Oxford Studies in Metaphysics: Volume 7Volume 7 [1 ed.] 9780199659081, 9780199659074, 0199659087

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OXFORD STUDIES IN METAPHYSICS

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

Managing Editor Matthew Benton (Rutgers University)

OXFORD STUDIES IN METAPHYSICS Volume 7

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 2012 The moral rights of the authors have been asserted First Edition published in 2012 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 British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 978–0–19–965908–1 (Hbk.) 978–0–19–965907–4 (Pbk.) Printed in Great Britain by MPG Books Group, Bodmin and King’s Lynn Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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

D.W.Z. New Brunswick, NJ

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CONTENTS Oxford Studies in Metaphysics: Younger Scholar Prize I.

ix

RELATIVE SAMENESS

1. Possibility Relative to a Sortal Delia Graff Fara 2. Reflections on Counterpart Theory Allen Hazen

3 41

II. ABSOLUTE GENERALITY 3. All Things Must Pass Away Joshua Spencer

67

4. Absolute Generality Reconsidered Agustín Rayo

93

III.

HUMEANISM AND LAWS OF NATURE

5. Goodbye, Humean Supervenience Troy Cross 6. “There sweep great general principles which all the laws seem to follow” Marc Lange IV.

129

154

CONTINGENT OBJECTS AND COINCIDENT OBJECTS

7. Relativized Metaphysical Modality Adam Murray and Jessica Wilson

189

8. Coincidence Through Thick and Thin Sydney Shoemaker

227

viii | Contents V.

THE OPEN FUTURE

9. The Real Truth about the Unreal Future Rachael Briggs and Graeme A. Forbes

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10. Presentism and Distributional Properties Jonathan Tallant and David Ingram

305

Name Index

315

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

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

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

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RELATIVE SAMENESS

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1. Possibility Relative to a Sortal Delia Graff Fara 1. THE LEADING IDEA There is a distinction between identity and sameness. There is only one identity relation but there are many sameness relations. As many have said, identity is that relation which everything bears to itself but to no other thing. But something may be the same, in a number of different respects, as something other than itself. Sameness relations fall under at least two categories. First, there are those that are relativized to a quality—e.g., same color or same height. Second, there are those that are relativized to a sort of thing—e.g., same person or same boat. Philosophers make the distinction between “qualitative identity” and “numerical identity.” Numerical identity is identity. Qualitative identity can be thought of as a special case of sameness relativized to a quality: sameness with respect to every quality. (Or perhaps the notion is usually relativized, in a context, to every salient or relevant quality.) Sameness with respect to a given quality obviously does not require numerical identity. For example, distinct things can be the same height as each other. But it is generally assumed that sameness with respect to a sort does require numerical identity. To ask whether the ship we are on now is the same ship as the one we were on last week is to ask, it is assumed, whether this ship and that one are ships that are numerically identical. I reject that assumption. The ship we were on last week might be the same ship as the one we are on now without being numerically identical to it. In my view, to say that they are the same ship is to say that if we were to count how many ships we have been on in the past week, the correct answer could be “one” even if the ship we were on last week is not numerically identical to the ship we are on now. Some people, when they are first introduced to the problem of “personal identity” say that they are not the same person as the one

4 | Delia Graff Fara they were when they were a child because their interests and values have changed drastically. In my view, such people are not thereby making a category mistake. They have not thereby confused qualitative identity with numerical identity. They have not made the mistake of thinking that significant qualitative change over time precludes numerical identity over time. Rather, they have made a mistake about what conditions are required for a particular sort-relativized sameness relation to hold over time. They mistakenly think that significant change with respect to interests and values precludes sameness of person over time. That latter relation, sameness of person, is not, on my view, the relation of numerical identity—even when restricted just to persons. In addition to the distinction between sameness relativized to a quality and sameness relativized to a sort, there is a distinction to be made even within the category of sort-relative sameness relations. There is sort-relative sameness of types and also sort-relative sameness of tokens. If you and I are both wearing blue platform wedges then there is a sense in which we are wearing the same shoes even though the ones on my feet are not on your feet. We are wearing the same type of shoes, but not the same individual (“token”) shoes Peter Geach would have thought that in the case of our shoes, not only are you and I wearing the same type of shoes, my shoes and your shoes are the same relative to a certain sort, namely, the sort shoe-type (Geach, 1962). Of course Geach did think that my shoes and your shoes were not the same shoe-tokens. I disagree with the former claim. We do not have a case here of things being the same relative to one sort of thing while not being the same relative to another sort of thing. Our shoes are not the same shoe-type while being different shoetokens. Although we are wearing the same type of shoes on our feet, neither of us is wearing a shoe-type on our feet. In other words, although we are wearing the same type of shoes, there is nothing on my feet that is the same, relative to any sort, as anything on your feet. On this point, I agree with William Alston and Jonathan Bennett (1984) and with John Perry (1970), as against Geach. From here onwards, I will ignore sort-relative sameness of types. I will also from here onwards ignore relations of sameness relativized to a quality. So as shorthand I will use the terms “relative sameness” and “sort-relative sameness”—even sometime just “sameness”—to mean sort-relative sameness of tokens.

Possibility Relative to a Sortal | 5 Some have used the term “relative identity” to mean what I am calling relative sameness. Some proponents of “relative identity” have denied that there is any relation of absolute identity, instead only same-F-as relations.1 For this reason it is worth emphasizing that in admitting the various relative-sameness relations, I in no way reject the absoluteness of identity. I am, however, driving a wedge between relative sameness and absolute identity—as concerns what was, what will be, or what might have been—that makes for a greater gap than philosophers typically allow space for.2 For example, I think that the boat that Michael just embarked on is the same boat as the one he will later row ashore (should all go as planned), even though the boat that he embarked on is not absolutely identical to the boat that he will row ashore. Like Theseus’s ship, Michael’s boat will have exchanged or lost some of its parts over the course of its long river crossing. This suffices to make the embarked-upon boat numerically distinct from the rowed-ashore boat. But it does not suffice to make them different boats.3 Modal variant: if Michael does not in the end row his boat ashore because it capsizes, then the boat that he would have rowed ashore—if only his boat had not capsized—would be the same boat as the one he in fact embarked on, even though the boat that he would have rowed ashore in that event would not be absolutely identical to the boat he did in fact embark on. I am not, however, driving so much of a wedge between sameness and identity that I would qualify as a “relative identity” theorist.

1

The view is most associated with Geach (1962). A number of philosophers do allow for the space: Peter Geach (1962, 1967, 1972); Roderick Chisholm (1973, 1976); and Anil Gupta (1980) are examples. 3 With respect to identity over time, I here agree with Roderick Chisholm (1976), who thought that this ship here now could be the same ship as that one there then even though they are not numerically identical. Like Chisholm, I also think that the same-ship relation, rather than the numerically-identical relation is the one that is relevant for counting ships. Following Bishop Butler, though, Chisholm accepted that it is only in a “loose and popular sense” that identity may hold between numerically distinct ships. On that view, for two objects to be the same ship they must each be a ship and must be identical in the loose and popular sense. But for the two objects to be numerically the same thing is for them to be identical in the “strict and philosophical” sense. We can speak of identity “in the loose and popular sense” if we like, but there is really no need to once we recognize that being the same F over time does not require numerical identity (identity in the strict and philosophical sense) over time. 2

6 | Delia Graff Fara Although I think that there is a distinction between sameness and identity, I do not think there is any world-time point at which distinct things can be the same with respect to any sort. Nor do I think that there is any world-time point at which identical things can be different with respect to any sort. I think rather that “transworld” or cross-time sameness can come apart from transworld or cross-time identity. A boat a at world-time point may be the same boat as the boat b at world-time point even though a ≠ b, as long as ≠ . The boat that I am actually on might have been a red boat even though it is in fact a blue boat. This is to say not that there’s a possible world in which there’s a red boat that is identical to the boat that I’m actually on (though that may be true). Rather, it is to say that there’s a possible world in which there’s a red boat that is the same boat as the blue one that I’m actually on. I hold that those two conditions can come apart. Correspondingly, I do not think that two things which are both F and both G can be the same F while not being the same G. This is to say that I reject another central thesis of the sort of relative-identity theory associated with Geach, though not with Roderick Chisholm. Since I do not think that there is any kind of identity other than absolute identity, I am happy to dispense with the qualification “absolutely.” The boat we’re on now is the very same boat as the one we were on last week. But they are not identical, for identity is “too unyielding a relation” (to borrow Robert Stalnaker ’s words) to hold of entities composed of different matter. (By the way, the use of the plural “they” here should not be taken to suggest that two different boats are being counted any more than two different planets are being counted when I use a plural verb form to say that Hesperus and Phosphorus were discovered to be the same planet. The plural is appropriate because two different noun phrases were used, not because more than one thing was mentioned.) I propose that transworld sort-relative sameness is the important relation for analyses of de re possibility—for analyses, that is, of what we mean when we say what a certain thing might have been or was or will be like.4 4 Such a relative-sameness counterpart theory is discussed briefly by Theodore Sider (1999), who attributes it to Michael Jubien (1993), although Jubien would disavow it. One finds a kernel of the same theory in Jubien (2001).

Possibility Relative to a Sortal | 7 For example, I propose that a certain woman might have worn green shoes today, even though she is in fact wearing blue ones, because there is a possible world in which someone who is the same person as her did wear green shoes today. There are two salient competing proposals. The first is associated with Saul Kripke (1963, 1972): transworld identity is the important relation for analyses of de re possibility. This is the view that it is true that a certain woman might have worn green shoes today just in case there is a possible world in which something that is identical to that woman did wear green shoes today. The metaphysical question of whether that thing itself need be a person may be left open. The second is associated with David Lewis (1968, 1971, 1986): transworld superlative qualitative similarity is the important relation for analyses of de re possibility. This is the view that it is true that a certain woman might have worn green shoes today just in case there is a possible world in which a thing wearing green shoes today is qualitatively similar to the woman and is no less qualitatively similar to her than is anything else in its world. Again, it is left open in principle whether that thing is a person. (I’ve used the term ‘superlative’ so that it allows for ties. Think of it as synonymous with the term ‘unbeaten’.) My proposal comes apart from the identity proposal since transworld sort-relative sameness does not require transworld identity. We might have been on the same ship now as the one we were actually on last week even if the ship we would have been on now, in that event, were not identical to the ship that we were actually on last week. My proposal comes apart from the qualitative-similarity proposal since transworld sort-relative sameness does not require transworld superlative qualitative similarity. The boat we are actually on might have (i) been a different color even while (ii) there were some different boat of the same color as the one we are actually on and qualitatively just like it—even in very extrinsic ways, such as having our qualitatively identical twins on it. We are actually on boat a. We might have been on the same boat b even while there were some different boat c that were qualitatively just like a—even in very extrinsic ways.

8 | Delia Graff Fara Granting this much, what do we say about the truth of the following modal and temporal claims? What conditions must be met in order for them to be true? (1) Angel, Michael’s boat, might have been rowed ashore. (2) Angel will be rowed ashore. The less popular answer, derived from Lewis (1968, 1971, 1986) for the modal case and from Katherine Hawley (2001) and Theodore Sider (1996, 2001) for the temporal case, is that these sentences are true just in case there is a possible world (in the case of (1)), or a future time (in the case of (2)), at which something that is a counterpart of Angel is rowed ashore. For the moment, let us leave behind the question of just what this counterpart relation is. The important point for us now is that it may hold between things that are not identical to each other. The more popular answer, derived from Kripke (1963, 1972), is that these sentences are true just in case: (A) There is a possible world, or future time, at which something identical to Angel is rowed ashore. Do we accept the popular answer, so phrased? For Kripke, and those who follow him on this question, there is no distinction between that condition and another: (B)

There is a possible world, or future time, at which something that is the same boat as Angel is rowed ashore.

But for us, the two conditions come apart. The leading idea for us is that (A) and (B) are different views. We may accordingly say that (B), the second phrasing of the popular view, is the correct one; while (A), the first phrasing, is incorrect. A de re modal claim “a might have been Φ” is true when there is something in some possible world to which a bears a certain relative-sameness relation and which is Φ in that world—whether or not there is something identical to a that is Φ in that world. Likewise, a de re future claim “a will be Φ” is true when there is something in the future to which a bears a certain relative-sameness relation and which is Φ at that future time—whether or not there is something identical to a that is Φ at that time. Which sameness relation is involved (same person, same

Possibility Relative to a Sortal | 9 body, same boat, et cetera) varies with the context.5 We will have to revise and sharpen this up in a bit. But before we do, I want to point out that this view about the truth conditions for de re predications (temporal or modal) has something in common with both the popular view and the Lewisian view. It shares with the popular view the idea that future and counterfactual possibilities for a thing a are determined by the ways that a thing b is at a future time, or possible world, whenever a bears a certain relative-sameness relation to b. Because the theory has this feature, I call it a relative-sameness theory of de re predication. It shares with the Lewisian view the idea that future and counterfactual possibilities for a can be determined by the ways that a thing b is at a future time, or possible world, even if a is not identical to b. Because the theory has this feature, I call it a counterpart theory of de re predication. Let me emphasize two things before moving on. One, I intend for relative-sameness counterpart theory to cover temporal predication (de re) as well as modal predication (de re). It simplifies matters, however, if I focus on just one of these at any given time. For the most part I focus on modal predication, but it should be somewhat clear how the view extends to temporal predication. Two, I use the term ‘counterpart theory’ liberally. To be a counterpart theorist is not to have any particular view about what it takes for an individual in one possible world to be a counterpart of some individual in some different possible world. Any theory of de re modality is a counterpart theory in my sense if it takes the transworld relation that is important for the analysis of de re possibility to be a relation other than identity. For example, someone who held the ridiculous view that it is true that a certain woman might have worn green shoes today just in case there’s a possible world in which she has a sister who is wearing green shoes today would be a counterpart theorist in my sense.

5 This appeal to contextual variation can be found in Lewis (1971) and Allan Gibbard (1975). Gibbard’s views on contingent identity inspired me to develop the view presented here. I differ from both Lewis and Gibbard on a number of important issues—among them, the question whether objects are extended in time.

10 | Delia Graff Fara 2. MOTIVATING THE IDEA Why develop a relative-sameness counterpart theory of de re modality? My aim is to provide a plausible account of possibility and identity claims while preserving the attractive metaphysical view that ordinary material objects are identical to the matter that constitutes them even though they might not have been constituted by that matter. The boat and the wood that it is made up of are one and the same thing—they are identical. But they might not have been. They are identical since if we were to count the things on the river we would have already counted at least one too many if we added the boat to our count and separately added the wood to our count. (This is not to say that we could, even in principle, count all of the things on the river.) Nevertheless, it might have been that that same boat was not constituted by that same wood. The boat would not have been constituted by that same wood if some of the boat’s planks had been exchanged during some previous river crossing. It might have been, therefore, that that same boat and that same wood were not one and the same thing. It also follows, moreover, that they might not have been identical. The boat’s name is ‘Angel’. ‘Dusty’ is the name of some wood that is a proper part of the wood that makes up Angel. We accept the following triad. (3) Angel and the wood that it is made up of are identical. (4) Angel might not have had Dusty as a part. (5) The wood that Angel is made up of cannot but have had Dusty as a part. One could reason for a contradiction from these three claims as follows. Appealing to Leibniz’s law and the identity statement in (3), substitute ‘Angel’ for ‘the wood Angel is made up of’ in (5) to yield (4†), which contradicts (4). (4†) Angel cannot but have had Dusty as a part. Alternatively, substitute ‘the wood Angel is made up of’ for ‘Angel’ in (4) to yield (5†), which contradicts (5). (5†) The wood that Angel is made up of might not have had Dusty as a part. I will show how this reasoning may be resisted: (3)–(5) do not jointly yield a contradiction.

Possibility Relative to a Sortal | 11 One could lamely reject the reasoning by noting that identity statements like (3)—those which have a definite description on one side of the identity sign—do not license substitution within the scope of a modal operator when the description involved is not necessarily true of the thing it is actually true of. If Barack Obama is identical to the president of the United States, and it is not possible for the president of the United States to have never been elected to a political office, it does not follow that it is not possible for Barack Obama to have never been elected to a political office. Why? Because the description ‘the president of the United States’ does not necessarily apply to Obama. But I do not want to reject the reasoning on those lame grounds. I want to promote the stronger claim that Angel, the boat, is identical to the wood that she is actually made up of—she is identical to that wood, to it. Give that wood the name ‘Lumber’. I want to promote the compatibility of this stronger triad of claims. ‘Dusty’ is the name of some wood that is a proper part of Lumber. (6) Angel and Lumber are identical. (4) Angel might not have had Dusty as a part. (7) Lumber cannot but have had Dusty as a part. One could now use the substitutivity reasoning made above to argue for conclusions that contradict those in the above triad. (4†) Angel cannot but have had Dusty as a part. This, again, contradicts (4). (7†)

Lumber might not have had Dusty as a part.

This contradicts (7). Generalizing from the claims in (3)–(7), I want to defend the following theses. (IT)

The Identity Thesis: Ordinary material objects are identical to the matter that makes them up.

(ACT) The Accidental Constitution Thesis: Ordinary material objects could be made up by matter that is different from the matter that actually makes them up. (NMT) The Necessary Matter Thesis: The matter that makes up an ordinary material object could not be made up by

12 | Delia Graff Fara matter that is different from the matter that actually makes it up. I call the combination of these three theses The Sensible View. The person who opposes the identity thesis has an argument for her view, namely, the one that proceeds by appeal to Leibniz’s Law. Rather than give a countervailing argument for my views, let me give some sense of my reasons for individually accepting the three theses. I just want to give some sense of why I hold them. I then go on to provide an account of de re possibility that blocks the Leibniz’s Law argument against The Sensible View. Some thoughts behind the Identity Thesis: If you were to tally the number of things on my desk, for example, you would have counted at least one thing too many if you counted the matter that makes up one of the books and separately counted the book itself. This is what David Lewis meant when he described this as “double counting” (1986, 252).6 Some thoughts behind the Accidental Constitution Thesis: Every ordinary material object has some material part, however small, that could be removed without the object itself either becoming a scattered object or ceasing to exist. If I bite off a small piece of my paper coffee cup and swallow it, nothing then in my stomach is then a part of the coffee cup, even though there is something then in my stomach that used to be part of the coffee cup. And what’s more obvious, I do not thereby destroy the coffee cup. Some thoughts behind the Necessary Matter Thesis: This is true by stipulation. I hereby stipulate that I am using the terms “constitutes” and “matter” so that whenever X is made up of some matter Y, and Y is made up of some matter Z, then Y is necessarily made up of Z.7 (I’m using the expression “makes up” interchangeably with “constitutes”. 6 Mark Johnston has argued that there is no plausible argument for the identity thesis (Johnston 1992). Harold Noonan (1993) argued, in response to Johnston, that not all of Johnston’s dismissals of arguments for the identity thesis are justified. That’s as may be. I do not take myself to be providing a direct argument for the identity thesis, nor even a rebuttal to Johnston’s arguments that there is no plausible such argument. Rather I take myself in what follows, to be giving a defense of the identity thesis—since it’s a thesis that I find attractive—by explaining how it does not lead to a contradiction. 7 I appreciate comments from Dean Zimmerman that led to this formulation of the stipulation.

Possibility Relative to a Sortal | 13 I use one or the other phrase according to what sounds more natural to my ear.) The leading idea of this paper enables us to coherently maintain The Sensible View. The leading idea is that sort-relative sameness does not require identity. This boat here now might be the same boat as that one there later even though they are not identical. Clearly, anyone who thinks this—that ordinary material objects, like boats, are identical to the matter that they are made up of even though the objects, unlike the matter itself, could be made up of different matter—must accept some statements of contingent identity. Angel is identical to Lumber, but she might not have been. The leading idea allows us to make sense of this (merely) seeming contingency of identity. It allows us to say that it is true that Angel might not have been identical to a thing x when there’s a possible world in which there’s some boat, Angel*, that is the same boat as Angel but which is not identical to x. Let us introduce a piece of terminology. We will call Angel* Angel’s “boat-mate” in that world. Since Angel is not identical to her otherworldly boat-mate, there is no contradiction in her being identical to Lumber even though Lumber is not identical to the boat-mate. The following are compatible: (8) Angel = Lumber; (9) Angel*, in the counterfactual world β, is the same boat as Angel; (10) Lumber ≠ Angel*. These are compatible since being a boat-mate does not require identity. The joint truth of (8)–(10) is what on my view renders the following true: (11)

Angel is identical to Lumber, but Angel might not have been identical to Lumber.

None of this really commits us to the contingency of identity, however. Identity is a relation that holds necessarily (and permanently). By this we mean that there are no things a and b which are identical in one possible world but not identical in some other possible world. (Similarly, there are no things a and b which are identi-

14 | Delia Graff Fara cal at one time but not identical at some different time.) Sort-relative sameness, in contrast, is not a relation that holds necessarily (or permanently). Sort-relative sameness does not coincide with identity, but it, rather than identity, is the important relation for analyses of de re modal (and temporal) claims—even when these are modal (or temporal) identity claims. That is why the truth of particular modal statements of contingent identity does not commit us to the contingency of identity.8 This thought will be developed and sharpened in what follows. 3. DEVELOPING THE IDEA Symmetry and transitivity are logical properties of relative sameness as well as identity. This is obviously true when the sameness is relativized to a quality: if I’m the same height as you then you are the same height as me (symmetry), and if you are also the same height as Maria then I’m also the same height as Maria (transitivity). It is also true when the sameness is relativized to a sort: if Rusty is the same boat as Scarab and Scarab is the same boat as Lancha, then Lancha is the same boat as Rusty (symmetry and transitivity). Unlike identity, however, relative-sameness relations are not quite equivalence relations, since they are not all reflexive. Since Barack Obama is not a boat, he is not the same boat as anything, not even himself. The only way for you to be the same boat as something is if you yourself are a boat. Only a boat can bear this relation to something. And, of course, every boat does bear this relation to itself. Since only boats bear the relation to anything, and all boats bear the relation to themselves, we may say that the same-boat relation is weakly reflexive. A relation is weakly reflexive when anything that bears the relation to at least one thing bears the relation to itself. Weak Reflexivity: ∀x(∃yRxy → Rxx).

We will say that a relation is a weak equivalence relation if it is symmetric, transitive, and weakly reflexive. Weak equivalence is a logical property of all relative-sameness relations.

8 This interpretation of what acceptance of contingent identity amounts to can be attributed to Gibbard (1975).

Possibility Relative to a Sortal | 15 Although I call my theory a counterpart theory, I do not accept all of the postulates that were originally stipulated by Lewis (1968) to constitute counterpart theory. In particular, I deny Lewis’s postulate P2, the stipulation that individuals are “world-bound”—that none exists in more than one possible world: P2:

∀x∀y∀z((x is in possible world y ∧ x is in possible world z) → y = z).

I reject P2. That is, I accept that there are transworld identities. Individuals are not world bound: an individual in one possible world may be identical to some individual in a distinct possible world. I also accept, remember, the following theses: (IT) The Identity Thesis: Ordinary material objects are identical to the matter that makes them up. (ACT) The Accidental Constitution Thesis: Ordinary material objects and their constituting matter might have been different things, however. (NMT)

The Necessary Matter Thesis: The matter that makes up an ordinary material object could not be made up by matter that is different from the matter that actually makes it up.

Along with these, I accept for now the following preliminary statement of an example of possibility relativized to a sortal (PRS). (PRS)

An ordinary material object, such as a boat, might have had properties that are different from the ones it in fact has only because it might have been that that very boat (something that is the same boat) did have the different properties.

The view as I’ve developed it so far might at this point seem bizarre. Here is one illustrative way to depict it. We have the actual world, α, and a counterfactual world, β.

α (actual)

β (counterfactual)

d

d

be = d

br ≠ d

be is the boat embarked upon in α; d is the wood constituting be in α; br is the boat rowed ashore in β; d does not constitute br in β; be and br are the same boat.

16 | Delia Graff Fara In the actual world, be is the boat that Michael embarked upon and it is constituted by some wood d. Where we have the boat and the wood that constitutes it we have just one thing, so in α, be = d. In β, br is the boat that Michael rows ashore. It is not constituted by the wood d. In β, d does not constitute any boat at all. So in β, d ≠ br. In the actual world, α, Michael does not row the boat ashore, since the boat he embarked on capsizes. Nevertheless, Michael might have rowed the boat ashore. In the part of modal space that I’m describing, that is because be , the boat embarked-upon in α, and br, the boat rowed ashore in β, are the same boat. We can see that be and br are nevertheless not identical, since be = d but d ≠ br. Now is anything wrong with this picture? We can see that it allows for transworld identities, which were proscribed by Lewis’s P2, since d exists in the actual world as well as in the counterfactual world β. But does be , the embarked-upon boat, exist in the counterfactual world β as well in the actual world α? Since it is identical with d it has to exist in β, since d exists there. And so it does. Identity is a necessary relation. So since the wood d, which in fact constitutes the embarked upon boat be , is not identical to the boat rowed ashore in β, be ≠ br. Although be and br are the same boat, they are not identical—not even in β. In view of the above, it might seem that we have incurred the following commitments, which cannot all be right: (12) (13) (14)

be = d. be is the same boat as br in β. d is not a boat in β and therefore not the same boat as anything in β.

These cannot all be right since it follows from these that be is the same boat as br in β but also not the same boat as anything in β. As we sharpen the idea, we will see that we are not in fact committed to this inconsistent triad. 4. SHARPENING THE IDEA Evidently, the relative-sameness relation same-boat must be relativized to worlds. Whether or not be in the actual world is the same boat as br in the counterfactual world depends not only on what br is like in the counterfactual world but also on what be is like in the actual world. At the very least, be must be a boat in the actual world

Possibility Relative to a Sortal | 17 and br must be a boat in the counterfactual world. Only boats can be the same boat as anything. Of course, there are stronger conditions in addition to that one. Switch for the moment to sameness over time. Here’s a plausible condition on cross-time sameness of boat. If I own a boat now and you own a boat later, then they are not the same boat unless I relinquished my right to it or was legitimately evicted from it.9 The transworld analog is not pretty to state: if I in fact owned a boat for the whole of last month but you might have owned that same boat by the end of the month, that can only be because I might have legitimately relinquished my right to it or have been legitimately evicted from it by the end of the month. Put another way, transworld boat-sameness requires any counterfactual transfer of ownership to have been done legitimately. This is not to say that you could not steal my boat. You might have stolen the very boat that I in fact own, but you would not in that case own the same boat that I in fact own. I will not make any further attempts to say what the necessary conditions are for transworld sameness of boat. For now, it is sufficiently illustrative to say something further about what necessary conditions there are not. Since the boat that Michael actually embarked on might have changed color during his river crossing, it is not required that the boat he rowed ashore in the counterfactual world be the same color in that world as the actually embarkedupon boat is in the actual world. Likewise, since the boat that Michael embarked on might have been manned by Gabriel by the time it was rowed ashore, it is not required that the boat rowed ashore in the counterfactual world be manned by the same person in that world that manned the embarked-upon boat in the actual world. And so on for any other accidental property of the boat that was actually embarked upon. I presume that there are necessary conditions for something in a counterfactual world to be the same boat as something in the actual world. This is just to presume that boats have essential properties. 9 Jubien makes roughly this same point (Jubien 2001). He, however, makes it as a psychological point about how we are inclined to think about the cross-time sameboat relation rather than as a metaphysical point about the cross-time same-boat relation itself.

18 | Delia Graff Fara What those conditions are is not per se part of the theory presented here.10 I am interested primarily in the formal properties of relative sameness and in applying these to certain metaphysical views about constitution that I happen to hold. I want to develop a relative-sameness counterpart theory whose formal properties allow for the combination of the Identity Thesis, the Accidental Constitution Thesis, and the Necessary Matter Thesis while providing a plausible analysis of de re modal (and temporal) predication. Someone with different metaphysical views about essence and accident—whether that be as it relates to material constitution or to something else, such as personal “identity”—might well find the formal analysis here congenial for her purposes. That would not be to accept a different view but rather to combine the view here with a different metaphysics.11 This is like saying that someone could accept Kripke’s identity analysis of de re possibility but reject his view that sameness of origin is required for transworld identity. Similarly, it would be like saying that someone could accept Lewis’s counterpart theory of de re possibility but think that similarity of origin is more important for transworld similarity than is similarity in any other respect.

4.1 World-Relativized Relative Sameness As we have said, whether or not x is the same boat as y will depend on what x is like and on what y is like. But, like all ordinary objects, boats might have been different from how they actually are. So if the transworld sameness of x and y is to depend on what they are like, it must depend on what they are like in a world. We therefore need to further relativize relative-sameness relations to worlds. The relative-sameness relations appealed to in our theory of de re modal predication will officially be understood as relations holding between thing-world pairs. In other words, whether bears a

10 Sider (1999, 289–90) makes essentially the same point in partial defense of Jubien’s (1993) version of counterpart theory. 11 Chisholm offered us criteria for cross-time sameness of table (Chisholm 1976, 218–21). But he need not have. I’m pointing out that we can present a theory of the kind presented here while having no obligation to analyze what’s required for crosstime or transworld sameness for each sort of thing that there is.

Possibility Relative to a Sortal | 19 relative-sameness relation to depends on what x is like at w and what y is like at w’. Of the many ways that x and y are in a possible world, which ones count will depend on which sortal the sameness is being relativized to. For instance, if x in w is to be the same boat as y in w’, then x and y will have to be boats in w and w’, respectively. Whether x and y, in their respective worlds, have all the same matter parts as each other will not be important. When the same-matter relation is at issue, however, sameness of matter parts is of the utmost importance. Since there are many different sorts to which a sameness relation may be relativized, we can indicate this by providing subscripts. Where F stands for a sort of thing, RF is to be that sort-relative sameness relation that is relativized to the sort F. For example, Rboat is the same-boat relation. Our relative-sameness counterpart theory of de re possibility can now be provisionally stated as follows: (RSC*◊) Relative to a sortal F, ‘◊Φx’ is true at a world w just in case there is a thing y and possible world w’ such that RF and such that ‘Φy’ is true at w’. Our instance of (RSC*◊) is this: Relative to the sortal boat, ‘Angel might have been rowed ashore’ is true at a world w just in case there is an object y and a world w’ such that Angel is the same boat as y, given the ways that Angel is in w and that y is in w’, and such that y was rowed ashore in w’. This allows us to rewrite the inconsistent (12–14) as the consistent (12*–14*): (12*) (13*) (14*)

be = d; be in α is the same boat as br in β; d is not a boat in β and therefore, in β, d is not the same boat as anything in β.

4.2 Formalizing World-Relativization We have now relativized the (sort-relativized) sameness of objects to worlds. This is not to admit funny objects—things-at-a-world, that are somehow world “slices” of (somehow) modally extended

20 | Delia Graff Fara objects. We are regarding these relations as relations on genuine pairs. The first member of each pair is a ordinary object. Ordinary objects do not have some of their parts in one possible world, some other parts in some other worlds. Rather, in each possible world in which an ordinary objects exists, there are ways that that object is. This is reflected in our taking relativized sameness to be a relation on thing-world pairs. These sameness relations are symmetric, transitive, and weakly reflexive as relations on pairs. With regard to symmetry, this is to say that if x is the same boat as y relative to the ways that x is in w and that y is in w’, then y is the same boat as x relative to the ways that y is in w’ and that x is in w. For example, if the boat that Michael embarked on in the actual word is the same boat as the one that Gabriel rowed ashore in world β, given the ways that these boats are in their respective worlds, then the boat that Gabriel rowed ashore in world β is the same boat as the one that Michael embarked on in the actual word, given the ways that these boats are in the respective worlds. In symbols: Symmetry:

RF → RF .

Transitivity and weak reflexivity are now given corresponding world-relative formalizations: Transitivity: ( RF ∧ RF ) → RF ; Weak Reflexivity: ∃y∃w’ ( RF ) → RF .

4.3 Relative Sameness and Identity Despite the need for these relativizations to possible worlds, we do not deny that there are normal relative-sameness relations that hold among things simpliciter, ones that are not further relativized to a world. I am the same person as myself, the same mother as myself, the same philosopher as myself, . . . simpliciter. It cannot be that there is something distinct from me that is the same woman as myself simpliciter, since no woman has properties that are incompatible with others of her properties. No woman is ever in two places at once, for example. We can put this point formally as follows. Simpliciter relative-sameness relations are not only weak equivalence relations, they are subsets of the identity relation. They will be proper subsets of the identity relation whenever

Possibility Relative to a Sortal | 21 they are relativized to a nonvacuous sort of thing—that is, to a sort of thing that some things are not. For example, I am not the same boat as myself since I am not a boat. Therefore, the simpliciter same-boat relation is a proper subset of the identity relation. The simpliciter same-boat relation is that subset of the identity relation which contains those pairs such that d is a boat. Vacuous sorts, if we admit them, might include thing or entity. The fact that simpliciter relative-sameness relations are always subsets of the identity relation can be represented formally as a kind of functionality condition on relative-sameness relations as they hold among thing-world pairs: Thing Functionality: RF → x = y. There is no possible world in which (or time at which) distinct things bear a relative-sameness relation to each other. This condition guarantees that we are not the same person (woman, philosopher, et cetera) as anything other than ourselves. So sameness really is a kind of identity after all—distinct things are never the same F, simpliciter, as each other. Relative sameness (simpliciter) does on our view, as on the standard view, require identity. We differ from the standard view in holding that transworld (or cross-time) relative sameness does not require transworld (or cross-time) identity. In saying this, we do not thereby reject the notion of transworld identity. I am happy to admit this relation, and without it could not state the difference between my theory of de re predication and the popular view. Just as there is only one thing in the actual world that Angel is the same boat as (viz., herself), she is also not the same boat as more than one thing in any counterfactual world. This is entailed by thing functionality together with symmetry and transitivity. For if there is a counterfactual world in which Angel is the same boat as both Cherub and Seraph (given the ways that they are in the counterfactual world and the way that Angel is in the actual world), then by symmetry, Cherub is the same boat as Angel. Then by transitivity, Cherub is the same boat as Seraph. But the distinctness of Cherub and Seraph is ruled out by thing functionality. We can call this stronger requirement strong thing functionality and formalize it as follows: Strong Thing Functionality: RF ∧ RF → y = z.

22 | Delia Graff Fara Unlike Lewis’s qualitative-similarity counterpart relation, the relative-sameness counterpart relation does not allow for a single individual to have more than one counterpart in some one possible world. In other words, unlike Lewis’s counterpart relation, our relation does not allow for multiple counterparts in some one other world.

4.4 Relative Sameness and Superlative Qualitative Similarity We have said that on our proposal, relative-sameness relations can be construed as a kind of counterpart relation given that (i) transworld relative sameness is the important relation for analyses of de re possibility, and that (ii) transworld sameness does not require transworld identity. But while relative-sameness counterpart relations are strongly thing functional, qualitative-similarity counterpart relations are not. Even once we fold in Lewis’s later (1971) relativization of the counterpart relation to salient respects of similarity, which vary from context to context, the reason that superlative qualitative similarity still allows for multiple counterparts in a world, relativized to a single respect of similarity, is exemplified by this example: genetically identical twins in one possible world may tie for most personsimilar with a single thing in some one other possible world. There might have been genetically identical twins, each qualitatively just like me, one of whom was born two minutes later than my actual birth-time, the other two minutes before my actual birth-time. So there is a possible world in which there are genetically identical twins who would tie for most person-similar to me. They would both be my qualitative-similarity personal counterparts. Because transworld relative sameness, when relativized to a single sortal F, does not allow for multiple counterparts in a world, we may speak of such a thing as the same-F counterpart of x in world w, if it has one. Since this is a functional expression, we may equivalently speak of x’s counterpart in w (if it has one). This uniqueness is relativized to a sortal, of course. When relativized to the sortal boat, for example, we called x’s counterpart in a world w (if it has one) x’s boat-mate in w. In the abstract, we call x’s counterpart in a world w (if it has one) x’s “F-mate” in w, when relativized to sortal F.

Possibility Relative to a Sortal | 23 (So there are different instances of F-mates: boat-mates; personmates; matter-mates; etc.) But whether an F-mate relation holds between objects x and y depends not only on what sortal F is, but also on the ways that x and y are in their respective possible worlds. Any “F-mate” relation is functional when doubly relativized to two worlds: given the way that x is in w, we may speak of such a thing as the F-mate of x in w’. In terms of our earlier notation, this would be the such that RF . So as to have a convenient, if cumbersome, functional expression, we use the functional expression ‘rFw, w’ (x)’: F-mate notation:

‘rFw,w′(x)’ denotes the such that RF ;

In other words:

‘rFw, w′(x)’ denotes the y such that x is the same F as y, given the ways that x is in w and that y is in w’.

The expression ‘rFw, w′(x)’denotes x’s F-mate in w’, given the way that x is in w. We can pronounce this functional expression as x’s Fw,w′ -mate—or just as x’s F-mate in w’, when the relativization of x to w may be left out without causing confusion; or simply as x’s F-mate, when the double relativization to both w and w’ can be thus left out. Given our entitlement to the use of this F-mate functional expression, we can restate our provisional version of relative-sameness counterpart theory of de re possibility using that functional expression as follows: (RSC◊)

Relative to a sortal F, ‘◊Φ’ is true at a world w just in case there is a possible world w’ such that ‘Φ(rFw,w’(x))’ is true at w’.12

For the Lewisian counterpart theorist, lack of uniqueness within a world for superlative qualitative similarity means that the Lewisian counterpart theorist cannot state his theory in the analogous way, replacing ‘x’s F-mate’ with ‘x’s F-ish counterpart’. On

12 We understand ‘Φ(rFw,w’ (x))’ to be the result of replacing each occurrence of ‘x’ in ‘Φ’ with ‘rFw,w’ (x)’. We assume that Φ has no unnecessarily repeated uses of ‘x’.

24 | Delia Graff Fara Lewis’s view, we cannot speak of the F-ish counterpart of x in w’, since x may have more than one counterpart there (x might have had an identical twin). So how must he state his theory? Lewis discussed two salient choices (Lewis, 1968). First: an individual might have been Φ just in case there is a possible world in which some counterpart of that individual is Φ. Second: an individual might have been Φ just in case there is a possible world in which every counterpart of that individual is Φ. The difference is whether in order to be possibly-Φ there just needs to be some possible world in which one of your counterparts is Φ, or whether there needs to be some possible world in which all of your counterparts are Φ. Lewis chose the first (Lewis 1968, 118f). Given his later (1971) relativization of the counterpart relation to salient respects of similarity—which could vary from context to context—we can give the following statement of his qualitative-similarity counterpart theory of de re possibility. It helps to use a variable-binding predicateabstraction operator and to assume that Φ contains no free variables other than x. QSC◊:

‘xˆ (◊Φ)’ is true of a thing, relative to respects of similarity F, just in case there is a possible world in which that thing has some F-ish counterpart of which ‘xˆ Φ’ is true.

5. APPLYING THE IDEA So how, after all this, do we do what we wanted? What we wanted was to preserve the coherence of the following: (15) (16)

Angel = Lumber; Possibly, Angel ≠ Lumber.

Angel and Lumber are one and the same thing since Angel is constituted entirely by Lumber. But they might not have been one and the same thing since Angel might not have been constituted entirely by Lumber. The view of de re modality offered here is that Angel might have been distinct from Lumber since there is a possible world in which

Possibility Relative to a Sortal | 25 Angel has a boat-mate and Lumber a wood-mate that are distinct from each other. The leading idea allows that this is compatible with the actual identity of Angel and Lumber since the transworld boat-mate of Angel and the transworld wood-mate of Lumber need not be identical to Angel. Once Lewis incorporated multiple counterpart relations into his semantics, he could say (and did say) just the same thing: Angel and Lumber are actually identical, but they might have been distinct since there is a possible world in which there is a boatish counterpart of Angel that is distinct from some woody counterpart of Lumber (Lewis 1971). It needn’t be that either of these things is identical to Angel/Lumber (in fact, they couldn’t be, given Lewis’s world-bound modal metaphysics), so there was no contradiction. But we have not yet made it explicit how relative-sameness counterpart theory allows for a single modal operator to be associated with more than one relative-sameness relation. Analogously, we have also not yet said how Lewis’s counterpart theory allows for a single modal operator to be associated with more than one qualitative-similarity relation. In each case, what we need is for distinct names within the scope of a single modal operator to each be associated with their own sortal relativization (in our case) or their own respect of similarity (in Lewis’s case). Within the scope of a single possibility operator, we need ‘Angel’ to be associated with the sortal boat and for ‘Lumber’ to be associated with the sortal wood. And this is so not just for names, but for variables as well. Just as we can say that Angel and Lumber are identical even though they might not have been, we can also say that there is a boat x and some wood y such that x and y are identical but might not have been. If we explicitly apply RSC◊ to the case of two variables, then we get the following statement of relative-sameness counterpart theory: RSC◊2 : Relative to sortals Fx and Fy, ‘◊Φ(x,y)’ is true at a world w just in case there is a possible world w’ such that ‘Φ(rFxw,w’ (x), rFyw,w’ (y))’ is true at w’.13

13 We understand ‘Φ(rFxw,w’(x),rFyw,w’ (y))’ to be the result of replacing each occurrence of ‘x’ in ‘Φ’ with ‘rFxw,w’ (x)’ and each occurrence of ‘y’ in ‘Φ’ with ‘rFyw,w’ (y)’. We assume that Φ contains no unnecessarily repeated uses of ‘x’ or ‘y’.

26 | Delia Graff Fara The idea here is that Fx might be the sortal boat while Fy is the sortal wood. This application to two variables can be straightforwardly extended to explicitly cover indefinitely many variables. The matter of how to extend this sort of generalization to qualitative-similarity counterpart theory is not quite so straightforward. (If there is a world in which I and my mother both have multiple personal counterparts, do we have our various counterparts independently? Or do we have our counterparts only as a pair? When Lewis originally presented his counterpart theory, he chose the former.) But we will come to this soon enough. The relevant instance of (RSC◊2) for us right now is this: Instance:

‘◊ Angel ≠ Lumber’ is true just in case there is a possible world w’ in which Angel has a boat-mate a in w’, Lumber has a wood-mate l in w’, and ‘a ≠ l’ is true at w’.

This condition is witnessed in any world in which Angel has a boatmate that is not made up of some wood-mate of Lumber.

6. DEFENDING THE IDEA The main theoretical advantage of counterpart theory (in my liberal sense of that term) is that it allows for the attractive combination of the Identity Thesis, the Accidental Constitution Thesis, and the Necessary Matter Thesis. The particular instances I have focused on are these: (IT)

Angel, Michael’s boat, is identical to Lumber, the wood that makes it up;

(ACT)

Angel might not have been made up of Lumber, the wood that actually makes it up;

(NMT) Lumber cannot but have been made up of the wood that actually makes up Angel. These jointly committed us to the following statement of contingent identity: (CI) Angel = Lumber, but ◊ Angel ≠ Lumber. The claim of possible distinctness (ʹ◊ Angel ≠ Lumber’) is true on our view not because there is some possible world in which iden-

Possibility Relative to a Sortal | 27 tical objects are distinct, but rather because the context dependency of modal predicates prevents co-referential names from being intersubstitutable in modal contexts—no matter whether they flank the identity symbol or not. Despite this, we can coherently affirm the necessity of identity by giving it a nonmodal, extensional expression. (NI)

x = y ⇒ there is no possible world in which x ≠ y.

While both of the counterpart theories we have discussed have the advantage of preserving (it), (act), and (nmt) there are nonetheless important differences between them. We will proceed, then, with our defense of relative-sameness counterpart theory by explaining its advantages over qualitative-similarity counterpart theory. We will subsequently turn to a discussion of Leibniz’s Law and explain why, although one version of Leibniz’s Law is not validated by our theory, there is a superior version of Leibniz’s Law that is. The paper concludes with a summary and in the end suggests why relativesameness counterpart theory provides for a more satisfactory response than qualitative-similarity counterpart theory does to Kripke’s famous Humphrey objection.

6.1 Comparison with Superlative Qualitative Similarity Given the equivalence between ‘¬◊¬Φ’ and ‘◽Φ’, relative-sameness counterpart theory and qualitative-similarity theories of de re necessity may be stated as follows. RSC◽: Relative to sortal F, ‘◽Φ’ is true at a world w just in case every possible world w’ is one in which the following is true: ‘Φ(x’s Fw,w′ -mate)’.14 QSC◽: Relative to respect of similarity F, ‘xˆ◽Φ’ is true of a thing just in case every possible world is one in which xˆΦ is true of all of that thing’s F-ish counterparts. Lewis remarked that his counterpart theory had the consequence that everything exists necessarily (Lewis 1968, 119). To exist neces14 For simplicity of exposition, we are temporarily prescinding from the need for distinct variables to have distinct sortal-relativizations.

28 | Delia Graff Fara sarily according to QSC◽ is for every possible world to be one in which all of your counterparts exist. In possible worlds where you have no personal counterparts, for example—such as those containing only barren deserts not populated by life of any kind—you will have no personal counterparts that do not exist. Even though there might have been no life at all, you exist necessarily. What is perhaps worse, since there is no possible world in which you have a personal counterpart that was never alive, you necessarily were alive at some point in time even though it might have been that nothing was ever alive. Since you are a person that was once alive in every possible world in which you exist (I may assume for the sake of argument) while there are possible worlds in which nothing was ever alive, the following is true according to qualitative-similarity counterpart theory: (17)

◊¬∃x x was once alive ∧ ◽ you were once alive.

Why does our version of counterpart theory allow for the falsity of (17), for things to have lived merely contingently? Worlds in which nothing is the same person as you are not worlds in which the following is true: the same person as you was once alive. In other words, if the functional expression ‘your person-mate’ does not denote anything that exists in some world w, then it does not denote anything that is in the extension of the predicate ‘was once alive’ in w. According to relative-sameness counterpart theory, you have been alive but it is not necessary that you were ever once alive. This, of course, is as it should be. The problem of necessary existence reflects a more general problem. As Allen Hazen in essence pointed out, Lewis’s theory makes no room for the difference between a property’s being among your essential properties and its being necessary that you have that property (Hazen 1979, 317–29). But there is indeed a difference. The contrast is displayed by the following schematic statements, where ‘E’ is the existence predicate. Essentiality: Necessity:

◽(Ex → Φx): ‘x is essentially Φ’. ◽Φx: ‘x is necessarily Φ’.

The difference is clear: I have a property essentially when I could not have existed without having the property. It is necessary that I have a property when it can only have been that I have the property.

Possibility Relative to a Sortal | 29 Take the property of being born a human. I was essentially born human. If a thing wasn’t born human, it most definitely is not me. Any counterpart theorist analyzes this as follows—whether her counterpart relation be sameness based or similarity based. There is no possible world in which I have a counterpart that wasn’t born human.

For the similarity-based counterpart theorist, who allows for multiple counterparts, this is equivalent to his most plausible analysis of what it is for it to be necessary that I was born human. According to him, its being necessary that I was born human is analyzed as follows. Every possible world is one in which all of my counterparts there were born human.

But the similarity-based counterpart theorist is clearly wrong on this score. I was essentially born human—if a thing wasn’t born human it definitely is not me. It is not possible that I was born a frog or a bird or any nonhuman animal. But it is not necessary that I was born human. I might never have been born, so it might have been that I was never around to be human at all. There might only ever have been barren deserts, in which case it would never have been that I was born human. The sameness-based counterpart theorist, whose counterpart relation does not allow for multiple counterparts, makes room for the distinction since the above conditions are not equivalent to the following gloss on her most plausible analysis of its being necessary that I was born human: Every possible world is one in which I have a counterpart that was born human.

As we said, similarity-based counterpart theory never makes room for the distinction between your having a property essentially and it is being necessary that you have it. In what may be the worst case, this includes existence. According to the similarity-based counterpart theorist, our having existence among our essential properties is equivalent to its being necessary that we exist. Let me now develop another of Hazen’s objections to superlativequalitative-similarity counterpart theory. Before I begin I must say something about the analysis of polyadic predicates in the scope of modal operators. We have not yet been explicit about how the simi-

30 | Delia Graff Fara larity-based counterpart theorist analyzes modal claims with polyadic predicates. In the case of modal claims involving only the monadic predicate ‘is a frog’, we said that the similarity-based counterpart theorist analyzes ‘◽(a is a frog)’ as being true just in case every possible world is one in which all of a’s counterparts there are frogs, and ‘◊(a is a frog)’ as being true just in case some possible world is one in which one of a’s counterparts is a frog. With relational expressions, such as ‘being friends with’, a similarity-based counterpart theorist might want to say that “◽(a is friends with b)” is to be analyzed as being true just in case every possible world is one in which every counterpart of a is friends with every counterpart of b. The problem with this analysis is that it renders necessity claims about relations too easy to falsify. There is some possible world in which Abel has two counterparts, Abel1 and Abel2, who tie for most similar with Abel and in which Adam has two counterparts, Adam1 and Adam2, who each tie for most similar to him. There is some such possible world in which Adam1 is father of Abel1 but not of Abel2 and in which Adam2 is father of Abel2 but not of Abel1. Given the aforementioned analysis, the existence of this possible world suffices to render it not essential to Abel that he have Adam as his father. But that should not suffice. We should consider rather the counterparts of Adam and Abel taken as a pair. Although Adam has both Adam1 and Adam2 as counterparts and likewise for Abel, the pair Adam–Abel has only the pairs Adam1–Abel1 and Adam2–Abel2 as counterparts—not the two other mix-and-matches of these. For this simple case, then, the similarity-based counterpart theorist will say that ‘xˆ yˆ◽Rxy’ is true of a pair just in case in every possible world, every counterpart of the pair is one of which ‘xˆ yˆ◽Rxy’ is true. As before, each argument place may be associated with its own respect of similarity. It might be that is a counterpart of the pair only if a’ is a bodily counterpart of a, and b’ is a personal counterpart of b. Now Hazen remarked that the following inference form is not valid on Lewis’s theory, even when Rab is an atomic sentence other than identity: (18) therefore

◽Rab,

Possibility Relative to a Sortal | 31 (19)

◽∃xRax.

Let us consider a concrete example. (20) (21)

◽ Adam is Abel’s father, therefore ◽ Adam is someone’s father.

Lewis analyses the first as true just in case every possible world is one in which every counterpart of is such that Adam’ is father of Abel’. This is to say that there is no possible world in which the pair of Adam and Abel have a counterpart pair with the first member of the pair not being father of the second. But there are possible worlds in which Adam has a counterpart but Abel does not, because there are possible worlds in which Adam has no children at all. But these possible worlds are effectively ignored for the purposes of evaluating (20). These are possible worlds in which the pair of Adam and Abel have no pair as counterpart. So these are possible worlds in which it is vacuously true that every counterpart of the pair of Adam and Abel have a first member that is father of the second member. But for the purposes of evaluating (21), we do not ignore such possible worlds. ‘Adam is someone’s father’ is analyzed as true on Lewis’s analysis just in case every possible world is one in which all of Adam’s counterparts have some child or other. This is false on Lewis’s analysis, since there are worlds in which Adam has a counterpart who has no child at all. But the inference should hold good, at least in the case of logically simple relations, as in (18)–(21) above.15 If it is necessary that a bear a certain logically simple relation to b, then surely it is necessary that a bear that relation to something or other. The failure of this inference to hold good within similarity-based counterpart theory stems at bottom from its failure to distinguish essence and necessity. Since (21) is false and the inference from (20) to (21) is a good one, (20) must be false. Here’s why it is false. There are possi-

15 Given that the atomic predication ‘F( f (x))’ is not true—discounting identity— whenever ‘f (x)’ has no denotation, the negation ‘¬F( f (x))’ is true whenever f (x) has no denotation. For this reason, the truth of ‘◽¬R(x, y)’ does not guarantee the truth of ‘◽∃y¬R(x, y)’. This is as it should be. It is necessary that I am not related to you, but it does not follow that it is necessary that there is someone to whom I'm not related. There are possible worlds in which I and my ancestors are the only humans that have ever existed.

32 | Delia Graff Fara ble worlds in which Abel was never born. So, contra (20), there are possible worlds that don’t contain both counterparts of Adam and Abel with the first being father of the second. It is, though, essential to Abel that he have Adam as a father. Nothing could be Abel if it didn’t have Adam as a father. For the similarity-based counterpart theorist, this is equivalent to its being necessary that Abel have Adam as a father. This in turn is equivalent to its being essential to Adam that he be father of Abel. But because there are possible worlds in which Adam has a counterpart who has no children at all, it is not necessary that Adam be father to some child or other. We want to be able to make the distinctions—and we can— between the pairs of sentences (22)–(23) through (26)–(27). (22)

◽(Abel exists → Abel’s father is Adam),

(true)

◽(Abel exists → ∃x Abel’s father is x).

(true)

therefore (23)

This is the sound argument from ‘Abel essentially has Adam as a father’ to ‘Abel essentially has someone as a father’. (22) is true on our theory because there’s no possible world in which there’s someone who is the same person as Abel but who does not have someone who’s the same person as Adam for a father. (23) is then guaranteed to be true on our theory since it is then guaranteed that there is no possible world in which there’s someone that is the same person as Abel but who does not have (in that world) anyone for a father.16 (24)

◽(Abel’s father is Adam),

(false)

◽(∃x Abel’s father is x).

(false)

therefore, (25)

This is the valid but unsound argument from ‘It is necessary that Abel’s father be Adam’ to ‘It is necessary that Abel have someone as a father’. The latter sentence, (25), is false since there are possible 16 We have not given truth clauses for all of the logical symbols. The details are slightly complex once spelled out fully. I refer readers to Fara (2008) for details.

Possibility Relative to a Sortal | 33 worlds in which nothing exists that is the same person as Adam or as Abel. (26)

◽(Adam exists → Adam is father of Abel),

(false)

◽(Adam exists → ∃x Adam is father of x).

(false)

therefore (27)

This is the valid but unsound argument from ‘Adam is essentially Abel’s father’ to ‘Adam is essentially the father of someone’. (27) is false on our theory because there is a possible world in which someone who is the same person as Adam fathers no-one at all. (26) is then false on our theory because such possible worlds are ones in which there’s someone that is the same person as Adam but who, given that Adam is not the father of anyone, is not the father of anyone who is the same person as Abel. Adam might have fathered only Cain. (There is a possible world in which someone who is the same person as Adam fathers only one son, and that son is the same person as Cain, and therefore not the same person as Abel.) The difference in truth value between and the pairs (22) and (24), (23) and (25), reflects the fact that sameness-based counterpart theory, unlike its qualitative-similarity rival, makes a distinction between essence and necessity. Moreover, each of these pairs represents one of two argument forms that are valid according to sameness-based counterpart theory. Those are the forms: (28)

◽(Ea → Rab) |= ◽(Ea → ∃xRax);

(29)

◽Rab |= ◽∃xRax.

and

6.2 Substitution Failures? No Problem One could complain, against the theory presented here, that it does not validate Leibniz’s Law. Once the complaint is accurately stated, however, it should not worry us too much. There are two different versions of Leibniz’s Law: (I) the sentence-schema version, in which a ‘Φ’ is a sentence-schema to be replaced by a sentence, and ‘Φb’ is

34 | Delia Graff Fara any sentence that results if ‘b’ is put in for ‘a’ in any or all occurrences of ‘a’; and (II) the property version. a

Leibniz Law I (a = b → (Φ → Φb)), Leibniz Law II ∀x∀y(x = y → ∀z(z is a property → (x has z → y has z))). These principles can be easily restated to cover relations of indefinitely large adicity. The first version of Leibniz’s Law is properly called “the principle of Substitutivity of Identicals” rather than Leibniz’s Law. The two may come apart since it may be that ‘a’ does not occur in a referential position in ‘Φa’. If we say “a is Φ,” we may not be attributing a genuine property to the thing named by ‘a’. Substitutivity of Identicals simply is not a valid principle. A classic counterexample is Quine’s: ‘Giorgione was so-called because of his size’. Whether a sentence ‘____was so-called because of his size’ is true once we put in a name for the blank depends not only on what the referent of that name is like, but also on which name we have used to refer to it. That relative-sameness counterpart theory invalidates the substitution principle is therefore not per se a problem with it. Moreover, we have an explanation for why it fails in cases of attributions using modal predicates. Our case is ‘____might have been constituted by different wood’. Whether this sentence is true when we put in a name for the thing that is Michael’s boat—e.g., ‘Angel’ or ‘Lumber’—depends on which sortal is associated with the name that is used to refer to the thing in question. What does it mean to say that different names for a thing can be associated with different sortals? Let me give an example. We have a woman, Larissa Seiger, who is both a ballerina and an elementaryschool teacher. Sometimes people ask, “What is Mrs. Seiger?” In those situations it is typically appropriate to answer by saying that she’s a teacher (my former teacher, my daughter’s teacher, et cetera). Sometimes people ask “What is Madame Larissa?” In those situations it is typically more appropriate to answer by saying that she’s a ballerina (the principle ballerina in company X, Odette in Swan Lake, et cetera). This is to say that different ones of her names are associated with different sortals—teacher or ballerina. So why is it that ‘Angel’ cannot always be substituted with ‘Lumber’ in the modal context in question? Why does this failure not

Possibility Relative to a Sortal | 35 falsify the property version of Leibniz’s law? One could say, with Allan Gibbard, that ‘____might have been constituted by different wood’ does not express a property (Gibbard 1975, 201). Or one could say instead, with Harold Noonan, that the predicate ‘____might have been constituted by different wood’ does express a property, but it is context dependent. Which property it expresses depends on which name is substituted for the blank (Noonan 1991, 188). I am inclined to side with Noonan, but his view would have to be modified so as to account for substitution failures with variables rather than with names. The satisfiability of the following illustrates this point: (30)

There is a boat x and some wood y such that x = y and x, but not y, is possibly not made up of z.

7. CONCLUSION The leading idea here was that relative sameness does not require identity. This was subsequently qualified by saying that transworld relative sameness does not require transworld identity. Michael’s boat actually had one third of its planks replaced by the time it was rowed ashore, but if one half of its planks had been replaced instead, it would still have been the same boat that was rowed ashore in that event, even though the boat rowed ashore in the counterfactual world is not (absolutely) identical to the boat rowed ashore in the actual world. Nonetheless, intraworld sameness—equivalent to what I called sameness simpliciter—does require identity. Whenever x is the same boat as y, x and y are identical. This was not to say that we never have identity between things in different possible worlds. In other words, it was not to say that individuals are “world bound.” Much less was it to deny that there is such a relation as transworld identity. In that sense, I rejected radical modal realism since I think that things in one possible world typically are identical to something in some other possible world. The very thing that is actually Michael’s boat, and also some wood, exists in many possible worlds in which it is not a boat at all. Given the distinction between transworld sameness and transworld identity, the following analyses of de re possibility come apart.

36 | Delia Graff Fara (A) (B)

Relative to a sortal F, x is possibly Φ just in case there is a possible world in which something identical to x is Φ. Relative to a sortal F, x is possibly Φ just in case there is a possible world in which something that is the same F as x is Φ.

The second of these analyses is the one that I favor. Whether it is true to say that x is possibly Φ depends on which sortal we are associating it with. For that reason we said that ‘being possibly Φ’ does not stand univocally for some one property any more than ‘being appropriately called ‘Mrs. Seiger’ ‘ does. Whether or not Mrs. Seiger is appropriately called “Mrs. Seiger” depends on the situation she is in, and on which of the many sortals that she falls under is most relevant for how she is referred to or addressed in that situation. At the dinner table, her daughter should address her as “mommy,” definitely not as “Mrs. Seiger.” One who accepts (B) but rejects (A) is a counterpart theorist in the sense that she thinks that the transworld relation that is involved in the analysis of de re modal claims is something other than identity. This does not mean that her counterpart theory is like Lewis’s in every respect. Like Lewis, I think that whether or not x in world w is a counterpart of y in world w’ depends on what x is like in w and on what y is like in w’. But there are many important respects of disagreement. First, Lewis thought—but I do not think—that if x is in world w and y is in a distinct world w’, then x and y are never identical. Lewis therefore could—but I cannot—think of the counterpart relation as not needing relativization to possible worlds. He said that x is an F-ish counterpart of y (simpliciter) just in case x and y are such-and-such a way simpliciter. I must rather say that x in w is an F-mate of y in w’ just in case x and y are such-and-such a way, relative to w and w’ respectively. Second, Lewis also thought—but I do not think—that whether x in w is a counterpart of y in w’ depends only on comparative similarities. For him, in order for x to be a counterpart of y it just has to be that x is similar to y and no less similar to y than anything else in x’s world, w, is. Moreover, Lewis thought that the relevant respects of similarity were purely qualitative—they never involved any other particular individuals. I think rather that whether x in w is a counterpart of y in w’ depends on sort-relative sameness. Sort-relative sameness is not a comparative relation. There is no sense in saying that x is more

Possibility Relative to a Sortal | 37 the same horse as y than z is. Furthermore, sameness relations are not purely qualitative. The boat that Michael rows ashore in the actual world might be the same boat as the one he rows ashore in some counterfactual world even if in that counterfactual world the boat that Gabriel rows ashore is qualitatively just like the one that Michael actually rows ashore—and even if Michael is, in that counterfactual world, qualitatively just like Gabriel is in the actual one. But like Lewis, I am able to maintain the sensible view that Michael’s boat and the matter that composes it are one, even though they might not have been, since the boat might have been composed of different matter, while the matter could not have been composed of different matter. In extensional possible-worlds talk in terms of relativized counterpart relations, Michael’s boat and the matter it is composed of are identical but they may have something in another world as a boat-counterpart (a “boat-mate”) without having that thing as a matter-counterpart (a “matter-mate”). We saw that a counterpart theorist gets into trouble if his counterpart relations allow for more than one counterpart in some one other possible world. This rendered relative-sameness counterpart theory far more attractive than superlative-qualitative-similarity counterpart theory. Unlike superlative qualitative similarity, sameness is a weak equivalence relation. Moreover, it is functional, albeit partial. Making explicit the relativization to sortal F and double world relativizations to w and w’, we expressed this latter fact by saying that x has at most one Fw,w’-mate. This allowed us to sharpen up (B) above as follows. RSC◊: When relativized to a sortal F, ‘◊Φ(x)’ is true at a world w just in case there is a possible world w’ in which ‘Φ(x’s Fw, w′-mate)’ is true. Treating the functional expression on the right-hand side of this truth clause as a singular term gave us this statement of truth conditions for claims of de re necessity. RSC□: When relativized to a sortal F, ‘□Φ(x)’ is true at a world w just in case every possible world w’ is one in which ‘Φ(x’s Fw,w’-mate)’ is true. A counterpart theorist without a functional counterpart relation is not entitled to the above analyses of de re modal claims. Instead,

38 | Delia Graff Fara he must quantify over a thing’s counterparts in a possible world. Lewis chose to existentially quantify in the case of possibility and therefore to universally quantify in the case of necessity: QSC◊:

Relative to respects of similarity F⃗, ‘◊Φ(x⃗)’ is true just in case some possible world is one in some F⃗-ish counterpart of x⃗ is Φ.

We use ‘x⃗’ to stand for the n-tuple of constants and variables in ‘Φ’ and we use ‘F⃗’ to stand for the series of the various respects of similarity associated, respectively, with each of these constants and variables.17 QSC□:

Relative to respect of similarity F⃗, ‘□Φ(x⃗)’ is true just in case every possible world is one in which every F⃗ish counterpart of x⃗ is Φ.18

The difference between the two counterpart relations enables the relative-sameness counterpart theorist to make a distinction between essence and necessity where the qualitative-similarity counterpart theorist cannot. There are other criticisms of qualitative-similarity counterpart theory that also turn on its allowance of multiple counterparts within a single world. I have in mind in particular those that criticize it for being unable to give a plausible semantics for a language enriched with the modal operator ‘actually’.19 And what of another famous objection to counterpart theory? Kripke complained about Lewis’s counterpart theory that according to it, whether or not Hubert Humphrey might have won the election does not depend on whether Humphrey himself won the election in some other possible world, but rather on whether someone else, a “counterpart,” won the election in some other possible world (Kripke 1980, nt. 13, p. 41). But while Humphrey does care whether he might have won the election, he has no concern whatsoever whether some different person—the counterpart—won the election. The relative17 The qualitative-similarity counterpart theorist who gives up world-boundedness needs to relativize his counterpart relation to pairs of worlds, just as we have done. 18 Murali Ramachandran (1998) has formalized Lewis’s version of sortal-relative counterpart theory by devising a “translation scheme” of the kind we find in Lewis (1968). 19 On this see Hazen (1979, 330) and also Michael Fara and Timothy Williamson (Fara and Williamson 2005).

Possibility Relative to a Sortal | 39 sameness counterpart theorist need not say anything as radical as this, however. The relative-sameness counterpart theorist says that winning the election is a possibility for Humphrey since there is a possible world in which he, Humphrey himself, that man, does win the election. For by this we mean that there is a possible world in which someone who’s the same man as Humphrey does win the election. Whether that man be identical to Humphrey or not, though, he will not be someone else. To be someone else would be to be a different man. Princeton University I intend for this paper to be an informal companion of my ’Relative-Sameness Counterpart Theory’ (Fara 2008). I’m grateful for discussion to Michael Graff Fara, Gilbert Harman, Ned Markosian, Theodore Sider, Robert Stalnaker, Jason Stanley, Judith Jarvis Thomson, and Timothy Williamson. I’m also grateful for discussion to participants of the New York Corridor Workshop as well as audiences at Rutgers University, the 2007 BIRS Mathematical Methods in Philosophy Conference, and the 2007 Arizona Ontology Conference, as well as to Berit Brogaard, my commentator on that last occasion. I’m also indebted to Allan Gibbard, whose (1975) work greatly influenced me and first got me to see that there could be any merit to the idea of contingent identity.

REFERENCES Alston, William and Bennett, Jonathan (1984), ‘Identity and Cardinality: Geach and Frege’, Philosophical Review 93(4): 553–68. Butler, Joseph, D.C.L. (1844), The Analogy of Religion, Samuel Hallifax, editor, Robert Carter & Brothers, New York. Chisholm, Roderick M. (1973), ‘Parts as Essential to their Wholes’, Review of Metaphysics 26: 581–603. —— (1976), Person and Object, Allen and Unwin, London. Fara, Delia Graff (2008), ‘Relative-Sameness Counterpart Theory’, Review of Symbolic Logic 1: 167–89. Fara, Michael and Williamson, Timothy (2005), ‘Counterparts and Actuality’, Mind 114: 1–30. Geach, Peter (1962), Reference and Generality, Cornell University Press, Ithaca. ——, ‘Identity’, Review of Metaphysics 21: 3–12. Reprinted in Geach 1972, pp. 238–47. ——, Peter (1972), Logic Matters, Blackwell, Oxford. Gibbard, Allan (1975), ‘Contingent Identity’, Journal of Philosophical Logic 4: 187–221.

40 | Delia Graff Fara Gupta, Anil (1980), The Logic of Common Nouns, Yale University Press, New Haven. Hawley, Katherine (2001), How Things Persist, Clarendon, Oxford. Hazen, Allen (1979), ‘Counterpart-Theoretic Semantics for Modal Logic’, Journal of Philosophy 76(6): 319–38. Johnston, Mark (1992), ‘Constitution is Not Identity’, Mind 101(401): 89–106. Jubien, Michael (1993), Ontology, Modality, and the Fallacy of Reference, Cambridge University Press, Cambridge. —— (2001), ‘Thinking About Things’, Philosophical Perspectives 15: Metaphysics pp. 1–15. Kripke, Saul (1963), ‘Semantical Considerations on Modal Logic’, in Proceedings of a Colloquium on Modal and Many-Valued Logics, Acta Philosophica Fennica, Helsinki, pp. 83–94. —— (1972), ‘Naming and Necessity’, in D. Davidson and G. Harman, eds., Semantics of Natural Language, Reidel, Dordrecht, pp. 253–355. Reprinted as Kripke (1980). —— (1980), Naming and Necessity, Harvard University Press, Cambridge, MA. Reprinted, with an added preface, from Kripke (1972). Lewis, David (1968), ‘Counterpart Theory and Quantified Modal Logic’, Journal of Philosophy 65(5): 113–26. —— (1971), ‘Counterparts of Persons and Their Bodies’, Journal of Philosophy 68(7): 203–11. —— (1986), On the Plurality of Worlds, Blackwell. Noonan, Harold (1991), ‘Indeterminate Identity, Contingent Identity and Abelardian Predicates’, The Philosophical Quarterly 41(163): 183–193. —— (1993), ‘Constitution is Identity’, Mind 102: 133–46. Perry, John (1970), ‘The Same F’, Philosophical Review 79: 181–200. Ramachandran, Murali (1998), ‘Sortal Modal Logic and Counterpart Theory’, Australasian Journal of Philosophy 76: 553–65. Sider Theodore (1996), ‘All the World’s a Stage’, Australasian Journal of Philosophy 74(3): 433–53. —— (1999), ‘Critical Study of Michael Jubien’s Ontology, Modality and the Fallacy of Reference’, NOUˆS 33: 284–94. ——(2001), Four-Dimensionalism: An Ontology of Persistence and Time, Clarenden Press, Oxford. Stalnaker, Robert (1986), ‘Counterparts and Identity’, Midwest Studies in Philosophy 11: 121–40. Page references are to reprinted version (Stalnaker 2003). —— (2003), Ways a World Might Be, Oxford University Press, Oxford.

2. Reflections on Counterpart Theory Allen Hazen I David Lewis’s project of analyzing modality de re in terms of a relationship of counterparthood between entities from different worlds has given rise to many applications, some more closely related to Lewis’s original intentions than others. At one extreme the additional degree of freedom allowed by a counterpart-theoretic interpretation of quantified intensional languages can be exploited for purely technical ends: Stalnaker 1995 proves the independence of a certain schema from certain axioms by use of a counterparttheoretic model satisfying formal constraints which have no Lewisian philosophical motivation. Other applications are closer to Lewis’s ideas. Delia Graff Fara has recently developed the suggestions in Lewis 1971 to give a general account of a variety of philosophical puzzles about identity. On this proposal there would be more than one counterpart relation, and terms (constants and variables) would each have a counterpart relation associated with them, with no requirement that terms assigned the same object as designatum have the same associated counterpart relation. The truth values of modalized formulas like ◊Fa would depend on the properties of those objects, in various possible worlds, related, by the counterpart relation associated with a, to the object assigned to a. Such a semantics would allow the satisfiability of formulas like a=b & ~◻a=b: the first conjunct would require that the two terms, a and b, be assigned the same object as designatum, but if the associated counterpart relations were different this single object could be related to distinct objects in another world. In Lewis’s original formulation, the counterpart relation was not required to be symmetric or transitive, an object in one world was

42 | Allen Hazen allowed to have multiple counterparts in some other world, and distinct objects in one world were allowed to share a counterpart in another. This is made plausible by his characterization of the counterpart relation in terms of comparative similarity. Many writers have felt, however, that it generates an implausibly weak modal logic, failing to validate a variety of intuitively valid formulas and inferences. (This is, of course, an objection that presupposes a choice of interpretation for the “modal” operators. The logic generated by Lewis’s counterpart theory is too weak as a logic of metaphysical or logical necessity and possibility: one can consistently allow this while claiming that the full generality of the counterpart relation is desirable for other applications. It seems to me at least initially plausible that a semantic analysis of fiction might say that Charles Martel and Charlemagne, though distinct entities in the actual world, have a common counterpart in a world of Medieval French chanson de geste.) To obtain the “standard” quantified modal logic for the logical or metaphysical modalities—a logic whose simplest model theory involves worlds with overlapping domains—on a counterpart-theoretic basis, we have to add non-Lewisian conditions on the counterpart relation: we require it to be an equivalence relation (reflexive, symmetric, transitive), with each of its cells having at most one member in the domain of a possible world. Fara explicitly disavows an analysis of her counterpart relations in terms of similarity, and imposes the equivalence and single representative per world restrictions on each counterpart relation. The resulting semantics is very similar to that of Gupta 1980: the extension of one of her counterpart relations and the corresponding sort in Gupta’s account can be defined from each other by elementary set theoretic means. It seems to capture many intuitions about identity in modal or temporal contexts. In the remainder of this essay I would like to look at modal semantics using a similarity-based counterpart relation. I do not mean thereby to criticize Fara’s work: there are different analytic tasks, and I think the issues about identity that she (and Gupta) address are in a sense orthogonal to those which motivate the project of defining counterparthood in terms of (something like some sort of) similarity. (Fara and Gupta’s project is descriptive, aimed at capturing and systematizing our “native speakers” intuitions. Lewis’s is more foundational or metaphysical, and so eschews

Reflections on Counterpart Theory | 43 metaphysical assumptions he found dubious, even if the validity concept for a quantified modal language derived from such assumptions does not jar our intuitions.)

II Let us consider some very general kind of alethic modality: perhaps what some philosophers have called metaphysical necessity and possibility, perhaps (if this is not the same thing) logical necessity and possibility. There is a certain amount of vagueness here: I doubt that the philosophers who have used and discussed these notions have said enough to single out a unique, precise, sense for their modal operators. I believe, however, that there are meaningful notions in the neighborhood, that at least some of the arguments that have been formulated in terms of metaphysical or logical modality make sense, even if the precise sense of their premises and conclusion may need a bit of further specification. Moreover, vague as it is, the sense that modal reasoning makes is one that claims truth values. Modal reasoning doesn’t have the status of a comprehensible fairy tale, one that can be rewritten with different endings, with the choice of ending responsive solely to aesthetic concerns: rather, some modal statements are right and others wrong (even if others are neither because they are in need of further clarification). And the truth values of modal statements are not isolated: they can have implications for other theoretical or practical questions. (A baseball fan would be a fool to bet that a team will win the pennant when the distribution of wins and losses in the league has made it “mathematically impossible” for the team to finish first: the modal claim supports a practically relevant prediction!) There may be other ways of explaining the status modal statements have, but the obvious one is realism: modal sentences are among those which have a real subject matter, and are true or false in virtue of the facts about this subject matter. Modal statements, in other words, have ontological commitments. Now, one of the useful techniques Quine has taught us is that clarity in the evaluation of ontological commitment is often best served by paraphrasing other kinds of discourse in the language of quantifiers. (I think Quine would be happy to see his

44 | Allen Hazen “criterion of ontological commitment” construed pragmatically in this manner. I am by no means certain that his chosen yardstick— classical first-order logic—is always the best one to use: questions about constructivity in the philosophy of mathematics, for example, may depend on a subtler analysis than the simple determination of what domain the quantified variables range over. Still, examination of explicit first-order formulations often seems illuminating.) So I think we should take talk of possible worlds seriously. There is, I think, compelling reason to take modal talk to be “about” possible worlds. As a logician, I am most impressed by the parallels between the rules of logical inference with modal operators and those for quantifiers: modal logic (particularly alethic modal logic, particularly S5) has a very simple and convenient natural deduction formulation, and the formal rules for the introduction and elimination of the necessity (resp. possibility) operator are, formally, simplified analogues of those for the universal (resp. existential) quantifier. (Simplified by doing without the machinery of variables.) If it walks like a quantifier and quacks like a quantifier . . . I find it very hard not to think of necessity and possibility operators as (lightly disguised) quantifiers. The logical analogy is not perfect: if we compare a formal language of quantified modal logic with a first-order language with quantified variables for possible worlds, interpreting them in the same models, we find that the modal language is expressively weaker, in the sense that every sentence in the modal language can be translated into the possibleworlds language, but some sentences about possible worlds are not (equivalent to) the translations of any modal sentences. Lewis and others, however, have pointed out many examples of informal, natural-language, modal locutions that also do not translate into the standard modal language with quantified variables for individuals, but can be expressed in the worlds language: ordinary modal discourse has a richness that may come closer to that of a formal language with quantified variables ranging over worlds! Indeed, the English expression of a piece of modal reasoning can contain locutions like “in that case” (or “then” in a non-temporal sense) functioning in ways similar to the English pronouns that corresponding to formal variables. Over the years there has been a fair amount of argle-bargle among philosophers as to the nature of possible worlds: what kind of entity

Reflections on Counterpart Theory | 45 are they? Lewis’s own view (which he himself dubbed “extreme” modal realism) was that the actual world—the world we find ourselves in, that we explore—was a typical example of a possible world, and that there was no reason to think that other worlds differed from it in fundamental metaphysical ways: in particular, no reason to think of them as abstract in any sense in which our own surroundings are concrete. Others preferred one or another conception of non-actual worlds as abstract entities: ersatz worlds in the terminology of Lewis 1986. I would argue that modal thinking fundamentally involves pretty much the same sort of abstraction as mathematical thinking: constructing a scenario to show that some proposition is not necessarily true seems fundamentally the same sort of intellectual activity as defining a weird space to refute a conjecture in topology. Epistemology and metaphysics ought to have something to do with one another, and it is tempting to think that subject matters we find out about in similar ways are of similar nature. My own preferred view, then, would be a variant of what Lewis 1986 calls linguistic ersatzism, with certain set-theoretic objects (viz, models in the sense of standard model theory) taking the place of his concrete possible worlds. In the bigger philosophical picture, I think such a view would represent progress in that, though ontological and epistemological problems about mathematics would remain, we would be rid of the separate ontological and epistemological perplexities faced by Lewis’s modal concretism. This is, however, an issue independent of the problem I wish to discuss here: I think what follows is equally applicable on my view or on Lewis’s as to the nature of non-actual worlds. In one respect, however, Lewis was an ersatzist. Our modal reasoning is often de re, and the simplest way to accommodate this in a possible-worlds framework is to allow the very same people (and other entities) who have one set of characteristics in one world to exist with different characteristics in other worlds. There is an obvious analogy, here, with temporal language: we make de re past and future tense statements, and their truth conditions rest on having people and things continue to exist, but with changing properties, over time. (The analogy, of course, has been successful and fruitful for semantic theory: David Lewis and Delia Graff Fara are only two of the many writers who have given parallel accounts, using the same or at least closely analogous technical concepts, of modal and

46 | Allen Hazen temporal semantics.) The criteria of personal or material object identity over time—memory, spatio-temporal continuity, assorted causal relations—just don’t apply from one possible world to another. Whether identity over time can be fully analysed or defined in terms of spatio-temporal continuity, etc. is of course a controversial issue. What should not be controversial is that there is a close conceptual connection: assertions that an object at one time is identical to one at another would be pointless if identity wasn’t related to these other things. Our understanding of identity through time depends on these connections; if we didn’t understand them we wouldn’t have the concepts of persons or material objects that we do. A theory of possible worlds simply postulating the literal identity of inhabitants of different worlds while admitting that this identity has no connections with “criteria” of identity would make identity (in the transworld case) into a completely “opaque” primitive, and leave it utterly mysterious how we could ever grasp the concept. So something else must take the place of the familiar criteria of identity. Similarity relations, in a suitably broad and abstract sense of similarity, are the obvious candidate. The entity in some other world that I take, in my modal thinking, for you will have to resemble you at least to the extent of sharing your essential properties. The obvious proposal for defining transworld identity is thus simply that, in the absence of anything else that looks hopeful, we should take this necessary condition as also sufficient. The problem is that this doesn’t work. Identity is an equivalence relation, and it is necessarily true (so: true in every possible world) that each object is identical to itself and not to anything else. But, however carefully we specify the sort of similarity between objects in different possible worlds we want to have be criterial of transworld identity, there is bound to be an object in (say) the actual world which bears this sort of similarity to two distinct objects in some other possible world, and this would force us to say that the object in the actual world was identical to two objects not identical to each other: a manifest repugnancy. (One could, I suppose, try saying that x, in world w, is identical to y, in world v, just in case they bear the appropriate sort of similarity to each other but x does not bear that sort of similarity to anything in v other than y and y does not bear that sort of similarity to anything in w other than x. This is an unattractive option, however: at least some worlds are big places, containing

Reflections on Counterpart Theory | 47 many similar objects, with the result that anything resembling one of them in some specified way would resemble many. And, indeed, we want to allow an object to be transworld identical to an object that closely resembles some other object in its world: I might have had an identical twin is intuitively plausible.) The logical point can be made without assuming “world-bound” individuals. Suppose, for the sake of argument, that individuals literally exist in multiple worlds. We can still define a domain of world-bound pseudo-individuals as, say, ordered pairs of a world and an individual existing at it. Identity of individuals induces a pseudo-identity relation over these pseudo-individuals, and this relation is an equivalence with no cell containing two distinct ordered pairs with the same world as a component. We can also define pseudo-similarity relations over them: relations hold between pairs and according to the manner and degree of similarity and difference between what i is like at w and what j is like at v. (Note that whether a pseudo-similarity holds between and is independent of whether i=j: our ability to say that someone “has changed,” or that he “looks just like his father at that age,” depends in effect on our understanding of the analogous intertemporal similarity relations as independent of the identity or distinctness of the terms.) Stated in this framework, the problem is that no relation between world-individual pairs defined in terms of pseudosimilarity relations will coincide with the relation induced by identity of individuals. All this should be acceptable even by the staunchest defender of transworld identity. For Lewis it was a starting point, providing (what must have been for him and certainly is for me conclusive) grounds for scepticism about literal transworld identity. So Lewis went ersatz. Instead of postulating the sort of multiverse our intuitive modal thinking seems to be about—one in which the same object often inhabits, and has different characteristics in, multiple worlds—he postulated a replacement multiverse in which no object exists in more than one world. (Those of us who prefer to think of worlds as set-theoretic entities can enforce the disjointness of worlds by stipulation: let non-actual worlds be models of the sort usual in model theory, but take the “objects” existing in them to be, not the entities in the domains of the models, but ordered pairs of these with the models themselves. Assorted ordinal numbers, for

48 | Allen Hazen example, can then be in the domains of many models, but when thought of as objects existing in worlds they are distinguished by using the models themselves as tags.) By itself, making the sets of objects inhabiting different worlds disjoint would trivialize de re modal locutions: since no object exists in any other than its “home” world, it would follow that if an object has some characteristic, it is a necessary truth that it has it. (Leibniz, of course, was tempted to say precisely this when speaking strictly, but most of us want a semantics for our vulgar modal language, a semantics that allows non-trivial de re modality.) Lewis’s counterpart theory was an attempt to provide a non-trivial semantics for the de re without assuming that individual objects could genuinely inhabit (and possess different properties in) multiple possible worlds. De re modal statements were explained in terms of a sort of resemblance—counterparthood, to give it its technical name—between objects in different worlds, but without taking it to define identity of objects. Unfortunately, as I argued in the critical part of my dissertation (basically reproduced in Hazen 1979), Lewis’s theory gave the wrong truth conditions, and even the wrong formal modal logic, for de re sentences. The (fundamental) problems stemmed from the fact that, on Lewis’s theory, an object might have multiple counterparts in some worlds. There are some minor problems—such as those of what I called “There but for the grace of God” examples: “I might have led a life like the one you have actually and vice versa”—that can be solved by fine-tuning the account of what sorts and degrees of resemblance make for counterparthood, but solving them if anything makes the main problems worse. The biggest problem, insoluble I think without some fundamental change to Lewis’s framework, has to do with relational de re statements. Intuitively we want to say that it is essential to some individuals that they be related in specific ways to specific other individuals: it is essential to Queen Elizabeth II, perhaps, that King George VI and no one else was her father. Consider, however, a possible world with two England counterparts (on planets in galaxies far, far apart), each with a pair of a royal father and royal daughter resembling the actual George VI and Elizabeth II enough to be their counterparts. On Lewis’s original formulation of the truth conditions of de re statements, such a world would prevent Elizabeth from essentially having George as her father: she has a

Reflections on Counterpart Theory | 49 counterpart in that world who is not the daughter of all the counterparts of George in that world. One can tinker with the truth conditions (substituting, say, existential for universal quantifiers, etc.), but nothing seems to work in general. What is needed, then, is something which yields what Fara, in her paper, has assumed: some way of picking out one of the perhaps many counterparts an object has in some world, and allowing the chosen counterpart, but not the others, to figure in our semantic account of de re modalities.

III When I wrote my dissertation, and for many years afterward, I thought I had solved the problem with what I called stipulational semantics. This took worlds as the basic truth-givers for modal locutions, like Lewis’s theory, but replaced the counterpart relation with a family of (partial) functions from the domains of worlds into the domains of other worlds. (As a first approximation, we can think of one of these functions as mapping an object in one world that has at least one counterpart in a second world to one of these counterparts. As a second approximation, note that, in order to give an account of essential relational predication, we may want the choice of a counterpart for one object to depend on which counterpart in the second world is chosen for another object, and this can be done by allowing the family of “counterpart functions” from one world to a second to be a proper subset of the set of all functions mapping objects in the first to counterparts in the second: Lewis, in the afterword appended to Lewis 1968 when it was reprinted in his Philosophical Papers, reformulates this idea in terms of a counterpart relation holding, not just between simple objects, but between fusions of objects.) Or rather, that is what it would have been for a simple modal language without nesting of modal operators. (This simplified modal language and its semantics are outlined in the appendix to Hazen 1979; much, though of course not all, intuitive modal reasoning can be expressed in this fragment of quantified modal logic, and I still think my semantics is perfectly adequate for it.) For the full language of standard first-order modal logic my semantics had more complicated set-theoretic machinery. What

50 | Allen Hazen played the semantic role of a possible world in my semantics—the entity “at” which a formula, on an assignment to its variables, is evaluated—was not the underlying world but an ordered pair , where S is a finite sequence of worlds (the sequence starting with the actual world), and I (an “identification”) is a partition of the union of the domains of the worlds in the sequence, with each cell of the partition containing at most one object in the domain of any particular world. (The partition I is obtained from an equivalence relation which itself is the union of one counterpart function for each ordered pair of worlds in S, but not all such unions will do: the different counterpart functions have to be “compatible.”) This I, obviously, takes the semantic place of the counterpart relation: if d is an object in the domain of one world of S, the truth values of modal formulas about it are determined by the properties in other worlds of the members (if any), in those worlds, of its I-cell. In a fully developed semantics (my discussion in my thesis didn’t go much beyond providing a model theory) conditions, formulated in terms of some sort of similarity between objects, would be placed on the class of admissible I, in analogy to Lewis’s proposal in (3) that a counterpart relation be defined in terms of degrees of similarity between objects. The “shortest” such pair—adequate for formulas containing no modal operators—would simply consist of the actual world (construed as a sequence of length + 1) and the discrete partition on its domain. For any pair with S of length = n, define its direct extensions to be the pairs , where S+ is the sequence (of length = n+1) obtained from S by adding one more world at the end, and I+ is an admissible “identification” partition of the union of the domains of worlds in S+ which agrees with I when restricted to the union of the domains of worlds in S. Where S is a finite sequence of worlds, let end(S) be the last world in S. (In particular, if S is the actual world, @, construed as a oneworld sequence, end(@) = @.) The basic idea of my semantics was that a formula embedded within the nested scopes of n modal operators should be evaluated at end(S) for a sequence of length = n+1. The quantificational logic the semantics was designed to validate was a version of Universally Free Logic with an existence predicate, so constants were allowed to be vacuous. Non-vacuous constants had designata in the domain of @. (It wasn’t clear to me then, and

Reflections on Counterpart Theory | 51 still isn’t, how to extend the semantics to allow constants to act as names of non-existent objects that exist in other worlds. I took this to be a desirable feature of the semantics from an analytic point of view: it provided an explanation (in some sense) for Kripke’s thesis that Sherlock Holmes couldn’t possibly have existed.) Variables were handled by assignments, which were partial functions from the set of variables into the union of the domains of the worlds. (Allowing assignments to be partial amounts to allowing free variables to be “non-denoting singular terms.” This was a departure from the literature on Free Logic I was familiar with, but seemed convenient in the context of modal logic.) For any assignment, a, and term, t, we define a(t), the denotation of t on a, by the clauses I. if t is a vacuous constant, a(t) is undefined, II. if t is a non-vacuous constant, a(t) is the designatum of t, III. if t is the variable x and a is undefined for x, a(t) is undefined, and IV. if t is the variable x and a is defined for x, a(t) is the value given by a for x. Given a pair , an assignment a, and a (Lewis-)world w, we define, for an expression d of the form a(t), rep(d,I,w) (“the representative, under I, of the object d in the world w”) to be the unique object in the domain of w which is in the same cell of I as d if there is such an object, and undefined otherwise. Note that rep(d,I,w) is undefined if and only if d is undefined, or w is not in S (since in that case I isn’t defined over the domain of w), or d is defined but stands for an object whose I-cell does not contain a member of the domain of w. Convention: an expression for an n-tuple of objects is taken to be undefined just in case one or more of the expressions d,e,f, . . . is undefined. Finally, we define by recursion what it is for a formula of the language of first-order modal logic to be true at a pair and an S-assignment a: i.

an atomic formula, F(t,u,v, . . .), with an n-ary predicate F and n terms t,u,v, . . . , is True(,a) if and only if the n-tuple

exists and is in the extension in end(S) of the predicate F,

52 | Allen Hazen ii. a conjunction, (A&B), is True(,a) if and only if both of its conjuncts, a and B, are True(,a) (and similarly for other truth-functional compounds), iii. a universal quantification, ∀xA, is True(,a) if and only if A is True(,a*) for every S-assignment a* which assigns x an object in the domain of end(S) and is otherwise like a (and dually for existential quantifications), and iv. a necessitation, NecA, is True(,a) if and only if A is True(,a) for every direct extension of (and dually for possibility formulas). (Note that clause (i) of the truth definition adopts the “falsity convention,” that atomic predicates are never true of non-existents. This seemed to me (and still seems to me) plausible for “ordinary” predicates: those expressing physical properties, for example. The analysis of “intensional” predicates (e.g. worships) seemed to me a topic for another thesis.) Note also that the truth definition would still make sense if we allowed the identification, I, to be defined over the domains of the worlds in a longer sequence than S: this makes possible a fairly simple extension to cover the Actuality operator (for which I could see no adequate treatment in Lewis’s original semantics). Abusing the notation a bit, v. an “actualitation,” AA, is True(,a) if and only if A is “True(,a)”: true, not in end(S), but in the actual world @, with the denotations of terms, however, taken as rep(d,I,@), where I is the identification from .

IV Before going on to discuss the possible inadequacy of even this complicated variant of Lewis’s semantics, let me say something about why I called it stipulational semantics. Many philosophers have been convinced by Kripke’s arguments in Naming and Necessity that “the problem of transworld identification” rested on conceptual confusion from the start: we don’t first discover objects in other possible worlds and then, after further

Reflections on Counterpart Theory | 53 investigation, discover which ones are to be identified with which actual objects: as Kripke put it, we stipulate possible worlds, and as part of this stipulate the identities of their inhabitants. There is no need, they may therefore be tempted to argue, for any special machinery of transworld identification, either in Lewis’s original counterpart relation or my more elaborate version. I find such a response to the problem unsatisfying. Kripke’s use of “stipulate” is metaphorical: in the literal use of the word we speak of stipulating deadlines or preconditions or conventions or definitions, not worlds. (The kind of stipulation most familiar in logical and philosophical discourse is, of course, stipulative definition. It seems to me that a rough characterization of the ordinary use of the verb, inspired by this case but capable of more or less neat extension to others would be: stipulation is a performative act—an exercitive in the terms of Austin’s (1960) classification—which establishes at least a temporary linguistic convention.) The analytic problem of transworld identification that remains, after the confusion is cleared away, amounts to that of saying what the literal content of Kripke’s metaphor is. We can’t just stipulate possible worlds by means of arbitrary descriptions: we cannot, for example, stipulate a world containing round squares. It is up to us to decide what sense of possibility—historical, physical, metaphysical, etc—we wish to express, but once we have made that decision it is a matter for investigation rather than for further stipulation what sorts of worlds are possible in that sense, and round squares are not historically or physically or metaphysically possible. The idea that there is a class of worlds (or perhaps, after further epistemological and ontological consideration, of suitable ersatzes) which is not subject to stipulation and which determines what the de dicto possibilities are is simply a recognition of this truism. The usual formal languages of quantified modal logic, however, allow the formulation of de re modal statements, and, as Kripke’s own discussion dramatically emphasized, it seems intuitively that many de re statements have non-trivial truth values: thus a full semantics for modal language needs something beyond the class of worlds. Now, what the essence of an object is, and so what de re statements about it are true, seems to depend on what sort of thing it is: on its intrinsic and relational properties. (Kripke does not disagree: it is, after all, because Queen Elizabeth II is a human person and has in fact

54 | Allen Hazen the relational property of being the daughter of George VI and Elizabeth the Queen Mother that she has this property essentially.) This, it seems to me, is an important truth, and Lewis’s proposal that the counterpart relation should be determined by similarities in respect of intrinsic and relational properties was an attempt to give a precise statement of its significance. Any adequate semantic account of de re modality will recognize this as part of the story, and as a part which constrains anything plausibly described as stipulation. Once we have decided to mean metaphysical possibility, in Kripke’s sense, by “possible,” we are not at liberty to “stipulate” a possible world in which Queen Elizabeth II has other parents. As Lewis pointed out (and as was implicit in other discussions of the so-called problem of transworld identification in the late 1960s) the definition of a counterpart relation based on similarity does not amount to a characterization of transworld identity in terms of similarity: if nothing else, an object can have many counterparts in a single world. And, as I argued in criticizing his proposal, the counterpart relation by itself does not suffice for an intuitively plausible semantics of de re modality. It is at this point, and only at this point, that something like stipulation enters the picture. When we say, for example, that Queen Elizabeth II might not have had any sons (in some chosen sense of “might”), this is true because there are possible worlds containing objects enough like Queen Elizabeth II in their intrinsic and relational properties to be admissible representatives of her— in a word, counterparts of Queen Elizabeth II—who have no sons. If, however, as seems overwhelmingly likely, some possible universe contains more than one such counterpart, there is no need to try to define which one is “really her” in that possible world. It is as if we could arbitrarily choose, or stipulate, one counterpart to represent her. But of course we don’t literally do this! We don’t spread out a possible world like a giant map and point to the counterpart we choose. A rigorous semantic theory (which is not what Kripke was trying to expound in the lectures published in Naming and Necessity) must not depend on such metaphors. I tried to outline such a semantics: I proposed that modal statements had the same truth conditions as certain general statements about abstract objects (the counterpart functions, the I components in the pairs ) which can be thought of as, well, counterparts of these imaginary stipulations.

Reflections on Counterpart Theory | 55 V The semantic machinery described in Section III is designed to cope with sentences containing nested modal operators. It now seems to me that it may fall down if the inner modal operators do not have nested scopes, but are allowed to occur “side by side.” For the sake of example, assume the correctness of Kripke’s thesis that human persons have their parentage essentially: if parents x and y have child z, then necessarily (if z exists, then x and y exist and have z). It follows that siblinghood is an essential relation: if x and y are siblings, then necessarily (if x and y both exist, they are siblings). On the other hand, it seems plausible that it is not essential to one of a set of siblings that the other(s) should exist. (One’s intuitions about this may be strained by monozygotic twins. Further, anyone who thinks the whole of history up to the time a contingent being comes to be is essential to that being—a view which has been suggested by some, but which seems extreme to me—would think that the existence of one’s older siblings is essential to one. For the sake of having an example, let us assume that neither of these worries is relevant.) Now, I am an only child, but I might have had siblings, siblings who might have existed without me. The following statement (of modal degree = 2 but with two inner modalities which are not nested) seems reasonable (if perhaps unreasonably complex!) on the assumptions of our example: (*) It holds necessarily of any two people x and y, that if it is possible that x and I both exist and are siblings and also possible that y and I both exist and are siblings, then x and y are siblings. Or, in symbols, with ◻ and ◊ for modal operators, S for the siblinghood relation, a for me, and leaving out the clauses about existence as made redundant by the falsehood convention on S: (*) ◻(∀x∀y((◊(S(ax)) & ◊(S(ay))) => S(xy))). On the semantics of my dissertation, however, (*) comes out false (or rather, more accurately, comes out dangerously close to coming out false). Evaluate at the actual world. Since it is a necessitation, what follows its initial operator must come out true at every of length =

56 | Allen Hazen 2, , if (*) is going to be true at @. But suppose (as seems likely) that there is a world, w1, in which I have no counterpart but in which there are two objects, x and y, which are not siblings (perhaps they live in different galaxies within w1) but which each have the appropriate (intrinsic and relational) properties to be siblings of me. Since the class of admissible identifications (like Lewis’s original counterpart relation) is supposed to be determined by the (intrinsic and relational) similarities between objects in different worlds, each will be a counterpart of some sibling of mine. Now the fact that the two possibility clauses in (*) are not nested becomes a problem. ◊(S(ax)) is true at because there is some world, w2, containing a pair of siblings s and t such that there is an identification I’ defined over the union of the domains of the worlds in the sequence whose restriction to is just I and which identifies me with s and x with t. ◊(S(ay)), on the other hand, is true at because there is some world, w3, containing a pair of siblings u and v such that there is an identification I’’ defined over the union of the domains of the worlds in the sequence whose restriction to is just I and which identifies me with u and y with v. (Perhaps, indeed, w3 = w2, and even s = u and t = v: all that is required is that I’’ not be identical to I’.) Which would make (*) false. Now, in fact the counterexample as described doesn’t quite work. The essentiality of the siblinghood relation, recall, is derivative from the essentiality of parentage. For I’ to identify object t with a sibling of an object, s, that it identifies with me, the domain of w2 must contain parents it identifies with my parents, and these parents must have both s and t. By the essentiality of parentage, any other world—in particular, w1—containing an object, like x, that I’ identifies with t must also contain parents I’ identifies with the parents of t, and which, therefore, it identifies with my parents. Mutatis mutandis for I’’, w3, and u. But, ex hypothesi, x and y are not siblings, so I (which agrees with both I’ and I’’ when restricted to the domains of @ and w1) cannot identify my parents both with the parents of x and with the parents of y. But let’s leave my parents out of this! Suppose there were a relation with the intuitive properties of siblinghood (viz. that it holds essentially between any objects it relates, and if it possibly relates an object to a second and possibly relates the second to a third,

Reflections on Counterpart Theory | 57 then necessarily, if the first and third both exist, it relates the first to the third: crossworld transitivity, to give it a name) but which was not derivative from another relation to further objects. I am not at all confident that there really are such relations (though, if your biological species is part of your essence, being conspecific with might be one), but I also don’t feel that I could, with any confidence, declare that there are none. So it seems to me that the modal semantics of (1) is, at least, exposed to a counterexample of the structure described here. Since one of my main criticisms, in both (1) and (5), of Lewis’s original semantics (and thus one of the main motivations for my complicated modification of it) was that it didn’t handle examples involving essential relations, it is acutely embarrassing to me that my own semantics may have problems with more complicated examples involving essential relations.

VI In Sections II and IV above, I argued that we need some account of the semantics of de re modality that, like Lewis’s counterpart theory, avoids an appeal to “brute” presence of the same objects in different worlds. Lewis’s original account is unsatisfactory, and we have just seen that the variant of it that I devised to repair its inadequacies may not work either. What to do? One obvious thought is to use “identifications” defined over the domains of bigger systems of worlds. The problem in the previous section was that I’ was an admissible “stipulation” of identities over the domains of worlds in the sequence and I’’ was admissible over but that they were incompatible. Why not require that an identifier be admissible over the full set of worlds relevant to the evaluation of the sentence: {@,w1,w2,w3}? There are minor and major technical difficulties, and a conceptual issue. In my proposed semantics, there was no need to specify which world in a set was relevant to the evaluation of a given subformula: evaluation is always at the last world in a sequence whose length is determined by the depth of modal embedding. When we go from sequences to sets we will have to index the worlds, and the identifications will have to be partitions, not of the simple union of their

58 | Allen Hazen domains but of their disjoint union. (Recall, from the discussion in Section III, that w2 and w3 can be the same world, with t (= v) identified with different objects in w1 for purposes of evaluating the two possibility subformulas.) This is ugly, but doable. More seriously, it is not at all obvious to me how to formulate the truth definition: somehow the definition will have to provide that the set of possible worlds is big enough, that adding a world (and extending the “stipulation” of identities to its domain) wouldn’t alter the truth value of the sentence. And—I think this leads to a serious conceptual issue— the set of worlds involved will generally have to be quite large. The discussion in Section V mentioned only four worlds in toto, but managed this by ignoring the fact that the embedded modal subformulas contained variables bound by quantifiers outside them. In the general case, quantifying in might force us to include a new world for each object in the domain of a given world. One way (but one I do not want to take) of not worrying about just what worlds have to be included would be to consider identifications defined over the union of the domains of all the worlds. (Dan Marshall, of Monash University, seemed to be proposing something like this a couple of years ago in work in progress.) My reason for not wanting to do this (both when I wrote my thesis and now) is ontological. My “working philosophy of mathematics,” for these purposes is that Zermelo–Fraenkel set theory (with urelements) is a true (though of course very far from complete!) description of the whole universe of abstract entities, and I wanted every entity mentioned in my semantic theory to be identifiable with something in the intended interpretation of ZF(U). (So in particular, my preferred candidates for ersatz worlds are particular sets.) On the other hand, I accept as a thesis of modal metaphysics that, for every cardinal number, it is (metaphysically) possible for there to be that number of concrete individuals. Thus the worlds themselves would not form a set. An identification defined over the union of the domains of all the worlds would have to be a proper class, and as a good ZF-ist I am willing to countenance proper classes only as a façon de parler, and only when they can be explicitly defined. (My view of proper classes is close to that presented in Parsons 1974.) Perhaps the issue can be avoided. I am at least somewhat attracted to the idea that it is essential to any concrete object in a given world that it inhabit a world whose fundamental physics is the same as

Reflections on Counterpart Theory | 59 that of the given world. If the fundamental physics of a world determines the structure of its space-time (or whatever takes the place of space-time in it), then this would put a limit on the number of concrete objects that could coexist with a given object: it will exist only in worlds with domains of some limited cardinality. And this, in turn, might allow us to restrict our attention to identifications defined over the unions of the domains of sets of worlds. I am not confident of the intuition here, however, and even if it is valid for some conception of essence, there may be some understandable and useful modalities requiring objects to inhabit . . . unusual . . . worlds. After all, there are non-trivial counterfactuals whose antecedents require an object to lack properties ordinarily thought essential to it: if Queen Elizabeth II had been the daughter of Rabbi and Mrs Kripke of Omaha. . . .

VII To summarize the conclusions of the last section, semantics would be easier if we could postulate an identifier (an equivalence relation, relating each object to at most one object in the domain of any possible world, associating unique “counterparts” in other worlds to objects) defined over the (union of the) domains of absolutely all possible worlds. Such a relation would have to be a proper class, not a set, but for proper classes, esse est definiri. And it was already clear to Lewis and others in the late 1960s that it was not possible to define a way of identifying objects with unique otherworldly counterparts on the basis of similarity (broadly construed). At this point it is tempting to try theft instead of honest toil: why not be Fictionalists about transworld identity? Interpret our modal discourse, that is, as based on the assumption that there is a properclass identifier, but don’t claim that the assumption is true. The problem here is common to many forms of fictionalism. Modal discourse is serious: modal reasoning can form part of our route to non-modal theoretical or practical conclusions, and a satisfying philosophical account of it will provide some assurance that the conclusions reached will be sound. A non-fictionalist semantic theory—based either on “extreme modal realism” or on some version of ersatzism that specifies particular abstracta as playing the role of

60 | Allen Hazen worlds—has a simple answer here: modal reasoning may be about a special domain, but it is nonetheless about a domain of actually existing entities, and the specific assertions it assumes about these entities are true. It may be, in the end, that a fictionalist account of transworld identity is the best we can hope for, but even fiction, if it is to play a role in serious reasoning, needs some sort of justification. The situation is analogous to one that arose in the foundations of mathematics early in the twentieth century. Modern mathematics makes strong assumptions about infinite sets, and uses them in deriving conclusions within “applicable” mathematics. Now, a naïve Platonism (such as the ZF-ism I described earlier as my working philosophy of mathematics!) will not find this disturbing: it holds that the infinities assumed actually exist and that the mathematical axioms appealed to in modern proofs are simply true. Such an attitude, however, was not convincing for critical minds, and did not even yield any conviction that the infinitistic axioms of set theory were consistent! Hilbert, in particular, argued at some length that there was no reason to believe in the actual reality of infinite sets. His view on infinitistic set theory can, I think, with some plausibility be described as fictionalistic: he did not hold out the hope of finding a semantic interpretation on which the axioms of set theory would be seen to be true. But— and this seems to me evidence that his views were deeper and more sophisticated than most of the fictionalisms proposed in the recent philosophical literature—he made a concrete proposal about how to justify the continued employment of “ideal” or fictional theories. Hilbert’s program of consistency proofs was meant to find proofs that the infinitistic theories of modern mathematics would have true consequences in applicable mathematics, proofs that did not depend on finding a true semantic interpretation for those theories. In Section III above, I argued that a certain restricted modal language—essentially the fragment of quantified modal logic in which no necessity or possibility operator stands within the scope of another—did have an adequate semantics in terms of “counterpart functions.” It is not just a formal language: it is an interpreted language, one whose sentences have (often unknown) truth values. We might take this fragment as the analogue, for modal discourse, of the applicable mathematics of which Hilbert wanted “ideal”

Reflections on Counterpart Theory | 61 mathematics to be a consistent extension. A fictionalist analysis of full quantified modal logic (or of some richer language with explicit quantification over possible worlds) proposes that this fragment be embedded in a theory postulating the existence of a “universal identifier”: an undefinable proper class of ordered pairs of objects in different worlds which, for each pair of worlds, coincides with some counterpart function between the domains of those worlds. This theory is not true, but we would like some assurance that it is consistent with all the truths of the restricted modal fragment, so that reasoning making use of its fictitious postulate will not lead to false conclusions in the fully interpreted part of its language. I think there is some reason to hope that a proof of this consistency or conservativity claim is possible. The usual axiom of choice for Zermelo-Fraenkel set theory (“Local Choice” in the jargon of the trade) guarantees that, for every set of non-empty sets, there is a selection function: a mapping that takes each set belonging to the given set to one of its members. Some formulations of set theory, however, have a stronger choice principle: Global Choice. This can be expressed in an extension of the usual language of ZF(U): add a function symbol σ (the selection operator or selector) with an axiom saying that if x is a nonempty set σ(x) is one of its members. Since the selection operator is taken to apply to all sets, σ, on a classical semantic interpretation of this stronger system, would have to denote a class function: a proper class of ordered pairs, but one which, restricted to any particular set of non-empty sets, would coincide with one of the selection functions for that set postulated by the Local axiom of choice. Now, the ZF-ist philosophy of mathematics I am presupposing here is thoroughly realist about ordinary set theory: set theory as expressed in the usual language of ZF(U). Even the most fanatically Platonistic upholder of this philosophy, however, should be a fictionalist about the extended version of set theory with a selection operator. Proper classes are façons de parler, and should only be asserted to exist when they can be defined, and there is no reason to suppose that a class suitable to interpret σ can be defined. Global Choice, however, is a justifiable fiction: it can be proven that Zermelo-Fraenkel set theory with Global Choice is a conservative extension of ordinary ZF(U)C. The easiest proof is by a forcing construction. Cohen’s original application of forcing used finite sets, thought of as finite approximations

62 | Allen Hazen to an indefinable infinite set. The “finite” part of that isn’t essential: forcing constructions are possible in which some other limitation of size is imposed on the “small” approximations. A forcing construction for showing that ZFC + a selection operator is conservative over ZFC would use, as forcing “conditions,” selection operators defined over sets: thus, for example, we could allow as a condition any function, defined over sets of rank less than some ordinal, mapping each nonempty set in its domain of definition to one of its own members (and the null set to some throwaway value). So perhaps we can do something like this. Let a condition be an identification scheme defined over a set of worlds: an equivalence relation over the union of their domains, each cell of which has at most unit overlap with the domain of each of the worlds in the set. One condition will extend another if it operates over a superset of the other’s set of worlds and the two coincide over the worlds the smaller one operates on. We’ll want to add some further constraints: (i)

For some cardinal number it is the case that for every world with a domain of that or smaller cardinality there is an isomorphic world in the set the condition operates on.

There’s a metaphysical presupposition here: that the “signature” of a possible world—the length of the sequence of its “natural properties,” in terms of which its isomorphism type is defined—is bounded by some function of the cardinality of its domain. I think I’m willing, tentatively, to buy into this. Perhaps it makes sense to speak of a world in which very many distinct natural properties are coextensive, but in that case we can take a more abstract view of things: allow the Armstrongian universals existing at a world to be part of its domain, and limit the “signature” to properties of being an individual, being a universal, and instantiation. (ii) For any worlds W1 and W2 in the set on which a condition operates and any counterpart function F from the domain of W1 to the domain of W2, there is a world W3 in the set which is isomorphic to W2, and the restriction of the transworld identification relation to Dom(W1) X Dom(W2) is the composition of F with an isomorphism from W2 to W3.

Reflections on Counterpart Theory | 63 (Note that it is a consequence of the idea that counterparthood is somehow a matter of similarity that composition of a counterpart function with an isomorphism between worlds will always be a counterpart function.) This is very far from being a fully worked-out proof. The breeze you just felt was produced by hands waving. University of Melbourne REFERENCES Austin, J.L., How to Do Things with Words, (Oxford: Oxford University Press, 1960). Cohen, P.J., Set Theory and the Continuum Hypothesis, (New York: W.A. Benjamin, 1966). Gupta, A.K., The Logic of Common Nouns: an investigation of quantified modal logic (New Haven: Yale University Press, 1980). Hazen, A.P., “Counterpart theoretic semantics for modal logic,” Journal of Philosophy 76 (1979), pp. 319–38. Kripke, S.A., “Naming and necessity,” in G. Harman and D. Davidson, eds., Semantics of Natural Language (Dordrecht: Reidel, 1972); reprinted with new preface as S.A. Kripke, Naming and Necessity (Cambridge: Harvard University press, 1980). Lewis, D.K, “Counterparts of persons and their bodies,” Journal of Philosophy 68 (1971), pp. 203–11; repr. In D.K. Lewis, Philosophical Papers, vol. I, (New York: Oxford University Press, 1983). —— On the Plurality of Worlds (Oxford: Basil Blackwell, 1986). Parsons, C.,”Sets and classes,” Nous 8 (1974), pp. 1–12; repr. in his Mathematics in Philosophy (Ithaca: Cornell University Press, 1983). Stalnaker, R., “The interaction of modality with quantification and identity,” in W. Sinnott-Armstrong et al., eds., Modality, Morality and Belief: essays in honor of Ruth Barcan Marcus (Cambridge: Cambridge University Press, 1995), pp. 12–28.

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II

ABSOLUTE GENERALITY

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3. All Things Must Pass Away Joshua Spencer 0. INTRODUCTION The notion of all things must pass away from philosophical theorizing. By this, I do not mean that we must reject the set of all things. Rather, we must reject the notion that some things are such that any things are amongst them. This is a claim made in a purely plural language. It may be formulated as follows: (AT)

(∃xs)(∀ys)(ys are amongst xs)

Unfortunately, (AT) is false. The argument against (AT) is quite simple. It involves three premises: (1) There are two or more things. (2) For any things, there is a unique thing that corresponds to those things. (3) For any two or more things, there are fewer of them than there are pluralities of them. Given (3), if there are some things that are all things and there are two or more things, then there are fewer things altogether than pluralities of them. But, given (2), there is a unique thing that corresponds to each plurality of things. So, there are at least as many things as there are pluralities of things. So, either there are no things that are all things or there are fewer than two things. But, according to (1), there are two or more things. It follows that there are no things that are all things. That is (AT) is false. Before I defend the argument above, I would like to indicate just a bit of what is and is not at stake in a denial of (AT). Many people might worry, for example, that one consequence of my thesis is that unrestricted singular quantification is impossible. In section 1, I indicate that this worry is, to some extent, legitimate. Once we acknowledge that there are no things that are all things, we must

68 | Joshua Spencer recognize a limitation on usefulness of plurals in logic. In section 2, I show that there is a surprising metaphysical ramification as well. It seems that one formulation of Unrestricted Composition entails that everything is a proper part of something further. The remainder of this paper is devoted to defending the argument outlined above. In section 3, I argue that a certain cardinality thesis involving pluralities is true. That is, I argue for premise (3). In section 4, after indicating several metaphysical views that seem to entail both (1) and (2), I show that a commitment to propositions supports both of these premises. Finally, I consider several objections to my argument. First, in section 5, I consider an objection that attempts to undercut the support for (2) by endorsing coarse-grained views about entities like propositions. In sections 6 and 7, I consider and respond to an objection according to which my view leads to paradox.

1. SPEAKING OF ALL THINGS Some people might think a denial of (AT) entails that unrestricted singular quantification is impossible. After all, if there are no things that are all things, then there are no things that every single thing is amongst. But, one might think, in order for unrestricted quantification to be possible, it must be that there are some things that every single thing is amongst.1 At the very least, though, a denial of (AT) is consistent with a denial of the possibility of unrestricted quantification. That is, a denial of (AT) does not logically entail that our singular quantifiers are indefinitely extensible. Consider any universal statement of the following form: (US)

Everything is either identical to α or identical to β or . . .

Someone who believes that our singular quantifiers are indefinitely extensible is committed to saying that for any context in which a statement that has the form of (US) is true, there is another con1 This thought may simply be an instance of what Uzquiano calls The All-in-Many Principle. According to the All-in-Many Principle, “quantification over objects satisfying a certain condition presupposes that there are some objects which are all and only those objects that satisfy the condition” (2009, 312).

All Things Must Pass Away | 69 text in which that same statement is false. One might say that the domain of any putatively unrestricted singular quantifier is always extensible.2 On the other hand, consider someone who denies (AT). That is, consider someone who accepts the following: (~AT) ~(∃xs)(∀ys)(ys are amongst xs) This person might consistently hold that there is a sentence of the form (US) which expresses a truth in every context. Here is an example to show that this is consistent. Suppose we restrict our attention to things that are finite in number and each one of which is a positive integer. Now, it is clear that the following three claims are consistent: (i) There are no integers that are finite in number and are all integers. (ii) For every integer, either it is identical to 1 or identical to 2 or . . . (iii) There is no context in which the quantifier of (ii) is expanded to make (ii) false. Notice that (i) involves the plural quantifier ‘there are no integers’ and expresses a restricted variant of (~AT). Thus, under an appropriate interpretation of our plural language, (~AT) expresses (i). Moreover, (ii) involves the singular quantifier ‘for every integer’ and expresses a restricted variant of (US). Thus, under an appropriate interpretation of our plural language, (US) expresses (ii). Finally, (iii) is just a denial of the indefinite extensibility of the singular quantifiers in (ii). But, since (i)–(ii) are consistent and are appropriate restricted interpretations of (~AT) and (US), and since (iii) is a

2 This, though, may be mistaken. On one view, there are certain unrestricted quantifiers that fail to have a domain. Consider someone who believes that domains are sets of things and yet still believes that unrestricted singular quantification is possible. On this view, when an unrestricted singular quantifier is employed, it has no domain. This is because domains are sets and there is no set of all things. Moreover, since there is no set of all things, domains are indefinitely extensible even though the unrestricted quantifier is not. So, on this view, the claim that unrestricted quantifiers are indefinitely extensible comes apart from the view that domains are indefinitely extensible. The same may be true if domains are pluralities rather than sets and there are no things that are all things.

70 | Joshua Spencer denial of indefinite extensibility under that interpretation, we must conclude that the impossibility of unrestricted quantification is not a consequence of (~AT). The denial of indefinitely extensible quantifiers and the denial of the existence of some things which are all things, is a coherent position.3 One lesson we might draw from this is that although we may quantify over absolutely everything, there need not be some things that are all things. However, this lesson may make us worry about the prospects of giving a model theoretic account of logical truth and logical consequence for various languages using plurals instead of sets. After all, if there is no modal theoretic interpretation of an unrestricted quantified sentence that appropriately corresponds to the intended meaning, then perhaps we cannot be sure that the sentence is true even if it is true in all models. In the case of first-order logic, this worry is misplaced. Completeness results for first-order logic show that if a sentence is true in all models, then it is provable and, on the assumption that the axioms of that language are true and the inferences truth preserving, we get that all provable sentences are true. Hence, any sentence of such a language which is true in all models is simply true. Of course, the worry is not misplaced when one considers second-order languages. This is because completeness results have not been obtained for various second-order languages. Hence, one might legitimately worry that the prospects for an account of logical truth in a secondorder language may be encumbered by the fact that there are no things that are all things.4

3 Here is another way to make the same point. The way that I understand a plurality, a single thing is merely a very sparse plurality. Given this fact, we can simply introduce singular quantification as a restriction on plural quantification. Singular quantification is merely plural quantification restricted to those things that are one in number (McKay 2006). But the argument only shows that there are no things that are all things. As long as there is more than one thing, the argument doesn’t show anything about things that are one in number. Moreover, since the argument relied on the premise that there are two or more things, it doesn’t show anything about things that are one in number even if there is only one thing. 4 For more details see Rayo and Uzquiano (1999) and Uzquiano (2009). Also see Williamson (2009) for related problems for unrestricted quantification. The limitations placed on plural languages by a denial of (AT) may impede certain attempts to solve the problems posed in Williamson (2009) using plurals.

All Things Must Pass Away | 71 2. UNRESTRICTED COMPOSITION AND JUNK Unrestricted Composition is the thesis that for any things there is something that composes them. This thesis is supposed to be incompatible with the claim that everything is a proper part of something else (that the world is junky).5 That is to say that these two theses are supposed to be incompatible with one another: (UC) For any xx, there is a y such that y is composed of xx (Junk) For any x, there is a y such that x is a proper part of y. The argument seems to be straightforward. If (UC) is true, then any things compose something. But, that means that all things compose something, namely U. If (Junk) is true, then the thing that all things compose must be a proper part of something else. That is, there must be an object, O, that U is a proper part of. But, since U is composed of all things, it follows that O is a part of U. It is clear, though, that nothing can be a proper part of one of its parts.6 So, either (UC) is false, or (Junk) is false. However, this argument clearly relies on the premise that there are some things that all things are amongst. If there are no such things, then the argument above is unsound. In fact Unrestricted Composition and (~AT) together entail (Junk). I have two arguments for this claim. The first is as follows: assume (UC) and (~AT). Now arbitrarily choose an object, hereby named ‘Galileo’, and consider some xs amongst which can be found every part of Galileo. Given that (AT) is false, there must be some zs that those xs are amongst and that are not amongst those xs. That is, there must be some zs that those xs are properly amongst. Again, by (UC), there is something composed of those zs. Call that thing ‘Jupiter’. But, the following principle seems pretty plausible, if some xs are properly amongst some ys, then anything every part of which is amongst those 5 Jonathan Schaffer (2010), for instance, says “Classical mereology—with its axiom of unrestricted composition—guarantees the existence of a unique fusion of all concrete objects. Thus there are gunky models of classical mereology, but no junky models. Indeed, a mereologically maximal element is the only individual that classical mereology guarantees on every model.” Einar Duenger Bohn, in both (2009a) and (2009b), has made similar remarks. 6 Some people might reject this premise if they think it’s possible for something to be shrunk down and sent back in time to become a part of itself. Although I am a fan of the possibility of such science fiction examples, I will not be discussing the implications of such fantastical possibilities here.

72 | Joshua Spencer xs is a proper part of anything those ys compose.7 It follows that Galileo is a proper part of Jupiter. But, since Galileo was arbitrarily chosen, we may conclude that everything is a proper part of some further thing. That is, (Junk) is true. So, (UC) and (~AT) entail (Junk). Some people might be reluctant to accept the principle that if some xs are properly amongst some ys, then anything every part of which is amongst those xs is a proper part of anything those ys compose. Someone might hold that Galileo is not a part of Jupiter, but rather just shares some parts with Jupiter. Perhaps this kind of view would allow someone to hold (UC) without (Junk). This seems like a plausible position, but my next argument shows that it is mistaken. The second argument is as follows: Consider some arbitrary thing. Call that thing ‘Chunk’ and call all of its parts ‘Bits’. Now, if (AT) is false, there are some things that Chunk’s parts are properly amongst. Call those things ‘Bits+Pieces’. Now, there are some things amongst Bits+ Pieces that are not parts of Chunk. Those things that are amongst Bits+Pieces which are not parts of Chunk are Pieces. According to (UC) there is something composed of Chunk and Pieces, namely Big Chunk. Chunk is a part of Big Chunk. Moreover, since Big Chunk has parts that are not parts of Chunk, namely Pieces, it follows that Chunk is a proper part of Big Chunk. So, there is something that Chunk is a proper part of. But, since Chunk was arbitrarily chosen, it follows that for anything whatsoever, there is something that it is a proper part of. So, (Junk) is true. Again, (UC) and (~AT) entail (Junk). Both of these arguments rely on the assumption that for any composite object, there are some things that are all of its parts. One may, of course, deny this assumption and hold that (UC) and (~AT) are both true and yet (Junk) is false. But, of course, the denial of this assumption is, in itself, an interesting mereological result.

3. “THE MORE YOU APPROACH INFINITY, THE DEEPER YOU PENETRATE TERROR” 8 I would like to start by considering the third premise of my argument first: (3) for any two or more things there are fewer of them than there are pluralities of them. This is just as true of pluralities as 7 8

This principle is entailed by Classical Extensional Mereology. Gustave Flaubert (or so I've heard).

All Things Must Pass Away | 73 it is of sets. However, unlike sets, there really aren’t any entities which are the pluralities. Admittedly, I seemed to reify pluralities to state this cardinality thesis.9 But, we can write out, individually, each claim if we have to. For example, if we have three things, A, B, and C, then in addition to the three singulars (which are merely very sparse pluralities) there are the things: A and B; the things: A and C; the things: B and C; and the things: A, and, B and C. Here, A, B, and C are fewer in number than the items on a list of things amongst which just A, B, and C are individually found; there are only three things, whereas there are seven items on the list. If we help ourselves to a perplural language, we can say that there are three things and there are seven thingses total.10 In fact, we can make our general claim in a language of perplurals: For any two or more things, there are fewer of those things than there are thingses amongst which just those things are found. It may sound weird to talk about thingses, but I think it makes the cardinality thesis rather clear.11 Formally, the cardinality thesis can be stated in a secondorder language with plurals as follows: There is no relation R, such that for any xs there is a y such that all and only those xs stand in R to y. However, in the remainder of this section, I will speak informally of lists that pair-up pluralities with individuals. 9 Literally speaking, something is a cardinal number only if it numbers the members of some set. I am going to use the notion of a cardinal number in an extended sense. On the notion I will be employing, something is a cardinal number if it numbers some things, whether or not they form a set. 10 Hazen (1993) introduced a language of perplurals. Hazen, however, tried to use the sensibility of such a language to argue against plural quantification. However, I think we can understand the language and need not accept Hazen’s rejection of plural quantification. Perhaps, we can understand the language through a convenient fiction according to which pluralities are real entities over which perplural quantifiers range. The fictional interpretation of the language need not lead to inconsistencies as long as we stipulate that the plural quantifiers of such a language range only over the non-fictional entities of the universe. The plural quantifiers will have their ordinary meaning whereas the perplurals will have a fictional meaning that merely helps us in our counting. 11 Something like it follows from a plurals version of Cantor’s theorem. Stewart Shapiro (1991) has a proof of a second-order version of Cantor’s theorem and George Boolos (1984) has famously suggested that we can reinterpret second-order statements in a language of plurals. If Boolos’s method of doing so is adequate, then Shapiro’s proof and statement of Cantor’s theorem should be reinterpretable in a language of plurals. Something like the cardinality claim immediately follows. In the remainder of this section, I present a Cantorian argument for the cardinality principle. See also Rayo (2002).

74 | Joshua Spencer It will be worthwhile to present an argument for the cardinality claim. The argument has two steps. We begin by assuming that there are at least two things and attempting to make a list that contains in one column all the things that are one in number, all the single things, and contains in the second column all the things that are any number whatsoever. If the cardinality thesis is false, then there should be such a list, each individual uniquely paired with a plurality and no plurality left unpaired. Our first step is to show that some individual on that list must be paired with a plurality that that very individual is not amongst. Assume that there are at least two things and that we can uniquely pair-up each individual with a plurality, leaving no plurality unpaired. Arbitrarily choose a very sparse plurality consisting of one thing and name it ‘Tom’. Either Tom is paired with itself or it is not. If it is not, then there is at least one individual that is paired with a plurality that that individual is not amongst (namely the individual paired with Tom). So, assume that Tom is paired with itself. Now arbitrarily choose another sparse plurality and name it ‘Dick’. Again, either Dick is paired with itself or it is not. If it is not, then there is at least one individual paired with a plurality that it is not amongst. So, assume Dick is paired with itself. Now, consider Tom and Dick. They cannot be paired with either Tom or Dick, since Tom is already paired with Tom and Dick is already paired with Dick. So, they must be paired with some new object, ‘Harry’, which is not amongst Tom and Dick. Hence, if there are more than two things and there is a pairing of individuals to pluralities, then there is at least one individual that is paired with a plurality that it is not amongst.12 The second step in our argument is to show that if there are more than two things, then there cannot be a list that pairs every plurality with an individual and leaves no plurality unpaired. We know from the first step that if there is such a list, then there must be at least one individual paired with some things that it is not amongst. Suppose there is such a list and let those individuals that are not amongst the pluralities they are paired with be called ‘sinful individuals’. Consider all and only the individuals that are not amongst the pluralities they are paired with; all and only the sinful individuals. Call that plurality ‘The Fallen’. The Fallen must also be paired 12

Thanks to Karen Bennett for this argument.

All Things Must Pass Away | 75 with an individual, call that individual, ‘Eve’. Now we may ask: Is Eve amongst The Fallen or not? If Eve is amongst The Fallen, then Eve must be a sinful individual. This is because The Fallen are a plurality of only the sinful individuals. But, if Eve is a sinful individual, then Eve is not amongst The Fallen. Because sinful individuals (by definition) are not amongst the pluralities they are paired with. So, if Eve is amongst The Fallen, then Eve is not amongst The Fallen. On the other hand, if Eve is not amongst The Fallen, then Eve is a sinful individual. This is because Eve is not amongst the plurality it is paired with. But, if Eve is a sinful individual, then Eve is amongst The Fallen. After all, The Fallen are a plurality of all the sinful individuals. So, if Eve is not amongst The Fallen, then Eve is amongst The Fallen. It clearly follows that Eve is amongst The Fallen iff Eve is not amongst The Fallen. But, of course, either Eve is amongst The Fallen or Eve is not amongst The Fallen. If the first disjunct is true, then Eve both is and is not amongst The Fallen. If the second disjunct is true, then, again, Eve both is and is not amongst The Fallen. Either way, we are led to a contradiction. So, it must be that our original supposition is false. But, our original supposition was that there is a list that pairs every plurality with a unique individual. So, there cannot be such a list. As I mentioned before, if the cardinality thesis is false, then there must be a list that pairs every plurality with an individual and leaves no plurality unpaired. Since we have just shown that there cannot be any such list, we must conclude that the cardinality thesis is correct. So, for any things, there are fewer of them than there are pluralities of them. Hence, premise (3) of the argument against all things is true.

4. SOMETHING FOR EVERYONESES 13 Certain metaphysical views commit us to the existence of two or more things and to a unique thing for each plurality of things. That is, if certain metaphysical views are true, then premises (1) and (2) 13 The argument of this section is similar to, but distinct from, arguments I presented in my (2006). However, I am indebted to Greg Fowler for many helpful discussions about some puzzling aspects of my previous argument. These discussions helped me to see some of my underlying assumptions and develop the argument that appears here.

76 | Joshua Spencer are true as well. Which metaphysical views have such consequence? There are lots of them. Some people, for example, believe in states of affairs. If there are states of affairs, then there are at least two of them, one that obtains and one that does not. Moreover, if there are states of affairs, then for any things, there is a state of affairs of just those things existing. Other people believe that there is an omniscient God. If there is an omniscient God, then there are at least two things, God and God’s first person existential thought. Moreover, if there is an omniscient God, then for any things, God has some thought about just those things (the thought that they exist, for example). There are even those who believe in possible and impossible worlds, or ways for the world to be. If there are possible and impossible worlds, then there are at least two worlds, one possible and one impossible. Moreover, if there are such worlds, then for any things, there is a world where just those things exist.14 Views like these commit one to the thesis that there are no things that are all things.15 I’d like to consider, in more detail, one view that seems to commit us to (1) and (2). Consider the view that there are truths and falsehoods; the view, in other words, that there are propositions. If there are propositions, then there are at least two things. First, there is the proposition that some propositions are true. Second, there is the proposition that every proposition is false. These are two distinct propositions because they must have different truth values; the first must be true whereas the second must be false. So, on the view that there are propositions, there are at least two propositions.16 So, on the view that there are propositions, statement (1) is true.

14 Many of these worlds will be impossible worlds since many things cannot exist without other things also existing. For example, I cannot exist without the number two also existing. 15 One more view that commits us to two or more things and a unique thing associated with any things is a robust view of properties. According to a robust view of properties, there are many properties and for any things, there is a property that they and only they have in common. 16 Moreover, there is also at least one more proposition. Not every proposition has its truth value necessarily. Some propositions are contingent. But, since each of the two propositions above have their truth values necessarily, it follows that there is at least one more proposition. So, on the view that there are propositions, there are at least three things.

All Things Must Pass Away | 77 Now, for any things, there is a proposition just about those things. I think it is fairly clear what it means to say of some things that a proposition is about them. The proposition that Nicholas and Brie both exist is, for example, about Nicholas and Brie. Some philosophers might use the word ‘about’ in an extended sense. They might say that, in addition to being about Nicholas and Brie, the proposition that Nicholas and Brie both exist is also about the property of existence. But, I do not intend to use the word in this extended sense. The way I am using ‘about’, the proposition that Nicholas and Brie both exist is about Nicholas and Brie and is not at all about the property of existence. There are, of course, propositions that are about properties (or at least I think there are). The proposition that existence is monadic is about the property existence. But, it should be clear, now, that it is not also about the property of being monadic. It is obvious that for any things there are some propositions about them. After all, for any things, it is obvious that there are some truths about them; that they exist, perhaps, or have some kind of being. But, I want to make a slightly stronger claim. I believe that for any things there is a proposition about them. Moreover, I believe that for any things, there is a proposition just about them. That is to say that for any things there is at least one proposition about those things and there are no further things that that proposition is also about. We can state this Propositions Thesis a bit more formally as follows: (PT)

for any xs, (i) there is a proposition about those xs and (ii) for any ys which are not the same as the xs, there is a different proposition about those ys.

If (PT) is true, then statement (2) is true as well. One way to support (PT) is to indicate a certain type of proposition that is such that for any things whatsoever, some proposition of that type is just about those things. That is, indicate some propositions that correspond, in the appropriate way, to all the various pluralities in the world. It seems to me that existential propositions are good candidates for the appropriate correspondence. If that is correct, then the following Existential Propositions Thesis might be true. (EPT) For any xs, (i) there is the proposition that those xs exist and (ii) for any ys which are not the same as the xs, the proposition that those ys exist is distinct from the proposition that those xs exist.

78 | Joshua Spencer If (EPT) is true, then for any things there is a proposition, an existential proposition, just about those things.17 Hence, (PT) is true and so is (2). In addition to existential proposition, there are also compositional propositions. That is, propositions that say of some things that they compose a further thing.18 Of course, some people believe that some things cannot compose a further thing. However, even if there are some things which cannot compose a further thing, there is still a compositional proposition just about them. After all, if it is necessarily false that they compose some further thing, then some proposition about them is necessarily false. So, perhaps the following Compositional Propositions Thesis is true: (CPT)

For any xs, (i) there is the proposition that those xs compose something and (ii) for any ys which are not the same as the xs, the proposition that those ys compose something is distinct from the proposition that those xs compose something.

If (CPT) is true, then (again) for any things, there is a proposition, this time a compositional proposition, just about those things. Hence, (PT) is true and so is (2). Finally, there are doxastic propositions, propositions that say of some things that someone or other is thinking about them. Again, it seems intuitively correct that for any things there is a proposition that someone is thinking about those things. Moreover, this seems right even if it turns out that most such propositions are false. So, it seems that the following Doxastic Propositions Thesis is true: (DPT)

For any xs, (i) there is the proposition that those xs are thought about by someone and (ii) for any ys which

17 Admittedly, there are some who deny the first tenet of (EPT). Some people think that existence is not a first order property of either individuals or pluralities. Rather, existence is a higher order property of properties. Thus, we cannot, for any individual x, say that x exists. Rather, we can only say, for some property F, that there are Fs. Furthermore, some people might claim that for all we know, there may be some things that share no properties. Thus, we cannot say of them that they exist because we cannot say of some property they all share that some things have that property. However, this worry should not be too worrisome since any things have the property of being just those things. This is a property that they have and any other things lack. So, even if existence is a second-order property, we can say of any things that existence applies to the property of being just those things. 18 Even if existence is not a first-order property of plurals, surely the property of composing something is such a property.

All Things Must Pass Away | 79 are not the same as the xs, the proposition that those ys are thought about by someone is distinct from the proposition that those xs are thought about by someone. If (DPT) is true, then for any things there is a proposition, a doxastic proposition, just about them.19 Hence, (PT) is true and so is (2). So, if any of (EPT), (CPT), or (DPT) are true, then (PT) is true as well. That is, each of the three theses above supports the claim that for any things there is a proposition just about those things. Of course, if (PT) is true, then so is (2). But, we need not appeal to the existence of particular types of propositions to support (PT). It is easy to show that for any things, there is a proposition just about them; that is, it is easy to show that (PT) is true. Suppose, for reductio, that there are some things such that there is no proposition just about them. Let’s say that a1 . . . an are some such things. Then, it is true that a1 . . . an are such that there is no proposition just about them. But, since every truth is a proposition, it is clear that there is at least one proposition about them, namely the proposition that a1 . . . an are such that there is no proposition just about them. Moreover, this proposition is just about them.20 So, there is at least one proposition just about them. But, this contradicts our claim that 19 Notice that the denial of (DPT) entails that there can be no omniscient being. For, if there could be an omniscient being (even a non-actual one), then for any things whatsoever, that being would believe that they possibly exist. 20 One might wonder whether this proposition really is just about a1 . . . an. I have to admit that I have no criteria for determining whether or not a proposition is just about some things. However, the following seems intuitively correct to me. P is about some xs if those xs are some of the things that make P true and they are the only such things that are such that P logically implies that they have some feature or other. For example, Nicholas and the property of being tall both help to make the proposition that Nicholas is tall true. But, the proposition that Nicholas is tall logically implies that Nicholas has some feature or other. It does not logically imply that the property of being tall has some feature or other. Moreover, there are no other things that help to make that proposition true. So, the proposition is about Nicholas and not tallness. Similarly, being blue and being a color are the only things that help to make it true that blue is a color. However, the proposition that blue is a color logically implies that blue has some feature or other but does not logically imply that being a color has some feature or other. So, that proposition is about the property of being blue and not about the property of being a color. Now consider the proposition that a1 . . . an are such that there are no propositions about just them. It seems that a1 . . . an are things that make that proposition true and that proposition logically implies that a1 . . . an have some property or other (namely being such that there are no propositions about them). Moreover, there are no other things that help to make that proposition true which are also such that that proposition logically implies they have some property or other.

80 | Joshua Spencer there is no proposition just about them. Hence, the supposition that there are some things of which there is no proposition just about them is false. So, for any things, there is a proposition just about them. That is, (PT) is true. So, it seems clear that the thesis that there are propositions commits us to rejecting the claim that there are some things that are all things. For, if there are propositions, then there are two or more of them. Hence, statement (1) is true. But, if there are propositions, then (PT) is true. Moreover, if (PT) is true, then for any things, there is a unique thing that corresponds to just them. Hence, statement (2) is true. Moreover, the cardinality thesis I argued for in the last section says that for any things there are fewer of them than pluralities of them. Hence, statement (3) is true. But, as I have shown, from (1)–(3) it follows that there are no things that are all things. That is, it follows from (1)–(3) that (AT) is false. Hence, (AT) is false. The remainder of this paper is devoted to defending this argument from objections. In section 5, I consider a popular view about propositions that might undermine the support for premise (2) in the argument. In section 6, I consider an objection that says premise (2) leads to paradox. Finally, in section 7, I consider an objection according to which my response to the paradox of section 6 undermines my argument.

5. ON THE PROPOSITION THAT THERE IS A PLURALITY OF WORLDS In addition to the claim that for any things there is a proposition about them, I also endorse the claim that for any things there is a proposition just about them. That is to say that for any things there is a proposition about those things and there are no further things that that proposition is also about. Some people, though, believe in a sparse view of propositions according to which propositions are merely sets of possible worlds.21 A proposition, P, is true at a world, w, just in case P is identical to a set of possible worlds and w is a 21 A better view might be that propositional attitudes (and other properties and relations that seem to take propositions as objects) are really irreducibly plural. The fundamental belief relation, on this view, is irreducibly plural in its second place and, hence, it will have the form ‘S believes those xs’ where the plural variable is satisfied

All Things Must Pass Away | 81 member of that set. On this view, for any propositions P and Q, if it is necessary that P is true iff Q is true, then P=Q. But, now consider the proposition that the number 2 exists and the proposition that the number 7 exists. Since numbers are necessarily existing entities, it is impossible for one of those propositions to be true without the other proposition also being true. But, then it follows, on this view, that the proposition that the number 2 exists is the same as the proposition that the number 7 exists. Hence, the proposition that the number 2 exists is not just about the number 2, it is also about the number 7. On this view, it might turn out that although it is true that for any things there is a proposition about them, it is false that there is a proposition just about them. Hence, if this sparse view is true, then (PT) may be false and part of the support for (2) may be undermined. There are two things that I would like to say in response to this objection. First, the best sparse theory of propositions is one according to which propositions are sets of worlds. But, it seems to me that this view of propositions is mistaken. In what follows, I will say a bit about why I think this sparse view of propositions is mistaken. I recognize, however, that there are some who have very

by worlds. Consider the sentence ‘grass is green’. On this view, that sentence picks out, plurally, all those worlds where grass is green. Moreover, if someone believes the content of that sentence, then she stands in the belief relation to those worlds. This view might be favored by those who believe that some worlds are too numerous to form a set. If propositions are sets, then there is no proposition that corresponds to some worlds that are too numerous to form a set. However, if (speaking vulgarly) propositions are pluralities, then those worlds that are too numerous to form a set are still propositions and might still be expressed and believed. One downside of this view is the following. There are some truths about propositions that seem to be irreducibly plural. For example, when I say that Nicholas’s beliefs are consistent, I seem to be saying something irreducibly plural about propositions. However, if propositions are pluralities, then I must be saying something irreducibly perplural. As I mentioned before, some people think that there’s no way to make sense of a perplural language without reifying pluralities (which is exactly what we want to avoid). The only hope for a defender of this view is to find a way to paraphrase away those claims that seem to be irreducibly plural; like the claim that Nicholas’s beliefs are consistent. Gabriel Uzquiano (2004) has a good discussion of the prospects for paraphrasing seemingly irreducible plural claims in another context. Much of what Uzquiano says will apply in this circumstance as well. Luckily, nothing that I say in the remainder of this section depends on whether propositions are sets of worlds or pluralities of worlds. So, I will simply focus on the former view.

82 | Joshua Spencer reasonable defenses of the sparse view of propositions. So, I will close this section by noting that there are many metaphysical views that support premise (2) of my main argument against all things. Endorsing a sparse view of propositions in order to avoid commitment to (2) is merely the first step on a dark path to a generally sparse metaphysics. Let’s start by considering a sparse view of propositions. Let’s consider a bare-bones view according to which (i) for any proposition P, P is possibly true iff there is a possible world, w, such that P is true-at-w, and (ii) for any possible world, w, and any proposition, P, of the form there are Fs, P is true-at-w iff P is identical to a set of F-worlds and w is a member of that set. At the very least, if the sparse view of propositions is correct, then (i) and (ii) must both be true. Unfortunately, (i) and (ii) are not both true. Let’s suppose that necessarily, for any proposition, P, P is possibly true just in case there is a possible world, w, and P is true-at-w. But, it is necessary that there are merely possible truths; claims that are not true but are possibly true. It follows from these last two claims that necessarily, there is a merely possible world; worlds, distinct from the actual world, at which the merely possible truths are true.22 Since, necessarily, there is a world that is not merely possible, namely the actual world, it follows that necessarily there is more than one possible world. That is, the following claim turns out to be necessarily true: (PW) There is (quantifiers wide open) a plurality of possible worlds. So, if (i) and (ii) are both true, then (PW) is necessarily true. But, if (i) and (ii) are both true and (PW) is necessarily true, then (PW) is identical to the set of all worlds. After all, given that a necessary proposition is one the negation of which is not possibly true 22 Strictly speaking this does not follow from the last two claims. On Lewis’s view, a merely possible de re modal truth about me may be true because there is some other actual individual who is my counterpart. However, it is necessary that there are merely possible de dicto truths. The claim that there is a world distinct from the actual world follows from the claim that there are merely possible de dicto truths and the claim that necessarily, for any proposition, P, P is possibly true just in case there is a possible world, w, and P is true-at-w.

All Things Must Pass Away | 83 (that is, given the duality of necessity and possibility), it follows from (i) that a necessary truth is true at all worlds. But, according to (ii), if (PW) is true at all worlds, then (PW) is identical to the set of all worlds. If (PW) is identical to the set of all worlds, then (PW) is identical to any other proposition that is true at all worlds; in particular, (PW) is identical to the proposition that arithmetic is incomplete. (PW), however, is not identical to the proposition that arithmetic is incomplete. Many of us believe the latter but disbelieve the former and, moreover, Gödel proved the latter without also proving the former. So, (PW) is not the set of all worlds. So, either (i) or (ii) is false. Since the sparse view of propositions is true only if both (i) and (ii) are true, it follows that the sparse view is mistaken. As I mentioned before, although I take the above objection to be sound, I recognize that there are some who have very reasonable defenses of the sparse view of propositions. Many of the defenders have plausible things to say about objections like the one above. But, there is something that I find more troubling about this style of response to my argument against all things. This style of response seems to be merely one step down a dark path to a generally sparse metaphysical view. As I noted at the beginning of section 4, there are several metaphysical views that seem to support premise (2) of my argument. I chose to focus on the view that for any things there is a proposition just about them. However, I could have easily focused on the view that for any things, there is a state of affairs of just those things existing; or the view that for any things there is a property of being just them; or the view that for any things there is a possible or impossible world were just they exist. Any of these metaphysical views supports premise (2) in my argument against all things. So, anyone who wishes to avoid the conclusion of my argument must accept not only a sparse view of propositions, but also a sparse view of states of affairs, properties, and worlds; one must accept a generally sparse metaphysics. Moreover, if one accepts a generally sparse metaphysics in response to my argument, then one must also hold that how these entities are sparsely distributed over individuals must match up. That is, one must reject the view that for any things there is either a proposition just about them or a state of affairs of just those things existing or a property that just they have, etc. In other words, in

84 | Joshua Spencer order to undermine all hope of support for premise (2), one must accept a generally sparse and rather radical metaphysics.23

6. WE MUST PASS OVER IN SILENCE In the last section, I considered a sparse view of propositions which some might have taken to be both true and inconsistent with (PT). If such a view were true, then the primary support for premise (2) would be undermined. However, it seems that the sparse view is false. Moreover, regardless of whether or not it’s false, one would have to accept a generally sparse and radical metaphysics to undermine all potential support for premise (2). In this section, I consider what I take to be the strongest objection to (PT). According to this objection (PT) entails a contradiction all by itself, and hence must be false. However, if (PT) is false, then our primary support for (2) is undermined. Moreover, the form of this objection can be applied to any robust metaphysical view. So, if this objection were successful, it would indicate a path in support of a generally sparse metaphysics. The objection to (PT) is fairly clear.24 First, assume that (PT) is true; that is, assume that for any things there is a proposition just about them. Now, it is clear that some propositions are not about

23 To make the darkness of this path more acute, note that one who accepts that propositions are merely sets of possible worlds should say exactly what a possible world is. But, the standard views of possible worlds cannot be held given a sparse metaphysical view. We cannot, for example, say that a possible world is a complex state of affairs or a complex property or even a complex sentence in a lagadonian language (where everything is its own name and every property is a predicate that expresses itself). For if any of these resources are rich enough to use in our construction of possible worlds, then they are also rich enough to generate support for premise (2). I see three options available to the defender of a sparse metaphysics. First, one could, of course, simply endorse Lewis’s unorthodox view according to which possible worlds are concrete things composed of individuals which are appropriately related. Second, one could accept Magical Ersatzism according to which there is no true account of how possible worlds represent ways the world could be (i.e. they just do). Finally, one could accept a poor world-making language and allow most representation to be implicit. However, I think few of us are willing to accept the counterintuitive costs of Lewis’s concrete modal realism or Magical Ersatzism (though see van Inwagen (1986) for a defense of the latter). So, the third option seems most plausible. 24 A version of this argument is presented by Rayo and McGee (2000).

All Things Must Pass Away | 85 themselves. For example, the proposition that Parmenides was born in Elea is not about itself (at most it is about Parmenides and Elea). Let all and only those propositions that are not about themselves be called ‘humble propositions’. Since there are some humble propositions, by (PT), there must be a proposition that is just about the humble propositions (perhaps the proposition that they exist or the proposition that they are humble). Arbitrarily choose any proposition that is just about the humble propositions and call it ‘Confusion’. Given the reference-fixing description used to name Confusion and the definition of ‘humble proposition’, the following two claims are true. A. For any proposition P, Confusion is about P iff P is humble. B. For any proposition P, P is humble iff P is not about P. But, by universal instantiation on (A) and (B) respectively, we get (C) and (D). C. Confusion is about Confusion iff Confusion is humble. D. Confusion is humble iff Confusion is not about Confusion. And, of course, (E) follows from (C) and (D). E. Confusion is about Confusion iff Confusion is not about Confusion. But, (F) is a logical truth and (G), (H), and (I) all follow by classical logic. F. Either Confusion is about Confusion or Confusion is not about Confusion. G. If Confusion is about Confusion, then Confusion both is and is not about Confusion. H. If Confusion is not about Confusion, then Confusion both is and is not about Confusion. I. So, Confusion both is and is not about Confusion. This contradiction followed from the supposition that for any things there is a proposition just about those things. So, there must be some things that no proposition is about. That is, (PT) must be false. But,

86 | Joshua Spencer since (PT) was our primary support for premise (2), it looks like our support for (2) has been undermined. This is a very powerful argument against (PT). However, I do not believe the argument is sound. I do not accept (F).25 This may seem shocking, given that (F) seems like a logical truth. But, it turns out that propositions of the form P or not-P are not logical truths. This is one lesson we should take away from the Liar Paradox. Consider the English sentence, hereby named ‘(LIAR)’: “(LIAR) does not express a truth”. If we accept that either (LIAR) expresses a truth or it does not, then contradiction immediately follows. So, we must not accept that either (LIAR) expresses a truth or it does not. But, that means that there is at least one proposition of the form p or notp which is unacceptable and hence, not a logical truth. Now, I said that I do not accept (F). But, I do not accept that Confusion is neither about itself nor not about itself either. To move from not accepting (F) to accepting the negation of (F) is to make the same mistake as before. Such a move presupposes that (F) or not-(F) is true. Rather, we must remain silent about whether (F) is true and we must remain silent about whether Confusion is about itself.26 It turns out that ‘is not about itself’ and ‘does not express a truth’ express partial properties. Let’s call those things that are instances of a property the ‘metaphysical extension’ of that property.27 Partial properties are properties that have a definite metaphysical extension and a definite metaphysical anti-extension. An atomic proposition of the form a is F is true if a lies within the metaphysical extension of the property F. The negation of an atomic proposition of the form a is F is true if a lies within the metaphysical anti-extension of the property F. However, there are some things that lie outside both the metaphysical extension and the metaphysical anti-extension of partial properties. If a lies outside both the metaphysical extension and

25 The view that I present is very similar to the view presented by Field (2008) and is inspired by Soames’s (1999) view of truth. 26 If we were to model a language that behaves this way, then we would assign to each predicate an extension and an anti-extension. However, some predicates, such as ‘is not true’ and ‘is not about itself’, will be such that the union of their extension and anti-extension fails to include the entire universe. 27 Here I am following Salmon (1981 pp. 46) in distinguishing a metaphysical extension from a semantic extension. Properties have metaphysical extensions whereas predicates have semantic extensions.

All Things Must Pass Away | 87 the metaphysical anti-extension of F, then we must remain silent about whether a is F; that is, a is F is unacceptable. There are lots of partial properties and it is very easy to introduce a predicate that expresses such a property into a language. Suppose I say “being a humanoid who is over 10 meters tall is sufficient for being a scholossal and being a humanoid under 200 centimeters tall sufficient for not being a scholossal. Furthermore, no non-humanoids are scholossals.” Suppose Breetai comes from a species of humanoids who are between 10 and 15 meters tall. If Breetai is over 10 meters tall, then clearly he is a scholossal. If, on the other hand, Max is an ordinary human who is under 200 centimeters tall, then clearly he is not a scholossal. What, however, about someone who is 210 centimeters tall? Such a human falls outside the metaphysical extension of the property of being scholossal. In this case, we must remain silent about whether such a human is a scholossal. The properties expressed by ‘is not about itself’ and ‘does not express a truth’ are partial properties. It is clear that some propositions fall within the metaphysical extensions of these properties and some propositions fall inside the metaphysical anti-extensions of them as well. However, there are a few propositions that are neither in the metaphysical extension nor in the metaphysical anti-extensions of these properties. (LIAR) is a sentence that lies outside both the metaphysical extension and the metaphysical anti-extension of the property expressed by ‘does not express a truth’.28 Similarly, any propositions that are about all and only the humble propositions, including Confusion, lie outside of the metaphysical extension and metaphysical antiextension of the property expressed by ‘is about itself’.

7. THE BOUNDS OF SILENCE Some may have noticed that the argument presented in the last section against (PT) is similar in structure to one part of the argument

28 Moreover, given the truth conditions for negation, (LIAR) lies outside of the metaphysical extension and anti-extension of the property expressed by ‘is true’ as well. This means that the proposition, hereby named (TRUTH), that this proposition is true is also outside the metaphysical extension and anti-extension of the property expressed by ‘is true’. Hence, we must remain silent about whether the truth teller is true. See Soames (1999) for a discussion of this consequence.

88 | Joshua Spencer used to defend the cardinality thesis in section 1. Remember that I called the plurality of all and only those individuals that are not amongst the pluralities they are paired with ‘The Fallen’. I also called the individual paired with The Fallen ‘Eve’. I then argued for the claim that Eve is amongst The Fallen iff Eve is not amongst The Fallen. Finally, I claimed that either Eve is amongst The Fallen or Eve is not amongst The Fallen and derived a contradiction. At that point, I concluded that the supposition that some things are such that there are as many of them as there are pluralities of them is false. That is, I concluded that the cardinality thesis is correct. However, when I considered an argument of the same form in section 6, I refused to accept premise (F), the disjunctive premise that either Confusion is about itself or it is not. Why, one might legitimately ask, did I decide to remain silent about whether Confusion is about itself but I did not decide to remain silent about whether Eve is amongst The Fallen? How could the premises of the first argument be acceptable yet the premise of the second not? The latter question is difficult to answer. I do know that some arguments of the form we are considering are sound. For example, suppose I say “let ‘Mark Barber’ name the barber of Syracuse who shaves all and only those who don’t shave themselves.” The proposition that Mark Barber shaves himself iff he does not shave himself is inconsistent with the proposition that either Mark Barber shaves himself or he does not shave himself. However, I do not remain silent over the claim that either Mark Barber shaves himself or he doesn’t shave himself. Rather, I accept that premise, reject the biconditional and conclude that Mark Barber does not exist. I also know that some arguments of this form are not sound. Obviously, (LIAR) expresses a truth iff it does not express a truth. However, I will not accept that either (LIAR) expresses a truth or it does not. Moreover, I cannot accept that premise lest contradiction ensues. The difference between the case of Mark Barber and the case of (LIAR) is that we must accept the existence of (LIAR). Here it is before us on this very page! (LIAR): “(LIAR) does not express a truth.” I can think of no clearer proof that (LIAR) exists than its very presence before my eyes. Mark Barber, on the other hand, is not presenting himself so clearly.

All Things Must Pass Away | 89 Eve is like Mark Barber. Eve’s existence is not foisted upon us and we are free to avoid contradiction by denying that Eve exists. This route leads us to the cardinality principle. Confusion, on the other hand, is like (LIAR). We know that Confusion exists because we know that there are some humble propositions. Moreover, I am convinced by the discussion of section 2 that there must be a proposition just about the humble propositions. The existence of Confusion forces us into silence. That is, we are forced to remain silent about whether Confusion is about itself. There is also an independent reason for avoiding silence about the claim that either Eve is amongst The Fallen or Eve is not. Suppose the claim that either Eve is amongst The Fallen or Eve is not amongst The Fallen is unacceptable. That is, suppose that we must remain silent about that disjunction. If that is the case, then we must also remain silent about each of the disjuncts. So, the claim that Eve is amongst The Fallen is a claim that we must remain silent about. But, we know that the amongst relation is transitive. But, that means that if Eve is amongst any things that are amongst The Fallen, then Eve is amongst The Fallen as well. So, if we must remain silent about whether Eve is amongst The Fallen, then we must remain silent about whether Eve is amongst any things that are amongst The Fallen. Some things that are amongst The Fallen are individuals and individuals are simply pluralities that are one in number. So, for any things that are one in number and amongst The Fallen, we must remain silent about whether Eve is amongst those things. One interesting discovery of the logic of plurals is that there is a relationship between the amongst relation and the identity relation. In particular, the following Identity Principle is true: (IP) (∀xs)(∀ys) (xs=ys ↔ (xs are amongst ys & ys are amongst xs & xs are one in number)) That is, one plurality is identical to a second plurality iff each is amongst the other and neither is more than one in number.29 But, supposedly we must remain silent about whether Eve is amongst any things that are amongst The Fallen (including individuals). Moreover, Eve is definitely one in number. It follows from these two claims that for any individual amongst the fallen, we must remain 29

See, for example, McKay (2006, 129).

90 | Joshua Spencer silent about whether it is identical to Eve. This suggests that there are some identity claims that we should remain silent about and the root of that silence is in the identity relation. But, if we must remain silent about any identity claim, the root of that silence is not in the identity relation. So, our supposition must be mistaken. That is, the claim that either Eve is amongst the Fallen or Eve is not amongst the Fallen is perfectly acceptable. But, if that is right, then the proof of the cardinality principle is sound.

8. CONCLUSION Although the claim that there are some things that any things whatsoever are amongst seems intuitively plausible, I believe this thesis must be rejected. Those who disagree must accept widespread metaphysical limitations, not only with respect to propositions, but also with respect to states of affairs, properties, divine thoughts, and impossible worlds (to name a few). Moreover, they must accept that there are some things that are such that there are is no proposition just about them and there is no state of affairs involving just them and there is no property had by just them etc. On the other hand, those who reject all things, must face certain limitations on the usefulness of plurals and perhaps accept certain surprising metaphysical theses. It seems to me that the costs of rejecting all things are less drastic than the costs of sparse metaphysics.30 University of Wisconsin, Milwaukee BIBLIOGRAPHY Bohn, Einar Duenger (2009a) “An Argument Against the Necessity of Unrestricted composition,” Analysis Vol. 69: 27–31. —— (2009b) “Must there be a Top Level?” The Philosophical Quarterly Vol. 59: 193–201. Boolos, George (1984) “To Be is to be the Value of a Variable (or to be Some Values of Some Variables),” The Journal of Philosophy Vol. 81: 430–49. 30 Thanks to Karen Bennett, Gregory Fowler, Mark Heller, Hud Hudson, Kris McDaniel, Tom McKay, Chris Tillman, and Gabriel Uzquiano, for discussing these issues and reading earlier drafts of this paper. Thanks also to audiences at The University of Manitoba and Syracuse University for helpful comments and discussion.

All Things Must Pass Away | 91 Cartwright, Richard (1994) “Speaking of Everything,” Nous Vol 28: 1–20. Field, Hartry (2008) Saving Truth from Paradox (Oxford: Oxford University Press). Grim, Patrick (2000) “The Being that Knew Too Much,” International Journal of Philosophy of Religion Vol 47: 141–54. —— (1991) The Incomplete Universe (Cambridge, MA: MIT Press). —— (1984) “There is No Set of All Truths,” Analysis Vol. 44: 206–8. —— (1983) “Some Neglected Problems of Omniscience,” American Philosophical Quarterly Vol. 20: 265–76. Leonard, Henry and Goodman, Nelson (1940) “The Calculus of Individuals and its Uses,” Journal of Symbolic Logic Vol. 5: 45–55. Lewis, David (1991) Parts of Classes (Oxford: Basil Blackwell). —— (1986) On the Plurality of Worlds (Oxford: Basil Blackwell). McKay, Thomas (2006) Plural Predication (Oxford: Clarendon Press). Plantinga, Alvin and Grim, Patrick (1993) “Truth, Omniscience, and Cantorian Arguments: An Exchange,” Philosophical Studies Vol. 71: 267–306. Rayo, Agustín (2002) “Word and Objects,” Nous Vol. 36: 436–64. —— and McGee, Van (2000) “A Puzzle about De Rebus Belief,” Analysis Vol. 60: 297–9. —— and Uzquiano, Gabriel (1999) “Toward a Theory of SecondOrder Consequence,” Notre Dame Journal of Formal Logic Vol. 40: 315–25. Rosen, Gideon (1995) “Armstrong on Classes as States of Affairs,” Australasian Journal of Philosophy Vol. 73: 613–25. Salmon, Nathan (1981) Reference and Essence (Princeton: Princeton University Press). Schaffer, Jonathan (2010) “Monism: The Priority of the Whole,” The Philosophical Review Vol. 119: 31–76. Shapiro, Stewart (1991) Foundations Without Foundationalism: A Case for Second-Order Logic (Oxford: Clarendon Press). Simons, Peter (1987) Parts: A Study In Ontology (Oxford: Clarendon Press). Soames, Scott (1999) Understanding Truth (Oxford: Oxford University Press). Spencer, Joshua (2006) “Two Mereological Arguments against the Possibility of an Omniscient Being,” Philo Vol. 9: 62–71. Tarski, Alfred (1935) “On the Foundations of the Boolean Algebra,” reprinted in Tarski, Alfred (1983) Logic, Semantics, Metamathematics (Indianapolis, IN: Hackett Publishing Company). Uzquiano, Gabriel (2004) “Plurals and Simples,” The Monist Vol. 87: 429–51.

92 | Joshua Spencer Uzquiano, Gabriel (2009) “Quantification Without a Domain,” New Waves in Philosophy of Mathematics edited by Otávio Bueno and Øystein Linnebo (New York: Palgrave Macmillan). van Inwagen, Peter (1986) “Two Concepts of Possible Worlds” Midwest Studies in Philosophy Vol. 9: 185–213. Varzi, Achille (2004) “Mereology,” The Stanford Encyclopedia of Philosophy (Fall 2004 Edition), Edward N. Zalta (ed.), URL = .

4. Absolute Generality Reconsidered Agustín Rayo Years ago, when I was young and reckless, I believed that there was such a thing as an all-inclusive domain.1 Now I have come to see the error of my ways. The source of my mistake was a view that might be labeled ‘Tractarianism’. Tractarians believe that language is subject to a metaphysical constraint. In order for an atomic sentence to be true, there needs to be a certain kind of correspondence between the semantic structure of the sentence and the ‘metaphysical structure’ of reality. The purpose of this paper is to explain why I think Tractarianism is mistaken, and what I think an anti-Tractarian should say about absolutely general quantification.

1. THE PLAN ‘Just is’-statements will be important in what follows, so let me start by giving you some examples: 1. Sibling For Susan to be a sibling just is for her to share a parent with someone. 2. Water For the glass to be filled with water just is for it to be filled with H2O. 3. Physicalism For such-and-such a mental state to be instantiated just is for thus-and-such brain-state to be instantiated (and for the environment to be thus-and-so). 4. Properties For Susan to instantiate the property of running just is for Susan to run. 5. Death For a death to take place just is for someone to die. 1

Relevant texts include Rayo (2002), Rayo (2003), and Rayo and Williamson (2003).

94 | Agustín Rayo 6. Tables For there to be a table just is for there to be some things arranged tablewise. 7. Dinosaurs For the number of the dinosaurs to be Zero just is for there to be no dinosaurs. Statement 1 is utterly uncontroversial. Statement 2 should be pretty uncontroversial too, at least if we ignore certain complications (such as the possibility of impurities). Statement 3 is not totally uncontroversial (Chalmers (1996)), but it seems to be the dominant view amongst philosophers. Statements 4–7, on the other hand, are all highly controversial metaphysical theses. My own view is that they are all true, but I won’t try to convince you of that here. The aim of this paper is to argue that they shouldn’t be rejected on general linguistic or metaphysical grounds. I will proceed by defending a conception of language I call compositionalism, and showing that it makes room for Statements 4–7. I will then argue that a compositionalist who accepts Statements 4–7 is left with an attractive metaphysical picture of the world. The plan for the paper is as follows. I will start by explaining how I think the ‘just is’-operator should be understood (section 2). I will then introduce my foil: Tractarianism. I will explain why I think Tractarianism is bad philosophy of language (section 3), and develop compositionalism as an alternative to Tractarianism (section 4). Attention will then turn to metaphysics. I will argue that compositionalism does not lead to untoward metaphyical consequences, even if one accepts ‘just is’ statements such as 4–7 (section 5). I will conclude by addressing the problem of absolute generality from the perspective of the compositionalist (section 6).

2. THE ‘JUST IS’-OPERATOR Before mounting my defense of compositionalism, it will be useful to say something about how I will be understanding the ‘just is’ -operator, as it occurs in Statements 1–7. Consider Sibling as an example. What it takes for Sibling to be true is for there to be no difference between Susan’s having a sibling and Susan’s sharing a parent. If Susan is a sibling it is thereby the

Absolute Generality Reconsidered | 95 case that she shares a parent, and if she shares a parent it is thereby the case that she is a sibling. More colorfully: when God created the world, and made it the case that Susan shared a parent, there was nothing extra She had to do, or refrain from doing, in order to ensure that Susan was a sibling. She was already done. And when God created the world, and made it the case that Susan was a sibling, there was nothing extra She had to do, or refrain from doing, in order to ensure that Susan shared a parent. She was already done. In the special case in which Susan is, in fact, a sibling, there is an additional way of clarifying the meaning of Sibling. For Sibling to be true is for ‘Susan is a sibling’ and ‘Susan shares a parent’ to be full and accurate descriptions of the same feature of reality. Other ‘just is’-statements should be understood in the same sort of way. For Death to be true is for there to be no difference between someone’s dying and a death’s taking place. When someone dies it is thereby the case that a death takes place, and when a death takes place it is thereby the case that someone dies. The feature of reality that is fully and accurately described by saying ‘A death took place’ is also fully and accurately described by saying ‘Someone died’. It is useful to compare Sibling and Death with an identity statement such as ‘Hesperus is Phosphorus’. If you accept ‘Hesperus is Phosphorus’, you believe that there is no difference between traveling to Hesperus and traveling to Phosphorus. Someone who travels to Hesperus has thereby traveled to Phosphorus, and someone who travels to Phosphorus has thereby travelled to Hesperus. The feature of reality that is fully and accurately described by saying ‘A Soviet spaceship traveled to Hesperus’ is also fully and accurately described by saying ‘A Soviet spaceship traveled to Phosphorus’. Since ‘just is’-statements are treated as equivalent to the corresponding ‘no difference’ statements, the ‘just is’-operator is treated as symmetric. There is a different reading of ‘just is’ on which it fails to be symmetric. One could suggest, for example, that a ‘just is’statement should only be counted as true if the right-hand-side ‘explains’ the right-hand-side, or if it is in some sense ‘more fundamental’. This is not the reading that will be relevant for present purposes. If you find the asymmetric reading more natural than the symmetric reading, please substitute a suitable ‘no difference’statement for each ‘just is’-statement in the text.

96 | Agustín Rayo There is a lot more to be said about ‘just is’-statements.2 But we had better plunge ahead. Otherwise we’ll never get to absolute generality.

3. TRACTARIANISM In this section I will introduce my foil: a view that will be referred to as Tractarianism.3 It makes no difference for present purposes whether there are any actual Tractarians. The point of introducing Tractarianism is that it’ll make it easier to explain what compositionalism amounts to, and why it is an attractive thesis. A Tractarian believes that in order for an atomic sentence to be true, there needs to be a certain kind of correspondence between the semantic structure of the sentence and the ‘metaphysical structure’ of reality. Consider ‘Susan runs’ as an example. Let us agree that in order for this sentence to be true, it must supply a full and accurate description of some feature of reality. As long as one is suitably deflationary about fact-talk, one might think of the relevant feature of reality as a fact: the fact that Susan runs. What is distinctive about Tractarianism is the claim that ‘Susan runs’ can only supply an accurate description of the fact that Susan runs if the semantic structure of ‘Susan runs’ is in sync with the particular way in which the world’s metaphysical structure carves up the fact that Susan runs. In particular: metaphysical structure must have carved up the fact into an object and a property such that ‘Susan’ refers to the object and ‘runs’ expresses the property. More generally, Tractarianism is the view that in order for an atomic sentence ⌜F(t1, . . . ,tn )⌝ to constitute a full and accurate description of a given feature of reality, the following three conditions must hold: 1. The world’s metaphysical structure carves up the relevant feature of reality into the objects a1, . . . ,an and the property P. 2. For each i, the singular term ⌜ti⌝ refers to ai. 3. The predicate F expresses P.

2 See Rayo (forthcoming) "Neo-Fregeanism reconsidered". For a different way of thinking about ‘just is’-statements, see Bennett (2009). 3 For further discussion of Tractarian conceptions of language, see Heil (2003). For criticism, see Eklund (2009).

Absolute Generality Reconsidered | 97 Tractarianism is a substantial view. Notice, in particular, that Tractarians are immediately barred from accepting certain ‘just is’-statements. Consider ‘for Socrates’s death to take place just is for Socrates to die’. The embedded sentences are both atomic, but they have different semantic structures. Suppose for reductio that they both describe the same feature of reality, as the ‘just is’ statement would have it. How does the relevant feature of reality get carved up by the world’s metaphysical structure? At most one of the following can be true: • It gets carved up into Socrates and the property of dying. • It gets carved up into Socrates’s death and the property of an event’s taking place. If the former is true, then the relevant feature of reality can be accurately described by ‘Socrates is dying’, but not by ‘Socrates’s death is taking place’. If the latter is true, then the relevant feature of reality can be accurately described by ‘Socrates’s death is taking place’, but not by ‘Socrates is dying’. Either way, the ‘just is’ statement turns out to be false. Here I am taking for granted that the logical form of a sentence can be read off more or less straightforwardly from the sentence’s surface grammatical structure. This is a non-trivial assumption. Say you believe that proper logical analysis of ‘Socrates is dying’ reveals it to have the same logical form as ‘Socrates’s death is taking place’. Then you should think that the Tractarian could accept ‘for Socrates’s death to take place just is for Socrates to die’ after all. Contemporary linguistics does suggest that there are certain cases in which there is a real mismatch between surface structure and semantically operative lower-level syntactic structure. But, as far as I can tell, it is not the sort of mismatch that would offer much comfort to the Tractarian. (See, for instance, the treatment of semantics in Heim and Kratzer (1998).) If this is right, then the assumption that logical form can be read off more or less straightforwardly from grammatical structure is a harmless simplification in the present context. There are certain ‘just is’-statements that the Tractarian is in a position to accept. She is free to accept ‘for Susan to be a sibling just is for Susan to share a parent’, for example. For, as long as she is happy to identify the property of being a sibling with the property of sharing a parent, she will be in a position to claim that the seman-

98 | Agustín Rayo tic structures of both ‘Susan is a sibling’ and ‘Susan shares a parent’ are in sync with the metaphysical structure of the fact that Susan is a sibling. For similar reasons, the Tractarian is free to accept Water and Physicalism, from section 1. The Tractarian is, however, barred from accepting Properties. And, on reasonable assumptions about the treatment of non-atomic sentences, she is also barred from accepting Dinosaurs and Tables. As I noted above, these are all controversial metaphysical theses. What is striking about Tractarianism is that it rules them out merely on the basis of syntactic considerations.

Metaphysical Structure Tractarianism is a hybrid of linguistic and metaphysical theses. It deploys a metaphysical assumption—the existence of metaphysical structure—to impose a constraint on linguistic theorizing. I suspect that the notion of metaphysical structure is not in good order, and I would like to make a few remarks about why I think this is so. Let me start by talking about objectivism. Most of us are objectivists about truth. We believe that it makes sense to speak of what is objectively the case, as something over and above what is true according to one person or another. Many philosophers, but not all, are objectivists about morality. They believe that it makes sense to speak of what is objectively good, as something over and above what would be good with respect to some value system or other. Few philosophers, if any, are objectivists about fashion. You may think that ascots are fashionable. But it would be preposterous to suggest that they are objectively fashionable: fashionable over and above the tastes of some community or other. Objectivism comes at a cost. An objectivist about fashion, for example, would be faced with the awkward task of elucidating a non-trivial connection between what is objectively fashionable and what various communities take to be fashionable. She would also have to choose between coming up with an explanation of what it takes for something to be objectively fashionable and burdening her picture of the world with the view that there are brute facts about objective fashion. And the rewards for her efforts would be

Absolute Generality Reconsidered | 99 decidedly meager. For it is not clear what theoretical advantages fashion objectivism could bring. As far as I can tell, interesting theoretical questions concerning fashion can all be addressed by using a community-relative notion of fashionability. When the price of objectivism is not worth paying, one should do more than simply deny that the relevant objectivist notion has any instances. One should deny that the notion makes sense. Someone who claims to understand the notion of objective fashionability faces the burden of elucidating the connection between objective and community-relative fashionability, whether or not she thinks the world happens to contain any instances of objectively fashionable outfits. For what gives rise to the explanatory burden is the concept of objective fashionability, not the assumption that it has any instances. In some cases, of course, the price of objectivism is worth paying. The notion of objective truth is fruitful enough that few would feel unduly burdened by the need to explain the connection between objective truth and truth according to an agent, or by a picture of the world whereby there are brute facts about what is objectively true and what is not. Metaphysics is filled with objectivist views. There are metaphysicians who believe that it makes sense to speak of objective similarity, as something over and above what might strike an agent as similar.4 There are metaphysicians who believe it makes sense to speak of objectively fundamental vocabulary, as something over and above the role a piece of vocabulary plays in some scientific theory or other.5 And—most relevantly for present purposes—a metaphysician might think that it makes sense to speak of the objectively correct way of carving up reality into objects, as something over and above the syntactic properties of the various representations one might use to describe the world. Before embracing a form of metaphysical objectivism, it is important to be mindful of the costs. My own view is that when it comes to metaphysical structure, the price is not worth paying. For I suspect that many of the most interesting metaphysical questions can be addressed without having to appeal to the notion of metaphysi4 5

See Lewis (1983) and Lewis (1984). See Fine (2001), Schaffer (2009), and Sider (typescript).

100 | Agustín Rayo cal structure. Because of this, the need to elucidate the connection between an objectively correct way of carving up reality and the ways in which reality gets carved up by our representations strikes me as too high a price to pay for the resulting theoretical benefits. (At the same time, I don’t think it would be irrational to think otherwise.6)

Bad Philosophy of Language Even though I suspect that the notion of metaphysical structure makes no sense, I will not be relying on this assumption anywhere in the paper. My argument against Tractarianism will be based on the claim that Tractarianism is bad philosophy of language. As the name suggests, Tractarianism is a close cousin of the ‘picture theory’ that Wittgenstein advocated in the Tractatus.7 And it ought to be rejected for just the reason Wittgenstein rejected the picture theory in his later writings. Namely: if one looks at the way language is actually used, one finds that usage is not beholden to the constraint that an atomic sentence can only be true if its semantic structure is in suitable correspondence with the metaphysical structure of the world. It is simply not the case that ordinary speakers are interested in conveying information about metaphysical structure. The sentences ‘a death took place’ and ‘someone died’, for example, are used more or less interchangeably in non-philosophical contexts. An ordinary speaker might choose to assert one rather than the other on the basis of stylistic considerations, or in order to achieve the right emphasis. But it would be tendentious to suggest that her choice turns on her views about metaphysical structure. It is not as if an ordinary speaker would only be prepared to assert ‘a death took place’ instead of ‘someone died’ if she has a certain metaphysical view about events: that they are amongst the entities carved out by the world’s metaphysical structure. Think about how inappropriate it

6 See, for instance, Schaffer (2009) and Sider (typescript). For a critique of certain forms of metaphysical objectivism, see Hofweber (2009). 7 Here I have in mind a traditionalist interpretation of the Tractatus, as in Hacker (1986) and Pears (1987). See, however, Goldfarb (1997).

Absolute Generality Reconsidered | 101 would be to respond to an assertion of ‘a death took place’ in a nonphilosophical context by saying ‘I am certainly prepared to grant that someone died, but I just don’t think that the world contains events amongst its ultimate furniture.’ One’s interlocutor would think that one has missed the point of her assertion, and gone off to a different topic. If ordinary assertions of ‘a death took place’ are not intended to limn the metaphysical structure of the world, what could be the motivation for thinking that the truth-conditions of the sentence asserted play this role? As far as I can tell, it is nothing over and above the idea that semantic structure ought to correspond to metaphysical structure. Remove this idea and there is no motivation left. To buy into Tractarianism is to start out with a preconception of the way language ought to work, and impose it on our linguistic theorizing from the outside—from beyond what is motivated by the project of making sense of our linguistic practice.

Moderate Tractarianism There is a moderate form of Tractarianism according to which the constraint that there be a correspondence between semantic structure and metaphysical structure applies only to assertions made by philosophers in the ‘ontology room’. When I use the term ‘Tractarianism’ here, the view I have in mind is always non-moderate Tractarianism. My arguments for the claim that Tractarianism is bad philosophy of language do not apply to moderate Tractarianism. For all I know, there is a special convention governing discourse in the ontology room, which demands correspondence between semantic and metaphysical structure. If you are sympathetic towards moderate Tractarianism, that’s fine. Just make sure you don’t interpret me as a moderate Tractarian. A moderate Tractarian is free to accept a ‘just is’-statement such as ‘for a death to take place just is for someone to die’. All she needs to do is insist that at most one side of the ‘just is’-statement is taken in an ontology-room spirit. To avoid confusion, moderate Tractarians might consider introducing a syntactic marker for ontologyroom discourse, as in Fine (2001). They could then say: What it really is for a death to take place is for someone to die

102 | Agustín Rayo or: What it is, in fundamental terms, for a death to take place is for someone to die to indicate that the feature of reality described by ‘a death takes place’ gets carved by the world’s metaphysical structure in a way that corresponds to the semantic structure of ‘someone dies’. Just to be clear: this is not what I intend when I use ‘just is’-statements here.

4. COMPOSITIONALISM I will now defend an alternative to Tractarianism: the view I shall refer to as compositionalism. Suppose you introduce the verb ‘to tableize’ into your language, and accept ‘for it to tableize just is for there to be a table’ (where the ‘it’ in ‘it tableizes’ is assumed to play the same dummy role as the ‘it’ in ‘it is raining’). Then you will think that what would be required of the world in order for the truth-conditions of ‘it tableizes’ to be satisfied is precisely what would be required of the world in order for the truth-conditions of ‘there is a table’ to be satisfied. In both cases, what would be required is that there be a table (equivalently: that it tableize). So you will think that—for the purposes of stating that there is a table—object-talk is optional. One can state that there is a table by employing a quantifier that binds singular term positions—as in ‘there is a table’—but also by employing an essentially different syntactic structure—as in ‘it tableizes’. If object-talk is optional, what is the point of giving it a place in our language? The right answer, it seems to me, is ‘compositionality’. A language involving object-talk—that is, a language including singular terms and quantifiers binding singular-term positions—is attractive because it enables one to give a recursive specification of truth-conditions for a class of sentences rich in expressive power. But there is not much more to be said on its behalf. In setting forth a language, we want the ability to express a suitably rich range of truth-conditions. If we happen to carry out this aim by bringing in singular terms, it is because they supply a convenient way of specifying the right range of truth-conditions, not because they have some further virtue.

Absolute Generality Reconsidered | 103 A proponent of this sort of view will disagree with the Tractarian about what it takes for a singular term to succeed in referring. Whereas the Tractarian would insist that a singular term can only succeed in referring if it is paired with one of the objects carved out by the world’s metaphysical structure, proponents of the present view will claim that all it takes is a suitable specification of truthconditions for sentences involving the term. More specifically, we shall let compositionalism be the view is that all it takes for a singular term t to refer is for the following three conditions to obtain: 1. Truth-conditions have been specified for every sentence involving t that one wishes to make available for use. 2. The assignment of truth-conditions respects compositionality, in the following sense: If ϕ is a syntactic consequence of ψ, then the truth-conditions assigned to ψ impose at least as strong a requirement on the world as the truth-conditions assigned to ϕ. 3. The world is such as to satisfy the truth-conditions that have been associated with the sentence ‘∃x(x = t)’. Compositionalism is a substantial view. The best way to see this is to imagine the introduction of a new family of singular terms ⌜the direction* of a⌝, where a names a line. The only atomic sentences involving direction*-terms one treats as well-formed are those of the form ‘the direction* of a = the direction* of b’, but well-formed formulas are closed under negation, conjunction, and existential quantification. A sentence ϕ is said to have the same truth-conditions as its nominalization [ϕ]N, where nominalizations are defined as follows:8 • • • • • •

[⌜the direction* of a = the direction* of b⌝]N = ⌜a is parallel to b⌝. [⌜xi = the direction* of a⌝]N = ⌜z i is parallel to a⌝. [⌜xi = xj⌝]N = ⌜zi is parallel to zj⌝. [⌜∃xi(ϕ)⌝]N = ⌜∃zi([ϕ]N)⌝. [⌜ϕ ∧ ψ⌝ ]N = the conjunction of [ϕ]N and [ψ]N. [⌜_ϕ⌝]N = the negation of [ϕ]N.

8 It is worth noting that the nominalizations of open formulas turn out to be open formulas, and therefore lack truth-conditions. Fortunately, all that is required for present purposes is an assignment of truth-conditions to well-formed sentences.

104 | Agustín Rayo It is easy to verify that every condition on the compositionalist’s list is satisfied. Notice, in particular, that since [‘∃x(x = the direction* of a)’]N is ‘∃z(z is parallel to a)’, and since every line is parallel to itself, all that is required for the truth-conditions of ‘∃x(x = the direction* of a)’ to be satisfied is that a exist. Accordingly, the compositionalist will claim that the existence of a is enough to guarantee that the singular term ‘the direction* of a’ has a referent. Moreover, by employing the newly introduced vocabulary in the metalanguage, the compositionalist will claim that the referent of ‘the direction* of a’ is the direction* of a, and therefore that the existence of a is enough to guarantee the existence of the direction* of a.9 How is this possible? How could a linguistic stipulation, together with the existence of lines, guarantee the existence of directions*? There is nothing deep or mysterious going on. As a result of the linguistic stipulation, the fact that is fully and accurately described by saying ‘a is parallel to b’ can now also be fully and accurately described by saying ‘the direction* of a = the direction* of b’. Similarly, the fact that is fully and accurately described by saying ‘a is parallel to a’ can now also be fully and accurately described by saying ‘the direction* of a = the direction* of b’. So of course the existence of lines is enough to guarantee the existence of directions*: for the direction* of a to be self-identical (equivalently: for the direction* of a to exist) just is for a to be self-parallel (equivalently: for a to exist). Needless to say, the Tractarian would insist that such a linguistic stipulation is inadmissible. She would insist, in particular, that ‘a is parallel to b’ and ‘the direction* of a = the direction* of b’ cannot be full and accurate descriptions of the same fact. For they are atomic sentences with distinct semantic structures. So they cannot both be in sync with the way in which the fact that a is parallel to b gets

9 For closely related views, see Frege (1884), Wright (1983), and Rosen (1993). Discussion of compositionalism amongst contemporary metaphysicians in the United States has tended to focus on the work of Eli Hirsch, who draws on earlier work by Hilary Putnam. (See, for instance, Putnam (1987) and Hirsch (2002); for criticism, see Eklund (2008) and Bennett (2009).) It is worth keeping in mind, however, that some of Hirsch’s linguistic theses go significantly beyond anything defended here, and that his general attitude towards metaphysics is profoundly different from my own (see, in particular, the discussion in section 4.1 of the present text).

Absolute Generality Reconsidered | 105 carved up by the world’s metaphysical structure. According to the Tractarian, there can’t be directions* unless some fact gets carved up into directions* by the world’s metaphysical structure. And even if the existence of lines is taken for granted, no stipulation can tell us that directions* are amongst the objects carved out by the world’s metaphysical structure. The compositionalist, on the other hand, believes that an atomic sentence can be true even if there is no correspondence between semantic structure and metaphysical structure. So there is no immediate obstacle for atomic sentences with different semantic structures to deliver full and accurate descriptions of the same fact. We needn’t check whether directions* are amongst the objects carved out by the world’s metaphysical structure in order to determine whether there are any directions*. It is enough to observe that ‘a is parallel to a’ and ‘the direction* of a = the direction* of a’ are full and accurate descriptions of the same fact, and that a is indeed parallel to itself. (Notice, incidentally, that it is no part of compositionalism that there is no such thing as metaphysical structure. The point is simply that the notion of metaphysical structure does not figure in a proper account of the reference of singular terms.10) It is important to be clear that compositionalism does not entail any interesting ‘just is’-statements unless it is supplemented with further claims. Notice, in particular, that compositionalism does not entail that all it takes for the direction of a (as opposed to the direction* of a) to exist is for a to exist. In order to get that conclusion we would need a substantial hypothesis about the truth-conditions of sentences involving the ordinary word ‘direction’. In particular, we would need to help ourselves to the claim that the ordinary sentence ‘the direction of a exists’ has the same truth-conditions as ‘a exists’. And this is non-trivial assumption. We would, in effect, be assuming that for the direction of a to be identical to the direction of b just is for a to be parallel to b, which is a controversial metaphysi10 It is also worth noting that compositionalism is not in tension with the view— first suggested in Lewis (1983) and Lewis (1984)—that problems of referential indeterminacy can sometimes be resolved by attending to metaphysical naturalness. Compositionalism is a view about what it takes for a singular term to be in good order, not about the sorts of considerations that might be relevant to fixing the reference of singular terms. There is room for thinking that Lewis himself was a compositionalist: see Lewis (1980).

106 | Agustín Rayo cal claim. It is true that we made the analogous assumption in the case of directions*. But back then we were introducing a new term, and were therefore free to introduce truth-conditions by stipulation. Although compositionalism does not commit one to the acceptance of any interesting ‘just is’ statements, it does eliminate an obstacle for the acceptance of ‘just is’-statements. One is no longer barred from accepting a ‘just is’-statement merely on the basis of syntactic considerations. But there is substantial work to be done before one can make a case for accepting any particular statement.

4.1 ‘Just is’-Statements in Metaphysics Some of the ‘just is’-statements a compositionalist is in a position to accept constitute interesting metaphysical theses: for the number of the dinosaurs to be Zero just is for there to be no dinosaurs; for Susan to instantiate the property of running just is for Susan to run. Let me say something about the sorts of considerations that might be relevant to deciding whether to accept ‘just is’-statements such as these. (My discussion is very much indebted to Block and Stalnaker (1999) and Block (2002).) When one accepts a ‘just is’-statement one closes a theoretical gap. Suppose you think that for a gas to be hot just is for it to have high mean kinetic energy.11 Then you should think there is no need to answer the following question: ‘I can see that the gas is hot. But why does it also have high mean kinetic energy?’ You should think, in particular, that the question rests on a false presupposition. It presupposes that there is a gap between the gas’s being hot and its having high kinetic energy—a gap that should be plugged with a bit of theory. But to accept the ‘just is’-statement is to think that the gap is illusory. There is no need to explain how the gas’s being hot might be correlated with its having high mean kinetic energy because there is no difference between the two: for a gas to be hot just is for it to have high mean kinetic energy. The decision whether to treat the gap as closed is partly a terminological issue. (How should we use the word ‘heat’?) But in interesting

11 This is a badly inaccurate statement of the thermodynamic theory of heat. Fortunately, the inaccuracies are harmless in the present context.

Absolute Generality Reconsidered | 107 cases the terminological issue is tied up with substantial theoretical issues. (Is the thermodynamic theory of heat superior to the caloric theory of heat?) And it isn’t always easy to separate the two. Rejecting a ‘just is’-statement comes at a cost, since it increases the range of questions that are regarded as rightfully demanding answers (why does this hot gas have high kinetic energy?), and therefore the scenarios one treats as intelligible (there is a hot gas with low mean kinetic energy). But having extra scenarios to work with can also prove advantageous, since it makes room for additional theoretical positions, some of which could deliver fruitful theorizing. (A proponent of the caloric theory of heat, for example, would want to make room for a scenario in which a substance is hot because it contains high quantities of caloric fluid, even though it is not made up of particles with high kinetic energy.) Disagreement about whether to accept a ‘just is’-statement is best thought of as disagreement about whether the additional theoretical space would be fruitful enough to justify paying the price of having to answer a new range of potentially problematic questions.12 We have been focusing on an example from the natural sciences, but our conclusions carry over to ‘just is’ -statements in metaphysics. One has to balance the cost of rejecting the relevant statement— an increase in the range of questions that are regarded as rightfully demanding answers—with the cost of accepting the statement—a decrease in the range of theoretical resources one has at one’s disposal. There is no quick-and-easy criterion for determining whether the extra theoretical space is fruitful enough to justify paying the price of having to answer a new range of potentially problematic questions. The only reasonable way to proceed is by rolling up one’s sleeves and doing metaphysics.13 Suppose we are considering whether to accept ‘for a time to be present just is for it to have a certain relational property’. By accepting the ‘just is’-statement one would eliminate the need to answer an awkward question: what does it take for a time to be present simpliciter, as opposed to present relative to some time or other? But there is 12 For a more detailed discussion of these matters, see Rayo (forthcoming) "The Contruction of Logical Space" 13 Here I am indebted to Andrew Graham’s PhD thesis.

108 | Agustín Rayo a price to be paid, because it is not immediately obvious that one will have the theoretical resources to explain the feeling that there is something special about the present. By rejecting the ‘just is’-statement, on the other hand, one would be left with a gap to fill—one needs to explain what it is for a time to be present simpliciter as something over and above being present relative to some time or other. One could try to fill the gap by saying something like ‘to be present simpliciter is to be at the edge of objective becoming’, and thereby introduce a new theoretical resource. It is not immediately obvious, however, that such a move would lead to fruitful theorizing, or be especially effective in explaining the feeling that there is something special about the present. The decision whether to accept the ‘just is’-statement is a decision about how to best negotiate these competing theoretical pressures. Here is a second example. Suppose we are considering whether to accept ‘to experience the sensation of seeing red just is to be in a certain brain state’. What sorts of considerations might be used to advance the issue in an interesting way? Jackson’s Knowledge Argument immediately suggests itself:14 Mary is confined to a black-and-white room, is educated through blackand-white books and through lectures relayed on black-and-white television. In this way she learns everything there is to know about the physical nature of the world . . . If physicalism is true, she knows all there is to know. For to suppose otherwise is to suppose that there is more to know than every physical fact, and that is just what physicalism denies . . . It seems, however, that Mary does not know all there is to know. For when she is let out of the black-and-white room or given a color television, she will learn what it is like to see something red, say. (Jackson (1986))

What Jackson’s argument brings out is that physicalists face a challenge. They must somehow accommodate the fact that it seems like Mary acquires information about the world—information she did not already have—when she first experiences the sensation of seeing red, even though physicalism appears to entail that she does not. My own view is that the challenge can be met.15 But someone who thinks that the challenge cannot be met might see the argument as motivating the introduction of possibilities that a physical14 See Jackson (1982) and Jackson (1986); for a review of more recent literature, see Byrne (2006). 15 See Rayo (forthcoming) “Neo-Fregeanism reconsidered”, ch. 4.

Absolute Generality Reconsidered | 109 ist would regard as unintelligible. According to the physicalist, to experience the sensation of seeing red just is to be in a certain brain state. So it makes no sense to consider a scenario in which someone is in the brain state but lacks the sensation. If, however, one were to give up physicalism and countenance the intelligibility of such a scenario, one might be able to relieve some of the pressure generated by Jackson’s argument. For one could claim that, even though Mary knew all along that she would be in the relevant brain state when she was first shown a ripe tomato, she did not yet know if she would also experience the relevant sensation. It is only after she is actually shown the tomato, and experiences the relevant sensation, that she is in a position to rule out a scenario in which she is in the brain state without having the sensation. And this ruling out of scenarios substantiates the claim that Mary does indeed acquire information about the world when she is first shown the tomato. I think there are good reasons for resisting this way of addressing the puzzle. (See, for instance, Lewis (1988).) But suppose one takes it to work. Suppose one thinks that by creating a gap between being in the relevant brain state and experiencing the relevant sensation— and thereby making room for the possibility of being in the brain state without having the sensation—one can adequately account for a case like Mary’s. Then one will be motivated to give up the ‘just is’- statement that keeps the gap closed (‘to experience the sensation of seeing red just is to be in a certain brain state’). But doing so comes at a cost because it opens up space for awkward questions. For instance: ‘I can see that Mary is in the relevant brain state. What I want to know is whether she is also experiencing the relevant sensation. I would like to understand, moreover, how one could ever be justified in taking a stand on this issue, given that we would find Mary completely indistinguishable from her zombie counterpart, or from someone with “inverted” sensations.’

4.2 Avoiding the Tractarian Legacy The Tractarian can be expected to reject Statements 4–7 from section 1. But it seems to me that they are all cases in which the advantages of accepting the ‘just is’-statement far outweigh the disadvantages. Consider Tables. By accepting the claim that for there to be a table just is

110 | Agustín Rayo for there to be some things arranged tablewise, one eliminates the need to address an awkward question: what would it take for a region that is occupied by some things arranged tablewise to also be occupied by a table? It is true that one also loses access to a certain amount of theoretical space, since one is no longer in a position to work with scenarios in which there are things arranged tablewise but no tables. It seems to me, however, that this is not much of a price to pay, since the availability of such scenarios is not very likely to lead to fruitful theorizing. (Not everyone would agree; see, for instance, van Inwagen (1990).) For similar reasons, it seems to me that Properties, Death, and Dinosaurs are all eminently sensible ‘just is’-statements. Again, not everyone will agree. But I hope to have convinced you that these statements shouldn’t be rejected merely on the basis of syntactic considerations. They should be rejected only if one thinks that the resulting theoretical space leads to theorizing that is fruitful enough to pay the price of answering awkward questions. And the relevant questions can be very awkward indeed. By rejecting Dinosaurs, for example, one is forced to concede that the following is a legitimate line of inquiry: I can see that there are no dinosaurs. What I want to know is whether it is also true that the number of the dinosaurs is Zero. And I would like to understand how one could ever be justified in taking a stand on the issue, given that we have no causal access to the purported realm of abstract objects.16

If, on the other hand, you accept Dinosaurs you will think that such queries rest on a false presupposition. They presuppose that there is a gap between the non-existence of dinosaurs and dinosaurs’ having Zero as a number—a gap that needs to be plugged with a philosophical account of mathematical objects. Dinosaurs entails that the gap is illusory. There is no need to explain how the non-existence of dinosaurs might be correlated with dinosaurs’ having Zero as a number because there is no difference between the two: for the number of the dinosaurs to be Zero just is for there to be no dinosaurs.17 16 That this is a legitimate line of inquiry is famously presupposed by Benacerraf (1973). 17 For an account of mathematics along these lines, see Rayo (2009) and Rayo (forthcoming) The Construction of Logical Space.

Absolute Generality Reconsidered | 111 Of course, you won’t see the closing of this theoretical gap as a real benefit unless you think that the resulting theory is consistent with a sensible metaphysical picture of the world, and unless you think that it gives rise to a sensible philosophy of mathematics. The remainder of this paper will be devoted to addressing the first of these two challenges: I will argue that a compositionalist incurs no untoward metaphysical commitments by accepting a ‘just is’-statement like Dinosaurs.18

5. LIFE AS AN ANTI-TRACTARIAN Let an anti-Tractarian be a compositionalist who accepts some metaphysically contentious ‘just is’-statements. (Like Tractarianism, but unlike compositionalism, anti-Tractarianism is a hybrid of linguistic and metaphysical theses.19) Anti-Tractarianism has a distinguished provenance. When Frege claims that the sentence ‘there is at least one square root of 4’ expresses the same thought as ‘the concept square root of 4 is realized’, and adds that ‘a thought can be split up in many ways, so that now one thing, now another, appears as subject or predicate’ (Frege (1892) p. 199), it is natural to interpret him as embracing the ‘just is’-statement: For the concept square root of 4 to be realized just is for there to be at least one square root of 4.

And when he claims, in Grundlagen §64, that in treating the judgement ‘line a is parallel to line b’ as an identity, so as to obtain ‘the direction of line a is identical to the direction of line b’, we ‘carve up the content in a way different from the original way’, it is natural to interpret him as embracing the ‘just is’-statement:

18 I address the second challenge in Rayo (forthcoming) The Construction of Logical Space, where I develop a semantics for mathematical discourse and an account of mathematical knowledge. 19 Tractarianism and anti-Tractarianism are incompatible with each other, but they are not contradictories. A compositionalist who accepts no metaphysically contentious ‘just is’-statements would reject them both. So would someone whose views of reference fall somewhere between Tractarianism and compositionalism.

112 | Agustín Rayo For the direction of line a to equal the direction of line b just is for a and b to be parallel.

More recent texts with broadly anti-Tractarian sympathies include Parsons (1974), Wright (1983), Rosen (1993), Stalnaker (1996), and Burgess (2005).20 My impression is that many contemporary metaphysicians are nonetheless suspicious of anti-Tractarianism. The purpose of this section is to get clear about what the view entails, and what it does not.

Realism A Tractarian might be tempted to complain that if anti-Tractarianism were correct, there would fail to be a definite fact of the matter about how the world is. I have sometimes heard arguments such as the following: Say you believe that for the number of the dinosaurs to be Zero just is for there to be no dinosaurs. You believe, in other words, that a single fact can be described fully and accurately by asserting ‘the number of the dinosaurs is Zero’ and by asserting ‘there are no dinosaurs’. This presupposes that a single fact can get carved up into objects and properties in different ways. When the fact is described by asserting ‘the number of the dinosaurs is Zero’, it gets carved up into an individual (the number Zero), a first-order property (the property of being a dinosaur), and a second-order function (the function taking first-order properties to their numbers); when it is described as ‘there are no dinosaurs’, it gets carved out into a first-order property (the property of being a dinosaur) and a second-order property (non-existence). But if this is so, there can’t be an objective, language-independent fact of the matter about whether there are numbers. It all depends on how we choose to describe the world.

I am happy to grant everything in the first paragraph of this argument (as long as the metaphor of fact-carving is spelled out properly; see section 6). The argument’s second paragraph, on the other hand, strikes me as deeply misguided.

20 Hirsch is a compositionalist (see Hirsch (2002)). But it is not clear to me that he is also an anti-Tractarian.

Absolute Generality Reconsidered | 113 The anti-Tractarian is certainly committed to the view that a single feature of reality can be fully and accurately described in different ways. But this does not entail that there is no fact of the matter about how the world is. On the contrary: it is strictly and literally true that the number of the dinosaurs is Zero, and therefore that there are numbers. And this is so independently of which sentences are used to describe the world—or, indeed, of whether there is anyone around to describe it. The point is simply that the relevant feature of the world could also be fully and accurately described in another way: by asserting ‘there are no dinosaurs’. Moral: If realism is the view that there is a definite, subjectindependent fact of the matter about how the world is, then antiTractarianism is no less of a realist position than Tractarianism.

The World as a Structureless Blob ‘Wait a minute!’—you might be tempted to reply—‘Isn’t the antiTractarian still committed to the view that the world is a structureless blob?’ Absolutely not. The anti-Tractarian believes that it is strictly and literally true that there are tables, that a death took place, that the number of the dinosaurs is Zero, and so forth. So if the strict and literal existence of tables, deaths, and numbers is enough for the world not to be a structureless blob, then it is no part of anti-Tractarianism that the world is a structureless blob. Perhaps what you mean when you say that the world might be a structureless blob is that the world might fail to be endowed with metaphysical structure. In that case, you should think that antiTractarianism is neutral with respect to the question of whether the world is a structureless blob. It is compatible with anti-Tractarianism that reality be carved up by the world’s metaphysical structure. The point is simply that such a carving is not presupposed by ordinary language. A brief aside: I don’t really understand what Putnam has in mind when he talks about Internal Realism. But perhaps one could interpret some of what he says as an endorsement of an anti-Tractarian form of realism. (See, for instance, Putnam (1987) pp. 18–19.)

114 | Agustín Rayo Comprehensivism Comprehensivism is that view that it is in principle possible to give a comprehensive description of the world—a description such that: (1) there is precisely one way for the world to be that would satisfy the description, and (2) the world, as it actually is, satisfies the description. A critic might be tempted to think that anti-Tractarianism is incompatible with comprehensivism. ‘According to anti-Tractarianism’—the critic might argue—‘the same fact can be described in many different ways. One can say that there is a table, or that some things are arranged tablewise, or that the world tableizes, or that tablehood is instantiated, or that two half-tables are put together in the right sort of way, and so forth, with no natural end. But one hasn’t given an exhaustive description of the world until one has described it in all these ways. So the anti-Tractarian could never give a comprehensive description of the world.’ To see where the critic goes wrong, it is useful to consider an example. Suppose I hand you a box and ask you to give me a comprehensive description of its contents. You examine it and say: ‘There is a hydrogen-1 atom in such-and-such a state, and nothing else.’ It would be inappropriate for me to respond by complaining that your answer is incomplete on the grounds that it failed to mention at least two objects: a proton and an electron. Such a response would be guilty of double-counting. Part of what it is for there to be a (non-ionized) hydrogen-1 atom is for there to be a proton and an electron. So when you mentioned that there was a hydrogen-1 atom, the presence of protons and electrons was already included in the information you gave me. It is true that you never mentioned protons and electrons explicitly. But that was not required for your description to be comprehensive. All that comprehensiveness requires is that there be precisely one way for the contents of the box to be such that it would satisfy your description. Moral: Anti-Tractarianism does not entail that comprehensivism is false. What it entails is that there could be more than one way of giving a fully comprehensive description of the world.

Paraphrase It is tempting to think that in accepting a ‘just is’-statement one commits oneself to the availability of a paraphrase-method for translating

Absolute Generality Reconsidered | 115 vocabulary that appears on one side of the statement into vocabulary that appears on the other. Consider, for example, an anti-Tractarian who accepts every instance of the following schema: Numbers For the number of the Fs to be n just is for there to be exactly n Fs.

It is tempting to think that she should also be committed to the claim that arbitrary arithmetical statements can be paraphrased as statements containing no mathematical vocabulary. It seems to me that this would be a mistake. The availability of a suitable paraphrase-function depends on the expressive richness of one’s non-mathematical vocabulary. And the decision to accept Numbers should be based on a cost–benefit analysis of the sort suggested above, not on whether one has access to a powerful enough stock of non-mathematical linguistic resources. It is easy to overestimate the importance of paraphrase when one sees things from the perspective of a nominalist: someone who thinks that numbers don’t exist. For a nominalist might think that non-mathematical paraphrases are needed to give an adequate statement of our best scientific theories, and of the real content of our mathematical accomplishments. But a friend of Numbers is no nominalist, and would see little advantage in stating our scientific theories or mathematical accomplishments in a non-mathematical language. Suppose, for example, that ‘there is an even number of stars’ can be paraphrased as a non-mathematical statement, ϕ. In all likelihood, ϕ will be significantly more cumbersome than its mathematical counterpart. And a friend of Numbers will think that its truth-conditions impose no less of a demand on the world, since she will think that for ϕ to be the case just is for there to be an even number of stars.21 She will therefore see little point in reformulating her scientific or mathematical theorizing in terms of ϕ. The question of whether it is possible to paraphrase arbitrary mathematical statements as statements containing no mathematical vocabulary is an interesting one, and I take it up in Rayo (forthcoming) The Construction of Logical Space. The present point is simply that one should not confuse Numbers with the view that a suitable paraphrase-function exists. Accepting a ‘just is’-statement is one 21

For a related point, see Alston (1957).

116 | Agustín Rayo thing; committing oneself to the availability of paraphrase-functions relating vocabulary on either side of the statement is another.

6. ABSOLUTE GENERALITY In this section I will consider the question of whether an anti-Tractarian should think that there is such a thing as an all-inclusive domain. The first thing to note is that there are several different ways of cashing out the claim that there is such a thing as an all-inclusive domain: • First Reading [Realism + Comprehensivism] There is a definite fact of the matter about how the world is, and it is in principle possible to give a fully comprehensive description of its contents. • Second Reading [Metaphysical Absolutism] There is a ‘fundamental domain’—a domain consisting of the entities that are carved out by the world’s metaphysical structure. • Third Reading [Recarving-Absolutism] There is a ‘maxi-domain’—a domain consisting of the entities that result from every possible way of carving up the world into objects. What should the anti-Tractarian say about the existence of an allinclusive domain, on each of these readings? We have seen that anti-Tractarianism is compatible with both Realism and Comprehensivism. So, on the first of the three readings, there is no tension between anti-Tractarianism and the existence of an all-inclusive domain. What about the second reading? Anti-Tractarianism is neutral with respect to the existence of a ‘fundamental domain’. To address the issue of a fundamental domain would require deploying the notion of metaphysical structure, and anti-Tractarianism does no such thing. Let us therefore turn our attention to the third reading. The antiTractarian believes that there are tables. So a ‘maxi-domain’ would have to include tables. But according to the anti-Tractarian, the fact that there are tables could also be described by saying that there are half-tables put together in the right sort of way, or that the property

Absolute Generality Reconsidered | 117 of tablehood is instantiated, or that some mereological simples are arranged tablewise, or that the set of tables is non-empty, or that the number of tables is greater than Zero. So the maxi-domain would also have to include half-tables and instantiated properties of tablehood and mereological simples arranged tablewise and non-empty sets and numbers greater than Zero and Zero itself, and so forth. Could such a list ever be completed? It seems to me that antiTractarians should be skeptical about the claim that it could. It is not that an anti-Tractarian should think that the world is somehow incomplete. The problem is that there is no reason to think that our concept of ‘carving the world into objects’ is determinate enough to allow for a final answer to the question of how it might be possible to carve up reality into objects. Let me explain.

Unpacking the ‘Carving’ Metaphor As I understand it, a ‘carving’ of the world is nothing more than a compositional system of representation for describing the world. In the most familiar case, a carving is a compositional system of linguistic representation: a language in which the truth-conditions of sentences are generated recursively from the semantic values of a restricted set of basic lexical items. To say that a subject carves the world into objects is simply to say that she represents the world using a language that contains singular terms, or variables that take singular term positions. Similarly, to say that a subject carves the world into properties is simply to say that she represents the world using a language that contains predicates, or variables that take predicate positions. Carving up the world is not like carving up a turkey. For the purposes of spelling out the carving-metaphor, one is not to think of the world as a big object—the mereological fusion of everything there is—and of a carving as a way of subdividing the world into smaller parts. The world, for these purposes, is to be thought of as ‘the totality of facts, not of things’, and a carving is to be thought of as a compositional system for describing these facts.22

22

Compare Eklund (2008).

118 | Agustín Rayo When the carving metaphor is spelled out in this way, the existence of a maxi-domain would require a final answer to the question of what counts as a possible system of compositional representation. And I see no prima facie reason to think that our notion of representation (and our notion of linguistic representation, in particular) are constrained enough for this question to have a definite answer. From the perspective of Tractarianism, the range of admissible compositional languages is restricted by metaphysical structure, since only languages whose semantic structure is in correspondence with the metaphysical structure of the world are potential vehicles for truth. From the perspective of anti-Tractarianism, on the other hand, the only constraint on semantic structure is that it deliver an assignment of truthconditions to sentences from the semantic values of basic lexical items. So it is hard to say in advance what would count as a possible compositional language. Whenever we dream up a new mechanism for representing reality, the potential for a new compositional language—and hence for a new way of carving up the world—will be in place.

An Analogy An analogy might be helpful. Suppose you are told that the ordinals are built up in stages. One starts with a ‘base’ ordinal, and at each stage one gets a new ordinal by pooling together all the ordinals that have been constructed so far. The process is to be carried out indefinitely. In the absence of further constraints, your understanding of ‘ordinal’ will be hopelessly incomplete. It will be consistent with taking the ordinals to be isomorphic with the natural numbers. But it will also be consistent with taking the ordinals to be isomorphic with the natural numbers followed by an additional copy of the natural numbers—or two additional copies, or three, or as many copies of the natural numbers as there are natural numbers. In fact, one’s understanding of ‘ordinal’ will be consistent with taking the ordinals to be isomorphic with any limit von Neumann ordinal.

Absolute Generality Reconsidered | 119 Notice, moreover, that assuming that there is a definite plurality of von Neumann ordinals wouldn’t bring a natural end to the process. For although your understanding of ‘ordinal’ is consistent with taking the ordinals to be isomorphic with the von Neumann ordinals, it is also consistent with taking the ordinals to be isomorphic with the von Neumann ordinals, followed by an ω-sequence of additional objects—or two ω-sequences of additional objects, or an additional ω-sequence for each von Neumann ordinal. And so on. If you give me a definite characterization of ‘ordinal’, I can use it to supply a significantly more generous one. (I can say, for instance, ‘the ordinals are isomorphic to the structure you just articulated followed by a copy of the structure you just articulated for every point in the structure you just articulated’.) And, crucially, I am not able to say anything definite about what it would mean to continue this sort of process ‘all the way up’—anything significantly more illuminating than the original invitation to carry on the process ‘indefinitely’. The upshot is that there is no sense to be made of an absolutely general ordinal-quantifier. But this is not because of some dark metaphysical thesis about the nature of ordinals, and it is not because of some mysterious limit on our referential abilities: it is simply that the notion of ordinal is not well enough defined.23 I think something similar holds for our general notion of linguistic representation. If you give me a definite characterization of ‘linguistic representation’, I suspect I’ll be able to use it to supply a significantly more generous one. If you give me a first-order language, I can give you a second-order language. If you give me an α-level language for some ordinal α, I can give you an α + 1-level language.24 But I’m not able to say anything very informative about what it would mean to iterate this process ‘all the way up’—anything significantly more illuminating than the vague idea that it could be carried out ‘indefinitely’. And, of course, the order of the quantifiers is not the only dimension along which the expressive power of a language might be expanded. If you give

23 24

For more on this sort of picture, see Parsons (1974). For more on languages of transfinite order, see Linnebo and Rayo (typescript).

120 | Agustín Rayo me a definite system of linguistic representation, there may be other ways in which I can use it to supply a significantly more generous one.

A Light-weight Conception of Objecthood? You may be worried that my way of cashing out the carving metaphor is too lightweight. ‘If the only relevant difference between asserting “there are tables” and asserting “some things are arranged tablewise” is to do with the system of compositional representation one chooses to employ’—you might be tempted to complain—‘then someone who asserts “there are tables” hasn’t really committed herself to the existence of tables. For what she says could be true even if there are really no tables.’ As far as I’m concerned, all it takes for there to really [table thump!] be tables is for an English sentence like ‘there are tables’ to be strictly and literally true. And all it takes for ‘there are tables’ to be strictly and literally true is that there be some things arranged tablewise (equivalently: that the property of being a table be instantiated; equivalently: that there be two half-tables put together in the right sort of way; equivalently: that there be tables). Perhaps you mean something different by ‘really’. Perhaps what you have in mind is that in order for something to really exist, it must figure in a ‘fundamental’ description of the world. It must, in other words, be carved out by the world’s metaphysical structure. In this sense of real existence, the view defended in this paper is neutral on the issue of whether there is anything that exists but doesn’t ‘really’ exist.

A Language-infused World? ‘Wait a minute!’—you might be tempted to complain—‘Are you setting forth a view according to which the existence of objects is somehow constituted by language?’ Absolutely not. What is ‘constituted by language’ is the use of singular terms. If we had no singular terms (or variables taking singular-term positions) we wouldn’t be able to describe the

Absolute Generality Reconsidered | 121 world in a way that made the existence of objects explicit. But there would be objects just the same. Speakers of a language with no singular terms can say things like ‘Lo, tableization!’. But for it to tableize just is for there to be a table. So even without singular terms, they would be in a position to convey information about tables. For the anti-Tractarian, the existence of tables depends entirely on how the non-linguistic world is. If there are things arranged tablewise (equivalently: if it tableizes; equivalently: if there are tables), then there are indeed tables. If no things are arranged tablewise (equivalently: if it fails to tableize; equivalently, if there are no tables), then it is not the case that there are tables. The Tractarians’ mistake is to conflate form and content. They think there is a difference in content (i.e. truth-conditions) between ‘there are tables’ and ‘some things are arranged tablewise’, when in fact there is only a difference in form (i.e. semantic structure).

7. CONCLUSION I have argued that a ‘just is’-statement like Tables or Dinosaurs shouldn’t be rejected on general linguistic or metaphysical grounds. I argued, first, that Tractarianism is bad philosophy of language, and suggested compositionalism as a promising alternative. Compositionalists are in a position to accept a number of interesting ‘just is’-statements. The possibility of accepting such statements can be extremely valuable, because by accepting a ‘just is’-statement one eliminates the need to address a certain kind of awkward question. (By accepting Dinosaurs, for instance, one eliminates the need to answer a question that many philosophers have found troubling: ‘I can see that there are no dinosaurs, but why is it also true that the number of dinosaurs is Zero’?) I then argued that anti-Tractarians—compositionalists who accept metaphysically contentious ‘just is’-statements—are not saddled with an unattractive metaphysics. They are not committed to anti-realism, or to the view that the world is a structureless blob, or to the view that the existence of objects is constituted

122 | Agustín Rayo by language. I noted, finally, that anti-Tractarians have reasons to resist the claim that there is such a thing as a maxi-domain—a domain consisting of the entities that result from every possible way of carving up the world into objects.25 Massachusetts Institute of Technology

REFERENCES Alston, W. (1957) “Ontological Commitments,” Philosophical Studies 8–17. Beall, J., ed. (2003) Liars and Heaps, Oxford University Press, Oxford. Beaney, M., ed. (1997) The Frege Reader, Blackwell, Oxford. Benacerraf, P. (1973) “Mathematical Truth,” Journal of Philosophy 70, 661–79. Reprinted in Benacerraf and Putnam (1983). —— and H. Putnam, eds. (1983) Philosophy of Mathematics, Cambridge University Press, Cambridge, second edition. Bennett, K. (2009) “Composition, Colocation and Metaontology.” In Chalmers et al. (2009). Block, N. (2002) “The Harder Problem of Consciousness,” The Journal of Philosophy 99, 391–425. —— and R. Stalnaker (1999) “Conceptual Analysis, Dualism, and the Explanatory Gap,” Philosophical Review 108, 1–46. Bueno, O. and Ø. Linnebo (2009) New Waves in Philosophy of Mathematics, Palgrave Macmillan. Burgess, J. (2005) “Being Explained Away,” Harvard Review of Philosophy 13, 41–56.

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For their many helpful comments, I would like to thank Karen Bennett, Ross Cameron, Matti Eklund, Andrew Graham, Caspar Hare, Jeremy Hartman, Øystein Linnebo, Ricardo Mena, David Nicolas, Damien Rochford, Brad Skow, Bob Stalnaker, Pedro Stepanenko, Eduardo Villanueva, Steve Yablo, and Dean Zimmerman. I would also like to thank seminar participants at MIT, and audiences at the Paris Absolute Generality Workshop, the MIT Work in Progress Seminar, the First Annual Meeting of the Latin American Association for Analytic Philosophy, the Institut Jean-Nicod, the University of Leeds, the University of Connecticut, and Institute for Philosophical Research at the National Autonomous University of Mexico. Portions of this paper were written during the tenure of an ACLS Fellowship, for which I am extremely grateful.

Absolute Generality Reconsidered | 123 Byrne, A. (2006) “Review of There’s Something about Mary,” Notre Dame Philosophical Reviews 2006.01.20. Available at . Chalmers, D. (1996) The Conscious Mind: In Search of a Fundamental Theory, Oxford University Press, New York. —— D. Manley, and R. Wasserman, eds. (2009) Metametaphysics, Oxford University Press, New York. Eklund, M. (2008) “The Picture of Reality as an Amorphous Lump,” In Sider et al. (2008). —— (2009) “On Some Recent Criticisms of the ‘Linguistic’ Approach to Ontology,” Dialectica 63, 313–23. Fine, K. (2001) “The Question of Realism,” Philosophers’ Imprint 1, 1–30. Frege, G. (1884) Die Grundlagen der Arithmetik. English Translation by J.L. Austin, The Foundations of Arithmetic, Northwestern University Press, Evanston, IL, 1980. —— (1892) “On Concept and Object,” Vierteljahrsschrift für wissenschaftliche Philosophie 16, 192–205. The English translation by Peter Geach is reprinted in Beaney (1997). Goldfarb, W. (1997) “Metaphysics and Nonsense: On Cora Diamond’s The Realistic Spirit,” Journal of Philosophical Research 22, 57–73. Hacker, P. (1986) Insight and Illusion: Themes in the Philosophy of Wittgenstein, Oxford University Press, Oxford. Heil, J. (2003) From an Ontological Point of View, Clarendon Press, Oxford. Heim, I. and A. Kratzer (1998) Semantics in Generative Grammar, Blackwell Textbooks in Linguistics, Blackwell, Oxford. Hirsch, E. (2002) “Quantifier Variance and Realism,” Philosophical Issues 12, 51–73. Hofweber, T. (2009) “Ambitious, Yet Modest, Metaphysics,” in Chalmers et al. (2009). Jackson, F. (1982) “Epiphenomenal Qualia,” Philosophical Quarterly 32, 127–36. —— (1986) “What Mary Didn’t Know,” The Journal of Philosophy 83, 291–5. Kanger, S. and S. Öhman, eds. (1980) Philosophy and Grammar, Reidel Publishing Company, Dordrecht. Lewis, D. (1980) “Index, Context, and Content,” in Kanger and Öhman (1980) pp. 79–100. Reprinted in Lewis (1998). —— (1983) “New Work for a Theory of Universals,” Australasian Journal of Philosophy 61, 343–77. Reprinted in Lewis (1999). —— (1984) “Putnam’s Paradox,” The Australasian Journal of Philosophy 62, 221–36. Reprinted in Lewis (1999).

124 | Agustín Rayo —— (1988) “What Experience Teaches,” Proceedings of the Russellian Society 13, 29–57. Reprinted in Lewis (1999). —— (1998) Papers in Philosophical Logic, Cambridge University Press, Cambridge. —— (1999) Papers in Metaphysics and Epistemology, Cambridge University Press, Cambridge. Linnebo, Ø. and A. Rayo (typescript) “Hierarchies Ontological and Ideological.” Morton, A. and S. Stich, eds. (1996) Benacerraf and his Critics, Basil Blackwell, Oxford. Parsons, C. (1974) “Sets and Classes,” Noûs 8, 1–12. Reprinted in Parsons (1983), pp. 209–20. —— (1983) Mathematics in Philosophy, Cornell University Press, Ithaca, NY. Pears, D. (1987) The False Prison: A Study of the Development of Wittgenstein’s Philosophy, volume 1, Oxford University Press, Oxford. Putnam, H. (1987) The Many Faces of Realism, Open Court, La Salle, IL. Rayo, A. (2002) “Word and Objects,” Noûs 36, 436–64. —— (2003) “When does ‘Everything’ Mean Everything?” Analysis 63, 100–6. —— (2009) “Towards a Trivialist Account of Mathematics,” In Bueno, O. and Ø. Linnebo (2009) New Waves in Philosophy of Mathematics, Palgrave Macmillan. —— (forthcoming) “Neo-Fregeanism reconsidered,” in Abstractionism, edited with Philip A. Ebert. Oxford University Press. —— (forthcoming) The Construction of Logical Space, Oxford University Press. —— and T. Williamson (2003) “A Completeness Theorem for Unrestricted First-order Languages,” In Beall (2003), pp. 331–56. Rosen, G. (1993) “The Refutation of Nominalism (?),” Philosophical Topics 21, 149–86. Schaffer, J. (2009) “On What Grounds What,” in Chalmers et al. (2009). Sider, T. (typescript) Writing the Book of the World. —— J. Hawthorne, and D. Zimmerman, eds. (2008) Contemporary Debates in Metaphysics, Blackwell, Cambridge, MA. Stalnaker, R.C. (1996) “On What Possible Worlds Could Not Be,” in Morton and Stich (1996), 103–19. Reprinted in Stalnaker (2003), 40–54. —— (2003) Ways a World Might Be: Metaphysical and Anti-Metaphysical Essays, Clarendon Press, Oxford.

Absolute Generality Reconsidered | 125 van Inwagen, P. (1990) Material Beings, Cornell University Press, Ithaca, NY. Wittgenstein, L. (1921) Tractatus Logico-Philosophicus, London. Published as “Logisch-Philosophische Abhandlung,” in Annalen der Naturphilosophische Vol. XIV, 3/4, 1921, pp. 184–262. Wright, C. (1983) Frege’s Conception of Numbers as Objects, Aberdeen University Press, Aberdeen.

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III

HUMEANISM AND LAWS OF NATURE

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5. Goodbye, Humean Supervenience Troy Cross THE MARCH OF THE CAUSAL The 1960s witnessed a remarkable string of causal theories. Functionalism—in essence a causal theory of mental states—was first on the scene (Putnam 1960) followed by a causal theory of perception (Grice 1961), a causal theory of action (Davidson 1963), a causal theory of knowing (Goldman 1967), a causal theory of events (Davidson 1969), and finally in 1970, a causal theory of reference (Kripke 1980). Though the most famous causal theories were already in play by the end of the decade, the trend had not run its course. In 1978, David Armstrong defended the Eleatic Principle, which might be characterized as a causal theory of being and followed it two years later with the somewhat less dramatic causal theory of object persistence (1978; 1980). It was in that same volume—Time and Cause: Essays Presented to Richard Taylor—that Sydney Shoemaker published his causal theory of properties, which is the focus of the present essay (1980). Why the causation fixation? Each of these causal theories, of course, had roots in its local dialectical history. Goldman was responding to the Gettier problem (1963), Kripke to the failures of descriptivism, and so on. But there’s also no denying that causation was in the air like a fine mist. It was the default analysans, followed closely in popularity by related notions like counterfactual dependence, chance, and nomic necessitation. Thus, Nozick rejected the causal theory of knowing only to try a counterfactual theory of knowledge instead, and Armstrong did likewise with nomic necessitation (Nozick 1981; D.M. Armstrong 1973). But causation, counterfactuals, laws, and chance all belong to the same tightly knit family. The question, suitably qualified, remains: why were these causal notions so popular? One reason was the rise of naturalism in the period. Naturalism attempts to shoehorn all meaningful talk of action, knowledge,

130 | Troy Cross mind, reference, and the like into a non-primitively-mentalistic world. It’s a tight fit. Most of the philosophical subjects up for naturalistic treatment involve relations between persons and their environments. To ground such relations appropriately, theorists had to find topic-neutral substitutes, relations that could apply univocally across the categories of mind and body, and of these there are precious few with any prospect of success: causation, counterfactual dependence, and nomic necessitation nearly fill out the roster. So, the ubiquity of causal analyses in philosophy during the rise of naturalism is no surprise. The surprise is how brilliantly philosophers managed (and still manage) with such crude implements. In any case, naturalism did not halt its advance upon reaching causal theories of mind, knowledge, action, reference, and the like, but turned upon its own devices, the very ideas like causation, law, chance, dispositions, and counterfactual dependence, which were themselves marked as spooky theological vestiges in need of some kind of reductive or eliminative treatment (Goodman 1983, 31–40). The modest strategy here was to pick the most central of these notions and use it to explain the others. But the more popular tradition, tracing back, perhaps, to David Hume, and culminating in David Lewis, aimed for more. Throughout the 1970s, Lewis offered ingenious reductions for the entire family of causal notions to facts about the distribution of what he called “the perfectly natural properties”. Lewis reduced the laws of nature to maximally simple and informative facts about the distribution of these properties (2001a, 73), reduced counterfactuals to complicated similarity relations (1979), and reduced causation to chains of counterfactual dependence (1973). Here is his purist vision in canonical form: Humean supervenience is named in honor of the greater denier of necessary connections. It is the doctrine that all there is to the world is a vast mosaic of local matters of particular fact, just one little thing and then another. (But it is no part of the thesis that these local matters are mental.) We have a geometry: a system of external relations of spatiotemporal distance between points. Maybe points of spacetime itself, maybe point-sized bits of matter or aether or fields, maybe both. And at those points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated. For short: we have an arrangement of qualities. And that is all. There is no difference without difference in the arrangement of qualities. All else supervenes on that. (Lewis 1987, ix–x)

Goodbye, Humean Supervenience | 131 Lewis’s conjecture holds undeniable “desert landscape” appeal: the distribution of point-sized qualities is a strikingly sparse ontological base. I think Shoemaker’s causal theory of properties, however, was no less ambitious. Rather than trying to reduce away all of the causal notions to something else, Shoemaker instead anchored to the idea of a causal power itself. So far, this sounds like the modest strategy mentioned above, but Shoemaker also used causal powers to regiment something lying outside the small causal circle, viz., the notion of a natural property, which Lewis himself could only regard as an untamed primitive. In this way, Shoemaker claimed equal rights, alongside Lewis, to the coveted title of “most naturalistic”, and—perhaps along with Armstrong’s Eleatic pronouncements— brought the sixties-era celebration of the causal to its climax. I’m going to discuss Shoemaker’s paper in this essay, but my goal is not intellectual history or critical exegesis. My goal, rather, is to offer a new argument for Shoemaker’s theory. Like Shoemaker, I will argue that the intrinsic properties of concrete objects are each necessarily correlated with a unique causal power, i.e., that properties have dispositional shadows. But unlike Shoemaker, I will make no use of (even slightly) controversial premises in my argument. Furthermore, I will note that if we take necessarily co-extensive properties to be identical then it follows that the intrinsic properties of concrete objects are identical to powers, i.e., that properties are nothing distinct from their dispositional shadows, i.e., that properties are dispositions. Though my argument is perfectly general, it’s most instructive to consider it as a direct attack on Lewis’s reductive project. If I am right, then though Lewis never saw this implication, his Humean accounts of the dispositions, laws, chances, and the like, together with his theory of property identity, commit him to saying that each of his beloved qualities—his purely intrinsic, categorical, local properties—is in fact a disposition. The famous “Humean supervenience” excerpt above would represent Lewis’s metaphysics with equal veracity if it were to substitute “dispositions” for “qualities”. (The horror!) In order to resist my conclusion, Lewis must either abandon his reductionism about the dispositional/causal/nomic/subjunctive in favor of full-bore eliminativism or else abandon his modal criteria of property individuation and take refuge in hyperintensional

132 | Troy Cross distinctions, i.e., of cases where a disposition and a quality are necessarily co-extensive, Lewis would have to say those dispositions and qualities are nevertheless wholly distinct, and that only the qualities are perfectly natural. Either strategy would represent not only a departure from the letter of Lewis’s metaphysics, but also a betrayal of its spirit.

SHOEMAKER’S THEORY Shoemaker’s thesis in “Causality and Properties” is that the intrinsic properties of concrete objects are uniquely and essentially correlated with the powers those properties bestow (1980). The view was not entirely novel, prefigured as it was in the work of Achinstein (1974), Harre and Madden (1975), Kneale (1952), Putnam (1970), Sellars (1948), and, indeed, perhaps in nearly every philosopher’s work before the early occasionalists, al-Ghazali and Gabriel Biel, wrested powers out of ordinary things and granted them to God alone (Freddoso 1986). But Shoemaker improves upon his predecessors in two noteworthy respects. First, he defines the view with a new measure of precision, identifying powers by appeal to a function from circumstances to effects, such that objects possessing the power and in ci “have a certain effect”: ei, then claiming that every property necessarily and uniquely correlates with a function from other properties to these powers: Just as powers can be thought of as functions from circumstances to causal effects, so the properties on which powers depend can be thought of as functions from properties to powers or, better, as functions from sets of properties to sets of powers. (Shoemaker 1980)

Since the function by which Shoemaker individuates properties itself takes properties as arguments, and the circumstances and effects are a matter of which properties are instantiated where, he is forced to acknowledge, and embrace, the non-well-foundedness of his proposal. Still, just as the individuation of sets by their elements and the individuation of ur-elements by the sets to which they belong is informative, though ultimately circular, Shoemaker’s theory promises us a non-trivial, substantive constraint on the identity of properties. (If we’re willing to allow that circumstances are not merely extrinsic, but can include intrinsic components as well, we may simplify

Goodbye, Humean Supervenience | 133 Shoemaker’s formulation, neatly folding the “other properties” into the circumstances that identify a power. For example, if F is identified by the function from another property, G, to the power to bring about E in circumstances C, we can call the conjunction of C’s holding, together with the object in question’s having G, “C*”. Now, F can be identified by appeal to the function from C* to E.) The second notable advance in “Causality and Properties” is the bevy of arguments, both epistemic and semantic, that Shoemaker marshaled in support of the newly clarified view. If properties were untethered from the powers of objects, he claimed, we could neither know much of anything nor successfully refer to the properties of things. Since we do have knowledge of the world, however, and also manage to refer to the properties of objects, properties are, in fact, tied to the causal powers they bestow in just the way his theory describes. Though much discussed in the three decades since the publication of “Causality and Properties,” Shoemaker’s epistemic and semantic considerations have not fared as well as his articulation of the view itself. To the contemporary philosophical ear they manifest a peculiar, twentieth-century anxiety about individuation, knowledge, and reference. Thus, although the theory itself boasts many fans among contemporary metaphysicians, it is rare indeed to find an approving citation of Shoemaker’s arguments for it (Bird 2007, 263). I shall briefly discuss the arguments, and in the predictably disapproving manner. But recall that my purpose in so doing is not refutation. Rather, it is to expose a new and powerful line of reasoning for Shoemaker’s view.

EPISTEMIC ARGUMENTS AND REPLIES Shoemaker’s central argument in “Causality and Properties” is a sort of reductio. He first asks us to assume that properties are not tied to powers in the manner he proposes and then points out that all sorts of new skeptical scenarios are possible. For instance, suppose there could be causally inert properties. Then two very similar-appearing objects—say, two ballpoint pens from the same package—may in fact be vastly dissimilar, because

134 | Troy Cross one is stacked with multitudes of inert properties that the other lacks. This may be the case despite the pens having exactly the same causal powers throughout their lifetimes. Likewise, two very dissimilar-appearing objects—say, a pen and a person—may in fact be highly similar, by virtue of sharing a multitude of inert properties. Further, if properties can endow different powers at different times, it may be that an object appearing to change—say, undergoing a pen-to-person transformation—actually isn’t changing its intrinsic properties at all; the intrinsic properties are remaining fixed, while the property/power relation shifts. And finally, it may be that what appears to be unchanging is, in fact, undergoing radical change in its intrinsic properties while the relation between properties and powers evolves in a disguising, compensating fashion. According to Shoemaker, if such skeptical scenarios were possible we could not have any knowledge of similarity and change. But we do have such knowledge. Therefore, the scenarios are impossible. It’s an odd way of doing modal metaphysics. After all, it’s never argued that the envisioned scenarios embed any hidden conceptual confusion. So it’s not clear how ruling them out metaphysically is supposed to do any epistemic work. Let me put the problem as a dilemma. Upon first considering the scenarios, we either know we’re not in them or we don’t know we’re not in them. In the former case, we needn’t worry about whether they’re metaphysically possible, so long as they’re not actual (or nearby). But in the latter case, it seems we also don’t know Shoemaker’s metaphysics is correct. That is to say, either Shoemaker’s ad hoc modal surgery is unnecessary or else we lack the nerve to perform it. Fortunately, standard epistemology itself saves us from systematic ignorance, and without a modal metaphysics driven by wish fulfillment. As many have pointed out (Cross 2004; Hawthorne 2001; Schaffer 2004), standard anti-skeptical theories of knowledge, or of “knowledge”, of the traditional fallibilist, reliabilist, subject-sensitive invariantist, contextualist, and relativist varieties, all address Shoemaker’s skeptical challenge with exactly the same methods—and just as successfully (or unsuccessfully)—as they address Descartes’ evil genius scenario. Epistemically speaking, there is nothing to see here, nothing new. Move along, everyone!

Goodbye, Humean Supervenience | 135 Although the foregoing replies deflect Shoemaker’s argument, the most satisfying is a tu quoque. Analogous skeptical worries about similarity and change arise even if properties are linked to powers in exactly the way that Shoemaker suggests. To see how this is so, first observe that one of the central motivations for dispositionalism is the thought that dispositional differences between objects need never be manifest in the actual world. Moreover, these dispositions might contain wholly novel manifestations, alien to the actual world. For example, two fundamental particles might never actually meet, but if they did, their collision would result in a brand new kind of fundamental particle, an “alien” property (C.B. Martin 1993; C.B. Martin 1997). Of course, this alien property itself would have powers, perhaps powers that would be manifest only upon reacting with other fundamental kinds of particles in the actual world, perhaps powers to produce further fundamental alien properties, and so on. Thus, actual properties may be sensitive to aliens; the actual world may contain any number of properties with potential effects that would be manifest only in the company of an alien. Call such properties “alien-sensitive powers”, and call their triggers “alien catalysts”. Now we’re in a position to mirror Shoemaker’s skeptical scenarios. Suppose a property is sensitive only to an alien catalyst. While not strictly speaking inert, it is for all practical purposes undetectable, requiring, as it does, an alien trigger. Call such properties “quasi-inert”. Now we can load up one ballpoint pen, but not the other, with multitudes of these quasi-inert properties, each of which has different alien sensitivities. The two pens will seem, under any actual test, to be exactly alike, while in fact being vastly dissimilar. Quasi-inert properties can also make a pen and person more alike than a pen and another pen, by the same means, mutatis mutandis. We can also mimic property/power swapping, as follows. Let F and F’ differ only with respect to (actual) catalysts C and C’. In the presence of C, F endows power P but F’ does not. In the presence of C’, F’ endows power P but F does not. Imagine that at time t, everything F switches to F’ and everything C switches to C’. Though everything will change, no change will be detected. Likewise for the case of objects seeming to change (pen-to-person, say) when

136 | Troy Cross they are in fact static, which could be the result of the introduction of new catalysts, rather than an actual intrinsic change. Though Shoemaker’s envisioned skeptical scenarios and the analogues that arise from within dispositionalism itself are subtly different, standard epistemologies will not draw a bright line between them. Shoemaker may have called our attention to a new and devastating species of skeptical argument, but he has failed to make a convincing case for dispositionalism.

THE ARGUMENT THAT LEWIS’S LOCAL QUALITIES SATISFY SHOEMAKER’S CONDITIONS The purpose of the previous section was not primarily to review Shoemaker’s epistemic arguments, but to introduce and motivate the notion of alien sensitivity, the idea that dispositions may have radically non-actual activation conditions. Once we see that Shoemaker’s criterion distinguishes properties whose dispositional differences are only revealed under alien circumstances, we are very close to recognizing that even paradigmatically categorical properties are powers. Begin by asking yourself whether Lewis thinks there are possible conditions such that, for arbitrary perfectly natural property F, F endows some power to objects in those conditions. The answer, quite clearly, is yes. Lewis’s view is not that perfectly natural properties cannot endow powers. Rather, it is that they endow different powers in different circumstances. Exactly how the powers granted by F vary with circumstances is a complicated matter. A nomic theory of powers would say something like the following: in worlds where it follows from the laws of nature that Fs become Gs, F endows the power to become a G. And in general, in worlds where it is nomically necessary that Fs do something, then F endows the power to do that thing. There is reason to think Lewis does not hold the nomic theory of powers, but it is nevertheless very close to his view, and helpful to bear in mind. But we needn’t worry about precisely which conditions are required for F to endow the power to become G. What matters is that there are possible conditions in which F disposes things to become Gs. These conditions may be states of the whole world, as

Goodbye, Humean Supervenience | 137 on the nomic theory of powers, or states of subregions of a world, if, for instance, F disposes things to become Gs in one part of the world but not in another. The important thing is that the regions of modal space at which F disposes things to become G are not random, but a matter of how powers generally supervene on the distribution of Humean facts. Let “C” name the set of conditions under which F endows the disposition to become G, (or the conditions under which F-ness disposes things to become G). Lewis should accept the following without objection: (1) In C, F disposes things to become Gs. Now to the substantive claim. Having said (1), I think Lewis must also say that: (2) F disposes things to become G in C. The step from (1) to (2) seems to add metaphysical weight, because (1) contingently correlates F with a disposition, while (2) pushes that contingency into the very specification of the disposition itself, yielding a necessary connection between F and a disposition. While, according to (1) there are certain conditions under which F disposes things to become G, according to (2), F, everywhere in modal space, disposes things in a certain way: to become G under conditions C. The transition is hard to resist. When someone says, “In the Sun, ice is disposed to melt”, we paraphrase them seamlessly as saying that ice is disposed to melt in the Sun. The same can be said for the habitual, e.g., ice melts in the Sun. Here it is even clearer. One does not hesitate in the inference from “In the Sun, ice melts,” to “Ice melts in the Sun”. Yet, on at least some theories of dispositions, the habituals are true only if being in the Sun ice is disposed to melt, and ice is disposed to melt in the Sun, respectively (Fara 2005). Compare also: (a) This joke is funny in England; and (b) In England, this joke is funny. It is very hard to see the difference. Suppose (a) and (b) are uttered in the US. (a) directly attributes a power to the joke: it has the power to make people laugh in England. (b) does not directly attribute a power, but says that the joke has a power under other conditions: being in England. Yet, it would be absurd to halt the inference from (a) to (b) or vice versa. “Yes, of course, if the joke is told in England, then it has the power to amuse audiences.

138 | Troy Cross But don’t go thinking that means that here in the US, the joke has any power to amuse audiences in England!” Bizarre. I think this is just a general feature of power and disposition talk. F disposes things to become G in C iff in C, F disposes things to become G. And it is easiest to see for habituals: Fs G in C iff in C, Fs G. It is worth asking why the analogous inference is invalid for subjunctive conditionals. At least according to the standard semantics for subjunctives, it does not follow from (x)(Fx >cf (Gx > cf Hx)) that (x) (Gx > cf (Fx > cf Hx)). It may be true that all soluble things are such that if they were immersed in water, they would dissolve, but false that all things actually immersed in water would dissolve if they were soluble. Suppose the only soluble things are not in water, and that nothing would prevent their dissolving if put in water, but that objects in the water are such that if they were to become soluble, they would be prevented from dissolving. Perhaps a sorcerer guards over all of the immersed objects, and would cast a spell instantly removing them from water, or else in some way interfering with the dissolution process in some way at the microphysical level, should they become soluble. This sorcerer, however, may be perfectly content to let soluble objects that are not actually immersed dissolve, should they be put in water. He watches over only the immersed objects. So in general, it would be a bad idea to infer from the fact that if something were F, then if it were G, it would be H to the fact that if something were G, then if it were F, it would be H. Given the undeniably close relation between disposition attributions and subjunctive conditionals, we must take care to avoid that fallacious inference in another guise. Yet it seems undeniable that if in water, soluble objects are disposed to dissolve, then soluble objects are disposed to dissolve in water, and also that if in water, soluble objects dissolve, then soluble objects dissolve in water. Why is this so? I think the cases where the relevant inference fails for subjunctives are precisely those cases where disposition ascriptions succeed and counterfactual or subjunctive analyses appear to falter. So-called “finkish” dispositions are dispositions that disappear in their stimulus conditions (C.B. Martin 1994). Lewis imagines a sorcerer guarding a fragile chalice, poised to cast a spell making it not fragile if it is going to be dropped (1997). The chalice’s disposition is finkish, vanishing exactly when called upon. It’s fragile, intuitively, but would not break if dropped. Relatedly, Bird notes

Goodbye, Humean Supervenience | 139 that “antidotes” are external factors that disrupt and prevent the manifestation of dispositions in the stimulus conditions, e.g., the antidote to a poisonous venom (1998). Here the venom has a disposition to kill when ingested, even though it may be ingested without causing death because of the interference of the antidote. Finally, we can generate the reverse phenomenon, where a certain subjunctive conditional holds without the associated disposition, by appealing to what Mark Johnston calls “mimics” (1992). Imagine a non-fragile piece of clay attached to a motion-sensing machine that, if it detects jarring or dropping of the clay, will instantly stream liquid nitrogen over the clay, causing it to shatter. This is my claim: all counterexamples to the pattern of subjunctive inference above are also cases of finks, antidotes, or mimics. That is why the analogous form of reasoning is permissible in the case of powers, dispositions, and habituals. I do not have a proof that all of the cases in which the relevant inference pattern for subjunctive conditionals fails are cases of finks, antidotes, or mimics, but you’re invited to submit counterexamples. My only goal is to explain away the threat that swapping the conditions in and out of the scope of the power operator inherits the danger of the analogous operation for subjunctives. To dismiss this threat, it’s enough to have strong initial intuitions about the legitimacy of the inference and a plausible explanation of the difference between the good case (dispositions) and the bad case (subjunctives), and that much, I think we have. Even on a pure subjunctive conditionals account of dispositions, Lewis is in trouble. While, in general, the form of inference with subjunctives noted above is invalid, it may still be valid for a subset of cases. For instance, suppose something is an F-detector iff it is disposed to register “1” if in a world with an F. Now, we can make this disposition to register “1” if in a world with an F as strong as we like. So let’s make it a full-strength, deterministic, unconditional power. Lewis might argue that there couldn’t be any such power. But I don’t see why he would say that. For him, it would just be a highly non-natural property. Now, isn’t every F such that, if it were in a world with an F detector, the F detector would register “1”? If so, then F underwrites a non-trivial subjunctive conditional. So, on a subjunctive conditional account of dispositions, F underwrites a disposition.

140 | Troy Cross But perhaps we can deny that F entails the conditional. Perhaps some Fs are guarded over by F-detector-averse sorcerers waiting to cast spells should an F-detector threaten. If this supposition is coherent, what we have here is a paradox equivalent to the existence of two omnipotent beings—the F-detector and the F-detectoraverting-sorcerer: they aren’t jointly possible. But now consider the conditions of being alone in a world with an F-detector. If an F were in those conditions, it wouldn’t have its protective sorcerer with it, and the F detector would register “1”. Let’s recap. If there are possible conditions, C, under which a “categorical” property F disposes things to become G, then throughout modal space, F disposes things to become G in C. The conditions may be strange. They may involve non-actual laws. But dispositions may have alien triggers, as we have seen, so this is nothing distinctive. Some parts of the pluriverse are poised to send the property into action. By the same token, the property, wherever instantiated, is poised to go into action in those parts of the pluriverse, and not in others. In this way, categorical properties, far from the inert, modally innocent creatures they purport to be, are in some sense modal monads, representing the full range of possible conditions (but unlike monads, causally interacting as well). That was the first question: do properties endow powers? It seems they do. In fact, it is difficult to see how one could offer any theory of how powers supervene on properties without giving us a recipe for such endowment. Now comes the second question: is there a unique power endowed by each natural property? After the last question, this should be relatively easy. Suppose we start with two arbitrary properties, F and F*. Would Lewis think there are any conditions under which F and F* would endow different powers? Of course. Again, the nomic account of powers would say that in worlds where it’s a law that Fs become Gs and F*s do not, F endows the power to become G and F* does not. Lewis’s theory may be a bit subtler than that, but it doesn’t matter. Given the wealth of the pluriverse, he will have such distinguishing conditions, for any two properties. Thus, we can find a single power that is both necessarily correlated with F and unique to it by taking the union of each of the powers (sets of pairs of circumstances and effects) that distinguishes F from F*, F from F**, and so on. For rhetorical purposes, my discussion has meandered through some of the specifics of Lewisian reductive metaphysics, but it must

Goodbye, Humean Supervenience | 141 be noted that my central claim in this section—that natural properties can be paired one-to-one with powers—depends on none of the implementational detail. It depends only on each perfectly natural property making a potential difference to the powers of things instantiating that property, and on no two perfectly natural properties making all the same potential differences. I simply note a general feature of powers, viz., that whatever the potential differences in powers that a natural property contributes to its bearers, those potential differences are also actual differences in potential. This is simply what powers do: they code potential differences as actual ones. Since Lewis aims to explain, rather than eliminate, causal powers, it should be uncontroversial that for him, every natural property potentially contributes to the powers of things, and that no two perfectly natural properties make all the same potential contributions. If one simply attends to what powers are, one can see that every perfectly natural property is shadowed by a unique power, i.e., Shoemaker’s thesis from “Causality and Properties” is true.

ANTI-HYPERINTENSIONALITY Observe that for Lewis, the perfectly natural properties, like all other properties, are sets of possibilia: “. . . the property of being donkey comes out as the set of all donkeys, the donkeys of other worlds along with the donkeys of ours” (Lewis 2001b). The natural properties may be specially marked by perfectly resembling tropes, universals, or simply primitive naturalness, but properties, whether natural are not, are sets, and sets are, by the axiom of extensionality, individuated by their membership. Thus if a perfectly natural property P and a power P’, which is endowed by P, are necessarily coextensive, then P=P’. Given the argument above for Shoemaker’s thesis, it follows that for Lewis the perfectly natural properties—all of them—are powers. And if the perfectly natural properties, the local qualities of which Lewis speaks, are all powers, then Humean Supervenience isn’t worthy of its name. Lewis might as well have said that everything supervenes on powers, or that there is just one little disposition and then another.

142 | Troy Cross NON-NOMIC, NON-GLOBAL CONDITIONS The heart of my strategy is to take global states of the world such as possible laws of nature and treat them as circumstances which, together with the original property, would trigger an effect. Just as placing salt in water or dropping a glass reveals its disposition to dissolve, or to break, so the laws of nature reveal the causal powers inherent in so-called categorical properties. One might think that this is cheating. It is certainly unorthodox. One might think that laws of nature are of an entirely different category from standard activation conditions such as being put in water or being dropped. Laws concern the relations between ordinary properties, and being in a world with such-and-such laws is itself a property only in a highly attenuated, highly derivative sense. (So goes the objection.) Standard categoricalist opposition to Shoemaker’s individuation and essence claims rested on everyone tacitly ignoring my strategy—which certainly would explain why no one has called attention to it—and so a repair should be easy enough. All we have to do is make that tacit assumption explicit. Very well. Let’s admit that Shoemaker is correct if we allow that the triggering circumstance in the specification of a power can be any sort of state, but contend that his view is only interesting if we restrict the relevant circumstances to exclude nomic facts, facts about what the laws are. The proposal, then, is that for Shoemaker, but not Lewis, property possession entails having a power to bring about certain effects under certain non-nomic conditions. And remember: this is supposed to be no substantial departure from Shoemaker’s or Lewis’s intentions, but rather a matter of noting a previously tacit restriction. To my mind, the proposed revision is grossly ad hoc. The power to bring about a G in a world where it’s a law that Gs follow Fs is still a power, and it’s a power that characterizes F in particular, and not properties generally. But I won’t presume my audience shares that immediate judgment of arbitrariness. The “non-nomic conditions” restriction has a more blatant flaw. Since Shoemaker thinks the laws of nature are necessary, every circumstance is in a sense nomic: it entails the laws of nature, simply because everything does. So Shoemaker, on this proposal, would end up agreeing with Lewis

Goodbye, Humean Supervenience | 143 that properties do not necessarily endow powers with non-nomic activation conditions, because on Shoemaker’s view, there aren’t any non-nomic conditions. Nor is it useful to stipulate that the circumstances identifying the relevant powers cannot “mention” the laws, even if they entail them. This is a debate about properties and not predicates. The circumstances are a matter of which things have which properties. They don’t “mention” anything. And remember that we are working in an anti-hyperintensional environment. Any attempt to get fine-grained enough to block the entailment of the laws by Shoemaker-envisioned circumstances will likely require distinctions too medieval for Lewis. If Lewis cannot distinguish his view of the nature of properties from Shoemaker’s without recourse to hyperintensionality, the game is already over. A related strategy would be to forbid the use of triggering circumstances that are temporally later than the effect in the specification of powers. Since the Humean conditions for F’s endowing the power to become G will undoubtedly involve large swathes of the future, this would rule out the troublemaking powers. But this is overly restrictive, particularly for someone of Lewis’s methodological bent. The recent theory that the Large Hadron Collider’s many failures and delays are due to time travel and the universe’s “hatred of the Higgs” should be ruled out on physical, not metaphysical grounds (Nielsen and Ninomiya 2008).

DLEWIS Before we try in earnest to engineer a better restriction on the conditions, or to find some way to block the inference pattern, let’s look ahead to the end game. The problem is not with formulating a proper version of the restriction, or with blocking a pattern of inference for that matter. The problem is deeper. Consider the following fictional philosopher: David K. Dlewis. Dlewis is much like Lewis. He thinks that all of the possible worlds exist as spatiotemporally disconnected concreta. He thinks properties are sets of possibilia. He endorses counterpart theory with table-banging glee. He strokes his beard. He likes trains. He inspires students. He writes beautifully. And he’s really really smart.

144 | Troy Cross Dlewis agrees with Lewis about which properties are the perfectly natural ones. That is, when Lewis and Dlewis survey the pluriverse from their God-like pedestals, there is never an occasion on which Lewis says some set of possibilia, S, is natural and Dlewis does not, or vice versa. They also agree on principles of recombination. They even agree on counterfactuals. There are notable differences in their metaphysics, however. Dlewis says that all of the natural properties are dispositions to have certain effects, conditional on global states of the world. Which global states? Precisely the ones Lewis says are the conditions in which the perfectly natural properties endow the powers to bring about those effects. In other words, Dlewis does not fight the general inference I was trying to draw earlier, but embraces it. Lewis’s actual views are a bit more complex, but if we think about the (very nearby) nomic theory of powers discussed earlier, we can simplify and say that F endows the power to become G just in the circumstances in which it is nomically necessary that Fs become Gs. Thus, for Dlewis, F-ness is (in part) the power to become G if in the following condition: the global state of the world is such that the simplest and most informative axiomatic description of that world entails that Fs become Gs (i.e., it’s a Lewis-law that Fs become Gs). For Dlewis, such global states of the world do not constitute laws. He thinks, like a typical dispositionalist, that laws reflect the dispositional essences of properties. He thinks laws are necessary truths and that (deterministic) dispositions metaphysically necessitate their effects. Yet, he agrees with Lewis that there are worlds where Fs do not become Gs. As Dlewis sees it, those are worlds where the crucial (global) activation conditions, e.g., the supervenience basis for its being a “Lewis law” that Fs become Gs, are absent, and thus fail to trigger the disposition into action. What Lewis sees as contingency in which powers are granted by F, Dlewis sees as contingency in the existence of the activation conditions for various F-involving dispositions. For Dlewis, that contingency in no way counts against the metaphysical necessity of the relation between a disposition, its activation conditions, and its manifestation. And Lewis agrees that each connection Dlewis thinks is necessary is indeed necessary. Suppose Lewis somehow resists my earlier argument. Dlewis’s metaphysics still looks coherent, and, when it comes to the charac-

Goodbye, Humean Supervenience | 145 terization of natural properties, only nominally different from Lewis’s own. The differences between Lewis and Dlewis are not differences in the modal behavior of the fundamental properties. So it’s hard to see what Lewis’s insistence that his properties are “genuinely” non-dispositional, his disavowals of Dlewisian metaphysics, amount to. Is Dlewis’s view somehow conceptually confused? How so? The position isn’t terribly attractive, but it does seem coherent. The incoherent-seeming position is Lewis’s. It is just very hard to imagine, of all people, David Lewis, insisting that although he and Dlewis agree about the modal extension of all the natural properties, all of the necessities, and all of the counterfactuals, that in fact, their views about the nature of fundamental reality are radically different, that Humeanism amounts to refusing the label “dispositional” to something that behaves, modally, just like a disposition. Intuitively, what Lewis should say to Dlewis is this: You’re right that the perfectly natural properties, in some sense, correspond to these funny powers with highly non-local activation conditions. But the perfectly natural properties are not themselves powers. Rather, they are perfectly qualitative, intrinsic features of point-sized things. Any powers to which they correspond are, in the end, a matter of how these purely qualitative intrinsic features are globally distributed. Powers are not primitive. They are highly unnatural, and exist only in virtue of the distributions of qualities, which are wholly non-dispositional in character. The purely qualitative properties do all of the deep explanatory work. They lie at the bottom of things. And they do not have dispositional essences.

Well and good. Except that for Lewis to say such a thing, he would also have to say that properties can be non-identical even if they’re necessarily co-extensive. That would mean that properties are not sets of possibilia, that naturalness is neither a primitive feature of sets, nor a matter of whether sets have members instantiating a single universal nor perfectly resembling tropes (Lewis 1983). Hyperintensionalism, then, is a perfectly acceptable counter to my argument, and I would be content if it were widely recognized as a presupposition of categoricalism. But there is reason to assume that Lewis himself would not abandon his set-based theory of properties. One reason, though certainly not the only one, is that the concession would undermine part of Lewis’s argument for modal realism, namely, his objection to building a semantics for modal

146 | Troy Cross talk, a la Robert Stalnaker, on an ontology of properties rather than on concrete, spatiotemporally disconnected things (Lewis 2001b, 3.4; Stalnaker 1976). If properties are individuated primitively, rather than by their transworld extensions, they could achieve the theoretical payoff of possible worlds, without a plentitude of spatio-temporally disconnected concreta. If hyperintensionalism is not an option, though, how can Lewis meet the challenge of my argument? Only, so far as I can see, to trade his reductivism for expressivism or eliminativism about powers. Suppose Lewis were to say that there is only really the Humean base—just a spread of qualities—and that all talk of powers, dispositions, potentialities, chances, and the like, while perhaps permissible or impermissible, felicitous or infelicitous, is either false or non-truth-evaluable (Ward 2002). Then he could deny the central premise of my argument. He could say that different properties do not make different potential contributions to the powers of objects, because objects are all—whatever their qualities—impotent, speaking strictly. However, this would trade his somewhat plausible supervenience claim for a far more controversial eliminativism.

OBJECTIONS AND REPLIES 1. Significance Some philosophers have granted the central argument of this paper, but charged it with triviality. According to this objection, the Humean should simply accept my conclusion—that natural properties are all powers—without batting an eyelash. After all, the powers identified with Lewisian natural properties are highly unusual, and easily screened off from the powers endorsed by mystery mongering dispositionalists: the Lewisian is committed only to perfectly local powers, instantiated by point-sized objects, and sensitive only to global states of the world. What’s wrong—so goes the objection—with saying that qualities are, in fact, identical to these kinds of powers? Yes, ontologically speaking, Lewis is Dlewis, but so what? We can resolve the matter by careful attention to the dispositionalist/categoricalist dialectic. The expressions “quality” or “categorical property” are typically used by categoricalists to mark

Goodbye, Humean Supervenience | 147 something that is not a disposition, and typically in order to analyze away the latter in terms of the former. In particular, Lewis’s iconic statement of Humean supervenience, quoted above, makes no sense if we allow that qualities can be dispositions. Lewis seems to be claiming that everything supervenes on non-dispositional properties, i.e., that the dispositional, nomic, chancy, subjunctive, and the like, are all fixed by the non-dispositional, non-nomic, nonchancy, non-subjunctive, and the like. But according to the present objection, Lewis would happily gloss his doctrine as, in part, saying merely that some dispositions fix others, which is hardly visionary. The problem is not that Lewis’s ontology is filled with dispositions. That is hardly disturbing, so long as the dispositions all supervene on a purely Humean base. The problem is rather that we’ve shown that there are no non-dispositional grounds for the supervenient dispositions; it’s dispositions “all the way down”. Even though, strictly speaking, if we read “quality” as compatible with “disposition”, dispositions will supervene on qualities (and vice versa), the lack of a non-dispositional base should be troubling to committed Humeans. In any case, suppose we take the non-standard reading on which “quality” is taken to be compatible with “dispositional”. Lewis’s view then looks to be in broad agreement with C.B. Martin and John Heil’s (Martin 1997; Heil 2005; Martin and Heil 1998), on which the natural properties are all both dispositional and categorical. Martin and Heil would only take issue with Lewis about which dispositions are perfectly natural. This is, at the very least, news. Moreover, once the disagreement is recast as about which dispositions are perfectly natural, the Martin Heil view seems to have a clear advantage over Lewis’s. The powers that shadow—and are in fact identical to—Lewis’s qualities are an odd choice for fundamentalia. They are, if you recall, globally sensitive, dispositions that specify how an object possessing the property would be poised to act under every possible circumstance, even the counter-nomic ones. Why would a metaphysician start here, rather than with the fundamental properties of physics, understood as dispositions to resist acceleration, warp spacetime, alter electro-magnetic fields, and such, with no reference to other possible laws? Note that Lewis cannot answer this question by appealing to the spookiness of dispositions generally, or the need to reduce the dispositional to the

148 | Troy Cross non-dispositional. Once we have admitted that the world is, indeed, composed of dispositions all the way down, it’s unclear why those dispositions would have only point-sized instances, and why they would all be globally sensitive, and never purely locally-sensitive, why they would always vary so radically when embedded in nonactual global patterns in the way that Lewis prefers. Given that the global distribution of (non-dispositional) qualities does not explain the powers of objects, Lewis’s view looks highly idiosyncratic and lacks its standard atomistic, anti-modalist motivations.

2. Symmetry Another objection charges me with favoritism. The Lewis/Dlewis scenario, as depicted, has no obvious asymmetries, so if Lewis is in trouble, then so, it seems, is Dlewis. In particular, Dlewis, as a fellow anti-hyperintensionalist, is just as incapable of distinguishing himself from Lewis as Lewis is incapable of distinguishing himself from Dlewis. So, why shouldn’t we conclude by extension that dispositionalists generally, at least if they are anti-hyperintensionalists, are in the same boat as categoricalists? The objection has a grain of truth: anti-hyperintensionalism can create a problem for categoricalists and dispositionalists alike. Dlewis is a case in point. However, actual dispositionalist philosophers differ from Dlewis in at least three respects. First, they tend not to be anti-hyperintensionalist, at least not explicitly. Perhaps this is because anti-hyperintensionalism is a remnant of nominalism and dispositionalists tend to be less squeamish about property realism, but the literature simply does not hold any loudly anti-hyperintensionalist dispositionalists. Second, dispositionalists do not attempt to reduce or explain the categorical in terms of a purely dispositional, non-categorical base. Rather, they are often friendly to the idea that dispositions are themselves categorical. It is categoricalists, historically, who have insisted on the non-categoricity of dispositions, in an effort to banish them from ontology along with the rest of the merely possible. If anything, dispositionalists tend to applaud C.B. Martin’s insistence that dispositions are as actual, as categorical, as anything else (1993, 75; 1996). Though there is a literature on the plausibility of pure

Goodbye, Humean Supervenience | 149 dispositionalism (Bird 2007, 6; Holton 1999; McKitrick 2003), it is difficult to find actual advocates of the view. Shoemaker, to whom the pure dispositionalist theory is often attributed, originally held that the dispositional/categorical distinction applies to predicates, and not to properties, thus leaving it open that a dispositional predicate and a categorical predicate might share the very same semantic value (1984). Third, Dlewis’s dispositional properties are tailored precisely to fit Lewis’s pure qualities and his reductive account of powers. Even setting aside the two disanalogies listed above, to force Dlewis’s problem on actual dispositionalists we would have to find a categoricalist who has a unique and necessarily correlated purely nondispositional quality for each of the dispositionalist’s fundamental dispositions. It is not clear to me that this can be done. Say we are searching for a categorical property to match the modal profile of the disposition to resist acceleration, where that disposition is not taken to be conditional on the laws or the global state of the world. What is the categorical shadow of this disposition? How does it not violate categoricalist principles of recombination? (Remember: this power is not conditional on laws or global regularities of any kind.) As such, how can it be a candidate for a perfectly natural categorical property? At the very least, the search for a categoricalist shadow for every disposition is a different project from the one going in the other direction, and it is not obvious that it will be successful. This should not be surprising. Lewis’s account of dispositions in terms of the distribution of qualities allows us to easily locate the dispositional shadows of his properties and conjure up a philosopher who takes just those dispositional shadows to be fundamental: Dlewis. But dispositionalists do not give us the analogous conjuring recipe. As I said, the symmetry objection has some genuine import. Though Dlewis thinks of his view as a radically different ontology from Lewis’s, he is mistaken, by his own lights. Just like Lewis. But Dlewis is a very peculiar dispositionalist, purpose-built as a foil. He thinks dispositions cannot also be categorical, he is anti-hyperintensionalist, and his fundamental powers, oddly enough, all share modal profiles with Lewis’s local qualities. Actual dispositionalist theories lack some or all of these features, and are therefore immune from the parallel criticism.

150 | Troy Cross CONCLUSION When a property potentially makes a difference to its bearers, it makes an actual difference to the potential of its bearers. This simple observation dooms Lewis’s grand vision of resting all dispositions upon a foundation of pure qualities. For it is easy enough to see, given Lewis’s own reductive theories, that for arbitrary quality F, being an F makes a difference to the powers of objects across the modal pluriverse. These potential differences in powers are, in fact, powers that F actually endows, and when the powers are considered collectively, powers that only F endows. That is enough to establish Shoemaker’s thesis in “Causality and Properties”. If, like Lewis, we then identify necessarily co-extensive properties, it follows that Lewis’s qualities are, one and all, dispositions. The small story here is that Shoemaker, not Lewis, turns out to be the hero of the naturalistic mini-epic recounted in the introduction of this essay. But there is a larger story too. My argument is easiest to process as a direct attack on Lewis, but it poses a perfectly general dilemma for realists about dispositions: either the world is dispositional at its fundamental level or else it is hyperintensional. Reed College WORKS CITED Achinstein, Peter. 1974. The Identity Conditions of Properties. American Philosophical Quarterly 11: 257–75. Armstrong, D. M. 1973. Belief, Truth and Knowledge. Cambridge University Press. —— . 1978. A Theory of Universals: Volume 2: Universals and Scientific Realism. Cambridge University Press. —— . 1980. Identity Through Time. In Time and Cause: Essays Presented to Richard Taylor, ed., P. van Inwangen, 67–78, Dordrecht: D. Reidel. Armstrong, D. M., C. B. Martin, U. T. Place, and Tim Crane. 1996. Dispositions: A Debate. Psychology Press. Ashwell, Lauren. 2010. Superficial Dispositionalism. Australasian Journal of Philosophy 88: 635–53. Bird, Alexander. 1998. Dispositions and Antidotes. Philosophical Quarterly 48: 227–34. —— . 2007. Nature’s Metaphysics. New York: Oxford University Press.

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6. “There sweep great general principles which all the laws seem to follow” Marc Lange My title is taken from a passage in Richard Feynman’s classic book, The Character of Physical Law: When learning about the laws of physics you find that there are a large number of complicated and detailed laws, laws of gravitation, of electricity and magnetism, nuclear interactions, and so on, but across the variety of these detailed laws there sweep great general principles which all the laws seem to follow. Examples of these are the principles of conservation . . . (Feynman 1967, 59)

My aim in this paper is to understand this conception of the conservation laws and to examine what it reveals about the character of physical law generally—in particular, about the laws’ modality, their explanatory role, and the adequacy of the dispositional essentialist conception of laws as metaphysical necessities arising from the causal powers essential to the sparse fundamental properties of physics. My main positive result is that science recognizes an important distinction: between conservation laws as constraints on the fundamental forces there could be, on the one hand, and conservation laws as coincidences of the fundamental forces there happen to be, on the other hand. In the above passage, Feynman characterizes conservation laws as constraints. I do not argue that they are constraints; on my view, this is a matter for science, not metaphysics, to decide. However, I argue that some conservation laws have sometimes been taken (with good reason) to be constraints, that their status as constraints would make an important difference to their role in scientific explanations, and that the distinction between constraints and coincidences applies to other laws besides the conservation laws. I ultimately cash out the distinction between constraints and coincidences in terms of the truth of various counterfactual

"There sweep great general principles" | 155 conditionals, and I briefly sketch how this way of elaborating the distinction relates to my broader account of natural law (Lange 2009). My main negative result is that metaphysical pictures along the lines defended recently by Alexander Bird (2007), Brian Ellis (2001, 2002), and Stephen Mumford (2004)—for which, despite some disagreements among these authors, I will use the catch-all term “dispositional essentialism”—cannot accommodate the distinction between constraints and coincidences. Such a picture must portray all conservation laws as coincidences. It thus forecloses options that science has (with good reason) taken seriously. This failure is a weighty count against dispositional essentialism.

1. CONSTRAINTS VERSUS COINCIDENCES Consider the law of energy conservation. (I could just as well have chosen any of the other conservation laws that have been proposed in the history of physics, such as the conservation of linear momentum, angular momentum, electric charge, mass, parity, baryon number, or lepton number.) As Feynman emphasizes, though the various kinds of fundamental interaction differ in a host of ways (in their range, their strength, the kinds of bodies that participate in them, and so forth), they are all alike in conserving energy. As convenient examples of kinds of fundamental interactions, I shall follow Feynman and take gravitational and electric interactions as described in classical physics by Newton’s gravitational-force law and (in the static case) Coulomb’s law, respectively. Despite their differences, these two types of interactions are alike in both conserving energy. Of course, gravity is not in fact a force at all according to general relativity, and electric and magnetic forces are not actually distinct kinds of force according to special relativity. But none of this matters to my argument. I shall be appealing to these two forces only to illustrate my claim that physical theory recognizes an important distinction between two different ways in which a law like energy conservation could hold: as a constraint or as a coincidence. I contend that any metaphysical account of natural law should leave room for both of these possibilities. The same distinction must be drawn whatever the fundamental forces actually

156 | Marc Lange are—indeed, even if there is in fact only a single kind of fundamental force (a “grand unified field”). Why are gravitational and electric interactions alike in conserving energy? Here are two possible explanations. 1. Gravitational interactions conserve energy because the gravitational force law requires them to. Electric interactions conserve energy because the electric force law requires them to.1 The two interactions are therefore alike in conserving energy—but for separate reasons. 2. Both kinds of interaction conserve energy for the same reason: because the law of energy conservation requires them to. On the first option, it is just a coincidence that these two different forces conserve energy, since there is no common explanation of their doing so. Just as it would be a coincidence for two friends both to be in Chicago on the same day if there was no important common reason for their both being there then (e.g., they had made no plans to meet there, they were not both attending the same convention), so likewise it is a coincidence for various distinct forces all to conserve energy if there is no important common “cause”, but rather each does so for a substantially separate reason.2 On the second option, in contrast, the law of energy conservation is not a coincidence. Rather, the various fundamental kinds of interaction all have a common reason for conserving energy: the conservation law. 1 The force law alone is not enough to entail that the interaction will conserve energy. The explanation must also appeal to the fundamental dynamical law: the law relating forces to the motions they cause (in classical physics: Newton’s second law of motion). 2 I say “important” and “substantially” in order to acknowledge that two components of a coincidence may have some explainers in common—as long as they are beside the point in the context in which an explanation of the two components is being demanded. For instance, suppose that the two friends both happened to travel to Chicago on the same airplane flight. Then there would be some common explainers of their both being there (e.g., the flight, the natural laws governing jet engines). But these are not the sorts of explainers that we would (ordinarily) be asking for in asking “What brings you to Chicago?” Likewise, although the fundamental dynamical law (see note 1) is common to the explanation that gravitational interactions conserve energy and to the explanation that electric interactions conserve energy, it is incidental; the force laws involved would (typically) be the focus of our explanatory demand. Hence, it is a coincidence that both forces conserve energy.

"There sweep great general principles" | 157 It is a constraint on the forces. That is, the law of energy conservation limits the kinds of forces there could have been. The only kinds of force there could have been are forces that conserve energy, and that is why every kind of force there actually is conserves energy. The difference between constraint and coincidence is a difference in what is explanatorily prior to what. If energy conservation is a coincidence, then the various force laws are explanatorily prior to the law of energy conservation. On the other hand, if energy conservation constrains the force laws, then the conservation law is explanatorily prior to them. It does not entail the particular force laws there are, but it explains why they each exhibit a certain feature. These two options (constraint or coincidence), then, are mutually exclusive.3 However, these two options are alike in one important respect: whichever option holds, the law of energy conservation is physically necessary—a law rather than an accident. As a coincidence, the conservation law is physically necessary in virtue of following exclusively from laws, such as the gravitational-force law, the electric-force law, and the law that all fundamental forces are gravitational or electric or . . . (a “closure law”). As a constraint on the force laws, the conservation law transcends the grubby, pedestrian details of the various particular force laws. It is a higher-order law, as Feynman suggests. It does not depend on the kinds of forces there actually happen to be. It limits the possible kinds of forces. Since the difference between constraint and coincidence is a difference in explanatory priority, the conservation law’s status as con3 They are not collectively exhaustive. Rather, they are the extremes; there are intermediate cases. For instance, suppose that some fundamental kinds of interactions have a certain feature (e.g., are capable of both attraction and repulsion) whereas others (namely, interactions A, B, and C) do not have this feature. Suppose it is a law that every kind of interaction with that feature conserves energy, and suppose that law is a constraint. Then the fact that every kind of interaction conserves energy might be explained by this constraint together with A’s force law, B’s force law, and C’s force law (along with the fundamental dynamical law). In that case, energy conservation is neither a complete coincidence nor a constraint. As another kind of intermediate case, the law of energy conservation might follow from exactly two separate constraints (e.g., that all of the forces capable of both attraction and repulsion must conserve energy, and that all of the forces capable only of attraction or only of repulsion must conserve energy). For the sake of simplicity, I shall not return to these intermediate possibilities, but I believe that it is clear how my remarks apply to them.

158 | Marc Lange straint or coincidence makes a difference to whether certain arguments carry explanatory power. It makes a difference to the success of many putative explanations well beyond whether the conservation law explains why gravitational and electric interactions both conserve energy. Consider the fact that an ideally incompressible, nonviscous fluid in a container at rest in a uniform downward gravitational field is not undergoing any circulation; none of its parcels at the top feels an unbalanced force pulling it downward, nor do any bottom parcels feel unbalanced forces pushing them upward. Why is that?4 If energy conservation is a constraint, then it explains why. A force arising from no outside agency that would make the fluid parcels begin to circulate from rest would violate energy conservation: in beginning to circulate, the parcels’ kinetic energy would increase but their total potential energy would be unchanged. (As ascending parcels gain gravitational potential energy, descending parcels lose an equal quantity of it.) Energy conservation as a constraint rules out any circulation-inducing force. However, as a coincidence, energy conservation cannot supply this explanation. If energy conservation is a coincidence, then the reason why the fluid undergoes no circulation is that electric forces fail to induce circulation (because of the electric-force law), gravitational forces fail to induce circulation (because of the gravitationalforce law), and so forth for all of the actual kinds of forces experienced by the fluid parcels. This is a “bottom-up”, causal/mechanical explanation. As a coincidence, the general principle of energy conservation cannot figure in such an explanation. It cannot explain why various kinds of fundamental force are alike in failing to induce fluid circulation, since as a coincidence rather than a constraint, it is not explanatorily prior to the force laws. The reason why electric forces fail to induce circulation (and the reason why they conserve energy) is not the coincidence that all forces conserve energy; it is the electric-force law. Suppose that, instead of trying to take the comprehensive conservation law and slot it into the explanation somewhere explanatorily prior to the force laws, we try to place it somewhere explanatorily 4 The explanandum is a scientifically significant fact; it is not a fact that only a philosopher would inquire into. (Don’t pretend that you don’t know what I mean!) For example, it is central to the reason why Archimedes’ Principle holds.

"There sweep great general principles" | 159 posterior to the force laws. Then we encounter a different problem. The fluid parcels feel gravitational forces, so the fact that gravitational forces conserve energy may help to explain why there are no circulation-inducing forces. But if the fluid parcels feel no magnetic forces, then the fact that magnetic forces conserve energy does not help to explain why there are no circulation-inducing forces. If the conservation law is just a coincidence, then it is effectively the fact that gravitational forces conserve energy, magnetic forces conserve energy, and so forth for all of the actual kinds of forces. But if some of these forces are not experienced by the fluid parcels, then the fact that they conserve energy is not explanatorily relevant, and so neither is the general principle of energy conservation. For a coincidence to be explanatorily relevant to an outcome, all of its components must be relevant. For instance, the reason why you and I ran into each other at the mall this afternoon might be the coincidence that you and I both chose this day to go shopping there— but the coincidence that you, I, and Frank all chose this day to shop there does not explain why you and I encountered each other there. As a coincidence of the various kinds of fundamental forces, the conservation law explains only if all of those forces are explanatorily relevant. Suppose, then, that the fluid parcels feel every species of fundamental force so that every component of the energy-conservation coincidence is explanatorily relevant. Then the resulting explanation from energy conservation would still not be a top-down explanation. Rather, it would have to include the fact that each of these forces is actually felt by the fluid parcels. The top-down explanation does not specify which kinds of fundamental force the fluid parcels experience. Its point is that the outcome does not depend on what possible forces are actually at work; no matter which possible forces were operating on the fluid parcels, the fluid would inevitably still fail to circulate. If energy conservation is a coincidence, then the reason why the fluid undergoes no circulation is that electric forces fail to induce circulation (because of the electric-force law), gravitational forces fail to induce circulation (because of the gravitational-force law), and so forth for all of the kinds of forces actually experienced by the fluid parcels. In contrast, if energy conservation is a constraint, then this bottom-up argument (though, of

160 | Marc Lange course, still sound) does not explain why the fluid is not circulating because it would then inaccurately depict the explanandum as depending on the particular kinds of forces that happen to be acting on fluid parcels. The only explanation is a top-down explanation: that any circulation-inducing force would violate energy conservation, which is impossible. Here is an analogy. Consider Jones and Smith, each convicted in separate trials before separate judges of possessing (independently) 100 kilograms of marijuana. Why did each of them receive a sentence exceeding five years? The reason is not that Smith’s judge passed this sentence because he believed that Smith’s crime rose to a certain level of seriousness because of various factors including . . . and Jones’s judge passed this sentence because he believed that Jones’s crime rose to a certain level of seriousness because . . . if the two judges were constrained by a mandatory minimum sentencing law to pass sentences of at least five years for the possession of 100 kilograms of marijuana. If there is such a law, then it is no coincidence that the two judges handed down sentences that are alike in this respect. Rather, the law is a common explainer—and any account is mistaken if it depicts the two sentences as the products of independent judicial decisions that weighed the particulars of the individual cases. The success of various proposed top-down scientific explanations, then, depends upon the status of energy conservation as a constraint. Even if energy conservation is a coincidence, the law that gravitational forces conserve energy could still be used to help explain why the fluid does not circulate. But this explanation would simply be a bottom-up account that portrays the fact that the fluid does not circulate as arising from the coincidence that each of the particular kinds of forces acting on the fluid conserves energy. In contrast, if energy conservation is a constraint, then the fact that the fluid fails to circulate does not depend on the particular kinds of forces at work on it. Here is another way to bring out this contrast. Consider a wooden block (of any shape) sitting on top of a post, and suppose that across the upper surface of the block is laid part of a uniform loop of rope (or chain), while the rest of the loop hangs below the block, experiencing uniform downward gravity. Why does the rope loop, having been laid across the block, not spontaneously begin to turn round

"There sweep great general principles" | 161 and round the block?5 If energy conservation is a constraint, then it explains why no force puts the rope loop into circulation: any such force is ruled out by energy conservation for exactly the same reason as it precludes a force inducing spontaneous fluid circulation. However, if energy conservation is a coincidence, then this explanation is unavailable. Energy conservation does not help to explain features of the force laws. Instead, the explanation is that one kind of force felt by the rope fails to induce circulation, another also (for independent reasons) fails to do so, and so forth for all of the kinds of forces at work on the rope. This explanation may involve different kinds of fundamental forces from the corresponding explanation of the fluid’s behavior; different forces may be at work on ropes and fluids. Therefore, the bottom-up explanations do not unify these two cases. In contrast, the top-down explanations not only unify these two phenomena under the same explainer (the law of energy conservation), but also unify them further by giving them explanations of the very same form. My point is that whether energy conservation is a constraint or a coincidence makes a big difference to features of the world that science cares greatly about: to the kinds of explanations that there are and to the unifications that those explanations bring.6 These are matters for empirical work to discover. I am not arguing that if every single fundamental kind of force conserves energy, then this “conspiracy” is unlikely to be a coincidence – that it probably has a 5 The explanandum is a very important fact. For example, it is central to Simon Stevin’s 1586 clootcrans explanation of the law of the inclined plane (Stevin, 1955, Vol. 1, 178). 6 If we believe that energy conservation is a constraint if it is true, then we are prepared to confirm the hypothesis that energy is conserved very differently than if we believe it is a coincidence if it is true. Roughly speaking, if we believe that energy conservation is a coincidence if it is true, then we regard the fact that one fundamental kind of interaction conserves energy as no evidence that another kind does (just as we take my being in Chicago as no evidence that you are there, too, if we believe that our both being there would be coincidental). However, if we believe that energy conservation might be a constraint, then we may take the fact that one fundamental kind of interaction conserves energy as some evidence that another kind also does. Feynman (1967, 76) says that we are “confident that, because we have checked the energy conservation here, when we get a new phenomenon we can say it has to satisfy the law of conservation of energy.” A good example of such a new phenomenon was radioactive decay which physicists believed to conserve energy before they had any significant confidence in any theories regarding the particular force(s) involved.

162 | Marc Lange common “cause”. I insist only that the hypothesis positing such a constraint is sometimes a reasonable one to entertain, that science has frequently taken such hypotheses seriously, and that therefore metaphysics should not foreclose such hypotheses. A conservation law need not be a brute fact in order for it to be a constraint. It may have an explanation. In fact, one way for a conservation law to be a constraint is for it to arise from a symmetry principle, since if it so arises, then each of the actual forces conserves the relevant quantity for the same reason: because of the symmetry principle. As is well known, various classical conservation laws follow from various space-time symmetries within a Hamiltonian dynamical framework: energy conservation follows from the laws’ invariance under arbitrary temporal displacement, linear momentum conservation from their invariance under arbitrary spatial displacement, and so forth. If these derivations explain why the conservation laws hold (as they are often said to do), then the conservation laws are constraints, not coincidences. As Eugene Wigner says: [F]or those [conservation laws] which derive from the geometrical principles of invariance it is clear that their validity transcends that of any special theory—gravitational, electromagnetic, etc.—which are only loosely connected. (Wigner 1972, 13)

In other words, Wigner contends that those symmetries are not coincidences of the particular kinds of forces there happen to be, and so the associated conservation laws transcend the idiosyncrasies of the force laws figuring in bottom-up explanations.

2. OTHER POSSIBLE KINDS OF CONSTRAINTS BESIDES CONSERVATION LAWS Conservation laws are not the only “great general principles” that have sometimes been reasonably thought to “sweep” across the various force laws, explaining why all of those laws share certain features. One candidate proposed by Heinrich Hertz may not turn out to succeed. But an adequate metaphysical account of natural law must at least leave room for explanations of the kind Hertz proposed.

"There sweep great general principles" | 163 Newton’s gravitational-force law is an inverse-square law. So is Coulomb’s law for the electric force between two point charges at rest. So is Ampere’s law for the magnetic force between two electriccurrent elements. In his 1884 lectures delivered at Kiel, Hertz said that (as far as science has been able to discover) all fundamental force laws are inverse-square—and that this regularity has never been thought coincidental [zufällig] (Hertz 1999, 68). What could Hertz have meant by this? Presumably, he did not mean that no one has thought this regularity to be physically unnecessary—since although this is true, it is a trivial remark: obviously, no regularity among the force laws could be accidental. The laws alone must suffice to logically entail any such regularity. Rather, I suggest, Hertz meant that the inverse-square character of all of the fundamental forces has always been considered to be a constraint, not a coincidence. In other words, Hertz meant that there is (according to widespread consensus) a common explanation for each force’s being inverse-square; they are not independently inverse-square. This interpretation of Hertz’s remark is confirmed by his characterizing this regularity among the various fundamental forces as too remarkable for its instances not to have a common explainer: “Is it not marvelous [wunderbar] that all long-range forces follow [an inverse-square] law?” (Hertz 1999, 68). Indeed, Hertz immediately suggests one possible common explainer: “Kant and many others before and after him have tried to relate this feature [the inversesquare character of the force laws] to the three-dimensional nature of space.” But whereas Kant offers the inverse-square character of forces as explaining why space is three-dimensional (see Callendar 2005), Hertz proposes that explanatory priority runs in the opposite direction. Hertz’s proposed explanation begins with another regularity among the fundamental forces that he takes to be a constraint on any force there might have been rather than a coincidence of the various forces there actually are: that every fundamental force acts by contact—that is, by a field acting at the same point in spacetime as the force that it causes, so that the field causally mediates between the two, perhaps spatiotemporally widely separated bodies thereby interacting (see Lange 2002). In other words, Hertz’s explanation begins with the premise that none of the fundamental kinds of

164 | Marc Lange interaction constitutes action at a spatiotemporal distance. In fact, Hertz presents himself as arguing for this premise by inference to the best explanation: the most plausible explanation of the “marvelous” fact that all fundamental forces are inverse-square is that all fundamental forces must operate by contact action. In his Kiel lectures, Hertz does not say anything about why, in turn, all fundamental forces must operate by contact action. But to fund his explanation of the inverse-square character of all fundamental forces, this contact-action regularity must also be a constraint rather than a coincidence. For if it were a coincidence, then it could not be a common reason why every force is inverse-square. At best, the electric-force law would be inverse-square because electric charges interact by contact (i.e., through the electric field at each charge’s location), the gravitational-force law would be inversesquare because gravity acts by contact, and so forth. In that case, it would be a coincidence that all of the fundamental forces are inversesquare, contrary to Hertz’s view (following a broad consensus, Hertz says) that this regularity is no coincidence. How is the constraint that all forces be inverse-square supposed to be explained by the constraint that all forces act by contact (in three-dimensional space)? Consider a configuration of bodies and any imaginary surface enclosing them. If a given sort of influence operates by contact action, then the influence of those bodies on any body outside of the surface must pass through the intervening surface (rather than hop “over” it). Therefore, the field at all points on the surface must fix the influence of the enclosed bodies on any body outside of the surface. Hence, any two configurations with the same field at all points on the surface must have the same field everywhere outside of the surface. The existence of such a “uniqueness theorem” (as it is commonly called today) imposes strict limits on the form that the force law can take. As Hertz (1999, 68) rightly notes, the requirement that there be a uniqueness theorem rules out a force that declines linearly with distance or with the cube of the distance. Indeed, though Hertz does not mention this result explicitly, it is a mathematical theorem that for a 1/rn force, a uniqueness theorem is possible (in three-dimensional space) only for n = 2 (Bartlett and Su 1994). That is why (according to Hertz) all of the various fundamental forces are inverse-square forces.

"There sweep great general principles" | 165 On Hertz’s view, at least two constraints are not conservation laws: that all fundamental forces are inverse-square and that all fundamental forces act by contact. Notice once again that a constraint need not be a brute fact; on Hertz’s view, the inverse-square constraint is explained by the contact-action constraint. It seems to me that Hertz’s explanation cannot be correct as it stands since an inverse-square force is not quite the only kind of central force (with a force law consisting of an analytic function that is real-valued except, perhaps, at isolated singularities) that permits a uniqueness theorem.7 Rather, a uniqueness theorem holds for such a force if and only if it is proportional to 1/(ekr r2) for some real k. This is called a “Yukawa force law” (or a force with a “Yukawa potential”). An inverse-square force is the special case where k = 0. More about the various types of forces is known today than in Hertz’s day. Not all of the forces that physicists today look upon as perhaps fundamental are inverse-square. However, if all actual fundamental forces are Yukawa forces, then perhaps an argument like Hertz’s explains why this is so. A Yukawa force was famously posited by (can you guess?) Yukawa in 1935 as the strong nuclear force (i.e., the force holding protons and neutrons together in atomic nuclei). However, even if Hertz’s proposed explanation fails because not all actual fundamental forces are governed by Yukawa force laws, my point still stands. An adequate metaphysics must not foreclose explanations of the sort Hertz proposes on pain of failing to do justice to the fact that science has rightly taken such proposals seriously. Many sorts of regularities among the various forces could be constraints rather than coincidences.

3. CONSTRAINTS AS MODALLY MORE EXALTED THAN THE FORCE LAWS THEY CONSTRAIN I have suggested if top-down explanations appealing to conservation laws succeed, then they work because those laws constrain the lower-level laws figuring in bottom-up explanations—namely, by limiting the kinds of forces and force laws there could possibly be.

7 A “central force” is a force directed along the line joining the body exerting it and the body feeling it.

166 | Marc Lange In that case, the kinds of forces and force laws there could have been go beyond the kinds there actually are. For gravitational forces to exist and to diminish with the inverse-cube of the distance, for example, is physically impossible (i.e., is logically inconsistent with some laws of nature—in this case, the gravitational-force law) but nevertheless possesses some broader species of possibility in being logically consistent with the various constraints on the force laws. In contrast, if energy conservation is a constraint, then energy’s conservation is not just physically necessary, but also possesses an even stronger kind of necessity (that is, one that applies to some but not all of the physical necessities). On this view, a top-down explanation may proceed by expressly considering hypothetical states of affairs that the lower-level laws rule out (and that are not even approximations to or idealizations of some physical possibility). A top-down explanation may succeed even if it appeals to a physical impossibility, as long as that hypothetical state of affairs is not ruled out by the constraints on the lower-level laws. The top-down explanation exploits this broader species of possibility since it works by showing the explanandum to possess the corresponding species of necessity (stronger than physical necessity). Here is an example of such an explanation: the standard textbook explanation (originating with J. Willard Gibbs) of the entropy of a mixture of two non-interacting ideal gases. The explanation uses energy conservation to account for the expression for ΔS: the difference between the mixture’s entropy and the entropy of the gases when separated. Suppose NA molecules of gas A occupy volume VA (the left side of a container) and NB molecules of gas B occupy volume VB (the right side); the container is isolated and the two gases have the same pressure P and temperature T. Suppose gas A is confined behind a freely moveable membrane permeable to B but not to A, and gas B is similarly confined behind a membrane permeable to A but not to B. Initially, the two membranes divide the container along the same plane, so the gases are entirely separated. Then the membranes are allowed to move slowly, each gas expanding quasistatically, so that ultimately the two membranes reach opposite ends of the container and both gases fill the entire container (volume VA + VB ). Each gas’s expansion is a reversible isothermal process. Let W be the total work done on the system:

"There sweep great general principles" | 167 W = − ∫ VA

PdV − ∫ VB

VA +VB

− ∫ VBA

VA +VB

VA +VB

N B kT

P dV = − ∫ VA

VA +VB

N A kT

dV V

dV VB VA = N A kT ln + N B kT ln V VA + VB VA + VB

By energy conservation (i.e., the first law of thermodynamics), the change ΔU in internal energy and the heat Q absorbed are related by ΔU = Q + W Since the gases expand isothermically, ΔU = 0 , so Q = −W. Thus Q = N A kT ln

VA + VB V + VB + N B kT ln A VA VB

Then since ΔS = Q/T, ΔS = N A k In

VA + VB V + VB + N B k In A VA VB

which is the explanandum: the formula for the entropy of a mixture of two non-interacting ideal gases. Crucially, this explanation does not presuppose that the lowerlevel laws make it possible for there to exist a pair of membranes, one permeable to A but not to B, the other permeable to B but not to A. Whether there are any possible materials that could constitute such membranes depends on the particular gases involved. Generally, such membranes are impossible. For instance, if molecules of A are small and uncharged whereas molecules of B are large and charged, then typically there is nothing that could form a membrane permeable to B but not to A according to the lower-level laws (which specify the molecular constitution of A and B, as well as the behavior of physically possible membrane materials). But remarkably, the thermodynamic explanation is not thereby undermined. That is because it proceeds entirely from constraints on possible lower-level laws. As far as those constraints are concerned, such membranes are possible for any molecular species. As Max Planck said in 1891 in commenting on Gibbs’ derivation of this equation: The enormous generalization that Gibbs has given to this tenet and which must, in and of itself, appear irresponsibly daring, rests clearly on the

168 | Marc Lange self-evident thought that the validity of so fundamental a tenet as that of the entropy of a mixed ideal gas, cannot depend on the arbitrary circumstance of whether we really have available in each individual case a suitable semi-permeable membrane. (translated in Seth 2010, 108)

It is not entirely an “arbitrary circumstance” since, after all, it is a matter of physical necessity. Yet it is “arbitrary” as far as the constraints are concerned, since whether any such membranes are possible for a particular pair of gases is a matter of what the fundamental force laws happen to be. Because the top-down explanation shows the explanandum to depend on thermodynamics alone, the explanation can afford to posit membranes that are impossible according to lower-level laws.8 The laws of thermodynamics transcend the laws concerning various particular physically possible kinds of gas and kinds of materials out of which membranes could be constructed. That is because the laws of thermodynamics (and any explanandum they entail) would still have held, whether or not the lower-level laws allow a suitable pair of membranes for a particular pair of gases—an “arbitrary circumstance”, as Planck says. Similarly, to explain various laws concerning dilute solutions, Planck in 1887 (cf. Fermi 1956, 115) considered what would happen were the temperature so high and the pressure so slight that the solute and solvent vaporized into a mixture of ideal gases. After explaining the equations in this way, Planck wrote: [I]ncidentally, it is completely inconsequential [gleichgultig] if the given state can really be arrived at experimentally, and certainly whether it represents a stable state of equilibrium or not; because these expressions [the explanandum] are completely independent of this [question]. (translated in Seth 2010, 102)

Later he elaborated:

8 Many textbooks dance lightly over the fact that these membranes are generally physically impossible, characterizing the two gases as separated “conceptually” (Yourgrau, Van der Merwe, and Raw 2002, 235) or “hypothetically” (Annamalai and Puri 2002, 145) without elaborating any further. Fermi (1956, 101) is admirably forthright: “We should notice . . . that in reality no ideal semipermeable membranes exist. The best approximation of such a membrane is a hot palladium foil, which behaves like a semipermeable membrane for hydrogen.”

"There sweep great general principles" | 169 In reality, such a process [vaporizing a dilute solution into a mixture of ideal gases] will admittedly often not be realizable, because in many cases, at high temperatures, as are necessary here, chemical transformations occur, and the molecules are thereby altered. (translated in Seth 2010, 108)

Seth (2010, 102) comments: [Planck’s point], introduced so casually, was, in fact, anything but incidental. Planck had clearly seen an apparent objection, and an obvious one at that. If one considers the case described (a dilute solution, say, of NaCl in water), lowering the pressure and increasing the temperature does not automatically produce the result required [a mixture of ideal gases, one of the solute and one of the solvent]. In most common cases, the water will vaporize, leaving a solid salt. For Planck’s process, however, one requires both the salt and the water to vaporize and to maintain their molecular integrity as compounds. Whether it was at all possible to carry out such a procedure cannot have been clear to Planck. . . . The argument, however, was a thermodynamic one and the details of the process, including the very possibility of its experimental realization, did not matter for Planck. It was thermodynamically possible and hence the result followed.

“Thermodynamic possibility”, as Seth nicely terms it, is broader than “physical possibility” because the laws of thermodynamics constrain the lower-level laws; like logical possibility, thermodynamic possibility includes more than just the physical possibilities. Planck’s explanation succeeds, despite trafficking in physical impossibilities, because it works by showing the explanandum to be thermodynamically necessary, not merely physically necessary. No bottom-up explanation could explain the explanandum by showing it to be inevitable in just this respect because no bottom-up explanation, rooted in the various particular lower-level laws, could show that the explanandum would still have held, had the lowerlevel laws been different. The lower-level laws do not entail what the lower-level laws would have been like, had they been different. I shall now focus on such subjunctive conditionals.

4. MY ACCOUNT OF THE DIFFERENCE BETWEEN CONSTRAINTS AND COINCIDENCES Let’s now try to be more precise about what it would take to make a conservation law into a constraint rather than a coincidence. As

170 | Marc Lange I hinted at the close of the previous section, I suggest that this distinction be elaborated in terms of subjunctive conditionals. Energy conservation constrains the possible force laws exactly when energy would still have been conserved even if there had been an additional kind of force (that is, a force that is not electric or gravitational or any of the other actual kinds) acting together with the various actual kinds—that is, even if there had been an additional kind of interaction experienced by some of the same entities undergoing some of the actual kinds of interaction. (If the additional kinds of force were uninstantiated, then they would obviously pose no threat to energy conservation. If forces of the additional kinds were not influencing any of the actual sorts of entities, then they would pose no threat to the conservation of quantities possessed exclusively by those entities.) The subjunctive fact associated with energy conservation as a constraint is supposed to be roughly that energy’s conservation is resilient: that energy would still have been conserved even if there had been additional kinds of force threatening to undermine its conservation. On the other hand, to say that energy conservation is a coincidence of the actual force laws is to say that it is not the case that energy would still have been conserved, had there been additional kinds of force. Rather, energy is conserved because as it happens, each of the actual kinds of force conserves energy as a result of its own particular force law. So had there been additional kinds of force, energy might still have been conserved, but then again, it might not have been, depending upon the force laws of the additional forces. This means of distinguishing constraints from coincidences portrays constraints as like “higher-order” laws. The lawhood of Coulomb’s law is traditionally thought to be associated with the fact that Coulomb’s law would still have held, had there been additional charged bodies. Similarly, the accidental character of the fact that each of the families on my block has exactly two children is associated with the fact that it is not the case that had there been an additional family on my block, it would still have been true that each of the families on my block has exactly two children. My account draws the same sort of distinction at a “higher order”: energy conservation is a constraint exactly when energy would still have been conserved, had there been additional kinds of forces. This means of distinguishing constraints from coincidences fits nicely into my more general account of natural law. I have presented

"There sweep great general principles" | 171 this account in detail elsewhere (Lange 2009). Within the confines of this paper, I am obviously not able to offer much in the way of argument for my general account. Rather, my main aim is to argue that any account of natural law must recognize the distinction between constraints and coincidences. This cannot easily be done by the accounts of natural law currently on the market (as I will illustrate in the following section on dispositional essentialism). Here I offer my account simply as an example of how it is possible for an analysis of lawhood to leave a natural place for the distinction between constraints and coincidences (as I have just drawn it) and thereby to recognize the important role that this distinction plays in science (as I have suggested in the preceding sections). As I just mentioned, laws of nature have traditionally been thought to differ from accidents in having greater perseverance under counterfactual suppositions. For instance, since it is a law that no body is accelerated from rest to beyond the speed of light, this cosmic speed-limit would not have been broken even if the Stanford Linear Accelerator had now been cranked up to full power. On the other hand, if it is just an accident that all gold cubes are smaller than a cubic meter, then had Bill Gates wanted a gold cube larger than a cubic meter, I dare say there would have been one. Of course, laws are unable to persist under counterfactual suppositions with which they are logically inconsistent. This suggests the following proposal: m is a law if and only if in any conversational context, under any counterfactual supposition p that is logically consistent with all of the laws, m would still have held (i.e., p □→ m).

In this proposal and until further notice, I reserve letters like “m” for “sub-nomic” claims, i.e., for claims such as “The emerald at spatiotemporal location . . . is 5 grams” or “All emeralds are green” as contrasted with “nomic” claims such as “It is an accident that the emerald at spatiotemporal location . . . is 5 grams” or “It is a law that all emeralds are green”. (On my view, a claim is “sub-nomic” exactly when in any possible world, what makes the claim hold (or fail to hold) is not that a given fact in that world is a law or that a given fact in that world is an accident.) Let me also note that the account of laws I am sketching here presupposes that every logical consequence of laws qualifies as a law and that every broadly logical

172 | Marc Lange truth (e.g., every truth holding with narrowly logical necessity, metaphysical necessity, mathematical necessity, moral necessity, etc.) is by courtesy a natural law. (These convenient simplifications could be dropped, however.) Although the above proposal captures an important difference between laws and accidents in their behavior in counterfactuals, this proposal uses the laws themselves to pick out the relevant range of counterfactual suppositions. This is problematic since if there is no prior, independent reason why this particular range of counterfactual suppositions is special, then the laws’ invariance under these particular suppositions fails to make the laws special. They merely have a certain range of invariance (just as a given accident has some range of invariance). This problem can be avoided. Let’s start by characterizing what I shall call “sub-nomic stability”: Consider a non-empty set Γ of sub-nomic truths containing every subnomic logical consequence of its members. Γ possesses sub-nomic stability if and only if for each member m of Γ and for any p where Γ∪{p} is logically consistent (and in every conversational context), it is not the case that ~m might have held, had p held (i.e., ~ (p ◊→ ~ m)).9

Notice that ~ (p ◊→ ~ m) logically entails p □→ m. Therefore, a set of truths is sub-nomically stable exactly when its members would all still have held (indeed, not one of their negations might have held) under any counterfactual supposition with which they are all logically consistent. So in contrast to the earlier proposal, stability does not use the laws to pick out the relevant range of counterfactual suppositions. Rather, each set picks out for itself the range under which it must be invariant in order for it to be stable. This suggests my proposal for distinguishing laws from accidents: that the set Λ of all sub-nomic truths m where it is a law that m is subnomically stable, whereas no set containing an accident is sub-nomically stable (except perhaps for the set of all sub-nomic truths, considering that the range of counterfactual suppositions under which this “maximal” set must be preserved in order to qualify as stable does not include any false suppositions since no falsehood is logically 9 For the sake of simplicity, this definition of “sub-nomic stability” omits some details from my (2009) that will not make any difference here.

"There sweep great general principles" | 173 consistent with all of this set’s members). For instance, the set spanned by the fact that all gold cubes are smaller than a cubic meter is unstable because this set’s members are all logically consistent with Bill Gates wanting a gold cube larger than a cubic meter, yet the set’s members are not all invariant under this counterfactual supposition. It is a law that m, then, exactly when m belongs to a (non-maximal) sub-nomically stable set. Now let’s show that this account leaves a natural place for the distinction between constraints and coincidences. Are there any other non-maximal sub-nomically stable sets besides Λ? The sub-nomic broadly logical truths form a subnomically stable set. I’ll now show that for any two sub-nomically stable sets, one must be a proper subset of the other: 1. Suppose (for reductio) that Γ and Σ are sub-nomically stable, t is a member of Γ but not of Σ, and s is a member of Σ but not of Γ. 2. Then (~s or ~t) is logically consistent with Γ. 3. Since Γ is sub-nomically stable, every member of Γ would still have been true, had (~s or ~t) been the case. 4. In particular, t would still have been true, had (~s or ~t) been the case. That is, (~s or ~t) □→ t. 5. So t & (~s or ~t) would have held, had (~s or ~t). Hence, (~s or ~t) □→ ~s. 6. Since (~s or ~t) is logically consistent with Σ, and Σ is subnomically stable, no member of Σ would have been false had (~s or ~t) been the case. 7. In particular, s would not have been false, had (~s or ~t) been the case. That is, ~((~s or ~t) □→ ~s). 8. Contradiction from 5 and 7. Thus, the sub-nomically stable sets must form a nested hierarchy. Since no non-maximal superset of Λ is stable (since it would include an accident), any other stable sets must be among Λ’s proper subsets. Many of them are clearly unstable. For instance, the set spanned by a restriction of Coulomb’s law to the past is unstable since had Coulomb’s law been violated sometime in the future, then (with Coulomb’s law “out of the way”) it might have been violated sometime in the past. However, some of Λ’s proper subsets may be stable, and I suggest that any constraint must belong to at least one such set. Other

174 | Marc Lange members of such a set plausibly include the fundamental dynamical law, the law of the parallelogram of forces, the space-time transformations, and other laws that “transcend” the particular kinds of forces there happen to be; the various force laws and the “closure law” specifying the actual kinds of forces are excluded from such a set. In other words, if a conservation law belongs to no non-maximal stable set besides Λ, then it is a coincidence. If energy conservation belongs to a stable proper subset of Λ that omits the various force laws and the closure law, then the set’s stability requires the subjunctive fact that (I proposed) distinguishes constraints from coincidences (that energy would still have been conserved, had there been additional kinds of forces) since the supposition that there are additional kinds of forces is logically consistent with each of the set’s members. Energy conservation’s status as a constraint is then associated with its invariance under a certain range of counterfactual antecedents, and that range consists of those antecedents that are logically consistent with every member of a stable subset of Λ to which energy conservation belongs. For instance, if the various particular force laws are all omitted from that set, then in connection with its status as a constraint, energy conservation would still have held, had gravity not been an inverse-square force. This approach leaves room for multiple levels of constraints on the force laws, each one associated with a stable set that occupies a spot in the nested hierarchy of stable sets somewhere between the set of broadly logical truths and Λ. Moreover, this approach accounts for the role of constraints as higher-order laws—that is, laws that are modally more exalted than the force laws they constrain, and so able to explain why all of the forces share certain features. It accounts for the way in which the constraints carve out a species of possibility that is more inclusive than physical possibility (as we saw in connection with “thermodynamic possibility”, which embraces some physical impossibilities). The members of a stable set would all still have held under any counterfactual supposition with which they are all logically consistent—that is, under which they could (i.e., without contradiction) all still have held. In other words, a stable set’s members are collectively as resilient under counterfactual suppositions as they could collectively be. They are maximally resilient—that is to say, necessary. Accord-

"There sweep great general principles" | 175 ingly, I suggest that a sub-nomic truth has a species of necessity exactly when it belongs to a non-maximal sub-nomically stable set, and that for each of these sets, there is a distinct species of necessity that is possessed by exactly its members. On this view, then, whereas constraints and coincidences are both physically necessary, a constraint also possesses a species of necessity (a stronger cousin of physical necessity) that the coincidences and force laws lack. Thus, we are entitled to say that a constraint limits the kinds of forces there could have been, whereas a coincidence merely reflects the kinds of forces there happen to be. The actual inventory of forces is a matter of physical necessity and yet also a matter of happenstance in that it lacks the stronger necessity possessed by a constraint. Finally, this view explains why any conservation law that follows from a symmetry principle within a Hamiltonian dynamical framework constitutes a constraint rather than a coincidence and so (as Wigner says) “transcends” the various force laws. A symmetry principle, such as the fact that the laws are invariant under arbitrary temporal displacement, is not expressed by a sub-nomic claim. Rather, a symmetry principle is made true by which facts are laws. It is expressed by a “nomic” claim, i.e., a claim that purports to describe which truths expressed by sub-nomic claims are (or are not) matters of law. So to characterize the invariance that is characteristic of symmetry principles as “meta-laws”, we need an analogue of sub-nomic stability that applies to sets of claims that are either sub-nomic or nomic. Here it is (now allowing letters like “p” to stand for claims that are either sub-nomic or nomic): Consider a non-empty set Γ of truths that are nomic or sub-nomic containing every nomic or sub-nomic logical consequence of its members. Γ possesses nomic stability if and only if for each member m of Γ (and in every conversational context), ~ (p ◊→ ~m) for any p where Γ∪{p} is logically consistent.

The symmetry principles are meta-laws in that they form a nomically stable set more exclusive than the set spanned by all truths about which sub-nomic claims are laws and which are not. Moreover, if the conservation laws are logically entailed by symmetry meta-laws within a Hamiltonian dynamical framework, then they belong to a sub-nomically stable set that is more exclusive than Λ

176 | Marc Lange and therefore are constraints. That is because for any nomically stable set, its sub-nomic members must form a sub-nomically stable set. Here is the proof: 1. If p (a sub-nomic claim) is logically inconsistent with a nomically stable set Γ, then Γ must entail ~p (also subnomic), and so p is logically inconsistent with the set Σ containing exactly Γ’s sub-nomic logical consequences. 2. Conversely, if p is logically inconsistent with Σ, then obviously p is logically inconsistent with Γ. 3. By Γ’s nomic stability, Σ is preserved under every subnomic antecedent p that is logically consistent with Γ—which (we have just shown) are exactly those subnomic antecedents that are logically consistent with Σ. Hence, Σ is sub-nomically stable. Therefore, if the symmetry meta-laws (forming a nomically stable set) entail that a given conservation law holds under the Hamiltonian dynamical framework, then the conservation law’s holding if the Hamiltonian dynamical law holds belongs to a sub-nomically stable set that is more exclusive than Λ (since presumably, not all of Λ’s members follow from the symmetry meta-laws’ nomically stable set). Hence, if the Hamiltonian dynamical law is not a member of that set but (in transcending the various particular force laws) belongs to another sub-nomically stable set that does not include the force laws, then (since the sub-nomically stable sets form a nested hierarchy) the fact that the conservation law holds under the Hamiltonian framework must also belong to that set, and hence (by the set’s logical closure) the conservation law must belong, too. So it constitutes a constraint. In other words, that the conservation law would still have held, even if the force laws had been different, follows from the fact that not only the fundamental dynamical law, but also the symmetry meta-law would still have held had the force laws been different. Of course, I cannot do more here than sketch the relevant parts of this conception of natural law. But it is worth seeing how an account of lawhood can incorporate the constraint/coincidence distinction in a natural way. Now I shall conclude by turning to an approach to natural law that is ill-equipped to have the same success.

"There sweep great general principles" | 177 5. DISPOSITIONAL ESSENTIALISM RULES OUT CONSTRAINTS According to Alexander Bird (2007), every sparse fundamental property of physics is constituted by one or more dispositions. On Bird’s view, the association between a fundamental property of physics and some disposition is a matter of metaphysical necessity; moreover, the identity of a given fundamental property of physics is exhausted by its dispositional character. Therefore, it is metaphysically necessary that any entity possessing a certain sparse fundamental property of physics exhibit certain further properties if suitably stimulated. These regularities, or the corresponding relations among properties, are the laws of nature. Although metaphysically necessary, the laws do not perform the explanatory heavy-lifting. The motor and cement of the universe are the dispositional essences of the fundamental properties of physics. Views along roughly similar lines have been proposed by Brian Ellis (2001; 2002) and Stephen Mumford (2004), among others. These views differ in some details from Bird’s: for example, Ellis takes the dispositional essences responsible for laws to be the essences of the natural kinds rather than of the sparse fundamental properties of physics, whereas Mumford holds that since any fundamental property of physics is constituted by a cluster of causal roles and other connections to other properties (such as its excluding certain properties and being compatible with certain others), there are no laws because nothing governs property instances in the manner traditionally ascribed to laws. However, these differences will make little difference here. A conservation law does not say that any entity possessing a certain sparse fundamental property of physics exhibits certain further properties if suitably stimulated. Unlike (for instance) the occurrence of a certain force, energy’s remaining conserved is not the manifestation of a particular disposition. Therefore, it is difficult for views like Bird’s to accommodate conservation laws, as Bird (2007, 211) himself notes. It is worth distinguishing three forms that this objection can take. The first form is that energy’s remaining conserved under any interaction of a given kind is simply not the manifestation of any disposition constituting any of the sparse fundamental properties of

178 | Marc Lange physics manifested in such interactions. The dispositions associated with electric charge, for instance, may manifest themselves in various accelerations or various contributions to the electric field. Thus, the laws specifying that charges manifest themselves in certain ways under certain conditions do not include among these manifestations anything about energy being conserved. The law that electric interactions conserve energy therefore seems to have been left out of the account. However, it seems to me that the law has not been neglected. From the laws concerning electric interactions, it follows logically that such interactions conserve energy. So on this account, energy’s conservation in electric interactions is a metaphysical necessity arising from the dispositions essential to various fundamental properties of physics, including charge. There is no problem yet for Bird’s view. The second form of the objection is that the law of energy conservation does not follow simply from the law that energy is conserved in electric interactions, the law that energy is conserved in gravitational interactions, and so forth. There must be a further premise, since these force laws taken together do not preclude the existence of another kind of process that fails to conserve energy. One premise that would close the gap is a “closure law”: that electric interactions, gravitational interactions, and so forth are all of the kinds of interactions there are, i.e., that every fundamental natural process belongs to one of these kinds. The objection, then, is that this “closure law” does not reflect any property’s dispositional essence. Hence, the overall conservation law is not the reflection of the dispositional essences of the fundamental properties of physics. Ellis’s view purports to be invulnerable to this form of the objection. Ellis takes the laws to be grounded not in the dispositional essences of various properties, but rather in the essences of the various natural kinds. He suggests (following Bigelow, Ellis, and Lierse 1992) that the world is the only member of a certain natural kind and that the essence of this kind includes various quantities being conserved (Ellis 2001, 205 and 250). Its essence also includes its being populated by exactly certain sorts of particles and fields undergoing exactly certain sorts of interactions, thereby accounting for the “closure laws”. I have replied elsewhere that this move “seems like a desperate attempt to find something the essence of which could be responsible for various

"There sweep great general principles" | 179 laws. Even if gravity and the electron have essences, it is not obvious that ‘the world’—reality—does” (Lange 2009, 83). Bird also finds this move “somewhat ad hoc” (2007, 213). However, I shall not pursue this concern here. This form of the objection may turn out not to pose a terribly severe challenge for Bird’s view. One option that Bird might explore is to say simply that although the conservation “law” is not a metaphysical necessity, the fact that energy is conserved is no accident either. Rather, it is grounded in the sparse fundamental properties of physics, i.e., in the world’s repertoire of fundamental causal powers. But this option would require Bird to allow different repertoires in different possible worlds, which Bird is reluctant to do (since in that case the actual laws, though still true in all worlds, are not laws in some of them). Another option for elaborating Bird’s view accords metaphysical necessity to the conservation law and allows every world to have exactly the same laws. As I just explained, energy’s conservation in electric interactions is a metaphysical necessity arising from the dispositions essential to various fundamental properties of physics. So likewise is energy’s conservation in gravitational interactions, and similarly for any other actual type of interaction. On Bird’s view, the various fundamental properties of physics would be different properties if they bestowed additional powers and susceptibilities—for instance, the power to exert and the susceptibility to feel some alien force that fails to conserve energy. Presumably, no metaphysical necessity precludes the instantiation of alien fundamental properties of physics. But “electric charge” (or any other non-alien property) would be a different property if it bestowed susceptibility not just to the influence of other electric charges, but also to some other alien influence. What if we go further and suppose that all of the non-alien properties (or at least enough of them that all non-alien kinds of entities must possess one) have as part of their essences that they bestow immunity to being influenced by any alien property (and so their possession by an entity precludes its also possessing any alien property that would bestow susceptibility to being so influenced)? Then the possession of any alien properties (by entities not possessing any of the nonalien properties) could not influence the behavior of any entities possessing non-alien properties and so could not disturb the conservation of the quantities that are in fact conserved under each of

180 | Marc Lange the various actual types of interaction, as long as those quantities are possessed only by entities possessing the non-alien properties. This is obviously the case for electric charge and, by the dispositional essentialist’s lights, arguably the case for energy and momentum, since these properties (the essentialist might say) are constituted by various combinations of non-alien fundamental properties such as mass, velocity, charge, separation, and so forth. In that case, although these quantities are not instantiated in every possible world, they are conserved in any world where they are instantiated, since the non-alien properties constituting these quantities bestow immunity to any other influences besides the various actual types of forces. Energy conservation is then metaphysically necessary even if the “closure law” is not and even though the conservation law is not the manifestation of any single fundamental property’s dispositional essence. However, let’s now consider a third form of the original objection that views like Bird’s cannot properly accommodate conservation laws. This form of the objection will prove more difficult for views like Bird’s to answer. On Bird’s picture, even if energy conservation is metaphysically necessary, it is the product of the various particular types of interactions there are: that gravitational forces conserve energy, electric forces conserve energy, and so forth. The various dispositions essential to various fundamental natural properties are responsible for energy’s conservation. So although energy conservation is metaphysically necessary, it is a coincidence rather than a constraint. To appreciate this, recall the subjunctive fact that (I have suggested) is what it is for energy conservation to be a constraint: that energy would still have been conserved even if there had been an additional kind of interaction experienced by some of the entities undergoing some of the actual kinds of interaction. This conditional’s antecedent is a countermetaphysical according to the view I have just offered Bird in response to the previous form of the objection. Even without that response, on Bird’s view there is no disposition (or collection of dispositions) essential to some sparse fundamental property of physics (or collection of such properties) that is available to underwrite this conditional’s truth. On Bird’s view, the causal powers constituting those properties do the metaphysical heavy lifting, but these causal powers cannot underwrite

"There sweep great general principles" | 181 this conditional’s truth; none of them has as its manifestation that certain alien fundamental natural properties are instantiated or has as its manifestation-eliciting stimulus that some alien kind of interaction occurs. The conditional associated with being a constraint does not concern what the actual causal powers would produce under suitable stimulation, but rather concerns what causal powers there would be if there were more besides the actual ones—so the actual causal powers do not sustain this conditional’s truth. (The same applies to other subjunctive conditionals whose truth would reflect a conservation law’s status as a constraint, on my view of constraints as belonging to stable proper subsets of Λ. For example, the fact that energy would still have been conserved, had gravity not been an inverse-square force, posits a countermetaphysical and is underwritten by no causal powers, on Bird’s view.) Ellis’s view faces a similar problem: there is no natural kind whose essence can step in to sustain the conditional needed to make energy conservation a constraint. Even the world’s essence cannot do the job since the conditional’s antecedent is inconsistent with that essence. On Ellis’s view, had there been additional kinds of interactions, there would have been a different kind of world; it might have been a world where energy is conserved, but then again, it might not. Energy conservation is therefore a coincidence, not a constraint. On a view like Bird’s, a conservation law cannot be a constraint because a conservation law cannot be explanatorily prior to the force laws—or, rather, to the various dispositions responsible for the force laws and essential to the various sparse fundamental properties of physics. Gravitational interactions conserve energy not because the law of energy conservation requires them to, but because of the causal powers involved in gravitational interactions (which the gravitational-force law reflects). The conservation laws cannot explain why (e.g.) no fluid circulates; rather, the explainers must be the particular powers involved. Even if a given power’s conserving energy when it is manifested suffices to entail the explanandum, energy conservation as a comprehensive law covering all of the actual powers is explanatorily irrelevant; all that matters to the outcome is the fact that the particular power at work in this case conserves energy. This is how all scientific explanations must work, according to Ellis—from the bottom up:

182 | Marc Lange Essentialists seek to expose the underlying causes of things, and to explain why things are as they are, or behave as they do, by reference to these underlying causal factors. Consequently, explanations of the sort that essentialists are seeking must always have two parts. They must contain hypotheses about the underlying structures or causal powers of things, and hypotheses about how things having these structures and powers must behave in the specific circumstances in which they exist. (Ellis 2002, 159–60)

This approach to scientific explanation leaves no room for the topdown explanations that constraints supply. Why are two forces alike in conserving energy? Why do no kinds of interaction put fluids spontaneously into circulation? That energy conservation constrains the kinds of forces there could have been is not the kind of explanation Ellis allows. On Ellis’s picture, the explananda are merely upshots of the various particular kinds of interaction built into the world’s essence, all of which conserve energy. Bird likewise seems to find top-down explanations from conservation laws problematic: [T]here is something mysterious about conservation laws. They seem to require explanation . . . How does a system know that energy should be conserved? . . . It is not clear how these could be fundamental laws—they seem to stand in need of a deeper explanation. (2007, 213)

The deeper explanation Bird has in mind would be in terms of the particular causal powers at work in the system. It is these powers that allow a system to “know that energy should be conserved”. This is the order of explanatory priority that is characteristic of conservation laws as coincidences: the various force laws (or, for the essentialist, the various particular causal powers responsible for the force laws) come before the conservation law. But as we saw earlier in this paper, conservation laws as constraints provide explanations that are in some respects “deeper” than any explanations appealing to the particular powers involved. By constituting common explainers, conservation laws as constraints unify the non-circulation of ropes and fluids, for example, and likewise unify various otherwise unrelated forces as all inverse-square (or all subject to uniqueness theorems) or all conserving energy for the same reason. These explanations reveal that the fact to be explained would still have held even if different fundamental forces had been at work.

"There sweep great general principles" | 183 That conservation laws must be coincidences rather than constraints on his account seems to be roughly what Bird has in mind when he acknowledges that his account cannot accommodate conservation laws: Conservation and symmetry laws tell us that interactions are constrained by the requirement of preserving, e.g., mass-energy or momentum . . . [T]he dispositional essentialist holds that the laws are necessary. If that is correct there is no room for further constraints. Properties are already constrained by their own essences and so there is neither need nor opportunity for higher-order properties to direct which relations they can engage in. (Bird 2007, 211 and 214)

Bird’s thought seems to be that as constraints on (the causal powers responsible for) the various force laws, conservation laws would have to impose limitations on the possible manifestations of the powers associated with electric charge, mass, and so forth, limiting the interaction laws to those that conserve energy. But these interaction laws are already fully determined by the properties involved in the interactions since those properties are essentially constituted by various causal powers, which fix their own possible manifestations. So there is no further constraining for a conservation law to do. It seems to me that there is more that the conservation law could constrain: what powers there would be, if additional fundamental natural properties were instantiated. But this would require that the conservation law do some metaphysical heavy lifting—a job that, on Bird’s picture, is reserved for the causal powers. In short, a conservation law as a constraint is a higher-order law; it has greater necessity than any of the motley collection of various particular force laws and thus explains why those laws are all alike in exhibiting certain features. But on views like Bird’s and Ellis’s, the particular force laws hold as matters of metaphysical necessity. No greater necessity is left for a constraint to possess; necessity has already been maxed out in the force laws. So there is no genuine “thermodynamic possibility”—no sense in which various actual properties could have been associated with certain alien powers but not with others (namely, with only those alien powers that, in being manifested, would conserve energy, momentum, and so forth). Therefore, conservation laws cannot be constraints; they can only be coincidences.

184 | Marc Lange Bird seems content with this result: The dispositional essentialist ought to regard symmetry principles as pseudo-laws. . . . So it may be that symmetry principles and conservation laws will be eliminated as being features of our form of representation rather than features of the world requiring to be accommodated within our metaphysics. (Bird 2007, 214)

But I have argued (in the opening sections of this paper) that the price of adopting this view is too high. It precludes top-down explanations of a kind that science has reasonably taken seriously— indeed, on which science has often placed great importance. Views like Bird’s rule out scientifically respectable theories. Perhaps Bird’s prognostication regarding future science will be proved right; perhaps symmetry principles and conservation laws, along with the alleged top-down explanations they supply, will ultimately be eliminated from physics—and their place not be taken by other top-down explanations (like Hertz’s) employing other sorts of constraints. Personally, I doubt it. But more importantly, a metaphysics that cannot do justice to top-down explanations and the constraints they require is at a serious disadvantage even if as a matter of fact, there turn out to be no such explanations. Room should still be left for them; it should be up to science rather than metaphysics to foreclose them. Consider, for instance, the conservation of baryon number in contemporary physics. By energy conservation, an isolated proton can decay only into particles that have less rest-mass than it does, and the proton is the lightest baryon (i.e., the lightest particle with non-zero baryon number). The conservation of baryon number thus entails that the proton is stable (radioactively, I mean—not “subnomically!). Does baryon-number conservation explain why the proton is stable? This remains controversial. It is no explanation if the conservation of baryon number is a socalled “accidental symmetry” (a term that was introduced by Steven Weinberg; see Weinberg 1995, 529). An accidental symmetry reflects merely the particular forces in action at lower-energy regimes rather than some deeper “symmetry of the underlying theory” (Weinberg 1995, 529). Accordingly, if baryon-number conservation is an accidental symmetry, then it may not even hold at higher energies (and so the proton may turn out not to be stable, but

"There sweep great general principles" | 185 rather to have an extremely long half-life). But even if an accidental symmetry is unbroken, it would still be a coincidence of the particular kinds of interactions written “by hand” into the underlying theory, and so it would fail to explain. (The stability of the lightest baryon would then help to explain why the baryon number turns out to be conserved, not vice versa.) On the other hand, baryon-number conservation may turn out to be a consequence of a more fundamental symmetry, in which case it would help to explain the proton’s stability. Thus, whereas some physicists cite baryon-number conservation as explaining why the proton is stable (e.g., Davies 1986, 159; Duffin 1980, 82), other physicists put scare-quotes around “explain” (Lederman and Teresi 1993, 303) or say that the jury is still out (Ne’eman and Kirsh 1996, 150–1). The uncertainty regarding the conservation law’s explanatory power is matched by the uncertainty regarding the law’s status as constraint or coincidence. But this live scientific controversy would be settled outright by views like Bird’s. I do not think that metaphysics should prejudge the outcome. If the laws are the upshot of the inventory of powers, and hence (according to dispositional essentialism) of the sparse fundamental properties of physics, then any regularity in those powers (even if metaphysically necessary) is coincidental; no law transcends the various powers, constraining those there could have been. However, an adequate metaphysics should allow the “great general principles which all the laws seem to follow” to be constraints; it should not require them to be coincidences. University of North Carolina at Chapel Hill REFERENCES Annamalai, K. and Puri, I. (2002), Advanced Thermodynamic Engineering. Boca Raton, CRC Press. Bartlett, D.F. and Su, Y. (1994), “What Potentials Permit a Uniqueness Theorem,” American Journal of Physics 62: 683–6. Bigelow, J., Ellis, B., and Lierse, C. (1992), “The World as One of a Kind: Natural Necessity and Laws of Nature,” British Journal for the Philosophy of Science 43: 371–88. Bird, A. (2007), Nature’s Metaphysics. Oxford: Oxford University Press.

186 | Marc Lange Callendar, C. (2005), “Answers in Search of a Question: ‘Proofs’ of the Tri-dimensionality of Space,” Studies in History and Philosophy of Modern Physics 36: 113–36. Davies, P. (1986), The Forces of Nature. Cambridge: Cambridge University Press. Duffin, W.J. (1980), Electricity and Magnetism, 3rd ed. London: McGraw-Hill. Ellis, B. (2001), Scientific Essentialism. Cambridge: Cambridge University Press. —— (2002), The Philosophy of Nature: A Guide to the New Essentialism. Montreal & Kingston: McGill-Queen’s University Press. Fermi, E. (1956). Thermodynamics. New York: Dover. Feynman, R. (1967), The Character of Physical Law. Cambridge: MIT Press. Hertz, H. (1999), Die Constitution der Materie, ed. Albracht Fölsing. Berlin: Springer-Verlag. Lange, M. (2002), An Introduction to the Philosophy of Physics: Locality, Fields, Energy, and Mass. Malden, MA: Blackwell. —— (2009), Laws and Lawmakers. New York: Oxford University Press. Lederman, L. and Teresi, D. (1993), The God Particle. New York: Dell. Mumford, S. (2004), Laws in Nature. London and New York: Routledge. Ne’eman, Y. and Kirsh, Y. (1996), The Particle Hunters. Cambridge: Cambridge University Press. Seth, S. (2010), Crafting the Quantum. Cambridge: MIT Press. Stevin, S. (1955), The Principal Works of Simon Stevin. Volume 1. Amsterdam: C.V. Swets & Zeitlinger. Weinberg, S. (1995), The Quantum Theory of Fields, volume 1. Cambridge: Cambridge University Press. Wigner, E. (1972), “Events, Laws of Nature, and Invariance Principles,” in Wigner, Collected Papers, Part A, volume 3. Berlin: Springer, 185–96. Yourgrau, W., Van der Merwe, A., and Raw, G. (2002), Treatise on Irreversible and Statistical Thermophysics. New York: Dover.

IV

CONTINGENT OBJECTS AND COINCIDENT OBJECTS

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7. Relativized Metaphysical Modality* Adam Murray and Jessica Wilson† INTRODUCTION Metaphysical necessity and possibility are commonly supposed to be necessity and possibility in the broadest, not merely syntactically logical, sense. Hence it is that metaphysical modality is often contrasted with other, restricted forms of modality, as when Burgess (2009) says: [W]e may distinguish the species of physical necessity, or what could not have been otherwise so long as the laws of nature remained the same, from metaphysical necessity, what could not have been otherwise no matter what. (46)

In quantificational terms, the supposition is that a single domain of possible worlds enters into metaphysical modal evaluation; a claim is metaphysically necessary just in case it is true in every possible world in the domain, and metaphysically possible just in case it is true in some possible world in the domain. We argue here that the standard understanding is strictly incorrect; rather, whether a given claim is metaphysically necessary or possible depends on which world is, as we put it, ‘indicatively actual’. In brief: metaphysical necessities and possibilities are relativized to indicative actualities. The proper understanding of metaphysical modality thus takes modal space to have a complex, relativized structure. The sense in which the standard view is correct concerns its coinciding with metaphysical modality when relativized to our very own indicatively actual world; the sense in which the standard view is incorrect concerns its failing to be sensitive to the more complex relativized structure of metaphysical modality. * Special thanks to Karen Bennett and Dean Zimmerman for extensive comments which greatly improved this paper. Thanks also to Benj Hellie, Phil Kremer, Dan Rabinoff, Chris Tillman, and audience members at the University of North Carolina and the University of Miami, for helpful comments and questions, and to Brent Cromwell for assistance in constructing the figures. † Department of Philosophy, University of Toronto; [email protected], [email protected]

190 | Adam Murray and Jessica Wilson We motivate the alternative proposal by attention to discussions in Salmon (1989) and Fine (2005). In each discussion, the author canvasses data which he takes to support a certain thesis—in Salmon’s case, that the transitivity of the accessibility relation between possible worlds, and associated systems of modal logic S4 and S5, should be rejected as characteristic of metaphysical modality; in Fine’s case, that nomological and metaphysical modality should be taken to be distinct and equally basic (as per “modal pluralism”). We argue that the data in each case can be accommodated, compatible with transitive accessibility and with modal monism, if metaphysical necessities and possibilities are relativized to indicative actualities; and we offer two ways of implementing the relativized conception within a possible-worlds semantics for metaphysical modal logic. We also note, for heuristic purposes, a formal analogy between the relativized conception and a thoroughly metaphysical interpretation of the ‘secondary’ or ‘horizontal’ intensions associated with the two-dimensional semantic framework, which intensions may be seen as representing what is counterfactually possible relative to each indicatively actual world. We close by observing the neutrality of our conception as regards the actualist/possibilist and trans-world identity/counterpart theory distinctions.1

1 SALMON’S ‘WOODY’ CASE 1.1 Possible Worlds Semantics and Transitive Accessibility It is intuitively natural and historically familiar (following Leibniz 1686) to characterize modal claims in quantificational terms, where the evaluation of such claims reflects the spectrum of truths across a given range of possible worlds. Such a characterization is formally vindicated in possible worlds semantics for modal logics (Kripke 1963). A modal logic extends the usual propositional (or predicate; here we follow Salmon in focusing on the simpler case) logics by introducing symbols ‘□’ and ‘◊’, along with certain 1 Our project here bears some similarity to but is in key respects different from projects of the sort at issue in Crossley and Humberstone (1977) and Davies and Humberstone (1979), in which the standard modal logic(s) are supplemented with an actuality operator (‘A’) and associated modal operators (‘fixedly’, ‘fixedly actually’). In the earlier paper, the motivation for an actuality operator is that scope interactions between quantifiers and standard modal operators fail to allow expression of claims like ‘It is possible that everything that is in fact red is shiny’, and the additional modal

Relativized Metaphysical Modality | 191 rules or axioms supposed to govern expressions containing these symbols on any of a wide range of interpretations, which include the necessitation rule (if p is a theorem of the logic, infer □p), and the distribution axiom (K) (□(p → q) → (□p → □q)).2 Creating a model for such a logic then involves two steps. The first step involves specifying a frame: a set W of possible worlds, along with a relation R between worlds; the desired features of the relation are encoded in certain axioms, to be discussed shortly. The second step involves specifying a valuation function v setting up the basic nonmodal facts in each world; truth clauses for expressions prefaced with one or other modal symbol are then added to the usual truth clauses in such a way that the truth of all basic and non-basic claims in the model is determined. The relevant modal clause (also determining, given that necessity and possibility are duals, the clause for claims involving the possibility operator) is then schematically as follows: v(□p, w) = T iff for every world w’ in W such that wRw’, v(p, w’) = T. operators are briefly introduced in order to respond to concerns about the validity of axioms of the form Aα→□ Aα on the semantics offered for ‘A’, that ‘“actually a” need not have been true because another world might have been actual’ (1979, 2); in the later paper, these new modal operators are applied in service of accommodating putative cases of the contingent a priori and necessary a posteriori, with the assistance of certain theses about names and natural kind terms. Our motivations and target applications are different. We aim to show that certain puzzles, having to do with natures or essences as opposed to scope or semantics (or epistemology), ultimately arise from a failure of standard metaphysical modal logics to incorporate relativization to indicatively actual worlds, and to argue that proper incorporation of a relativized structure makes for better resolutions of these puzzles than those currently on offer. Our suggested ways of making sense of such relativization within a possible worlds semantics for modal logic do not involve any additional operators, and though in the course of explicating our view we heuristically appeal to properly metaphysical interpretations of the notions, familiar from 2-D semantics, of considering worlds ‘as actual’ or ‘as counterfactual’, unlike Davies and Humberstone we are officially neutral on both the semantics and epistemology of names and natural kind terms. These differences aside, in arguing that the traditional modal operators should be relativized to indicatively actual worlds we are on the same side as these other authors; our contribution here is, first, to offer distinctively metaphysical reasons for incorporating such relativization, and second, to show that this can be done in ways minimally departing from standard modal logic(s). 2 More precisely, modal logics that include the distribution axiom (□(p → q) → (□p → □q)) are known as normal modal logics. We assume in what follows that the modal logics under discussion are normal.

192 | Adam Murray and Jessica Wilson As above, different systems of modal logic impose different constraints on the relation R at issue, which are encoded, either explicitly or indirectly, in certain axioms. Typically, the relation R is understood as an ‘accessibility’ relation; in the case of metaphysical modality, the features of this relation are intended to ensure—again reflecting the standard understanding of metaphysical necessity as necessity in the broadest sense—that the facts holding at any and all possible worlds are relevant to metaphysical modal deliberation. Such an accessibility relation is standardly supposed to be reflexive, such that any world is accessible to ‘(can “see”)’ itself; this requirement is encoded in axiom (T): (T): □p → p (for any necessarily true proposition p, the proposition that p is true). The resulting system (i.e., the system imposing no further constraints on R) is system T. The accessibility relation is also standardly supposed to be transitive, which requirement is satisfied by adding axiom (4) to system T: (4): □p → □□p (for any necessarily true proposition p, the proposition that p is necessarily true is itself necessarily true). The resulting system is system S4. Finally, the accessibility relation is standardly supposed to be symmetric, which requirement is satisfied by adding axiom (5) to system T: (5): ◊p → □◊p (for any possibly true proposition p, the proposition that p is possibly true is itself necessarily true). The resulting system is system S5. Since in S5 the accessibility relation R is also reflexive and transitive, R in S5 is an equivalence relation. It is commonly assumed that S5 is the correct logic for metaphysical modality (see Sider 2010).3 3 This assumption reflects, in part, that the theorems of S5 coincide with those on a modal logic where the accessibility relation is ‘total’, such that (as per the standard conception) every world is accessible to every other. To prefigure a bit: one way of implementing our transitive relativized conception of metaphysical modality takes advantage of the fact that, notwithstanding the coincidence of theorems, S5 is compatible with modal space being partitioned into non-overlapping equivalence classes (see S2.2).

Relativized Metaphysical Modality | 193 1.2 Salmon’s Rejection of Transitive Accessibility Salmon (1981, 1984, and 1989) argues (following Chandler 1976) that this is a mistake: axiom (4) has false instances, and so S4 and the stronger system S5 are fallacious logics of metaphysical modality. Salmon takes this result to follow from consideration of a case where a table (‘Woody’) could have originated from matter m’ slightly different from the matter m it actually originated from, but could not have originated from some matter m” very different from its actual originating matter:4 Wherever one may choose to draw the line between what matter Woody might have originated from and what matter Woody could not have originated from, it would seem that [. . .] we may select some [. . .] matter m” such that, although Woody could not have originated from m”, m” is close enough to being a possibility for Woody that if Woody had originated from certain matter m’ that is in fact possible for Woody—matter differing in as many molecules from the actual original matter m as possible, and sharing as many molecules with m” as possible, while remaining a possibility for Woody—then it would have been possible for Woody to have originated from m”, even though it is not actually possible. [As such] the conditional claim (which is an axiom of S4) that if Woody necessarily does not originate from m”, then it is necessary that Woody necessarily does not so originate fails. [. . .] S4 modal logic is fallacious. (1989, 5)

Somewhat more formally, Salmon’s argument is as follows: 1. 2. 3. 4.

Woody originates from matter m. It is possible that Woody originates from matter m’. It is not possible that Woody originates from matter m”. If Woody had originated from matter m’, then it would have been possible for Woody to originate from matter m”. 5. It is possible that it is possible that Woody originates from matter m”. (2, 4) 6. It is not possible that Woody originates from m”, but it is possible that it is possible that Woody originates from matter m”. (3, 5) 7. It is necessary that Woody does not originate from matter m”, but it is not necessary that it is necessary that Woody does not originate from matter m”. (6) 4 For simplicity we have altered the indexing on the hunks of matter at issue, and will later do so for associated worlds.

194 | Adam Murray and Jessica Wilson The last claim expresses that a certain claim is necessary, but not necessarily necessary, contra axiom (4). Salmon thus concludes that neither S4 nor S5 is the correct logic for metaphysical modality.5 We find the data that Salmon canvasses in the Woody case to be intuitively compelling. But do the data really establish, as Salmon maintains, that axiom (4) should be rejected as a general constraint on metaphysical modality?

1.3 The Woody Data, Specified In fact, the data do not clearly establish this. As we now show, one of the premises of Salmon’s argument is under-specified, and the argument must be reformulated accordingly. This discussion serves two purposes. First and most importantly, it makes explicit that accommodating the data of the Woody case requires that metaphysical possibilities and necessities be relativized to indicative actualities; as we will see, this is a claim with which Salmon arguably agrees. Second, the discussion reveals two objections to Salmon’s argument for the rejection of transitivity; these objections will make room for our preferred treatment of the data, and an associated ‘S4-friendly’ relativized conception of metaphysical modality. (Before continuing, a small caveat. We present our concern with Salmon’s argument in terms of premise 4’s having two ‘readings’; but by this we do not mean to imply that this premise or any of its constituting bits of language are ambiguous (perhaps the premise is not ambiguous, but its assessment is sensitive to certain presuppositions— namely, concerning which world is held fixed as indicatively actual). What we are mainly concerned to do is make explicit, via the comparatively coarse-grained means of a difference in truth-value of the two readings of premise 4, the role that relativization to indicatively actual worlds plays in appropriately accommodating the Woody data. Similar remarks will apply to our reformulation of Salmon’s argument.) To start, note that there are two readings of premise 4. The first reading follows premises 2 and 3 in presupposing (or as we’ll put it, 5 Salmon’s discussion focuses on the normal modal logical systems S4 and S5. However, insofar as Salmon’s target is the transitivity of the accessibility relation, as characterized by axiom (4), his conclusion plausibly extends to simpler modal logics such as K4, which also include (4). Thanks to Phil Kremer for discussion of this point.

Relativized Metaphysical Modality | 195 ‘holding fixed’) that Woody actually originates from matter m. As a first pass, the first reading of premise 4 might be expressed as follows: Holding fixed that Woody actually originates from matter m: if Woody had originated from matter m’, then it would have been possible for Woody to originate from matter m”.

The first pass is not yet sufficiently specified, however, since it fails to express the sense in which Woody’s actual origins in matter m are ‘held fixed’, notwithstanding that in evaluating the antecedent of the conditional, it is supposed that Woody actually originates in some different matter. What is needed is the distinction between a given state of affairs (or whatever) being indicatively vs. its being counterfactually actual. To prefigure our heuristic analogy: such a distinction is operative when we allow that, holding fixed that ‘water is H2O’ is (‘indicatively’) true in our very own actual world, ‘water is H2O’ would remain true were a world where the watery stuff is XYZ to be (‘counterfactually’) actual. The first reading of premise 4 should mark this distinction, as follows: Holding fixed that Woody indicatively actually originates from matter m: if Woody had originated from matter m’, then it would have been possible for Woody to originate from matter m”.

The second reading of premise 4 does not follow premises 2 and 3 in holding fixed that Woody indicatively actually originates from matter m. Rather, on this reading the premise is read as presupposing that which world is held fixed as indicatively actual is one where Woody originates from matter m’. Making this explicit, we might express premise 4 as follows: Holding fixed that Woody indicatively actually originates from matter m’: if Woody had originated from matter m’, then it would have been possible for Woody to originate from matter m”.

These two readings of premise 4 are not equivalent, of course. On the first reading, premise 4 is false. Here, that Woody indicatively actually originates from matter m is held fixed; hence even if the antecedent of the embedded conditional is counterfactually true (as it might be, as per premise 2) the consequent of this conditional will be false, since (as per premise 3) the fact that Woody (‘indicatively actually’) originates from matter m places constraints on the mate-

196 | Adam Murray and Jessica Wilson rial Woody could have (‘counterfactually actually’) originated from instead.6 On the second reading, however, premise 4 is (under the relevant assumptions) true. Here, that Woody indicatively actually originates from matter m is not held fixed; rather, in evaluating the embedded conditional the indicatively (as opposed to merely counterfactually) actual world is taken to be one where Woody originates from matter m’. Whether the conditional is true will then depend on whether Woody’s (counterfactually actually) originating from matter m″ is possible given that Woody (indicatively actually) originates from m’. And as Salmon points out, for properly chosen m’ and m″, the conditional will indeed be true. How does the fact that premise 4 has distinct readings, only one of which makes sense of the Woody data, bear on Salmon’s argument against axiom (4)? This fact indicates, at a minimum, that the truth-values of modal claims are, somehow or other, sensitive to which world is held fixed as indicatively actual. As such, we need to rewrite Salmon’s argument in a way that respects this sensitivity, which we will do by appending subscripts to the modal operators indicating which world is held fixed as indicatively actual when the possibility or necessity at issue is evaluated. (Again, in presenting this reformulation, we do not mean to commit ourselves or Salmon to any particular semantics of the modal operators; we append subscripts to operators here merely to make explicit the need for relativization of modal claims to indicative actualities. Later, we’ll offer two ways of accommodating the needed relativization within a possible worlds semantics for modal logic.) So, for example, premise 2 should be rewritten to indicate that the possibility at issue is evaluated given that w1 is held fixed as indicatively actual: 2’. It is possiblew1 that Woody originates from matter m’. And premise 4, if it is to be true, should be rewritten to reflect that, given the background suppositions noted above, the possibility at issue is evaluated holding w2 fixed as indicatively actual: 6 Note that the claim here is not that premise 4 is false because the consequent of the embedded conditional is actually false; the claim is rather that, holding fixed that Woody indicatively actually originates in matter m, the consequent would be false in a world where the antecedent is (counterfactually) true. In other words: the constraints imposed by Woody’s indicatively actual origin are in force even in contexts (e.g., w2) where Woody counterfactually originates from different matter.

Relativized Metaphysical Modality | 197 4’. If Woody had originated from matter m’, then it would have been possiblew2 for Woody to originate from matter m”. Correspondingly, premise 5 should now reflect that the possibilities at issue are evaluated with respect to different indicatively actual worlds: 5’. It is possiblew1 that it is possiblew2 that Woody originates from matter m”. (2’, 4’) Properly specified, then, Salmon’s argument is as follows: 1’. 2’. 3’. 4’.

Woody originates from matter m. It is possiblew1 that Woody originates from matter m’. It is not possiblew1 that Woody originates from matter m”. If Woody had originated from matter m’, then it would have been possiblew2 for Woody to originate from matter m”. 5’. It is possiblew1 that it is possiblew2 that Woody originates from matter m”. (2’, 4’) 6’. It is not possiblew1 that Woody originates from matter m”, but it is possiblew1 that it is possiblew2 that Woody originates from matter m”. (3’, 5’) 7’. It is necessaryw1 that Woody does not originate from matter m”, but it is not necessaryw1 that it is necessaryw2 that Woody does not originate from matter m”. (6’)

Note that the last claim is no longer a clear counter-instance to axiom (4). As per the standard assumption that metaphysical modality is modality in the broadest (non-syntactically logical) sense, previous discussions of the axiom have not incorporated the need for relativization to an indicatively actual world. How the axiom should be understood in light of the need for relativization—and in particular, whether it should be understood to apply to modal claims involving iterated or (as we’ll put it) ‘in situ’ shifts in which world is held fixed as indicatively actual (of the sort occurring in 5’–7’) remains to be seen. This result constitutes our first objection to Salmon’s argument that the Woody case motivates the rejection of transitive accessibility: if we appropriately attend to which worlds are held fixed as indicatively actual in this case, no clear violation of axiom (4) results. More generally, the need to incorporate facts about which world is held fixed as indicatively actual in order to appropriately express the

198 | Adam Murray and Jessica Wilson data indicates that the standard conception of metaphysical modality is strictly incorrect. Hence though we disagree with Salmon’s diagnosis and treatment of the Woody case, we agree with him that the data here motivates a revision of the standard conception in the direction of relativization. (The need for relativization is an underappreciated insight of Salmon’s discussion, which has been, we speculate, obscured by his rejection of transitivity and endorsement of so-called ‘impossible worlds’.) More specifically, a proper understanding of the data concerning Woody indicates that metaphysical modal reasoning and any associated modal logics must be able, first, to make room for different worlds to be indicatively actual; and second, to keep track of which world is held fixed as indicatively actual. That said—and here we raise our second objection to Salmon’s argument against transitive accessibility—we do not think that the best way to implement the needed relativization is to make sense, one way or another, of metaphysical modal reasoning that involves in situ shifts in which world is held fixed as indicatively actual, of the sort which explicitly occurs in the disambiguated premise 5’ of Salmon’s argument. On the contrary, we are inclined to see something defective in such claims. To again prefigure, compare (what we will later argue is) the formally analogous epistemic interpretation of the two-dimensional (2D) semantic framework, endorsed by, e.g., Jackson and Chalmers (2001). Epistemic two-dimensionalism distinguishes between ‘considering as actual’ and ‘considering as counterfactual’; it makes room for our being able to ‘consider as actual’ either a world where water is H2O, or a world where water is XYZ, and to go on to ‘consider as counterfactual’ other worlds against the assumption that one or other world has been considered as actual. Yet within this framework there is little motivation to accommodate the following sort of claim: Considering as actual a world where water is H2O: considering as actual a world where water is XYZ, then necessarily, water is XYZ.

We similarly do not see any motivation for a revision of metaphysical modal logic or associated semantics on which claims such as Salmon’s disambiguated premise 5’ are accommodated (indirectly, on Salmon’s view, by relaxing constraints on accessibility, or directly, by revising rules of modal logical inference so as to explicitly incorporate, e.g., double indexing). As we see it, such claims illegitimately shift indicatively actual horses in modal mid-stream.

Relativized Metaphysical Modality | 199 Our view is rather as follows: metaphysical modal claims and associated reasoning need to be appropriately sensitive to which world is held fixed as indicatively actual, primarily in order to avoid such illegitimate in situ shifts in which world is held fixed as indicatively actual. From this perspective, Salmon’s argument against transitive accessibility is problematic not just in that no clear counter-instance to axiom (4) follows from its (properly specified) premises, but also in that the argument, once specified, relies on the ill-formed premise 5’. In response to our objections, Salmon might maintain that, though no counter-instance of axiom (4) is explicitly entailed by the specified data, the axiom’s rejection, and associated acceptance of claims involving in situ shifts in which world is held fixed as indicatively actual, are required to accommodate the data concerning Woody. As we’ll argue in S2, such maneuvers are unnecessary. First, though, we present Salmon’s treatment of the data; this will serve to highlight how our relativized conception of metaphysical modality differs from Salmon’s, and to flag certain general concerns with his treatment which our preferred conception avoids.

1.4 Salmon’s Intransitive Relativized Conception Salmon understands possible worlds as maximal abstract ways for goings-on to be, and endorses ‘the standard identification of necessity with truth in every possible world and possibility as truth in at least one possible world’ (1989, 5). He maintains that these commitments are compatible with the data concerning Woody, if one accepts impossible worlds—‘total ways things cannot be’—and allows that a world that is impossible ‘relative’ to one world may be possible relative to another. So, for example, a world where Woody originates in matter m” is such a world. That world is not possible relative to the actual world, but it is possible relative to a world in which Woody originates from matter m’. Relative to the actual world, it is merely possibly possible. Salmon suggests that ‘other impossible worlds may not be even possibly possible, but only possibly possibly possible, and so on; hence the binary relation between (possible or impossible) worlds of relative possibility—the modal relation of accessibility—is not transitive’. (1989, 7). Beyond the relativization of what is possible and necessary to which world ‘obtains’, and the rejection of transitive accessibility,

200 | Adam Murray and Jessica Wilson Salmon notes two related ways in which his treatment departs from the standard understanding of metaphysical modality. First, ‘[i]f worlds include ways things metaphysically cannot be in addition to ways things metaphysically might have been, then the idea that metaphysical necessity corresponds to truth in every world whatsoever is flatly mistaken’ (1989, 15). Second, given that ‘[a] possible world is a total way for things to be that conforms to metaphysical constraints concerning what might have been [. . .] metaphysical modality is definitely not an unrestricted limiting case’ (1989, 12–13). The conception of metaphysical modal space that is in the first instance suggested by Salmon’s treatment of the Woody case is of a single space of (‘maximal’, abstract) worlds, whose status as possible or impossible is relative to whatever world is supposed to obtain,7 and which are connected by an intransitive accessibility relation. In pictorial terms: the standard conception of metaphysical modal space—the conception that Salmon, and we, reject—has the following structure:

w2

w1

w3

Figure 1. The Standard Conception: Unrelativized Transitive Accessibility.

(Here, and in the figures to follow, arrows point towards worlds accessible to the origin world.) And the conception of metaphysical modal space suggested by Salmon’s treatment of the Woody case is as follows: 7 We uniformly interpret Salmon’s talk of metaphysical possibilities and necessities as relative to which world ‘obtains’ (in our terms: is indicatively actual), such that, e.g., w3 is possible relative to w2 when the latter obtains in just the same way that our very own actual world obtains. Though Salmon’s emphasis on impossible worlds might be thought to suggest that he sees a substantive difference between whatever world in fact obtains and other worlds (e.g., w2) that merely hypothetically obtain, Salmon seems to reject such a privileging of our world when disparaging what he calls the ‘ostrich approach to modality, with its consequent misconstrual of “necessarily” as meaning actual necessity and “possibly” as meaning actual possibility”’ (1989, 29). In any case, a restricted understanding would still have the structure to follow.

Relativized Metaphysical Modality | 201

w1

w2

w3

Figure 2. Salmon’s Intransitive Relativized Conception. Relative to w1, w2 is possible, and w3 is not possible; relative to w2, w3 is possible.

(Here, and in the figures to follow, dotted lines around a world indicate that the world is held fixed as indicatively actual. Solid-line ovals around worlds represent the relativization of accessibility due to shifts in which world is held fixed as indicatively actual.8) Again, we emphasize that in our view, Salmon’s conception is on the right track, in recognizing the need for metaphysical modal deliberation to be sensitive to which world is supposed to ‘obtain’ (be indicatively actual). Still, Salmon’s approach to a relativized conception is revisionary, in departing from the usual assumption of transitive accessibility and associated systems of modal logic; and many have found the posit of metaphysically impossible worlds problematic (see Lewis 1986, 7, fn. 3 and 246–8).

2 THE TRANSITIVE RELATIVIZED CONCEPTION (RELATIVIZED METAPHYSICAL MODALITY) Our approach to a relativized conception accommodates the Woody data, compatible with both transitive accessibility and the rejection of impossible worlds. Schematically, our conception is one naturally seen as involving not a single space of mutually accessible worlds (as on the standard conception), nor a single space of intransitively accessible worlds (as on Salmon’s conception), but rather multiple spaces, each containing one indicatively actual world, along with whichever worlds are (one might reasonably 8 Our purpose here is merely to sufficiently distinguish Salmon’s intransitive relativized conception from the standard, transitive conception of metaphysical accessibility (see Figure 1). Plausibly, on Salmon’s intransitive conception, had w2 obtained (in our terminology, been ‘indicatively actual’) then w1 as well as w3 would have been accessible/metaphysically possible from w2. In this sense, Figure 2 as it stands represents the relations of accessibility that obtain between worlds w1, w2, and w3 in a fairly abbreviated form.

202 | Adam Murray and Jessica Wilson qualify: counterfactually) metaphysically possible relative to that indicatively actual world. In fact, there are two ways of implementing our conception, reflecting two ways of locating the relativization at issue. In presenting these versions and their application to the Woody case, we will speak freely (as we have been doing all along) of possible worlds as located in spaces; depending on one’s further commitments, one may or may not understand such talk as metaphorical. Either way, the structural distinctions we aim to identify will be clear enough to get a feel for our proposal(s), and— most importantly—to see that endorsement of either version of the relativized conception is compatible with taking the correct logic of metaphysical modal reasoning to conform to S4 or S5.

2.1 The Overlapping Spaces Interpretation: (Some) Truth Relativized to an Indicatively Actual World On the first implementation, the spaces of worlds associated with different indicatively actual worlds overlap. Here it is assumed that, prior to identification of any world as indicatively actual, there is a single space of (what we will call) ‘basically individuated’ possible worlds, connected, we assume, by a transitive accessibility relation. The principle of basic individuation of worlds might be primitive; or it might proceed by way of, e.g., ‘semantically stable’ (Bealer 2000) or ‘canonical’ (Chalmers 2006) descriptions, or some combinatorial function of basic elements (as per Lewis 1986 or Armstrong 1989). While the individuating principle suffices to distinguish worlds, such individuation, we assume, leaves open the truth values of various claims at a given world. For example, in the pre-relativized space there will be a world, w3, containing a table-shaped hunk of matter m”. In w3, is it true or false that Woody originates from m”? In the pre-relativized space, there is no answer to this question, nor to many other questions, whose answers depend on which world is (held fixed as: henceforth we take this qualification as understood) indicatively actual.9 Pictorially, the pre-relativized space of basically individuated worlds is structured according to the standard conception of metaphysical modal space discussed above: 9 Hence, on this implementation, worlds are not alethically ‘maximal’ prior to relativization; accordingly, in the pre-relativized space, v(p) (where p is the proposition that Woody originates in matter m”) is indeterminate.

Relativized Metaphysical Modality | 203

w2

w3

w1

Figure 3. A pre-relativized space of basically individuated worlds. w1 contains a table-shaped hunk of matter m, w2 contains a table-shaped hunk of matter m’, w3 contains a table-shaped hunk of matter m”

For each world in the pre-relativized space, there is an associated relativized space, containing the same worlds as in the pre-relativized space, but with the truth values of at least some previously undetermined truths—namely, those dependent on which world is indicatively actual—now determined. In other words, on the present interpretation (some) truth is relative to which world is indicatively actual. Implementing this interpretation requires that the valuation function v assigning truth values to propositions be sensitive to which world is indicatively actual, for both non-modal and modal clauses. We might do this, for a language, by incorporating such a reference in an additional argument place in these clauses (as we do below: see Figures 3.1 and 3.2), or we might keep the usual semantic clauses, and distinguish valuation functions for each indicatively actual world. Now, when the worlds in the pre-relativized space are relativized to w1, then the proposition that Woody originates from m” is thereby rendered false in w3:

w2

w1

w3

Figure 3.1 The space of basically individuated worlds, relativized to w1 (w1 is indicatively actual). p = the proposition that woody originates from matter m” v(p, w3, w1) = F

204 | Adam Murray and Jessica Wilson When the worlds in the pre-relativized space are relativized to w2, however, the proposition that Woody originates from m” is thereby rendered true in w3:

w2

w1

w3

Figure 3.2. The space of basically individuated worlds, relativized to w2 (w2 is indicatively actual). p = the proposition that woody originates from matter m” v(p, w3, w2) = T

Crucially, the need for such relativization is no barrier to the standard assumption that modal reasoning proceeds as per S4 or S5; we may continue to assume, as per usual, that the worlds in any postrelativized space are mutually (totally) accessible. In particular, as we’ll now show, on our understanding of the data of the Woody case, no violation of axiom (4) is forthcoming. Again, we start with a pre-relativized space of possible worlds. Let’s start by relativizing to w1—that is, holding w1 fixed as indicatively actual; we want to consider the relevant semantic clauses concerning the proposition p, that Woody originates from matter m”. Given that Woody actually originates from matter m in w1, p will be false in w3: v(p, w3, w1) = F (Read: the semantic value of p in w3 relative to w1’s being indicatively actual is F.) More generally, given that w1 is indicatively actual, v will assign F to p in every world in the relativized space. Hence v will assign F to p in every world accessible to w1; hence v will assign F to the proposition that p is possible in w1; and similarly for w2: v(◊p, w1, w1) = F v(◊p, w2, w1) = F For the same reason, given that w1 is indicatively actual, v will assign F to the proposition that p is possible in every world accessible

Relativized Metaphysical Modality | 205 to w1. Consequently, v will assign F to the proposition that p is possibly possible in w1: v(◊◊p, w1, w1) = F When w1 is indicatively actual, no violation of axiom (4) is in the offing. And similarly if we relativize to w2 (hold w2 fixed as indicatively actual).10 We can now be more explicit about how our characterization of what it is for a proposition p to be possible at a world contrasts with Salmon’s. Salmon and we agree that propositions such as ◊p, for p as above, are (to speak in purposely rough terms) only relatively true. Salmon takes this to mean that ‘◊p’ is true in some worlds but not others. This suggests, on the standard semantics, that some worlds can access a p world and others cannot; it follows that not all worlds can access the same worlds, contra S5. The rejection of transitive accessibility is thus built into Salmon’s understanding of the rough thought that some modal claims are only relatively true. We, on the other hand, do not take the rough thought to mean that ‘◊p’ is true in some worlds and not in others. Rather, we take it to mean that, though ‘◊p’ is either true in all worlds or true in none, which truth value is assigned to ‘◊p’ in all worlds is relative to which world is indicatively actual. (Similarly for the implementation presented in S2.2.) Relative to some indicatively actual worlds, ‘◊p’ is true in all worlds; relative to other indicatively actual worlds, ‘◊p’ is false in all worlds. Either way, transitive accessibility is maintained. Two questions remain. The first concerns what justifies thinking of worlds in differently relativized spaces as ‘the same’, such that, e.g., we may speak of w3 as existing both in a space relativized 10

In this case it will rather be true at w3 that Woody originates from matter m”: v(p, w3, w2) = T

Furthermore, given that w2 is indicatively actual (together with the fact that, in w2, Woody originates in matter m'), v will assign T to the proposition that p is possible in both w1 and w2: v(◊p, w1, w2) = T v(◊p, w2, w2) = T Moreover, since w2 is accessible to w1, v will assign T to the proposition that p is possibly possible in w1: v(◊◊p, w1, w2) = T Again, no violation of transitivity results.

206 | Adam Murray and Jessica Wilson to w1 and in a space relativized to w2. A fairly straightforward answer is that worlds in different relativized spaces may be considered to be ‘the same’ notwithstanding that they differ on the truth values of certain claims (e.g., that Woody originates in matter m”) in virtue of being basically the same—that is, strictly the same at the level of basic individuation, understood in one or other of the primitive or non-primitive ways mentioned earlier. The second question concerns what justifies thinking of individuals in differently relativized spaces as in some sense ‘the same’, so that, e.g., we may speak of Woody as existing both in a space relativized to w1 and in a space relativized to w2. The answer here is less straightforward, and will depend on further details concerning the metaphysics and individuation of material objects. One way to go here, putting things in the formal mode, would be to suppose that expressions denoting material objects (e.g., ‘Woody’) are relevantly like expressions for natural kinds (e.g., ‘water’), in having something like a descriptive or reference-fixing sense, whose association with a referent is dependent upon which world is indicatively actual, and once fixed, is metaphysically necessary (allowing for some counterfactual flexibility). Another way to go, putting things in the material mode, would be to suppose that material objects such as Woody have a sort of ‘relativized essence’, such that, as a primitive or nonprimitive matter, Woody exists and has certain metaphysically necessary features at some relativized worlds, but either fails to exist or has different metaphysically necessary features at others.11 2.2 The Non-overlapping Subspaces Interpretation: Domain Relativized to an Indicatively Actual World On the second implementation of our relativized conception, the spaces of worlds associated with different indicatively actual worlds do not overlap. Rather, the single space of worlds is partitioned into non-overlapping subspaces of finely individuated worlds, each of 11 What if some claims concerning Woody—e.g., that Woody is a table—are necessary relative to each indicatively actual world? We may define, if we like, a notion of ‘absolute’ metaphysical necessity. But given the desirability of avoiding shifts in indicatively actual worlds, we should see the absolute notion as grounded in the relativized notion—as tracking a uniform pattern of variation in what is relatively metaphysically necessary—as opposed to taking ‘absolute’ metaphysical necessity to be either prior to or distinct from relativized metaphysical necessity.

Relativized Metaphysical Modality | 207 which is, unlike the basically individuated worlds of the overlapping subspaces implementation, alethically ‘complete’; and each subspace corresponds to a space of worlds relativized to a single indicatively actual world (also contained in the subspace). Again, the principle of individuation here may be primitive or otherwise. Within a subspace, worlds are (we may assume) mutually accessible; but worlds are not accessible across subspaces.12 On this implementation, it is domains that are relativized to indicatively actual worlds. Here, the relativization happens at the level of the frame—that is, in the selection of one subspace of worlds from among the many subspaces. On the standard conception, a frame contains a single set of worlds W. On our conception, the frame contains not a set, but a partition, of worlds:

w2

w1

w2⬘

w3 S1

w2⬘⬘

w3 ⬘

w1⬘ S2

w3 ⬘⬘

w1⬘⬘ S3

Figure 4. Non-Overlapping Subspaces. In subspace S1, w1 is indicatively actual; in subspace S2, w2’ is indicatively actual; in subspaces S3, w3” is indicatively actual. Each ‘w1-type’ world contains a tableshaped hunk of matter m; each ‘w2-type’ world contains a tableshaped hunk of matter m’; each ‘w3-type’ world contains a table-shaped hunk of matter m”. p = the proposition that woody originates from matter m” v(p, w3, w1) = F v(p, w3’, w2’) = T 12 Even though worlds in different subspaces are not identical on this view, worlds in different partitions may be taken to be ‘basic’ or ‘canonical’ counterparts of each other, in that (lifting some of the structure of the overlapping spaces interpretation) worlds in different spaces may be basically alike; such similarity may serve as the basis for loose (as opposed to strict) identification of worlds across subspaces. So, for example, distinct subspaces might each contain worlds that are basically, canonically, or qualitatively similar, in containing, e.g., a table-shaped hunk of matter m”; such worlds, we might say, are of type w3. In a world of type w3, is it true or false that Woody originates from m”? That depends. In a subspace where the indicatively actual world is (of type) w1, this is false; but in a subspace where the indicatively actual world is (of type) w2, this is true.

208 | Adam Murray and Jessica Wilson Upon relativization, one of the subspaces in the partition is selected as a basis for subsequent modal and semantic deliberation. So, for example, one might select the subspace in which w1 is indicatively actual; or one might select the subspace in which w2 is indicatively actual. Having so chosen, all the usual logical axioms and semantic clauses can remain exactly as per the standard conception. In particular, the valuations of iterated and non-iterated modal clauses involving the proposition p in a given post-relativized space will, as in the ‘overlapping spaces’ interpretation, conform to the assumption of transitive accessibility.13 As such, the need for relativization is clearly no barrier to the standard assumption that modal reasoning proceeds as per S4 or S5; we may continue to assume that the worlds in any given subspace are mutually accessible. Indeed, given that the accessibility relation on S5 is an equivalence relation, it is well suited to characterize, not just the mutual accessibility of worlds in a single space, but also the relativized mutual accessibility of worlds partitioned by such an equivalence relation into subspaces. Summing up: a conception of metaphysical possibilities and necessities as relative to indicative actualities can accommodate the data at issue in the Woody case without incurring the revisionary costs associated with Salmon’s treatment of the data.

13 Indeed, given the satisfaction of transitive accessibility within a particular subspace (either overlapping or non-overlapping), the evaluation of claims of necessity at a world w in a subspace in which some distinct world w' is indicatively actual is straightforward (thanks to Phil Kremer for pressing us on this point). Given transitivity, if some proposition A is necessary at a world w, A is necessarily necessary at w, and hence necessary at every world in the subspace accessible from w; thus, in particular we have:

v(□A,w,w') = T iff v(□A, w',w') = T (A is necessary at w, given that w' is indicatively actual, if and only if A is necessary at the indicatively actual world w' of the subspace). Necessary truth at the indicatively actual world of a subspace can then be given its familiar analysis in terms of truth at all accessible worlds: v(□A, w',w') = T iff ∀w[wRw' → v(A, w) = T] Combining these results yields the truth-clause for necessary propositions evaluated at worlds accessible from the indicatively actual world of a particular subspace: v(□A,w,w') = T iff ∀w[wRw' → v(A, w) = T]

Relativized Metaphysical Modality | 209 2.3 Heuristically Situating the Transitive Relativized Conception: A Properly Metaphysical Interpretation of the 2-D Semantic Framework Our proposal is not as unusual as it might appear, for it may be naturally situated in a thoroughly metaphysical interpretation of the sort of 2-D semantic framework developed by Kaplan (1979, 1989), Stalnaker (1978), and others. Although our interest here is metaphysics, not meaning, we take the formal analogy between our relativized conception of metaphysical modality and (a metaphysical interpretation of ) the 2-D framework to be heuristically useful. The so-called ‘epistemic’ interpretation of the 2D semantic framework (see, e.g., Chalmers and Jackson 2001, Chalmers 2006), commonly applied to natural kind terms and associated Kripkean necessities (e.g., ‘water’, ‘water is H2O’), appeals to a distinction between possible worlds, as either considered as actual or considered as counterfactual. One starts by constructing a 2-D matrix listing worlds potentially considered as actual in the far-left column, and worlds potentially considered as counterfactual along the top row; conventionally, the first world in each list is our very own actual world, and the expression whose meaning is at issue appears in the top left-hand corner. The basic suggestion, then, is that (at least some especially salient) aspects of meaning may be represented by intensions, understood as functions from worlds to extensions. More specifically, the suggestion is that aspects of meaning may be represented by functions taking as arguments two worlds (hence ‘2-D’)—one considered as actual (drawn from the leftmost column), one considered as counterfactual (drawn from the topmost row); different aspects of meaning are then associated with different 2-D intensions. On the epistemic interpretation, one salient aspect of meaning corresponds to metaphysical reference. This aspect is associated with the function which takes our very own world as its first argument, and a world considered as counterfactual as its second argument; this function is sometimes called the ‘secondary’ or ‘horizontal’ intension (we’ll use the latter terminology, as visually more evocative), and for relevant expressions, is understood to encode (or as we’ll loosely say, ‘represent’) what is metaphysically necessary. Consider a portion of the 2-D matrix associated with ‘water is H20’, where an H2O-world is one where the predominant liquid falling from the sky, found in lakes, etc. (the ‘watery stuff’, for short) is

210 | Adam Murray and Jessica Wilson H2O, and an XYZ-world is one where the watery stuff is XYZ; and where it is furthermore assumed that our very own actual world is an H2O-world: ‘water is H2O’ H2O-world

H2O-world T

XYZ-world T

The horizontal intension for the expression ‘water is H2O’ here reflects, among other relevant facts, that the horizontal intension of ‘water’ is sensitive to the way the world actually turns out to be, so that, given that the actual world is an H2O-world, ‘water is H2O’ will be true in every world considered as counterfactual. As such, the horizontal intension of ‘water is H2O’ returns ‘T’ for every world considered as counterfactual (if an XYZ-world were to be counterfactually actual, then ‘water is H2O’ would have been true, etc.), consonant with Kripke’s claim that ‘water is H2O’ is metaphysically necessary. Other modally implicated intensions may be defined within this framework. Chalmers, Jackson and others maintain that terms such as ‘water’ have an aspect of meaning corresponding to epistemic sense, an aspect of meaning supposed to be a priori accessible, in being independent of details about which world is considered as actual, and which is posited as explaining, e.g., intuitions that it might have turned out that water was not H2O. The associated function is sometimes called the ‘primary’ or ‘diagonal’ intension (again, we’ll use the latter terminology), and takes as arguments pairs of identical worlds . Consider a portion of the 2-D matrix associated with ‘water is the watery stuff’: ‘water is the watery stuff’ H2O-world XYZ-world

H2O-world T

XYZ-world T

We want to call attention to a third, underappreciated class of intensions associated with the 2-D framework, that is required if the matrix is to be appropriately ‘filled in’. Note that the diagonal intension, but not the horizontal intension, takes as input worlds other

Relativized Metaphysical Modality | 211 than our very own actual world ‘considered as actual’. We may also define generalized horizontal intensions, where a generalized horizontal intension is a function which takes a world considered as actual as its first argument, and a world considered as counterfactual as its second argument; we call such secondary intensions ‘generalized’ in that the world considered as actual need not be our very own actual world. So, for example, consider the generalized horizontal intension associated with ‘water is XYZ’, when an XYZ-world is considered as actual: ‘water is XYZ’ H2O-world XYZ-world

H2O-world

XYZ-world

T

T

On the usual epistemic interpretation, such a generalized horizontal intension is taken to represent a merely epistemic necessity: following the usual interpretation of Kripke’s results, only the non-generalized horizontal intension, taking as its first argument our very own actual world, is capable of representing what is genuinely metaphysically necessary (or possible). But as we see it, there is good reason to interpret the necessities represented by generalized horizontal intensions as genuine. After all, the actual world might have turned out to be an XYZ-world, in which case the horizontal intension associated with ‘water is XYZ’ would have represented a genuine metaphysical necessity. Rather than obscure this fact by treating the represented necessity as epistemic, why not treat all horizontal intensions on a par as representing genuine relativized metaphysical necessities—that is, as representing (for relevant expressions) what is metaphysically necessary relative to a given indicatively actual world? Under a properly metaphysical interpretation of the generalized horizontal intensions, the 2-D framework is well suited for such representation; and more generally is structurally analogous to the relativized conception, in encoding what is counterfactually the case, relative to each indicatively actual world. Let’s make the structural analogy explicit. To accommodate the data concerning Woody, we represent two ways in which the counterfactual possibilities for Woody might depend on Woody’s actual

212 | Adam Murray and Jessica Wilson origin, one of which holds fixed that Woody indicatively actually originates from matter m, and the other of which holds fixed that Woody indicatively actually originates from matter m’. We do this by letting entries in the leftmost column represent which world is held fixed as indicatively actual; entries along each row then represent what is the case, as regards Woody’s origin, in worlds that are counterfactual relative to the associated indicatively actual world (and where an ‘m-world’ is a world where the salient candidate for being Woody originates in matter m, etc.). ‘Woody does not originate from m”’ m-world m'-world

m-world

m'-world

m”-world

T T

T T

T F

As desired, the structure allows us to represent the dependence of what is metaphysically possible and necessary concerning Woody on which world is indicatively actual. At this point we want to revisit our earlier (S1.3) observation that attention to the 2-D framework supports thinking that in situ shifts in which world is held fixed as indicatively actual, of the sort associated with Salmon’s disambiguated 5’, are in some sense defective or ill-formed. As above, one of the salient intensions associated with the 2-D framework is the diagonal intension, which for each world w returns the extension of the relevant expression at w when w is considered as actual, and which on the epistemic interpretation is taken to represent epistemic sense—an aspect of meaning that is supposed to be a priori accessible, in being independent of details about which world is considered as actual, and which allows for representation of certain modal truths, such as ‘necessarily, water is the watery stuff’. As an extension of our ‘metaphysical’ interpretation of generalized horizontal intensions, constant diagonal intensions might be taken to represent genuine facts—namely, those independent of which world is indicatively actual. Such independence, in turn, might be understood in terms of a notion of ‘absolute’ metaphysical modality, tracking patterns in what is the case, either nonmodally or relatively modally, when different worlds are held fixed as indicatively actual. (See note 11.)

Relativized Metaphysical Modality | 213 Does the fact that (as attention to the diagonal and generalized horizontal intensions suggests) we can make sense of claims involving shifts in indicatively actual worlds pose a problem for our thinking that some such shifts are illegitimate, from the perspective of metaphysical modal reasoning? No. Certain claims and associated reasoning involving shifts in which world is (considered as) indicatively actual are, on our view, perfectly legitimate—namely, those which are appropriately seen as ‘meta-modal’, in tracking patterns of what is the case, non-modally or relatively modally, when different worlds are held fixed as indicatively actual. We see nothing defective in claims like the following, tracking patterns in what is non-modally the case across different worlds considered (held fixed) as indicatively actual: Considering as (indicatively) actual a world where water is H2O: water is the watery stuff; and considering as (indicatively) actual a world where water is XYZ: water is the watery stuff.

Nor do we see anything defective with claims like the following, tracking patterns in what is modally the case across different worlds considered (held fixed) as indicatively actual: Considering as (indicatively) actual a world where water is H2O: it is necessary that water is H2O; and considering as (indicatively) actual a world where water is XYZ: it is necessary that water is XYZ.

Again, the sort of claims that we maintain are in some sense defective, from the point of view of metaphysical modal reasoning, are those involving iterated or ‘in situ’ shifts in which world is indicatively actual, of the sort characteristic of Salmon’s disambiguated premise 5’, or of the following sort of claim: Considering as (indicatively) actual a world where water is H2O: considering as (indicatively) actual a world where water is XYZ, then it is necessary that water is XYZ.

Such in situ shifts are not motivated by diagonal intensions, or the associated generalized horizontal intensions, whether these are understood to involve merely epistemic or properly genuine necessities. Relatedly, we have no deep complaint against claims, e.g., to the effect that some world w* different from our very own actual world

214 | Adam Murray and Jessica Wilson ‘could have been’ indicatively actual—so long as such claims are interpreted as meta-modal claims, whose evaluation requires looking at the space of worlds or subspaces ‘from the outside’, as it were.

3 FINE’S ‘SCHMASS’ CASE 3.1 Fine’s Rejection of Modal Monism In ‘The Varieties of Necessity’ (2002), Fine notes that there appear to be different ways in which a claim might be said to be necessary or possible, reflecting, e.g., logical, conceptual, mathematical, metaphysical, nomological, or normative necessity or possibility; he then considers whether any of these can be defined in terms of the others, and if so, which are most basic. Fine characterizes metaphysical necessities as necessities which hold in virtue of the natures and identities of the entities at issue, and takes it to be plausible that logical, conceptual, and mathematical necessity may be defined in terms of metaphysical necessity, with the former varieties of necessity being defined as restrictions on the latter. So, for example, the logically necessary claims are those that are, first, metaphysically necessary and second, true in virtue of the nature of logic. Fine does not think, however, that metaphysical necessity is the only basic variety, but rather argues that nomological necessity is also basic, in not being appropriately seen as a restricted form of metaphysical necessity. The focus of Fine’s discussion is the view, typically endorsed by those (Shoemaker 1980, Ellis 2001, and Bird 2007) taking powers or laws to be essential to properties, according to which nomological necessities are metaphysical necessities (as Shoemaker puts it, are ‘necessary in the strongest sense’), based in the nature or identity of laws of nature or natural kinds. While Fine is inclined to agree that some nomological necessities (e.g., that electrons are negatively charged) are metaphysically necessary, certain other nomological necessities, he claims, are such that their denials are metaphysically possible. Suppose, for example, that it is a law of nature that massy entities attract according to an inverse square law. As Fine notes, the neces-

Relativized Metaphysical Modality | 215 sitarian may plausibly maintain that it is metaphysically necessary that massy entities so attract, reflecting the nature or identity of the property of being massy. Still, Fine continues, there is a nomological necessity in the vicinity that is not metaphysically necessary, to which necessitarians appear to be committed. He reasons as follows. Among the necessitarian’s burdens is to explain away intuitions that massy entities might have entered into different laws—say, an inverse cube rather than an inverse square law. One might maintain that the intuition is merely epistemic—the content is not really ‘imaginable’—and there is no genuine possibility corresponding to the intuition of contingency at issue; but given the important role intuitions play in supporting modal claims there is a case to be made that this line is unprincipled. Hence it is that Kripke prefers to treat intuitions of the contingency of certain identities as tracking possibilities that are genuine but misdescribed. Necessitarians about laws also typically implement such a redescription strategy, as in Shoemaker’s treatment of intuitions that strychnine might not be fatal to humans: Let the law be that strychnine in a certain dosage is fatal to human beings. We can grant it to be imaginable that ingesting vast amounts of what passes certain tests for being strychnine should fail to be fatal to what passes certain tests for being a human being, but deny that this amounts to imagining a human being surviving the ingestion of that much strychnine. (1998, 62)

Applying the redescription strategy to the case at hand would allow the necessitarian to maintain that massy entities necessarily attract according to an inverse square law; but on the other hand such an implementation is puzzling, in seeming to undermine the necessitarian’s core claim that nomological necessities are metaphysically necessary. As Fine notes, the strategy requires commitment to there being some property—call it ‘schmass’—which enters into the redescription of the purported counterexample to the nomological necessity of the inverse square law. Given that schmass enters into the redescribed scenario in this way, however, it follows that a world containing schmass is metaphysically possible. Furthermore, Fine surmises, the proposition that there is no schmass is nomologically necessary, given that mass exists in the actual world and the existence of schmass is nomologically incompatible with it. In that case, there appear to be some nomological necessities—e.g., ‘There

216 | Adam Murray and Jessica Wilson is no schmass’—that are not metaphysically necessary. Hence, Fine continues, nomological necessity cannot be seen as a species or restricted form of metaphysical necessity, contra the usual necessitarian line. To be sure, the necessitarian has certain options for response: they may maintain (as Shoemaker does) that intuitions of the metaphysical possibility of schmass are mistaken; or they may deny that the non-existence of schmass is nomologically necessary, on grounds that the incompatibility of mass and schmass is due, e.g., to nomologically contingent initial conditions. Such responses do not seem fully principled, however. Given that many intuitions of contingency admit of genuine redescription, why not the one concerning schmass? And given that many claims are nomologically necessary as a matter of nature, why not the one concerning the non-existence of schmass? Indeed, attention to redescriptive strategizing isn’t necessary to see that it is problematic to suppose that nomological necessity is a restricted form of metaphysical necessity, when the latter is characterized, as per usual, as involving a single space of mutually accessible possible worlds. After all, necessitarians are typically not modally nomocentric; as Shoemaker says, ‘Nothing I have said precludes the possibility of there being worlds in which the causal laws are different from those that prevail in this world.’ (1980, 248). Such worlds must involve completely alien properties, but no matter—such alien worlds can serve as witness to the general claim that some nomological necessities are not necessary tout court. But how are such alien possibilities not precluded, one wonders, if ‘nomological necessity is necessity in the strongest sense’? However one interprets the data concerning ‘schmass’, it remains to make sense of why necessitarians like Shoemaker, on the one hand, subsume nomological under metaphysical necessity; yet on the other, allow that some nomological necessities are not metaphysically necessary. Fine interprets the data concerning ‘schmass’ along the following lines. Insofar as some nomological necessities (‘There is no schmass’) are not metaphysically necessary, it follows that nomological necessity is not a restricted form of, and more generally cannot be defined in terms of, metaphysical necessity. Meanwhile (though this step is implicit in his discussion) other forms of neces-

Relativized Metaphysical Modality | 217 sity (e.g., conceptual necessity) to which nomological necessity might be reduced are plausibly taken to be restricted forms of metaphysical necessity, so that nomological necessity cannot be defined in terms of these other forms of necessity either. Putting the previous results together, Fine concludes that nomological necessity is a basic form of necessity—as basic as metaphysical necessity. We have three concerns with Fine’s proposal. First, the supposition that nomological modality is fundamentally distinct from metaphysical modality fails to sync with the fact that, as Fine grants, it is natural to see many nomological necessities (e.g., that massy entities attract as per an inverse square law) as grounded in the natures and identities of the entities at issue, and hence as metaphysically necessary.14 Second, if nomological necessities aren’t grounded in the natures or identities of the entities at issue, then what are such necessities grounded in? In virtue of what are they true? As it stands, Fine’s proposal to take nomological modality as basic is unilluminating. Third, Fine’s proposal fails to illuminate how it could be—as necessitarians typically allow— both that nomological necessities are metaphysically necessary and that some nomological impossibilities are metaphysically possible. To be sure, as above, necessitarians haven’t explained how this could be, either. But a more satisfying treatment of the data concerning ‘schmass’ would show how, when properly understood, the necessitarian’s seemingly contradictory claims might jointly make sense.

3.2 An Alternative Treatment of the ‘Schmass’ Case: Relativize Necessities to Indicative Actualities Taking metaphysical necessities to be relative to indicative actualities makes sense of the data concerning schmass and provides the

14 Moreover, qua natural property schmass appears to be on a par with mass: it too is a property that, in appropriate circumstances, lawfully influences the motion of entities having the property. Hence considerations rendering it natural to think that necessities involving mass are grounded in its nature and identity would seem equally to motivate thinking that necessities involving schmass would be grounded in its nature and identity.

218 | Adam Murray and Jessica Wilson basis for a consistent necessitarianism, while avoiding concerns associated with taking nomological necessity to be a basic form of necessity. (Either implementation of the relativized conception will do for purposes of treating the ‘schmass’ case.) According to the relativized conception, what is nomologically possible or necessary is, like what is metaphysically possible or necessary, relative to which world is indicatively actual. Since different worlds may be indicatively actual, the necessitarian can avoid being nomocentric. The necessitarian may rather suppose that relative to the world which is in fact actual (that is, our very own world), it is nomologically necessary that there is no schmass; but allow that relative to another indicatively actual world, it might rather be nomologically possible that there is schmass. If we are operating with the domain-relative version of the relativized conception, for example, then on the necessitarian’s view, there may be multiple subspaces of possible worlds, associated with different laws of nature. Moreover, the relativized conception can accommodate the core necessitarian claim that the laws are metaphysically necessary: here it will be supposed that the laws of nature operative at each indicatively actual world impose constraints on the laws at all the other worlds in the associated subspace—namely, that these laws be the same as (or relevantly similar to) the laws operative at the indicatively actual world. Relativized metaphysical modality thus has the resources to reconcile the basic necessitarian claim that what is nomologically necessary is metaphysically necessary with Fine’s observation (with which Shoemaker agrees) that it is metaphysically possible that there be worlds governed by entirely different laws: the first claim may be accommodated by supposing that, relative to a given indicatively actual world, every nomological necessity is metaphysically necessary; while the second claim (like the data concerning schmass) may be accommodated by supposing that indicatively actual worlds may differ with respect to what is metaphysically, hence nomologically, necessary. Here again the real culprit giving rise to the seemingly problematic nature of the data is the insensitivity of the standard conception of metaphysical modality to the need for relativization to indicatively actual worlds. Insofar as (a live interpretation of ) the

Relativized Metaphysical Modality | 219 data concerning schmass indicates that some nomological impossibilities are metaphysically possible, there is no way, on the standard conception, to reconcile the data concerning schmass with the necessitarian view that nomological necessity is a species of metaphysical necessity. However, on the supposition that metaphysical modalities are relative to indicative actualities, this relativization may be interpreted in necessitarian-friendly fashion as indicating that relative to an indicatively actual world, every nomologically necessary claim is metaphysically necessary. As with the Woody case, the key moral of the schmass case, in the first instance, is that appropriately accommodating the data requires that metaphysical modal space have a relativized structure. More generally, to return to Fine’s deeper concern, this structure illustrates how nomological necessity might be, in an appropriately relativized sense, a restricted form of metaphysical necessity. Every nomological necessity is a metaphysical necessity, relative to some indicatively actual world. As such, on the relativized conception nomological necessity need not be seen as a basic form of necessity, but rather may be seen, in a fashion desirably unified with the other non-metaphysical forms of necessity, as ultimately grounded in the natures or identities of the entities at issue in nomological claims.

4 THE BROAD NEUTRALITY OF RELATIVIZED METAPHYSICAL MODALITY In closing, we want to briefly flag the broad neutrality of relativized metaphysical modality with respect to the actualist/possibilist and transworld identity/counterpart theory distinctions and associated debates. We can’t do full justice to the options here, but will try to illustrate the flexibility of the relativized conception, and note a couple of choice points, by attention to how the conception might accommodate certain standard positions in these debates.

220 | Adam Murray and Jessica Wilson 4.1 The Actualist/Possibilist Debate Actualists subscribe to the thesis that everything that exists is actual. Possibilists disagree. According to the possibilist, in addition to the actual world and actual individuals, there exist other, merely possible worlds and individuals. Relativized metaphysical modality is broadly neutral with respect to this debate: actualists and possibilists alike can in principle help themselves to either the overlapping spaces or non-overlapping subspaces implementations of the view. We say ‘in principle’, though, since depending on how a given version of actualism or possibilism is spelled out, one or other implementation of a relativized conception might be thought a better fit. On a standard actualist treatment, possible worlds are identified with some sort of actually existing abstract entity—a complex property (Stalnaker 1976), a complex state of affairs (Plantinga 1976), or a set of propositions (Adams 1974); the actual world is distinguished from merely possible worlds as being the world that is instantiated, obtains, or is such that the constituent propositions are true, respectively. Can abstractionist actualists endorse worlds of the sort entering into either implementation of the relativized conception? First, note that abstractionist actualists typically assume that possible worlds are ‘maximal’, which assumption might be thought to fit better with the non-overlapping subspace implementation, on which worlds, both pre- and post-relativization, are maximally characterized; on the overlapping spaces implementation, worlds are incomplete prior to relativization, hence (in abstractionist terms) represent only the ‘canonical’ or basic truths (e.g., as a non-maximal set of propositions or complex property). Still, on either implementation worlds post-relativization will be maximal; and since there is no in-principle problem with abstract entities’ being non-maximal, it seems the abstractionist actualist can go either way. The question remains: does the actualist supposition that everything that exists is actual make good sense in a context where different worlds can be indicatively actual? Again, we see no conflict here. On one reading, the concern is that the relativized conception can’t make sense of the intuition, sometimes seen as supporting the actualist view, that the actual world is somehow ‘special’ as compared to other merely possible worlds. As above, for the abstrac-

Relativized Metaphysical Modality | 221 tionist actualist, the special nature of the actual world is reflected in one of the worlds being instantiated or obtaining; as such, that some other world might instead be indicatively actual is no more problematic than that some other properties than those that are actually instantiated might instead be instantiated. In any case, one needn’t insist that making sense, e.g., of the Woody data requires that w2 ‘really’ (somehow or other) be instantiated; if no worlds besides our very own actual world can be so lucky, then one may rather understand the relativized conception as tracking a certain complexity in our hypothetical deliberations (as involving consideration of not just counterfactually actual, but also hypothetically indicatively actual, goings-on). On another reading, the concern is that relativization to a world different from our very own actual world would introduce a non-actualist domain. But the relativized conception, while making room for worlds relevant to modal deliberation to involve non-actual individuals, does not require any such thing. Here the action is in the further details of what worlds the modal theorist thinks exist; given that the actualist constructs merely possible worlds from actualia, as per the abstractionist and other standard ‘domain-inclusion’ versions of actualism, then the relativization to such a world as indicatively actual will not introduce a non-actualist domain. Indeed, it is worth pointing out that either implementation of relativized metaphysical modality is broadly consistent with versions of actualism according to which possible worlds and individuals do not exist at all, but merely could exist, as on the ‘non-domain-inclusion’ actualism recently developed in Bennett (2005). For example, holding fixed our own world as indicatively actual, the non-domain-inclusion actualist may consistently hold both that everything that exists is actual and that each of the other possible worlds in the post-relativized space does not exist, but merely could exist, as a matter of fact not grounded in any existing entity. The relativized conception can also accommodate standard accounts of possibilism. Broadly conceived, possibilism allows, contra the actualist view, that possible worlds and their occupants may not actually exist. So broadly characterized, possibilism is compatible with either implementation of a relativized conception; indeed, one standard way to make out the view is as extending the

222 | Adam Murray and Jessica Wilson sort of abstractionist actualist account to allow that some abstracta corresponding to possible worlds may advert to alien individuals or properties (see Menzel 2008). Here again the usual supposition of the maximality of worlds may be accommodated, post-relativization, on either implementation; and either implementation may accommodate the status of worlds as constructed from possibilistfriendly resources. On a more specific, and more notorious, approach to possibilism, this view is combined with the thesis that possible worlds and their occupants are in some sense ‘concrete’ (see Lewis 1986, Bricker 2008). Supposing, as in Lewis (1986) and McDaniel (2004), that concrete worlds are determinate with respect to all matters of particular fact, the non-abstractionist possibilist will find the overlapping subspaces implementation unappealing, in requiring a pre-relativized space of worlds individuated at the level of canonical or semantically stable descriptions which leave out, as above, many (perhaps most) truths of matters of particular fact. Possibilists who take on this additional (but by no means mandatory) metaphysical constraint on the nature of possible worlds will then presumably find the non-overlapping spaces implementation more amenable.

4.2 The Transworld Identity/Counterpart Debate A similarly broad neutrality applies to the issues of transworld identification and representation de re. Does a given individual ever literally exist at more than one possible world, in the sense defended in Kripke (1972), Plantinga (1973), and van Inwagen (1985)? Or do worlds represent that something is possible or necessary, for an individual i, in virtue of containing a numerically distinct counterpart of i which resembles i in certain (typically contextually determined) respects? Relativized metaphysical modality does not force a choice here: each implementation of this view is consistent with either literal transworld identity across worlds or its denial in favor of some counterpart-theoretic means of de re representation. Paradigmatic of accounts that reject transworld identity is Lewis’s treatment according to which representation of an individual in modal claims involves not (necessarily) that individual itself, but rather the individual’s counterparts at various possible worlds,

Relativized Metaphysical Modality | 223 where the notion of a counterpart is defined in terms of overall (see Lewis 1968) or context-dependent (see Lewis 1971) similarity. So, for example, given that Woody actually originates from matter m in world w1, what it would be for a possible world w2 to represent that Woody originates from some different matter m’ would be for Woody to have a counterpart in w2 that originates from m’. In the same vein are accounts on which representation de re is based in sameness of maximally specific qualitative roles (see McMichael 1983). Here, for distinct worlds w1 and w2 to represent de re facts concerning Woody is for a certain qualitative role to be exemplified at both w1 and w2 (exemplification of the role will presumably make appropriate room for the flexibility of Woody’s origins). This proposal, like Lewis’s, is typically offered against a background where individuals are strictly speaking world-bound. The relativized conception, in either version, has no trouble accommodating the failure of individuals in different worlds to be strictly identical; even on the first implementation of the conception, talk of overlapping worlds might be understood as involving type rather than token identity of basically individuated worlds. To be sure, if the modal facts are context-dependent in the way that counterpart theory supposes, then this will introduce another degree, so to speak, of relativization: rather than the metaphysical modal facts being relative just to which world is indicatively actual, such facts will also be relative to which counterpart relation is in place. In any case, nothing in either version of the relativized conception rules out incorporating further contextual aspects, in line with counterpart theory. On the other hand, one might rather treat de re representation in terms of literal transworld identity. Here again there are options. One might suppose (following Kripke) that in considering what is possible or necessary for a given individual, one may stipulate that it is that very individual that one has in mind—notwithstanding, of course, that we cannot stipulate (assuming the falsity of modal conventionalism) what is modally the case with the individual in question. Alternatively, one might suppose that Woody’s existence across various possible worlds is grounded in the exemplification of a haecceity (roughly, the property of being identical to Woody), as in Plantinga (1974). Or, as per Spencer (2006), one might endorse an intermediate position and treat representation de re in

224 | Adam Murray and Jessica Wilson terms of counterpart relations that are restricted so as to model the formal properties of the identity relation (i.e., symmetry and transitivity). Each of these options is formally compatible with either implementation of relativized metaphysical modality, as developed so far. This conception simply leaves open such further details concerning how and why de re representation is to proceed. That each implementation of relativized metaphysical modality is compatible with either transworld identity or counterpart theory leads to a final moral of the Woody case; namely, that debates over the viability of transworld identity are largely orthogonal to debates over essentialism and the sorts of flexibility in material origins brought out by the data in that case.15 This orthogonality is liable to be overlooked, given Salmon’s own treatment of the data in terms of literal transworld identity, and Lewis’s subsequent (1986) reply and critique, couched entirely in the language of counterpart theory. The choice presented by the Woody case is not, as the Salmon– Lewis debate suggests, between a view on which transworld identity is retained by accepting intransitive accessibility (along with the ‘impossible’ worlds that gave Lewis such pain), and a view on which transitive accessibility between worlds is retained by accepting counterpart theory between individuals. Indeed, on relativized metaphysical modality, the Woody case can be closed while leaving all these options open. University of Toronto REFERENCES Armstrong, D.M., 1989. A Combinatorial Theory of Possibility. Cambridge: Cambridge University Press. Bealer, George, 2000. ‘A Theory of the A Priori’. Pacific Philosophical Quarterly 81 (1):1–30. Bennett, Karen, 2005. ‘Two Axes of Actualism’. Philosophical Review 114(3):297–326. Bird, Alexander, 2007. Nature’s Metaphysics: Laws and Properties. Oxford: Oxford University Press.

15

Thanks to Laurie Paul for calling this advantage to our attention.

Relativized Metaphysical Modality | 225 Bricker, Phillip, 2008. ‘Concrete Possible Worlds’. In Contemporary Debates in Metaphysics, J. Hawthorne, T. Sider, and D. Zimmerman (eds.). Oxford: Blackwell Publishers. Burgess, John P., 2009. Philosophical Logic. Princeton: Princeton University Press. Chalmers, David J., 2006. ‘The Foundations of Two-dimensional Semantics’. In Two-Dimensional Semantics: Foundations and Applications, Manuel Garcia-Carpintero and Josep Macia, (eds.). Oxford University Press. Chalmers, David and Jackson, Frank, 2001. ‘Conceptual Analysis and Reductive Explanation’. Philosophical Review 110: 315–60. Chandler, Hugh, 1976. ‘Plantinga and the Contingently Possible’. Analysis, 36:106–109. Davies, Martin and Humberstone, Lloyd, 1980. ‘Two Notions of Necessity’. Philosophical Studies. 38:1–31. Ellis, Brian, 2001. Scientific Essentialism. Cambridge: Cambridge University Press. Fine, Kit, 2005. ‘The Varieties of Necessity’. In Modality and Tense. Oxford University Press. Kaplan, David 1979. ‘On the Logic of Demonstratives’. Journal of Philosophical Logic. 8. —— 1989. ‘Demonstratives’. In Themes from Kaplan, Almog, J., Perry, J., Wettstein, H., and Kaplan, D. (eds.). Oxford: Oxford University Press. 481–564. Kripke, Saul, 1963. ‘Semantical Considerations on Modal Logic’. Acta Philosophica Fennica, 16:83–94. —— 1972. Naming and Necessity. Cambridge MA: Harvard University Press. Leibniz, Gottfried, 1686. Discourse on Metaphysics and the Monadology. Drier Publications. Lewis, David, 1968. ‘Counterpart Theory and Quantified Modal Logic’. Journal of Philosophy 65:113–126. —— 1971. ‘Counterparts of Persons and their Bodies’. Journal of Philosophy. 68:203–11. —— 1986. On the Plurality of Worlds. London: Blackwell. McDaniel, Kris, 2004. ‘Modal Realism with Overlap’. Australasian Journal of Philosophy 82(1): 137–52. McMichael, Alan, 1983. ‘A Problem for Actualism about Possible Worlds’. Philosophical Review, 92(1): 49–66. Menzel, Christopher, 2008. ‘Actualism’. Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), http://plato.stanford.edu/entries/actualism. Plantinga, Alvin, 1973. ‘Transworld Identity or Worldbound Individuals?’. In Logic and Ontology, Milton Munitz (ed.). New York: New York University Press.

226 | Adam Murray and Jessica Wilson Plantinga, Alvin, 1976. ‘Actualism and Possible Worlds’. Theoria (42): 139–160. Salmon, Nathan, 1981. Reference and Essence. Princeton: Princeton University Press. —— 1984. ‘Impossible Worlds’. Analysis, 44:114–17. —— 1989. ‘The Logic of What Might Have Been’. Philosophical Review, 98:3–34. Shoemaker, Sydney, 1980. ‘Causality and Properties’. In Time and Cause, Peter van Inwagen, (ed.). 109–35. Dordrecht: D. Reidel. Sider, Theodore, 2010. Logic for Philosophy. Oxford: Oxford University Press. Spencer, Cara, 2006. ‘Keeping Track of Objects in Conversation’. In TwoDimensional Semantics: Foundations and Applications, Manuel GarciaCarpintero and Josep Macia, (eds.). Oxford: Oxford University Press. Stalnaker, Robert, 1976. ‘Possible Worlds’. Noûs 10 (1): 65–75. —— 2008. ‘Assertion’. In Formal Semantics: The Essential Readings, P. Portner and B. H. Partee, (eds.). Blackwell Publishers. van Inwagen, Peter, 1985. ‘Plantinga on Trans-world Identity’. In Alvin Plantinga, van Inwagen and Tomberlin, (eds.). Dordrecht Publishers, 101–120.

8. Coincidence Through Thick and Thin Sydney Shoemaker I Coincident objects are numerically different objects that are composed of the same matter and so occupy the same portion of space. Supposing this is possible, the coincidence could be either temporary or permanent; two objects might coincide for a while and then part company and go their separate ways, or they might coincide throughout their careers. Whether coincidence is possible is disputed. Some philosophers think it obviously is possible, and cite as examples a statue and the piece of clay of which it is composed, and persons and their bodies. The usual reason for holding this is that such pairs of entities appear to differ in their historical and modal properties. Others think it is not possible, and claim that the view that it is possible has counterintuitive consequences, raises unanswerable questions, and involves a severe case of overcounting. Important issues turn on whether coincidence is possible. Perhaps the most serious of these concerns the nature of personal identity. On psychological accounts of personal identity, sometimes called neo-Lockean accounts, it is possible for persons to change bodies, via a brain or cerebrum transplant or via some sort of brain reprogramming.1 This seems to have the consequence that persons are not identical with their bodies. Yet persons and their bodies are composed of the same matter. This seems to give us coincident 1 The idea that a person could change bodies by way of a brain transplant (see Shoemaker (1963)) takes its inspiration from Locke’s example in which the soul of a prince is transferred to the body of a cobbler. Eric Olson (1997) has claimed that since the whole brain includes the brainstem which is the control center for the biological life of a person, the claim that the person would go with the brain in a brain transplant is compatible with the rejection of the psychological view; it is compatible with personal identity consisting in biological continuity rather than psychological continuity. If this is right, the psychological account should say that the transplantation of just the cerebrum could be person preserving. For a case in which the change of bodies is via brain reprogramming, see Perry (1972).

228 | Sydney Shoemaker entities. It also seems that on such accounts persons are not identical with the biologically individuated animals with which they share their space and matter. If the body of a person can survive death, as a corpse, and if both the person and the biological animal cease to exist at death, then wherever there is a person there are (at least) three different coincident entities – the person, the person’s body, and the biological animal that shares the space and the matter of both.2 Advocates of perdurance (four-dimensionalist) views of the persistence of objects are committed to a kind of coincidence. They are committed to the coincidence of temporally extended objects with their temporal parts, and to the coincidence during the period of overlap of temporally extended objects that share temporal parts. For some perdurance theorists this is the only sort of coincidence that is possible. And they see it as a benign form – since they think that in the first instance it is the momentary temporal parts of objects that have the properties things have at times, and since these parts are shared in cases of temporal overlap, they are not committed to what may seem an obnoxious consequence of coincidence, namely there being multiple instantiations of the same property at the same time and place. E.g., they are not committed to there being two instances of weighing one hundred pounds where a coincident statue and piece of clay, each weighing one hundred pounds, are located, and the apparent consequence of this that a scale ought to read two hundred pounds when the statue is placed on it. If the only coincidence allowed is in cases of temporary overlap, the perdurance theorist will not allow that there can be pairs of objects that coincide throughout their careers. However, it appears that a perdurance theorist could allow this for the same reasons that others do, namely modal reasons – if the careers of the statues and the piece of clay could come apart, even if in fact they don’t, it would appear that the statue and the piece of clay must be different. In any case, I will not be concerned here with the sort of coincidence that all perdurance theorists allow – that which consists in the sharing of temporal parts. I don’t myself believe that persisting 2 By a “biological animal” I mean an animal with biological persistence conditions. A neo-Lockean can hold that persons are animals, but not that they are biological animals.

Coincidence Through Thick and Thin | 229 objects, as opposed to events and processes, have temporal parts. The sort of coincidence I will be concerned with is a sort that can occur in objects as conceived by endurance theorists, i.e. by threedimensionalists. I shall not here argue for the three-dimensionalist view. And it may be that some of what I say can be incorporated into a four-dimensionalist account. My aim in this paper is to give an account of what coincidence would consist in, and why we should allow that it is both possible and actual. It is with respect to a certain kind of objects, or a certain conception of objects, that I think it is possible; and it is natural to think of this conception in three-dimensionalist terms.

II Let me start by sketching the conception of objects I will be working with here. The conception is one I think is satisfied by the objects that figure in the commonsense picture of the world, and in a good part of the scientific picture as well – whether it fits the objects that figure in microphysics I am not competent to say. I do not claim, in any case, that all of the objects there are fit this conception. If there are objects that are arbitrary chunks of spacetime, composed of whatever matter is contained in those chunks, or are mereological sums of the members of arbitrary sets of familiar objects, they do not satisfy the conception. I am not denying (or affirming) that there are such objects. It is essential to objects on this conception that they have careers, which can be thought of as causally unified series of sets of simultaneous property instances. The property instances in each of the sets in the series will be unified by synchronic unity relations, and the sets in the series will be unified by diachronic unity relations. There being such a career, a series of sets of property instances standing in appropriate unity relations, will be sufficient for there existing an object having that career. But this should not be understood as amounting to a reduction of objects to property instances, a theory which makes objects bundles of property instances. A property instance is the instantiation of a property in a thing at a time, and the notion of a property instance involves the notion of objects in which properties can be instantiated. What we have here is a “package deal” – the

230 | Sydney Shoemaker notion of an object and that of a property instance are internally related. The synchronic unity relation for property instances involves the instances being so related that they jointly produce certain effects or bestow certain powers – what effects and powers these are being determined by the causal profiles that individuate the properties. So, for example, an instance of a certain mass and one of a certain shape jointly bestow, when synchronically unified, the power to make a certain sort of impression on things made of certain materials when applied to them with appropriate force, and instances of certain beliefs and certain desires that bestow, when synchronically unified, a disposition to engage in certain actions. There is a two-way connection here. If the property instances are synchronically unified, they jointly produce the relevant powers and effects. But also, if the property instances are so related that they jointly produce the relevant powers and effects, they are synchronically unified. The diachronic unity relation for property instances involves the instances being so related, causally, that the instances that occur in the series at one time generate, in combination with input from the outside, the instances that occur in it at subsequent times. This does not require that every instance that occurs in the series at one time is causally connected to every instance, or even to any instance, that occurs at some subsequent time. Property instances occurring at different times can be diachronically unified in virtue of the fact that each is synchronically unified with a member of a pair of property instances whose members are causally related, the earlier one being a partial cause of the later one. There is, I believe, an internal relation between the causal profiles of properties and the persistence conditions of the things that have them.3 Transtemporal identity requires that the successive stages in the career of a thing be causally related, where this involves there being relations of counterfactual dependence of later stages on earlier ones – often later stages inherit their properties from earlier ones, and even when they don’t it will be true that if the earlier stages had been different in certain ways the later stages would have been correspondingly different (a growing tree changes its size and shape, but what its size and shape are at a later time is partly determined by what they

3

See Shoemaker (1979).

Coincidence Through Thick and Thin | 231 were earlier). These relations of causal and counterfactual dependence hold in virtue of the causal profiles of the properties instantiated in the course of the object’s career. But the persistence conditions of the subjects of the properties figure in the specification of the causal profiles. For in some cases what is built into the causal profile of a property is not just that instances of it will contribute to the production of certain effects but that they will contribute to the production of those effects in the future career of the thing in which that property instance occurred. This is clearly the case with many dispositional properties. For something to be combustible is for it to be such that if it is subjected to a certain degree of heat it, that same thing, will burn. For something to be elastic is for it to be such that if subjected to certain forces it, that same thing, will undergo a change of shape and then, when the force is removed, revert to its (that same thing’s) original shape. And this will be reflected in the causal profiles of the properties that realize these dispositional properties. We have here the same sort of two-way connection we noted earlier in the case of synchronic unity – the holding of certain causal relations is implied by, but also implies, diachronic unity and the transtemporal identity it brings with it. This is all part of the “package deal” mentioned earlier – the internal relation between the notion of an object and the notion of a property instance. The objects that have careers of the kind I have characterized include ourselves. If one thinks that there are objects that are not of this kind, one might speculate that it is because we ourselves are objects of this kind that our commonsense ontology is built around such objects. One might even think that there could be knowers and conceivers that are not of this sort, and that these would have a different conception of the objects that surround them, taking these to be entities like themselves. But on reflection it seems that it is not possible for there to be conceivers and knowers that are not of this sort. Maybe there are objects that are simply chunks of spacetime, or are mereological sums of ordinary objects. But these could not have the sorts of properties that knowers and conceivers must have.

III Assuming physicalism, as I will do throughout this paper, the mental properties of persons are realized in physical properties. And as

232 | Sydney Shoemaker we will see shortly, all of the physical properties that figure in our ordinary thought and discourse are realized in other physical properties. It is useful to distinguish two different sorts of realization.4 One is the sort that figured in the preceding paragraph, where the realizer of a property instance is another property instance. Call this property realization. I favor an account of this on which one property realizes another just in case the forward-looking causal features of the realized property are a subset of the forward-looking causal features of the realizer, and, accordingly, one property instance realizes another just in case the realized properties are so related. In the other sort of realization what realizes a property instance is a microphysical state of affairs – its being the case that certain micro-entities are propertied and related in a certain way. Here too the notion of realization involves the notion of a causal profile; microphysical states of affairs have causal profiles, consisting in their aptness to cause or contribute to causing certain effects, and the causal profile of the microphysical state of affairs that realizes a property instance must match the causal profile of the instantiated property.5 Call this microphysical realization. In both cases there will be a multiplicity of ways in which the instantiation of a given property can be instantiated – in both there will be “multiple realization” – but in both the existence of a realizer of a property instance constitutes that property instance. Realization of the second of these sorts, microphysical realization, brings with it a special case of realization of the first sort, property realization. If a thing’s career embeds a microphysical state of affairs at a given time, the thing thereby has at that time the property of having a career that at that time embeds such a state of affairs. Call this a microphysical-state-of-affairs-embedding property, or, for short, an MSE property. There is a one–one relation between types of microphysical states of affairs and MSE properties. Since every physical property instance has a microphysical state of affairs realizer, every physical property instance has a property realizer that is an instance of an MSE property. MSE properties are not properties that figure in ordinary thought and discourse. They are too fine-grained for that. 4 5

See Shoemaker (2007). For a discussion of this sort of matching, see Shoemaker (forthcoming).

Coincidence Through Thick and Thin | 233 The slightest difference between two microphysical states of affairs, e.g., a slight difference in the location of an electron, makes for a difference in MSE properties. So it cannot be known in any ordinary way whether two things share an MSE property, or that something has an MSE property it had a moment before (although in both cases it can be known that such sameness of MSE properties is highly unlikely).

III As a final preliminary to presenting my account of coincidence, I distinguish two sorts of properties of things. There are properties that can belong to things of very different kinds. Examples would be colors, shape properties, mass, and electric charge. I call these thin properties. Their thinness is with respect to causal features in their causal profiles – their causal profiles lack features that limit their instantiation to things of particular kinds. This distinguishes them from what I call thick properties, which can only belong to things of certain kinds. Sortal properties, like being a tree or being an automobile, are of course thick. But so also are properties that imply sortal properties, like being in bloom or being in first gear. Thickness varies in degree. The property of being an oak tree is thicker than the property of being a tree, and the property of being a 1974 Chevy is thicker than the property of being an automobile. But for present purposes I will ignore such differences in degree of thickness. I mentioned earlier that the causal profiles of properties include causal features that are such that the effects of their activation necessarily includes effects on the future career of the thing that has them. It is such properties whose natures are internally related to the persistence conditions of the things that have them, and so to the persistence conditions of particular kinds of things. And it is especially in the case of thick properties that we find such causal features. Thick properties are associated with kinds of things, and kinds of things are individuated, at least in part, by their persistence conditions. I take mental properties to be thick. Their functional roles give them causal profiles of the sort just mentioned, ones whose specification

234 | Sydney Shoemaker refers to effects on the future careers of their possessors. What beliefs and desires a person has at a time will be to a considerable extent determined by what beliefs and desires the person had at earlier times, although they of course will be partly due to inputs to the person’s mind subsequent to those earlier times. The dispositions associated with these states are ones that are largely played out within the careers of their possessors. On psychological, neo-Lockean, accounts of personal identity it is relations between states of this kind occurring at different times, relations that reflect their causal profiles, that constitutes the persistence of persons over time.

IV It is evident that the thin physical properties true of something do not realize any of its thick properties, since a thin property can have only a proper subset of the causal features belonging to a thick property. It follows from this that the thin physical properties belonging to a person cannot be property realizers of the person’s mental properties. If all physical properties were thin, this would have the consequence that mental properties cannot be realized by physical properties. So if one is a physicalist, one must hold that there are thick physical properties, these including the physical realizers of mental properties. This relates to an objection sometimes raised against psychological accounts of personal identity. Such accounts are committed to the view that persons are not identical to their bodies. But persons and their bodies are composed of the same matter. It might seem to follow that persons and their bodies are physically exactly alike, and share all of the same physical properties. If this were so, then if the mental states of persons are realized in physical properties it must be the case that those mental states are also possessed by the bodies of persons. They must also be possessed by the biologically individuated animal that shares its matter with the person and the body. This gives us “too many minds,” or “too many thinkers.” So it might seem that the proponent of the psychological view must choose between rejecting physicalism, i.e., rejecting the claim that the mental states of persons are physically realized, and accepting the claim that where we take there to be one person there are two or

Coincidence Through Thick and Thin | 235 three psychologically identical subjects. The latter view would of course undermine the psychological view – for if the body and the biological animal share the mental properties of persons they should themselves count as persons, and as persons that lack the persistence conditions that the psychological view ascribes to persons. The former view will also be unacceptable to psychological theorists who are physicalists, as most recent psychological theorists have been. This objection of course collapses if we allow that the only physical properties shared by coincident entities are thin properties, that there are thick physical properties, and that it is only the latter that realize mental properties. But this leaves us with the task of explaining how coincident objects can differ in some of their physical properties, i.e., of how one of a pair of coincident objects can have thick properties the other lacks, this despite the fact that the coincident objects are composed of the same matter and share the same thin physical properties. A related task is that of explaining how the coincidence view can be reconciled with the view, central to physicalism, that the distribution of fundamental physical particles in the world determines the distribution of mental properties. That physicalist view implies that the distribution of fundamental physical particles determines what microphysical states of affairs there are, and so what microphysical realizers of mental states there are. But, as we have seen, to each type of microphysical state of affairs there corresponds a physical property, an MSE property, that is instantiated in a thing just in case its career embeds a microphysical state of that type. These MSE properties appear to be thin properties (though we will see later that there are also thick MSE properties). The same microphysical states of affairs can exist in the careers of objects of different types, and, more to the point, a microphysical state of affairs that occurs in the career of a thing will occur in the career of an object coincident with that thing. Because of this, things of different types can share the same MSE properties, and coincident objects will share the same (thin) MSE properties. All of this suggests that the distribution of MSE properties in the world determines the distribution of mental properties in the world, and so that the distribution of mental properties is determined by the distribution of thin physical properties. How can this be if mental properties are not realized in thin physical properties?

236 | Sydney Shoemaker V There is a difference between some properties being instantiated in a thing guaranteeing that some other property is instantiated in that thing, and their guaranteeing simply that that other property is instantiated in something composed of the matter of which the thing is made. If the distribution of MSE property instances in the world determines the distribution of mental property instances, then for each mental property instance there will be a set of MSE property instances whose members jointly necessitate its existence. One such set, of course, would be the set of all MSE-property instances. But there will be much smaller sets whose members jointly necessitate its existence, and some of these will be sets whose members occur in a single thing and where their occurring in a thing does not entail that that thing has the mental property in question, but does entail that some related thing has it. To put the matter in terms of microphysical states of affairs, for any mental property instance there will be a microphysical state of affairs whose existence entails the existence of that property instance but which can be embedded in the career of a thing that does not have the mental property in question. I have said that an MSE property is one that a thing has in virtue of a microphysical state of affairs being embedded in its career. But we need to distinguish different ways in which a microphysical state of affairs can be embedded in the career of a thing. One sort of embedding, call this weak embedding, simply consists in the fact that the microentities involved in the state of affairs are all ones that are, at the time in question, among those that make up the object in whose career they are embedded. A property’s being embedded in this way amounts to its being a microstructrual property, and any microphysical state of affairs that is embedded at all is embedded in at least this sense. Such embedding gives us an MSE property that is a thin property that anything having it shares with any object coincident with it. But the microrealization of a thick property requires a stronger sort of embedding, call it strong embedding. A microphysical state of affairs being strongly embedded in a career at a time requires that it be related in certain ways to states of affairs occurring in that career at other times. As I said earlier, we can think of the career of a thing

Coincidence Through Thick and Thin | 237 of a certain sort as a series of sets of property instances that are so related, causally, that the persistence conditions for things of that sort, together with the causal profiles of the properties instantiated, make that series the career of a thing of that sort. Assuming physicalism, this will be realized in a series of microphysical states of affairs that realize the property instances. Let’s say that a microphysical state of affairs that is a member of such a series at a time is a momentary stage of the thing at that time. This stage will be made up of smaller states of affairs that are realizers of property instances occurring in that thing at that time. What makes the stage made up of the states of affairs the stage of a particular thing is the fact that the causal profiles of the states of affairs that make it up are such that their occurrence contributes to the implementation of the persistence conditions for things of the sort in question. This will involve some of the states of affairs being realizers of thick properties that can be instantiated only in careers of things of that sort. A microphysical state of affairs will be strongly embedded in the career of a thing of a given sort at a time if it is such a part of the microphysical state of affairs that is the momentary stage of the thing at that time, i.e., if it has a causal profile that makes the requisite contribution to the implementation of the thing’s persistence conditions, qua thing of that sort. In the case of coincident entities, like me and my body, there will be two different careers that involve, during the period of coincidence, the same micro-entities. These will have the same microphysical states of affairs weakly embedded in them. Some of these states of affairs will be strongly embedded in the career of one of the coincident entities, and some will be strongly embedded in the career of the other. Which is to say that one of these objects will have one set of thick MSE properties, and the other will have a different set of such properties. A thick MSE property will be a property something has in virtue of its career strongly embedding a microphysical state of affairs. The things will of course share a number of thin properties, and the thin properties will be property realized by thin MSE properties. Here the MSE properties will be realized by states of affairs that are weakly embedded in the careers of their possessors. Some of these states of affairs will also be strongly embedded in the career of one or the other of the coincident entities.

238 | Sydney Shoemaker Consider a microphysical state of affairs that realizes an instance in me of a mental property, say thinking of Vienna. This state of affairs occurs in me, and it also occurs in my body. But while it is strongly embedded in my career, in the sense just explained, it is not strongly embedded in the career of my body, although it is weakly embedded in it. And so my body does not have the thick MSE-property constituted by the strong embedding of that state of affairs. The occurrence of that state of affairs in my body’s career does guarantee that something has that mental property. But that something is me, not my body. What its occurrence in my body entails is that there is something coincident with my body whose career strongly embeds it and so has the mental property it realizes.

VI I should clarify what I mean when I speak of the microphysical states of affairs that make up a momentary stage contributing to the implementation of the persistence conditions for things of a certain sort. Microphysical states of affairs have causal profiles, which they have in virtue of the laws governing the microentities that enter into them. So the microphysical states of affairs that make up a momentary stage will contribute to generating subsequent states of affairs – they will generate them in conjunction with other microphysical states of affairs that lie outside the career of which the stage is a stage. Normally a subset of these subsequent states of affairs will form a series of sets of synchronically unified states of affairs that form thing-stages and together constitute part of a career to which our initial stage belongs.6 The states of affairs that make up these subsequent stages will realize property instances that stand in relations of diachronic unity to property instances realized by states of affairs occurring in other stages of the series, including those in our initial stage. These property instances will include instances of thick properties that can only be instantiated in things that are of a certain sort and have certain persistence conditions; the persistence 6 I say “normally” because there is the exceptional case in which the stage is the final stage in the thing’s career, because its propensity to contribute to a future career is overridden by the causal efficacy of other states of affairs that leads to the termination of the career.

Coincidence Through Thick and Thin | 239 conditions will be reflected in the relations of diachronic unity that unite the series. Since the series of stages will be generated in part by states of affairs external to our initial stage, and these can be different in different possible worlds in which that stage occurs, the future shape of the career to which that stage belongs will be different in those different possible worlds. But in all those worlds there will be future stages with which it is diachronically unified, unless there occurs something that prevents that stage from making its contribution to the future – something that ends the existence of the thing of which it is a stage. It is in virtue of the causal profiles of the states of affairs that make it up that the stage contributes to generating a future career of a thing of a certain sort and having certain persistence conditions. Given that there can be coincident objects, the states of affairs in our stage may contribute to the generation of two or more future careers. Here it will be the causal profiles of different states of affairs among those that make up the stage that contribute to the generation of the different careers. The careers may coincide for a while and then go their separate ways. But they may coincide as long as they exist. In the latter case the diachronic unity of the series of stages making up the career will be overdetermined; one subset of the property instances realized by the states of affairs in these stages will be unified by one set of diachronic unity relations, and another will be unified by a different set. And the same will be true during the interval in which two careers coincide. I have spoken of careers as series of stages, and have spoken of coincident objects as sharing stages. That would suggest that coincident objects have the same career as long as they coincide. But if, as I said earlier, stages are sets of synchronically unified property instances, the coincident objects will have different stages and their careers will be different even when they coincide. This seems the preferable meaning of “stage,” and of “career.” But when things coincide there will be at a given time a single set of microphysical states of affairs whose members realize the property instances that make up the two stages, and we can allow a secondary sense of “stage” in which this counts as one. (Note that stages of coincident objects will overlap in what property instances they contain; they will contain the same thin property instances.) They will differ only in their thick property instances, and this will be a difference with

240 | Sydney Shoemaker respect to which of the microphysical states of affairs are strongly embedded in which career.

VII Corresponding to the distinction between strong and weak embedding of states of affairs is a distinction between two sorts of property realization. So far I have taken realization to be a same-subject relation: the realized property instance and the property instance that realizes it must belong to the same thing. Call this realization1. But there seems to be room for a notion of realization that permits a property instance in one thing to be realized by a property instance in a numerically different thing. E.g., assuming that I and my body are numerically distinct, it nevertheless seems reasonable to say that my mental states are realized in states of my body. This requires a new realization relation, call it realization2. This has to be explained in a way that is compatible with the fact that no property of my body realizes1 any mental property and that the only properties I share with my body are thin properties. The motivation for saying that properties of my body realize my mental properties is the acceptance of physicalism together with the fact that there is a good sense in which I and my body are physically identical. Of course, our being physically identical consists in our sharing the same thin physical properties. So if there is a sense in which the properties of my body realize my mental properties, it had better be a sense in which my thin physical properties realize my mental properties. My thin physical properties do not by themselves realize1 my mental properties. But in conjunction with the fact that I have the persistence conditions of a person – the fact that I have the sortal property of being a person – they do realize1 them. That is to say, they are conjuncts of conjunctive properties that do realize1 them. So we can define realization2 in terms of realization1 as follows: where A and B are coincident objects, the instantiation of thin property P in A realizes2 the instantiation of thick property Q in B just in case B has a sortal property S such that the conjunction of P and S realizes1 Q. Taking everything to be coincident with itself, this gives the result that the instantiation of my thin properties in me realizes2 the instantiation of my mental properties. But it also

Coincidence Through Thick and Thin | 241 gives the result that the instantiation of these thin properties in my body realizes2 the instantiation of my mental properties. If we like we can introduce different notions of microphysical realization corresponding to the distinction just made between different notions of property realization. We can say that a microphysical state of affairs in the career of a thing realizes1 a thick property instance in that thing when it has the persistence conditions required for the instantiation of that thick property, and that a microphysical state of affairs in the career of a thing realizes2 a thick property instance in the career of a coincident thing if the latter has the persistence conditions required for the instantiation of that thick property.

VIII The case of coincidence I have mainly used as my example is that between a person and that person’s body. I will now discuss how my account applies to the case of coincidence most cited in the literature, that between a statue and the piece of clay that constitutes it. It is not immediately clear what count as thick properties of statues and, especially, of pieces of clay. Presumably the sortal properties being a statue and being a piece of clay count. But what else? In the case of pieces of clay, I think that the only other thick properties are modal properties having to do with the object essentially having such and such microentities among those that compose it. In the case of statues we can cite such properties as having a slender waist and being in a sitting position, and perhaps aesthetic properties like being graceful. Matters are complicated here by the fact that statues are artifacts, and both the properties and the persistence conditions of artifacts essentially involve the social context in which they exist, including the intentions and interests of those who make them, maintain them in existence, and use them. Few if any of their thick properties are intrinsic – typically the thick properties involve relations to the social context. A property like having a slender waist is partly a representational property; it is the property of representing, by resemblance, a person or other animal with a slender waist. The relational, or partly relational, nature of these properties means that

242 | Sydney Shoemaker the microphysical realizers of instances of them extend far beyond their boundaries, and are highly complex – in the case of statue properties they may include the realizers of the mental states of sculptors, collectors, and museum curators. Still, the account I have given applies. Statues and the pieces of clay that constitute them share thin properties like size, shape, color, and mass. They also share relational properties that must count as thin. It is true of the piece of clay, as it is of the statue, that people in its presence tend to have feelings of admiration, or of revulsion, although the admiration or revulsion is directed at the statue rather than at the piece of clay per se. Assuming physicalism, instances of these properties are realized in microphysical states of affairs. And some of these states of affairs are strongly embedded in the career of the statue and only weakly embedded in the career of the piece of clay. Their being strongly embedded in the career of the statue intimately involves the relation between the causal profile of the property and the persistence conditions of the statue. The kind of interest people have in statues leads them to count certain changes in them, e.g., the replacement of some of the matter inside their boundaries, as ones the statue survives, and others, e.g., the compression of the same matter into an altogether different shape, as amounting to its destruction. And here – as is the case generally with artifacts – what people’s interests lead them to count as persistence is partly constitutive of the persistence. In the case of the piece of clay, the microphysical states of affairs that are strongly embedded are of course those whose causal profiles are such as to make it a piece of clay and, what goes with this, to make it have the persistence conditions of a piece of clay. How can a microphysical state of affairs do this? It does it by having a causal profile that makes it part of a stage that is a stage in the career of a piece of clay (other microphysical states of affairs that are parts of this stage will make it also a stage in the career of a statue). Such a stage must be apt to generate, or contribute to the generation of, other stages that stand in the diachronic unity relation to it. Presumably the persistence conditions for a piece of clay require that it be composed of the same clay (or mostly the same clay) throughout its career, and at each time during that career this clay fills a continuous portion of space, and that the career be spatiotemporally continuous. Or something like that. The microphysical state of affairs that

Coincidence Through Thick and Thin | 243 realizes a piece of clay must be such as to guarantee that the object in which it occurs is made of clay, and it must be such as to generate, or contribute to the generation of, subsequent thing-stages containing microphysical states of affairs of the same sort. Of course, the full story would have to be more complicated than this, and would have to take into account the fact that the persistence conditions of such things as pieces of clay are somewhat vague.

IX It is natural to say that the piece of clay constitutes the statue, and that the body, or at any rate the biological animal, constitutes the person. The account I have given of coincidence suggests an account of constitution. As a first pass, let’s say that object A constitutes object B if A and B are coincident and the states of affairs that are strongly embedded in the career of B, and so realize its thick properties, are only weakly embedded in the career of A. This faces a difficulty. I have taken it that when coincident objects are of different kinds they have different persistence conditions, and that something’s being of a kind and having the persistence conditions that go with that kind is a thick property of it. This means that each member of a pair of coincident objects has a thick property the other lacks and is realized by a microphysical state of affairs that is strongly embedded in its career and only weakly embedded in the career of the other. So on the proposed account of constitution each member of a pair of coincident objects would constitute the other. While the piece of clay would constitute the statue, it would also be the case that the statue constitutes the piece of clay. Similarly, the person would constitute, as well as being constituted by, the body and the biological animal. Yet we presumably want the constitution relation to be asymmetrical. If we could be sure that the objects we want to count as constituting other objects have no thick properties other than sortal properties that they don’t share with the objects they constitute, then we could modify the account so as to give us the asymmetry we want: A will constitute B just in case A and B are coincident and B has thick properties not shared by A that are realized by microphysical states of affairs strongly embedded in B’s career and only weakly

244 | Sydney Shoemaker embedded in A’s, and A has no thick properties other than sortal properties that are not shared by B. But it would seem that some of the objects we want to count as constituting other objects have thick properties that are not sortal properties and are not shared by the things they constitute. I have already mentioned that the modal compositional properties of a piece of clay count as thick properties of it. Another example is found in the biological animals that constitute persons. The same biological predicates can be ascribed both to persons and to biologically individuated animals, but it is arguable that the properties are somewhat different in the two cases, and that persons have the biological properties they have in virtue of being coincident with biological animals having the corresponding biological properties. On a psychological account of personal identity, a biological property like being anemic could be lost by a person who acquires a new body in a cerebrum transplant while the samenamed property would not be lost by the biological animal left behind as a human vegetable. This implies that the causal profile of the property of the person and that of the property of the biological animal are somewhat different, making these different properties. If it is the case that the person’s being anemic is just a matter of his being coincident with a biological animal that is anemic, this would suggest that we have here a case in which a microphysical state of affairs is strongly embedded in the career of the biological animal and only weakly embedded in the career of the person. Which on the proposed account has the undesired result that the biological animal does not constitute the person. I have noted that where As constitute Bs it is true both that the microphysical states of affairs that realize the sortal property of being a B are weakly embedded in the careers of As and that the microphysical states of affairs that realize the sortal property of being an A are weakly embedded in the careers of Bs. But there is the following asymmetry. Where a particular A constitutes a B, the B could go out of existence simply in virtue of the A’s career undergoing a change consisting in its ceasing to embed the state of affairs that is the microphysical realizer of the property of being a B, while it is not true that the A could go out of existence simply in virtue of the B’s career undergoing a change consisting in its ceasing to embed the state of affairs that is the microphysical realizer of the property of being an A. A person can go out of existence simply in

Coincidence Through Thick and Thin | 245 virtue of the continuing career of its coincident biological animal ceasing to embed the microphysical realizer of the property of being a person, and a statue can go out of existence simply in virtue of the continuing career of its coincident piece of clay ceasing to embed the microphysical realizer of the property of being a statue. But a biological animal cannot go out of existence simply in virtue of the continuing career of its coincident person ceasing to embed the microphysical realizer of the property of being a biological animal; and a piece of clay cannot go out of existence simply in virtue of the continuing career of its coincident statue ceasing to embed the microphysical realizer of the property of being a piece of clay. This underlies the truth of Frederick Doepke’s observation that “if x constitutes y at time t, then the fact that y could perish at t is explainable by describing a change, beginning at t, in which x would be the substratum,” and that (still in Doepke’s words), “if x constitutes y then y exists because x accidentally has a certain property.”7

X It is central to this account that the microphysical states of affairs involved in microphysical realization are shared by the coincident objects. While a microphysical state of affairs may be strongly embedded in the career of one of a pair of coincident objects and not in the career of the other, it will be weakly embedded in the careers of both if it occurs in the career of either. This enables us to reply to one objection sometimes raised against the notion of coincidence. Suppose that a statue weighs one hundred pounds. Its coincident piece of clay, which on the coincidence view is numerically different, will also weigh one hundred pounds. So their combined weight will be two hundred pounds. This seems to generate a false prediction about what will happen when this pair of objects is placed on a scale, namely that the scale will register two hundred pounds. Similar difficulties can be raised about electric charge; it would seem that the combined charge of a pair of objects should be greater than the charge of either one taken separately, but this not what our instruments record. 7

Doepke (1982), p. 54 and p. 57.

246 | Sydney Shoemaker One reply to the weight objection rests on the claim that the coincident objects are composed of the same matter. What determines the reading on the scale is the weight of the matter placed on it, and in the example this is one hundred pounds, not two hundred pounds. I have no objection to this reply, but I doubt if it can be applied in the case of electric charge. I think that we can give a better reply by appealing to the fact that the microphysical states of affairs in the careers of the objects are identical. The influence of the piece of clay’s presence on the scale is due to the causal profile of a microphysical state of affairs it contains; and since the very same microphysical state of affairs occurs in the statue, the influence of the statue’s presence on the scale is already included in the influence of the piece of clay’s presence. The same point applies to electric charge; there is a single microphysical state of affairs that determines the effects of both coincident objects on our instruments, and its occurring in two objects shouldn’t be expected to double its effects.

XI Perhaps the most frequently raised objection to the view that there are coincident objects is that given that such objects share the same matter, arranged in the same way, it is impossible to explain how they can be of different sorts and have different persistence conditions and different modal properties. Michael Burke asks “Given the qualitative identity of these objects, what explains their alleged difference in sort?”8 Addressing Alan Gibbard’s Lumpl and Goliath, a supposedly coincident lump and statue that come into existence and go out of existence at the same time, Karen Bennett asks “What grounds the difference between Lumpl and Goliath, given that they are otherwise so alike? They are the same shape, the same size, made of the same parts, have the same history and future, are the same distance from the bagel store, and so on and so forth. So what exactly makes it the case that they could have different shapes and sizes, etc.?”9 And Eric Olson poses what he calls 8

Burke (1992), pp. 14–15. Bennett (2004), pp. 339–40. Gibbard (1974). Bennett is addressing here only the case of coincident objects that come into existence and go out of existence at the same time. But she clearly does not think that the problem arises only for such cases. 9

Coincidence Through Thick and Thin | 247 the “indiscernibility problem” as follows: “By definition materially coinciding objects are made up of exactly the same particles, arranged in exactly the same way, in identical surroundings… How could they have the qualitative differences that constitutionalists say they have?”10 This objection, or one aspect of it, was briefly addressed in section IV as the “too many minds” or “too many thinkers” problem. My response to it there invoked the distinction between thick and thin properties. Let me expand on that response. As we have seen, the careers of two coincident objects weakly embed the same microphysical states of affairs, and in virtue of this the objects have the same thin MSE properties, and in virtue of this they have the same thin properties. (We can take it that all thin properties of physical things are realized by thin MSE properties.) And the properties they share include the realizers2 of all of the thick properties possessed by either of them. This is the extent to which they are alike in virtue of being composed of the same matter. The microphysical realizers of some of these thick properties, while weakly embedded in the careers of both coincident objects, are strongly embedded in only one of them – some in one, some in the other. It is this that accounts for how the objects differ in their modal properties, their sortal properties, and their persistence conditions. As I have pointed out (section II), there is an internal relation between the causal profiles of properties – especially of thick properties – and the persistence conditions of the things that have them. The causal features in the causal profile of a property include the aptness of instances of the property to contribute to the generation of successor states of certain sorts later in the career of the same object, and the exercise of such aptnesses is partially constitutive of the persistence of the object. What we have in a case of coincidence is the instantiation in the same volume of space, realized by different arrangements of the micro-particles located in that volume of space, of properties whose causal profiles constitutively require that their possessors have different persistence conditions. The difference in persistence conditions brings with it a difference in sortal properties and a difference in modal properties. The instances of 10 Olson (2001), p. 339. Note that Olson takes the constitutionalists to hold that coincident objects have a qualitative difference while Burke takes it to be agreed that they are qualitatively identical. I think this is only a terminological difference.

248 | Sydney Shoemaker these properties will be synchronically unified with the instances of the thin properties instantiated in that volume of space, so the possessors of the thick properties will share the same thin properties, and will be coincident objects. At bottom, then, the source of the difference in modal properties, sortal properties, and persistence conditions lies in differences in the causal profiles of properties instantiated in the same place. There can’t be instantiations of these properties without there being things that have them, and because of these differences their possessors must be different. It should not be surprising that differences in causal profiles account for differences in modal properties, since a causal profile is at the same time a modal profile – what causal profile a property has is partly a matter of what effects it can contribute to causing, and what sorts of things it can be caused by. It should be noted that it is not only “constitutionalists” who require an explanation of how things have the modal properties, sortal properties, and persistence conditions they do. It often seems assumed in the literature that if we reject the possibility of coincident objects we have a ready explanation of this – somehow the physical makeup, or qualitative character, of an object determines its modal and sortal properties and its persistence conditions. But how does it do so? That is a question any theorist of these matters, not just constitutionalists, must address. It will have to be agreed that there are different sorts of things that differ in their modal properties and their persistence conditions, and differ in what kinds of properties they can have. Somehow the nature of the properties must enter into the explanation of this. If the explanation is in terms of the causal profiles of the properties, we get the sort of account I have offered. Olson grants that my account of properties, and their relation to identity conditions, provides an explanation (not one he accepts) of the difference between persons and human animals, but he says that even if it is right “it offers no solution to the indiscernibility problem. Like the others, it must begin with the assumption that Person is a person (or thinking thing) and not an animal while Animal is an animal and not a person. Without this starting point the account has nothing to work on. But it offers no explanation of the difference. Shoemaker too must deny that properties of and relations among a thing’s parts, even in conjunction with their surroundings, explain, entail, or even cause its higher-level properties, such as its ability or

Coincidence Through Thick and Thin | 249 inability to think, or its being an animal or an organism” (p. 352). (Here “Person” and “Animal” name a pair of coincident things which a constitutionalist must claim are distinct.) It should be apparent from what I have said that my account does not begin with the assumption that Olson says it must begin with. It begins, rather, with the assumption that where Person and Animal are there is a set of microphysical states of affairs that realize the properties Person and Animal have. It is then argued that given the nature of these properties there must be two different objects there, one having properties distinctive of persons and one having properties distinctive of biological animals. Having established that, we can if we want give the name “Person” to one of them and the name “Animal” to the other.

XII What I have said so far may have left the impression that on my view the coincidence of two objects requires that they be of different sorts and have different persistence conditions. That is, in fact, a common view among believers in coincidence; citing Locke they hold that it is impossible for two objects of the same kind to share their matter and occupy the same place at the same time. But that is not my view. I think it is possible for there to be coincident objects that are of the same kind and have the same persistence conditions. We need to see how this can be. Kit Fine has offered one simple example of such intra-kind supervenience.11 Someone writes a note on a piece of paper. The recipient turns it over and writes a reply on the other side. There are two notes, but just one piece of paper. The notes coincide with the piece of paper, but they also coincide with each other. In several place I have offered a different example, in which the coincident objects are both persons.12 The case is similar to that of Dr. Jekyll and Mr. Hyde. (I got it from Eric Olson, who puts a different interpretation on it.13) We are to imagine that two persons 11

Fine (2000). See my 2003 and my 2007. 13 Olson presented his example at a conference on the Self at the University of Arkansas in September 1999. What I take to be a case of coincident persons he takes to be a case of one person with a highly unusual history. 12

250 | Sydney Shoemaker alternate in controlling a single body, each having control of it for the period of a day. While one of them is conscious, and manifesting his mental states in the behavior of the body and responding to the effects of its surroundings on its sense organs, the other sleeps. But both of them have mental states – different mental states – at all times. The one who is asleep has beliefs, desires, intentions, memories, etc. in the way a normal person has such states while asleep. We can suppose that their mental states are realized in different hemispheres of their shared brain, but this is not essential. The states of each form a career in which the successive states are causally connected in such a way as to exhibit diachronic unity, but the total series of states occurring in the body over an extended interval does not form such a career, since there are no relations of “immanent” causal relations between the states of one of the persons and those of the other. Here the coincident objects are of the same kind – both are persons – and have the same persistence conditions. What makes it the case that there are two persons associated with the body is that two sets of thick properties are realized by microphysical states of affairs occurring in it, where each of those sets is unified by the synchronic and diachronic unity relations for persons, while the total set is not so unified. The microphysical states of affairs that are strongly embedded in the career of each of the persons are weakly embedded in the careers of both, and in virtue of this the persons share the same thin properties.

XIII I suspect that one reason for the skepticism many philosophers feel about the idea that there are coincident objects is the paucity of plausible examples of coincidence. The most frequently cited examples are the two I have discussed – the case of the statue and the piece of clay, and the case of persons and their coincident bodies and biological animals. Some believers in coincidence cannot avail themselves of the latter example because they deny that persons and their bodies are distinct; thus Judith Thomson, a believer in coincidence, says ”I take cats and people to be their bodies.”14

14

Thomson (1998), p. 169.

Coincidence Through Thick and Thin | 251 It is natural to say that where there is a statue there is a piece of clay (or of bronze, etc.) that constitutes it, and that where there is a person there is a human body that constitutes it. But for most ordinary language sortal terms S there is not an idiomatic completion of “Where there is an S there is a . . . that constitutes it.” There is no ordinary language sortal term with which we can complete “Where there is an automobile, there is a…that constitutes it.” And the same is true if for “automobile” we substitute “house,” “computer,” “river,” or “tree.” To be sure, we can complete such statements with the likes of “portion of matter.” But portions of matter, or quantities of matter, or collections of molecules are not natural candidates for being kinds of objects. We would be hard pressed to say what, other than being a portion of matter, is a thick property of a portion of matter. I suppose that a portion of matter can be said to have the shape, size, and mass of the object that is composed of it, but speaking this way can seem artificial. Animals other than human beings can be said to have bodies, and there are, I think, the same reasons for denying that dogs or horses are identical with their bodies as there are for denying that persons are identical with their bodies.15 But we don’t speak of trees, or other plants, as having bodies. Just why this is so is difficult to say. It must have to do with the ways these different sorts of entities are related to our interests, and in particular to the interest we have, in the different cases, in the distinction between being dead and being alive. It must be for similar reasons that we have no colloquial counterparts of “human body” in the case of houses, computers, and rivers. I think that portions or quantities of matter are entities in good standing, and that there being these makes coincidence ubiquitous. It is where it is most interesting that there are the resources in our language for designating the coincident entities. But it should be noted that even there we need a term of art, “biological animal” (animal individuated biologically), to designate one of the entities persons are coincident with, given that there is a good sense in which persons are animals.

XIV To sum up, I believe that there are cases of coinciding objects and that such cases are common. I have restricted my attention to how 15

See Unger (2000).

252 | Sydney Shoemaker there can be coincidence in a physicalist world – a world in which all property instances are physically realized. The key to its possibility is the distinction between properties that are thin, those that must be shared by coincident objects, and properties that are thick, those with respect to which coincident objects can differ. And the key to how that distinction applies to physicalist worlds is the distinction between two ways in which a microphysical state of affairs can be embedded in the career of an object, namely what I have called weak embedding and strong embedding, together with the role of the causal profiles of properties in determining whether a state of affairs realizing a property instance is strongly embedded in the career of an object or only weakly embedded in it. A state of affairs that is weakly embedded in an object’s career will be weakly embedded in the career of any object coincident with that object, while such a state of affairs can be strongly embedded in the career of an object without being strongly embedded in the career of an object coincident with it. This difference has to do with the fact that strong embedding requires that the state of affairs be suited for playing a role in implementing the persistence conditions for a particular sort of object, while weak embedding leaves it open what the persistence conditions are for the objects in whose career they are embedded. Thin properties are realized by weakly embedded microphysical states of affairs, while thick properties are realized by strongly embedded microphysical states of affairs. The account solves the “too many minds” problem by saying that mental properties are thick, and so can belong to an object of a certain sort, a person, without belonging to objects of other sorts, bodies or biologically individuated animals, that are coincident with it. And it explains how things made of the same matter, identically arranged, can differ in their modal and sortal properties, and in their persistence conditions.16 Cornell University REFERENCES Bennett, K. (2004). “Spatio-Temporal Coincidence and the Grounding Problem.” Philosophical Studies, 118: 339–71. 16

Thanks to Karen Bennett and Carl Ginet for comments on an earlier draft.

Coincidence Through Thick and Thin | 253 Burke, M. (1992). “Copper Statues and Pieces of Copper: A Challenge to the Standard Account.” Analysis, 52: 12–17. Fine, K. (2000). “A Counterexample to Locke’s Thesis.” The Monist, 84: 357–61. Gibbard, A. (1975). “Contingent Identity.” Journal of Philosophical Logic, 4: 187–221. Olson, E. (1997). The Human Animal – Personal Identity without Psychology. Oxford: Oxford University Press. —— (2001). “Material Coincidence and the Indiscernibility Problem.” The Philosophical Quarterly, 51: 337–55. Shoemaker, S. (1963). Self-Knowledge and Self-Identity. Ithaca: Cornell University Press. —— (1979). “Identity, Properties, and Causality.” Midwest Studies in Philosophy, 4. —— (1999). “Self, Body, and Coincidence.” Proceedings of the Aristotelian Society, Supplementary Volume, 73: 287–306. —— (2003). “Realization, Micro-Realization, and Coincidence.” Philosophy and Phenomenological Research, 67: 1–23. —— (2007). Physical Realization. Oxford: Oxford University Press. —— (forthcoming). “Physical Realization Without Preemption.” In a volume, to be published by Oxford University Press, on the ontology of the mental causation debate. Thomson, J. (1998). “The Statue and the Clay.” Nous, 32: 148–73. Unger, P. (2000). “The Survival of the Sentient.” Nous, 34: 325–48.

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V

THE OPEN FUTURE

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9. The Real Truth about the Unreal Future* Rachael Briggs and Graeme A. Forbes Recent times have been very much focussed on the future. The election of Barack Obama in America was accompanied by a wave of optimism. The Global Financial Crisis and the challenge of climate change caused many to descend into pessimism. It is not whether our glass is half-empty or half-full that we worry about, but whether it will be empty or full. As philosophers, naturally, we want to illuminate such concerns, so that we might understand them better and subject them to rational scrutiny. According to the Growing-Block view of time, most famously put forward by C.D. Broad (1923), the flow of time consists in events coming into existence, so that past things and events are real, while future things and events are not. While the Growing-Block view is often considered intuitively appealing, some are concerned it has little to say about the future it denies the existence of. We show how Growing-Block theorists can assign meaningful, mind-independent truth conditions to sentences about the future.

1. INTRODUCING THE GROWING-BLOCK THEORY Asymmetries between past and future abound. The past, many of us think, is fixed and determinate; the future is open and indeterminate. The arrows of time and causation point from past to future, not from future to past. The Growing-Block view explains the difference between the fixed past and the open future in terms of ontological commitment:

* Thanks to Jamin Asay, David Braddon-Mitchell, Mark Colyvan, Alison Fernandes, Patrick Greenough, Dominic Hyde, Jenann Ismael, Mark Jago, Rosanna Keefe, Kristie Miller, Jonathan Payne, Huw Price, Lionel Shapiro, Nicholas J.J. Smith, Agustín Rayo, and Richmond Thomason for their helpful questions and comments.

258 | Rachael Briggs and Graeme A. Forbes the view is committed to the (tenseless) existence of past objects and events, but not to the (tenseless) existence of future objects or events. It treats the present like the past, not like the future: present things are the last things the Growing-Block theory takes seriously. The commitments of the Growing-Block theory change—or more precisely, increase—as time passes. Though the Growing-Block theory takes only the past and present seriously, it holds that in time, there will be more to the past and present, because more will have happened, even though the events that will happen are not ones to which the Growing-Block view is (tenselessly) committed. Claims about the future create a puzzle for the Growing-Block theory. We ordinarily think that there are truths about the future— for instance, it is true that there will be a lunar eclipse on January 21, 2019. (Astronomers know that there will be a lunar eclipse on January 21, 2019, and what is known is surely true.) But if the future is unreal, then all claims about the future would seem to be trivially false. It will not help to stipulate that all sentences about the future are trivially true or trivially truth valueless, instead of trivially false. The proposition that there will be a lunar eclipse on January 21, 2019 is true, and the proposition that the moon will be replaced with a hunk of green cheese on January 21, 2019 is false. Everyone— Growing-Block theorists included—should distinguish between truths about the future and falsehoods about the future. We claim that although Growing-Block theorists must deny the reality of the future, they can still countenance non-trivial truths and falsehoods about future events. In the remainder of part 1, we develop a metaphysical and semantic framework for discussing modality in the context of the Growing-Block theory. This stage lays the groundwork for an account of future possibility, or feasibility, which can in turn be used to define a concept of future truth. In part 2, we propose three ways in which Growing-Block theorists might deploy our modal framework to give semantics for sentences about the future. Each proposal is associated with a different non-classical logic. Rather than choose among these proposals, we state the advantages and disadvantages of each, and leave the reader to decide.

The Real Truth about the Unreal Future | 259 In part 3, we respond to two general problems, having to do with expressive power, that face three all of our semantic proposals. Finally, in part 4, we explain the advantages of the Growing-Block theory over similar theories proposed by Craig Bourne, Storrs McCall, and John MacFarlane. Before we continue, one word about our starting point is in order. We will treat propositions as tensed—as having truth values relative to a time and to a possible world. We will use P and F as past and future operators on tensed sentences. These are to be read ‘At some past time’ and ‘At some future time’ respectively. This clarification out of the way, we are ready to develop the Growing-Block theory in more detail.

1.1 Times (Genuine and Ersatz) Our first task is to develop an ontology of possible worlds suited to the purposes of the Growing-Block theory. This ontology is a type of linguistic ersatzism: it presupposes that possible worlds are not concrete hunks of matter like the actual world, but descriptions of the world in a logically idealized language. We follow the approach of Craig Bourne (2006, 52– 4), who builds ersatz worlds from ersatz times. Just as an ersatz world is a complete representation of a world from a tenseless perspective, an ersatz time is a complete representation of a single time in a world. Each ersatz time is a maximal consistent set of tensed propositions which do not themselves contain any tense operators. (Ultimately, we will develop a way of assigning truth values to sentences containing tense operators—so the propositions contained in an ersatz time comprise only a proper subset of the propositions which are true at that time.) We take propositions to be interpreted sentences in an idealized logical language, and we will switch freely between talk of propositions and talk of sentences. We will make some further assumptions about the structure of ersatz times. We assume that each ersatz time e is associated with a vocabulary of names De, which (in a slight abuse of the use–mention distinction) we will call a ‘domain’. The domains of different ersatz times may overlap partially, completely, or not at all. We assume that if e contains a sentence of the form ∃xϕ(x), then it also contains

260 | Rachael Briggs and Graeme A. Forbes a sentence of the form ϕ(a) for some a ∈ De. (Both these assumptions are claims about what it is for e to be a complete representation of a time: e must specify exactly which things exist; and if it says that something satisfies a particular open sentence, then it also specifies which thing satisfies that open sentence.) Finally, we assume a ‘one name, one object’ rule: each name has at most one referent, and refers to the same object across times and possible worlds, while each object has at most one name. Since they are abstract representations, ersatz times contain no tables and chairs, donkeys and cats, or quarks and electrons. Nonetheless, they can be instantiated by concrete times: maximal simultaneous hunks of spatially connected stuff that do include tables, chairs, donkeys, cats, quarks, and electrons as parts. For a concrete time t to instantiate an ersatz time e, we claim, is simply for e to be a complete and true description of t. The concrete present instantiates one ersatz time; various concrete past times have instantiated other ersatz times. Still other ersatz times are not instantiated, and never have been. The Growing-Block theorist should claim that concrete times (past and present) are ordered from earlier to later. We will assume that this ordering relation is total, transitive, irreflexive, and asymmetric, and that it has all four properties necessarily. Following Bourne (2006, 53–5), we can represent possible timelines as sets of ersatz times ordered by a total, transitive, irreflexive, and asymmetric relation R.1, 2 (The same ersatz time may appear more than once in the ordering.) Having defined timelines, we can now explain what it is for a timeline to be instantiated. The whole of reality instantiates a time-

1 Perceptive readers will note that we have just ruled out a number of apparent possibilities, including closed time-like curves and branching concrete universes. We are not at all convinced these apparent possibilities are genuine possibilities. We believe that our theory could be expanded to accommodate these scenarios, but to do so here would require adding to an already long and complex paper. 2 In addition to the ordering relation, one might think that there are meaningful distance relations between ersatz times. Given two times such that one is later than the other, it makes sense to ask how much later—an hour, a minute, a year? We will assume that facts about the distances between times supervene on the facts about which concrete times there are and how those times are ordered, and do not need to be added to possible timelines as extra information.

The Real Truth about the Unreal Future | 261 line T just in case there is a function f mapping the ersatz times in T onto concrete times such that i. for every concrete time t, there is some e ∈ T such that f(e) = t, ii. for every e ∈ T, f(e) concretely instantiates e, and iii. for every e, e’ ∈ T, eRe’ iff f(e) is earlier than f(e’). Timelines, then, are sequences of ersatz times. They are possible world histories, or put another way, they are ersatz possible worlds. The Growing-Block theorist may want to place additional restrictions on which timelines represent genuine metaphysical possibilities. For instance, perhaps the properties of objects must change in a more-or-less continuous fashion, so that no possible timeline can represent a person as having one head at one time and three heads at a time immediately afterward. (Note that this continuity constraint is available, but not mandatory.) Only a subset of the possible timelines will be physically possible, biologically possible, and so forth. Saying exactly which timelines are possible in which ways is a serious philosophical problem, but it is not a problem peculiar to the Growing-Block view. We will not address it here, except to make explicit our assumption that what is possible according to T supervenes on the ersatz times T contains, together with the ordering of those times.

1.2 Truth (Relative and Absolute) Now that we have developed an ontology of possible worlds, our next step is to define truth at a possible world. Or rather, since we are working with tensed propositions, to define truth at a time at a possible world. A complete account of truth will require a thorough discussion of how to treat the nonexistent future, and we will not be in a position to give such an account until part 2. In the meantime, though, we can give the beginnings of a definition of truth at a time in a timeline. These beginnings will help us to clarify the question of how to treat the nonexistent future, which in turn will help us to further clarify the concept of truth. We will write veT(ϕ) for the truth value of ϕ according to e in T. Defining truth for atomic propositions is straightforward.

262 | Rachael Briggs and Graeme A. Forbes Where p is an atomic proposition, veT ( p) =

1 if p ∈ e 0 if p ∉ e

The definitions of truth-functional connectives are similarly straightforward. veT (¬f ) =

1 if veT (f ) = 0 0 if veT (f ) = 1

veT (f ∧y ) =

1 if veT (f ) = 1 and veT (y ) = 1 0 if veT (f ) = 0 or veT (y ) = 0

veT (f ∨y ) =

1 if veT (f ) = 1 and veT (y ) = 1 0 if veT (f ) = 0 or veT (y ) = 0

We assume that the material conditional ϕ ⊃ ψ is equivalent to the disjunction ¬ϕ ∨ ψ, at least until section 2.3, where we present an alternative definition of the material conditional. We will assume that ϕ ≡ ψ is equivalent to the conjunction (ϕ ⊃ ψ) ∧ (ψ ⊃ ϕ). We can define tensed quantifiers ∃ and ∀, which range (at e in T) over all and only the objects whose names belong to De. veT (∃xf x ) =

1 if for some a ∈De , such that veT (f x / a) = 1 0 if for all a ∈De , veT (f x / a) = 0

veT (∀xf x ) =

1 0

if for all a ∈De , veT (f x / a) = 1 if for some a ∈De , veT (f x / a) = 0

We can also partially define tenseless quantifiers Σ and П, which range over everything there was, is, or will be. veT Σx(f x ) =

1 if for some e’ in T , there is an a ∈De ’ such that ve ’T (f x / a) = 1

veT Πx(f x ) =

0 if for some e’ in T , there is an a ∈De ’ such that ve ’T (f x / a) = 0

We have not fully defined Σ or П, because we have not yet established a way of assigning truth values to statements about the

The Real Truth about the Unreal Future | 263 future. Where there is no object x such that ϕx, it may be the case that there will be an object x such that ϕx (once more times have become instantiated by the Growing-Block), or it may not. In order to settle the matter, we will need to say more about the future. We can also fully define the tense operator P, and partially define the tense operator F. veT (Pf ) =

1 if T contains an e’ such that e’Re and ve ’T (f ) = 1 0 else

veT (Ff ) = 1 if T contains an e’ such that eRe’ and ve ’T (f ) = 1 F is only partially defined for the same reason that Σ and П are only partially defined. We’ve said nothing yet about how to ascertain veT(Fϕ) when T fails to contain an e’ such that eRe’ and ve’T(ϕ) = 1. We do not want to say that veT(Fϕ) is invariably 0 under these circumstances, because we want to allow for the possibility that ϕ will become true in the future that does not yet exist. We can introduce an additional class of tense operators—one for each temporal distance d. (We will write the distance from e to e’ as positive where eRe’, and as negative where e’Re.) 1 if T contains an e’ such that e’Re , the distance from e’ to e is d , veT (Pdf ) = and ve ’T (f ) = 1 0 else 1 if T contains an e’ such that eRe’, the distance from e to e’ is d , and ve ’T (f ) = 1 veT ( Fdf ) = 0 if T contains an e’ such that eRe’, the distance from e to e is d , and ve ’T (f ) = 0 Fd is only partially defined, for the same reason that F was only partially defined. Since Σ, П, F, and the Fd operators are only partially defined, the valuation function as a whole is only partially defined. The fragment

264 | Rachael Briggs and Graeme A. Forbes of the language containing only atomic sentences, ∧, ∨, ¬, the quantifiers, and P behaves classically. But the valuation function as a whole is non-classical. We will discuss three different (non-classical) ways of extending the valuation function in sections 2.1–2.3. In addition to the concept of truth at a time in a timeline (which we have yet to fully define), we can define a number of less relativized types of truth: truth at a time, truth in a timeline, and truth, full stop. There is one timeline that represents the world as it really is—that is, it correctly represents all the times that concretely exist, and it does not represent the world as containing any times that don’t concretely exist. This is the actualized timeline. We can say that what is true at a (concrete) time t is simply what is true according to the ersatz time e in the actualized timeline (an ordered set of ersatz times) instantiated by a concrete time t in the actual world (a temporally ordered hunk of concrete future stuff). To define truth at a timeline, we will need the concept of an absolute present. Many theorists of time treat ‘present’ as an indexical, so that a token of ‘it is raining at present’ is true just in case it is raining at the time when the sentence is uttered. But the Growing-Block theorist can also pick out a non-indexical or absolute present: the last time in existence. Furthermore, each timeline has its own (ersatz) absolute present: the last time in it. The ersatz absolute present in a timeline T would, if T were actualized, be instantiated by the concrete absolute present. What is true at a timeline T is simply what is true at T’s absolute present. Some complete timelines will have no absolute present, since they will have no last time. There is nothing that is true according to these timelines (though there are still propositions that are true at times in them). What is true full stop is what is true at the absolute present in the actualized timeline. That is to say, what is true full stop is what is true of the concrete hunks of stuff that exist at the current stage of the Growing-Block. We have four concepts of truth in all, corresponding to cells in the following table. Timeline

Time

Relative

Absolute

Relative

Truth at e in T

Truth at t

Absolute

Truth in T

Truth

The Real Truth about the Unreal Future | 265 The cells on the left-hand side of the table correspond to types of truth that are wholly relative to representations, while the cells on the right-hand side correspond to types of truth that are somehow grounded in concrete existence. The cells in the top row correspond to tensed types of truth, while the cells in the bottom row correspond to tenseless types of truth. (Notice that we have used a single, tensed type of proposition throughout, but that our tensed propositions admit of both tensed and tenseless truth.) Given our fourfold definition of truth, how shall we define validity? A valid sentence is necessarily true, while a valid argument is one that is necessarily truth-preserving. We need a univocal way of understanding ‘true’ and ‘truth’ (and also a univocal way of understanding ‘necessarily’). Which sentences and arguments are valid should not depend on what concretely exists. Rather, questions of validity must be evaluated by considering all possibilities—both those that are concretely instantiated and those that are not. Possibilities, in our framework, are timelines. Therefore, we should define a valid argument as one that preserves one of the timeline-relative types of truth on the lefthand side of the matrix: either truth at e in T or truth in T. We have two appealing conceptions of validity: Validity eT

Φ ⊧ Ψ just in case, for every time e and timeline T, if veT(ϕ) = 1 for all ϕ ∈ Φ, then veT(ψ) = 1 for at least one ψ ∈ Ψ.

Validity T

Φ ⊢T Ψ just in case, for every time e and timeline T such that e is the absolute present in T, if veT(ϕ) = 1 for all ϕ ∈ Φ, then veT(ψ) = 1 for at least one ψ ∈ Ψ.

Validity eT is the stronger notion: an argument or sentence that is valid eT must also be valid T, but the converse does not hold.3 We will take validity to be validity eT, rather than the weaker validity T.4 Likewise, we will take necessity to be truth at every time in every timeline—not just truth in every timeline. 3

See Section 3.1. Still other interpretations of validity are possible in the supervaluationist framework we develop in section 2.1; see (Varzi 2007) for a detailed discussion. 4

266 | Rachael Briggs and Graeme A. Forbes The distinctions among the four relativized types of truth can be brought to bear on a recent objection to the Growing-Block view. We discuss this objection and our solution to it in the next section.

1.3 How Do We Know it is Now Now? Craig Bourne (2006) and David Braddon-Mitchell (2004) have independently raised the concern that, if there are two interpretations of ‘the present’ on the Growing-Block view (viz. the indexical and the absolute), then it is possible that, say, Marie Curie might ask if, in addition to being at the indexical present (i.e. being at the time she is at) she is at the absolute present (i.e. at the time succeeded by nothing). If we accept that Marie Curie could ask such a question, despite not being at the absolute present, then we must explain why Marie Curie would be wrong to think she is at the absolute present, while we are right to think we are at the absolute present. One solution, put forward by Peter Forrest, is to deny the possibility of Marie Curie asking such a question, so long as she is located in the past. According to him “the past, although real, is lifeless and (a fortiori?) lacking in sentience” (Forrest 2004, 358). More generally, “Life and sentience are . . . activities not states. Activities only occur on the boundary of reality, while states can be in the past” (Forrest 2004, 359). For Forrest, the distinction between the living present and the dead past is grounded in their different causal properties—activity requires a ‘causal frisson’ that can occur only at the absolute present. We think that the distinction between the living present and the dead past is better understood in terms of the distinction between what is true at a past time (i.e. what is true according to the ersatz time e that corresponds to t in the actualized timeline), and what is simply true (i.e. true at the absolute present in the actualized timeline). This distinction is meant to be applied concerning not the content of Marie Curie’s beliefs, but our attribution of those beliefs to her. It is true at some past time (e.g. 1903) that Marie Curie falsely believes she is (absolutely) present, but it is not true (full stop) that she falsely believes she is (absolutely) present. This is because it is not true (full stop) that Marie Curie believes anything at all. In the actual world, at the absolute present, Curie is dead. Dead people lack beliefs.

The Real Truth about the Unreal Future | 267 Whether the concept of absolute truth aligns with any causal concept is a further question, and one that the authors of this essay disagree about. One of us thinks Forrest’s concept of a causal frisson captures something important about the sense in which the present is ontologically privileged; the other thinks that it is unlikely to satisfy any of the useful roles carved out for causation in metaphysics and the philosophy of science.

1.4 Feasibility So far, timelines have behaved somewhat like the possible worlds of Robert Merrihew Adams (1974). We should, however, mark several important disanalogies between timelines and traditional possible worlds. First, unlike traditional possible worlds, timelines are not sets of propositions, but ordered sets of such sets. Second, discerning what is true at a time (i.e., a set of tensed propositions) in a timeline, unlike discerning what is true at a traditional possible world (i.e., a set of tenseless propositions), is not simply a matter of checking to see which propositions the relevant set contains. Every proposition contained in a time in a timeline is true at that time in that timeline, but a proposition may be true (within a timeline) at times that do not contain it. Third, unlike traditional possible worlds, timelines stand in interesting parthood relations. In formal terms, we can say that timeline T is an initial segment of timeline T’ or that T’ is an extension of T, just in case every ersatz time in T also occurs in T’, if eRe’ in T, then eRe’ in T’, and if eRe’ in T’, then either eRe’ in T, or for all e* ∈ T, e*Re’ in T’. (We could define final-segment and middle-segment relations between timelines in a similar way, but the initial-segment relation is the one that will turn out to be most theoretically useful.) With the apparatus set forth so far, the Growing-Block theorist is equipped to start talking about the fixed past and the open future. The actual world concretely instantiates a timeline—the actualized timeline. There are a number of physically possible extensions of the actualized timeline. (Call these feasible timelines.) Feasible timelines,

268 | Rachael Briggs and Graeme A. Forbes like the actualized timeline, correctly represent the denizens of the actual world. Furthermore, like the actualized timeline, they evolve in a way compatible with the dynamic physical laws that govern the universe. So in a sense, they accurately represent the world— that is, everything in the world, they represent accurately. But unlike the actualized timeline, they represent some things that never happened. So in another sense, they fail to accurately represent the world—they represent it as containing things it does not in fact contain. Feasible timelines are possible timelines, in some very strong sense of ‘possible’. For all that the world determines, any feasible timeline might be actualized in the future, but the world does not determine which feasible timeline will be actualized in the future. Feasibility can be understood as a possibility relation among timelines. ‘It is feasible that’ behaves as a type of possibility operator, and its dual as a necessity operator. We will assume that the feasibility relation is transitive (what is feasibly feasible is feasible) and reflexive (every timeline is feasible relative to itself). Furthermore, feasibility is antisymmetric (no two distinct timelines are feasible relative to each other). The antisymmetry of the feasibility relation follows from the antisymmetry of the extension relation, together with the fact that what is feasible relative to a timeline must be an extension of it. A set of feasible timelines for a time can be represented as a tree structure, like the one in Figure 1. Notice that Figure 1 depicts a great many timelines, since any initial segment of a timeline is itself a timeline. Some of those timelines, which represent particular kinds of possibilities, will play a special role in the semantics of sentences about the future. We can make the following distinctions between three types of timelines. Some timelines end in a way that (together with the laws of nature) precludes any future happenings; they contain a final ‘big crunch’, after which all space, time, and matter disappear, or their laws of nature permit a certain boundless span of time—which is filled up with events that carry on indefinitely—and nothing more. Call these timelines complete. Other timelines seem to end abruptly, with tigers frozen mid-leap, the planets stopped in their orbits, and human actors posed in lifelike tableau. The laws of nature require that these timelines go on into the future; call them incom-

The Real Truth about the Unreal Future | 269

Key: concrete times ersatz times

earlier

later

Figure 1. The Growing-Block and Ersatz Tree

plete. Finally, there may be timelines that the laws of nature permit, but do not require, to extend into the future. Call these semi-complete. We will leave open the question of how to distinguish among the three types of timelines, the question of whether there are any complete timelines, and the question of whether there are any semicomplete timelines. We will, however, assume that every timeline has either a feasible complete extension or a feasible semi-complete extension—that it is possible to tack on enough time to reach an acceptable stopping point. The concepts of completeness and semi-completeness are crucial because they are linked to the concept of an exhaustive description—

270 | Rachael Briggs and Graeme A. Forbes one that details everything that ever has happened, is happening, or will happen. A complete or semi-complete timeline can (in some appropriate nomological sense) be exhaustive in this sense, while an incomplete timeline cannot. When it comes to assigning truth values to sentences about the future, based on a timeline, it will be important to know whether the timeline is exhaustive—whether it catalogues everything that will be, in addition to everything that is. The terms ‘complete’ and ‘incomplete’ have logical connotations: a classically complete valuation function is one that assigns value 0 or 1 to every sentence in some language, while a classically incomplete valuation function fails to assign values to some some sentences in the language. We welcome these connotations: it will turn out that every complete timeline is associated with a classically complete valuation function, incomplete timelines tend to be associated with classically incomplete valuation functions, and semi-complete timelines play two roles—one in which they must be associated with classically complete valuation functions, and one in which they may be associated with classically incomplete valuation functions.

2. EVALUATING FUTURE CONTINGENTS Consider the tensed propositions expressed by the following sentences. 1. Exactly one day into the future, there will be a sea battle. (FdS) 2. Exactly one day ago, there was a sea battle. (PdS) 3. Exactly one day into the future, it will be the case that: either there is a sea battle, or there is not. (Fd(S ∨ ¬S)) 4. Either there will be a sea battle exactly one day into the future, or there will be no sea battle exactly one day into the future. (FdS ∨ Fd(¬S)) 5. Either there will be a sea battle exactly one day into the future, or it is not the case that there will be a sea battle exactly one day into the future. (FdS ∨ ¬Fd(S)) 6. Someone will win tomorrow’s lottery drawing (though not necessarily anyone in particular). (Fd(ΣxLx)) 7. There is someone (in particular) who will win tomorrow’s lottery drawing. (Σx(FdLx))

The Real Truth about the Unreal Future | 271 How should one ascertain the truth values of these propositions? Do all of them have truth values? Do any? One option for the Growing-Block theorist is to claim that, since all of the sentences except (2) make apparent reference to a future that does not, strictly speaking, exist, all except (2) are either necessarily false or necessarily truth valueless. We think this option should be rejected. (1) looks like be the sort of proposition that can be known, at least under the right circumstances. If I know that the ships have already been launched on a collision course, and that there is no nomologically possible way of preventing the battle, then I appear to be in a position to know (1). But surely what can be known can be true. This argument is not unanswerable. Someone who believes that all the example sentences but (2) are truth valueless might reply that when one knows the ships have been launched beyond recall, one’s attitude toward (1) is properly described not as knowledge, but as justified belief. (There will have to be some further story about what justified belief amounts to when the proposition believed is truth valueless, and knowably truth valueless.) But on the face of it, there appear to be circumstances in which (1) is true, which can usefully be contrasted with circumstances in which (1) is false (e.g., circumstances in which no one has any ships, and it is not physically possible to make any by tomorrow). So (1) seems to be true at some times in some timelines, and false at other times in other timelines. Are there also times when (1) is neither true nor false? The Growing-Block theorist has good reasons to say “yes”. Sometimes, the past, the present, and the laws of nature are not enough to settle whether there will be a sea battle one day into the future.5 But according to the GrowingBlock theory, there is nothing but the past, the present, and the laws of nature. If something is not settled by the past, the present, and the laws of nature, then it is not settled. Where nothing in the world settles whether a sentence is true or false, that sentence must not have a truth value—there is no truth without some sort of truthmaking. 5 Even if the actual world is deterministic, we would not want to rule out the possibility of indeterministic worlds.

272 | Rachael Briggs and Graeme A. Forbes Again, our argument is not unanswerable. Roy Sorensen (2001) and Patrick Greenough (2008) defend the position that all propositions are either true or false, that some true propositions are not made true by anything in the world, and that cases of indeterminacy are cases of truthmaker gaps rather than truth value gaps. The Sorensen–Greenough view has a notable payoff: its proponents can keep classical logic, while proponents of truth value gaps must give classical logic up. Still, contingent truths that are not grounded in reality seem puzzling and spooky. We think they are so spooky that on balance, it is best to assume that (1) is sometimes truth valueless (as well as sometimes true and sometimes false). Hardcore adherents of the Sorensen–Greenough approach, however, can re-interpret our discussion of truth and falsity as a discussion of grounded truth and grounded falsity. (2), unlike (1), is about past events. The past—no matter what it was like—seems to be enough to fix the truth value of (2).6 This suggests that unlike (1), (2) is always either true or false according to the Growing-Block theory. The possible indeterminacy of (1), when taken together with the necessary determinacy of (2), is one way of cashing out the idea that the future is open while the past is fixed. Sentences (3)–(5) all appear to be logical consequences of (1). We will assume that appearances are right, although they could conceivably be disputed. (Disputation is easiest in the case of (3), since (4) and (5) follow from (1) by tautological consequence alone.) One might wonder whether in addition to being entailed by (1), (3)–(5) are logically valid—that is, whether they are entailed by the empty set. (3) looks valid at first glance—it is simply an instance of the law of excluded middle embedded in the scope of a future operator. And isn’t tomorrow bound to obey the law of excluded middle, just like yesterday and today? On second thought, though, what if there is no tomorrow? (3) is arguably false at the last moment of the universe.

6 Here and throughout, we ignore the complications caused by unrelated phenomena such as vagueness and the Liar. A theory that addressed these complications might replace our ersatz times—assumed to be classical—with ersatz times that were complete and consistent by the lights of some suitable non-classical logic.

The Real Truth about the Unreal Future | 273 (4) seems to say something stronger than (3)—not just that the future obeys the law of excluded middle, but that it is determined by the past and the present, at least as far as sea battles are concerned. For (4) is a disjunction, and each of its disjuncts entails that the future is determined, at least as far as sea battles are concerned. One could object to the inference from the claim that each of (4)’s disjuncts entails that the future is determined to the claim that (4) itself entails that the future is determined. The supervaluationist view we discuss in section 2.1 takes this objection seriously. Still, there is at least a prima facie case for thinking that (4) might not be true, even when (3) is. (5) is a classical tautology. This suggests that (5) is valid. But to all three of the proposals we consider, there are situations where the sentences FdS and ¬FdS are both untrue (that is, not true—either false or indeterminate). If (5) is valid, then it is sometimes true without having a true disjunct—a puzzling state of affairs! (6) says simply that a prize will be given out tomorrow. (7) appears to say something stronger—that the lottery is a fix. The relationship between (6) and (7) is closely analogous to the relationship between (3) and (4). (3) says that a particular disjunction will be true tomorrow, while (4) says that one of its disjuncts will be true tomorrow. Likewise, (6) says that a particular existentially quantified statement will be true (of something that did, does, or will exist) tomorrow, while (7) says that there did, does, or will exist something that will witness that existentially quantified statement tomorrow. (4) and (7) seem to claim that certain questions (whether there will be a sea battle, who will win the lottery) are settled, while (3) and (6) seem not to make such claims. We have chosen to translate both (6) and (7) using the tenseless quantifier Σ rather than the tensed quantifier ∃. Our translation of (6) does not require that the winner of tomorrow’s lottery be someone who will exist tomorrow. (For all the logic says, the winner may exist today but be dead by tomorrow’s drawing, or may not be born until several years into the future.) Likewise, our translation of (7) does not require that the future winner of tomorrow’s lottery be someone who exists today. (For all the logic says, the winner may be dead already, or not yet born.) The aim of this choice is simply to avoid the complications that would result from changes in the domain of quantification.

274 | Rachael Briggs and Graeme A. Forbes We will consider three proposals for assigning truth values to sentences like (1)–(7): a supervaluationist proposal, a proposal inspired by Łukasiewicz’s three-valued logic, and an intuitionist proposal. Each proposal has different strengths and requires different tradeoffs. We endorse the intuitionist proposal, but think the other two proposals are serious contenders worthy of attention. Proponents of any non-classical logic must reject some intuitively plausible claims. All of the following claims are highly plausible, but there are numerous incompatibilities among them, which illustrate the tradeoffs available to the Growing-Block theorist. (a) (b) (c) (d) (e)

(f) (g) (h)

There are some propositions such that (it is at least possible that) neither they nor their negations are true. A disjunction can be true only if one of its disjuncts is. All classical tautological inferences are valid. All other things being equal, truth values should be assigned to as many propositions as possible. A language for describing the open future should be sufficiently powerful to include an expression for determinate truth. Reductio ad Absurdum, Contraposition, and Conditional Proof are valid natural deduction inferences. Truth is determinate truth: a sentence is true just in case it is guaranteed by history together with the laws of nature. Truth is disquotational: if a truth operator Tr is introduced into the language, every instance of the schema ϕ ≡ Trϕ should be valid.7

All three of our proposals are committed to (a). They count (1) as sometimes true, sometimes false, and sometimes indeterminate. They also agree in counting (2) as always either true or false, and (3) as true in circumstances where the universe is guaranteed to persist until tomorrow, false in circumstances where the universe is guaranteed not to persist until tomorrow, and indeterminate otherwise. For reasons we discussed at the beginning of this section, Growing-Block theorists should accept (a). There are (or could be) propositions about the future whose truth values are not settled by the 7 Some exceptions will have to be made in cases of self-reference; we ignore these complications here.

The Real Truth about the Unreal Future | 275 past, the present, and the laws of nature. For Growing-Block theorists, this means that there are (or could be) propositions whose truth values are not settled at all. Philosophers who accept (a) must make tradeoffs elsewhere. First, they must choose between giving up (b) and giving up (c). By (a), there is some disjunction ϕ ∨ ¬ϕ, neither of whose disjuncts is true—(5) is such a disjunction under some possible circumstances. By (b), (5) is sometimes untrue. But since (5) is tautologically valid, by (c) it is always true. So rejecting either (b) or (c) is mandatory. The supervaluationist proposal gives up (b) and keeps (c).8 It treats (3) and (4) as equivalent, and (6) and (7) as equivalent. The Łukasiewicz and intuitionist proposals, on the other hand, give up (c) and keep (b). They are able to treat (4) as stronger than (3) and (7) as stronger than (6). (d) need not be accepted or denied wholesale, but can be accepted to a greater or a lesser extent, depending on how important one thinks it is to assign truth values to as many sentences as possible. If we treat (d) as a desideratum for theories, the supervaluationist semantics does better than the intuitionist semantics, which in turn does better than the Łukasiewicz semantics. All three proposals are outdone by classical logic, but we think the cost of classical logic for the Growing-Block theorist—spooky truths without truthmakers— is excessive. In addition to the tradeoff between (b) and (c), and the decision about how seriously to take (d), accepting (a) necessitates a tradeoff between (e) and (f). In any sufficiently powerful semantics that allows for truth value gaps, one can formulate propositions for which Reductio ad Absurdum, Contraposition, and Conditional Proof fail. (A reminder: these three inferences are as follows:

Reductio ad Absurdum

If ϕ ⊢ ⊥, then ⊢ ¬ϕ.

Contraposition

If ϕ ⊢ ψ, then ¬ψ ⊢ ¬ϕ.

Conditional Proof

If ϕ ⊢ ψ, then ⊢ ϕ ⊃ ψ.

8 At least, it keeps (c) in the sense that it validates all classically valid single-conclusion arguments. Classically valid multiple-conclusion arguments are supervaluationistically invalid.

276 | Rachael Briggs and Graeme A. Forbes Since entailment is transitive and ¬⊥ follows from the empty set, Reductio ad Absurdum is the special case of Contraposition where ψ = ⊥.) The above inferences fail in three-valued logics because they rely on the assumption that there is no gap between truth and falsity. If ϕ entails ψ, then there is no time in any timeline where ϕ is true and ψ is false. But where there are truth value gaps, ϕ’s entailing ψ is compatible with there being times in timelines at which ϕ is indeterminate and ψ is false. In any sufficiently powerful language, one will be able to define a ϕ and a ψ such that ϕ entails ψ, and ϕ may be indeterminate where ψ is false. Possibilities in which ϕ is false and ϕ is not true can be redescribed as possibilities in which ¬ψ is true and ϕ is not true. In such possible circumstances, ϕ ⊃ ψ—a conditional with a false consequent and an indeterminate antecedent— should not count as true. Even if ϕ entails ψ, ¬ψ does not necessarily entail ¬ϕ, nor is the conditional ϕ ⊃ ψ necessarily valid. Finally, accepting (a) means trading off between (g) and (h). If (a) is correct, then there are some sentences that are indeterminate—not false, but not true. It is false that these sentences are determinate. So, where ϕ is a proposition with indeterminate truth value, and Δ is a determinacy operator, Δϕ is false. When a biconditional has an indeterminate sentence on one side and a false sentence on the other, the entire biconditional should not count as true—it is either indeterminate or false. (This claim could be disputed, but it holds according to all the interpretations of the biconditional we consider.) Therefore, where ϕ is indeterminate, Δϕ is false, and so the biconditional ϕ ≡ Δϕ is not true.

2.1 The Supervaluationist Semantics Richmond Thomason (1970) has proposed a supervaluationist semantics for future contingents, which can easily be adapted to suit the Growing-Block theorist’s ontology. Our discussion of Thomason will be somewhat technical, but the gist of his theory is this: ϕ is true if every possible future course of events will make ϕ true, false if every possible future course of events will make ϕ false, and indeterminate otherwise. More technically, for Thomason sentences receive truth values at times in model structures. Each of Thomason’s times is associated

The Real Truth about the Unreal Future | 277 with a classical valuation function over sentences without tense operators—our ersatz times are therefore well suited to play the role of Thomason’s times. A model structure, for Thomason, is a set of times partially ordered by a relation R. Thomason assumes that R is ‘treelike’—meaning that each time has a unique past, or more formally, that for all times e, e’, and e* in the model structure, if e’Re, e*Re, and e’ ≠ e*, then either e’Re* or e*Re’. Truth at a time in a model structure is defined in terms of truth at a time in a history—a maximal set of times h such that for any two distinct times in the set, one bears the R relation to the other. (One can picture histories as branches of the tree stretching all the way from root to leaf-tip.) A proposition ϕ, says Thomason, is true at e according to a model structure M if veh(ϕ) = 1 for every h in M such that e ∈ h, false at e according to M if veh(ϕ) = 0 for every h in M such that e ∈ h, and indeterminate at e in M otherwise. The next step, then, is to define truth at a time in a history. Where e is any time and h is any history such that e ∈ h, e and h are associated with a classical valuation function veh, built up in a familiar fashion. For any atomic sentence p, veh ( p) =

1 if p is true according to e 0 else

Thomason adds the usual clauses for the truth-functional connectives (he does not discuss quantification) and the following clauses for tense operators P and F. veh (Pf ) =

1 if h contains an e’ such that e’ Re and ve ’ h (f ) = 1 0 else

veh (Ff ) =

1 if h contains an e’ such that eRe’ and ve ’ h (f ) = 1 0 else

Notice that so far, the definition of veh looks exactly like the definition of veT, except that the F operator is fully rather than partially defined. F(ϕ) is false at e in h whenever it is not true at e in h. Before we go on defining veh for a larger range of operators and quantifiers, it is worth pausing to think about how Thomason’s system maps onto ours.

278 | Rachael Briggs and Graeme A. Forbes We have already said that Thomason’s times correspond to our ersatz times. In order to finish adapting Thomason’s semantics to our ontology, we will need to find objects in our ontology to play the role of his model structures and histories. Sets of the form {T’: T’ is a feasible extension of T} are ideally suited to play the role of model structures. These sets, as we pointed out in section 1.4, have a tree-like structure. Furthermore, they are exactly the right sorts of things to relativize truth to. Each set is uniquely associated with a timeline, and timelines represent possibilities. Since our ultimate aim is a concept of something like truth in a possible world, we want to relativize truth to timelines (or to entities that admit of a natural one-to-one mapping to timelines). Given a timeline and the tree structure consisting of its feasible extensions, we will need to pick out some entities to serve as its histories. Histories in Thomason’s system must be something like exhaustive catalogues of events. This must be the underlying rationale for the semantic rule that treats F(ϕ) as false at e in h whenever h fails to contain an e’ later than e at which ϕ is true. If h does not explicitly represent the universe as containing a time at which ϕ, then h represents the universe as not containing a time at which ϕ. Complete timelines are well-suited to play the role of histories in Thomason’s system. They are exhaustive catalogues of events, or to put it another way, they describe ways the Growing Block might finish up, if it were allowed to grow until it could grow no more. What about semi-complete timelines? There seem to be two possible ways of considering them. On the one hand, one can consider them under the assumption that they are exhaustive catalogues of the universe’s events—complete descriptions of the way the Growing Block will end up. (After all, it is nomologically possible for the Growing Block to grow until it actualizes a semi-complete timeline, and then stop growing.) In other words, semi-complete timelines can be considered as histories. This is the appropriate stance if one is thinking from the perspective of a particular timeline, considering all possible exhaustive ways things might go next. The semicomplete extensions of the original timeline are indeed exhaustive ways things might go next. On the other hand, one can take semi-complete timelines at face value, not assuming that they are exhaustive. (After all, it is nomologically possible that the Growing Block to grow until it actualizes

The Real Truth about the Unreal Future | 279 a semi-complete timeline, and then keep growing.) This is the appropriate stance to take if one is thinking from the perspective of a semi-complete timeline, wondering what is true according to it. It is a way of thinking of the semi-complete timeline in itself. Depending on how we think of a semi-complete timeline—either as a history or in itself—different sentences will come out as true at various times in that timeline. Where e is a time in a semi-complete timeline T, what is really true at e in T is what is true at e according to T considered in itself. But for the purposes of divining what is true at times in other timelines, it may be necessary to consider what is true at e according to T considered as a history. One must treat complete timelines as histories, since it is nomologically necessary that they be exhaustive. Unlike the case of semicomplete timelines, there is no difference between considering complete timelines as histories and considering them in themselves. And one may not treat incomplete timelines as histories—they cannot be exhaustive catalogues of the world’s events. Growing Blocks that actualize incomplete timelines will have an irresistible impetus to keep growing, adding more to the catalogue of the world’s events. Incomplete timelines are not exhaustive, and should only be considered in themselves. Since histories are timelines considered as exhaustive representations of the universe, we can add the following clauses to the definition of veh. 1 if h contains an e’ such that eRe’, the distance from e to e’ is d , veh (Fdf ) = and ve ’ h (f ) = 1 0 else veh Σx(f x ) =

1 if for some e’ in h , there is an a ∈De ’ such that ve ’ h (f x / a) = 1 0 else

0 if for some e’ in h, there is an a ∈De ’ such that ve ’ h (f x / a) = 0 veh Πx(f x ) = 0 else So truth for sentences whose main operators are F, Fd, Σ, and П is defined for histories exactly as it was defined for incomplete and semi-complete timelines considered in themselves. But for histo-

280 | Rachael Briggs and Graeme A. Forbes ries, the definition of falsity is more extensive: sentences are false whenever they are not true. In incomplete timelines, and in semicomplete timelines considered in themselves, there may be truth value gaps; in histories there are none. How do propositions about the future get their truth values in incomplete timelines, or in semi-complete timelines considered in themselves? For us, each timeline is associated with a model structure—the set of its complete or semi-complete extensions. We can say that ϕ is true at e in T just in case it is true at e in the model structure associated with T. Now it only remains to interpret the concept of truth in a model structure, according to our conception of model structures as sets of extensions of individual timelines. Every model structure contains histories: complete timelines and semi-complete timelines which can be considered as histories. So, mapping our ontology onto Thomason’s semantics yields the following inheritance clause: Where T is an incomplete timeline, or a semi-complete timeline considered in itself, 1 iff for every extension h of T such that h is either a complete timeline or a semi-complete timeline treated as a history, veh (f ) = 1 veT (f ) = 0 iff for every extension h of T such that h is either a complete timeline or a semi-complete timeline treated as a history, veh (f ) = 0 If the supervaluationist semantics is to be consistent, the new inheritance clause cannot assign values to compounds which are incompatible with the values assigned to those compounds by the definitions of conjunction, negation, and the tense operators. Fortunately, the types of sentences introduced so far have a convenient heredity property (proved in the Appendix). Any sentence that receives value 0 or value 1 at e in T must receive the same value at e in every feasible extension of T. So the inheritance

The Real Truth about the Unreal Future | 281 clause cannot generate any inconsistency in the valuation function—at least for the types of sentences considered so far. (For a discussion of some sentences that lack the heredity property, see section 3.1.) How does the supervaluationist semantics handle sentences (1)–(7)? (1) comes out true at e in T whenever each complete or semi-complete extension of T contains a sea battle exactly one day after e. In other words, (1) is true whenever a sea battle is guaranteed true by the absolute past and present together with the laws of nature. Likewise (1) is false at e in T whenever no complete or semi-complete extension of T contains a sea battle exactly one day after e—whenever the absolute past and present, together with the laws of nature, guarantee that there will be no sea battle. (1) is indeterminate whenever the absolute past, the absolute present, and the laws of nature fail to determine whether there will be a sea battle exactly one day after e. (2), unlike (1), always comes out as either determinately true or determinately false, since all feasible timelines agree about what happened exactly one day before e. (3) comes out as true when all feasible complete timelines are ones according to which there is an e’ exactly one day after e—when there is guaranteed to be a tomorrow. Likewise, (3) comes out as false when there is guaranteed not to be a tomorrow, and indeterminate when tomorrow’s existence is uncertain. (4) comes out as true in exactly the same circumstances as (3). This is somewhat surprising. (4) seems to say that the future is determined, at least as far as sea battles are concerned. After all, (4) is a disjunction, each of whose disjuncts says that the future is determined in a particular way. But (4) can be true at e in T even when some feasible extensions of T contain a sea battle a day after e and others contain no sea battle a day after e. The trick is that on the supervaluationist picture, (4) can be true without either of its disjuncts being true. Consider the following example, depicted in Figure 2. e is the absolute present of an incomplete timeline T, which has two feasible complete extensions and no semi-complete extensions. According to one feasible history that extends T, FdS is true at e; according to the other, Fd¬S is true at e.

282 | Rachael Briggs and Graeme A. Forbes

A Timeline and Two Extensions

T1

T

e

e⬘ such that Ve⬘T1(S)=1

T2

Figure 2. A Timeline and Two Extentions

Sentence (4), the disjunction FdS ∨ Fd¬S, is true at e according to both histories that extend T. So by the inheritance clause, (4) is true at e in T. But neither disjunct of (4) is true at e in both histories that extend T—different disjuncts are true in different histories. So neither disjunct of (4) is true at e in T. (5) is a classical tautology—an instance of the Law of Excluded Middle—and therefore true at every time in every timeline. Like (4), (5) can be true without either of its disjuncts being true. (In fact, the reader can confirm that according to the proposed semantics, (5) will be true without either of its disjuncts being true whenever (4) is true without either of its disjuncts being true.) It seems like a point in favour of supervaluationism that it treats (5) and all other classical tautologies as valid. (6) comes out as true at e in T whenever, according to all feasible timelines, there is a lottery one day after e, and history together with the laws of nature guarantee that the lottery will be won. (Perhaps this scenario is farfetched when it comes to real lotteries, which typically stand some chance of being called off. But we will avoid tinkering with qualifiers in order to keep the logical point as clear as possible.) (7) comes out as true whenever (6) does. This putative equivalence is surprising, just as the putative equivalence between (3) and (4) was surprising. (7) appears to say something stronger than (6)—that the lottery is a fix. But (7) is true so long as the absolute past, the absolute present, and the laws of nature guarantee that a prize will be given out. In such circumstances, it is

The Real Truth about the Unreal Future | 283 true according to every history that extends the actualized timeline that there is someone who will win the lottery. There need not (tenselessly) exist anyone of whom it is true that they will win the lottery—who wins the lottery on every history that extends the actualized timeline. Just as (4) was a disjunction that could be true without having any true disjunct, so (7) is an existentially quantified statement that can be true without having any witness. 2.2 The Łukasiewicz Semantics Growing-Block theorists who find the supervaluationist treatment of (4) and (7) unsatisfactory might try a different tradeoff between (b) and (c). Instead of emphasizing the link between what is true and what is guaranteed to become true no matter how the future goes, they might emphasize the importance of preserving the meanings of certain logical connectives—in particular, the truth-functionality of of disjunction and the extensionality of existential quantification. Supervaluationists exploited a useful insight about the future: what can be truly said of future events is roughly what is guaranteed to be true by the past, the present, and the laws of nature. But it was a mistake, opponents of supervaluationism might claim, to treat this insight as though it applied to all propositions. Propositions whose main operator is F are true whenever history and the laws of nature guarantee that they will become true—i.e., a proposition of the form Fϕ is true at e in T whenever it is true at e in all feasible extensions of T. But no such claim holds for propositions in general. (In particular, no such claim holds for disjunctions and existentials.) Instead, the supervaluationist’s inheritance clause should be restricted to sentences whose main operators are tense operators, as follows. Where ϕ’s main operator is a tense operator, 1 if for every complete or semi- complete extension T ’ of T , veT ’ = 1 veT (f ) = 0 if for every complete or semi- complete extension T ’ of T , veT ’ = 0

284 | Rachael Briggs and Graeme A. Forbes The other connectives can be defined exactly as before, and veT can be taken to be the smallest fixed point satisfying the recursive definition of the value function. This way of defining the connectives yields an extension of the logic advocated by Jan Łukasiewicz (1970). (Łukasiewicz outlines definitions for truth-functional connectives and tense operators, but does not discuss quantifiers.) Unlike the supervaluationist semantics, the Łukasiewicz proposal treats the connectives as truth-functional: the truth of a disjunction (existential) will always be appropriately grounded in the truth of one or more disjuncts (witnesses). The benefits of the Łukasiewicz system are not without their costs, however. In particular, Łukasiewicz’s logic does much worse than supervaluationism on criterion (d)—it assigns truth values to far fewer sentences. As a result, it treats fewer sentences as valid— every classical tautology has untrue instances in the Łukasiewicz logic, including the law of excluded middle. For instance, (5) is indeterminate at e in T whenever some but not all semi-complete extensions of T contain an e’ exactly one day after e at which there is a sea battle. The situation is not quite as bad as one might expect from the observation that Łukasiewicz’s truth-functional logic has no theorems, however. Sentences without F operators or tenseless quantifiers behave classically, so that every sentence that is both valid in first-order logic and free of F operators and tenseless quantifiers is a theorem. Since the Łukasiewicz semantics agrees with the supervaluationist semantics in its treatment of the operator, it agrees with the supervaluationist semantics in its treatment of (1) and (2). (1) is indeterminate at T in whenever T‘s feasible semi-complete extensions disagree about whether there is a sea battle exactly one day after, while (2) is determinate at every time in every timeline. The Łukasiewicz semantics also agrees with the supervaluationist semantics in its treatment of (3) and (6). (3) is true at e in T provided there will be a future—i.e., provided all of T‘s feasible complete or semi-complete extensions contain an e’ exactly one day after e. And (6) is true at e in T whenever a prize will be given out for the lottery—i.e., whenever all of T’s feasible complete or

The Real Truth about the Unreal Future | 285 semi-complete extensions contain an e’ exactly one day after e at which the lottery is won. But in the Łukasiewicz semantics, unlike the supervaluationist semantics, (4) and (7) are stronger than (3) and (6). (4) is true only if one of its disjuncts is true—that is, (4) is true at e in T only if either all of T’s feasible complete or semi-complete extensions, or none of them, contain an e’ exactly one day after e at which there is a sea battle. So (4) says that the past, the present, and the laws of nature settle whether there will be a sea battle tomorrow. Likewise (7) is true only if one if its instances is true— that is (7) is true at e in T only if the same person wins the lottery a day after e in all of T’s feasible extensions. So (7) says that the lottery is a fix. 2.3 The Intuitionist Semantics Łukasiewicz’s semantics can be strengthened by making slight changes to the definitions of negations and the tenseless operators, and by adding a non-truth-functional conditional. if for all complete or semi- complete extensions T ’ of T , veT (¬f ) = veT ’ (f ) ≠ 1 0 if veT (f ) = 1 1

if for all complete or semi- complete extensions T ’ of T , veT (f ⊃ y ) = veT (f ) ≠ 1 or veT (y ) = 1 0 if veT (f ) = 1 and veT (y ) ≠ 1 1

if for some e’ in T , there is an ∈De ’T such that ve ’T (fx/a) = 1 veT ∑ x(fx ) = 0 if for every extension T ’ of T and every e’ in T ’, there is no a ∈De ’T ’ such that ve ’T (fx/a) = 1 1

1 if for every extension T ’ of T and every e’ in T ’, there is no a ∈De ’T ’ such that ve ’T (fx/a) = 0 veT Πx(fx ) = 0 if for some e’ in T , there is an a ∈De ’T ’ such that ve ’T (fx/a) = 0

286 | Rachael Briggs and Graeme A. Forbes Both changes can be reasonably motivated. In the case of negation, it seems sufficient for ¬ϕ’s being true that ϕ should be guaranteed never to become true—even if ϕ is not guaranteed to become false. Consider the example of a universe that is divided into epochs, such that the laws of nature permit it to end after any finite number of epochs, but forbid its going on for more than finitely many epochs. Consider the sentence ‘There will be a last time’ (F¬F⊤), as uttered sometime e in the middle of the first epoch, in a timeline T where e is the absolute present. This sentence is not false, but it is not true, and can never become true. (After every epoch there always might be another.) So it seems reasonable to say that there won’t be a last time—that ¬F¬F⊤ is true at e in T. Likewise, in the case of the quantifiers, it seems sufficient for Πx(ϕx)’s truth (Σx(ϕx)’s falsity) that the laws of nature and guarantee that there will be no counterexamples (witnesses). In order to make a true universal generalization, we needn't guarantee, for all past, present, and future things, that they support the generalization. We need only guarantee that all past, present, and future things support the generalization, whatever they happen to be. The new conditional is really a strict conditional: ϕ ⊃ ψ is true at e in T just in case, at all feasible timelines, ¬ϕ ∨ ψ is true at e. As a strict conditional, it captures the idea that the antecedent’s truth necessitates the truth of the consequent, where the kind of necessity is feasibility. Notice that ¬ϕ is equivalent to ϕ ⊃ ⊥. The result of adding the strengthened negation and conditional operators is something very close to the semantics for intuitionism proposed by Saul Kripke (1963). Our feasibility relation satisfies the requirements on the accessibility relation in the Kripke semantics for intuitionism—it is transitive, reflexive, antisymmetric, and (as proved in the Appendix) hereditary—and our semantics defines truth for conjunctions, disjunctions, negations, conditionals, and quantified sentences in the same way as Kripke’s. There are, however, a number of notable differences between our semantics and Kripke’s. First, our language contains the F operator, which is undefined in Kripke’s semantics. Second, we give falsity conditions in addition to truth conditions for our propositions. (Kripke uses the word ‘false’ for the sentences that we are calling ‘untrue’.) Third, our incomplete timelines generate a slight complication. While incomplete timelines are feasible extensions of themselves,

The Real Truth about the Unreal Future | 287 they are not complete or semi-complete feasible extensions of themselves. T’ is accessible to T, in the sense required by Kripke’s semantics, just in case T’ is a complete or semi-complete extension of T. So even though the feasibility relation is reflexive, our analogue of Kripke’s accessibility relation is not. (It is reflexive only when we restrict our attention to complete and semi-complete timelines.) This complication, however, makes little logical difference—we can simply declare each incomplete timeline to be Kripke-accessible relative to itself, without changing any logical features of the semantics. The intuitionist semantics does better than the Łukasiewicz semantics on criterion (d). It is stronger than the Łukasiewicz semantics in two notable senses. First, at any time in any timeline, the set of true sentences according to the Łukasiewicz semantics is a subset (and sometimes a proper subset) of the set of true sentences according to the intuitionist semantics. Second, the set of theorems according to the Łukasiewicz semantics is a proper subset of the set of theorems according to the intuitionist semantics. (The set of inferences validated by the intuitionist semantics is neither larger nor smaller than the set of inferences validated by the Łukasiewicz semantics—they are simply incomparable.) Our motivation for the proposed intuitionist view differs from that of traditional intuitionism, and from the intuitionism about the future (and the past) advocated by Michael Dummett (1978a; 1978b). Traditional intuitionists treat mathematical truth as essentially constrained by evidence, in the form of constructive proof. Dummett extends the idea of evidential constraints on truth to empirical propositions, including propositions about times other than the present. A proposition is true for Dummett insofar as it admits of constructive proof from available evidence, and false for Dummett insofar as it admits of constructive disproof from available evidence. For Dummett, these evidential constraints on truth spring from considerations about the meanings of logical and non-logical terms—meanings are truth conditions; hence one knows the meaning of a term only insofar as one can apply it in various evidential situations. Our approach emphasizes not evidence and meaning, but truthmaking and grounding. A sentence about the future of the form Fϕ is true only if its truth is grounded by what exists, i.e., past and present things and the laws of nature. A sentence with any other logically complex form is true only if its truth is appropriately

288 | Rachael Briggs and Graeme A. Forbes grounded in the truth of simpler statements—disjuncts, conjuncts, or instances. We see no reason to require that all truths about the future be knowable, or that they be grounded by knowable truths about the past and present. Our version of the intuitionist proposal requires only that they be grounded in past and present reality.

2.4 Taking Stock So far, we have shown how supervaluationism, Łukasiewicz logic, and intuitionism—all ways of cashing out (a)—require different tradeoffs between (b), (c), and (d). Supervaluationists reject (b) for (c), and do better than advocates of Łukasiewicz logic and intuitionism by the lights of (d). Advocates of Łukasiewicz logic and intuitionism reject (c) for (b). Intuitionism outperforms Łukasiewicz logic by the lights of (d), but is outperformed in turn by supervaluationism. Logicians who reject (a) must make additional tradeoffs. The choice between supervaluationism, Łukasiewicz logic, and intuitionism has consequences for (f)—the claim that Contraposition, Reductio ad Absurdum, and Conditional Proof are valid inferences. Choosing Łukasiewicz logic means rejecting (f). Reductio ad Absurdum and Contraposition fail since classical contradictions, such as FS ∧ ¬FS, may be truth valueless rather than false, and may therefore have untrue negations. Conditional proof fails for much the same reason: FS ⊧ FS, but FS ⊃ FS is truth valueless whenever FS is. So advocates of Łukasiewicz logic must give up (f). Intuitionists and supervaluationists have not yet been forced to give up (f), but they are not yet in the clear. Like most theorists who countenance truth value gaps, they must eventually choose between (f) and (e)— the claim that a language for the open future should be sufficiently powerful to include an expression for determinate truth. Furthermore, advocates of all three logics must choose between (g), according to which truth is determinate truth, and (h), according to which truth is disquotational. The tradeoffs between (e) and (f) and (g) and (h) will be illustrated in section 3.1, where we define an operator that captures the concept of determinate truth. Although we have more logical work to do in part 3, we are ready to conclude our discussion of the tradeoffs between the supervalu-

The Real Truth about the Unreal Future | 289 ationist, Łukasiewicz, and intuitionist proposals. We believe that all three proposals are viable responses to the situation that faces the Growing-Block theorist, and that we have presented the reader with the materials necessary to make an informed choice. We now turn our attention to two logical issues that affect all three proposals equally.

3. GENERAL LOGICAL ISSUES The two issues to which we now turn are closely bound up with the concept of expressive power. In section 3.1, we consider how Growing-Block theorists might enrich their language to express the idea of determinate truth. Our results shed further light on the tradeoffs between (e), (f), (g), and (h) discussed in part 2. In section 3.2, we consider the prospects for expressing the GrowingBlock theorist’s ontological commitments in a formal language.

3.1 Determinacy Our metalinguistic discussion of truth at times in timelines has made extensive use of the distinction between sentences that are true (at times in timelines) and sentences that are not true (at times in timelines)—either because they are false or because they are indeterminate. How can the Growing-Block theorist express the thought that a sentence is true, rather than false or indeterminate, using an object language expression in one of the idealized languages we have developed? The usual response to this question is to define a determinacy operator. How exactly should this be done? Thomason defines a type of determinacy operator—the inevitability operator L—as follows: 1 if veh ’ (f ) = 1 for every h’ in the model structure containing e veh (Lf ) = 0 else How should this inevitability operator be understood in terms of truth at times in timelines? The question is somewhat thorny. For any sentence not containing the L operator, we could define truth at a history, and more

290 | Rachael Briggs and Graeme A. Forbes generally, truth at a time in a timeline. But for a sentence whose main operator is L, we can only define truth at a time in a history in a model structure (and derivatively, truth at a time in a timeline in a model structure). Different choices of model structure give different truth conditions for sentences of the form Lϕ. In order to translate L into our formalism, then, we need a method for picking out the relevant model structure in any given situation. One option, in keeping with the spirit of Thomason’s proposal, is to index the relevant model structure (for time e in timeline T ) to the initial segment of T containing e and all the times before it. Interpreting L this way gives us: veT (Lf ) =

1 if veT ’ (f ) = 1 0 else

where T’ is the initial segment of T containing only e and the e’s such that e’Re This interpretation of L makes the model structure drop out of the semantics again—we can assign a univocal truth value at each time in each timeline to sentences of the form Lϕ. Thomason claims that L has the following properties. L1. L2. L3. L4. L5.

ϕ ⊧ Lϕ Fϕ ⊧ LFϕ ⊨ ϕ ⊃ Lϕ ⊭ Fϕ ⊃ LFϕ ⊭ PFϕ ⊃ PLFϕ

(Notice that if Thomason is right, adding L to the language makes it possible to generate counterexamples Conditional Proof, since Fϕ ⊧ LFϕ, but ⊭ Fϕ ⊃ LFϕ. Adding L to the language also makes it possible to generate counterexamples to Contraposition, since ϕ ⊧ Lϕ, but presumably ¬Lϕ ⊭ ϕ) According to our semantics for L, L4 and L5 are invalid, just as Thomason claims. Furthermore, they are invalid regardless of whether we use the supervaluationist semantics, the Łukasiewicz semantics, or the intuitionist semantics. We can produce the following countermodel for both sentences. Let T be a timeline with absolute present e, and some e* such that e*Re. And suppose T has exactly two feasible complete extensions—T1 and T2. T1 contains an e’ such

The Real Truth about the Unreal Future | 291

A Timeline and Two Extensions

T1

T

e*

e

e⬘ such that Ve⬘T1(S)=1

T2

Figure 3. A Timeline and Two Extentions

that eRe’ and the atomic sentence S is true at e’; T2 contains no such e’. Furthermore T has no feasible semi-complete extensions. The example is depicted in Figure 3. In this example, veT2 (FS ⊃ LFS) = 0 , so that veT(FS ⊃ LFS) ≠ 1. Similarly, veT2 (PFS ⊃ PLFS) = 0 , so that veT(PFS ⊃ PLFS) ≠ 1. But according to our semantics for L, and contrary to Thomason’s claims, L1–L3 are also invalid. Once again, they are invalid regardless of whether we use the intuitionist semantics, the supervaluationist semantics, or the Łukasiewicz semantics. In the example just considered, veT1 (FS) = 1 , while veT1 (LFS) = 0; this constitutes a counterexample to L1 and L2. Similarly, veT1 (FS ⊃ LFS) = 0 ; this constitutes a counterexample to L3. Although L1–L3 are not valid in general—that is, not valid eT—they are valid T. Thomason’s model structure runs validity eT and validity T together. For the fragment of the language without the L operator, this conflation is harmless: validity eT and validity T coincide. But in a language containing the L operator, the difference matters. Despite initial appearances to the contrary, the introduction of L does not provide counterexamples to Conditional Proof or Contraposition. Enriching a language with the L operator preserves the heredity property (see Appendix for proof). And in any supervaluationist, Łukasiewicz, or intuitionist language where all sentences satisfy the heredity property, Conditional Proof and Contraposition are valid. Distinguishing between validity T and validity eT also reveals that L is a poor candidate for a truth operator. Not everything true at a time is inevitable at that time, as the invalidity of inference L1

292 | Rachael Briggs and Graeme A. Forbes shows. What is true full stop coincides with what is inevitable at the absolute present, but it would be a mistake to confuse truth at a time with inevitability at that time. We can define a determinacy operator that does a better job of capturing the idea of determinate truth by making a different choice about how to pick out the relevant model structure, given a time and a timeline: veT ( Δf ) =

1 if veT (f ) = 1 0 else

While Lϕ expresses the thought that ϕ is determinately true at e from the perspective of the initial segment of the actual timeline ending at e, Δϕ expresses the thought that ϕ is true at e from the perspective of the actual timeline. Determinate truth at e and inevitability at e correspond when e is the absolute present, but come apart when e is in the absolute past. This definition of the Δ operator creates some trouble for the supervaluationist and intuitionist proposals, in which certain connectives are defined intensionally rather than truth-functionally. Sentences containing Δ violate the heredity condition—a sentence may have a truth value in one timeline without having that truth value in all of its extensions. In supervaluationism, the failure of the heredity condition leads to inconsistent value assignments. In the example shown in Figure 3, veT1 ( ΔFS ∨ ΔF¬S) = veT2 ( ΔFS ∨ ΔF¬S) = 1 —and so by the supervaluationist inheritance clause, veT(ΔFS ∨ ΔF¬S) should be 1 as well. But veT(ΔFS) = veT(ΔF¬S) = 0—and so by the semantic rule for disjunction, veT(ΔFS ∨ ΔF¬S) should be 0. In intuitionism, the failure of the heredity condition leads to highly counterintuitive value assignments. In the example shown in Figure 3, veT(ΔFS) = 0—it is false at e in T that determinately, there will be a sea battle. Yet veT(¬ΔFS) ≠ 1. This is because at e in T, there is some feasible timeline—namely T1—according to which ΔFS is true. Flat-footedly applying the intuitionist semantics yields counterintuitive results. The trouble can be easily resolved by making the following adjustment to both the supervaluationist and the intuitionist semantics. To evaluate sentences containing determinacy operators at a time in T, first set veT ’ ( Δf ) equal to either 1 (if veT(ϕ) = 1) or 0 (other-

The Real Truth about the Unreal Future | 293 wise) for every sentence ϕ and feasible extension T’ of T. This ensures that the heredity condition is satisfied. Then, and only then, evaluate complex sentences in the usual way. Δ, so defined, violates the T schemas. The example in Figure 3 illustrates why. In this example, veT(FS) ≠ 1, so veT ( ΔFS) = 0. But then veT1 ( ΔFS) = 1, so that veT(FS ≡ ΔFS) = 0 according to the intuitionist semantics, and veT(ϕ ≡ ψ) ≠ 1 according to the supervaluationist semantics and Łukasiewicz semantics. We claim that Δ behaves like an actuality operator rather than a disquotational truth operator. The values of sentences containing the Δ operator can take on different values at the same time in the same timeline, depending on which world is considered as actual. This is unsurprising, since Δ denotes a type of truth that is grounded in concrete existence. One cannot tell what is true at a time simply by considering the representational features of a possible world; one needs to know which world is actualized. No wonder, then, that there is a tradeoff between (g), the determinacy conception of truth, and (h), the disquotational conception of truth. Determinacy, like actuality, and unlike disquotational truth, violates the T schema. With determinacy, as with actuality, one might be misled by the fact that the determinacy operator satisfies the T rule. For arbitrary ϕ, ϕ ⊧ Δϕ, and Δϕ ⊧ ϕ. But since Conditional Proof is invalid in a language with a determinacy operator, this does not allow us to infer instances of the T schema. This would be just as invalid as inferring the validity of the biconditional ⌜ϕ ≡ actually ϕ⌝ from the fact that ϕ and ⌜actually ϕ⌝ always take the same truth value at the actual world. 3.2 Quantifier Trouble The eternalist quantifiers Σ and П create trouble for the GrowingBlock theory. Suppose that in T, the actualized world, e is the absolute present. And let B be the property of being born after e. Suppose the absolute past and present, together with the laws of nature, determine that someone will be born after e. What are we to make of the sentence ΣxBx? The supervaluationist semantics counts it as determinately true that ΣxBx. But the Growing-Block theorist is committed to the claim that only past and present things exist. Things look just as bad for

294 | Rachael Briggs and Graeme A. Forbes the intuitionist and the advocate of Łukasiewicz logic if the past and present, together with the laws of nature, decide who will be born after e. In these situations, they too will have to affirm that ΣxBx. Our semantics for the tenseless quantifiers appears to be at odds with our explicit commitments. We reply that ontological commitment cannot be read off the tenseless existential quantifier Σ. Quantifying over everything that was, is, or will be is like quantifying over all detectives, real and fictional—a useful logical device, but not a way of stating one’s ground-level commitments, since it involves quantifying over such unreal entities as Sherlock Holmes and Hercule Poirot. How, then, should Growing-Block theorists state their groundlevel ontological commitments? We suggest that a Growing-Block theorist is committed to everything that exists according to the actualized timeline. One can quantify over things that exist according to T’ in the following way.

veT ∑ T ’ x(fx ) =

veT Π T ’ x(fx ) =

1 if for some e’ in T ’, there is an a ∈De ’ such that veT (fx/a) = 1 0 If for every e’ in T ’, there is an a ∈De ’ such that veT (fx/a) = 0

1 if for some e’ in T ’, there is an a ∈De ’ such that veT (fx/a) = 0 0 if for every e’ in T , there is an a ∈De ’T such that ve ’T (fx/a) = 1

To say that ΣTx(ϕx) is to claim that some object in T‘s ontology is ϕ (at the indexical present, in the actual world). We claim that a Growing-Block theorist is ontologically committed to everything that (by her lights) actually exists. That is, a GrowingBlock theorist is ontologically committed to the existence of an object a just in case she accepts that ΣTx(x = a) and T is the actualized timeline. Is there a way of introducing a formal expression for the concept of ‘actualized timeline’? We fear not. The formal semantics gives us a way of determining what is true at a time in a timeline, and a way determining which timeline is actualized according to each time-

The Real Truth about the Unreal Future | 295 line (each timeline is actualized according to itself), but no way of determining which timeline is actualized full stop. Being actualized is not a matter of being actualized according to any timeline. 4. WHY THE GROWING-BLOCK THEORY OUTPERFORMS SIMILAR RIVALS So far, we have developed a metaphysical and semantic framework for explicating modality in the context of the Growing-Block theory, used this framework to distinguish between true sentences about the future and false ones, developed three semantic proposals for handling the indeterminacy about the future that is likely to accompany the Growing-Block theory, and addressed questions that this indeterminacy raises about the expressive power of the languages available to Growing-Block theorists. We now turn to some of the Growing-Block theory’s close relatives: Craig Bourne’s ersatzer Presentism, Storrs McCall’s Shrinking-Tree view, and John MacFarlane’s relativistic semantics for future contingents. These theories are rivals to the Growing-Block theory, since they share many of its appealing features. We argue that where the Growing-Block theory differs from these three proposals, it outperforms them. 4.1 Bourne’s Ersatzer Presentism Bourne accepts a theory of branching time much like our own, in which ersatz possible worlds consist of sequences of ersatz times. Unlike us, however, Bourne thinks that only one time—the present— is concretely instantiated. Sentences of the form Pp are made true by the existence of ersatz times which represent p as being true, and which are appropriately related to the present by the ordering of ersatz time. So if it is now true that Bill the Brontosaurus ate an oak leaf, then what makes this true is the existence of an ersatz time, appropriately related to the ersatz present, which represents Bill as eating an oak leaf. We believe that Bourne’s theory provides insufficient grounding for truths about the past. What makes that ersatz time appropriately related to the present? Why isn’t the present related exclusively to ersatz times according to which Bill failed to eat oak leaves? Bourne thinks there is no issue here. “[A]ccording to ersatzer presentism,” he writes,

296 | Rachael Briggs and Graeme A. Forbes what makes ‘It was the case that p‘ true is an actually R-related ordered triple, whereas according to the tenseless theory, what makes it true is an actually earlier than-related concrete fact. Now to ask why these ordered triples are actually R-related is about as fair as asking why the concrete facts are actually earlier than-related in the tenseless theory, i.e., not at all— they just are.9

But this misses the point of the objection. Truth should supervene on being—on the concrete things that tenselessly exist, the properties and relations those things instantiate, and the laws of nature. According to Bourne’s theory, however, there could be two possible worlds exactly alike with respect to the concrete things they contained, the properties and relations those concrete things instantiated, and the laws of nature, in which the ersatz present was nonetheless R-related to different ersatz past times. This, we think, grants the abstract realm of ersatz times far too much spooky autonomy. The Growing-Block theory does better. Which timeline is actualized is determined by which concrete things tenselessly exist, the properties and relations those concrete things instantiate, and the laws of nature. Which complete timelines are feasible are determined by the whole of concrete reality. There are no brute facts about which ersatz times are appropriately related to the present: where concrete reality and the laws of nature are insufficient to uniquely determine the future, all candidate futures are possible.

3.2 McCall’s Shrinking-Tree View If we are concerned that Presentism has too much autonomy from concrete existence, we might consider Storrs McCall’s view (1994, 3) known as ‘the Shrinking Tree’. The Shrinking-Tree view, like the Growing-Block view, posits a moving absolute present and a concrete past. Unlike the Growing-Block view, the Shrinking-Tree view holds that all the future possibilities are concrete, with branches ceasing to (tenselessly) exist as they cease to be feasible possibilities: The universe, then, has in this model the shape of a tree, with a single four-dimensional trunk for the past and a densely branching set of four9 In the interest of notational uniformity, we have replaced Bourne’s ‘E-related’ with our ‘R-related’.

The Real Truth about the Unreal Future | 297 dimensional manifolds for the future. Each of these manifolds in turn branches, so that the branching pattern is very complex and the number of branches very large.

McCall’s tree-like structure is like our tree-like structure generated by a timeline and the set of its feasible extensions, but McCall’s tree is made up of space-time, whereas the our tree is made up of ordered sets of propositions. A view where space-time has many future branches but only a single past trunk might seem a good way of capturing the intuitive view that the future is pregnant with possibility and the past is implacably fixed. There is no question, on McCall’s view, of the truths not being grounded in concrete existence. McCall’s view is not, then, vulnerable to the sort of objection that affects Bourne. There are some reasons for concern, however. Michael Tooley (1997, 239) argues that there are two considerations that count against McCall’s view: Broad’s model of a dynamic world [the Growing-Block view] seems preferable to McCall’s, for at least two reasons. First, it allows one to make sense of the idea of dynamic worlds that are deterministic, and this seems desirable, since deterministic worlds can, no less than indeterministic ones, be worlds where states of affairs come into existence. Secondly, given that a world containing no future states of affairs at all is rather more austere than one that contains states of affairs corresponding to all future possibilities, Broad’s model is also to be preferred on grounds of simplicity.

McCall defines the present as the point at which branching first occurs. So, unless there is branching, there is no absolute present. This rules out the possibility of there being only one feasible future, on McCall’s view, which means, as Tooley points out, the Shrinking-Tree view is incompatible with determinism. More than that, the Shrinking-Tree view cannot cope with a world which is deterministic for five minutes, since time could not pass for those five minutes, but rather would skip until the first branching-point. McCall’s view rests on a strong claim, then: The laws of nature are, and always will be, indeterministic. Such assumptions should be avoided if we can do the same work without them. When Tooley describes the Growing Block as being more austere than the Shrinking Tree he seems to understate the case. It is not just that McCall is committed to more than a Growing-Block theorist, but that McCall is committed to any number of as yet

298 | Rachael Briggs and Graeme A. Forbes unborn children, who continually drop out of existence when they cease to become feasible. On both the Growing-Block and the Shrinking-Tree views, something ceasing to be feasible is a question of ontological change. On the Growing Block something comes into existence which rules out the possibility, whereas on the Shrinking Tree the possibility literally ceases to exist. This lends the Shrinking Tree a destructive air. It might seem as though we are all guilty of mass-murder for wiping all the possibilities that did exist out of the universe. It is not clear that we ought to treat possible future persons as having the same moral status as present persons, nor that this is a consequence of McCall’s view, but it seems odd that we should treat our prize-winning first novels, which we have not yet thought of and may never write as being just as real, as the first short-stories that we wrote at school. That is the sense in which it seems the Growing Block is more austere than the Shrinking Tree. You might worry that we face the same sort of objection from McCall that we put to Bourne. Do our ersatz times not have the same sort of spooky autonomy that Bourne’s did? In short, they do not. Our ersatz times, unlike Bourne’s are grounded in what (tenselessly) exists and the laws of nature. To explain what will be true we need no more than that what has already happened and the laws of nature. It is then left as a question for science whether the laws of nature and the history of the universe allow of a number of possible futures or not. The Growing-Block view does not need as much, ontologically, as the Shrinking-Tree view, nor does it rest on tendentious assumptions about the indeterministic nature of the universe. Our view also has explanatory advantages over the ShrinkingTree view. The inevitability operator generates a convenient way of talking about lost possibilities, which McCall cannot easily replicate. For McCall the flow of time meant that branches of space-time dropped out of existence when they were no longer feasible. In our model, there exist not only those possibilities which, given the state of the universe and the laws of nature, are feasible, but those which were feasible and are no longer. Thus not only optimists and pessimists, but also historians, can be subjected to rational scrutiny through our account. Was it inevitable that the Government forces would win at Culloden? Well, if the Government forces win according to all feasible extensions of

The Real Truth about the Unreal Future | 299 the timeline the Growing Block instantiates up to the start of the battle, then yes, it was inevitable. If there were feasible extensions of the timeline instantiated up to the start of the battle according to which the Government lose, then it was not inevitable. Unlike McCall, we need not have a further mechanism for accounting for possibility in addition to the one we use for feasible futures. We get an account of this ordinary sort of counterfactual possibility included in the package.

4.3 MacFarlane’s Relativism John MacFarlane’s (2003) view, unlike the Shrinking-Tree view or Presentism, is concerned primarily with semantics rather than ontology. He argues that the best way to capture the idea that some sentences about the future are truth valueless is to embrace relativism about truth. On MacFarlane’s theory, the truth values of tensed sentences must be relativised not only to the time of utterance, but to the time of assessment. MacFarlane asks us to consider a particular utterance of 1. Exactly one day into the future, there will be a sea battle. (FdS) We will call the utterer, as he does, Jake. Imagine Jake utters (1) on a Monday, and the world’s history up to Monday together with the laws of nature fails to determine whether there will be a sea battle on Tuesday. MacFarlane thinks that on Monday, we should regard Jake’s utterance as having an indeterminate truth value. MacFarlane accepts, as we do, that there can be two possible futures (one on which there is a sea battle and one on which there isn’t), and no reason to think that one of these futures is ‘singled out’ as the actual one. In such a case “symmetry conditions seem to rule out saying either the utterance is true or that it is false. Thus, it seems, we must count it as neither true nor false. This is the indeterminacy intuition” (MacFarlane 2003, 323). Next, imagine it is Tuesday, the day after Jake’s utterance of (1) on Monday. Imagine also that on this particular Tuesday a sea battle is raging. MacFarlane argues we should be inclined to agree to the following:

300 | Rachael Briggs and Graeme A. Forbes Jake asserted yesterday that there would be a sea battle today There is a sea battle today So Jake’s assertion was true. When we take this retrospective view, we are driven to assign a determinate truth value to Jake’s utterance: this is the determinacy intuition. (325)

The indeterminacy intuition is in tension with the determinacy intuition. MacFarlane argues that we can accommodate both by making the truth of Jake’s utterance relative to the context of assessment, so that his utterance has a different truth value depending on when you ask if it is true: it is indeterminate when assessed on Monday but true when assessed on Tuesday. The idea that Jake’s utterance is true, false or indeterminate full stop, MacFarlane claims, should be rejected. We (the authors) do not have to accept that there is no such thing as truth full stop: we already have the resources to accommodate both the determinacy intuition and the indeterminacy intuition. We needn’t relativize truth to contexts of assessment; we can simply relativize truth to timelines. (Since timelines are possible worlds, this amounts to evaluating truth in different possible worlds.) (1) is indeterminate on Monday in the relevant timeline where Monday is absolutely present—in that timeline, then, Jake’s utterance of (1) is true. But (1) is false on Monday in the relevant timeline where Tuesday is absolutely present—in that timeline, Jake’s utterance of (1) is false. The difference in truth values can be put down to a difference between possible worlds, rather than a difference between contexts. (Which world is actualized will change, of course, as time passes. This, on the Growing-Block view, is just what the passage of time is.) On Tuesday, observers can capture the sense in which Jake’s utterance of (1) was indeterminate by noting that, although the proposition Jake asserted then was true (PdΔFdS), it was not inevitably true (¬PdLΔFdS). There is no need to abandon truth full stop in favour of truth relative to context of assessment. What is true at a given time is dynamic, and can change as the Growing Block grows, because which world is actualized changes as the Growing Block grows. The continual change in which world is actualized is what C.D. Broad (1973, 766) described as the “rock-bottom peculiarity” of time.

The Real Truth about the Unreal Future | 301 5. CONCLUSION We have, above, considered what view we should have if we wanted to talk about the future. We have argued that someone inclined to the Growing-Block view could give a semantics that involves no commitment to anything more than what they are ontologically committed to already, but, nevertheless, can accept that there are non-trivial future truths, and that some options are feasible while others are not. Three options have been given about how this semantics might go; one supervaluationist, one Łukasiewicz, and one intuitionist. We have also considered some similar rivals, to show that our theory outperforms them. We have not argued for the Growing-Block view itself. Equally we have not tried to deal with all of the common objections to the Growing-Block view. These topics would extend the scope of this paper beyond that which can reasonably be dealt with in a single article. We are, however, optimistic about the fortunes of what we take to be a plausible position in the philosophy of time, and believe we have shown that claiming the future does not exist, rather than rendering future possibility mysterious, allows fruitful analysis. University of Sydney University of Sheffield REFERENCES Robert Merrihew Adams. Theories of actuality. Noûs, 8(3): 211–31, 1974. Craig Bourne. A Future for Presentism. Oxford University Press, Oxford, UK, 2006. David Braddon-Mitchell. How do we know it is now now? Analysis, 64(3): 199–203, July 2004. C.D. Broad. Scientific Thought. Kegan Paul, London, 1923. —— A reply to my critics. In P.A. Schlipp, editor, The Philosophy of C.D. Broad, pages 711–830. Tudor, New York, 1959. Michael Dummett. The philosophical basis of intuitionistic logic. In Truth and Other Enigmas, pages 215–47. Harvard University Press, 1978a. —— The reality of the past. In Truth and Other Enigmas, pages 358–74. Harvard University Press, 1978b.

302 | Rachael Briggs and Graeme A. Forbes Peter Forrest. The real but dead past: a reply to Braddon-Mitchell. Analysis, 64(284): 358–9, 2004. Patrick Greenough. Indeterminate truth. Midwest Studies in Philosophy, 32: 213–41, 2008. Saul Kripke. Semantical analysis of intuitionistic logic. In Michael Dummett and J. N. Crossley, (eds.), Formal Systems and Recursive Functions. pages 92–130. North-Holland, Amsterdam, 1963. Jan Łukasiewicz. On determinism. In Ludwik Borkowski, editor, Selected Works. North-Holland, Amsterdam, 1970. John MacFarlane. Future contingents and relative truth. The Philosophical Quarterly, 52(212): 321–36, 2003. Storrs McCall. A Model of the Universe. Clarendon Press, Oxford, UK, 1994. Roy Sorensen. Vagueness and Contradiction. Oxford University Press, Oxford, 2001. Richmond Thomason. Indeterminist time and truth-value gaps. Theoria, 36: 264–81, 1970. Michael Tooley. Time, Tense, and Causation. Oxford University Press, Oxford, 1997. Achille C. Varzi. Supervaluationism and its logics. Mind, 116(463): 633–76, July 2007.

APPENDIX: PROOF OF THE HEREDITY CONDITION Heredity Condition: Any sentence that receives value 0 or value 1 at e in T must receive the same value at e in every feasible extension T ’ of T. Base Case: The heredity condition holds for atomic propositions (whose truth values at e in T are solely a matter of e’s representational features). Inductive Hypothesis: The heredity condition holds for propositions ϕ and ψ. Inductive Steps: Likewise, the heredity condition holds for ∃xϕ(x/a) and ∀x(ϕ(x/a), since these quantifiers depend only on the extension of De (which stays constant between T and T’s extensions) and the truth values of the ϕ(x/a)'s for various values of a (assumed to be herdiatary by the inductive hypothesis). The heredity condition holds for ϕ ∨ ψ and ϕ ∧ ψ, since ∨ and ∧ are truth functions. To inherit veT(ϕ) and veT(ψ) is automatically to inherit veT(ϕ ∨ ψ) and veT(ϕ ∧ ψ). Likewise, the heredity condition holds for ∃xϕ(x/a) and ∀x(ϕ(x/a), since these quantifiers depend only on the extension of De (which stays constant between T and T’s extensions) and the truth values of the

The Real Truth about the Unreal Future | 303 ϕ(x/a)’s for various values of a (assumed to be hereditary by the inductive hypothesis). The heredity condition holds for ¬ϕ, according to both the Łukasiewicz definition and the intuitionist definition of negation. On the Łukasiewicz definition, ¬ is a truth function, so that to inherit veT(ϕ) is automatically to inherit veT(¬ϕ). On the intuitionist definition, we have two cases to consider: the case where veT(¬ϕ) = 0, and the case where veT(¬ϕ) = 1. If veT(¬ϕ) = 0, then by the semantic rule for negation, veT(ϕ) = 1. By the inductive hypothesis, for every feasible extension T’ of T, veT ’ (f ) = 1 . By the intuitionist semantic rule for negation, veT(¬ϕ) = 0. If veT(¬ϕ) = 1, then by the semantic rule for negation, at every feasible extension T’ of T, veT ’ (f ) ≠ 1 . Since feasibility is transitive, for every feasible extension T’ of T, for every feasible extension T* of T’, veT* ≠ 1. By the semantic rule for negation veT ’ (¬f ) = 1. The heredity condition holds for ϕ ⊃ ψ in the intuitionist semantics. Whenever veT(ϕ ⊃ ψ) = 1, by the semantic rule for ⊃, every feasible complete or semi-complete extension T* of T is such that veT*(ϕ) ≠ 1 or veT*(ψ) = 1. Since feasibility is transitive, for every feasible extension T’ of T, every feasible complete extension T* of T’ is such that either veT*(ϕ) ≠ 1 or veT*(ψ) = 1. So by the semantic rule for ⊃, veT’(ϕ ⊃ ψ) = 1. Whenever veT(ϕ ⊃ ψ) = 0, by the semantic rule for ⊃, veT(ϕ) = 1 and veT(ψ) = 0. By the inductive hypothesis, where T’ is any feasible extension of T, veT ’ (f ) = 1 and veT ’ (y ) = 0. By the semantic rule for ⊃, veT ’ (f ⊃ y ) = 0 . The heredity condition holds for P(ϕ) and F(ϕ). On all three of our logical proposals, this follows from the definition of an extension. By clauses (a) and (b) of the definition, whenever a timeline T contains an e and an e’ such that e’Re and ve ’T (f ) = 1 , so does any extension—and hence, any feasible extension—of T. Likewise, whenever a timeline T contains an e and an e’ such that eRe’ and ve ’T (f ) = 1 , so does any (feasible) extension of T. Therefore, whenever veT assigns value 1 to P(ϕ) or F(ϕ), so must veT ’ for any feasible extension T’ of T. By clause (c) in the definition of an extension, whenever a timeline T fails to contain an e and an e’ such that e’Re and ve ’T (f ) = 1 , so does any (feasible) extension of T. Therefore, whenever veTP(ϕ) = 0, veT ’P(f ) = 0 for any feasible extension T’ of T. Finally, whenever veT (Fϕ) = 0, T has no feasible complete or semi-complete extension T' containing an e' such that eRe' and ve'T' (ϕ) = 1. Since feasibility is transitive, no feasible extension of T has a feasible complete or semi-complete extension T' containing an e' such that eRe' and ve'T' (ϕ) = 1. Thus, in every feasible extension T* of T, veT* (Fϕ) = 0. The heredity condition holds for Pd and Fd. This follows from the definition of an extension, together with the assumption that the distance between two ersatz times depends only on the ordering, together with the representational features, of the times between them. If T’ is

304 | Rachael Briggs and Graeme A. Forbes an extension of T, and T contains times e and e’, T’ must contain e and e’, and will contain all the times between them in exactly the same order. Since the distance between two times supervenes on their contents and the contents and orderings of the times between them, if the distance between e and e’ is d according to T, then the distance between e and e’ is d according to T’. Thus, whenever veT(Pd(ϕ)) = 1 or veT(Pd(ϕ)) = 0, for any extension T’ of T, veT ’ (Pd (f )) = veT (Pd (f )) , and likewise, whenever T contains an e’ such that eRe', e' is distance d from e, and v eT( F d( ϕ )) = 1 or v eT( F d( ϕ )) = 0, for any extension T ’ of T , veT ’ ( Fd (f )) = veT ( Fd (f )). If T contains no e' such that eRe' and e' is distance d from e, we can see that for all extensions T' of T, veT' (Fdϕ) = veT (Fdϕ) by observing that veT (Fdϕ) takes on a value if and only if veT' (Fdϕ) takes on the same value for all complete or semi-complete extensions T' of T. The proof is then exactly as for the above case of Fϕ. The heredity condition holds for Σxϕ and Πxϕ. In the intuitionist and Kleene semantics as well as the supervaluationist semantics. First, consider only the case where veT(Σxϕ) = 1, and the case where veT(Πxϕ) = 0. Where veT(Σxϕx) = 1, there is an e’ ∈ T such that for some a ∈De ’ , ve ’T (f x / a) = 1. Any extension T’ of T will also contain e’, so that by the semantic rule for Σ , veT ’ (Σxf x ) = 1. Likewise, where veT(Πxϕx) = 0, there is an e’ ∈ T such that for some a ∈De ’ , ve ’T (f x / a) = 0 . Any extension T’ of T will also contain e’, so that by the semantic rule for Π, veT ’ (Πxf x ) = 0 . Next, for the intuitionist semantics, consider the case where veT Σx(ϕx) = 0, and the case where veT Πx(ϕx) = 1. (We needn’t consider this case in the Kleene semantics, since it never arises.) In both the semantic rule for Σ guarantees that for all extensions T' of T, e' in T', and De’ such that e' ∈ T', and all a ∈ De', we have that ve’T’ (ϕx/a) = 0. Once again, since feasibility is transitive, this suffices to guarantee for every extension T* of x, for all extensions T' of T*, e' in T', and De’ such that e' ∈ T', and all a ∈ De’, we have that ve’T’(ϕx/a) = 0. Thus, by the semantic rule for Σ, veT Σx(ϕx/a) = 0. An exactly parallel argument works for the case where veT Πx(ϕx) =1. The heredity condition holds for ΔTϕ and TrTϕ. By the semantic rule for ΔT, veT(ΔT*ϕ) and veT(TrT*ϕ) are functions of veT*(ϕ). Likewise, where T’ is any feasible extension of T, veT ’ ( Δ T* f ) and veT ’ (TrT* f ) are the same functions of veT*(ϕ). By the transitivity of identity, veT ( Δ T *f ) = veT ’ ( Δ T *f ), and veT (TrT *f ) = veT ’ (TrT *f ) . The heredity condition holds for Lϕ. By the semantic rule for L, veT(Lϕ) depends only on veT*(ϕ), where T* is the initial segment of T containing only e and the e’ such that eRe’. But for any extension T’ of T, T* is also the initial segment of T’ containing only e and the e’ such that eRe’. So

veT ’ Lf = veT Lf.

10. Presentism and Distributional Properties* Jonathan Tallant and David Ingram 0. INTRODUCTION Presentism is the thesis that everything that exists, exists now. The view faces a familiar problem. Most are inclined to think that there are truths about the past. However, if the past does not exist, then what is it that makes true our talk about the past? To borrow from Armstrong: What truthmaker can be provided for the truth ? The obvious truthmaker, at least, is Caesar himself. But to allow Caesar as a truthmaker seems to allow reality to the past, contrary to [presentism]. (2004: 146)1

There are a variety of suggested solutions to the ‘truth-maker problem’. We focus upon a particular solution offered by Cameron (2011). Briefly, Cameron claims that the truth-makers for truths about the past are distributional properties, instantiated by present entities. Thus, to say of Barack Obama, , is true, because Barack Obama instantiates a distributional property, a part of which is being-a-boy-at-t* (where t* is earlier than the utterance). We argue that Cameron’s proposed solution fails.

1. DESIDERATA The type of solution offered by Cameron is one that we call a ‘properties solution’: one that involves the instantiation of properties by objects in order to do truth-making work. This type of solution is developed by Bigelow (1996) and termed ‘Lucretianism’. Since * We are very grateful to Karen Bennett, Ross Cameron, Jonathan Curtis, Suki Finn, Daniel Nolan, Harold Noonan, and Dean Zimmerman for comments and feedback on previous drafts. 1 We follow the convention, as Armstrong does, that ‘

’ denotes ‘the proposition that p’.

306 | Jonathan Tallant and David Ingram Cameron’s proposed solution is a variation on a Lucretian theme, we now outline Lucretianism and Cameron’s adaptation.

1.1 Lucretianism An adequate solution to the truth-maker problem must, according to Cameron, satisfy four conditions, which are: (1) presentism; (2) truthmaker theory (that truths about the past are made true by some existing element of our ontology); (3) realism about the past – simply, that there are (evidence-transcendent and objective) truths about the past; and, (4) we should not posit ‘suspicious’ properties, where a property is suspicious iff it makes no difference to the present intrinsic nature of its bearer.2 To illustrate how we might apply these conditions, consider the true proposition . The Lucretian claims that there is a property being-such-as-to-have-contained-Socrates and the state of affairs of the world instantiating it. This makes true . Thus, the world instantiates present, past-directed properties under Lucretianism, and these necessitate truths about the past. Lucretianism therefore satisfies each of (1)–(3). The Lucretian endorses (1) by affirming that only the present exists. Lucretians endorse (2): what makes true our talk about the past are present properties, and so the Lucretian can thereby satisfy (3): there are truths about the past. But the Lucretian property of being-such-as-to-have-contained-Socrates tells us nothing about the present intrinsic nature of the bearer, i.e. the world. According to Cameron, the property being-such-as-to-havecontained-Socrates makes no difference to the present intrinsic nature of its bearer. Lucretian properties are therefore suspicious, because they fall foul of condition (4), and should be rejected. We are prepared to concede, here, that (4) is plausible enough. The Lucretian is not obviously compelled to agree. Indeed, so far as we can tell, nothing in Cameron’s paper compels a Lucretian who does not feel the force of the ‘suspicion’ intuition to concur with Cameron’s rejection of Lucretianism.3 2 This is based on Cameron’s view that a property is suspicious iff it fails to satisfy his principle of Intrinsic Determination: ‘for all objects x and properties F and times t, if x instantiates F at t, then x has the intrinsic nature at t that it has partly in virtue of instantiating F at t’ (2011: 61). 3 The seriousness of this charge is discussed in Tallant (2009). Our preferred reason for rejecting Lucretianism is given by Sanson and Caplan (2010) and Merricks (2007: 136–7): what we might crudely call the ‘aboutness’ objection.

Presentism and Distributional Properties | 307 1.2. Distributional Properties Cameron attempts to adapt Parsons’ (2004) spatial distributional properties (SDPs), an example of which is as follows. Consider an object, O, which is white with black polka-dots, and its corresponding property being-polka-dotted. This property tells us how O is across a region of space. O can be seen as both wholly white at places, and wholly black at others. While nothing can be both wholly white and wholly black, instantiating the SDP, being-polkadotted, is enough to explain exactly how O is across the region of space it occupies.4 Cameron argues that, in the same way, the appropriate temporal distributional properties (TDPs) explain how objects are across time. We borrow a story from Cameron: Consider a simple world consisting of just one spatial dimension and one temporal dimension. There is one entity in this world – Flatty – who starts off his life at time t as a point, but who as time progresses grows continuously in one direction of the one spatial dimension he occupies. After the beginning of this life, then, he is no longer a point but a line; and at each moment he is a longer line than he has ever been previously. Exactly one year later, at t*, Flatty tragically ceases to be, and the world is empty. (2011: 63)

On this account Flatty is a point at t and a line at t*.5 Flatty’s spatial dimensions are distributed a certain way across a period of time. Flatty is a point at one moment in time, and a line at others in virtue of instantiating the relevant TDP. However, the TDP alone does not give us enough to explain how Flatty is at a particular moment in time. In order to explain this, we must appeal to a further property, Flatty’s age at the moment in question. Given this account, we can fix how Flatty is, was, and will be, by appealing to the TDP and Flatty’s age. TDPs and the age property both satisfy (4). A TDP is a single property, one that charts the nature of its bearer throughout the

4 Being-polka-dotted is just one example of an SDP, a colour-distributional property. Other dialectically respectable SDPs include being-hot-at-one-end-and-cold-at-the-other (a heat-distributional property), having-a-uniform-density-of-1kg/m3-throughout (a densitydistributional property), and so forth. 5 It is more accurate to say that Flatty is a line at the moment immediately preceding t*, because at t* Flatty ceases to be. However, for simplicity, we will assume that it is true of Flatty that he was a line at t*, and then ceases.

308 | Jonathan Tallant and David Ingram whole of history. At every moment, the TDP makes a contribution to the present intrinsic nature of its bearer, and so satisfies (4). The property of age also clearly makes a difference to how old the object is now, thereby also satisfying (4). Thus the union of TDPs and ages ground truths about how their bearers were, providing present truth-makers for truths about the past, and avoiding charges of ‘suspiciousness’.

2. TENSE VS. TENSELESS Cameron’s preferred understanding of the TDP is that it is tenseless. By this Cameron (2011: 67) seems to mean that the TDP is akin to being-Fthen-G-then-H. Contrast this with a TDP that ‘builds in’ tense and is of the form having-been-F-being-G-and-going-to-be-H. As Cameron notes, if ‘tense’ were ‘built into’ the TDP, then which TDP were instantiated by an object would have to vary from moment to moment to reflect the fact that, given the passage of time, an object may change from going to be F, to being F, to having been F. Thus, an object, O, bearing a TDP would have to change from bearing the TDP having-been-not-Fand-being-not-F-and-going-to-be-F, to bearing the TDP having-been-notF-and-being-F-and-going-to-be-not-F, and so on. By contrast, an object instantiating the TDP being-F-then-G-then-H does not need to change which TDP it instantiates, over time, to generate the appropriate truthvalues. As we explained above, the ‘tenseless’ TDP and the property of age are sufficient to ground the tensed truths.

2.1. Changing TDPs We think it possible that an object instantiating one of Cameron’s (tenseless) TDPs could change over time in such a way that it instantiates different TDPs at different times and we also think that this is a problem for Cameron’s view.6 Suppose, to illustrate, that we have the following natural progression through time, where we allow that the underlined portions of the TDP are those that are ‘now’:

6

Cameron (2011: 76–7) considers this problem as his ‘Objection 5’.

Presentism and Distributional Properties | 309 (A) The rose is red and then the rose is dead.7 This would most naturally be followed by: (B) The rose is red and then the rose is dead. The progression described would simply be one in which we move from one portion of the TDP—the rose is red—being present, to another portion of the TDP—the rose is dead—being present. So far, this all looks as it should. But we maintain that the following sequence is possible. Begin as before: (A)

The rose is red and then the rose is dead.

And then suppose that this is followed by: (B*) The rose is green (not red) and then the rose is dead. This situation is one in which the property bearer—the rose— changes with respect to which TDP it instantiates; first one TDP is instantiated, and then another. This second TDP has a different first portion. We take it that this result would be undesirable.8 The scenario described above is one in which is true, but where it is never true, later than this, that . After all, given (B*) what will be true about the past is that . This, we think, is sufficient to motivate a rejection of Cameron’s view. We have spoken as if TDPs have ‘portions’ or ‘parts’. Cameron does not licence such talk, claiming that such properties cannot be broken down into simple components. If TDPs cannot be broken down, then how can we say that TDPs with different parts can be instantiated at different times? We are prepared to concede the point, but do not think it pressing to the aims of the paper. A moment ago, we described a sequence that involved a transition from (A) to (B*). We think it perfectly intelligible to think of the transition being generated by the replacement of one TDP (the TDP described in (A)) with another (the TDP described in (B*)). We therefore maintain that the objection stands. 7

These TDPs are ‘tenseless’ in the way described. This problem is discussed for presentist ersatzism (e.g. Crisp (2007)) by Tallant (2010). 8

310 | Jonathan Tallant and David Ingram Talk of ‘portions of TDPs’, though useful, is not essential to the setting up of the problem.

2.2. Cameron on change and the TDP Cameron, in part, anticipates our objection: It makes no sense to speak of an object changing its distributional properties. Why? Because what change is on the account being offered is to instantiate (at each moment of your existence) a non-uniform distributional property. Being red at one time and then orange at some later time, for example, is to be analysed as instantiating (at all times) the distributional property being red-then-orange. To speak of an object changing its properties is a loose way of saying something about the distributional property it has that says how it is across time; it makes no sense to speak of an object gaining or losing the property that says how it is across time. (2011: 77)

So far as we can tell, this contradicts Cameron’s earlier treatment of age properties. As we noted in 1.2, Cameron’s solution requires the union of a (tenseless) TDP and the property of age. Clearly, the property of age that is instantiated by any given object must change over time. We do not now bear the same age-property that we bore five years, five minutes, or even five seconds ago. Thus, to ‘speak of an object changing its properties’ is not merely a loose way of saying something about the distributional property it has across time. We present Cameron with the following dilemma: the property of age must be either a tenseless distributional property or some other kind of property. Suppose, taking the first horn, that the property of age is a tenseless distributional property. If that property is tenseless, then it will fail to reflect the fact that we are now one of those particular ages. Rather, the distributional property simply exists, describing how we are at all times. Suppose, to illustrate, that Cameron instantiates a TDP and a property of age. The TDP is of the form being-a-boy-and-then-being-a-man-and-then-being-an-oldman. To simplify, we suppose that Cameron exists for only three instants, and that each clause of the TDP makes true our talk about a distinct instant. We assume that a TDP age property would be something like the property of being one-instant-old-then-two-instants-old-and-then-three-

Presentism and Distributional Properties | 311 instants-old. But we think it plain that this property, in conjunction with the TDP being-a-boy-and-then-being-a-man-and-then-being-an-oldman, does nothing to fix the present intrinsic nature of Cameron’s person. The two properties simply serve to describe the whole lifespan of the person and are not sensitive to what is now the case. This is insufficient; nothing in either of the TDPs permits us to fix how the person is now. Since, as we explained above, this is essential to our fixing the truths about the past, so Cameron’s view fails. We now turn to the second horn: there is change that is not accounted for merely by the instantiation of a non-uniform TDP. If age is a property, e.g. the property being one instant old, but is not a distributional property, then change is not merely a matter of something bearing a distributional property. An object will have to change from being-now-five-years-old to being-now-six-years-old, for example; but this change will be due to an ever-changing, non-distributional property of age. Thus, some change would not be accounted for by TDPs. If that is right, then Cameron loses the leverage to insist that an object cannot change with respect to which TDP it instantiates. Cameron’s original line of resistance was that an object cannot change its TDPs because change just is the instantiation of distributional properties; to say that ‘distributional properties may change’ simply makes no sense. However, if a change in the property of age is not accounted for solely via the instantiation of a non-uniform TDP, then change does not consist solely in the instantiation of a non-uniform TDP. Clearly, it cannot then fail to make sense to say that ‘things change in ways other than by instantiating a nonuniform distributional property’.9 We therefore see no reason to deny that an object can change which TDP it instantiates, over time. If the idea is coherent and conceivable (and we think we have demonstrated that it is10) then we think it is possible.11

9 Cameron might reply that it is only a change in an object that requires the instantiation of the TDP, and once again attempt to run his argument. We do not see that this will work: age, surely, is a property of objects. 10 There is no obvious contradiction in the above representation of the change of TDPs over time. Recall, Cameron forced the contradiction by insisting that change was simply variation in the tenseless TDP. Since that is not the case, there is no longer any contradiction. 11 Elsewhere, Cameron (2010: fn.14) makes it clear that he thinks possibility is the ‘default mode’.

312 | Jonathan Tallant and David Ingram Nor can Cameron abandon the property of age. As Cameron has it: Flatty couldn’t have the particular distributional property and age that he has without being pointy at t and without being that length of line at t*, and it is because he instantiates this distributional property and age that he is pointy at t and that length of line at t*. And so the existence of the state of affairs of Flatty instantiating this distributional property and age at t* necessitates that Flatty was pointy at t, and can suffice as a truthmaker for that truth about the past. (2011: 65)

We share the thought that although TDPs themselves are sufficient for all of the tenseless truths, about (e.g.) Flatty, it is only when the TDP is conjoined with the property of Flatty’s age that we generate the claim that Flatty is now, and was and will be, any particular way. Since the tensed truths about the past and the future were the primary target for grounding, we cannot do very well without the property of age on the tenseless account of TDPs. Thus, denying the existence of the property of age, whilst preserving tenseless distributional properties, does not seem viable. So, instead of denying that there exists an age property, can Cameron claim that age properties are special; that they are the one property for which change is not a matter of instantiating a TDP? He might be able to deny that such a move would be ad hoc. After all, the age property is basically a surrogate for the ‘now’ of a moving spotlighter.12 We agree that Cameron could take such a position, but we are less certain as to how this will solve the problem. Cameron’s original view was that change just is variation in the TDP. This permitted him to argue that it would make no sense to say that objects undergo changes in which TDPs they exemplify. If Cameron then allows that there can be change that is not due to simply having a single changeimplying TDP throughout one’s career, then this line of argument cannot be mounted. Once Cameron concedes that there is a type of change that requires the gaining and loss of a particular property – the property of age – it becomes unclear why it makes no sense to describe objects as changing with respect to which TDP they exemplify. If objects can gain and lose age properties we see no reason that they not also gain and lose TDPs.

12

We owe this objection to Karen Bennett.

Presentism and Distributional Properties | 313 Cameron might then try to deny that it’s possible for objects to change with respect to which TDPs they exemplify, but we do not think Cameron should take this line. We take the view that possibility is the default status for a proposition that seems conceivable, and that if Cameron wishes to take the view that some element of our ontology is necessarily unchanging in some regard, then this generates particular dialectical requirements. Namely, Cameron must explain why objects cannot gain or lose TDPs. To see the explanatory burden more clearly, we think it is worth considering the claim that objects cannot change with respect to which TDPs they exemplify, in light of claims typically made about concreta and abstracta. We naturally think of concreta as capable of undergoing change. Were we to claim of any concrete object that it is impossible for it to undergo any change, then we think some explanation must be forthcoming as to why this is the case. For example, were Cameron to tell us that there was a concrete object that instantiates the property ‘being green’ and, of necessity, cannot lose this property and/or gain another colour property, despite the fact that this object can gain and lose other properties, we should want some explanation of why this is the case. It would not do to simply assert that this is how things are. Nor, then, should Cameron merely assert that an object cannot change which TDP it exemplifies, whilst simultaneously allowing that an object can change which age property it exemplifies. Now consider abstracta; putative examples of which include numbers, sets, and propositions. These entities are often thought to be necessarily unchanging. Certainly, we do not think that ‘2’ can undergo any change in its properties over time. However, the contrast between objects bearing TDPs and abstracta is marked. Abstracta are usually thought to exist ‘outside’ time, in some sense or other. Indeed, they lack any location in time. That abstracta lack temporal location renders them unchanging. Because abstracta do not reside at any point along the temporal dimension, we have an explanation of why it is that they cannot change their properties over time. Some similar explanation is required if we are to endorse Cameron’s position, but it is hard to see how the appeal to abstracta can help Cameron here. For, as already noted, the bearers of TDPs, unlike abstract objects, have a location in time. As a consequence of these arguments we think that Cameron’s view should be rejected. University of Nottingham

314 | Jonathan Tallant and David Ingram REFERENCES Armstrong, David (2004) Truth and Truthmakers (Cambridge: Cambridge University Press). Bigelow, John (1996) ‘Presentism and properties’ Philosophical Perspectives 10, 35–52. Cameron, Ross (2010) ‘From Humean truthmaker theory to priority monism’ Noûs 44:1, 178–98. —— (2011) ‘Truthmaking for presentists’ in Karen Bennett and Dean W. Zimmerman (eds.) Oxford Studies in Metaphysics, vol. 6 (Oxford: Oxford University Press), 55–100. Crisp, Thomas (2007) ‘Presentism and the grounding objection’ Noûs 41:1, 90–109. Merricks, Trenton (2007) Truth and Ontology (Oxford: Clarendon Press). Parsons, Josh (2004) ‘Distributional properties’ in Frank Jackson and Graham Priest (eds.) Lewisian Themes (Oxford: Oxford University Press), 173–180. Sanson, David and Ben Caplan (2010) ‘The way things were’ Philosophy and Phenomenological Research 81:1, 24–39. Tallant, Jonathan (2009) ‘Presentism and truth-making’ Erkenntnis 71:3, 407–16. —— (2010) ‘Time for presence?’ Philosophia 38:2, 271–80.

NAME INDEX Achinstein, P., 132 Adams, R. M., 220, 267 Adams, R., 220 Al-Ghazali, A., 132 Alston, W., 4, 115 n. 21 Ampere, A., 163 Annamalai, K., 168 n. 8 Armstrong, D., 62, 129, 131, 202, 305 Austin, J. L., 53 Bartlett, D., 164 Bealer, G., 202 Benacerraf, P., 111 n. 16 Bennett, J., 4 Bennett, K., 96 n. 2, 104 n. 9, 221, 246 Biel, G., 132 Bigelow, J., 178, 305 Bird, A., 133, 138–9, 148, 155, 177–85, 214 Block, N., 106 Bohn, E. D., 71 n. 5 Boolos, G., 73 n. 11 Bourne, C., 259–60, 266, 295–6, 298 Braddon-Mitchel, D., 266 Bricker, P., 222 Broad, C. D., 257, 300 Burgess, J., 112, 189 Burke, M., 246 Callander, C., 163 Cameron, R., 305–12 Caplan, B., 306 n. 3 Chalmers, D., 94, 198, 202, 209–10 Chandler, H., 193 Chisholm, R., 5–6, 18 n. 11, Cohen, P., 61 Coulomb, C., 155, 163, 169, 173

Cross, T., 134 Crossley, J., 190 n. 1 Davidson, D., 129 Davies, M., 185, 190 n. 1 Descartes, R., 134 Doepke, F., 245 Duffin, W. J., 185 Dummett, M., 287 Eklund, M., 96 n. 3, 104 n. 9, 117 n. 22 Ellis, B., 155, 177, 181–2, 214 Fara, D. G., 32 n. 16, 41–2, 45 Fara, M., 39 n. 19, 137 Fermi, E., 168 Feynman, R., 154–5, 161 Field, H., 86 n. 25 Fine, K., 99 n. 5, 101, 190, 214–17, 249 Flaubert, G., 72 n. 8 Forrest, P., 266 Freddoso, A., 132 Frege, G., 104 n. 9, 111 Geach, P., 4–5 Gettier, E., 129 Gibbard, A., 9 n. 5, 14 n. 8, 35, 246 Gibbs, J. W., 166, 167 Gödel, K., 83 Goldfarb, W., 100, n. 7 Goldman, A., 129 Goodman, N., 130 Graham, A., 107 n. 13 Greenough, P., 272 Grice P., 129 Gupta, A., 5 n. 2, 42

316 | Name Index Hacker, P., 100 n. 7 Harre, R., 132 Hawley, K., 8 Hawthorne, J., 134 Hazen, A., 28–9, 38 n. 19, 48–9, 73 n. 10 Heil, J., 96 n. 3, 147 Heim, I., 97 Hertz, H., 162–3, 164–5 Hilbert, D., 60 Hirsch, E., 104 n. 9, 112 n. 20 Hofweber, T., 100 n. 6 Holton, R., 148 Humberstone, L., 190 n. 1 Hume, D., 130

McCall, S., 259, 295–9 McDaniel, K., 222 McGee, V., 84 n. 25 McKay, T., 70 n. 3, 89 n. 29 McKitrick, J., 148 McMichael, A., 223 Menzel, C., 222 Merricks, T., 306 n. 3 Mumford, S., 155, 177 Ne’eman, Y., 185 Newton, I., 155, 162, 163 Nielson, H., 143 Ninomiya, M., 143 Noonan, H., 12, n. 6 Nozick, R., 129

Jackson, F., 108, 198, 209–10 Johnston, M., 12 n. 6, 139 Jubien, M., 6 n. 4, 17 n. 9, 18 n. 10

Olsen, E., 227 n. 1, 246, 247 n. 1, 248

Kaplan, D., 209 Kirsch, Y., 185 Kneale, W., 132 Kratzer, A., 97 Kripke, S., 7, 8, 18, 27, 38, 51–5, 129, 190, 193, 209, 211, 215, 222, 223, 286

Parsons, J., 112, 119 n. 23, 307 Pears, D., 100 n. 7 Perry, J., 227 n. 1 Planck, M., 167–9 Plantinga, A., 220, 222 Puri, I., 168 n. 8 Putnam, H., 104 n. 9, 113, 129, 132

Lange, M., 155, 163, 171, 179 Lederman, L., 185 Leibniz, G. W., 48, 190 Lewis, D., 7–9, 12, 15, 16, 18, 22–5, 27, 36, 38, 41, 44–6, 48–57, 59, 82 n. 22, 84 n. 23, 99 n. 4, 05 n. 10, 109, 130–2, 136–50, 201, 202, 222–4 Lierse, C., 178 Linnebo, Ø., 119 n. 24 Łukasiewicz, J., 274, 275, 284–8, 292, 293

Quine, W. V. O., 34, 43

MacFarlane, J., 259, 299, 300 Madden, E., 132 Marshal, D., 58 Martin, C. B., 134, 138, 147, 148

Ramachandran, M., 38 n. 18 Raw, G., 168 n. 8 Rayo, A., 70 n. 4, 73 n. 11, 84 n. 24, 93 n. 1, 96 n. 2, 107 n. 12, 108 n. 15, 111 n. 17, 115, 119 n. 24 Rosen, G., 104 n. 9, 112 Salmon, N., 86 n. 27, 87 n. 28, 190, 194–201, 208, 212, 213, 224 Sanson, D., 306 n. 3 Schaffer, J., 71 n. 5, 99 n. 5, 100 n. 6, 134 Sellars, W., 132 Seth, S., 168, 169 Shapiro, S., 73 n. 11

Name Index | 317 Shoemaker, S., 129, 131–6, 140, 141, 143, 149, 150, 214–16, 227 n. 1, 230 n. 3, 232 n., 248 Sider, T., 6 n. 4, 8, 18 n. 10, 99 n. 5, 100 n. 6, 192 Soames, S., 86 n. 25 Sorenson, R., 272 Spencer, C., 223 Stalnaker, R., 6, 41, 106, 112, 146, 209, 220 Stevin, S., 161 n. 5 Su, Y., 164 Tallant, J., 306 n. 3, 309 n. 8 Teresi, D., 185 Thomason, R., 276–8, 280, 289 Thomson, J., 250 Tooley, M., 297

Unger, P., 251 n. 15 Uzquiano, G., 68, 69 n. 4, 81 n. 21 Van der Merwe, A., 168 n. 8 Van Inwagen, P., 84 n. 23, 110, 222 Varzi, A., 265 n. 4 Ward, B., 146 Weinberg, S., 184 Wigner, E., 162 Williamson, T., 39 n. 19, 70 n. 4, 93 n. 1 Wittgenstein, L., 100 Wright, C., 104 n. 9, 112 Yourgrau, W., 168 n. 8 Yukawa, H., 165