Oxford Studies in Metaphysics, Volume 12 0192893319, 9780192893314

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OXFORD STUDIES IN METAPHYSICS

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OXFORD STUDIES IN METAPHYSICS Editorial Advisory Board Elizabeth Barnes (University of Virginia) Ross Cameron (University of Virginia) David Chalmers (New York University and Australasian National University) Andrew Cortens (Boise State University) Tamar Szabó Gendler (Yale University) Sally Haslanger (MIT) John Hawthorne (University of Southern California; Australian Catholic University) Mark Heller (Syracuse University) Hud Hudson (Western Washington University) Kathrin Koslicki (University of Alberta) Kris McDaniel (University of Notre Dame) Trenton Merricks (University of Virginia) Kevin Mulligan (University of Lugano ) Laurie Paul (Yale University) Jonathan Schaffer (Rutgers University) Theodore Sider (Rutgers University) Jason Turner (University of Arizona) Timothy Williamson (Oxford University) Managing Editors Christopher Copan (Rutgers University) Isaac Wilhelm (Rutgers University)

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Oxford Studies in Metaphysics Volume 12 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 2020 The moral rights of the authors have been asserted First Edition published in 2020 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 Library of Congress Control Number: 2020945205 ISBN 978–0–19–289331–4 DOI: 10.1093/oso/9780192893314.001.0001 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. 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.

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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 Sanders Prize in Metaphysics, a biennial award described within. K. B. & D. W. Z. New Brunswick, NJ

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Contents The Sanders Prize in Metaphysics

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PART I: EXISTENCE 1. Truthmakers and Easy Ontology

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Amie L. Thomasson

2. Easy Ontology, Two-Dimensionalism, and Truthmaking

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Ross P. Cameron

3. Ontological Pluralism and Notational Variance

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Bruno Whittle

P A R T I I : ME T A P H Y S I C S O F S C I E N C E 4. Some Consequences of Physics for the Comparative Metaphysics of Quantity

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David John Baker

5. How to Be a Relationalist

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Shamik Dasgupta

6. Nomothetic Explanation and Humeanism about Laws of Nature

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Harjit Bhogal

7. The Fundamentality of Physics: Completeness or Maximality? Alyssa Ney

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P A R T I I I: TI M E A N D P E R S I S T E N C E 8. A Little Puzzle about a Piece and a Puddle

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Mahrad Almotahari

9. Advanced Temporalizing

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Daniel Deasy

Author Index

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The Sanders Prize in Metaphysics Sponsored by the Marc Sanders Foundation* and administered by the editorial board of Oxford Studies in Metaphysics, this essay competition is open to scholars who are within fifteen years of receiving a Ph.D. or students who are currently enrolled in a graduate program. (Independent scholars should inquire of the editors to determine eligibility.) The award is $5,000, and the competition is biennial. 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 November 1. To be eligible for next year’s prize, submissions must be electronically submitted by January 31, 2022 . Refereeing will be blind; authors should omit remarks and references that might disclose their identities. Receipt of submissions will be acknowledged by email. 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 Sanders Prize are: Thomas Hofweber, “Inexpressible Properties and Propositions,” Vol. 2; Matthew McGrath, “Four-Dimensionalism and the Puzzles of Coincidence,” Vol. 3; Cody Gilmore, “Time Travel, Coinciding Objects, and Persistence,” Vol. 3;

* The Marc Sanders Foundation is a nonprofit organization dedicated to the revival of systematic philosophy and traditional metaphysics. Information about the Foundation’s other initiatives may be found at http://www.marcsandersfoundation.com/.

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     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; R.A. Briggs and Graeme A. Forbes, “The Real Truth about the Unreal Future,” Vol. 7; Shamik Dasgupta, “Absolutism vs Comparativism about Quantities,” Vol. 8; Louis deRosset, “Analyticity and Ontology,” Vol 9; Nicholas K. Jones, “Multiple Constitution,” Vol. 9; Nick Kroll, “Teleological Dispositions,” Vol. 10; Jon Litland, “Grounding Grounding,” Vol. 10; Andrew Bacon, “Relative Locations,” Vol. 11; T. Scott Dixon, “Plural Slot Theory,” Vol. 11; Harjit Singh Bhogal, “Nomothetic Explanation and Humeanism about Laws of Nature,” Vol. 12. Inquiries should be addressed to Dean Zimmerman at: [email protected].

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PART I

EXISTENCE

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1 Truthmakers and Easy Ontology Amie L. Thomasson

Quine’s “On What There Is” initiated a heyday for existence questions. For decades, the central task of ontology was seen as answering questions about what exists, by seeking a ‘best theory’ and determining its ontological commitments. And so there followed decades of inquiry into whether our best theory will tell us that there are possible worlds, persons, artifacts, mereological sums, works of art, and so on. But those days are fading. One reason existence questions are fading from interest is the increasing sympathy for the thought that many existence questions are easy to answer—and in the affirmative. Kit Fine, for example, accepts that “given the evident fact that there is a prime number greater than 2, it trivially follows that there is a number . . . and similarly, given the evident fact that I am sitting on a chair, it trivially follows that there is a chair” (2009a, 158). Jonathan Schaffer similarly accepts that questions about the existence of numbers, properties, mereological sums and the like, “are trivial, in that the entities in question obviously do exist” (2009a, 357). One reason many existence questions seem ‘easy’ to answer is that we can often make what seem to be trivial inferences from uncontested truths to claims that numbers, properties, or other ‘controversial’ entities exist. Bob Hale and Crispin Wright (2001) famously made the case in the philosophy of mathematics. An uncontroversial claim such as “The cups and saucers are equinumerous”, they argued, may be combined with Hume’s Principle (“The number of ns = the number of ms iff the ns and ms are equinumerous”) to derive the claim “The number of cups=the number of saucers”, in which we seem to have a true identity claim with

Amie L. Thomasson, Truthmakers and Easy Ontology In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Amie L. Thomasson. DOI: 10.1093/oso/9780192893314.003.0001

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singular terms that refer to numbers. Stephen Schiffer (2003) generalized the approach, arguing that what he called ‘something from nothing’ inferences could in each case take us from an uncontroversial claim that involved no terms for properties, propositions, states, etc. to one that does include such new terms, which are seemingly guaranteed to refer. So, for example, we can make the inference from ‘Snow is white’ to ‘Snow has the property of being white’—and while the latter is intuitively redundant with respect to the former, it apparently includes a new referring term for a property of being white. In Ontology Made Easy (2015), I further generalized such easy arguments for existence—expanding them to include arguments for concreta as well as abstracta—and defended them against a range of objections. One common response to such easy arguments is to allow that questions about what exists are easy to answer, and that, as a result, the proper, interesting project of ontology must lie elsewhere than in determining what exists. A common suggestion—particularly among those metaphysicians who follow the lead of Armstrong more than of Quine— is to say that the project of ontology should instead be thought of as the search for truthmakers. As Ross Cameron puts it: one particularly pleasing metametaphysical feature of the truthmaking account [is that] it allows us to recognize that existential questions can often times be easy—sometimes to the point of triviality—to answer, while still allowing that there is a deep, difficult and interesting ontological project to be undertaken. (forthcoming, 7)

That project is answering the truthmaker question: “What makes what true?” In many ways the truthmaker theorist and the easy ontologist are allies. Both reject the Quinean criterion of ontological commitment, on grounds that it makes irrelevant grammatical differences implausibly lead to huge differences in our ontological commitments (see Armstrong 2004, 23; Alston 1958, 13–14). Both deny that ontology should be concerned with deep questions about what exists, as existence questions are typically easy to answer, and both are happy to accept a broadly permissive view about what exists.

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The question I aim to address in this paper is this: Suppose we accept the easy approach to ontology.¹ Can we then legitimately turn to address instead the question what makes what true? Ross Cameron argues that the answer is yes: all this [easy ontology] just shows us that the project of ontology should not be understood in terms of cataloging what exists. I agree with Thomasson that existential questions are often easy to answer (Of course there are tables—look, my dinner’s on one), and that for many traditional ontological debates the existence of the entities in question follows trivially from obvious truths . . . But there is still a deep, hard to answer, ontological question: what makes what true? Do tables, numbers, properties, etc., figure amongst the [fundamental] truthmakers for the way the world is? (forthcoming, 10 in draft)

The plan for this paper is as follows. I will begin by briefly recounting some key theses of easy ontology, so that we can better see what its consequences are for the truthmaker approach. Then, I will consider two different versions of the claim that the proper question for metaphysics is ‘What makes what true?’. First, I will consider the strong claim (developed by Cameron) that a project remains of determining what the fundamental truthmakers are, with the goal of giving a uniquely true statement of what the fundamental entities are. Second, I will consider a weaker claim (endorsed in different forms by several truthmaker theorists), which holds that it is at least a constraint on metaphysics to give some good account of what the (fundamental) truthmakers are for claims we accept—and that wielding this constraint gives us a way of ‘catching cheaters’ in metaphysics. I will argue that, if we truly take on board some of the basic theses of easy ontology, we should have serious reservations about both of these projects. I will conclude by considering what may, nonetheless, remain to be done.

¹ I will not argue for the easy approach to ontology here, though I have done so extensively elsewhere (2015). Instead, the question to be addressed here concerns what follows if we accept easy ontology.

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1. Theses of easy ontology Easy ontology consists of two interrelated theses,² conceived as part of an overarching project: 1. All well-formed existence questions are easy to answer, in the sense that they may be answered by conceptual and/or empirical work (Thomasson 2015, 128).³ 2. At least some disputed existence questions may be answered by means of trivial inferences from uncontroversial premises (Thomasson 2015, 128). The project these theses are designed to support is demystifying metaphysics, by arguing that those metaphysical questions that are wellformed and answerable descriptive questions can be answered via nothing more mysterious than empirical and conceptual work.⁴ The theses support the project by showing how one prominent swath of metaphysical questions (existence questions) may be answered without relying on any work that is ‘epistemically metaphysical’ in Theodore Sider’s sense of being answerable neither by direct empirical methods nor by conceptual analysis (2011, 187). (I address another swath of metaphysical questions, the modal questions, in my (2020)—as another step on the way to making good on the general project.)

² At least as I use the term ‘Easy Ontology’. For the original discussion, see my (2015, 128). ³ Actually, thanks to discussion with Katherine Hawley, I have now softened this claim: all well-formed existence questions that can be answered at all may be answered by conceptual and/or empirical work. That is, for example, a question about whether there are concrete entities spatiotemporally and causally isolated from us may be unanswerable empirically (given the spatiotemporal and causal isolation), but there seems to be no reason to think that ‘serious metaphysics’ could give us knowledge of the answer either (see my 2019, following Hawley’s 2019). It may simply be unanswerable. This caveat will not play a role in what follows here, nor does it constrain what I have wanted to say about traditional metaphysical debates. ⁴ By ‘descriptive questions’ I mean questions about what is the case, rather than normative questions about what we should do. I have argued elsewhere (2016, 2017a, 2017b, 2018) that we can also reconceive of many debates in metaphysics as debates about what terms or concepts we should use or how we should use them.

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At least in some cases, allowing that existence questions are easy to answer, but insisting that there remains a ‘substantive metaphysical project’ is a way of accepting the theses, while rejecting the overall project by identifying remaining questions that are the target of serious metaphysical inquiry. As Cameron goes on to describe it, the question ‘What makes what true?’ “cannot be answered by the combination of conceptual analysis and empirical observation: it is a substantive metaphysical question” (forthcoming, 10). Some accept that existence questions are easy to answer, in the sense that they are (generally) obvious, without commitment to whether there are trivial or analytic inferences that take us to existential conclusions.⁵ But the form of easy ontology I am interested in here, and that I have defended elsewhere (2015), is one that gives a particular reason for thinking that many of the disputed ontological existence questions are easy to answer—and that comes from premise (2): that questions, say, about the existence of numbers, properties, propositions, facts, or even tables may be answered by making trivial inferences (such as those reviewed in the introduction) from premises that are not in dispute by the parties debating. Accepting that there are trivial inferences that enable us to answer disputed existence questions (on the basis of an uncontroversial premise combined with a conceptual truth) has the consequence that there may be trivial inferences across what we might call ‘ontologically alternative’ statements. Take a sentence to have as its ‘apparent ontological commitments’ those entities that would most directly correspond to its surface grammar: objects corresponding to its noun terms, properties corresponding to its predicates, etc. Say that a sentence ‘A’ is ‘ontologically alternative’ to a sentence ‘B’ when its apparent ontological commitments are distinct from those of B. Since easy ontologists accept that there are valid trivial inferences that take us, say, from “There are particles arranged tablewise” to “There is a table”, or from “There are three cups on the table” to “The number of cups on the table is three”, they accept that there are valid trivial inferences across ‘ontologically alternative’ ⁵ Jonathan Schaffer, for example, takes the existence of numbers as a ‘Moorean certainty’, “more credible than any philosopher’s argument to the contrary” (2009, 357).

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expressions. For the right side in each case involves introducing a new noun term that does not appear in the original; a noun term that (as Stephen Schiffer (2003) puts it) is apparently guaranteed to refer, given the truth of the original claim and the triviality of the inference.⁶ However, as I argue next, taking seriously the idea that there may be trivial inferences across ontologically alternative expressions undermines the idea that we can hope to use an appeal to truthmakers to get a uniquely true statement of what the fundamental truthmakers are.

2. Cameron’s project: finding the fundamental truthmakers As discussed in the introduction, a common response to the easy approach to ontology is to allow that existence questions are easy to answer, and yet to hold that a deep project remains for metaphysics. That project, as Cameron presents it, is finding the fundamental truthmakers for the claims we accept, and thereby determining what we should accept as being fundamental. The idea is roughly this: Quineans hold that we are ontologically committed to (only) whatever we must quantify over to render true the statements of a theory we accept, and that to determine what we should say exists, we should figure out our best total theory, and what it is ontologically committed to, and then allow that those (and only those) things exist. Truthmaker theorists tend to reject the Quinean criterion of ontological commitment (Armstrong 2004, 23–4; Cameron 2008) and the neo-Quinean focus on existence questions. Instead, Cameron aims to determine the fundamentality commitments of a sentence by asking what must exist to make the claims of our theory true. Then, to determine what we should say is fundamental, we should figure out our best total ⁶ As will be apparent, the easy approach also requires that we give up the Quinean criterion of ontological commitment. For there may also be valid trivial inferences across sentences that Quine would take to have different ‘ontological commitments’, for example, taking us from “Some dogs are white” to “there is a property of whiteness some dogs have”. For discussion see my (2015, ch. 1).

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theory, and what it is fundamentality-committed to, and allow that those (and only those) things are fundamental.⁷ But what are we fundamentality-committed to? We need some criterion—some way of deciding, given a sentence or theory one accepts, what one is—and is not—committed to saying is fundamental. Say that a sentence’s fundamentality commitments are whatever entities one is committed to saying are fundamental, in virtue of accepting that sentence. It seems that even if truthmaker theorists don’t give us a recipe for determining what we are fundamentality-committed to, they must at least provide a criterion that will tell us when we aren’t fundamentalitycommitted to entities of a kind apparently referred to in a given sentence or theory. Under what conditions can we say that an entity isn’t needed as a fundamental truthmaker and so deny the apparent commitment? The truthmaker theorist clearly needs such a principle to narrow things down—giving us a way to say that, while a sentence may have many apparent ontological commitments, we needn’t (in virtue of accepting that sentence) be committed to saying that all of those are fundamental (“Everything is Fundamental” being an even worse slogan than “Everything is Awesome”). Actual truthmaker theorists have not (to my knowledge) directly articulated such a principle, so it will take some reconstructive work to see what suggestions might be offered to provide a way of determining which of those entities we are apparently ontologically committed to (in accepting sentences of various forms) are not things we are fundamentality-committed to. We will begin by looking at some of the classical, simple, and most plausible cases in which truthmaker theorists claim that their approach entitles us to avoid certain apparent ontological commitments. But first, a little history is in order. In earlier literature (e.g., Armstrong 2004; ⁷ It is certainly not true, however, that all of those who appeal to truthmakers think that they can be used in aiming to get a uniquely true statement of what is fundamental. John Heil and Heather Dyke, for example, would clearly reject this goal. Jonathan Schaffer holds that the fundamentality commitments ‘are best viewed as constraints’ (2008, 19), and so is not committed to the idea that appeal to truthmakers can get us to a unique statement of what is fundamental (I return to this later in this section). Ross Cameron has argued most prominently for using the appeal to truthmakers to determine one’s fundamentality commitments. This section is largely directed toward his views.

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10  .  Cameron 2008), the truthmaker approach was often held to give an alternative to Quine’s approach to ontological commitment. As a result, truthmaker theorists tended to distinguish a sentence’s apparent ontological commitments from its genuine ontological commitments, as given by a ‘truthmaker criterion’ of ontological commitment. Mulligan, Simons, and Smith write “The glory of logical atomism was that it showed that not every kind of sentence needs its own characteristic kind of truthmaker” (1984, 289). As Cameron put it, “I am a truthmaker theorist: I hold that the ontological commitments of a theory are just those things that must exist to make true the sentences of that theory” (2008, 4). More recent versions of truthmaker theory often aim instead to distinguish a sentence’s apparent ontological commitments from its fundamentality commitments.⁸ I will translate the original work to the fundamentality phrasing, to streamline the exposition. Here are some classic, central cases in which truthmaker theorists argue that the apparent ontological commitments of a sentence can be avoided (or not treated as fundamentality commitments): • Disjunctive properties: “X is (A or B)” has an apparent ontological commitment to a disjunctive property (being A or B), as well as to X. But since “X is A” entails the truth of “X is (A or B)” and does not even appear to require a disjunctive property to make it true, we need not accept disjunctive properties [as fundamental] to accept the truth of the disjunctive statement. As Mulligan, Simons, and Smith put it, “Provided we can account for the truth and falsehood of atomic sentences, we can dispense with special truth-makers for, e.g., negative, conjunctive, disjunctive, and identity sentences” (1984, 289). • Determinables: “The cloth is colored” appears to commit us to a determinable property of being colored. But, as Armstrong argues, not all sentences of the form “X is Q” require the property of being

⁸ See Schaffer (2008) for arguments that appeal to truthmakers cannot “provide a viable measure of ontological commitment” but does “provide a needed constraint on what is fundamental” (2008, 7). More on this later in this section.

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Q as a [fundamental] truthmaker. For “The cloth is scarlet” entails “the cloth is red”, and “the cloth is colored”. So we can just accept the property of being scarlet as a (predicate-side) truthmaker for all of these sentences and can avoid [fundamental] ontological commitment to determinable properties.⁹ Using this approach, Armstrong held that he could maintain a sparse account of what properties we are committed to as truthmakers. But he did not extend the approach to object terms. Instead, Armstrong held that “for every truthmaker T, the truth has T as its unique minimal truthmaker” (2004, 23). Cameron extends the approach to object terms, arguing that even existence claims may have different (fundamental) truthmakers than at first appears (2008, 4). Thus, Cameron argues that we needn’t accept that an entity X is fundamental if we can show that the truth of “X exists” is grounded without appeal to X: If X is needed as the truthmaker for “X exists” then X really exists—it is part of fundamental reality. But if “X exists” is made true not by X but by Y then, while X exists, X does not really exist: It is Y that really exists; it is Y that is part of fundamental ontology, and which is the [fundamental] ontological commitment of “X exists”. (2008, 17, italics mine)

Later he puts it this way: if you want to hold that ‘there are Xs’ is strictly and literally true whilst resisting [fundamentality] commitment to the Xs, you should show that one can provide grounds for the truth of such claims without appealing to the Xs; that is, you should show that a [fundamental] ontology lacking in Xs can nonetheless make true a sentence proclaiming the existence of, or attributing features to, the Xs. The thought is

⁹ Mulligan, Simons, and Smith similarly acknowledge that one and the same truthmaker may make true sentences with different meanings (1984, 300).

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12  .  that the [fundamentality] commitments of a sentence are those entities that are needed as truthmakers for the sentence: those entities that must number amongst the [fundamental] ontology of the world if the world is to provide an adequate grounding for the truth [of] the sentence. (Cameron 2010, 249–50)

Thus, in addition to the above classic cases in which truthmaker theorists argue that various apparent commitments to properties can be avoided (or needn’t be treated as fundamental), Cameron adds the following for apparent commitments to certain objects: • Mereological sums: “The mereological sum of a, b, and c exists” is made true by a, b, and c (Cameron 2008, 5)—so we can avoid being fundamentality-committed to mereological sums. • Sets: “{Socrates} exists” is made true by Socrates (Cameron 2008, 10)—so we needn’t be fundamentality-committed to sets. • Symphonies: “Beethoven’s ninth symphony exists” is made true by the event of Beethoven indicating a certain abstract sound structure for performance (Cameron 2010, 261)—so we needn’t be fundamentality-committed to symphonies. What do these cases have in common that inclines us to accept the claim that we needn’t think that disjunctive properties, determinables, mereological sums, sets, or symphonies are fundamental, in order to accept that sentences about them are true? They all have in common the following feature: The original sentence ‘A’ (which has the relevant entities among its apparent commitments) is non-vacuously entailed by an ontologically alternative sentence ‘B’, which lacks those apparent commitments. It is, I submit, this relation that makes the examples plausible—for it seems that in cases where a sentence ‘A’ is non-vacuously entailed by an ontologically alternative sentence ‘B’, we are able to look for the truthmakers for ‘B’ to serve as the ultimate truthmakers for ‘A’ and thereby reduce our fundamentality commitments.¹⁰ So, for example, “a, b, and c ¹⁰ The restriction to cases of non-vacuous entailment is to rule out cases in which necessary truths, for example about the existence of numbers or properties, might be thought to be

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exist” is apparently committed (only) to a, b, and c; yet it entails “the mereological sum of a, b, and c exists”; “Beethoven indicated a certain abstract structure for performance” has as apparent ontological commitments only a man, a certain abstract structure, and an act of indication, but also seems to entail “Beethoven’s ninth symphony exists”. While truthmaker theorists do not explicitly propose such a criterion, and often resist thinking of truthmaking itself as entailment, they clearly need some criterion if they are to provide a way of narrowing down what must be counted as fundamental. What I am suggesting is that it is arguably the presence of these analytic entailments that implicitly makes their central and most plausible cases seem plausible. That is, arguably, it is these underlying analytic entailments that makes it seem plausible that we need not be fundamentality-committed to the objects that would be the apparent commitments of the original sentences (mereological sums, sets, symphonies), but only to the truthmakers for sentences asserting the existence other entities (individuals a, b, and c; Socrates; an indication event). At least in these most central and plausible cases (as in the cases describing disjunctive and determinable properties) the relevant entailments seem to reflect linguistic rules.¹¹ As Cameron himself puts it: that “{Socrates} exists” is made true by Socrates is “just a consequence of how we use the term ‘Socrates singleton’” (2008, 10). So one might think that a relevant criterion which would make sense of the examples commonly used to motivate the view would be the following: Negative methodological principle (NMP): if a true sentence “A” (non-vacuously) analytically entails an ontologically alternative

vacuously entailed by anything whatsoever—though it seems odd to count ‘there is a cup of coffee on my desk’ as a truthmaker for ‘there is a number greater than 23,000’. For discussion of the problem, see Restall 1996 (333–4). I will put those issues to the side here, since the problems I identify remain even if we restrict our interest to cases of entailments to contingent truths. ¹¹ Note that this doesn’t commit one to thinking that truthmaking is entailment (a position Heil argues against at length (2003, ch. 7)). It is simply a view that the presence of certain entailments from “A” to “B” entitle one to deny that we need to accept any new apparent commitments of “B” as genuine commitments (since one can make do with truthmakers for “A” instead).

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14  .  sentence “B”, then we are not fundamentality-committed to any apparent commitments of “B” that are not already among the apparent commitments of “A”.¹²

For, presumably, we can limit our fundamentality commitments to those entities needed to make “A” true, and (given the analytic entailment) we are assured that these can also make “B” true—without the need to treat any new entities “B” apparently commits us to as fundamental. The principle makes good sense of the truthmaker theorist’s standard and most plausible examples. It is important to note, however, that the principle is also quite weak—far less strong than most truthmaker theorists would want. For truthmaker theorists are keen to emphasize that there may be truthmaking relations even where there are not relations of analytic entailment (see Heil 2003, 55–6, 66; Cameron 2010). Heil, for example, explicitly denies that an account of truthmaking “requires an analytical path from truth-bearer to truth-maker” (2003, 66). To make good on the overall project, truthmaker theorists would (also) need some other principle to cover cases in which there are no such analytic entailments, and yet they think we can still avoid fundamentality commitment to the apparent commitments of “B” by appealing to other truthmakers. Nonetheless, the NMP might be thought to at least provide a start— one principle that would, for certain paradigm and most plausible cases, give the truthmaker theorist license to claim that a theory has fewer fundamentality commitments than its apparent ontological commitments would suggest. I will argue, however, that even such a weak principle is not one that anyone can accept, while also accepting (as Cameron does) the easy approach to ontology. For accepting easy ontological arguments, as noted in Section 1 above, commits one to accepting that there are often analytic entailments from a sentence with one set of apparent ontological commitments to a sentence with a different set of apparent ontological

¹² Note that entailments from, say, “P & Q” to “Q” would not, on this principle, relieve us of any fundamentality commitments, since the apparent commitments of Q are already among the apparent commitments of “P&Q”.

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commitments. And such entailments seem all too readily available. Ironically, part of what eliminativist neo-Quinean metaphysicians have taught us is how to make grammatical shifts that enable us to still say what we need to, without using noun terms to refer to the entities we don’t want. So, for example, we might (using the NMP) avoid fundamentality commitment to tables by saying that ‘there is a table here’ is entailed by ‘there are particles arranged tablewise here’, so that we are not committed to tables: particles arranged tablewise may serve as truthmakers for the claim about tables. But by the same token, the NMP suggests that we are not fundamentality-committed to particles. For as Hawthorne and Cortens (1995) have suggested, we can instead adopt an ontologically alternative feature-placing language and say ‘it’s particling around here’, which non-vacuously entails ‘there are particles’. Should we then say (using the NMP) that what we are really fundamentality-committed to is features, or ways the world is, not to any objects at all? (I suspect that this may be the impulse behind certain nihilist views.) But one must be careful with objectualizing here: to move from it is particling around here to: there is a world-feature of particling (or: this feature exists) also involves an analytic entailment from one sentence (which has no apparent commitments to things) to an ontologically alternative one, which seems to commit us to certain kinds of ‘things’: features or ways. But if we take the above interpretation of the truthmaking strategy seriously, that should also lead us to deny that we are fundamentality-committed to such world-features. Putting this grammatical difficulty aside, should we at least (following this principle) accept that we are fundamentality-committed only to whatever it takes to make sentences in a feature-placing language true? This would also seem to be the wrong move. For feature-placing sentences may also be (non-vacuously) entailed by ontologically alternative sentences. If we start from an objectual-language claim, like ‘there is a particle’, it seems we are entitled to introduce the feature-placing language and conclude ‘it’s particling around here’. And so, if we follow the spirit of the negative methodological principle, it seems we have no good reason to think that our fundamentality commitments are to the truthmakers of sentences of the feature-placing language rather than to sentences of the particle language.

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16  .  Much the same story could be even more readily told for the predicate-side truthmakers. For (as the easy approach to ontology again makes clear) there seem to be entailments from ‘the ball is red’ to ‘the ball has a redness trope’, ‘the ball instantiates the universal of redness’, ‘the state of affairs of the ball’s being red obtains’, and so on; and it seems we can also get trivial entailments from any of these back to ‘the ball is red’. That is, any of these claims may be non-vacuously entailed by an ontological alternative—and so accepting the negative methodological principle would entitle us to deny that we have fundamentality commitments to any of these. In short, as noted in Section 1, one consequence of the easy approach to ontology is that language is ontologically flexible: it seems that there are quite typically entailments to a target sentence from an ontologically alternative one (whether one we already have in the language or one we can invent). Cameron held that the [fundamentality] commitments of a sentence are those entities that “must number amongst the ontology of the world if the world is to provide an adequate grounding of the truth of the sentence” (2010, 249–50). What I have been arguing in essence is that—if we accept easy ontological inferences at all—we should also accept that, whatever our sentence, there is typically no unique statement of what entities must be in the ontology of the world to account for its truth: we can generally get an entailment up to the truth of the target sentence from an ontologically alternative one, which would apparently require a different fundamental ontology to make it true. As a result, if we adopt the negative methodological principle as a way determining what we are (not) committed to treating as fundamental, then it seems we must conclude that there is nothing we are fundamentality-committed to.¹³

¹³ The Quinean approach faces a similar problem if addressed at the level of determining what our theories commit us to, rather than of what a particular regimentation of a theory commits us to. For Quine insists that we can avoid an ontological commitment as long as we can paraphrase a sentence that quantifies over the suspect entities in a way that shows that the ‘seeming reference’ was an “avoidable manner of speaking” (1953, 13). Given the ontological flexibility of language, it seems that we can quite typically avoid apparent ontological commitments by paraphrasing a given sentence with an ontologically alternative one that shows the original to have been an ‘avoidable manner of speaking’.

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Now truthmaker theorists are a varied lot, and some might have been inclined from the start to reject the negative methodological principle. For those who reject the negative methodological principle, but think that the appeal to truthmakers can give us a method for determining what’s fundamental, the challenge is to articulate some substitute principle that can tell us when we are licensed in denying that the apparent ontological commitments of a sentence are fundamentality commitments. Some other prominent suggestions about how to understand truthmaking, however, are likewise unpromising as ways of telling us what a theory is committed to saying is fundamental: 1) Truthmaking as supervenience: there could be no difference in what things are true without a difference in what things exist (Bigelow 1988, 132). But, of course, the supervenience claim will remain true, in whichever of the ontologically alternative languages we describe what exists, so this does not tell us how to determine what is fundamental. 2) Truthmaking as necessitation, so that x is a truthmaker for T iff T is true and, necessarily, if x exists, then T is true (Armstrong 1997, 15). But again if we accept the trivial entailments endorsed by easy ontology, then we must accept the ontological flexibility of language. And that means acknowledging that there will often be many options for ways of describing our (fundamental) ontological commitments (to universals versus tropes, to enduring versus perduring objects), each of which is such that, necessarily, if x exists, then T is true. (Necessarily, if Fido’s whiteness trope exists, then ‘Fido is white’ is true. Necessarily, if the state of affairs of Fido exemplifying the universal of whiteness exists, then ‘Fido is white’ is true . . . ). The idea of truthmaking as necessitation does not enable us to narrow down which is the real truthmaker, or real ‘fundamental ontological commitment’, of T. 3) Truthmaking as grounding, such that x is a truthmaker for T iff T is true and, necessarily, T’s truth is grounded in x (or T is true in virtue of the existence of x). Here again, given the ontological flexibility of language, we will often have many candidates for the grounds of a single sentence T’s truth. (For “Fido is white”

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18  .  we might have as candidates: Fido’s having a whiteness trope, the state of affairs of Fido’s being white existing, Fido’s instantiating the universal of whiteness . . . and no obvious way to narrow them down, enabling us to say what the real fundamentality commitments are.) In short, whichever approach to truthmaking one takes, it is hard to see how to parlay an appeal to truthmakers into a criterion to tell us what we are committed to saying is (or is not) fundamental, given some theory or set of statements we accept. For—given the ontological flexibility of language that follows from accepting easy ontology—any statement we could give of what the fundamental truthmakers are would have a range of rival ontologically alternative statements. And so, if we say that we can deny an entity (or kind of entity) is fundamental when it isn’t needed as part of the supervenience base or to account for the necessitation of the truth, then we would again have to deny that there is anything we are committed to accepting as fundamental. Yet if we fail to provide a principle for identifying the fundamental truthmakers of our sentences or theories, it seems we have no way of using the truthmaker approach to provide a principle for getting a uniquely true statement of what we are (and aren’t) committed to treating as fundamental. Unless another principle is suggested,¹⁴ the appeal to truthmakers fails to provide a method for determining what our fundamentality commitments are. I actually think that, if one accepts easy ontological arguments, it’s right to deny that there is a uniquely true statement of the fundamental ontology that must be presupposed for certain statements to be made true. This conclusion in some ways echoes Jonathan Schaffer’s arguments that the appeal to a necessitation criterion of truthmaking “does not in general provide unique truth-necessitaters” (2008, 14). But there are two important differences. First, Schaffer’s argument against uniqueness was based in the problem that (while one clearly has to appeal ¹⁴ Armstrong avails himself of an entirely different approach, suggesting the Eleatic proposal as a way of trying to find the fundamental entities—they must be those that make a contribution to the causal order of the world. I have argued against this elsewhere (see Thomasson 2008; 2015, ch. 2).

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to minimal truthmakers to make the approach work) one can’t find a minimal truthmaker for certain truths (such as ‘there are denumerably many electrons’). The problem raised here is more general: that for any truth we will be able to give a statement of the truthmakers in ontologically alternative terms, so that we can get no good claim via the truthmaker route to have a uniquely true statement of the fundamental ontology required. Second, while Schaffer argued that the truthmaker approach cannot provide a viable approach to ontological commitment, he allowed that appeal to truthmaker commitments can be “re-targeted to the task of fundamentality commitments” (2008, 19). What I am arguing is that the problem of failing to delimit a unique statement of what the commitments are applies whether we aim to state the ontological commitments or the fundamentality commitments of a theory. More precisely, it applies if the goal is seen as giving a uniquely true statement of what a theory is committed to as being fundamental.¹⁵

3. What fundamentality projects might remain? There has been widespread and growing sympathy for the idea that fundamentality questions are the (or at least a) proper domain for metaphysics, and it is important to note that, despite the arguments of Section 2, certain versions of fundamentality questions may be pursued in complete consistency with the easy approach to ontology. Fundamentality questions that can be answered empirically, for example, are as much on the table as ever.¹⁶ As long as it is empirical, easy ontology raises no doubts about the work of physics, say, in telling us that atoms are more fundamental than molecules, that particles such as protons, and neutrons and electrons are more fundamental than atoms, and that quarks are more fundamental than protons and neutrons. ¹⁵ This is not a goal of Schaffer’s (see Section 3), who aptly treats the fundamentality commitments as merely providing constraints, and so this line of criticism does not apply to his work. (More on this in Section 3.) ¹⁶ Schaffer again provides an interesting contrast case to Cameron. For he makes use of both arguments from mereology and empirical reason, from quantum mechanics, to hold that the “physical story is best told in terms of fields pervading the whole cosmos, rather than in terms of local particles” (2010b, 51).

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20  .  The easy ontologist may also be open to addressing certain questions of relative fundamentality in conceptual terms.¹⁷ Perhaps one could use a restricted version of our negative methodological principle to offer at least some assessments of relative fundamentality: If “B” can be trivially inferred from an ontological alternative “A”, and the reverse is not the case, then “A” is a more fundamental sentence, and any apparent commitments of “B” that are not among the apparent commitments of “A” would not be fundamental truthmakers. That would entitle us to deny that disjunctive properties and determinables are fundamental. It might thereby enable us to get local orderings of some more and less fundamental sentences and corresponding entities (or perhaps, if you prefer, of dependence or grounding relations among entities). This might indeed be a worthwhile and manageable project, which can be carried on within a language, by assessing certain one-way conceptual relations. In this way, the appeal to truthmakers also might give us license to embrace a more parsimonious view of what’s fundamental—allowing that some things are things we should say exist, but deny are fundamental. These projects, of course, are not the same as Cameron’s project. For he insists that the metaphysician’s fundamentality question “cannot be answered by the combination of conceptual analysis and empirical observation: it is a substantive metaphysical question” (forthcoming, 10 in draft). And he seeks not merely local claims of relative fundamentality, but a uniquely true statement of what the fundamental entities are: “We are [fundamentality] committed to what must exist to make true the claims we make about the world, and there is a unique answer to the question of what those truthmakers are” (forthcoming, 16 in draft). In his most recent work,¹⁸ Cameron seems to accept that there is no criterion we can use to determine what is needed to make a claim true, and in response rejects the need for the truthmaker theorist to give “some general principle about what is or is not needed to make some claim true”. Instead, he suggests that we can discover what the unique fundamental truthmakers are only by “engaging in the highly speculative

¹⁷ Although we need not be committed to the idea that all such questions of the relative fundamentality (of Ps versus Qs) are answerable. ¹⁸ Which was responding to an earlier version of this paper.

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and highly fallible project of metaphysics . . . We have to get our hands dirty and engage with all the arguments metaphysicians give concerning the benefits of an ontology of tropes versus an ontology of states of affairs, or of the costs and benefits ontological nihilism” (forthcoming, 16 in draft). Whether such a project can be done, and, if so, how one could do it, would have to be topics for another occasion. It is worth noting, however, that adopting the easy approach to ontology should give us reason to hesitate before embracing this project. For if there are often mutual (rather than one-way) trivial entailments across sentences—all true— that have different apparent ontological commitments, what grounds can there be for thinking that one but not the other (of statements made in various ontologically alternative languages) gives us the uniquely true statement of what the fundamental ontology is? Taking on this project leaves us with a massive (and familiar) epistemological puzzle: how are we supposed to tell which of these is the real grounds of the truth? Avoiding such epistemological mysteries for metaphysics was, of course, a central goal of the project of easy ontology. One response to the epistemological mystery is to suggest that, while there are different ways of saying what the fundamental truthmakers are, we should simply weigh up the competing options by comparing theoretic virtues, and thereby arrive at the true fundamental ontology.¹⁹ There are, however, familiar grounds (which I will not rehash here) for doubting that we can appeal to theoretic virtues to answer serious metaphysical questions.²⁰ At this stage in the dialectic, we might expect some response to the criticisms of the epistemological mysteries surrounding such serious approaches to metaphysics. Another reason for doubting the viability of Cameron’s ambitious project comes from truthmaker theory itself. For many truthmaker theorists were originally motivated by criticisms of the so-called ‘picture theory’ of the relation between language and the world, according to

¹⁹ Cameron (forthcoming, 16 in draft) comes close to suggesting this, recommending that we simply engage in arguments assessing costs and benefits. Laurie Paul (2012) explicitly defends appealing to theoretic virtues as a method for metaphysics. ²⁰ For an overview of some of these epistemological difficulties, see my (2017b, 369–70).

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22  .  which “the character of reality can be ‘read off ’ our linguistic representations of reality—or our suitably regimented linguistic representations of reality” (Heil 2003, 6). A truthmaker theorist who aims to provide a uniquely true description of what’s fundamental denies that we can read features of reality off just any ways we have of representing reality. Nonetheless, in thinking that there is (among all the ontologically alternative ways in which fundamental reality might be described) one that is uniquely true in ‘corresponding to the ontological structure of the world’, Cameron still presupposes that there is some privileged representation of reality where the terms of the language ‘line up with’ the true fundamental ontology—and so seems committed to a version of the picture theory that many truthmaker theorists reject. Given the availability of ontologically alternative languages, however, it seems likely that any statement one could give of what the fundamental entities are could be rivalled by a statement in an ontologically alternative language. The idea that one but not the others of these ontologically alternative languages gives us the ‘true’ story about what the fundamental entities are seems to itself rely on a picture theory that would take the world to have to have a fundamental ontological structure that mirrors some but not other modes of expression. The real lesson of the need to abandon the picture theory, I would say, is that we should give up looking for a ‘best language’ in which one could then claim to have a correlation between the terms of the language and the ‘true ontology’—or even the ‘true fundamental ontology’ of the world. And that means giving up looking for a uniquely true statement of what the true ontology of the world is. For in whatever terms we describe the fundamental ontology, these could (assuming we accept easy ontological inferences) be expressed instead in some ontologically alternative language, with different apparent ontological commitments. And we are left with the usual crisis in the epistemology of metaphysics in asking by what methods we could possibly hope to determine which of these is ‘the correct’ one. (I hope I will be forgiven if I find ‘we just have to get our hands dirty and do some really hard metaphysics’ to be a less than satisfying response to the epistemological crisis.) All of this adds up to trouble for the view that one can both accept the easy ontologist’s trivial arguments for the existence of many sorts of

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entity, and yet still think that truthmaker theory will enable us to engage in a new ontological project: determining what (given our best theory) we are committed to saying is fundamental. As I have argued, if we accept the idea that language is ontologically flexible, then the appeal to truthmakers cannot give us any method for determining what the fundamental truthmakers are. Moreover, even if one allows that truthmakers give us no method, embracing the project of finding a uniquely true statement of what the fundamental truthmakers are leaves us in epistemological mystery, and leads us back to a version of the picture theory that the appeal to truthmakers was designed to free us from. Not all truthmaker theorists embrace the goal of getting a uniquely true statement of what’s fundamental.²¹ Early on, Armstrong acknowledged that “in metaphysics we have a certain latitude in what we assign as truthmakers for particular truths . . . There can very reasonably be disagreement among metaphysicians . . . about what the truthmakers for certain truths are” (2004, p. 33). Jonathan Schaffer explicitly notes that often a theory will only have disjunctive fundamentality commitments— say, to arrangements of particles, or objects, or a wave function, or . . . which simply act as constraints without telling us the true fundamental ontology (2008, 18). The theory that simply states ‘You exist’, for example: is fundamentality-committed to the existence of fundamental entities sufficient to ground the truth of the proposition that you exist. This does not tell us whether these fundamental entities are an arrangement of particles, or perhaps an effective wave-function abstracted from the wave-function of the whole universe, or anything else. But it does impose constraints. Supposing that particles are fundamental, it tells us that the fundamental entities include particles arranged you-wise. (2008, 18)

Schaffer is admirably upfront in rejecting the goal of finding a uniquely true statement of the fundamental ontology, and doesn’t propose any

²¹ It seems that Heil and Dyke would both reject such projects.

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24  .  principle to aid in determining what the fundamental ontology is.²² All he gives is a way of conceiving of truthmaking as grounding, according to which: For all p, for all w, (if p is true at w, then p’s truth at w is grounded in the fundamental features of w). (2008, 10)

But (given the openness of the truthmaking constraint) this does not even purport to give us guidance on determining which of various ontologically alternative statements gives us a statement of what those fundamental truthmakers are.²³ Schaffer’s own proposed answer to what the fundamental truthmakers are also remains admirably neutral among statements that could be made in ontologically alternative languages. His proposed answer is that the one and only truthmaker (in any given world) is the world (2010a, 307). But what is the world? At one point he speaks of it as a ‘big concrete object’, but quickly adds that this: will play no role in the discussion. It is merely grammatically impossible to speak of the truthmakers in a category-neutral way. The fan of facts may substitute a fact, namely, the way the world is . . . The fan of tropes may substitute the global tropes, namely, the ways that are the world. (2010a, 309)

If we do aim to determine what is fundamental by appealing to what the basic truthmaker(s) are, then it may indeed be that the most that

²² Heil also does not suggest that we can appeal to truthmakers to determine what the fundamental ontology is, saying for example “It is an open question what the ultimate truthmakers are for true descriptions of the world” (2003, 189). Instead, he appeals to truthmakers largely in opposing the ‘picture theory’, and insisting that statements about ordinary objects, mental states, and the like may be made true by the world, without requiring that we ‘posit’ ‘higher level’ entities as truthmakers (2003, 55). ²³ Rettler (2015, §5) similarly accepts that “we are not able to distinguish between whether tables or simples-arranged-tablewise are the truthmakers for sentences about tables”. His ‘general truthmaker view’ just maintains that we are committed to there being something that makes the statements we accept true, and so also seems to retain the needed ontological neutrality (2015, 16). As Rettler puts it, on his General Truthmaker View, ontological commitments “can’t be read off of sentences at all” (2015, 19).

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can be said is that the world—in this ontologically neutral sense—is what is fundamental.²⁴ For rival ontological descriptions of the world, in ontologically alternative languages, will always remain available, undermining the claim of any such description to provide the uniquely true (ontological) account of what the fundamental entities are. This does capture something of what truthmaker theorists wanted: the idea that, as Heather Dyke puts it, “what reality is like dictates which of our utterances are true. Truths are true in virtue of the way reality is” (2007, 80). And it does seem to me clearly a better alternative—and one that does not invite epistemic mysteries—to simply accept that the world (in an ontologically neutral sense) (Schaffer 2008) or being ‘something’ (in an ontologically neutral sense) (Rettler 2015) may make all of the relevant ontologically alternative statements true—there being no pressure to choose among them, or pretense that the world somehow makes this choice for us. Nonetheless, as I will turn to argue next, even accepting the generic idea that the world or ‘something’ makes all of our statements true may be accepting too much.

4. Truthmakers as a constraint As the discussion in Section 3 suggests, the truthmaker approach may be taken as having a more modest goal: simply imposing a constraint or requirement on any acceptable ontological view, that one must at least give some account of what the fundamental truthmakers are for the claims one accepts. One might, of course, accept this job for truthmakers without embracing the idea that there is a uniquely true statement of what the fundamental truthmakers are. Indeed this is, I think, the more standard view among truthmaker theorists.

²⁴ And this should be taken not as an endorsement of a ‘blobby’ ontology, of a world without ontological distinctions, but rather as a refusal to privilege any particular statement of the ontology in more ontologically specific terms.

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26  .  Truthmakers have often been invoked as a constraint in metaphysics. Both Armstrong (2004) and Heil (2003) trace the idea of truthmakers to C. B. Martin, who introduced the truthmaker principle that “when a statement concerning the world is true, there must be something about the world that makes it true” (Heil 2003, 61). As the idea of truthmakers was popularized by Armstrong, the primary role of the appeal to truthmakers was to serve as a constraint on metaphysics. Armstrong introduces his book on truthmakers by criticizing Ryle for failing to say what the truthmakers are for dispositional truths, and writing “the truthmaker insight . . . prevents the metaphysician from letting dispositions ‘hang on air’ as they do in Ryle’s philosophy of mind. That is the ultimate sin in metaphysics, or at any rate, in a realist metaphysics” (2004, 3). Thus, the appeal to truthmakers is often thought to play a central role in ‘catching cheaters’ and ruling out ‘dubious ontologies’. Theodore Sider writes that the point of the truthmaker principle (that “for every truth T, there exists an entity—a ‘truth-maker’—whose existence suffices for the truth of T”) is to “rule out dubious ontologies that posit ‘ungrounded’ truths’” (2001, 36), and to catch ‘cheaters’ who are “unwilling to accept an ontology robust enough to bear the weight of the truths [they feel] free to invoke” (2001, 41). And so, since its inception, a popular use of truthmaker theory has been the idea that the appeal to truthmakers can enable us to rule out certain ontological views: that if truthmakers are not provided, the view is illegitimate. This idea is enshrined in the doctrine known as ‘the truthmaker principle’: the view that every true proposition must be made true by something (Beebee and Dodd 2005, 1). Armstrong proposes the same idea under the name ‘Truthmaker Maximalism’—the idea that “every truth has a truthmaker” (2004, 5). Yet he admits that he does not “have any direct argument” for Truthmaker Maximalism, adding simply, “My hope is that philosophers of realist inclinations will be immediately attracted to the idea that a truth, any truth, should depend for its truth on something ‘outside’ it, in virtue of which it is true” (2004, 7). This hope has been borne out—many have been immediately attracted by it. But the question is whether that attraction comes from considering a too narrow diet of examples, and from insufficient

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appreciation of the strengths of alternative views, particularly nonRepresentational views of certain areas of discourse.²⁵ The Rylean example immediately shows the problem with the idea that appeals to truthmakers provides a suitable way of ‘catching cheaters’ and ruling out certain views. For Ryle (1949) is not properly criticized as failing to answer the question of what the truthmakers are for dispositional truths (about the mind or anything). The very point of Ryle’s discussion in The Concept of Mind (which Armstrong seemed to miss) was to argue against the assumption that all forms of discourse served the same functional role—of aiming to describe portions of reality (such that we can then aptly ask “what way is the world in virtue of which these truths are true?”). Instead, as Ryle (1949) argues at length, dispositional talk does not even aim to describe covert dispositional properties, categorical properties, laws of nature, or powers. Instead, it serves a very different function: licensing inferences about what might happen in a variety of circumstances. While there is not space to fully make this case here, the use of truthmakers even in the limited role of ‘catching cheaters’, by requiring metaphysicians to state the truthmakers for their claims, completely ignores the possibility of a functional pluralism in language: the idea that areas of discourse may have different functions, and that not all forms of discourse (not even all indicative utterances) serve the purpose of describing or tracking some portion of reality that must be a certain way to make the statement true. Truthmaker theorists assume that all true simple indicative sentences have truthmakers (Mulligan, Simons, and Smith 1984, 313–14)—but this is to assume a kind of functional monism. It is to assume something like what Huw Price calls a ‘Representationalist’ theory of language that assumes that all our “statements ‘stand for’, or ‘represent’ aspects of the world” (Price 2011, 5). This is an assumption Ryle fought against, Wittgenstein warned of, and that, more recently, such authors as Huw Price (2011, 14–33), Robert Brandom (1994), and Michael Williams (2011) have inveighed against in general terms.

²⁵ Blackburn (1993, 4) uses the term ‘non-descriptive’ rather than ‘non-Representational’ for such views.

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28  .  If we take on board the assumption that all true indicative statements require truthmakers, taken as “some portion of reality in virtue of which that truth is true” (Armstrong 2004, 5), then we presuppose that all such indicative statements have the function of tracking certain features of the world²⁶—such that the relevant statements are defective if there fails to be some feature of the world that can make them true. But there are welldeveloped, viable philosophical approaches to many areas of discourse that deny this (Price 2011). As an early example, one can find Ramsey’s (1931) view that universal generalizations (what he called ‘variable hypotheticals’) “are not judgments but rules for judging. ‘If I meet a phi, I shall regard it as a psi’”, where his reason for this is in part because asking ‘what would make [For all x, Px] true’ would “force us to make it a conjunction, and to have a theory of conjunctions which we cannot express for lack of symbolic power” (1929, 134). In more recent work, consider the expressivist views of moral discourse developed by Simon Blackburn (1993) and Allan Gibbard (2003); the treatments of modal discourse by Wilfred Sellars (1958), Robert Brandom (1994), and myself (2020); and many other cases. Such non-Representational approaches treat an area of discourse as fundamentally serving some other function—say, to express and coordinate attitudes, to endorse prescriptions, to make explicit the rules of use for our terms, or to license inferences. Fully developed,²⁷ alternative functional stories may also explain why we come to make the relevant statements in the indicative mood, and how they may be true, without that requiring that they be made true by some ‘portion of reality’ (Price 2011, 8–9). Here again, there is an important connection to easy ontology. For example, once we accept that there are easy inferences from any proposition to “It is a fact ”, we can be assured that we are entitled to infer, say, from ‘torturing kittens is morally wrong’ that ‘it is a fact that torturing kittens is morally wrong’—and so to acquire a commitment to what I have elsewhere called a form of ‘simple realism’ about moral facts (2015, ²⁶ That is, roughly, that they serve as e-representations in something like Price’s sense (2011, 20) of representations that have as their job to “co-vary with something else—typically some external factor, or environmental condition” (2011, 20). ²⁷ For example, in Simon Blackburn’s (1993) ‘quasi-realist’ program.

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145–58). This adds to the plausibility of alternative functional stories, by showing that they are perfectly consistent with beliefs that there are the relevant facts, properties, etc. There are many forms of speech from which we can get trivial inferences to the existence of the relevant facts and objects rather than requiring that we ‘posit truthmakers’ to explain what makes the relevant claims true. Once we (with the easy ontologist) see our talk about the relevant facts as derived by hypostatizations from other forms of speech, however, we can see that these facts are not suited to play the role of truthmakers. For truthmakers were supposed to play an explanatory role, to be that in virtue of which the truth is true. But if talk of such facts just arises via hypostatizations out of the relevant truths, then the facts posited can’t explain the truths except via a blatant dormitive virtue explanation.²⁸ Otherwise put, if saying ‘it is a fact that torturing kittens is morally wrong’ is just a hypostatization out of ‘torturing kittens is morally wrong’, licensed by the rules for introducing fact talk, then the former can’t explain what makes the latter true, any more than ‘poppies have the dormitive virtue’ (as a simple hypostatization out of ‘poppies make us sleepy’) can explain why poppies make us sleepy. So while we may start with an alternative functional story, and come to be entitled to say that such indicative statements in some deflated sense ‘describe facts’ about what is moral or what is possible, this is not a matter of treating them as having the relevant facts as truthmakers in the original sense given to the idea of ‘truthmakers’ as “that in the world in virtue of which sentences or propositions are true” (Mulligan, Simons, and Smith 1984, 289).²⁹ Accepting the easy approach to ontology thus both makes it clear how a non-Representational approach may lead to a form of (simple) realism and why it would nonetheless be a mistake to treat the relevant

²⁸ For further discussion of this point, see my (2015, 156–7). ²⁹ I have no objection if someone wants to introduce some deflated sense of ‘truthmaking’ according to which it only requires, for every truth P, that there be a fact that P (without requiring that it is in virtue of the existence of P that the statement is true). Nor do I object to asking questions of the form “What makes it true that P?” in the sense of what Price calls a “firstorder request for an explanation” of why P—for example, asking for a scientific explanation of why snow is white (rather than for a ‘metaphysical explanation’ of what makes “Snow is white” true) (Price 2011, 14).

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30  .  facts as truthmakers, or to demand that some proper truthmakers be presented in order to avoid ‘cheating’. I do not aim here to argue for any (local or global) nonRepresentational view. All I aim to note is that truthmaker theorists are not entitled to presuppose that all such views are false, untenable, or involve ‘cheating’. Accepting a ‘truthmaker principle’ as a way of ruling out certain views does just that, and blinds us to the possibilities that different areas of discourse may serve different functions, in ways that make it inappropriate to demand some account of what the ‘portion of reality’ is, in virtue of which they are made true. As Price, following a range of expressivists, urges, “issues that seem at first sight to call for a metaphysical treatment may be best addressed in another key altogether” (2011, 26). That is, before demanding truthmakers for all truths, we had better step back and settle whether we have an adequate understanding of how the discourse in question functions, whether it aims to track worldly features, and, if not, what functions it does serve in our lives and in our language, and how the discourse works. All of this requires far more extensive discussion, of course. The lesson here is simply that one is certainly not entitled to unilaterally invoke a ‘truthmaker principle’ as a way to rule out certain sorts of view without engaging with the contrary view that such a principle is based on a failure to acknowledge the possibility of functional pluralism in language, and without engaging with contrary proposals about how different areas of discourse work, what functions they serve, and how they may entitle us to refer to the relevant facts and objects without those having any prospect of serving as truthmakers. This gives us reason for pause in accepting an appeal to truthmakers as a constraint on ontologies, even if it is stated ontologically neutrally in terms that say only that all true propositions must be made true by ‘the world’ or (with Rettler 2015) say merely that all true propositions must have ‘something’ that makes them true.

5. Conclusions The easy approach to ontology was formulated in opposition to the neoQuinean approach. While it renders existence questions easy to answer,

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there is a common thought that it leaves open other difficult questions for ontology—including the question “What makes what true?” As I have argued, this question may itself be interpreted either in a strong sense (asking for a uniquely true statement of what the fundamental truthmakers are) or in a weak sense (simply requiring that metaphysicians give some account of the truthmakers for any truths they accept). I have argued, however, that those who accept the easy approach to ontology have reason to hesitate to pursue either of these questions. While truthmaker theorists in some ways appear as allies of easy ontology, the easy approach to ontology, more fully understood, also casts doubt on their ‘further projects’ for metaphysics. What else should we do? As Section 4 suggests, one thing we should do is to begin our inquiries a step back from metaphysics—not just jumping into the question of what the truthmakers are for a given area of discourse, but first asking questions about the functions the discourse fulfills, and the rules it follows.³⁰ As the easy approach to ontology shows, following those rules sometimes can entitle us to refer to the relevant objects and facts without ‘positing’ them as ‘truthmakers’. Another thing we can do is to reconceive of the work of metaphysics, thinking of it neither as in the business of discovering what exists or nor as aiming to discover what makes what true.³¹ Instead, I have argued elsewhere (2018, 2017a, 2017b) that we should reconceive the work of metaphysics as fundamentally conceptual work. Some metaphysical work may be better seen as involving descriptive conceptual work in investigating the workings and interrelations among our concepts. That may include work that could reveal relative fundamentality relations as well as modal truths, and work in determining whether and, if so, how parts of our everyday conceptual scheme may be reconciled with the conceptual schemes and empirical results from the natural sciences. In other cases, I think metaphysical work is better reconceived as involving normative conceptual work: not analyzing the concepts we do have, but ³⁰ At least some truthmaker theorists might be open to this move. John Heil, for example, at a couple of places speaks favorably of the idea that “the concepts we use have evolved to satisfy a variety of purposes” (2003, 43; cf. 49). ³¹ Though the empirical sciences may certainly address non-mysterious versions of existence and fundamentality questions.

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32  .  pressing for what concepts we should have, to serve a variety of purposes in the social and natural contexts in which we find ourselves. But all of that is the story for another day.³²

Works cited Alston, William P. (1958). “Ontological Commitments.” Philosophical Studies 9/1–2: 8–17. Armstrong, David M. (1997). A World of States of Affairs. Cambridge: Cambridge University Press. Armstrong, David M. (2004). Truth and Truthmakers. Cambridge: Cambridge University Press. Beebee, Helen and Julian Dodd, eds. (2005). Truthmakers. Oxford: Oxford University Press. Bigelow, John (1988). The Reality of Numbers: A Physicalist’s Philosophy of Mathematics. Oxford: Clarendon Press. Blackburn, Simon (1993). Essays in Quasi-Realism. New York: Oxford University Press. Brandom, Robert B. (1994). Making It Explicit. Cambridge, MA: Harvard University Press. Cameron, Ross (2008). “Truthmakers and Ontological Commitment: Or How to Deal with Complex Objects and Mathematical Ontology without Getting into Trouble.” Philosophical Studies 140: 1–18. Cameron, Ross (2010). “How to Have a Radically Minimal Ontology.” Philosophical Studies 151: 249–64. Cameron, Ross (forthcoming). “Truthmaker and Metametaphysics,” in Ricki Bliss and James Miller, eds., Routledge Handbook of Metametaphysics. London: Routledge. Dyke, Heather (2007). Metaphysics and the Representational Fallacy. New York: Routledge. Gibbard, Allan (2003). Thinking How to Live. Cambridge, MA: Harvard University Press.

³² See my (2016, 2017a, 2017b, 2018) for development of this idea.

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Hale, Bob and Crispin Wright (2001). The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon. Hawley, Katherine (2019). “Comments on Ontology Made Easy by Amie Thomasson.” Philosophy and Phenomenological Research 9/1: 229–35. Hawthorne, John and Andrew Cortens (1995). “Towards Ontological Nihilism.” Philosophical Studies 79: 143–65. Heil, John (2003). From an Ontological Point of View. Oxford: Oxford University Press. Mulligan, Kevin, Peter Simons, and Barry Smith (1984). “Truth-Makers.” Philosophy and Phenomenological Research 44/3: 287–321. Paul, L. A. (2012). “Metaphysics as Modeling: The Handmaiden’s Tale.” Philosophical Studies 160: 1–29. Price, Huw (2011). Naturalism without Mirrors. Oxford: Oxford University Press. Quine, W. V. O. (1953). “On What There Is,” in From a Logical Point of View. Cambridge, MA: Harvard University Press: 1–19. Ramsey, F. P. (1931). “General Propositions and Causality,” in R. B. Braithwaite (ed) The Foundations of Mathematics and Other Logical Essays with a preface by G. E. Moore. London: Kegan Paul, Trench, Trubner, & Co.: 237–55. Rettler, Bradley (2015). “The General Truthmaker View of Ontological Commitment.” Philosophical Studies 173: 1405–25. doi: 10.1007/s11098015-0526-x. Schaffer, Jonathan (2008). “Truthmaker Commitments.” Philosophical Studies 141: 7–19. Schaffer, Jonathan (2009). “On What Grounds What,” in David Chalmers, David Manley, and Ryan Wasserman, eds. Metametaphysics. Oxford: Oxford University Press: 347–83. Schaffer, Jonathan (2010a). “The Least Discerning and Most Promiscuous Truthmaker.” Philosophical Quarterly 60/239: 307–24. Schaffer, Jonathan (2010b). “Monism: The Priority of the Whole.” The Philosophical Review 119/1: 31–76. Schiffer, Stephen (2003). The Things We Mean. Oxford: Oxford University Press. Sellars, Wilfrid (1958). “Counterfactuals, Dispositions and the Causal Modalities.” In Minnesota Studies in Philosophy of Science, Volume 2:

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34  .  Concepts, Theories and the Mind-Body Problem, edited by Herbert Feigl, Michael Scriven, and Grover Maxwell, 225–308. Minneapolis: University of Minnesota Press. Sider, Theodore (2001). Four-Dimensionalism. Oxford: Oxford University Press. Sider, Theodore (2011). Writing the Book of the World. Oxford: Oxford University Press. Thomasson, Amie L. (2015). Ontology Made Easy. New York: Oxford University Press. Thomasson, Amie L. (2016) “Metaphysical Disputes and Metalinguistic Negotiation,” in Analytic Philosophy 57/4: 1–28. Thomasson, Amie L. (2017a). “What Can We Do, When We Do metaphysics?” in Giuseppina d’Oro and Soren Overgaard, eds. The Cambridge Companion to Philosophical Methodology. Cambridge: Cambridge University Press: 101–21. Thomasson, Amie L. (2017b). “Metaphysics and Conceptual Negotiation.” Philosophical Issues 27: 364–82. doi: 10.1111/phis.12106. Thomasson, Amie L. (2018). “Changing Metaphysics: What Difference Does It Make?” Philosophy, Supplement 82: 139–63. Thomasson, Amie L. (2019). “Replies to Comments on Ontology Made Easy.” Philosophy and Phenomenological Research 99/1: 251–64. Thomasson, Amie L. (2020). Norms and Necessity. New York: Oxford University Press. Williams, Michael (2011). “Pragmatism, Minimalism, Expressivism.” International Journal of Philosophical Studies 18/3:317–30.

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2 Easy Ontology, Two-Dimensionalism, and Truthmaking Ross P. Cameron

1. Easy ontology and two-dimensionalism You’re organizing an important event. You want to make sure there are enough chairs for everyone. So you ask your assistant how many chairs are in the room. Hope that they are not a metaphysician! Otherwise, you might receive the answer ‘None’, even though everyone would be able to find somewhere to sit, for this metaphysician doesn’t believe in chairs. It’s not that they think we’re suffering from an illusion, like a desert mirage. They agree that people aren’t going to fall to the floor when they attempt to sit themselves on what looks like a chair. They just hold that there aren’t actually any chairs. What’s keeping people from falling to the floor isn’t one thing, a chair, it’s the collective action of the many simple particles that are arranged in a chair-like fashion. Trenton Merricks, for example, would be one such metaphysician.¹ He thinks that for composition to occur, the resulting complex object would have to have some causal powers that cannot be reduced to the combined causal powers of the simple objects that compose it. Conscious objects have such causal powers, thinks Merricks, since the consciousness of a person cannot, he argues, be explained by the collective powers of the simple parts of a person. But chairs—were there such things—have no such causal powers, since their ability to hold people off the ground can be accounted for in

¹ Merricks (2001).

Ross P. Cameron, Easy Ontology, Two-Dimensionalism, and Truthmaking In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Ross P. Cameron. DOI: 10.1093/oso/9780192893314.003.0002

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36  .  terms of the collective action of the simple chair-parts. So Merricks thinks he has discovered via metaphysical inquiry that there are no chairs. (And yet he insists on drawing his salary as a chair of philosophy!²) Some philosophers are skeptical of metaphysicians’ claims to be able to discover that there are no chairs. Amie Thomasson, for example, thinks that it is easy to establish that there are chairs, and we don’t need any metaphysics to do it.³ She thinks that the question of whether there are chairs breaks down into two components. There is a conceptual component, whereby we discover the application conditions for the concept chair, and discover that the concept ought to be deployed to describe things that you can sit on as opposed to, say, large astronomical bodies orbiting stars. Then there is an empirical question as to whether those application conditions are met: in this case, that question is to be answered by looking in the room, as opposed to looking through a telescope pointed toward the night sky. And this is true in general of existential questions, thinks Thomasson. Whether there are Fs breaks down into the question of what the application conditions are for the concept of F-hood, and whether those application conditions are met (in a particular context). The first question is one of conceptual analysis, the second a straightforward empirical investigation into our surroundings. Neither leaves any work for the metaphysician. If Thomasson is correct, then existential questions are often easy to answer. Of course there are chairs: I am sitting on one as I write this. Of course there are planets: look through this telescope. Of course there are universities: my paycheck comes from one. Etc. I say ‘often’, for, of course, it is compatible with Thomasson’s metaontology that some existential questions remain hard to answer. Some concepts are harder to analyze than others. What are the application conditions for, say, ‘artwork’, or ‘disease’? And sometimes the empirical question of whether the application conditions are met is hard to answer. Are the application conditions for the concept intelligent alien life form met anywhere in the universe? But while ‘Is there an artwork?’ might be hard to answer when ² Fun fact: at the University of Virginia, you used to, on promotion to chair, receive an actual chair, with a plaque on it with your name and new title. As far as I know, nobody tested whether your continued status depended on the survival of the chair. ³ See especially Thomasson (2015).

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looking at Duchamp’s Fountain, and ‘Is there a disease?’ might be hard to answer when considering a 70-year-old with osteoarthritis, and ‘Are there intelligent alien life forms?’ might be hard to answer for anyone other than Fox Mulder, they’re not hard for metaphysical reasons. The first two questions are hard because those concepts, while each admitting clear cases and ruling out clear cases, allow for gray areas where the concept neither clearly applies nor clearly fails to apply; the third is hard because the universe can’t be easily empirically surveyed for life forms the way the room can be easily empirically surveyed for chairs. The metaphysician, qua metaphysician, can’t come to the rescue here. I have a lot of sympathy with Thomasson’s claim that many existential questions that metaphysicians appear to debate about are easily answered. Looking into a room, seeing lots of things you can sit on, and still wondering if there are chairs seems to require having a view of how language works that places too stringent a set of rules on sentences in order for them to be true. It seems to require something like what Agustin Rayo calls ‘Metaphysicalism’⁴ or what John Heil calls ‘The Picture Theory’⁵: the idea that, for a sentence to be true, there has to be the right kind of correspondence between the syntactic structure of the sentence and the metaphysical structure of the world. On this view, for ‘There is a red chair’, e.g., to be true, the ontological structure of the world had better contain certain individuals that are chairs, and the property of redness, and one of those individuals has to instantiate that property. Then it is open to the metaphysician to discover that, in fact, the ontological structure of the world contains only mereologically simple individuals, or mereologically simple individuals and conscious beings, etc., and hence that it doesn’t include chairs, and hence that this sentence is false after all. Like Rayo and Heil, I think that this is a bad view about how language works.⁶ Declarative sentences are used to convey information, and they convey the information they do as a result of our patterns of usage. The ⁴ Rayo (2013, esp. ch.1). ⁵ Heil (2003). ⁶ It’s also probably not a view anyone has explicitly held. However, I do think that something like it is an implicit commitment of many philosophers. Or more carefully, that something like this view of language would have to be true, if the things those philosophers take for granted are to be true.

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38  .  sentence ‘There is a chair’ is not, by most people, used to convey information about the metaphysical structure of reality⁷; it is used to convey information about whether or not we can sit down. ‘There are lots of chairs in the room’ is used to convey to people the information that the world looks like this (as I gesture around to a room in which many people can sit). It’s not that it’s impossible for us to be wrong about whether it’s true—our eyes could be deceiving us, e.g., and when we try to sit down, we’ll fall to the floor—but it’s hard for it to be false. If lots of people can go into the room and sit down without falling to the floor, then the sentence will be true, because that is the entirety of what that sentence aims to convey. Compare Eddington’s tables.⁸ On discovering that the thing in my kitchen upon which I place my dinner is mostly empty space, should I conclude that there are no tables? No, because ‘table’ is used by ordinary speakers to refer simply to things like this (gesturing at the thing on which I place my dinner), and whatever that turns out to be, that is what a table is. We can discover that tables are mostly empty space, that they are sentient, that they are communities of nanobots, that they are collections of ideas in minds, etc., but something really unusual has to be going on in the world for us to discover that they don’t exist. Let’s unpack that thought. Idealists like Berkeley think that there are no physical objects existing independently of perception. While we might have thought that the table is a physical being, capable of existing even in a world where nobody is around to see it, Berkeley denies this: to be is to be perceived, and everything is either a mind, or a collection of ideas in minds. As I see it, the materialist and Berkeley agree that ‘table’ picks out that thing in the kitchen, and they just disagree about the underlying metaphysics: they disagree on what things like that are ultimately like. Are they mereological sums of fundamental physical particles, each capable of existing in the absence of a perceiver, or are they a collection of ideas in minds, ontologically dependent on there being some mind to perceive them?

⁷ If you don’t understand all this ‘structure of reality’ talk, see Sider (2011). And then you might still not understand it, but you’ll have a better idea of what you’re not understanding. ⁸ Eddington (1928, xii).

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It is tempting to conclude that the materialist and Berkeley agree that there is a table; they just disagree on what a table is. But we must tread carefully here. David Lewis argues that a mere congeries of ideas does not deserve the name ‘table’, and thus that we should not take Berkeley at his word when he says that he believes in tables.⁹ If he is correct, then the materialist and Berkeley do not agree that there is a table, any more than the fisherman and the financial provider agree that they spend their day at the bank. I favor a two-dimensionalist treatment of this issue.¹⁰ Suppose the materialist is right about what the world is like, and in particular that when I sit down to dinner, I place my dinner on a physical object that is a sum of fundamental physical particles. Now consider a counterfactual world that is Berkeley’s world of minds and ideas in minds. There are no tables in that counterfactual scenario, since there are no physical objects at all, and tables are a kind of physical object. Considered counterfactually, Berkeley’s world is one bereft of tables. However, if we consider Berkeley’s world¹¹ as actual, I think we should say that tables are collections of ideas. After all, we are thereby considering a scenario in which we discover that the things that we use table talk to talk about— things like that (pointing to the thing on which I place my dinner)—are, in fact, merely collections of ideas. And whatever that turns out to be, that is surely what tables are, since ‘table’ was introduced into the language for the sole purpose of picking out things like that. Similarly, consider the sentence ‘There are chairs’. Suppose that, in fact, the world is as the mereological universalist sees it: there are a bunch of simple¹² fundamental physical particles, and for each such collection of simple particles there is something that is their mereological sum. Some simple particles are arranged in a chair-like fashion. The mereological sum of those simples is a chair. ‘There are chairs’ is true, then, ⁹ Lewis (1990). ¹⁰ For discussion of two-dimensionalism relevant to my treatment here, see Stalnaker (2002) and Chalmers (2005). ¹¹ I make no commitment to thinking that such a world is metaphysically possible. The twodimensionalist approach I favor requires at the most that the worlds in question be epistemically possible in the broadest sense. ¹² The existence of simples is not a commitment of universalism, of course, but let’s suppose our universalist believes in such things, just for ease of example.

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40  .  because there are such sums. Now consider the mereological nihilist world in which there are no sums; there are just the simples. Or the Merricks world, in which there are simples and conscious objects composed of simples, but no non-conscious complex objects, so no object composed of simples arranged chair-wise. We should not describe the nihilist world or the Merricks world as worlds in which it is true that there are chairs, since the things that are in fact chairs are absent in that world. However, as above, I think that if we consider those worlds as actual, we should say that it is true that there are chairs in those scenarios.¹³ For if our world is the nihilist’s world, then ‘There are chairs’ is used by people simply to describe scenarios in which some of the simples in the world are arranged in such a fashion that we can place ourselves in a seated position in relation to them and not fall to the ground. That would just be what it takes, in that world, for it to be true that there are chairs.¹⁴ And if we consider such a world as actual, then considering the universalist world as counterfactual, it will be true that there are chairs in those worlds as well, since there are, of course, still collections of simples arranged in that fashion in the universalist world. (There is also their mereological sum, but if we consider the nihilist world as actual, the existence of such things at a world is irrelevant to whether or not there are chairs at that world, considered counterfactually.) If this is right, then we can be very confident that there are chairs, because no matter whether this world is the universalist’s world, or the nihilist’s world, or something in between like Merricks’ world—or, indeed, if it’s something more radical like Berkeley’s world—it will turn out to be true that there are chairs, because whatever is going on metaphysically in those circumstances where it is appropriate to deploy

¹³ Cf. Cameron (2010a). ¹⁴ Of course, that makes it hard for the nihilist to truly state their view, since on the assumption that this is a nihilist world, it is simply false that everything is simple. But all that means is that the nihilist must have a way of speaking that forces a metaphysical interpretation of their words—a special language, or a special context in which language is used: a way of speaking whereby there is not a competing ordinary usage that trumps the metaphysical claim that these sentences are attempting to convey. That is all that metaphysicians mean when they speak about using language ‘in the ontology room’ (see the Introduction to van Inwagen (2014)), or about a special language of ‘Ontologese’ (Dorr 2005, Cameron 2008b, Sider 2011), or about what really exists (Cameron 2008a).

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 , -,  

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the concept chair, that is all that it takes for the world to make it true that there are chairs. This is why I agree with Thomasson that existential questions are often easy to answer, but it is also why I’m not ready to turn in my metaphysics club card just yet. For while it is easy to establish that there are chairs, what is not easy to establish is what makes that true. What worldly conditions is this obvious truth ultimately sensitive to? What is the underlying reality that, given our pattern of usage of chair talk, ‘There are chairs’ has latched onto as its truthmakers? That, I think, is a hard metaphysical question, not (wholly) answerable by a mixture of conceptual analysis and empirical investigation. This is the work for the metaphysician: not to say what there is, but to say what makes it true that there is what there is.¹⁵ I should note one place in which I depart from Thomasson’s easy ontology, however. Part of Thomasson’s case for thinking that ontological questions like ‘Are there chairs?’ are easily answered in the affirmative is that the existence of chairs follows trivially from something that is not in dispute by the metaphysicians who are party to the debate, namely that there are some things arranged chair-wise. Suppose we have a universalist like David Lewis,¹⁶ someone who thinks composition is restricted but that ordinary objects exist, like Dan Korman,¹⁷ and someone who thinks composition is restricted and something unusual has to happen for composition to occur, like Trenton Merricks¹⁸—they all agree that there are some fundamental physical particles arranged chair-wise. But while Lewis and Korman say that there are chairs, Merricks says that there are no chairs. Thomasson thinks this debate is a bad one because you can’t rationally accept that there are things arranged chair-wise and deny that there are chairs, because the existence of chairs follows trivially from the existence of things arranged chair-wise, just as it follows trivially from the fact that there is an even prime number that there are numbers.

¹⁵ My initial defense of this was in Cameron (2008a). For my most developed account of the truthmaker account of ontological commitment, see Cameron (2009). ¹⁶ Lewis (1986). ¹⁷ Korman (2016). ¹⁸ Merricks (2001).

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42  .  I think Thomasson might be right that there is a trivial inference from ‘There are things arranged chair-wise’ to ‘There are chairs’—but it depends on what the actual world is like. As a result of the twodimensionalist approach defended above, I hold that if our world is the nihilist’s world, or Merricks’ world, then it trivially follows, as Thomasson says, from the fact that there are things arranged chairwise that there are chairs. Because if that’s the way the world is, then ‘There are chairs’ is used simply to convey the information that the world is the way it is when there are things arranged chair-wise. Those two claims will be sensitive to the same worldly goings on, so if one is true, of necessity, so is the other. Considering Merricks’ world as actual (e.g.), we should hold that there is nothing more to it being the case that there are chairs than that there are some things arranged chair-wise, in which case Thomasson’s inference is indeed trivial, and so we should describe any world that contains things arranged chair-wise, considered counterfactually, as a world in which there are chairs. However, if our world is the universalist’s world, or Korman’s world, then things are more complicated. Considering the universalist’s world as actual (e.g.), there are in fact particular objects carved out of reality’s ontological structure that can serve as the reference of ‘chair’—namely, the mereological sum of the things arranged chair-wise. The simplest metasemantic story in such a world, then, yields the result that ‘chair’ refers to those individuals, and so ‘There are chairs’ will be sensitive to the existence of such mereological sums.¹⁹ In which case, when we take the universalist’s world as actual and consider Merricks’ world as counterfactual, we should not say that it is true that there are chairs in Merricks’ world, since it is lacking in those mereological sums. But there are, of course, things arranged chair-wise in Merricks’ world. And so, with the universalist world taken as actual, Merricks’ world is a counterexample to Thomasson’s inference. So it seems to me that the success of Thomasson’s ‘trivial inferences’ from agreed-upon existential data like ‘There are things arranged chairwise’ to contested existential claims like ‘There are chairs’ depends upon what the world is like in the first place. That inference is indeed trivial if we find ourselves in a nihilist world, or in Merricks’ world, because the ¹⁹ Cf. Williams (2010).

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 , -,  

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best metasemantic theory if that is our world is one in which ‘There are chairs’ turns out to require nothing more of the world than that there are things arranged chair-wise. But if we find ourselves in the universalist’s world, this inference is not necessarily truth-preserving, because the best metasemantic theory if that is our world is one in which ‘There are chairs’ turns out to require that the ontological structure of the world furnish us with certain complex objects. So whether or not Thomasson’s existential inferences are indeed trivial depends, I think, on what the actual world turns out to be like. So I claim that there are things we can discover about the world—that it is a universalist world, e.g.—that should lead us to hold that it is possible that there are things arranged chair-wise but no chairs. That is a big way in which I depart from Thomasson’s overall easy ontology project. However, this does not affect my agreement with one of her main claims. Whether or not those inferences are necessarily truthpreserving, it is still the case that it can be easily established that there are chairs (etc.). Because whether or not the existence of chairs follows trivially from the existence of things arranged chair-wise, we can still be sure, given that there are things arranged chair-wise, that there are chairs. That’s because either we’re in something like the nihilist’s world, where nothing more is required for there to be chairs than that there are things arranged chair-wise, and so the inference is indeed trivial, or we’re in something like the universalist’s world where something more is required precisely because that extra requirement is met— because the world presents us with things that are chairs. No matter which world we turn out to be in, then, it is true that there are chairs. And so it is easy to establish that there are chairs, and we don’t need to know that the inference from ‘There are things arranged chair-wise’ to ‘There are chairs’ is necessarily truth-preserving in order to easily establish this: we can know a priori that if that inference admits of counterexamples, they are not actual. Here’s another way to put the point. The status of the inference from ‘There are things arranged chair-wise’ to ‘There are chairs’, with the universalist world taken as actual, is tricky, because while there are counterexample worlds—worlds at which the premise is true and the conclusion false—it is nevertheless the case that we have a guarantee that our world is not a counter-example world. Even

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44  .  without knowing what the actual world is like, I know that I can rule out our world being one of the worlds in which there are things arranged chair-wise but no chairs, because if we’re in a world with only things arranged chair-wise, that’s a world in which that’s all it takes for there to be chairs, so it’s not a counterexample world, because there are no counterexample worlds; and if we’re in a world with chairs as well as things arranged chair-wise, then even though there are counterexample worlds, our world isn’t one of them, since there are chairs in our world. Either way then, this world isn’t a counterexample world. So while I’m not prepared to outright agree with Thomasson that the inference is trivial—since depending on what the world is like, it might not be necessarily truth-preserving—I am happy to agree with her that we’re rationally compelled to accept the conclusion, given acceptance of the premise,²⁰ and that’s all that is needed to render the existential question concerning whether there are chairs easy to answer in the manner Thomasson argues for.

2. Truthmaking and ontology So, existential facts are often easy to establish. It is easy to establish that there are chairs. But what is not easy, I think, is saying why there are chairs: what makes it the case. And so the metaphysical project of ontology, as I see it, is not to say what there is. It is (at least in part) to say what makes it true that there are universities, that there is only one even prime number, that there could have been more than ten planets orbiting our sun, that murder is wrong, etc. Given what is the case—including

²⁰ As Chalmers would put it, the premise doesn’t entail the conclusion, but it implies it (Chalmers 2005, note 7): it is not necessarily truth-preserving, but if you believe the premise, you ought to believe the conclusion. Cf. the argument from ‘There is a clear, odorless liquid that falls from the sky that quenches thirst’ to ‘There is water’. Acceptance of the premise rationally demands acceptance of the conclusion, because if you consider as actual a world in which a clear, odorless liquid that quenches thirst falls from the sky, it is also true at that world, considered as actual, that there is water, since ‘water’ just picks out whatever that substance is. However, no matter what world you consider as actual (as long as ‘water’ manages to have content at that world, at least), you will recognize that there are counterexamples to this inference: worlds that, considered as counterfactual, are worlds without water but in which a clear, odorless thirst-quenching liquid falls from the sky, i.e., worlds with ‘fool’s water’.

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 , -,  

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facts about what there is, many of which are very easy to establish—what must the world be like to make that the case? Is Berkeley right that a world of ideas and minds makes all this true? Is Lewis right that a world of point-sized (both spatially and temporally) objects and their mereological sums makes all this true?²¹ Is Descartes right that a bifurcated world of material bodies and immaterial minds makes all this true? Is David Armstrong right that a world of states of affairs of individuals instantiating properties and standing in relations makes all this true?²² Those difficult questions remain, even after all the existential questions have been solved, many of them easily. And they can only be answered— if they can be answered at all—by doing metaphysics. Thomasson argues against this idea that there is work remaining for the metaphysician after the existence facts have been easily settled.²³ The gist of her argument is this. Let’s say the metaphysician gives up on Quine’s project of taking a theory of the world, regimenting that theory in the language of first-order logic, determining what must be in the domain of quantification for that theory thus regimented to be true, and declaring: these are the ontological commitments of our theory—this is what we say exists, given that we say that this theory is true.²⁴ Nevertheless, I am urging that they do something different but similar: take a theory of the world and tell us: this is what makes it true that the world is this way—this is what we say is really out there, the fundamental ontology of the world—given that we say that this theory is true. And so, seemingly, I need a replacement for Quine’s dictum ‘To be is to be the value of a variable’. Thomasson asks: if there is this new work for the metaphysician to do, what is the criterion for when a thing is needed as a truthmaker for a claim or theory? Or, at the very least, what is a criterion for determining that a thing is not needed as a truthmaker for a claim or theory? After all, the truthmaker theorist happily asserts that various things are not needed as truthmakers that are ontological commitments by Quinean lights. While the Quinean sees the Eiffel Tower as an ²¹ See Weatherson (2015) for a discussion of Lewis’s doctrine of Humean Supervenience. ²² Armstrong (1996). ²³ Thomasson (Chapter 1 of this volume). All the quotes from Thomasson below are from this paper. ²⁴ Quine (1948).

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46  .  ontological commitment of ‘The Eiffel Tower is tall’, the truthmaker theorist may urge that all that is needed to make this claim true is the arrangement of simple particles in an Eiffel Tower-like pattern.²⁵ While the Quinean sees sets as an ontological commitment of ‘There is some way that A is’, the truthmaker theorist may urge that this is made true by a state of affairs involving A having some property, and not by anything involving sets.²⁶ While the Quinean sees numbers as an ontological commitment of ‘2+2=4’, the truthmaker theorist may urge that this is made true vacuously, and so requires nothing at all of ontology.²⁷ What criterion, then, is the truthmaker theorist relying on in making these judgments? How, asks Thomasson, do we determine when something is needed as a truthmaker, or at least how do we determine that it is not? The truthmaker theorist is explicit that we cannot simply look to the syntactic structure of the sentence and discover that we need an object for each of the singular terms and a property for each of the predicates. But what, then? Looking at what truthmaker theorists, including myself, have said in the past to justify the claim that they are not committed to certain things, Thomasson suggests the following principle on the truthmaker theorist’s behalf: Negative Methodological Principle (NMP): if a true sentence “A” (non-vacuously) analytically entails an ontologically alternative sentence “B”, then we are not fundamentality-committed to any apparent commitments of “B” that are not already among the apparent commitments of “A”.

The idea, on behalf of the truthmaker theorist, is that we can avoid commitment to complex objects, e.g., by showing that claims about them are entailed by claims that do not even on their surface talk about complex objects, such as claims about simple objects and the relations they stand in to one another. Or that we can avoid commitment to

²⁵ Cameron (2008a). ²⁶ Cameron (2009, forthcoming). ²⁷ Cameron (2010b). Cf. Rayo (2009), although Rayo would not wish to be saddled with the ideology of truthmaking.

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abstracta by showing that claims about abstract objects are entailed by claims that only talk about concreta, etc. However, Thomasson says that this is a recipe for never being committed to anything, for language is, she says, ontologically flexible: any sentence you have that includes terms that purport to pick out certain things, we can always find or invent a new sentence that entails it that includes no such terms. Worried about the existence of chairs? No problem, just talk about the fundamental physical particles arranged chair-wise. Worried about the existence of fundamental physical particles like electrons? No problem, just talk about how it is electroning chairly, where this is to be read much like ‘It is raining hard’: the ‘it’ is a dummy pronoun, not aiming to pick out some thing, electroning is a thing that happens, and one of the ways it can happen is in a chairly fashion. Feature-placing languages let us describe the world without ever purportedly talking about things, so if we can always replace an apparently ontologically committal sentence with a sentence in a feature-placing language that entails it, then, by NMP, we never need be committed to anything belonging to the fundamental ontology of the world that makes all that is true true.²⁸ But not only does that seem too easy; it seems unstable. For, as Thomasson says, why privilege the feature-placing sentence? Any sentence of a feature-placing language will, likewise, be entailed by sentences that do purport to talk about entities. Just as ‘It is electroning chairly’ entails ‘There is a chair’, so does ‘There is a chair composed of electrons’ entail ‘It is electroning chairly’. Why conclude that the former entailment shows us that we’re not committed to chairs, rather than that the latter entailment shows us that we are? I agree with Thomasson that NMP is no good. If we had some reason to think that, for any given sentence, we could trace back a chain of entailments to find some special sentence that wasn’t itself entailed by any other sentence with different apparent ontological commitments, then perhaps NMP could work, and we could take these sentences at the start of the chains to transparently reveal the genuine ontological ²⁸ See O’Leary-Hawthorne and Cortens (1995) and Turner (2011) for discussion of featureplacing languages and ontological nihilism. Turner argues that the existence of a feature-placing that does what Thomasson has in mind is by no means obvious, but let’s grant for the sake of argument that there could be such a language.

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48  .  commitments of all the claims we make about the world. But as Thomasson says, given the ontological flexibility of language, there is no reason to think that, and every reason to deny it. Thankfully, I don’t think the truthmaker theorist should have been attracted to NMP in any case. Put aside my worries above about whether or not ‘There are things arranged chair-wise’ entails ‘There are chairs’. Let’s grant for a moment that this entailment holds, for the sake of argument. That this entailment holds is not my reason for concluding that we are only committed to the things arranged chair-wise when we say that there are chairs. At most, what this entailment shows is that nothing more is required of the world for there to be chairs than that there are things arranged in a chair-wise fashion. But that leaves open what is required. Maybe the world needs to contain chairs for there to be simples arranged chair-wise. Noting the patterns of entailments won’t settle that. My reason for concluding in this case that there are no chairs, only the simple things, is that this is the simpler and more parsimonious account of the world,²⁹ and so, since it is not lacking in explanatory power over the non-nihilist alternative theories, we should conclude that it is true. To put it in the terms I prefer, without invoking the notion of entailment: if the two-dimensionalist reasons I gave for thinking that it is easy to establish that there are chairs are correct, then chairs would be a wholly redundant addition to the list of things that are really out there in the world. All the work that chairs do, on the assumption that a universalist world (e.g.) is actual, is done by simples arranged chair-wise, on the assumption that a nihilist world (e.g.) is actual. In that case, chairs look like metaphysical idlers. So don’t believe in them! In that sense, I’m very sympathetic to Merricks’s method when it comes to composition: only admit complex objects to your ontology if there is some work they do that wouldn’t be done by an ontology lacking them. I disagree with Merricks here only on what is true; he thinks that if the ontology of world doesn’t include chairs, then it is not true that there are chairs, and I disagree.

²⁹ This is explicit in Cameron (2008a).

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That there are chairs if there are things arranged chair-wise is, I agree with Thomasson, easily established as true. But it is not NMP that should lead us to conclude that our ontology is one lacking in chairs. It is the familiar weighing up of theoretical virtues and vices. In this case, it’s the prohibition against admitting ontological idlers to one’s ontology: things that do no work for you over what’s already there. Thomasson asks the truthmaker theorist: When are we, or when aren’t we at least, committed to certain entities that are apparent commitments of a claim? Well, I’ve already given the answer: You are committed to certain entities when those entities are needed to make true what you are claiming. You are not committed to certain entities when those entities are not needed to make true what you are claiming. In place of Quine’s ‘To be is to be the value of a variable’, I offer ‘To really be is to be a truthmaker for some truth’. In asking for something like NMP, Thomasson is asking for a principle that will let us look at a sentence and work out what is needed to make it true. Even if we can’t just look at the syntactic structure of a sentence and work out from that what we commit ourselves to in asserting it, she wants a simple rule, that will let us find another sentence to which we can apply a principle as simple as Quine’s and determine from that new sentence what we are committed to—or at least not committed to—in asserting the original. But I don’t think there is any such simple rule. There is nothing that does what NMP is trying to do. The whole point is that while existential questions are often easy to answer, properly ontological questions—questions about what makes what true—are often very hard. Encountered with a simple sentence like ‘Ball is red’, the Quinean has a simple method of determining its ontological commitments: it is Ball, for that is the only thing that needs to be in the domain of quantification in order for ‘Ball is red’ to be true. The truthmaker theorist makes life harder for themselves. Think of all the options metaphysicians have offered for what reality must ultimately be like in order for it to be true that Ball is red: a world of states of affairs, including that of Ball instantiating the property of redness³⁰; a world of tropes, one bundle of which is Ball, which includes Ball’s particular redness³¹; a world of bare ³⁰ Armstrong (1996).

³¹ Paul (2002).

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50  .  substrata, one of which is Ball, and properties, one of which is redness, with that property inhering in that substratum³²; a world of structureless Facts, one of which is like ‘Ball is round’ in one way and is like ‘Firetruck is red’ in another way³³; a world of minds, some of which share the idea of Ball’s being red; a world of nothing at all, in which it Balleth redly.³⁴ These candidate metaphysics offer wildly different accounts of what the world must be like to make it the case that Ball is red. So working out what we commit to, or at least don’t commit to, in saying that Ball is red involves working out which of these rival metaphysics is true, or at least ruling some of them out as false. That involves working out the costs and benefits of the different views, and weighing them against one another. And that is incredibly hard.

3. Hard ontology, epistemic humility, and the picture theory I say that while existential facts are often easily established, not only are ontological facts—what really exists in order to make true the facts concerning how things are—not easily established, they are very hard to answer. But the project of answering them, while difficult, is familiar: formulating rival metaphysical theories and comparing them at the tribunal of cost/benefit analysis. Thomasson is unhappy with this as a defense of there being a substantive ontological project that is distinctively metaphysical in nature. One objection she raises is that it threatens to render ontological questions epistemically intractable: [A]dopting the easy approach to ontology should give us reason to hesitate before embracing this project. For . . . what grounds can there be for thinking that one but not the other (of statements made in various ontologically alternative languages) gives us the uniquely true

³² See Sider (2006) for discussion of this historically popular view. ³⁴ O’Leary-Hawthorne and Cortens (1995), Turner (2011).

³³ Turner (2016).

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statement of what the fundamental ontology is? Taking on this project leaves us with a massive (and familiar) epistemological puzzle: how are we supposed to tell which of these is the real grounds of the truth? Avoiding such epistemological mysteries for metaphysics was, of course, a central goal of the project of Easy Ontology.

I grant, of course, that I am burdened with epistemological issues that Thomasson can avoid. And I’m prepared to grant that this is a pro tanto reason to think that her easy ontology project is the end of the story. Likewise, I will grant that avoiding epistemological questions concerning our access to normative facts is a pro tanto reason to be a moral expressivist. Nevertheless, I am a moral realist, because I think the Nazis were bad. (Yes, I understand that expressivists say they think this as well. Don’t @ me!) Likewise, I think there is a way the ontology of the world really is, which is why I can’t think that Thomasson is right to think that the easy-to-establish existential facts are all there is to ontology. Those of us with realist inclinations take on an epistemological burden. However, one benefit of agreeing that existential facts are often easy to establish is that it limits the epistemological threat. I might not know what makes it true that there are chairs, but I am pretty certain that there are chairs. I don’t need to solve the metaphysical question in order to be confident about the various ordinary facts that I rely on to navigate my ordinary life. It’s not clear to me that merely taking on a commitment to there being hard-to-establish facts is itself a particularly bad cost to a view, if those epistemic challenges don’t threaten our day-to-day ordinary knowledge. Of course, not only do I think that there is this additional hard-toanswer ontological question, but I also think that it’s worth spending time trying to answer it. That’s because I think—on my more optimistic days, at least—that considering the various benefits and costs of the various rival metaphysics can help us work out what the correct metaphysics is regarding the underlying ontology of the world. Or at the very least, that it can narrow down the options. However, on my more pessimistic days, I feel the pull of epistemic intractability. Maybe, I think to myself on my dark nights of the soul, there is just no way to discover what the world is really like in order to make it true that there

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52  .  are chairs. Karen Bennett thinks this.³⁵ She is a realist in that she holds that there’s a fact of the matter as to whether composition always occurs, or only sometimes occurs, or never occurs; and it’s not just a matter of how we choose to speak, or the concepts we deploy. But she thinks that this ontological fact is unknowable, because there is just no evidence that points to one view over another. None is simpler than another, none more powerful. This ontological fact just can’t be discovered. Perhaps Bennett is right. Perhaps it’s even worse. Perhaps there’s no evidence to point to the world even being a world of things that compose other things, as opposed to it being a world of unstructured facts, or a world or ideas in minds, or . . . etc. Maybe all these rival metaphysics are equally good theories, and there is not in principle any reason to believe one over another. That would be disappointing. But even were it so, I would still believe there was a fact of the matter, and I would still think it was worth our while to investigate the question and to debate the rival metaphysics. I think there is value in asking the question, and value in the development and discussion of the various metaphysical options, even if we cannot come to know which of them is true. Progress doesn’t need to mean getting closer to answering the question; our understanding of the world can be increased by understanding the metaphysical options. Let me deal with a final objection from Thomasson. I started this paper by declaring my sympathy for at least some of what Thomasson says about the ease of establishing existential facts, and I said that the alternative seemed to rely on a bad view about how language works: the kind of view Heil calls the Picture Theory and Rayo calls Metaphysicalism. Thomasson argues however that this anti-PictureTheory attitude does not sit well with the view that I have ended up defending. She says: Another reason for doubting the viability of Cameron’s ambitious project comes from truthmaker theory itself. For many truthmaker theorists were originally motivated by criticisms of the so-called ‘picture theory’ of the relation between language and the world, according ³⁵ Bennett (2009).

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 , -,  

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to which “the character of reality can be ‘read off ’ our linguistic representations of reality—or our suitably regimented linguistic representations of reality” (Heil 2003, 6). A truthmaker theorist who aims to provide a uniquely true description of what’s fundamental denies that we can read features of reality off just any ways we have of representing reality. Nonetheless, in thinking that there is (among all the ontologically alternative ways in which fundamental reality might be described) one that is uniquely true in ‘corresponding to the ontological structure of the world’, Cameron still presupposes that there is some privileged representation of reality where the terms of the language ‘line up with’ the true fundamental ontology—and so seems committed to a version of the picture theory that many truthmaker theorists reject. Given the availability of ontologically alternative languages . . . it seems likely that any statement one could give of what the fundamental entities are could be rivalled by a statement in an ontologically alternative language. The idea that one but not the others of these ontologically alternative languages gives us the ‘true’ story about what the fundamental entities are seems to itself rely on a picture theory that would take the world to have to have a fundamental ontological structure that mirrors some but not other modes of expression.

But the objection to the Picture Theory, or Metaphysicalism, was not meant to be the impossibility of language mirroring reality; it was to the requirement that it do so in order for you to speak truly.³⁶ If the way the world really is is describable by creatures like us, then there will be a way of describing it perspicuously. There will also be ways of describing it imperspicuously, but truly nonetheless. The point of rejecting Metaphysicalism/the Picture Theory is that truth does not demand a perspicuous representation. Insofar as you are trying just to communicate ordinary information about the world, the true imperspicuous representation is just as good—and, pragmatically, may be better—than ³⁶ Rayo, e.g., is explicit that the main reason for rejecting Metaphysicalism is that it isn’t true of every possible language that a sentence of the language must limn the structure of reality in order to say something true, and as such there is no reason to think that English must meet that constraint (Rayo 2013, 10–11). That, of course, allows that there are possible languages that do work as the Metaphysicalist demands.

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54  .  the true perspicuous one. Since natural languages like English evolved for the purpose of conveying ordinary information, and not the purpose of describing the fundamental nature of reality, we should not expect the structure of English sentences to mimic the metaphysical structure of the world. Limning the structure of reality is no requirement on truth, for a sentence of English. And so we should not, from the fact that ‘There are chairs’ is true, jump to the conclusion that the ontology of our world is one including chairs. But that doesn’t mean that there isn’t some way— perhaps in English, perhaps not in any natural language but in a language that we can come to understand—of perspicuously describing the world: of giving a description where the structure of the representation does limn the structure of reality. I don’t read any truthmaker theorist, including Heil who is probably the most forceful rejector of something like the Picture Theory, as objecting to the mere possibility of speaking a language whose sentences are true iff they correspond appropriately to the objective structure of reality. The objection is to thinking that this is what language must do if we are to speak truly, or to uncritically assuming that this is how a natural language like English is. I don’t reject the Picture Theory because I think reality can’t be pictured; I reject it because I think it doesn’t need to be pictured in order for you to truly describe it. To be sure, there is no guarantee that it is possible to perspicuously describe reality. Maybe the fundamental nature of reality is so strange as to resist perspicuous description by creatures like us. Maybe reality is truly ineffable, and no language we could understand limns the structure of reality. That would make the epistemological worries discussed previously even worse, for not only do we not have evidence to favor a unique metaphysic; we would not even be able to formulate the correct metaphysic. That would be disappointing. But we have no reason to suppose that this is case; and insofar as there is a threat here, it is one that arises in any area of inquiry for which we are realist about its subject matter. We have no guarantee that the scientific facts are in principle perspicuously describable either; but I think we should keep doing science, on the optimistic assumption that they are. In conclusion, then, I think that while existence facts can often be very easily established, there is work remaining for the metaphysician that is

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very, very hard: saying what the nature of reality is, to make true all that is true. That question may be so hard, indeed, as to be unanswerable. But it is worth pursuing nevertheless.

Acknowledgments Thanks to Elizabeth Barnes, Kris McDaniel, Trenton Merricks, Amie Thomasson, Jason Turner, and Robbie Williams for helpful discussion. This paper can usefully be read alongside Cameron (forthcoming), which is also in part a response to (an earlier version of) the paper of Thomasson’s (Chapter 1 of this volume) that this paper is in part a response to.

References Armstrong, D. M. (1996). A World of States of Affairs. Cambridge: Cambridge University Press. Bennett, Karen. (2009). ‘Composition, Colocation, and Metaontology,’ in David Chalmers, David Manley and Ryan Wasserman (eds.) Metametaphysics. Oxford: Oxford University Press, pp. 38–76. Cameron, Ross P. (2008a). “Truthmakers and Ontological Commitment: Or, How to Deal with Complex Objects and Mathematical Ontology without Getting into Trouble.” Philosophical Studies 140(1): 1–18. Cameron, Ross P. (2008b). “There Are No Things That Are Musical Works.” The British Journal of Aesthetics 48: 295–314. Cameron, Ross P. (2010a). “Quantification, Naturalness and Ontology,” in Allan Hazlett (ed.), New Waves in Metaphysics. New York: PalgraveMacmillan, pp. 8–26. Cameron, Ross P. (2010b). “Necessity and Triviality.” The Australasian Journal of Philosophy, 88(3): 401–15. Cameron, Ross P. (2009). “Truthmaking, Second-Order Quantification, and Ontological Commitment.” Analytic Philosophy, 60(4): 336–360. Cameron, Ross P. (forthcoming). “Truthmaking and Metametaphysics,” in Ricki Bliss and James Miler (eds.), The Routledge Handbook of Metametaphysics.

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56  .  Chalmers, David. (2005). “The Matrix as Metaphysics,” in Christopher Grau (ed.), Philosophers Explore the Matrix. Oxford: Oxford University Press, pp. 132–76. Dorr, Cian. (2005). “What We Disagree about When We Disagree about Ontology,” in Mark Kalderon (ed.), Fictionalist Approaches to Metaphysics. Oxford: Oxford University Press, pp. 234–86. Eddington, A. S. (1928). The Nature of the Physical World. New York: Macmillan. Heil, John. (2003). From an Ontological Point of View. Oxford: Oxford University Press. Korman, Dan. (2016). Objects: Nothing out of the Ordinary. Oxford: Oxford University Press. Lewis, David K. (1986). On the Plurality of Worlds. Oxford: Blackwell. Lewis, David K. (1990). “Noneism or Allism?.” Mind 99(393): 23–31. Merricks, Trenton. (2001). Objects and Persons. Oxford: Oxford University Press. O’Leary-Hawthorne, John and Cortens, Andrew. (1995). “Towards Ontological Nihilism.” Philosophical Studies 79(2): 143–65. Paul, L. A. (2002). “Logical Parts.” Noûs 36(4): 578–96. Quine, W. V. (1948). “On What There Is.” The Review of Metaphysics 2(1): 21–38. Rayo, Agustin (2009). “Toward a Trivialist Account of Mathematics,” in Otavio Bueno and Øystein Linnebo (eds.), New Waves in the Philosophy of Mathematics. New York: Palgrave-Macmillan, pp. 239–60. Rayo, Agustin. (2013). The Construction of Logical Space. Oxford: Oxford University Press. Sider, Theodore. (2006). “Bare Particulars.” Philosophical Perspectives 20: 387–97. Sider, Theodore. (2011). Writing the Book of the World. Oxford: Oxford University Press. Stalnaker, Robert. (2002). “What Is It like to Be a Zombie?,” in Tamar Szabo Gendler and John Hawthorne (eds.), Conceivability and Possibility. Oxford: Oxford University Press, pp. 385–400. Thomasson, Amie L. (2015). Ontology Made Easy. Oxford: Oxford University Press.

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Thomasson, Amie L. (2020). “Truthmakers and Easy Ontology,” in Karen Bennett and Dean W. Zimmerman (eds.), Oxford Studies in Metaphysics, Volume 12. Oxford: Oxford University Press, pp. 3–34. Turner, Jason. (2011). “Ontological Nihilism’ in Dean W. Zimmerman and Karen Bennett (eds.), Oxford Studies in Metaphysics, Volume 6. Oxford: Oxford University Press, pp. 1–50. Turner, Jason. (2016). The Facts in Logical Space. Oxford: Oxford University Press. van Inwagen, Peter. (2014). Existence: Essays In Ontology. Cambridge: Cambridge University Press. Weatherson, Brian. (2015). “Humean Supervenience,” in Barry Lower and Jonathan Schaffer (eds.) A Companion to David Lewis. John Wiley & Sons, pp: 99–115. Williams, J.R.G. (2010) “Fundamental and derivative truths.” Mind 119(473), pp: 103–41.

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3 Ontological Pluralism and Notational Variance Bruno Whittle

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy. For example, Aristotle seemed to endorse it when he said that ‘there are many senses in which a thing may be said to ‘be’’.¹ Although the view fell out of favour, there has recently been a resurgence of interest, sparked by defences from Kris McDaniel (2009, 2010a, 2010b) and Jason Turner (2010, 2012). Indeed, while the position may still have relatively few adherents in quite these terms, the influential Fregean approach to higher-order quantification—according to which this is over ‘concepts’ rather than objects—would seem to be an instance of it.² In contemporary presentations, the view is stated in terms of fundamental languages.³ That is, languages whose expressions ‘carve nature at the joints’, or whose meanings are natural in the sense of Lewis (1983, 1986). Thus stated, it is the claim that such languages have more than

¹ Metaphysics IV.2. For more on the history of ontological pluralism, see McDaniel (2009) and Caplan (2011). ² See Frege (1891, 1892). A similar point could be made about many other approaches to higher-order quantification, e.g. those of Russell (1908), Prior (1971), and Williamson (2003). On the relation between ontological pluralism and higher-order quantification, see Turner (2010: 12–13) and Caplan (2011). ³ See McDaniel (2009) and Turner (2010, 2012). On fundamental languages more generally, see Sider (2011).

Bruno Whittle, Ontological Pluralism and Notational Variance In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Bruno Whittle. DOI: 10.1093/oso/9780192893314.003.0003

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one type of quantification, ranging over different domains. For example, ∃a ranging over abstract objects, and ∃c ranging over concrete ones.⁴ There is, however, a worry that one might have about ontological pluralism. This is not a worry that it is false, but rather that it is not substantively different from its supposed rival, (ontological) monism. In particular, one might worry that the languages proposed by the pluralist are mere notational variants of those proposed by the monist. For example, one might fear that the pluralist language ℒP with quantifiers ∃a and ∃c is a mere such variant of the monist language ℒM , with the single quantifier ∃ but predicates Abstract and Concrete. By way of analogy, consider the languages ℒ◽ and ℒ◇ , identical except that the first contains ◽, but not ◇, while the second is the opposite. If one came across two people arguing over which of these is a fundamental language, one would presumably not think that this was an important philosophical question that had been too long neglected. Rather, one would think that it is not a substantive question at all. The worry is that ℒP stands in the same sort of relation to ℒM that ℒ◽ stands in to ℒ◇ , and thus that the debate between the pluralist and the monist is similarly insubstantive. However, Turner (2012) has recently given an ingenious response to this concern, employing a principle that he calls ‘logical realism’.⁵ According to this, if two fundamental languages are notational variants, under a given translation, then this translation must preserve logic. That is, a formula must be a consequence of a given set of formulas iff this is also true of their translations. But, Turner argues, unlike in the case of ℒ◽ and ℒ◇ , the translation that threatens to show that ℒP and ℒM are notational variants fails to preserve logic. ⁴ In fact, there is some debate about how exactly the position should be stated: see McDaniel (2009), Turner (2010), Caplan (2011), and Spencer (2012). For example, it has been suggested that Lewisian naturalness might be replaced by Schafferian fundamentality (see Caplan (2011: 91–3)), or that, in the spirit of Wittgenstein’s Tractatus, it be stated without reference to quantification at all, but rather ‘sorts’ of terms and argument places (see Turner (2010: 10–11)). Most of these considerations are orthogonal to those of this paper. The exception is the ‘Tractarian’ question: the arguments to follow are in terms of quantification, and it would take work to reframe them in other terms. However, almost all recent discussion of ontological pluralism has similarly assumed that is stated in quantificational terms. ⁵ Strictly speaking, Turner uses ‘logical realism’ for the view that motivates the principle, and (LR) for the principle itself. But for simplicity I use ‘logical realism’ for the principle.

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60   The purpose of this paper is to offer a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that ℒP and ℒM are notational variants. Indeed, there seems to be every reason to think that they are. The structure of the paper is as follows. Section 1 contains preliminaries. Section 2 gives the worry, and section 3 Turner’s response. Section 4 argues against this response. An appendix (section 5) extends the argument to a variety of pluralism not considered in the main text: that according to which the fundamental language is ‘sorted’.

1. Preliminaries Throughout the paper I focus on the example from the introduction: the instance of pluralism on which the fundamental language has exactly the quantifiers ∃a and ∃c . However, everything that I say carries straightforwardly over to other instances.⁶ More specifically, in the main text I assume that the pluralist proposes a language, ℒP , that is just like a standard first-order language, except that it has ∃a and ∃c in place of ∃. That is, ℒP is ‘unsorted’: with one sort of term and argument place, rather than distinct sorts corresponding to the two quantifiers. However, as I show in the appendix (section 5), a version of the argument to follow can also be given in the case where a sorted language is proposed. The instance of monism under consideration, then, is that on which the fundamental language, ℒM , is the standard first-order language that is just like ℒP , except that it contains ∃ and unary predicate symbols Abstract and Concrete, in place of ∃a and ∃c . Finally, I assume that ℒP and ℒM are languages with equality. To discuss the question of notational variance, we need the concept of a translation. Thus, if ℒ and ℒʹ are languages, a translation between ℒ and ℒʹ is a pair 〈t; tʹ〉 such that t is a function from formulas of ℒ to those of ℒʹ, and tʹ is one in the opposite direction. However, I often

⁶ This includes those in which the fundamental language has infinitely many quantifiers, as long as one is willing to countenance formulas of infinite length.

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abuse notation and use the same symbol for both functions, allowing context to disambiguate. A paradigm example of notational variants are thus a pair of languages along the lines of ℒ◽ and ℒ◇ , under the translation that sends ◽φ to ¬◇¬tðφÞ, and ◇ψ to ¬◽¬tðψÞ. Similarly, ℒ¬ ∧ and ℒ¬ ∨ , or ℒ∃ and ℒ∀ . For simplicity, I build into the definition of notational variance the requirement that the languages involved are fundamental, although one could also, if desired, give a more inclusive definition. Thus, I say that ℒ and ℒʹ are notational variants under translation t if: (i) ℒ and ℒʹ are fundamental languages; and (ii) for any formula φ of ℒ, tðφÞ ‘says the same thing’ as φ, and similarly in the other direction. This characterization is, of course, less than completely precise, but it will suffice for our purposes. I should, though, say something by way of explanation. Concerning (i): we need some clause along these lines because we want notational variants to be metaphysically on a par. But (ii) on its own would not establish this: for example, one might hold that there is a translation between ℒP and ℒM that satisfies (ii), but that nevertheless only one of these languages is fundamental; in which case one would be metaphysically superior, and the debate between the pluralist and the monist would be substantive. Concerning (ii): which notion of content is at issue? It cannot be an extremely fine-grained notion, requiring a perfect isomorphism between a formula and what it expresses: since that would rule out paradigm examples such as ℒ◽ and ℒ◇ . On the other hand, nor can it be a very coarse-grained notion, such as that which identifies contents with sets of possible worlds: for we would then get the result that any two mathematical languages that satisfy (i) are notational variants, which is surely unacceptable. What is needed is a middle way: perhaps the sort of notion Frege had in mind when he famously said that ‘the direction of a is the same as the direction of b’ has the same content as ‘a is parallel to b’.⁷ Such a notion is notoriously difficult to make precise. Fortunately, however, we can to a great extent sidestep this issue here: since the heart of the matter is whether the translation between ℒP and ℒM (to be given in ⁷ See (1884: 74–5). For discussion of this line of thought in Frege, see Hale and Wright (2001).

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62   section 2) satisfies the principle of logical realism (i.e. preserves logic); and this question can be addressed quite independently of that of how to explicate such a ‘medium-grained’ notion of content.⁸ Quite generally, if ℒ and ℒʹ are notational variants, then any debate between their proponents would seem to be insubstantive.

2. The worry Consider the translation t between ℒM and ℒP , defined by induction as follows. Going first from ℒP to ℒM : if φ is a formula of the form ∃a xψ, then tðφÞ is ∃xðAbstractðxÞ ∧ tðψÞÞ; and similarly if φ is ∃c xψ, but with Concrete in place of Abstract; formulas of other forms are handled in the obvious way.⁹ Going in the opposite direction: if φ is ∃xψ, then tðφÞ is ∃a x tðψÞ ∨ ∃c x tðψÞ; t(Abstract(s)) is ∃a x x ¼ s; and t(Concrete(s)) is ∃c x x ¼ s. Formulas of other forms are handled as before. Thus, for example, ∃a x SetðxÞ is translated as ∃xðAbstractðxÞ ∧ SetðxÞÞ, while ∃x x ¼ harry is translated as ∃a x x ¼ harry ∨ ∃c x x ¼ harry. The problem is that we seem to be dealing here with translations that say the same thing as the formulas translated—and with equal metaphysical transparency. Just as in the case of ℒ◽ and ℒ◇ , for example. But if that is right, then ℒP and ℒM are notational variants, and the debate between the pluralist and the monist would seem to be insubstantive.

⁸ Turner (2012: 423) gives a definition of notational variance that promises to avoid this issue. This is in terms of theories rather than languages, and (simplifying inessentially) is as follows. Theories T and Tʹ are notational variants under t if: (I) the languages of T and Tʹ are fundamental; and (II) φ is a theorem of T iff tðφÞ is one of Tʹ, and similarly in the other direction. The problem is that this definition is inadequate by Turner’s own lights, for it is incompatible with logical realism. This can be seen by letting T ¼ Tʹ be some theory in a fundamental language such that for some atomic sentence α ¼ Fs1 : : : sn , ¬α is a theorem of T, but α is logically contingent (i.e. neither logically true nor false). For consider the translation that sends atomic formulas φ of the form Fr1 : : : rn to φ ∧ ¬α; that is the identity on other atomic formulas; and that handles non-atomic formulas in the obvious way. This satisfies (II), but does not preserve logic: because the logically contingent α is sent to the contradiction α ∧ ¬α. There does not seem to be any way, therefore, of ultimately avoiding the difficult questions about content raised in the text. I should note, however, that this issue with Turner’s definition in no way undermines his argument, since he could just as well use the characterization that I have given. ⁹ i.e. t sends atomic formulas to themselves; tð¬ψÞ ¼ ¬tðψÞ; and tðψ ∧ χÞ ¼ tðψÞ ∧ tðχÞ.

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3. Logical realism Turner has a response, however, which uses the following principle (see (2012)): (LR) If ℒ and ℒʹ are notational variants under t, then t preserves logic. That is, for any set Γ ∪ fφg of formulas of ℒ, Γ ⊨ φ iff tðΓÞ ⊨ tðφÞ; and similarly for any set Γ ∪ fφg of formulas of ℒʹ.¹⁰ This principle is supported by the following, apparently very plausible line of thought (see Turner (2012: 426–7)). If Γ ⊨ φ, then this corresponds to an important metaphysical relation between the contents of these formulas—it is not a mere accident of notation. But then this relation should be captured by any metaphysically perspicuous expression of these contents. However, if ℒ and ℒʹ really are notational variants under t, then the members of tðΓÞ ∪ ftðφÞg offer just such an expression, and so we have tðΓÞ ⊨ tðφÞ. That is, if Γ ⊨ φ, then tðΓÞ ⊨ tðφÞ. Similarly for the converse: giving (LR). Armed with this principle, however, Turner has an answer to the worry of section 2. He points out that the translation t between ℒP and ℒM does not seem to preserve logic. For consider the following sentence αM of LM : ∀xðAbstractðxÞ ∨ ConcreteðxÞÞ.¹¹ This does not seem to be a logical truth. However its translation does seem to be one. For this (simplifying slightly) is αP : ∀a x ð∃a y y = x ∨ ∃c y y = xÞ ∧ ∀c x ð∃a y y = x ∨ ∃c y y = xÞ; which appears to be a logical truth in virtue of the fact that ∀a x ∃a y y = x and ∀c x ∃c y y = x are.¹² ¹⁰ Here Γ ⊨ φ means that φ is a logical consequence of Γ, and tðΓÞ = ftðψÞ : ψ ∈ Γg. ¹¹ Universal quantifiers are defined in terms of existential ones in the usual way. ¹² Is there a comparable example in the opposite direction, i.e. a set Γ ∪ fφg of formulas of ℒP such that we similarly seem to have Γ ⊨ φ ⇎ tðΓÞ ⊨ tðφÞ? This depends on whether ¬∃a x∃c yx = y is a logical truth. If it is, then we seem to have a case in which the translation from ℒP to ℒM fails to preserve logic: for the translation of this sentence is (equivalent to) ¬∃xðAbstractðxÞÞ ConcreteðxÞÞ, which doesn’t seem to be logically true. On the other hand, if ¬∃a x∃c yx = y is not a logical truth, then we don’t seem to have a failure to preserve logic in this direction. Ultimately, though, Turner’s argument is unaffected: a counterexample in one direction is enough. (Note that, even if there is no counterexample from ℒP to ℒM , we cannot ‘fix’ our translation from ℒM to ℒP exploiting this fact, because the function from ℒP to ℒM is not surjective.)

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64   Perhaps, then, we are not dealing with notational variants after all?¹³

4. Notational variance even so One way of trying to counter this response would be to insist that either Abstract or Concrete be defined in terms of the other: for example, that Abstract be defined as ¬Concrete. If such a definition was permitted, then αM would be a logical truth (a quantified instance of the law of excluded middle), and we would no longer have a violation of (LR). But there seem to be a number of drawbacks with this idea. For one thing, to give such a definition is to treat one of Abstract and Concrete as more fundamental than the other. But it is at least plausible that the properties of being abstract and concrete are equally fundamental. For another, this idea (at least on the most obvious implementations) requires that nothing can be within the range of more than one of the pluralist’s quantifiers (see Turner (2012: 428–9)). Perhaps in the case of ∃a and ∃c this is uncontentious. But there are other versions of pluralism where it is very far from so: e.g. that with a quantifier ∃p over the physical, and another ∃m over the mental. Thus, this way of trying to counter Turner’s response seems unsatisfactory. The strategy that I pursue is quite different. The basic idea is this. We only get Turner’s result—that αP is a logical truth, while αM is not—if we assume that ∃a and ∃c are logical constants, but Abstract and Concrete are not. (For it is only the thought that these are standard, non-logical, predicate symbols that justifies the claim that αM is not a logical truth.) However, the work that ∃a and ∃c do in ℒP is precisely that which is done by Abstract and Concrete (together with ∃) in ℒM . But, then, if we are treating the former as logical (as Turner quite naturally is), we should surely also so treat the latter. Indeed, this differential treatment seems

¹³ I should note that Turner (2012) in fact gives a whole range of arguments aimed at establishing that ℒP and ℒM are not notational variants. Specifically, aimed at establishing that there is no other translation under which they are such. However, these general arguments make the same assumptions about the logic of ℒP and ℒM that the particular one does—and that I argue against in section 4. Thus, if the argument of that section succeeds, then it refutes these general arguments as well.

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particularly problematic if one is trying to refute the suggestion that ℒP and ℒM are notational variants. For the whole thrust of that suggestion is that the relevant expressions are essentially interchangeable. But, then, if we are treating one set as logical, we must also treat the other as such. To argue as Turner does would seem akin to arguing against the claim that ℒ◽ and ℒ◇ are notational variants (under the standard translation), on the basis that if one counts ◽ as logical, but does not so count ◇, then this translation fails to preserve logic (since one would have ◽FðaÞ ⊨ FðaÞ but ¬◇¬FðaÞ ⊭ FðaÞ, for example). But that would, of course, be a very unpersuasive way to argue against this claim of notational variance. The contention that ∃a and ∃c , on the one hand, and Abstract and Concrete, on the other, are on a par as far as logicality is concerned can be further supported by considering standard accounts of logical constancy. There is, of course, no universally accepted such account.¹⁴ But two of the most widely invoked are those in terms of (a) topic neutrality or (b) permutation invariance.¹⁵ Consider (a) first: the claim is that an expression is a logical constant iff it can be used to talk about any subject matter. But, of course, the expressions in question are exactly tied in that regard: Abstract can be used to talk about abstract objects completely generally (but not others); exactly what is true of ∃a ; and similarly for Concrete and ∃c . As for (b), the usual statement of such an account is that an expression e is logical iff for any interpretation I with domain D, and any permutation π of D,¹⁶ the semantic value of e in I, e I , is unchanged under π. That is, πðeI Þ = eI .¹⁷ For example, = passes this test, because =I = f〈d; d〉 : d ∈ Dg = πð=I Þ (since π is a surjection). Similarly, the test is passed by ∃, given that ∃I = fX ⊆ D : X ≠ ∅g = fπðXÞ :

¹⁴ See MacFarlane (2015) for a useful survey. ¹⁵ For (a), see Ryle (1954: 111–29) and Peacocke (1976: 229), for (b), Tarski (1986) and McGee (1996). ¹⁶ i.e. any bijection of D into itself. ¹⁷ In the first instance, π is defined for members of D, but it can be extended to ordered tuples of members of D, subsets of D etc. in the obvious way: e.g. if a,b ∈ D, then πð〈a; b〉Þ = 〈πðaÞ; πðbÞ〉; if X ⊆ D, then πðXÞ = fπðdÞ : d ∈ Dg; and so on.

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66   X ⊆D ∧ X ≠ ∅g = πð∃I Þ.¹⁸ On the other hand, a standard non-logical unary predicate symbol F will not pass the test: just choose I with F I ∉ f∅; Dg, and π such that for some d ∈ F I , πðdÞ ∉ F I . We then have F I ≠ πðF I Þ. What happens when we apply this test to the expressions of ℒP and ℒM ? Consider first ∃a and ∃c . In any ℒP -interpretation I, each of these will be assigned its own domain, Da and Dc .¹⁹ If π is a permutation of the total domain, i.e. Da ∪ Dc , then: πð∃Ia Þ = ∃Ia and πð∃Ic Þ = ∃Ic iff for any d ∈ Da ∪ Dc , d ∈ Da ⇔ πðdÞ ∈ Da , and d ∈ Dc ⇔ πðdÞ ∈ Dc . That is, the semantic values of ∃a and ∃c are invariant under precisely those permutations that respect the ‘abstract’/‘concrete’ divide: i.e. that send ‘abstract’ objects of the interpretation to other such objects, and ‘concrete’ objects to other such ones. But the situation with Abstract and Concrete is exactly similar. If I is an ℒM -interpretation, then it will have a single domain D. And a permutation of D will leave AbstractI and ConcreteI unchanged iff it similarly respects the ‘abstract’/‘concrete’ divide, i.e. sends ‘abstract’ members of the domain, which are in this case the members of AbstractI, to other such objects, and similarly for ‘concrete’ ones. Again, then, the two sets of expressions would seem to be precisely tied as far as logicality is concerned. Thus, given that we are taking ∃a and ∃c to be logical, we should also so take Abstract and Concrete.²⁰ We will see, however, that once we do this, our translation t satisfies (LR) after all.

4.1. Logic for ℒP and ℒM I want to show, then, that once we take Abstract and Concrete to be logical, the translation t of section 2 does indeed preserve logic. As is standard, I assume that logical consequence for a language ℒ is defined in terms of interpretations of ℒ. ¹⁸ As usual, I take the semantic value of a quantifier Q to be a set of subsets of the domain. The idea is that Qxφ is true iff the extension of φ is in this set. ¹⁹ As with ∃; ∃Ia = fX ⊆ Da : X ≠ ∅g, and similarly for ∃c . ²⁰ What about the possibility of taking none of these expressions to be logical? I consider that towards the end of this section.

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Consider ℒP first. The natural notion of an interpretation of ℒP is as follows (see Turner (2012: 432)). An ℒP -interpretation I is a triple 〈Da ; Dc ; i〉 such that Da and Dc are non-empty²¹ sets (I use D ∪ for Da ∪ Dc ); and i is a function that sends every individual constant of ℒP to a member of D ∪ ; every n-ary ðn ≥ 1Þ function symbol to an n-ary function from D ∪ to D ∪ , and every n-ary ðn ≥ 1Þ predicate symbol to a subset of D∪n . One might also insist that Da and Dc are disjoint. And everything that I say would straightforwardly carry over, but for definiteness I assume that Da and Dc are allowed to overlap. An ℒP -valuation is an ℒP -interpretation together with an assignment of values to variables. Satisfaction is then defined in the obvious way: with ∃a taken to range over Da , and ∃c over Dc . Finally, for a set Γ ∪ fφg of formulas of ℒP , we say that Γ ⊨ φ if, whenever an Lp ‐valuation satisfies every member of Γ, it also satisfies φ. What, now, about interpretations of ℒM ? That is, what is the cash value of the claim that Abstract and Concrete are logical? I suggest that the natural definition of an interpretation of ℒM is, in fact, exactly the same as in the ℒP case. It is just that, now, Da is not the range of ∃a , but the extension of Abstract (and similarly for Dc ). Further, we do not need to supply an additional domain for ∃, because this is simply taken to range over D ∪ . This would seem to be the natural definition of an ℒM -interpretation, given that ∃ is intended to range over abstract and concrete things—and nothing else. Of course, there are distinct monist languages in which ∃ is also intended range over other sorts of thing (e.g. mental items). Such languages will still be notational variants of pluralist ones: but those with additional quantifiers beyond ∃a and ∃c (e.g. ∃m ). Here is another way in which one can justify this definition. One aspect of our treating ∃a and ∃c as logical is to insist that the other expressions of the language have their interpretations drawn from the ‘abstract’ and ‘concrete’ objects of this interpretation, i.e. the objects that constitute the interpretations of ∃a and ∃c , which is to say the members of D ∪ . Thus, we require that the interpretations of individual constants are members of D ∪ , the interpretations of n-ary predicate symbols are ²¹ We could also allow one or both to be empty, but for simplicity I do not do this.

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68   subsets of D∪n , etc. In the case of =, for example, this amounts to = I = f〈d; d〉 : d ∈ D ∪ g. The natural way of treating Abstract and Concrete as logical would seem to be by imposing a similar requirement. That is, to insist that the interpretations of other expressions be drawn from the objects that constitute those of Abstract and Concrete. In the case of =, this again amounts to = I = f〈d; d〉 : d ∈ D ∪ g. In that of ∃, it amounts to ∃I = fX ⊆ D ∪ : X ≠ ∅g. But then we have precisely the notion of an LM -interpretation that I have proposed, together with a definition of satisfaction on which ∃ ranges over D ∪ . We’re then home and dry, by the following result (which is proved by a straightforward induction on the degrees of φ and ψ):²² Proposition 1. Let σ be an ℒP -valuation (i.e. ℒM -valuation), φ a formula of ℒP , and ψ one of ℒM . Then: ðiÞ σ ⊨ φ iff σ ⊨ tðφÞ; ðiiÞ σ ⊨ ψ iff σ ⊨ tðψ). It follows immediately that (LR) is satisfied. For suppose that Γ ∪ fφg is a set of formulas of ℒP . Then: Γ ⊨ φ iff any ℒP -valuation satisfying Γ satisfies φ; which, by (i), holds iff any ℒM -valuation satisfying tðΓÞ satisfies tðφÞ. Thus, Γ ⊨ φ iff tðΓÞ ⊨ tðφÞ. The case where Γ ∪ fφg is a set of formulas of ℒM is exactly similar. As promised, then, (LR) is met after all. There is one final loose end to tie up. We saw that if we are taking ∃a and ∃c to be logical, then we should also so take Abstract and Concrete. Just as: if we take ◽ to be logical, then we should also so take ◇. In the case of ℒ◽ and ℒ◇ , however, (LR) is satisfied both when we treat ◽ and ◇ as logical and when we treat neither as. Might one insist that if ℒP and ℒM are genuine notational variants, then (LR) must similarly be satisfied both when we treat all of the relevant expressions as logical and when we treat none of them as?

²² If σ is an ℒP -valuation, and φ is a formula of ℒP , then σ ⊨ φ means that σ satisfies φ. And similarly for ℒM . Note that although any ℒP -valuation is an ℒM -valuation, and vice versa, ‘σ ⊨’ means something rather different, depending on whether we are treating σ as an ℒP -valuation or an ℒM - one. Thus, slightly more verbosely, (i), for example, might be written: σ ⊨P φ iff σ ⊨M tðφÞ.

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In fact, unlike in the case of ℒ◽ and ℒ◇ , (LR) is not satisfied when we treat none of these expressions—i.e. ∃a ; ∃c ; ∃, Abstract and Concrete—as logical.²³ On reflection, however, to insist that (LR) is satisfied even in this case is unreasonable. This can be seen by considering a simple variant of the modal example: ℒ◽◇ and ℒ◽ .²⁴ It seems clear that these are notational variants (under the obvious translation s). But (LR) is not satisfied if we treat neither ◽ nor ◇ as logical: since if δ is ¬◽¬FðaÞ and ζ is ◇FðaÞ, then δ ⊭ ζ; but sðζÞ = δ = sðδÞ and so sðδÞ ⊨ sðζÞ. It seems, then, that notational variance does not require (LR) to hold even in the case where we treat the expressions being (non-homophonically) translated as non-logical. Indeed, what the example of ℒ◽◇ and ℒ◽ seems to show is that we can only expect notational variants to satisfy (LR) if we are treating the expressions being translated as logical. A result that lends further—if slightly more indirect—support to the contention that when we are considering whether ℒP and ℒM are notational variants, we should treat all of the expressions being translated as logical. There seems, then, to be every reason to think that ℒP and ℒM are notational variants, and thus that the debate between the pluralist and the monist is not a substantive one.

5. Appendix: sorted pluralism In this appendix, I consider the possibility that the pluralist might propose a sorted language, P . That is, every term and argument place in P has a sort, a or c, and a string is a formula only if its terms and ²³ The following would be a counterexample to (LR). Let β be ∃xð¬AbstractðxÞ ∧ ¬ConcreteðxÞÞ, and γ, ∃x x ≠ x. If we are treating none of the expressions as logical, then β ⊭ γ. However, the ‘double’ translation of β, tðtðβÞÞ, is ∃x(Abstract(x) ∧ ¬Abstract(x) ∧ ¬Concrete(x)) ∨ ∃x(Concrete(x) ∧ ¬Abstract(x) ∧ ¬Concrete(x)), while tðtðγÞÞ is ∃xðAbstractðxÞ ∧ x ≠ xÞ ∨ ∃xðConcreteðxÞ ∧ x ≠ xÞ:

And tðtðβÞÞ ⊨ tðtðγÞÞ (since in each formula ∃x is applied to contradictions). ²⁴ Of course, these are identical except that while ℒ◽◇ contains ◽ and ◇, ℒ◽ contains only ◽.

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70   argument places match. Again, I assume that the quantifiers of P are ∃a and ∃c . In this case, too, there is a monist language that seems to be a notational variant of the pluralist one. As in the unsorted case, the main choice point in the logic of P is whether to allow the domains of ∃a and ∃c to overlap. Here it seems most natural not to allow this: the idea of a sorted language goes naturally with that to the effect that we are dealing with two fundamentally different and thus mutually exclusive types of things. Thus, that is the approach that I consider. But one could easily enough derive similar conclusions on the overlap approach. A P -interpretation is just like an ℒP - one, except that sorts are respected. That is, if I = 〈Da ; Dc ; i〉 is a P -interpretation, and ba is an individual constant of sort a, then iðba Þ ∈ Da ; and if Rac is a binary predicate symbol of sort 〈a; c〉 (the first argument place is of sort a, the second of sort c), then iðRac Þ ⊆ Da  Dc ; and so on. Let M be just like P , except that it contains Abstract, Concrete and ∃ in place of ∃a and ∃c , and variables are no longer sorted. Thus, e.g., Rac ðx; xÞ is a formula of M . In M individual constants and argument places are sorted: this is required if even the quantifier- and variable-free fragment of the language is to be translatable into P . However, the machinery of quantification—the quantifier and variables—is unsorted, and so it is in this sense a genuinely monist language. It will follow from that fact that M and P are notational variants that, given this quantifier- and variable-free base, the choice of whether to quantify in a monist or a pluralist fashion is not a substantive one. As in the unsorted case, a M -interpretation is simply a -interpretation. P How can we translate between these languages? To keep things simple, I give a translation only for sentences: this satisfies (LR) in the sense that the logic of sentences is preserved. If one thinks that true notational variance requires a translation that works for formulas more generally, then one could extend that which I give, but for reasons of space I do not do this here. The translation s is defined as follows. From P to M this works just as before (i.e. like t of section 2). The opposite direction is slightly more involved, given the fact that variables are sorted in P but not in M : if

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we tried to translate as before, we would take some formulas of M to ill-formed strings. Instead, I define a function s0 from formulas of M to strings of P as follows. If φ is Abstract(r), then s0 ðφÞ is ∃a xa xa =aa r; if φ is Concrete(r), then s0 ðφÞ is ∃c xc xc =cc r; if φ is atomic of some other form, then s0 ðφÞ = φ; if φ is ¬ψ, then s0 ðφÞ = ¬s0 ðψÞ; if φ is ψ ∧ χ, then s0 ðφÞ = s0 ðψÞ ∧ s0 ðχÞ; and if φ is ∃xψ, then s0 ðφÞ = ∃a xa ½s0 ðψÞa=x ∨ ∃c xc ½s0 ðψÞc=x , where χa=x is the result of replacing every occurrence of an atomic formula with x in an argument place of sort c with ⊥, and then replacing every remaining occurrence of x with xa (and similarly for χc=x ). It is then easy to see that if β is a sentence of M , s0 ðβÞ is one of P . We thus set sðβÞ = s0 ðβÞ. Although for reasons of space I will not make the case in any detail, it is at least very plausible that s preserves both content (in the sense of section 1) and metaphysical perspicuity, and thus that P and M are notational variants under s. Further, we have the following, which ensures that (LR) is indeed satisfied for sentences: Proposition 2. Let σ be a M . Then:

P -valuation,

β a sentence of

P

and γ one of

ðiÞ σ ⊨ β iff σ ⊨ sðβÞ; ðiiÞ σ ⊨ γ iff σ ⊨ sðγ).

Acknowledgements This work was supported by the Arts and Humanities Research Council [grant number AH/M009610/1]. For comments and discussion, I am grateful to Gary Kemp, Stephan Leuenberger, Adam Rieger, Ted Sider, Jason Turner, Nathan Wildman, an audience at Being in Ligerz, and two referees for this volume.

References Caplan, B. (2011). Ontological Superpluralism. Philosophical Perspectives 25: 79–114. Frege, G. (1884). The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number. Translated by J. L. Austin (1953). Oxford: Basil Blackwell.

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72   Frege, G. (1891). Function and Concept. In P. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege (1980): 21–41. Oxford: Basil Blackwell. Frege, G. (1892). On Concept and Object. In P. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege (1980): 42–55. Oxford: Basil Blackwell. Hale, B. and C. Wright. (2001). The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press. Lewis, D. (1983). New Work for a Theory of Universals. Australasian Journal of Philosophy 61: 343–77. Lewis, D. (1986). On the Plurality of Worlds. Oxford: Blackwell Publishing. MacFarlane, J. (2015). Logical Constants. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2015 Edition), http://plato.stanford.edu/ archives/fall2015/entries/logical-constants/, accessed July 15, 2020. McDaniel, K. (2009). Ways of Being. In D. Chalmers, D. Manley and R. Wasserman (eds.), Metametaphysics: 290–319. Oxford University Press. McDaniel, K. (2010a). Being and Almost Nothingness. Noûs 44: 628–49. McDaniel, K. (2010b). A Return to the Analogy of Being. Philosophy and Phenomenological Research 81: 688–717. McGee, V. (1996). Logical Operations. Journal of Philosophical Logic 25: 567–80. Peacocke, C. (1976). What is a Logical Constant? Journal of Philosophy 73: 221–40. Prior, A. N. (1971). Objects of Thought. P. T. Geach and A. J. P. Kenny (eds.). Oxford: Clarendon Press. Russell, B. (1908). Mathematical Logic as Based on the Theory of Types. American Journal of Mathematics 30: 222–62. Ryle, G. (1954). Dilemmas. Cambridge University Press. Sider, T. (2011). Writing the Book of the World. Oxford University Press. Spencer, J. (2012). Ways of Being. Philosophy Compass 7: 910–18. Tarski, A. (1986). What Are Logical Notions? History and Philosophy of Logic 7: 143–54. Turner, J. (2010). Ontological Pluralism. Journal of Philosophy 107: 5–34. Turner, J. (2012). Logic and Ontological Pluralism. Journal of Philosophical Logic 41: 419–48. Williamson, T. (2003). Everything. Philosophical Perspectives 17: 415–65.

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PART II

M E T A P H Y S I C S OF S C I E N C E

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4 Some Consequences of Physics for the Comparative Metaphysics of Quantity David John Baker

According to comparativist theories of quantities, their intrinsic values are not fundamental. Instead, all the quantity facts are grounded in scaleindependent relations like “twice as massive as” or “more massive than.” I show that this sort of scale independence is best understood as a sort of metaphysical symmetry—a principle about which transformations of the non-fundamental ontology leave the fundamental ontology unchanged. Determinism—a core scientific concept easily formulated in absolutist terms—is more difficult for the comparativist to define. After settling on the most plausible comparativist understanding of determinism, I offer some examples of physical systems that the comparativist must count as indeterministic although the relevant physical theory gives deterministic predictions. Several morals are drawn. In particular: comparativism is metaphysically contingent if true, and it is most natural for a comparativist to accept an at-at theory of motion.

1. Introduction The notion of a physical quantity or magnitude is, so far as we know, impossible to do without. Mass is one example, and it will function as my central example here. Mass is unlike simple monadic properties in that individuals don’t simply have it or lack it—it comes in degrees. These degrees, or values of mass, are commonly treated like monadic

David John Baker, Some Consequences of Physics for the Comparative Metaphysics of Quantity In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © David John Baker. DOI: 10.1093/oso/9780192893314.003.0004

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76    properties. Each individual either has or lacks a mass of one gram—but this sort of description leaves out a lot of relevant “mass facts.” One such fact is that nothing has both a mass of one gram and a mass of two grams. Also, there’s a special relationship between a one-gram object and a two-gram object, since the latter is twice as massive. We need a theory of quantity that can capture all of these mass facts. Indeed, as Mundy (1987, 29) has pointed out, such a theory is implicitly presumed by any quantitative scientific theory, and thus is essential to the empirical success of science. In this sense, a successful theory of quantity is one of the most valuable contributions metaphysics can offer to the scientific project.¹ To this end, two general theories have been proposed: the comparativist theory of quantity and the absolutist theory. The comparativist theory holds that relations like “more massive than” and “twice as massive as” give an exhaustive list of the fundamental mass facts. The absolutist view holds that there are some extra fundamental facts as well, namely which intrinsic mass property each object has, so that without intrinsic properties like “has a mass of one gram” the list of metaphysically basic mass facts is incomplete. If the above definition of comparativism seems imprecise, that’s because no fully precise definition of comparativism has yet been proposed. Instead, most of comparativism’s defenders have focused their efforts on motivating their own particular versions of the theory. But there is a common thread in the metaphysics of quantity proposed by Bigelow and Pargetter (1988), Arntzenius (2012, 49–59), and Dasgupta (2013), along with that presupposed by the approach to measurement theory outlined in Suppes and Zinnes (1963) and the nominalist physics of Field (1980). On all these comparativist accounts, the fundamental relations are independent of scale. That is to say, on these accounts, the fundamental relations holding between a one-gram massive object and a ten-gram object are exactly the same as those holding between a one-kilogram object and an object massing ten kilograms. This scale independence of

¹ See Eddon (2013) for a recent survey of the metaphysics of quantity.

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the fundamental relations is one point of overlap between comparativist theories of quantity. The best way to understand scale independence is as a sort of symmetry—a class of transformations which, when applied to the ontology of a possible world, leave the comparativist’s fundamental ontology unchanged. So the scale independence of the comparativist’s fundamental mass facts amounts to the fact that mass doubling, mass tripling and so on do not change these facts. I explain all this in detail in section 2. The correct definition of determinism for the comparativist is difficult to formulate, but this notion of scale independence allows us to settle on what is at least a plausible necessary condition. For a comparativist interpretation of physics to count as deterministic, the scale-independent facts about the past and present must fix the scale-independent facts about the future. This will be established in section 3. Given this definition of determinism, we can formulate a fairly quick and straightforward argument that any comparativist interpretation of Newtonian gravity must be indeterministic in a wide variety of cases. But this argument’s soundness depends in an interesting way on a metaphysical premise: it only works if velocity is taken to be a truly instantaneous quantity, and not a property of infinitesimal temporal neighborhoods. In section 4 I show that this means the comparativist should accept a theory of motion, like the popular “at-at” theory, according to which there are no truly instantaneous velocities. One can also construct more involved examples in which even an at-at theorist should agree that comparativist physics gives indeterministic predictions where absolutist physics does not. In some such cases, comparativist physics exhibits a peculiar sort of temporal action at a distance: the state of the entire past may be sufficient to determine the future, while the present state is insufficient. (In other cases, even the whole past history is insufficient.) These examples are explored in section 5. They have the flavor of idealized toy models, so I hesitate to count them as evidence against the truth of comparativism. But it seems to me that determinism and temporal locality, in these cases, should not be ruled out as metaphysically impossible. Consequently, if comparativism is true, it is contingently true.

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78   

2. Scale independence As I mentioned in section 1, the precise definition of comparativism is somewhat up for grabs. Relatively few metaphysicians have gone on record as comparativists, and of those who have, most have defended specific theories of quantity rather than comparativism in general. The only exception is Dasgupta (2013). Dasgupta is concerned with defending comparativism broadly speaking, rather than identifying some particular system of comparative relations as fundamental. So he characterizes comparativism in the following way: [T]hings 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 . . . [C]omparativism is the view that the fundamental facts about mass concern how material bodies are related in mass, and all other facts about mass hold in virtue of them. (Dasgupta 2013, 1)

Although Dasgupta gives several examples of relations that might count as fundamental for particular comparativists, he never sets down criteria for these relations to meet. In fact, he goes on to add: [T]he comparativist thinks that the fundamental, unexplained facts about mass are facts about the mass relationships between bodies, and all other facts about mass hold in virtue of those mass relationships. This leaves open what kinds of mass relations those fundamental facts concern: they might concern 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 an object lying between two others in mass. But this in-house dispute will not matter for our purposes. (Dasgupta 2013, 3)

This seems to indicate that for Dasgupta, any relation at all could count as fundamental as long as it expresses some comparison between different values of a quantity like mass. But when he moves on to consider arguments for and against comparativism, it becomes clear that this is not what Dasgupta has in mind.

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The first of several modal arguments he considers against comparativism is “that while it is possible for everything’s mass to double tonight at midnight, the comparativist cannot make sense of this since the mass relationships would be exactly the same tomorrow as they were today” (Dasgupta 2013, 7).² But this is only true if one restricts which comparative mass relationships are allowed to count as fundamental. If relations that depend on a choice of scale are left out, Dasgupta’s point is quite correct. For example, the fact that my brother’s mass is greater than mine will not change if every object’s mass is doubled. But, on the other hand, the fact that my brother’s mass is ten kilograms greater than mine will certainly change. So Dasgupta must intend to leave out relations like this family of two-place relations: a is n kilograms more massive than b. It thus appears that there must be some implicit restriction on which comparative relations can count as fundamental for Dasgupta’s comparativist. I agree with this implicit commitment of Dasgupta. There must be some restriction on which relations the comparativist may count as fundamental, or else the view threatens to collapse into an uninteresting variant of absolutism. If relations like Kn above are allowed to count as fundamental, comparativist possible worlds will be able to contain just as much ontological structure as absolutist worlds. There is nothing incoherent about a version of comparativism that recognizes as much structure as absolutism, but I don’t see any appeal in such a view. Why not just be an absolutist? Dasgupta’s main argument for comparativism is that the comparativist requires less ontological structure to “build” a possible world than the absolutist does. In effect, he means that the absolutist recognizes more differences between possibilities than the comparativist does. While the absolutist will count an otherwise empty world containing

² Note that this problem is easily solved, as Dasgupta points out, by allowing mass relations to hold between objects at different times.

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80    two one-kilogram brass balls as distinct from a world with a pair of fiveton balls, the comparativist identifies these two worlds as the same. But if the Kn relations are allowed into the comparativist’s roster of fundamental relations, almost all of this ontological parsimony disappears. If a privileged “zero mass” value were also picked out, the comparativist would recognize exactly as much structure (just as many differences between possible worlds) as the absolutist. It’s hard to see why anyone willing to recognize this much structure in a quantity would not just adopt absolutism about that quantity. Comparativism (at least when not used as a tool for implementing nominalism) is motivated by the thought that the scale-dependent features of a quantity are of no fundamental metaphysical importance. Part of what is distinctive and attractive about the comparativist picture is that these scale-dependent features are grounded in relations that are scale-independent (see Dasgupta 2013, 14–16). But the Kn relations seem to have a scale built into them. How should we define comparativism so as to rule out these problematic relations? I suggest the following modification (and generalization) of Dasgupta’s definition: comparativism about some quantity—or family of quantities—is the view that the fundamental facts about those quantities are given by the scale-independent relations comparing different objects’ values of the quantities. We may then define global comparativism to be comparativism about every quantity.³ This definition depends, of course, on a prior notion of what it is for a relation to be scale-independent. We could attempt to define this the way a physicist intuitively might, positing that the scale-independent relations are the ones that don’t depend on our choice of units, or something like that. But (as Ted Sider has pointed out to me) such an attempt would be doomed. Regardless of how we normally express differences in mass, there is nothing to stop us from adopting a unit-independent language in

³ I will mostly focus on global comparativism, which I think is the most interesting and unified thesis, and which comparativists like Dasgupta, Bigelow, and Pargetter endorse. But even on an absolutist approach, there remains the question, for any given quantity we use in physics, of whether that quantity’s absolute values are physically meaningful (see, e.g., Skow 2011). So something akin to comparativism about particular quantities may be warranted even if global absolutism is preferable to global comparativism.

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which the relation of being five kilograms more massive is expressed without mentioning kilograms. Calling that particular relation K, we might just say that aKb whenever a is five kilograms more massive than b. And, of course, this merely underscores that, while we normally represent the fact that a is five kilograms more massive than b in a way that varies depending on our choice of units, the fact that a is five kilograms more massive does not itself depend on our choice of units in any interesting way. Neither do the Kn relations. Yet there remains a sense in which the Kn relations depend, not on our choice of units, but on the scale of the mass quantity. The best way to express this dependence seems to be via Dasgupta’s observation that the comparative relations he regards as fundamental would not be changed by an operation like doubling the mass of every object in the universe. The Kn relations, on the other hand, would change. Dasgupta’s comparativism (as well as Field’s, and Bigelow and Pargetter’s) differs from absolutism, not merely in that the fundamental facts about quantities are given by relations rather than absolute values, but also in the following way: there are transformations we can perform on the numerical values of quantities that alter which absolutist possible world the values represent, but not which comparativist world they represent. When we double the values of mass, we have changed something fundamental about the world if the absolutist is right, but not if the comparativist is right. We may, therefore, say there is a sort of symmetry to the comparativist theory of quantity that the absolutist theory lacks: transformations multiplying every value of a quantity by some constant leave the comparativist’s fundamental ontology invariant, but not the absolutist’s fundamental ontology. We may thus adopt the following as a definition of scale independence: A comparative relation for a quantity like mass is scale-independent iff, when the quantity is represented numerically, multiplying its values by a constant cannot change whether the relation holds.⁴ Comparativism,

⁴ The reader may be concerned that representing a quantity’s values numerically is itself incompatible with comparativism, but the representation theorems discussed in section 3 establish that this is indeed possible for the comparativist.

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82    in its most interesting form (and the form that has appeared in the metaphysics literature to date), is best understood as the claim that the fundamental facts about quantity are given by scale-independent relations. Besides fitting with Dasgupta’s theory, this is very much in keeping with other authors identified as comparativist. When characterizing “the many relations which are associated with a quantity like mass” in their “relational” theory of quantity, Bigelow and Pargetter (1988, 298) include “relations like ‘more massive than,’ ‘half as massive as,’ and so forth,” but no scale-dependent relations. This variety of comparativism is obviously unsatisfactory for Field’s purposes, since describing it requires reference to numbers. Field wants to pursue a nominalist interpretation of Newtonian physics which can be formulated without referring to any mathematical objects. Thus, he defines a scalar physical quantity Q via the following system of primitive predicates: x 0 s value of Q is between y 0 s and z 0 s. The (absolute value of the) difference between x 0 s value of Q and y 0 s is the same as the difference between z 0 s value of Q and w 0 s. x 0 s value of Q is less than y 0 s value of Q. (Field 1980, 55–60) The important thing for present purposes is that all of these relations are scale-independent. Doubling the value of Q for every object in a world will lead to no change in any of these relations.⁵ The scale independence of the comparativist’s fundamental ontology will be crucial in examining the question of determinism. In particular, it will allow us to evaluate whether a deterministic comparativist version of a given absolutist scientific theory is possible. In some cases, this can be done even in the absence of a thorough definition of determinism for the comparativist—a definition which may be quite difficult to arrive at.

⁵ As Field notes, further primitives will be needed to define mass density (Field 1980, 55–60).

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3. Determinism and laws for the comparativist There is a venerable definition of what it takes for a world to be deterministic, due to Laplace. This requires that the state at a time determine the entire history: A world w is deterministic iff, for any time t, there is only one physically possible world whose state at t is identical to w 0 s. The state of w at t corresponds to what is normally called the initial conditions of a physical system. Although there are other definitions of determinism (Earman 1986, 6–22), this one is probably the most often used in physics, and it has obvious advantages. For example, by treating initial conditions as the state at a time (instead of, for example, the state of the whole past), Laplacean determinism doesn’t beg the question against the view that there is no fundamental arrow of time. This definition, however, is potentially mysterious if we assume comparativism. The comparativist’s fundamental ontology consists of comparative relations between objects at different times, as well as the present. It is this whole web of relations which grounds the (metaphysically non-fundamental) values physical quantities take on, according to the comparativist. But a physical theory is normally taken to be deterministic if the values of these quantities at a time (plus the laws) are sufficient to determine their values at all times. In other words, determinism in physics is normally defined in terms of entities that are fundamental for the absolutist, but not the comparativist. Moreover, the comparativist’s fundamental ontology is extremely spare when restricted to a single instant of time. For, as we saw before (n. 2), fundamental relations between objects at different times are crucial if the comparativist is to allow for metaphysical possibilities like the doubling of all objects’ masses. Nonetheless, on the most obvious picture of laws for the comparativist, it is natural to treat only the relations between objects at t as a world w 0 s initial conditions when we ask whether w is deterministic. For example, in his discussion of laws, Dasgupta (2013) reinterprets the absolutist laws featured in physics textbooks in terms of fundamental

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84    comparative relations. In particular, he offers a comparativist version of Newton’s Second Law, F = ma. He begins by identifying those parts of the law’s content that can be understood in the comparativist’s fundamental terms. For example, the Second Law entails that, since my brother is 1.16 times as massive as I am, whenever we’re both accelerating at the same rate, the force acting on him will be 1.16 times as great as the force on me. Generalizing from examples of this sort, Dasgupta captures the entire comparativist-friendly content of the Second Law as follows: (L2) For any material things x and y, For any reals r₁ and r₂, if x is r₁ times as massive as y and is accelerating r₂ times the rate of y, then x has r₁r₂ times as much force acting on it than y. For any real r₃, if x has r₃ times as much force acting on it than y, then there are reals r₄ and r₅ such that r₄r₅ = r₃, and such that x is r₄ times as massive as y and is accelerating r₅ times the rate of y (Dasgupta 2013, 18–19)⁶ Dasgupta then offers an argument that his comparativist law (L2) reproduces all of the measurable predictions made by the Second Law—a claim we’ll return to in section 5. But assuming he’s correct about this— and his argument generalizes to more complicated physics—the seeming absolutist character of the laws we find in our physics texts is evidently not necessary for their empirical success. Setting aside for a moment the empirical predictions of (L2), it’s pretty clear how we should apply the Laplacean definition of determinism to such a law. The state of the world at t is given by all the fundamental comparative relations that hold between objects at t (in Dasgupta’s example, all the ratios of forces to forces, masses to masses, distances to distances, etc.). A world is then deterministic iff the fundamental relations between objects at t, plus the laws, determine all the other fundamental relations (including the relations between objects at different times).⁷ ⁶ Note that additional direction relations between forces and accelerations will be needed in more than one dimension; Dasgupta ignores these for brevity. ⁷ As we will see in section 4, the initial conditions for instant-determinism may also have to be extended to include relations between objects at times falling in an infinitesimal neighborhood of t.

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The same general approach to laws is adopted by Field (1980). Although Field does not commit himself to a particular metaphysics of laws, he seems to recognize an imperative to state the laws of his nominalist physics in terms of his primitive comparative relations. For example, he formulates Poisson’s equation (for non-zero mass) as follows: “[A]t any two points where the mass density is not zero, the ratio of the Laplaceans of the gravitational potential is equal to the ratio of the mass densities” (Field 1980, 79). Although ratios (and Laplaceans, for that matter) are not among Field’s primitive relations, he shows how to formulate this sort of statement purely in terms of the congruence relations that he does count as primitive (Field 1980, 68–70). So Field’s laws could be written out in full detail using only his primitive (comparative) relations. The corresponding initial data for a possible world on Field’s theory would then be the comparative relations between spacetime points lying on one time slice. It is also possible for comparativist laws to make use of absolutist-style values, however, using some of the resources of measurement theory. Given a sufficiently rich and constrained system of comparativist relations for a quantity like mass, a representation theorem can be proven. Such a theorem establishes the existence, for every possible arrangement of fundamental comparativist relations between physical objects, a mapping or homomorphism from the objects into some mathematical structure—for example, the real numbers or a vector space. We call this mapping a “homomorphism” because it doesn’t just map the physical objects to the mathematical objects; it does so in such a way that the structure of the mathematical set parallels the structure of the fundamental relations between the physical objects. As a simple example, on one possibility according to Dasgupta’s theory of mass, there are three objects A, B, C, with B0 s mass being twice A0 s and C0 s mass being three times as great as B0 s. One homomorphism from these objects into the real numbers would take A to 1, B to 2, and C to 6. Another would take A to 2, B to 4, and C to 12. A representation theorem for mass on Dasgupta’s theory would establish the existence of such homomorphisms for any possible set of massive objects and fundamental mass relations. A couple of points are worth noting immediately. First, a homomorphism of this sort is not unique, and the resulting absolute values

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86    are unique only up to some factor, such as multiplication by a constant— an expression of the fact that comparativist relations are scaleindependent. Second, the domain of a homomorphism includes all of the objects in a comparativist world, at all times. Thus, the absolute value an object is mapped to will in general represent that object’s relations to past and future objects as well as present ones. That said, given a representation theorem for a particular comparativist theory, it is possible to characterize laws for that theory in a way that mimics absolutist laws as closely as possible. We can formulate a comparativist version of an absolutist physical law expressed by a mathematical equation, like F = ma, by positing the existence of a homomorphism such that the absolute values the objects are mapped to obey that equation. In the particular example of F = ma, we might state the comparativist law as follows: (L2*) For any physically possible world w, for any homomorphisms F(x), m(x),a(x) where F is a homomorphism for the force relations, m for the mass relations and a for the acceleration relations, F(x) = m(x)a(x) for all objects x ∈ w.⁸ The natural definition of determinism—and, in particular, of initial conditions—looks quite different on this picture of laws. For it seems natural to define the initial conditions at t as the values assigned to each quantity by the homomorphisms. (Note that this is not the same as the intrinsic state of the world at t, which includes only the comparative relations between objects at t. Rather, it includes information about relations with objects at other times as well.) On this approach, a world is deterministic iff the values assigned by homomorphisms to past and future objects are fixed by the laws (which will take the form of (L2*)) plus the values at t. There is no guarantee that these two definitions will agree about which worlds are deterministic. Which definition is superior, from the perspective of comparativism? In other words, should the comparativist believe in laws like (L2) which govern the fundamental relations, or laws like ⁸ This form for comparativist laws was suggested to me by Ted Sider.

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(L2*) which govern the (non-fundamental) values assigned to each quantity by homomorphisms?⁹ Laws like (L2*) will no doubt be useful for the comparativist in extracting predictions from physics, and in relating a comparativist reformulation of physics to existing absolutist theories. However, (L2*) seems to me a very poor candidate for a fundamental law of nature, and its corresponding definition of determinism seems inappropriate given comparativism. It is a familiar platitude that, while fundamentality may be a brute concept with no definition, the fundamental properties and relations “are the properties and relations that occur in the fundamental laws of physics” (Arntzenius 2012, 41). On Lewis’s popular Humean account of laws, for example, the fundamental laws are regularities in the instantiation of fundamental properties (Lewis 1983, 368). And altering this feature of Lewis’s system would rob it of much of its interest. Our best theories of physics have a particular mission: to describe the universe at its most fundamental level. Insofar as they fail to do so, either through inaccuracy or by failing to describe reality in fundamental terms, we should take that as a sign that the true fundamental laws have not yet been discovered.¹⁰ Moreover, although determinism in non-fundamental laws is a topic of great foundational interest, when we ask whether our world is deterministic, for purposes of metaphysics we are most interested in whether things are fundamentally deterministic. For example, if the fundamental facts about the present did not determine the fundamental facts about the future, one can certainly imagine how an incompatibilist would see a ray of hope for free will—even if some description of the present in nonfundamental terms did determine the corresponding non-fundamental facts about the future. Especially if these non-fundamental facts about the present were not, strictly speaking, intrinsic to the present, but included lots of information about the past and future as well. The facts expressed by an assignment of values to quantities via a ⁹ It may be more accurate to say that laws like (L2*) govern the fundamental relations indirectly, via posits concerning the existence of homomorphisms. ¹⁰ I take it that Field’s nominalist project, for one, arises from similar motives. If there is nothing wrong with formulating the fundamental laws in non-fundamental terms, it’s hard to see why the nominalist should be unhappy with laws of physics involving Platonist assumptions.

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88    comparativist representation theorem are exactly like that. To ask whether our world is deterministic when described in terms of those facts would be to change the subject, quite radically, away from the concept of determinism that matters for metaphysics. To cast the problem in more scientific terms, although determinism is a metaphysical thesis, it is one with epistemological implications. Determinism helps us derive predictions from a theory, and affects how we confirm the theory using these predictions (for example, the Principal Principle connecting chance with subjective credence is trivial for deterministic theories). It is a truism that we lack direct epistemic access to the fundamental facts about the future (unless something like time travel is allowed by the laws).¹¹ A notion of determinism in which the “initial conditions” include information about the future would thus appear remarkably ill-suited to the epistemic role determinism normally plays in science. That said, representation theorems are a useful tool for studying the relationships between comparativist and absolutist laws of nature. Most physical theories do not have extant comparativist formulations (although in many cases there are obvious ones in the offing), and none has a single canonical comparativist version. Yet we would like to answer questions of the following sort: Does a given absolutist scientific theory (e.g., special relativistic electromagnetism) admit a comparativist formulation which is also deterministic? Obviously this is not the sort of question that could be answered by examining comparativist versions of the given theory one by one, even if the theory possesses known comparativist versions. But there is a way to answer this general question in at least some cases. As we saw in section 2, the interesting varieties of comparativism are the ones that replace absolutist ontology with an ontology of scaleindependent relations. This places a limit on the amount of ontological structure a comparativist can ascribe to a world. A system of comparativist relations can only distinguish between two worlds—or (which is ¹¹ Since we also have no direct access to many facts about the present, at least according to relativity, the Laplacean definition of determinism may not be the ideal one for epistemic purposes. But my point that the ideal definition will not include facts about the future in the initial conditions surely stands.

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equally interesting for our purposes) two time slices of worlds—if there is some scale-independent difference between those worlds. And this is a question we can ask about absolutist worlds as well. Thus, if there is no scale-independent difference between two physical possibilities of a theory written in absolutist terms, any comparativist version of that theory must identify those two possibilities as the same. And if two sets of initial conditions—two time slices of worlds—are indistinguishable in scale-independent terms, no (interesting) comparativist theory can treat those time slices as distinct. Determinism is a thesis about which time slices can fit into which overall histories for physically possible worlds. A world is deterministic when no time slice of that world can fit into any other, different possible world while still obeying the laws. The comparativist can “tell the difference” between two worlds (or time slices) just in case the differences between those worlds (or slices) are scale-independent.¹² Put together, these facts give us a necessary condition for determinism under comparativism: A world w (described in absolutist terms) may be deterministic under comparativism only if, for any time t, the scale-independent facts at t physically necessitate all other scale-independent facts about w.¹³ What we have, then, is a formula for looking at a possible world described in terms of ordinary (absolutist-looking) physics and determining whether the scale-independent facts a comparativist would count as fundamental (or fixed by the fundamental) could evolve deterministically under a comparativist version of the relevant laws. With this formula in hand, let’s look at some cases where determinism comes under threat.

¹² Here I assume that the comparativist will want a system of fundamental relations that doesn’t leave out any scale-independent differences between worlds. For example, a comparativist theory of mass on which the only fundamental relation is “more massive than” would be unsatisfactory. There may be more debatable cases, but none of the cases arising in my examples will be debatable ones, I think. ¹³ To clarify: in the comparativist version of w, time itself will not be an absolutist quantity, so in a sense it will be illegitimate to talk about an absolute value like t. But it will still be possible to define a time slice of a comparativist world as a maximally large set of (temporal stages of ) objects in w, all of which are related by the “equal time” relation.

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4. Escape velocity and at-at motion I’d like to start out by looking at an instructive example from Newton’s theory of gravity—a case in which determinism is threatened but can be restored by adopting further metaphysical commitments. This example will serve to illustrate why the comparativist should adopt the at-at theory of motion, as well as underscoring some relevant complexities that apply in other cases. The informed reader may be puzzled that the question of determinism in Newtonian gravity is even on the table. After all, it is well known that Newtonian gravity is an indeterministic theory, in which (for example) swarms of massive objects may swoop in at any moment from infinitely far away (Earman 1986, 23–37), balls may—or may not—spontaneously slide down a hill if it’s shaped just right (Norton 2003), and so on (Earman 1986, 37–53). Why should it be surprising or interesting if the comparativist version of an indeterministic theory itself exhibits indeterminism? What is interesting, to me at least, is the possibility that indeterminism might be more widespread in comparativist physics than it is in absolutist physics. Indeed, this possibility takes on pivotal importance when viewed in light of the fact that, whatever the de jure status of determinism in Newtonian gravity, the theory is de facto deterministic as it is ordinarily applied. In other words, there is some vaguely defined set of implicit posits made by working physicists which serve to rule out the indeterminism in Newtonian gravity for purposes of deriving predictions from the theory. Moreover, there is some hope that these implicit restrictions could be made explicit and rigorous by means of imposing boundary conditions and restrictions on physically admissible initial conditions (Earman 1986, 37–9, 52–3).¹⁴ Given all this, it seems we should be very interested in examples where ordinary absolutist Newtonian gravity, plus

¹⁴ Another example of a restriction on initial conditions is the popular “past hypothesis” sometimes posited to explain the time-asymmetry of thermodynamics. So classical indeterminism is not the only foundational problem that seems to call for this sort of solution.

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the necessary extra posits, is deterministic while comparativist versions exhibit indeterminism.¹⁵ If there are such examples, presumably they will appear in processes where the scale of some quantity helps determine scale-independent features of the outcome. So let’s look at the theory and see if we can find any. The force law for Newtonian gravity is, of course, F =G

m1 m2 r2

ð1Þ

where F is the attractive force between objects with masses m₁ and m₂, m3 r is the distance between them, and G = 6:67428  10 − 11 kgs 2 is the gravitational constant. Since none of the terms in this equation is scaleindependent, this isn’t immediately helpful in our search for potential comparativist indeterminism. But when it is combined with F = ma, we can derive a law in which scale-independent features of an experiment’s outcome appear to depend on scale-dependent features of the initial conditions. This is the law governing escape velocity. Like many concepts from physics, escape velocity admits both an intuitive and a technical definition. In this case the two are pretty close together. The intuitive idea behind a planet’s (or other object’s) escape velocity is: the velocity needed to “escape” from its surface past its orbit. The technical version is as follows: An object’s escape velocity ve is the magnitude of the velocity directed away from their common center of mass that a projectile¹⁶ initially located at its surface would need to asymptotically approach an infinite distance from the object in the limit of infinite time, if the two were alone in an otherwise empty universe.¹⁷ In other words, a projectile initially located at the Earth’s surface will continue to move farther from the Earth forever, without any

¹⁵ If the reader remains nagged by the thought that indeterminism in Newtonian gravity is nothing new, section 5 will provide an example of comparativist indeterminism in classical electromagnetism, a theory whose deterministic credentials are hard to dispute. ¹⁶ By “projectile” I mean an object with some initial velocity which is never subject to any external force aside from gravity. ¹⁷ This is the definition of barycentric escape velocity (escape velocity relative to the common center of mass of two objects).

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92    limit to its eventual distance, if and only if its initial velocity¹⁸ exceeds the Earth’s escape velocity. Clearly—and crucially—whether a projectile escapes is a scale-independent fact. The important thing about escape velocity for our purposes here is that it is a function of the combined mass of the planet and projectile— which implies that its value will change if the mass of both objects is doubled. In effect, the force law (1) implies a physically necessary relationship between the total mass (and the planet’s radius) and the escape velocity. This means the theory can “tell the difference” between different initial conditions that the comparativist will identify as the same instantaneous state, since the differences between them are not scale-independent.

4.1. Earth and Pandora Let’s put a detailed example on the table. The escape velocity for a projectile of mass m from a planet of mass M and radius r is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2GðM + mÞ ; ve = r

ð2Þ

expressed in the rest frame of the planet.¹⁹ From this equation it follows that if we double the mass of both objects (transform M to 2M and m to pffiffiffi 2m), the escape velocity increases from ve to 2ve .²⁰ As an intermediary example, consider a universe containing two planets, Earth and Pandora, located far enough apart that for all practical purposes we can ignore their gravitational interaction. These two planets are identical in all their physical properties except that Earth’s mass (ME) is half of Pandora’s mass (MP). Suppose that initially each planet has a

¹⁸ That is, its initial velocity relative to the center of mass of the combined Earth-projectile system. ¹⁹ This is the solution to the gravitational two-body problem for such a system, which is qffiffiffiffiffiffiffi

different from the usual textbook escape velocity equation, ve = 2GM r —a useful approximation when m is much less than M. ²⁰ For a derivation of the escape velocity law, see Halliday, Resnick and Walker (1997, 331).

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projectile located at its surface with some initial outward velocity of magnitude v. We may stipulate that the projectile on Earth has mass m and the one at Pandora has mass 2m, so that the system of Pandora and its projectile is a mass-doubled duplicate of the system of Earth and its projectile. Then, as noted above, if vE is Earth’s escape velocity and vP is pffiffiffi Pandora’s, vP = 2vE . Now suppose that vE < v < vP , that is, the projectiles both have a velocity greater than Earth’s escape velocity but less than Pandora’s. Then the system of Earth and its projectile will behave quite differently from its mass-doubled counterpart Pandora. The distance between Earth and its projectile will keep growing forever as the projectile shoots off “to infinity.” Pandora’s projectile, on the other hand, will only travel finitely far away, remaining confined by the more massive planet’s gravity. The laws of gravity seem capable of telling the difference between Earth and Pandora, despite the fact that they are indiscernible except for the difference in their masses. And since we stipulated that there was no measurable interaction between them, it should be obvious that the laws will recognize the same difference between the initial state of a universe containing only Earth and its projectile, and an initial state containing only Pandora and its projectile.²¹ In the Earth universe, the projectile will fly off to infinity. In the Pandora universe, on the other hand, the projectile will go only finitely far before stopping.

²¹ I say this should be obvious because, for the practicing physicist, approximately isolated systems can generally be treated as if they were entirely isolated (where “approximately isolated” means they are subject to almost no external forces, or other non-force external influences like quantum entanglement). This principle is what allows us to idealize an approximately isolated system, treating the balls on our pool table as if they were colliding in a vacuum instead of surrounded by a bunch of other objects (for example). This is not to say that there couldn’t be a good scientific theory in which even isolated systems behave differently in the absence of their environment. But there’s an obvious scientific advantage to theories that do have this feature (they make it possible to idealize in useful ways that actually work in practice), so there is (and should be) a strong preference in favor of theories which predict that approximately isolated systems behave almost exactly the same as completely isolated systems. Now, as a metaphysical matter, under comparativism Pandora and Earth are far from “isolated,” insofar as each planet’s fundamental nature is partly constituted by its relations to the other planet. But as a matter of physics, if comparativism entails that approximately isolated systems (in the sense stipulated above) cannot be idealized as fully isolated, this saddles comparativism with a serious disadvantage relative to absolutism. In my discussion of this example, I will assume that the comparativist is successful in avoiding this disadvantageous commitment. (Thanks to Shamik Dasgupta for pressing me to articulate this premise.)

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94    This example threatens indeterminism for any comparativist interpretation of Newtonian gravity. We saw in section 3 that determinism only holds for the comparativist when the scale-independent facts about the world’s initial state determine the scale-independent facts about other times. But the (instantaneous) initial states of the Earth and Pandora worlds agree about all scale-independent facts. The two initial states are related by mass doubling, so by our definition of scale independence they must agree on all scale-independent comparative relations. For the comparativist, the initial state of the Earth universe and the initial state of the Pandora universe are really the same initial conditions. But these two worlds’ futures differ in physically significant ways that are clearly scale-independent. The differences between the future evolution of Earth’s universe and Pandora’s show up in the comparativist’s fundamental relations as well as the absolutist’s intrinsic values. Earth’s projectile will continue to move away from the planet forever without limit; Pandora’s will not. So if ratios or orderings of distances and times (for example) are among the fundamental relations, these will be quite different in the two cases. The distance between Earth and its projectile at some time will always be greater than the distance at an earlier time. The distance between Pandora and its projectile, on the other hand, will eventually begin to decrease.²² Since there are scale-independent differences over time between these worlds, the comparativist can certainly recognize that both Pandora’s universe and Earth’s are physically possible. But only at the expense of denying determinism. There can be no fundamental difference, for the comparativist, between the initial state of Earth’s universe and the initial state of Pandora’s. Since there is a difference between the futures of these two initial states, it must be that the same initial state can evolve into two

²² As noted on this page 6 in n. 3, the comparativist could in principle save determinism by positing a system of fundamental relations so austere that possibilities in which a projectile escapes are treated as equivalent to possibilities in which it does not. But this would require leaving out even relations as weak as “d is a greater distance than d,” thus leaving it unclear how the resulting ontology could even represent our ordinary acquaintance with the quantity of distance. This option is obviously unsatisfactory.

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or more distinct future states while obeying the comparativist’s laws. This is the very definition of indeterminism.²³ Or so it seems. But as Kenny Easwaran has pointed out to me, the argument of this section smuggles in a hidden assumption: that initial conditions are to be understood as truly instantaneous time slices. The comparativist can maintain determinism in the face of my example by denying this assumption—a plausible move on its own merits, as we shall see.

4.2. A hidden premise: instantaneous initial conditions The nature of the hidden premise will become apparent after a quick look at a puzzle surrounding the nature of velocity. In mechanics, the state of the universe at a time is normally thought to include the velocity of every material object at that time. But an object’s velocity is ordinarily defined as the derivative of its position with respect to time, v = dx=dt. And this quantity’s definition involves more than just the way the world is at the exact moment t. The derivative of an object’s position x at t is the limit: vðtÞ =

dx xðt 0 Þ − xðtÞ ðtÞ = lim t 0 !t dt t0 − t

ð3Þ

This limit is not a property intrinsic to t; it depends on the infinitesimal neighborhood of points in time before and after t.²⁴ So the notion of instantaneous velocity as a component of a scientific theory’s initial conditions would appear to be confused. There are a few options for making sense of this puzzle (Arntzenius 2000). One is to posit that velocity is a truly instantaneous, intrinsic ²³ Although the escape velocity example is particularly straightforward, a more general argument for indeterminism is also possible. If all masses in the universe are doubled, this will double the acceleration of all massive objects, which will have scale-independent effects on the ratios of present and future velocities. It is less clear, however, that the comparativist can’t get rid of this indeterminism by positing fewer fundamental relations, per n. 3 above. ²⁴ When I say the limit depends on t’s “infinitesimal neighborhood,” I do not mean to imply that there is any unique such neighborhood. Rather, I mean that the limit is not determined merely by the position at t, but is determined by the positions in any interval ðt − δt; t + δtÞ.

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96    property of the world at t. In that case, velocity is not identical with the time derivative of position, and Eq. (3) cannot be the definition of the velocity at t. Rather, it must express either a physical or metaphysical necessity. This view has a revisionary flavor that may seem troubling, since it leaves open at least the conceptual possibility that velocity and the derivative of position could disagree. An attractive alternative is the at-at theory. On this theory of motion, Eq. (3) is the definition of velocity, which is not truly an instantaneous quantity. Rather, an object’s velocity at t is a property extrinsic to t but intrinsic to t 0 s infinitesimal neighborhood. And since velocity is a part of the initial conditions for classical theories of mechanics, it must be that these initial conditions don’t correspond to a truly instantaneous state at a time. Rather, the “state at t” mentioned in our definition of determinism really refers to the state over a vanishingly small temporal neighborhood of t. While there are no true instantaneous velocities on this view, the limit of the velocities over smaller and smaller intervals around t serves the same purpose while maintaining the ordinary calculus definition of v(t). My argument from the Earth/Pandora examples to indeterminism covertly assumed the first, truly instantaneous picture of initial conditions. The argument proceeds by identifying the initial conditions at t with the scale-independent facts about position and velocity intrinsic to t0 s instantaneous time slice. But according to the at-at picture, and its associated picture of initial conditions, none of the velocity facts is intrinsic to this time slice, not even v(t). And velocity is a necessary component of the initial conditions for any theory of mechanics— specifying just the positions at a single time while leaving out the velocities is not sufficient to determine the future evolution of the state, even granting ordinary absolutist assumptions. So by identifying the initial conditions with the facts intrinsic to t, I have covertly assumed the instantaneous velocity view. What happens to the argument if we instead assume the at-at view? On that view there is really no fundamental quantity of velocity— position is the only fundamental quantity, and velocity is defined reductively from position via Eq. (3). But the initial conditions at t consist of any facts that obtain in t’s infinitesimal neighborhood—that is, facts that

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hold true of any finite interval ðt − δt; t + δtÞ before and after t, no matter how small. Suppose now that t is the initial stage of a universe like Earth’s or Pandora’s, with a projectile initially moving away from a planet of mass M. Assuming absolutism, of course, the facts about t’s neighborhood together with the gravitational force law allow us to predict whether the projectile will escape. But what if we assume comparativism? In other words, do the scale-independent facts about t’s infinitesimal neighborhood determine whether the projectile will escape? Consider, for simplicity’s sake, the special case where the projectile’s mass equals the planet’s mass (M = m). (The point that follows can be extended to the general case where M≠m.) In that case the escape velocity is rffiffiffiffiffiffiffiffiffiffi 4GM ve = : r From the gravitational force law (1) and F = ma, we know that the acceleration of the projectile at t is GM=r 2 . (The acceleration a is dv/dt, which is also determined by the projectile’s position r(t) in t’s infinitesimal neighborhood just as v is.) Comparing this with the escape velocity, pffiffiffiffiffiffiffi we see that ve = 4ar , so the projectile will escape if v2 > 4ar. Is it a scaleindependent fact whether this inequality holds? Multiplying mass by a scalar doesn’t change it, of course, since mass doesn’t appear. What if we change the scale of other quantities? In terms of the relevant fundamental quantity, the position r, the inequality is ðdr=dtÞ2 > 4ðd 2 r=dt 2 Þr:

ð4Þ

But if we multiply r by a scalar c, this will just multiply both sides of the inequality by c², which won’t change whether it holds. Whether the inequality holds or not is a scale-independent fact about t’s neighborhood. Since the truth or falsehood of the inequality determines whether v > ve , it determines whether the projectile will escape. Therefore, the scale-independent facts about the initial conditions determine whether the projectile will escape, if we understand initial conditions in the at-at theorist’s way. (The argument here is due to Easwaran.)

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98    So, if we leave out the hidden premise, the argument for indeterminism does not succeed. Does this give the comparativist good reason to deny the premise—to deny that initial conditions are instantaneous? I believe so. The goal of any comparativist interpretation of Newtonian gravity should be to achieve the same scientific aims as the absolutist version of the theory, without the absolutist’s more elaborate ontology. If the comparativist and absolutist theories differ in their predictions—if they differ about which physical possibilities are deterministic—the comparativist has failed in this goal. Indeterminism by itself is not so bad, of course. We’re happy to accept the possibility of objective chances in quantum theory. But indeterminism in classical physics is more troubling, since the theory includes no probability measure over outcomes to interpret as chance. In the case of escape velocity, an indeterministic comparativist interpretation will simply predict that a projectile near a planet’s surface will either escape its orbit or not, without offering any statistical predictions about the likelihood of these outcomes. So the comparativist should prefer the at-at theory of motion over the alternative view that there are instantaneous velocities. A way to preserve instantaneous velocities and initial conditions may seem to suggest itself: just make acceleration an instantaneous fundamental quantity as well. Whether a projectile will escape is determined by whether inequality (4) holds. It appears that knowing the scale-independent facts about position, velocity, and acceleration is sufficient to determine this fact. And the view that acceleration, as well as velocity, is an instantaneous fundamental quantity has been independently defended (Lange 2005). Adding instantaneous accelerations as well as velocities to the mix will not suffice to determine this, however. Not if they are fundamental quantities distinct from position, as the instantaneous view has it. For then their relationship with position is no longer definitional—velocity is not defined as the derivative of position, but rather there must be some law that makes this the case. And this means it doesn’t follow that when we change the scale of position, for example by doubling it, we must also double the value of velocity and acceleration. These are separate fundamental quantities, and thus it will make sense from the comparativist point of view to change the scale of one without changing the scale of

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the others. But it was the fact that changing the scale of position automatically changed the scale of velocity and acceleration by the same factor, on the at-at view, that allowed us to determine whether inequality (4) would be satisfied. That response to the Earth/Pandora argument does not succeed if we assume that position, acceleration, and velocity are separate fundamental quantities. We’ve seen that if the comparativist adopts the at-at picture of motion, as they should, the Earth/Pandora examples do not imply that a comparativist version of Newtonian gravity must be indeterministic. But more involved examples will bring back the specter of indeterminism.

5. Indeterministic examples and metaphysical modality What is the factor that prevents the Earth/Pandora examples from exhibiting indeterminism, on the at-at view? The fact that certain comparative relationships between the different derivatives of position—velocity and acceleration—are scale-independent is what sinks the otherwise promising argument for indeterminism. One way to make some of these relationships either trivial or undefined is to set one or both of these derivatives to zero. So, one way to start looking for indeterminism is to study systems with zero initial acceleration. The example I have in mind is one I’ll call Friction World (because the details can be filled in with an idealized Newtonian account of frictional forces). Think of the mass-m hockey puck in Figure 1 as sliding over some surface with an initial velocity v, feeling no forces at all. But upon entering the shaded region, it feels a constant force F in a direction opposite v. F = ma tells us that the resultant acceleration will slow the puck (reducing the value of v). The shaded region is only L meters wide, so if the puck can make it L meters without slowing to a stop, it will continue moving in the leftward direction forever afterward. Suppose we describe these initial conditions solely in scale-independent terms. Here’s a question we won’t be able to answer: will the puck make it past the shaded area and continue moving left? To answer this question, there would need to be a scale-independent difference between the different possible initial conditions for Friction World. But suppose that v is just

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100    L

v F

m

Figure 1 Friction World

barely great enough for the puck to make it past the shaded region. Then, if we double F—a transformation that makes no scale-independent difference—the puck will no longer be moving fast enough to make it. Or suppose the puck is just barely too slow to make it past. Halving F will halve the puck’s acceleration as it moves over the shaded region, permitting it to make it past.²⁵ As in the escape velocity case, the differences between the possible outcomes in which the puck stops or slides past are clearly scaleindependent. In some cases, the puck remains in the shaded area permanently. In others, it continues moving forever (which is to say, the scale-independent ratios between its distances from the shaded area’s edge at different times will keep increasing eternally). So here we have an indisputable case of comparativist indeterminism for a physical system that is deterministic on the ordinary, absolutist understanding of classical mechanics. Interestingly, the indeterminism in the Friction World example eventually “goes away” once the puck becomes subject to the force. There are stages of Friction World’s history whose infinitesimal neighborhoods do determine the scale-independent facts about the puck’s future. For the ²⁵ If we understand this example in terms of friction, these transformations correspond to doubling or halving the coefficient of friction.

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puck will immediately begin accelerating at a rate a = F/m once it enters the shaded area. At that point, we may ask again whether it will stop or not, and there is more scale-independent information to work with. Assume for the moment that the puck will eventually stop. It will take an amount of time equal to t = v/a to do so. Since the acceleration a will be constant, its average velocity will just be vavg = v/2. So it will cover a total distance vavg t = v2 =2a over the course of its journey. Clearly, this means it would have slid past the shaded area iff the area’s length L were less than this distance. So, in general, the puck will make it past iff v2 > L: 2a

ð5Þ

As we saw in our discussion of Earth and Pandora, assuming at-at motion, when we double the quantity of distance we must also double velocity and acceleration. So transforming the scale of distance will multiply both sides of this inequality by the same constant, leaving the same inequality. So it is a scale-independent fact whether (5) obtains. Thus, once the puck enters the shaded area and begins accelerating, the scale-independent data about Friction World’s present will determine the scale-independent facts about its future. The indeterminism disappears once the puck begins to experience the force.²⁶ Another peculiarity of this sort of example (understood the comparativist way): the indeterminism can actually reappear at later times if we change the example in the right way. For suppose we modify Friction World by adding a second, identical shaded region of force, some distance beyond the first (Figure 2). To make it past both regions, one after the other, will clearly require that v2 > 2L; 2a

ð6Þ

²⁶ This “disappearing indeterminism” is a feature shared with one of the (debatable) cases of indeterminism in classical mechanics. In the dome example described by Norton (2003), at the initial time it is indeterministic whether and when the ball will begin to slide down the dome (and which direction it will go). But if it does begin to slide down, it will move deterministically from then on (assuming we’ve ruled out other potential sources of indeterminism like “space invaders” from infinity).

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102    L

L

F

F

v m

Figure 2 Double Friction World

since the total velocity lost by the puck as it slides over two successive regions of length L will be the same as if it had to slide over one region of length L/2. At the time when it enters the first region, the scaleindependent facts will, as before, determine whether inequality (6) is satisfied and hence will determine the scale-independent facts about the future (such as whether the puck will pass the second region). But (assuming the puck gets past the first region) consider a time after it has left the first region and before it has entered the second region. At this point its acceleration will again be zero, and the scale-independent facts about the infinitesimal neighborhood surrounding this moment in time will not determine whether it will pass through the second region or be captured. The instantaneous situation will be exactly as in the original Friction World case. So the indeterminism has “gone away” while the puck was moving on the first surface, only to “come back” again between the two surfaces. A moment’s reflection will reveal another bizarre feature of this case. Suppose that, at the time it enters the first region of force, the puck does in fact satisfy inequality (6). Then, it is physically impossible for the puck to later stop within the second region. Its velocity is

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high enough that it must keep moving forever. But holding fixed only the instantaneous (neighborhood) state while it is somewhere between the two surfaces, it is physically possible for the puck to stop within the second region. (This is just another way of saying that the instantaneous state at the first time determines the future evolution, while the state at the second time does not.) Something very curious is happening: given the earlier state, it is physically necessary that the puck will keep moving forever, but given the later state, it is physically possible for the puck to stop moving. So the Double Friction case exhibits a sort of temporal action-at-a-distance. While the puck is in between the two regions, the entire past of the system physically necessitates an outcome that is not physically necessitated by the state at the present time. According to the comparativist, in a case like this the past can influence the future without said influence being mediated by the state of the world in the present.

5.1. A more realistic example These aspects of Friction World are odd and foundationally interesting. The reader may wonder whether they can be reproduced in a more realistic setting. Friction World is, after all, a toy model that bears little resemblance to any complete theory of physical forces. In fact, we can develop an analogous example in Newtonian gravity by modifying the escape velocity example, although idealizations will remain. The key to doing so is the gravitational Shell Theorem. This theorem has two consequences: the gravitational force outside a uniform spherical shell of mass M is the same as that from a mass-M particle located at the center of the shell; and a uniform, hollow spherical shell exerts no net gravitational force on any object inside it. Our new example, Shell World (Figure 3), is formed by modifying the Earth/Pandora-type examples. Instead of a planet, the initial state consists of a uniform, rigid shell of mass M, with a mass-m projectile initially located at the center of the shell with outward velocity v. By the Shell Theorem, the projectile will initially feel no force and its acceleration will be zero. Here comes the idealization: we must suppose that the projectile

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104    will somehow pass through the shell.²⁷ One possibility would be to just suppose that the objects in the example are permeable and only interact via the gravitational force. Another possibility would be to assume the projectile is a point particle and there is a point-sized gap in the shell at the place where the projectile’s trajectory will intersect the shell. This may be the best option, although the idea of a hole with no spatial extension may seem conceptually confused. Finally, we could make some assumptions about what will happen when the projectile collides with the shell (perhaps the shell will break apart?). I will assume one of these solutions is postulated. For our purposes it won’t matter which one. Regardless, once the projectile passes through the shell, it will immediately feel a gravitational force from the shell’s mass, just as if the shell were a solid planet with total mass M (this also follows from the Shell Theorem). At that point it will begin to accelerate toward the shell. The case will then become exactly parallel to the Earth and Pandora examples: the projectile will escape if v exceeds the shell’s escape velocity.²⁸ But think about the initial time, before the projectile exits the shell. At that point we may also ask whether the projectile will escape M

V R

m

Figure 3 Shell World

²⁷ A further significant idealization: we must ignore whatever forces hold the rigid shell together, preventing its gravitational collapse. This idealization strikes me as unproblematic, though, since filling in these details will not undermine the morals of the example unless they introduce some initial acceleration. ²⁸ If we postulate that the projectile collides with the shell and breaks it, the problem of whether it escapes will be more complicated to solve but not qualitatively different.

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or not. The ordinary way to find out, of course, is to compare the projectile’s velocity v with the shell’s escape velocity. But because there is no initial acceleration due to gravity, we may now pull the trick that made the Earth/Pandora cases look indeterministic: we may double the mass of every object in Shell World. And since the acceleration in Shell World is initially zero, the comparativist may not respond by using inequality (4) to determine whether the projectile will escape. The scale-independent facts about Shell World are not changed when we double the mass, even if we assume the at-at theory of motion. So the type of indeterminism that at first appeared to be present in the Earth/ Pandora examples is actually present in Shell World.²⁹ As in Friction World, the indeterminism is temporary: once the projectile passes through the shell and feels the force, the instantaneous state will determine the future evolution. What about temporal actionat-a-distance? Can we get the indeterminism to “come back” by modifying the Shell World case, the way we did with Friction World? This may seem challenging (what if we assume the shell breaks?), but actually it is easy to construct a case of temporal action-at-a-distance by modifying Shell World. The key ingredient is the time reversal invariance of classical mechanics: since the history of Shell World is physically possible, so is the time reverse of that history. Consider this timereversed copy of Shell World. In effect, the projectile will begin at the right of the diagram in Figure 3 and move to the left, entering the shell.³⁰ At the end of this process, it will be at the center of the shell, moving to the left with velocity v. In effect, this temporal stage of the time-reversed Shell World will be the mirror image of Shell World’s

²⁹ An anonymous referee has raised a concern about Shell World and related examples. Note that the initial conditions of Shell World are very unusual: only a tiny class of physical states will exhibit the sort of symmetry required to guarantee zero initial acceleration. Indeed, these states make up a set of measure zero in the space of all states. Does this undermine the foundational importance of the example? I don’t see why it should. The significance of some pathological or peculiar behavior of certain states is not lessened unless the states themselves can be ruled out as somehow unphysical or uninteresting. The fact that they are rare or atypical states should not by itself disqualify them in this way. ³⁰ Depending on how the collision is handled, the pieces of the shell may also converge to form the shell.

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106    initial conditions. As before, the projectile will escape iff v is greater than the shell’s escape velocity—which is not determined by the scaleindependent facts about this temporal stage. But if we rewind time to a point before the projectile entered the shell, the instantaneous state at that time will determine whether the projectile will eventually escape the shell’s orbit. For at that earlier time, we will have scale-independent data about the acceleration the moving object undergoes due to the shell’s gravity—and we know it will feel the same acceleration when it leaves the shell once more. More generally, since the state of the modified Shell World while the projectile is inside the shell is just the mirror image of the initial state of the original Shell World—and the past history of the modified Shell World is the time reverse of the original Shell World—we know that the future of the modified Shell World will necessarily be the mirror image of the time reverse of its past. So the past history of this modified Shell World physically necessitates its entire future, but the instantaneous state while the projectile is inside the shell does not. Just as in the Double Friction case, the comparativist must accept temporal action-at-a-distance. It is worth mentioning that a close analog of the Shell World example also exists in classical electromagnetic theory. Although the theory is relativistic, I will describe the initial conditions for this example in one frame (the rest frame of the shell). Since there is also an electromagnetic Shell Theorem, we may construct an example in which a positively charged projectile is initially located at the center of a negatively charged shell, with some initial outward velocity. All that’s needed is to modify the Shell World example so that the projectile has electric charge Q and the spherical shell has charge –Q. We can call the modified example Charge World. Once the projectile passes through the shell, it will be attracted to the shell’s charge by a force very closely approximated by the Coulomb force³¹:

³¹ This is the equation for electrostatic force, which is a close approximation to the electrodynamic Lorentz force law, F = qE + qv  B which holds exactly in this frame in special relativistic electromagnetism. The electromagnetic Shell Theorem is not an approximation, so the essential features of the example are unchanged by the Coulomb approximation. Here, the constant k = 8:988  10 − 9 N  m2 =C2 (in units of meters, newtons of force, and coulombs of charge).

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The scale-independent facts about the initial state of Charge World will not determine whether the projectile escapes this attractive force. This example is especially interesting because relativistic field theories like electromagnetism are entirely deterministic as ordinarily understood, unlike Newtonian gravity (Earman 1986, 55–78). So here we have a case in which comparativism and absolutism disagree, with no caveats needed, about whether a certain physical theory has indeterministic solutions. And since classical electromagnetism is also time-reversalinvariant, one may modify the example in the same way that we modified Shell World to introduce temporal action-at-a-distance.

5.2. Metaphysical modality In light of these examples, I want to consider the proposition that comparativism about quantity is not only true, but metaphysically necessary. I’ll proceed from the assumption that any reasonable system of physical laws is the laws of some metaphysically possible world(s). The motivation for this assumption is as follows: In the past, physicists considered seriously the possibility that Newton’s laws were true. Indeed, they remain interested in many counterfactuals concerning what would be true if Newton’s laws were the actual laws.³² It would be bizarre if, in doing so, past and present physicists were entertaining a hypothesis as impossible as the proposition that water is not H2 O, or the existence of an all-red, all-green object, or an object with a mass of both one gram and five grams. The view that these laws are metaphysically impossible but logically possible would entail this bizarre consequence. Moreover, the laws of our present-day best theories (general relativity and quantum field theory) are known to be false, although they approximate the truth

³² One reason for this is that Newton’s laws are very good at approximating the more accurate laws of relativity and quantum theory. Another reason is that they share many broad physical principles (which one might, following Lange (2007), call “meta-laws”), such as conservation laws, with more accurate theories.

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108    closely in broad domains. Again, it would be bizarre to suppose that we entertain a metaphysical impossibility every time we apply these laws.³³ By a reasonable system of laws, I mean a collection of putative laws that good physicists might hypothesize to be the fundamental laws of nature if presented with the right sort of experimental data. Newton’s laws are like this: faced with the experimental data available at that time, the best scientists of the early modern period hypothesized that they were the fundamental laws. The laws of Friction World—Newton’s laws of motion plus a toy force law—seem to me like reasonable laws in this sense. I am certain that the laws of Shell World and Charge World—the laws of Newtonian gravity or classical electromagnetism, plus perhaps some laws about collisions which I’ve neglected to fill in—are reasonable. I’ve been a bit unspecific just now. By “the laws of Friction World (or Shell/Charge World),” do I mean the absolutist version of these laws or some comparativist version of them? This question gets to the heart of the matter. I have no argument that comparativist laws for Friction World or Shell World are unreasonable. But it seems clear to me that the absolutist version of Friction World’s (or Shell/Charge World’s) laws—as distinct from any possible comparativist laws—are reasonable laws that ought to be true of some metaphysically possible world. For it would be reasonable, in a case like Friction World or Charge World, to posit that the laws of nature are deterministic. It would be entirely reasonable, after all, for physicists, faced with the sort of experiments that might lead them to accept classical electromagnetism, to suppose that the laws are deterministic under initial conditions like those of Charge World. All the observable predictions of classical electromagnetism are consistent with determinism, and so it would seem rather ad hoc to postulate (as the comparativist must) that the theory is indeterministic under certain narrow special conditions

³³ Arguments for so-called causal structuralism have been thought to imply that non-actual but reasonable-seeming laws are metaphysically impossible (Shoemaker 1998). The thought is that, for example, it is essential to the quantity of charge that opposite charges attract, and so a universe in which opposite charges repel is a metaphysical impossibility. But as Fine (2005, Sec. 3) has shown, this implies only that non-actual laws may have to involve alien properties, or the absence of familiar fundamental properties (e.g., a world with no charge), not that they are metaphysically impossible.

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despite otherwise exhibiting determinism.³⁴ Even more plausibly, a theory that avoids temporal action-at-a-distance under any conditions might be preferred. Lange (2002, 7–17), for example, has argued for a principle of temporal locality according to which there must be no temporal gaps between an event and its causes. If we understand causation to involve physical necessitation, this principle is violated by comparativist physics in cases like Double Friction and the analogous modification of the Shell World example. Again, I’m not claiming that the comparativist’s laws for Friction/Shell/Charge World are unreasonable laws. But I do maintain that some deterministic system of laws governing these cases must be reasonable—and since any such laws will be absolutist, I conclude that absolutism must be a metaphysically possible thesis. It is, of course, open to the comparativist to maintain that his view is metaphysically necessary by denying my principle that reasonable laws must be metaphysically possible.³⁵ At this point it becomes difficult to respond, beyond registering my disagreement. But readers who trust their modal intuitions about specific cases more than I do may find themselves filled with the intuition that it is metaphysically possible for Friction World, Shell World, or Charge World to exhibit deterministic, temporally local laws. Such readers will then be obliged to agree with the thesis of this section, even if they disagree with my principle about the metaphysical possibility of reasonable physical laws.

6. Conclusions I have argued that comparativism is not metaphysically necessary, and that it brings with it certain theoretical commitments. Namely, comparativists should accept the at-at picture of motion (or some close cousin of it), and should also accept that temporal action-at-a-distance is possible according to their view (unless they want to deny that Friction World, ³⁴ An alternative would be to stipulate that Charge World’s initial conditions are physically impossible, but this appears equally ad hoc given the straightforwardness of the example. ³⁵ I expect this would be Dasgupta’s position, since he takes seriously the proposition that only the actual world is metaphysically possible (Dasgupta 2016).

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110    Shell World and Charge World are real possibilities). What does this imply about the truth or falsehood of comparativism? Do we have any new reasons for rejecting the view? To begin with, some will be attracted to the view that the true theory of quantity must be metaphysically necessary. If the argument of section 5.2 is correct, comparativism cannot succeed on these terms. But for my part, I don’t see why a theory of quantity should have to be metaphysically necessary in order to be true of our world. It seems entirely plausible to me that the space of possibilities includes both absolutist and comparativist worlds. Kripke has taught us, of course, that certain scenarios which are logically and (in some sense) conceptually possible are nonetheless not intelligible ways a universe could be. But it seems obvious, from the existence of both comparativism and absolutism as formal theories of quantity with well-defined models, that absolutism and comparativism are both intelligible ways for a universe to contain quantities. I take this to be a strong indication that both views are metaphysically possible. If one is persuaded by arguments like those of Lange (2005) that velocity must be an instantaneous fundamental quantity, one should probably reject comparativism, to avoid accepting indeterminism in a wide variety of physically important cases like the example of escape velocity. But at-at motion is pretty plausible on its own merits, so I don’t see this as a huge theoretical cost for comparativism. The sort of disappearing, reappearing indeterminism exhibited by the comparativist versions of Friction World and Shell World is pretty peculiar, as is the temporal action-at-a-distance that arises in these examples. On the other hand, these two examples are rather idealized (especially Friction World) and don’t much resemble any of the physical possibilities according to our actual best theories. So, at most, they commit the comparativist to the metaphysical possibility of temporal action-at-a-distance. Again, this does not strike me as a huge cost— especially given that determinism in Newtonian physics is already known to be a vexed issue. The most troubling example is Charge World. Here we have an example where the comparativist must disagree with the absolutist—and with physics as ordinarily understood—about whether a relativistic theory of

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force is deterministic and temporally local. Classical electromagnetism is non-fundamental, which draws some of the sting. By itself, Charge World shows at most that comparativism would lead to unwelcome pathologies if electromagnetism were the fundamental theory of our universe. But classical electromagnetism is also a limiting case of quantum electrodynamics, one of our most fundamental quantum field theories. This raises the uncomfortable possibility that our most fundamental theories might also exhibit chanceless indeterminism and temporal action-at-adistance under comparativism. This is a difficult question to address, given the outstanding controversy over which quantities are fundamental in our most successful quantum theories. And even if those theories were better understood, they are known not to be truly fundamental. But given comparativism’s track record with determinism and temporal locality in classical theories, I would not want to bet on its success in quantum theory. Comparativism should be explored further, but metaphysicians who accept the theory run the risk that it might lead to pathologies in fundamental physics as bad as the ones it causes for classical physics. Absolutism is a safer option in this regard.

Acknowledgments First off, major thanks are due to Shamik Dasgupta for a long and deeply edifying correspondence over the entire course of this project, which included many helpful comments on previous drafts. Thanks to Kenny Easwaran for catching a crucial error in a previous version, and to Charles Sebens and Ted Sider for big piles of cogent comments that led to major changes in my approach here. Comments and corrections from Gordon Belot and Nick Stang were also very valuable. Thanks also to the Crop Circlers who met in April 2011 to discuss an early ancestor of this paper: David Manley, Sarah Moss, and Eric Swanson.

References Arntzenius, Frank. (2000). “Are There Really Instantaneous Velocities?” The Monist 83: 187–208. Arntzenius, Frank. (2012). Space, Time, and Stuff. Oxford: Oxford University Press.

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112    Bigelow, John and Robert Pargetter. (1988). “Quantities,” Philosophical Studies 54: 287–304. Dasgupta, Shamik. (2013). “Absolutism vs Comparativism about Quantity,” in Karen Bennett and Dean W. Zimmerman (eds.), Oxford Studies in Metaphysics, Volume 8, Oxford: Oxford University Press, 105–50. Dasgupta, Shamik. (2016). “Metaphysical Rationalism,” Noûs 50: 379–418. Earman, John. (1986). A Primer on Determinism. Dordrecht: D. Reidel Publishing Company. Eddon, M. (2013). “Quantitative Properties,” Philosophy Compass 8: 633–45. Field, Hartry. (1980). Science without Numbers: A Defense of Nominalism. Princeton, NJ: Princeton University Press. Fine, Kit. (2005). “The Varieties of Necessity,” in Modality and Tense: Philosophical Papers. Oxford: Oxford University Press, 235–61. Halliday, David, Robert Resnick, and Jearl Walker. (1997). Fundamentals of Physics, 5th ed., New York: Wiley. Lange, Marc. (2002). An Introduction to the Philosophy of Physics. Oxford: Blackwell. Lange, Marc. (2005). “How Can Instantaneous Velocity Fulfill Its Causal Role?” The Philosophical Review 114: 433–68. Lange, Marc. (2007). “Laws and Meta-Laws of Nature: Conservation Laws and Symmetries,” Studies in History and Philosophy of Modern Physics 38: 457–81. Lewis, David. (1983). “New Work for a Theory of Universals,” Australasian Journal of Philosophy 61: 343–77. Mundy, Brent. (1987). “The Metaphysics of Quantity,” Philosophical Studies 51: 29–54. Norton, John D. (2003). “Causation as Folk Science,” Philosopher’s Imprint 3: 1–22, http://www.philosophersimprint.org/003004/, accessed July 15, 2020. Shoemaker, Sydney. (1998). “Causal and Metaphysical Necessity,” Pacific Philosophical Quarterly 79: 59–77. Skow, Bradford. (2011). “Does Temperature Have a Metric Structure?” Philosophy of Science 78: 472–89. Suppes, Patrick and Joseph L. Zinnes. (1963). “Basic Measurement Theory,” in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (eds.), Handbook of Mathematical Psychology, vol. I. New York: Wiley, 1–76.

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5 How to Be a Relationalist Shamik Dasgupta

1. Relationalism In Dasgupta (2013) I argued that mass is fundamentally relational. On this “relationalist” view, the physical state of a system of bodies vis-à-vis mass consists at bottom just in facts about how they are related in mass. This is in contrast to the “absolutist” view that there are further facts about their intrinsic masses.¹ But David Baker (2020) has argued that relationalism implies that classical physics is indeterministic. Baker’s stated aim is just to point out this consequence, but it’s not hard to imagine someone complaining that the consequence is objectionable. A relationalist should, therefore, have something to say in response. As Baker recognizes, there are complications with some of his examples—particularly the example of escape velocity—so a relationalist might quibble over details. But that would be a distraction, for I believe Baker is right that relationalism leads to indeterminism. I believe it leads to a kind of non-locality too, and indeed I now believe that a host of other relationalist views to which I am sympathetic also lead to indeterminism and non-locality. Consider relationalism about motion, the view that all motion is motion relative to another body, as opposed to the absolutist view that there are extra facts about whether something is “really” moving independently other bodies. This relationalist view also implies that classical physics is indeterministic and non-local—I’ll argue in ¹ In Dasgupta (2013) I called the relational view ‘comparativism’. This is because in the case of other quantities like distance, even the absolutist agrees that the fundamental facts are relational; hence it seemed inappropriate to contrast absolutism with relationalism. Still, the multiplication of terms might be confusing in this paper, so here I revert to the terms ‘absolutism’ and ‘relationalism’. Shamik Dasgupta, How to Be a Relationalist In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Shamik Dasgupta. DOI: 10.1093/oso/9780192893314.003.0005

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114   section 2 that this is the lesson of Newton’s bucket argument. A third example is relationalism about handedness, the view that the fundamental facts about handedness consist in relations of congruence—that this hand is congruent with that one, incongruent with this other. This is in contrast to the absolutist view that there is a further property that distinguishes the left hands from the right hands. It turns out that relationalism about handedness also leads to a physics that’s indeterministic and non-local.² If these relationalist views lead to indeterminism and non-locality, the question is whether this is objectionable. Here I argue that it is not. This will involve distinguishing two senses in which a theory can be indeterministic and non-local. The relationalist views do lead to indeterminism and non-locality in one sense, but section 3 argues that this is a virtue, not a vice. There is a second sense of the terms in which indeterminism and non-locality would be a vice, but sections 5–10 argue that relationalist views do not lead to indeterminism or non-locality in that sense. With respect to determinism and locality, then, these relationalist views get things exactly right. Determinism and locality are intimately connected to metaphysical possibility. Thus, to distinguish the two senses of determinism and locality we’ll need to distinguish two species of metaphysical possibility (sections 7 and 8). This distinction may be of interest to modal metaphysicians regardless of its bearing on determinism and locality. In what follows I focus on the case of handedness because it is free of needless complications and so illustrates the main ideas more perspicuously. I’ll then apply my approach to the case of mass at the end (section 10). I’ll discuss the case of motion at times as we go along, but I leave a complete discussion of that case for another time. What exactly is the issue of relationalism vs. absolutism about handedness? Consider a pair of gloves: one right-handed, one left-handed. They are known as “incongruent counterparts.” They are counterparts because they share the same intrinsic geometry: each contains a thumb

² For an introduction to the issue of absolutism vs. relationalism about motion, see Sklar (1974), Maudlin (2012), and Dasgupta (2015). For an introduction of the issue about handedness, see Earman (1989) and Pooley (2003).

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and forefinger standing in the same angular relation, the same distance apart, etc. But they are incongruent insofar as there is no way of translating and rotating one glove through space in such a way that it exactly superimposes over the other. So defined, whether they’re congruent depends on the geometry of space: they’re incongruent if space is Euclidean but not, for example, if it’s a Mobius strip. I’ll assume for simplicity that space is Euclidean. Consider some gloves divided into two equivalence classes under the relation of congruence. Call one class ‘left-handed’ and the other ‘righthanded’. The question is whether there’s a further physical property, above and beyond their relations of congruence, that distinguishes members of one class from members of the other. Absolutism is the view is that there is, but this view comes in many varieties. One variety states that there’s a primitive intrinsic property that all and only the lefthanded gloves have. Another variety posits a pervasive oriented field, so that all and only the left-handed gloves are aligned with the field. And a third variety states that the gloves are situated in substantival space. On this third variety, the set of all glove-shaped regions of space can be divided into two equivalence classes under the relation of congruence; call the regions in one class the L-regions. Then, the idea is that all and only the left-handed gloves are located in L-regions.³ By contrast, relationalism is the view that there is no physical property that distinguishes hands in one class from those in the other. The hands in one class are congruent with each other, and incongruent with hands in the other class, and that is all there is to it. The clearest version of this view is the anti-substantivalist view that physical reality consists just in material bodies standing in spatiotemporal relations to one another. Suppose, as the anti-substantivalist must, that the geometry of space is then fixed by (actual or possible) spatiotemporal relations between bodies. Then, whether two gloves are congruent is ultimately fixed by the spatial relations between their parts together with the geometry of

³ To be clear, this third variety of absolutism assumes that there is trans-world identification of regions of space. So-called “sophisticated substantivalism” denies this, but that view does not yield an absolutist view of handedness as I understand the term.

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116   space. For the relationalist, these relational facts of congruence are all the facts of handedness there are.⁴

2. Indeterminism and non-locality I said that a relationalist physics of handedness is indeterministic and non-local. It’s worth seeing why before asking whether this is a problem. Earman (1989) thought that the situation for the relationalist is in fact more dire—he claimed that she can offer no physics of handedness at all. But I’ll argue that Earman overstated his case: the relationalist has a welldefined physics; it’s just indeterministic and non-local. Earman’s argument starts from the observed fact that some things behave differently depending on whether they’re left- or right-handed. Examples include neutral hyperons and the cobalt atoms, but to keep things simple I’ll work with a fictional example drawn from Pooley (2003). Imagine we discover that the fundamental components of matter are not point particles or strings, but little hand-shaped things. Call them handrons, and imagine they move around and collide in accordance with deterministic laws of motion. Suppose, further, that they come in two colors, red and green, and suppose we see that they sometimes change color when they collide. Looking closer, we see that it’s all and only the left-handrons that change color. These observations confirm the following “First Law” of handrons: (F) Whenever a handron collides with another, it changes color iff it is left-handed. Earman’s idea is that this would refute relationalism: since left- and right-handed handrons behave differently, there must be some real

⁴ See Brighouse (1999) and Pooley (2003) for discussion of how an anti-substantivalist is to understand relations of congruence. I will assume for the sake of argument that the relationalist recognizes facts of cross-temporal congruence. This entails admitting some kind of crosstemporal spatial relation, contrary to what is sometimes known as “Leibnizian relationalism”, but I bracket exactly how this is best understood.

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difference between them. As he put it, the relationalist ‘does not have the analytic resources for expressing’ a law like (F) (1989, 148). It will help to compare this with Newton’s analogous bucket argument against relationalism about motion. Newton observed that bodies behave differently depending on their state of rotational motion: water in a bucket sloshes up the side when spinning but stays flat when not. He argued that the difference is not whether the water is spinning relative to the bucket, or relative to the laboratory, but whether it is spinning absolutely; that is, independently of its motion relative to other bodies. Thus, observations of water in buckets confirm a theory that appeals to absolute acceleration—a theory that relationalists “lack the analytical resources to express”, as Earman might put it.⁵ But both objections are too quick, for in each case the relationalist does have the analytic resources to express an alternative theory of the phenomena. In the case of motion, Mach took this approach when he proposed that water sloshes up the side of a bucket when it spins relative to the fixed stars.⁶ The general approach here is to choose some body (or bodies) and say that water sloshes up the bucket when it spins relative to it (or them). One could take the same approach in the case of handedness. Suppose that a particular handron was observed to change color upon collision; call it ‘Changy’. Then a relationalist could offer the following “Machian” alternative to (F): (F-Machian) Whenever a handron collides with another, it changes color iff it is congruent with Changy. Still, these Machian theories are unattractive. Some object to Mach’s theory of motion on the grounds that he never developed it with a precision to rival Newton’s. Others might object that fundamental physics should not make reference to particular entities like Changy or the fixed stars. But let me emphasize a third objection, which is that these

⁵ Newton’s text can be read in a number of ways; my summary is just one reading. For more on the bucket argument, see Sklar (1974, ch. 3), Maudlin (2012), and Dasgupta (2015). ⁶ At least, this is the view typically attributed to Mach; I won’t comment on whether this interpretation is accurate.

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118   Machian theories fly in the face of scientific practice. If our basic theory of motion were Mach’s, we could never use it to model hypothetical physical systems in which the fixed stars do not exist; yet we appear to do this all the time. Just remember high-school physics problems in which you predict the behavior of a harmonic oscillator: on the face of it, you’re reasoning about a hypothetical physical system containing just an oscillator. More seriously, cosmologists routinely model counterfactual scenarios in which heavy elements, and hence the fixed stars, never form. The same goes for any Machian theory formulated with reference to some special entity like Changy: the theory can’t be used to model counterfactual systems in which the special entity doesn’t exist. Is this so bad? Being a positivist, Mach may not have cared whether his theory can model far-out counterfactual scenarios. Still, it would be nice to avoid the problem if we can. In the case of handedness this is straightforward. Perhaps the relationalist can’t say that the handrons that change color are all left-handed. But she can say that they’re all congruent with one another, and incongruent with those that don’t change color. Thus, as Pooley (2003) points out, she can offer the following “minimalist” theory of handrons: (F-Minimalist)

(i) If x and y are congruent handrons, then x changes color on collision iff y does too. (ii) If x and y are incongruent handrons, then x changes color on collision iff y does not.

Unlike (F-Machian), (F-Minimalist) is not tied to any particular body and so can be used to model counterfactual systems that don’t contain Changy. Sklar (1974) proposed an analogous minimalist theory of motion in response to Newton’s bucket argument. True, the relationalist cannot say that flat bodies of water are absolutely unaccelerated. But she can say that they’re all unaccelerated relative to one another and accelerated relative to water that goes concave. Sklar’s idea was that the relationalist should simply offer that as her “minimalist” law of motion. As he put it, the law would state ‘(1) that objects in relative motion vary in the inertial forces they suffer and that (2) objects in uniform motion with respect to one

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another suffer similar inertial forces’ (1974, 230).⁷ On this view, the water in Newton’s bucket goes concave because it’s unaccelerated relative to other concave bodies of water. This minimalist theory stands to Newton’s theory just as (F-Minimalist) stands to (F). Unlike Mach’s theory, it is not tied to any particular body and so can model counterfactual systems in which the fixed stars don’t exist.⁸ If there’s an objection to these relationalist views, then, it’s not that they can offer no physical theory of the phenomena—they can offer the minimalist theories just described. Still, it turns out that these theories are indeterministic and non-local in a way that the original theories were not. To see why (F-Minimalist) is indeterministic, consider a world containing just two left-handrons about to collide. Will they change color? (F) implies that they will. But (F-minimalist) does not, it just implies that either both or neither will change color. Hence there are two possible futures consistent with (F-minimalist)—the mark of indeterminism. I said that the world contains two left-handrons, but can a relationalist legitimately describe counterfactual worlds in terms of ‘left’? Perhaps not; we will discuss this later. But for now it doesn’t matter, for we may instead describe it as a world containing two congruent handrons about to collide. For the relationalist this is a complete description of the world vis-à-vis handedness. And (F-Minimalist) is still consistent with two possible futures, one in which both change color and one in which neither does. The absolutist avoids the problem because he distinguishes two worlds that fit this description: one in which the handrons have that physical property that (on his view) distinguishes the left-handrons, and another in which they don’t. And the absolutist reads (F) as stating that handrons change color if and only if they have that property. (F) then

⁷ Note that by ‘relative motion’ he means relative acceleration. Obviously this is just a rough gloss of what should ultimately be expressed with mathematical precision, but the kind of theory Sklar has in mind is clear enough. ⁸ Sklar (1974) is often said to have responded to the bucket argument with an absolutist theory that posits a primitive property of absolute acceleration. This is some truth to this; see his discussion on pp. 230–1. But strangely, his discussion slurs that absolutist theory together with the minimalist theory described in the text, and the subsequent literature latched onto to the former at the expense of the latter. I find that unfortunate, for his minimalist proposal is far more intriguing.

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120   implies that the handrons will change color in the first world but not the second, so in each world there is only one possible future. I’ve been assuming the following standard definition of determinism: A theory is deterministic iff any two metaphysically possible worlds in which it obtains, and which agree at one time, agree at all times.⁹

The point is that the absolutist’s physics satisfies this definition while the relationalist’s physics does not.¹⁰ Note that the indeterminism here is unusual: (F-minimalist) does not assign probabilities to possible futures; it just stays silent on the matter. Say that a theory is complete iff it is deterministic or assigns a probability to each possible future given the state of the world at a time. Then, the point here is that (F-minimalist) is incomplete.¹¹ What about the claim that (F-Minimalist) is non-local? Well, suppose that a handron is about to collide. Will it change color? According to (F-Minimalist), this depends on the results of other collision events which may occur thousands of miles away. So, as Pooley (2003) noted, whether it changes color is a function not of its local environment but of its relation to potentially far-off events—the mark of non-locality. By contrast, the absolutist’s physics is local in this respect. For according to (F), whether a handron changes color depends on whether it’s lefthanded, and for the absolutist this depends on whether it possesses the physical property that distinguishes the left-handrons. And on all the ⁹ See Earman (1986) for definitions along these lines. His definitions quantify over physically possible worlds; I assume for now that these are metaphysically possible worlds in which the actual physical laws obtain. Earman and others have argued that this definition must be refined, but the issues they raise are not relevant for our purposes. ¹⁰ I’m sacrificing precision for the sake of brevity. Even as the absolutist interprets it, (F) isn’t deterministic because on its own it doesn’t determine anything about how the handrons move. More accurately, then, the claim is that the conjunction of (F) and the laws of motion that we’re assuming to be deterministic vis-à-vis motion is a deterministic theory, while the result of replacing (F) with (F-minimalist) is not. But I’ll bracket this complication for brevity in the text. ¹¹ Compare the “Hole” argument, which purports to show that substantivalism renders General Relativity indeterministic. In that case too the indeterminism is incompleteness, not stochasticity. See Brighouse (1994) for some of the issues involved there. Of course, (FMinimalist) yields implications about all collision events when conjoined with information about a single collision event, for example that Changy changed color. But that conjunction is just (F-Machian).

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absolutist views surveyed earlier, this is a function of its local environment such as whether it’s aligned with the oriented field—at any rate, it doesn’t depend on its relation to far-off collision events. Later I’ll be more precise about what non-locality amounts to, but this gloss will do for now. The upshot is that the relationalist’s minimalist physics of handrons is indeterministic and non-local in a way that the absolutist’s physics is not. The same goes in the case of motion. Given a world with just two buckets of water at rest relative to one another, Sklar’s minimalist theory does not determine whether the water in either bucket will slosh up the side. All it says is that either both will remain flat or both will go concave, but it does not determine which (indeterminism). And whether a body of water goes concave depends on its motion relative to potentially far-off bodies of water (non-locality). This, I suggest, is the real moral of Newton’s bucket argument and Earman’s analogous argument about handedness. Properly understood, the argument in each case is not that the relationalist can provide no physical theory of the phenomena whatsoever. Rather, it’s that her best theory is indeterministic (in the sense of being incomplete) and nonlocal in a manner that the absolutist’s theory is not.¹²

3. Turning the tables The question now is whether this is objectionable. The consensus seems to be that it is; that completeness and locality are weighty, perhaps nonnegotiable constraints on an adequate physical theory.¹³ For example, having noted that (F-Minimalist) is non-local, Pooley (2003) immediately concludes that this is a serious strike against relationalism.

¹² The dialectic is a little different in the case of the physics of cobalt atoms that Earman discussed. In that case, the absolutist theory he starts with is a quantum theory and hence is indeterministic anyway (at least, on some interpretations). But then the current point is that its minimalist correlate would introduce a further degree of indeterminism, in the manner of incompleteness. ¹³ Skow (2007) is explicit about these constraints, and notes that they are shared by theorists of many different kinds, including relationalists such as Barbour (1999).

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122   But I think this consensus gets things exactly back to front: incompleteness and non-locality are virtues of the relationalist’s physics, not vices. After all, why should we think that completeness or locality is a constraint on an adequate physics? Here we must distinguish a priori from empirical reasons. In the first case, the idea is that these constraints are justified independently of observation. For example, one might argue on a priori grounds for an anti-Humean view of laws on which they “govern” events, and then argue that they can’t govern if they’re incomplete or non-local. Or one might claim that a preference for complete and local laws is a primitive epistemic norm governing theory choice. Whatever the details, the result would be an objection to relationalism based on a priori constraints. But this kind of objection should not convince. The history of physics—in particular the history of geometry—shows that we shouldn’t put much weight on a priori principles to the effect that the world has this or that spatiotemporal or causal structure. Conflicting with such “principles” is not such a big deal. It would be a much bigger deal, in my view, if completeness and locality were empirical findings, principles that constrain an adequate physics because of how things look. For example, if handrons were observed to behave deterministically and relationalism implies that they do not, then relationalism would be disconfirmed by observation. This objection has more teeth, for on the face of it handrons were indeed observed to behave deterministically. When our fictional scientists observed handrons colliding, they didn’t just see 80 percent of the left-handrons change color. No, fully 100 percent of the observed left-handrons changed color upon collision, and 100 percent of the observed right-handrons did not. It was on the basis of these observations that they proposed the deterministic theory (F), rather than a stochastic theory like (F*) When a left-handron collides with another, there is an 80 percent chance that it changes color. This is not an a priori requirement that the process is deterministic, but an observation that it looks to be that way.

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Similarly in the case of motion. Why is it objectionable that Sklar’s minimalist theory of motion is indeterministic? Not because we know a priori that the laws of motion are deterministic, for we know no such thing. Rather, it’s because, when we observe water spinning in buckets, it goes concave 100 percent of the time. Thus, it’s an observed fact that water behaves in the deterministic fashion described by Newton’s theory. Sklar’s minimalist theory, on which water does not behave deterministically, therefore, flies against an empirical finding. Seen like this, what we have is an empirical objection to relationalism: it’s an observed fact that handrons and water behave deterministically; yet relationalism implies that they do not. But what exactly is the observed fact? Here we must take care. To be sure, the absolutist’s theory of handrons is deterministic and local; but did we observe handrons behaving in accordance with that theory? We did not. For let us be clear about its content. The absolutist interprets ‘left’ as referring to that physical property L that distinguishes left from right-handrons, and for concreteness let’s suppose that L is the property of being aligned with some oriented field. So interpreted, (F) states that the handrons that change color are those that are aligned with the field. This theory is indeed deterministic and local, but our observations did not confirm this theory. We never observed that the handrons that changed color were all aligned with some field. After all, everything would look (and smell, and taste) exactly the same in a mirror world that differs only in that every handron is flipped from left to right and vice versa; that is, a world in which the handrons that change color are all anti-aligned with the field and the ones that don’t change color are all aligned. What we observed is no reason to think that we live in one world over its flipped cousin. All we really observed is the relational facts common to both worlds, namely that the handrons that changed color were all congruent with one another. This confirms (F-Minimalist) but does nothing to confirm the absolutist’s interpretation of (F).¹⁴

¹⁴ One might suggest that observation at least provides reason to believe that there exists a field, such that the handrons that change color are either all aligned or anti-aligned with it. The idea might be that this theory better explains the observations than (F-Minimalist), and so is justified on the basis of inference to the best explanation. But even if so, this theory is

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124   Sklar made this point when discussing his minimalist theory of motion. He emphasized that we never directly observe whether a body of water is spinning absolutely; we just observe (i) whether it goes concave, and (ii) its state of motion relative to other bodies (Sklar 1974, 230). Everything would look (and smell, and taste) exactly the same in a world in which water stays flat when spinning at some absolute rate and goes concave when spinning relative to that absolute rate. Thus, Sklar claimed, his minimalist theory, not the absolutist’s theory, is exactly what gets confirmed by what we actually observe. I’m making the same point in the case of handedness. With regard to this empirical objection, then, the tables are turned: it is the relationalist’s physics, not the absolutist’s, that gets things right with regard to observations of determinism and locality. The relationalist’s theory is indeed indeterministic and non-local, but such a theory is precisely what observation confirms. If anything, it’s the absolutist who has the problem here, proposing a theory that’s not confirmed by the evidence. If this is surprising, it might be because we always observe subsystems of the universe, not the entire universe itself. If I see a handron h about to collide and I know it’s congruent with another handron that changed color, then (F-Minimalism) plus my background knowledge implies that h will change color. Likewise, if I observe a single bucket of water and I know that it’s at rest relative to another bucket of water that remained flat, then Sklar’s minimalist law plus my background knowledge implies that the water will remain flat. This gives the appearance of determinism: the behavior of these subsystems can indeed be deduced from minimalist laws plus information about how the subsystem relates to the outside world. What has not been observed, however, is that the universe as a whole evolves in the deterministic and local manner described by the absolutist. So far, then, it is the relationalist who has the upper hand.

indeterministic and non-local in just the same way that (F-Minimalist) is. Thus, the point remains that the theory justified by observation, whatever it is, is indeterministic and non-local.

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4. Determinism in practice Still, there remains something to the empirical objection. When observing handrons our fictional scientists did, after all, find that 100 percent of the left hadrons and 0 percent of the right-handrons changed color upon collision. So, I’ll argue, they would write down (F) as their physics of handrons and reason with it in a deterministic and local manner. At least, they would if our fiction is to reflect actual physics. For when investigating chiral objects like cobalt atoms, physicists use seemingly absolutist language like ‘left’ and ‘right’ to formulate theories and reason with them in seemingly absolutist ways. We must, therefore, imagine that our fictional scientists do the same, which in this case means reasoning with (F) in a deterministic and local manner. The question is whether we can make any sense of this practice if, as the relationalist thinks, the world is fundamentally indeterministic and non-local. As we will see, this is surprisingly difficult to do. To my mind this is the real challenge facing relationalism; the rest of the paper is an attempt to address it. Let us start by describing the practice at issue. First, it involves using ‘left’ and ‘right’ to record observations of handrons. After all, physicists investigating cobalt atoms don’t limit themselves to talk of congruence; they routinely use ‘left-handed’ and ‘right-handed’ (or mathematical cognates) to sort the atoms. Thus, we must imagine that our fictional physicists do the same and record their observations by writing statements like This left-handron changed color. This right-handron did not. Already there’s a question of what ‘left’ and ‘right’ could mean on the relationalist’s view, but put that aside until later; here I’m just describing the practice. Second, the practice involves taking these statements to confirm a theory they write down as:

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126   (F) Whenever a handron collides with another, it changes color iff it is left-handed. Again, practicing physicists don’t hesitate to use such language (or cognates) to express theories of cobalt atoms, so we must imagine that our fictional physicists do the same. Finally, the practice includes reasoning with (F) as if it were deterministic and local. I’ll focus on determinism for now. To illustrate, consider how (F) would be used to predict actual events. Imagine a fictional physicist teaching a student about (F), and suppose they come across a left-handron about to collide. Sensing a teachable moment, the professor sets a pop quiz: Question 1: According to (F), will this left-handron change color? The right answer, of course, is that it will; that given (F) it must change color; that (F) rules out any other possible outcome. The professor should give this answer full marks. But this is to reason deterministically with (F). To emphasize the point, just imagine the professor had asked what will happen according to (F*). In that case the correct answer is that the handron will probably change color but might not; that (F*) leaves open both possibilities. Thus, our physicists reason differently with (F) and (F*); only with the former do they reason deterministically. The same goes when reasoning about hypothetical events. Imagine that our fictional professor sets another pop quiz: Question 2: Consider a possible world in which there are just two lefthandrons about to collide. According to (F), what will happen? Again, the right answer is that both will change color; that (F) does not leave open any other possibility. Once again, this is to reason with (F) deterministically. The professor may then set a third pop quiz: Question 3: Consider a possible world in which there are just two righthandrons about to collide. According to (F), what will happen?

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This time, the students can easily calculate that neither will change color; that according to (F) this is the only possible outcome. Yet again, this is paradigmatic deterministic reasoning. Questions 1–3 are known as “initial value problems”: one is given the initial state of a hypothetical physical system and must solve for what happens. Note that in these problems the initial states are characterized in terms of ‘left’ and ‘right’. Note also that the left–right distinction is treated as an independent variable, something whose initial value can be set independently of other quantities so that the effects of each value can be investigated—witness how Questions 2 and 3 differ only in a uniform switch of this value. The point is that initial value problems like these are standard in actual physics: the initial state of some cobalt atoms will be characterized in terms of ‘left’ and ‘right’ (or cognates), and a theory expressed in such terms is used to solve for what happens. The question is whether the relationalist can make sense of this practice. Of course, she’ll regard this talk of ‘left’ and ‘right’ as nonfundamental in some sense—fundamentally speaking the only truths about handedness concern relations of congruence. But to account for scientific practice she must make sense of the talk nonetheless. What could ‘left’ and ‘right’ mean such that we can reason about initial value problems in the deterministic manner just described? And how is this deterministic reasoning consistent with her view that fundamentally speaking—under the hood as it were—handrons behave indeterministically as described by (F-Minimalist)? It’s not clear what the relationalist can say. There are two worries here. The first is whether she can legitimately characterize problems like Questions 2 and 3 in terms of ‘left’ and ‘right’. Suppose that the only meaning the relationalist could attach to ‘x is left-handed’ is to treat it as synonymous with ‘x is congruent with S’, where S is some standard object like Changy, or a set of objects like hands on our heart side or physical particles of some kind; and conversely for ‘right’. On this semantics she can meaningfully describe actual handrons like the one in Question 1 as left- or right-handed: they are being described as congruent or incongruent with S. But this semantics is unable to account for the practice of characterizing the counterfactual handrons in Questions 2 and 3 in terms of left and right. For the standard object(s)

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128   S do not exist in the worlds described by Questions 2 and 3, and so on this semantics it cannot be true that those worlds contain left or righthandrons.¹⁵ More generally, the worry is that relationalism implies Incommensurability: There is a class C of metaphysically possible worlds such that, for any possible world W in C, and any handron x in W, there is no fact of the matter whether x is left- or right-handed in W.¹⁶ Different specific semantics of ‘left’ will differ on what the class C is, but the worry is that any relationalist semantics will entail that it is nonempty and includes worlds like those of Questions 2 and 3. And if the handrons in Question 2 can’t be said to be left-handed—if all we can say is that they are congruent—then (F) won’t entail that they’ll change color and our deterministic reasoning falls apart. But put this aside—suppose the relationalist can legitimately characterize the world in Question 2 as containing two left-handrons. The second worry is whether she could then characterize the world in Question 3 differently, as containing two hands. For the two worlds agree on all facts about congruence, and surely—the worry goes— relationalism implies that any two worlds agreeing on all facts about congruence cannot disagree on any further facts about handedness precisely because on her view there are no further facts. Specifically, the worry is that relationalism implies Relational Supervenience: Metaphysically possible worlds agreeing on all relational matters of congruence agree on all matters of handedness. Thus, if we grant that the world in Question 2 contains two lefthandrons, Relational Supervenience implies that the world in Question 3 ¹⁵ This is, in effect, our objection to the Machian theory from section 2. For if the relationalist defines ‘left’ as ‘congruent with Changy’, then in her mouth the theory (F) becomes synonymous with (F-Machian). And as we saw, this theory makes no predictions about worlds in which Changy does not exist. ¹⁶ This idea that there is “no fact of the matter” can be understood in a number of ways. One way is to admit of truth value gaps at W, and say that instances of “x is left-handed” and “x is right-handed” are neither true nor false at W. But those who dislike truth value gaps can understand the idea differently and say all such instances are false at W. It will not matter for our purposes which version we pick, though I will sometimes talk as if there are truth value gaps for simplicity.

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also contains two left-handrons; hence the relationalist cannot distinguish Questions 2 and 3 or say that they have different answers. She cannot, that is, make sense of the practice of treating left–right as an independent variable in these problems. If this is right, the relationalist can’t make sense of the use of ‘left’ and ‘right’ we see in physical practice. In effect, there’s a kind of modal breakdown: thanks to Incommensurability she can’t characterize initial value problems like Questions 2 and 3 in those terms, and thanks to Relational Supervenience she can’t distinguish them. By contrast, the absolutist faces no such breakdown. He interprets ‘left’ as denoting the special physical property L that all and only the left-handrons have. There is, then, no problem with characterizing worlds like Questions 2 and 3 in terms of ‘left’ and ‘right’: he is characterizing the distribution of L in those worlds. Nor is there any problem in distinguishing the worlds: they differ in their distribution of L. And we know that (F), so interpreted, is deterministic. But the relationalist can’t interpret ‘left’ like this—and in any case she doesn’t want to, since (as we saw) on that interpretation (F) expresses a theory for which there is no evidence. So the question is whether she can understand this talk of ‘left’ and ‘right’ in some other way. If not, she must bite the bullet and say that scientific practice must be revised. But I find this unacceptable—who would seriously tell their colleagues in the physics department to revise their physics of cobalt atoms? The point here is particularly perspicuous in the case of mass, where the analogue of ‘left’ and ‘right’ is talk of mass in kilograms (or some other unit), and the analogue of Questions 2 and 3 is initial value problems that differ only in a uniform doubling of mass in kilograms. As I’ll discuss in section 10, what Baker (2020) shows is that standard scientific practice distinguishes these problems and predicts a different evolution for each problem. It would be outrageous to tell the physics department that talk of kilograms must be banished. So I think the relationalist must make sense of this practice or go home.¹⁷ ¹⁷ It might be thought that initial value problems like these can be interpreted as descriptions of possible subsystems that exist alongside various reference points such as standard objects in Paris, laboratory equipment, and so forth, so that Incommensurability will never be a problem. But even if this is an option with some initial value problems, the solution clearly does not

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130   This then is the challenge. The relationalist must interpret talk of ‘left’ and ‘right’ in such a way that makes sense of the deterministic just described, yet does not collapse into the absolutist’s interpretation on which (F) expresses a theory for which there is no evidence.

5. Devices of coherence In what follows I will show how the relationalist can do this. I will not argue that it is the only way, or even the best way, just that it works. I proceed in two stages. First, I’ll describe how a relationalist might use the terms ‘left’ and ‘right’. On the basis of that usage, I’ll then develop a theory of metaphysical possibility that avoids the modal breakdown and makes sense of the deterministic reasoning with (F) just described. So, how might a relationalist use ‘left’ and ‘right’? The obvious way is to define ‘left’ as ‘congruent with Changy’ or some other standard, but we know that this leads straight to Incommensurability. Thankfully, there is another way to use the terms. Let me illustrate with a fictional community of relationalists who lack the terms ‘left’ and ‘right’. They have only a two-place relational predicate ‘x is congruent with y’ with which to talk about handedness, and they wish to introduce one-place predicates to help them store and communicate information about congruence more efficiently. To this end, they introduce two new marks, # and *. Their idea is to use these marks in such a way that if two hands are given the same mark, this means they are congruent, while marking them differently means that they are incongruent. More precisely, they stipulate that the marks are monadic predicates governed by the following three rules of inference: (R1)

(R2)

(R3)

x is #, y is # x is *, y is * x is #, y is * x and y are congruent. x and y are congruent. x and y are incongruent.

generalize. As I mentioned in section 2, cosmologists think about initial value problems in which heavy elements, and hence our planet and all the reference points it contains, never exist.

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But they stipulate nothing else. Note that these are all exit rules; there are no introduction rules that specify sufficient conditions for concluding (say) that x is #. Thus, they have not given explicit definitions of the predicates in terms of standard reference bodies. Still, this is enough to ground a practice that encodes information about congruence. When they first encounter a handron h₁, they can call it # or *, since either is consistent with the rules. Suppose they call it #. And suppose they then encounter a handron h₂ that is incongruent to h₁. If they call it #, then by (R1) they could infer a falsehood. So, insofar as they aim to utter things that imply truths, their hand is forced: they must call it *. Thus, when a speaker uses the marks, her aim is to “cohere” with other accepted uses in the community, in the sense that their combined uses yield truths about congruence via (R1)–(R3). Once their usage becomes entrenched—imagine them using these marks for a few decades—there will be a clear distinction between correct uses that cohere and incorrect uses that do not. In this way, they can use # and * to communicate information about congruence. Cohering with one’s community is not easy, especially if the community is large and dispersed. To aid coherence it may help to use a standard glove displayed in a public place: if each speaker ensures that she coheres with the sentence ‘The standard glove is #’, they will all cohere with one another. But—and this is important—they need not define the predicate ‘x is #’ to be synonymous with ‘x is congruent to the standard glove’. The standard glove is functioning just as a practical aid to help them cohere, not as a definition (or as a reference fixer). If they discover by surprise that all their interactions with the standard glove were subject to some massive and systematic illusion, so that the standard glove is in fact incongruent with the gloves they call #, they would report this discovery by saying that the standard glove is in fact *. So long as their other applications of # and * still cohere with one another, there is no need for further revision. I think that that our words ‘left’ and ‘right’ are in all important respects like # and *. They are devices of coherence: our aim in uttering ‘x is lefthanded’ is to cohere with other utterances in our linguistic community in the above sense. On this view, reference objects like the hand on our heart side, or certain kinds of molecules, or standard gloves in Paris, do

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132   not serve to define ‘left’ and ‘right’ but are practical aids to help us cohere with one another. But the claim that we use ‘left’ and ‘right’ as devices of coherence is an empirical claim that I won’t defend. Here I just make the uncontroversial claim that a community of relationalists could use ‘left’ and ‘right’ this way. What I’ll argue is that if they use the terms like this, that would explain the practice described in section 4. It is easy to see that it would account for the practice of using ‘left’ and ‘right’ to record their observations and advance theories like (F). Imagine that a community has used ‘left’ and ‘right’ as devices of coherence for generations, so that there is a clear distinction between correct uses that cohere and incorrect uses that do not. They will then recognize left hands from right hands in much the same way that we do, say, by making an “L” shape with the hand on their heart side. For them, these methods are not an attempt to determine whether a given hand satisfies a definition of ‘left’, but an attempt to ensure that their use of the terms cohere. So, when observing handrons, they’ll fill their notebooks with inscriptions like This left-handron changed color. This right-handron did not. And they would take these to confirm a theory they’d write down as (F). Since their use of ‘left’ is not tied to any particular standard object, (F) in their mouths is not equivalent to a Machian theory. To be clear, this is not to settle all semantic questions about this practice. What propositions are expressed by the inscriptions in their notebooks, or by (F)? What are their truth conditions? I haven’t said, though I’ll discuss this further in section 8. For now, the claim is just that a relationalist using ‘left’ and ‘right’ as devices of coherence can record her observations of handrons in these terms and will take those observations to confirm a theory she’d write down as (F).

6. Modal correspondence That much is straightforward. The more difficult question is whether the relationalist can make sense of the deterministic reasoning in Questions 2

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and 3. Here she faced the “modal breakdown”: thanks to Incommensurability she couldn’t characterize the worlds in Questions 2 and 3 in terms of left and right, and thanks to Relational Supervenience she couldn’t distinguish them. Perhaps we’ve already made some progress here. For if ‘left’ and ‘right’ are devices of coherence, why can’t they be used to characterize those worlds? By labeling the handrons in Question 2 ‘left’ one can’t derive anything false with (R1)–(R3); all one can derive is that they’re congruent.¹⁸ Thus, nothing about the rules governing the terms prohibits their application in Question 2. This is in stark contrast to the semantics on which ‘left’ is defined as ‘congruent with Changy’, which entails that ‘left’ has no application to worlds in which Changy doesn’t exist. It’s tempting to wonder, then, whether we have a solution to the problem of Incommensurability. The same goes for Question 3: nothing in the rules (R1)–(R3) prohibits us from labelling those handrons as ‘right’. But if we label the handrons in Question 2 as ‘left’ and those in Question 3 as ‘right’, we appear to be distinguishing the initial value problems after all. It’s tempting to wonder, then, whether we have a solution to the problem of Relational Supervenience too. The idea can be modeled by representing the initial value problems with pairs of the form , where W is a possible world and f is a function assigning the word ‘left’ to one congruence class of handrons in W and the word ‘right’ to the other. Even if a relationalist can’t distinguish two possible worlds that differ only in a uniform flip of left to right and vice versa, she can easily distinguish and , where f₁ and f₂ differ only in flipping which congruence class is mapped to the word ‘left’. Thus, while Relational Supervenience might be true of the possible worlds, it certainly isn’t true of the pairs. More generally, it’s clear that the space of pairs recognized by the relationalist corresponds one-to-one with the space of metaphysically

¹⁸ I’m assuming that, like any set of exit rules, they are only applicable within a world. After all, suppose it’s an exit rule governing ‘bachelor’ that from ‘x is bachelor’ one can infer ‘x is unmarried’. If an individual a is a bachelor in W, one can infer by this rule that a is unmarried in W but not that a is unmarried in a distinct world W*.

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134   possible worlds recognized by the absolutist. The tempting idea, then, is that anything the absolutist does with possible worlds, the relationalist can mimic with the pairs instead. Thus, while the absolutist interprets an initial value problem as characterizing a possible world, the relationalist might interpret it as characterizing a pair and solve it in exactly the same way. Et voilà: she’s made sense of the reasoning involved in Questions 2 and 3. Similarly, recall our definition of determinism from section 2, which quantified over indices we called “possible worlds”. The absolutist will take the definition at face value, and we saw that (F)—as he reads it— satisfies it. But the relationalist might now interpret the indices as pairs, not possible worlds, and so understand a theory to be deterministic iff any two pairs in which it obtains and which agree at one time, agree at all times. Et voilà again: (F), even in her mouth, is deterministic in this sense. But is this legitimate? The worry is that it’s just trickery; that the pairs are cheap, formal objects with no bearing on the philosophical issues at hand. For the pairs are not themselves metaphysically possible worlds, so what they represent isn’t metaphysical possibility in the standard sense. Given what philosophers typically mean by “metaphysical possible world”, there is I think no question that relationalism implies Relational Supervenience. The pairs are formal objects that make that problem go away, but the cost is that we’re no longer talking about metaphysical possibility in the standard sense. Is it a significant cost? Perhaps not, for it’s an open question whether initial value problems must be understood in terms of the possible worlds of the philosopher. When a high-school physics teacher assigns her students initial value problems, it’s far from clear that she’s asking them to think about “possible worlds” in the contemporary metaphysician’s sense of the term. Indeed, physicists rarely use the term themselves, talking rather of “models” or “hypothetical physical systems” or “fictional situations”. The same goes for the indices quantified over in the definition of determinism: physicists typically call them “models”, or “closed systems”, or something of that ilk, and it’s not at all obvious that these are alternative labels for what philosophers call “metaphysically possible worlds”. So, there must be some leeway to diverge from the philosopher’s preferred notions.

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But not unlimited leeway. For the pre-theoretic idea behind determinism is that a system must evolve in a certain way, given the laws. And it’s widely agreed that this is not the logical ‘must’: it’s not that the present state of the system and the laws logically imply its future state. Nor is it an epistemic ‘must’: it’s not that an ideal agent could infer how the system will evolve, given complete knowledge of its initial state and the laws— this notion of “Laplacian determinism” is not of interest to contemporary philosophers of physics. Rather, the consensus is that the relevant sense of determinism involves a metaphysical ‘must’. It’s hard to say exactly what this means, but the rough idea is that metaphysical possibility is constrained not by the logical constants (logical possibility), or by states of knowledge (epistemic possibility), or by the meanings of words (conceptual possibility), but by “the world”; by the things it contains and what they’re like. Thus, when a deterministic system must evolve in a certain way, this is due not to states of knowledge or the logical properties of a formal language or the meanings of our words, but to the nature of the system itself. Our definition of determinism attempts to capture this by quantifying over a set of indices, the “possible worlds”. But this will succeed only if each index is a distinct possible world in some recognizably metaphysical sense of the term. So it’s not enough to just produce formal objects like the pairs and use them to define determinism; it must also be shown that they represent a genuinely metaphysical sense of possibility. I will argue that they do. Now, obviously the kind of possibility they represent isn’t what philosophers ordinarily mean by “metaphysical possibility”, for this creates the very problems that the former is meant to solve. So what I’ll argue is that we must distinguish two varieties of metaphysical possibility. One variety is what philosophers typically mean by the term—the familiar variety on which it’s uncontroversial that relationalism implies Relational Supervenience. Call this “strict possibility”. But I’ll argue that there’s another notion of metaphysical possibility, “loose possibility”, on which relationalism does not imply Relational Supervenience or Incommensurability. This is the notion of possibility represented by the pairs. If that’s right, our hope is vindicated after all: the relationalist can legitimately use the pairs—or, more accurately, the

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136   loosely possible worlds they represent—to mimic the absolutist and account for deterministic reasoning involving (F).¹⁹ For this to work we must establish two things about loose possibility. First, we must show that the relationalist’s conception of loose possibility accurately mimics the absolutist’s conception of strict possibility. Thus, where the absolutist distinguishes two strictly possible worlds that differ only in a uniform flip of left and right, the relationalist must distinguish two loosely possible worlds that differ in the same way. More precisely: Modal Correspondence Thesis: There is a one-to-one correspondence between the space of strictly possible worlds recognized by the absolutist and the space of loosely possible worlds recognized by the relationalist that preserves all relational facts of congruence and facts about left and right. Second, we must show that loose possibility really is a species of metaphysical possibility. But what exactly is meant by “metaphysical” possibility? I glossed is as possibility that’s constrained by “the world”, rather than our concepts or states of knowledge or the logical properties of a formal language, but that’s hardly precise. Still, we can sidestep the question by taking strict possibility as our fixed point. For there’s no doubting that it is a species of metaphysical possibility—it’s the paradigm case. I’ll sketch a theory of strict possibility on which strict possibility is understood in terms of fundamentality, which in turn is understood in terms of metaphysical explanation (section 7). I’ll then distinguish a slightly different notion of metaphysical explanation and show that it yields a slightly different notion of possibility; this will be loose possibility (section 8). Since strict and loose possibility are both understood in terms of two closely related senses of explanation, that

¹⁹ To be clear, my claim is not that physicists use ‘possibility’ to mean loose possibility, not strict possibility (these are fine distinctions of modal metaphysics, so there may be no fact of the matter what they mean). My claim is rather that a relationalist can explain why the practice of reasoning deterministically with (F) makes sense by interpreting it as reasoning over a domain of loosely possible worlds. This is more along the lines of “rational reconstruction”, not psychological description.

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will suffice to show that each is as “metaphysical” as the other, whatever that means.²⁰

7. Possibility and explanation So, let me sketch a theory of strict possibility, the notion of metaphysical possibility familiar to philosophers. There won’t be space to defend it fully here; my aim is just to outline the basic idea and lend it some plausibility. The theory consists in two claims. The first is that a strictly possible world is a way of reorganizing fundamental matters. For example, consider a relationalist view on which, fundamentally speaking, there are just handrons spatially related to one another—ignore any nonspatiotemporal properties and relations. A strictly possible world, then, is a way of reorganizing the spatial relations between handrons. This explains why relationalism implies Relational Supervenience when it comes to strictly possibility. For, if a strictly possible world just is a way of spatially relating handrons, then strictly possible worlds agreeing on those relational respects are one and the same. This conception of a possible world is, I think, ubiquitous throughout philosophy, albeit implicitly. Consider the physicalist view that the world is, fundamentally, just physical. It is widely presumed that this has the consequence that worlds that are physically identical, and which contain no “alien” fundamentalia, agree in all other respects concerning consciousness, normativity, and so on. This is, indeed, a consequence of physicalism ²⁰ In Dasgupta (2013) I tried to distinguish strict from loose possibility somewhat differently. The idea was to mimic Lewis’s (1986) “cheap haecceitism”, where he uses counterpart theory to distinguish between two modal indices: possible worlds and possibilities. I showed that this approach can be extended to the case of mass, and one could apply it to the case of handedness too. But counterpart theory (of any form) now strikes me as an inadequate way of understanding metaphysical possibility, so here I explore a different strategy that understands possibility in terms of explanation instead. This strategy is structurally analogous to Bhogal (2020). There he distinguishes metaphysical explanation from scientific explanation, and he then argues that each notion yields a corresponding notion of possibility, metaphysical and scientific possibility respectively. But his notion of scientific explanation is different from the notion I’ll distinguish, and hence the notions of possibility we end up with are different too. As he emphasizes, his notion of scientific explanation is explicitly non-metaphysical and hence his notion of scientific possibility is non-metaphysical too. Here I employ the same strategy but my aim is to distinguish two notions of possibility that I suspect would both count as metaphysical even by Bhogal’s lights.

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138   if, as the first claim states, a world just is a way of recombining fundamental matters. This first claim does not imply “combinatorialism”, the view that every reorganization of fundamental matters is a strictly possible world. Suppose distance-in-feet is fundamental. Then one reorganization of fundamental matters puts x 1 foot from y, y 1 foot from z, and x 50 feet from z—a violation of triangle inequality. If you think that’s impossible, you might propose that principles like triangle inequality restrict which ways of reorganizing fundamental matters are genuinely possible. I’ll call these “metaphysical principles”, though I leave open whether there are any and what they might be. Thus, the intended content of the first claim is that a strictly possible world is a reorganization of fundamental matters that’s consistent with the metaphysical principles (whatever they are). This first claim connects strict possibility with fundamentality, but what is fundamentality? The second claim is that something is fundamental iff it is unexplained in a “metaphysical” or “constitutive” sense of the term. To illustrate, suppose that the fundamental matters just concern atoms arranged in space. Still, there are chairs; it’s just that chairs are things that exist when and because atoms are arranged in a certain way. The fact that there are chairs holds in virtue of the arrangement of the atoms; that arrangement makes it the case that there is a chair. This mode of explanation is not causal but “constitutive”. We can be ecumenical in how we understand this notion. Perhaps it is (or tracks) a primitive relation of “grounding” between facts (Rosen 2010), or perhaps it is better analyzed in terms of dependency relations (Schaffer 2016, 2017), metaphysical laws (Wilsch 2015), or unifying patterns (Kovacs 2020). Or we might understand the explanation semantically, as reporting that “There are chairs” is made true by the arrangement of atoms.²¹ Indeed, for our purposes we could understand it in a deflationary manner congenial even

²¹ This is close to Sider’s “metaphysical semantics” (2011, ch. 5). While Sider himself doesn’t endorse this, one could in principle say that P explains Q iff the correct metaphysical semantics implies that ‘Q’ is true if P. But this bears considerable refinement, and, of course, if one wants to join me in defining fundamentality in terms of explanation, one had better not join Sider in defining metaphysical semantics in terms of fundamentality. But I will not elaborate on these issues here.

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to a logical positivist. For even a positivist can accept that John is a bachelor in virtue of being an unmarried male; she just understands this as meaning that “John is a bachelor” follows logically from “John is an unmarried male” together with the analytic definition of “bachelor”.²² All I require is that we do not understand this notion of explanation in terms of possibility. Quite the opposite: I want to understand possibility in terms of this notion of explanation. According to the second claim, then, the fundamental matters are those that are unexplained in this constitutive sense: if they obtain, there’s nothing in virtue of which they obtain. This conception of fundamentality is controversial (see Wilson 2016 for objections) but I won’t defend it here. If you like, you can read this second claim as stipulating the sense of “fundamentality” meant in the first claim. Either way, putting the two claims together yields a theory of strict possibility in terms of explanation: a strictly possible world is a way of reorganizing unexplained matters. One nice feature of this theory is that it explains why the space of possibilities is as broad as it is. Handrons could have been arranged differently in space, let’s agree, but why? Our theory offers an answer. For there is, I think, a constitutive connection between explanation and possibility: roughly, that if something has no explanation, then it could have been otherwise. The idea is that if there’s no reason why something is the way it is—if there is nothing making it be that way—then it needn’t have been that way. This is why (ignoring the metaphysical principles for a moment) if you reorganize those matters for which there is no explanation, such as how handrons are arranged in space, you get a way things could have been; a possible world. Of course, as stated, this constitutive connection between explanation and possibility is no more than an aphorism and needs considerable refinement. Still, like all good aphorisms it contains a germ of truth—enough to shed some light on why it makes sense to think of a strict possibility as a reorganization of unexplained matters.²³ ²² See Dasgupta (2017) for various “deflationary” ways of understanding this notion of constitutive explanation. ²³ What about the metaphysical principles themselves? Why can’t they be otherwise? As stated, our aphorism implies that, being necessary truths, they must have an explanation. But

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140   Indeed, the connection between possibility and explanation goes deeper. Suppose the fundamental matters concern atoms arranged in space, and suppose that W is a world in which atoms are arranged chairwise. Arguably, W is also a world in which there is a chair. But why? Why isn’t it a world in which there’s a penguin? The question is what constrains what’s the case at W above and beyond the fundamental matters. One natural answer is that whatever’s the case at W must hold in virtue of the fundamental matters at W. The existence of a chair would hold in virtue of the arrangement of atoms in W but the existence of a penguin would not; that is why W is a world in which there is a chair. The general idea, then, is that the space of possibility is limned by explanation. If something is constitutively explained by other matters, it is thereby constrained: it cannot be otherwise so long as those other matters remain unchanged. But if it has no constitutive explanation, then it lacks this constraint and can therefore vary freely.²⁴ To implement this more precisely we need a nonfactive notion of constitutive explanation on which one state of affairs, which may or may not obtain, is explicable by others, which may or may not obtain. This could be regimented with an operator T, U, . . . settle that S glossed as “if T, U, . . . , that would make it the case that S”, where this doesn’t imply that T, that U, . . . or that S.²⁵ For convenience I will

the explanans must presumably be necessary too, so do we have an infinite regress? Here is one place the aphorism needs refinement. One solution is to restrict it to matters that I have elsewhere called “substantive”; that is, matters that are “apt for being explained” in the constitutive sense. If so, then “autonomous” matters—matters that are not apt for being explained—fall outside the scope of the aphorism and can then be necessary and true for no reason. See Dasgupta (2014b, 2016) for more on the distinction between ‘substantive’ and ‘autonomous’ truths. We can then say that the metaphysical principles, if any, are autonomous and hence also outside the scope of the aphorism. But it would detract from the main thread to refine the aphorism in detail here. ²⁴ Ignoring the metaphysical principles, of course. They are, in effect, a second source of modal constraint alongside explanation. ²⁵ Since statements of this form help generate the space of worlds, they don’t vary in truth value from world to world; they are just true or false simpliciter. Does this mean that they’re necessary? We could say that, but it would be misleading, since their truth or falsity is prior to

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sometimes abbreviate this with quantification over states of affairs, or facts, or “matters”, saying that some matters (facts, states) settle others. Our theory, then, is that a strictly possible world is a way of reorganizing those matters that aren’t settled by anything. If those matters just concern handrons spinning in a void, then a strictly possible world W is a way of reorganizing how the handrons spin in the void. And W is a world in which there is a chair just in case the handrons in W spin in such a way that constitutes (that is, settles that there is) a chair.

8. Strict and loose possibility We’ve just understood strict possibility in terms of being unexplained. But I’ll now argue that there are two related notions of being unexplained. If ‘left’ and ‘right’ are devices of coherence, then there is one sense in which matters of left and right have an explanation and another sense in which they don’t. The former sense yields the notion of strict possibility; the latter yields the notion of loose possibility we’re looking for. The key is to recognize that if ‘left’ and ‘right’ are devices of coherence, then they’re not fully factual. Lee and Yalcin (n.d.) also argue for a nonfactualist view about left and right; here I’ll argue that non-factualism follows from the claim that they’re devices of coherence. By calling an expression factual I mean, roughly speaking, that its function is descriptive. The word ‘green’ is factual insofar as it is (typically) used to describe what something is like, i.e., as being green. By contrast, ‘Hooray!’ is non-factual insofar as its function is not descriptive but emotive—one uses it to express joy. Likewise, on a simple expressivist view of moral discourse, ‘good’ is non-factual insofar as it is used not to describe something but to express a pro-attitude toward it.²⁶ the worlds they generate. I prefer to think of them as “aworldly” in something akin to Fine’s (2005) sense. ²⁶ I just explained the factual vs. non-factual distinction in terms of a word’s function, but some philosophers express puzzlement about the idea that words have functions. I confess that I’m puzzled about their puzzlement. Like cups and saucers, words are things we use for certain purposes; their function can be thought of as whatever we use them to do. This is not the only

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142   An utterance can be evaluated as to whether it fulfills its function. For example, since an utterance of (1) Grass is green. functions to describe what grass is like, it fulfills its function iff, and because, grass is as (1) describes it to be; that is, iff, and because, grass is green. We therefore evaluate (1) positively by calling it ‘right’ or ‘correct’. More generally, a factual utterance fulfills its function iff, and because, it is true. Hence the primary standard of evaluation for a factual utterance is truth: it is correct iff, and because, it is true. By contrast, since nonfactual utterances have different functions, they may have different standards of evaluation.²⁷ Is ‘left’ factual? It would be if defined as synonymous with ‘congruent with Changy’, for its function would then be to describe things as being congruent with Changy. Thus, given a particular glove Gary, an utterance of (2) Gary is left-handed. would be correct if, and because, Gary is congruent with Changy. The same goes for an absolutist who interprets ‘left’ as expressing that physical property L that distinguishes the left from right hands. For then the function of ‘left’ would be to describe things as having that property L, and an utterance of (2) would be correct iff, and because, Gary has L.

sense in which words have functions, but it’s a particularly unproblematic sense and is all I need here. ²⁷ In saying that truth explains the correctness of factual utterances, am I assuming an inflationary theory of truth? Strictly speaking, yes. But this is just a convenience, and the main point can be put in deflationary terms. For a deflationist can agree that (1) is correct because snow is white, and so can mimic our account of factual utterances with the following scheme: if ‘S’ is factual, then ‘S’ is correct if, and because, S. There is, of course, more to say about the factual vs. non-factual distinction, but the rough idea just glossed is clear enough for our purposes. For more, see Gibbard (2003) and Yalcin (2012), though I stress that the distinction I’m drawing here may not coincide exactly with theirs.

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But if ‘left’ is a device of coherence, then it is not factual in this sense. For one’s aim in uttering (2) is then not to describe Gary but to cohere with one’s linguistic community—to utter something which, along with other accepted applications of ‘left’ and ‘right’, would yield truths about congruence via the rules (R1)–(R3). So, what makes the utterance correct is not that it is true but that it coheres: coherence, not truth, is the primary standard of evaluation. Remember, the rules (R1)–(R3) are just exit rules; they do not define a property of being left-handed, like L, possession of which by Gary would make (2) true. Rather, the correctness of (2) is fully explained by the fact that it coheres. To be clear, to say that this talk is non-factual is not to deny that it is constrained by the world. For its correctness consists in coherence, and utterances cohere iff they imply truths about congruence via the rules (R1)–(R3). Thus, to say that (2) is non-factual is not to say that its correctness depends solely on one’s subjective mental state or such like, for it depends largely on the worldly facts about congruence. So this is a non-factualism of a somewhat mild stripe, but it’s non-factualism nonetheless.²⁸ Now, suppose (2) is correct: Gary is left-handed. What constitutively explains why Gary is left-handed? I suggest that, on this non-factualist view, there is no answer: there is nothing about the underlying facts of congruence in virtue of which it is left-handed. Again, things would be different if ‘left’ were defined to be synonymous with ‘congruent with Changy’. For in that case it would be natural to say that Gary is lefthanded in virtue of being congruent with Changy. Here there is a kind of semantic transparency: that which explains why (2) is correct—namely, Gary’s being congruent with Changy—also explains why Gary is lefthanded. But on the non-factualist view this transparency appears to break down. There is a constitutive explanation of why my utterance of (2) is correct: it is correct because it coheres with my neighbor’s utterances. But this is not an explanation of why Gary is left-handed. To think so would be to mistake (2) with the factual assertion that Lefty is ²⁸ Moreover, note that the non-factualist need not deny that there are truths or facts of left and right in a deflated sense. If (3) is correct, then it’s assertable that Gary is left-handed; hence it’s assertable that (3) is true in a disquotational sense of ‘true’ (see Field 2001). But its truth in this sense is explanatorily derivative; it doesn’t enter into the explanation of why (3) is correct.

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144   congruent with hands my neighbors call ‘left’. And that mischaracterizes the phenomena: when ‘left’ is used as a device of coherence, the aim is not to describe Gary as having that property, but to cohere with other utterances.²⁹ This breakdown in transparency is familiar from other cases of nonfactual discourse. Consider again a simple expressivist view on which in uttering ‘Charity is good’, I express my pro-attitude toward charity. On this view, the content or meaning of the utterance will involve that pro-attitude—that is what’s expressed, after all. But it is not part of this expressivist view that charity is good in virtue of my pro-attitude toward charity—even in my mouth this explanation would not be right. To think otherwise would be to confuse expressivism with a subjectivist view on which my saying ‘Charity is good’ means that I have a pro-attitude toward charity.³⁰ What we see here is a distinction between a (non-homophonic) explanation of what makes an utterance of ‘S’ correct, and an explanation of what makes it the case that S. With factual utterances they coincide precisely because correctness amounts to truth. But with non-factual utterances they can differ: that which makes (2) correct—that it coheres—does not make it the case that Gary is left-handed. The upshot is that if a relationalist uses ‘left’ and ‘right’ as devices of coherence, then there is a sense in which matters of left and right have a constitutive explanation and another sense in which they do not. On the one hand, the correctness of all talk of ‘left’ and ‘right’ is explained in terms of congruence: my utterance of (2) is correct in virtue of the fact that Gary is congruent with gloves my neighbors call ‘left’. In this sense, matters of left and right do have a constitutive explanation in relational terms. On the other hand, there is nothing in virtue of which Gary is left-handed—in that sense it’s a brute, unexplained matter whether Gary is left-handed. If it sounds odd to hear a relationalist saying this, remember that being left-handed is, on her view, a non-factual matter. Thus, it ²⁹ To be clear, it is not impossible to speak a language in which ‘left’ is defined as ‘congruent to what members of my community call “left” ’. My claim is just that a community using ‘left’ and ‘right’ as devices of coherence do not speak this language. ³⁰ What does explain why charity is good, on a simple expressivist view? I won’t try to answer; see Berker (2020) for more discussion.

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remains the case on her view that all the factual matters that are unexplained (in the constitutive sense) are relational; that is what makes her view count as relationalist. We can characterize the two senses of being unexplained as follows: It is loosely fundamental whether S iff nothing settles whether S. It is strictly fundamental whether S iff (i) nothing settles whether S and (ii) it is factual whether S. Put in these terms, the question of absolutism vs. relationalism concerns the strictly fundamental matters of handedness: Which factual matters of handedness are explanatorily basic? The absolutist thinks they consist in the distribution of the physical property L that distinguishes the left from right hands, while the relationalist thinks they consist just in relational matters of congruence. The relationalist I envisage then adds that matters of left and right are loosely fundamental, but that does not contradict her relationalism.³¹ We therefore have two senses in which something can be fundamental (unexplained). Insofar as a possible world is a way of reorganizing fundamental (unexplained) matters, we have two corresponding notions of a possible world. The strictly possible worlds from section 7, we now see, are ways of reorganizing strictly fundamental matters. For the relationalist, the strictly fundamental matters concerning handedness just concern relational matters of congruence; hence a strictly possible world just is a way of reorganizing relational matters of congruence (and other strictly fundamental matters concerning mass, etc.); hence strictly possible worlds agreeing on relational matters of congruence agree on all matters of handedness per Relational Supervenience. But for the absolutist, ³¹ In Dasgupta (2014a) I tried to get at this distinction between two senses of being “unexplained” in terms of plural grounding. Suppose a plurality of facts including X have an explanation when taken together, but X has no explanation on its own. Then there’s one sense in which X has an explanation—it is explained as part of a plurality—and another sense in which it does not—it has no explanation on its own. But I’ve come to think that the framework of plural grounding doesn’t helpfully illuminate the phenomena. The phenomena were the use of ‘kilogram’ predicates described in §9 of that (2014a) paper, which in essence is the idea that such talk is a device of coherence like ‘left’ and ‘right’. I still believe we use ‘kilogram’ talk that way (see section 10 of this chapter), but I now think this usage is better understood in terms of non-factualism than plural grounding.

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146   the strictly fundamental matters include the distribution of the physical property L that distinguishes left from right; hence a strictly possible world on his view is a way of reorganizing L (as well as other strictly fundamental matters); hence the absolutist can distinguish strictly possible worlds that differ only in a uniform flip of left to right and vice versa, contra Relational Supervenience. This is all as expected, for strict possibility is the normal sense of metaphysical possibility already familiar to philosophers. By contrast, say that a loosely possible world is a way of reorganizing loosely fundamental matters. On the relationalist view under discussion, the loosely fundamental matters include relational matters of congruence and matters of left and right, so a loosely possible world is a way of reorganizing those relations and, in addition, reorganizing what’s leftand right-handed. Even for a relationalist, then, matters of left and right can be freely stipulated of a loosely possible world; there is no need to require that they be fixed or settled by the relational facts at that world, so there is no danger of Incommensurability when it comes to loosely possible worlds. For the same reason, there can be loosely possible worlds that agree on all relational facts, yet disagree only in a flip of left to right and vice versa. Thus, when it comes to loosely possible worlds, relationalism does not imply Relational Supervenience. Again, if this sounds odd, just remember what loosely possible worlds are. Whereas strictly possible worlds represent how factual matters could have differed, loosely possible worlds represent how all matters could have differed—including non-factual matters. Thus, for the relationalist, the loosely possible worlds that differ only in a flip of left to right agree in all factual respects; they disagree only in the non-factual respect of left and right.³² In this way the relationalist can recognize a distinct loosely possible world for each strictly possible world recognized by the absolutist. But is the converse also true? One might think not. After all, the absolutist (as ³² This distinction between loose and strict possibility is closely related to Russell’s (2015) distinction between possibility and factual possibility: in both cases the former represents how all matters, including non-factual matters, could have differed. However, he explicitly rejects the idea that there are two distinct notions of possibility in play; his aim instead is to understand Lewis’s idea that distinct possibilities can “correspond” to the same possible world. I’m not exactly sure what this Lewisian idea amounts to and so am unsure to what extent our views really differ.

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we are imagining him) thinks that the only strictly fundamental matters of handedness are matters of left and right, and matters of congruence are settled by them. By contrast, the relationalist thinks that matters of left and right and matters of congruence are loosely fundamental. Does that mean there are weird loosely possible worlds that the absolutist makes no sense of, for example one in which there are two incongruent left-handrons? In principle, yes. But more likely, the relationalist will think that the conditionals encoded in the exit rules (R1)–(R3)—for example, that if two handrons are both left-handed, then they are congruent—are metaphysical principles. This is not ad hoc: on her view those exit rules are constitutive of meaning, and hence the encoded conditional is plausibly a conceptual or analytic truth. If that is right, the relationalist does not in the end recognize these weird worlds that the absolutist cannot make sense of. Hence, we have Modal Correspondence Thesis: There is a one-to-one correspondence between the space of strictly possible worlds recognized by the absolutist and the space of loosely possible worlds recognized by the relationalist that preserves all relational facts and facts about what is left and right. Which is precisely what we were after.

9. How to be a relationalist We are finally in a position to see how the relationalist can account for the use of ‘left’ and ‘right’ in scientific practice, as described in section 4. That practice included expressing observations of handrons in those terms and writing down their theory of handrons as (F) Whenever a handron collides with another, it changes color iff it is left-handed. We saw in section 5 that if ‘left’ and ‘right’ are devices of coherence, this practice makes perfect sense to a relationalist. The residual question was how to make sense of the deterministic reasoning in which (F) is used to

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148   solve initial value problems like Questions 2 and 3. That reasoning involved characterizing the possible worlds in terms of ‘left’ and ‘right’ and distinguishing worlds that are mirror images of one another, and the worry was that the relationalist can’t make sense of that. The solution is now clear: the relationalist can interpret initial value problems as involving loosely possible worlds. After all, we know that the absolutist can interpret Question 2 as involving a strictly possible world containing two left-handrons, and Question 3 as involving a distinct strictly possible world containing two right-handrons. By the Modal Correspondence Thesis, the relationalist can do exactly the same with loosely possible worlds instead. More generally, the relationalist can now say that (F) is deterministic. Recall from section 2 that we defined determinism in terms of possible worlds thus: A theory is deterministic iff any two possible worlds in which it obtains, and which agree at one time, agree at all times.

So we must now distinguish two notions of determinism, strict and loose, depending on whether “possible world” is understood in the strict or loose sense. We said in section 2 that (F), as the absolutist understands it, is deterministic, and in retrospect what we meant was that it is strictly deterministic—we were working with the standard sense of possibility, which we now call “strict possibility”. By the Modal Correspondence Thesis it follows that (F), as the relationalist understands it, is loosely deterministic.³³ But if the relationalist mimics the absolutist so closely, what, then, is her advantage? The answer is that she avoids the epistemic problem plaguing absolutism from section 3. Remember, the absolutist interprets

³³ Or more cautiously, (F) is loosely deterministic on the relationalist’s interpretation if and only if it is strictly deterministic on the absolutist’s. The more cautious statement is more accurate because, of course, (F) itself isn’t really deterministic at all. One has a deterministic theory only by combining (F) with a determinist theory of motion; see section 2. But the point remains that, on the absolutist’s view, any two strictly possible worlds that agree on (F) and agree on all facts of left and right at a time, agree on which handrons will change color upon collision. The point is that the same is true on the relationalist’s view vis-à-vis loosely possible worlds.

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(F) as stating that the handrons that change color are those with some physical property L, such as being aligned with some oriented field. But we never observed that handrons with that property change color. Everything would look exactly the same in a strictly possible world in which all handrons are flipped, so that the ones changing color are antialigned with the field. For the absolutist, the difference between these worlds is factual: there is a genuine fact about whether the handrons changing color are aligned with the field, a fact that can’t be known. But on the relationalist view there is no such fact. True, she can distinguish loosely possible worlds that differ only in a uniform flip of left to right, but the difference is merely non-factual, a difference in labels (as it were) but not in fact. As emphasized in section 6, this approach is adequate only if loose possibility is a genuine species of metaphysical possibility. Otherwise we’ve not shown how the relationalist can say that (F) is deterministic in the right sense. The argument of sections 7 and 8 is that it is indeed a genuine kind of metaphysical possibility. The key lies in the connection between metaphysical possibility and constitutive explanation: that if something is constitutively unexplained—if there is nothing making it be that way—it could have been otherwise. What I then argued is that there are two related notions of being constitutively unexplained: strict and loose fundamentality. Each notion then gives rise to a corresponding notion of possibility: strict and loose possibility, respectively. Thus, loose possibility is as much a genuine notion of metaphysical possibility as strict possibility is, and for the same reason. This approach will be rejected by those seduced by the idea that there is a clear, univocal notion of “metaphysical possibility” that we grasp independently of its connection to fundamentality and explanation. On this view, metaphysical possibility may (as it happens) correspond to ways of reorganizing strictly fundamental matters, but no notion of metaphysical possibility corresponds to ways of reorganizing loosely fundamental matters. This view is antithetical to the approach I take here, on which there is no content to talk of metaphysical possibility apart from its connection to explanation. There is no space to settle this deep issue of modal metaphysics in this paper; I can only be explicit about what my approach is.

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150   Still, even granting that loose possibility is an appropriately metaphysical modality, one may still object that it is not the right kind of metaphysical modality to use when defining notions like determinism. The worry would be that there is no reason why the relationalist should use it, rather than strict possibility, other than the ad hoc reason that things work out nicely for her if she does. But this objection gets things exactly back to front. After all, the difference between strictly and loosely possible worlds is that the former represent how the factual matters could have differed, while the latter represent how all matters could have differed. Since the relationalist thinks that matters of left and right are non-factual, it follows that questions of how such matters could have differed are intelligible for her only as questions about loose possibility. It would, therefore, be perverse to expect her to model counterfactual reasoning about left and right with strictly possible worlds. To the contrary, loosely possible worlds are precisely what it makes sense for her to use when thinking counterfactually about left and right, independent of the fact that things work out nicely for her if she does. Even granting this, one might still worry whether loose determinism counts as determinism in any serious sense of the term. The worry is that in the definition of determinism it is important that worlds agree at a time if they agree in all respects intrinsic to that time. Otherwise, one could say that agreement at a time requires agreement in what will happen from that time onward, in which case determinism becomes trivial. And the worry would be that matters of left and right are not intrinsic to a time, on the relationalist’s view, since they serve to summarize relational matters of congruence that may hold between times. In response, I won’t quibble over whether right and left count as intrinsic for the relationalist (I suppose we could distinguish loose and strict senses of ‘intrinsic’ . . .) The important point is that allowing agreement at a time to include agreement in matters of left and right does not render determinism trivial. For the theory (F*) When a left-handron collides with another, there is an 80 percent chance that it changes color.

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is not deterministic in either the loose or strict sense, so there is no danger that the notion of loose determinism is trivial. Of course, ‘left’ and ‘right’ is just loose talk on the relationalist’s view; it doesn’t get at how things are at the (strictly) fundamental level. When doing physics in strictly fundamental terms, she will just talk in terms of congruence. Thus, the strictly fundamental First Law of handrons will be: (F-Minimalist)

(i) If x and y are congruent handrons, then x changes color on collision iff y does too. (ii) If x and y are incongruent handrons, then x changes color on collision iff y does not.

I argued in section 2 that this is indeterministic, and that is true in both the strict and loose senses. But as we saw in section 3, this is a virtue of relationalism since, fundamentally speaking, the indeterministic behavior described by (F-Minimalist) is all that we observe. Here, then, is the relationalist picture. At the strictly fundamental level, the world is an indeterministic system governed by (F-Minimalist). But we rarely represent matters of handedness in strictly fundamental terms; we typically use ‘left’ and ‘right’ as devices of coherence to characterize the world more efficiently. And characterized like that, handrons behave in the loosely deterministic manner expressed by (F). Thus, once the relationalist has introduced talk of ‘left’ and ‘right’, she’ll reason about handrons in the same deterministic manner that the absolutist will; the only difference will be their interpretation of the reasoning. Much the same goes for locality. We saw in section 2 that the absolutist’s physics of handrons is local. For (F) says that whether a given handron will change color depends on whether it is left-handed, and for the absolutist this depends on whether it has that property L that distinguishes left from right-handrons, not on its relation to far-off handrons. This was in contrast to (F-Minimalist), which says that whether a handron will change color depends on the results of other collision events that may occur thousands of miles away. But our mistake was to conclude that the relationalist’s physics in toto is non-local. For once she introduces talk of ‘left’ and ‘right’, her physics will include (F) and she can agree that (F) is local. After all, she agrees that whether a handron is

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152   left-handed doesn’t depend on its relation to far-off handrons—on her view being left-handed is loosely fundamental and so doesn’t depend on anything. True, the correctness of calling a handron ‘left’ may depend on congruence relations to far-off handrons, but that it is left-handed does not. In that sense she can agree that, according to (F), whether a handron changes color does not depend on the results of far-off collision events. If one defines locality in terms of possible worlds, this point can be put in terms of strict and loose possibility just as it was with determinism. How might such a definition work? As Baker explains, the idea behind locality is that “approximately isolated systems can generally be treated as if they were entirely isolated” (2020, n. 21). This is what allows us to model our solar system to a high degree of accuracy while ignoring the gravitational effects of Alpha Centauri. The idea is that if you ignore everything outside the system and pretend that it’s its own possible world, this should have negligible effect on what the theory predicts about the system. Thus, if S is an approximately isolated subsystem of a world W, locality amounts to the following idea: that the result of first taking a world that’s an intrinsic duplicate of S at a time t0 and then evolving the duplicate forward to t1 according to the theory is approximately the same as first evolving the entire world W from t0 to t1 according to the theory and then taking a world that’s an intrinsic duplicate of S at t1. Thus, locality amounts to a kind of commutativity between the operation of taking a world that’s an intrinsic duplicate of a subsystem at a time and the operation of evolving a world over time according to the theory. Admittedly, precisifying this idea is somewhat complex and there is room to quibble over the details, but those details don’t matter to the general point. Take whatever possible worlds definition of locality you like; we can then distinguish strict vs. loose locality depending on whether the worlds are read as strict or loosely possible worlds. By the Modal Correspondence Thesis, (F) is loosely local on the relationalist’s interpretation if and only if it is strictly local on the absolutist’s.³⁴ ³⁴ Here is one way to precisify a possible worlds definition of locality. Given a theory T and a subsystem S of a world W, call the worlds in which T is true and whose complete state at t duplicates the intrinsic state of S at t the isolated ST,t worlds. These worlds represent how S can evolve from t, according to T, if we ignore the rest of the world W. Then take the worlds that

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And the same goes for other modal reasoning about left and right concerning counterfactuals, causation, explanation, and so on: the relationalist will exactly mimic the absolutist’s reasoning, differing only in her interpretation of what it means. Thus, suppose an absolutist asserts a counterfactual such as ‘Had two left-handrons collided, they would have changed color’. And suppose he interprets this as having a possibleworlds truth condition à la Lewis-Stalnaker. Then the relationalist can agree, adding just that she interprets the truth condition as involving loose rather than strict possible worlds. This might sound odd. If being left-handed is non-factual, how could there be scientifically respectable counterfactuals regarding how lefthandrons would behave? How could being left-handed play a role in scientific explanation or causation? The worry would be that only factual matters can be “real” pushes and pulls. But remember, the relationalist is not trying to say that being left-handed does any work at the strictly fundamental level. She is rather earning the right to speak like an absolutist, with no pretense that this is getting at how things are most fundamentally in the strict sense. Thus, she will happily say that a handron changes color because it is left-handed, so long as it is noted that this is not getting at the strictly fundamental pushes and pulls.

10. Mass That is how to be a relationalist about handedness. Let me now explain how this approach carries over to the case of mass and discuss Baker’s arguments I mentioned at the beginning. As with handedness, the issue of absolutism vs. relationalism concerns the strictly fundamental facts about mass. The relationalist thinks they

agree with W at t and in which T is true, and call the subsystem S in each world an embedded ST,t system. These represent how S can evolve from t, according to T, if we take into account the rest of the world W. Finally, say that the embedded ST,t systems match the isolated ST,t worlds iff there’s a one-to-one correspondence between them that maps each system to a world that’s an approximate intrinsic duplicate. Then we can define a theory T to be local iff, for any world W, any approximately isolated subsystem S of W, and any time t, the embedded ST,t systems match the isolated ST,t worlds. This can then be interpreted over strictly or loosely possible worlds as indicated in the text.

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154   just concern how bodies are related in mass, for example that x is less massive than y, or twice as massive as z. Relationalists may disagree on which relations are strictly fundamental—orderings, ratios, or something else—but for our purposes nothing hangs on this and I’ll assume they’re ratios for simplicity. By contrast, the absolutist thinks that the strictly fundamental facts concern which intrinsic mass each body has, and these then settle their mass relationships: if x is more massive than y, this is because of their intrinsic masses. In the case of handedness a central question was the interpretation of ‘left’ and ‘right’. Here the corresponding question is the interpretation of units such as ‘kilogram’. For the absolutist, a natural interpretation is that terms like ‘1 kg’ and ‘2 kg’ directly refer to intrinsic masses. She might add that their referent is fixed by a description involving a standard object—perhaps ‘1 kg’ is to refer to that intrinsic mass possessed by the standard kilogram in Paris. But however reference is fixed, the view is that kilogram terms make direct reference to strictly fundamental, intrinsic properties.³⁵ How might the relationalist interpret kilogram talk? One option is to define ‘1 kg’ to be synonymous with ‘equal in mass to the standard kilogram in Paris’—this is like defining ‘left’ as ‘congruent with Changy’. But the better option is to understand ‘kilogram’ as a device of coherence, just as we did with ‘left’ and ‘right’. On this view, predicates of the form ‘x is r kilograms’ are stipulated to be governed by the following exit rule: (K) x is r kilograms, y is s kilograms x is r/s times as massive as y But one stipulates nothing else: there are no introduction rules that specify sufficient conditions for concluding (say) that a given body is 2 kg, so the predicates have no explicit definition. Nonetheless, this is sufficient

³⁵ While the term ‘1 kg’ contains a numeral, on this view it makes direct reference to a particular intrinsic mass and not the number. Thus, this is not the “Pythagorean” view on which numbers are fundamentally enmeshed in matter.

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to ground a meaningful practice in which the predicates are used to record information about mass-ratio. When first applying the predicates to a body one has free reign—one can call it 1 km, 2 kg, or whatever. But if one calls it 2 kg then subsequent usage is constrained: given another body twice as massive as the first one must call it 4 kg else one could infer a falsehood via (K). Thus, one’s aim in applying these predicates of mass-inkilograms is to cohere with one’s neighbors in the sense that the combined uses imply truths about mass-ratio via the rule (K). On this view, the standard kilogram in Paris plays no semantic role in defining or fixing the referent of ‘kilogram’. Its role is rather to help billions of language users worldwide cohere: if we all cohere with the statement ‘The standard kilogram is 1 kg’, then we’ll all cohere with each other. I believe that this is, in all important respects, how our talk of mass in kilograms actually functions.³⁶ But that is an empirical hypothesis that I will not defend here. All I need is the uncontroversial claim that it is possible for a community of relationalists to use ‘kilogram’ like this. If they do, their kilogram talk will be non-factual for the same reason as with ‘left’ and ‘right’. For in uttering ‘x is 2 kg’ their aim is not to describe but to cohere; hence the utterance is correct not because it is true but because it coheres. Thus, matters of mass in kilograms are loosely fundamental, again for the same reason: there’s no constitutive explanation of why x is 2 kg, just an explanation of why ‘x is 2 kg’ is correct. When it comes to counterfactual reasoning, then, the relationalist will distinguish strict from loose possibility just as before. The strictly fundamental matters about mass just concern mass relationships; and so a strictly possible world is just a reorganization of mass relationships (along with other strictly fundamental matters unrelated to mass); hence strictly possible worlds agreeing on all mass-relational matters must agree on all matters of mass. This is the analogue of Relational Supervenience in the case of mass. But the loosely fundamental matters include matters of mass in kilograms; hence a loosely possible world includes a reorganization of the mass in kilograms of each thing; hence there can be loosely possible worlds that agree on all relations of mass and disagree only in a uniform doubling of mass in kilograms. Indeed, ³⁶ See Dasgupta (2013, §4) and (2014, §9) for arguments to this effect.

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156   this space of loosely possible worlds will correspond one-to-one with the space of strictly possible worlds recognized by the absolutist, for the same reasons as before. Thus, we have: Modal Correspondence Thesis for Mass: There is a one-to-one correspondence between the space of strictly possible worlds recognized by the absolutist and the space of loosely possible worlds recognized by the relationalist that preserves all mass-relational matters and matters of mass in kilograms. We can now see how a relationalist can interpret a physics of mass. Imagine that some fictional physicists observes the behavior of massive bodies and records their measurements in units like mass in kilograms and acceleration in m/s². And imagine their observations confirm a classical theory consisting of f = ma and various force laws, where f = ma is to be understood as (N) The total force in Newtons acting on a body = its mass in kilograms times its acceleration in m/s² and the equations describing force laws are understood similarly. Imagine further that they use this theory to solve initial value problems like Baker’s cases of escape velocity. One initial value problem involves a planet with a rocket on its surface, each with a specified mass in kilograms, and the rocket is fired upward at a specified velocity. The question is whether the rocket will escape the planet’s gravitational field, and the answer is that it does. A second problem is just like the first except the planet and rocket are double in mass what they were in the first, and Baker shows that in this case the rocket does not escape. Note that the initial states are characterized in terms of kilograms and they’re distinguished only by a uniform doubling thereof. These are, therefore, analogous to Questions 2 and 3 concerning handedness, which were characterized in terms of left and right and distinguished only by a uniform flip (indeed, I designed Questions 2 and 3 to be analogous to Baker’s cases). Baker shows that the theory behaves deterministically insofar as it entails a unique solution for each problem. Indeed, let’s

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imagine that the theory satisfies the standard possible worlds definition of determinism from section 2.³⁷ To the absolutist this all makes perfect sense. Measurements of mass in kilograms are measurements of a strictly fundamental quantity, namely intrinsic mass, and equations like (N) describe how this quantity relates to others. Initial value problems characterize the initial state of a strictly possible world with respect to this quantity, and since on his view strictly possible worlds can differ in a uniform doubling of mass, there is no problem distinguishing the two problems. And the theory is strictly deterministic insofar as any two strictly possible worlds in which it holds, and which agree at one time, agree at all other other times—that’s why the theory yields a unique solution in each case. The relationalist can make good sense of this too, but her interpretation is different. On her view measurements of mass in kilograms aren’t descriptions of a strictly fundamental quantity; they’re statements in a derivative and non-factual vocabulary designed to conveniently store information about the underlying mass relationships. Thus, an equation like (N) isn’t a strictly fundamental physics of mass; it’s what physics looks like when presented in this derivative and non-factual vocabulary of mass in kilograms. Since talk of kilograms is non-factual, reasoning about how things could have differed in kilograms must involve loose, not strict possibility—after all, strictly possible worlds don’t represent how nonfactual matters could have differed. Thus, the relationalist will naturally interpret Baker’s initial value problems as characterizing the initial state of two loosely possible worlds. By the Modal Correspondence Thesis for Mass, the relationalist can distinguish two such problems differing only in a uniform doubling of mass, just as the absolutist can. Indeed, the correspondence shows that the relationalist’s theory must be loosely deterministic if and only if the absolutist’s theory is strictly deterministic. Why, then, did Baker think that relationalism about mass leads to indeterminism? He noted that the initial states of the two problems agree on all mass relationships (and in all respects other than mass); hence on the relationalist’s view their initial states agree in all respects simpliciter. ³⁷ To be clear, classical mechanics is arguably not deterministic; see Norton (2008). But these kinds of failures of determinism are not germane to our discussion so I’ll bracket them here.

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158   Since they diverge thereafter, this is indeterminism. Moreover, he argued ingeniously that the relationalist must agree that there are indeed two ways for that initial state to evolve.³⁸ But the mistake was to think that since they agree on all mass relationships, they must agree in all respects of mass. That would be right, on the relationalist’s view, if they are strictly possible worlds. But on her view they’re not: they’re loosely possible worlds, and, as we know, she can distinguish two such worlds that differ only in a uniform doubling of mass in kilograms. Thus, there’s no failure of determinism: there are, indeed, two ways for the initial state characterized relationally to evolve, but each way corresponds to a different initial state vis-à-vis mass in kilograms. Nonetheless, Baker is right that indeterminism lurks in the vicinity. For what would the relationalist’s physics look like if expressed in strictly fundamental terms? She would restrict herself to talk of mass relationships and so couldn’t propose (N). In its place, one option would be a Machian alternative that contains reference to a particular body, such as: (N-Machian) The total force in Newtons acting on a body = its mass ratio with the standard kilogram in Paris times its acceleration in m/s². But this inherits the same problems with Machian laws discussed in section 2: it doesn’t apply to worlds in which the standard object in Paris doesn’t exist. Better is a minimalist theory that stands to (N) just as (F-Minimalist) stands to (F). Such a theory is complex to write down in full generality, but the idea is that it would imply statements like (N-Minimalist) If one body is r times as massive as another and they are both subject to the same force, the first will accelerate at 1/r the rate as the second.³⁹ ³⁸ The argument is this. Consider a third initial value problem containing two planet-rocket systems with one system double the mass of the other, and let the systems be sufficiently far apart that they’re more or less isolated. And suppose the rocket in the less massive system escapes, while the other rocket does not. The relationalist must agree that this is possible. But by locality, the behavior of each system doesn’t depend on its relation to the other, so each would behave similarly were it its own possible world. Thus, we have two worlds that agree initially in all mass relationships (and all respects other than mass), yet diverge thereafter. ³⁹ See Field (1980) for a proper articulation of what a set of minimalist laws like this might look like.

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And (N-Minimalist) is indeterministic and non-local in both the strict and loose sense, just as (F-Minimalist) was. After all, it implies nothing about how a world containing just one body will evolve over time (indeterminism), and it implies that a body’s rate of acceleration depends on how other bodies many thousands of miles away are accelerating (non-locality). But as with handedness, this is a virtue: the indeterministic and nonlocal behavior described by (N-minimalism) is all we really observe. In particular, we did not observe the deterministic behavior described by the absolutist’s interpretation of (N). Recall that on that interpretation (N) states how a body with a particular intrinsic mass M would accelerate under a given force. But we never observed that that particular mass M affects acceleration in that way, since everything would look (and smell, and taste) exactly the same in a uniformly mass-doubled world. In that world, the mass that’s half M would affect acceleration in the same way that M actually does, and the behavior of all the other intrinsic masses would be similarly transformed.⁴⁰ What we actually see gives no reason to think we live in the one world over its doubled cousin. All we really observe are the relational facts common to both worlds, namely that bodies accelerate at a rate inversely proportional to their mass. But that just confirms (N-Minimalist), not the absolutist’s interpretation of (N). The upshot is this. When it comes to strictly fundamental physics, the relationalist proposes (N-Minimalist), while the absolutist proposes (N) interpreted as stating how particular intrinsic masses affect acceleration. The absolutist’s theory goes beyond what’s confirmed by observation; the relationalist’s does not; that’s a point in favor of relationalism. True, the relationalist’s theory is indeterministic and non-local, but that’s a virtue, since that’s all that observation confirms. Still, when reasoning about mass, the relationalist will find great utility in using kilograms as a device of coherence, and when she does physics in these terms, she’ll propose (N) and reason counterfactually in terms of loose possibility. Her physics ⁴⁰ In the doubled world, would (N) be true? Yes and no. What (N) actually states, in our mouths, is false at that world: the intrinsic masses line up with rates of acceleration differently than (N) states. But our counterparts in the doubled world, using the terms as we do, would express a truth with (N). For their term ‘1 kg’ would pick out a different intrinsic mass than ours and therefore what (N) means in their mouth would be true.

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160   will then behave on the surface just like the absolutist’s: it is deterministic and local (in the loose senses), and initial value problems are solved in just the same way as the absolutist thought. The only difference lies in the interpretation of this practice. For the absolutist it gets at how things are at the strictly fundamental level, while for the relationalist it is, ultimately, just a convenient shorthand.

11. Conclusion In both handedness and mass, then, the success of relationalism hangs centrally on the interpretation of the vocabulary of ‘left’ and ‘right’, and ‘kilograms’, respectively. This issue has received insufficient attention. It is sometimes presumed that the relationalist has no right to such vocabulary—witness Earman saying that the relationalist cannot so much as express the law (F) since it contains the term ‘left’. Other times the relationalist is allowed the vocabulary but it’s presumed to behave modally much like other vocabulary—witness Baker’s assumption that if two initial value problems agree in all mass-relational respects, then the relationalist must count them as agreeing in all respects of mass. But both presumptions are wrong. The relationalist can interpret the contested talk as a device of coherence, and if she does, its modal behavior will mimic how the absolutist always thought it behaved. Indeed, this may explain why absolutism initially strikes many as the more plausible view. For being left- or right-handed and having a particular mass in kilograms certainly appear to be independent variables, states that hold independently of relations of congruence and mass ratio respectively. Relationalism has always been thought to deny this appearance; hence the plausibility of absolutism. But we now know that a relationalist needn’t deny the appearance. They are independent variables, she can say, insofar as something’s being left-handed and being 2 kg doesn’t hold in virtue of anything. But the idea that a relationalist (of all people) can say this is only apparent once we reflect on her interpretation of the relevant vocabulary and recognize it as a device of coherence. I’ve applied this approach to the cases of handedness and mass; could it also apply to other domains in which there is an analogous dispute

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between absolutist and relationalist views? These include the case of motion discussed at the beginning, and also disputes about the interpretation of ‘gauge’ theories. The question is whether the language used by the absolutist to describe his extra feature of reality can be interpreted by the relationalist as a device of coherence, in which case the relationalist could largely mimic the absolutist in the way I’ve described in this chapter. But I leave a discussion of these other cases for another time.

Acknowledgments This paper has been in the making far too long, and I’ve received feedback from more people than I recall. I do remember that Daniel Berntson, Neil Dewar, Niels Martens, Michaela McSweeney, John Morrison, Brad Skow, Jack Spencer, and Nat Tabris all suffered through early drafts and responded with undue politeness; I thank them all for their helpful feedback. I also thank audiences at the University of Oslo, Rochester University, Columbia University, Princeton University, Brown University, the National Autonomous University of Mexico, the Philosophy of Science Association 2018 meeting, and North Carolina State University for engaging with this material at various stages of its development. Most of all, I thank David Baker for raising these issues of determinism and locality and for discussing this material with me over the years, during which time he has invariably been a model philosophical interlocutor.

References Baker, David. (2020). “Some Consequences of Physics for the Comparative Metaphysics of Quantity.” In Karen Bennett and Dean W. Zimmerman (eds.), Oxford Studies in Metaphysics: Volume 12, 75–112. Oxford: Oxford University Press. Barbour, Julian (1999). The End of Time. New York: Oxford University Press. Berker, Selim. (2020). “Quasi-Dependence.” In Russ Shafer-Landau (ed.), Oxford Studies in Metaethics: Volume 15, 192–218. Oxford: Oxford University Press. Bhogal, Harjit. (2020). “Nomothetic Explanation and Humeanism about Laws of Nature.” In Karen Bennett and Dean W. Zimmerman (eds.), Oxford Studies in Metaphysics: Volume 12, 164–202. Oxford: Oxford University Press.

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162   Brighouse, Carolyn. (1994). “Spacetime and Holes.” PSA: The Proceedings of the Biennial Meeting of the Philosophy of Science Association 1: 117–25. Brighouse, Carolyn. (1999). “Incongruent Counterparts and Modal Relationalism.” International Studies in the Philosophy of Science 13(1): 53–68. Dasgupta, Shamik. (2013). “Absolutism vs Comparativism about Quantity.” Oxford Studies in Metaphysics: Volume 8: 105–48. Dasgupta, Shamik. (2014a). “On the Plurality of Grounds.” Philosophers’ Imprint 14(14): 1–28. Dasgupta, Shamik. (2014b). “The Possibility of Physicalism.” The Journal of Philosophy 111(9): 557–92. Dasgupta, Shamik. (2015). “Substantivalism vs Relationalism about Space in Classical Physics.” Philosophy Compass 10(9): 601–24. Dasgupta, Shamik. (2016). “Metaphysical Rationalism.” Noûs 50(2): 379–418. Dasgupta, Shamik. (2017). “Constitutive Explanation.” Philosophical Issues 27(1): 74–97. Earman, John. (1986). A Primer on Determinism. D. Reidel. Earman, John. (1989). World Enough and Space-Time: Absolute versus Relational Theories of Space and Time. Cambridge, MA: MIT Press. Field, Hartry. (1980). Science without Numbers. Oxford: Oxford University Press. Field, Hartry. (2001). “Deflationist Views of Meaning and Content.” In Truth and the Absence of Fact, 104–56. Oxford: Oxford University Press. Fine, Kit. (2005). “Necessity and Non-Existence.” In Modality and Tense: Philosophical Papers, pp. 321–54. Oxford, UK: Oxford Uuniversity Press. Gibbard, Alan. (2003). Thinking How to Live. Cambridge, MA: Harvard University Press. Kovacs, David. (2020). “Metaphysically Explanatory Unification.” Philosophical Studies 177: 1659–83. Lee, Geoffrey and Seth Yalcin. (n.d.). “Finding Meaning in a Disoriented World.” MS. Lewis, David. (1986). On The Plurality of Worlds. Wiley-Blackwell. Maudlin, Tim. (2012). Philosophy of Physics: Space and Time. Princeton: Princeton University Press.

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Norton, John D. (2008). “The Dome: An Unexpectedly Simple Failure of Determinism” Philosophy of Science 75(5): 786–98. Pooley, Oliver. (2003). “Handedness, Parity Violation, and the Reality of Space.” In K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, 250–80. Cambridge: Cambridge University Press Rosen, Gideon. (2010). “Metaphysical Dependence: Grounding and Reduction.” In B. Hale and A. Hoffman (eds.), Modality: Metaphysics, Logic, and Epistemology, 109–36. New York: Oxford University Press. Russell, Jeff. (2015). “Possible Worlds and the Objective World.” Philosophy and Phenomenological Research 90(2): 389–422. Schaffer, Jonathan. (2016). “Grounding in the Image of Causation.” Philosophical Studies 173(1): 49–100. Schaffer, Jonathan. (2017). “The Ground between the Gaps.” Philosophers’ Imprint 17: 1–26. Sider, Ted. (2011). Writing the Book of the World. Oxford: Oxford University Press. Sklar, Larry. (1974). Space, Time, and Spacetime. Berkeley, CA: University of California Press. Skow, Brad (2007). “Sklar’s Maneuver.” British Journal for the Philosophy of Science 58(4): 777–86. Wilsch, Tobias. (2015). “The Nomological Account of Ground.” Philosophical Studies 172(12): 3293–312. Wilson, Jessica. (2016). “Grounding-Based Formulations of Physicalism.” Topoi 37 (2): 1–18. Yalcin, Seth. (2012). “Bayesian Expressivism.” Proceedings of the Aristotelian Society 62: 124–60.

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6 Nomothetic Explanation and Humeanism about Laws of Nature Harjit Bhogal

Imagine the mosaic of events that occur in this world, spread out across the totality of space and time. Within this mosaic, certain patterns are discernible. The core idea of Humeanism about laws of nature is that laws of nature are just patterns, or ways of describing patterns, in this mosaic. Laws of nature are not entities that stand above the mosaic, governing the patterns of events. More precisely, take Humeanism about laws of nature to be the view that the laws of nature reduce to the Humean Mosaic—that is, the intrinsic physical state of each spacetime point (or each pointlike object) and the spatiotemporal relations between those points—and that the Humean Mosaic is not further reduced to anything else. (From now on I’ll refer to the view simply as ‘Humeanism’.) Humeanism is a popular view, but currently existing versions face serious problems. In particular, there are a series of powerful objections to existing versions of the view. Many of these objections share a guiding thought, though; they are based on the idea that there is a certain kind of divergence between the practice of science and the metaphysical picture suggested by Humeanism. In particular, I’m going to focus on three such objections: the non-supervenience objection, the non-fundamentality objection, and the explanatory circularity objection.¹ ¹ One common objection to Humeanism that I won’t be considering is about the consistency of the view with quantum mechanical entanglement phenomena (see, for example, Maudlin (2007, 50–61)). Some recent work has shown how Humeanism need not be in conflict with quantum mechanics (see Bhogal and Perry (2017), Miller (2014), Esfeld et al. (2014)).

Harjit Bhogal, Nomothetic Explanation and Humeanism about Laws of Nature In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Harjit Bhogal. DOI: 10.1093/oso/9780192893314.003.0006

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I’m going to suggest, however, that this divergence is something that the Humean can make sense of. Implicit in a certain strand of Humean thinking, I claim, is the idea that some of the aims of science differ from the aims of metaphysics. For example, a certain type of Humean might naturally think that scientific explanation aims to unify in a way that metaphysical explanation does not. Because of this, I claim, the Humean can argue that it is appropriate for there to be a divergence between certain elements of metaphysical and scientific practice. I develop this thought by sketching a novel version of Humeanism which accepts this kind of divergence and arguing that this version deals with the objections better than traditional Humean accounts. And further, I argue that my approach can help the Humean give better accounts of other physical modalities—like counterfactuals, chance, and physical possibility—than they have been able to give so far. Clearly, doing this all in the same paper is very ambitious, perhaps to an extent that’s ill-advised. But it is important, I think, to see that many of the objections that Humeanism faces are closely related to each other and, consequently, we can deal with the objections in a unified way. I’m not giving piecemeal responses to the objections that the Humean faces. Rather, I’m going to motivate and sketch a single Humean picture that deals with the objections together. So there is value in discussing them all in the same place. This does come at a cost, though. Since I’m dealing with multiple issues for the Humean which have traditionally been discussed separately, I won’t be able to discuss each issue as deeply. And I certainly won’t be able to fully discuss all the literature on these issues. Rather, the aim is just to sketch a certain Humean picture and show that this picture has promise for dealing with the objections to Humeanism. In section 1 I describe the three objections to Humeanism that I’m focusing on. In section 2 I argue that the Humean, or at least a certain type of Humean, should expect a divergence between scientific and metaphysical practice and that seeing this allows us to respond to the objections discussed in section 1. This comes in two steps. Firstly, I develop Loewer’s (2012) suggestion of distinguishing between scientific and metaphysical

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166   explanation, stressing the different aims of scientific and metaphysical explanation. And secondly, I extend this to a divergence between scientific and metaphysical notions of fundamentality and possibility. In section 3 I argue that this distinction between scientific and metaphysical possibility feeds through to other scientific notions. In particular, recognizing this different space of possibility motivates certain approaches to counterfactuals, physical possibility, and chance.

1. Problems for Humeanism In this section I’m going to briefly describe three objections to Humeanism and quickly look at some of the responses that are common in the Humean literature. As will become clear, these objections are all driven by the idea that there is a divergence between scientific practice and the practice of Humean metaphysics.

1.1. Non-supervenience The most common objection to Humeanism is the non-supervenience objection. Humeanism implies that the laws supervene upon the Humean Mosaic. But this appears to be false; there appear to be worlds that match in their mosaics but differ in their laws. There are many putative examples of this—see, for example Tooley (1977, 669) and Carroll (1994, 57–67)—but here is a particularly simple one due to Maudlin (2007, 67–8): Consider a universe that just consists in an empty Minkowski spacetime. Such a spacetime is a model of General Relativistic laws.² So it appears that there could be a world like this where the laws are General Relativistic. But, such a spacetime is also consistent with Special Relativity being the full story about spacetime and there being a different set of laws about gravitation. So it appears that

² That is, the truth of the propositions that are the laws of General Relativity are consistent with an empty Minkowski spacetime.

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there could be a world like this where Special Relativity is true and there is some non-General Relativistic set of laws. But now we have two worlds that match in their mosaics (both consist in an empty Minkowski spacetime) but differ in their laws. Call these two worlds the empty-SR world and the empty-GR world. There are a couple of distinct ways of pressing this objection, which are not often distinguished in the literature. The first version of the objection is that Humeanism says false things about metaphysical possibility. It says that Humeanism implies that at least one of the two worlds described above is metaphysically impossible, but that is not true—both are metaphysically possible. The second version of the objection is that Humeanism does not do justice to the scientific practice of reasoning about these cases. When scientists reason about GR and SR, the two empty universe cases are both taken seriously and are treated as distinct. In fact, there look to be substantial differences between the cases. For example, there are counterfactuals that differ in truth value between the worlds—in the emptyGR world if there was a massive object then spacetime would be curved. This is because the laws of General Relativity associate masses with curvature of spacetime. Not so in the empty-SR world. Explanations differ between the worlds too. For example, the (lack of) curvature of regions of spacetime is explained differently in the empty-GR world and the empty-SR world. The objection is that the Humean cannot make sense of how these cases are treated differently in scientific practice because, according to them, there are not two distinct worlds here which both have the same mosaic but differ in the laws; rather, there is only one world with that mosaic.

1.2. Non-fundamentality Maudlin (2007, 105) objects to Humeanism as follows: [N]othing in scientific practice suggests that one ought to try to reduce fundamental laws to anything else. Physicists simply postulate fundamental laws, then try to figure out how to test their theories;

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168   they nowhere even attempt to analyze those laws in terms of patterns of instantiation of physical quantities. The practice of science, I suggest, takes fundamental laws of nature as further unanalyzable primitives. As philosophers, I think we can do no better than to follow this lead.

One way (though not the only way) of understanding this passage is as an objection that Humeanism does not respect the practice of science. Scientists don’t attempt to reduce away (certain) laws; rather, they take them as fundamental. The Humean, however, does not take laws as fundamental. So Humeanism conflicts with scientific practice over what is fundamental. Call this the non-fundamentality objection.³

1.3. Explanatory circularity Laws can explain particular events; specifically, they can explain features of the Humean Mosaic. But, the explanatory circularity objection says, Humeanism explains what the laws are in terms of the mosaic. So we have a circularity. For definiteness, I’m going to assume in this paper that this explanation of the laws in terms of the mosaic goes by way of a version of the Best System account of laws (see Lewis, 1983, 42–3). This is the most popular Humean view of laws. On this account the laws are given by systemizing all of the facts about the mosaic in a way that best balances simplicity and informativeness. Slightly more precisely, consider sets of axioms and the deductive closure of those axioms. Some sets of axioms are informative about the mosaic—their deductive closure tells us a lot about the facts about the mosaics. Some sets of axioms are simple, in the sense of being syntactically simple when written down—there are few axioms and they are syntactically short. The axioms that best balance simplicity and informativeness best systemize the mosaic. The

³ Perhaps you don’t find this objection pressing because it seems clear that the sense of fundamentality relevant to the scientist is not the same as the philosophical sense relevant to the debate over Humeanism. Such a thought turns out in my favor, since I will end up encouraging you to have similar thoughts with respect to explanation and possibility.

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axioms of the best system (or some privileged subset thereof) count as the laws.⁴ So, the Humean has to accept a certain kind of explanatory circularity—the laws explain particular events, and in turn those particular events partially explain the laws. But explanations cannot be circular in this way, so Humeanism is false. Versions of this objection are also stated by Armstrong (1983, 40), Bird (2007, 86), Maudlin (2007, 172), and Hajek (2010, 219–20), among many others.

1.4. Responses to the problems Humeans have had disappointingly little to say about the nonsupervenience objection. Commonly, Humeans have just bitten the bullet and accepted that it is not the case that both the empty-GR world and the empty-SR world are possible (see Earman and Roberts (2005, §2), Loewer (1996, 192–4), Roberts (1998, 428–33), Beebee (2000, §§5–6)).⁵ In fact, the typical move is to claim that neither is possible. Biting the bullet seems reasonable enough as a response to the first version of the objection—rejecting an intuition about metaphysical possibility is perhaps not too big a loss for the theory, especially given that all the authors cited above go on to argue that we shouldn’t be confident in the reliability of such intuitions in this context. But this doesn’t help with the second version of the objection. The Humean still has to make sense of how, in our scientific reasoning, we recognize these two cases as distinct and as saying very different things— for example with respect to explanations and counterfactuals—about them. It seems plainly true that the empty-SR world and the empty-GR world differ, for example, in their explanations of the lack of curvature of ⁴ There are lots of further issues here, for example, about what language the axioms can be formulated in, about what exactly ‘informativeness’ means, and about how to weigh informativeness and simplicity. But these don’t matter for our aims. ⁵ Roberts (2008, 358–61) is an exception to this. In effect, he claims that the empty-SR and empty-GR worlds exist and are the same world. This is possible because he has a very nonstandard conception of laws—he calls it the meta-theoretic conception. I’m going to ignore this response and focus on the mainstream conception of law.

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170   spacetime. In the empty-GR world spacetime is not curved because there happen to be no massive objects in the world; in the empty-SR world spacetime is not curved because the basic geometry of spacetime does not allow it. It doesn’t seem open for the Humean to reject this in the way they rejected the intuitions about metaphysical possibility. The Humean literature has been largely silent about such cases. But without a response, this is an extremely serious objection to Humeanism. Responses to the non-fundamentality objection are not common either. This, though, is more understandable than the lack of response to the non-supervenience objection, given the relative rate at which the objections are pressed in the literature. There has been more progress with respect to the explanatory circularity problem. In particular, Loewer (2012) has influentially suggested a response based on distinguishing ‘scientific’ and ‘metaphysical’ explanation. In section 2 I’m going to develop this response and show how it is motivated from a certain Humean way of looking at the world. Further, I’m going to suggest that we can leverage this response into responses to the non-fundamentality and non-supervenience objections.⁶

2. The view The objections of section 1 all point to a divergence between the practice of science and that of Humean metaphysics. The explanatory circularity objection says that scientific explanations run contrary to the explanations given by Humeanism. The non-fundamentality objection says that Humeanism disagrees with scientific practice about what is fundamental. The non-supervenience objection says that scientific practice regards as possible situations that Humeanism does not. My strategy is not to deny this divergence, but to suggest that it is appropriate. In particular, the Humean should accept a distinction ⁶ Another approach to the circularity problem which is gaining in popularity says that laws don’t actually explain events in the mosaic; rather, they explain why one event explains another (Skow (2016); Hicks (2020)). This response deserves much more attention that I can give it here, but it’s rather unclear whether the view does, in fact, avoid problematic circularity (see Lange (2013, 2018); Hicks (2020)).

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between the scientific and the metaphysical notions of explanation, fundamentality, and possibility. Each of these three distinctions can be used to respond to one of the three problems discussed in section 1. Ultimately, these distinctions are motivated by a Humean view where there are differing aims of science and of metaphysics.

2.1. Explanatory circularity and two types of explanation Loewer (2012) suggests, but does not fully develop, a response to the explanatory circularity objection. The idea is to accept both that the laws explain elements of the mosaic and those elements explain the laws but to avoid circularity by claiming that there are two different senses of ‘explain’—the mosaic metaphysically explains the laws, but the laws scientifically explain the mosaic. Loewer’s terminology is apt to cause confusion though, because, as we will see, some metaphysical explanations are relevant to science. So, instead of distinguishing metaphysical and scientific explanation I’m going to distinguish metaphysical and nomothetic explanation. In this section I will consider how the Humean can use this distinction to respond to the circularity objection and I’ll discuss how this response is motivated from within the Humean picture. This response doesn’t come for free; it requires commitments about the nature of nomothetic and metaphysical explanation. But these commitments naturally flow from the Humean viewpoint. (One quick clarification: In what follows, I take explanation to be a ‘mind-independent’ relation between facts⁷—one that holds in the world and is, in general, not determined by us. This is to be sharply distinguished from an act of explanation, which is typically a speech act.) Let’s start by considering the distinction between nomothetic and metaphysical explanation. What is the difference between them? Consider nomothetic explanation first. I’m taking nomothetic explanation to be a type of scientific explanation—they are those where the laws ⁷ Sometimes, for simplicity, I’ll talk about events being explained. Strictly I mean that the fact that the event occurs is explained.

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172   of nature play an explanatory role. For example, an explanation of the velocity of a billiard ball from the facts about its recent collision and the Newtonian laws would count as a nomothetic explanation (if Newtonian mechanics were true). There may be other types of scientific explanation too, like pure causal explanations where the laws do not play an explanatory role. But I’m restricting my attention to nomothetic explanation because the circularity issue is about the laws of nature—whether laws can explain without circularity. We have good reason, independently of Humeanism, to think that a relation of metaphysical explanation exists and is different from nomothetic explanation. In particular, many authors recently have noted that an asymmetric relation of metaphysical explanation is central to philosophical theorizing in many areas (for example, Fine (2001), Schaffer (2009), Rosen (2010)). This is the relation that holds between the fact that there is a table and the fact that there are atoms arranged table-wise (if certain views about composition are true); between the facts about mental states and the facts about brain states (if certain kinds of physicalism about mind are true); and between moral facts and the facts about God’s commands (if certain kinds of divine command theory are true). This type of explanation has some distinctive features. It is very closely connected to metaphysical dependence relations, like grounding or constitution. (Perhaps those relations ‘back’ metaphysical explanations, or perhaps those relations just are metaphysical explanations.) Further, the explanandum in a metaphysical explanation is nothing over and above the explanans, or, at least, the explanandum is, in some sense, not a substantial addition to the ontology when we already have the explanans. (For more substantial discussion, see Bennett (2017, §8.2.2) and Schaffer (2015).) Further, as Loewer (2012, 131) notes, it looks as if the relation of metaphysical explanation cannot be probabilistic. Clearly, there are nomothetic explanations which aren’t metaphysical explanations. So the relations of nomothetic and metaphysical explanation are not identical. Notice that, given the distinction between metaphysical and nomothetic explanation, the thesis of Humeanism—that laws are reduced to the Humean mosaic—is naturally precisified in terms of metaphysical explanation. In particular, we can say that the core thesis of Humeanism

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is that the laws are metaphysically explained by, and thus are reduced to, the Humean mosaic. The strategy, then, is to say that the mosaic metaphysically explains the laws and the laws nomothetically explain the mosaic, thus avoiding circularity. This, in effect, was Loewer’s (2012) suggestion. Loewer, though, doesn’t develop this strategy further. But further development is needed, since it is just at this point that things start to get complicated. To see why, notice that there are some nomothetic explanations which involve metaphysical explanations. Consider this case: Interlevel Temperature: We are aiming to explain T, the fact that the temperature in this room now is 71 degrees. The explanation cites P, the state of the particles in this room and the surrounding area 10 minutes ago, and the fundamental laws. The way this explanation works is that P and the laws together imply E, a fact about the kinetic energy of the particles in this room now. And, given the reduction of facts about temperature to facts about kinetic energy then T is nothing over and above E.⁸

This is a nomothetic explanation of T from P and the laws. It is made up of a metaphysical explanation of T from E and a nomothetic explanation of E from P and the laws. This case points to an important issue with the Loewer response: Certain metaphysical explanations can be part of larger nomothetic explanations, but the Loewer response relies on there being some metaphysical explanations that are not part of nomothetic explanations in this way. In particular, the metaphysical explanation of the laws from the mosaic cannot be part of nomothetic explanations. For example, the set of facts about the mosaic metaphysically explains the laws, and the laws nomothetically explain (at least partially) some particular fact about the mosaic F. But we can’t chain together these explanations. If we did, we would be left with the set of facts about the mosaic explaining a particular fact about the mosaic F. But clearly this is unacceptably circular—a set of facts containing F cannot explain F. ⁸ I also discuss this case in Bhogal (2017, §2).

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174   So, for the Loewer response to be successful we need a reason why the metaphysical explanation of the laws from the mosaic cannot be part of nomothetic explanations, even when other metaphysical explanations can. To put it another way, we need a reason why we cannot chain this particular metaphysical explanation with nomothetic explanations. (So, the Loewer response needs to reject the ‘Transitivity Condition’ of Lange (2013) which implies that you can always chain metaphysical and nomothetic explanations. But more than this, it needs to reject chaining in this particular case. Miller (2015) and Hicks and van Elswyk (2015) provide some reasons to think that the Transitivity Condition fails, but we need more than this—we need an account of how and why this chaining can occur in some cases but not in the cases that would cause problems for Loewer-style responses to the circularity worry.) The key to locating such a reason is to say more about the natures of nomothetic and metaphysical explanation. In particular we need to say more about the aims of such explanations—about what epistemic value those explanations yield. Grasping an explanation of a fact seems to put us in an epistemically better position with respect to that fact—but what is this benefit? Let’s look at the case of metaphysical explanation—I’m going to work with a very simple account of metaphysical explanation: A set of facts S (wholly) metaphysically explains F if and only if S (wholly) grounds F. What is the epistemic value of such explanations? The natural answer, I take it, is that such explanations elucidate the underlying structure of the world. So, in grasping an explanation we gain knowledge of this structure—the grounding structure of the world. Perhaps if we didn’t take talk of grounding, or related notions, seriously, then we would have some other story about what the value of metaphysical explanation is. But I am taking such talk seriously. I’m assuming that there really is a grounding structure to the world in the way that people like Fine (2001), Schaffer (2009), and Rosen (2010) suggest. Given that there is this structure, then it looks that the value of metaphysical explanations comes from limning this structure. Anti-Humeans will likely think that nomothetic explanations work in a similar way. They think that the world comes with a built-in nomic structure, so it is reasonable to claim that the value of nomothetic

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explanations comes from elucidating this structure. For example, imagine that we explain the momentum of a particle by citing its collision with another particle and the laws governing that collision. In doing so we gain knowledge of the nomic structure of the world that gave rise to that event. Things look different for Humeans, though. It would be strange for them to think that the epistemic value of nomothetic explanations, like the particle collision explanation, consists in elucidating nomic structure. Or, more carefully, the Humean should not think that elucidating nomic structure has substantial non-instrumental epistemic value. This is because, for the Humean, such nomic structure isn’t a deep part of the world. Nomic structure, for the Humean, is defined up using a somewhat complicated systemization procedure. If the Humean thinks that knowledge of the output of the best system procedure is particularly epistemically important, it has to be because it helps us with some other epistemic end. It is not plausible for the Humean to take knowledge of the output of this complex systemization procedure to have substantial non-instrumental epistemic value. If, on the other hand, there were deep, primitive governing laws, as certain anti-Humeans think, then it is much more plausible to take knowledge of these to have substantial noninstrumental epistemic value. The Humean, then, should say something different about the epistemic value of nomothetic explanations. What they should do, I claim, is to appeal to an important strand of thinking in the Humean tradition, and in the philosophy of science more generally—a strand of thinking that stresses the value of unification. Stressing the importance of unification in science, and in explanation in particular, has a long tradition. Friedman (1974, 16), for example, gives an account of explanation in terms of unification, where we unify by ‘reducing the total number of independent phenomena’ that we have to accept. Kitcher (1981) gives an account in a similar spirit. But these accounts also locate themselves in a broader tradition, one that includes, for example, Kneale (1949), who says that ‘an explanation must in some sense simplify what we have to accept’. Similarly, Feigl (1970) claims that ‘the aim of scientific explanation throughout the ages has been unification, i.e., the comprehending of a maximum of facts and regularities in

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176   terms of a minimum of theoretical concepts and assumptions’. And Hempel (1966) argues that ‘what scientific explanation, especially theoretical explanation, aims at is . . . an objective kind of insight that is achieved by a systematic unification, by exhibiting the phenomena as manifestations of common, underlying structures and processes’. Further, unification seems to fit very naturally with the Humean picture. The mainstream Humean view on laws is the best system approach, where the laws are given by a systemization procedure—the laws are the facts that best balance simplicity and informativeness about the mosaic—that is extremely closely related to unification. And Loewer (1996, 113), in his discussion of the best system account, commits to the view that such laws explain via unification. So, I claim that the Humean should say that the epistemic value of nomothetic explanations stems from unification. This suggestion seems very natural, and maybe even obvious. But surprisingly, Humeans have not committed to it in the literature, apart from the brief mention in Loewer (1996). Perhaps Humeans have always had this view in mind (though nothing Lewis, for example, said suggested that), or perhaps not. Either way, I’m going to commit to this view of the value of nomothetic explanation. And I’m going to understand unification in the Friedman way—we gain unification by reducing the number of phenomena that we need to accept independently. So, for example, the development of the kinetic theory of gases allowed us to give a single explanation of phenomena that we previously had to accept independently, like phenomenon to do with the relationship between the heat and pressure of gases, and to do with gaseous diffusion. In this way we reduce the number of phenomena that we need to accept independently. Of course, there are questions about how to make this idea precise: How do we individuate phenomena? And what are the conditions under which phenomena can be accepted independently? But, given the large amount of ground that this paper covers, I can’t discuss this in detail here. In any case, we won’t be relying on fine judgments about unification—in the cases we consider the judgments about unification will be (hopefully) clear. Let’s look at an example. When we explain the momentum of a particle by citing the collision and the laws, we see how the fact about

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the momentum fits into the most general patterns of the world because, for the Humean, the laws are the most general patterns of the world. In showing how this event fits into the general patterns of the world we are gaining unification by reducing the total number of independent phenomena that we have to accept. Instead of accepting all of the instances of collisions and the corresponding momentum changes separately, we accept them together by identifying a general pattern. So, in explaining the fact of the particle’s momentum using the collision and the laws, we are gaining unification by assimilating the fact to this general pattern. Again, the proposal is that the Humean should take the epistemic value of nomothetic explanation to consist in unification. Notice, though, that this does not force the Humean to accept the sorts of accounts of explanation that have been labeled ‘unificationist’ in the literature on explanation. For example, we don’t have accept Friedman’s (1974) account of explanation (even though we are working with his gloss on unification) or Kitcher’s (1981) account. My suggestion is that the underlying epistemic aim of nomothetic explanation is unification. The surface-level account of explanation doesn’t have to appeal to unification as long as the account, together with the Humean view of laws, validates this conception of the underlying aim. The core idea, then, is that there is a difference, for the Humean, or at least a certain type of Humean, between the epistemic value of metaphysical and nomothetic explanations. Metaphysical explanation aims at elucidating underlying structure. Nomothetic explanation instead aims at unification. This divergence is why some metaphysical explanations can be part of nomothetic explanations and some cannot. In particular, explanations that contain as a part the explanation of the laws from the mosaic don’t look as if they will help achieve the aim of unification. In fact, it looks as if they will significantly hinder it. Imagine that the facts about the mosaic metaphysically explain the laws, and the laws nomothetically explain (at least partially) a particular fact about the mosaic F. Chaining these together wouldn’t leave us with a unificatory explanation—it wouldn’t reduce the number of independent phenomena we have to accept. Rather, it tells us to understand a particular fact F by reference to an incredibly large number of distinct facts—all the facts about the mosaic.

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178   Chaining the metaphysical explanation of the laws with a nomothetic explanation of any particular fact, then, will not yield explanations that help the cause of unification.⁹ Here’s another way to think about why the metaphysical explanation of the laws from the mosaic doesn’t help the cause of unification: When we unify, we are trying to reduce the number of phenomena we accept independently by assimilating specific events to more general patterns. But the metaphysical explanation of the laws starts from the general patterns—the laws themselves—and reduces them to large numbers of specific facts—the facts about the mosaic. Clearly this procedure will not help unification. Any account of nomothetic explanation that fits with the aim being unification will not allow the metaphysical explanation of the laws from the mosaic to chain with any other nomothetic explanation. On the other hand, some other metaphysical explanations can chain with nomothetic explanations because the resulting explanation would help the cause of unification. Interlevel Temperature is an example of this. The metaphysical explanation of facts about temperature in terms of facts about the energy of particles helps us assimilate the facts about temperature to the more general patterns about the movement of particles. And thus, such an explanation does help the cause of unification. The thought that the reduction of facts about temperature to facts about energy helps the cause of unification is extremely common. (See, for example, Sklar (1993, 333), Oppenheim and Putnam (1958, 6), Friedman (1983, 239–41), among many others.¹⁰) When we reduce facts about temperature to the facts about the energy of particles, then the theory about energy can now capture all the phenomena about temperature because the facts about temperature are nothing over and

⁹ The idea that it is the differing aim of scientific and metaphysical explanations that is the key to avoiding circularity for the Humean was independently developed by Dorst (2019)— though our stories about what those aims are very different (Dorst has a conception of ‘scientific explanation as fundamentally aimed at predictive utility’ (p. 2675)). But it’s somewhat unclear what, for him, determines when a scientific explanation can chain with a metaphysical one and when it cannot. ¹⁰ Though, traditionally, the type of reduction considered in these contexts has not been metaphysical explanation, but rather a more linguistic conceptions of reduction. But the same ideas apply to metaphysical explanation.

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above the facts about energy. We no longer need to accept facts about temperature independently of the facts about energy, and so we reduce the number of phenomena that we need to accept independently. To summarize, following Loewer, I claim that we avoid the initial circularity concern by distinguishing between metaphysical and nomothetic explanation—the facts about the mosaic metaphysically explain the laws, while the laws nomothetically explain facts about the mosaic. However, if we could chain these two explanations together, we would reintroduce circularity. The reason we cannot chain these explanations, I suggest, is because there is a difference in the underlying value of metaphysical and nomothetic explanation. The ultimate epistemic value of nomothetic explanation is unification, while the ultimate epistemic value of metaphysical explanation is elucidating metaphysical dependence structure. We can’t chain the metaphysical explanation of the laws with nomothetic explanations of particular facts to form a larger nomothetic explanation because doing so would hinder the cause of unification. Or, to put it another way, scientists should ignore the reduction of laws to the mosaic when giving nomothetic explanations because it is not unificatory. But other metaphysical explanations—ones that do help the cause of unification—can be part of nomothetic explanations. Again, the key idea here is that the ultimate value of nomothetic explanation is unification, while the ultimate aim of metaphysical explanation is elucidating metaphysical dependence structure. I don’t take these claims to be uncontroversial; they are significant commitments of my view. But they are commitments that are antecedently attractive to a certain type of Humean. The idea that the force of explanation in science comes from unification has a long tradition, and naturally flows from the Humean worldview. And the idea that the value of metaphysical explanation comes from elucidating metaphysical structure is a very natural one if we take that structure seriously.¹¹

¹¹ Notice that the Humean will not think that the aim of metaphysical explanation is also unification because, as we have seen, the metaphysical explanation of the laws from the mosaic is not unificatory. So, again, it is a commitment of my view that the aim of metaphysical explanation is illuminating metaphysical structure, not unification.

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2.2. Two types of fundamentality So, a Humean understanding of the value of nomothetic explanation, and how it differs from metaphysical explanation, allows us to respond to the explanatory circularity objection. Given this picture of the different aims, it is reasonable for the Humean to think that metaphysical explanation diverges from the explanations given in science. That response motivates a similar response to the non-fundamentality objection. The distinction between nomothetic and metaphysical explanation induces a distinction between nomothetic and metaphysical fundamentality. This is because there is a link between fundamentality and explanation: A fact is fundamental if and only if it is unexplained. The Humean takes the elements of the mosaic to be metaphysically unexplained. In doing so they take the elements of the mosaic to be metaphysically fundamental.¹² The laws are metaphysically explained and thus metaphysically non-fundamental. Analogously, I claim, something is nomothetically fundamental if and only if it is nomothetically unexplained. Some laws are nomothetically unexplained. Plausibly, there are laws that are nomothetically explained by other laws, but some—like the most basic laws of physics—are not explained in this way. So, some laws are nomothetically fundamental. Given our conception of nomothetic explanation as being tied to unification, the nomothetically fundamental is the unificatory base—it is the sparse set of facts from which we can unify all the other facts in the world. To say that a law is nomothetically fundamental, then, is to say that it is at the end point of unification—it is not one of the facts that we fit into a more general pattern; rather, it is the pattern that we fit facts into. This, I think, is a natural way of interpreting claims about fundamentality of laws, when made in a scientific and not a metaphysical context. When scientists take a particular law to be fundamental, they are not claiming that it is metaphysically fundamental—it’s unlikely that they care at all about metaphysical fundamentality—rather they are saying ¹² This is generally accepted in the literature, though there is some disagreement, notably Fine (2001).

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that it is a fact that is taken as basic; we will not unify it with, or assimilate it to, any other facts. It’s a fact that, in our unified understanding of the world, stands alone. This, then, is the response to the non-fundamentality objection: The Humean can respect scientists’ claims about the fundamentality of certain laws—interpreting their claims as about nomothetic fundamentality. There is a divergence between the scientific notion of fundamentality and the metaphysical notion.

2.3. Non-supervenience and two types of possibility I argued that distinguishing between two types of explanation could allow the Humean to respond to the circularity problem. Further, I suggested that we can extend this response to the non-fundamentality problem by distinguishing between two types of fundamentality. Similarly, I will argue distinguishing between metaphysical and nomothetic possibility is a promising way for the Humean to respond with the non-supervenience objection. Or rather, they can respond to the second version of the nonsupervenience objection in this way. The first version of the objection is that the Humean says the wrong things about what is metaphysically possible—for example, that the Humean has to say that either the emptyGR or the empty-SR world is metaphysically impossible. The Humean just has to bite the bullet here and say, along with the Humeans we discussed in section 1.4, that our intuitions about metaphysical possibility are misleading in this case. But the second version of the objection—that the Humean cannot make sense of the way that our scientific reasoning treats the cases like the empty-GR and empty-SR worlds as distinct—is the more pressing version. This is the version of the objection I’m considering. There are two steps to the response that I will suggest. Firstly, I’ll argue that in the sense of possibility relevant to scientific practice both the worlds are possible, even though at least one is metaphysically impossible. I call this sense of possibility nomothetic possibility. The Humean can say that supervenience does not hold in this space of

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182   possibility, whilst it does hold in the space of metaphysical possibility. (The view remains distinctively Humean because the laws of nature are reduced to the Humean mosaic.)¹³ The second step in this response is to say something about counterfactuals and explanations (and related notions) that allows us to understand how they can differ between these worlds. But before that, in order to ward off confusion, it is very important to be clear that nomothetic possibility is not the same as the more familiar physical possibility. A proposition is physically possible relative to a world if it is consistent (in some sense) with the laws of that world. It is typically thought of as a restriction of metaphysical possibility.¹⁴ Nomothetic possibility, in my sense, does not have this structure. So, central to this response is an appeal to a sense of possibility that is broader than metaphysical possibility—there are worlds that are metaphysically impossible but nomothetically possible. This is clearly a relatively unfamiliar idea (though appeals to senses of possibility that outrun metaphysical possibility are becoming more common (e.g. Jenkins and Nolan (2012); Spencer (2017)). Further, I claim that this sense of possibility is the relevant sense for scientific practice. So, we should start by considering why scientific practice would countenance possibilities that outrun the space of metaphysical possibility. In general, I take it that the point of scientific reasoning about possibility is to explore the implications of scientific theories by looking at what alternative situations are consistent with the theory. Doing this can shed light on the theory we are considering but also it can shed light on the actual world. Let’s look at a few examples of such scientific reasoning about possibility. Imagine that the theory that we are investigating is Newtonian mechanics. We might, for example, investigate what the theory says about

¹³ Notice that responding to the second version of the non-supervenience objection—by saying that that the Humean should expect there to be a divergence between the metaphysical possibility and the possibilities countenanced in science—also weakens the first version. After this response to the second version of the objection the opponent of Humeanism cannot defend their assertion that both the empty-GR and the empty-SR worlds are both metaphysically possible by appealing to the way that those possibilities are countenanced in scientific practice. ¹⁴ Though in section 3.3 I will argue that the Humean should not think of it that way.

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objects falling to earth. A natural place to start is by considering a situation where one object falls toward a much more massive object, and apart from those two objects, there is a vacuum. It is simple for us to use the theory to calculate what happens in this possibility. If we carry on this investigation, we find a variety of interesting things about what possibilities the theory allows. We find, for example, that in every situation where small objects fall toward earth in the absence of air resistance, they accelerate at almost exactly the same rate. This sheds light on the theory. But also, exploring these possibilities—especially simple possibilities like worlds containing only two objects—tells us a lot about how objects fall in the actual world. For example, by comparing actual cases of objects falling to these very simple situations, we can see what the effect of air resistance is in our world. In particular, we find that air resistance is a lot more significant than we might have initially thought—it is the difference between the way a feather falls to the ground and the way a hammer does. More generally, scientific modeling often works like this—we look at simple cases which are consistent with the theory in question, seeing what happen in these cases is informative about the mechanics of the actual world. Here’s a very different example of scientific reasoning about possibility. When we explore the possibilities left open by non-inflationary Big Bang theory we find that nearly all of those possibilities lead to situations which are inconsistent with striking cosmological phenomena that we observe (like the uniform temperature of the microwave background radiation). Only a very specific choice of initial conditions allows the theory to be consistent with such phenomena (Maudlin (2007, 40–5)). This is generally taken to be powerful evidence against the theory, and a key part of the motivation for the development of inflationary theories. Exploring the space of situations consistent with the theory, then, gives us reason to suspect that the theory is false. Here’s another case. Krauss (2012) argues that relativistic quantum field theory is inconsistent with stable situations where there are no particles. This is unlike, for example, standard general relativistic theories which are, as we noted, consistent with empty worlds. Plausibly, this has implications for what the explanation of the existence of particles

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184   is.¹⁵ Seeing the differences between the theories that are consistent with there being no particles and those that are not is revealing both about the theories and about the nature of particles in our world. All of this reasoning involves looking at what situations are logically consistent with particular scientific theories. This, I take it, is what scientific reasoning about possibilities ultimately consists in. Such reasoning can be enlightening in a variety of ways. This suggests an understanding of what possibilities are relevant to scientific practice: they are those situations where the events in the world—i.e., the Humean mosaic—are logically consistent with the scientific theory. That is, where the Humean mosaic is consistent with the laws of that world and the patterns of explanations vindicated by those laws. This might all seem pretty uncontroversial and obvious, but it has important implications. This is because the metaphysical explanation of the laws from the mosaic is not part of the scientific theory of the world. As we saw in section 2.1, scientific practice has a good reason to ignore such an explanation. And so the mosaic being consistent with the scientific theory of that world does not require that it is consistent with the scientific theory together with the metaphysical explanation of the laws from the mosaic. Consider an example. Assume that the best system of the world is given by Newtonian mechanics. So, on the Humean account we are working with, the correct scientific theory of the world is given by the laws of Newtonian mechanics. But the metaphysical explanation of the laws given in terms of facts about the mosaic is not part of the scientific theory—it is not part of Newtonian mechanics. Scientists ignore this metaphysical explanation (at least when doing science), and they are right to do so because this explanation runs contrary to the aim of unification. So, when we consider what situations are consistent with Newtonian mechanics, we find that very simple situations like those we considered earlier in the section—for example, the case where there is one body ¹⁵ Though it does not seem, contra Krauss, to have implications for why there is something rather than nothing.

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falling toward another much larger body, and those are the only two things in the world—are consistent with the theory. In fact, Newtonian mechanics has some precise predictions for how such a situation would unfold. It is true that this situation is inconsistent with Newtonian mechanics and the metaphysical explanation of the laws from the mosaic because the best system for a world where those are the only two objects in the world would not output Newtonian mechanics. But this fact is irrelevant for scientific appeals to possibility. It would clearly be inappropriate for a philosopher to complain to a scientist that we shouldn’t model objects falling by appealing to a two-object Newtonian world because such a world is metaphysically impossible. Again, the metaphysical explanation is not part of the scientific theory, and so it is not relevant for scientific judgments of possibility. Similar reasoning applies to the empty-GR world. The empty mosaic is logically consistent with the theory of the world being given by general relativity, and thus the empty-GR world is a possibility that is relevant to scientific practice. This is true even though the empty mosaic is inconsistent with the theory of the world being given by the laws of GR together with the metaphysical explanation of the laws in terms of the mosaic (and similarly with the empty-SR world). We now have a picture of why scientific practice would countenance situations that are not metaphysically possible. The suggestion is that the metaphysical explanation of the laws from the mosaic is not part of the scientific theories considered by scientists, and there is a principled reason for this—such an explanation acts contrary to the goal of unification that is central to nomothetic explanation. Consequently, that metaphysical explanation does not constrain the possibilities that are relevant to scientific practice.¹⁶

¹⁶ The relation between scientific and metaphysical practice here is somewhat analogous to the relation between higher- and lower-level sciences. Just as scientific practice countenances possibilities that are metaphysically impossible, so higher-level sciences often countenance possibilities that are physically impossible. For example, economists often consider what their theories say about situations where information moves instantaneously and costlessly. The fact that such possibilities are physically impossible because they involve superluminal signaling does not stop them from being relevant for the practice of economics.

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186   Let’s give a name to this space of possibility that is relevant to science. Call it nomothetic possibility because it is importantly connected to nomothetic explanation. We have seen that the empty-GR world counts as nomothetically possible, even though it is metaphysically impossible (clearly the same reasoning applies to the empty-SR world).¹⁷ We should pause at this point and recap some important features of nomothetic possibility: (1) The key feature of nomothetic possibility for our purposes is that the laws do not supervene upon the mosaic in the space of nomothetic possibility, whilst they do in the space of metaphysical possibility. Again, the reason that laws supervene upon the mosaic in the space of metaphysical possibility is that the laws are metaphysically explained by the mosaic. But this explanation is ignored by scientific practice, because it hinders the cause of unification—and so is not relevant for what is nomothetically possible. Rather, the nomothetic possibilities are the cases where the mosaic is logically consistent with the laws. This allows for non-supervenience in the space of nomothetic possibility. (2) More specifically, both the empty-SR and empty-GR worlds are nomothetically possible, even though they are not metaphysically possible. The empty Minkowski spacetime is logically consistent with the laws of GR, so the empty-GR world is nomothetically possible. The empty Minkowski spacetime is not logically consistent with the laws of GR and the metaphysical explanation of

¹⁷ In this volume, Dasgupta (2020), defends relationalism about motion and about quantities like mass, and in doing so develops a strategy that is structurally very similar to the one implemented here. He distinguishes two senses of explanation and has that feed through to different senses of fundamentality and possibility. In fact, this general structure could apply to many cases where we might reasonably think that there is more than one type of explanation at work. For example, the relation between the higher- and lower-level sciences which seem to explain in importantly different ways is one possible case. Further, some meta-ethical views might want to distinguish two types of explanation. Some views reduce moral facts to some other facts, say, facts about mental states or facts about our society. But defenders of those views might want to say that, nevertheless, certain facts are morally unexplained—they have no explanation from within our first-order moral theory; rather, they are foundational to the theory. We could then distinguish different senses of fundamentality and possibility. In fact, I think this could be an attractive way to develop those meta-ethical views—having benefits that are closely analogous to the ones I describe for the Humean here.

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the laws in terms of the mosaic, so the empty-GR world is metaphysically impossible. (3) So, some worlds are nomothetically possible but not metaphysically possible. (4) Nomothetic possibility, for the Humean, in some ways mirrors what the non-Humean thinks is metaphysically possible. The non-Humean thinks that laws do not supervene on the mosaic in the space of metaphysical possibility and that things like the empty-SR and empty-GR worlds are possible because the mosaic is consistent with the relevant laws. My suggestion is that the Humean should think the analogous things about nomothetic possibility. Of course, this isn’t a full characterization of nomothetic possibility— there are a lot of questions still open. To pick just one, is it the case that everything that is metaphysically possible is nomothetically possible? Or are there certain metaphysical possibilities which just aren’t relevant for the practice of science? There is a lot more work to be done on filling out the idea of nomothetic possibility. But hopefully we can start to see how such a notion could help with respect to the non-supervenience problem. In fact, before we move on, we can motivate this sense of nomothetic possibility in a slightly different way as well, though it ultimately relies on the same ideas. The key is to consider the links between explanation and possibility. We can use the distinction between nomothetic and metaphysical explanation to motivate an analogous distinction in possibility—just as we leveraged the distinction between two types of explanation into a distinction between two types of fundamentality. There is a link between explanation and possibility—explanation has a distinctive modal signature.¹⁸ If A explains B, then this has implications for how A and B can covary. More specifically, if A wholly explains B, then B supervenes upon A. The motivation for this is pretty simple: If I find a case where A is present but B is not, then it looks as if A is either wrong, or not complete, ¹⁸ I’m ignoring probabilistic explanations for now. Soon it will become clear why this does not significantly affect my argument.

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188   as an explanation of B.¹⁹ This thought is common (though not totally uncontroversial) in the case of metaphysical explanation.²⁰ And it also seems very intuitive when considering nomothetic explanations. Different types of explanation imply different grades of supervenience, though. Consider an analogy with moral explanation. Imagine that utilitarianism is true. The fact that an action A is wrong is wholly morally explained by its consequences for utility. The wrongness of the action will supervene upon the facts about utility, then, but only in the range of worlds where utilitarianism is true. If we are considering worlds where utilitarianism is false, we should not expect supervenience to hold across this range of worlds. The supervenience holds over the morally possible worlds.²¹ Similarly, nomothetic explanation and metaphysical explanation imply supervenience over different spaces of worlds—nomothetic explanation constrains the nomothetically possible worlds; metaphysical explanation constrains the metaphysically possible worlds. More generally: If A wholly explains B, then B supervenes on A in the corresponding space of possibility. Metaphysical possibility is constrained by the metaphysical explanation of laws from the mosaic. Since the laws are metaphysically explained by the mosaic, then the laws supervene on the mosaic in the space of metaphysically possible worlds. But, as we have seen, there is no nomothetic explanation of the laws from the mosaic. And so, there is no reason for there to be supervenience of the laws on the mosaic in the space of nomothetically possible worlds. Thus, the space of nomothetic possibility contains worlds which are not metaphysically possible.²² ¹⁹ DeRosset (2010, §4) explicitly defends such a principle, arguing that it is central to our practices of reasoning with, and about, explanations. ²⁰ Most notably, people who deny grounding necessitism (e.g., Leuenberger (2014); Skiles (2015)) will reject this claim. Bennett (2017, §3.3) discusses this in more detail, suggesting that our principle would need to be adapted slightly if we accept that there are background conditions required for A to explain B but which are not part of the explanation of B. ²¹ Which may or may not, depending on your view, overlap with the metaphysically possible worlds. ²² We can now see why ignoring the possibility of probabilistic explanation is not problematic. There are no probabilistic or indeterministic metaphysical explanations (see Bennett (2017, §3.3.1)), and the key claim about nomothetic explanation is that there is no supervenience of laws on the mosaic in the space of nomothetic possibility. Clearly, countenancing indeterministic explanation will not allow us to recover any supervenience claim.

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In particular, worlds like the empty-GR and empty-SR worlds are ruled out from both being metaphysically possible by the metaphysical explanation of the laws from the mosaic. But since there is no nomothetic explanation of the laws from the mosaic there is nothing to stop both of them being nomothetically possible. Just as before, the central idea is that the metaphysical explanation of the laws from the mosaic is not relevant for what situations are taken as possible in scientific practice.²³ So, we have the first step in the response to the non-supervenience problem. The empty-GR and empty-SR worlds are not both metaphysically possible, but they are both possible in the sense relevant for scientific practice. The second step is to describe what is going on with counterfactuals and explanations (and related concepts) in these worlds. The story here is very simple. Most accounts of explanations and counterfactuals (and other scientific modal notions, like causation) in the literature are given in terms of the mosaic and the laws. I can just appeal directly to such accounts. The reason that I have freedom to appeal to such accounts is because I countenance worlds, like the empty-GR world, where the laws and the mosaic are not connected via the Humean story about laws. Such worlds are metaphysically impossible, but they are nomothetically possible and important for the practice of science. The traditional Humean—one who does not countenance an important space of possibility that is broader than metaphysical possibility—has problems here because they ignore such worlds and so cannot appeal to these accounts. I do not face these problems. Standard accounts of explanations and counterfactuals are going to have the result that there is a difference in the truth value of explanatory and counterfactual claims between the empty-GR world and the emptySR world because there is a difference in the laws. So my view, being able

²³ Dasgupta’s (2020) story about how two distinct types of possibility arise from two distinct types of explanation is superficially different, but, I think, rests on the same ideas. He takes the two different spaces of possibility to be generated via a recombination procedure from the two different fundamental bases. But, applying this to our setting, this relies on the idea that the metaphysical explanation of the laws from the mosaic constrains the ways in which the metaphysically fundamental can recombine to create metaphysical possibilities worlds, but it doesn’t constrain the way that the nomothetically fundamental can recombine to create nomothetic possibilities.

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190   to appeal to such accounts, will also accept that there is a difference in these respects between these nomothetically possible worlds.²⁴ In summary, I suggest that the Humean should respond to the nonsupervenience problem by saying that there is a divergence between the space of possibility relevant for scientific practice and metaphysical possibility—there is a space of nomothetically possible worlds that is broader than the space of metaphysically possible worlds. The Humean has reason to accept that supervenience of laws on the mosaic does not hold in this space of worlds. In particular, the empty-SR and empty-GR worlds are both possible in this sense. It’s important to note that the view is still distinctively Humean, though, because the laws of nature are metaphysically explained by, and thus are nothing over and above, the Humean mosaic. As I noted, there is still a lot more work to be done on the nature of nomothetic possibility if a fully satisfying Humean account along these lines is to be developed. But hopefully we have seen the outlines of how such a project might go.

3. Nomothetic possibility and scientific modalities I have sketched a new way of developing Humeanism, one that involves a divergence between elements of scientific and metaphysical practice. In particular, I distinguished nomothetic and metaphysical versions of explanation, fundamentality, and possibility and outlined how we might use these distinctions to deal with the circularity, non-fundamentality and non-supervenience problems. ²⁴ Perhaps you might worry that, given the introduction of nomothetic possibility and my argument that this sense of possibility plays a central role in science, metaphysical possibility becomes, in some sense, devalued; it no longer plays as central a role as it used to. I think there is some truth in this, but I don’t think it’s a bad result. For example, Sider (2020) discusses what he calls the ‘postmodal’ approach to philosophy, and to metaphysics in particular. This approach takes the central philosophical tools to be concepts like metaphysical explanation, or fundamentality, rather than modal notions. Like many philosophers, I am independently attracted to this approach. For example, I think that many philosophical positions—like physicalism, say— are best cashed out by appealing to metaphysical explanation rather than by appealing to modal notions. So I don’t think it’s problematic that my account here de-emphasizes metaphysical possibility. Clearly there is much more to say about this, but equally clearly, this is not the right place to do that.

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In section 2.3 I argued that the Humean should take there to be a divergence between the type of possibility appealed to in science and metaphysical possibility. I’m going to finish in this section by noting how this distinction between metaphysical and nomothetic possibility has consequences for other parts of scientific practice. In particular, I’ll argue that countenancing nomothetic possibility allows the Humean to give accounts of various physical modalities—like chance, counterfactuals, and physical possibility—that are better than they could previously. The Humean can avoid problems that they face with these notions, and can do so in unified way—not by dealing with each in a piecemeal fashion.

3.1. Counterfactuals Here is a problem that is closely related to the non-supervenience problem. The Humean appears to evaluate the truth of certain nested counterfactuals wrongly. Assume that the laws of GR hold in our world. Then consider the following counterfactual: If the world were empty, then (if there were a massive object in the world, then spacetime would be curved) This counterfactual is true. The rationale is fairly simple. If the world were empty, then it would be the empty-GR world. But in any world where the laws are those of GR it’s true that if there were massive objects, then spacetime would be curved. But the traditional Humean fails to get this result. For them, if the world were empty, then we would not be in a GR-world; we would be in a world where the laws were something much simpler. The laws of that world would almost certainly not associate masses with curvature of spacetime. And so in that world it is not true that if there was a massive object, then spacetime would be curved. Lange (2009, 54), Sklar (2014, 79–80), and Hall (n.d., 32–3) develop versions of this objection.²⁵ ²⁵ Hall (2015) is a much shorter version of the unpublished Hall paper just mentioned that doesn’t include the content discussed here. Confusingly, though, both papers have the same title

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192   Hall suggests that the Humean should respond by tweaking the standard possible world semantics for counterfactuals to get the desired result for the nested counterfactual. He suggests that in order to evaluate a counterfactual like ‘if the world were empty, then (if there were a massive object in the world, then spacetime would be curved)’ the procedure is as follows: The first step is to find an empty world w₁ where the mosaic is a model of the actual laws. That is a world where the mosaic is consistent with the propositions that make up the actual laws, though not necessarily with the fact that those propositions are laws. Then we find the world w₂ closest to w₁ where the mosaic is again a model of the actual laws and there is a massive object in the world. If in w₂ spacetime is curved, then the whole nested counterfactual is true. It’s not totally clear that this procedure gets the right result here—it depends on fine details of the closeness relation. But even assuming it does, there’s a concern that this tweak to the semantics is somewhat ad hoc. In particular, the account in effect implies that we should evaluate a counterfactual differently when it is embedded in a larger counterfactual than when it is not. It’s not clear why we should change the semantics in this way just to save Humeanism.²⁶ My account, on the other hand, gets the right result for the nested counterfactuals without such an ad hoc tweak to the semantics. All we need to do is to recognize that the relevant space of possibility is nomothetic possibility, not metaphysical possibility. Standard accounts of counterfactuals say that if the world were empty, then the world would be an empty-GR world. And my account can appeal to those standard accounts because it accepts that there is an empty-GR world—a nomothetically possible one. The traditional Humean does not accept that there is a possible empty-GR world, and so had to deny that if the world were empty, then it would be an empty-GR world—hence they fail to get the right result for the nested counterfactual. (though the longer, unpublished version is sometimes referred to as the ‘Director’s Cut’), and it is the unpublished version that is more influential in the literature. My discussion here is of the unpublished paper, not the 2015 version. ²⁶ Loew and Jaag (2019) argue that a certain type of Humean does have motivation for accepting a semantics for counterfactuals such that the actual laws remain held fixed in nested counterfactuals. But their story doesn’t motivate the Hall approach over mine.

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So we have some reason to think that my account deals better with the nested counterfactual problem than that of traditional Humeans. But there’s a much more powerful reason for this to do with the way in which the nested counterfactual problems is part of a larger class of issues for the Humean. I’ll describe this more soon.

3.2. Chance Consider cases of ‘undermining’. These are cases where a probabilistic law assigns a positive chance to a situation in which it is not a law. It is often argued that the Humean has to accept the existence of such cases (Lewis (1994); Thau (1994); Arntzenius and Hall (2003)). Imagine, for example, that there is a probabilistic law L: coin tosses have a 50% chance of landing heads. (Imagine too that coin tosses are irreducibly chancy.) Then consider a mosaic where every coin toss in the history of the world lands heads (and nothing about it violates any of the other laws). Assuming there are a finite number of coin tosses, then the proposition P describing this mosaic will be assigned a positive probability. But it is very plausible that if P is true, then the Humean recipe for constructing laws from the mosaic will not have the result that L is a law. More likely, a law that assigns coin tosses a higher chance of landing heads would result. It seems undeniable, though, that there is a positive chance of P. This is puzzling, because it looks as if the laws shouldn’t assign a positive chance to propositions that are inconsistent with the laws.²⁷ On one understanding this thought about consistency is in conflict with the case described—there is no metaphysically possible world where L is a law and P is true; yet L assigns P a positive probability. However, there is a nomothetically possible world where L is a law and P holds. More generally, all cases of undermining involve laws assigning probabilities to situations which are models of the laws. That is, all cases ²⁷ Much of the debate about undermining has been about what principles for how chance guides rational credence are plausible in the presence of undermining. I’m not going to consider this question here, but I hope, in future work, to take on this question in light of what I say about undermining here.

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194   of undermining involve the laws assigning probabilities to situations which are logically consistent with the laws (though not logically consistent with the laws along with the metaphysical explanation of the laws from the mosaic). So, all cases of undermining involve laws assigning chances to (parts of) mosaics such that the law and the mosaic hold together in a nomothetically possible world. The Humean can, therefore, retain a thought about the laws not assigning a positive chance to situations which are impossible given the laws, as long as they understand it in terms of nomothetic, not metaphysical possibility. More precisely, the Realization Principle of Schaffer (2007) (a strengthening of the Basic Chance Principle of Bigelow et al. (1993)) says that if a proposition has a chance of more than zero, there must be a possible world where the history is the same as what it actually is, the laws are the same, and where the proposition is true. If we understand this principle as appealing to metaphysically possible worlds, then the possibility of undermining cases shows that this principle rules out Humeanism about probabilistic laws. But the Humean can accept this principle on the understanding the relevant grade of possibility is nomothetic, not metaphysical. In fact, the considerations in this section suggest something more. They suggest that chances are defined over (restrictions of) the set of nomothetically possible worlds, not metaphysically possible worlds; the algebra over which chance is defined is made up of nomothetically possible worlds. Consequently, chance doesn’t always assign probability 1 to the metaphysically necessary facts. I think this is the right thing for the Humean to say about chance, though I don’t have space to discuss it further here.

3.3. Physical possibility There are related issues with the traditional Humean account of physical possibility. Here are two natural and commonly used accounts of physical possibility (see, for example, Roberts (1998, 433); Schneider (2007, 312); Hall (n.d., 32–3)). The first account says that a proposition is

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physically possible relative to a world w if and only if it is true in a (metaphysically possible) world that has the same laws as w. The second account says that a proposition is physically possible relative to a world w if and only if it is true in a (metaphysically possible) world that is a model of the laws of w. The traditional Humean move is to accept the second account (see, again, the authors cited above). The first account fails for the Humean because it implies that an empty mosaic is not physically possible relative to both GR and SR worlds (because for the empty mosaic to be physically possible relative to both GR and SR worlds there would have to be a metaphysically possible empty-GR world and a metaphysically possible empty-SR world). But, nevertheless—as, I think, has gone unrecognized in the literature—this second account fails too. The reason it fails is because it construes physical possibility as a restriction of metaphysical possibility when, for the Humean, there are some propositions that are intuitively physically possible but not true in any metaphysically possible world. Here is a case. Imagine the actual world is Newtonian. The following situation then seems physically possible: There are only three bodies in the world. Two bodies orbit a central body such that at all times a straight line connects the centers of mass of all the bodies. The outer bodies are the same mass and are always at the same distance from the central body. Call this setup S. The gravitational forces in this case are such that the outside bodies orbit, whilst the central body does not ever move, because it is always equally attracted by the two outside bodies. So, here is a causal claim, call it C, that seems physically possible: S holds and the distribution of the outside bodies causes the central body to remain motionless forever. But C is, very plausibly, not true in any metaphysically possible world for the Humean. Because for C to be true not only must it be the case that the mosaic is the way that was described in the last paragraph, but it must also be the case that something like the laws of gravitation hold so that the central body is held in balance by the outside bodies. And it is implausible that any such law would follow from the best system account (or some alternative Humean story about laws) applied to such a mosaic. (Rather, the laws of that mosaic would likely say that outside bodies orbit

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196   the middle, or something similar.) So C is not going to be judged to be physically possible on any account that involves a restriction of metaphysical possibility. Rather than a restriction of metaphysical possibility what we need is an account where physical possibility is a restriction of nomothetic possibility. For example, consider an account which says that a proposition is physically possible relative to a world w if and only if it is true in a nomothetically possible world that has the same laws as w. There is a nomothetically possible world in which the mosaic is as described above and the laws are Newtonian. And C is true in such a world (since we can appeal to standard accounts of causation to get this result). So such an account would evaluate the physical possibility of C correctly.

3.4. A class of counterexamples In fact, this novel problem about physical possibility is an instance of a larger class of counterexamples to Humeanism, one which subsumes the problems about nested counterfactuals and undermining discussed in sections 3.1 and 3.2. Any time there are claims that hold in virtue of the mosaic and the laws of a different world, then the traditional Humean is prone to evaluate such claims wrongly (because of the restrictions the traditional Humean has on how mosaics can combine with laws). The claim about the physical possibility of C is an example of this—the physical possibility of the causal claim C holds in virtue of the mosaic and the laws at other worlds. The nested counterfactual objection has the same structure. The nested counterfactual we considered was: If the world were empty, then (if there were a massive object in the world, then spacetime would be curved). The truth of this depends upon the laws of another world. We evaluate this counterfactual by looking at the closest empty world and then seeing what the laws are in that world. If the laws are those of General Relativity, then it is true that if there were a massive object in the world, then spacetime would be curved, and so the whole nested counterfactual is true. But, of course, the Humean will not think that the laws of GR hold in any empty world. They will take the laws to be much

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simpler; thus, the counterfactual in the consequent would be false, and the whole nested counterfactual is false. So, the nested counterfactual holds in virtue of the mosaic and the laws of a different possible world. In this case, the closest empty world. The Humean evaluates this proposition wrongly because of the restrictions it puts on how laws can combine with mosaics. The non-supervenience objection is also of this kind. Consider the proposition that it is possible that there is an empty-GR world and an empty-SR world. This proposition seems to be true, but the Humean incorrectly evaluates it because it depends on the laws, as well as the mosaics, of other possible worlds. Looking at these cases, we have almost a recipe for generating counterexamples to Humeanism. Take a proposition that has a component that pushes us to look at another possible world, and a component that relies upon the laws as well as the mosaic of that different world. Put these components together, and we have a proposition that holds in virtue of the mosaic and the laws of a different possible world—the traditional Humean will misevaluate some of these propositions. So, for example, we can easily, by slightly adapting the cases we have already seen, generate counterexamples to traditional Humeanism where the Humean misevaluates certain counterfactual claims about explanations; counterfactual claims about chances; claims about the physical possibility of chances; claims about the chances of chances; claims about the physical possibility of laws; and so on. I’ll leave it to the reader to generate these, and other, counterexamples. The point of all this is that since there is this unified class of counterexamples to Humeanism, responding in a piecemeal way to the counterexamples—for example, by tweaking the semantics of counterfactuals to deal with the nested counterfactual objection—is unsatisfactory. The Humean needs a unified response. And the appeal to nomothetic possibility provides this response—in the space of nomothetically possible worlds there are looser restrictions on how mosaics can combine with laws, and so counterexamples cannot be generated in the same way. Expanding the relevant space of worlds from metaphysical to nomothetic defuses this class of counterexamples.

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198   The fact that it allows this unified response to a variety of counterexamples is perhaps the most powerful reason for the Humean to countenance nomothetic possibility.

4. Conclusion I have outlined a new way of developing Humeanism—one that develops Loewer’s response to the circularity problem and extends it in order to deal with the other problems that Humeans face. This way of developing Humeanism points toward certain novel responses to the nonfundamentality and non-supervenience problems. In particular, recognizing that there are important types of fundamentality and possibility that are distinct from metaphysical fundamentality and possibility allows us to respond to the non-fundamentality and non-supervenience problems. Further, this new grade of possibility seems to give the Humean certain advantages with respect to their accounts of counterfactuals, chance, and physical possibility. This picture is motivated by the idea that the aims of science diverge from the aims of metaphysics. While metaphysicians want to understand the deep dependence structure of the world, the aim, or at least one aim, of science is to unify. This warrants a certain kind of divergence between the practice of metaphysics and that of science—the aim of unification means that it is not part of the practice of scientific explanation to reduce the laws to anything else (and this is true even if particular scientists, in their philosophical moments, were to accept Humeanism). This feeds through to the other elements of the practice of science—like the possibilities that scientists countenance diverging from metaphysical possibility. The divergence between metaphysical and scientific practice that the objections to Humeanism make salient is not something that refutes Humeanism. Rather, it is something that follows from a natural Humean understanding of science. Of course, a lot more work is required in order to fully flesh this out. What I have done is just provide the outline for a larger Humean project. But I think that the picture outlined here is promising.

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One final point: I expect a certain type of reaction from a totally committed anti-Humean. They will think their view fits better with scientific practice and will consider the view developed in this paper as ad hoc or overly complicated machinery that is supposed to replicate what their view gets naturally. One part of this is right. Often antiHumeans build their metaphysics in order to mirror scientific practice—Maudlin, for example, is pretty explicit about this approach. And if the anti-Humean builds their metaphysics in order to mirror scientific practice, then anyone who wants to capture the practice will replicate, in some sense, elements of the anti-Humean view. For people who see no attraction at all in the Humean picture this will be enough to reject the view. But this paper is not for them. For people who find some attraction in the austerity and elegance of the Humean picture, then there is value in the account developed here—it allows you to enjoy those attractions whilst avoiding unpalatable consequences.

Acknowledgments Thanks to Karen Bennett, Thomas Blanchard, Eddy Keming Chen, Anthony Dardis, Marco Dees, Tom Donaldson, Cian Dorr, Chris Dorst, Alison Fernandes, Kit Fine, Laura Franklin-Hall, Martin Glazier, Michael Townsen Hicks, David Kovacs, Barry Loewer, Tim Maudlin, Michaela McSweeney, Zee R. Perry, Erica Shumener, Michael Strevens, Trevor Teitel, Dan Waxman, Mike Zhao, and audiences at the Midsummer Philosophy Workshop, the Society for the Metaphysics of Science Annual Conference, and NYU Thesis prep for extremely helpful feedback and discussion.

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200   Bhogal, H. (2017). Minimal Anti-Humeanism. Australasian Journal of Philosophy 95(3), 447–60. Bhogal, H. and Z. R. Perry. (2017). What the Humean Should Say about Entanglement. Noûs 51(1), 74–94. Bigelow, J., J. Collins, and R. Pargetter. (1993). The Big Bad Bug: What Are the Humean’s Chances? British Journal for the Philosophy of Science 44(3), 443–62. Bird, A. (2007). Nature’s Metaphysics: Laws and Properties. New York: Oxford University Press. Carroll, J. (1994). Laws of Nature. Cambridge: Cambridge University Press. Dasgupta, S. (2020). How to Be a Relationalist. In K. Bennett and D. Zimmerman (ed.), Oxford Studies in Metaphysics, Volume 12 New York: Oxford University Press.. deRosset, L. (2010). Getting Priority Straight. Philosophical Studies 149(1), 73–97. Dorst, C. (2019). Humean Laws, Explanatory Circularity, and the Aim of Scientific Explanation. Philosophical Studies, 176(10), 2657–79. Earman, J. and J. T. Roberts. (2005). Contact with the Nomic: A Challenge for Deniers of Humean Supervenience about Laws of Nature Part I: Humean Supervenience. Philosophy and Phenomenological Research 71(1), 1–22. Esfeld, M., D. Lazarovici, M. Hubert, and D. Dürr (2014). The Ontology of Bohmian Mechanics. British Journal for the Philosophy of Science 65(4), 773–96. Feigl, H. (1970). The “Orthodox” View of Theories: Remarks in Defense as Well as Critique. In M. Radner and S. Winokur (eds.), Analyses of Theories and Methods of Physics and Psychology, Minnesota Studies in the Philosophy of Science, Vol. IV, 3–16. Minneapolis, MN: University of Minnesota Press. Fine, K. (2001). The Question of Realism. Philosophers’ Imprint 1(2), 1–30. Friedman, M. (1974). Explanation and Scientific Understanding. The Journal of Philosophy 71(1), 5–19. Friedman, M. (1983). Foundations of Space-Time Theories. Princeton, NJ: Princeton University Press. Hajek, A. (2010). Mises Redux-Redux: Fifteen Arguments against Finite Frequentism. In A. Eagle (ed.), Philosophy of Probability: Contemporary Readings, 209–27. New York: Routledge.

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Hall, N. (n.d.). Humean Reductionism about Laws of Nature. MS. Hall, N. (2015). Humean Reductionism about Laws of Nature. In B. Loewer and J. Schaffer (eds.), A Companion to David Lewis, 262–77. Malden, MA: Wiley. Hempel, C. G. (1966). Philosophy of Natural Science. Englewood Cliffs, NJ: Prentice-Hall. Hicks, M. T. (forthcoming). Breaking the Explanatory Circle. Philosophical Studies. Hicks, M. T. and P. van Elswyk. (2015). Humean Laws and Circular Explanation. Philosophical Studies 172, 433–443. Jenkins, C. S. and D. Nolan. (2012). Disposition Impossible. Noûs 46(4), 732–53. Kitcher, P. (1981). Explanatory Unification. Philosophy of Science 48(4), 507–31. Kneale, W. (1949). Probability and Induction. Oxford: Oxford University Press. Krauss, L. M. (2012). A Universe from Nothing: Why There Is Something rather than Nothing. London: Simon and Schuster. Lange, M. (2009). Laws and Lawmakers: Science, Metaphysics, and the Laws of Nature. New York: Oxford University Press. Lange, M. (2013). Grounding, Scientific Explanation, and Humean Laws. Philosophical Studies 164, 255–61. Lange, M. (2018). Transitivity, Self-Explanation, and the Explanatory Circularity Argument against Humean Accounts of Natural Law. Synthese 195(3), 1337–53. Leuenberger, S. (2014). Grounding and Necessity. Inquiry 57(2), 151–74. Lewis, D. (1983). Philosophical Papers, Volume 1. Philosophical Papers, Volume 1. Oxford: Oxford University Press. Lewis, D. (1994). Humean Supervenience Debugged. Mind 103(412), 473–90. Loew, C. and S. Jaag (2019). Humean Laws and (Nested) Counterfactuals. The Philosophical Quarterly 70(278), 93–113. Loewer, B. (1996). Humean Supervenience. Philosophical Topics 24(1), 101–27. Loewer, B. (2012). Two Accounts of Laws and Time. Philosophical Studies 160(1), 115–37. Maudlin, T. (2007). The Metaphysics within Physics. New York: Oxford University Press.

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202   Miller, E. (2014). Quantum Entanglement, Bohmian mechanics, and Humean Supervenience. Australasian Journal of Philosophy 92(3), 567–83. Miller, E. (2015). Humean Scientific Explanation. Philosophical Studies 172 (5), 1311–32. Oppenheim, P. and H. Putnam. (1958). Unity of Science as a Working Hypothesis. Vol. 2. Minneapolis, MN: University of Minnesota Press. Roberts, J. (1998). Lewis, Carroll, and Seeing through the Looking Glass. Australasian Journal of Philosophy 76(3), 426–38. Roberts, J. T. (2008). The Law-Governed Universe. New York: Oxford University Press. Rosen, G. (2010). Metaphysical Dependence: Grounding and Reduction. In Bob Hale and Aviv Hoffmann (eds.), Modality: Metaphysics, Logic, and Epistemology, 109–36. New York: Oxford University Press. Schaffer, J. (2007). Deterministic Chance? British Journal for the Philosophy of Science 58(2), 113–40. Schaffer, J. (2009). On What Grounds What. In D. Manley, D. J. Chalmers, and R. Wasserman (eds.), Metametaphysics: New Essays on the Foundations of Ontology, 347–83. New York: Oxford University Press. Schaffer, J. (2015). What Not to Multiply without Necessity. Australasian Journal of Philosophy 93(4), 644–64. Schneider, S. (2007). What Is the Significance of the Intuition That Laws of Nature Govern? Australasian Journal of Philosophy 85(2), 307–24. Sider, T. (2020). The Tools of Metaphysics and the Metaphysics of Science. Oxford: Oxford University Press. Skiles, A. (2015). Against Grounding Necessitarianism. Erkenntnis 80(4), 717–51. Sklar, L. (1993). Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge: Cambridge University Press. Sklar, L. (2014). Physical Theory: Method and Interpretation. New York: Oxford University Press. Skow, B. (2016). Reasons Why. New York: Oxford University Press. Spencer, J. (2017). Able to Do the Impossible. Mind 126(502), 466–97. Thau, M. (1994). Undermining and Admissibility. Mind 103(412), 491–504. Tooley, M. (1977). The Nature of Laws. Canadian Journal of Philosophy 7(4), 667–98.

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7 The Fundamentality of Physics Completeness or Maximality? Alyssa Ney

1. Completeness physicalism Why is it reasonable to try to explain consciousness in physical terms, in terms of information integration, ion channels, or resonant frequencies? Or to search for the origins of life in the principles of thermodynamics? To look to the spin states of elementary particles for a means of early cancer detection? Or to think that what is accelerating all galaxies away from each other must be some kind of physical force? Although the answers to all of these questions vary in their details, there is a working assumption underlying them all. This is an assumption the scientist (the neuroscientist, biologist, medical researcher, or cosmologist) takes for granted and rarely if ever will explicitly discuss. The assumption is that our world is fundamentally physical, that physics is a fundamental science and so there are physical truths that can serve to explain even the most complex (and animate) scientific phenomena. As the philosopher would put it, the working assumption behind this and so much else of scientific research is that some sort of physicalism is true. My aim in this paper is to put this basic assumption under philosophical scrutiny and ask what is the right way to understand physicalism. There is a standard way of interpreting it, certainly in the philosophical literature, but also I think more broadly in the scientific community. This is as a completeness thesis of some kind. Let’s characterize the view I will call completeness physicalism disjunctively in the following way:

Alyssa Ney, The Fundamentality of Physics: Completeness or Maximality? In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Alyssa Ney. DOI: 10.1093/oso/9780192893314.003.0007

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204   Completeness physicalism: all facts or entities consist of or are dependent on or supervene on or are realized by or may be completely explained by or grounded in the facts or entities of physics.

Completeness physicalists believe that there is or in principle could be some future physics that plays this role of providing a complete explanatory or ontological basis for our universe.¹ And this provides a basis for claiming that physics is special among the sciences, that it is fundamental. There is in principle some physical theory that alone provides supervenience bases or realizers or grounds for all facts or entities, or that describes a class of independent entities on which all else depends, or that is explanatorily privileged in some way, in being explanatorily complete. My main aim in this paper will be to show why we as physicalists should move beyond completeness interpretations of physicalism and the completeness of physics. Typically those who have raised critiques of positions like completeness physicalism do so in order to motivate some version of dualism, pointing to phenomena like phenomenal consciousness that seem to resist explanation in physical terms. Yet it is easy to show that completeness physicalism is unjustified, if not outright false, without making any appeal to consciousness or other intractable mental phenomena. Completeness physicalism is problematic already for its reliance on questionable assumptions about physics, many of which have been widely recognized as questionable in the philosophy of science for decades. Moreover, completeness physicalism is untenable in failing to provide the physicalist with any usable guide to ontology or metaphysical commitments. This undermines the entire point of adopting a position like physicalism: to give one an empirically motivated metaphysical framework that can then be put to work in directing one’s philosophical and scientific projects. Before I develop these points, I want to make it clear that my aim in criticizing completeness physicalism, unlike that of others who have raised some of these concerns (Crane and Mellor 1990, Koons and Bealer 2010, Stoljar 2010), is not to try to convince us to discard ¹ Or, perhaps, our concrete universe.

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physicalism. I am a physicalist, and I think physicalism is an important position worth defending because it is a position that has done a lot of good for us, motivating philosophical programs, bodies of scientific research, and technological innovations that have improved our lives.² Thus, it is important for us as physicalists to be clear about the flaws with the standard completeness interpretations of physicalism so that we may move past them and formulate versions of physicalism that can withstand critical scrutiny. To that end, I will propose a formulation of physicalism that could be used to replace the standard completeness interpretations. I will contrast the standard completeness physicalism with what I regard as a more plausible maximality physicalism. While completeness physicalism asserts the ontological or explanatory completeness of some future or in principle formulable physical theory, maximality physicalism instead only requires the ontological or explanatory maximality of our current physical theories. That is, it requires the ontological or explanatory superiority of physics, in certain respects, over all other scientific theories or epistemic frameworks. I think the maximality physicalism I develop in Section 3 is a promising way to go, but my primary goal in this paper will not be to convince the reader to adopt my specific form of maximality physicalism. The main point here is rather that completeness physicalism ought to be replaced, and something like my maximality physicalism takes us in a more promising direction, more in line with what we know about physics and what we want from physicalism than completeness physicalism does. I will rely on one fixed point in the discussion that follows since it is important to have some fixed point when we are asking questions about how to interpret some key concept or position. This is a claim I have already made and now want to underline about the practical import of physicalism: physicalism is worth defending for its success and future promise in motivating explanatory, predictive, and engineering projects in philosophy, science, technology, and public life that have in the past

² I will not, therefore, be advocating we reject physicalism and replace it with some kind of dualism. I am optimistic, on the basis of progress in the philosophical and scientific study of consciousness, that we will be able to explain conscious experience in physical terms.

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206   improved and continue to improve our lives in many ways. The adoption of physicalism drives ways of understanding ourselves and many previously puzzling aspects of reality, motivating an impressively broad range of explanations. It motivates frameworks for modeling and predicting the behavior of complex systems, including biological systems, with an extraordinary level of precision, leading to innovations with medical and other practical benefits too numerous to mention.³ This is the physicalism we are trying to characterize.

2. Hempel’s dilemma We can begin to see the problems with completeness physicalism by considering a question about the formulation of physicalism raised by Hempel (1980). This is commonly called Hempel’s dilemma. Hempel asked, if we are to be physicalists and claim that physics occupies a privileged position among the sciences—in the positivistic terms of his day, that it be regarded as the unitary language of science—then which “physics” are we talking about? The physicalistic claim that the language of physics can serve as a unitary language of science is inherently obscure: The language of what physics is meant? Surely not that of, say, 18th century physics; for it contains terms like ‘caloric fluid’, whose use is governed by theoretical assumptions now thought false. Nor can the language of contemporary physics claim the role of unitary language, since it will no doubt undergo further changes, too. The thesis of physicalism would seem to require a language in which a true theory of all physical phenomena can be formulated. But it is quite unclear what is to be understood here by a physical phenomenon. (1980: 195)

Hempel’s dilemma is the problem that if physicalism is understood in terms of current physics, then it is false, because current physics will ³ See Elpidorou and Dove (2018), and also Melnyk (2009) on naturalism, both of whom also emphasize the positions’ roles as research programs.

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likely be replaced with a better theory, and physics presently doesn’t have the resources to characterize all phenomena. But if physicalism is understood in terms of future physics, then it is difficult to know what physicalism comes to, because we don’t yet know what the future completed theory is.⁴ Most physicalists who address this issue today do not regard it as much of a problem because they think the answer is obvious. Surely physicalism is a claim about the ontological or explanatory completeness of some future physics. Past physical theories were all in some way false, and current physics, if it isn’t false, is at least incomplete.⁵ So, physicalists ought to characterize their position in terms of the completeness of some future physical theory. For example, according to Loewer, “physicalism claims that all facts obtain in virtue of the distribution of the fundamental entities and properties—whatever they turn out to be—of completed fundamental physics” (2001; see also Dowell 2006b, Pettit 1993). And Lewis defined physicalism (he preferred the term ‘materialism’) as the view that “physics—something not too different from present-day physics, though presumably somewhat improved—is a comprehensive theory of the world, complete as well as correct” (1983: 361). We can see Lewis as being cautious here, hoping that this future completed physical theory is close enough on the scientific horizon that we already have some idea of its theoretical commitments. Lewis’s characterization of physicalism thus attempts to navigate between the two horns of the dilemma. But given the magnitude of the open problems in current physics, it is not likely physics will reach completion without significant revolutions. To cite just two examples, physics still has no idea what makes up dark matter, which is supposed to constitute 85% of the total mass in our universe (Duda and Garrett 2011). There was an early near-consensus that dark matter could be explained by the postulation of a supersymmetric particle, the neutralino, but as of yet, there has been no evidence for supersymmetry at the Large Hadron ⁴ For discussions of Hempel’s dilemma, see Papineau and Spurrett (1999), Ney (2008c), Stoljar (2010), and the essays in Dowell (2006a). ⁵ Only Melnyk (1997, 2003) seems to recommend viewing physicalism as the view that current physics provides a complete explanatory or ontological basis for reality. He takes current physics to provide a complete set of realizers for all entities.

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208   Collider (Redlinger and de Jong 2017). So, the empirical evidence does not seem to point toward understanding dark matter as constituted by supersymmetric particles. In addition, although physicists would very much like to have a quantum theory of gravity, there is no consensus here either of what is even the right starting point from which to develop such a theory. String theories, strategies based on canonical quantum gravity, and approaches like causal set theory all have very different theoretical starting points and arrive at very different fundamental ontologies ranging from strings to spin foams to causal sets (Smolin 2001). Given the significance of these problems, a completed physics seems very much in the future, and its nature obscure.⁶ I want us to be clear now just how problematic this is for the kind of physicalist Hempel has in mind, one whose claim is that physics should serve as the unitary language of science. This is a physicalist like Carnap (1934, 1938) or Neurath (1931), whose core claim is that one should aim to translate all other sciences into the one language of physics, or the physical language, thus promoting the unity of science.⁷ Since a formulation in a single language makes more transparent the connections between different fields, and translation into the language of physics in particular allows a science’s claims to be intersubjectively testable, this sort of physicalism was not intended merely as a linguistic claim, but one that had the potential to be practically useful in improving science as we know it.⁸ How is this physicalist supposed to go about her task of translating the statements of all other sciences into the language of physics, if the relevant physics is one of the distant future? The problem is, Hempel’s physicalist is trying to do something with physics. And she can’t do something with a physics she can’t get her mind around. If we are going to take seriously the idea of physics as a unitary language of science, then we have to be talking about some version of physics we have access to. This is current physics. But this then takes us back to the first horn of the dilemma. Current physics is likely to be replaced. It isn’t the final theory. It isn’t a complete theory. But now it ⁶ See Smolin (2007) for an overview. ⁷ Carnap and Neurath went back and forth over the years discussing whether it was the language of physics or some other language that should serve as the unitary language of science. ⁸ See Ney (2008a) for further discussion and references.

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becomes natural to ask: For the role Hempel’s physicalist wants physics to play, does it need to be a final theory? Does it need to be a complete theory? To these questions, the answer is clearly “No.” Physics doesn’t need to be final or complete for it to be reasonable for us to begin the process of unifying science with it. For it to be “the best we have” in certain salient respects is enough to motivate us to use it in this respect.⁹ And so the first horn of Hempel’s dilemma isn’t a problem after all, at least for the version of physicalism Hempel was concerned with. It is only a problem if we make the false assumption that for the language of physics to be the unitary language of science, physics must be complete or a final theory. I hope this is clear enough for the version of physicalism that Hempel had in mind. What should we say about more contemporary versions of physicalism, versions that instead take physics to be a fundamental science in some respect, where this doesn’t mean that we should try to translate the statements of all other sciences into the language of physics? My claim is that we reach a similar conclusion. If physicalism motivates us to do something with physics, then the second horn of the dilemma is still a problem: we can’t do anything with a theory we can’t get our minds around. Indeed, as we will see, there are more problems with taking the “physics” in physicalism to be some future completed physical theory than this. And yet the first horn does not present a problem. For what the contemporary physicalist needs physics to do, it does not need to be a final or complete theory.

3. Maximality physicalism It is now time to put a proposal on the table for what a more promising and useful formulation of physicalism could look like, a position I call maximality physicalism. Maximality physicalism gets us what we want from physicalism without facing the problems completeness physicalism ⁹ Carnap (1934) and Neurath (1931) certainly didn’t think a theory needed to be final or complete to play this role. Both often took seriously the idea that the unitary language of science should be the language we use to describe ordinary material objects. But the folk theory of ordinary material objects is surely not a final or complete theory.

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210   faces. However, I recognize the proposal I will make is only a start. Moreover, it is just one way to go in developing a version of physicalism that is well supported and can play the roles the physicalist needs it to play. According to maximality physicalism, physics holds a privileged status among the sciences not in being ontologically or explanatorily complete, but in being ontologically or explanatorily maximal, or superior in some respect. The version I recommend takes physics to be maximal in the sense that it provides a successful class of explanations that are broader, deeper, and more precise than those of any other science or explanatory scheme.¹⁰ Its explanations are broader in the sense of covering more phenomena. Its explanations are deeper in tracing the constitutive bases of phenomena further than other explanatory schemes. Its explanations are more precise in having more mathematical specificity (e.g., are given to more decimal places) than those of other explanatory schemes. Note that to say that physics is fundamental in this sense, that it is explanatorily maximal, is not to say that its explanations are, all things considered, better than the explanations provided by other sciences or explanatory schemes, nor, of course, that the other sciences should or could be eliminated. It is a complicated and vexed issue what makes for the best explanation of a given phenomenon; indeed it is a complicated and vexed issue what makes for an explanation of a given phenomenon. In claiming that the explanations of physics are maximal, my claim is only that as a whole they are broader, deeper, and more precise than those of other sciences. To claim that physics is fundamental is not, therefore, to claim that physics is better than other sciences. When one holds the claim that physics is privileged or fundamental in the sense of being explanatorily maximal, this will then justify a set of attitudes that make up what I have elsewhere called the physicalist attitude (Ney 2008b; see also van Fraassen 2002 for a predecessor position and Stoljar 2015 for critique). For the completeness physicalist, the physicalist attitude is just the belief that the world is the way (the completed) physics says it is, or that everything supervenes on the physical

¹⁰ There may be a way of developing an ontological sense of maximality for physics, a way that improves upon supervenience, realization, or grounding formulations of completeness physicalism. However, I will focus only on an explanatory construal here.

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facts, or is grounded in the physical facts, or . . . But given the maximality physicalist’s assessment that the world isn’t simply the way physics says it is (nor does it supervene on or is wholly grounded in the physical facts or . . . ), she won’t have this sort of belief. For her, the physicalist attitude will amount to something different. As a first pass, we may characterize the physicalist attitude in the following way so as to include: • The disposition to take on commitment to the kinds of things our best current physical theories say exist, that is, to use them in one’s philosophical and scientific projects; • The disposition to not take on commitment to the kinds of things that one thinks won’t be explained by current physical theories¹¹; • The expectation (given that current physics is not yet complete) that near future physical theories will continue to improve in their ability to guide explanatory, predictive, and technological projects. These are all attitudes we expect of a physicalist reasonably informed about the character of current physics.¹² Although the completeness physicalist attempts something more, this something more is not reasonable in light of the arguments I will provide in sections 4–6. Hempel’s dilemma is avoided when one claims only the maximality, not the completeness of physics. To the question of what should be the unitary language of science or what should be considered the fundamental science, the answer for the physicalist is, of course, the language of current physics, the only physics that is presently formulated and reasonable to use in one’s projects. And, as Melnyk notes: Physicalists who hold, as I do, that current scientific findings provide support for physicalism must at the least have a formulation of

¹¹ This disposition is made plausible by the vast scope of physical explanations, their depth in providing constitutive explanations of a diverse range of phenomena, and a kind of exclusion reasoning that it would be unreasonable to believe in phenomena that aren’t explained physically (cf. Kim 2005). For more, see section 7. Further development of this connection between the maximality thesis and the physicalist attitude is work in progress. Again, my main aim here is to show why completeness physicalism should be rejected and to give an initial sketch of what a reasonable replacement position would look like. ¹² See Maddy (2007) for an approach to naturalism in a similar spirit.

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212   physicalism whose content is determinable by us now. But it is hard to see how else they can get one other than by defining “physical” by appeal to current physics; so that is what I shall do. (Melnyk 2003, 14)

Which part or version or interpretation of current physics provides a metaphysical framework for one’s projects is a matter left up to the individual physicalist. Scientific practice underdetermines the content of current physics in several ways (French 2014, ch. 2). So there are many satisfactory ways to be a maximality physicalist.

4. The vacuity objection Now that we have these two contrasting versions of physicalism to consider, we can begin to see the significant problems facing completeness physicalism and how moving in the direction of something more like maximality physicalism can help to better capture the view the physicalist is trying to put forward. I am going to start by returning to the challenge for completeness physicalism raised on the second horn of Hempel’s dilemma because it has not been recognized by most physicalists just how serious this challenge to their position is. Indeed, it has been used by philosophers such as Crane and Mellor (1990), Van Fraassen (2002), and Stoljar (2010) to argue that we should not be physicalists. Crane and Mellor use it in part to argue that “physicalism is the wrong answer to a meaningless question,” Stoljar to advise us that the projects in philosophy for which we have used physicalism to try to state a thesis or motivate a position would be more successful if we avoided talk of physicalism altogether. As I’ve already said, I am convinced that adopting physicalism and using it to motivate work not just in philosophy, but in science, engineering, and public life is very much a good thing. The physicalist attitude has yielded for those who adopt it many epistemic and practical benefits. So we should not be so quick to give it up. Nonetheless, the philosophers I just mentioned are all correct that the current dominant form of physicalism faces a significant challenge.

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The challenge again is that we don’t know what a future completed physics looks like and so this makes physicalism framed as the view that a future completed physics provides a complete explanatory or ontological basis vacuous, or at least lacking in sufficient content. To this one might reply that, of course, the phrase ‘a future completed physics’ has content. We know what physics is, what ‘future’ and ‘completed’ mean. What this shows is that the vacuity objection requires a bit more spelling out so that we may see the problem. In my view, there are two significant issues raised by the vacuity objection. First, our ignorance of what a future completed physics will look like undermines the ability of completeness physicalism to play the role in guiding philosophical, scientific, and engineering projects that physicalism is supposed to play. Second, this ignorance undermines the justification for completeness physicalism. Again, physicalism is good and worth defending because of the role it plays in motivating projects that have enhanced our understanding of ourselves and other organic and inorganic systems, as well as our place in the universe, in promoting advancements in scientific research, and in providing a framework that guides us toward engineering strategies that have improved our lives in numerous ways. For this to work, research and development must begin with certain facts about what our physical theories posit, the kinds of principles they employ, as well as those they do not. Thus, Hempel’s concerns are just as relevant for the contemporary understanding and use of physicalism, not just the form of physicalism explored by the logical positivists. Just as much today as in the past, physicalism guides us to make use of physics because it has some special features other theories do not have. But we can only make use of a physics whose formulation we have at hand. Second, physicalism is a position that is empirically justified. It is a myth that some anti-physicalists use in their polemics that physicalism is nothing more than a dogma arising due to some kind of unreasonable physics fetish. As Papineau showed in his 2001 paper “The Rise of Physicalism,” physicalism is not a dogma. It is a position supported by empirical argument, one citing the predictive, explanatory, and other scientific successes of our current physical theories, successes that have

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214   not similarly been achieved by other epistemic frameworks.¹³ These successes are, of course, the successes of physical theories that have actually been formulated and put to work. They are not successes of some future completed physical theory. And so we can only build a case for the special explanatory or ontological status of our current physical theories, for the fundamentality of our current physical theories, not for any future ones, because they haven’t had any successes. To remain committed to some unformulated physical theory, one that hasn’t met any empirical successes, should strike one as deeply unmotivated. This plays right into the hands of the anti-physicalists who accuse physicalists of clinging to dogma. Here I should comment on some alternative strategies that have been used to address the vacuity objection, different than my own proposal, which is to characterize physicalism in terms of the maximality of current formulated physics, rather than the completeness of future as yet unformulated physics. Some try to respond to the vacuity objection by filling in the conception of what a future completed physics will look like, saying physical theories are theories of a certain kind: theories that postulate a certain class of entities or theories that engage in explanations of a certain kind. One might then try to use such a more substantive characterization of physical theory in order to formulate responses to these two vacuity-related concerns about completeness physicalism. There are three kinds of characterization of physical theories that recur in the literature: (a) those that characterize physical theories as theories that provide microscopic bases for other phenomena (e.g., Pettit 1993, Dowell 2006b), (b) those that characterize physical theories as theories that describe a class of entities spread out somehow in spacetime (e.g., Poland 1994, Dowell 2006b, Howell 2013), and (c) those that characterize physical theories as those that make appeal only to entities that are (fundamentally) nonmental (e.g., Montero and Papineau 2005, Wilson 2006). Using one of these characterizations, one might respond ¹³ I have raised issues for the details of the argument that Papineau formulates in that paper (Ney 2016, 2019), relying as it does on a claim about the causal explanatory completeness of physics, but the big point Papineau is making in that paper about physicalism being supported by the empirical track record of physics in explaining a broad and diverse range of phenomena is correct. See section 7 for more discussion.

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to my first concern by saying it is true, one doesn’t know what the final completed physics will look like in its details, but since physical theories are theories that characterize phenomena ultimately in terms of [microscopic bases or entities in spacetime or nonmental phenomena], then completeness physicalism recommends projects that start from a class of [microscopic constituents or entities in spacetime or nonmental entities] because we know now that is what a final, completed physics will postulate. To answer the second worry, one would point to the empirical support that has accrued to theories formulated in terms of [microscopic bases or entities in spacetime or fundamentally nonmental phenomena]. One can then claim that this provides empirical support for a completed theory that is a theory of [microscopic bases or spacetime entities or fundamentally nonmental phenomena] and this in turn can provide empirical support for completeness physicalism.¹⁴ Although the response to the first worry is interesting and worth spending some time on, the response to the second clearly fails, for any such strategy of filling in the notion of the physical. For it just isn’t the case that all or even most theories formulated in terms of microscopic bases or spacetime entities or fundamentally nonmental phenomena have empirical support. Some do. Some do not. Just the fact that a theory is formulated in these terms doesn’t on its own serve to garner that theory any empirical support. And so there is no empirical support for the claim that any as yet unformulated future theory describing microscopic bases or entities in spacetime or fundamentally nonmental phenomena will be empirically supported. So, any such characterization of the physical will not suffice to answer the vacuity objection. Nonetheless, let’s address the response to the first worry. This was that we can see completeness physicalism as recommending particular kinds of philosophical and scientific projects in the following way. Since we know that a final, completed physics will describe the world in terms of microscopic constituents or entities in spacetime or fundamentally nonmental entities, we can see that completeness physicalism recommends projects that address the world in these terms. The ¹⁴ This is indeed what is done in Papineau and Spurrett (1999), appealing to a notion of the physical as (c) the fundamentally nonmental.

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216   problem is we can’t know that the final, completed physics will describe the world in these terms. Indeed there are physical theories today, widely accepted physical theories, that fail to meet each of these criteria of what physical theories look like. In physics, wholes are not always explained in terms of the features of microscopic parts. Appeals to emergence are rampant. Physics challenges the notion that spacetime is fundamental and routinely appeals to more basic frameworks that can explain the appearance of spacetime in certain regimes (Huggett and Wüthrich 2013). And irreducible mental phenomena are appealed to throughout physics, not merely in claims of consciousness collapsing the wave function, but more widely in the use of unexplained notions of information and anthropic principles. There are certainly physicalists who frown upon or lament these facts, especially the last. But they are facts about what real, mainstream physics looks like. And so the claim that the proposed criteria correctly characterize what it is to be a physical theory simply fails because the criteria fail to characterize actual physical theories in use by actual physicists.¹⁵ This is an illustration of the point put well by Van Fraassen that: Whenever philosophers take some general feature of physics and use it to identify what is material, what happens? Physics soon goes on to describe things that lack that feature and are altogether different. (Van Fraassen 2002)

Moreover, such characterizations aren’t sufficient for a theory to be a physical theory. A theory that the world was fundamentally built out of tiny nonsentient amoebae, a theory derived from a drug-induced hallucination, would satisfy all three criteria for what it is to be a physical theory; yet this seems obviously not to be what the physicalist is after.

¹⁵ This isn’t to say that there aren’t positions in the neighborhood of physicalism that one might want to defend. In the spirit of seventeenth-century corpuscularianism, one might want to advocate for the use of microscopic explanations or explanations in terms of spatiotemporal or nonmental entities. But one shouldn’t confuse this with physicalism, a view that takes physics to have some privileged ontological or explanatory status among the sciences, since physics frequently violates such restrictions.

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Again, we shouldn’t take any of this to lead us to reject physicalism, because to be physicalists we don’t need a robust characterization of what it is to be physical or to be a physical theory. We just need a sense of what the physical theories we have look like, and which are empirically well supported.

5. No positive argument for a future completed physics We have just considered the first concern with completeness physicalism: the lack of an adequate conception of “physics” with which to evaluate what is meant by a future completed physics. However, even if the concerns of section 4 could be addressed, there is still a question of why one should grant the assumption that there ever will be a completed physics one day in the future, or that such a theory is in principle possible. This section will consider different arguments that might be used to support the completeness physicalist’s assumption that there will be, or in principle could be, such a thing as a completed physics. First, logic or meaning alone doesn’t compel us to believe that physics will one day be complete. As Chomsky once noted, simply defining the fundamental physics as completed, true science makes physicalism trivial: “the material world is whatever we discover it to be, with whatever properties it must be assumed to have for the purposes of explanatory theory” (1998: 144). It is not trivial that we are able to give an account of phenomena as diverse as galactic expansion, the origin of life, and consciousness in terms of a few fundamental features and principles. A better strategy for supporting the claim that there will one day be a complete physical theory is to look for an inductive argument. But note: the way inductive arguments work is by seeing that there were some cases we observed in the past that all had something in common, and they all or generally turned out to be a certain way, and from there we infer that unobserved things that have that feature in common will also turn out to be that way. So, for example, we note that every raven we have observed up to now has been black and so we infer that the next one will be black too, or that they will all be black, or that ravens are generally black. In the present case, to give an inductive argument for the future

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218   completeness of physics, we would have to say something like every time we have observed something needing an explanation, it has been given a physical explanation; therefore, eventually everything will have a physical explanation. The trouble though (setting aside the fact that the premise is false) is that we don’t have a similarity class to support the induction. When we are talking about absolutely everything and anything, there is no similarity that ties together the group of observed instances.¹⁶ So one can’t argue inductively to the conclusion that there will be a physical explanation for all facts. So one can’t argue inductively to the eventual existence of a complete physical theory. To exhaust all of the options, we should also consider what may be said for an abductive argument for the claim that there will one day be a completed physics. This would require showing that the assumption of a future completed physics provides the best explanation of some fact. But what fact? One might say it provides the best explanation of the fact that we have been able to give physical explanations for a diverse class of phenomena in living and nonliving systems, on Earth and elsewhere in the cosmos. But does the hypothesis that there will be a complete physics provide the best explanation of the breadth of successful physical explanations? Isn’t rather the approximate truth of our current incomplete physical science a better, safer explanation of this success? I submit that it is. The hypothesis of a complete theory is much more than is needed to explain the history of successful physical explanations. We have now seen that there is no good argument for the claim that there will one day be a completed physics. We have no positive reason to believe in a future true and completed physical theory. One might respond that we don’t need an argument that there will actually be in the future a complete physics, but only that there could be such a physics in principle (in a world like ours in certain relevant respects). But similar points apply. This isn’t something that is true by definition. A modified inductive argument of the form “All observed phenomena have been given physical explanations; therefore, all phenomena could in principle

¹⁶ It has been suggested we restrict the induction to the contingently existing things. But one may dispute whether this makes for a similarity class. Are all contingently existing things intrinsically similar in virtue of their contingency?

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receive a physical explanation” is blocked again by there being no similarity class on which to base the induction. And the assumption of a possible complete physics is no better at explaining data than the assumption of an actual complete physics. It would thus be better to interpret physicalism in such a way that it is independent of such a completeness assumption. I stress my claim here is not that we know now there won’t be a completed physics. I do think there are reasons to be skeptical of this claim. But I am only saying we don’t have any positive justification for thinking there will be or in principle could be a completed physics. And so the assumption shouldn’t be built into the very meaning of physicalism. Physicalism is better supported and more reasonable without it.

6. Tensions with philosophy of science We may now turn to the third and final argument against completeness physicalism. There are several reasonable claims that mark important milestones of late twentieth-century philosophy of science. When physicalism is viewed as a kind of completeness claim, these can prove disastrous for physicalism. A physicalist ought to provide an interpretation of her position and of the fundamentality of physics that is reasonable in light of these lessons. I will argue this is another reason to prefer maximality physicalism in the sense I have proposed. The first lesson of late twentieth-century philosophy of science is that often the best explanation of a phenomenon is not the microphysical explanation, but rather some “higher-level” or “special-science” explanation. A classic illustration of this point comes from Putnam (1975), who asked us to consider the best explanation for why a certain peg is incapable of entering a hole: Very often we are told that if something is made of matter, its behavior must have a physical explanation. And the argument is that if it is made of matter (and we make a lot of assumptions), then there should be a deduction of its behavior from its material structure . . . On the other hand, if you are not ‘hipped’ on the idea that the explanation must be at

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220   the level of the ultimate constituents, and that in fact the explanation might have the property that the ultimate constituents don’t matter, that only the higher level structure matters, then there is a very simple explanation here. The explanation is that the board is rigid, the peg is rigid, and as a matter of geometrical fact, the round hole is smaller than the peg, the square hole is bigger than the cross-section of the peg . . . That is a correct explanation whether the peg consists of molecules, or continuous rigid substance, or whatever. (Putnam 1975: 296)

Putnam actually takes a rather hard line here, insisting that the physical explanation is the wrong explanation, because it appeals to features that aren’t, in his words, relevant. This attitude has been agreed to by many philosophers of the special sciences. However, a more moderate position, one that is favored by other philosophers of science who have been influenced by Putnam’s example, is not that the physical explanation is the wrong explanation and the nonphysical explanation, appealing to higher level structural features, is the only explanation, but rather that the physical explanation provides a worse explanation and the nonphysical structural explanation provides the better explanation. Both the completeness and the maximality physicalist should agree that the physicalist claim that physics provides fundamental explanations doesn’t mean these explanations are better than any others in the senses one might care about for all purposes. That a higher-level explanation is better in a given context does not by itself undermine the fundamentality of physics. A metaphysical claim to fundamentality should not be confused with a claim of superiority in all respects or importance. And so if the lesson one wishes to draw from Putnam’s example is that often the physical explanation is not the optimal one to use in a given context, then this is compatible with either form of physicalism. On the other hand, if the lesson is supposed to be not that the nonphysical explanation is sometimes the better one, but instead that it is the correct one, then this does present a challenge. The difference between the higher-level explanation being the right one vs. only the better one makes a difference to the viability of completeness physicalism. If the nonphysical explanation is the right explanation, then physics is not explanatorily complete.

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Note the maximality physicalist need not take a stand on this issue. She doesn’t need physics to be complete, only maximal. Her claim is only that physics provides precise and deep explanations of a wide range of phenomena. And so even if physics does not provide the right explanation of why a certain peg won’t go through a certain hole, there will be a host of related facts that the physical details do explain. That is what makes physics maximal. So this first milestone of late twentieth-century philosophy of science supports maximality physicalism over completeness physicalism. Another lesson has to do with the form and intended scope of our scientific theories, including our best physical theories. Philosophers of science in practice note that scientists, physicists included, rarely try to formulate theories of everything from which we could derive all true facts of the universe. Rather, their aims are generally to model some local phenomenon or other. This is true even of the most fundamental physical theories, quantum field theories or cosmological theories. This second point about most physical theories being local theories, however, provides a challenge to completeness physicalism. In the event that these many local theories (or models) cannot be patched together to form some one complete theory—and why think that they would? why would there not be gaps?—then this straightforwardly undermines the claim that the world is the way some true completed physics says it is. Yet this does nothing to undermine maximality physicalism which relies only on the success, depth, breadth, and precision of physical explanations, not their completeness. I would take issue with the trope one finds in contemporary philosophy of science that physics is just one among many special sciences, that it does not have some special status among the sciences, of being fundamental. This is no doubt caused by the assumption that fundamentality must be cashed out in some notion of completeness. I, of course, am arguing here that there is a more realistic interpretation of the fundamentality of physics that does not require its completeness. And so the physicalist can uphold the “locavorism” defended by many philosophers of science (Ruetsche 2015), while maintaining the view that physics occupies a privileged status among the sciences, that it’s fundamental.

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222   A third milestone comes from feminist philosophy of science, which has questioned the reductionism implied in claims of the fundamentality of physics. To focus on one strand of argument, claims of the fundamentality of one science have been shown to lead to a potentially dangerous monopolization of resources that might be better used on projects that would have a more beneficial impact on our world. As Cartwright puts it: theories that purport to be fundamental—to be able in principle to explain everything of a certain kind—often gain additional credibility just for that reason itself. They get an extra dollop of support beyond anything they have earned. (Cartwright 1999)

Cartwright argues that we should move beyond viewing some theories or branches of science as fundamental and instead recognize that the reliability of any theory, including those offered as “theories of everything,” have only limited applicability within a circumscribed domain. The social consequences of claims of the fundamentality of physics are relevant to this issue of the best interpretation of physicalism and should not be ignored, as they often are in metaphysical discussions. The physicalist should think through what a claim of the fundamentality of physics implies for the privileging of certain research projects over others. But it is possible to say a lot about the practical benefits that come with the funding of projects, even very expensive projects, in physics (Ney 2019). My point here, however, is that it is difficult to even begin to formulate these issues if we are taking the fundamentality of physics to imply the truth and completeness of some far distant, perhaps unrealizable theory. And so if we are going to have a responsible defense of the claim of the fundamentality of a particular theory, then this ought to be a currently formulated theory we can evaluate for its practical consequences.

7. The inductive arguments again Taking stock, I have noted two ways in which the fundamentality of physics can be interpreted: as a claim about the completeness of some (at

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least in principle) future physical theory and as a claim about the maximality of current physical theory. I have defended the latter interpretation, arguing that physicalism is best interpreted as the claim that physics is fundamental in that sense, combined with the adoption of a set of attitudes. Physicalism should not be interpreted as the thesis that the world is exhaustively and completely the way some physical theory says it is. Part of my argument against the completeness approach was that there is no positive argument for there ever being a completed physics (now or in the future). However, one might ask whether my points in section 6 undermine not just completeness physicalism, but any sort of physicalism, including the version I am advocating here. For the inductive argument I was criticizing looked like this: 1. Many observed phenomena have received physical explanations. Therefore, 2. All phenomena will receive physical explanations. (Completeness physicalism) My complaint was that since there is no unified class of phenomena that is the subject of the induction, any argument like this is bound to fail. So it seems the following argument with a substantially weakened conclusion would be equally bad: 1. Many observed phenomena have received physical explanations. Therefore, 2. The next unexplained phenomenon will receive a physical explanation. The failure of this argument looks to be a problem even for the weaker maximality physicalism. A maximality physicalist will hold the view that physical explanations should be sought in general. And this is supposed to be an empirically based view, one that is reasonable in light of the past reductive successes of physical science.

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224   But there is no need to panic. Maximality physicalism is fine. To address this concern, we must distinguish between two types of phenomena the physicalist may encounter that do not yet have explanations in terms of physical science: those that are in a genuine sense like those that have already been explained, and those that are unlike those that have already been explained. For the phenomena that are genuinely like those that have already been given physical explanations, there will be an inductive argument available. These arguments will have a narrower scope than those we considered above, e.g.: 1. Many observed macroscopic features of living things have received physical explanations. Therefore, 2. All macroscopic features of living things will receive a physical explanation.¹⁷ This sort of inductive argument can be successful to the extent that the premise concerns a unified class of phenomena. I believe that it does. As for phenomena that are unlike those that have already received physical explanations, here there won’t be an inductive argument we can use to underwrite the case for looking for a physics of those phenomena. But that is OK. After all, there isn’t an inductive case for looking for an alternative theory of phenomena like those either. Instead, what we can say is that since physics has shown itself already to be capable of giving successful explanations of a very wide range of phenomena, it is a good starting point. It’s a practical point of the “only game in town” variety that supports the development of physical explanations of the unknown and radically unlike what has already been explained (cf. Dawid 2013). There is no need for the maximality physicalist to hold that physics will in the end explain everything. For reasons I’ve already mentioned, that claim is unreasonable. But at the same time, the physicalist should be optimistic about current physics, and believe it is the right place to start. ¹⁷ Note this is very much like the inductive argument for physicalism Papineau (2001) considers.

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8. Conclusion Although the standard interpretation of physicalism is problematic in the many ways I’ve noted, this doesn’t mean we should discard physicalism, discard the view that physics has a special status among the sciences, that it is fundamental. This would be an overreaction. There is a way of capturing physicalism and underwriting reductive philosophical and scientific projects that doesn’t rely on unmotivated assumptions and an outdated philosophy of science. I’ve called this maximality physicalism. I’ve focused above on physicalism’s role as a framework guiding certain research projects, those seeking to explain a diverse and initially disunified class of phenomena in physical terms. I don’t mean here to say definitively that there is no purpose for which the claim that there will be a completed physical theory may be useful. There is no harm in physics (in some branches) trying to achieve that goal. But this claim is both stronger and less useful than what is needed for the practical aims for which one should promote physicalism.

Works cited Carnap, Rudolf. (1934). The Unity of Science. London: Kegan Paul, Trench, Trubner, and Co. Carnap, Rudolf. (1938). Logical Foundations of the Unity of Science. Foundations of the Unity of Science: Toward and International Encyclopedia of Unified Science. O. Neurath, R. Carnap, and C. Morris, eds. Chicago: University of Chicago Press: 42–62. Cartwright, Nancy. (1999). The Dappled World: A Study of the Boundaries of Science. Cambridge: Cambridge University Press. Chomsky, Noam. (1988). Language and Problems of Knowledge. Cambridge: MIT Press. Crane, Tim and D. H. Mellor. (1990). There Is No Question of Physicalism. Mind. 99: 185–206. Dawid, Richard. (2013). String Theory and the Scientific Method. Cambridge: Cambridge University Press.

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226   Dowell, Janice, ed. (2006a). Special Issue on Physicalism. Philosophical Studies. 131. Dowell, Janice. (2006b). The Physical: Empirical, Not Metaphysical. Philosophical Studies. 131: 25–60. Duda, G. K. and K. Garrett. (2011). Dark Matter: A Primer. Advances in Astronomy. 2011. doi.org/10.1155/2011/968283. Elpidorou, Andreas and Guy Dove. (2018). Consciousness and Physicalism: A Defense of a Research Program. New York: Routledge. French, Steven. (2014). The Structure of the World. Oxford: Oxford University Press. Hempel, Carl. (1980). Comments on Goodman’s Ways of Worldmaking. Synthese. 45: 193–200. Howell, Robert J. (2013). Consciousness and the Limits of Objectivity: The Case for Subjective Physicalism. Oxford: Oxford University Press. Huggett, Nick and Christian Wüthrich. (2013). Emergent Spacetime and Empirical (In)Coherence. Studies in History and Philosophy of Modern Physics. 44(3): 276–85. Kim, Jaegwon. (2005). Physicalism, or Something Near Enough. Princeton: Princeton University Press. Koons, Robert and George Bealer, eds. (2010). The Waning of Materialism. Oxford: Oxford University Press. Lewis, David. (1983). New Work for a Theory of Universals. Australasian Journal of Philosophy. 61(4): 343–77. Loewer, Barry. (2001). From Physics to Physicalism. Physicalism and Its Discontents. C. Gillett and B. Loewer, eds. Cambridge: Cambridge University Press: 37–56. Maddy, Penelope. (2007). Second Philosophy: A Naturalistic Method. Oxford: Oxford University Press. Melnyk, Andrew. (1997). How to Keep the ‘Physical’ in Physicalism. Journal of Philosophy. 94(12): 622–37. Melnyk, Andrew. (2003). A Physicalist Manifesto: Thoroughly Modern Materialism. Cambridge: Cambridge University Press. Melnyk, Andrew. (2009). Naturalism as a Philosophical Paradigm. Philo. 12(2): 188–99. Montero, Barbara and David Papineau. (2005). A Defence of the Via Negativa Argument for Physicalism. Analysis. 65: 233–7.

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Neurath, Otto. (1931). Soziologie im Physikalismus. Erkenntnis. 2: 393-431. Ney, Alyssa. (2008a). Reductionism. Internet Encyclopedia of Philosophy. J. Fieser and B. Dowden, eds. https://www.iep.utm.edu/red-ism/, accessed July 18, 2020. Ney, Alyssa. (2008b). Physicalism as an Attitude. Philosophical Studies. 138: 1–15. Ney, Alyssa. (2008c). Defining Physicalism. Philosophy Compass. 3(5): 1033–48. Ney, Alyssa. (2016). Microphysical Causation and the Case for Physicalism. Analytic Philosophy. 57(1): 141–64. Ney, Alyssa. (2019). The Politics of Fundamentality. What is Fundamental? A. Aguirre, B. Foster, Z. Merali, eds. Cham: Springer, 27–36. Papineau, David. (2001). The Rise of Physicalism. Physicalism and Its Discontents. C. Gillett and B. Loewer, eds. Cambridge: Cambridge University Press, 3–36. Papineau, David and David Spurrett. (1999). A Note on the Completeness of ‘Physics.’ Analysis. 59(1): 25–9. Pettit, Philip. (1993). A Definition of Physicalism. Analysis. 53: 213–23. Poland, Jeffrey. (1994). Physicalism: The Philosophical Foundations. Oxford: Oxford University Press. Putnam, Hilary. (1975). Philosophy and Our Mental Life. Mind, Language, and Reality. Cambridge: Cambridge University Press, 291–3. Redlinger, George and Paul de Jong. (2017). Broken Symmetry: Searches for Supersymmetry at the LHC, https://atlas.cern/updates/atlas-feature/super symmetry, accessed July 18, 2020. Ruetsche, Laura. (2015). The Shaky Game +25, Or: On Locavoracity. Synthese. 192(11): 3425–42. Smolin, Lee. (2001). Three Roads to Quantum Gravity. New York: Basic Books. Smolin, Lee. (2007). The Trouble with Physics. New York: Houghton Mifflin. Stoljar, Daniel. (2010). Physicalism. London: Routledge. Stoljar, Daniel. (2015). Physicalism. The Stanford Encyclopedia of Philosophy. E. Zalta, eds., https://plato.stanford.edu/archives/win2017/ entries/physicalism/, accessed July 18, 2020. Van Fraassen, Bas. (2002). The Empirical Stance. New Haven, CT: Yale University Press. Wilson, Jessica. (2006). On Characterizing the Physical. Philosophical Studies. 131: 61–99.

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PART III

TIME AND PERSISTENCE

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8 A Little Puzzle about a Piece and a Puddle Mahrad Almotahari

Imagine Descartes’s study. On a table next to his armchair there’s a small, solid piece of wax—the handiwork of a local Parisian craftsman, from whom it was purchased. Now imagine that Descartes melts the whole thing, thus forming a single cohesive wax puddle. Two equally compelling lines of thought suggest themselves: Thesis. The puddle of wax was made by Descartes, but the initial piece of solid wax wasn’t. Furthermore, the puddle was made by melting the initial piece of solid wax, while the initial piece of solid wax wasn’t made by melting the initial piece of solid wax. Finally, Descartes didn’t purchase the puddle of wax, but he did purchase the initial piece of wax. Plausibly, then, the initial piece of wax is distinct from the puddle of wax. Antithesis. The puddle of wax is a liquified piece of wax, and a liquified piece of wax is a piece of wax—a piece of wax now in liquid form. The piece of wax now in liquid form wasn’t made by Descartes (though he is responsible for its liquid form), and Descartes did purchase the piece of wax now in liquid form (though it wasn’t in liquid form when he purchased it). The piece of wax now in liquid form acquired its present form by melting its past solid form (though its solid form wasn’t acquired in that way). Thus the initial piece of solid wax became the liquified piece of wax, and didn’t cease to exist

Mahrad Almotahari, A Little Puzzle about a Piece and a Puddle In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Mahrad Almotahari. DOI: 10.1093/oso/9780192893314.003.0008

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232   in the process. Plausibly, then, the “two” pieces are in fact one and the same piece of wax. Therefore, the initial piece of wax is identical to the puddle.¹

What gives? More to the point, is the initial piece of solid wax identical to the wax puddle, or isn’t it? I’ll have more to say about this question in the discussion to come. Before we charge ahead, though, I’d like to outline a plan of attack. In sections I–III I explain the dialectical significance of our little puzzle. Specifically, I argue that it puts a great deal of pressure on a certain kind of view about material reality, and I explain why there are no quick and easy solutions if one accepts the kind of view I’m targeting. For advocates of the target view, the puzzle is indeed a puzzle. If you think

¹ One might think that it doesn’t follow from something’s being a liquified piece of wax that it’s a piece of wax, just as it doesn’t follow from something’s being a fake Rolex that it’s a Rolex. Or one might challenge the inference from ‘liquified piece’ to ‘piece’ on the grounds that ‘liquified piece’ implies ‘liquid’, whereas ‘piece’ sans modifier implies ‘solid’. Four points should be borne in mind, however. First, pointing out that something is a fake Rolex is a way of denying that it’s a Rolex. In contrast, saying of something that it’s a liquified piece of wax is not a way of denying that it’s a piece of wax. Consider how odd it would be for me to challenge the truth of your claim that this thing here is a piece of wax by saying, “False! It’s a liquified piece of wax.” Insofar as there’s a disagreement between us at all, I’m tempted to say it’s metalinguistic: the consideration to which I appeal doesn’t challenge the truth of your statement, merely the felicity of how it was expressed. This understanding of our hypothetical exchange is supported by the observation that it would have been a perfectly good response to your initial statement to say instead, “I agree, though I would just add that it’s a liquified piece of wax.” So the inference from ‘liquified piece’ to ‘piece’ is crucially different from ‘fake Rolex; therefore, Rolex’. Second, it’s a popular misconception that glass is a liquid—so popular, in fact, that Scientific American devoted an article to debunking it (Ciara Curtin 2007). Although the full story about the thermodynamics of glass is far too complicated to adequately summarize in a footnote (Philip Gibbs 1997), the popularity of the misconception, and how it’s typically corrected (not by conceptual analysis, but by more information), indicates that the thought, Glass is a liquid, is merely an empirical falsehood. Even those of us who once thought that glass is a liquid had no problem with the phrase, ‘piece of glass’. In other words, some of us once thought that pieces of glass are liquid pieces of matter, and we weren’t conceptually confused in thinking so; we were simply ignorant. Consequently, ‘piece’ sans modifier doesn’t imply ‘solid’. Third, even if we grant that ‘piece’ sans modifier implies ‘solid’, the whole puzzle can be reformulated in terms of ‘bit’, as in ‘the bit of wax’, etc. It’s absolutely unproblematic to say, ‘I drank the bit of water’, which clearly indicates that a bit of something may well be liquid. Now imagine we have some ice. Well, the ice is a bit of solid water. If melted, we obtain a liquified bit of water; and a liquified bit of water is indeed a bit of water. Similarly, a liquified bit of wax is a bit of wax. Corresponding adjustments to my Thesis and Antithesis would generate our puzzle, though it would require an unhappy change to my title. Finally, even if the objection stands, the central claim under Antithesis is left intact, namely, that the initial piece of solid wax became the liquified piece of wax, and didn’t cease to exist in the process. As long as this claim is left unchallenged, we arrive at the puzzling conclusion. I’ll have more to say about the central claim under Antithesis in the discussion below.

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you’ve already hit on a response that affords a quick and easy solution, I ask that you bear with me for the nonce. In section IV I explain the theoretical significance of our little puzzle. In particular, I argue that our ordinary modal thought and language is too messy to be metaphysically probative without some degree of disciplinary intervention. This theoretical point is an old one,² but I hope to present a more persuasive case for it. If sound, then we should expect our theorizing about modality to be more or less rationally reconstructive, by which I mean that an analysis deserves to be regarded as an adequate interpretation of ordinary modal thought and language, not because it vindicates all or even many of our ordinary modal beliefs, assertions, and patterns of reasoning (though hopefully it won’t do too poorly on this count), but because it enables us to achieve many of the legitimate purposes for which we use modal concepts and terms. Achieving these purposes needn’t require that our one-off modal judgments and patterns of thought be sound. It doesn’t even require that large fragments of our ordinary ways of thinking be accurate. No doubt theorists can disagree about what our “legitimate” purposes are in this context, and I certainly won’t say anything conclusive about the matter, but for the sake of concreteness I will identify one or two purposes that seem to me to be especially important. To put the point in a somewhat different and increasingly popular way, interpretations of modal thought and language should be ameliorative, not explicative.³ I conclude in section V by returning to the original question—is the initial piece of solid wax identical to the wax puddle, or isn’t it?—and spelling out what I suspect the correct answer is. I’d like to emphasize, though, that my dominant attitude is one of puzzlement, not confidence. This will be apparent in the tone I take, if not what I say. ² It is, I think, one way of interpreting Quine’s famous argument involving the person who’s both a mathematician and a cyclist. The mathematician is alleged to be necessarily rational but not necessarily two-legged; the cyclist is alleged to be necessarily two-legged but not necessarily rational. We’re told that neither description allows us to represent this person’s essence any better than the other. So quantified modal logic, and the essentialism it requires, is problematic, because it involves “an invidious attitude toward certain ways of uniquely specifying [things] and favoring other ways . . . as somehow better revealing the ‘essence’ of the object” (W. V. Quine 1953, p. 155; 1960, p. 199). The problems with this argument are well known, so I won’t linger (Kit Fine 1989; David Kaplan 1986; Saul Kripke 1980). ³ See Sally Haslanger (2012).

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234  

I. To better understand the significance of our little puzzle—in particular, how it differs from the more familiar sort of case that guides theorizing about material identity and constitution—let’s quickly compare it to Allan Gibbard’s classic example, Lumpl and Goliath: I make a clay statue of the infant Goliath in two pieces, one the part above the waist and the other the part below the waist. Once I finish the two halves, I stick them together, thereby bringing into existence simultaneously a new piece of clay and a new statue. A day later I smash the statue, thereby bringing to an end both statue and piece of clay. The statue and the piece of clay persisted during exactly the same period of time. Here, I am tempted to say, the statue and the piece of clay are identical. . . . If indeed the statue and piece of clay are the same thing, then their identity is contingent. . . . the statue I shall call ‘Goliath’; the piece of clay, ‘Lumpl’. (Gibbard 1975, p. 191)

I agree with Gibbard’s judgment about this case,⁴ but not everyone does. Many philosophers have drawn a different conclusion, according to which Lumpl and Goliath are numerically distinct despite their spatiotemporal coincidence. Advocates of this view, or pluralists, say that Lumpl merely “constitutes” Goliath.⁵ More generally, pluralism is the doctrine that spatiotemporal coincidence doesn’t entail identity. As one might expect, then, monism is the thesis that spatiotemporal coincidence entails identity. A particularly extreme form of monism is that spatial coincidence by itself entails identity. Gibbard, however, is a moderate monist; he believes that, in the absence of temporal coincidence, a statue (of an elephant, say) and its constituent hunk of matter will have different ⁴ I’ve explained why elsewhere. See Mahrad Almotahari (2014a; 2014b; 2017; 2019). See also David Lewis (1971) and Delia Graff Fara (2012). ⁵ See Lynne Baker (2007), Simon Evnine (2016), Kit Fine (2003; 2006; 2008), John Hawthorne (2008), Mark Johnston (1992; 2006), Daniel Korman (2015a), Kathrin Koslicki (2008), Thomas Sattig (2015), Judith Jarvis Thomson (1983; 1998), Amie Thomasson (2007), David Wiggins (2001), and Stephen Yablo (1987; 2015).

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temporal properties, such as “the property of being elephant-shaped as long as it exists” (1975, p. 190). Pluralism, too, has a more extreme relative, according to which not even necessary spatiotemporal coincidence entails identity.⁶ For the most part, I’ll ignore the extreme variants of each thesis. Pluralism is typically conceived as part of a larger Aristotelian framework,⁷ appropriately dubbed hylomorphism, according to which being (οὐσία, ousia) is somehow constituted by both matter (ὕλη, hyle) and form (μορφή, morphē). Differences between materially identical beings, such as Lumpl and Goliath, are explained in terms of their form. The precise shape of the explanation varies from theorist to theorist, but in each case an appeal is made to a fundamental bifurcation whose history can be traced to Aristotle’s remarks about form, matter, and their role in explanation.⁸ Two different kinds of considerations are cited in favor of pluralism. The first kind is modal. Although Lumpl was destroyed when Goliath was smashed, things might have been different: Goliath might have been destroyed in a way that wouldn’t have destroyed Lumpl—perhaps by squashing, or melting. According to this line of thought, Lumpl and Goliath are modally discernible and therefore numerically distinct. Approaching the matter from a somewhat different direction, the modal predications that allegedly distinguish Lumpl from Goliath rely on the thought that pieces of matter tolerate changes due to cohesion-preserving transformative processes. (Typically, the persistence of an artwork imposes different requirements.) Let’s give this thought a name, to make discussion of it easier. Call it the transformative tolerance principle: (TTP) Pieces of matter would survive cohesion-preserving transformations, if they underwent such a process.⁹

⁶ Another view could be characterized as “extreme” pluralism, namely, what Karen Bennett (2004) calls plenitudinous primitivism. The two views are importantly different and shouldn’t be confused. ⁷ See especially Simon Evnine (2016), Kit Fine (2008), Mark Johnston (2006), Kathrin Koslicki (2008), and Thomas Sattig (2015). ⁸ See, in particular, Metaphysics Zeta. ⁹ Actually, I think TTP may need further refinement, but the additional complexity wouldn’t change the shape of my argument; it would only delay its presentation. So I’ll suppress any additional discussion.

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236   Although this principle is implicated in the pluralist’s modal argument, it fundamentally specifies a condition on diachronic identity. And that’s why it bears on our little puzzle, as I’ll argue below. The second kind of consideration in favor of pluralism is non-modal. For example, it’s compatible with their complete spatiotemporal coincidence that Goliath is badly made for a thing of its kind, while Lumpl isn’t. Gibbard could stipulate that Lumpl and Goliath don’t differ in this particular way, but it doesn’t really matter. We could go fishing for another non-modal difference: Goliath represents the infant Goliath but Lumpl doesn’t; Goliath is Romanesque but Lumpl isn’t, etc. Take your pick.¹⁰ Given that we’re interested in artifacts and pieces of matter for different reasons, pluralists are reasonably confident that a categorical difference is inevitable. For the sake of argument, though, let’s stack the deck in favor of the pluralist by reimagining Gibbard’s example so as to make it seem that the dyadic predicate, ‘___ was made by ___’, can be filled in to distinguish Goliath from Lumpl. One might initially think this can’t be done, given that Lumpl and Goliath begin to exist at precisely the same time, but I think it can. And seeing that it can will be quite useful later in this section. Suppose that two smaller pieces of clay are each connected to the arms of two separate mechanical devices. For mnemonic (certainly not stylistic) reasons, call these devices The Lumper and The Shaper. Each device operates independently of the other. When The Lumper is turned on, it pushes the two pieces of clay together, eventually forming a single and comparatively large lump of clay. When The Shaper is turned on, it begins to mold the two pieces of clay so that one represents the lower half of the infant Goliath and the other the upper half. Now suppose you press the button that initiates The Shaper. Some time passes. I then press the button that starts The Lumper. Now suppose that the two devices complete their tasks at precisely the same time, and let ‘Goliath’ name the resulting statue and ‘Lumpl’ the new lump of clay. Finally, suppose that an hour later Goliath is smashed, destroying both statue and lump simultaneously. We thus have complete spatiotemporal coincidence. But, plausibly, Goliath was made by The Lumper and The Shaper ¹⁰ See Fine (2003; 2006).

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collectively, though Lumpl wasn’t; it was made by The Lumper alone.¹¹ (Or, if you’re reluctant to say that a mechanical device can make anything, perhaps because making is essentially intentional, then plausibly Goliath was made by you and me collectively, though Lumpl wasn’t; it was made just by me.) According to the pluralist, then, the mere fact that Lumpl and Goliath share the same matter for the same period of time is perfectly compatible with a difference in origin—in one sense of the word, ‘origin’, which I will isolate with the phrase, ‘origin-as-maker’.¹² Generalizing the point, we might say that material identity is compatible with non-modal (that is, categorical) differences.¹³ We thus obtain the categorical contrast principle: (CCP) Sameness of matter is compatible with categorical (that is, non-modal) differences. This principle isn’t restricted to synchronic applications. But not just any categorical difference over time makes for material non-identity. For example, Lumpl might be spherical and then cubical. A change in shape over time is consistent with being Lumpl throughout. So the applicability of CCP in an argument for the non-identity of things at different times requires that one select the right sort of categorical difference. Plausibly, a difference in origin is of the right sort. I believe that TTP and CCP conflict. Our little puzzle about Descartes’s puddle makes the conflict explicit. Have another look at the reasoning under Thesis. The initial piece of wax and the puddle are materially identical. They’re made from the same matter. But, given CCP, they can differ in certain non-modal respects. My story ensures that this possibility is realized in Descartes’s study, and it’s realized in a way that supports non-identity: the piece and the puddle differ with

¹¹ A monist might reasonably challenge this claim; for on her view Lumpl and Goliath are numerically identical. But on what plausible basis might a pluralist reject it? ¹² Evnine (2016) distinguishes between origin-as-matter and origin-as-act. ¹³ Often enough, the term, ‘categorical’, is used for properties whose possession by a thing is determined by the way the thing is or was, as opposed to the way it would or could have been. The various ways the thing would or could have been are its hypothetical properties. The categorical/hypothetical distinction is to modality as the occurrent/non-occurrent and intrinsic/ extrinsic distinctions are, respectively, to time and space (Yablo 1992).

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238   respect to who made them and how they were made. So, anyone inclined to think that Lumpl and Goliath are distinct because of a non-modal difference between them ought to conclude, by parity of reasoning, that the piece and the puddle are distinct too. In fact, the non-modal considerations under Thesis are identical to the non-modal considerations for distinguishing statue and lump that Jeffrey King (2006, p. 1057, n. 52) attributes to Timothy Williamson. I don’t know whether Williamson endorses those considerations, or whether he’s a pluralist, but the parallel is interesting. Furthermore, King indicates that these considerations, which rely on the predicate, ‘___ was made by ___’, are more worrisome for monists than arguments involving ‘well made’, ‘Romanesque’, and ‘represents’. The reason is that they don’t seem to involve any of the familiar sort of inference-undermining phenomena (gradability; conversational implicature; referential opacity) that seem to vitiate the other arguments. Now have another look at the reasoning under Antithesis. The crucial step is the claim that the initial piece of solid wax became the liquified piece of wax, and didn’t cease to exist in the process. To be absolutely explicit, the rationale for this claim is that the transformative process (melting) left intact the cohesion of the underlying matter. But this is merely an application of TTP, with which pluralists justify the nonidentity of Lumpl and Goliath. In our little puzzle, however, consistency requires that pluralists treat TTP as a reason to identify the piece and the puddle. But that yields an explicit contradiction, since we concluded earlier that the piece and the puddle are distinct. Thus, our puzzle brings into conflict the considerations that are cited in support of pluralism. The onus is on the pluralist to explain how her take on Lumpl and Goliath can withstand this conflict. One might think that TTP should be understood, not as a universal generalization, but as a generic truth (e.g., ‘Ravens are black’ (but albinos aren’t); ‘Dogs have four legs’ (but unfortunate specimens don’t); ‘Sea turtles are long-lived’ (though the vast majority die young)). The potential worry is that, if TTP is a generic truth, then it doesn’t necessarily apply to the situation in Descartes’s study, and therefore doesn’t justify the reasoning under Antithesis. I’m confident that our little puzzle can’t be dissolved by appeal to genericity. Generic truths are exception-tolerant generalizations as long

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as the exceptions are somehow abnormal or unsystematic (Bernhard Nickel 2016). Linguists and philosophers disagree about how to properly analyze the relevant notion of normality or systematicity. Is it statistical (Ariel Cohen 1999), normative (Michael Thompson 2008), or nomological?¹⁴ Could it be altogether psychological, and not the sort of phenomenon that calls for a metaphysical account (Sarah-Jane Leslie 2008)? Let the chips fall where they may. There’s nothing abnormal or uncharacteristic about the transformative process that occurs in Descartes’s study. What I’ve described is an ordinary case of melting, relevantly like any other boring case of melting. If my little story presents an exception to TTP, it’s not the benign sort which is compatible with generic truth; it must be a genuine counterexample. So, for my purposes, it doesn’t matter whether TTP is generic or universal in form.

II. One might be tempted to think that our little puzzle has an easy fix—so easy, in fact, that it might seem perverse for me to have presented the problem with all of the fuss and fanfare of a genuine philosophical conundrum. After all, it seems as though the conflict between TTP and CCP would be resolved if the pluralist simply rejected one kind of consideration in favor of non-identity. She might choose to embrace TTP and the modal contrast between Lumpl and Goliath but abandon CCP and the categorical differences between them. Or she might reject the modal contrast and decide that Lumpl and Goliath are distinct because of their inevitable categorical differences. Problem solved, right? Wrong. TTP and CCP derive an equal measure of support from the same source, namely, pre-theoretical intuition.¹⁵ Neither exerts more intuitive pressure than the other, and each derives whatever epistemic ¹⁴ Thompson’s view is not an analysis of generic statements per se, but a subclass of them that he calls “natural-historical judgments”. It might be that genericity has a normative ground in some cases, a statistical ground in others, etc. However, if the phenomenon is too disjunctive, it may be less misleading to say there’s no such thing as genericity. ¹⁵ See Fine (2003, p. 207; 2006, pp. 1069–70), Korman (2015a, pp. 204–5, n. 4), and Yablo (1987; 2015), all of whom accept the modal and non-modal considerations in favor of pluralism, emphasizing their intuitive appeal as weighty grounds for acceptance. Not one of these authors,

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240   authority it has from the intuitive pressure it exerts. Without some further story, plumping for one principle over the other would be entirely unmotivated and objectionably arbitrary.¹⁶ To be clear, my point isn’t that a plausible story can’t be told (although I’m highly skeptical, for reasons that will emerge momentarily). What I’m claiming, rather, is that one needs to be told. Otherwise, those of us who agree with Gibbard could reasonably reject both TTP and CCP, thus blocking the pluralist’s challenge in one swift stroke. There’s no advantage to being the more intuitive view if the intuitions promoting the view are both contradictory and on an equal footing, epistemically speaking.¹⁷ as far as I’m aware, gives any indication that they take one kind of intuitive consideration to be more or less epistemically secure than the other. ¹⁶ Furthermore, why would it be unreasonable to draw a much stronger conclusion than the one I’ve been suggesting? In particular, what would be wrong with thinking that the little puzzle about Descartes’s puddle undermines the reliability of our intuitions regarding the nature of material objects? There’s a pleasing irony in this potential conclusion, since Descartes himself relied on a series of intuitive judgments about his wax to motivate the idea that the nature of matter is extension. ¹⁷ It may be illuminating to compare the structure of my argument with Jennifer Saul’s defense of a naive Russellian account of content, according to which (i) and (ii) express the very same proposition (see Saul 1997; 2007). (i) Hammurabi believes that Hesperus is visible in the evening. (ii) Hammurabi believes that Phosphorus is visible in the evening. This commitment of the naive Russellian view is often considered to be highly implausible. To address the worry, naive Russellians have developed sophisticated ways of explaining how an utterance of (ii), on a particular occasion, might nevertheless convey different information from an utterance of (i) on that very same occasion, and how, on account of that difference in conveyed information, uninitiated language users intuitively (though incorrectly) judge that (i) and (ii) express different propositions (see Nathan Salmon 1986 and Scott Soames 2002). Many theorists take the naive Russellian’s response to be inadequate (see, among others, William Taschek 1992). It is here that Saul’s defense is relevant. Abbreviating quite a lot, Saul argues that no one is in a better theoretical position than the Russellian, because we’re all forced to tell a similar story—one in terms of a difference in conveyed information rather than a difference in proposition expressed—about (iii) and (iv): (iii) Clark Kent went into the phone booth, and Superman came out. (iv) Superman went into the phone booth, and Superman came out. Saul’s point is that if we’re all in the uncomfortable position of having to tell such a story about (iii) and (iv) anyway, then it isn’t so bad to extend that story to (i) and (ii). (This isn’t quite right, in light of Saul (2007, ch. 6), but the subtlety can be ignored for the sake of brevity.) The upshot is supposed to be that considerations about the substitution of coreferential terms, contrary to popular belief, actually support naive Russellianism. Similarly, I’ve been arguing that we’re all in the position of having to deal with the conflict between Thesis and Antithesis, which is structurally parallel to, and indeed motivated by, the intuitive considerations that are supposed to justify pluralism. Therefore, pluralists can’t claim to be better off, dialectically, than their rivals. They are, in fact, worse off.

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In a brief exchange about our little puzzle, one respondent drew my attention to this passage from John Hawthorne (2008, p. 269), suggesting that it provides a principled reason to epistemically privilege the modal considerations over the non-modal, and thus to opt for TTP over CCP: Turning to an example from the literature, it is true that ‘That statue is well made’ sounds acceptable while ‘That lump is well made’ sounds awkward (even in a case where the statue and lump come into existence at the same time). But notice that there is a similar contrast between ‘That actress is well trained’ and ‘That person is well trained’, but we hardly wish to infer from this that the actress is not identical to the person. The Leibniz Law arguments from modal predication are much more powerful. Here we seem to have positive and conflicting modal intuitions about the statue/lump . . . , even in a case where their paths actually coincide. (Notice that while such sentences as ‘The statue but not the lump is well made’ . . . merely sound awkward and strange, the sentence ‘The lump but not the statue could have survived crushing’ sounds straightforwardly true. This is essentially why the Leibniz Law arguments from modal predication are more promising.)

I’d like to make three points in response. First, it’s unclear (at least initially) whether Hawthorne is making a psychological/dialectical observation or an epistemological one. What is clear is that Hawthorne takes one particular pair of non-modal considerations in support of pluralism to be “inconclusive”, less “powerful”, and less “promising” than arguments that rely on modal predication. I find myself wanting to ask, less promising for what, exactly? Presumably, for convincing the unconvinced. That sounds like a dialectical point to me. But a consideration in favor of some doctrine can be regarded as inconclusive, or less powerful, or less promising in virtue of the rules of Much more could be said along these lines, and saying it, I believe, would only support my central argument. But it would require that we explore the literature on belief and belief ascription in the sort of detail that would only be of interest to an even narrower readership. For a discussion that’s heavily influenced my thinking on these matters, see Richard Heck (2014).

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242   disputation and the background commitments of one’s interlocutor, even though the consideration isn’t in any worse epistemic standing for it. Recall, for example, Moore’s response to the Cartesian skeptic: here is a hand, and here is another. Moore’s argument seems not at all promising, from a dialectical point of view. Certainly no skeptic is likely to find it conclusive (nor, for that matter, are some non-skeptics) but the relevant consideration is, I believe, pretty well justified. It’s epistemic standing, in other words, isn’t diminished by its dialectical inefficacy. Are we to interpret Hawthorne as suggesting that the dialectical inefficacy of a certain non-modal predication is grounds for treating the corresponding claim as epistemically worse off? That strikes me as an uncharitable interpretation—uncharitable because obviously fallacious. If Hawthorne’s point is dialectical and not epistemological, then it doesn’t bear on my rationale for thinking that TTP and CCP are epistemically on a par. But let’s assume that Hawthorne’s point in the passage I’ve quoted is epistemological and not merely dialectical. Still, it seems unable to provide an easy fix to our little puzzle. The reason is that—and here I’m offering my second response—even if Hawthorne is right about ‘well made’, he hasn’t offered any reason at all to think that other non-modal predications are bound to be similarly problematic. It seems to me that the deck-stacking argument I presented earlier, in terms of the predicate, ‘___ is made by ___’, can’t be dismissed as quickly as Hawthorne dismisses the argument in terms of ‘well made’. And I’m not alone here; recall King (2006, p. 1057, n. 52), who shares Hawthorne’s opinion about ‘well made’ but who explicitly acknowledges the force of arguments based on discernibility with respect to origin-as-maker (crediting Timothy Williamson with the observation). In any event, my point about the epistemic parity of TTP and CCP boils down to this: the claim that Lumpl would, but Goliath wouldn’t, survive melting and the claim that Lumpl and Goliath differ with respect to origin-as-maker are equally well justified. And nothing that Hawthorne (or anyone else I’m familiar with) has said challenges that. Third, the modal considerations in support of the pluralist’s take on Lumpl and Goliath are, in a certain respect, weaker than the non-modal considerations. For the proper interpretation of modal speech and thought is no less controversial than the issue between the pluralist and

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her rival. And there are sophisticated techniques for analyzing modal claims so that they cohere with the identity of Lumpl and Goliath. Hawthorne criticizes these interpretive techniques because they rely on forms of context dependence that aren’t paradigmatic. He recommends (and I agree) that these techniques are better thought of as “rational reconstructions” of our ordinary modal speech and thought, but he wonders “exactly what the motivation is supposed to be for the recommended departures from ordinary, natural ways of thinking. The burden certainly seems to be squarely on the [monist] to find a powerful motivation for it” (2008, p. 269). Fair enough; I’ll take up this burden in section IV. What I want to do here is simply observe that the interpretive strategies that straightforwardly apply to the modal considerations seem not at all to get a grip on at least some of the non-modal considerations. Kit Fine (2003) argues for this point at great length. To that extent, then, some of the non-modal considerations are more powerful. Each kind of consideration has its strength and its weakness; I doubt one can definitively say whether one kind of consideration is in better epistemic standing than the other.¹⁸ I’ve been criticizing a response to our little puzzle that rejects CCP but maintains TTP. I know of no one who has rejected TTP while affirming CCP, but let’s entertain the possibility for the sake of argument. A pluralist who opts for this response would no doubt want to maintain that Lumpl and Goliath are modally discernible, just not in the sort of way that an application of TTP would imply. Specifically, she would think that Lumpl, like Goliath, would not have survived melting, squashing, etc., but Lumpl could have existed without Goliath existing. This position strikes me as unstable. What justifies the thought that Lumpl could have existed without Goliath existing? Whatever the story may be—intuition? brute plausibility? conceivability? the intelligibility of certain why-questions?—a parallel story will surely apply to the thought that Lumpl, but not Goliath, would survive cohesion-preserving transformations. The challenge for the pluralist is to tell a story that doesn’t

¹⁸ Even if Hawthorne is right, so many prominent pluralists endorse both modal and nonmodal considerations that it would still be interesting to learn that there isn’t enough room in logical space for this position to be occupied. For references, see n. 15.

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244   have this consequence—that doesn’t treat the two sorts of modal judgments as epistemically on a par—but this challenge is of a piece with the original problem: to tell a story that explains why TTP and CCP don’t enjoy the same epistemic standing. The new challenge is no less challenging, it seems to me, than the original one. And without an adequate response, the resulting position seems no less stipulative than plumping for one principle with no explanation at all. In short, our hypothetical critic seems not to have made any progress by opting for CCP over TTP. The temptation to think that our little puzzle has a quick and easy fix is, I have found, hard to shake. Another way it manifests itself is in the thought, shared by some respondents, that the situation in Descartes’s study can be given a consistent description. The details of the description don’t really matter for now—though I myself offered one in the opening paragraph of this essay, and I’ll consider another one in section III. What I want to emphasize, rather, is a valuable lesson that Kripke taught us in the context of a different puzzle, one about his fictional character, Pierre, and whether he does or doesn’t believe that London is pretty. I’ve taken the liberty of streamlining Kripke’s discussion, and substituting expressions that make it bear on our subject: It is no solution in itself to observe that some other terminology, which evades the [puzzle] may be sufficient to state all the relevant facts. I am fully aware that complete and straightforward descriptions of the situation are possible and that in this sense there is no paradox. . . . But none of this answers the original question. . . . As in the case of the logical paradoxes, the present puzzle presents us with a problem for customarily accepted principles and a challenge to formulate an acceptable set of principles that does not lead to paradox, is intuitively sound, and supports the inferences [pluralists] usually make. Such a challenge cannot be met simply by a description of [the] situation that evades the question. (Kripke 1979, p. 147)

A consistent description of puzzling Pierre’s situation doesn’t, by itself, answer the question, does he or doesn’t he believe that London is pretty?

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A fortiori, it doesn’t answer the question in such a way as to reconcile the interpretive principles that Kripke relies on to generate an inconsistency. And, as long as it fails to do that, it fails to properly engage with Kripke’s puzzle. Similarly, a consistent description of the situation in Descartes’s study doesn’t, by itself, answer the question, is the initial piece of wax identical to the wax puddle, or isn’t it? A fortiori, it doesn’t answer the question in such a way as to reconcile the principles that I rely on to generate an inconsistency. As long as it fails to do that, it fails to properly engage with my little puzzle. We’ve looked at two allegedly quick and easy fixes. The first was premised on the suggestion that one of the two puzzling principles can simply be given up; the second on the thought that there are consistent ways of describing Descartes’s study. I’ll have more to say about this second strategy in section III, where I’ll evaluate one detailed way of redescribing the situation. I’d like to close this section, however, by looking at a third consideration that might tempt one to think there’s a quick fix, and hence no genuine puzzle for pluralists. One might feel inclined to respond by pointing out that there are other ways of arguing for pluralism which don’t require appealing to TTP. (I’ve encountered this reaction more than once.) For example, one might say that Lumpl could have existed even if Goliath hadn’t.¹⁹ I wonder whether this thought really is all that different from TTP. After all, the thought is just an instance of the more general principle that pieces of matter could have existed even if the artifacts they constitute hadn’t. This more general principle specifies an existence condition, just as TTP specifies an identity condition. It may be that material existence and identity are grounded in a common source, and that the clash between TTP and CCP is just as much a challenge for that more basic source from which TTP

¹⁹ One respondent suggested that the pluralist should rely on the thought that Goliath can survive the destruction of smallish parts, but the mereological sum of clay particles spatially coincident with it can’t. This is, of course, the sort of argument with which Thomson (1983) begins her classic discussion. However, this response misses the point. Gibbard and Lewis are moderate monists; they already grant that Goliath and the mereological sum of clay particles with which it’s spatially coincident are numerically distinct. Thomson’s argument is no good against them. After all, Goliath and that mereological sum of clay particles aren’t temporally coincident. What’s at issue is whether one can argue against Gibbard and Lewis without relying on TTP.

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246   and the existential principle derive their truth or credibility. In any case, we needn’t dwell on this possibility, because there’s a deeper problem for the suggestion, namely, that it’s completely beside the point. Just imagine a similar response to Kripke’s puzzle about belief: but the Fregean has other arguments for her view—arguments that are independent of substitution in belief contexts.²⁰ Okay, sure, but does Pierre, or doesn’t he, believe that London is pretty? Furthermore, are the interpretive principles (translation and disquotation) that give the question urgency coherent or not? The imagined response bears not at all on the central question of Kripke’s puzzle. Similarly, the response we’re now entertaining on behalf of the pluralist bears not at all on the central questions of our little puzzle. Is the initial piece of solid wax identical to the wax puddle, or isn’t it? Are TTP and CCP compatible or aren’t they? The response is, therefore, entirely unhelpful.²¹

²⁰ Recall that one of Kripke’s stated aims in his classic discussion is to demonstrate that substitution arguments in terms of belief provide no support for Fregean theories of content. There are deeper problems about belief and belief ascription that everyone must face, and they’re independent of the debate between Fregeanism and Russellianism. ²¹ I can think of one more response on behalf of the pluralist. I’m reluctant to call this a “quick and easy” fix, since it relies on a highly controversial theory that not all of the pluralists within the blast zone of my bomb would be happy to grant, so I relegate discussion of it to this footnote. Let a temporal profile be a function from times to occupied regions of space. Some pluralists believe that every temporal profile represents an existing object. On this view, there are many more things coincident with Goliath than just Lumpl. There’s the statue-like object, Lefty, that ceases to exist when moved ever so slightly to the right but persists when moved ever so slightly to the left. There’s also the statue-like object, Righty, that ceases to exist when moved ever so slightly to the left but persists when moved ever so slightly to the right. (There’s a temporal profile that represents both Lefty and Righty.) Pluralists of this sort believe in a plenitude of entities. Recently, Shamik Dasgupta (2018) has relied on this plenitude to address the nonidentity problem in ethics. His proposal assumes that, sometimes, unbeknownst to us, our referring terms pick out some of the temporally fragile objects that we ordinarily ignore. I can imagine someone tempted by a similar thought vis-à-vis our little puzzle. Perhaps she might say that the phrase, ‘the initial piece of wax’, undergoes a kind of reference shift at some point that we fail to detect. Its referent in Thesis is something relatively fragile; its referent in Antithesis is something comparatively robust. So the identity that Thesis denies is compatible with the identity that Antithesis affirms. If this proposal is sound, then why can’t I detect a true reading of the sentence, ‘The initial piece of solid wax is not identical to the initial piece of solid wax’? Furthermore, many sentences beginning with ‘the initial piece of solid wax’ would be semantically defective (either “gappy” or false), because the uniqueness presupposition would fail. But no such sentence, evaluated with respect to Descartes’s study, sounds as bad to me as, say, ‘The present king of France is bald’. In other words, the plenitude-lover is committed to there being a form of presupposition failure that’s not at all paradigmatic.

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III. Three points are worth bearing in mind. First, my puzzle about Descartes’s puddle is not a mere repackaging of Gibbard’s example involving Lumpl and Goliath. The pair of considerations that motivate the pluralist’s take on Gibbard’s case is the very source of my puzzle. Second, neither one of these considerations enjoys more plausibility, or greater intuitive appeal, than the other. They are, in short, epistemically equivalent. Rejecting one while holding onto the other would be an act of desperation. Finally, however the pluralist chooses to resolve this new little puzzle, her story had better satisfy an important dialectical constraint: it had better not undercut the rationale for treating Lumpl and Goliath as distinct. It might help to linger on this last point for a bit. I’ll clarify what I mean here by considering a potential response to the puzzle that, I believe, violates the constraint. One common reaction to the puzzle is that it somehow equivocates: Thesis encourages us to conceive of the piece and the puddle as artifacts, and, qua artifacts, they are discernible; Antithesis invites us to think of them as quantities of matter, or mere occupants of spacetime, and, qua matter, the piece literally became the puddle. If TTP is understood so as to apply only to quantities of matter, and CCP is interpreted to be about countable objects, such as artifacts, we obtain the beginnings of a story, or perhaps a redescription of my earlier story, that appears to resolve the inconsistency—a story or redescription according to which trouble arises from a failure to detect the underlying equivocation. I say we have the “beginnings” of an account because the basic idea— that our puzzle exploits an illicit conflation—can, I believe, be worked out in a number of different ways. One way of working out the strategy involves the postulation of an ambiguity in the phrase, ‘piece of wax’. Respondents often rely on this thought, but it seems to me to have very little going for it. It fails all of the tests for ambiguity that I’m aware of (Arnold Zwicky and Jerrold Sadock 1975; Adam Sennet 2016). For instance, the sentence, ‘Pieces of wax are not pieces of wax’, has no coherent reading. If ‘piece of wax’ were ambiguous, one would expect the sentence to have a coherent reading, just as the sentence, ‘Bats are not bats’, has a coherent reading. Of course, there may be subtle ambiguities

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248   that escape detection even by our most reliable linguistic tests. But these ambiguities aren’t paradigmatic, and the pressure to acknowledge them should, I believe, come from theoretical demands internal to the empirical study of natural language. At any rate, pluralists often criticize monists for positing forms of context sensitivity or referential opacity that aren’t paradigmatic (Fine 2003; Hawthorne 2008; Korman 2015a). It would be unprincipled to now rely on a form of ambiguity that displays the same theoretical shortcoming. David Lewis considers a similar strategy in a different context. His criticism of it is relevant here: “I don’t object to the strategy of claiming ambiguity. As you’ll see, I shall defend a version of it. But it’s not plausible to cook up an ambiguity ad hoc . . . . It would be better to find a widespread sort of ambiguity . . . and show that it will solve our problem” (1980, p. 230).²² One advantage of the monist-friendly response to the case of Lumpl and Goliath is that it identifies a widespread sort of ²² Another way of working out the equivocation strategy in detail yields a view called perspectival hylomorphism (Thomas Sattig 2015). This view takes to heart Obi-Wan Kenobi’s advice to a certain boy from Tatooine: “Luke, you’re going to find that many of the truths we cling to depend greatly on our own point of view.” Abbreviating quite a lot, the view has two components. The first is that ordinary material objects are hylomorphic wholes (in some sense, compounds of matter and form). Second, speakers and thinkers employ two different “modes of predication” when speaking and thinking about ordinary material objects—a material mode of predication to speak and think about the material component of the hylomorphic whole, and a formal mode of predication to speak and think about the formal component. This view is noteworthy because it’s tailor-made for pluralists. For a nice criticism of it, however, see Korman (2015b), who persuasively argues that there’s no semantic or cognitive evidence for positing distinct modes of predication, and that even if there were distinct modes of predication, our actual intuitions can’t be explained by reference to them. A more radical way of implementing the equivocation strategy involves the idea of sameness/ difference in a respect: the piece is the same mass of wax as the puddle (Antithesis), but not the same artifact (Thesis); the appearance of a genuine puzzle is due to an illicit conflation of logically different identity claims. This implementation is no solution at all if the phrase, ‘same F ’, is analyzable in terms of predication, absolute identity, and conjunction. For then we would obtain Mass-of-Wax(the piece) & Mass-of-Wax(the puddle) & the piece = the puddle & Artifact(the piece) & Artifact(the puddle) & the piece ≠ the puddle which is contradictory. The phrase, ‘same F’, must be understood as an unanalyzable unit, which carries a commitment to the doctrine of relative identity (Peter Geach 1962). There are numerous technical problems with this doctrine. For a survey of them, see Hawthorne (2003), in which the author concludes that “it is no mere artefact of philosophical fashion that Geach’s relative identity approach has few adherents” (p. 23). But even if we conceive of these problems as research opportunities, and take the idea of relative identity to be understood well enough for use, the use of it here doesn’t make much dialectical sense for a pluralist. There isn’t any reason to apply it here while withholding its application from Lumpl and Goliath, in which case the whole motivation for pluralism would be undermined. More on this point to come.

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meaning variation (call it “ambiguity” if you like) in virtue of which the equivocation strategy can be more plausibly implemented. According to Gibbard and Fara, de re modal predications are, in general, sortalrelative; according to Lewis, the evaluation of a de re modal claim is always sensitive to a counterpart relation. Zoltán Gendler Szabó (2003) takes this sort of relativity a step further and suggests (albeit tentatively, it seems) that even non-modal predications are implicitly qualified, or semantically relativized, in a way that’s compatible with Lumpl and Goliath’s identity. Pluralists are reluctant to accept this widespread sort of meaning variation in their treatment of Lumpl and Goliath. They view the move as empirically objectionable,²³ and dialectically confused.²⁴ But if, in light of our new puzzle, a pluralist were now inclined to accept it, then the rationale for her treatment of Lumpl and Goliath would be undermined. For why should the pluralist not just embrace this reliance on sortal-sensitive evaluation a step sooner in the dialectic, thus taking Lumpl and Goliath to be identical, and the considerations that appear to militate against the identity to involve a similar kind of equivocation? If the pluralist concedes that predication is sensitive to a context-variable sortal or counterpart relation, which neutralizes the apparent tension between TTP and CCP, then how confident can she now be about her earlier assessment of Lumpl and Goliath? What reason has she to think that her take on Gibbard’s case isn’t similarly based on equivocation? An adequate solution to the initial puzzle shouldn’t be self-undermining in this way. In particular, it shouldn’t lend credence to the charge of tu quoque.

IV. So far, I’ve been arguing for the dialectical significance of our little puzzle. But I’ve also claimed that it has theoretical import. We’re now in a position to say what the latter consists in.

²³ That is, it yields false semantic predictions (Hawthorne 2008; Korman 2015a, pp. 207–9). ²⁴ “Philosophers of a monist persuasion have been content to show that we might talk as if their views were correct rather than that we actually do so talk” (Fine 2003, p. 202).

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250   Both TTP and CCP are modal claims, and the initial plausibility of each is revealed by ordinary modal thought about particular cases. Thus, our little puzzle entails that some of our ordinary modal judgments are inconsistent. What are we to make of this? I think it’s reasonable to attribute the inconsistency we’ve uncovered to the inherently undisciplined character of ordinary modal thought. One response, then, would be to eschew modal concepts altogether. I regard this option as a massive overreaction. A more plausible response is to simply abandon the presumption against “revisionary” ways of understanding ordinary modal thought and speech. In particular, we should be open to interpretive projects, such as the sortal-relative semantics of Gibbard (1975) and Fara (2012), and the context-dependent counterpart theory of Lewis (1971), which arguably fail to accommodate certain ordinary modal claims but which impose a kind of rigorous flexibility on our modal thoughts and statements so as to render them coherent. Of course, this sort of “revisionary” approach is subject to an important challenge that deserves a response: Why isn’t a revisionary take on ordinary modalizing simply eliminativism in sheep’s clothing? Jennifer Hornsby (2001) raises a version of this challenge quite explicitly. To help fix ideas and facilitate a clearer exposition of the issues, it will be useful to sharpen our formulation of the potential threat: (1) Folk modalizing (the application of modal concepts and terms in ordinary speech and thought) is pluralistic about material reality.²⁵ (2) So, whatever else they may be, analyses that purport to have modality as their subject, but which are not pluralistic, can’t be theories of what folk-modal concepts and terms represent.²⁶ (3) Modality is what folk-modal concepts and terms represent.

²⁵ Kit Fine (2003), John Hawthorne (2008), Eli Hirsch (2003), and Daniel Korman (2015a) defend this claim at great length. ²⁶ What justifies the inference from (1) to (2)? The thought is that any account of modality that isn’t pluralistic will have to abandon too many platitudes about what would be, or what might have been, for the subject matter to be preserved. Taken individually, no one platitude makes the difference. I’m imagining an interlocutor who’s sufficiently skeptical of the analyticsynthetic distinction, and who’s savvy enough to acknowledge that the reference-fixing, or “metasemantic”, story here is bound to be messy.

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(4) Therefore, analyses that purport to have modality as their subject, but which are not pluralistic, can’t be theories of modality. I don’t know of any explicit formulation of this exact argument, but it strikes me as a more perspicuous statement of the worry that Kit Fine was voicing when he wrote, “Philosophers of a monist persuasion have been content to show that we might talk as if their views were correct rather than that we actually do so talk” (2003, p. 202). If Fine had something like the argument above in mind, then it would be clear why he takes the interpretive frameworks of Gibbard, Lewis, and Fara to be languages that differ from our actual modal language: these frameworks don’t represent modality; they have an altogether different subject matter.²⁷ I’m inclined to think that the inference from (1) to (2) is really quite bad and ought to be resisted. (Correlatively, any metasemantic theory that supports the inference from (1) to (2) is false.) The best way of demonstrating the mistake is by considering a parallel argument that’s more clearly problematic in the relevant respect.²⁸ Imagine someone who uncritically assumes that terms like ‘parent’, ‘mother’, ‘father’, ‘son’, ‘daughter’, etc., and the corresponding concepts, denote biological categories—they represent relationships that are grounded in the inheritance of genes. It wouldn’t surprise me if many people actually believe something like this, nor would it surprise me if this sort of assumption controls patterns of use in speech and thought. For the sake of argument, let’s say the assumption is widely shared and normatively influential. No doubt many parents of adopted children are ²⁷ I think Nathan Salmon’s harsh review of Lewis (1986) can be interpreted in the same way, though I won’t pause to do so. See Salmon (1988, pp. 127–8). ²⁸ Let me acknowledge upfront that there are important substantive differences between the central argument about modality, (1)–(4), and the analogue argument about family kinds, (5)–(8). For example, I’m urging us to reject the former on broadly logical grounds, whereas the latter is problematic for ethical reasons. Furthermore, there seems to be a deeper tension between monism and pluralism than there is between biological and constructionist theories of family. (One can imagine a kind of contextualism to resolve the latter; in fact, at various points Haslanger seems sympathetic to such context sensitivity. Her view is certainly more nuanced than my brief characterization below suggests.) What matters for my purpose is not that there be a substantive analogy between the two arguments, but that there be a formal analogy between them, and that they both fail because the reference of our terms and concepts is partially determined by facts about the purpose for which we use them.

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252   made to feel awkward and uncomfortable by such assumptions on a regular basis.²⁹ So there’s an ethical reason to disrupt or critique the assumption. But suppose the person we’re imagining tried to convince us that any modification to a biological understanding of folk family concepts and terms would merely change the subject. Coincidentally, this fictional character’s imaginary argument looks rather familiar: (5) Folk family classifications are biological. (6) So, analyses that purport to have parenthood, motherhood, etc. as their subject, but which don’t represent these relationships as being grounded in the inheritance of genes, can’t be theories of what folk family classifications represent. (7) Parenthood, motherhood, etc. are what folk family classifications represent. (8) Therefore, analyses that purport to have parenthood, motherhood, etc. as their subject, but which don’t represent these relationships as being grounded in the inheritance of genes, can’t be theories of parenthood, motherhood, etc. Given the prevalence of the belief that terms like ‘parent’, ‘mother’, ‘father’, ‘son’, ‘daughter’, etc., and the corresponding concepts, denote biological categories, and given the influence this belief has on patterns of use, denying (5) seems gratuitous. At any rate, we’re supposing for the sake of argument that (5) is true. Moreover, (7) seems too plausible to credibly challenge. But—and here’s the important point I’ve been building up to—we certainly aren’t forced to accept an ethically objectionable understanding of family classifications. The reason, as Sally Haslanger (2006; 2012) has convincingly argued at length, is that enough of the legitimate purposes for which we encode family classifications in thought and language are served by an understanding of parenthood, etc. according to which parents are either primary caregivers or immediate progenitors that a change in one’s understanding which represented parenthood

²⁹ “ . . . my own experience as an adoptive mother has convinced me that at least in many contexts the dominant understanding of ‘parent’ frames it as a biological notion” (Haslanger 2012, p. 389–90).

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as (at least partially) a social classification would still deserve to be an analysis of parenthood. So, by merely acknowledging that we presently live in an unfortunate time when a certain assumption about the biological basis of family uncritically regulates the use of family classifications, we aren’t committed to viewing alternative analyses of family notions as failing to be about what our folk classifications represent. In short, (5) doesn’t entail (6). Here the rationale for treating an analysis of parenthood which fails to vindicate certain folk claims and patterns of reasoning as nevertheless having the same subject matter as the folk conception is based on the attractive thought that such an analysis would still serve the point of family classifications. My suggestion is that the inference from (1) to (2) is problematic for a similar reason. Analyses of modal thought and language that are pluralistic may superficially appear to do better at accommodating a wider range of individual judgments and inferences,³⁰ but if these patterns of thought turn out to be inconsistent, as I’ve argued they are, then such analyses do worse relative to perhaps the most important purpose of representation: truth and knowledge. Coherence is a minimal requirement for truth and knowledge, and paradigm forms of representation (belief and assertion) aim at these values. It is in relation to truth and knowledge that the whole point of belief and assertion is defined. Insofar as “revisionary” analyses render folk modalizing coherent, and still enable us to achieve other legitimate purposes for which we modalize, they have a better claim to being about what folk-modal representations are about. What might some of these other purposes be? Foremost in my mind is an account of informational content, linguistic communication, and rational agency (Robert Stalnaker 1984; Frank Jackson 1994). Modal notions figure indispensably in such theorizing. Insofar as an understanding of necessity and possibility in terms of context-dependent counterpart theory, for instance, can satisfy these theoretical purposes, then it has a good claim to be a theory of modality. It isn’t merely changing the subject. It would require a substantial amount of independent argumentation to convince me that an understanding of necessity ³⁰ For arguments to this effect, see Fine (2003), Hawthorne (2008), Hirsch (2003), and Korman (2015a).

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254   and possibility in terms of counterpart theory is incapable of satisfying these broader theoretical aims. I’m not saying it can’t be done (but I will register my deep skepticism on this score); I’m saying that unless it’s done, one can reasonably reject (2) while granting (1). And, of course, if such an argument were to succeed, then (1)–(4) would be otiose.

V. You may have noticed that I haven’t yet directly answered the question I keep incessantly asking: Is the initial piece of solid wax identical to the wax puddle, or isn’t it? This is no accident. Philosophical problems are often more irritating than even the most promising responses are comforting. The problems have a way of lingering, dulled somewhat by the therapeutic treatment of an illuminating theory, but perennially threatening to flare up if the treatment is scrutinized. Why risk almost inevitable failure? Be that as it may. I think I’ve said enough to hint at my preferred response, at least in outline: some form of monism, à la Gibbard, Lewis, and Fara, is probably right, and it ought to be applied here. Of course, the monist’s treatment of our little puzzle doesn’t involve any special pleading; it applies to both the piece and the puddle and to Lumpl and Goliath. I take this to be a theoretical virtue—one that pluralism lacks; for we ought to prefer theories that treat likes as likes. And it’s my hope that the foregoing discussion has convinced you that the two issues—my puzzle and Gibbard’s case—ought to be treated as likes. Furthermore, if the monist’s treatment of these cases is based on considerations that are largely independent of the issues in this paper—for example, if she’s a sortalist or counterpart theorist as part of a larger project to, say, vindicate a thoroughgoing Humeanism—then her application of the view doesn’t merely accommodate our puzzle and Gibbard’s case; it predicts the monist-friendly position. And predictive theories have an epistemic edge over theories that merely accommodate the phenomena.³¹ This means the ³¹ See Roger White (2003) on the epistemic advantage of prediction over mere accommodation.

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Ludovician package of metaphysical and semantic views derives an extra bit of support for its ability to handle our puzzle, since it’s ultimately motivated by a Humean outlook (Lewis 1983). But I prefer to think in Ludovician terms primarily for autobiographical reasons (it was how I was raised), not because I’m a Humean. So I’ll proceed accordingly. CCP is, at bottom, a claim about possibility: it’s possible that material objects with the same matter are categorically different. Insofar as this is true, it means that there’s a way the world might have been such that if the world were that way, then artifactual counterparts of actual material objects would be categorically different despite having the same matter. Artifactual counterparts are objects that resemble each other along dimensions of resemblance appropriate for comparing artifacts (aesthetically, functionally, representationally, etc.). Applying this interpretation of CCP to the situation in Descartes’s study, we arrive at the thought that, in the relevant way our world might have been (namely, the way it actually is), the piece and the puddle have artifactual counterparts (themselves) that differ with respect to origin-as-maker. Now, interpreting TTP along similar lines, we obtain the principle that every way the world might have been (quantifier suitably restricted) is such that if the world were that way, then material counterparts of actual material objects would undergo a cohesion-preserving transformation and would survive. Material counterparts are objects that resemble each other along dimensions of resemblance appropriate for comparing matter (physically, mereologically, quantitatively, etc.). Applying this interpretation to Descartes’s study, we’re able to say that in all of the relevant ways our world might have been, the piece and the puddle have one and the same material counterpart: a single measurable quantity or mass of wax, conceived not as a countable individual but as a mere occupant of spacetime, which is identical neither to the artifactual counterpart of the piece in those ways nor to the artifactual counterpart of the puddle.³² (This is one of the chief virtues of counterpart theory: the counterpart ³² I’m assuming a “multiple-category” theory of masses (Dean Zimmerman 1995). If this sort of theory turns out to be false, then my little puzzle wouldn’t be so little. It would pose a serious challenge to monists, as well, and we would all be forced to make some hard decisions. It’s for this reason that I began the section with a pessimistic tone: I have no independent reason to be confident in the ultimate success of a multiple-category conception of masses.

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256   relation isn’t transitive.) And this mass of wax does persist through the melting. One more question deserves our attention. The story I just told grants that the piece and the puddle are distinct on the basis of a difference with respect to origin-as-maker. But the same kinds of considerations are at play in my reimagining of Gibbard’s example about Lumpl and Goliath—what I earlier advertised as an argument that “stacks the deck” in favor of pluralism. How can a monist consistently say that, in our puzzle, these considerations entail non-identity, but that, in our reimagining of Gibbard’s example, considerations of the same kind don’t? Am I not more or less contradicting myself? No, not if we extend counterpart theory to tensed, as well as modal, thought and speech (Hugh Chandler 1971). Consider, first, the claim that Goliath was made by The Lumper and The Shaper collectively. This means, roughly, that there’s a time, t, earlier than the present such that The Lumper and The Shaper collectively make the artifactual counterpart of Goliath at t. But the artifactual counterpart of Goliath at t just is the artifactual counterpart of Lumpl at t. (I am, after all, an avowed monist; I believe that Lumpl is Goliath.) So it mustn’t be relative to the artifactual counterpart of Lumpl/Goliath that the claim, ‘Lumpl wasn’t made by The Lumper and The Shaper collectively’, is evaluated; for if it were, the claim would be false. But it isn’t false; it’s true. The claim, ‘Lumpl wasn’t made by The Lumper and The Shaper collectively’, must therefore be evaluated relative to some other counterpart, namely, the material counterpart of Lumpl/Goliath, which is just some mass of clay, not a countable individual. And it’s true of the material counterpart of Lumpl/Goliath—a certain mass of clay—that, at t, The Lumper alone makes it. The difference, then, between the situation in Descartes’s study and our reimagining of Gibbard’s example is that, in the former, there are two artifactual counterparts relative to which the relevant claims are evaluated and, in the latter, only one relative to which the corresponding claims are evaluated. My preferred response to our little puzzle about the piece and the puddle ends here. If I had a story to tell on behalf of the pluralist, believe me, I would tell it. But I don’t. So I invite others to do better than I’m able.

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Acknowledgments For their encouraging feedback and constructive criticism, I’m grateful to Karen Bennett, Min Buchanan, Aidan Gray, Mahmoud Morvarid, Bernhard Nickel, Will Small, Dean Zimmerman, and the anonymous reviewers for this volume of Oxford Studies in Metaphysics. I’m especially grateful to Min for the look of incredulity that prompted me to write n. 1.

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9 Advanced Temporalizing Daniel Deasy

0. Introduction There is a widespread assumption that B-theorists—according to whom there is no fundamental distinction between present and non-present times—should interpret tense operators such as ‘It was the case that’ and ‘It will be the case five minutes hence that’ as implicit quantifierrestrictors, so that (for example) an utterance at the present time n of the sentence ‘It was the case that there are dinosaurs’ is true just in case there are dinosaurs located at some time t earlier than n. However, it is easy to show that this interpretation of the tense operators causes problems for B-theorists when combined with certain other natural B-theoretic commitments. In this paper, I argue that a good way for B-theorists to avoid these problems is to treat the tense operators as redundant when the sentences in their scope are qualitative—that is, not about any particular individual(s). The paper is structured as follows: in section 1, I describe the B-theory. In section 2, I show how the standard interpretation of the tense operators as quantifier-restrictors causes problems for B-theorists. I also describe the well-known analogous problem for Modal Realists, according to whom there is no fundamental distinction between actual and merely possible worlds. In section 3, I show that B-theorists can avoid the problems described in section 2 by rejecting the standard interpretation of the tense operators as quantifier-restrictors in favour of the view that the tense operators are redundant when the sentences in their scope are qualitative. I then describe and respond to what I take to be the most serious objection to this view, namely, that it has highly implausible Daniel Deasy, Advanced Temporalizing In: Oxford Studies in Metaphysics Volume 12. Edited by: Karen Bennett and Dean W. Zimmerman, Oxford University Press (2020). © Daniel Deasy. DOI: 10.1093/oso/9780192893314.003.0009

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consequences given the B-theory. Finally, in section 4 I describe four alternative B-theoretic strategies for avoiding the problems generated by the standard interpretation of the tense operators. I describe objections to each of these strategies. My aim is not to provide a decisive argument in favour of the ‘redundancy’ view described in section 3; rather, it is to establish the more modest conclusion that that view ought to be taken seriously as an alternative to the strategies described in section 4.

1. The B-theory The ‘B-theory of time’ is sometimes characterized informally as the view that ‘time is like space’, that ‘all times are on a par’, that ‘time does not flow’, or that ‘tense is unreal’.¹ While there is something to be said for each of these slogans, we can make more progress by thinking of the B-theory as combining the following two theses:  : There is no fundamental distinction between present and non-present times²  : Every proposition is if true always true

A few words on each of these. Temporal Parity is supposed to read as implying that being present doesn’t metaphysically distinguish this time n from other times. However, it is not supposed to be read as implying that there is nothing special about n, or indeed nothing metaphysically special about n—for example, it is consistent with Temporal Parity that n is God’s favourite time, and therefore (plausibly) metaphysically special relative to every other time. But even if n is God’s favourite time, it ¹ B-theorists include Beer (2010), Deng (2013), and Sider (2001). ² I assume that there are such things as times (or ‘moments’). Many contemporary B-theorists identify times with maximal simultaneous regions of spacetime (or ‘hyperplanes’). See, for example, Sider (2001) and Skow (2015). Strictly speaking, for B-theorists something is a time only relative to a frame of reference—given the Special Theory of Relativity, there is no nonframe-relative foliation of spacetime into hyperplanes. For ease of exposition, in what follows I write as if there are times simpliciter according to the B-theory, as this makes no important difference to the arguments in what follows.

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264   doesn’t follow that being present metaphysically distinguishes n from other times. What does make n present according to the B-theory? The best way to answer this question is to ask a different but closely related question: what does ‘the present time’ mean given the B-theory? According to the standard B-theoretic account, ‘the present time’ is an indexical term like ‘here’, and means the same as ‘this time’—it refers directly to the time of utterance on any occasion of use. It follows that, given the B-theory, an assertive utterance at the present time n of the sentence ‘n is the present time’ expresses the proposition that n = n. In that sense, for B-theorists the question ‘What makes this time the present time?’ is like the question ‘What makes this place here?’—just as being here doesn’t metaphysically distinguish this place from other places, being present doesn’t metaphysically distinguish this time from other times. B-theorists defend Temporal Parity. A-theorists, in contrast, defend Temporal Disparity:  : There is a fundamental distinction between present and non-present times

However, A-theorists disagree among themselves about what metaphysically distinguishes the present time from other times. For example, some Presentists identify times with maximal, consistent, sometime-true propositions, and hold that for a time t to be present is just for t to be true.³ Among non-Presentist A-theorists, some hold that the present time is just the time than which there is no later; some that it is the time that instantiates fundamental presentness; and some that it is the accurate time, where an time t is accurate iff for all propositions p, p is true at t iff p is true simpliciter.⁴ Propositional Eternalism is the view that every proposition is if true always true or, in other words, that every proposition is permanent. For example, according to the standard B-theoretic account, the sentence ³ See, for example, Bourne (2006), Crisp (2007), and Markosian (2004). ⁴ The first view is held by some defenders of the Growing Block Theory, such as Broad (1923); the second view is held by some Moving Spotlight Theorists, such as Deasy (2015); and the third view is held by, e.g., Bacon (2018).

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(1) It is raining in Cork as uttered at the present time n expresses the permanent proposition that it is raining in Cork at n. Some B-theorists may argue that they reject Propositional Eternalism, as follows: propositions are properties of times, so that, e.g., the proposition that there are dodos is just the property G such that for any time t, G(t) just in case there are dodos located at t.⁵ A permanent proposition is a property of times that is true at—i.e. possessed by—either every time or no times, and a temporary proposition is a property of times that is true at some but not all times. It follows that, given the B-theory, there are temporary propositions—e.g. the proposition that there are dodos, which is true at some but not all times—and therefore Propositional Eternalism is false. However, Propositional Eternalism is not intended to be read as inconsistent with the view that there are ‘temporary propositions’ in the sense just described. Assuming that there are such things as propositions, Propositional Eternalism should be read as the view that for all x, if x is a proposition, then if x is true, always, x is true—where the predicate ‘is true’ expresses a monadic property, rather than a dyadic property (i.e., relation) such as the true-at relation between propositions and times described above. And on this reading, Propositional Eternalism is true given the B-theory. We saw above that according to the standard B-theoretic account, the predicate ‘is the present time’ as uttered at the present time n expresses the property of being identical to n. Notice that the property of being identical to n is a permanent property—a property that is never gained or lost over time—and therefore that according to the standard B-theoretic account, the proposition that n is the present time is a permanent proposition. In contrast, consider the A-theorist, according to whom for n to be present is for n to possess some metaphysically special property F (such as being true, or being accurate). For the A-theorist, F had better be a temporary (indeed, instantaneous) property of times— otherwise, she is open to the charge of defending a view according to ⁵ See, e.g., Sider (2001, 20–1) and Zimmerman (2005).

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266   which presentness is ‘frozen’ at a certain time. It follows that if the A-theory is true, there is at least one temporary proposition—namely, the proposition that n is F—and A-theorists must accept Propositional Temporalism:  : Some propositions are sometimes true and sometimes false

Finally, given that Temporal Disparity plausibly implies Propositional Temporalism, it follows (contraposing) that Propositional Eternalism implies Temporal Parity. Therefore, all that is required in order to be a B-theorist is to accept Propositional Eternalism (correctly interpreted). I have characterized the B-theory in terms of Temporal Parity and Propositional Eternalism. But is there more to being a B-theorist? According to Sider (2001, 13–14), the B-theory implies ‘reductionism about tense’, the thesis that ‘tokens of tensed sentence types . . . can be given tenseless truth-conditions’. What exactly does this mean? Think of a ‘tensed sentence type’ as a sentence type whose natural regimentation is in the language of Quantified Tense Logic (QTL). QTL is the result of adding tense operators such as ‘P’—pronounced ‘It was the case that’— and ‘F’—pronounced ‘It will be the case that’—to standard first-order predicate logic. Given ‘P’ and ‘F’, we can define the further tense operators ‘H’ (‘It always has been the case that’), ‘G’ (‘It is always going to be the case that’), ‘A’ (‘It is always the case that’), and ‘S’ (‘It is sometimes the case that’) as follows: Hφ = def ¬P¬φ Gφ = def ¬F¬φ Aφ = def Hφ ∧ φ ∧ Gφ Sφ = def Pφ ∨ φ ∨ Fφ For example, the sentence (2) There used to be dinosaurs

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is a tensed sentence—it is naturally regimented in QTL as follows (where ‘D’ expresses the property of being a dinosaur): ð3Þ

P∃xDx

A simple way to understand the claim that tensed sentences such as (2) can be given ‘tenseless truth conditions’ is as the claim that sentences such as (2) express (relative to contexts of utterance) permanent propositions. In that case, ‘reductionism about tense’ follows straightforwardly from Propositional Eternalism. However, for most B-theorists, there is more to ‘reductionism about tense’ than Propositional Eternalism. For example, consider the sentence (4) Sometimes, there are dinosaurs (4) is a tensed sentence in the sense described above—it is naturally regimented in QTL as follows: ð5Þ

S∃xDx

Moreover, the proposition that, sometimes, there are dinosaurs is a permanent proposition. However, B-theorists standardly reject the claim that the relevant ‘tenseless truth condition’ for (4) is that sometimes, there are dinosaurs. The reason is that when B-theorists say that tensed sentences such as (2) and (4) can be given ‘tenseless truth conditions’, what they typically mean is that the truth conditions for such sentences can be stated in a language that is entirely free of tense operators such as ‘S’. Insofar as B-theorists take this language to be more ‘metaphysically perspicuous’ than QTL, this reflects a B-theoretic commitment to the thesis of Anti-tensism: -: Tense operators are metaphysically non-fundamental⁶ ⁶ Note that Anti-tensism neither implies nor is implied by Propositional Eternalism. For example, in the left-to-right direction, Deasy (2015) defends a view that combines Anti-tensism with Propositional Temporalism. In the right-to-left direction, one could (for example) defend a view that combines Propositional Eternalism with the thesis that the tense operator ‘S’ is metaphysically fundamental.

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268   For example, here is Sider (2011, 24; read ‘B-theorist’ for ‘Spatializer’): Spatializers do not admit tense operators into their fundamental ideology, since they can describe temporal reality without them—by quantifying over past and future entities and predicating features of them relative to times. Spatializers may use tense operators in their nonfundamental languages, since they can give a metaphysical semantics for the language of quantified tense logic in their tense-operator-free fundamental language.

In other words, ‘reductionism about tense’ implies that the truth conditions for tensed sentences such as (2) and (4) can in principle be stated in a ‘fundamental language’ that is free of tense operators (given Antitensism), all of the sentences of which express—relative to contexts of utterance—permanent propositions (given Propositional Eternalism). It follows that the relevant ‘tenseless truth condition’ for sentence (4) cannot be that sometimes, there are dinosaurs. More generally, it follows that, for B-theorists, QTL is ‘metaphysically second-rate’, as it contains expressions—in particular, tense operators such as ‘P’ and ‘F’—which fail to ‘carve reality at the joints’. However, as Sider indicates in the above quotation, this does not mean that B-theorists can simply bypass QTL. Rather, an important part of the B-theoretic project is to provide (as Sider 2011 puts it) a ‘metaphysical semantics’ for QTL in the B-theorist’s fundamental, tense operator-free language. The question of how to do this—and in particular, of how to interpret the tense operators when the sentences in their scope are not about any particular individuals—is at the heart of this paper.

2. Locator Consider the following characterization of the B-theory (“eternalism”) due to Sider (2006, 77–8): For the eternalist, past- and future-tensed claims are ultimately made true by claims that quantify over past and future times and entities. For

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instance, an assertion of ˹It was the case that φ˺ is true iff φ is true at some time located before the assertion. Construing (2) [‘Dinosaurs once existed’] (somewhat artificially) as having this form, the eternalist thinks of (2) as amounting to: (2E) There exist dinosaurs, located temporally before us. ∃x(Dx&Bxu) Note that (2E) entails that there exist dinosaurs (∃xDx). Presentists, on the other hand, deny that past-tense statements give way to statements quantifying over past entities. Rather, such statements involve primitive, unanalyzeable tense operators. The presentist’s rendition of (2) is this: (2P) It was the case that: there exist dinosaurs. P∃xDx ‘P’ symbolizes the past-tense operator it was the case that. (Other tense operators include it will be the case that, and it is always the case that.) Inside the scope of such a tense operator, the existential quantifier is not existentially committing; that is why the truth of (2P) is consistent with presentism.

According to Sider, if the sentence ‘Dinosaurs once existed’ is regimented as ð3Þ

P∃xDx

then given the B-theory, (3) ‘amounts to’—i.e. expresses the same state of affairs as—the sentence (where ‘T’ expresses the property of being a time; ‘