of possible worlds and truth-at-a-world is false; rather, it is that once we take his package of views on board, we cannot go in for a particular kind of analysis of the counterfactual. We can, instead, take the counterfactual in (18) as primitive, and this is in fact something Soames explicitly takes on board.25 Not all theorists will be happy with this last option, however, so it is worth asking whether there are other options for analyzing truth-at-a-world. Section 4 explores this question in the hope of finding an analysis of truth-at-a-world that can feature in a plausible second-order account of the modal operators.

4. HYPERINTENSIONAL CONNECTIVES The primitive counterfactual, as we have said, has one thing in its favor from the perspective of the second-order view. Since it can be put to use in an analysis of truth-at-a-world even when there is no ontology of worlds to appeal to, the second-order account could in principle analyze ‘◇p’ as follows: 19.

∃F: if things were F, then it would be the case that p.

This works because the primitive counterfactual doesn’t require the existence of a world where things are F. But since the counterfactual construction must be taken as primitive in (19), there is some reason to look elsewhere for another resource that can do the same job. This will show that analyzing ◻ and ◇ in second-order terms is not committed to a primitive counterfactual connective. Any alternative resource should be one that is independently motivated—i.e. it should not be a resource we appeal to simply because it is needed for an analysis of ◻ and ◇ in second-order terms.26 For this reason, I will sketch one possible motivation: that adequately expressing the determinate/determinable relationship 25

See Soames (2011: 126 fn. 2). This is for two reasons: first, it ensures that the analysis of ◻ and ◇ does not, by itself, bring in an extra ideological commitment. And second, it guarantees that we have a working grasp on the key notion. If our favored locution appears in a wide range of linguistic contexts (i.e. not just those that feature in an analysis of modality), we can arrive at a grasp of the intended meaning of the notion without the help of an explicit definition—cf. Williamson’s “direct method.” 26

168 | Billy Dunaway requires a hyperintensional connective like ‘in virtue of’ in our primitive ideology. (By ‘hyperintensional,’ I mean simply that substituting co-intensional arguments of the connective does not necessarily preserve truth value.) Some might not find it compelling that this particular case calls for a hyperintensional connective, but will find it plausible that we need such a resource for other purposes.27 The essentials of the second-order account of ◻ and ◇ will not depend on which motivation we accept for a primitive ‘in virtue of’-like connective—my primary aim will be to show in section 5 that, once we have it, we can use it to do the necessary work in explicating the notion of truth-at-a-world. There are, however, some especially interesting theoretical connections to be made if the notion we use to explicate truth-at-a-world is the same as the one used to analyze the determinate/determinable relationship. I mention these briefly at the end of the present section. Some things are both blue and colored, or both dogs and mammals, and thereby instantiate what we can call a determinate and its corresponding determinable.28 Cases of this kind are ripe for treatment with hyperintensionality—in particular, we should want to use expressions that are not definable in terms of the standard modal resources to capture the determinate/determinable relationship.29 Here are two candidate analyses of the determinate/determinable relationship in purely intensional terms: 20. ◻∀x (x is blue x is colored). 21. ◻∀x (x is blue x is colored) ∧ ◇∃y (y is colored ∧ y is not blue). (20) is too weak: necessarily, every blue thing is shaped, but this isn’t an instance of the determinate/determinable relationship. (21) is also too weak, as someone might think that necessarily, if anything is a tree, then God loves it (because he loves everything he

27

Some discussion of other motivations can be found in Fine (2001) and Schaffer (2009). 28 See, for instance, Yablo (1992). 29 There may be other metaphysical phenomena that likewise need hyperintensional expressions to be fully analyzed; the case of Socrates and his singleton from Fine (1994) comes to mind. One reason for not discussing these cases alongside the determinate/ determinable relationship is that it would require the (perhaps unwarranted) assumption that the same notion should be used to explain both cases—see Manley (2007).

Modal Quantification without Worlds | 169 creates) and possibly God loves something that isn’t a tree (because God might create more than trees). But plausibly, being a tree and being loved by God isn’t a way to instantiate a determinate and its corresponding determinable. This suggests that purely intensional idioms are not capable of capturing the phenomena, and naturally points toward analysis in hyperintensional terms. But before approaching this idea, we should ward off a different direction of thought, according to which a broader analysis in terms of conceptual necessity might do the job. This suggestion has some initial plausibility when we consider the present examples, as someone who has the concepts blue and colored must appreciate the relevant entailments.30 But there are instances of the same phenomena that are discovered empirically: having atomic number 13 and being a metal seem to be instances of a determinate and corresponding determinable, yet someone could have both concepts and not know that the entailment from the former to the latter holds. Adding a hyperintensional connective to our primitive ideology gives us the resources to make the relevant distinctions. The details of exactly which connective we should use here are tricky, but we can introduce the contours of how such a notion would help by starting with ‘thereby,’ as follows: 22. If x is blue, then x is thereby colored; 23. If x is a dog, then x is thereby a mammal.31 So long as we don’t require the use of ‘thereby’ to have a further definition in intensional terms, we can rely on our intuitive understanding of the term to capture the relevant relationship. For it is quite natural to say that if something is blue, it isn’t thereby shaped, even though this follows with necessity. This isn’t the place to offer an extended argument that ‘thereby’ or something similar should appear in our primitive ideology, however. Of more interest is the

30

Conceptual entailment by itself cannot be sufficient for the determinate/determinable relationship, as being a bachelor and being unmarried are not instances of the relationship. And the notion of conceptual entailment that this account appeals to needs to be made more precise, so that it is clear that there is no entailment of the relevant kind between blue and shaped. We will not pursue these issues further here, however, as the following example in the text suggests that conceptual entailment is not even necessary for the determinate/determinable relationship. 31 See Teichmann (1992) for a similar suggestion.

170 | Billy Dunaway question of what we might say about truth-at-a-world if we capture the determinate/determinable relationship with sentences like (22) and (23). Before turning to this question, there are some issues about how exactly to proceed, once we go in for use of a hyperintensional connective. The use of ‘thereby’ in (22) and (23) would appear to be embedded in a conditional construction, as it occurs in a sentence with ‘if’ and ‘then.’ We might, in this case, want to know how this construction is composed from its constituent parts. But it can’t contain a material conditional; surely the analogue of (22), ‘if x is blue, then x is thereby a giraffe,’ is not true of a non-blue thing.32 A stronger reading of the conditional component is needed, but natural ways of strengthening it will invoke possible worlds. This would make the conditional ineligible for analyzing truth-at-aworld for the same reasons that the counterfactual interpreted along the Lewis-Stalnaker lines was ineligible.33 The expression ‘in virtue of’ is another option: 24. 25.

x is colored in virtue of x’s being blue x is a mammal in virtue of x’s being a dog.

‘In virtue of,’ like ‘thereby,’ appears to have the needed force: it is implausible to say that something is loved by God in virtue of its being a tree, as treehood consists in being a member in a certain phylogenetic group—a biological property. What stands out about the ‘in virtue of’ construction, however, is that it is grammatically required to be followed by a gerund phrase like ‘x’s being blue’; ordinary sentential constructions like ‘x is blue’ are ungrammatical in the second argument place. We might then worry that ‘in virtue of’ requires an ontology of some kind to supply the referents of 32 I am supposing that claims like (22) and (23) are about a particular object x. If these are instead read as schemas or sentences bound by a wide-scope universal quantifier, then the objection becomes that ‘if x is blue, then x is thereby a giraffe’ should not be true at (or acceptable for) worlds containing no blue things. 33 This point of contact with the counterfactual analysis suggests another option, which is to hold that (22) and (23) do not involve complex connectives after all, and to take ‘if . . . then thereby . . .’ to be a single, unanalyzable connective. As with the primitive counterfactual, this is a legitimate option that is not entirely theoretically satisfactory; the construction certainly seems to involve the familiar ‘if . . . then . . .’ that appears in other analyzable English constructions without the company of ‘thereby.’

Modal Quantification without Worlds | 171 these gerund phrases. And it would be disappointing if, in the course of carrying out a project where the ontology of modality is our primary concern, we jettisoned worlds from the ontological commitments of the quantificational expressions, only to introduce some very similar entities elsewhere in the analysis. There are other options still; perhaps we could use one of the following: 26. For it to be that x is blue just is for it to be that x is colored; 27. For x to be blue is for x to be colored. Unfortunately, the most natural reading of (26) is one on which it is exhaustive, entailing that there is nothing more to being colored than being blue. But of course this is false: red things are also colored (and in the same sense that blue things are colored). (27), moreover, isn’t obviously different from (26) in this respect. Perhaps adding a ‘thereby’ as in (27) removes the feeling of exhaustivity from (26): 28.

For it to be that x is blue is thereby for it to be that x is colored.

Adding ‘thereby,’ in other words, removes the suggestion in (26) that being blue is the only way to be colored. But some readers may find this construction burdensome, and the use we will put it to in an analysis of ◻ and ◇ will only make the problem worse (as the full analysis would require several embeddings of the ‘for it to be that . . .’ connective, which we quantify into using primitive secondorder quantifiers). There is no obvious answer as to which hyperintensional connective is best: some may prefer the linguistic simplicity of ‘thereby’ or ‘in virtue of’ as in (22)–(25). They might be willing to pay for this simplicity by refusing to view the ‘if . . . then thereby . . .’ construction as analyzable into further component parts. Or they might insist that, despite its superficial form, the gerund argument for ‘in virtue of’ has the ontological commitments of an ordinary sentence, and requires no special ontology. Another approach would be to find an accessible reading of (26) or (27) that is not exhaustive. And of course we could work with the length and awkwardness of (28). Instead of legislating on the issue here, let us instead adopt a piece of notation and introduce the connective ⇒ as a stand-in for

172 | Billy Dunaway our favored English expression in the family of hyperintensional connectives that appear in (22)–(28). We can then write 29. 30.

x is blue ⇒ x is colored; x is a dog ⇒ x is a mammal

to express our hyperintensional analysis of the determinate/determinable relationship. Importantly, the ⇒ connective is not interpreted as merely meaning whatever has the right properties to capture the determinate/determinable relationship. Rather, it is interpreted in terms of one of the expressions in (22)–(28) that we, as competent speakers of English, are antecedently familiar with. The formalism is simply a shortcut, allowing us to avoid taking a stand on exactly which expression this should be. If a hyperintensional ‘in virtue of’-like connective is needed for an analysis of the determinate/determinable relationship, then it is very natural to think that it can contribute to an analysis of truth-ata-world. Recall our problem was that of explaining what truth-at-aworld is for a view which takes “possible worlds” to be maximal properties that might be instantiated, without appealing to a primitive counterfactual construction, or to structural features of the worlds. What is it for p to be true at such a property? Take a worldproperty S: for p to be true at S is for the following to obtain: 31. S is instantiated ⇒ p. The introduction of ⇒ in terms of the determinate/determinable relationship suggests an important connection between truth-at-aworld claims and the determinate/determinable relationship. S is a maximal property, which roughly means that its instantiation determines all of the (non-modal) facts about the world which instantiates it. (See section 5 for more on the notion of maximality.) And for the most part, sentences like p that are true at a world are less than fully maximal in this way: there are multiple world-properties with respect to which p is true. In these cases, we might think of the claim that S is instantiated as the determinate to the corresponding determinable of p’s being true. Being blue is one of many ways to be colored, and likewise the instantiation of S is one of many ways for p to be true. In the special case where p is itself equivalent to the claim that some world-state property is instantiated, we can treat sentences like (31) as trivially true if p is equivalent to the antecedent,

Modal Quantification without Worlds | 173 and false otherwise. Truth-at-a-world claims are then special cases of claims about a determinate and a corresponding determinable.34 5. A SECOND-ORDER ANALYSIS OF ◻ AND ◇ The pieces are now in place for giving a fully worked-out analysis of ◻ and ◇ in terms of primitive second-order quantification and the hyperintensional ⇒. We noted in the beginning of section 3 that simply inserting second-order quantifiers into the traditional analysis of ‘◇p’ to obtain (17) will not do: 17. ∃F: p is true at F. We need to make revisions in other parts of the analysis to accommodate the move to second-order quantifiers. ⇒ is helpful for nominal analyses that similarly lack structure to define truth-at-a-world, and it can also help with understanding the grammatical form of (17), a problem unique to the second-order view. Begin with the notion of truth-at-a-world: (17) contains a construction of the form 32.

p is true at G,

where G is a predicate.35 Following the analysis of truth-at-a-world when world-properties are in play, we can say that (32) is to be analyzed in the following terms: 34 The connection between these two cases might fall apart if we adopt a different motivation for an ‘in virtue of’-like hyperintensional connective. Whether this is so depends on the details; other motivations for such connectives may permit structurally similar analyses of both determinate/determinables and truth-at-a-world, but are not guaranteed to do so. The cases of metaphysical dependence from Fine (1994), exemplified by Socrates and his singleton, give another candidate motivation for a hyperintensional connective. Using ‘depends’ to capture the relevant relationship between Socrates and his singleton, it is plausible that the fact that p is true does not depend on S’s being instantiated, but rather that S’s being instantiated depends (in part) on the fact that p is true. In this case, we should interpret the connective ⇒ in our analysis of truth-at-a-world to mean something like ‘depends partly on,’ and take truth-at-a-world claims to be special cases of claims about this kind of partial dependence. (One exception: given the “monism” of Schaffer (2010), p’s being true depends on the global fact that S is instantiated, and so the original direction of explanation would be preserved.) 35 (17) itself doesn’t contain a predicate, but merely a second-order variable, which occurs in predicate position. To separate issues, we will first consider the kind of claim made by sentences containing ‘true at’ followed by a predicate. We will then consider the more general case when the predicate is replaced by a bound variable as in (17).

174 | Billy Dunaway 33.

things are G ⇒ p.

That is, on one way of interpreting ⇒, (32) claims that for it to be that things are G is thereby for it to be that p is true. Quantificational analyses of ◻ and ◇ are simply generalizations on truth-in-a-world claims like (33), where the predicate is replaced with a bound variable (and where the quantifier is appropriately restricted—see below). This gives us an analysis of ‘◇p’ in terms of (34): 34.

∃F: things are F ⇒ p.

‘◻p’ is similar, with the existential second-order quantifier replaced with a universal second-order quantifier: 35.

∀F: things are F ⇒ p.

This is a step toward showing that traditional quantificational analyses can be given in second-order terms, but there is more to be done. In ordinary possible worlds analyses of ◻ and ◇ that quantify in a nominal position, the quantifiers are restricted to range over possible worlds. Exactly how this restriction is to be accomplished depends on the nature of the worlds in question (whether they are sets of states of affairs, or sets of propositions, etc.) but, with one exception, the restriction is accomplished in part by the use of a modal language, claiming that the relevant entities are all possible. The exception is a “realist” view of the Lewisian variety, where the quantifiers are supposed to range over maximal isolated regions of spacetime.36 The prospects for a complete reduction of this kind 36 Forrest (1986) claims to put forward a quantificational analysis over worldproperties (in the style of Stalnaker and Soames) that does not require a modal notion like ‘possible’ to restrict the range of the quantifiers (Forrest (1986: 24)). We should question whether he genuinely succeeds at this, however. There are lots of abstract objects which are not world-properties; the Forrest analysis requires that the quantifiers in the analysis of ◻ and ◇ do not range over these; otherwise the theory would predict the wrong truth conditions for modal sentences. What in his analysis guarantees this? What Forrest has in mind is that we could, in principle, restrict the quantifiers simply by enumerating all of the properties we wish to quantify over, and thus enumerate all and only world-properties. ‘◇p’ would then be analyzed as follows:

∃x [x = w1 or x = w2 or . . . ]: p is true at x (where w1,w2, etc. are all of the possible world-natures, though the analysis doesn’t say this). Very briefly, I think the problem with this approach is that it fails to explain why the modal notions expressed by the operators ◻ and ◇ are metaphysically

Modal Quantification without Worlds | 175 seem dim, however, so the question for the second-order analysis should be whether we can use the predicate ‘possible’ to restrict the second-order quantifiers in (34) and (35) in a manner that approximates its use in standard non-Lewisian analyses. The notion of maximality is also present in the quantifier restrictions of standard quantificational analyses of ◻ and ◇. If, for instance, worlds are sets of states of affairs, the quantifiers don’t range over every possible state of affairs; just those that are “maximal” in some appropriate sense. There is one difference between the notions of possibility and maximality when they are used in restrictions on second-order quantifiers, however: in the standard nominal analyses, ‘possible’ and ‘maximal’ are predicates that apply to names for the world-like entities in the domain of the quantifiers. In the present setting, these are plural predicates that take plural arguments like ‘things’ and plural variables. These predicates convey how things are—i.e. whether they are maximal or possible. And together with the hyperintensional ⇒, they can be used to formulate the following restriction on the quantifiers in (34) and (35): 36. things are possibly F ∧ (things are F ⇒ they are maximal). Roughly, then, using (36) as a quantifier restriction makes instances that generate truths when plugged into (36) the admissible substitution instances for the bound variables in (34) and (35). This is only a rough explanation—we will return to give the official non-substitutional explanation of quantifier restriction in a second-order setting at the end of this section. We should also note that it is possible to reduce the number of primitives in (36) by defining ‘maximal’ in terms of other primitives we are already committed to, namely possibility, primitive second-order quantification, and ⇒, as follows: 37.

∀G: things are possibly G ((things are F ⇒ they are G) ∨ (things are F ⇒ things are not G)).

significant. For example, the Forrest analysis does not explain why possibility so defined is more metaphysically fundamental than (for example) the highly gerrymandered and metaphysically inconsequential notion that is defined in the same way except that it is missing the first disjunct in the restriction. See also Forbes (1989: 80–2) for a related discussion.

176 | Billy Dunaway Informally, maximality on this understanding takes a stand on any possible way for things to be (maximal or otherwise). By substituting (37) for the conjunct containing ‘maximal’ in (36), we arrive at a restriction on the quantifiers ∀F and ∃F that appeals to one less primitive. The final question is how to incorporate the condition (36) rigorously as a restriction on the quantifiers in (34) and (35). Of course, we cannot adopt a simple notion of restrictions on which they characterize the entities in a domain of quantification—there are no domains that the quantifiers in (34) and (35) range over. But so long as the quantifiers in question are second-order ∀ and ∃, we can use familiar truth-functional connectives to restrict the quantifiers in the desired way. To start, suppose that the quantifiers in 38. ∃x: x is in the room; 39. ∀x: x is in the room, are restricted to range over people with red hair. (37) and (38) are then equivalent to the more complex sentences (39) and (40), where the quantifiers are unrestricted: 40. ∃x: x has red hair ∧ x is in the room; 41. ∀x: x has red hair x → is in the room. Things can go similarly when the quantifiers are second-order: a restriction-condition can be embedded in the antecedent of a material conditional to restrict ∀, and can be embedded in a conjunction to restrict ∃.37 Using the restriction (36) (or its more complicated cousin with ‘maximal’ defined away) to restrict (34) and (35), we then arrive at the following final analysis of ‘◇p’ and ‘◻p’: 42.

∃F: (things are possibly F ∧ (things are F ⇒ they are maximal)) ∧ (things are F ⇒ p).

37 Thus, if Φ is a formula with a free second-order variable F, ‘∃F: [Φ] . . .’ (where Φ functions as a second-order restriction) is equivalent to

∃F: Φ∧ . . . , and ‘∀F: [Φ] . . .’ is equivalent to ∀F: Φ → . . . .

Modal Quantification without Worlds | 177 43.

∀F: (things are possibly F ∧ (things are F ⇒ they are maximal)) (things are F ⇒ p).38

One lesson to take from the last two sections is that primitive second-order quantification is not a device that gives us a cheap and easy way to reduce our ontological commitments. In spite of their ontological innocence, there are many settings where second-order quantifiers are not themselves sufficient to accomplish the tasks that ordinary quantification into the nominal position (with its attendant ontology) is well suited to do. This is for two reasons: first, these quantifiers bind variables in places that are syntactically distinct from the variables bound by ordinary quantifiers into the nominal position, and so we often need to rework other aspects of our analysis to accommodate the syntactic changes that accompany a move to bound predicate-position variables. Moreover, the reduction in ontology brings with it a potential loss of structure (settheoretic or otherwise) that is useful in formulating analyses. We can handle these complications, but this requires some substantial reworking of the original analyses containing nominal quantifiers, plus an extra resource in the form of a hyperintensional connective. The benefits of primitive second-order quantifiers are therefore part of a more expensive package—a package some philosophers will not be willing to pay the extra costs for. It should be clear, however, that if we are willing to countenance the extra constructions in our primitive ideology, there is a non-obvious yet substantial benefit in ontology to be had.39 38

It is well known that there is no truth-functional operation that can accomplish the effects of restriction on natural language quantifiers like ‘most,’ ‘few,’ etc. So, if there are second-order analogues of these quantifiers, the present strategy for second-order quantifier restriction cannot be extended to these cases. Two options would then be available: one is to add an additional primitive restriction construction for second-order quantification. (Perhaps we can begin to grasp this primitive construction by understanding that its function, in the case of ∃ and ∀, is equivalent to a construction containing ∧ and →, as in the move from (38–9) to (40–1). Of course, these cases would fall short of a definition.) Another alternative, which is consistent with the needs of our present project, is to deny that quantifier restriction can be extended to second-order ‘most.’ 39 At this point a potential criticism should be mentioned: the goal of eliminating the ontological commitments of modality seems extremely narrow-minded, since we will need propositions and/or properties as tools of analysis in other areas. The fact that these entities are unneeded for an analysis of modality is of little interest; since we will need entities of a certain kind elsewhere, we might as well help ourselves to them here.

178 | Billy Dunaway 6. REFORMULATING GLOBAL SUPERVENIENCE Our main concern has been to develop a second-order quantificational analysis of ◻ and ◇. But there is a reason why an analysis in quantificational terms is to be preferred over taking ◻ and ◇ as primitive: even though quantificational analyses in their typical form take on additional ontological commitments, they are also potent explanatory tools. Lewis (1986: ch. 1) makes the case that, in addition to providing an analysis of ◻ and ◇, quantification over worlds is needed for formulating other metaphysical claims. Global supervenience theses are one example: For a case where the distortion is more serious, take my second example: the supervenience of laws. We wanted to ask whether two worlds could differ in their laws without differing in their distribution of local qualitative character. But if we read the ‘could’ as a diamond, the thesis in question turns into this: it is not the case that, possibly, two worlds differ in their laws without differing in their distribution of local qualitative character. In other words: there is no world wherein two worlds differ in their laws without differing in their distribution of local qualitative character. That’s trivial—there is no world wherein two worlds do anything. At any one world W, there is only one single world W. The sentential modal operator disastrously restricts the quantification over worlds to something that lies within its scope. Better to leave it off. But we need something modal—the thesis is not just that the one actual world, with its one distribution of local qualitative character, has its one system of laws!40

If Lewis is right, there is a significant theoretical benefit to quantifying over possible worlds: by refusing to analyze ◻ and ◇, we deprive ourselves of important resources for capturing the

There are two ways to respond to this line of thought. The first is to hold out hope that the tools we use to eliminate ontological commitments in our analysis of modality will also make available a reduction in ontology in other domains. (To take one example: so-called “indispensability” arguments for the existence of properties of the kind found in van Inwagen (2004) are suspect once primitive secondorder quantification is in the picture—see Manley (2009) for discussion.) Second, even if we are in the end saddled with the existence of abstracta, we might nonetheless hold that they are low-grade entities that are not very fundamental. By avoiding an analysis of modality that requires the existence of these things, we can consistently hold that the modal idioms express highly fundamental notions without upgrading the metaphysical status of abstracta. Thanks to Daniel Fogal for discussion of this issue. 40

Lewis (1986: 16).

Modal Quantification without Worlds | 179 supervenience of laws on local qualitative character.41 But suppose we avoid commitment to worlds in our analysis of the operators by using second-order quantifiers: will commitment to worlds arise again once we turn our attention to global supervenience theses? No, for the second-order analysis is a quantificational analysis, and we can use the quantificational apparatus to capture the relevant supervenience theses without commitment to worlds. We need, however, to be careful about what exactly the benefit in formulating global supervenience theses is. In many cases, global supervenience claims have been put forward to capture a kind of dependence relation between two domains.42 But with the resources of a hyperintensional connective in place, global supervenience claims are plausibly dispensable for this purpose.43 Still, other jobs remain for global supervenience. First: there might be domains that globally supervene on others without the former depending on the latter or vice versa. (As an example, we might take a Leibnizian who thinks that the mental globally supervenes on the physical owing to a “pre-established harmony,” or a Moorean about the normative who holds that rightness supervenes on, but is importantly independent of, the natural. Such theorists plausibly reject the presence of any dependence between the domains in question.) In this case, we need global supervenience claims to capture modal covariation between domains without dependence. This leads to a second point: even when we are dealing with domains related by a kind of dependence—such as, perhaps, the laws and local qualitative character—it is nonetheless still true that the two domains bear a weaker relation of global supervenience. In short, once we relinquish the claim that global supervenience captures an interesting sort of metaphysical dependence, it still captures a theoretically interesting relation between domains, and we should like to have the resources to express this relation. 41 Nothing in what follows will hinge on the assumption that Lewis is right about the supervenience between laws and distribution of local qualitative character. I will work with the example here, but readers who disagree with Lewis on this count might accept other global supervenience theses (such as the supervenience of the moral on the descriptive, or of the mental on the physical) and are free to substitute their favored examples in the rest of this section, without loss of substance. Use of quantification in formulating global supervenience theses is common outside of Lewis; see also Kim (1984: 168) and Sider (1999: 915). 42 See Kim (1984: 175), Stalnaker (1996: 230), and Bennett (2004: 507 ff ). 43 Thanks to Karen Bennett for raising this question.

180 | Billy Dunaway To formulate Lewis’s claim of the global supervenience of laws on local qualitative character, we can begin with the following canonical statement of the thesis, which uses nominal quantifiers ranging over worlds:44 GS

∀w1 , w2: if w1 and w2 are the same in distribution of local qualitative character, then w1 and w2 do not differ in their laws.

Can we use the resources of sections 2–5 to formulate a secondorder equivalent for GS? I will not spell out precisely what “equivalence” amounts to here, but our earlier discussion highlights one desideratum for a second-order replacement of GS: it should not commit us to holding that laws obtain in virtue of a particular distribution of local qualitative character. For even if we share Lewis’s views about laws and wish to claim in addition that laws depend on distribution of local qualitative character, the statement of global supervenience should not commit us to this; it should entail a kind of modal covariance between the two domains and nothing more. We know how to simulate quantification over possible worlds using second-order quantifiers and ⇒; I will use the subscripted second-order variables F1 and F2 under the assumption that they are appropriately restricted. To replace the notion of two worlds having the same distribution of local qualitative character, let us first introduce a second-order variable Q, and restrict it so that, in effect, its admissible instances allow for only one way for things to have a maximal local qualitative character. In other words: 44. things are Q ⇒ things have a particular maximal local qualitative character. Using a clause like (44) as a restriction (indicated here by square brackets—cf. fn. 38.), the second-order equivalent of a condition requiring that w1 and w2 are identical in distribution of local qualitative character is the following: Same LQC ∃Q [things are Q ⇒ things have a particular maximal local qualitative character]: (things are F1 ⇒ things are Q) ∧ (things are F2 ⇒ things are Q). 44 More could be said about what it is for two worlds to be the same in distribution of local qualitative character and the same in laws; I will briefly address these complications in fn. 45.

Modal Quantification without Worlds | 181 We next need a second-order equivalent for the condition that w1 and w2 are governed by the same laws, and this can be accomplished by using a restriction structurally similar to (44) (where L is a secondorder variable and l1, l2 . . . are variables ranging over specific laws): Same Laws ∃L ∃l1 , l2 ... [things are L ⇒ things are governed by the laws l1,l2 . . . and there are no other laws]: (things are F1 ⇒ things are L) ∧ (things are F2 ⇒ things are L). We now have the resources to state a second-order equivalent of GS: we just need to add universal second-order quantifiers to bind F1 and F2 and a material conditional, to say that for any two “worlds,” if they satisfy Same LQC, then they satisfy Same Laws. The result is GS: GS+ ∀F1 , F2: (∃Q [things are Q ⇒ things have a particular maximal local qualitative character]: (things are F1 ⇒ things are Q) ∧ (things are F2 ⇒ things are Q)) (∃L ∃l1 , l2 ... [things are L ⇒ things are governed by the laws l1,l2 . . . and there are no other laws]: (things are F1 ⇒ things are L) ∧ (things are F2 ⇒ things are L)). Crucially, GS+ doesn’t entail that the laws depend on local qualitative character—GS+ uses only a material conditional to connect a claim about the local qualitative character at a pair of worlds and a claim about the laws at those worlds. There are further questions about whether GS+ is an adequate substitute for GS in every respect.45 45 One of these further questions is the following: in formulations of global supervenience that quantify over worlds, what it is for two worlds to be the same in some respect can be given a further gloss. In particular, we can follow the characterization Stalnaker (1996: 227) gives of what it is for two worlds to be the same (or indiscernible) with respect to some class of properties:

[T]wo worlds w and z are B-indiscernible iff there is a 1–1 correspondence between the domains of w and z, and any individual in the domain of w has the same B-properties in w as the corresponding individual from the domain of z has in z. This gives rise to different versions of global supervenience, which are often called “strong” and “weak” global supervenience (for more on these notions, see Sider (1999), Shagrir (2002), and Bennett (2004)). Can clauses like Same LQC be adapted to accommodate this rendering of what it is to be the same in some respect? Here is one (admittedly sketchy) suggestion. Let µ be a 1:1 mapping from the domain of one “world” to the domain of another. We can then introduce a variable “ranging over” local qualitative characteristics of individuals as follows: ∀x: if x is Q ⇒ x has a local qualitative character. Two “worlds” that preserve local qualitative characteristics, then, satisfy the following: ∃µ ∀y ∀Q [∀x: x is Q ⇒ x has a local qualitative character]: (things F1 ⇒ y are is Q) ∧ (things are F2 ⇒ μ(y) is Q).

182 | Billy Dunaway But we have a promising start, and it is an open question whether other discrepancies between GS and GS+ would reflect metaphysical shortcomings of GS+, or whether they would serve to bring out mere artifacts of GS.

7. CONCLUSION: THE PLACE OF WORLDS IN A HYPERINTENSIONAL SETTING Sections 2–5 show how someone who accepts the resources of primitive second-order quantification and an ‘in virtue of’-like connective can analyze ◻ and ◇ without committing to an ontology of possible worlds. When we take into account considerations that might motivate a hyperintensional ideology, however, this result is not entirely surprising, and perhaps even to be expected. The motivations for theorizing with a primitive hyperintensional connective often derive from the inadequacy of reference to possible worlds as an analytical tool. We have already encountered several examples of this: the discussion of determinates and determinables in section 4 is one. Global supervenience claims, of the kind discussed in section 6, provide another example; modal claims like GS (or GS+) plausibly need explanations and do not provide them.46 We saw how determinate/determinable claims can be captured with hyperintensional devices. And hyperintensional notions as they appear in some recent work in metaphysics seem readymade to explain global supervenience claims.47 There are other examples which fit the same pattern. Here is one: some philosophers have suggested that what distinguishes realist ethical theories from their irrealist counterparts is their consequences for the modal profile of ethical facts. Since irrealists typically accept that the ethical is mind-dependent in some way, one This reformulation of Same LQC is not without questions: for instance, the presence of a nominal quantifier outside the scope of a hyperintensional connective raises the question of whether it requires an ontology of possibilia. Answering this question while remaining faithful to the ontological aims of the present paper is a further project, which I cannot undertake here. Thanks to Karen Bennett for raising this question. 46

See Kim (1984: 174) for discussion. For instance, the notion of ground in Schaffer (2009) can do the job: it is plausible to say that if A is grounded in B, then it follows that A supervenes on B. 47

Modal Quantification without Worlds | 183 might think that the distinctive feature of the irrealist view is that it entails that in possible worlds where the mental facts are different (say, worlds where everyone endorses murdering), the ethical facts are likewise different (and so in these worlds murdering is permissible). The realist, according to this line of thought, rejects that ethical facts have this modal profile.48 Such a proposal runs into trouble when we note that sophisticated expressivist views like that of Gibbard (2003) do not entail that the ethical counterfactually depends on facts about the mental states of agents, yet expressivism of this kind does not seem to be a realist view. In light of considerations like these, Fine (2001) goes hyperintensional, holding that what separates these views is that the ethical realist alone holds that ethical facts hold “in Reality.” This kind of proposal at least has the right structure to explain the difference between the realist and Gibbard-style expressivist: even if they agree on the distribution of ethical facts across the space of metaphysical possibility, they might disagree over the Reality of these facts.49 Thus it might be that an ontology of possible worlds is not the explanatory paradise that Lewis and others took it to be; talk of worlds instead provides a rough approximation of deeper and more powerful explanatory notions which are hyperintensional. If this is right—and I have not provided here anything more than a few examples that gesture in this direction—then we could hope that, just as explanations in terms of possible worlds are supplanted by hyperintensional explanations, the work done by possible worlds might itself be grounded in the hyperintensional. Sections 2–6 give one idea of how this might go. Of course, we needed an extra resource in the form of primitive second-order quantification to do away with the worlds. This is another resource that some philosophers will be reluctant to accept on the grounds that they don’t understand it well enough to use it in philosophical theorizing.50 I have tried to show in section 2 that there are legitimate avenues for understanding these quantifiers, and to indicate the work they can do. But we can’t argue someone into acquiring a grasp of the relevant concepts, and some might still 48 49 50

This seems to be the position in Dworkin (1996). See Dunaway (MS: ch. 1) for discussion and refinement of this idea. This is the sentiment expressed by the earlier quote from van Inwagen (2004).

184 | Billy Dunaway complain that they haven’t been sufficiently helped—after all, by taking second-order quantifiers to be primitive, we thereby refuse to define the notion in familiar terms involving quantification into nominal positions. In this respect, the situation is similar to the original move away from the kind of skepticism about modality voiced in Quine (1953b). The movement in this case was plausibly not a result of wholly new explanations of the meaning of ‘possible’; rather it came about as philosophers saw modal notions prove fruitful in providing rigorous analyses of a wide range of further phenomena. Languages bereft of modal force are too impoverished to do the work of metaphysics; new tools for analysis were needed. Expanding our conceptual resources to include primitive secondorder quantifiers and hyperintensional connectives might represent a similar improvement in our analytical toolbox.51 University of Oxford REFERENCES Adams, Robert Merrihew (1974) Theories of actuality. Nous 8(3): 211–31. Armstrong, D. M. (1978) A Theory of Universals, volume 1. Cambridge: Cambridge University Press. Bennett, Karen (2004) Global supervenience and dependence. Philosophy and Phenomenological Research 68(3): 501–29. Boolos, George (1984) To be is to be a value of a variable (or to be some values of some variables). The Journal of Philosophy 81(8): 430–49. Devitt, Michael (1980) “Ostrich Nominalism” or “Mirage Realism”? Pacific Philosophical Quarterly 61: 433–9. Dunaway, Billy (MS) Realism and fundamentality in ethics and elsewhere. PhD thesis, University of Michigan, Ann Arbor. Dworkin, Ronald (1996) Objectivity and truth: You’d better believe it. Philosophy and Public Affairs 25(2): 87–139. Fine, Kit (1994) Essence and modality. Philosophical Perspectives 8: 1–16. —— (2001) The question of realism. Philosophers’ Imprint 1(1): 1–30. Forbes, Graeme (1989) Languages of Possibility. Oxford: Basil Blackwell. Forrest, Peter (1986) Ways worlds could be. Australasian Journal of Philosophy 64(1): 15–24. 51 Thanks to Daniel Fogal, Shen-Yi Liao, Scott Soames, James Van Cleve, the editors at Oxford Studies in Metaphysics, and especially David Manley for comments on earlier drafts of this paper.

Modal Quantification without Worlds | 185 Gibbard, Allan (2003) Thinking How to Live. Cambridge, MA: Harvard University Press. Goodman, Nelson (1951) The Structure of Appearance. Cambridge, MA: Harvard University Press. Hirsch, Eli (1997) Dividing Reality. Oxford: Oxford University Press. Kim, Jaegwon (1984) Concepts of supervenience. Philosophy and Phenomenological Research 45(2): 153–76. Lewis, David (1973) Counterfactuals. Oxford: Blackwell Publishers. —— (1986) On the Plurality of Worlds. Oxford: Basil Blackwell. Manley, David (2007) Review of Existential Dependence and Cognate Notions by Fabrice Correia. Notre Dame Philosophical Reviews. . —— (2009) When best theories go bad. Philosophy and Phenomenological Research 78(2): 392–405. Melia, Joseph (1992) Against modalism. Philosophical Studies 68(1): 35–56. Plantinga, Alvin (1978) The Nature of Necessity. Oxford: Clarendon Press. Prior, Arthur (1971) Objects of Thought. Oxford: Oxford University Press. Quine, W. V. O. (1953a) On what there is. In W. V. O. Quine, From a Logical Point of View. Cambridge, MA: Harvard University Press. —— (1953b) Two dogmas of empiricism. In W. V. O. Quine, From a Logical Point of View. Cambridge, MA: Harvard University Press. —— (1986) Philosophy of Logic, 2nd edition. Cambridge, MA: Harvard University Press. Rayo, Agustin and Stephen Yablo (2001) Nominalism through denominalization. Nous 35(1): 74–92. Schaffer, Jonathan (2009) On what grounds what. In David Chalmers, David Manley, and Ryan Wasserman (eds), Metametaphysics. Oxford: Oxford University Press. —— (2010) Monism: The priority of the whole. The Philosophical Review 119(1): 31–76. Shagrir, Oron (2002) Global supervenience, coincident entities and antiindividualism. Philosophical Studies 109: 171–96. Sider, Theodore (1999) Global supervenience across times and worlds. Philosophy and Phenomenological Research 59(4): 913–37. —— (2012) Writing the Book of the World. Oxford: Oxford University Press. Soames, Scott (2007) Actually. Proceedings of the Aristotelian Society Supplementary Volume 81(1): 251–77. —— (2011) True at. Analysis Reviews 71(1): 124–33. Stalnaker, Robert (1968) A theory of conditionals. In Nicholas Rescher (ed.), Studies in Logical Theory. American Philosophical Quarterly Monograph Series. Oxford: Blackwell.

186 | Billy Dunaway Stalnaker, Robert (1976) Possible worlds. Nous 10: 65–75. —— (1996) Varieties of supervenience. Philosophical Perspectives 10: 221–41. Teichman, Robert (1992) Abstract Entities. Basingstoke: Macmillan. Van Cleve, James (1994) Predication without universals? A fling with ostrich nominalism. Philosophy and Phenomenological Research 54 (3): 577–90. Van Inwagen, Peter (2004) A theory of properties. In Dean Zimmerman (ed.), Oxford Studies in Metaphysics, volume 1. Oxford: Oxford University Press. Williamson, Timothy (2003) Everything. Philosophical Perspectives 17: 415–65. Wright, Crispin (2007) On quantifying into predicate position. In Mary Leng, Alexander Paseau, and Michael Potter (eds), Mathematical Knowledge. Oxford: Oxford University Press. Yablo, Stephen (1992) Mental causation. The Philosophical Review 101(2): 245–80.

5. Slots in Universals Cody Gilmore 1. INTRODUCTION I am attracted to a pair of theses about universals. One of them is popular, though controversial. The other is rarely discussed, so its popularity is hard to gauge. The popular thesis is Platonism: Properties and relations are abstract (non-spatiotemporal) entities and are not sets or ordered sequences. They are abundant, in the sense that almost1 every open sentence expresses one, and they are hyperintensionally individuated, in the sense that necessarily equivalent properties and relations are sometimes non-identical. (Being a triangle ≠ being a trilateral.) There are haecceitistic properties and relations (being Socrates and being introduced to Socrates by), there are uninstantiated properties and relations (being a golden mountain and having given a golden mountain to), and there are necessarily uninstantiated properties and relations (being a round square and being both larger and smaller than).2

The other thesis is Slot Theory: There are such things as argument places, or “slots,” in universals; in particular, for any universal u and number n, u is n-adic if and only if there are n slots in u. Slots are presumably abstract entities, and they are perhaps ontologically dependent upon the universals that host them, but this does not entail that there are no such things.3

1 “Almost” to avoid a commitment to such universals as being a thing that does not instantiate itself. 2 See, e.g. van Inwagen (2006a) for a defense of this view. van Inwagen uses “platonism” for the weaker thesis that there are some abstract objects. However, he argues not merely for that weak view but also for the stronger view that I am calling “Platonism.” Other Platonists (give or take a bit) include Bealer (1982), Zalta (1988), Jubien (1993 and 2009), Menzel (1993), Horwich (1998), Wetzel (2009), and Carmichael (2010). 3 Many philosophers apparently find slot theory natural and put it to work in a larger theoretical apparatus, though typically without stopping to spell out the adverse effects of rejecting slots. See Williamson (1985: 257), Zalta (1988: 52), Menzel (1993: 82), Armstrong (1997: 121), Swoyer (1998: 302), Yi (1999: 168 ff), Newman (2002: 148), McKay (2006: 13), King (2007: 41), Gilmore (forthcoming), and especially Grossmann (1983: 200 and 1992: 57) and Crimmins (1992: 99–140). (Crimmins accepts both slots—which he calls “arguments”—in universals, and associated entities that he calls “roles” in propositions.)

188 | Cody Gilmore In this paper I note that slots are invoked by a natural account of the notion of the adicy of a universal: for a universal to be n-adic just is for it to have exactly n slots in it. I then consider a series of “slotfree” accounts of adicy and argue that each of them has significant drawbacks, at least given a sufficiently abundant ontology of hyperintensionally individuated universals. For those of us who accept such a theory of universals, this is a prima facie motivation for realism about slots. My goal, then, is limited: it’s to show that slot theory has a certain virtue, one that has so far gone unmentioned and perhaps unnoticed. I do not try to tally up all the virtues and vices of slot theory and argue that the former outweigh the latter. The plan is as follows. In section 2 I briefly discuss a direct argument for slots, along the lines of familiar arguments for numbers, properties, holes, fictional characters, and so on. Section 3, which constitutes the bulk of the paper develops a less direct argument: it presents an account of adicy in terms of slots and goes on to criticize the ten most natural slot-free accounts. Section 4 responds to an objection to slot theory—namely, that it’s in tension with the existence of multigrade universals.

2. EXPLICIT QUANTIFICATION OVER SLOTS Here is one very simple and direct route to slot theory. In discourse about properties and relations, we often speak as if we believed in slots. We might say that there are three argument places in the relation of being between, two argument places in the relation of loving, and just one in the property of roundness. On its face, such talk is ontologically committing. The relevant sentences seem to express propositions that entail that slots exist. Cast as an argument, the thought goes like this:

In “Propositions: what they are and how they mean” (1956: 286, originally published in 1919), Russell speaks of positions in facts, and in “Compound Thoughts” (1984: 398–9, originally published in 1923), Frege speaks of positions in senses. Paul Horwich (1998: 91) speaks of positions in propositional structures (which he apparently takes to be sui generis abstract entities). Linda Wetzel (2009: 134) accepts even such sui generis abstract entities as places in flag types (e.g. the position in the flag type Old Glory occupied by the third red stripe from the bottom).

Slots in Universals | 189 (1) There are three slots in the relation of being between. (2) Therefore, there are slots. Thus, at first glance, we have an analogy between an argument for slots on the one hand and familiar arguments for numbers, properties, holes, and fictional characters, on the other (see Table 1). As far as I am aware, however, no one has advanced an argument of this sort for slots. A frequent assumption, I suspect, is that any apparent commitment to slots can be “paraphrased away” much more easily than can commitment to holes, fictional characters, and so on. The opponent of slots can say that what’s obvious is not that (1) is true, but rather that it’s “in the vicinity” of a truth, and that it’s better-with-respect-to-truth than such sentences as

Table 1. Arguments for numbers, properties, holes, and fictional characters

Numbers

Properties

Holes

Fictional Characters

There are prime numbers.

There are properties that you and I share.

There are remarkably many holes in this piece [of Gruyere] (Lewis and Lewis 1970: 206).

There are characters in some nineteenthcentury novels who are presented with a greater wealth of physical detail than is any character in any eighteenthcentury novel (van Inwagen 2001: 43).

Therefore, there are numbers (Schaffer 2009: 357).

Therefore, there are properties (Schaffer 2009: 358).

Therefore, there are holes.

Therefore, there are characters.

190 | Cody Gilmore (3) There is exactly one slot in the relation of being between; (1) at least gestures in the direction of a truth, whereas (3) does not even do that. In particular, the opponent of slots can say that (1), though false, gestures in the direction of (1*) The relation of being between is triadic, which is true (and does not entail (2)), whereas (3) gestures in the direction of (3*) The relation of being between is monadic, which is false. Thus one can respect the Moorean fact that (1) is better than (3) without incurring any commitment to (2) or the existence of slots. Further, it’s hard to see what is lost when the friend of universals drops ontically-loaded slot-talk in favor of ontically-lessloaded adicy talk. When we do indulge in slot-talk, its only purpose seems to be that of specifying—in a picturesque and perhaps metaphorical way—the adicy of universals. This suggests that any Moorean truths that can be expressed or gestured toward by sentences that explicitly quantify over slots can also be expressed by sentences that use adicy predicates and do not explicitly quantify over slots. So, anyway, the opponent of slots will be inclined to argue. (Presumably the arguments for numbers, properties, holes, and fictional characters cannot be dealt with quite so easily, since satisfying paraphrases of their premises are harder to formulate.)

3. ACCOUNTS OF ADICY The argument in the previous section for slots takes, as its starting point, a sentence that explicitly quantifies over slots. This limits its appeal. There are Platonists who are antecedently skeptical of slots and who exhibit no tendency to accept such sentences in the first place. Other things being equal, a better strategy would be to start with something closer to the core of Platonism, acceptable even to skeptics about slots, and argue that this generates pressure toward slots. I think that certain facts about adicy fit the bill. Some universals are monadic, others are dyadic, still others are triadic, and so on. These and related facts lie at the heart of any form of realism about

Slots in Universals | 191 universals, not merely the Platonist’s extreme realism.4 What is it for a universal to be, say, dyadic? As I mentioned earlier, the slot theorist will find it natural to answer: for a universal u to be dyadic is for there to be exactly two slots in u and, more generally, for a universal u to be n-adic is for there to be exactly n slots in u. If we could show that this is the best answer to the question, we would have a promising new argument for slot theory. I won’t attempt anything so ambitious. What I will do is argue that the leading slot-free answers have some significant—and heretofore unmentioned—drawbacks, which the slot theorist’s answer avoids. I leave the weighing of costs and benefits to others.

3.1. First slot-free account: Fundamental one-place adicy predicates One can give an account of something without giving an analysis. Accordingly, the first slot-free account takes “is dyadic” to be a primitive one-place predicate that expresses a fundamental, unanalyzable property, being dyadic. Parallel treatment is given to “is monadic,” “is triadic,” and so on. This avoids any commitment to slots, but it faces three problems.5 First, it leads to an unwelcome inflation of our ideology. We get a new fundamental adicy predicate for each number that specifies the adicy of a universal. Second, it apparently prevents us from offering any explanation of the fact that necessarily, no universal is both monadic and dyadic. Without an analysis of the relevant properties, we seem forced to accept all such incompatibility facts as brute. Third, the present account makes it a complete mystery why our predicates for adicy properties incorporate number prefixes such as “mon,” “dy,” “tri,” and so on. To borrow some language from Sider (which he uses in the context of making a different point), the prefix “dy” would be semantically inert in “dyadic,” “like the occurrence of ‘nine’ in ‘canine’” (2009: 389–90). Given the fundamentality and unanalyzability of the relevant properties, we might just as well have coined a bunch of syntactically simple predicates (“blorgs,” “fooms,” “kibs,” . . . ) for those properties! Relatedly, in the case of 4

Thanks to Ted Sider for suggesting that I frame the issue this way. These three problems are analogous to those facing the corresponding account of perforatedness properties in Lewis and Lewis (1970). 5

192 | Cody Gilmore certain apparent truths about adicy that are not simple ascriptions of determinate adicy properties, the present account leaves us with no way even to express these truths. For example: (4) The number that specifies the adicy of loving is greater than the number that specifies the adicy of being triangular. (5) The number that specifies the adicy of loving is equal to the number of characters in the paper “Holes.” (6) For any u, x, and n, if x is an ordered n-tuple and x instantiates u, then u is n-adic. (7) For any u, x, and n, if x is an ordered n-tuple and there is an atomic proposition that predicates u of the first item in x, . . . , and the nth item in x, in that order, then u is n-adic. The fundamental language of the first account includes one-place predicates for determinate adicies (“is monadic,” “is dyadic,” . . . ). Thus it can be used to formulate sentences like “if u is monadic, then it is not dyadic.” But that language does not include any twoplace predicate such as “__specifies the adicy of . . . ” or “ . . . is __-adic” that are satisfied by ordered pairs whose members are universals and numbers. And there doesn’t appear to be any satisfactory way to define these predicates in terms of the fundamental language of the first account. Thus the first account is left with no way to express (4)–(7). Perhaps the friend of the first account can bite the bullet on (4) and (5). They may not seem especially fundamental, and they don’t obviously do work in a Platonist theory of universals. But (6) and (7) do seem relatively fundamental and most Platonists presumably will see them as doing work in a theory of universals. At least in some cases, the reason why there is no proposition that predicates a certain universal u of some things in a certain order is that the adicy of the universal rules it out. To make this style of explanation explicit, we need (7) and its association of numbers with adicy properties. I suspect that similar remarks apply to (6). The slot theorist has none of these problems. Concerning the first, his account invokes just one distinctively “adicy-related” piece of fundamental ideology: the two-place predicate “is a slot in.” Each oneplace adicy predicate gets defined in terms of “is a slot in” together with further general purpose fundamental ideology that everyone already employs: “u is monadic” gets defined as “there is exactly one

Slots in Universals | 193 slot in u,” “u is dyadic” as “there are exactly two slots in u,” and, more generally, “u is n-adic” as “there are exactly n slots in u.” (Alternatively, the slot theorist might define “u is n-adic” as “n numbers the slots in u” or as “n is the cardinality of the set {x: x is a slot in u}.”) Concerning the second problem, the slot theorist sees each of the relevant “incompatibility facts” as an instance of the following general schema, already accepted by everyone on independent grounds: necessarily, for any u, any n, and any n*, if n≠n*, then if there are exactly n entities that R u, it’s not the case that there are exactly n* entities that R u.

Finally, concerning the third problem, the number prefixes in “monadic,” “dyadic,” etc., are obviously not semantically inert for the slot theorist; rather, they have the same numerical content in those words as they do elsewhere. And the slot theorist’s language, with its fundamental “is a slot in” predicate, permits natural definitions of “__ specifies the adicy of . . . ” and “ . . . is __-adic,” which allow him to express (6) and (7) more or less as written. 3.2. Second slot-free account: A primitive two-place “specifies-theadicy-of” predicate A second strategy for the opponent of slots is to take “__ specifies the adicy of . . . ” or “ . . . is __-adic” as primitive and fundamental, rather than as being analyzed in terms of “is a slot in.” Again we avoid any commitment to slots, and this time we economize on fundamental adicy predicates, making do with just one (“__ specifies the adicy of . . . ”) where the first account required a great many (“is monadic,” “is dyadic,” “is triadic,” . . . ). Moreover, the friend of the second account can express (4)–(7) in his fundamental language just as easily as the slot theorist can. The key feature of those sentences is just that they use a two-place adicy predicate expressing a relation that holds between universals and numbers. The second account is tailor-made for these sentences. This account faces two potential problems of its own, however. 3.2.1. First problem: Brute facts Advocates of the second account will apparently be forced to take it as a brute fact that

194 | Cody Gilmore (8) necessarily ∀x∀y∀z[(x specifies the adicy of z and y specifies the adicy of z) → x = y] and that (9) necessarily ∀x∀y[x specifies the adicy of y → x is a cardinal number]. According to (8), nothing can have its adicy specified by more than one thing. (Or, if you like, nothing can bear the adicifying relation to more than one thing.) According to (9), the only entities that can specify the adicies of things are cardinal numbers. (In other words, nothing can bear the adicifying relation to anything but a cardinal number.) Thus, while there might be a universal whose adicy is specified by 6 or even ℵ0, there couldn’t be a universal whose adicy is specified by 2.5, p, or the Eiffel Tower. If, as the slot theorist is free to claim, the adicy of a universal is just the number of slots in the universal, then (8) and (9) are easy to explain. In speaking of the number of slots in a universal, we are speaking of the cardinality of the set of those slots, and it is independently known that, necessarily, each set has only one cardinality (thus explaining (8)), and that, necessarily, only cardinal numbers can be cardinalities of sets (thus explaining (9)).6 Everyone—slot theorists and their opponents alike—already accepts these latter 6 Alternatively, the slot theorist is free to avoid talk of sets and instead paraphrase “there are two slots in loving” in plural terms, as “2 numbers the slots in loving.” In that case, he could explain (8) by appeal to the general principle that (8*) necessarily, for any y, any z, and any X, if y numbers X and z numbers X, then y = z, and he could explain (9) by appeal to the general principle that (9*) necessarily, for any y and any X, if y numbers X, then y is a cardinal number. Raul Saucedo has suggested a potential counterexample to this latter principle, (9*). Yesterday I ran exactly 1.5 miles. So 1.5 numbers the miles that I ran yesterday. But 1.5 is not a cardinal number. In response, I want to suggest that the sentence

(R)

1.5 numbers the miles that I ran yesterday

is either false or irrelevant to (9*). On a reading that makes it relevant to (9*), (R) entails (R.1)

∃X [1.5 numbers X & the miles that I ran yesterday = X],

where “=” expresses plural identity. But (R.1) is implausible. Surely it is not the case that there are some things such that 1.5 numbers them. (If 1.5 numbers them, then there is more than one of them, but fewer than two of them. If there is more than one of them, then (R.2) ∃x1∃x2[x1 is one of them & x2 is one of them & x1≠x2], but if there are fewer than two of them, then

Slots in Universals | 195 necessities, so, although they may be brute, the slot theorist is at no disadvantage vis-à-vis his opponent in appealing to them. On the other hand, if “specifies the adicy of” is fundamental and sentences like “2 specifies the adicy of loving” incur no commitment to slots, then the foregoing explanation is no longer available. Why then couldn’t more than one thing specify the adicy of a given universal?7 And why couldn’t the Eiffel Tower or p specify the adicy of a universal? As far as I can tell, the friend of the second account strategy has no answer, and must take the facts in question as rockbottom, admitting of no explanation at all. Whereas the slot theorist derives (8) and (9) from necessities that everyone already accepts, the opponent of slots (if he opts for the second account) sees (8) and (9) as additional brute necessities. This is not an automatic disqualification, but surely it is a vice. Objection. True, the slot theorist can explain (8) and (9) whereas the friend of the second account must take them as brute. But presumably the slot theorist must posit brute necessities of his own governing “is a slot in.” For example: (10) necessarily ∀x∀y[x is a slot in y → ¬ y is a slot in x] (Asymmetry of slot-in) So it hasn’t been shown that the friend of the second account is any worse off with regard to positing brute necessities than is the slot theorist. Reply. In the end, this objection may be correct. But there are two points in response to it that deserve to be aired. First point. For what it’s worth, there are a number of other arguments to which a very similar objection can be made. Consider a (R.3)

¬∃x1∃x2[x1 is one of them & x2 is one of them & x1≠x2],

which contradicts (R.2). There may be a reading on which (R) is true, but on such a reading (R) is just a stilted variant of “I ran 1.5 miles yesterday,” which, I take it, attributes a certain determinate length property, denoted by “1.5 miles” to my run. So understood, it does not entail (R.1) or generate a counterexample to (9*). 7 It is no answer to say that “specifies the adicy of” is properly symbolized as a functor rather than as a predicate. True, if we opted for this, then our formalization of “If x specifies the adicy of z and y specifies the adicy of z, then x = y” would be a logical truth (it would be “[x = adicy(z) & y = adicy(z)] → x = y”), but what justifies the assumption that “specifies the adicy of” is properly symbolized as a functor? This seems just to presuppose, rather than to explain, the fact in question. Moreover, the “functor” suggestion leaves (9) untouched.

196 | Cody Gilmore “van Inwagen-esque” argument for fictional characters. van Inwagen considers the sentence (11) there are characters in some nineteenth-century novels who are presented with a greater wealth of physical detail than is any character in any eighteenth-century novel (2001: 43), which, taken at face value, is committed to characters. He then notes that one might paraphrase away this commitment by introducing a primitive two-place predicate, “dwelphs,” that is satisfied by an ordered 〈x, y〉 pair just in case: x and y are classes of novels and, as we might intuitively put it, “there are characters in some member of x who are presented with a greater wealth of physical detail than is any character in any member of y,” so that (12) the class of nineteenth-century novels dwelphs the class of eighteenth-century novels turns out to be necessarily equivalent to (11) but is understood in such a way that it does not entail that there are characters. So far, so good for the opponent of characters. But regardless of whether one accepts characters, one will agree that the relation of dwelphing is transitive: (13) Necessarily, for any x, y, and z, if x dwelphs y and y dwelphs z, then x dwelphs z.8 8

van Inwagen himself does not discuss (13). Instead, he claims that (11*)

every female character in any eighteenth-century novel is such that there is some character in some nineteenth-century novel who is presented with a greater wealth of physical detail than she is (2001: 46)

is a logical consequence of (11) and argues that the anti-realist about characters who takes “dwelphs” as primitive will be unable to explain this fact. An analogous argument for slots runs as follows. Start with the apparent truth that (A1)

there are exactly two slots in the relation expressed by “is one of,”

which the friend of the second account will paraphrase as (A2)

2 specifies the adicy of the relation expressed by “is one of.”

Then note that (A3) if only one of the slots in the relation expressed by “is one of” is singular, then there is a slot in the relation expressed by “is one of” that is not singular is a logical consequence of (A1), and argue that if we take “specifies the adicy of” as primitive and paraphrase (A1) as (A2), we cannot explain this fact.

Slots in Universals | 197 If one rejects characters and takes “dwelphs” as primitive, one will see (13) as a brute fact. If on the other hand one is a realist about characters and defines “dwelphs” in terms of them in the natural way, then one can derive (13) from logic plus the transitivity of being-presented-with-a-greater-wealth-of-physical-detail-than, which is independently plausible. This would seem to speak in favor of realism about characters. However, just as the realist about slots has her primitive “is a slot in” predicate, governed by certain brute facts such as the Asymmetry of slot-in, the realist about characters has his own primitive predicate. For van Inwagen, it’s the three-place “ - - - is ascribed to ___ in . . . ,” which he takes to be satisfied only by ordered 〈property, character, work-of-fiction-or-part-of-a-work-of-fiction〉 triples. And this predicate is, no doubt, governed by its own group of unexplained necessary truths. So the realist about characters, like the realist about slots, avoids brute necessities in one place only by positing them in another. Perhaps this point undermines both arguments. But perhaps there is room to claim that certain principles are more appropriately taken as brute than others. Second point. In a very different vein, the slot theorist might try to argue that the fundamental relation expressed by “is a slot in,” and the brute principles governing it, are things that we already have reason to accept, even apart from considerations about universals and their adicies. In that case, though the principles would be brute, the slot theorist would be at no disadvantage vis-à-vis her opponent in positing them, whereas the opponent of slots in universals would be at a disadvantage when he takes (8) and (9) as brute. What might be the independent motivation for accepting such a fundamental relation and its associated principles? Here’s the idea. First, one might think that holes (e.g. holes in pieces of cheese) can neither be eliminated nor reduced to more familiar entities, such as material “hole-lining” objects or regions of space or spacetime. (See Casati and Varzi 1994.) In that case, one is likely to take the predicate “is a hole in” as a primitive that expresses a fundamental relation holding between holes and their hosts. Second, one might take this relation to be topic-neutral, much like identity and—according to some—parthood. (The identity relation that material objects bear to themselves is the same as the identity relation that abstract objects bear to themselves; the fundamental part–whole relation that holds

198 | Cody Gilmore between my hand and my body is the same, some say, as the fundamental part–whole relation that holds (a) between the property being a hydrogen atom and the property being a methane molecule and (b) between the semantic content of “John” and the semantic content of “John loves Mary.”) Third, on grounds of parsimony, one might simply identify the fundamental relation expressed by “is a slot in” with the fundamental relation expressed by “is a hole in.” A less misleading predicate for such a topic-neutral relation would be “is hosted by.” (Similar language is already used by Casati and Varzi 1994.) When a “hosted” entity is hosted by a concrete particular, we call it a hole (or a depression, indentation, crack, tunnel, etc.). When a hosted entity is hosted by a universal or concept, we call it a slot (argument place, argument position, etc.). But the fundamental relation between hosted entity and host is the same in both cases. Or so one might be tempted to claim. In any event, this would open up the possibility that the slot theorist’s crucial relation, and the bruteness of the principles governing its behavior, are independently motivated and therefore do not count against slot theory in the way that the bruteness of (8) and (9) would count against the second slot-free account of adicy. 3.2.2. Second problem: Ungrounded numeric adicy facts Here is a second problem for the proposal that “__ specifies the adicy of . . . ” is primitive and fundamental. Presumably, any adicy fact about a given universal should be grounded by non-numeric facts about that universal, facts that are not about numbers.9, 10 The slot theorist can respect this. Admittedly, the slot theorist does say that the property being dyadic is a numeric property: it is the property kx[2 is the cardinality of {y: y is a slot in x}], that is, the property being an x such that 2 is the cardinality of the set of slots in x.11 This directly involves the number 2. Hence any atomic fact to the effect that a certain universal is dyadic will itself be a fact about a number. However, any such fact will be grounded by a fact that is not about any number. For example, the slot theorist will say that 9 Together, perhaps, with general principles that are not about the universal in question. 10 Thanks to Ted Sider for this point. 11 Alternatively, the slot theorist may say that it is the property being an x such that 2 numbers the slots in x.

Slots in Universals | 199 (F1) the fact that loving is an x such that 2 is the cardinality of the set of slots in x which is about the number 2, is grounded by (F2)

the fact that loving is an x such that ∃y∃z[y is a slot in x & z is a slot in x & y≠z & ∀w[w is a slot in x → (w = y ∀ w = z)]],12

which is not about 2 or any number. The friend of the second account has no comparable story to tell. He will posit (F3) the fact that loving is an x such that 2 specifies the adicy of x, which is a numeric fact about the number 2. But given that specifying the adicy of is taken as fundamental and unanalyzable, it’s hard to see what non-numeric fact about loving might ground (F3).

3.3. Third slot-free account: Define “specifies the adicy of” in terms of a non-distributive instantiation predicate So let us consider a third slot-free account of adicy. As with account two, we employ the predicate “specifies the adicy of,” but now, instead of taking it as primitive and fundamental, we define it. An initial thought is that “n specifies the adicy of u” means something like “u can be instantiated by n things.” If we treat “instantiate” as a non-distributive predicate, we can sharpen this suggestion with the following definition: (D0)

x specifies the adicy of y = df. y is a universal13 & ∀Z[Z instantiates y → x numbers Z].

The idea is that for one entity, n, to be the adicy of another, u, is for u to be a universal that is instantiated only by “n-membered pluralities.” Thus redness has 1 as its adicy because, for any Z, if they instantiate redness, then 1 numbers them (there is exactly one of 12

Together, perhaps, with general principles, not about loving, that link non-settheoretical, non-numerical, purely quantificational claims (like those in (F2)) to settheoretical and/or numerical claims (like those in (F1)). 13 Without this clause, (D0) would count any non-universal (e.g. my computer) as having everything as its adicy. Nothing instantiates my computer and, a fortiori, nothing instantiates my computer that is not a seven-membered plurality.

200 | Cody Gilmore them); and being taller than has 2 as its adicy because, for any Z, if they instantiate it, then 2 numbers them (there are exactly two of them).14 One obvious problem for (D0) arises from non-asymmetric dyadic relations, such as identity (symmetric) and being at least as tall as (non-symmetric). Identity is dyadic but is only ever instantiated by one-membered “pluralities.” Being at least as tall as is dyadic but is in some cases instantiated by two-membered pluralities, in other cases by one-membered pluralities. It is in part because of examples like these that we speak of an n-adic universal as being instantiated by an ordered n-tuple of entities or, alternatively, by some entities in a given order (where the order is “of length n”). This suggests a natural improvement on the third account.

3.4. Fourth slot-free account: Define “n is the adicy of u” as “u is instantiated only by n-tuples” Again we define “specifies the adicy of” partly in terms of “instantiates,” but now we eliminate the plural quantifier and variable and treat “instantiates” as a distributive predicate that is satisfied only by ordered 〈ordered-tuple, universal〉 pairs, and we appeal to another two-place predicate, “ . . . is an ordered ___-tuple,” understood as being satisfied only by 〈ordered n-tuple s, positive integer n〉 pairs: (D1) x specifies the adicy of y = df. y is a universal & ∀z[z instantiates y → z is an ordered x-tuple].15

14

Here “instantiates” needs to understood in such a way that “a instantiates u” and “b instantiates u” do not jointly entail “a and b instantiate u.” Otherwise (D0) would fail to count redness as monadic, given that that stop sign instantiates it, this book instantiates it, and that stop sign ≠ this book. Likewise “instantiates” needs to be understood in such a way that (ii) “a and b instantiate u” entails neither “a instantiates u” nor “b instantiates u.” Otherwise (D0) would fail to count being taller than as dyadic. 15 Those who wish to avoid talk of n-adic universals being instantiated by n-tuples can replace (D1) with (D1*)

x is the adicy of y = df. y is a universal & ∀w∀Z[Z instantiate y in order w → w is of length x],

provided that they are able to make sense of the somewhat mysterious-sounding predicates involved. Intuitively, the thought underlying (D1) is that instantiation

Slots in Universals | 201 Again we get the result that 1 specifies the adicy of redness, since redness is instantiated only by 1-tuples, and that 2 specifies the adicy of being taller than, since it is instantiated only by 2-tuples. But now identity is correctly classified as dyadic, since it is instantiated only by 2-tuples. Likewise for being at least as tall as. Whatever its virtues may be, this strategy obviously has limited appeal. In particular, it will be rejected by those who believe that there are uninstantiated universals, such as the relation having given a golden mountain to (or R, for short). Intuitively, the only entity that specifies the adicy of R is the number two: R is dyadic. According to (D1), however, R has everything as its adicy. After all, nothing instantiates R, and so it is vacuously true that every x is such that, for any z, either z does not instantiate R or z is an x-tuple. Of course this poses no problem for those (e.g. Armstrong 1997) who reject uninstantiated universals, but the rest of us will want to find an alternative to (D1).

3.5. Fifth slot-free account: Define “n specifies the adicy of u” as “u can be instantiated only by n-tuples” One potential fix is to say that what it is for a universal u to be n-adic is for u to be such that, not merely in fact, but as a matter of necessity, the only things that instantiate it are n-tuples. This gives us: (D2)

x specifies the adicy of y = df. y is a universal and necessarily, for any z, if z instantiates y, then z is an x-tuple.

The uninstantiated dyadic relation R is no problem for (D2). For R is a universal, and although it is in fact uninstantiated, it is not necessarily so: it is possible for someone to have been given a golden mountain by someone. Moreover, it is plausible that the number 2 is the one and only entity that satisfies the open sentence “necessarily, might be a three-place relation that can hold between (i) some things, (ii) an n-adic universal, and (iii) an entity—call it an “order”—that specifies an order of length n in which the given things can be “taken.” Thus a given plurality, say, the surviving Beatles, might instantiate loving in w but not in order w*, where w specifies that Ringo comes first, Paul second, and w* specifies that Paul comes first, Ringo second. Orders will need to bear some “length” relation to entities, and presumably it will need to turn out that (a) if w is an order, then w bears the length relation to exactly one thing, and that (b) if w bears the length relation to y, then y is a cardinal number.

202 | Cody Gilmore for any z, if z instantiates R, then z is an x-tuple.” For it is possible that there be an ordered pair that instantiates R, and it is natural to think that it is not possible that there be anything other than an ordered pair that instantiates R. In that case (D2) yields the intuitively correct result that 2 is the one and only entity that R has as its adicy. (D2) does face an obvious problem, however. Just as (D1) falters on the case of uninstantiated universals whose adicies are specified by exactly one entity, (D2) falters in the case of uninstantiable universals whose adicies are specified by exactly one entity. Here I have in mind things like the monadic property being both round and square or the dyadic relation being both larger and smaller than. Since these are necessarily uninstantiated, (D2) tells us that they have everything as their adicy. But this is incorrect: there are many things, such as the number 12 and the Eiffel Tower, that do not specify the adicy of either of the universals just mentioned.16 There is a further problem for (D2) that arises even if there are no uninstantiable universals. Consider the fact, concerning a given universal u and number n, that u cannot be instantiated by anything other than n-tuples. This is a de re modal fact about u. Prima facie, it may seem desirable to treat the de re modal facts about universals as being grounded by non-modal facts about those universals. In particular, some may have found it plausible that the modal fact that u cannot be instantiated by anything other than n-tuples is grounded by (among other things, perhaps) the non-modal fact that u is n-adic. That is, it may seem plausible that the given modal fact obtains because u is n-adic. According to (D2), however, the fact that u is n-adic just is the given de re modal fact, and so cannot ground it. In somewhat different terms, if adicy facts are metaphysically prior to de re modal facts about instantiation, then the former cannot simply be defined as a species of the latter. Call this the “priority problem.” 16 There is a second objection against (D2) that deserves a brief mention, though I do not endorse it. Just as one might say that some universals have certain of their causal powers accidentally, one might wish to hold that some universals have even their adicies accidentally. Thus, e.g. one might say that there is a universal u that is in fact dyadic but that could be triadic (and instantiated). (D2) rules this out. For if u could be triadic and instantiated, then presumably it could be instantiated by a 3-tuple. But in that case u is not necessarily such that the only things that instantiate it are 2-tuples, and hence (D2) denies that u is in fact dyadic. Of course, this objection disappears if (as most philosophers seem to think, and as I tend to agree) universals have their adicies essentially.

Slots in Universals | 203 (Consider an analogy. Suppose that one thinks that a certain statue, Goliath, cannot survive being squashed. Further, suppose that Goliath cannot survive being squashed because Goliath is a statue. In that case one must not define “is a statue” as “is a thing that cannot survive being squashed.”)

3.6. Sixth slot-free account: Adicies of uninstantiable universals explained by appeal to facts about the universals in terms of which they are analyzed One might claim that uninstantiable universals are always analyzable in terms of instantiable universals. One might then suggest that the facts about the adicy of an instantiable universal can be explained by appeal to something like (D2), and that the facts about the adicy of an uninstantiable universal u can be explained by appeal to facts about the adicies of the instantiable universals in terms of which it is analyzed, together with facts about the manner in which those universals are combined in u.17 Consider the uninstantiable universal being both round and square. It is natural to think that this universal results from applying a certain logical operation, call it conjunction, to two instantiable universals, namely being round and being square.18 Since being round is instantiable, the present strategy tells us that the facts about its adicy can be explained by appeal to (D2); and since being round can be instantiated by 1-tuples only, (D2) tells us that 1 is the adicy of being round. Likewise for being square. Finally, it is presumably a truth about the conjunction operation that for any x, any y, and any 17 Alternatively, rather than applying (D2) directly to any instantiable universal, one might prefer to apply it only to “simple” or unanalyzable universals (which according to the present suggestion will all be instantiable). The variant proposal suffers from the same problems that I raise for the fifth strategy, plus one further problem of its own: it fails if all universals are analyzable into further universals (a possibility that some philosophers—notably Armstrong—have been unwilling to rule out). 18 The basic conjunction operation is often regarded as one that takes in an m-adic universal u and an n-adic universal u* as arguments and yields an m + n-adic universal u** as value. See Menzel (1993), Swoyer (1998), and footnote 19. Accordingly, in the main text, when I use the term “conjunction,” I do not refer to that basic operation, but rather to one that takes being round and being square as arguments and yields being both round and square as value.

204 | Cody Gilmore z, if y is monadic, z is monadic, and x = conjunction(y, z), then x is itself monadic.19 This yields the desired result: being both round and square is monadic.20 Of course, no help has been offered with the priority problem that afflicts the simple, purely modal approach of the previous strategy. But the sixth account also generates three additional complaints. (We should also note that so far, the sixth account merely gestures in the direction of a definition of “is the adicy of,” rather than actually giving one.) First, the sixth account strikes me as being analogous to the definition of “is red” as “is either scarlet or crimson or . . . .” To oversimplify, the sixth account says something like this: 19 Menzel (1993) develops a formal language that uses lambda abstracts as singular terms that refer to properties, relations, and propositions (PRPs): thus “[kx(x is round)]” is a singular term, in the same grammatical category as “John”, that refers to being red, “[kx(x is round & x is square)]” is a singular term that refers to being both round and square, etc. In sketching the semantics for this language, Menzel employs notions for various logical operations—e.g. reflection, conjunction—by means of which PRPs can be “combined.” In the closely related terminology of Swoyer (1998: 303), the reflection1,2 operation takes in an n-adic universal (n≥2) u as argument and yields an n−1 adic universal u* as value, where, intuitively, u* is what results from “identifying the first and second argument places of u.” To illustrate, being self-identical is the reflection1,2 of identity: [kx(x = x)] = reflection1,2([kxy(x = y)]). The basic conjunction operation takes in an m-adic universal u and an n-adic universal u* as arguments and yields an m + n-adic universal u** as value. The dyadic relation being an x and a y such that x is red and y is blue is the (basic) conjunction of being red and being blue: [kxy(x is red & y is blue)] = conjunction([kx(x is red)], [kx(x is blue)]). With these notions in hand, we can say, first, that [kx(x is round & x is square)] = reflection1,2([kxy(x is round & y is square)]), and second, that [kxy(x is round & y is square)] = conjunction([kx(x is round)], [kx(x is square)]). Hence we get the result that [kx(x is round & x is square)] = reflection1,2(conjunction([kx(x is round)], [kx(x is square)])). Since, for any x, y, z, if y is 1-adic, z is 1-adic, and x = reflection1,2(conjunction(y, z)), x is 1-adic, and since [kx(x is round)] and [kx(x is square)] are both 1-adic (according to (D2)), we get the result that [kx(x is round & x is square)] is 1-adic too. Menzel’s and Swoyer’s treatment of the given operations is based on that of Bealer (1982). Bealer mentions some analogies between his operations and the predicate functors discussed by Quine (1995, originally published in 1960). 20 One way to make this line of thought a bit more precise and general is via the following trio, the first two of which are definitions and the third of which is a definition schema:

(D3.1)

x specifies the adicy of y = df. either (i) y is an uninstantiable universal and x specifies the adicyu of y, or (ii) y is an instantiable universal and x is the adicyi of y. (D3.2) x is the adicyi of y = df. (i) y is instantiable and (ii) necessarily, for any z, if z instantiates y, then z is an x-tuple. (D3.3) x is the adicyu of y = df. (i) y is uninstantiable and (ii) ______,

Slots in Universals | 205 for a universal u to be such that its adicy is specified by (say) the number 2 is for either (i) u to be instantiable but only by 2-tuples, or (ii) u to result from applying operation o1 to a universal of type A, or (iii) u to result from applying operation o2 to a pair of universals of types B and C, or (iv) u to result from applying operation o3 to a universal of type D, or . . . .

I find both definitions implausible, and for similar reasons. Just as redness is intuitively a relatively simple property (unlike the property of being either scarlet or crimson or . . . ), adicy seems relatively simple as well—in any case it seems much simpler than the sixth account makes it. Second, it is preferable, other things being equal, to give a fully uniform account of adicy, rather than making its definition into a disjunctive affair, with a modal part that applies to instantiable universals, and a very different non-modal part that applies to uninstantiable universals. After all, being dyadic is, I take it, a relatively natural, nongerrymandered, non-disjunctive property of universals: when two universals both have the given property, they will genuinely resemble each other in at least one respect, even if one of them is instantiable and the other is not. A disjunctive definition such as the one suggested earlier makes a mystery out of this evident fact. Third, it is unclear how much weight we (as lovers of abundant, hyperintensionally individuated universals) should be willing to rest on the hope that all uninstantiable universals are analyzable. True, no one has produced a very persuasive example of an uninstantiable universal that appears to be unanalyzable,21 but the non-existence of such universals is not the only plausible explanation for this. One might where the blank will be filled in roughly as follows: There is an n-tuple of instantiable universals 〈u1, . . . , un〉 of adiciesi a(u1), . . . , a(un) respectively, and either • y is formed by performing operation o1 on the given universals in the given order and x results from performing operation o*1 to 〈a(u1), . . . , a(un)〉, or • y is formed by performing operation o2 on the given universals in the given order and x results from performing o*2 on 〈a(u1), . . . , a(un)〉, or . . . The basic idea, of course, is just that (D2) works for instantiable universals and that some other account can be given for the uninstantiable ones. 21 Though here are a few tries. (1) Perhaps it is impossible that there be a unicorn (as argued by Kripke 1980), and yet there is such a thing as being a unicorn, where this is not merely uninstantiable but also unanalyzable (as typical examples of properties corresponding to species appear to be) and monadic (i.e. monadic only, not dyadic, etc.). (2) Perhaps eliminativist theories of color (or value or morality) are necessarily true, so that necessarily, nothing instantiates any color (or value or moral) properties.

206 | Cody Gilmore think, e.g. that the only universals that we can grasp or refer to are (i) those whose instances have affected us, or (ii) those that are sufficiently similar to the universals in group (i) as to be graspable by something like “extrapolation,” or (iii) those that are analyzed in terms of the universals in (i) or (ii). On the assumption that any universal that bears the relevant degree of similarity to an instantiated universal must itself be instantiable, this view would predict that we do not grasp any uninstantiable unanalyzable universals, whether or not there are any.22

3.7. Seventh slot-free account: Counterfactual definitions of “specifies the adicy of” Accounts five and six attempted to define “specifies the adicy of” by appeal to the necessity operator, together with the notion of instantiation, the notion of an n-tuple, and, in the case of the sixth strategy, notions for a variety of logical operations, such as conjunction. Perhaps the necessity operator is too blunt an instrument for the task at hand. A natural alternative is to appeal to the counterfactual conditional operator, perhaps as follows: (D4) x specifies the adicy of y = df. (i) y is a universal,23 (ii) if y were instantiated by something, it would be instantiated by an ordered x-tuple.24

It might still be the case that there is such an entity as (e.g.) the property of redness (or wrongness or goodness), and that is monadic (only) despite being unanalyzable and uninstantiable. Indeed, we might even be acquainted with it in experience, perhaps by standing in a certain relation to a proposition that (falsely) predicates redness of some object in our visual field. (3) Perhaps some version of presentism that entails that nothing is earlier than anything is a necessary truth, and yet the relation earlier than exists and is dyadic (only) despite being unanalyzable and uninstantiable. Again we might be acquainted with such a relation in experience. (4) Perhaps some version of the B-theory of time that entails that nothing instantiates any A-properties is necessarily true, and yet there is an unanalyzable monadic property of presentness with which we are acquainted by standing in certain relations to propositions that (falsely) predicate this property of certain events. 22 We might also add that (D3) is no improvement over (D2) with respect to the “problem” of universals that have a single adicy accidentally. 23 Without this clause, we might get the result that I am (say) monadic. 24 For those who want to avoid speaking of n-adic universals being instantiated by n-tuples, this definition can be restated in the manner suggested in footnote 15. The discussion to follow would then need to be restated accordingly.

Slots in Universals | 207 The thought here is that even if a universal is uninstantiable, it can still be non-vacuously true that if it were instantiated, it would be instantiated by, say, an ordered pair.25 Consider being both larger and smaller than. Now, although it is impossible for this universal to be instantiated, it nevertheless seems true that if it were instantiated, it would be instantiated by an ordered pair, i.e. by a 2-tuple. According to (D4), then, the universal in question is dyadic: it bears the adicifying relation to the number 2. This provides a uniform, nondisjunctive account of the adicies of all universals, instantiable and uninstantiable alike. It may, however, be vulnerable to counterexamples. Suppose that there is such a universal as being an x such that: x is round and x is such that anything that is instantiated is instantiated only by ordered 2-tuples,26 and call it U4 for short. Intuitively, U4 is monadic and uninstantiable, much like being an x such that: x is round and x is square. It is not dyadic or triadic or . . . . But in order for (D4) to yield the intuitively correct verdict that U4 is not 2-adic, it needs to turn out that (14) it’s not the case that: if U4 were instantiated, it would be instantiated by an ordered 2-tuple. But it’s not clear to me that (14) is true. For suppose that, per impossibile, U4 were instantiated. Then, given the nature of U4, everything that is instantiated, including U4 itself, would be instantiated by an ordered 2-tuple. So U4 would be instantiated by a 2-tuple. This makes me doubt (14).27 Now, since U4 is uninstantiable, (14) is the negation of a counterpossible. Perhaps, even if some counterpossibles are non-vacuously true and others non-vacuously false, there is something distinctive about them that undermines my case against (14). Alternatively, perhaps even friends of abundant, hyperintensionally individuated 25 Contrary to Stalnaker (1968) and Lewis (1973), both of whom say that counterfactual conditionals with impossible antecedents are vacuously true. For opposition to Lewis and Stalnaker on this point, see, e.g. Nolan (1997), Kim and Maslen (2006), and Dorr (2008). 26 In the language of Menzel (1993), we could refer to U4 with the expression “[kx(x is round & ∀y∀z (z instantiates y → z is a 2-tuple))].” 27 Similar problems arise if we define “n specifies the adicy of u” as “u is a universal, and if u were instantiated, it would be instantiated only by n-tuples” or “u is a universal, and n is the one and only entity that is such that: if u were instantiated, it would be instantiated by n-tuples.”

208 | Cody Gilmore universals have independent reason for denying the existence of U4 and other universals that would generate counterexamples to (D4). Maybe—but for the moment I am unable to see how to fill out these replies, and accordingly I am persuaded by the argument against these definitions.28 Finally, we should note that counterfactual definitions of “specifies the adicy of” make no progress on the priority problem. Suppose that one thinks that facts of the form u cannot be instantiated by anything other than n-tuples are grounded by facts of the form u is n-adic. Further, suppose that one takes it to be non-vacuously true that (C) if being both larger and smaller than were instantiated, it would be instantiated by a 2-tuple. In that case, one ought to find it plausible as well that the relevant counterfactual fact is grounded by the fact that that given universal is dyadic, and one ought to take this latter fact to be non-modal and non-counterfactual. This is in tension with counterfactual definitions of “specifies the adicy of.”

3.8. Eighth account: Defining “specifies the adicy of” in terms of essence If the necessity operator is too blunt, perhaps the right response is to appeal to a notion of essence that is not analyzed in modal or

28 A very different counterfactual definition of “x specifies the adicy of y” runs roughly as follows:

(D4c)

x specifies the adicy of y = df. if there were such things as slots in universals, then x would number the slots in y.

But if one were willing to paraphrase away “slot talk” in terms of “counterfactualized slot talk,” presumably one should also be willing to paraphrase away “property and relation talk” in terms of “counterfactualized property and relation talk” in the manner advocated by Dorr (2008). (For example, Dorr paraphrases “spiders and insects share some anatomical properties” as “if there were abstract objects and the concrete world were just as it actually is, then spiders and insects would share some anatomical properties.” He says that the former is true, taken superficially, iff the latter is true, taken fundamentally.) My target audience in this paper is confined to Platonists, who I take it have some reason for being dissatisfied with Dorr-style paraphrases of property and relation talk. I assume that any such reason will apply with equal force to (D4c).

Slots in Universals | 209 counterfactual terms (Fine 1994).29 We might try using such a notion alone or in combination with those expressed by modal and/or counterfactual operators. The simplest definition along these lines runs as follows: (D5) x specifies the adicy of y = df. it is essential to y that: for any z, if z instantiates y, then z is an x-tuple.30 To see why such a definition might seem promising, consider the following schema. (Italicized expressions are to be understood as schematic.) U-Essence Necessarily, for any u, if u = [kx1 . . . xnφ], then it is essential to u that: ∀y [y instantiates u if and only if ∃y1 . . . ∃yn (i) y = 〈y1, . . . , yn〉, and (ii) x . . . x φy . . . y ]31 1

n

1

n

29

Thanks to Brad Skow for suggesting something in this neighborhood. As before, those who want to avoid talk of n-adic universals being instantiated by n-tuples can restate this definition, and the discussion that follows, in the manner suggested in footnote 15. On a related point, some might be tempted to object to (D5) on the grounds that, quite generally, no non-logical truth concerning sets or ordered n-tuples is part of the essence of anything but sets or n-tuples themselves. One might take this to be the lesson of Fine’s example concerning Socrates and {Socrates}: although it is necessary that if Socrates exists, he belongs to {Socrates}, it is not essential to Socrates that he belongs to {Socrates} (or that if he exists, then he so belongs). Likewise, the objection runs, it may be necessary that if being taller than is instantiated, it is instantiated only by 2-tuples, but it is not essential to being taller than that if it’s instantiated, it’s instantiated only by 2-tuples. Two points are worth making in response. First, I suspect that the principle driving the objection overgeneralizes on Fine’s example. The property being a unit set is not itself a set or an ordered n-tuple, but it would be surprising if its essence didn’t somehow involve set-theoretical notions. Further, it might turn out that for each property and relation, its essence involves some notion concerning n-tuples, though admittedly this would be a surprise. Second, since (D5) can be restated in the manner suggested earlier, one gets the feeling the objection turns on an idiosyncrasy of my formulation of the underlying idea, rather than on any core feature of the idea itself. Accordingly, I will not pursue the objection any further. 31 Instances of U-Essence are formed as follows. There is a variable vx such that each occurrence of “x1” is to be replaced by an occurrence of vx , . . . , and there is a variable vx such that each occurrence of “xn” is to be replaced by an occurrence of vx ; there is a variable vy such that each occurrence of “y1” is to be replaced by an occurrence of vy , . . . , and there is a variable vy such that each occurrence of “yn” is to be replaced by an occurrence of vy , where vx , . . . , vx , vy , . . . , and vy are 2n pairwise non-identical variables. Further, “φ” is to be replaced by some formula fφ in 30

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210 | Cody Gilmore Here is an instance of the schema: U-Essence1 Necessarily, for any u, if u = [kx1x2(x1 is larger than x2 & x1 is smaller than x2)], then it is essential to u that: ∀y [y instantiates u if and only if ∃y1∃y2 (i) y = 〈y1, y2〉, and (ii) y1 is larger than y2 & y1 is smaller than y2]. This says, roughly, that it belongs to the essence of the universal being both larger and smaller than that it’s instantiated by a thing just in case that thing is an ordered pair whose first member is both larger and smaller than the thing’s second member.32 It is natural to think that parallel remarks apply to other universals, and indeed that every instance of the given schema is true. Further, this might seem to harmonize with (D5). Let S be an instance of U-Essence, and suppose that the lambda abstract A in S contains n variables bound by the initial occurrence of “k.” This lets us make two further claims. (i) The universal u of which A is a “canonical name” will be intuitively n-adic, and (ii) it is essential to u that u be instantiated by something y only if there are things y1 . . . yn such that y = 〈y1, . . . , yn〉.33 And of course it will be at least necessary that 〈y1, . . . , yn〉 here be an n-tuple. This makes it natural to think that for any instance of U-Essence, if the lambda abstract in that instance refers to a universal u that is intuitively n-adic, then the instance will tell us that it’s essential to u that it be instantiated only by n-tuples, as predicted by (D5). (There is conceptual space to grant (i) and (ii) while still denying that it will always be essential to which no variables other than vx , . . . , and vx occur free, and in which vy , . . . , and 1 vy do not occur at all. Finally, “x . . . x φy . . . y ” is to be replaced with the formula fψ that results from replacing each free occurrence of vx in fφ with an occurrence of vy , . . . , and replacing each free occurrence of vx in fψ with an occurrence of vy . Thanks to Linda Wetzel for catching some mistakes in an earlier formulation of the schema. n

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32 I suspect U-Essence and its instances could be restated so that they do not take instantiation as a relation that holds between an n-adic universal and an n-tuple, but rather take it to hold between (i) some things, (ii) an n-adic universal, and (iii) an order, where the order is “of length n,” as suggested in note 15. 33 I temporarily assume that “essential properties are closed under logical consequence”: if the proposition that ψ is a logical consequence of the proposition that φ and it is part of the essence of x that φ, then it is also part of the essence of x that ψ. More on this later.

Slots in Universals | 211 U that it be instantiated only by n-tuples,34 but the motivation for such a denial is unclear.) However, (D5) may run into trouble with U4, the same universal that generated apparent counterexamples to the counterfactual definition. U4, recall, is the universal being an x such that: x is round and anything that is instantiated is instantiated by ordered 2-tuples. Using machinery from Menzel (1993), we can refer to it with the lambda abstract “[kx(x is round & ∀y∀z (z instantiates y → z is a 2-tuple))],” which is grammatically a singular term. Now, to see why this universal might pose a problem for (D5), consider the following instance of U-Essence: U4-Essence

Necessarily, for any u, if u = [kx(x is round & ∀y∀z (z instantiates y → z is a 2-tuple))], then it is essential to u that: (E4) ∀v [v instantiates u if and only if ∃v1 (i) v = 〈v1〉, and (ii) v1 is round & ∀y∀z (z instantiates y → z is a 2-tuple)].

If essential properties are closed under logical consequence, then U4-Essence causes trouble for (D5) right away. For it is a logical consequence of (E4) that (E4.1) ∀v [v instantiates u → ∀y∀z (z instantiates y → z is a 2-tuple)] and hence that (E4.2) ∀v [v instantiates u → v is a 2-tuple]. So, if essential properties are closed under logical consequence, then U4-Essence tells us that it is essential to U4 that for any z, if z 34

There is conceptual space to grant, e.g. that (2*) it is essential to u that, for any y, if y instantiates u, then ∃y1∃y2 y = 〈y1, y2〉,

while denying that (3*)

It is essential to u that, for any y, if y instantiates u, then OrderedTuple (y, 2),

where “OrderedTuple(,)” is a two-place predicate that is satisfied by an 〈x, n〉 pair just in case x is an ordered n-tuple. For it might be doubted that “OrderedTuple(y, 2)” is a logical consequence of “∃y1∃y2 y = 〈y1, y2〉,” even together with definitions.

212 | Cody Gilmore instantiates U4, then z is a 2-tuple. And in that case (D5) incorrectly declares that 2 is the adicy U4. (Intuitively U4 is monadic and not dyadic.)35 Admittedly, those who embrace Fine’s notion of essence will be likely to deny that essential properties are closed under logical consequence. It is essential to me that I am human; the proposition that either snow is white or snow is not white is a logical consequence of the proposition that I am human; but, plausibly, it is not essential to me that I am such that either snow is white or snow is not white. So not all the logical consequences of a thing’s essence themselves belong to its essence. But it doesn’t follow that none of them do, or that we shouldn’t expect some of the more direct and obvious ones to do so. Suppose that it is essential to a that: a is F & a is G. Then it seems that, other things being equal, we should expect it to be essential to a that: a is F. Or, suppose that it is essential to b that: all Fs are Gs & b is F. Then it seems that, other things being equal, we should expect it to be essential to b that: b is G. Now one might think that the case of U4 is more similar to these simple cases than it is to the case involving me and snow’s being either white or not white. That is, even if there is no closure principle in place to guarantee that it is essential to U4 to be instantiated only by 2-tuples, it still seems natural to expect that this belongs to U4’s essence, given what else we know about its essence (via U4Essence). Its essence involves the relation of instantiation, the relation of being an ordered __-tuple, and the number 2 (or the associated concepts). So it’s not as if we are pulling extraneous concepts into its essence willy-nilly by the use of logical trickery. Rather, in the manner of the simple examples from the previous paragraph, we’re drawing straightforward consequences from U4-Essence while working exclusively with the concepts that are already invoked by that principle. This is far from a knock-down argument. But it does 35 To be sure, one can accept all of this and at the same time argue (again by appeal to U4-Essence) that it is also part of the essence of U4 that it be instantiated only by 1-tuples. In that case one would say that, according to (D5), U4’s adicy is specified by both 1 and 2. This is cold comfort to the friend of (D5). I take it to be intuitively clear not merely that 1 does specify the adicy of U4 but also that 2 does not. Again, even lovers of abundant fine-grained universals might have independent reason to deny the existence of U4.

Slots in Universals | 213 show, I think, that if (D5) has an advantage over (D4) vis-à-vis U4, the advantage is not clear-cut. Further, (D5) sheds no light on the apparent fact that adicies are intrinsic properties.36 Essences need not be intrinsic:37 Why then think that being an x such that it is essential to x that x is instantiated only by 2-tuples should turn out to be intrinsic? There is no obvious reason. (More on the intrinsicness of adicies later on.)38 As I mentioned earlier, (D5) is just the simplest and most naïve attempt to define “specifies the adicy of” in terms of essence. Perhaps some other definition of its kind would fare better. But since no alternative “essentialist” definition suggests itself as especially promising, and since a broad survey would be tedious and probably not very illuminating, these brief remarks will have to suffice.

3.9. Ninth account: Define “specifies the adicy of” in terms of canonical names In my discussion of the previous strategy, I made use of the notion of a canonical name of a universal.39 I said that if a given universal had, as a canonical name, a lambda abstract containing n variables bound by the initial occurrence of the lambda operator, “k,” then

36 In section 3.10, I offer two considerations in favor of the view that adicies are intrinsic. The second of these (the only consideration remotely resembling an argument) is that the view plays a role in a natural explanation of the fact that for each universal u, if u is n-adic, then necessarily, if u exists, it’s n-adic. But those who embrace U-Essence seem to have an alternative explanation of the given fact: if being n-adic just is being an x such that it is essential to x that it is instantiated only by n-tuples, then presumably if a thing is in fact n-adic, it is essentially n-adic. (I assume that if a thing x is essentially F, then necessarily, if x exists, it’s essentially F.) No appeal is made to the intrinsicality of adicies in this explanation. So friends of U-Essence may be under less pressure than the rest of us to treat adicies as intrinsic. Thanks to Carrie Ichikawa Jenkins for pointing this out to me. 37 I have the property being an x such that it is essential to x that: Gene Gilmore is a parent of x. But perhaps I could have an intrinsic duplicate who has different parents and so lacks this property. If so, then the relevant essence property is extrinsic. 38 Perhaps there is also room to raise a version of the priority problem for (D5). One might think that, if it is part of the essence of u that it be instantiated only by n-tuples, then this is because u is n-adic. 39 See van Inwagen (2006b) for an extensive discussion of the notion of canonical names for properties, relations, and propositions.

214 | Cody Gilmore the universal would be intuitively n-adic. Christopher Menzel describes these lambda abstracts as follows: The more formal counterpart to the property denoting form above is ⌜being an object v such that φ⌝ where (typically, but not necessarily) φ contains a free occurrence of the variable v. We will symbolize this expression more concisely with the term ⌜[kvφ]⌝. Thus, for instance, “being something such that someone desires it” translates into “[kx∃yDyx].” (So in this context “kx” plays the role of the variable binding operator “being an object x such that.”) The form of expressions denoting n-place relations generally, then, is ⌜being objects v1, . . . , vn such that φ⌝ (e.g., “being objects x, y, and z, such that x loves y but not z”), which we symbolize with the term ⌜[kv1 . . . vn φ⌝ (e.g., “[kxyz(Lxy ∧ ~Lxz)]”). Where n≥0, this is the general form of all PRP denoting expressions. Such expressions will also be known as complex terms, or lambda abstracts. (1993: 67–8)

Given that our intuitions about the adicy of a given universal are so closely tied to the facts about the number of variables bound by the initial lambda operator in its canonical name, why make a detour through counterfactuals about instantiation, or through claims about essence? Why not just define “is the adicy of” directly in terms of facts about lambda abstracts?40 As a first try, we might say that n specifies the adicy of u just in case there are n variables bound by the initial occurrence of the lambda operator in the lambda abstract that names u. But of course we want to leave open the possibility that (i) some universals that have an adicy are named by more than one lambda abstract and that (ii) other universals that have an adicy are not named by any lambda abstracts. So, to leave these possibilities open without making any overly controversial claims about the modal profiles of lambda abstracts, we can offer the following definition: (D6) x specifies the adicy of y = df. (i) y is a universal and (ii) necessarily, ∀z [(z is a lambda abstract & z names y) → x numbers the variables bound by the initial occurrence of the lambda operator in z].41 40

Thanks to Brad Skow for suggesting this. Is it possible that a lambda abstract whose initial “k” binds two or more variables (say “[kxy(x is larger than y)],” names a monadic universal? (Perhaps this would be so if, in addition to being a lambda abstract in some language that allows the formation of such complex terms, the expression were also used as an idiom.) If such a situation is possible, presumably we could fix (D6) by requiring that the 41

Slots in Universals | 215 What does it mean to say, e.g. that 1 specifies the adicy of U4? According to (D6), it means that U4 is a universal that can be named by a lambda abstract only if the initial occurrence of the lambda operator in that abstract binds exactly one variable. I find it plausible that U4 could not be the referent (as determined by the appropriate kind of compositional semantics) of a lambda abstract of any other sort. So it seems to me that (D6), unlike any of our previous definitions, delivers the right verdict in this case. Indeed, it is easy to see that (D6) handles a wide range of cases successfully. Still, if my own reaction to (D6) is any guide, most philosophers— especially those with Platonist leanings—will find (D6) bizarre, though perhaps for reasons that are not easy to articulate immediately. For what it’s worth, (D6) initially seems to just change the subject. It takes us from talk about universals themselves and the things that have them—a broad and fundamental topic, even by metaphysical standards—to talk about linguistic expression types, their syntax, and their semantic properties—a rather narrow and superficial corner of reality. It’s natural to think that the syntax of a given lambda abstract is a guide to the adicy of the universal (if any) that it names, but this is a far cry from saying that facts about the syntax of linguistic expressions are in any way constitutive of that universal’s adicy! Let me try to sharpen all this up a little, by casting it in the form of three distinct objections. First, (D6) is incompatible with the plausible view that adicy properties, such as being dyadic, are extralinguistic properties: they do not “constitutively depend” upon facts about languages, expressions, words, etc. Second, (D6) yields implausible counterpossibles. This objection can be framed as follows: The Counterpossible Argument C1 If any version of the Ninth Account is true, then if it were impossible that there be linguistic expressions, there wouldn’t be any exclusively monadic universals (or exclusively dyadic universals or. . . .).

lambda abstract name the relevant universal in virtue of an appropriate sort of compositional semantics governing the complex terms of the language to which the lambda abstract belongs.

216 | Cody Gilmore C2 It’s not the case that: if it were impossible that there be linguistic expressions, there wouldn’t be any exclusively monadic universals. C3 Therefore, no version of the Ninth Account is true. According to (D6), to say that n specifies the adicy of u is to say that u is a universal that can be named only by lambda abstracts that have n variables bound by the initial occurrence of the lambda operator. Suppose (D6) were true and that, surprisingly, linguistic expressions were impossible. What else would then be true? In particular, which entity or entities would specify the adicy of roundness? Well, presumably roundness would still be a universal. But since it would be impossible for there to be expression types, it would also be impossible for there to be lambda abstracts, and hence it would turn out that: for every x, it is necessarily such that for any z, if z is a lambda abstract (which nothing could be) and z names roundness, then x numbers the variables bound by the initial occurrence of the lambda operator in z. In other words, roundness would turn out to have its adicy specified by each and every entity. Accordingly, roundness would not be “exclusively monadic” (in the sense of having its adicy specified by 1 and only 1). Since this argument doesn’t turn on any special features of roundness, we can generalize: there would, given the suppositions stated earlier, be no exclusively monadic universals or exclusively dyadic universal, or universals of any fixed, exclusive adicy. I assume that parallel remarks could be applied to other versions of the Ninth Account. This gives us C1. Of those who take counterpossibles seriously,42 most I suspect will want to reject the claim that if linguistic expression types had been impossible, there would be no exclusively monadic universals. Granted, it would be strange if there couldn’t be linguistic expressions. But on its face, this would have relatively little bearing on the world of universals. Maybe it would result in the non-existence of haecceitistic universals such as being identical with a, where a is the word “the.” But most universals would survive unscathed.

42 I.e. of those who think that some counterpossibles are true and that some are false.

Slots in Universals | 217 And in particular, there would still be such things as roundness and being taller than, and the facts about their adicies would still be just as they actually are. There would still be monadic universals, there would still be dyadic universals, and so on. For this reason, it seems false that if linguistic expressions were impossible, there wouldn’t be any monadic universals. So the Counterpossible Argument looks sound. Now let me turn to a third objection to (D6) and the Ninth Account: it faces an especially serious priority problem. Consider some universal u, and suppose that necessarily, it is not named by any lambda abstract that doesn’t contain n variables bound in the right way. Surely this is the wrong place for a brute fact about u! To take this as brute would be like taking it as brute that I cannot be expressed by a predicate. Although it is hard to say exactly why, I take it to be relatively uncontroversial that this latter fact cries out for explanation along the following lines: (i) I am a particular, (ii) all particulars are essentially particulars, and (iii) necessarily, predicates do not express particulars. The facts corresponding to (i)–(iii) ground the fact that I cannot be expressed by a predicate. Similarly, the fact that u cannot be named by any lambda abstract that doesn’t contain n variables—call it u-fact—cries out for explanation, and grounding, in terms that mention u, certain nonlinguistic properties that are essential to u, and certain de dicto necessities about how the syntactic properties of a lambda abstract must be mirrored by certain non-linguistic properties of any entity it names. In particular, I assume that the explanation ought to run as follows: (i) u is an n-adic universal, (ii) all n-adic universals are essentially n-adic universals, and (iii) necessarily, lambda abstracts that contain exactly n variables bound in the right way (compositionally) name only n-adic universals. These facts ground u-fact. So, contrary to (D6), u-fact must not be identified with the fact corresponding to (i). 3.10. Tenth account: Saturation One might wish to define “specifies the adicy of” not in terms of “instantiates” or via the notion of a canonical name, but rather in terms of “saturates,” roughly as follows:

218 | Cody Gilmore (D7) x specifies the adicy of y = df. (i) there is at least one x-tuple that saturates y, and (ii) anything that saturates y is an x-tuple.43, 44 Loosely put, whereas the n-tuple 〈a1, . . . , an〉 instantiates the universal R just in case the proposition that Ra1, . . . , an is true, that n-tuple saturates the given universal just in case there is such a thing as the proposition that Ra1, . . . , an, regardless of whether it’s true. This approach, like the canonical names approach, fares well in terms of its verdicts on cases. It has no apparent trouble, for example, even with the universal U4, being an x such that: x is round and anything that is instantiated is instantiated only by 2-tuples. Presumably this is saturated by 1-tuples and 1-tuples only. Take the 1-tuple 〈Obama〉. Since there is such a thing as the (false) proposition that Obama is round and such that anything that is instantiated is instantiated only by 2-tuples, the given 1-tuple intuitively does saturate the given universal. It should be easy to convince oneself, by appeal to related considerations, that nothing but 1-tuples saturate this universal. 43 Just as one might prefer to avoid talk of n-adic universals being instantiated by n-tuples, one might wish to avoid talk of such universals being saturated by n-tuples. One might therefore prefer to replace (D7) with

(D7*)

x is the adicy of y = df. (i) ∃Z[Z saturate y & x numbers Z] and (ii) ∀Z[Z saturate y → x numbers Z].

But this gets the adicy of identity wrong. Identity has 2 as its only adicy, but it’s not the case that identity is saturated only by 2-membered pluralities. This problem could be avoided if clause (ii) were omitted, but then we would have no explanation of the fact that being monadic and being dyadic, e.g., are incompatible. To deal with these problems without taking universals to be saturated by n-tuples, the friend of the saturation approach could appeal to a three-place saturation predicate, “Z saturate y in order w” and a two-place length predicate “w is an order of length x.” The idea would be to give a treatment of saturation that mirrors the treatment of instantiation sketched in footnote 15. (D7) could then be restated accordingly. I suspect that none of these variant definitions does much to help with the central problems for (D7) discussed in the main text. 44 Why no modal operators in this definition? The reason is just that we don’t need them. For most universals, whether or not the given universal is instantiated (by a given n-tuple, or by anything at all) is a contingent matter. But saturation is different. Suppose that u is n-adic. Then, intuitively, it is a necessary truth that if u exists, u is saturated by every n-tuple. And I assume that, for every positive integer n, it is a necessary truth that there is at least one n-tuple. (I assume that pure sets, numbers, and many—though perhaps not all—universals and propositions exist necessarily.) So I don’t think that there will be any “variation from world to world” in the facts about which sorts of n-tuples a given universal is saturated by.

Slots in Universals | 219 Further, the saturation approach may do better with respect to the priority problem than do some of the other strategies. Consider some universal u, and suppose that s-fact u is saturated by n-tuples and only by n-tuples. Is s-fact grounded by the fact that u is n-adic? For what it’s worth, although I do find this plausible,45 I see nothing absurd in its denial. Saturation facts strike me as relatively fundamental, certainly more fundamental than facts about “de re modal instantiation profiles” or than facts about canonical names. Given the absence of modal operators in (D7), one cannot accuse the saturation strategy of treating adicy properties as modal properties. Thus it leaves space to ground the de re modal properties of universals in their non-modal properties. This is good. What then is not to like about the saturation strategy? First, the approach introduces what appears to be a new primitive, fundamental predicate, “saturates,” and so it incurs some cost in ideology.46 Second, and more significantly, it’s in tension with the principle that adicy properties are intrinsic properties of universals. In other words, friends of the saturation approach face pressure to deny that Intrinsicality of Adicy (IA)

for any n, if there is such a thing as the property being n-adic, then that property is intrinsic.

45 Those who take counterpossibles seriously may think that the following argument for this carries some weight: if there hadn’t been any 2-tuples, being taller than wouldn’t have been saturated by any 2-tuples but it still would have been dyadic; so being dyadic is not the same thing as being saturated by 2-tuples and only 2-tuples. 46 As Chad Carmichael pointed out to me, the friend of saturation might define it in terms of a primitive functor, “pred,” as follows:

(D7)

x saturates y = df. ∃z z = pred(y, x)

Intuitively, the predication of a given universal to a given ordered -tuple is the proposition that predicates that universal of the members of that -tuple, in the order specified by that -tuple. E.g. the predication of being taller than to 〈Obama, Putin〉 is the proposition that Obama is taller than Putin. In symbols: pred(being taller than, 〈Obama, Putin〉) = the proposition that Obama is taller than Putin. The slot theorist will be tempted to gloss “pred” roughly as follows: the predication of a universal u to an n-tuple 〈a1, . . . , an〉 = the proposition that results from plugging a1 into the first slot in u, . . . , and plugging an into the nth slot in u. But the friend of the saturation strategy who endorses (D7) will say that “pred” is well understood even without being defined in slot-theoretic terms. My criticism of the “saturation strategy” applies equally to this variant.

220 | Cody Gilmore Let me start by saying something about how I understand intrinsicality. In the first place, I won’t offer any precise definition of the notion. I assume that the notion is fairly easily grasped, at least with the help of some examples together with a rough gloss. This assumption is standard in the literature on defining “intrinsic.” The participants in that literature are not offering stipulative definitions of technical terms. Rather, they are trying to analyze an intuitive notion that we grasp pre-analytically. Even before one arrives at a settled view on the analysis of intrinsicality, one is in a position to know, for example, that being an x such that x and its parts are the only contingent objects that exist is not intrinsic, and hence one is in a position to know that any analysis that calls the given property intrinsic is incorrect (Lewis 1999: 111–115). The notion of intrinsicality invoked by IA is the same one that serves as the target of most of the literature on defining “intrinsic.” Here is a standard informal characterization of that notion: to say that a property F is intrinsic is to say that whether or not a thing o has F depends only on what o is like in itself (“internally”) and is independent of how o is related to things outside of (or “external to”) itself. It is an open question as to which, if any, familiar properties of material objects will ultimately turn out to be intrinsic, but prima facie, it is natural to think that shapes and masses are intrinsic, as are the mereological properties being simple and being composite. Being two miles from a lake, by contrast, is extrinsic (non-intrinsic). So understood, IA is plausible on its face. If the notion of intrinsicality can ever be sensibly applied to properties that are instantiated only by abstract entities, surely it ought to turn out that adicies are intrinsic. Consider the property being either red or round. This property has many other properties. Some of these seem to be extrinsic: being used as an example by philosophers, being such that 2 + 2 = 4, being instantiated by more than one thing. Others seem to be intrinsic: being self-identical, being identical to being either red or round, being a universal, being logically complex (i.e. having an analysis), being disjunctive, being unnatural, involving a shape and a color. One further property of being either red or round is being monadic. With which group does this property belong? I submit that it belongs with the latter. It’s intrinsic.

Slots in Universals | 221 I find the considerations just given persuasive by themselves. But if an argument for IA is wanted, one option would be to cite it, in conjunction with the fact that universals are abstract objects and the fact that abstract objects have their intrinsic properties essentially,47 as constituting the best explanation of the relatively uncontroversial fact that universals have their adicies essentially.48 So let us agree that IA is true. Now let me say why I think the saturation strategy is in tension with IA. I’ll start with an analogy. The saturation strategy tells us that the property being dyadic is the property being saturated by a 2-tuple and nothing other than 2-tuples. Intuitively, this latter property is like being an egg carton that is completely filled by some 6-membered set of eggs, and by no set of eggs that has a cardinality other than 6. Call this fullness for short. It is obviously extrinsic. The egg carton in my refrigerator is a piece of packaging. It is made out of recycled paper products. This piece of packaging does not become any less massive, e.g. as the eggs are used up. Whether or not a given carton has the property fullness is a matter of how the carton is related to things that are, in the relevant sense, “external” to itself—namely, eggs and sets of them. A carton that had the given property wouldn’t change intrinsically if it lost that property. It would go from being full to being empty, but this would be an extrinsic change. According to the slot theorist, by contrast, the property being dyadic is the property being an x such that there are exactly two slots in x.49 In terms of the egg carton analogy, this is like being something

47 48

This view figures prominently in Jubien (2009: 93). More carefully, the thought is that IA, together with

(i) universals are not possibly concrete, and (ii) for any x, if x is not possibly concrete, then for any property F, if x has F and if F is intrinsic, then necessarily: if x exists, then x has F constitutes the best explanation of the fact that (iii) for any n and any universal u, if u has being n-adic, then necessarily: if u exists, u has being n-adic. It is an interesting question whether adicies (or perhaps all intrinsic properties of notpossibly-concrete objects) are essential in Fine’s stronger sense, but I take no stand on that here. Further, I take no stand on the question of whether there are contingently non-concrete entities. For arguments that there are such entities, see Linsky and Zalta (1996) and Williamson (2002). 49 Or perhaps being an x such that 2 numbers the slots in x, or being an x such that the cardinality of {y: y is a slot in x} is 2.

222 | Cody Gilmore that has exactly six holes in it. This seems intrinsic. Whether or not a given egg carton has this property is a matter of how it is related to things—holes—that are suitably “internal” to itself. A carton that had this property would change intrinsically if it lost it.

3.10.1. Inward-looking relations This analogy points toward an explanation of why the saturation approach is in tension with IA, and of why slot theory harmonizes with that principle. Some intrinsic properties are non-relational: roughly, no relations are involved in their analysis. Perhaps the maximally determinate masses are non-relational intrinsic properties. But other intrinsic properties are clearly relational. Being composite, for example, is just being an x such that for some y: ~y = x and y is a part of x. This involves the relation of parthood and the relation of identity, but the given property still seems intrinsic. Of course, not all relational properties are intrinsic. The paradigm examples of extrinsic properties are relational: being exactly two miles away from Barack Obama is both extrinsic and relational. It involves the spatial relation being exactly two miles from. There is a general lesson that we can extract from these examples. Consider some intrinsic property F, and suppose that F is a relational property of the form • being an x such that Rxa (e.g. being an x such that x = Obama) • being an x such that for some y, Rxy (e.g. having a hole) • being an x such that for all y, if Rxy then Gy (e.g. being such that all of one’s parts are negatively charged) • being an x such that for exactly n y, Rxy (e.g. having exactly 5 holes) • being an x such that for some y and z, Rxy & Rxz & R*yz (e.g. having parts that are exactly two feet away from each other) or something along these lines. Informally, suppose that having F is a matter of bearing relation R to some particular thing, or to at least one thing, or only to things of a certain kind, or to exactly 5 things, etc. In

Slots in Universals | 223 that case, R had better be the right sort of relation. Not just any relation can be used as the “primary relational ingredient” in the construction of an intrinsic relational property. Only certain special relations will do. Roughly, any such relation must be one that a given thing can bear only to entities that are suitably “internal” to that thing. I will call such relations inward-looking. I won’t try to list all the inwardlooking relations or give a definition of “inward-looking.” But I will set out a pair of plausible principles that state necessary conditions on inward-lookingness. Schematically, they are: IL1 If [kxyRxy] is an inward-looking dyadic relation and [kxFx] is an intrinsic property, then the property [kx∃y(Rxy & Fy)] is intrinsic.50 IL2 If [kxyRxy] is an inward-looking dyadic relation, then, for any cardinal number n, the property [kx there are exactly n things y such that: Rxy] is intrinsic.51 The following are instances of IL1 and IL2, respectively:52 IL11 If having as a part is an inward-looking dyadic relation and being round is an intrinsic property, then the property having a round part is intrinsic.

50 Instances of IL1 are formed by finding some dyadic predicate R* and some monadic predicate F* and replacing each occurrence of “R” with an occurrence of R* and each occurrence of “F” with an occurrence of F*. Roughly put, IL1 is a generalized version of what Sider calls the Inheritance of Intrinsicality: “if property P is intrinsic, then the property having a part that has P is also intrinsic” (2007: 70). Sider’s principle concerns parthood only. IL1 concerns inward-looking relations (of which having as a part is an example) more generally. Other plausible candidates for being inward-looking are being an x and a y such that y is a hole in x, i.e. having as hole, perhaps identity, perhaps having as a member. For discussion of Sider’s principle, see Gilmore (2010). Weatherson’s (2001: 373) principle (M) is also closely related to both IL1 and IL2. 51 Instances of IL2 are formed by finding some dyadic predicate R* and replacing each occurrence of “R” with an occurrence of R*. 52 More formally:

IL11 If [kxy y is a part of x] is an inward-looking dyadic relation and [kx x is round] is an intrinsic property, then the property [kx∃y(y is a part of x & y is round)] is intrinsic. IL21 If [kxy y is a part of x] is an inward-looking dyadic relation, then, for any cardinal number n, the property [kx there are exactly n things y such that: y is a part of x] is intrinsic.

224 | Cody Gilmore IL21

If having as a part is an inward-looking dyadic relation, then for any cardinal number n, the property having exactly n parts is intrinsic.

There are other plausible necessary conditions on inward-lookingness, but these are the only two that we’ll need here. Informally, IL1 says that if I bear an inward-looking relation to a thing y, then y’s intrinsic nature is reflected somehow in my intrinsic nature; and IL2 says that if R is an inward-looking relation, then the facts about how many things I bear R to are relevant to my intrinsic nature. IL1 and IL2 can serve as tests for inward-lookingness. If the intrinsic nature of the things that I bear R to is not relevant to my intrinsic nature, then R is not an inward-looking relation. Likewise, if the facts about how many things I bear R to are not relevant to my intrinsic nature, then again R is not inward-looking. (In the other direction, if a relation R passes both tests, this is some evidence that R is inward-looking.) Since they can serve as tests for inward-lookingness, IL1 and IL2 can also serve, indirectly, as tests for intrinsicality. For let F be a relational property, and suppose that we want to find out whether or not F is intrinsic. Further, suppose that the “primary relational ingredient” in F is the dyadic relation R. Finally, suppose that R flunks one or both of the tests for inward-lookingness. Then we can conclude that F, the relational property built up from R, is not intrinsic. (And in the other direction, if R passes both tests, that’s some evidence that R is inward-looking. Hence it’s some evidence that if F is built up from R in an appropriate way, then F is intrinsic.)

3.10.2. Applying the test to being saturated by a 2-tuple and only by 2-tuples I suggest that we apply this test to the property being saturated by a 2-tuple and only by 2-tuples. If we let “Sxy” abbreviate “x is saturated by y” and let “TTz” abbreviate “z is a 2-tuple,” then we can refer to the given property with the following lambda abstract: [kx∃y(Sxy & ∀z(Sxz → TTz))] I assume that the “primary relational ingredient” in the given property is the dyadic relation being saturated by, i.e. [kxySxy]. Having

Slots in Universals | 225 the given property is a matter of bearing the given relation to at least one thing of a certain kind, and only to things of that kind. Is being saturated by an inward-looking relation? If it is, then (i) we should expect the intrinsic nature of the things that saturate a given universal to be somehow reflected in the intrinsic nature of the universal itself (by IL1), and (ii) we should expect the facts about how many things saturate a given universal to be relevant to the intrinsic nature of that universal (by IL2). But it seems to me that the second prediction, at least, is false. (I have serious doubts about the first prediction as well.53) Consider the universal loving. It is instantiated by ordered 2-tuples and only by 2-tuples. How many 2-tuples is it instantiated by? Equivalently, how many 〈x, y〉 pairs are there such that x loves y? Some fairly large finite number, probably. Maybe it’s between ten and twenty trillion. Call this number n. So loving has the property being instantiated by exactly n things. But this is clearly a contingent

53

The first prediction is that: IL1P If [kxFx] is intrinsic, then [kx∃y(Sxy & Fy)] is intrinsic.

This says that if being an F is intrinsic, then being saturated by an F is intrinsic. To see why this might seem doubtful, consider: Saturationists need to say that for any positive integer n, the property being an n-tuple is intrinsic. (If these properties were not intrinsic, then being saturated by a 2-tuple and only by 2-tuples would be like having a part that is two miles from a lake, and being such that each of one’s parts is two miles from a lake, i.e. obviously extrinsic.) But if the given properties are intrinsic, that’s presumably because n-tuples bear some inward-looking relation to their members. And in that case, the intrinsic properties of the members of an n-tuple are relevant to the intrinsic nature of the n-tuple itself. So, e.g. the 2-tuple 〈Plato, Shaquille O’Neal〉 differs intrinsically from the 2-tuple 〈Plato, Aristotle〉: the former but not the latter has the property being such that its second member is over 7 feet tall (Fs1 for short), which on the current view would be intrinsic. From here it is a short step to the view that IL1P is false. In the actual world, loving has the property being saturated by something whose second member is over 7 feet tall (Fs2 for short). There are other possible worlds in which nothing is over 7 feet tall. In those worlds, there are no 2-tuples whose second member is over 7 feet tall. Accordingly, in those worlds, loving lacks the property being saturated by something whose second member is over 7 feet tall; it lacks Fs2. Since loving has exactly the same intrinsic properties in all possible worlds but has Fs2 in some possible worlds and not in others, it follows that Fs2 is extrinsic. But Fs1 is intrinsic. This yields a counterexample to IL1P. If being saturated by is either a dyadic relation that holds between a universal and some things (“a plurality”) or a triadic relation that holds between a universal, some things, and an order, this argument would need to be restated, but I suspect that some version of it would still go through.

226 | Cody Gilmore fact about loving. It could have been instantiated by more things, or by fewer, and if it had, it wouldn’t have been any different intrinsically. It’s a Platonic entity, after all, and it has exactly the same intrinsic properties in every possible world in which it exists. So being instantiated by exactly n things is an extrinsic property, and being instantiated by is not an inward-looking relation. Similar remarks apply to being saturated by. Loving is saturated by 2-tuples and only by 2-tuples. How many 2-tuples is it saturated by? This depends on how many 2-tuples there are. Some very large infinite number, probably. Call it n*. So loving has the property being saturated by exactly n* things. Admittedly, this may be a non-contingent fact about loving. Whatever n* is, it may well be a necessary truth that there are exactly n* 2-tuples. But it’s still clearly a fact “external” to loving. The intrinsic nature of loving in no way depends upon how many 2-tuples there are, even if it is necessary that there are exactly n* 2-tuples. Suppose (per impossibile perhaps) that there were fewer 2-tuples, but things were otherwise largely as they actually are. In particular, suppose that it were still true that loving was saturated by infinitely many 2-tuples and only by 2-tuples. In that case, loving wouldn’t have been any different intrinsically; it wouldn’t have had any intrinsic property that it actually lacks. It’s just that there would have been fewer things out there for it to be saturated by. Intuitively, just as the facts about how many 2-tuples a dyadic universal is instantiated by are not relevant to its intrinsic nature, the facts about how many 2-tuples it’s saturated by are likewise irrelevant to its intrinsic nature. One might initially think that what kinds of -tuples a universal is saturated by is relevant to the universal’s intrinsic nature. But surely one ought to admit that how many -tuples of that kind the universal is saturated by is not relevant. So it seems to me that being saturated by exactly n* 2-tuples is an extrinsic property, and hence that being saturated by is not an inward-looking relation.54

54 As indicated in footnote 53, if being saturated by is either a dyadic relation that holds between a universal and some things (“a plurality”) or a triadic relation that holds between a universal, some things, and an order, this argument will need to be restated. Intuitively, the thought would be that the facts about how many “n-membered pluralities” a given n-adic universal is saturated by are irrelevant to that universal’s intrinsic nature.

Slots in Universals | 227 Accordingly, being saturated by a 2-tuple and only by 2-tuples fails our test for intrinsicality.

3.10.3. Applying the test to having exactly 2 slots How does the slot theorist’s proposal fare by the lights of this test? I think it fares well. For each cardinal number n, the slot theorist identifies the property being n-adic with the property having exactly n slots.55 So, like the saturationist, he takes adicies to be relational properties. According to the slot theorist, however, the primary relational ingredient in adicy properties is the relation having as a slot (the converse of being a slot in). And this relation passes the tests for inward-lookingness. Consider the first test. It’s plausible that for any cardinal number n, the property having exactly n slots is intrinsic. This is confirmed by reflecting on the appropriate counterpossibles. Suppose that loving is dyadic, i.e. that it has two slots in it. It seems to me that if, per impossibile, loving were to lose one of its slots, it would change intrinsically. We might also compare the principle about slots with the analogous principle about holes: for any cardinal number n, the property having exactly n holes is intrinsic. This too is plausible. Objects with different numbers of holes in them will be differently shaped, and so will have different intrinsic properties. Now consider the second test. It’s plausible that if being an F is an intrinsic property, then having a slot that is an F is an intrinsic property too. Byeong-Uk Yi suggests that certain relations have multiple slots some of which are singular and others of which are plural. Call an argument place of a relation plural if it admits of many objects as such; singular otherwise. Accordingly, call a relation singular if all of its argument places are singular; plural otherwise. (1999: 169)

According to Yi, the predicate “is one of” expresses a dyadic relation (being one of) whose first slot is singular and whose second slot is plural. If one embraces something like Yi’s view, one might think, further, that being a plural slot and being a singular slot are intrinsic properties of slots. In particular, one might think that if a certain 55 Or perhaps being an x such that n numbers the slots in x, or being an x such that the cardinality of {y: y is a slot in x} is n.

228 | Cody Gilmore slot s “admits of many objects as such,” that’s because it has a certain intrinsic property, being a plural slot. Assume that this view is correct. In that case, if having as a slot is inward-looking, then it should turn out that (Y) having a plural slot and having a singular slot are themselves intrinsic properties. Is (Y) true? I don’t know how to prove that it is, but I do find (Y) highly plausible, and I can’t think of any reason to doubt it. It seems to me that any two universals that have plural slots are thereby intrinsically similar in that respect; likewise for those that have singular slots. (The analogy with holes might again be helpful: being large and triangular is intrinsic, and so is having a large, triangular hole. Any two objects that have large triangular holes in them are thereby intrinsically similar in that respect.) Further, it seems to me that if, per impossibile, one of the slots in loving became plural, loving would change intrinsically. In sum, then, all signs point to the conclusion that having as a slot is inward-looking, and that properties of the form having exactly n slots are intrinsic. Slot theory harmonizes with IA, the principle that adicies are intrinsic properties.56

56 At this point one might reply with the following slot-free explanation of why adicy properties are intrinsic. If it works, it is equally available to any of the ten slotfree accounts of adicity.

A property is intrinsic if and only if it never differs between possible duplicates. This is a very well-entrenched principle in discussions of intrinsicality; call it ID. Now suppose, as some believe, that universals have no possible duplicates (aside from themselves): if universal u in possible world w is a duplicate of u* in possible world w*, then u = u*. Then ID tells us that every property that can be possessed only by universals, and that is possessed necessarily by any universal that possesses it, is intrinsic. Further, since adicy properties can be possessed only by universals, and since they are had necessarily by anything that has them, ID tells us that they are intrinsic. So there is our explanation of the relevant fact, and it has nothing to do with slots. The problem with this explanation is that relies on ID. ID may seem plausible when one restricts one’s attention to concrete particulars and the properties thereof (though see Eddon (2011) for a convincing criticism that applies even there), but it quickly loses its appeal when one turns to the realm of abstracta. For example, ID tells us that being an x such that necessarily, x is instantiated by 〈2, 1〉 is an intrinsic property. Intuitively, however, the given property is clearly extrinsic. It is had by being less than, and necessarily so, but it is not an intrinsic property of that relation.

Slots in Universals | 229 4. A PROBLEM FOR SLOTS? Kit Fine (2000) has raised a number of worries for views in the vicinity of slot theory. I cannot discuss them all here.57 Nor is it the goal of this paper to give a comprehensive defense of slots. But there is one consideration that I would like to address before concluding. Fine puts it thus: The antipositionalist view has another, related, advantage over the positionalist view. For it is able to account for the possibility of variable polyadicity. It is plausible to suppose that certain relations are variably polyadic in the sense that they can relate different numbers of objects (and not merely through some of those objects occurring several times as a relatum). There should, for example, be a relation of supporting that holds between any positive number of supporting objects a1, a2, . . . and a single supported object b just when a1, a2, . . . are collectively supporting b. Under the positional view, it is hard to see how any relation could be variably polyadic, for the number of argument places belonging to a relation will fix the number of relata that may occupy them. Under the antipositionalist view, however, there is no impediment to a relation being variably polyadic, since there are no preordained positions by which the number of arguments might be constrained. (2000: 22)

Two responses are available. The singularist response takes supporting to be a dyadic relation that holds between, on the one hand, the sum, or aggregate, or set, or . . . of supporting objects and, on the other hand, the supported object. (See McKay (2006) for a survey of these views.) This is consistent with the view that supporting is a dyadic relation with exactly two slots in it, each of them singular. The pluralist response, by contrast, takes supporting to be a relation that holds between, on the one hand, some things (the supporting objects) and, on the other hand, the supported object. This view is also consistent with the view that supporting has exactly two slots in it. Indeed, the most natural way of explaining this view is to say, following Yi (1999: 169) and McKay (2006: 13), that supporting has two slots in it, but that these slots are qualitatively different from each other: the first is plural, the second is singular.58 On this view, when

57 See also MacBride (2005: 588–9) and (2007), Fine (2007), Wieland (2010), and Orilia (2011). 58 Suppose that a1, a2, and a3 support b, and that supporting has two slots, the first plural, the second singular. In that case, what instantiates supporting? Presumably, if we initially took instantiation to be a dyadic relation between an n-adic universal u and an n-tuple of things that “stand in” u, then we should now say that 〈{a1, a2, a3}, {b}〉

230 | Cody Gilmore we give the logical form of a universal, it is not enough to say that it is monadic, dyadic, etc. To give a more complete and fine-grained characterization, we should say that it is, e.g. dyadic and plural-withrespect-to-its-first-slot but singular-with-respect-to-its-second-slot. Not only is such a view compatible with the existence of slots, the view is very hard to understand without them!59

5. CONCLUSION There are further slot-free accounts of adicy, but those that I’ve discussed are probably the ones that have the most prima facie appeal.60 In light of the problems with these strategies, it seems increasingly likely that Platonists face real pressure to be slot theorists. Whether instantiates supporting. So we should now think of instantiation as a dyadic relation between an n-adic universal u and n-tuple of sets of things whose members “stand” in u. If u is singular with respect to its ith slot, then it will be instantiated only by n-tuples whose ith members are singleton sets. See MacBride (2005). 59 In the following passage, Fraser MacBride considers a view like Yi’s and McKay’s. He then goes on to argue that there may be universals that are variably polyadic in a way that is more radical:

From this point of view each universal has a fixed number of argument places: form a circle has one argument place, causation has two argument places, and so on. But even though the number of places is fixed, different numbers of individuals may occupy each place. . . . However, it does not follow from the fact that some universals have a fixed degree (in the sense of having a fixed number of argument positions) that all universals have a fixed degree. . . . The multiple relation of belief applies differentially to different numbers of objects, properties, and relations. For example, Iago may believe that Roderigo loves Desdemona whilst not believing that Desdemona loves Roderigo. It follows that the objects, properties, and relations the belief relation relates cannot fall within a single undiscriminating position. Rather to account for the differential application of the belief relation, the related items must be slotted into different argument positions of the relation. Then since the number of objects, properties, and relations related by belief varies—Iago may simply believe that Roderigo is a fool—it follows that the number of argument positions in the belief relation must vary too. (2005: 588–9) Might this suggestion constitute a problem for slot theory? I think not. In the first place, I doubt that there are any varigrade universals. But even if there are, I don’t see why this poses a threat to slot theory. After all, the most natural way to describe the scenario that MacBride has in mind is to say that believing has different numbers of slots in it relative to different propositions in which it occurs. Far from undermining slots, this description presupposes them. 60 Further accounts are given by Bealer (1982: 83) and Hossack (2007: 67). I hope to discuss these in future work.

Slots in Universals | 231 these pressures override the equally real countervailing motivations to reject slots is a difficult question, and one that I have not tried to answer here.61 University of California, Davis REFERENCES Armstrong, D. 1997. A World of States of Affairs (Cambridge: Cambridge University Press). Bealer, G. 1982. Quality and Concept (Oxford: Oxford University Press). Carmichael, C. 2010. “Universals,” Philosophical Studies 150(3): 373–89. Casati, R. and A. Varzi. 1994. Holes and Other Superficialities (Cambridge, MA: MIT Press). Crimmins, M. 1992. Talk about Belief (Cambridge, MA: MIT Press). Dorr, C. 2008. “There are No Abstract Objects,” in Theodore Sider, John Hawthorne, and Dean W. Zimmerman, eds., Contemporary Debates in Metaphysics (Oxford: Blackwell). Eddon, M. 2011. “Intrinsicality and Hyperintensionality,” Philosophy and Phenomenological Research 82: 314–36. Fine, K. 1994. “Essence and Modality,” in James Tomberlin, ed., Philosophical Perspectives, 8, Logic and Language: 1–16. —— 2000. “Neutral Relations,” The Philosophical Review 109: 1–33. —— 2007. “Response to Fraser MacBride,” Dialectica 61: 57–62. Frege, G. 1984. [originally published 1923] “Compound Thoughts,” in B. McGuinness, ed., Collected Papers on Mathematics, Logic, and Philosophy (Oxford: Basil Blackwell). Gilmore, C. 2010. “Sider, the Inheritance of Intrinsicality, and Theories of Composition,” Philosophical Studies 151: 177–97. —— forthcoming. “Parts of Propositions,” in Shieva Kleinschmidt, ed., Mereology and Location (Oxford: Oxford University Press). Grossmann, R. 1983. The Categorial Structure of the World (Bloomington: Indiana University Press). —— 1992. The Existence of the World: An Introduction to Ontology (London: Routledge). 61 Thanks to Ross Cameron, Ben Caplan, David Copp, Andrew Cortens, Greg Damico, Scott Dixon, Kit Fine, Michael Glanzberg, Robbie Hirsch, Carrie Ichikawa Jenkins, Brian Kierland, Seahwa Kim, David Liebesman, Kris McDaniel, Bernard Molyneux, Brad Morris, Andrew Newman, Laurie Paul, Raul Saucedo, Adam Sennet, Brad Skow, Mark Steen, Paul Teller, Amie Thomasson, Jason Turner, Gabriel Uzquiano, Dean Zimmerman, and especially Chad Carmichael, Paul Hovda, Ted Sider, and Linda Wetzel for helpful comments and criticism on earlier versions of this paper.

232 | Cody Gilmore Horwich, P. 1998. Truth, 2nd edition. (Oxford: Oxford University Press). Hossack, K. 2007. The Metaphysics of Knowledge (Oxford: Oxford University Press). Jubien, M. 1993. Ontology, Modality, and the Fallacy of Reference (Cambridge: Cambridge University Press). —— 2009. Possibility (Oxford: Oxford University Press). Kim, S. and C. Maslen. 2006. “Counterfactuals as Short Stories,” Philosophical Studies 129: 81–117. King, J. 2007. The Nature and Structure of Content (Oxford: Oxford University Press). Kripke, S. 1980. Naming and Necessity (Cambridge, MA: Harvard University Press). Lewis, D. 1973. Counterfactuals (Oxford: Blackwell). —— 1999. “Extrinsic Properties,” in D. Lewis, Papers in Metaphysics and Epistemology (Cambridge: Cambridge University Press). —— and S. Lewis. 1970. “Holes,” Australasian Journal of Philosophy 48: 206–12. Linsky, B. and E. Zalta. 1996. “In Defense of the Contingently Non-concrete,” Philosophical Studies 84: 283–94. MacBride, F. 2005. “The Particular-Universal Distinction: A Dogma of Metaphysics?,” Mind 114: 565–614. —— 2007. “Neutral Relations Revisited,” Dialectica 61: 25–56. McKay, T. 2006. Plural Predication (Oxford: Oxford University Press). Menzel, C. 1993. “The Proper Treatment of Predication in Fine-Grained Intensional Logic,” Philosophical Perspectives, 7, Logic and Language: 61–87. Newman, A. 2002. The Correspondence Theory of Truth (Cambridge: Cambridge University Press). Nolan, D. 1997. “Impossible Worlds: A Modest Approach,” Notre Dame Journal of Formal Logic 38: 535–72. Orilia, F. 2011. “Relational Order and Onto-thematic Roles,” Metaphysica 12: 1–18. Quine, W. V. O. 1995 [originally published 1960]. “Variables Explained Away,” in W. V. O. Quine, Selected Logic Papers, enlarged edition (Cambridge, MA: Harvard University Press). Russell, B. 1956. Logic and Knowledge (London: Routledge). Schaffer, J. 2009. “On What Grounds What,” in D. Chalmers, D. Manley, and R. Wasserman, eds, Metametaphysics. Oxford: Oxford University Press. Sider, T. 2007. “Parthood,” The Philosophical Review 116: 51–91. —— 2009. “Ontological Realism,” in D. J. Chalmers, D. Manley, and R. Wasserman, eds., Metametaphysics (Oxford: Oxford University Press).

Slots in Universals | 233 Stalnaker, R. 1968. “A Theory of Conditionals,” in Nicholas Rescher, ed. Studies in Logical Theory (Oxford: Blackwell). Swoyer, C. 1998. “Complex Predicates and Logics for Properties and Relations,” Journal of Philosophical Logic 27: 295–325. van Inwagen, P. 2001. “Creatures of Fiction,” in van Inwagen, Ontology, Identity, and Modality. Cambridge: Cambridge University Press. —— 2006a. “Properties,” in Thomas Crisp, Matthew Davidson, and David Vander Laan, eds., Knowledge and Reality: Essays in Honor of Alvin Plantinga (Dordrecht: Springer). —— 2006b. “Names for Relations,” in John Hawthorne, ed., Philosophical Perspectives, 20, Metaphysics: 453–77. Weatherson, B. 2001. “Intrinsic Properties and Combinatorial Principles,” Philosophy and Phenomenological Research 63: 365–80. Wetzel, L. 2009. Types and Tokens: On Abstract Objects (Cambridge, MA: MIT Press). Wieland, J. 2010. “Anti-positionalism’s Regress,” Axiomathes 20: 479–93. Williamson, T. 1985. “Converse Relations,” The Philosophical Review 94: 249–62. —— 2002. “Necessary Existents,” in A. O’Hear, ed., Logic, Thought, and Language (Cambridge: Cambridge University Press). Yi, Byeong-Uk. 1999. “Is Two a Property?,” The Journal of Philosophy 96 (4): 163–90. Zalta, E. 1988. Intensional Logic and the Metaphysics of Intensionality (Cambridge, MA: MIT Press).

MEREOLOGY

6. Against Parthood Theodore Sider In this paper, I will defend what Peter van Inwagen calls nihilism: composite entities (entities with proper parts) do not exist.1 This formulation will need to be refined, and, at the very end of the paper, softened a little. But let us stick to the simple, strong version for now. Nihilism may seem absurd. For the world of common sense and science consists primarily of composite entities: persons, animals, plants, planets, stars, galaxies, molecules, viruses, rocks, mountains, rivers, tables, chairs, telephones, skyscrapers, cities . . . According to nihilism, none of these entities exist. But it is not absurd to reject such entities if one accepts their noncomposite subatomic particles. Consider three subatomic particles, a, b, and c, arranged in a triangular pattern. According to some, there exists in addition a fourth thing, T, which contains a, b, and c as parts. According to me, this fourth thing does not exist. Picture the disagreement thus:

a

b

T

a

c

According to me

b

c

According to my opponents

1 van Inwagen (1990). “Proper parts” of x are parts of x other than x itself (it is customary to count entities as being parts of themselves). By “composition” I have in mind only mereological composition, i.e. composition by parts, though I do discuss sets in the final section. Other nihilists include Dorr (2002) and Cameron (2010b); see also Dorr (2005). See Dorr and Rosen (2002) for a defense—partly overlapping mine— of nihilism against objections. For stylistic reasons I often speak of existence, but as a good Quinean I intend this to be recast in terms of quantification.

238 | Theodore Sider (But take the picture with a grain of salt: my opponents don’t think that T is encircled by a faint aura, or accompanied by a ghostly “T”.) My opponents and I agree on the micro-description of the situation: on the intrinsic states of the particles (such as their charges and masses) and their spatial arrangement. Our sole disagreement is over whether these particles are accompanied by a further object that is composed of them. Since I accept the existence of the particles, my denial of an object composed of them isn’t absurd. Denying that T exists in addition to a, b, and c is no more absurd than denying that holes exist in addition to perforated things, or denying that smirks exist in addition to smirking faces. Similarly, denying the existence of persons, animals, plants, and the rest is not absurd if one accepts subatomic particles that are “arranged personwise” (to use van Inwagen’s phrase), animal-wise, plant-wise, and so on. Indeed, it would seem that ordinary evidence is neutral over whether composite objects or merely appropriately arranged particles exist. Which hypothesis is correct is thus an open philosophical question, like the question of whether there exist holes and smirks. That is just the first skirmish; a series of battles is yet to be fought. Some say that the existence of persons and other composites is common sense; others say that we know of composites through perception; still others say that the dispute between nihilists and their opponents is merely verbal. But before discussing these and other challenges, I should say why I think that nihilism is true.

1. THE ARGUMENT FROM IDEOLOGICAL PARSIMONY 2 Quine famously distinguished between ideology and ontology.3 A theory’s ontology consists of the objects that the theory posits—the

2 This argument was inspired by Dorr’s (2005) claim that nihilists ought to regard ‘part’ as a failed natural kind term. 3 (1951a). The argument from parsimony is akin to Quine’s own approach to ontology; see (1948, 1951b, section 6, 1960, chapter 7, 1976).

Against Parthood | 239 range of its quantifiers, if the theory is to be true. Its ideology consists of the undefined notions it employs, both logical and extra-logical. In addition to eliminating composite objects from our ontology, nihilism also allows us to eliminate the extra-logical (or perhaps quasi-logical) notion of ‘part’ from our ideology, and this kind of ideological simplification is an epistemic improvement. Nihilism is an ideologically simpler theory, and so is more likely to be true.4 This argument from ideological parsimony is, I think, more powerful than the argument that nihilism is ontologically parsimonious. Many agree that simply cutting down on the number of entities one posits isn’t particularly important.5 Also, many defenders of parts say that there is something distinctive about parthood which makes commitment to mereologically complex entities somehow “innocent”,6 a thought which perhaps defends against the argument from ontological parsimony, but not at all against the argument from ideological parsimony. The argument presupposes an epistemic principle: ideologically simpler theories are more likely to be true.7 The intuitive basis of the principle is the vague but compelling idea that simplicity is a guide to truth, together with the thought that eliminating primitive notions makes a theory “structurally” simpler. A theory’s one-place predicates correspond to the kinds of things it recognizes, and its multi-place predicates to the kinds of connections between things that it recognizes; cutting down on kinds or connections is one way of making a theory structurally simpler. The epistemic principle is most naturally paired with a metaphysical realism about ideology. Ideologically simpler theories aren’t just more convenient for us. The worlds that they purport to describe are objectively simpler, contain less structure. Ideology is a worldly matter, not about ideas at all.8 4 Notice that since “semi-nihilists” like van Inwagen (1990) and Merricks (2001) admit some composites, they cannot eliminate parthood. 5 See Lewis (1973, p. 87), although see Nolan (1997). 6 See Lewis (1991, section 3.6); see also Armstrong (1997, section 2.12). 7 Huemer (2009) considers various ways to justify principles of parsimony, and argues that none of them underwrites the use of parsimony in philosophy. I doubt that the ways to justify parsimony that Huemer considers are adequate to all the uses of parsimony in science, and suspect that principles of parsimony cannot be derived from more fundamental epistemic principles. 8 See Sider (2011).

240 | Theodore Sider I am writing from a nominalist point of view when I formulate the epistemic principle in terms of ideological simplicity, but a realist about properties could say something similar. The thought behind the principle is that “structurally simpler” theories are more likely to be true; a realist would simply need to understand structural simplicity as being a matter of the properties and relations included in the theory’s ontology, as well as the theory’s ideology. Thus the realist would be arguing for nihilism on the grounds that it does not require a relation of parthood in ontology. The epistemic principle should be restricted to theories about the fundamental nature of the world (such as physics and, by my lights, mathematics and fundamental metaphysics). Only for fundamental theories does simple ideology correlate directly with worldly simplicity; and it is far less clear that lean ideology is truth-conducive in biology, economics, and geology, let alone in everyday nonscientific contexts. Thus it is no objection that nihilists must use ideology like ‘arranged plant-wise’, ‘arranged dollar-bill-wise’, ‘arranged river-wise’, and so forth to describe reality’s biological, economic, and geological features—these predicates are not part of the nihilist’s theory of fundamental matters.9 When the principle is restricted in this way, the argument from ideological parsimony rests on the claim that nihilism allows us to eliminate ‘part’ from the ideology of our fundamental theories. And this claim seems correct. If one’s theory of fundamental matters included an ontology of composite objects, then that theory would presumably also need a predicate of parthood to connect those composites to their parts (since there do not seem to be more fundamental predicates in terms of which ‘part’ could be defined10); but without the composites, the predicate isn’t needed. 9

Thus I can reply to Bennett (2009, p. 64). Objection: parthood could be defined in terms of a fundamental predicate of spatial (or spatiotemporal) location: x is part of y = df for every point p of space (or spacetime), if x is located at p, then y is located at p. Replies: i) this gives us no account of parthood relations over space (or spacetime) itself; ii) this presupposes the falsity of supersubstantivalism (see section 9); iii) this presupposes that fundamental theories include a predicate for location that applies to composite as well as simple objects; and if I am right that fundamental theories do not need composites or parthood, then surely they do not need such a notion of location either. 10

Against Parthood | 241 Simplicity is not the only epistemic virtue. Choiceworthy theories must also be compatible with our evidence and predict as much of it as possible. It is only when multiple theories fit the evidence that we turn to simplicity and other epistemic virtues. But this is exactly the situation with nihilism and its competitors, since our best theories of fundamental matters—physics and, I say, mathematics and fundamental metaphysics—have no need for composite objects. Physics, for example, makes predictions based on laws governing simple entities like subatomic particles. Deleting ‘part of’ and all reference to composite objects in these theories does not weaken their predictive power.11 So ideological parsimony gives us a reason to accept nihilism. Given an expansive conception, the realm of the fundamental might include chemical, biological, and other macro-phenomena, in which case fundamental theories could not so easily rid themselves of composite objects and parthood. This is a big issue; here I will say simply that I presuppose a more restrictive conception: despite the existence of genuine explanations in chemistry, biology, and other higher-level sciences, such phenomena are not fundamental.12 The principle that ideologically simpler theories of fundamental matters are more likely to be true needs to be further qualified. First, the principle should say that other things being equal, the ideologically simpler theory is more likely to be true. For as just noted, we turn to simplicity only when multiple theories fit the evidence; moreover, there may be further super-empirical virtues other than simplicity; and moreover, there is more to simplicity than ideological simplicity—simplicity of laws counts as well, for instance. Second, merely counting primitive notions is too crude a measure of ideological simplicity, since one can always replace many predicates with a single many-placed predicate; the many-placed predicate would be, in an intuitive but elusive sense, a highly complex notion despite being one in number.13 Counting primitive notions is a better measure when the theories are comparable in other respects—when their laws are equally simple and when their notions are equally simple in the 11

Although see section 11. The restrictive conception is best coupled with an account of the relation between fundamental and nonfundamental that is neither semantic (in the ordinary sense) nor epistemic; see Sider (2011, sections 7.3–7.8). 13 See Goodman (1951, chapter 3) for a heroic attack on this problem. 12

242 | Theodore Sider elusive but intuitive sense—but these further comparisons of simplicity can be difficult to assess. Fortunately, the argument from ideological parsimony relies only on a quite straightforward comparison of ideological simplicity, that “mere deletion” makes a theory ideologically simpler. Fundamental theories do not need composites or parthood in order to predict the evidence, I have said. Any talk of parthood in fundamental theories is explanatorily superfluous, so that one can simply delete the predicate ‘part’ from a fundamental theory that contains it, together with any laws in which ‘part’ figures, without sacrificing predictive power. And even though comparisons of ideological complexity are generally fraught, it’s comparatively safe to regard this sort of mere deletion as reducing ideological complexity. (The deletion also simplifies the laws, if any of the original laws involved ‘part’. Eliminating the need for fundamental laws of mereology is a further epistemic benefit of nihilism.14) The epistemic principle on which I have relied may be further supported by considering how it illuminates the much-discussed case of neo-Newtonian spacetime. Neo-Newtonian spacetime is a spacetime in which (frame-independent) temporal distances and spatial distances between simultaneous points are well defined, and in which paths of unaccelerated particles through spacetime are well defined, but in which spatial distances between nonsimultaneous points are not well defined, and hence in which absolute velocities are not well defined. Newtonian spacetime is an otherwise similar spacetime but in which the notion of remaining at the same place—and hence notions of distance between nonsimultaneous points, absolute rest, and absolute velocity—are well defined. Philosophers of physics generally agree that if Newtonian mechanics had been right, it would have been more reasonable to think that spacetime was neo-Newtonian than to think spacetime was Newtonian. But there is no consensus over exactly why that is so.15

14

Relatedly, consider the objection that adopting parthood in fundamental theories allows the elimination of identity from ideology via the definition “x = y =df x is part of y and y is part of x”. The savings in ideological parsimony would be outweighed by increased complexity in the laws, which I take to include laws of logic and metaphysics. The logical laws governing ‘=’ must now be rewritten in terms of the proposed definition, making them more complex; and further, laws of mereology will be needed. Thanks to Steve Steward. 15 See Dasgupta (2011, section 6) for an overview.

Against Parthood | 243 One popular idea is that the demerit of Newtonian spacetime is epistemic: its facts about absolute velocity would be undetectable. But verificationism is long dead; why should this epistemic fact in itself count against the theory?16 A better—and more directly metaphysical—argument is that Newtonian spacetime’s undetectable absolute velocities are not themselves problematic, but rather are a sign of an intrinsic defect that is problematic: the theory’s spacetime is overly complex. Newtonian spacetime contains more “structure” than is required for the theory to fit the evidence—in particular, more structure than is needed for the formulation of Newton’s laws of motion.17 Neo-Newtonian spacetime is more choiceworthy because it lacks that excess structure. The principle that ideologically simpler theories are more likely to be true gives us a particularly straightforward way to cash out the thought that neo-Newtonian spacetime is preferable because it contains “less structure”. Describing neo-Newtonian spacetime requires a certain ideology, such as the notion of three points being on a straight line through spacetime.18 Describing Newtonian spacetime requires this ideology and then some further ideology as well: the notion of two points of spacetime being at the same absolute position. Further, Newton’s laws, as optimally formulated in the context of Newtonian spacetime, do not mention the notion of being at the same absolute position. Thus the neo-Newtonian theory results from the Newtonian theory via a mere deletion of the notion of being at the same absolute position; and so, given the principle, is less choiceworthy for that reason.19

16 Dasgupta (2009) argues convincingly that the epistemic argument in favor of neo-Newtonian spacetime should, if accepted, be pursued much further than is customary. Since individual points are in the relevant sense undetectable, he says, we should reject their existence and defend an individuals-free metaphysics (a descendent of the bundle theory of particulars). In my view, we should turn this argument around (in part because I doubt that the individuals-free metaphysics is a stable stopping point), and reject the epistemic argument. 17 Earman (1989, p. 46); North (2009, p. 9). 18 See Field (1980, chapter 6). 19 This argument does not go through if we are comparing neo-Newtonian spacetime to Newton’s own theory, which was not a spacetime theory, but rather a theory of time plus enduring space: neither Newton’s own nor the neo-Newtonian ideology is a proper subset of the other. The question of simplicity in this case is far less straightforward.

244 | Theodore Sider So the situation is this: i) ordinary evidence seems to leave open whether composite objects exist in addition to appropriately arranged subatomic particles; and ii) ideological parsimony (and also simplicity of laws) gives us a positive reason to reject parthood, and thus composites. Does anything counterbalance this case for nihilism? Many arguments for parts-based ontologies are really just arguments against other parts-based ontologies, and so do not support their intended ontologies any better than they support nihilism. For example, David Lewis’s argument from vagueness for unrestricted composition is really just an argument against middling views according to which some collections of objects compose a further object and some do not; it does not threaten the nihilistic view that no collections of objects compose a further object.20 And the familiar paradoxes of coinciding objects, which are so nicely resolved by a temporal parts metaphysics combined with composite objects, are resolved just as well by a nihilist metaphysics.21 Still, some arguments are genuinely directed against nihilism, including:22 1. nihilism goes against common sense 2. knowledge of composites is given in perception 20

Lewis (1986a, 212–13). See also Sider (2001, chapter 4, section 9). Merricks (2001, pp. 38–47). See Sider (2001, chapter 5) for a survey of the issues. McGrath (2005) argues that since nihilists regard claims about composites as at least being correct in the sense of section 3 (his word is ‘factual’), they still face the paradoxes at the level of correctness. But the shift to correctness (or to nonfundamental languages—again, see section 3) dissolves the paradoxes if some of the claims generating the paradoxes have force only when read as claims about fundamental truth. Consider, for example, those paradoxes that appeal to the principle that composition is unique—that no collection of objects composes more than one thing. The appeal of this principle is “theoretical”: it is based on a putative insight into the fundamental nature of the part–whole relation. The principle loses its appeal if it is taken as being merely correct (or as being in a nonfundamental language). For correctness (or truth in nonfundamental languages) is more closely tied to ordinary speech, and ordinary speech is fine with there being particles that, say, compose both a statue and a distinct lump of clay. 22 There is also the argument that composites are required to support emergent properties. The argument would need to assume that “emergent properties” are perfectly fundamental (otherwise claims about those properties could be “correct” in the sense of section 3 or true in a nonfundamental language) and incapable of being reconstrued as relations over simples (perhaps because the putative relations would have no fixed -adicy). I doubt such properties exist; but if they do, they present a challenge like that discussed in section 11. 21

Against Parthood | 245 3. the existence of composites is part of our evidence, given Timothy Williamson’s conception of evidence 4. we are entitled for Cartesian reasons to believe in our own existence 5. the denial of composite objects is conceptually incoherent 6. nihilism is incompatible with “atomless gunk” 7. parts and composite objects are required by spacetime physics In what follows I will rebut these arguments. The final argument is the most powerful one, and my response will be tentative. In fact, my response will be to soften the nihilist position a bit: although there do not exist composites in the mereological sense—i.e. objects with proper parts—there do exist “composites” in the set-theoretic sense—i.e. objects with members; i.e. sets. (Also, my response will be conditional on certain issues in the philosophy of mathematics and physics.) But my rebuttals of the earlier arguments are to be independent of this concession; so forget sets until section 11.

2. MOOREANISM Recent metaphysics, especially in the tradition of David Armstrong, Saul Kripke, and David Lewis, has been dominated by a sort of “Mooreanism”, according to which being “common sense” counts in favor of truth.23 Theories that are consistent with common sense are

23

This view seems more prevalent amongst metaphysicians than epistemologists. But I do not attribute the view to Armstrong, Kripke, or Lewis (or Moore, for that matter) themselves. Excepting a few passages (notably Naming and Necessity, pp. 41–2), what has been most influential in their writings is not explicit endorsement of Moorean epistemic principles, but rather a pervasive attitude of respecting common sense (think of the phrase “Moorean fact”). And let me also distinguish what I am calling Mooreanism from some alternatives. Alternative 1 insists merely on the propriety of performances like this: “I have reason to reject the conclusion of your argument, and thus, reason to believe that at least one of its premises is false”. This innocuous point about argumentative dynamics does not threaten nihilism; one would need to establish independently that nihilism’s implications are reasonable to reject. Alternative 2 says that the claims of common sense are justified, but not because they’re common sense. My response here depends on the alleged source of justification (if it’s perception, for example, see section 5). Alternative 3 says that common sense beliefs are pragmatically, not epistemically, justified. 3a: We should believe them because we could not get along without them. 3b: We may continue to

246 | Theodore Sider preferable to those that contradict common sense; common sense is an epistemic difference-maker. According to some it is nearly decisive; according to others it is one factor among many. Either way, Mooreanism seems to give us an (at least prima facie) argument against nihilism, since the existence of tables, chairs, and other composites is as commonsensical as it gets. But on the face of it, Mooreanism is utterly implausible. Why should the inherited prejudices of our forebears count for anything? We should, of course, trust common sense in some particular domain if there is independent reason to think that it is reliable about that domain. But there is no independent reason to think that common sense is reliable about whether there exist tables and chairs as opposed to there merely existing suitably arranged particles. Our forbears presumably did not even consider the latter possibility. After all, the issue is a subtle one, makes little practical

believe them because we already believe them and we can’t start from scratch (compare Lewis (1986a, pp. 134–5)). But we can get along without belief in tables and chairs, if we believe instead in particles arranged table-wise and chair-wise; and this doesn’t require starting from scratch. Alternative 4 is Gilbert Harman’s (1986) conservatism. Conservatism is less of a threat to nihilism than is Mooreanism, because of two points. First point: although conservatism says that one may carry on believing what one already believes even in the absence of positive reasons to do so, it does not prohibit radically rethinking one’s beliefs in order to facilitate global improvement in one’s belief state, such as the sort of global improvement promised by the argument from parsimony. And for a nihilist engaged in such a rethink, conservatism does not say that being previously believed gives one a lingering reason to believe the proposition that there exist tables and chairs, whereas Mooreanism says that one always has a reason (defeasible, of course) to believe such commonsensical propositions. Still, even though conservatism allows nihilists the rethink, it also allows their opponents to decline the rethink on the grounds that the promised global improvement of parsimony isn’t worth the disruption. But a second point defuses even this threat. Harman’s conservatism, it seems to me, is appealing insofar as norms of reasoning are conceived in a distinctive way: as being practically implementable. Consider, for example, Harman’s argument that alternatives to conservatism implausibly require us to keep track of all of our justifications. This is convincing only if we conceive of norms of reasoning as practical. Now, I agree that some norms should be conceived in this way, but this does not prevent us from recognizing other norms that are conceived differently. In foundational inquiries, for example, such as those undertaken by philosophers, we submit ourselves to demands that would be out of place in ordinary epistemic life, in full realization of the practical difficulties of doing so. Even a conservative might admit that in foundational contexts—such as the one we are in now—we cannot decline the rethink on the grounds that it would be too disruptive. In such contexts, we are governed by norms of reasoning that are less practical, more ambitious (though perhaps quixotic).

Against Parthood | 247 difference, and can even seem empty (see section 8).24 The Mooreanism I oppose says that we should trust common sense even in the absence of independent reason to think that it is reliable. And that seems no better than the absurd: “believe the masses”. Why are so many metaphysicians Mooreans? Partly because they fear that if we reject common sense, there will not be enough to go by.25 Both metaphysics and inquiry generally, it is thought, would be paralyzed. Without Mooreanism, we could not reply to the external-world skeptic, for example; we could not dig in our heels and say: of course there is an external world! But again a flat-footed answer tempts. The dictates of common sense are often independently reasonable, and when they are, they do not need backing from common sense. Reason can stand on its own. Consider, for example, Russell’s and Quine’s answer to the external-world skeptic: it is reasonable to posit a world of external objects because this posit best explains our sensory experiences.26 However exactly we cash out the notion of inference to the best explanation, this sort of inference need not be rooted in its commonsensicality. Inference to the best explanation is just: reasonable! The objection to Mooreanism is not that common sense must be shown to be reliable before it can justify. We should not require the reliability of all sources of justification (such as inference to the best explanation) to be antecedently demonstrable; that would apparently lead to skepticism. The objection is a simpler one: commonsensicality is just not a source of justification. Some Mooreans disavow the form of argument It is common sense that ϕ Therefore, (probably) ϕ Instead, they simply insist on: ϕ

where ϕ is in fact common sense. They do not infer that tables exist from the fact that common sense says that tables exist; so, it may be 24 I would also stress these facts in an answer to Korman’s (2009) problem of reasonableness. 25 See, e.g. Zimmerman (2007). 26 Russell (1912); Quine (1948). See also Vogel (1990).

248 | Theodore Sider thought, they do not rely on the prejudices of our forebears. Rather, they simply take as a premise: there are tables. This maneuver is just a fig leaf. These propositions that Mooreans simply take as premises exhibit a striking pattern: they include all the dictates of common sense. If Mooreans realize this but are unwilling to regard common sense as a source of justification, it would be unreasonable (and un-self-aware) for them to continue insisting on the premises, unless they have reason to believe that there is another source (or sources) of justification for the premises. Consider a man who believes (perhaps defeasibly) whatever his father believes, about a wide range of subjects. He doesn’t cite his father’s beliefs as evidence, but we detect this pattern in what he’s saying, and point it out to him. If he is unwilling to accept that being believed by his father confers epistemic worth, he must surely then accept that there is some other positive epistemic status or statuses shared by these beliefs. If it is unreasonable to accept that there is some other such status or statuses, he shouldn’t just continue with his pattern of believing whatever his father believes. There is a further reason to dislike the Moorean approach to metaphysics; but first we must consider the relationship between nihilism and ordinary language.

3. ORDINARY AND FUNDAMENTAL LANGUAGES The Moorean argument assumes that nihilism contradicts commonsense beliefs about composites. This assumption is incorrect if the ordinary believer and the nihilist mean different things by sentences like ‘there are tables and chairs’, so that the nihilist’s denial of such sentences is compatible with the believer’s assertions. And this may well be the case. Consider Nihilo, god and creator of a world comprised solely of subatomic particles. On the first day Nihilo creates some particles and arranges them in beautiful but lifeless patterns. He becomes lonely, so on the second day he creates some minions (or rather, particles arranged minion-wise). On the third day he tries to teach his minions to speak. But this goes badly. The dim-witted minions struggle to understand Nihilo’s talk of subatomic particles and their physical states. So on the fourth day he teaches them an easier way

Against Parthood | 249 to speak. Whenever an electron is bonded (in a certain way) to a proton, he teaches them to say “there is a hydrogen atom”; whenever some subatomic particles are arranged chair-wise he teaches them to say “there is a chair”, and so on. (Pretend that electrons and protons have no proper parts.) When the minions utter sentences like ‘there is a hydrogen atom’, do they speak falsely? They do if their language is the same as the language I used to describe the example, since I described Nihilo as having created a world comprised solely of subatomic particles. But perhaps the minions’ language is different; perhaps what the minions mean by ‘there is a hydrogen atom’ is consistent with I meant in my description of the example when I said “the world is comprised solely of subatomic particles”. Perhaps, for example, by ‘there is a hydrogen atom’ the minions mean a proposition that is true if and only if, as I (and Nihilo) would put it, some electron is bonded to some proton.27 In that case the minions speak truly. So there’s a question of whether the minions speak truly or falsely. But even if the minions speak falsely, there is an important distinction to make amongst their falsehoods. Nihilo taught them to utter ‘there is a ϕ’ in certain circumstances; call such utterances correct if and only if the specified circumstances in fact obtain. Correct utterances, even if untrue, play a role in communication and thought that is similar to the role played by true ones. For example, telling a visiting philosopher-minion from Iowa riding the N train that “The NYU philosophy department is near the 8th street stop” will have the desired effect (since the particles arranged NYUphilosophy-department-wise are indeed near the particles arranged 8th-street-stop-wise); telling her that “The NYU philosophy department is near the Astoria-Ditmars Boulevard stop” would not. Neither sentence is true, but the first and only the first is correct; usefulness here tracks correctness, not truth. Again: if confronted, in ideal perceptual conditions, by particles arranged chair-wise, a minion would be warranted in thinking to himself “there is a chair”, and saying this to others—or at least, more warranted than

27 If Nihilo had created a world more like the world I believe in (see section 11), containing impure sets as well as subatomic particles, then another possibility would be that the minions’ sentence is true iff there is a set containing an electron and a proton bonded to each other.

250 | Theodore Sider thinking and saying various alternatives, such as “there is an elephant”. If nihilism is true, we speakers of ordinary language are like Nihilo’s minions. We’re trying to find our way in a world whose ontology is minimal, we know little if any particle physics, and we certainly don’t have enough computational power to derive useful conclusions from what we do know about particle physics. It’s useful to say things like “there is a chair” when there are some subatomic particles arranged chair-wise, even if there really aren’t any chairs—just as it’s useful for the minions to speak as instructed on the fourth day. Indeed, it would be sensible for creatures like us to adopt a system of conventions or norms that prescribe saying things like “there is a chair” in appropriate circumstances. Perhaps we speak falsely (though correctly) when we say such things. But perhaps instead we speak truly. Just as there’s a question of whether the minions’ sentence ‘there is a hydrogen atom’ is true if and only if (as Nihilo and I would put it) some electron is bonded to some proton, so there’s a question of whether the same is true for English. My last few sentences threaten paradox. I defend nihilism, which I defined as the claim that there are no composite entities. Wasn’t I speaking English when I said this? If so, how can I be open to the possibility that English sentences like ‘there is a hydrogen atom’ and ‘there is a chair’ are true? Distinguish what ‘there is’ and other quantifiers mean in English from what ontologists use them to mean. In my definition of nihilism, ‘there is’ was intended in the ontologist’s sense. If the ontologist’s sense differs from the ordinary English sense, then nihilism is consistent with the claim that ‘there is a hydrogen atom’ is true in ordinary English. Perhaps, for example, the truth condition for this sentence in ordinary English is that it be true in the ontologist’s sense that some electron is bonded to some proton. More and more ontologists are coming around to the view that taking their subject seriously requires making some sort of distinction between ordinary and ontological understandings of existenceclaims.28 It’s not only defenders of minimal ontologies who find the 28 See Dorr (2005); McDaniel (2009); Sider (2009, 2011, 2012); Cameron (2010a, b); Turner (2010). A seminal work is Fine (2001), which argues that a related distinction is needed throughout metaphysics; see also Fine (2009).

Against Parthood | 251 distinction useful. Even defenders of fuller ontologies sometimes deny the existence of some ordinary things, so to speak, such as holes and shadows (McDaniel (2010)), propositions and numbers, or economies and organizations, and may wish to say that ordinary claims about such ordinary things are true. It’s not enough merely to distinguish ordinary language from the ontologist’s language; ontologists also need an asymmetry between them. If there’s nothing special about the ontologist’s language—if it’s just one language among many—then why make such a big deal over what’s true in it? Ontologists have therefore tended to say that their language is distinguished by being fundamental. It gets at the facts more “directly” or “perspicuously” than do nonfundamental languages; it expresses the facts that “underly” all other facts. (What “underlying” amounts to is a complex issue. Here I will say only that ordinary speakers needn’t have any idea of what unfathomably complex reality underlies their ordinary utterances, just as they needn’t have any idea of the fundamental physics that underlies their ordinary utterances.) It might be objected that since ontology has traditionally been about what there is—i.e. what there is in the ordinary sense—I have simply changed the subject. But I think that fundamental ontology is what ontologists have been after all along. It’s what they’ve been fumbling for with misguided talk of what “strictly” or “literally” exists. And it’s certainly in line with the traditional conception of metaphysics as inquiry into the ultimate nature of reality. There are subtle questions about how exactly to understand this notion of fundamentality (see Sider (2011, chapter 7)). Although I will generally remain neutral on such questions, I should mention one conception of fundamentality, and one construal of the dispute over nihilism, that I reject. Jonathan Schaffer (2009) construes ‘fundamental’ as a predicate of entities: some entities are fundamental and others are not. (He defines this predicate in terms of ontological dependence: fundamental entities are those that do not depend on other entities.) Moreover, according to Schaffer, in disputes over ontology, all sides ought to accept that the disputed entities exist; the only question is whether the entities are fundamental. So on Schaffer’s construal, all participants in the dispute over nihilism agree that there are composite entities; what nihilists think is that

252 | Theodore Sider only simple entities are fundamental.29 For me, on the other hand, ‘fundamental’ is not a predicate of entities, but rather attaches to concepts such as quantification; and for me, the question of nihilism is whether, under the fundamental sort of quantification, there are composite entities. So my picture is not that there exist both composite and noncomposite entities, with only the latter enjoying a certain status; it is rather that there are different ways to take ‘exists’, and in the fundamental sense of ‘exist’, there simply do not exist any composite entities. To reject nihilism thus construed, you don’t have to think that “tables and chairs are fundamental entities”. You just have to think that in the fundamental sense of ‘exists’, there exist tables and chairs. You might even combine this with the view that tables and chairs are in some sense “nonfundamental entities” (perhaps in the sense that all of their properties are nonfundamental).30 The distinction between existence in the ordinary sense and existence in the fundamental sense should not be thought of as “arising from within natural language”, so to speak. It is to be drawn with distinctively metaphysical concepts, not everyday linguistic concepts or concepts from empirical semantics. For example, the distinction is not supposed to derive from any ambiguity or context-sensitivity of natural-language quantifiers. Any such ambiguity or contextsensitivity (such as contextual variation of quantifier domains) is irrelevant. For another example, “exists in the fundamental sense” should not be equated with “strictly and literally exists”, as it’s often put. If ‘I exist’ is true in English, then its truth is both strict and literal, in any normal sense. “Literal” is opposed to things like metaphor and hyperbole; “strict” casts off things like quantifier domain restriction and loose talk (as when people who live in Cherry Hill, New Jersey say they’re “from Philadelphia”31); and ‘I exist’ is neither metaphorical nor hyperbolic nor restricted nor loose. The distinction is metaphysical. There are two quite different sets of facts one might 29 Schaffer (2009, p. 361). This is not the only possible construal, given the conception. Someone who regarded ‘fundamental’ as a predicate of entities, and did not admit a distinction between fundamental and nonfundamental quantification as I do, could still hold that, under the one and only sort of quantification, there simply are no composite entities. 30 See Sider (2009, section 9). 31 See Sperber and Wilson (1986); Wilson and Sperber (2004).

Against Parthood | 253 be getting at with talk of ‘existence’. The facts in one set are what we express with ordinary talk of what “exists”; the facts in the other set are much more fundamental, and may only be expressible by shifting to an entirely different language introduced with stipulations like this: “Quantifiers are not to mean what they mean in English, but rather are to mean something perfectly fundamental, albeit with a similar inferential role”.32 In light of this section, then, we should reformulate nihilism as the view that in the fundamental sense, there are no composite entities.33

4. MOOREANISM AGAIN Mooreanism assumes that inconsistency with common sense makes an epistemic difference. Common sense consists of propositions believed by ordinary people; and the propositions that ordinary people believe are those expressed by ordinary sentences. In the case at hand, these are ordinary sentences like ‘there are tables’. So according to Mooreanism, in order to decide whether to accept 32

See Sider (2012). The language of this formulation cannot be a perfectly fundamental one since it contains ‘composite’, which is defined in terms of ‘part’. The language must instead be a mixed one, with fundamental quantifiers but nonfundamental predicates. But there are arguments purporting to show that all such languages are suspect (see my 2007a, section 2.7, and 2011, section 9.6.1). Also, in this mixed language, ‘part’ might be semantically empty, since it might have no suitable basis in the fundamental. (Entire sentences containing ‘part’ in the language of the minions can be given a basis in the fundamental, but that language’s quantifiers are not fundamental.) The latter concern could be addressed by making the claim metalinguistic: “‘there are composites’ is not true”. This is an improvement but doesn’t capture the form of nihilism discussed at the very end of the paper, which identifies ordinary objects with sets: on this view, ordinary objects like tables and chairs do exist in the fundamental sense (since they’re sets), and they satisfy the nonfundamental predicate ‘composite’ (their “parts” are their subsets). So perhaps we should return to a perfectly fundamental language, give up on a general formulation of nihilism, and instead formulate particular nihilistic ontologies, such as “Everything is a fundamental particle”, “Everything is a point of spacetime”, “Everything is either a point of spacetime or a set”, and so on. (These formulations must be cleaned up since ‘point of spacetime’, ‘fundamental particle’, and ‘set’ are probably not fundamental predicates. The third one, for example, can be cleaned up as follows: “There is exactly one thing that has no members but is not a member of any open thing [this is the null set]; everything else either has a member [and so is a set], or is a member of some open thing [and so is a point of spacetime]”.) 33

254 | Theodore Sider nihilism, we must ascertain whether nihilism allows such ordinary sentences to be true. If it does, then it passes the Moorean test, and we have no common-sense-based reason to reject it. But if nihilism prohibits their truth (albeit allowing their correctness), then it fails the Moorean test, and we have our common-sense-based reason to reject nihilism. Whether nihilism allows these sentences to be true turns on a difficult issue in metasemantics.34 Recall Arthur Eddington’s (1928) claim that because of the mismatch between our ordinary conception of solidity and the scientific fact that matter is largely empty space, the ordinary notion of solidity has no application in our scientific world—ordinary objects like tables aren’t really solid. Most philosophers nowadays agree with L. Susan Stebbing’s (1937) reaction at the time: Eddington was wrong about the table; tables are indeed solid; it’s just that common sense was wrong about what it takes to be solid. Of course, even Stebbing and her contemporary followers will admit that an Eddingtonian stance is sometimes appropriate. However commonsensical it was that mental illness is caused by demonic possession, that simply wasn’t (and isn’t) true.35 The Stebbingsonian will not say: “Mental illness is caused by demonic possession, it’s just that common sense was wrong about what it takes to be possessed by a demon”. The difficult issue in metasemantics is this: how distant from our ordinary ways of talking can the underlying facts get, before what we say counts as false? As with Eddington’s table, nihilism implies a mismatch between our ordinary conception and the underlying reality. According to our ordinary conception of existence, simple and composite things

34 Caveat: suppose “there exists” in ordinary English functions analogously to theoretical terms in science—it is intended to mean something fundamental, whatever fundamental is “in the vicinity”, regardless of whether it satisfies our ordinary conception of the term. I doubt this is the case; but if it is, then given nihilism, ‘there are tables’ will be false regardless of how much metasemantic tolerance there is, and the argument of this section won’t apply. (But Mooreanism is even less plausible for claims phrased using theoretical terms.) 35 Nor was it or is it correct. This case is not meant to be analogous to the case of solidity, in which the metasemantic conservative ought to say that truth and correctness come apart. It is rather meant to be a case in which reality differs so drastically from our ordinary conception that even the metasemantic liberal will think that truth is not present; and in such a case, the conservative should not claim that correctness is present.

Against Parthood | 255 exist in the same way. We ordinarily think of “there are tables” and “there are subatomic particles” as getting at facts that are similar save that one concerns being a table and the other concerns being a subatomic particle. But according to the nihilist, “there are subatomic particles” gets at a fact of fundamental singular existence, whereas there are no such facts in the vicinity of “there are tables” (the only facts in the vicinity are facts such as that there are things arranged table-wise). Thus our ordinary conception, which embraces similar macro- and micro-existential facts, fails to match the underlying nihilist reality. As I say, the general issue of how much mismatch it takes to undermine truth is a hard one. Now, one response to the Moorean argument against nihilism would be to take a stand on this hard issue, argue for a liberal conception of when ordinary sentences are true in a hostile metaphysical environment, and conclude that nihilism doesn’t after all conflict with common sense.36 But this is not how I want to respond to the Moorean argument (though I wouldn’t be surprised to learn that the liberal conception is right). My response is rather that what we say about the hard issue cannot possibly have the epistemic significance that Mooreanism requires it to have. The question of when Eddingtonian views are true is of no deep epistemic importance; so the important question of whether it’s reasonable to believe nihilism can’t turn on how we resolve it; so Mooreanism can’t be right. It’s an interesting question whether Eddington was right that the ordinary sentence ‘tables are solid’ is falsified by modern atomic theory. But how we resolve this question surely carries no weight when one is deciding whether to believe modern atomic theory. It is intuitively clear that, rather than using our prior beliefs about whether tables are solid to decide what to believe about the atomic theory, we ought instead to decide on independent grounds whether the atomic theory is correct, and whether Eddington was right about the connection between the atomic theory and solidity; and we ought then to use our answers to those questions to decide whether to believe that tables are solid. The Eddingtonian question is that of how much “metasemantic tolerance” there is—how much error there can be in our ordinary 36

Cameron (2010b) and perhaps van Inwagen (1990, chapter 10) take this approach.

256 | Theodore Sider conception of a term before paradigmatic sentences containing the term become false. Its answer lies in metasemantics, in how semantic facts are determined. Consider how we determine how much metasemantic tolerance there is. We think about our reactions to Eddington’s argument, and our reactions to metasemantic thought experiments (like: if the things we think are cats were discovered to be robots, would they still be rightly called “cats”? (Putnam, 1962)). Surely our reactions to these thought experiments carry no weight when it comes to deciding what to believe about the atomic theory, or about nihilism. It might be objected that my argument illegitimately semantically ascends. I construed the Moorean as demanding consistency with the truth of certain sentences. But, it might be claimed, what she demands is rather consistency with my having a hand, with the existence of tables, with murder being wrong. . . . If so, Mooreanism does not concern sentential truth, and so, it may be thought, Mooreanism does not make epistemic value depend on metasemantics. This response is like the fig-leaf maneuver at the end of section 2. Mooreanism says that reason demands that we accept certain propositions p1, p2 . . . Although these propositions are exactly the propositions that are expressed by the sentences of common sense, the Moorean now insists that it’s not under this description that reason demands that we accept them. But then, under what description does reason demand that we accept them? To refuse to answer would be unsatisfying. And to answer that the are justified by some further feature they have—that they best explain our evidence, say— would be to give up on Mooreanism. The Moorean might answer that the are justified because they’re propositions of common sense, where this status attaches directly to the propositions, and is not due to the fact that they’re expressed by commonsensical sentences. But this would render Moorean justification implausibly precarious in the following way. Suppose for the sake of argument that nihilism is true and that English is a metasemantically intolerant language, so that ‘there are tables’ is in fact false. The Moorean says that the proposition that there are tables is a proposition of common sense, and that we therefore have a common-sense-based reason to reject nihilism. But imagine we had spoken a slightly different language, L, which is a lot like English but

Against Parthood | 257 differs in its metasemantic tolerance, so that ‘there are tables’ is true in L (this sentence is true in L if and only if there are things arranged table-wise). We could, I think, easily have spoken such a language simply by using ‘true’ and other semantic vocabulary more liberally in conjunction with reflective discussions of Eddington’s table, thought experiments about robot cats, and the like (while continuing to use such semantic vocabulary disquotationally, insofar as we actually do). The difference between being a speaker of L and being a speaker of English would only show up in highly theoretical contexts, for example contexts in which the speaker is aware of the question of nihilism and the distinction between fundamental and nonfundamental uses of language. Think, now, of the plight of speakers of L. They are cut off from the justification to reject nihilism that we speakers of English possess. For that justification comes from the fact that nihilism is inconsistent with the proposition that there are tables; and ordinary speakers of L have no sentences that express this proposition. (Their sentence ‘there are tables’ is true if and only if there are things arranged tablewise, and so does not express that proposition.) Only philosophically sophisticated speakers of L could even formulate the proposition that there are tables (using the sentence ‘there are, in the fundamental sense, tables’), and it’s hard to see why they should recognize the proposition in this guise as one of common sense. The problem with this Moorean answer, then, is that it makes our access to Moorean justification implausibly precarious; we could not have accessed it if we had used semantic vocabulary in an innocuously different way. At the beginning of this section I refrained from defending nihilism by appealing to a liberal view about metasemantic tolerance. I did so not only because I am not sure whether liberalism is correct, but also because I doubt that the reasonableness of nihilism could turn on whether it is. For the remainder of the paper, I will continue to not appeal to liberalism in my defense of nihilism, but rather will assume the conservative view for the sake of argument. This gives nihilism the strongest possible defense (since the objections typically presuppose the conservative view), and it avoids the risk of overinflating the significance of metasemantics to epistemology.

258 | Theodore Sider 5. THE PERCEPTUAL ARGUMENT A further objection to nihilism is that we have perceptual evidence for the existence of composite things like tables and chairs: we see, hear, smell, touch, and taste them. A natural first reply is that we have no such perceptual evidence because our perceptual experiences would be exactly as they are in fact if subatomic particles were arranged as they actually are but composed nothing.37 Perceptual experiences are determined by interactions between subatomic particles (those in our sensory apparatus, the perceived object, and the environment); and these interactions are unaffected by whether the particles compose further entities.38 But this first reply is not decisive, since it might be argued that it’s just a fact about justification that we are justified in believing our senses—and this despite the fact that things would appear the same even if our senses were deceived. A recent view of this sort has been put forward by James Pryor (2000).39 According to Pryor, if it perceptually seems to me as if p, then I have an “immediate justification” for believing p—immediate in that the justification doesn’t rest on any further evidence or justification. In particular, I needn’t be able to independently rule out alternative hypotheses that are also consistent with my perceptual experiences, such as the hypothesis that I am a brain in a vat that is stimulated to have those experiences. Given this view, someone might argue that seeming to see a table40 immediately justifies believing that there is a table, even if one can’t rule out the nihilistic hypothesis that the visual experience is caused by particles arranged table-wise rather than by a table.

37 See, for example, Merricks (2001, pp. 8–9). Note that this claim might be false if the contents of perceptual experiences include singular propositions about particular external objects. For an overview of issues about the contents of perception, see Siegel (2005). 38 Even a dualist about consciousness can accept this since the states of subatomic particles can include the holding of irreducibly phenomenal relations. 39 See also Pollock (1974); Huemer (2001); Burge (2003). 40 Note that Pryor construes the contents of perception “thickly”, so that they include propositions about physical objects (and not sense-data, say); see (2000, pp. 538–9).

Against Parthood | 259 To be sure, this immediate justification for believing in the table might be outweighed by other evidence (such as philosophical arguments in favor of nihilism). But at least it provides some evidence against nihilism, according to the objector. (And, the objector might say, philosophical arguments are invariably weaker than evidence supplied by perception.) My response to this argument will be based on examples like the following. Suppose you are just learning of the scientific evidence for the modern atomic theory of matter. And suppose further that— and you know this—Eddington was right that the atomic theory implies that tables are not solid. You then walk into your kitchen and perceive a table as being solid. It would be closed-minded and irrational to say: “The table looks solid, so the atomic theory must be wrong!” Rather, to the extent that the scientific evidence for the atomic theory is strong, you should take that evidence to show that your perception of solidity is unreliable. Further, the scientific evidence doesn’t merely outweigh the perceptual evidence in favor of solidity in the overall balance of reasons. Rather, the original perceptual evidence simply “vanishes”. For imagine that the scientific case for the atomic theory had been equivocal. Your degree of confidence in the atomic theory should not then have been attenuated because the table looked solid! It should have been as high as the scientific case warranted. To put it in terms of all-or-nothing belief: no matter how weak the scientific case had been, provided it was stronger than the opposing case, you should have believed (albeit tentatively) the atomic theory. A mediocre scientific case for the atomic theory could not have been overcome by the fact that the table looks solid.41 This example does not conflict with the idea that perception is a source of immediate justification. It merely shows that the notion of immediate justification must be properly understood, so as to allow immediate justification to be capable of vanishing in this way. Pryor 41 Could this be because sources of justification are lexically ordered, with scientific evidence outranking perceptual evidence? But “scientific evidence” is partially constituted by perceptual evidence; moreover, perceptual evidence is normally thought to be very strong. Further, even if the example is changed so that the scientific evidence in favor of the atomic theory is replaced with some other form of evidence— testimonial evidence, say—it still seems that the evidence, however weak, would not be overcome by the perception of solidity.

260 | Theodore Sider himself says that the immediate justification delivered by perception is merely prima facie, and that prima facie justification can be “defeated or undermined by additional evidence” (2000, p. 534). Perhaps the atomic theory’s challenge to the apparent perception of solidity is akin to John Pollock’s (1986) notion of undercutting defeat. For present purposes, it isn’t important exactly how the vanishing is conceptualized;42 what is important is that no matter how weak the scientific evidence gets, if it favors the atomic theory, that is what we should believe. The conflict between apparently perceiving a solid table and the atomic theory of matter is, I think, analogous to the conflict between apparently perceiving composite objects and nihilism. To anyone who understands the challenge of nihilism and takes it seriously, any prior perceptual justification in favor of tables vanishes. Arguing against nihilism on the basis of perception is no better than arguing that the atomic theory of matter must be false because tables look solid. It might be objected that the cases are disanalogous because the scientific evidence for the atomic theory was so much stronger than the alleged philosophical evidence for nihilism. But recall how the perceptual evidence for solidity vanished, and was not merely outweighed, once the atomic theory was on the scene. No matter how weak the scientific case for the atomic theory had been, I claimed, it would not have been overturned by the apparent perception of solidity. (Notice that this is so even if the scientific case for the atomic theory relied heavily on super-empirical considerations such as simplicity.) So the strength of the philosophical case for nihilism does not matter. Regardless of its strength, overturning it because of perceptual experience is no better than overturning a scientific case with similar strength for the atomic theory because “tables look solid!” Here are some further examples to bolster my response to the argument. • An astronomer considers the theory that a certain star has just gone nova. Then she looks into the nighttime sky, and it visually

42 Caveat: like many forms of epistemic defeat, it is unclear whether this vanishing can be construed in Bayesian terms. See Pryor (2011).

Against Parthood | 261 seems to her that the star is now twinkling. She realizes that light takes time to reach Earth from distant stars, and hence that the star would appear to twinkle even if it no longer existed. Nevertheless her experience as of the star now twinkling persists. • A physicist considers the special theory of relativity. Then he seems to perceive two events as being simultaneous. He understands the relativistic explanation of what is going on; nevertheless his experience as of simultaneity persists. • A student begins to rethink her racist upbringing. Nevertheless she still seems to perceive The Other as inferior. (The belief is not inferential; it forces itself on her immediately, as with more mundane perceptual beliefs.) She understands that deeply ingrained prejudice can be slow to dissipate; nevertheless, her racist experience persists.43 In each example, it seems to me, the perceptual experiences have no justificatory force, not even outweighed force. In each example, no matter how weak we imagine the conflicting evidence to be, it would not be overcome by the perceptual experience. In the first example, for instance, it would be absurd to try to overturn a weak but winning case from astronomy by pointing out that the star appears now to be twinkling. These examples—and that of the atomic theory of matter—are, I say, analogous to the situation we are in with nihilism. The important points of analogy seem to include (but may not be exhausted by) the following. (1) A proposition is given in perception but conflicts with a theory. (2) The theory is one that we’re taking seriously—we aren’t merely idly considering its possibility. (3) The theory has some evidential support.44 (4) The theory provides a specific, reasonable account of why perception is unreliable in the case at hand. In the examples, it is intuitively clear that any perceptual justification in the proposition vanishes, and is not merely outweighed. I conclude that the same is true with nihilism. Pryor says that although perceptual justification can be defeated by certain ordinary challenges, skeptical challenges don’t defeat 43 Pryor’s view is limited to what he calls perceptually basic beliefs. Someone might argue in this case (or even others) that the beliefs in question are not perceptually basic. 44 This is arguably inessential.

262 | Theodore Sider perceptual justification (2000, p. 534). Suppose it appears to me that a computer screen is in front of me, but a skeptic points out that my experiences would be the same if I were a brain in a vat. In the face of this skeptical challenge, even though I have no independent reason for thinking that the vat scenario is not actual, I can, according to Pryor, continue to justifiably believe in what I perceive (hence his name for his position: dogmatism). Might it be argued that the nihilistic hypothesis is a skeptical challenge to perceptual beliefs in composites, not an ordinary challenge, and hence that perceptual justification in composites does not vanish in the way that I have been arguing? There’s no reason to think that Pryor intended the notion of a skeptical challenge to be understood in this way; but might this position be defended? The nihilist’s challenge differs from the skeptical one in that, intuitively, it is a real contender to be believed, whereas the brain in a vat hypothesis is a mere possibility—something that is hard to independently rule out but for which we have no positive evidence. It is hard to make this distinction precise, but the following factors seem relevant. (1) By ordinary standards, nihilism is supported reasonably well by the evidence (so long as that evidence is construed neutrally; but see the discussion of Williamson in section 6), whereas the vat hypothesis is not. (2) Nihilism gives a satisfying explanation—again, by ordinary standards—of our perceptual experiences; the vat hypothesis does not. (3) There are positive reasons to believe nihilism, but not the vat hypothesis.45 Are these differences enough to rebut the idea that the nihilist’s challenge is a skeptical one? The notion of a “skeptical challenge” is not a precise one, so it is hard to say anything definitive here. But perhaps the following bird’s-eye remarks constitute progress. There is a point to having a concept of justification that allows skeptical challenges to be summarily dismissed: namely, to avoid the stultification of inquiry. If explanation-givers needed to be able to answer the challenge posed by each and every alternate explanation, including the brain-in-the-vat “explanation”, then we would never get

45 Compare Pryor: “I don’t want to claim that you never have to rule out skeptical hypotheses . . . [Prima facie justification for perceptual beliefs] can be undermined or threatened if you gain positive empirical evidence that you really are in a skeptical scenario” (2000, pp. 537–8, my emphasis).

Against Parthood | 263 anywhere. But a concept allowing nihilism to be dismissed in this way would also be stultifying. It would encourage tunnel-vision, limiting our attention to the familiar, and discouraging the consideration of radically new approaches to old problems. (Indeed, openness to unfamiliar viewpoints is part of what philosophy is most concerned to teach.) I don’t think our existing concept of justification is stultifying in this way, so I don’t think the nihilistic challenge can be dismissed in the way that the skeptical challenge can. But if this is wrong as a descriptive matter, then so much the worse for our existing concept of justification. We ought then to adopt a better concept that is more tolerant of challenges to the status quo.46 Incidentally, the preceding discussion yields a defense against the following thought: philosophy is less secure than science and ordinary thinking—so much so that it couldn’t possibly overturn scientific or ordinary beliefs in composites. The defense is this: once the question of nihilism has been seriously engaged, it becomes an open question just what science and ordinary thinking deliver. Before nihilism was in question one might be forgiven for assuming that verdicts on whether tables and chairs exist are delivered. But once nihilism is in question, one can no longer assume this—to do so would be like continuing to assume that perception favors the solidity of tables once the atomic theory of matter is in question. One must instead treat what is secure in science and ordinary thought as being more neutral propositions, such as the proposition that there exist things arranged table-wise.

6. WILLIAMSON In chapter 7 of The Philosophy of Philosophy, Timothy Williamson addresses the question of what our evidence is, when we ask philosophical questions. This is relevant to our discussion since nihilism would be refuted if the evidence we must accommodate in philosophy included such propositions as that there are tables. Williamson’s discussion is compelling in many ways; and it is clear that his sympathies do not lie with radical philosophical 46 Someone who regarded epistemic justification as metaphysically fundamental, or close to it, might feel less free to so cavalierly consider changing our existing concept.

264 | Theodore Sider views like nihilism. But in the end, Williamson’s arguments do not refute nihilism, since a nihilist can consistently embrace Williamson’s conclusions. One of Williamson’s central aims is to oppose the psychologizing of philosophy. The following trend, Williamson argues, is common but misguided. A radical philosophical position like nihilism is under discussion. In an attempt not to “beg questions”, only “neutral” evidence is admitted. Propositions such as that there are tables are not neutral (since they immediately rule against nihilism); so philosophers turn instead to certain propositions about mental states, such as the proposition that there appear to be tables. And in addition to psychologizing the evidence, some philosophers go further and psychologize the very question under discussion, construing it as being about language or concepts rather than the external world. Williamson argues that we should psychologize neither the evidence nor the subject matter. The question under discussion squarely concerns the external world: Do there exist composite entities? And while psychologizing the evidence would protect nihilism from immediate refutation, it would also, Williamson argues, lead to skepticism. Thomas Kelly (2008) bolsters Williamson’s case here by appealing to a general requirement of total evidence: One ought to form beliefs based on all of one’s evidence. Mundane examples show that forming conclusions based on only some of one’s evidence leads to trouble; but if the requirement of total evidence is generally correct, then it remains so even in philosophy. Rather than following the “Cartesian” procedure of using only propositions meeting some higher, more rarified standard deemed more appropriate for philosophical questions, we ought always to utilize all of our evidence, even if the evidence concerns “dialectically inappropriate” propositions such as the proposition that there are tables. Moreover, Williamson argues elsewhere (2000) that every proposition one knows is part of one’s evidence. So if objectors to nihilism, and uncommitted but interested bystanders, do indeed know that there are tables, then this becomes part of the evidence that their philosophical theories must accommodate, and for them, the case against nihilism is immediate and decisive. These claims—that we should psychologize neither the evidence nor the subject matter, and that all known propositions should enter

Against Parthood | 265 into the evidence used to decide philosophical questions—threaten nihilism only if the objectors and bystanders do in fact know that there are tables. And why think that they do? After all, they have no independent reason to reject nihilism. The mere fact that they have no independent reason to reject nihilism does not by itself show that they do not know there are tables. We apparently have no independent reason to think we are not brains in vats, but this does not deprive us of all knowledge of the external world.47 To put it vaguely, knowledge does not require the ability to independently rule out all conflicting hypotheses. But it surely requires the ability to independently rule out a great range of conflicting hypotheses. Pointing out certain alternatives that an opponent has not and cannot independently rule out is a paradigmatic, perfectly ordinary way of showing that one’s opponent does not know. Imagine a scientist who has put forward a theory to explain certain data, but then discovers a rival theory, put forward by a respectable colleague, that she cannot rule out. The scientist does not know that her theory is true. This kind of undermining of knowledge is utterly ordinary and commonplace. Granted, one needn’t be able to independently rule out all the conflicting claims of skeptics, cranks, and perhaps even nonskeptical, noncranky alternative hypotheses that one simply hasn’t considered, in order to know.48 But nihilism is not a skeptical hypothesis, it’s not the claim of a crank, and here we are considering it. It’s hard to know how to define “skeptical hypothesis” or “crank”. But rather than tackling such difficult general questions, just consider the analogies between the challenge that nihilism poses to ordinary claims to knowledge (such as the claim to know that there are tables), on the one hand, and perfectly ordinary, nonskeptical challenges to knowledge like the example of the unexcluded alternative scientific theory just mentioned. Or consider again the analogy between nihilism and the challenges to the status

47 In general, Williamson (2007, chapter 7) draws on the many analogies between radical philosophical positions like nihilism, on the one hand, and skepticism on the other, to defend ordinary claims of knowledge from attacks. 48 And one needn’t be able to independently rule out the “conflicting hypothesis” of p’s negation in order to know p.

266 | Theodore Sider quo considered in the previous section (the atomic theory of matter, the twinkling star, the perception of simultaneity, and the racist). Or better, consider the even closer analogy between nihilism and certain challenges to the status quo presented by physicists and philosophers of physics. One example is like the example of simultaneity considered earlier. According to the special theory of relativity, on its Minkowskian conception anyway, physical reality consists of matter in four-dimensional spacetime, rather than consisting of matter in three-dimensional space as we used to think. It would have been inappropriate for a turn-of-the-twentieth-century curmudgeon to object to Einstein and Minkowski by claiming that her evidence includes the proposition that two finger-snaps are objectively simultaneous. Moreover, surely the curmudgeon did not know that proposition. This is not merely because Einstein and Minkowski were right. For imagine that they are wrong; there is such a thing as objective simultaneity after all. Still, assuming the curmudgeon understood what Einstein and Minkowski were saying, she surely did not know that the finger-snaps were objectively simultaneous. For a second example, consider “configuration space realism”, a serious theory about the foundations of quantum mechanics according to which reality ultimately unfolds, not in a space of three or four dimensions, but rather in the many-dimensional configuration space of quantum mechanics, where that space is conceived not as an abstract mathematical formalism but rather as the concrete space of reality.49 This view is perhaps more threatening to ordinary beliefs about physical objects than nihilism is, for no part of configuration space can be straightforwardly identified with ordinary three- or four-dimensional space or spacetime.50 Yet it seems clear that the view cannot be refuted simply by appeal to knowledge of the existence of ordinary three- or four-dimensional things. Once the view has been taken seriously, we no longer know that such things exist.

49 See Albert (1996); North (2012). Actually the term “configuration space realism” includes also a view that is perhaps less threatening to the status quo: the view that both ordinary space and configuration space are fundamental. See Dorr (2009). 50 See Ney (2012).

Against Parthood | 267 Nihilism’s challenge to the status quo is like the challenges in these examples, which cannot be answered by citing the status quo as evidence, and which are not like the challenges posed by skeptics and cranks. There is another argument against nihilism that can be based on Williamson’s views about evidence. We noted earlier Williamson’s claim that every proposition one knows is part of one’s evidence. Williamson (2000) also accepts the converse: All evidence must be known.51 Since ‘knows’ is factive, it would follow that evidence must be true. But given nihilism, one might think, most if not all of our perceptual beliefs are false. So nihilism implies that we have little if any perceptual evidence.52 A nihilist could, of course, respond by challenging the claim that all evidence is known. Alternatively, a nihilist could argue that the contents of perceptual experiences do not concern ordinary external objects. Perhaps they are nonpropositional or perhaps they concern appearances or sense-data; either way, nihilism would allow plenty of perceptual evidence after all. But a nihilist can concede more to the objection and still escape. The trick is to extend the distinction between truth and correctness. Correctness was claimed in section 3 to be an adequate substitute for truth in our epistemic and cognitive lives. We can introduce corresponding substitutes for other factive concepts such as knowledge and (given Williamson’s view) evidence. “Quasi-knowledge”, let us say, is the substitute for knowledge: Quasi-knowledge is to knowledge as correctness is to truth. Similarly, “quasi-evidence” is the substitute for evidence. I cannot define these notions, but I hope the intuitive idea is clear: The conceptual or theoretical role of these concepts is to be like that of the originals, except with correctness substituted everywhere for truth. (For instance, if part of the knowledge role is that knowledge be “safe” from error, as Williamson (2000) says elsewhere, part of the quasi-knowledge-role is that quasi-knowledge be safe from incorrectness.) Even if nihilism

51

For similar views see Meyers and Stern (1973); Unger (1975, chapter 5). A cautious person might (try to) form only nihilist-friendly perceptual beliefs (e.g. that some things are arranged table-wise). But the case of less cautious people remains. 52

268 | Theodore Sider precludes most perceptual evidence, then, it allows us a rich array of quasi-evidence.

7. THE CARTESIAN ARGUMENT A familiar Cartesian idea is that one can be certain of one’s own existence. Given the further (and much less Cartesian!) premise that one is not mereologically a simple entity, one can infer that nihilism is false. Nihilism allows sentences about our own existence to be correct even if they are untrue, just as it allows sentences about hydrogen atoms to be correct even if untrue. So the alleged certainty cannot be of the mere correctness of the claim that we exist. The Cartesian objector must claim to be certain that, in addition to there being particles arranged thinkingcogito-wise, she herself exists. It’s hard to see why she should be so certain—or even justified. The preceding sections establish, I take it, that we are not entitled to conclude on Moorean, perceptual, or Williamsonian grounds that ordinary things like tables and chairs exist. What further grounds are there for concluding that we ourselves exist, as opposed to there merely existing appropriately arranged particles? Van Inwagen (1990, chapter 12) seems to suggest that further grounds lie in the nature of mentality. He concedes that the correctness (or truth, given a liberal metasemantics) of ‘there is a table’, ‘there is a hydrogen atom’, and so on, demand nothing more than appropriately arranged particles; but, he says, the correctness (or truth) of ‘I am thinking’ demands more. It demands that there be a thinker that is me. Mentality is metaphysically singular. But why think this? What is wrong with saying that the correctness (or truth) of ‘I think’ is a matter of arrangements of particles? It’s not enough to emphasize how justified or certain ‘I think’ is, or the immediacy of our awareness of it. The arrangement of particles constituting its correctness (or truth) might be one that is especially immediate, both epistemically and psychologically. Rejecting materialism about the mind would not on its own support metaphysical singularity. Irreducible or nonsupervenient mentality could consist of irreducible or nonsupervenient mental relations which relate many subatomic particles, rather than irreducible or nonsupervenient mental properties that are instantiated by single entities.

Against Parthood | 269 Perhaps van Inwagen’s belief in metaphysical singularity has something to do with the character of conscious experience? A subject’s simultaneous experiences are experienced by that subject as being in some sense part of one conscious episode, and as experienced by a single subject. But it is unclear why these aspects of phenomenology could not be due, metaphysically, to states of particles.

8. THE DEFLATIONARY ARGUMENT Certain “ontological deflationists” argue that the dispute over nihilism is not a substantive one. Rather than concerning the objective nature of the world, it is merely verbal or conceptual or notational. Some even claim that nihilism is conceptually incoherent, on the grounds that it’s a conceptual truth that composites exist if subatomic particles are appropriately arranged.53 Ontological deflationism challenges all philosophical ontology, not just nihilism. If it’s a conceptual truth that composites exist if subatomic particles are appropriately arranged, then it’s presumably also a conceptual truth that holes exist if objects are perforated, that propositions exist if sentences are synonymous, that directions exist if lines are parallel, and so on. But then it’s incoherent to deny the existence of holes while accepting perforated objects, to deny the existence of propositions while accepting synonymous sentences, to deny the existence of directions while accepting parallel lines, and so on. The practice of ontology presupposes the coherence of such denials, and so is quite generally undermined by ontological deflationism. A full discussion of this issue would take us too far afield and would repeat what has been said elsewhere.54 But in brief: My reply to the deflationist is that even if sentences like ‘composites exist if subatomic particles are appropriately arranged’ are conceptual truths of ordinary languages, they’re not conceptual truths of the ontologist’s fundamental language. And so, since nihilism is formulated in a fundamental language, it is not conceptually incoherent. 53 Writings in this ballpark include Carnap (1950); Putnam (1975, 1987); Thomasson (2007, 2009); Chalmers (2009); Hirsch (2011). 54 Dorr (2005); Hawthorne (2006, 2009); Eklund (2007, 2009); Sider (2009, 2011, chapter 8, 2012).

270 | Theodore Sider 9. GUNK An object is “gunky” if and only if each of its parts (including itself) has proper parts. Nihilists must obviously reject talk of gunk in fundamental languages, since they think those languages do not contain ‘part’. But they must also reject talk of gunk in non fundamental languages. For talk of composite objects in nonfundamental languages rests on fundamental talk about simple subatomic particles; recall how the minions were taught to speak of hydrogen atoms when protons are bound to electrons. The rules given to the minions make no provision for talking about proper parts “all the way down”. Similarly, since the correctness (section 3) of talk of parthood and composite objects rests on fundamental talk of simple subatomic particles, nihilists cannot admit that talk of gunk is correct or that talk of tables and chairs could be correct despite gunk. Nihilists simply cannot admit gunk. But is there any reason to think that gunk exists? (I.e. that gunk actually exists; the next section discusses the possibility of gunk.) Traditional arguments that point-sized things are somehow conceptually incoherent are unconvincing since we now know that theories of point-sized things are mathematically coherent (and anyway, the arguments wouldn’t immediately imply gunk—mereologically simple things might be larger than point-sized). But there is a more compelling recent argument in favor of gunk. Frank Arntzenius (2008) argues that a physics based on a gunky space or spacetime has the advantage of collapsing certain distinctions to which the laws of nature are insensitive, for example the distinction between open and closed regions. (Arntzenius doesn’t regard this as a decisive case for gunk, but rather as a reason to pursue a certain formal project, namely, that of seeing whether physical theories based on gunk can be fully developed.) I have no particular response to this argument, except to say that the added complexity of countenancing a fundamental part– whole relation must be weighed against the benefit Arntzenius adduces, and moreover, that the attraction of the gunk-based theory will depend in part on the simplicity of its ideology and also on how simply the laws of nature may be formulated in terms of that ideology. So my case for nihilism must be tentative at this stage.

Against Parthood | 271 There is a further argument one might offer in favor of gunk: an inductive argument that there are no smallest particles.55 Historically, the following pattern has been repeated several times: a type of particle was discovered; the particles were first thought to be simple; but scientists later concluded that the particles are in fact made up of smaller particles. Physicists first posited molecules as the ultimate particles, but molecules gave way to atoms, which gave way to protons, neutrons, and electrons, which have given way to quarks, leptons, and gauge bosons. Each time a new type of particle was discovered, physicists posited new features of the newly discovered particles, whose distribution accounted for, but could not be accounted for in terms of, the distribution of the distinctive features of the older, larger particles. This historical progression of theories will probably continue forever, the argument continues, so there are no ultimate particles on whose features everything depends. But this argument is bad, for a number of reasons. First, induction from four cases is unimpressive. Second, the argument at best supports the claim that there are no smallest bearers of physical magnitudes; but there might yet be smallest things. Third, by moving from initial “finite” observations to an “infinite” conclusion, the argument makes a big leap. Compare it to the argument that there must be infinitely many people, since for each person we’ve observed, there exists a taller person.56 Fourth, the argument assumes a particle ontology. I have been writing as if a particle ontology were indeed correct; but a better approach, I think, rejects particles in favor of points of spacetime (or points of some “higher-order” space such as configuration space). Spacetime (or some higher-order space) must be posited regardless in order to support fields. But then the particles are gratuitous; and moreover one would need additional ideology, such as the predicate ‘particle x is located at point p’, to connect the particles to spacetime. (Although I believe this “supersubstantivalist” view to be correct, I’ll go back to writing as if what exists fundamentally are subatomic particles.)

55 56

The remainder of this section overlaps Sider (2011, section 7.11.2). Thanks to Jason Turner and Cian Dorr for these last two points.

272 | Theodore Sider There is also a fifth and subtler problem, though it depends on certain assumptions about the nature of fundamentality. Assume that (i) we can speak of the fundamentality of features (such as the property of having unit negative charge), and that (ii) fundamentality is all-or-nothing, rather than a matter of degree. The historical progression of physical theories that is cited by the inductive argument may then be formulated as follows: Theory 1: The fundamental features are those of molecules. Theory 2: The fundamental features are not those of molecules, but are rather those of atoms. Theory 3: The fundamental features are not those of atoms, but are rather those of protons, neutrons, and electrons. Theory 4: The fundamental features are not those of protons, neutrons, and electrons, but are rather those of quarks, leptons, and gauge bosons. These theories, notice, concern which features are fundamental, and not just the existence of composite entities. (It might be objected that the inductive argument could ignore the facts about fundamentality and consider merely a progression of theories about the existence of composite entities; but it is bad inductive practice to draw conclusions based on arbitrarily selected subsets of one’s evidence.) Now, we are asked to inductively draw a certain conclusion from the fact that scientists have been led to accept, and then subsequently reject, Theories 1–4. But what conclusion? Two possibilities suggest themselves: Conclusion 1: It’s parts all the way down, but there is some mereological level at which all the fundamental features reside. The features of all other objects (including objects at mereologically smaller levels) depend on these fundamental features. Conclusion 2: It’s parts all the way down, and there is no such level. For every mereological level, mereologically smaller parts have distinctive fundamental features. But neither conclusion is inductively suggested by the initial pattern. Conclusion 1’s postulation of smaller objects beyond the level on which everything depends is gratuitous, so it’s hard to see why it would be inductively suggested.

Against Parthood | 273 Conclusion 2 is a very bizarre hypothesis (a kind of infinite ideological complexity). And it isn’t inductively suggested by the initial pattern. Conclusion 2 might seem at first to be suggested because it has the superficial appearance of a kind of limit point of the initial pattern, if that pattern were infinitely extended. By moving through Theories 1–4, so the idea goes, scientists have been moving closer and closer to Conclusion 2. But this impression vanishes upon closer inspection. Each Theory in the progression does not add a new layer of fundamental features, but rather replaces the previous Theory’s layer (since it regards the previous layer as just depending on the newly hypothesized layer). Extending the pattern indefinitely results in a series that simply has no intuitive limit. For comparison, imagine a countably infinite series of chairs: c1, c2, . . . Suppose first that in scenario 1, c1 is filled; in scenario 2, chairs c1 and c2 are each filled; in scenario 3, chairs c1, c2, and c3 are each filled; and so on. I suppose there’s a sense in which the limit of this series is a scenario in which all the chairs are filled. But consider a second series in which only c1 is filled in scenario 1, only c2 is filled in scenario 2, only c3 is filled in scenario 3, and so on. This series has no intuitive infinite limit, and certainly not one in which all the chairs are filled. The imagined infinite extension of the progression through Theories 1–4 is like the second series. It has no intuitive infinite limit, and certainly not Conclusion 2. The assumption of all-or-nothing fundamentality is crucial to this criticism. If fundamentality came in degrees, we could re-describe Theories 1–4 as follows: Theory 1a: Molecules have certain distinctive features. Theory 2a: toms have certain distinctive features, which are more fundamental than those of molecules. Theory 3a: Protons, neutrons, and electrons have certain distinctive features, which are more fundamental than those of atoms. Theory 4a: Quarks, leptons, and gauge bosons have certain distinctive features, which are more fundamental than those of protons, neutrons, and electrons. If continued infinitely, this progression does seem to have an infinite limit, namely:

274 | Theodore Sider Conclusion 3: It’s parts all the way down, and for every mereological level, mereologically smaller parts have distinctive features that are more fundamental than the features of the previous level. A full discussion here would require delving into difficult questions about the nature of fundamentality. Here I’ll make just two brief points. First, there are reasons to reject comparative fundamentality (Sider 2011, chapter 7). Second, the friends of comparative fundamentality are likely to argue that comparative fundamentality must be well founded; it cannot be that for each feature there is a more fundamental feature. (This stance does not on its own rule out gunk. Gunk is infinite descent in the part–whole relation; the stance rules out infinite descent in the fundamentality-over-features relation.57) Given well-foundedness, Conclusion 3 is guaranteed, on independent grounds, to be false.

10. POSSIBLE GUNK So with the possible exception of Arntzenius’s argument, I don’t think there are good arguments that gunk is actual. But the alleged possibility of gunk is sometimes thought to threaten nihilism.58 Gunk is, I suppose, epistemically possible. Maybe scientists will one day tell us that there is gunk after all; or maybe Arntzenius’s argument will prove decisive. I don’t pretend to know that these things won’t happen. But defenders of nihilism can happily grant that nihilism itself is epistemically possibly false. Substantive metaphysics is not a search for epistemic first principles, compatible with whatever the future might bring; it can be held hostage to empirical fortune. This is the price a metaphysician pays for regarding her speculations as substantive hypotheses about the real world. If the future brings evidence for gunk, I will reduce my degree of belief in nihilism accordingly.

57 If one construed fundamentality as applying to facts rather than features, then the assumption that relative fundamentality is well founded might prohibit gunk on its own. See Sider (2011, section 7.7). 58 See, I’m afraid, Sider (1993).

Against Parthood | 275 A quite different threat comes from the alleged “metaphysical” possibility of gunk. If gunk is metaphysically possible, then nihilism is not metaphysically necessarily true (let all modalities be understood as metaphysical henceforth). But nihilism is a “proposition of metaphysics”; and such propositions are noncontingent; they are necessarily true if true and necessarily false if false. So nihilism is necessarily false, and so it is actually false.59 I have no clear definition of ‘proposition of metaphysics’, but I have in mind propositions about abstract and general questions that metaphysicians debate, such as “numbers exist”, “any charged object instantiates the property of being charged”, “time is like space”, and so on. The argument from the possibility of gunk faces a challenge. Consider this argument for the opposite conclusion: “nihilism is possibly true; nihilism is a proposition of metaphysics and hence is noncontingent; so nihilism is necessarily true; so nihilism is true”. This argument assumes the possibility of nihilism and concludes that nihilism is true; the previous paragraph’s argument assumes the possibility of gunk and concludes that nihilism is false. Anyone who wants to defend the previous paragraph’s argument needs an asymmetry between gunk and nihilism, a reason to think that gunk is genuinely possible but nihilism is not.60 But I won’t press this point, since in my view the argument fails for a more basic reason. All such arguments from possibility are undermined by what I believe to be the correct metaphysics of modality: modal “Humeanism”.61 The Humean theory assumes that necessity and other modal notions are not fundamental. It further gives the following reduction of necessity: to be a necessary proposition is to be a proposition that is (i) true, and (ii) of an appropriate type, where the appropriate types are given by a list that I will specify in a moment. More carefully, to be necessary is to be a logical consequence of the true propositions of the types on the list. For example, one of the types on the list is the type mathematical proposition (i.e. proposition purely about 59

The alleged possibility of emergent properties raises some of the same issues. Our “intuition” of possibility might be claimed to be stronger in the case of gunk. Alternatively, the possibility of nihilism might be rejected on the grounds that it clashes with the principle of universal composition, a principle that may be alleged to flow from the very nature of the part–whole relation (see Sider (2007b)). 61 See Sider (2011, chapter 11) for a fuller and more careful presentation. 60

276 | Theodore Sider mathematics); thus, the necessity of the proposition that 2 + 2 = 4 involves nothing more than the fact that (i) two plus two in fact equals four, and (ii) the proposition that 2 + 2 = 4 is a mathematical proposition. What types of propositions go on the list? The list is given by our use of ‘necessary’; nothing metaphysically deep unifies it. (Thus the Humean theory is in a sense deflationary: it says that there is much less to modality than most philosophers think.) Given the way ‘necessarily’ is typically used—by philosophers, in the sense of metaphysical necessity anyway—the list clearly includes at least these types: 1. 2. 3. 4.

propositions expressed by analytically true sentences propositions of mathematics “natural kind” propositions (such as: all water is made of H2O) propositions of metaphysics

Consider, now, type 4: propositions of metaphysics. (Similar remarks apply to types 2 and 3.) The inclusion of this type on the list corresponds to the dogma mentioned ealier: the metaphysical is a noncontingent subject matter.62 The truth of this dogma is a shallow matter, according to the Humean; it is simply the result of our decision to mean by ‘necessary’ a property of true propositions given by a list including the type proposition of metaphysics.63 (This is not to say, where M is a metaphysical proposition, that M itself, or the proposition that M is necessarily true, is about that linguistic decision, or that its truth is counterfactually dependent on the decision.) I won’t try to defend the Humean theory here, except to say that it seems to me the most promising form of modal reductionism. The 62 There are some dissenters. According to Cameron (2007), for example, it’s contingent whether mereological composition is unrestricted. On my view, the list of types of propositions can vary with the speaker’s context; Cameron’s statement is true in a somewhat nonstandard but still linguistically allowable context, namely, one in which the kind proposition of metaphysics is dropped from the list. In this context the argument from the possibility of gunk remains unsound, now because the premise that nihilism is noncontingent is false. The argument is unsound for the same reason if the Humean decides that since some metaphysical disputes are contingent (over Humean supervenience or dualism, say), not all propositions of metaphysics go on the list, provided atomism-related propositions are thus left off the list. 63 And also that this type is closed under negation.

Against Parthood | 277 leading alternatives are Lewisian modal realism and conventionalism. But modal realism is very hard to believe, and conventionalism requires the discredited notion of truth by convention.64 Assume for the sake of argument that the Humean theory is true. The problem for the argument from the possibility of gunk is then intuitively the following (I will lay out the argument more carefully in a moment). Given the Humean theory, to be necessary is to be true and to fall under a type on the list. But proposition of metaphysics is one of the types on the list. So for a proposition of metaphysics such as nihilism, necessity just boils down to truth. But then, the only way to support the claim that nihilism isn’t necessary is to argue directly that nihilism is false, in which case the argument from possibility plays no distinctive role. No one argues from possibility for mathematical propositions; no one tries to argue against Goldbach’s conjecture by asserting its possible falsity and then citing the noncontingency of mathematics. Perhaps this is because we realize that in mathematics there is no distance between truth and necessity, and so we cannot support the possible falsity of Goldbach’s conjecture except by directly supporting its actual falsity. At any rate, this is the situation that I think obtains for propositions of metaphysics. More carefully. The objector’s premise is that gunk is possible; or, equivalently, that (mereological) atomism is not necessary. Now, consider the dialectical situation in which the objector and the nihilist both accept the Humean theory of modality. It is then common ground between them that what it is for the objector’s premise to be true is for the following to be true: (P)

atomism is not a logical consequence of any true propositions that fall under one or more of the kinds on the list

I will argue that there is no distinctively modal way for the objector to support (P). Only by arguing directly that atomism (or some related proposition) is actually false can she support (P). But if she could do that, she would have no need for the argument from possibility, since she could argue directly from the actual falsity of atomism to the falsity of nihilism. The argument from possibility is superfluous.

64

See Sider (2003).

278 | Theodore Sider The point is clearest if atomism is itself a proposition of metaphysics. In that case atomism itself falls under a kind on the list, in which case it’s hard to see how (P) could be supported other than by directly arguing that atomism is false. (If atomism is true, then atomism would be a true proposition that falls under a kind on the list, and which implies atomism.) And if we had a direct reason for thinking that atomism is (actually) false, that would on its own give us a reason to reject nihilism, without the need for modal considerations. The situation is a little more complex if atomism isn’t a proposition of metaphysics.65 To reduce complexity, let me make a few assumptions, which I’ll take to be common ground between the nihilist and the objector. First, this discussion is being conducted in a nonfundamental language (so that talk of parts is dialectically appropriate). Second, this language contains the means to state nihilism (recall that nihilism is a thesis about what is fundamentally the case). And third, if nihilism is true, then the following conditional is analytically true: “If nihilism is true, then atomism is true”. (Recall the point from the beginning of section 9: If nihilism is true, then nonfundamental talk of composites is governed by rules of use that rest all talk of parts on talk of simple subatomic particles.) Given these assumptions, the only way to support (P) would seem to be by directly arguing that atomism is false. For both sides agree that if nihilism is true, then nihilism and the proposition expressed by “if nihilism is true, then atomism is true” are both true propositions that fall under some kind on the list. Return to the first, simpler, case in which atomism itself is assumed to be a proposition of metaphysics. I claimed that the only way to support (P) in this case is to argue directly that atomism is false. This is not merely because atomism implies that (P) is false, or even that it’s common ground in the dialectical situation that this is so. (It can be common ground that the conjunction of the premises of an argument implies its conclusion without its being the case that the only way to support the conjunction of the premises is to directly 65 Not that anything deep is at stake in the question of whether atomism is a proposition of metaphysics; the notion of a proposition of metaphysics is vague and not particularly fundamental.

Against Parthood | 279 support the conclusion.) It is rather based on inspection of (P). All (P) says is that atomism isn’t a consequence of a certain class of propositions, a class that is simply defined as the class of all true propositions of a certain type T; and atomism is admitted by all hands to be of type T. Compare (P) to the claim: The number 2 is not identical to any member of the set of numbers that are both (i) amongst Ted’s favorite numbers, and (ii) even It’s hard to see how one could support this claim without directly arguing that 2 isn’t one of Ted’s favorite numbers; similarly, it’s hard to see how the objector could support (P) without arguing directly that atomism is false. How else could the objector support (P)? What reason could the objector offer for thinking that atomism is not a logical consequence of any true propositions falling under some kind on the list? I suppose the objector might claim that her belief that atomism is not necessary is such a reason (since for atomism to fail to be necessary is precisely for it to fail to be a logical consequence of these propositions), thereby reversing what I think is the proper way to form beliefs here. Relatedly, someone might criticize me for recasting the question of whether atomism is nonnecessary as the question of whether (P) is true. The Humean theory of modality says that what it is for atomism to not be necessary is for (P) to be true; but, the critic might point out, epistemic features do not in general transmit across “what it is to be F is to be G”; thus the mere truth of the Humean theory does not imply that the only way to support the claim that atomism is not necessary is to support (P). To be fair, I began by assuming that the Humean theory is “common ground” between the nihilist and the objector, and not merely that it is true. But, it might be objected, even if one believes a metaphysical analysis, it does not follow that one’s attitudes towards the analysandum are equivalent to one’s attitudes towards the analysans. (Perhaps this follows if one knows the analysis; but surely no one knows that the Humean analysis is right.) These concerns are serious, but ultimately not compelling. For a rational approach to modal argumentation surely should be informed by reasonable beliefs about the underlying nature of modality. Imagine a time before you ever considered the question

280 | Theodore Sider of the nature of modality. Suppose you then thought that atomism might well be true, or at any rate took yourself to have no reason to think that atomism is false; but nevertheless, you also believed that atomism is not necessary. (Why? Perhaps you noted that the falsity of atomism is conceivable, doesn’t seem like standard examples of conceivable impossibilities, and even seems epistemically possible, and so you concluded that it’s probably possible.) At that time, you didn’t believe (P), nor did you take yourself to have any reason to believe (P). After all, you would have said, atomism is a proposition of metaphysics, and might well be true, in which case atomism itself would be on the list, in which case (P) would be false. But now, suppose you later come to believe the Humean view, and hence that what it is for atomism to not be necessary is just for (P) to be the case. Unless your reasons for coming to believe the Humean view somehow give you reason to believe (P) (and how could they?), you should surely then abandon your former belief that atomism isn’t necessary. Insofar as your belief in the Humean theory is tentative, the abandonment should be tentative; but the stronger you reasonably believe the Humean theory, the stronger the abandonment ought to be.66 Taking a step back: Given the Humean theory, conceivability is no guide to possibility when it comes to propositions of metaphysics. For as we have seen, the necessity of such propositions boils down to truth; and conceivability is no guide to the truth of propositions of metaphysics. (Conceivability might yet be a guide to possibility for certain other types of propositions. Perhaps our ability to conceive of a proposition’s being false is good evidence that it isn’t expressed by any analytically true sentence; in that case, this ability would be good evidence for its not being necessary if we know that the proposition doesn’t fall under any of the other types on the list.) Taking a further step back: I believe that those who argue from possibility for propositions of metaphysics typically make two presuppositions. First, modal facts are “further facts”: a proposition’s being necessary involves its possessing some further fundamental 66 My argument here is what Mark Johnston (1997) calls an “argument from below”; it assumes that, in this case anyway, we should look to the underlying metaphysical nature of a proposition to decide what attitude to take toward that proposition. (Johnston criticizes certain other arguments from below.) Acceptance of my argument here does not require accepting all arguments from below.

Against Parthood | 281 feature, beyond its merely being the type of proposition that it is. And second, conceivability gives (defeasible) evidence concerning the presence of this further feature. Perhaps such arguments have weight, given these presuppositions. But given the Humean theory of modality, the presuppositions are mistaken—there is no such further feature. My reply to the argument from the possibility of gunk has assumed the Humean theory of modality. Here are some brief remarks about how a nihilist might reply without making that assumption. First, the defender of the argument from the possibility of gunk must overcome the apparent symmetry mentioned earlier between gunk and nihilism itself: each is apparently conceivable, so why is only the first possible? Second, it is particularly difficult to maintain both the conceivability/possibility link in fundamental metaphysics, and the claim that propositions of fundamental metaphysics are noncontingent. Consider the fundamental nature of time: is a spatializing theory correct, or a presentist theory, or some other theory? For each theory, we can apparently conceive of its falsity; if conceivability implied possibility in fundamental metaphysics, then each theory would be possibly false, leading via the noncontingency assumption to the absurd conclusion that each theory is actually false. Third, I have been writing as if propositions of metaphysics are universally regarded as noncontingent. But two exceptions from David Lewis’s writings come to mind: Humean Supervenience (1986b, introduction) and materialism (Lewis (1983)). Lewis’s claim of Humean Supervenience was a contingent one, made only with respect to an “inner sphere” of possible worlds that include no fundamental (“perfectly natural”, in his terms) properties or relations beyond those that are actually instantiated; and his claim of materialism was similarly qualified. Lewis regarded these views as contingent because they are claims that the actual world lacks fundamental properties or relations of certain sorts (nonlocal qualities, fundamental mental properties), and because he had a generous view about what fundamental properties and relations are possible. But now notice that nihilism—as I have developed it, anyway—could be regarded as being contingent in exactly the same way. The key nihilist claim that ‘part’ is not fundamental ideology—or to put it less nominalistically, that there is no fundamental part–whole relation—could be restricted

282 | Theodore Sider to the inner sphere of worlds. Outside the inner sphere, a nihilist could say, there are fundamental two-place relations that are distinct from all actual fundamental relations, and which play “the parthood role”—they obey suitable axioms of mereology and otherwise behave the way that opponents of nihilism think the part– whole relation behaves. And in some worlds outside the inner sphere, the nihilist could say, one of these relations is “nonatomic”: each thing in that world bears the relation to some distinct thing in that world. In this sense the nihilist could allow that nihilism is contingent and gunk is possible.67

11. COMPOSITES NEEDED IN PHYSICS The final argument I will consider is, to my mind, the most formidable one. Our best physical theories include a physical geometry— a theory of the intrinsic structure of physical space or spacetime (or some higher-order space). But, it might be argued, physical geometries quantify over paths and regions, not just points; paths and regions are composite objects containing points as parts; and so, such theories must employ ‘part’. Our best physical theories are our best guide to the correct fundamental ontology and ideology. So, ‘part’ is likely part of fundamental ideology, and composites are likely part of fundamental ontology. Let’s look more closely into the alleged need for parthood. Some geometric notions apply only to points, and thus are nihilist-friendly. In Tarski’s axiomatization of solid geometry, for example, the primitive predicates ‘between’ and ‘congruent’ relate only points (Tarski and Givant (1999)). But consider the topological notion of an open region: a region where, intuitively, each point is surrounded in all directions by further points in the region. (An example of an open region is the spherical region consisting of all points strictly less than 67

Someone channeling Dorr (2005) might object that ‘part’ is a failed natural kind term and is therefore semantically empty; so the outer-sphere relations playing the parthood role are not really parthood; so the “nonatomic” world does not really contain gunk. (There is a parallel argument, in the case of materialism, that nonphysical fundamental properties playing “the mentality role” in the outer sphere are not really mental properties.) I’m not sure whether the argument is sound; but if it is, there is surely no good reason to think that the genuine claim, as opposed to the mere role-claim, is possible.

Against Parthood | 283 one meter from a given point p. The “closed sphere” consisting of all points less than or equal to one meter from p would not be open, since the surface of the sphere is part of the closed sphere, and points on the surface are not surrounded by points in the closed sphere. The first, “open”, sphere includes points that are arbitrarily close to the surface, but not points on the surface itself.) The predicate ‘open’ is a predicate of regions, and thus is not nihilist-friendly. And it cannot be replaced with a multi-place predicate relating the points that are part of that region, since open regions can contain infinitely many points. (One might formulate topology in a language with plural quantifiers, and take ‘open’ to be an irreducibly plural predicate. That would eliminate the need for composites and ‘part’, but at the cost of increased complexity in logical ideology. We will return to this trade-off later.) It is common to divide geometric structure into different “levels”: topological, differentiable, affine, and metric. We have been discussing an apparent need for regions at the topological level. However, on some views, there is no need to regard topological structure as metaphysically fundamental. For example, Tarski’s predicates ‘between’ and ‘congruent’, which generate affine and metric structure, induce a topology. So if an adequate geometry for physics could be built solely on Tarski’s predicates, there would be no need for composites of points. However, Tarski’s approach—which is designed to apply to flat spaces—takes metric facts to be direct connections between distant points. Such facts emerge from the holding of Tarski’s congruence predicate (‘congruent(x,y,z,w)’ means that the distance from x to y is the same as the distance from z to w). But in the curved spacetimes of general relativity, the metrical facts are normally taken to be pathdependent. That is, the distance between two points is not taken to be a direct connection between those points, but is rather defined in terms of path-length: the distance between two points is the length of the shortest path connecting them.68 The anti-nihilist argument would be immediate if distances under the path-dependent conception emerged from something as simple as comparative predicates of paths (such as: ‘path p1 is longer than path p2’). But the mathematics in question—the mathematics of 68

See Bricker (1993); Maudlin (1993, p. 196).

284 | Theodore Sider tensor fields on differentiable manifolds—is not that simple. Moreover, this mathematics characterizes geometry in highly “extrinsic” terms, using mathematical objects that are, intuitively, not part of the underlying geometric facts. What is needed is a “synthetic” development of differential geometry. We need what Tarski gave us for flat spaces: a theory using purely geometric predicates that can be regarded as underlying (via representation theorems) the usual mathematical development of differential geometry. Given such a synthetic development, we could look to see whether predicates of regions (or other composites of points) are needed. Unfortunately, very little work has been done on this topic. But it seems inevitable that a synthetic development of differential geometry will require predicates of regions, given the centrality of the notion of the geometry of “infinitesimal neighborhoods” around points. Moreover, the only attempt at constructing a synthetic differential geometry of which I know—Arntzenius and Dorr (2011)—does indeed employ predicates of regions and a predicate for parthood. So I will assume that the geometry of curved spacetimes does indeed call for something like predicates of regions. In sum: to do physical geometry we need a way to attribute a feature (such as openness or path-length) to a collection of infinitely many points. And the natural way to do this is to posit a “gathering entity”, an entity that somehow incorporates those points, and then attribute the features to the gathering entity. (The paths and regions mentioned earlier were gathering entities.) If the gathering entity must be a composite that contains the points as parts, then we need composites and parthood to do physical geometry. But the gathering entity could be a set instead. We could construe fundamental geometric features of paths and regions as being features of sets, not mereological composites, of points. In topology, for instance, we can take ‘open’ as a predicate of sets. It’s of course commonplace to take ‘open set’ as the undefined expression in purely mathematical topology; but what I am recommending is taking ‘open’ as the metaphysically fundamental notion for physical topology—the topology of physical space. My reason for preferring the set-theoretic conception of physical topology is that it is ideologically more parsimonious since we need set-theory anyway in our fundamental theory of the world. And my reason for thinking the latter is just the familiar indispensability

Against Parthood | 285 argument,69 but construed as an argument for a set-theoretic fundamental ideology in addition to an ontology of sets. According to this argument, our best fundamental theory includes mathematical physics; and the best theory of the foundations of mathematical physics—that is, our best theory of the fundamental nature of the world that contains mathematical physics (or something very much like it)—is set-theoretic. The set-theoretic foundational theory is a theory whose ontology includes sets and whose ideology contains, in addition to the ideology of first-order predicate logic, a primitive predicate for set-membership, ∈. Given this ontology and ideology, set-theoretic topology then requires only a single added primitive predicate, namely ‘open’, whereas mereological topology also requires the predicate < for parthood. The addition of < would be gratuitous. I won’t defend this form of the indispensability argument in detail, but let me comment briefly on two issues. First, the present perspective on ideological parsimony is that ideologically simpler theories aren’t just more convenient for us. They’re more likely, other things being equal, to be true. So when evaluating a competitor to the proposed (first-order) set-theoretic foundation for mathematical physics, one must take into account the competitor’s ideology as well as its ontology. A modal structuralist such as Hellman (1989), for example, saves on the ontology of sets but requires a primitive ideology of modal operators and higher-order quantifiers. Relatedly, compare my approach, in which ∈ is fundamental ideology, with Lewis’s (1991) structuralist mereological approach to set theory, which replaces ∈ in fundamental ideology with . Frege, G. The Foundations Of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd edn. Evanston, IL: Northwestern University Press, 1980. Gillon, B. Towards a common semantics for English count and mass nouns. Linguistics and Philosophy 15, 6 (1992), 597–639. Landman, F. Groups I. Linguistics and Philosophy 12, 5 (1989), 559–605. Laycock, H. Words without Objects. Oxford: Clarendon Press, 2006. Lewis, D. Parts of Classes. Oxford: Basil Blackwell, 1991.

42 This paper arose out of two excellent seminars I took while a graduate student at the University of Connecticut: Marcus Rossberg’s seminar on higher-order and plural logics, and Don Baxter’s seminar on identity in metaphysics. Thanks to the following for many helpful discussions on these issues: Andrew Bacon, Ralf Bader, Jc Beall, Don Baxter, Einar Bøhn, Colin Caret, Michael Della Rocca, Tim Elder, Tamar Gendler, Katherine Hawley, Michael Hughes, Philip Kremer, Jon Litland, Doug Owings, David Nicholas, Agustín Rayo, Marcus Rossberg, Yael Sharvit, Zoltan Szabo, John Troyer, Gabriel Uzquiano, Achille Varzi, Sam Wheeler, Bruno Whittle, and Jessica Wilson. Finally, I wish to thank four anonymous referees for comments that led to many improvements of this paper.

322 | Aaron J. Cotnoir McDaniel, K. Composition as identity does not entail universalism. Erkenntnis 73, 1 (2010), 97–100. Nicolas, D. Mass nouns and plural logic. Linguistics and Philosophy 31, 2 (2008), 211–44. Nicolas, D., and Linnebo, O. Superplurals in English. Analysis 68, 3 (2008), 186–97. Oliver, A., and Smiley, T. Is plural denotation collective? Analysis 68, 1 ( January 2008), 22–34. Rayo, A. Word and objects. Noûs 36, 3 (2002), 436–64. Rayo, A. Beyond plurals. In A. Rayo and G. Uzquiano (eds), Absolute Generality. Oxford: Oxford University Press, 2006, pp. 220–54. Saucedo, R. Composition, identity, and the number of things. Unpublished manuscript. Schlenker, P. Properties, plurals, and paradox. Unpublished manuscript: . Schwarzschild, R. Pluralities. Dordrecht: Kluwer, 1996. Sharvey, R. A more general theory of definite descriptions. The Philosophical Review 89 (1980), 607–24. Sider, T. Parthood. The Philosophical Review 116 (2007), 51–91. Uzquiano, G. The supreme court and the supreme court justices: A metaphysical puzzle. Noûs 38, 1 (2007), 135–53. van Inwagen, P. Composition as identity. In J. Tomberlin (ed.), Philosophical Perspectives, vol. 8. Ridgeview, Atascadero CA, 1994, pp. 207–20. Varzi, A. Mereological commitments. Dialectica 54 (2000), 283–305. Wallace, M. Composition as identity: Part I. Philosophy Compass 6 (2011a), 804–16. Wallace, M. Composition as identity: Part II. Philosophy Compass 6 (2011b), 817–27. Yi, B. Is mereology ontologically innocent? Philosophical Studies 93 (1999), 141–60.

8. Cut the Pie Any Way You Like? Cotnoir on General Identity Katherine Hawley 1. INTRODUCTION Aaron Cotnoir does all sorts of interesting things in his contribution to this volume. He makes a helpful distinction between syntactic and semantic objections to the thesis that composition is identity, and outlines some empirical points relevant to the syntactic issue. But the centrepiece is his development of a formal framework for addressing the semantic objections. Cotnoir articulates a general notion of ‘identity’ which can hold one-one, one-many, or many-many (where the identical manys don’t have to be equinumerous). The necessary and sufficient condition in each case for general identity is that the same portion of reality appear on each side of the identity sign; it doesn’t matter whether that single portion of reality is counted in different ways on each side, perhaps as a copse on one side and several trees on the other, or as three string quartets on one side and two ice-hockey teams on the other. Any such attempt to generalize identity must show how the more general relation is still an identity relation, and in particular how it conforms to Leibniz’s Law that identicals must be indiscernible. So Cotnoir offers us two alternative ways of preserving Leibniz’s Law, either by introducing an index, or by using subvaluational techniques. There is a lot to like and a lot to think about here. If Cotnoir’s generalization of both identity and Leibniz’s Law succeeds, this may have consequences for other philosophical puzzles which turn on worries about discernibility. Perhaps we now have space for a novel account of the relationship between (e.g.) the statue and the lump of clay, at least where these permanently coincide. They are surely the same portion of reality in the relevant sense, albeit ‘counted as’ a statue and ‘counted as’ a lump, respectively, and so perhaps they are generally identical despite their apparent

324 | Katherine Hawley differences. And might we now have a new solution to the problem of temporary intrinsics? The unripe green banana and the ripe yellow banana are the same portion of reality (counted at different times?), so they may be generally identical without being identical in the narrowest one-one way. Now, these and other applications require a distinction amongst one-one identities which Cotnoir does not make in his paper: that is, a distinction between those one-one identities which are governed by Leibniz’s Law in the strictest sense (‘numerical identities’, in Cotnoir’s terms), and those which are governed by Leibniz’s Law only in its indexed or subvaluational version. Moreover, considering such extensions of Cotnoir’s framework forces us to think harder about what it is for some objects to be the same portion of reality as each other: perhaps the unripe banana is the same portion of reality as the later ripe banana, but can we say the same of objects (like living organisms) which undergo very significant turnover of material parts even while continuing to exist? In this brief paper, I will focus on the notion of ‘same portion of reality’: what metaphysical assumptions must we accept if we are to acknowledge Cotnoir’s general identity as a genuine identity relation? We need to understand the metaphysics behind the semantics so that we can judge the significance of the claim that composition is general identity; moreover, we need to understand the metaphysics so that we can understand how, and whether, Cotnoir’s framework can be extended to address other philosophical problems. 2. WHY ANTIPODEAN COUNTERPARTS ARE NOT EVEN SLIGHTLY IDENTICAL As Cotnoir indicates, the notion of being the same portion of reality is crucial to his picture: In order to take many-one identity seriously, we need to suppose that we can refer to a portion of the world singularly or plurally, and that our way of referring to this portion of the world does not change the fact that it is the same portion either way. (p. 302)

Given a single portion of reality, different partitions capture different ways of dividing the portion into mutually-disjoint individuals:

Cut the Pie Any Way You Like? Cotnoir on General Identity | 325 your very own portion of reality can be partitioned as you, or as your right half with your left half, or as your top half with your bottom half, or as your cells, and so on. Each such single object (you), or plurality (the cells), results from partitioning the same portion of reality—is the same portion of reality—and this is why the relation ≈, the ‘general notion of identity’, holds between them. You ≈ your cells. The ‘same portion of reality’ constraint is also crucial to Cotnoir’s ingenious retooling of Leibniz’s Law. How can you be identical to your cells, when they are unthinking microscopic cells and you are a conscious, all-too-macroscopic human being? Cotnoir suggests two different answers to this question, drawing on independent considerations about the logic of plurals. The indexical option is to say that your cells are a conscious, macroscopic human being, relative to the single-human way of partitioning their portion of reality, and that you are unthinking microscopic cells, relative to the manycells way of partitioning your portion of reality. (Your portion of reality just is their portion of reality, of course.) Moreover Leibniz’s Law holds so long as we are careful to index to the same way of partitioning on each side of the general identity relation. The alternative subvaluational option is to say that it’s true that your cells are a conscious, macroscopic human being because there is some way of partitioning their portion of reality such that the resulting object(s) satisfies ‘is a conscious, macroscopic human being’. In the same way, it’s true to say that you are unthinking microscopic cells. (I am neglecting the difference between partitions and covers, along with some other subtleties.) Without the ‘same portion of reality’ constraint, these strategies can be generalized to absurdity. Consider the antipodean counterpart of a given object: that is, the object, if any, which is located on the exact opposite side of the Earth from that given object. It might seem that I and my antipodean counterpart are discernible: I am in Scotland, he is in New Zealand; I am female, he is male; I eat porridge for breakfast, he prefers kiwi fruit.1 But we could cook up a

1 In fact, if I have an antipodean counterpart right now, she/he/it is floating in the Pacific. But if I were in Gibraltar I might have an antipodean counterpart on Te Arai beach near Auckland. Thus the sun never sets on the Commonwealth (according to Wikipedia at least).

326 | Katherine Hawley semantics according to which it’s true to say that I am male (relative to my antipodean counterpart), or else true to say that I am male so long as either I or my antipodean counterpart is male. If we become intensely relaxed about discernibility, we can regard my antipodean counterpart and me as ‘indiscernible’, because there are ways of ‘attributing’ our properties to one another. Now, no one would mistake this for real indiscernibility, and the transworld antipodean-counterpart relation is no species of identity. This is because it does not satisfy the ‘same portion of reality’ constraint. There is no relevant sense in which I and my antipodean counterpart are the same portion of reality, even though the antipodean-counterpart relation is somewhat natural. So there is no sense in which we really partake in one another’s properties, no matter what semantics is cooked up. As Cotnoir makes clear, the plausibility of his claim that general identity is an identity relation rests upon the fact that general identity is governed by the ‘same portion of reality’ constraint. In this he follows Lewis’s lead in Parts of Classes: ‘Take them together or take them separately, [they] are the same portion of reality either way’ (1991, 81), and ‘the many and the one are the same portion of reality’ (1991, 87). To appreciate the force of the claim that composition is a kind of identity, we must therefore understand what it is for objects to be the same portion of reality as one another.

3. PORTIONS OF REALITY DISTINGUISHED FROM OBJECTS? Our task is to understand what it is for objects to be the same portion of reality as one another, in a way that shows why this is a genuine identity relation and the relation of antipodean counterparthood is not. Here is one picture: there are entities called ‘portions of reality’, each individual object or plurality of objects is associated with one such entity, and facts about these associations ground facts about which objects are the same portion of reality as one another. We might link this with a ‘stuff ontology’ according to which portions (or quantities) of stuff (or matter) are more fundamental than the individual objects (or pluralities of such objects) they constitute.

Cut the Pie Any Way You Like? Cotnoir on General Identity | 327 This two-level ontology promises to vindicate the claim that being-the-same-portion-of-reality is indeed an identity relation. The fundamental entities are the portions of reality, so the fundamental identity facts are facts about the identities of portions of reality. The dependency relationship between an object and its portion of reality is sufficiently intimate for the object to inherit its identity relations from those of its portion of reality. So far, so good. In places, Cotnoir beckons us towards the idea that portions of reality are fundamental, and that individual objects (single or plural) are mere aspects of our conceptual scheme, or in some other way non-fundamental. The intuitive pull behind composition as identity is the thought that we may ‘carve up’ reality however we like. But no matter whether we carve a portion of it as one individual or many, it is still the same bit of reality. (p. 302) Recall, the intuitive idea behind many-one identity is that identities are insensitive to our ways of counting things. In other words, what there is, and hence what is generally identical to what, does not depend on our practices of counting. (p. 302, Cotnoir’s italics) [Quoting Lewis] . . . [T]he many and the one are the same portion of reality, and the character of that portion is given once and for all whether we take it as many or take it as one . . . [But it] does matter how you slice it—not to the character of what’s described, of course, but to get the form of the description. (Cotnoir 307; Lewis 1991, 87) The count-sensitive predicates like ‘is a copse’ and ‘are five trees’ are true in virtue of the form of our description of the world . . . [Count-insensitive predicates] are true in virtue of the character of the world. (Cotnoir 313, his italics)

It sounds as if there are portions of reality, which have their characters independently of us. We slice, dice, or carve these portions in different ways, resulting in object(s) which may satisfy formally different descriptions, but only within the constraints imposed by the character of the underlying portion of reality; in particular, the truth of identity claims about these objects is governed by underlying facts about portions of reality. The tone suggests that objects like copses and trees are somehow the joint product of portions of reality and our ways of describing or counting. The ‘differences’ between copses and trees generated by our ways of counting are not the kind of deep-seated differences which can prevent copses and trees from being identical to one another.

328 | Katherine Hawley In his final paragraph, Cotnoir describe his framework as compatible with a more realist attitude to carving portions of reality into objects: The composition as identity theorist is free to endorse a single way of counting as the correct one, and in so doing would give an answer to the special composition question. And that answer need not be a universalist answer . . . But composition as identity theorists are also free to claim that all ways of counting are equally good . . . There are also intermediate views according to which some but not all ways of counting are correct. (317, Cotnoir’s italics)

And he keeps the options open elsewhere: It may even be controversial as to whether there is any mind-independent fact of the matter as to whether the referent of a term is many or one. (297, Cotnoir’s italics)

But even if it is the nature of reality, not our conceptual schemes, which determines how we ought to carve portions of reality into objects, the two-level picture can nevertheless help explain why objects which are the same portion of reality as one another are identical.

4. PORTIONS OF REALITY ARE OBJECTS But this two-level picture doesn’t really capture what’s going on in Cotnoir’s paper, for several reasons. First, for Cotnoir a copse simply is a portion of reality, the same portion of reality as the trees, so both copse and trees a fortiori have the same metaphysical status as the portion of reality. Objects are not merely associated with or constituted by portions of reality, they are portions of reality. This is not a two-level ontology, and portions of reality are not mere portions of stuff or matter. Second, Cotnoir does not need to quantify over or refer to portions of reality as such in his framework. The key notion is that of some object(s) being the same portion of reality as some object(s), but this does not need to be cashed out in terms of each object(s) standing in some relation to some particular portion of reality. Roughly speaking, for Cotnoir objects are the same portion of reality as one another only if they are ultimately composed by the same

Cut the Pie Any Way You Like? Cotnoir on General Identity | 329 atoms. (More precision is needed to ensure that we are dealing only with respectable, i.e. exhaustive but non-redundant, ways of dividing up a portion of reality into object(s). Cotnoir achieves the required precision in set-theoretic terms, but he warns us against reading ontological consequences off his decision to work with a set-theoretic rather than, for example, a higher-order plurals framework for his semantics.) Cotnoir’s strategy here shows that he is not trying to reduce object identity wholesale to sameness of portion of reality. Rather, the idea is to take ordinary, one-one, numerical identity between objects as well understood, then to generalize this, relying upon a notion of ‘same portion of reality’ which is defined in terms of the numerical identity of each atom with itself. Correlating ‘are the same portion of reality’ with ‘are composed of the same atoms’ makes the application of the framework to permanent statue-lump coincidence almost irresistible. But any application to the problem of temporary intrinsics (or accidental intrinsics) would require a different story about what it is for objects to be the same portion of reality as one another. Recall our task: to understand what it is for objects to be the same portion of reality as one another, in a way which shows why this is a genuine identity relation, unlike the relation of antipodean counterparthood. This task is of interest in its own right, but it is also an important first step towards applying Cotnoir’s framework to other philosophical puzzles. Can this be done without reifying portions of reality as a metaphysically distinctive category of entity, underlying the more familiar category of individual objects? Cotnoir has given us some grip on the being-the-same-portionof-reality relation by correlating it with the being-composed-ofthe-same-atoms relation. But anyone prima facie sceptical about composition as identity will still want to know why being-composed-of-the-same-atoms is a genuine (if somewhat loose) identity relation. To answer this question, we must widen our gaze, to consider the role which the identity relation is expected to play in metaphysics and elsewhere, to think about criteria of identity, the necessity of self-identity, the connection between identity and ontological innocence, and so on. Cotnoir has already considered perhaps the most important of these aspects of the identity-role—

330 | Katherine Hawley namely, the connection between identity and Leibniz’s Law—but establishing that being-the-same-portion-of-reality-as can do the theoretical work expected of an identity relation more generally would help to establish its credentials. University of St Andrews REFERENCES Cotnoir, Aaron J. (2013) ‘Composition as General Identity’. This volume. Lewis, David (1991) Parts of Classes. Oxford: Blackwell.

THE A-THEORY OF TIME

9. Living on the Brink, or Welcome Back, Growing Block! Fabrice Correia and Sven Rosenkranz In this paper, we clarify what proponents of the Growing Block Theory (GBT) should and should not say, and what they consistently can say. At some stage, our discussion will be premised on a simplifying ontological assumption, viz. that when it comes to metaphysics all we ever quantify over, besides purely abstract objects such as numbers, are instantaneous entities. Its controversial nature notwithstanding, this ontological assumption is not a priori incoherent, and so in order to show that GBT is a tenable view it will do to consider a version of GBT that subscribes to it. In fact, talk about a block to which more and more layers are added as time goes by strongly suggests that such a version of GBT is very much in the spirit of C. D. Broad’s original proposal (Broad 1923). Since this ontological assumption is likewise open to GBT’s main rivals, presentism and eternalism, we prejudge no issues in our attempt to bring out what is distinctive of GBT. Once all the central tenets of the view are on the table, we address both David Braddon-Mitchell’s and Trenton Merricks’s recent eulogies for GBT, based on what is representative of a certain type of argument meant to show that GBT is internally incoherent (Braddon-Mitchell 2004, Merricks 2006; cf. also Bourne 2002). This type of argument at no stage presupposes that our simplifying ontological assumption is false, and so it should retain its dialectical force, if any, even after this ontological assumption has explicitly been made. However, we argue that this type of argument proceeds from a mistaken assumption about GBT’s core, something which becomes most obvious in the light of the particular version of GBT we consider. We conclude that this type of argument misfires and that for all we know, we might be living on the brink of reality.

334 | Fabrice Correia and Sven Rosenkranz I We conceive of the debate to which GBT and its rivals, presentism and eternalism, contribute as centring on the question of what there is, where this question is couched in terms of the, metaphysically speaking, most fundamental notion of quantification (hence the small caps). This is quite consistent with the idea that GBT, presentism, and eternalism disagree about the facts that make up reality, and not just about the objects populating it. For, any such disagreement can be recast as one about what facts there are. Thus, ‘what there is’ is meant to refer to what there fundamentally is, or what there is in reality, and is meant to be absolutely unrestricted (at least as far as things in time are concerned, see next paragraph). This meaning is common to all the uses made of it, whether they are made by proponents of GBT, presentists, or eternalists. The parties disagree, though, on principles essentially involving such quantification. In other words, they tell different stories as to what there fundamentally is. The disagreement does not turn on whether in reality there are any purely abstract objects and, if so, what ontological status they enjoy at different times, where purely abstract objects are those abstract objects, like numbers but unlike certain sets, that do not depend for their existence on any concrete things. After all, GBT, presentism, and eternalism disagree about what there is in time; and most plausibly (and intuitionistic considerations aside), purely abstract objects, if any, are not in time. Consequently, our discussion of their disagreement will not in any way be affected if, for the sake of simplicity alone, we take the most fundamental notion of quantification, whatever its proper conception, to exclude purely abstract objects from its range. This restriction will come in handy later, when we use metric tense-logical operators and quantify over numbers to measure temporal distances. We hasten to add that quantification over numbers for such purposes alone may well be conceived not to involve any serious ontological commitment (cf. Prior 1971). The disagreement between GBT, presentism, and eternalism over what there is is best formulated in terms of a tensed notion of present existence (or being present). All parties agree that there are entities which presently exist. Presentists hold in addition that

Living on the Brink, or Welcome Back, Growing Block! | 335 there are only presently existing entities. Proponents of GBT and eternalists alike deny the presentists’ claim: eternalists hold that there are entities which were present but are no longer so, as well as entities which will be present but are not yet so, while friends of GBT accept the first but reject the second of these two eternalist claims. As we shall see in due course, characterizing GBT by means of a tensed notion of present existence will have the consequence that on this view (just as on the presentist view), the metaphysically most fundamental notion of quantification is likewise tensed. This, as we shall now argue, is indeed a welcome consequence that is faithful to the basic idea behind GBT. The basic idea behind GBT is the combination of two thoughts: reality constantly grows so that (i) at any moment in time, a new layer is added to it that did not form part of it before, and (ii) at any moment in time, there is no layer of reality in the future of the layer added to reality at that moment. Consequently, at any moment in time, there is a last layer of reality that is, as it were, on the edge of reality (Broad 1923: 87–9).1 Often, the view is merely presented as holding that reality consists of all the layers added to it up until the present moment and of no other layers. This claim, while certainly part of GBT, fails to capture the idea that the block is constantly growing. It at most captures the idea of a grown block, and not that of a block that continues to grow. But on closer inspection, it even fails to capture the idea that the block has grown to be what it is at the present moment, for it is so far consistent with the contravening thought that at any past moment, reality is exactly as it is today, never mind that the addition of layers occurs at different moments. Thus, to capture the dynamic aspects of the view, GBT must be conceived to operate with a tensed notion of reality and so of what there is.2 This should already make plain that, contrary to what 1 (Page references are to the reprint; see References section.) On pages 87–8, Broad (1923) speaks of ‘fresh slices of existence’ being added to ‘the total history of the world’ and defines the present as ‘the last thin slice’ added to it ‘that is succeeded by nothing’. For more pertinent quotes, see next footnote. 2 On page 88, Broad (1923) explicitly states that on his view of becoming, ‘the sum total of existence is always increasing’ so that ‘when we say that the red section [of a signal lamp’s past history] precedes the green section, we mean that there was a moment when the sum total of existence included the red event but did not include the

336 | Fabrice Correia and Sven Rosenkranz authors like Braddon-Mitchell and Merricks suggest, GBT does not appropriate eternalism with respect to the present and past layers of reality that it posits: unlike GBT, eternalism operates with a notion of what there is that is untensed.3 Eternalists often speak of their conception of reality as that of a block universe. This terminological coincidence might be taken to suggest that proponents of GBT and eternalists agree on their conception of the past (see Braddon-Mitchell 2004: 199–200; Merricks 2006: 104). But block isn’t block. If we think of past world-slices as those succeeded by others, then according to GBT, we can look back at a world-slice that has become past ever since the block continued to grow beyond it and can correctly identify it as past in this sense. What according to GBT we cannot do, however, is perform the mental operation of ‘going back’ to that past world-slice and once we have done so, ‘back there’ correctly identify it as past in this sense. But neither should we want to, if we believe in GBT: on this view, unlike the eternalists’ view, in performing that mental operation we consider what reality was like at a time when the block still had not grown beyond that world-slice, and evidently, reality was then such that that world-slice was on its edge and so was not succeeded by any other. It might be complained that a catalogue of all that there is should not, on anyone’s conception of it, come with a sell-by date. But any such complaint would not be suasive. For, as long as it is part and parcel of the view under discussion that reality changes over time in the sense that always, reality comprises more than what, at some

green one, and that there was another moment when the sum total of existence included all that was included in the first moment and also the green event’. This is clear evidence that Broad had a tensed conception of what there is. See also footnote 12. 3 Here a comparison with the debate between actualism and possibilism might be helpful. It is standard to characterize these views by saying that the actualist claims that absolutely everything there is is actual, whereas the possibilist says that there are things which are not actual—where the quantifiers are understood as absolutely unrestricted. Now the typical possibilist will hold that ‘there is’, taken in this sense, is not ‘modally tensed’, where this means that the quantifier is not sensitive to embedding by modal operators (e.g. she will hold that ‘There is an x such that possibly p’ is equivalent to ‘Possibly, there is an x such that p’). By contrast, the typical actualist will claim that ‘there is’ is modally tensed (e.g. she will hold that possibly, there is a human being which is four meters tall, but deny that there is a human being which could have been four metres tall).

Living on the Brink, or Welcome Back, Growing Block! | 337 past moment, was real, such a complaint will beg the question against that view and so already presuppose that the view is incorrect. Accordingly, one might ask back on what grounds that presupposition can legitimately be made.

II In what follows, we lay down principles governing the tensed notion of what there is at work in GBT. We will use ‘something’ and ‘∃x’, ‘∃y’, etc., in order to express this notion, and ‘everything’ and ‘∀x’, ‘∀y’, etc., as their respective duals. The principles further involve quantification over numbers suitable for measuring temporal distances. In line with our policy to exclude purely abstract objects from the range of what there is, we will use different quantifiers and variables for this purpose, viz. ‘Σn’, ‘Σm’, etc., for existential quantification over numbers, and ‘Πn’, ‘Πm’, etc., for universal quantification over numbers. As already adumbrated, the principles also involve a tensed notion of present existence, which we express by using the verb phrase ‘presently exist’ or sometimes simply the verb ‘exist’, as well as the predicate symbol ‘P’.4 Given an additional assumption about the kinds of entities quantified over, which we make explicit in due course, this tensed notion of present existence neatly corresponds to the notion of being on the edge of reality. It will prove useful to work with quantifiers defined by restricting the metaphysically most fundamental quantifiers to presently existing entities. The restricted existential quantifier will be expressed by means of ‘something’, ‘there is’, and variants thereof (all in normal font), and is defined as follows: Something φs iffdf something which presently exists φs. The restricted universal quantifier is simply the dual of the quantifier just defined, and it will be expressed by means of ‘everything’, ‘for all’, and variants thereof (again, all in normal font). It is defined thus: Everything φs iffdf everything which presently exists φs. 4 Note that ‘presently exist’ does not function in the way ‘exists now’ does, where ‘now’ is a temporally rigid indexical which is insensitive to tense-logical embedding.

338 | Fabrice Correia and Sven Rosenkranz The restricted existential quantifier will be symbolized by ‘Ex’, ‘Ey’, etc., and its dual by ‘Ax’, ‘Ay’, etc. In order to state the principles characterizing GBT’s conception of what there is, we shall adopt the following convention governing the metric tense-logical operator ‘n days from the present’: it is equivalent to ‘presently’ if n = 0, to ‘−n days ago’ if n < 0, and to ‘n days hence’ if n > 0. We symbolize ‘n days from the present’ by ‘Dn’. As advertised, we will make the simplifying ontological assumption that at least in the context of metaphysics, all we ever quantify over by means of ‘∃x’, ‘∃y’, etc., or ‘Ex’, ‘Ey’, etc., are entities that are instantaneous. So in particular, the following will be assumed to always hold: (0) ¬ΣnΣm ≠ 0DnExDmP(x). Needless to say, this assumption is controversial. However, it is not a priori incoherent, and it is likewise open to both presentists and eternalists.5 Accordingly, for the purposes of showing that GBT is a consistent view, it will do to focus on a particular version of it that is committed to (0); and for the purposes of bringing out what is distinctive of GBT, it will do to contrast that version with versions of presentism and eternalism that are likewise committed to (0). According to GBT, everything is such that for some n, with n ≤ 0, n days from the present, it then exists: (1) ∀xΣn ≤ 0DnP(x). 5 The claim that everything is instantaneous can be formulated by an eternalist as follows:

(a)

For all x, there is a unique moment t at which x exists (or ‘is located’, as stage theorists like to say).

An eternalist will typically hold that (0) is true at a moment t0 iff (b) is true: (b)

¬(ΣnΣm ≠ 0 such that at the moment t n days from t0, there is an x which exists and at the moment m days from t, x also exists).

For her, (b) will be equivalent to (c), which is in turn equivalent to (d): (c)

¬ (there is an x and ΣnΣm ≠ 0 such that at the moment t n days from t0, x exists and at the moment t' m days from t, x also exists).

(d)

¬ (there is an x and there are two distinct moments t and t' such that at t, x exists, and at t', x also exists).

(d) is clearly acceptable to an eternalist.

Living on the Brink, or Welcome Back, Growing Block! | 339 This at most tells us that everything either exists or used to exist, but is as yet silent on whether all that either exists or used to exist is something. According to GBT, however, reality certainly is not less than what exists or used to exist. So according to GBT, for all n, with n ≤ 0, n days from the present, anything that then exists is, −n days from the then present, something: (2) Πn ≤ 0DnAxD−n∃y(y = x). The idea of a grown block certainly also suggests that reality is such that at any moment present or past, something then exists that forms a layer of it. So we take it that the following too is an essential ingredient of GBT: (3) Πn ≤ 0∃xDnP(x). It follows from (1) that nothing is such that for some n, with n > 0, n days from the present, it then exists but for no m ≤ 0, m days from the present, it does—more perspicuously, that nothing will exist that neither already exists nor already existed in the past: (4) ¬∃x(Σn > 0DnP(x) & Πm ≤ 0¬DmP(x)). In conjunction with (0), (4) implies that nothing is such that for some n, with n > 0, n days from the present, it then exists—i.e. that nothing will exist: (5) ¬∃xΣn > 0DnP(x). We now claim that ‘alwaysations’ of (1) to (3) are quite sufficient to express the core of GBT.6 These ‘alwaysations’ are obtained by prefixing either principle with ‘ΠkDk’, where this operation can be reiterated.7 Unlike (1) to (3) themselves, these ‘alwaysations’ nicely capture

6 The ‘alwaysation’ of (3) implies that for any n, n days from the present, something exists. This is a substantial claim, but the issue of whether this claim is true is orthogonal to the debate between GBT and its rivals, and hence we do not prejudge any issues by conceiving of GBT as implying it. 7 The principle that what always holds always always holds, though plausible, is not uncontroversial. However, any doubts about its validity will be independent from the controversy between GBT and its rivals, presentism and eternalism. So we prejudge no issues by here assuming it.

340 | Fabrice Correia and Sven Rosenkranz the dynamic aspects of the view. The version of GBT we will consider when responding to Braddon-Mitchell’s and Merricks’s objections includes the core of GBT, and in addition the ‘alwaysation’ of (0). However, given only minimal factual assumptions, (1) to (3) themselves are already sufficient to mark the contrast with both presentism and eternalism. Thus, assuming that sometimes in the past, something existed which does not now exist, presentists will deny (2), and assuming that sometimes in the future, something will exist which neither exists now nor existed before now, eternalists will reject (1). The ‘alwaysations’ of (1) to (3) furthermore entail that always, whatever was something still is something: (6) ΠkDkΠn ≤ 0Dn∀xD−n∃y(y = x), which is precisely what one would expect GBT to imply. Once the ‘alwaysation’ of (0), and so of (5), is added to GBT’s core, the resulting view furthermore entails that always, there is something which was nothing earlier than then: (7) ΠkDk∃xΠn < 0Dn¬∃y(y = x). It likewise follows that the open sentence ‘Πn < 0Dn¬∃y(y = x)’ is always equivalent to ‘P(x)’. Accordingly, given the ‘alwaysation’ of (5), whenever something presently exists, it was nothing earlier than then and nothing exists later than then, and vice versa. Thus, there is a clear sense in which whenever something presently exists it is on the edge of reality but was not on the edge of reality before, and vice versa. The latter equivalence will prove an asset when it comes to defusing Braddon-Mitchell’s and Merricks’s arguments.

III Before we turn to these arguments, however, let us make clear what GBT does not involve. To begin with, note that (2) differs from the stronger claim that for all n, with n ≤ 0, n days from the present, anything that then exists is such that −n days from the then present, it likewise exists: (*) Πn ≤ 0DnAxD−nP(x). GBT’s core does not entail (*). (*) in fact negates one of the minimal factual assumptions made earlier. It is ruled out by (3) in conjunction

Living on the Brink, or Welcome Back, Growing Block! | 341 with (0) and so proves false on the particular version of GBT that assumes (0) to always hold. Note also that nothing said so far implies that, according to GBT, for all n, with n > 0, n days from the present, nothing then exists: (**) Πn > 0Dn¬∃xP(x). (**) is indeed inconsistent with GBT. For (3) entails ‘∃xP(x)’, and so assuming that (3) always holds, ‘Πn > 0Dn∃xP(x)’ will hold as well, and this is inconsistent with (**). Proponents of GBT should not accept that nothing is such that, for some n, with n > 0, n days from the present it is something: (***) ¬∃xΣn > 0Dn∃y(y = x), which is stronger than (5) and so isn’t implied by it. Actually, the negation of (***) is entailed by the claim that everything will still be something at any future moment: (8) ∀xΠn > 0Dn∃y(y = x), which itself follows from (6)—which is all to the good, since plausibly, GBT implies that the block will never erode. That (8) follows from (6) can be shown as follows. (6) is obviously equivalent to ‘ΠkDkΠn ≥ 0D−n∀xDn∃y(y = x)’, which entails ‘Πk > 0DkΠn > 0D−n∀xDn∃y(y = x)’. From this we can infer ‘Πk > 0DkD−k∀xDk∃y(y = x)’, and then ‘Πk > 0∀xDk∃y(y = x)’, which is obviously equivalent to (8). So much for our characterization of GBT’s core and of the particular version of GBT that accepts the ‘alwaysation’ of (0) and so of (5). We now turn to Braddon-Mitchell’s and Merricks’s swan songs for GBT. If their arguments are successful at all, they should affect all versions of GBT. IV Braddon-Mitchell (2004) argues against GBT as follows. Suppose that this moment, and all that exists at this moment, are on the edge of reality, where ‘this moment’ functions as an indexical. Then, if at some moment in time earlier than this moment, Caesar is crossing the Rubicon and judges his crossing to be on the edge of reality, Caesar’s judgement is false. Of course, when that earlier moment

342 | Fabrice Correia and Sven Rosenkranz was on the edge of reality, Caesar’s judgement was true; however, if this moment is on the edge of reality, that earlier moment no longer is so. But now, by parity of reasoning, if some moment in time later than this moment is on the edge of reality, then if at this moment we are talking philosophy and judge our talking to be on the edge of reality, our judgement is likewise false. Of course, if this moment is on the edge of reality, then our judgement is true. But we have no means to find out whether this moment in fact is on the edge of reality. We do know that this moment is this moment and also that our talking philosophy occurs at this moment, but to know this is not to know anything that is only momentarily the case, whereas it is only momentarily the case that this moment is on the edge of reality. So, for all we know, this moment might be on the edge of reality, and for all we know, some moment later than this moment might be on the edge of reality. This moment is only ever on the edge of reality for the briefest of moments; and the probability that it is not so now far outstrips the probability that it is so now. In so far as, according to GBT, to be present is to be on the edge of reality, ‘we should regard the hypothesis that the current moment is present as only one among very many equally likely ones’ and therefore conclude ‘that the current moment is almost certainly in the past’. This, however, amounts to a reductio ad absurdum of GBT (Braddon-Mitchell 2004: 200–1). In a similar vein, Merricks (2006) argues that proponents of GBT must distinguish between what he respectively calls ‘the subjective present’ and ‘the objective present’. While ‘the objective present’ is meant always to refer to the edge of reality as postulated by GBT, wherever it happens to be located, the reference of ‘the subjective present’ varies across time in that at any moment of time, that moment is the subjective present. Merricks’s reason for saying that GBT forces this distinction is that, on the one hand, Nero’s judgement e, that e exists at the present, which occurred sometime in the past, does not exist on the edge of reality, while on the other, there is also a clear sense in which e is true. Now, if e related to the objective present, then we would have to conclude that e is false. Therefore, e is best construed as being about the moment at which Nero makes that judgement—what, at that moment, is the subjective present as opposed to the edge of reality. By parity of reasoning, or so Merricks argues following Braddon-Mitchell (2004), our own

Living on the Brink, or Welcome Back, Growing Block! | 343 judgements j, made today, to the effect that j exists at present, are best construed as being about our subjective present and not about the edge of reality: since our judgements are located on the edge of reality only for the briefest of moments, the probability of their being true would be ‘vanishingly small’ otherwise. But now, once the objective present is thus dissociated from what, at any given moment including today, is the subjective present, it becomes entirely unclear what the objective present is and so where the edge of reality is located. For all we know, it might lie in the future of today. Thus, it now becomes difficult to see on what grounds we can distinguish between GBT and what Merricks calls the theory of the unmotivated growing hunk according to which the edge of reality lies ten years ahead. Merricks concludes that, therefore, GBT ultimately proves to be incoherent (Merricks 2006: 105–10). Since Braddon-Mitchell is ready to concede to the proponents of GBT an ontology of world-slices, his argument is clearly intended to refute the particular version of GBT which is committed to the ‘alwaysation’ of (0) and which we outlined in previous sections (Braddon-Mitchell 2004: 199). Although Merricks considers the failure of (0) in a footnote, his reasoning nowhere presupposes that (0) might fail: for argument’s sake, the judgements in question are treated as instantaneous (Merricks 2006: 105n). So if successful at all, it too ought to count against that particular version of GBT.8 It should be evident that Braddon-Mitchell’s and Merricks’s swan songs are variations on an identical theme. However, if our characterization of GBT is correct, nothing of this should carry any conviction. To begin with, note that according to GBT as characterized, always, there are past moments which we can continue to refer to by means by which we could likewise refer to them when they were on the edge of reality. Tenseless date terms are a common means to that end. What we can never do, though, is continue to refer to past moments by means of ‘this moment’, by means of which we could

8 Our response to Braddon-Mitchell and Merricks on behalf of GBT will be premised on an ontology of instantaneous entities. But for all we can see, ultimately, all that is strictly speaking needed is the assumption that judgements can be instantaneous; and as long as there can be trivially correct judgements of the form ‘The present moment is present’, this assumption should not be contentious.

344 | Fabrice Correia and Sven Rosenkranz refer to them when they were on the edge of reality. For, always, ‘this moment’ refers to the present moment, and to no past moment. This is a semantic platitude, on anyone’s account. The same applies, mutatis mutandis, to ‘what exists at this moment’: always, ‘what exists at this moment’ refers to all and only those things that exist at the present moment, and not to all and only those things that exist at some past moment. Now, it may readily be agreed that we might not now know truths of the form ‘d is the moment on the edge of reality’ (or of the form ‘What exists at d exists at the moment on the edge of reality’), where ‘d’ is some tenseless date term that refers to t0, where t0 is this, the present, moment. But it does not follow from this that we do not know, of t0 and all that exists at t0, that they are on the edge of reality under the modes of presentation respectively encoded by ‘this moment’ and ‘what exists at this moment’. For that to follow, it must furthermore be assumed that it is epistemically possible that sometimes ‘the moment on the edge of reality’ does not refer to what then is the present moment. Both Braddon-Mitchell and Merricks suggest that, on condition that GBT holds, we might indeed be in a position in which we are left to wonder whether this moment is the moment on the edge of reality or is rather earlier than the moment on the edge of reality. Given the triviality that if t is on the edge of reality, at t, ‘the moment on the edge of reality’ picks out t, this makes sense only if GBT allows that at this moment, ‘the moment on the edge of reality’ might refer to a moment it could likewise refer to at a later moment. Both Braddon-Mitchell and Merricks furthermore assume that according to GBT, for any moment t, if t and all that exists at t are on the edge of reality, then for any moment t' earlier than t, at t', t and all that exists at t are on the edge of reality. For unless this was so, t’s being on the edge of reality would not imply that if at an earlier moment t', one judged t' or something that exists at t' to be on the edge of reality, one would be making a false judgement; and the basis for the authors’ negative induction would crumble.9 Yet, according to GBT, and in stark contrast with eternalism, if at some moment t, we ‘go back’ in thought to an earlier moment t' and ask what is the case at that earlier moment t'—an operation effected 9 Throughout, we treat ‘at t’ as equivalent to ‘it is the case at t that’. This is a tense logical operator (of the sort used in so-called hybrid logic, for instance) which one could in the present context define as ‘Always, if t is present, then’.

Living on the Brink, or Welcome Back, Growing Block! | 345 by the use of ‘at t' ’—we ask what the block included when t' was sitting on its edge; and always, when something is on the edge of reality, there is nothing which exists later than then. A fortiori, at t', t does not exist and so is not on the edge of reality. It is only if one assumes that what, at t', there is includes what there is at t, that one might think otherwise. Accordingly, both arguments presuppose that GBT operates with an untensed notion of what there is. As we have argued, this is a mistake. Proponents of GBT do want to say that something is on the edge of reality and that something, e.g. Caesar’s crossing the Rubicon or Nero’s past judgement e, used to be on the edge of reality but no longer is. But they do not want to say that there is any edge of reality such that something used to be on it but no longer is. Nor is GBT helpfully construed as claiming that there is some edge of reality on which something was which no longer is on any edge of reality today. To say that something used to be on the edge of reality but no longer is, while something is on the edge of reality today, need not commit one to the claim that there are many edges. Instead, on any version of GBT committed to the ‘alwaysation’ of (0), being on the edge of reality is best conceived as a property such that always, something has it but did not have it before, always, something used to have it but no longer does, and always, nothing is going to have it. This property is obviously tensed and, most plausibly, versions of GBT committed to the ‘alwaysation’ of (0) will identify it with the property of present existence which we expressed by means of ‘P(x)’.10 Even those who reject GBT may accept that there is a tensed property of present existence, but they will anyway deny that it satisfies the description GBT gives of it. But this is no surprise, because that description involves use of the metaphysically most fundamental notion of quantification and whether the property of present existence satisfies it will depend on the principles which GBT claims to hold for that notion.11 10 Of course, to say that nothing is going to presently exist is not the same as to say that there is going to be nothing that presently exists. According to the version of GBT under consideration, the former holds, while the latter fails to hold. 11 Proponents of the version of GBT under consideration may wish to hold that there are facts involving this property which, in the past, made Nero’s judgement true and so existed back then, but do no longer exist today. But as long as proponents of that version of GBT are not committed to (*), and they aren’t, this claim is quite consistent with anything they are in fact committed to. See footnote 14.

346 | Fabrice Correia and Sven Rosenkranz In fact, we can prove that on the version of GBT we characterized, ‘The present moment and all that exists at the present moment are on the edge of reality’ is always true. The proof makes use of two principles both of which are acceptable to friends and foes of GBT alike, viz. (P) Always, there is a unique moment which is present. (Q) Always, for all moments z and z', if z is present and z' is later than z, then z' will be present. On the version of GBT we characterize, we can take ‘x is on the edge or reality’ to be always equivalent to ‘It is not the case that there is a moment y such that y is later than x’. This said, let us turn to the proof. By (P), there is a unique present moment. Let then t0 be the present moment. Assume, for reductio, that t0 is not on the edge of reality. Then for some moment y, y is later than t0. Let then t1 be such a moment. By (Q), it follows that t1 will exist, and so, that something will exist. But given our principle (5), nothing will exist. Contradiction! We can therefore conclude that t0 is on the edge of reality. Let now a be something which exists at t0. Notice that, by (0), since t0 is present, a will not exist anymore in the future. Assume, again for reductio, that a is not on the edge of reality. Then for some moment y, y is later than a. Let then t1 be such a moment. Since a exists at t0 and will not exist anymore in the future, then t1 is not only later than a, it is also later than t0. But, as we saw before, this is impossible. Therefore, a is on the edge of reality. Since a was arbitrary, we can conclude that all that exists at t0 is on the edge of reality. Thus, we have proved that ‘The present moment and all that exists at the present moment are on the edge of reality’ is true. Notice that besides (P) and (Q), the only other metaphysical principles at work in this proof are principles which our version of GBT assumes to always hold, viz. (0) and (5). The logical principles used in this proof can safely be taken to always hold as well. As a consequence, what the proof establishes is not only that ‘The present moment and all that exists at the present moment are on the edge of reality’ is true. The proof furthermore establishes that this statement is always true. With this result being in place, the answer to Braddon-Mitchell’s question ‘How do we know that it is now now?’ is straightforward.

Living on the Brink, or Welcome Back, Growing Block! | 347 For if I know that GBT is true, on the particular version of it that we have characterized in this paper, then I know, or at least am in a position to know, that ‘The present moment is on the edge of reality’ is always true, and hence presently true, which is to say that I know, or am in a position to know, that now (i.e. the present moment) is now (i.e. the moment that is on the edge of reality). Suppose I am now making judgement j to the effect that j is on the edge of reality. Can I exclude the possibility that the proposition I thereby judge is false? This is to ask whether it is epistemically possible for me at the time of making judgement j, that something earlier or later than that time is on the edge of reality. Again, the answer is now straightforward. Suppose that I know GBT, on the version of it we suggest. Then I know, or at least am in a position to know, that always, the present moment and all that exists at the present moment are on the edge of reality. Given that I know that j is present, I know that j exists at the present moment. Putting these two pieces of knowledge together, I am accordingly in a position to know that j is on the edge of reality. Therefore, it is not epistemically possible for me at the time of making judgement j, that the proposition I thereby judge is false. Both arguments presuppose that I can be credited with knowledge of the relevant version of GBT. But this presupposition should be considered harmless, if the task is to show how GBT can ensure that one might know that the present moment and one’s present judgements are on the edge of reality. Where does this leave Nero? According to the particular version of GBT under consideration, to say that something, e.g. Nero’s past judgement e to the effect that e exists at present, does not exist at present, amounts to saying that e is not such that nothing exists later than the moment at which it exists. But this is quite consistent with saying that at the moment Nero made his judgement, he related to what then was on the edge of reality and so judged the moment at which e was made to be such that nothing existed later than it. The fact that even today, both Nero and e are something evidently does not imply that even today, Nero is judging the moment at which e was made to be such that nothing exists later than it.12 Nero’s judgement is about the time it is made, and as Prior 12 Although, presently, Nero’s judgement e still is a judgement made by Nero and so has not changed in this respect, still, pace Merricks 2006: 105 (cf. also Bourne

348 | Fabrice Correia and Sven Rosenkranz reminds us, even if the time it occupies is an instant, one should not conflate the history an event is with the history that event has (Prior 2003: 10). Evidently, the same considerations apply to Caesar’s past judgement that his crossing of the Rubicon existed (!) on the edge of reality. Call the proposition that e exists at the present ‘Eddie’, and take Eddie to say of e that it exists at the objective present in Merricks’s sense (i.e. that it is on the edge of reality). Eddie is a tensed proposition which is true at the moment at which e occurs and false at all others. So Eddie is false most of the times. Does this imply that e is a false judgement most of the times at which e is something? No, because for all present-tensed propositions p of the kind Eddie exemplifies, the following principle may be assumed to (always) hold: (9) ∀x(x is a judgement that p → (x is true ↔ ΠnDn(P(x) → p))), where it is presupposed, in line with the ‘alwaysation’ of (0), that always, for every judgement, there is a unique n such that n days from the present, it then exists. Accordingly, unlike Eddie, e never ceased to be true.13 So from the fact that Eddie is false most of the 2002: 364) it is no longer presently the case that Nero is making that judgement. One should not lump these things together. Sure, Nero is still the author of the judgement and will forever be the author of that judgement. But in order for it to be presently the case that Nero is making that judgement, Nero would have to be presently alive and that judgement would have to be such that, presently, nothing succeeds it—neither of which is the case. According to Broad, changing from present to past involves a change in relational properties: ‘When an event, which was present, becomes past, it does not change or lose any of the relations which it had before; it simply acquires in addition new relations which it could not have before, because the terms to which it now has these relations were then simply non-entities’ (Broad 1923: 87). Accordingly, the tenses can be understood to encode possession of, lack of, or a change in, relational properties of this kind. (Note that what Broad (1923: 87–8) lists as the fundamental third type of change besides change in attributes and receding into the past, i.e. Becoming (present), can also be counted as a change in relational properties of this kind, because if something is becoming present which, before, was nothing, it is entering into the relation of succession to something else.) 13 Here an analogy might help. Suppose passing through Chicago on a train to New York, I judge that I am presently passing through Chicago. Then I arrive at Grand Central. Even if my judgement does not cease to exist in New York, and even if the egocentric proposition that I am presently passing through Chicago is false in New York, it would be rather unorthodox to conclude that my judgement is false in New York. This is not how we evaluate judgements of that type. Perhaps a different example

Living on the Brink, or Welcome Back, Growing Block! | 349 times, it does not follow that most of the times at which e is something, e is false.14 These considerations likewise confute Merricks’s and BraddonMitchell’s suggestion that since my present judgement j, to the effect that j exists at present, is located on the edge of reality only for the briefest of moments, the probability of its being true would have to be regarded as vanishingly small should it be understood to attribute to j that it is on the edge of reality. At last, what about Merrick’s theory of the unmotivated growing hunk? Given that the block constantly grows, at any moment the subjective present coincides with the objective present. So given the ‘alwaysation’ of (5), at no moment does anything exist ten years later than then. Consequently, the theory of the unmotivated growing hunk is always false. If some version of GBT implies that that theory is always false, it is hard to see how that version of GBT might nonetheless be indistinguishable from it. We conclude that neither Braddon-Mitchell nor Merricks succeeds in showing that GBT is incoherent. At least pending further involving spatial indexicals might make things even clearer. Suppose the present moment is t, that I am currently in Chicago and make a judgement whose content is the indexical proposition It is raining at t here. Call the corresponding judgement ‘k’. k is an event that is located (exists) in Chicago and not in New York; still, we assume that in New York, there is such a thing as k. Also assume that it is not raining in New York. Then the proposition It is raining at t here is not true in New York. What about k? Judgement k is true simpliciter, and so—if we want to talk about judgements being true at places—true at all places, in particular in New York. The same applies, mutatis mutandis, to judgements of the type exemplified by Nero’s judgement e: they too retain their value, true or false, depending on the circumstances that obtain simultaneously with their being made, even if the tensed propositions that are their contents do not retain their truth value, and everyone should agree that it is not the case that everything is simultaneous with everything else. (9) corresponds to what John MacFarlane (2003) calls ‘the absoluteness of utterance-truth’, which is, as he points out, the orthodox view. 14 Assume that if e is true there is a fact that made e true at the moment e occurred and so existed back then. Call that fact ‘Fred’. In so far as Fred still is something, the question arises of how to describe Fred today. We know that at the moment e occurred, Fred was the fact that e exists at present. We now face two options: we may either say that, today, Fred still is the fact that e exists at present, or we may say that, today, Fred is the fact that -n days from the present, e exists, where e occurred n days ago (cf. Correia and Rosenkranz 2011: 55–70 and 2012. In either case, (0) forces us to deny that, today, Fred exists; and since Fred can make statements true only as long as Fred exists, on either reading, Fred does not make Eddie true today. See footnote 11.

350 | Fabrice Correia and Sven Rosenkranz argument, we might, for all we know, be living on the brink of reality.15 Université de Genève University of Barcelona REFERENCES Bourne, C. 2002: ‘When am I? A Tense Time for Some Tense Theorists?’, Australasian Journal of Philosophy 80, 359–71. Braddon-Mitchell, D. 2004: ‘How Do We Know that it is Now Now?’, Analysis 64, 199–203. Broad, C. D. 1923: Scientific Thought. London: Routledge. Relevant excerpts reprinted in: van Inwagen, P. and Zimmerman, D. (eds), Metaphysics: The Big Questions. Oxford: Blackwell, 1998, 82–93. Correia, F. and Rosenkranz, S. 2011: As Time Goes By. Paderborn: Mentis. —— —— 2012: ‘Eternal Facts in an Ageing Universe’, Australasian Journal of Philosophy 90, 307–20. MacFarlane, J. 2003: ‘Future Contingents and Relative Truth’, The Philosophical Quarterly 53, 321–36. Merricks, T. 2006: ‘Goodbye Growing Block’, in: Zimmerman, D. (ed.), Oxford Studies in Metaphysics Vol. 2. Oxford: Oxford University Press, 103–10. Prior, A. N. 1971: Objects of Thought. Oxford: Oxford University Press. —— 2003: Papers on Time and Tense, new edition. Oxford: Oxford University Press.

15 We would like to thank Carl Hoefer, Dan López de Sa, Manolo Martínez, Giovanni Merlo, Moritz Schulz, Albert Solé, Stephan Torre, and Dean Zimmerman for helpful discussions. The research leading to these results has received funding from the European Community’s Seventh Framework Programme under grant agreement PITN-GA-2009-238128, and was also partially funded by the Consolider-Ingenio project CSD2009-0056, the projects FFI-2008-06153 and HUM2007-61108, all financed by the Spanish Ministry of Science and Innovation (MICINN), as well as by the projects PP001-114758, PP00P1-135262, and CRSI11-127488, financed by the Swiss National Science Foundation.

10. Fighting the Zombie of the Growing Salami1 David Braddon-Mitchell Correia and Rosenkranz (in this volume) offer a suggestive attempt to raise the growing block model of time from the grave. The thought is something like this: the opponents of the block use an epistemic argument that mistakenly deploys an untensed notion of existence. The epistemic argument assumes, they say, that if the block ends at, say, 2015, one can go back ‘mentally’ to 2013 and note that the benighted authors writing at that time think that they are in the present but are mistaken. But this is not so, they argue: for at every time it is the final slice of the block. At 2015, to think of 2013 is to think of the time when 2013 was the last slice of the block. Before attempting to put a stake through the lumbering zombie of the growing block theory (henceforth usually GBT), I’ll clear up a couple of things which might be misunderstandings about my argument against it (Braddon-Mitchell 2004).

1. SOME PRELIMINARY CLARIFICATIONS First, Correia and Rosenkranz (henceforth C&R) characterize my view as an attempt to show that GBT is incoherent. That would be a tall order, and I don’t in fact attempt it. Instead the thought was that the model, while coherent, divorces the indexical conception of ‘now’—Nowindex—from the objective one given by the metaphysics of the model, namely, the fact of being located on the last slice— Nowls. This leaves us open to the unwelcome likelihood that, Nowindex, it is not Nowls. This is an undesirable outcome in my view, reason enough to reject GBT; reason beyond the mere surprisingness of the view. 1 I’m greatly indebted to a series of comments on earlier drafts of this paper from Dean Zimmerman.

352 | David Braddon-Mitchell I think it undermines the motivation for holding it in the first place. But it is nevertheless far from rendering it incoherent. Committed growing blockers sometimes accept the argument but embrace it as demonstrating surprising evidence of our epistemic limitations, even if none have yet done so in print. If the best metaphysical model of time tells us that we can’t be sure that we are in the present, and in fact are very likely to be in the past, then so be it, runs the thought. However that view, while perhaps coherent, is pretty unpalatable, hence perhaps the reluctance to swallow it publicly. A second misunderstanding: C&R say that I adopt an untensed notion of existence simpliciter in characterizing GBT. But this is also not right. While there are entirely untensed ways of characterizing GBT2 they are not really in the spirit of GBT. Instead I think of it as a hybrid. The way I think of GBT is as an A-series, and what exists at each A-time is a block universe. What exists at every moment of true time or A-time is that block. At later times these blocks are larger. At every A-time the present is, according to the theory, the last slice. But equally at every A-time there is a volume of worldslices which are at least quasi-B-related, and most of the slices are in the ‘past’ at that A-time. Quasi-B-relatedness is an ordering imposed by the geometry of a spacetime; although it is a geometrical relation, one constraint on quasi-B-relatedness should be that it puts the slices of spacetime in the same order in which they were, successively, present. The idea is just that world-slices are ordered, but it is left open whether there is a privileged direction, and it is left equally open whether the ordering is genuinely temporal. In my original paper I simply called this ‘B-relatedness’. I did so in part because, on some views about time to which I am sympathetic, all it takes to be genuinely B-related is to be quasi-B-related. Going into the back-block is not going back in A-time. But it’s the existence of the back-block at every A-time which is what gives rise to the epistemic challenge. For whatever objective A-time it is, if the back-block exists at that time, then there are agents in the back-block mistaken about what time is truly now. Of course for 2 For example, as a series of increasingly larger worlds all of which exist, and which are ordered externally by the B-relations of the last slices, and internally by the B-relations amongst them.

Fighting the Zombie of the Growing Salami | 353 the GBT so construed there is a difference at every spatiotemporal location between the true time—the slice which is the present, and which marks where the A-series has got up to, and which is given by the last slice in a B-series at that A-time—and where that spatiotemporal location is to be found within what exists simpliciter (within what exists, in C&A’s terminology and henceforth) at that A-time. Finally, one desideratum for a GBT: it would count as a real advantage if objective nowness could be reductively explained in terms of which slice is the final slice of being. If the location of nowness and the cutting edge of the block were merely correlated, then much of the motivation for having a growing block view in the first place would be undermined.

2. THE FIRST STAKE What I take to be the misunderstanding of my objection provides the nub of one reply to C&R. So to make this reply I’ll first state how one might take C&R’s own view about what’s wrong with the epistemic argument. The epistemic argument requires that from the standpoint of a given moment, you can look back on an earlier moment and see that, at that moment, being extends beyond it. So from our current perspective we can look back at the time at which Prior was writing ‘The Syntax of Time Distinctions’ and see that at that moment in time he would think that he was in the present but be mistaken, and in fact be in the past. But, C&R say, this is a mistake, because it is part of the most charitable formulation of GBT that at every moment in time that moment is on the edge of being. At every moment, and a fortiori at all the moments when Prior wrote, the past exists but not the future. This they say is guaranteed by the tensed notion of existence that they work with. It seems to me that I can accept all of that, with some terminological clarification, while leaving the sting of the epistemic argument unchanged. Consider the sense in which existence, and moments, are tensed on my reading of the GBT. There is an A-series, which is of course fully tensed. Call positions in the A-series moments for maximum

354 | David Braddon-Mitchell consistency with C&A’s terminology. At every moment in the A-series, the spatiotemporal hyperplane3 that is the latest in the block is objectively now. So existence is entirely tensed. If I look back in the A-series to the time when Prior was writing, I find a block which terminates at the hyperplane where he writes, and so at that time he is in the present. What exists at each moment is a block with a different last hyperplane from any distinct moment. That last hyperplane is what grounds the facts about what is true at the present, and the back-block which exists at that moment grounds truths about the past at that moment. So far, perhaps, no disagreement with C&R. But C&R say something else. When we are quantifying over times, we only ever quantify over moments. But I can accept that too. The time 1900 is that moment in time when a certain hyperplane—the 1900 one—was last. The hyperplane in the block which exists as at 1900 which makes true the claim that at 1900 certain things were true in 1800, is not a moment in the relevant sense and thus not a time; it’s a location in the back-block of the moment 1900. We can, if they exist, quantify over the hyperplanes in the backblock which exists at any time. When quantifying over times, however, I quantify over moments—when I talk of 1900 as a time I’m talking about the last hyperplane of the 1900 block. So it’s always now, in the sense that at every location in the block that moment is present, since the present moment is the last slice of the block. But this still allows us to formulate the epistemic argument. For although I cannot quantify over a moment or a time when that time is not on the edge of being, I can quantify over hyperplanes which aren’t the last hyperplane at that time. So now I can talk about the hyperplane in which Prior is writing, a hyperplane which exists in spacetime (understood as a physical notion) located in one spatiotemporal direction (i.e. in the direction of one of the quasi-B-relations,

3 I set aside considerations that are nevertheless important: talk of the last hyperplane implies objective facts about simultaneity that are additions to physics. Some might think that the growing block actually helps, because the fact that being has an edge might be used to define that hyperplane of simultaneity. But of course that edge is only a boundary, and being has other boundaries, so perhaps the GBT may still require extra resources to stipulate what the right simultaneity relations are.

Fighting the Zombie of the Growing Salami | 355 equally understood as a physical notion, and not as going back in time). I could equally have said here that we are talking about slices that are earlier in the B-series which exists at that point in the A-series. But that would, because of the associations of slices earlier and later in the B-series with times, make it sound as though the hyperplanes are times. However, nothing hangs on calling them times. I am allowing that the tensed part of the theory exhausts times. The physical hypothesis about the existence of a back-block at every time can be neutrally described as a hypothesis about the existence of hyperplanes in physical spacetime that are merely quasi-B-related to one another and to the present slice. Assuming that consciousness supervenes on physical structures in spacetime, there’s something back there (I think it’s Prior, but that isn’t germane) who thinks he’s Prior, and who thinks that the present is the hyperplane which is part of what he calls ‘1954’ and in which he is located. He’s wrong about that. He’s not located in the past in the A-theoretic sense—he’s located in that part of the back-block of 2013. Now this formulation has certain advantages. It allows it to be the case that, at every moment, being the last slice plays a role in marking or constituting (take your pick) the present. It has the advantage of taking on a standard interpretation of part of the scientific story about what exists at any time: a partial block universe account. And it does indeed have the feature that C&R say my reading doesn’t have. It’s genuinely dynamic—the universe grows as we move forward in the A-series—as time passes. But all this is at the price of there being two ways to gloss ordinary talk about past moments. Strictly past moments (when the block was smaller), and parts of the universe in the backwards direction in the back-block which are not in the true past (in the A-series), but just distant parts of the blocks that exist at each time.4 This is in part why it remains vulnerable to the epistemological argument. So the idea behind the epistemological objection I am pressing is that the indexical use of ‘now’ does not pick out a moment or a time, but rather a location in spacetime construed as a physical entity. 4 I leave out here considerations of whether the ‘early’ components of the block are identical in successive larger blocks.

356 | David Braddon-Mitchell Thus using ‘moment’ or ‘time’ in the A-theoretic way, the issue is not that ‘this moment is not now’ but rather that ‘this location in spacetime is likely not at the present’. If you think this is an objectionable use of the expression ‘now’ because that should be tensed, I’m happy to replace it with ‘here in spacetime’. The slogan for the objection would then become ‘how do we know that here in spacetime the events are present’. This would, I trust, just be a terminological variant of what I said in my original paper, where I was using ‘moment’ to mean something like ‘a hyperplane in spacetime’ and ‘objective present’ to mean the last such moment, but one which makes it plain that my view can obey the constraints that C&R offer, while leaving the epistemological objection alive.

3. A SECOND STAB This first formulation of the GBT has benefits, and is consistent with C&R’s stipulations. The way it accepts a fully tensed version of the growing block is that there are in effect different blocks of varying size at each A-time, but the locations in the back-block are not A-times. If the objective A-time is, say, 2013, then the back-block contains many locations spatiotemporally connected to 2013 which are not themselves A-times. At what one might call the ‘2010’ location, the true time is not 2010, it is 2013. The real 2010 was when 2010 was the last slice. It’s a different A-time, which has associated with it a smaller back-block. What some growing blockers might not like about this view (though perhaps that just is an inevitable result of accommodating blocks and growing) is that if, for expositional reasons, we allow ourselves to consider a God’s-eye perspective from outside time, intrinsic duplicates of times5 will appear over and again. First, as the last slice of a block—when they are ‘the time’ as it were—and again embedded as locations in the back-block of later times. My second stab (which will matter only for those who prefer the second view which is so stabbed—if you think the view criticized in the first stab is the best understanding of the growing block, then 5 I leave it open here if they are identical—i.e. if they are the same thing persisting over time, but losing certain temporal properties. In second stab I foreclose that openness.

Fighting the Zombie of the Growing Salami | 357 you can stop now without loss) removes this feature. It has genuinely temporal, not merely physical, relations between the locations in the blocks. The 2010 location in the back-block of 2013 is the time which is 2010. It is unqualifiedly identical to the location which is on the bleeding edge of 2010. If you could move from the 2013 location of the block at 2013 to the (at 2013) existing 2010 you would arrive when it was 2010, and your place of origin (2013) would not exist. So, on this interpretation of GBT, what exists simpliciter (what exists) is not utterly unqualified. existence is not confined to what exists in a time—it’s about what exists in every time; however, existence is relativized to different times, because at different times, different other times exist. At 2013, 2010 exists; but at 2010, 2013 does not. I’ll call a version of the growing block that has these features the purely tensed growing block (PTGB) since it features no untensed quasi-B-relations. Understanding things this way perhaps makes it easier to reply to my epistemic objection, but doesn’t come without its own costs. Here’s the plan: first I’ll explain why it might fare better against the objection. Then I’ll briefly argue that, for all this, my objection still works. Then I will make two remarks about unattractive features of PTGB independently of whether the objection alone is fatal to it. So let’s consider Julius Caesar again. It’s now 2013. It’s true that Julius Caesar exists2013 even though he does not exist in 2013. Does he falsely believe that he is in the present? No, because the content and truth of beliefs should be assessed at the location of the beliefs themselves. At 60 bc, 2013 doesn’t exist. So the fact that 2013 does not exist60 bc is what is relevant to assessing the content of Caesar’s belief that he is present. His belief that he is present is true just if he is in a time-slice (60 bc) such that, at that time-slice, no later ones exist60 bc. Thus it looks like the epistemic worry is defeated. For there is no Julius Caesar in the past who mistakenly thinks that he is in the present. For at the time in which he exists he is right to think that he is in the present. I’m not convinced that this does evade the worry. This is because it’s not clear how to justify the principle of content attribution on which the evasion depends. From the perspective of 2013, there are things I can say about 60 bc that are indexed to 2013. That is, after

358 | David Braddon-Mitchell all, how I get to say that Caesar exists: he exists2013. Saying that the content of his beliefs and so forth have to be evaluated at his location is equivalent to saying that what’s relevant is what he believes60 bc; specifically that he believes60 bc that he is present60 bc. It’s the fact that he does believe60 bc that he is present60 bc, and that that is true, which is what appears to defuse the problem. But I’m allowed to say that he exists2013 in 60 bc. So why can’t I ask whether he believes2013 that he is present2013 in 60 bc? That would be a false belief. To deny that, at 2013, Caesar believes2013 anything would be to treat existence very differently from other attributions when we look back at the past: at 2013 Caesar exists2013 in 60 bc, but Caesar does not believe2013 that he is present20136 at 2013, and this starts to look close to various solutions that I’ve argued against elsewhere— solutions that make the past very different from the present, in that the contents of beliefs are different, or the past is populated by philosophical zombies.7 I think this likely settles the matter. But if you are not persuaded, there are a couple of independent strikes against PTGP that might make it not worth adopting. The first is that it seems hard to say how this view differs from presentism with a fixed past and open future. Existst amounts to existence in t or earlier than t. It’s certainly isomorphic to a presentist account according to which there is an important notion of what is fixed at t—that which exists or did exist. So the relativized notion of existencet behaves very much like the presentist’s notion of what is fixed (about existence) at t. What distinguishes my version of the GBT from presentism is the genuine existence, from the perspective of each time, of multiple hyperplanes that timelessly and tenselessly exist in the way that presentists can’t countenance. How does PTGB differ from presentism? There is an incantation we can chant. According to PTGB it’s now the case that Julius Caesar and indeed 60 bc exist in the most unrestricted sense of quantification. So Julius Caesar exists. According to presentism, on the

6 A related worry is that if we are allowed to index presentness in this way, it will become a triviality that one is presentt at t, and the view will be in danger of looking like a more cumbersome way of expressing the indexical view of presentness. 7 This last way of putting things was suggested by Dean Zimmerman, who charitably expressed it as a reading of an earlier draft of this reply.

Fighting the Zombie of the Growing Salami | 359 other hand, 60 bc does not exist in the most unrestricted sense of quantification. The worry which often arises in disputes like this is that there is quantifier variance at work. How do we know that it’s the same quantifier being used in both statements? As I’ve intimated, the presentist certainly accepts that 60 bc did exist. And given that the PTGB behaves in a way remarkably like presentism with respect to what it says about past times, there is even more reason than usual to doubt that there is a real distinction—reason to doubt that what PTGB means when it says, ‘60 bc exists2013, and at it there are no slices past 60 bc’, and what presentism means when it says, ‘The facts about what existed at 60 bc are fixed, and when it did exist there were no times after it’, are the same. It boils down to difficult issues about fixing the meaning of the quantifier in such a way as to be sure that each theorist means the same thing when they say ‘unrestrictedly quantify’. Of course this issue bedevils more than just the growing block. Some think (Meyer forthcoming) that the distinction between presentism and other views is hard to make out for these kinds of reasons. But the striking isomorphisms between PTGB and presentism make them a very likely candidate for such treatment if anything ever was. The final independent (and to my mind greatest) worry for PTGB is simply the oddity of the relativization that it requires. The relativization produces a kind of asymmetry of existence. Let’s suppose that it is now 2013, and the past we are talking about is 59 bc (time to move a little further along in Caesar’s biography). Let’s now introduce some events at these times: the crossing of the Rubicon (at 59 bc) and the publication of Prior’s Wellington Address (at 1954). Henceforth I’ll call them Publication and Crossing. Now at Publication it’s true that amongst what other events exist2013 is Crossing. But at Crossing—from its perspective, so to speak—it’s the case that Publication does not exist60 bc. Unlike the growing block of the first stab, where there is no asymmetry of existence (at Publication, Crossing exists, but at the back-block Crossing that exists, Publication exists too), there is a real asymmetry. It can be the case that if B exists at A, A need not exist at B. Now if this is equivalent to the thought that at Publication it’s true that Crossing existed (but not that it exists), but that when it did, Publication had not yet come into existence, it’s fine and there isn’t the relevant asymmetry. But here’s the dilemma: either PTGB is so

360 | David Braddon-Mitchell equivalent, in which case it is not distinct from presentism, or it is somehow distinct from presentism but committed to asymmetries of existence between different parts of being. It’s possible for things to exist from the perspective of one part of being—2013—although, from the perspective of some of those existing things, 2013 does not exist. Of course the idea that there might be two such events such that at one of them they both exist, but at the other only one of them does can be made technically coherent with the appropriate handling of accessibility relations in a logic. But understanding the metaphysics so described is another matter. Those asymmetries of existence are not just odd. They take out the block from the growing block—we have gone far from the idea of trying to add dynamism to a block universe. It’s a strange volume of spacetime that has locations at which other locations exist, but at those locations the first location doesn’t! The thought would be that there is at each A-time a block universe of different sizes, where each of these whole blocks exists only from the perspective of its last slice. It’s true of the blocks that exist at every A-time that from the perspective of almost all of its parts the entire block doesn’t exist. That’s a strange mereology indeed: strange enough to suggest that this is not a view which really has block universes in it at all.

4. CONCLUSION Two stabs, and I think the growing block can return to its grave. The interpretation that I gave of the growing block in my first stab preserves the point of a growing block account, and also possesses the features C&R take to be crucial for an adequate account of it. I’ve made it more explicit that on this interpretation of GBT existence simpliciter—existence—is wholly dynamic, tensed, and A-theoretic. Nevertheless the epistemic objection still survives. Another understanding of GBT—the one I call PTGB—might at first look as though it fares better against the epistemic objection. But on closer inspection this is not at all clear. And there is a final worry: the view countenances a strange asymmetry of existence, one which can be removed only by understanding it in a way which may not make it distinct from presentism. University of Sydney

Fighting the Zombie of the Growing Salami | 361 REFERENCES Braddon-Mitchell, David (2004) ‘How Do We Know it is Now Now?’, Analysis 64: 199–203. Meyer, Ulrich (forthcoming) ‘The “Triviality of Presentism”’. In Roberto Ciuni, Kristie Miller, and Giuliano Torrengo (eds), New Papers on the Present—Focus on Presentism. Munich: Philosophia Verlag.

11. Changing Truthmakers: Reply to Tallant and Ingram Ross P. Cameron 1. THE THEORY AND THE PROBLEM In Cameron (2011) I proposed a solution to the truthmaker problem for presentism: that problem being, if there are no past entities, what makes it true that things happened as they did? Jonathan Tallant and David Ingram (2012) object to this proposal. In this reply, I will make a partial concession, but argue that the theory can nonetheless be defended in the face of their objection. First, a reminder of (or introduction to) my proposal. The challenge I set myself was to postulate properties that would fulfil a dual role: they would settle how their bearers were in the past, but they would also contribute to how their bearers are now. Fulfilment of the first half of this dual role lets them act as presentistfriendly truthmakers for truths about the past: it is in virtue of things having these properties now (hence the proposal is presentist-friendly) that the bearers of these properties were such-andsuch a way (hence the proposal grounds historical truths, solving the truthmaker problem). Fulfilment of the second half of the dual role, I argued, means that the properties in question are not objectionably ‘suspicious’. The thought here was that what is intuitively unsatisfying about simply appealing to my presently having the property having been a child to ground the truth of the historical claim is that my having this property makes absolutely no difference to how I am now.1 Intuitively, properties make a difference to their bearers; so they should make a difference to the intrinsic nature of their bearers at the time they’re being

1 Care is needed in spelling this demand out, but I won’t do it here. See Cameron (2011) for details.

Changing Truthmakers: Reply to Tallant and Ingram | 363 instantiated. Mere past-directed properties don’t do this: they only make a difference to how their bearers were, not to how they are at the time of instantiation. This, I argued, is why they strike us as suspicious—this is the sense in which, in Ted Sider’s words, they ‘point beyond themselves’ in an objectionable way.2 And so the challenge is to locate properties that do two things at once: make a difference to how their bearers are while they are instantiated, thus being unsuspicious, but also settle how they were, thus providing the required grounding for historical truths. What properties could do both things at once? I proposed that each thing has the following two properties: a temporal distributional property (TDP), and an age. A TDP describes how its bearer is over time, just like a spatial distributional property describes how its bearer is across space.3 If something has the spatial distributional property is polka-dotted, that tells you about the look of the thing across space; similarly, if something has the TDP grows from a child to an adult, that tells you about how the thing is across time. A thing’s having the TDP it has settles the B-theoretic truths about it: that it is at some time this way, and before that it is another way, and after that it is another way, etc. But of course, this is not enough: we need to be able to say how a thing is now, and how it was and will be. And so I proposed combining these TDPs with ages: a property that says of a thing simply how far along in its life it is.4 The TDP tells you about the nature of a thing across its lifespan, and its age tells you where in that lifespan we are; hence, together they tell you how a thing is now, but also how it was and how it will be. The properties combined solve the grounding problem, by giving a present truthmaker for past (and future5) truths, but they also avoid the charge of suspiciousness, I claim, because it is in virtue of having these properties that things are the way they are at the time they instantiate them. So together, the properties fulfil the dual role, thus meeting the challenge that was set. 2

Sider (2003, p. 185). For discussion of distributional properties see Parsons (2004). 4 This is compatible with there being things that have been around for an infinitely long period. Again, see Cameron (2011) for details. 5 You might think you shouldn’t want the properties to ground future truths as well as past ones, since the future is open. I deny this: see Cameron (2011) for my account of how to handle the open future in this setting. 3

364 | Ross P. Cameron Here’s a problem I considered in the paper.6 Suppose that something changed which TDP it instantiated. So suppose I currently have a TDP that describes my height as growing from 2ft to 6ft over a certain period of time, but that I am about to lose that property and have it replaced with a TDP that describes my height as growing from 2ft to 5ft and then halting there. In that case, while I am now 6ft tall, it will be the case, once my TDP has changed, that I have never been and never will be 6ft tall, since the new TDP has my height vary only between 2ft and 5ft over time. But this violates a massively plausible principle of tense logic: that if p is the case, then it always will be the case in the future that p was the case in the past. What’s true now cannot come to never have been—what’s true now will, no matter how else the future unfolds, always remain a part of the past. If things can change their TDPs, then this principle of temporal logic can be violated. So things must not be able to so change. But is that plausible, given that these are accidental properties? Here’s what I said earlier to justify the claim that things do not change with regard to their TDPs7: It makes no sense to speak of an object changing its [TDPs]. Why? Because what change is on the account being offered is to instantiate (at each moment of your existence) a non-uniform distributional property. Being red at one time and then orange at some later time, for example, is to be analysed as instantiating (at all times) the distributional property being red-then-orange. To speak of an object changing its properties is a loose way of saying something about the distributional property it has that says how it is across time; it makes no sense to speak of an object gaining or losing the property that says how it is across time.

Tallant and Ingram are unsatisfied. They say8: Cameron’s solution requires the union of a TDP and the property of age. Clearly, the property of age that is instantiated by any given object must change over time. We do not now bear the same age-property that we bore five years, five minutes, or even five seconds ago. Thus, to ‘speak of an object changing its properties’ is not merely a loose way of saying something about the distributional property it has across time.

6 7 8

The challenge originally arose in conversation with Tallant. Cameron (2011, p. 77). Tallant and Ingram (2012, p. 310).

Changing Truthmakers: Reply to Tallant and Ingram | 365 In essence, the complaint is: if ages can change (as they must), why not temporal distributional properties? And if the latter can change: trouble! Here’s the partial concession I promised: my earlier defence of the claim that objects don’t change their temporal distributional properties is no good. Tallant and Ingram are right that I can’t simply say that what change is is having a certain kind of TDP, for there’s at least one kind of change—change with respect to age—that is not to be analysed thus. And so, if my account is to be acceptable, I must say something else. 2. WHOSE PROBLEM IS IT ANYWAY? But before I offer my defence, let me note that the problem Tallant and Ingram are raising is by no means a problem only my view faces. Any version of presentism faces a version of this puzzle. Suppose we are presentists and truthmaker theorists. In that case, we think that present reality—which is reality simpliciter—contains entities which ground the historical truth that, e.g. Caesar crossed the Rubicon. There’s disagreement as to what kind of thing does the grounding, but if we’re truthmaker theorists, then some such thing does. But if we are presentists, then we also think that what there is—by which I mean, what there is simpliciter—changes: being is present being, and things come into and go out of being. Caesar existed, but he exists no more; and it’s not just that he presently existed but doesn’t exist presently any more (everyone agrees with that!)—it’s that he existed, simpliciter, but now doesn’t exist, simpliciter (and not everyone agrees with that). But if what there is simpliciter can so change, then what is to stop the presently existing truthmaker for going out of existence and being replaced by an entity that makes it the case that Caesar never crossed the Rubicon? But if that will happen, then something that is now a part of history will not in the future be a part of history: and so we would have a violation of the principle cited earlier—that if something is now the case, it always will be that it was the case. So Tallant and Ingram ask: given that things can change their properties, why can’t they change their TDPs, thus leading to a violation of this principle of tense logic? But really, their challenge

366 | Ross P. Cameron is an instance of a more general question that faces any presentist truthmaker theorist: given that what exists can change, why can’t it change what truthmakers for historical truths there are, thus leading to a violation of this principle of tense logic?9 And the puzzle doesn’t just arise for the presentist who is a truthmaker theorist. It is more vivid in the truthmaker case, but the puzzle faces any version of presentism. Suppose you eschew truthmakers and think that there are simply brute tensed facts. So I say that it is true that Caesar crossed the Rubicon, and this is simply a brute fact about how reality is, with no further ground. Still, what is true changes, according to the presentist, so why not this? While it is now the case that is brutely true, what ensures that tomorrow it won’t be brutely true that Caesar did not cross the Rubicon? But were this so, our principle of tense logic would again fail. The moral of the story: all presentists must rule out certain changes from occurring if they are to ensure the truth of plausible principles of tense logic. If you believe in brute tensed facts, you must place constraints on how those tensed facts can change; if you believe in truthmakers for tensed truths, you must place constraints on what changes in being can occur. My ban on things changing their temporal distributional properties is simply an instance of what every presentist must do.

3. THE SOLUTION The presentist should say that it is indeed impossible for the tensed facts, or the existence of the truthmakers for the tensed truths, to change in the objectionable manner, and that the argument to the contrary illegitimately attempts to invoke a kind of tensed claim that goes beyond what is sanctioned by the presently obtaining brute tensed facts, or the presently extant truthmakers for tensed truths. Consider again the brute-truth presentist. She thinks that reality looks something like this: there are cars and cats, but no dinosaurs 9 Of course, not any such change in what truthmakers there are would lead to such a violation. The presentist only needs to rule out certain changes from occurring. That’s what I’ll be arguing she can do, but sometimes for ease of presentation I’ll speak of her simply ruling out changes in what truthmakers there are.

Changing Truthmakers: Reply to Tallant and Ingram | 367 or lunar colonies. But it is brutely true that there were dinosaurs and will be lunar colonies. But the brute tensed facts that partly make up present reality (which is reality simpliciter) don’t end with these relatively mundane past and future facts concerning dinosaurs and lunar colonies: they will also include facts about what were and will be the past and future truths. Just as (present) reality includes the brute tensed fact that there were dinosaurs, so does it include the brute tensed fact that it always will be the case that there were dinosaurs. So what stops the brute tensed fact that there were dinosaurs changing and it becoming the case that there never were any dinosaurs? The presently obtaining brute tensed facts stop that, for one of them says precisely that this change won’t happen. Similarly with the truthmaker presentist. She thinks reality looks something like this: there are cars and cats, but no dinosaurs or lunar colonies; but there are things that make it true that there were dinosaurs and things that make it true that there will be lunar colonies. But the presently existing truthmakers for tensed truths don’t end there: there is also presently a thing that makes it true that there will always be a thing that makes it true that there were dinosaurs. So what stops the truthmaker for going out of existence and a truthmaker for coming into existence? The presently existing thing that makes it true that there will always be a thing that makes it true that there were dinosaurs stops that. We look at, e.g. the presentist’s ontology of presently existing truthmakers and ask: but what if there’s a change in what truthmakers exist? Well, what is change? Change is a matter of something being true at one time and not at another, and it is these very truthmakers that we’re asking about that account for such facts. So if there is to be a change in which of these truthmakers exist, it’s because one of them makes it true that there will be such a change. That’s the only place change can come from, because all tensed truths come from the existence of these truthmakers. So to see how the existence of the truthmakers will change, we need only look to those very truthmakers. And so to rule out violation of the principle of tense logic—to guarantee that if something is the case, it always will be that it was the case—the presentist merely has to be careful when she specifies her ontology. If she postulates something that makes it true that it was the case that p, she had better be careful to

368 | Ross P. Cameron also posit something that makes it true that it will always be the case that there is something that makes it the case that it was the case that p. So long as she is so careful, there can be no violation of the principle that appeared under threat. And likewise, mutatis mutandis, for what brute tensed facts the non-truthmaker presentist postulates in reality. I see nothing methodologically suspect about the presentist simply postulating that this is how things are, in order to uphold the relevant principle of tense logic. Of course she should let her beliefs about what kinds of change are possible guide her theory about what tensed truthmakers10 exist (or what brute tensed facts obtain), given that it is these tensed truthmakers (or brute tensed facts) that account for change, on her metaphysic. And whatever strength she thought attached to the principle of tense logic, she should think the same strength attaches to the principles concerning tensed truthmakers or brute tensed facts. If all she is concerned with is the truth of the material conditional that if something is true now then it always will be the case that it was true, then her theory merely needs to entail the truth of material conditionals like: if there’s a truthmaker for , then there is a truthmaker for . If, on the other hand, you think the principle of tense logic holds with metaphysical necessity, then your theory should claim that this connection between what truthmakers there are is a metaphysically necessary one. If you think there is some special status of ‘being a theorem of tense logic’ that the principle has, then you should think the link between truthmakers has this special status also.11 10

By ‘tensed truthmaker’ I mean simply a truthmaker for a tensed truth. This may be where the strongest objection can be made. Suppose you held that tense logic is really a logic, and not just a description of certain facts about how time in fact is, and that it is a theorem of this logic that if p is now true then it always will be the case that p was true. My suggestion is that you then also take it to be a theorem of tense logic that if there’s a truthmaker for p, there’s also a truthmaker for . But, you might object, there cannot be theorems concerning what exists: ontological claims like this are never a matter of logic. On these grounds, an objector might see a problem that arises particularly for truthmaker versions of presentism. I am unsympathetic to this objection. Firstly, I don’t believe that there is this special logic: ‘tense logic’, to my mind, is merely a description of how time works—calling something a theorem of tense logic is just to say that it is a true description of how time works, in which case the link between tensed truthmakers just needs to be true for it to have the same 11

Changing Truthmakers: Reply to Tallant and Ingram | 369 It’s tempting to see a threat to the principle of tense logic because it’s tempting to think that there can be changes that are not grounded by the presently obtaining brute tensed facts or the presently extant tensed truthmakers. We see this truthmaker for , and we worry about what might happen if it fails to be around. We’re told not to worry, because here’s another truthmaker that says that the first one will always be around, thus securing the truth of . But intuitively, that doesn’t address the worry. For what if neither of these truthmakers is around in the future? What if all the truthmakers for tensed truths disappear? And it’s no good, so the thought goes, to be told that there’s now something that makes it true that this won’t happen, for what happens if that goes out of existence too? This is a seductive line of thought, but it simply has to be abandoned. It attempts to invoke tensed claims that are not grounded in the presently existing truthmakers for tensed truths (or the presently obtaining brute tensed facts), and the presentist should simply deny that this makes any sense. The sole source of tensed truth, she claims, are the presently existing tensed truthmakers (or the presently obtaining brute tensed facts), and these tell you everything there is to know about what was or will be the case, including the facts about what tensed truthmakers did or will exist (or what brute tensed facts did or will obtain). Any attempt to ask a question about what was or will be the case that is not to be settled by what tensed truthmakers presently exist (or what brute tensed facts presently obtain) is simply illegitimate by the presentist’s lights. And so, as long as she is careful, in the sense spelled out earlier, about what

kind of status. Secondly, even if there is this special kind of logic, and our principle has a special kind of status of theoremhood that is more than mere truth (and more than mere metaphysical necessity, even), I don’t see a good reason for denying that the link between tensed truthmakers can have the same status. Why can’t there be theorems of tense logic that concern what exists, when what exists includes tensed truthmakers? One might hold that in ordinary (non-tense) logic, there are no theorems concerning what there is. (In fact, I think we should reject even this, since it’s simply a bad prejudice left over from logical positivism; but let’s grant it for the sake of argument.) But if so, that is because ordinary logic is topic neutral. But of course, tense logic, if there is such a thing, is not topic neutral: it’s about time. So just as it can include theorems about time, I don’t see why it can’t also include theorems about tensed truthmakers.

370 | Ross P. Cameron tensed truthmakers (or what brute tensed facts) she posits, then there is simply no threat here. So to return to my preferred metaphysic, I say I have a TDP and an age: together, they make it true that I am 32 years old and 6ft tall, and that I was 10 years old and 5ft tall. Tallant and Ingram ask: how can I rule out my changing my TDP, and coming to instantiate one that makes it true that I was never 5ft tall? Answer: my having the TDP I presently have makes it true that I will always have it! My present age makes it true what ages I had and will have. Since I have the property being 32 years old, this makes it true that I will have the property being 33 years old a year from now. And just as the age I have now grounds the facts about what ages I had or will have, so does the TDP I have now ground the facts about what TDPs I had or will have. And in particular, it grounds the fact that I always had and always will have (for as long as I exist, anyway) the one I presently have. There was something right about what I said in the passage quoted earlier from my original paper. I said: ‘It makes no sense to speak of an object changing its distributional properties. Why? Because what change is on the account being offered is to instantiate (at each moment of your existence) a non-uniform distributional property.’ As Tallant and Ingram point out, this can’t be quite right: it can’t be simply that this is what change is, given that ages change. But I was on the right lines. You can’t vary in what TDP you have from one time to another, because that would be to vary across time in how your intrinsic nature is across time. Of course that can’t happen, because once you’ve settled how your intrinsic nature varies across time, that also settles how at other times your intrinsic nature varies across time. Compare the spatial case. Consider a really long bar that stretched the length of Britain, and which is black-and-white striped, with stripes thick enough to be the length of cities. Looking at the bar in Leeds, I can say: (i) the bar is black here, (ii) the bar is white in Nottingham, and (iii) the bar is black-and-white striped across space. (iii) describes how the bar varies across space here in Leeds, but it also truly describes how the bar varies across space in Nottingham, and indeed anywhere. When we’re talking about how the bar is across space, it doesn’t matter where we’re assessing things from. When claiming that it’s black here, or white here, my claim is sensitive to

Changing Truthmakers: Reply to Tallant and Ingram | 371 where ‘here’ is, since how the bar is with respect to colour varies across space; but when claiming that it’s such-and-such a way across space, the location of assessment just drops out. How things are across space isn’t the kind of thing that can vary from one place to another. Likewise, how things are across time isn’t the kind of thing that can vary from one time to another: settle how a thing is across time and you thereby settle how it is across time at other times. Which is just to say that having a particular TDP at a time makes it true that you always had, and always will have (so long as you exist(ed), at least), that very TDP. And that ensures that our principle of tense logic will never be violated.

4. THE SOLUTION MADE MORE PALATABLE: ILLUSIONS OF POSSIBILITY AND ILLUSIONS OF CHANGE One more thing must be done before the problem can be considered adequately addressed. I said earlier that the appearance that the principle of tense logic might be violated was due to a seductive, but illegitimate, thought: that there could be changes concerning what truthmakers for tensed truths exist (or changes concerning what brute tensed facts obtain) that are not themselves simply a result of what truthmakers for tensed truths there are (or what brute tensed facts obtain). That thought is illegitimate, but it is so seductive that I think we cannot rest until we have accounted for it. This is what I will attempt to do in this final section. I aim to borrow a methodological trick from Kripke: that when you are denying the possibility of a scenario that really seems possible, you should identify a genuinely possible scenario that you are confusing with the scenario deemed impossible. So, for example, Kripke tells you that it’s impossible that water is not H2O; but he accounts for intuitions to the contrary by claiming that you’re confusing this scenario with the genuinely possible scenario that something has all the phenomenal surface qualities of water but is not H2O.12 I deny that it’s possible that the truthmaker for go out of existence and be replaced by a truthmaker for . And yet I concede that it 12

Kripke (1980).

372 | Ross P. Cameron genuinely seems possible. But don’t worry: there’s a genuinely possible scenario that you’re confusing with this impossible one. The genuinely possible scenario is that there’s now a truthmaker for but that there hyperwill be no truthmaker for and instead a truthmaker for . Here I’m invoking the notion of hypertime, which is what one needs to make sense of the past changing.13 While ordinary tenses let us describe our past and future, hypertenses let us describe a kind of change to history as a whole. So while it will always be the case that Caesar crossed the Rubicon, this is compatible with it being true that it hyperwill be the case that Caesar never crossed the Rubicon. Likewise, while what tensed truthmakers there presently are determines what tensed truthmakers there were and will be (since those tensed truthmakers determine all the facts concerning what was and will be), there now being these truthmakers does not determine what tensed truthmakers there hyperwere or hyperwill be. So there’s a truthmaker for , and there always will be such a truthmaker: but maybe it hyperwill be the case that there is no such truthmaker. I’m not claiming that there is such a thing as hypertime. But I think it is metaphysically possible, and that is all I need. While it seems possible that what tensed truthmakers there are can change in a way not governed by those tensed truthmakers, what’s really possible is merely that there are such tensed truthmakers, and that history is (thereby) thus-and-so, but that there hyperwill be different tensed truthmakers, hence that it hyperwill be the case that history is different from how it now is. So on my own preferred metaphysic: it might be that I hyperwill have a different TDP, but that’s no problem, for it leads to no violation of any plausible principle of tense logic. Changing my TDP would cause trouble; but I won’t change my TDP, and I know that this is the case because my having my TDP makes this the case. So we have both an explanation for why Tallant and Ingram’s problematic scenario cannot arise, and also an explanation for the seductiveness of the intuition that it may.14 University of Leeds 13 14

See van Inwagen (2010) and Hudson and Wasserman (2010). Thanks to Elizabeth Barnes, Karen Bennett, Jason Turner, and Dean Zimmerman for helpful comments.

Changing Truthmakers: Reply to Tallant and Ingram | 373 REFERENCES Cameron, Ross (2011) ‘Truthmaking for presentists’, in Karen Bennett and Dean W. Zimmerman (eds), Oxford Studies in Metaphysics Vol. 6. Oxford: Oxford University Press, pp. 55–100. Hudson, Hud and Wasserman, Ryan (2010) ‘van Inwagen on Time Travel and Changing the Past’, in Dean W. Zimmerman (ed.), Oxford Studies in Metaphysics Vol. 5. Oxford: Oxford University Press, pp. 41–9. Kripke, Saul (1980) Naming and Necessity. Cambridge, MA: Harvard University Press. Parsons, Josh (2004) ‘Distributional Properties’, in Frank Jackson and Graham Priest (eds), Lewisian Themes. Oxford: Oxford University Press, pp. 173–80. Sider, Theodore (2003) ‘Reductive Theories of Modality’, in Michael J. Loux and Dean W. Zimmerman (eds), The Oxford Handbook of Metaphysics. Oxford: Oxford University Press, pp. 180–208. Tallant, Jonathan and Ingram, David (2012) ‘Presentism and Distributional Properties’, in Karen Bennett and Dean W. Zimmerman (eds), Oxford Studies in Metaphysics Vol. 7. Oxford: Oxford University Press, pp. 305–14. van Inwagen, Peter (2010) ‘Changing the Past’, in Dean W. Zimmerman (ed.), Oxford Studies in Metaphysics Vol. 5. Oxford: Oxford University Press, pp. 3–28.

AUTHOR INDEX Adams, Robert 152, 164 Albert, David 135, 142, 266 Armstrong, David 81, 85, 86, 87, 92, 107, 161, 187, 245 Arntzenius, Frank 12, 63, 270, 283 Baker, David 106, 109, 132 Baxter, Don 295, 303–5, 308, 311, 317–18 Bealer, George 197, 204, 230 Bennett, Karen 72, 179, 181, 240 Bigelow, John 89, 92, 107 Bøhn, Einar 317 Boolos, George 154 Braddon-Mitchel, David 333, 336, 241–50, 351–61 Bricker, Phillip 283 Broad, Charles 333, 335, 348 Burge, Tyler 258

Dworkin, Ronald 183 Dunaway, Billy 151–86 Earman, John 137, 243 Eddington, Arthur 254 Eddon, Maya 17, 46, 78–104, 113, 119, 228 Effingham, Nick 318 Eklund, Matti 269 Ellis, Brian 80, 84, 107 Field, Hartry 30, 38, 84, 87, 90–1, 107, 131, 133, 243, 286 Fine, Kit 81, 108, 109, 168, 173, 183, 209, 229, 240 Forbes, Graeme 153 Forge, John 92–3 Forrest, Peter 174 Frege, Gottlob 168, 302 Francescotti, Robert 17

Cameron, Ross 31, 238, 250, 255, 276, 318, 362–74 Casati, Roberto 197–8 Carmichael, Chad 187 Carnap, Rudolf 5, 269 Chalmers, David 8, 18, 121, 269 Correia, Fabrice 333–50 Cotnoir, Aaron 294–322, 323–32 Crimmins, Mark 187

Gärdenfors, Peter 23 Gibbard, Allan 183 Gilmore, Cody 102, 187–234 Gillon, Brendan 309 Givant, Steven 282 Goodman, Nelson 4, 152, 241 Grossmann, Reinhardt 187

Dasgupta, Shamik 88, 105–50, 243–4 Denby, David 80, 95, 97 Descartes, Rene 61 Devitt, Michael 161 Dorr, Cian 3–77, 207–8, 238, 250, 266, 269, 282, 283, 286

Harman, Gilbert 246 Hawley, Katherine 323–32 Hawthorne, John 3–77, 84, 88, 96, 113, 269 Hellman, Geoffrey 285 Huemer, Michael 239, 258

376 | Author Index Hirsch, Eli 152, 269 Holly, Jan 83 Hossack, Keith 230 Horwich, Paul 187–8 Hudson, Hud 362 Ingram, David 362 Johnston, Mark 280 Jubien, Michael 187, 221 Kelly, Thomas 274 Kim, Jaegwon 11, 179, 182, 207 King, Jeff 187 Korman. Daniel 247 Krantz, David 88 Kripke, Saul 120, 215, 245, 361 Ladyman, James 137 Landman, Fred 309 Langton, Rae 18 Laycock, Henry 298 Lewis, David 3–74, 78, 79, 80, 87, 89, 93, 95, 96, 98, 99, 100, 107, 118, 119, 127, 151, 152, 178, 180, 207, 238, 244, 245–6, 281, 285, 294–5, 300, 302, 307, 313, 326 Liggins, David 88 Linnebo, ØYstein 309 Linsky, Bernard 213 Macbride, Fraser, 229, 230 Malament, David 286 Manley, David 158, 168, 178 Marshall, Dan 17 Maslen, Cei 207 Maudlin, Tim 64, 135, 146, 283 McDaniel, Kris 250–1, 318 MacFarlane, Ian 249 McGrath, Matthew 254

McKay, Thomas 187, 229 Melia, Joseph 88, 153 Menzel, Christopher 187, 203, 204, 207, 211, 213 Merricks, Trenton 244, 258, 333, 336, 242–50 Meyer, Glen 88, Meyer, Ulrich 259 Meyers, Robert 267 Milne, Peter 84 Moss, Sarah 47 Moore, George Edward 245–6 Mundy, Brent 66, 80, 82, 84, 88, 91, 92, 105, 107 Newman, Paul 187 Ney, Alyssa 266 Nicolas, David 309 Nolan, Daniel 207 North, Jill 137, 243, 266 Oliver, Alex 299, 309 Orilia, Francesco 229 Pargetter, Robert 89, 92, 107 Parsons, Josh 17, 64, 363 Pieri, Mario 73 Plantinga, Alvin 152, 164 Pollock, John 258–60 Price, Henry Habberly 51 Prior, Arthur 153, 334 Pryor, James 258–62 Putnam, Hilary 256, 269, 285 Quine, Willard van Orman 155, 170, 184, 238, 247, 285 Rayo, Agustin 154, 157–62, 299, 301, 308 Roberts, John 135

Author Index | 377 Rodriguez-Pereyra, Gonzalo 51 Rosen, Gideon 4, 108, 238 Rosenkrantz, Sven 333–350 Ross, Don 137 Russell, Bertrand 51, 178, 247 Saucedo, Raul 194, 300 Schaffer, Jonathan 18, 72, 168, 182, 189, 251–2 Schlenker, Phillipe 299, 300 Schwarzschild, Roger 309 Shagrir, Oron 181 Shapiro, Stewart 286 Sharvy, Richard 299 Sider, Ted 7, 21, 24, 29, 36, 40, 51, 65–68, 70, 71, 100, 179, 181, 190, 223, 235–93, 295–6, 307–8, 313–6, 263 Siegel, Susanna 258 Skow, Brad 64 Smiley, Timothy 299, 309 Soames, Scott 20, 152, 163, 166–7 Sperber, Dan 252 Stalnaker, Robert 152, 163, 179, 181, 207 Stebbing, Susan 254 Stern, Kenneth 267 Swoyer, Chris 80, 187, 203, 204 Tallant, Jonathan 262 Tarski, Alfred 63, 282–3 Teichmann, Robert 169 Thomasson, Amie 269

Turner, Jason 250 Unger, Peter 267 Uzquiano, Gabriel 298 Varzi, Achille 197–8, 303, 308, 311 Van Cleve, James 161–2 Van Fraassen, Bas 56 Van Inwagen, Peter 157, 161, 178, 183, 187, 196, 213, 237, 255, 296, 362 Vogel, Jonathan 247 Wasserman, Ryan 362 Wallace, Megan 317 Weatherson, Brian 32, 82, 223 Wedgewood, Ralph 55 Weiland, Jan Willem 229 Wetzel, Linda 187–8 Williams, Donald 51 Williams, Robbie 29 Williamson, Timothy 11, 18, 31, 159, 163, 187, 221, 245, 263 Wilson, Jessica 87, 252 Wright, Crispin 161 Yablo, Stephen 154, 157–62, 168 Yi, Byeong-Uk 187, 227, 229, 294, 308, 313 Zalta, Edward 187, 221 Zimmerman, Dean 247