Oxford Studies in Metaphysics [5] 0199575797, 9780199575794, 9780199575787, 0199575789

Table of contents :
Contents......Page 8
The Oxford Studies in Metaphysics: Younger Scholar Prize......Page 10
TIME-TRAVEL, à la VAN INWAGEN......Page 12
1 Changing the Past......Page 14
2 Can a Soufflé Rise Twice? Van Inwagen’s Irresponsible Time-Travelers......Page 40
3 Van Inwagen on Time-Travel and Changing the Past......Page 52
PERSISTENCE THROUGH TIME......Page 62
4 Location and Perdurance......Page 64
5 Coinciding Objects and Duration Properties: Reply to Eagle......Page 106
6 Duration in Relativistic Spacetime......Page 124
7 Strange Kinds, Familiar Kinds, and the Charge of Arbitrariness......Page 130
8 Many as One......Page 156
TIME, SPACE, AND LOCATION......Page 188
9 Extrinsic Temporal Metrics......Page 190
10 Parthood and Multi-Location......Page 214
THE METAPHYSICS OF SOUNDS......Page 256
11 Constructing a Theory of Sounds......Page 258
12 Hearing Sounds......Page 282
13 What Sounds Are......Page 290
14 Sounds and Temporality......Page 314
H......Page 332
T......Page 333
Z......Page 334

Citation preview

OXFORD STUDIES IN METAPHYSICS

OXFORD STUDIES IN METAPHYSICS Editorial Advisory Board: Karen Bennett (Cornell University) David Chalmers (Australasian National University) Andrew Cortens (Boise State University) Tamar Szabo Gendler (Yale University) Sally Haslanger (MIT) John Hawthorne (Oxford University) Hud Hudson (Western Washington University) Kathrin Koslicki (Tufts University) E. J. Lowe (University of Durham) Brian McLaughlin (Rutgers University) Trenton Merricks (University of Virginia) Kevin Mulligan (Universit´e de Gen`eve) Theodore Sider (New York University) Timothy Williamson (Oxford University)

Managing Editor Matthew Benton (Rutgers University)

OXFORD STUDIES IN METAPHYSICS Volume 5

Edited by Dean W. Zimmerman

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Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © the several contributors 2010

The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2010 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Laserwords Private Ltd., Chennai, India Printed in Great Britain on acid-free paper by MPG Books Group, Bodmin and King’s Lynn ISBN 978–0–19–957578–7 (Hbk.) 978–0–19–957579–4 (Pbk.) 1 3 5 7 9 10 8 6 4 2

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 (such as modality, ontology, and mereology) but also metaphysical questions that emerge within other subfields (such as 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. Dean W. Zimmerman New Brunswick, NJ

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CONTENTS The Oxford Studies in Metaphysics: Younger Scholar Prize

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TIME-TRAVEL, a` la VAN INWAGEN 1 Changing the Past Peter van Inwagen 2 Can a Souffl´e Rise Twice? Van Inwagen’s Irresponsible Time-Travelers Peter Forrest 3 Van Inwagen on Time-Travel and Changing the Past Hud Hudson and Ryan Wasserman

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PERSISTENCE THROUGH TIME 4 Location and Perdurance Antony Eagle

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5 Coinciding Objects and Duration Properties: Reply to Eagle Cody Gilmore

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6 Duration in Relativistic Spacetime Antony Eagle

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7 Strange Kinds, Familiar Kinds, and the Charge of Arbitrariness Daniel Z. Korman

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8 Many as One Thomas Sattig

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TIME, SPACE, AND LOCATION 9 Extrinsic Temporal Metrics Bradford Skow

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10 Parthood and Multi-Location Maureen Donnelly

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Contents THE METAPHYSICS OF SOUNDS

11 Constructing a Theory of Sounds Casey O’Callaghan

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12 Hearing Sounds Roger Scruton

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13 What Sounds Are Matthew Nudds

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14 Sounds and Temporality Jonathan Cohen

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Index

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THE OXFORD STUDIES IN METAPHYSICS YOUNGER SCHOLAR PRIZE Sponsored by the Ammonius Foundation∗ and administered by the editorial board of Oxford Studies in Metaphysics, the 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 2009. To be eligible for next year’s prize, submissions must be electronically submitted by 15 January 2010 (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 late February 2010. 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; Cody Gilmore, ‘‘Time Travel, Coinciding Objects, and Persistence’’, Vol. 3; Stephan Leuenberger, ‘‘Ceteris Absentibus Physicalism’’, Vol. 4;

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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 http://www.ammonius.org/.

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Younger Scholar Prize Jeffrey Sanford Russell, ‘‘The Structure of Gunk: Adventures in the Ontology of Space’’, Vol. 4; and Bradford Skow, ‘‘Extrinsic Temporal Metrics’’, Vol. 5

Enquiries should be addressed to the Editor: [email protected]

TIME-TRAVEL, a` la VAN INWAGEN

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1. Changing the Past Peter van Inwagen There are two kinds of time-travel: travel to the past and travel to the future. Travel to the future raises no conceptual problems. A ‘‘time-machine’’ for travel to the future need do no more than uniformly slow down the physical processes that go on inside it relative to physical processes external to the machine. If all you want of a time-machine is that it be capable of taking you to the future, a spaceship that can reach speeds near the speed of light will suit your purposes very well. To ‘‘travel to 2071’’ in your spaceship/time-machine, simply leave the Earth, keep the throttle open till you have reached a speed near the speed of light, travel at that speed for a while (relative to the stellar background), and then decelerate in such a way that you come to rest (relative to your point of arrival) in 2071 (you will, of course, have had to travel in a big loop1 —you will have had to follow a trajectory that ended at the point on the surface of the earth at which you wanted to ‘‘arrive in 2071’’). If you follow the right schedule of acceleration and deceleration—a very demanding one, to be sure—, the trip will take only half an hour (as measured by onboard clocks—either the digital display on the control panel or the clock constituted by your own metabolic processes). I suppose that even the chair I am sitting on can be regarded as a limiting case of a machine for travel to the future: if I sit on it long enough, shall find myself in 2071. Travel to the past is another story, and it is the story I am interested in. Accordingly, in the sequel I shall use ‘time-travel’ to mean ‘travel to the past’, unless otherwise noted.

1 If you ‘‘depart for 2071’’ in 2010, the loop will be sixty-one light-years in length (assuming that you reach your ‘‘cruising speed’’—very near the speed of light relative to the stellar background—pretty quickly and subject yourself to the necessary fierce episode of deceleration only for a short period at the end of the trip). Since you will be traveling in a loop, you will be constantly changing your direction, and thus be subject to ferocious g-forces even when your speed is constant relative to the stellar background.

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I have no idea whether the (actual) laws of physics permit time-travel. My best layman’s guess is that it will turn out that time-travel is physically impossible for extended objects with an internal structure (such as a human being or a machine). I am much less confident, however, that it will turn out that it is physically impossible to send electrons or photons into the past—one at a time, one after the other. (The best bet for time-travel is a wormhole, and it is far more likely that we shall one day open or discover a zerowidth wormhole whose ‘‘other end’’ is in, say, 1920—an electron or photon might pass through a zero-width wormhole—than one so wide that a proton or a cat could pass through it and retain its internal structure.) If a sequence of electrons or photons can be sent into the past, then information about the present can be sent into the past2 , and the possibility of sending information into the past creates the same paradoxes (whether they are real paradoxes or only apparent ones) as the possibility of sending macroscopic physical objects like human beings or Terminators into the past. (Tim sends information into the past in the hope that this information will convince the recipient to kill his, Tim’s, grandfather . . . . ) I am going to assume that ‘‘extended physical object’’ timetravel is physically possible—but if time-travel of that sort is physically impossible, and it is nevertheless physically possible to send sequences of individual elementary particles (and hence information) into the past, what I shall say can be easily adapted to the philosophical problems raised by that case (for the simple reason that the problems are essentially the same in both cases). Travel to the past is (like time-travel simpliciter) of two kinds. I will call them Ludovician and non-Ludovician time-travel.3 Ludovician time-travel does not involve changing the past. Consider the following story. Tim the time-traveler enters a time-machine in 2020 2 I am thinking of something like Morse Code. But even a single particle sent into the past could carry information if a proper ‘‘code’’ had been agreed on: ‘‘Observe the particle detector at noon the day before election day, 2020. On noon the day after the election I shall send an electron forty-eight hours into the past if Blenkinsop wins and a muon if Entwhistle wins.’’ 3 In honor of David Lewis. See his ‘‘The Paradoxes of Time Travel,’’ American Philosophical Quarterly (1976), pp. 145–52. Reprinted in David Lewis, Philosophical Papers, Volume II (New York: Oxford University Press, 1986), pp. 67–80, and in Peter van Inwagen and Dean W. Zimmerman, eds., Metaphysics: The Big Questions, 2nd. edn. (Oxford: Blackwell, 2008), pp. 224–35.

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and travels to 1920; he spends a month in 1920, during which month he has a series of adventures; then he returns to 2020. For this story to be a story of Ludovician time-travel, the events related in the following story must have been a part of the historical record, a part of the one true history of the world, at every moment from the moment of Tim’s arrival in 1920 to the moment at which he entered the time-machine to travel to 1920—and at every moment thereafter to boot4 : At a certain moment in 1920 a marvelous machine popped into existence5 —apparently ex nihilo—and a man who said his name was Tim disembarked from it and had a certain series of adventures; a month later, this fellow Tim once more entered his ‘‘time-machine’’—that is how he referred to it—and he and the machine vanished like a soap bubble. My topic is non-Ludovician time-travel: time-travel that involves changing the past. An episode of time-travel is non-Ludovician if it is not a part of the historical record at the moment the time-traveler entered the time-machine for a trip to the past that he or she arrived in the past—not a part of the record that a marvelous machine with (very nearly) the same intrinsic properties as the ‘‘departing’’ machine popped into existence at the ‘‘target date’’—and so on. Note that non-Ludovician time-travelers change the past by the mere fact of their arrival in the past—however careful they may be thereafter ‘‘not to do anything to change the past.’’ (I see no reason to suppose that a world in which there were episodes of nonLudovician time-travel would have to be a world in which there were no episodes of Ludovician time-travel: it might be that while certain rogue time-travelers—‘‘time bandits’’—engaged in nonLudovician time-travel, responsible, professional time-travelers were careful to travel to the past only if they could first find a

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The historical record as God knows it: human historians, whose knowledge of the past is of course fragmentary, may or may not have been aware at any time during this interval that history contained events of this description. 5 In the illustrative stories I shall be telling I shall assume that the time-machine is a ‘‘Wellsian vehicle’’: the time-traveler travels to the past in it. In some time-travel stories, the time-machine is more like a projectile launcher than a vehicle: it ‘‘sends’’ the traveler to the past, but does not itself travel to the past. No point of principle is affected by supposing that the time-machine is a vehicle as opposed to a launcher.

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trustworthy historical record of ‘‘their visit to the past.’’ I will not discuss the possibility of a ‘‘mixed’’ time-travel world.) I want to present a ‘‘model’’ that permits non-Ludovician timetravel without paradox.6 The model I will present presupposes the ‘‘growing-block’’ theory of time. I take no stand on the question whether the growing-block theory is metaphysically possible, on the question whether it is consistent with present-day physics, or even on the question whether it is meaningful (whether there is such a theory). I will assume—for the sake of argument, as it were—that the growing-block theory is at least meaningful. I will not develop 6 There are other ‘‘models’’ that may have this feature than the one I shall present in the text. Here is a very simple one. When the ‘‘depart’’ button in the time-machine is pressed, the machine and its passengers leave our physical universe (perhaps for some region of spacetime unconnected with the spacetime of the physical universe). The machine ‘‘leaves behind’’ in our physical universe a perfect atom-for-atom duplicate of itself and its passengers (maybe it borrows the mass-energy contained in the duplicate from the quantum vacuum or some such jargon). Then the direction of the momentum of every particle in the physical universe is reversed, and every particle moves ‘‘backward’’ along its historical trajectory (each of the particles that compose the newly created duplicate of the time-machine moves backward along the historical trajectory of the particle which it replaces) until the physical universe has returned to the state it was in in, say, 1920. Then the time-machine and its passengers return to the physical universe (managing somehow to push the matter at their arrival point out of the way) and ‘‘everything goes forward again.’’ A ‘‘return to the future’’ is accomplished by the same means as in the model presented in the text. Episodes of this kind are episodes of non-Ludovician time-travel (without paradox)—provided they are episodes of time-travel at all. The only defect in this model is that it is not entirely plausible to suppose that it really does involve ‘‘traveling to the past,’’ that it provides the reality, and not merely the appearance, of time-travel. (Note the occurrence of phrases like ‘has returned’ and ‘the state it was in’ and ‘everything goes forward again’ in the description of the alleged episodes of time-travel.) A different sort of complaint can be brought against ‘‘branching histories’’ models of changing the past: although such models involve real time-travel, it is not entirely plausible to suppose that they really represent the past as being changed. It is plausible to say that, on a branching-histories model, when Tim murders grandfather, the ‘‘original past,’’ the past in which grandfather eventually dies, full of years and wickedness, is still ‘‘there.’’ It is plausible to say that what Tim’s action has accomplished, according to a branching-histories model, is the creation of a ‘‘new past’’ in which he cuts his grandfather’s career short, a past that somehow runs parallel to the original past—and which is ‘‘no less real’’ than the new past. Or to put the matter another way. Suppose I know that you plan to travel to the past and, as you put it, ‘‘murder van Inwagen when he was twenty.’’ If the successful accomplishment of the undertaking you have so described would really be a case of ‘‘changing the past,’’ I ought to be worried. But, on the branching-histories picture of ‘‘murdering van Inwagen when he was twenty,’’ I have nothing to worry about: you will get into your time-machine and vanish, never to be seen again, and my life will go on much as before.

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this theory in the detail that would be necessary if my interest in it were other than instrumental—if I were interested in the theory ‘‘for its own sake,’’ as a theory of the nature of time. I have relegated many nice points—ones I could not resist making—to the notes. The reader should feel free to skip the notes to the following section of the chapter.

1. the growing-block theory The following statement of the growing-block theory is my own. It differs from the statements of Broad and other writers7 in detail, but not, I think, in any essential way.8 For the sake of simplicity, the ‘‘time’’ of which this theory treats will be supposed to be Newtonian: I will assume the existence of absolute or observerinvariant simultaneity. And—also for the sake of simplicity—I will suppose that the past and future are infinite. (It might be thought that, since ‘The future is infinite’ implies that there are future times, it contradicts the growing-block theory. We shall see that this is not so.) The theory comprises the following eight theses. (1) Timeless or tenseless or ‘‘pure’’ predication is possible. (And timeless predication that applies to ‘‘temporal’’ subjects like horses 7 See Broad’s ‘‘The General Problem of Time and Change’’ in C. D. Broad Scientific Thought (London: Routledge & Kegan Paul, 1923). An excerpt from this paper is printed in van Inwagen and Zimmerman, Metaphysics: The Big Questions, 2nd. edn., pp. 141–59. There has been extensive discussion of the growing-block model in the recent philosophical literature. See Michael Tooley, Time, Tense, and Causation (Oxford: Oxford University Press, 1997), esp. ch. 6, ‘‘Tensed Accounts of the Nature of Time’’; Craig Bourne, ‘‘When am I?’’, Australasian Journal of Philosophy 2002, pp. 359–71; David Braddon-Mitchell, ‘‘How Do We Know it is Now Now?’’, Analysis 2004, pp. 199–203; Peter Forrest, ‘‘The Real but Dead Past: A Reply to BraddonMitchell,’’ Analysis 2004, pp. 358–62; Trenton Merricks, ‘‘Good-Bye Growing Block,’’ in Dean Zimmerman (ed.) Oxford Studies in Metaphysics (Oxford: Oxford University Press, 2006). 8 Many recent critics of the growing-block theory contend that growing-block theorists should be skeptics about whether it is now the present time. More exactly, that adherents of the growing-block theory are committed to a skepticism of the sort that would be expressed by the utterance of sentence-tokens of the following types: ‘‘For all I know, this utterance occurs in the remote past. For all I know, the present moment is a moment approximately thirty-seven thousand years after the moment at which this utterance occurs.’’ (See the essays by Bourne, Braddon-Mitchell, Forrest, and Merricks cited in the previous note.) I will not address this criticism of the growing-block theory.

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and explosions and not only to timeless things like numbers and attributes and the God of the Philosophers.) There is a sense of ‘is’ in which the sentence ‘Bucephalus is a horse’ expresses a truth, even though Bucephalus does not now exist. And there is a sense of ‘are’ in which the sentence ‘Bucephalus and Street Smart are horses’ expresses a truth, despite the fact that there is no time at which both horses exist. For the sake of simplicity, we shall consider timeless predication only of ‘‘full career’’ predicates like ‘is a horse’ (in the sense of ‘horse’ in which the word applies to foals as well as to adult horses). That is, predicates that apply to things over the whole span of their existence if they apply to them at all. (2) Timeless or tenseless quantification—over temporal things—is possible. I should like to explain what this means, but I find that I cannot offer an explanation of timeless quantification that is consistent with another aspect, an essential aspect, of the growingblock theory. I will explain my difficulty presently (in note 13), after that ‘‘aspect’’ has been introduced. Timeless quantification will therefore be taken as an unexplained notion in this statement of the growing-block theory. The predications in the open sentences bound by timeless quantifiers will, at least in most cases, be tenseless predications: If one says ‘There is, timelessly, a horse’ one almost certainly does not mean ‘There is, timelessly, a thing that is-atpresent a horse’. In this statement of the growing-block theory, we shall consider timeless quantification only on open sentences like ‘x is a horse’; that is, on open sentences that instances of full-career predicates. (3) Physical reality (hereafter, Reality) is to be thought of as comprising physical events (hereafter, events).9 Events, pace Chisholm, are concrete particulars, not universals: events cannot ‘‘recur,’’ and we make no distinction between an event’s happening at a time and its existing at that time.10 Reality is the ‘‘grand event,’’ the mereological sum of all events—where ‘all’ is a timeless quantifier. 9

Dualists may wish to replace my phrase ‘physical reality’ with ‘temporal reality’. It will be observed that we quantify not only over events but also over the times at which events occur. These ‘‘times’’ are to be understood as abstract objects of some sort, but I do not much care exactly what abstract objects they are. Here is one example of what they might be. Pick some ‘‘benchmark’’ event to serve as a ‘‘reference point’’—the first Olympic games, the founding of Rome, the birth of Christ (if past time is infinite, any choice of a benchmark will be arbitrary). Call it B. 10

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(We assume that just any events have a unique sum. The sum of the Battle of Waterloo and the Battle of Stalingrad, for example, existed and is [timeless predication] a gappy event, an event—it would not be right to call it a battle—that happened or existed partly in 1815 and partly in 1942: at just those times at which one battle or the other was being fought.) In saying this, we do not deny the existence of physical substances or continuants. Nor do we deny that at least some events ‘‘involve’’ continuants (Caesar’s death presumably has Caesar as a constituent in some sense of ‘constituent’, and could not exist if Caesar did not exist). The present statement is no more than a stipulation that establishes the way we shall use ‘(physical) reality’. (4) As the previous statement suggests, we are assuming that at least some events have proper parts and that all events (other than Reality) are proper parts. We in fact assume that only events are parts of events, and that events are parts only of events. Every part of an event is a ‘‘sub-event’’ of that event: the event ‘‘the Battle of Stalingrad’’ was a part of the event ‘‘the Second World War.’’ (Again, we do not deny that there are physical substances or continuants, but we so use ‘part’ that no continuant is, in the strict and philosophical sense, a part of an event: no German tank or Russian soldier was—strictly speaking—a part of the Battle Let ‘‘times’’ be such properties as ‘‘is an event that is one year later than B,’’ ‘‘is an event that is one billion years earlier than B’’, and so on. Such ‘‘times’’ have an obvious ordering, and there is an obvious measure of the ‘‘interval’’ between two times. For an event to ‘‘happen at’’ a time is for it to have (the property that is) that time. Given a robust platonism about properties, it follows that for any temporal interval and any event E, there is a time ‘‘that long before’’ E and a time ‘‘that long after’’ E. In the text, we have assumed for the sake of convenience that the past is infinite (we have assumed there was no Big Bang or other ‘‘downwhen terminus’’—to borrow a term from a novel by Isaac Asimov), but I should point out that there is a good sense in which this conception of times does not imply an infinite past. True, this conception of times implies that if there was a Big Bang there are ‘‘times’’ earlier than the time at which that event occurred. But those times are times at which nothing happened. If ‘‘times’’ are the properties with which I tentatively identified them above (if the founding of Rome is our benchmark event, they would be properties like ‘‘is an event twenty billion years earlier than the founding of Rome’’), the times earlier than the time of the Big Bang are uninstantiated properties: on the ‘‘property’’ conception of times, times at which nothing happens are uninstantiated properties. To say that the past is finite is not to say that there is a time such that there is no earlier time; it is rather to say there is a time such that there is no earlier time at which anything happened.

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of Stalingrad. And if Caesar was in some sense a constituent of Caesar’s death, he was nevertheless not a part of that event. But we assume that if substances or continuants exist at a time, then events exist or happen at that time—changes or ‘‘unchanges’’ in the substances that then exist, if no other events.) Reality, then, is (the predication is timeless) the event of which all (timeless quantifier) events are parts (and all of whose parts overlap some event—but it is not necessary to add this customary clause, owing to our stipulations concerning the ‘‘parts’’ of events). (5) There are, timelessly, events that have happened and there are events that are happening, but there are (timelessly) no events that have not happened yet11 but will happen.12 If Jones died 11 The qualification ‘have not happened yet’ is not idle, at least on one conception of the identity of events across time. Consider, for example, the longish event ‘‘my life.’’ According to the growing-block theory, there is, timelessly, such an event. And this event will still be happening—at least I hope so—ten minutes from now. In a sense, therefore, the growing-block theory implies that there can be, timelessly, events that will happen in the future—it implies, at any rate, there can be events that will be happening at various future times. Understand the words ‘events that have not yet happened yet’ in the text in this sense: ‘events that are not happening now and were not happening at any times in the past (but which will be happening at some times in the future)’. For some points related to the ‘‘dating’’ of events and the use of the ‘‘ongoing aspect’’ of the verb ‘to happen’, see the note that follows. (There is another way to solve the problem to which this note is addressed, a very simple and elegant way: the event I now call ‘my life’ is not the event that admiring historians of philosophy after my death will call ‘van Inwagen’s life’: the event I now call ‘my life’ will be only a proper part of any event that historians refer to post mortem meam—unless they should happen to use phrases like ‘the part of van Inwagen’s life that occurred before 6:10 p.m., GMT, 14 August 2008’.) 12 We shall say that an event is or was happening at a time if any part of it is or was happening at that time. This statement is not a definition of ‘is or was happening at t’—for, if it were a definition, it would be circular. The point of the statement is this. I assume that some events are of such short duration that it is unproblematical to say that those events happen at a particular time. (Only event-slices, events of zero duration, could, strictly speaking, have this feature if a ‘‘time’’ is an instant.) What the statement comes to is a stipulation. And the stipulation is this: If such an ‘‘unproblematical’’ event happens or happened at a time, any ‘‘longer’’ event of which it is—timeless predication—a sub-event will be said also to be or to have been ‘‘happening’’ at that time. Thus the event ‘‘my life’’ is happening now and the Second World War was happening on 21 September 1942. If, moreover, there will be unproblematical sub-events of a ‘‘current event’’ that (will) happen at a future time t, that current event will (still) be happening at t. My life, for example, will—I hope—still be happening ten minutes from now, despite the fact that it is now true that there are, timelessly, no unproblematical sub-events of my life that will happen ten minutes from now.

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yesterday, there is (timelessly) such a thing as Jones’s death. But if (it is now inevitable that) Smith will die tomorrow, there is nevertheless (timelessly) no such event as Smith’s death. Now these three sentences contain indexical elements (if that is what tenses are; at any rate, the verb-phases ‘will happen’ and ‘died’ and ‘will die’ are tensed). And these three indexical, or at any rate tensed, sentences would express truths whenever they were uttered or written. But the point I wish to make has nothing to do with indexicality or tense. To convey the point I really wish to make, I must ask you, the reader of these words, now—at the moment you are reading them—to perform the following two linguistic acts: first, give the moment you now call ‘the present moment’ the proper name ‘Nunc’; then, having done that, utter the sentence, ‘There are (timelessly) no events that happen later than Nunc’. Have you done those things? You have? Good. In so speaking, you expressed a truth.13 And so would anyone who performed those two linguistic acts (mutatis mutandis) at any time. (6) One might have supposed that timeless quantifications could not have different truth-values at different times—that either truthvalues can be ascribed to such propositions only ‘‘timelessly,’’ or at any rate, that every such proposition is either unalterably true or unalterably false. But statement (5) implies that this is not the 13 It is this ‘‘aspect’’ of the growing-block theory—the aspect presented in statement (5)—that has forced me to conclude that the growing-block theory must treat timeless quantification as primitive. The obvious definition of ‘There is, timelessly, an F’ (in terms of temporal or tensed quantification) would treat this schema as equivalent to the following schema: ‘Either there has been an F or there is at present an F or there will be an F’. (The disjunction, of course, is inclusive. The quantifications are understood to be over temporal objects, and F is understood to represent a full-career predicate.) This, I say, is the obvious definition—and I can see no other. If the definition is obvious, however, it is equally obvious that it cannot be accepted by the growing-block theorist. Suppose, for example, that the earth will one day be sterilized by a supernova. Let us express this baleful supposition in these words: a ‘‘sterilization event’’ will one day occur. (Suppose, for good measure, that it is now causally inevitable that a sterilization event will occur.) But this implies that there will be a sterilization event (although there never has been one). The proposed definition of timeless quantification obviously implies that, in the circumstances imagined, there is, timelessly, a sterilization event. But the growing-block theory implies that there is, timelessly, no sterilization event. I myself doubt whether there is any possible sense of ‘There is, timelessly’, such that a statement of the form ‘There is, timelessly, no F’, understood in that sense, is consistent with the corresponding statement of the form ‘There will be an F’. But, of course, a doubt is not an argument.

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case. Timeless quantifications over events—even ones expressed by sentences that contain no indexical element whatever—can change their truth-values. The most important case of this phenomenon is this: Timeless existential quantifications over events that have been false can become true. (But existential quantifications over events obey the rule, ‘‘Once true, always true.’’) For example, the proposition that there is (timelessly) such a thing as the death of Caesar was once false, but it became true at a certain moment in the Ides of March, 44 bc (and it was thereafter unalterably true).14 It follows that timeless universal quantifications over events can be true and later become false, and that they obey the rule, ‘‘Once false, always false.’’15 For example, the proposition expressed by the ‘‘Nunc’’ sentence that—obedient reader that you are—you uttered a moment ago was true at the moment you uttered that sentence and immediately thereafter became false and is destined to be forever false. (7) An event is or was past at a time t if it is or was true at t that there are (timelessly) events that occur after it.16 An event is or was present at a time t if it is or was false at t that there are (timelessly) events that occur after it.17 Thus, it is true at every time that every event—timeless quantification—is either past or present. If, at any time, an event is then a future event if it is then neither past nor present, it follows from these definitions that it is true at every time that there are (timelessly) no future events. But our having 14 Suppose that in 45 bc, Brutus had said—using tensed language—that it was then false but would eventually be true that there was such an event as Caesar’s death—or would eventually be true that there was an event that was a death of Caesar. Was the proposition that he expressed true at the time he spoke? Was it true timelessly but false at the time he spoke? Was it false, both timelessly and at the time he spoke? Did it perhaps have no truth-value of either sort, although it was destined to become timelessly false? A proponent of the growing-block theory will have to answer questions of this sort. I am content to leave them open. 15 Ted Sider has pointed out to me that the two rules, ‘‘once true, always true’’ and ‘‘once false, always false’’ require some sort of restriction. Consider, for example, ‘There is a sunrise that is followed by no sunset’. This is true at the moment at which I write (according to the growing-block theory), but it will become false this evening when the sun sets. I have no idea how to formulate the appropriate restriction. 16 ‘‘Is or was’?’’ Well, the Battle of Waterloo is past now and was past in 1915. 17 My writing the words you have just read is present at 1:02 p.m., Eastern Daylight Saving Time, 4 August, 2008—so I say at 1:02 p.m., Eastern Daylight Saving Time, 4 August, 2008. You would say that it was present at that time.

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offered definitions of ‘past’ and ‘present’ and ‘future’ that have this consequence should not be regarded as an attempt to make the substantive thesis (5) above true by definition. It should rather be said that we have offered a definitions of ‘past’ and ‘present’ and ‘future’ that presuppose the truth of the substantive thesis. We say, moreover, that an event is present (without qualification) if it is now present. It follows that an event is present if and only if it exists now—as opposed to ‘it is now true that it exists timelessly’. And it follows that an event is past if and only if it (exists timelessly but) does not exist now. A time is present (or is the present time, for all present events are simultaneous) if and only if some event that happens at that time is present.18 A time is past if and only if it is earlier than the present time, and a time is future if and only if it is later than the present time. (Every future time is such that the proposition that there are, timelessly, events that will happen at that time is now false. Our supposition that the future is infinite could be expressed this way. Every future time t has this property: the currently false proposition that there are, timelessly, events that happen at t will become—unalterably—true at t. If there were—contrary to what we are supposing—a ‘‘beginning of time,’’ then some past time would be such that it and every earlier time had this property: the proposition that there are, timelessly, no events that happen at that time is and always has been unalterably true.) (8) Define ‘x exists timelessly at t’ as follows: ‘the proposition that there is, timelessly, something identical with x is true at t’. Then: For any times t1 and t2 (t2 later than t1 ), every event that exists timelessly at t1 exists timelessly at t2 , and some events that exist timelessly at t2 do not exist timelessly at t1 .19 18 If times are understood as in n. 10, we may say that t is the present time just in the case that an event that has the property t is present. 19 There are confusions of tense in this statement. I let the statement stand in the text because a careful statement of the intended thesis is so complicated as to be, to say the least, hard to take in at a glance. Strictly speaking, we should regard ‘x exists timelessly at t’ as a present-tense statement, and go on to define ‘x existed timelessly at t’ as ‘the proposition that there is, timelessly, something identical with x was true at t’ (and similarly for ‘x will exist timelessly at t’). Thus, the Battle of Waterloo exists timelessly now, existed timelessly in 1915, did not exist timelessly in 1715, and will exist timelessly in 2115. The carelessly stated thesis may now be carefully stated as the following tensed statement:

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These eight statements comprise the growing-block theory. And that name is appropriate, for there is an obvious intuitive sense in which the theory represents Reality as a ‘‘growing block.’’ It is a ‘‘block’’ because the B-relations among its parts are (once established) unalterable. It is growing, because, if we think of the temporal axis as a dimension, then Reality is, as we might say, growing along the temporal dimension: at every time it contains—timelessly—all the parts it contained at earlier times and other parts as well. (Of course, if past time is infinite, Reality is growing only in the sense in which a railway that is infinitely long in one direction is ‘‘growing’’ as new track is laid. We need not worry about the question whether Reality is ‘‘really the same object’’ from moment to moment—a question raised by the fact that, at any two moments, the word ‘Reality’ denotes the mereological sum of different sets of events at those moments.) Some philosophers have supposed that if Reality is indeed a growing block, there must be a ‘‘second sort of time’’ in which it does its growing. Other philosophers have denied this. Let us suppose that whether or not the existence of this second sort of time follows from or is essential to the growing-block theory, this other time—hyper-time, I shall call it—does exist. (In my view, for what it is worth, the existence of hyper-time does not follow from the growing-block theory; not, at any rate, from the theory as I have stated it. But, if the concept of hyper-time is meaningful at all, the existence of hyper-time does seem to be consistent with the theory.) Let us provide this hyper-time with an intuitive representation by imagining that there is an immaterial rational being, an (a) If t1 is a past time and t2 is a later past time, all the events that existed timelessly at t1 existed timelessly at t2 ; and some events that existed timelessly at t2 did not exist timelessly at t1 . (b) If t is a past time, all the events that existed timelessly at t exist timelessly at the present time; and some events that exist timelessly at the present time did not exist timelessly at t. (c) If t1 is a past time and t2 is a future time, all the events that existed timelessly at t1 will exist timelessly at t2 ; and some events that will exist timelessly at t2 did not exist timelessly at t1 . (d) If t is a future time, all the events that exist timelessly at present will exist timelessly at t; and there will be events that exist timelessly at t that do not exist timelessly at present. (e) If t1 is a future time and t2 is a later future time, there will be no events that exist timelessly at t1 and will not exist timelessly at t2 and there will be events that exist timelessly at t2 that will not exist timelessly at t1 .

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Intelligence, that exists outside ‘‘our sort of time,’’ and whose conscious experience comprises the successive awareness of events in the order in which they occur in hyper-time. Let us imagine that the Intelligence can actually watch Reality (physical reality) hyper-become temporally longer as hypertime passes—in a sense analogous to the sense in which human beings can watch a railway become spatially longer as new track is laid. Reality, therefore, not only grows with the passage of time (at a rate of so many standard events per second20 ), but hyper-grows with the hyper-passage of hyper-time (at a rate of so many seconds per hyper-second).21

2. the model Hyper-time is the key to our model of non-Ludovician time-travel. That is to say, the model presupposes a version of the growing-block theory that includes hyper-time. Unless the growing-block theory actually entails the existence of hyper-time (I have said that I do not think it does) postulating its existence does nothing for the theory as a theory of time. If we were interested in the growing-block theory only as a theory of time, hyper-time would be an otiose postulate. But suppose that there in fact are episodes of time-travel. What does the Intelligence see when it examines Reality at a hypertime at which it contains (timelessly) one of these episodes? If the episode is Ludovician, something of the following sort. (I shall tell two stories about what the Intelligence sees as it examines the hyper-growing-block that is Reality.22 In these stories, the 20 A ‘‘standard event’’ would be something like a single vibration of a cesium atom. If the number of cesium atoms is always the same—let us suppose this to simplify the example—, then, for some n, n standard events become ‘‘new parts’’ of Reality every second, and we may say that Reality grows at a rate of n standard events per second. 21 The Intelligence could use Reality as a hyper-clock. It might in fact use Reality as the ‘‘standard’’ hyper-clock and, for some m, define a hyper-second as the amount of hyper-time in which m standard events (see the previous note) are added to Reality. If this m is equal to the n of the previous note, then Reality hyper-grows at a rate of one second per hyper-second. If m = 2n, then Reality hyper-grows at a rate of two seconds per hyper-second, and so on. In the text, I will assume that Reality grows at the rate of one second per hyper-second. 22 From our point of view, Reality is a growing block: it is constantly gaining new parts and never loses any parts. From the point of view of the Intelligence, it is a hyper-growing-block: it is a certain number of seconds ‘‘longer’’ every hyper-second.

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apparent tenses of verbs are really hyper-tenses. If you like, when I tell these stories, I am imagining the words of a speaker of hyper-English—another inhabitant of hyper-time—who is telling the story of the observations of the Intelligence. I will, however, scatter a few ‘hyper-’s around in the stories, more or less at random, simply to keep the sort of stories they are before the reader’s mind. My stories have the following defect. The block the Intelligence is watching is a block: apart from its hyper-growth, it is entirely hyper-static. But I represent the stories as being in large part records of things happening. The first story, for example, contains the words ‘The human beings then left the machine’. My only excuse is that it is not possible for me to tell the story in any other way.) When the block had grown to the point at which its internal calendars read ‘1920’, an object (a machine with human beings inside it) having the set of intrinsic properties F appeared, thrusting aside the matter that had occupied the region of space that it, the machine, now occupied; this event had no temporal causal antecedents. The human beings then left the machine. As hyper-time passed and the block grew a certain number of seconds longer, there came a point at which those human beings got back into the machine. At that moment, the machine-cum-passengers composite had the set of intrinsic properties G; it promptly disappeared, leaving no physical residue whatever. When the block had grown another hundred years and its internal calendars read ‘2020’, an object having (approximately) the set of intrinsic properties F disappeared without leaving a trace. A few moments later, an object having (approximately) the set of intrinsic properties G appeared in a manner similar to the way in which the object with the set of properties F appeared in 1920. It should be clear that postulating the existence of hyper-time contributes nothing to our understanding of Ludovician time-travel. Our hyper-temporal—the strictly correct word would be ‘hyperchronic’—observer does no work toward that end. (A long-lived and omnisicent Intelligence who was, like ourselves, an inhabitant of time, would have described this episode in almost the same words. To turn the story above into the story such an observer would tell is a purely mechanical task.)

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But suppose we try to imagine a hyper-temporal Intelligence watching an episode of non-Ludovician time-travel. What would it see? No doubt this is an open question, but, on the model of non-Ludovician time-travel I propose, the answer is as follows. The Intelligence sees the block grow beyond the 1921-NewYear’s point with no apparently ex nihilo appearance of any sort of machine having been included in the block. After the block has grown about a century longer, the Intelligence sees one of its inhabitants, Tim, enter an elaborate machine. At the moment t, Tim presses a button (at t, the machine-Tim composite has the set of intrinsic properties F)—and suddenly, all in a hyper-instant, the futuremost century-long part of the block vanishes. The events that hyper-now constitute the ‘‘leading face’’ of the block occur at a moment t0 in 1920. But those events are hyper-now a bit different from the way the t0 -events hyper-were before the block lost its century-long terminal segment, for the leading face hyper-now contains the appearance of a machine-Tim composite with (approximately) the set of intrinsic properties F. (The matter that had, just before t0 , occupied the region of space the composite now occupies has been somehow thrust aside into the space surrounding that region.) Thereafter, the block grows at the same rate (so many seconds per hyper-second) it hyperalways has. But it does not grow in quite the same way it did following the previous hyper-occasion on which it passed the point t0 —not even if the laws of physics that govern the ‘‘ordinary’’ evolution of the block are deterministic. It grows in a different way because conditions at t0 are hyper-now different from what they were the previous hyper-time the block reached t0 : on this hyper-occasion, the leading face contains the appearance of the machine-Tim composite. A month later, Tim re-enters the time-machine and presses a button. At that moment, the machine-Tim composite has the set of intrinsic properties G. The physical processes inside the time-machine then slow down, relative to external physical processes—almost, but not quite, to a standstill—till the block has grown a bit past the moment t, and then the composite begins once more to ‘‘age’’ at the normal rate. (Let us suppose that when the machine is in its ‘‘slow’’ state

18

Peter van Inwagen it has essentially no causal interaction with the rest of the world; it is, for all practical purposes, at least, undetectable.) At that moment, the composite has (approximately) the set of intrinsic properties G.23

This is part of the story. But, we may well ask, what happens then? How should the story be continued? This is a matter of conjecture—even if the ‘‘ordinary’’ growth of the block is deterministic. I am inclined to think that the world is sufficiently chaotic—you know: butterfly wing-flaps and hurricanes two weeks later—that Tim will find that he ‘‘has never been born,’’ that ‘‘no one knows who he is,’’ that there is no record of anyone who bears his name and looks like him and shares his DNA and has the same set of great-grandparents. And I suspect that he would find that the same thing was true, mutatis mutandis, with respect to ‘‘everyone he ever knew or knew of’’: that there was no one who corresponded to the friends and relations and public figures that he (would think he) remembered.24 He may well, in fact, discover that time-travel ‘‘has never been invented.’’ Non-Ludovician time-travel entails changing the past; if the world is chaotic, a very small change, even the arrival of a time-machine in an isolated area and its ‘‘departure’’ a fraction of a second later, might well lead to immense changes in 23 Does this story represent Tim as a mass-murderer? In a way, it does. But only, I think, in virtue of being a story in which a century-long segment of the past is changed. ‘‘Mass-murder’’ in this sense would seem to be, of logical necessity, a feature of any story in which the past is changed to that extent: If Tim changes the past in the way I have imagined, he will have to be a ‘‘mass-murderer’’ according to any theory or model of changing the past. 24 It is a commonplace that time-travel, even Ludovician time-travel, creates difficult problems of personal identity. In the end, these problems cannot be evaded, since—for one thing—if it is not Tim the time-traveler who ‘‘arrives’’ in 1920, the same Tim who (timelessly speaking) departs for 1920 in 2020, then we do not really have a case of a time-traveler. (But if the person who pops into existence in 1920 is not Tim but is a qualitative duplicate of Tim—as he hyper-was at the moment he pressed the ‘‘depart’’ button—we should still have a case of sending information into the past, and thus the same problems with ‘‘the paradoxes of time-travel.’’) For example, if Tim travels one hour into the past and meets himself (or meets his earlier self, as some writers like to say)—well, what do we say about that? Is he then bi-located—or what? Would not bi-location imply a violation of Leibniz’s Law? I am not going to try to ‘‘talk round’’ all those difficult problems, since my project is to deal with the paradoxes of time-travel (which are raised simply by sending information into the past) and not with all philosophical problems raised by time-travel.

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the course of history—the history of the human Lebenswelt, that is: no doubt the Earth’s orbit and the shapes of the continents would remain the same—within a few years or even a few months.25 But the ‘‘what happens then?’’ question, interesting as it is, is not really relevant to our central problem, the paradoxes of time-travel, for if there are such paradoxes, they should have shown up by the time (that is narrative time) the story has got to this point.

3. changing the past without paradox Our model represents the growing block as having a ‘‘hyperhistory.’’ Normally it grows at a steady rate, but hyper-occasionally it instantly ‘‘shrinks’’ (loses a terminal segment) and then begins to grow again, and grow in a different way from the way it grew the previous hyper-time. The hyper-history of the block comprises all the episodes of normal growth and sudden ‘‘snappings-back’’ that have hyper-ever occurred. Describing the hyper-history of the growing block by adopting, in our imaginations, the point of view of an imaginary hyper-temporal Intelligence has its intuitive and expositional advantages. But we are going to examine a very complex episode in hyper-history, and the story of what the Intelligence observed in the course of that episode would be very hard to follow indeed. Since one picture is worth a thousand words, I will use a different descriptive device. I will use what I shall call ‘‘hyper-temporal diagrams.’’ (One might think of them as notes that the Intelligence uses to record its observations of the growing block.) A hyper-temporal diagram is a diagram that has the following features. (The reader is advised first to skim through this list of features and then to consult it frequently while working through the examples—diagrammatic representations of episodes of time-travel—that follow the list.) —Each hyper-temporal diagram (‘diagram’ for short) consists of a vertical sequence of horizontal lines. Each line is a ‘‘time-line’’ of 25 If the world is both indeterministic (if its ‘‘normal evolution’’ is indeterministic) and chaotic, the vanishing of a ‘‘terminal’’ segment of the block even without the ‘‘arrival’’ of a time-machine at its hyper-new leading face might well lead to immense changes in the course of history in a short time. The arrival of a time-machine in such a world would, to a near certainty, magnify these changes.

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the familiar kind used to display the order of events in history or in a narrative. Each point on each line represents both a time and the events that are occurring or have occurred or hyper-have occurred at that time. (If a vertical line is drawn through a diagram, each of the points it intersects represents the same time. Recall that ‘‘times’’ are abstract objects of some sort, and that the existence and identity of a time are therefore unaffected by the changes the growing block hyper-undergoes.) —The top line of a diagram represents the growing block as it was (in hyper-time) just before (in hyper-time) the first-ever (in hypertime) episode of time-travel.26 The events belonging to a world in which there were no episodes of time-travel would be represented by a ‘‘unit diagram’’—a diagram that consisted of a single line. And so would a world that contained only Ludovician episodes. A ‘‘purely Ludovician’’ diagram is distinguishable from a ‘‘no time-travel’’ diagram only in that some of the events represented in the former will have rather bizarre causal properties (machines pop into existence and these sudden apparitions are in principle causally inexplicable by any appeal to the prior state of the world; machines vanish into thin air leaving no trace of themselves behind).27 —The number of lines in a diagram equals the number of episodes of time-travel there hyper-have been plus 1. (The number of episodes of time-travel there hyper-have been should not be confused with the number of episodes of time-travel there have been. For one thing, if an ‘‘episode’’ of time-travel has both a departure and an arrival as sub-events, there never have been any episodes of time-travel because there never have been any departures. But even the number of arrivals that there hyper-have been is not necessarily 26 I will assume—once more, for the sake of simplicity—that there hyper-was a hyper-first episode of time-travel (despite the fact that this is of 0 probability if the past is infinite and the large-scale features of the physical world have always been much as they are now). There would be no fundamental difficulty in extending the notion of a hyper-temporal diagram to include diagrams in which every line has a line above it. They would be hard to draw, of course. 27 The single line that represents the sequence of events in no-time-travel and Ludovician-time-travel worlds terminates in a moving point, the present; that is to say, a normal sort of time-line requires continual revision as new things happen. Only the bottom line of a non-Ludovician diagram has this feature: all the other lines terminate in a (hyper-) fixed point, the moment at which a time-traveler departed for the past.

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identical with the number of arrivals that there have been. For it might be, for example, that there has been only one arrival (B), but there hyper-have been two arrivals, A and B—owing to the fact that B changed the past in such a way that the hyper-earlier arrival A never occurred. That is: A was contained in the segment of the block that vanished when B occurred. —The lowest line represents the past and present (all the events that are occurring or have ever occurred), the present being its endpoint. That is, the lowest line represents the hyper-latest history of the growing block: by ‘the past and present’ we mean the past and present as they are at hyper-present. The second line from the bottom represents the hyper-previous history of the block, the line above that represents the history hyper-previous to that one, and so on. The diagram as a whole represents the hyper-history of the block: the entire hyper-temporal sequence of histories that it hyper-has had. —Each line but the top one displays at least one time-machine arrival. If line 16 displayed seven time-machine arrivals, and if the ‘‘new’’ (in hyper-time) arrival displayed in line 17 is after four of them (in time), line 17 will display five time-machine arrivals: the first four that were displayed in line 16, and the new one—the last three arrivals displayed in line 16 having been in the segment of the block universe that was annihilated by the new arrival.28 —The events represented in line n to the left of (before—in time) the latest time-machine arrival are numerically identical with the events before that time in line n − 1; the events represented by the two lines after that time are numerically distinct (if a time-machine arrived ten minutes ago, no event happening now is represented in any line but the last), and, as we have seen, qualitatively or descriptively different to at least some extent—probably to a very great extent. 28 If an ‘‘episode’’ of non-Ludovician time-travel is an event of which a departure and an arrival are both sub-events, it cannot be identified with any event displayed on any one of the lines in a hyper-temporal diagram. The arrival sub-event will be represented on one (or more) lines and the departure sub-event will be represented on the next line up—the line immediately above the highest line on which the arrival is represented.

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Let us look at some examples of hyper-temporal diagrams. The simplest case is of course the diagram that represents a world in which there is no time-travel, or in which there are some episodes of Ludovician time-travel and no episodes of non-Ludovician timetravel. The diagram for such worlds is, as I have said, a single time-line of the sort one finds in history textbooks: 1066

1492

1815

↓

↓

↓

1.

Now let us consider a very simple episode of non-Ludovician timetravel and see how it would be represented in a hyper-temporal diagram. Suppose that Tim presses the ‘‘depart’’ button in his timemachine in 2020 (he is then twenty-five years old), and that the machine ‘‘then arrives’’ in 1920, and that this is the only episode of time-travel there has ever been (the only one that there has ever been in hyper-time:). And let us suppose that subsequent to his arrival in 1920, eighty-eight years have passed and it is now, as I write this, 2008. In the following diagram, x represents ‘‘pressing the ‘depart’ button,’’ and y represents the arrival of the time-machine. 1920 ↓

x

1.

2.

2008 2020 ↓ ↓

y

(In the diagrams that follow, we omit the year-annotations.) The events to the left of point y in line 2 are the same as the events to the left of the same point in time (the moment in 1920) in line 1. But the events after y are different in the two lines, both numerically and descriptively—in all probability very different descriptively. (The ‘‘double’’ line represents the events contained in the part of the block that vanished: events that hyper-have occurred but have never occurred.)

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Note that this is how we draw the diagram in 2008. If we had drawn it in 1920 (or if Tim had drawn it the moment he stepped out of the time-machine), it would have looked like this: x

1.

2.

y

If it had been drawn in 1950, it would have looked like this: x

1.

2.

y

An episode of non-Ludovician time-travel annihilates the part of the block after the arrival29 ; then the block commences re-growth, but, as happens with starfish that re-grow a severed limb, the re-grown part will not be exactly like the original. (In the case of time-travel, however, this is of metaphysical necessity. In the present example, the ‘‘new growth’’ starts with the appearance, apparently ex nihilo, of a time-machine at a certain point in 2–1920 (that is, the second hyper-time the year was 1920: the point in line 2 that represents 1920), and no such event occurred in 1–1920. Now let us suppose that very soon after Tim got out of the timemachine in (2–)1920, he killed his mother’s paternal grandfather. (Tim’s mother hyper-was born in 1–1960. In 2–1920, she was born sixty years ago in hyper-time; for, at the moment when Tim was

29 What would happen if two people in different time-machines pressed their respective ‘‘depart’’ buttons simultaneously and had both set the same arrival date into their machines? Let us suppose that some factor—possibly mere chance—would have the following consequence: at most one would arrive; either both would find that nothing happened when the button was pressed, or one or both of them would be annihilated.

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Peter van Inwagen

about to press the ‘‘depart’’ button, she was sixty years old.30 ) This creates no paradox. Tim simply continues to exist in the normal way after he kills great-grandpa. But why did he exist ‘‘in the first place’’? (Imagine that someone asks this question in 2–1920.) There’s no answer to this question in time (he and the time-machine have recently popped into existence for no reason, no reason to be found in time.). There is, however, an answer in hyper-time: he was conceived and born in the normal sort of way twenty-five hyper-years ago—in 1–1995, in the part of the growing block that hyper-no-longer exists. And the time-machine exists for no reason to be found in time, but rather because Tim built it over the course of the last two hyper-years—between 1–2018 and 1–2020, in the part of the growing block that hyper-no-longer exists.31 We must therefore distinguish events that occurred n years ago in time and events that occurred n years ago in hyper-time. To find (in a diagram) the events that occurred n years ago in time, go back one hundred years from the present along the lowest line. To find the events that occurred n years ago in hyper-time go back one hundred years—but switch to the next line up at every time-machine arrival (in the present example, when you have reached point y, proceed ‘‘directly’’ to point x). Thus, the events that occurred one hundred years ago in hyper-time are represented by the point z in the following diagram: z

1.

2.

x

y

The underlining represents the last one hundred years of events in hyper-time—as opposed to the last one hundred years 30 We assume that a year of time ‘‘equals’’ a year of hyper-time. That is, we assume that the block is hyper-growing at a rate of one year per hyper-year. More exactly, we assume that the hyper-temporals use the growing block as their standard hyperclock and have defined a hyper-year as the amount of hyper-time in which 3.16 × 10 exp 7 × n standard events are added to the hyper-growing-block. (See n. 21.) 31 These statements, of course, presuppose that causal relations can hold between events between which no temporal relations hold. But we had already supposed this: the departure of a time-machine is part of the cause of its arrival in the past, and any two such events are temporally unrelated.

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of events in time, whose representations are underlined in this diagram: x

1.

2.

y

(The events represented in line 1 to the left of the point above y are underlined because they are numerically identical with the events represented in line 2 by points to the left of y.32 ) Thus, if Tim is asked, immediately after his arrival in 1920, how long he has existed, there are two things he can say truly: ‘I have existed for just a few minutes (in time)’ and ‘I have existed for twenty-five years (in hyper-time)’. Now let us look at a world with a more complex hyper-history and draw a diagram that represents the events comprised in that hyper-history. Tim has built a time-machine. He presses the ‘‘depart’’ button in (1–) 2020 (x), hyper-then arrives in 2–1950 (y), and soon thereafter kills his maternal grandfather (k); thereafter, he does no more time-traveling and continues to exist in the normal sort of way. Seventy-two years later, in 2-2022, he confesses this act to his young friend Teresa (c). (Of course, he is a very old man then—seventy-two years old in time, and ninety-seven years old in hyper-time—and thus ninety-seven years old ‘‘physiologically.’’) Teresa decides to undo Tim’s temporal meddling—the murder at least. She, too, has built a time-machine. She enters it and presses the ‘‘depart’’ button (b). She arrives in 1950 very soon after Tim’s arrival33 but before he 32 A hyper-temporal diagram might more perspicuously have been drawn in the form of a ‘‘multi-branched Y,’’ the fork occurring at the earliest temporal date on which an arrival has hyper-ever occurred. But such diagrams are hard to draw—and they might be somewhat misleading because they might suggest a ‘‘branching history’’ account of changing the past. (The resemblance would, however, be superficial since a hypertemporal diagram drawn in the contemplated style would have only one fork even if it represented the past as having been changed hundreds of hyper-times.) 33 Let ty be the date of point y. Hyper-after Teresa has arrived in 1920, shall we say that Tim arrived in the past at 2-ty or at 3-ty ? The answer is 2-ty . Remember that

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has got round to murdering grandpa (a). (A simpler way to undo the murder would have been to arrive before Tim’s arrival—for in that case Tim would, to a near certainty, never have existed.34 But Teresa is fond of Tim.) She prevents the murder (p). Tim and Teresa continue to exist in the normal sort of way for fifty-eight years—no time-travelers arrive during that period—and it is 2008 (for the third hyper-time). Grandpa, who also continued to exist in the normal sort of way, died of natural causes in 1980 (d), but Tim and Teresa are still alive. Here is the diagram (some intervals are exaggerated): 1.

x

2.

y

3.

y a p

c b

k

d

Now let us ask this. How old are Tim and Teresa (now, in 2008)? Well, Teresa, who was twenty years old when she pressed the button, is fifty-eight years old and seventy-eight hyper-years old. No problem there, for ‘seventy-eight’ is what she would give as her ‘‘true age,’’ although she would concede that, strictly speaking, she had existed for only fifty-eight years. And Tim is also fifty-eight 3-ty is short for ‘the third hyper-time the time ty has occurred’—and there is no such hyper-time (not hyper-yet, anyway), for ty hyper-has occurred only twice: its first occurrence is displayed in line 1, and its second in both line 2 and line 3. 34 In a non-Ludovician time-travel world, a person can have come into existence (in time) in one of two ways: by having been born or by having arrived by timemachine. If Teresa had traveled to, say, 1919, Tim would, to a near certainty, not have come into existence in either way. If Teresa had arrived in 1919 Tim might have been born: although vastly improbable, that is a possibility. And if Tim did somehow manage to be born (in 1995 or thereabouts) despite the disruption of history caused by Teresa’s arrival in 1919, he might have gone on to travel to a point in the past subsequent to her arrival in 1919 and have encountered her in, say, 1920. But while Teresa would know that if she were to travel to 1919, she might conceivably encounter time-traveling Tim in 1920 or might conceivably encounter non-time-traveling Tim by living to be a very old woman and witnessing his birth, she would also know that if she were to travel to 1919, Tim’s very existence, his being ‘‘available’’ for her to encounter in either way, would be a vastly improbable circumstance.

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years old (he is a little older than Teresa, since his arrival in 1950 was earlier than hers). But his birth, which occurred at a point twenty-five years before the end of line 1 was 155 hyper-years ago. (Fifty-eight hyper-years of line 3, seventy-two hyper-years of line 2 and twenty-five hyper-years of line 1.) If B represents Tim’s birth, here (underlined) are the points at which the events contained in his hyper-biography occurred: B

1.

2.

y

3.

y a p

x

c b

k

d

And yet, Tim is physiologically in his early eighties, and if he had been carrying a chronometer that measured elapsed time since the moment of his birth it would register eighty-three years. (Let us use the readings displayed by such imaginary chronometers as the definition of the bearer’s ‘‘true age.’’) We can, therefore, identify ‘‘one’s true age’’ and ‘‘the amount of hyper-time that has passed since one’s birth’’ only for the latest time-traveler (the time-traveler whose arrival is the latest in both time and hyper-time). Consider Tim, who, in our present example, is not the latest time-traveler. The underlining in the following diagram marks the set of events whose collective hyper-duration we must measure to determine Tim’s true age: B

1.

2.

y

k

3.

y a p

x

c b

d

The ‘‘personal biography’’ of a time-traveler (for the sake of simplicity, we consider only ‘‘one-trip’’ time-travelers), therefore, should

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be identified with the sum of the events contained in his or her hyper-latest post-arrival life and the events contained in his or her pre-departure life. The part of one’s life between one’s birth and one’s departure will be displayed in the line immediately above the line that displays one’s arrival only if one is the latest time-traveler. This is what ‘‘changing the past’’ comes to on the ‘‘growing block with hyper-time’’ model. Note that the model does not grant to the non-Ludovician time-traveler all the powers in respect of changing the past that someone with an interest in changing the past might want a time-traveler to have. I am thinking of those time-travel stories in which a conscientious time-traveler changes the past in such a way that ‘‘time-travel has never existed.’’ It is certainly possible, on the growing-block model, to change the past in such a way that time-travel has never been invented35 —that is, that its invention exists only in the hyper-past. But, on the growing-block model, once (hyper-once) an episode of non-Ludovician time-travel has occurred, there is no way to get rid of time-travel altogether, since the bottom line of a two-or-more-line hyper-temporal diagram must contain the arrival of a time-traveler.36 University of Notre Dame 35 For example: It is now the year 2500. History records that in 2018, a Terminator popped into existence out of thin air and murdered Alice, a brilliant physicist, and burned her laboratory and all her notes. (Alice was working in secret, and no one else knew anything about her research.) The Terminator then destroyed itself, but left a note saying that it had been sent back in time from the year 2040 to prevent Alice from constructing the first time-machine (which, according to the note, she would have succeeded in doing in 2020)—owing to the fact that time-travel had turned out to have disastrous consequences. No one, in all the intervening centuries, has been able to construct a time-machine. 36 I am grateful to Ted Sider for comments on a draft of this essay, which have led to changes that I hope he will regard as improvements.

2. Can a Souffl´e Rise Twice? Van Inwagen’s Irresponsible Time-Travelers1 Peter Forrest If physicists send a particle or two back in time, I fear they will make snide remarks about philosophers in their armchairs claiming to know a priori that this is impossible. Peter van Inwagen’s account (in this volume) of non-Ludovician time-travel, hereafter referred to as VIT travel, should forestall that criticism by providing an interpretation of what the physicists will have done, and why, if his interpretation is correct, it is irresponsible. It might seem, therefore, to be quibbling for me to argue that VIT travel should be redescribed as a variant on the branching time hypothesis, according to which ‘timetravellers’ cause there to be a new branch of time (or more accurately a new branch of 4-space) when they get out of their machines. In this chapter I first explain why this is not a mere quibble. Then I argue for the redescription thesis. Finally I anticipate objections.

1. why it matters that we cannot alter the past We have a strong intuition that we cannot alter the past, even if we can affect it. Block theorists have a straightforward explanation of this truth, assuming it is a truth. For if, per impossibile they say, the past is altered then the result of the alteration is just how it always was, so this is a case of affecting the past, not altering it. Now, if one theory provides a straightforward explanation of something that is intuitively taken to be true, then it has a prima facie advantage 1 I am grateful to Peter van Inwagen for raising the taboo topic of altering the past, because thinking about this topic has helped clarify the growing-block theory. I am also grateful to Dean Zimmerman for helpful advice on an earlier draft. I attribute the question, ‘Can a souffl´e rise twice?’ to the then Australian Treasurer, Paul Keating, in 1989.

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over other theories that make a mystery of these truths. So those who hold rivals to the block theory should be dismayed if their account seems to permit past-altering time-travel. Consider, for instance, Arthur Prior’s (1957: 18–28) account of past and future as tense-modalities, and suppose the need for further grounding of past tense truths is resisted. Then the impossibility of altering the past is expressed by saying that what is past will be past. In terms of the operators Pt , that is, t seconds ago, and Ft , that is, t seconds in the future, we have the following schema: for positive s and t and for any q, Ps q ⊃ Ft (Pt+s q).2 It is tempting to defend this principle on the grounds that for any t and q, Ft q = P –t q, but defenders of dynamic theories should resist any such uniform treatment of past and future. To be sure, the principle can be commended on grounds of theoretical simplicity, but that merely alleviates rather than removing the mystery.3 Growing-block theorists like myself might be complacent about the contrast between the block theory and the Prioran primitive modality theory, concerning the impossibility of altering the past—surely, we share the block theorists’ easy explanation. VIT travel should, therefore, give us pause. For if the growing block is compatible with altering the past, then the block theory has a significant advantage over the growing-block. My claim that VIT travel is a variant on the branching time hypothesis has as a corollary the thesis that the past is never altered, not even, it turns out, by the addition of new branches. In this way I defend the growing-block from a new objection. My defense of the growing-block from the threat of VIT travel is required only if we take the hyper-time hypothesis seriously. Now van Inwagen does not think that the growing block requires hyper-time (this volume, p. 14). Nonetheless, it is natural to think of 2 This differs from Prior’s notation not only in the substitution of seconds for days and its non-Polish character but also in the raising of the time coordinate, to bring out the analogy between Pt Ps = Pt+s and the algebra rule: xn xm = xn+m . This matters if we generalize from time to spacetime, in which case the mapping t → Pt is a homomorphism from the semi-group of forwards time-like translations under addition to operators under composition. Likewise for the Ft operators. If the identity Ft = P-t holds then this extends to a homomorphism of groups from the group of all spacetime translations to the operators. 3 We could prove that the identity Ft = P –t holds if time is circular, but that conflicts with the intuitions on which dynamic theories are based.

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the block as growing with respect to hyper-time. Likewise on Storrs McCall’s falling branches theory it is natural to think of the branches as falling in hyper-time (1994). Therefore I am reluctant to abandon hyper-time in order to defend the growing-block, even though J. J. C. Smart has criticized hyper-time on the grounds that it leads to an infinite regress: if time requires hyper-time, then hyper-time requires hyper-hyper-time, and so on. (Smart 1965: 137).4 The redescription that I provide has the additional advantage of avoiding the regress.

2. the case for redescription Roughly speaking, I say that van Inwagen’s hyper-time just is time, and what he calls time is, if you like, hypo-time, namely the t-dimension of 4-space, where by ‘4-space’ I mean either Minkowski space or its General Relativistic analog. Although 4-space is a hyperspace it differs from other hyper-spaces that have been proposed by string-theorists or metaphysicians (Hudson 2005) in that no new dimension is posited. Instead the t-dimension has now been re-classified as a spatial dimension. Although this is proposed in order to redescribe VIT travel, it might help readers if I provide a context for this program of combining 4-space with a ‘hyper-time’ that is in fact time itself. Prior to Minkowski’s spacetime interpretation of special relativity, our conception of time included two components: a metric component, measured say in seconds, and for which I use the symbol ‘t’; and an order component, which, for definiteness I take as the ordering of states of affairs with respect to priority—where to say state of affairs X is prior to state of affairs Y (X < Y) is not to say anything about any t coordinate involved in these states of affairs but rather that X itself is before Y. Because our intuitions about priority do not concern the t coordinate, they are undefeated by relativity and so the ordering is necessarily linear.5 Formally: (1) the relation < is 4 Interestingly Smart (1965: 137) anticipates my thesis that hyper-time is ordered but has no metric. 5 It might turn out that for a suitable choice of the t coordinate if X and Y are existential states of affairs and if X < Y then for some z X is about a part of t < z and Y about a part of t > z, in the sense defined below. But even if this is no accident I doubt that it would hold of metaphysical necessity.

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transitive, (2) the relation ≈ defined by X ≈ Y iff (X < Y)&(Y < X) is an equivalence relation and (3) the relation ≤ defined by X ≤ Y iff X ≈ Y or X < Y is transitive. The difference between the metric and the priority ordering may be noted in two ways—one inspired by medieval angelology, the other by logical positivism. Angels were sometimes considered aeviternal; that is, their lives have a temporal ordering but the only temporal metric is that derived from the ordering.6 Thus, the angel’s dargue (list of the ‘day’s’ jobs) might be to leave Heaven, inspire first Kate then Pierre and after that Natasha, before returning to Heaven. Of these events the first and last might lack any Earthly temporal correlate to within ten thousand years, while inspiring Kate might be five hundred years before inspiring Pierre, which is thirty seconds before inspiring Natasha. The angel might know of these intervals of time but does not experience anything other than a succession of three jobs. The positivist story is that although there must be a metric we may rescale so that if t is the standard metric in seconds we could as easily measure time using, say, t-cubed. The constraint is that the order must be preserved but the metric need not be. My proposal is to use Minkowski space to unify space with the metric component of time to obtain a non-Euclidean 4-space, but to take the order component of time as our new conception of time. I take the block to be the sum of the states of affairs that there are. The label ‘block’ is justified because many of these are existential states of affairs, of the form: [there is an F in u], where I use the square brackets to denote a state of affairs, and where u is a region, maybe a point location.7 I say that the state of affairs [there is an F in u], is about region u. As a growing-block theorist I propose that, for a suitable t coordinate, all existential states of affairs are about regions that are parts of the region t ≤ tn , where tn is the t-coordinate of your reading this sentence.8 6 If the ordering is like that of the natural numbers in that each event has an immediate successor then the number of events between two given ones is a derived metric, the unit of which we may call the chronon. 7 In the case in which p has a unique minimal ground then [p] is that ground. But if p is ‘There is an F in u’, p may well have several minimal grounds in which case I stipulate that [p] is the sum of all the minimal grounds for p. 8 (1) My conjecture is that the t coordinate is one with respect to which the universe increases in volume at the same rate everywhere. (2) Quantum theory might complicate the growing block by giving it a fuzzy edge.

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Among the states of affairs there are some of the form [X < Y], where X and Y are themselves states of affairs. Such higher-order states of affairs are all the grounds we need for past truths; that is, truths of the form ‘It was true that p’. We may characterize the time T(X) at which a state of affairs X occurs as the sum of all the states of affairs Z such that Z ≤ X. It is plausible that T(X) is itself a state of affairs . Then ‘It was true at time T that p’ holds if T grounds p. Here the grounding relation is such that if S grounds a proposition p, then p is true, provided every state of affairs is part of S. That proviso does not hold for propositions that were but are no longer true. I am therefore a presentist, according to the letter but not the spirit. The account I have given is, fortunately, incompatible with the destruction of states of affairs. For that would require there to be some present grounds for the truth that a certain state of affairs used to, but does not now exist. It is not easy to think what such grounds might be. What about the state of affairs [X < Y] without the state of affairs [X]? That suggestion fails because either, as I would hold, priority is a genuine relation incapable of having a non-existent relatum or it can relate fictional states of affairs, as in [ [that Merlin lives before 1000 ad] < [that St Francis lived after 1000 AD] ]. I conclude that the attempt to ground a truth about altering the past collapses into fiction. It might be objected that 4-space is a naïve idea—the preserve of block-heads—for we all know that space is quite unlike time. My response is that this common knowledge concerns the order, not the metric, aspects of time. Again it might be objected that intuitively space is isotropic, every direction being of the same kind as every other direction. My response is that geocentrism was not considered counter-intuitive, but it implies a distinction between up (away from the center of the Earth) and down (towards the center of the Earth), a distinction exploited by Aristotelian physics. Hence any intuition we now have that isotropy must hold is an extrapolation from post-Aristotelian physics, combined with the assumptions that space has three dimensions and that the dimension represented by coordinate t in the physics is time. So there is no objection to 4-space as counter-intuitive. It is now easy to resist Smart’s regress: it is not time that requires hyper-time for its passage, but 4-space that requires time. Moreover

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I have tamed hyper-time: all I have done is distinguish the ordinal from the metric aspects of our pre-theoretic concept of time. The former make up hyper-time, which I take as true time; the latter is unified with 3-space to make 4-space. On this re-interpretation, examples of VIT travel are cases in which the block grows for a while from t = 0 to t = k; then it stops growing in its usual way but sprouts a new branch from t = 0. The state of affairs S that such a sprouting occurs is about a region near t = 0, but nonetheless the state of affairs about the original growth of the block from t = 0 to t = k is prior to S. Hence VIT travel does not alter the past in the sense of destroying or even adding past states of affairs. The new branch then grows until some other time-traveler meddles with the structure of 4-space. This complicates the description of the growing block because the t coordinate has become ambiguous. In its place of a single coordinate we require a t-coordinate for each branch, t(1) , t(2) etc. How far the block has grown is then specified by t(1) n , t(2) n etc.9 Whether the past can be altered depends, then, on the following question: which is genuine time, van Inwagen’s time, namely the t-coordinate, or his hyper-time? If time is whatever clocks measure in the actual world then van Inwagen’s time is genuine time. For clocks are the rulers for the t-coordinate of 4-space. I submit, though, that time is whatever clocks were intended to measure in the physical world, during the long historical period when the metric and order-aspects of time were linked except perhaps in a non-physical realm. Therefore the function of clocks should not be used to decide whether van Inwagen’s time or his hyper-time is the genuine time. My case for my redescription is based on the premiss that our pre-theoretical distinction between time and space is not due to their metric properties, which are similar, but due to the ordered character of time. The nearest spatial analog to temporal ordering is the three-place metathety (between) relation, in terms of which we may analyse the apparent ordering of being higher than, by noting 9 Some of which might be +∞. These would be branches for which the block theory holds. Since I hold that normal human consciousness is always restricted to the specious present, which is a thin surface layer of the block, these branches now lack normal human consciousness.

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that z is higher than y if y is between z and the center of the Earth. This premiss supports, in an intuitive sort of way, the assimilation of ‘hyper-time’ to time and the joining of the t-coordinate to 3space to result in 4-space. My main argument for redescription, however, is not this direct appeal to intuition, but that hyper-time is essentially ordered but 4-space is not, and so hyper-time is the better candidate for genuine time, because time is, intuitively, essentially ordered. Hyper-time is essentially ordered because there is nothing to it except its order. That 4-space is not essentially ordered is a controversial thesis among block theorists, but an examination of the controversy in the context of the block theory shows why it should not be controversial in the context of the growing block. The controversy arises because at each point in 4-space we may distinguish (locally anyway) a forward from a backward light cone. Some block theorists such as Huw Price argue that which cone counts as forwards and which as backwards depends on some contingent feature of the universe, and maybe a local one (Price 1996). The usual candidate for this feature is the direction of the overall increase in entropy (Price 1996: 22–38). The reason why Price takes this to be contingent is that entropy will decrease if the universe is (locally) evolving towards a low-entropy state. Price considers a Gold universe in which a Final Crunch mirrors the Big Bang (1996: 78–103). Current physical cosmology suggests there will be no Final Crunch, so my preferred example is the diabolo universe, in which the currently visible part of the universe contracts to, say, a grapefruit-sized, high temperature, low entropy region, which then expands out again in what we call the Big Bang. In that case the direction of increasing entropy before the Big Bang is the opposite of that afterwards. Given that there is a contingent ordering due to a low-entropy state for some t-coordinate, why would philosophers posit an essential ordering to 4-space? Precisely because, I say, time is intuitively essentially ordered. Therefore, to preserve as much as we can of our intuition that there is an essential ordering of time, we are tempted to posit an essential ordering for 4-space. That reason is one that growing-block theorists should ignore. For we can do better than block theorists when it comes to preserving intuitions about time, by positing that the priority relation is essentially

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linear, whereas the block theorist has to make do with the nonlinear ordering resulting from a choice of which are the forward light cones.

3. objections to the redescription I anticipate the objection that once the metric and order aspects of time are separated then time becomes an ambiguous concept, and hence I have no right to identify it with the ordinal aspect, even though I have made a case that if we have to disambiguate then time is better explicated as ‘hyper-time’. Suppose I conceded this objection. In that case I would note that my case for redescription has been in the context of the firmly held intuition that the past is unalterable. This intuition is about ordering, and so even if the concept of time is ambiguous we should explicate it as hyper-time; that is, as the ordering, when applying that intuition. Moreover, because some other metaphysical intuitions about time concern the ordering not the metric, I would strongly urge that explication to be adopted generally.10 The other objection that I anticipate is that there are scientific, empirical, reasons for assuming that 4-space is essentially ordered in addition to metaphysical, intuition-driven ones, and that growingblock theorists should respect these scientific reasons even if the metaphysical ones are undermined. Price has discussed some of the purported scientific reasons for the essential ordering of 4-space (1996). Rather than argue from Price’s authority, but acknowledging his influence, I shall briefly discuss, and respond to, the three main reasons we might have for thinking that the sciences are best interpreted as implying an essential ordering of 4-space. The first is the need for causal hypotheses, especially in astronomy, geology, and biology. We might argue that futurewards is either always or for the most part the direction of causation, so that if, per impossibile, backwards causation were the rule rather than the exception then we would say that we had labeled the directions incorrectly. Hence, it may 10 Among other intuitions there is that which Zeno’s paradox of Achilles and the Tortoise exploits, namely that an infinite sequence of events one after the other cannot be completed. This provides grounds for holding that hyper-time is discrete but not that 4-space is.

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be objected, science relies upon a temporal ordering because it relies upon causation. My reply to this objection is a dilemma: Is backwards causation possible? If not, then the direction of time is essentially that of causation. That is, I grant, a reasonable hypothesis, which requires further discussion, but if backwards causation is impossible, then there is no time-travel and a fortiori none that alters the past. So, for the purposes of this chapter, I shall assume that backwards causation is exceptional but not impossible.11 In that case, there is a continuity between hypothetical universes in which backwards causation is a rare exception, through universes with increasing proportions of backwards causation to one in which no one direction predominates. We should not posit arbitrary boundaries to what is essential if we can avoid it. Hence we should avoid analysing the temporal order in terms of the causal relations between events. The second argument relies on the premiss that the Second Law of Thermodynamics, telling us that entropy cannot decrease, is a genuine law of nature and not, pace Boltzmann, Reichenbach (1957) and more recently Price (1996: 22–48), just the result of a very low entropy state in the past. In that case, it could be said that there has to be an ordering of 4-space for there to be the distinction between increase and decrease. My response is that this might persuade a block theorist but there is no reason why the Second Law should not relate entropy to ‘hyper-time’; that is, time, rather than the t-coordinate. For a start, there is nothing in the Second Law that tells us the rate of increase of entropy, so it merely concerns the ordering. But to settle the question of whether the t-coordinate or ‘hyper-time’ is involved, we need to consider again the diabolo universe, in which the order of entropy change reverses at t = 0. This seems a coherent model of the universe and we should try to reconcile it with the Second Law if we can. And we can, for I say that the block grows from an initial state when the currently visible portion is grapefruit-sized, and that it grows in both the +t and –t directions, with entropy increasing in both directions. The third argument from science is based on quantum theory. If we interpret it so that there is a ‘collapse of the wave-packet’ on 11 That is, I assume there can be an effect located in the backwards light cone from its cause.

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various occasions including observation, then quantum theory requires a temporally asymmetric process, and so, it is said, distinguishes pastwards from furturewards. My response is that the collapse may be taken as the change from one wave-function to another, where both of them are defined over the whole of 4-space.12 Hence this change does not just affect the forwards cone (from the point at which the wave-packet collapses) but also the backwards cone. Suppose for instance that an electron passes through the famous twin slits and is detected on a screen. Prior to detection, the wave-function for the electron fails to specify its position precisely. The detector then collapses the wave-packet so the electron is at point P some distance from point Q, where it might have been detected. Hence the wave-function has changed from one with rather imprecise location on the screen to one of more precise location (but less precise velocity). Because the electron is moving at less than the speed of light its location just before detection must be in the backwards light cone from P and not in the backwards light cone from Q. Yet the wave-function prior to detection allowed it to be in either of the backward light cones. What this shows is that there is a change of wave-function on detection even though both the earlier and later wave-functions govern the electron’s position in both the forward and backwards light cones. Therefore the ‘collapse of the wave-packet’ does not imply an essential distinction between pastwards and futurewards in 4-space. To sum up: my case for redescription is that once we grant the occurrence of hyper-time we no longer have any good reason to think that 4-space has an essential past-to-future ordering. In terms of coordinates we may say that the transformation by which is sent to preserves the essential structure of 4-space, contrary to there being an essential ordering. Therefore, because the essential ordering is intuitively the feature that distinguishes time from space we should identify ‘hyper-time’ with time. Hence the ‘past’ changed by VIT travel is not the past 12 If indeterminacy in re is too counter-intuitive, then we may interpret the collapse in terms of ‘many worlds’; that is, parallel universes. This interpretation of the ‘collapse of the wave-packet’ does not require the growing block, but it coheres well with that theory, if we allow that some but not other universes grow. (On a variant they grow at different rates.) The wave-function alters because some of the universes are terminated (Forrest 2007).

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but just a region of 4-space for which t < tn . To alter the ‘past’ in this way by VIT travel is no more puzzling than altering the east by travelling east. In van Inwagen’s description there are truths about what (hyper-) was the past (hyper-)before the time travel that are altered. For example, we may suppose there are truths about the winning numbers in lotteries, and that some time-travellers announce these numbers beforehand to prove that they have come from the future. The disturbance due to their presence then alters these winning numbers, resulting in scepticism about their claims. As a presentist, that is a ‘hyper-presentist’, I say that these truths are true simpliciter, and the apparent contradiction between what is true before the VIT travel and what is true afterwards is resolved by disambiguating the description ‘t < tn ’, into ‘t(1) , < t(1) n ’ and ‘t(2) < t(2) n ’ that is, by taking 4-space, but not time, to branch. More important, I have also concluded that the past cannot be altered. University of New England, Australia

references Forrest, Peter (2007) ‘The Tree of Life: Agency and Immortality in a Metaphysics Inspired by Quantum Theory’, in Persons: Human and Divine, eds. Peter van Inwagen and Dean Zimmerman (Oxford: Oxford University Press), ch 13. (2008) ‘Relativity, the Passage of Time and the Cosmic Clock’, in The Ontology of Spacetime II, ed. by Dennis Dieks (Amsterdam: Elsevier), 245–53. Prior, Arthur (1957) Time and Modality (Oxford: Oxford University Press). Reichenbach, Hans (1957) The Direction of Time (Cambridge: Cambridge University Press). Smart, J. J. C. (1963) Philosophy and Scientific Realism (London: Routledge & Kegan Paul). van Inwagen, Peter (2009) ‘Changing the Past’, Oxford Studies in Metaphysics (Oxford: Oxford University Press), 3–28.

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3. Van Inwagen on Time-Travel and Changing the Past Hud Hudson and Ryan Wasserman Peter van Inwagen’s discussion of time-travel (‘‘Changing the Past’’, in this volume) invokes both hypertime and the growing-block theory to provide us with a model for changing the past that is both rigorous and ingenious.1 We are impressed. But we are not yet convinced. In this essay, we present three potential objections for van Inwagen’s model (section 1), and then show how his model can be adapted to avoid those worries (sections 2–4).

1. three potential objections for van inwagen’s model The reader is directed to van Inwagen’s essay for a full and careful presentation of the model in question. Our purposes here mainly require taking note of the following components of that presentation. Like van Inwagen, we are interested in discussing ‘‘non-Ludovician time-travel’’—time travel that involves changing the past. Throughout our discussion, we will accept van Inwagen’s (restricted) characterization of the growing-block thesis, his identification of times with certain properties, and his claim that the existence of hypertime is consistent with (even if not required by) the growing-block theory of time. We begin with three potential objections. The first potential objection, which neither author endorses, is that van Inwagen is guilty of false advertising. One of the most salient features—perhaps the most salient feature—of van Inwagen’s time-machines is that they obliterate an awful lot of present and past objects and events. Because of this, some might be 1 For a similar discussion see G. C. Goddu (2003) ‘‘Time Travel and Changing the Past: (Or How to Kill Yourself and Live to Tell the Tale)’’, Ratio vol. 16: 1, pp. 16–32.

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tempted to say that such ‘‘time-machines’’ are really nothing more than annihilation machines and that van Inwagen offers not an account of time-travel but of temporal annihilation.2 We think that this temptation should be resisted. Granted, van Inwagen’s machines obliterate many present objects, but they are also able to relocate present objects in the past and, for this reason, we count his time-machines as worthy of the name. That being said, we do have one puzzling—and potentially worrisome—question to ask about van Inwagen’s obliteration-and-relocation devices. Suppose that uncountably-many would-be time-travelers all set their dials to different times in the past (such that no time is left unselected) and all put their machines into motion at the exact same moment.3 Question: What happens hypernext? The second potential objection, which at least one author endorses, is that van Inwagen’s model does not provide for the possibility of time-travel that changes the future. Many of the most familiar time-travel stories involve changing the past—the past was one way, then the time-traveler did his thing, and now the past is different.4 But other time-travel stories tell the same kind of tale in the opposite direction—the future was one way, then the time-traveler did his thing, and now the future is different.5 However, if there are no merely future objects or events (as the growing-block theory tells us) and every episode of time-travel-with-change involves the annihilation and eventual replacement of some past objects and events (as van Inwagen tells us), then it is impossible to travel to the future and change what will be. Some will find this objectionable.6 The third potential objection, which both authors endorse, is that van Inwagen’s account includes the growing-block theory, which 2 This objection was suggested to us by both Frances Howard-Snyder and Ned Markosian. 3 Taken together, the uncountably many machines might be called a Boojum, after the Snark who shares its power. 4 See, for example, Back to the Future I and III. 5 See, for example, Back to the Future II. 6 Note, however, that if we adapted van Inwagen’s story so that it made use of a ‘‘shrinking-block’’ theory of time (i.e., one in which the present and the future exist but not the past), then we would have an account of time-travel to the future and of changing the future every bit as good (or bad) as van Inwagen’s account of time-travel to the past and of changing the past. Of course, we then would complain about the asymmetry again, just in the other direction. Happily, we believe we have a resolution to this problem below.

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we take to be the least plausible—and least popular—theory of time. Actually, van Inwagen says that his model ‘‘presupposes’’ a version of the growing-block theory.7 This is careful wording. Weakly interpreted, we take him to be announcing that he will present his model by way of that particular theory of time, remaining neutral about whether it could be presented in other ways. Strongly interpreted, we take him to be announcing that the growing-block theory of time is required for his model. Setting aside hard questions about identity conditions for models, we think the strong interpretation is the natural one. But either way, we think it would be a significant advantage to show how van Inwagen’s general approach could be combined with other, more popular theories. In the next three sections, we attempt to do exactly this.

2. the growing block, time-travel, and changing the past To prepare the way for the discussion of the eternalist’s block theory and for the presentist’s slice theory of time, allow us to note some variants on van Inwagen’s own use of the growing-block theory. First a refresher on a van Inwagenian version of events. Suppose, for convenience, that hypertime is finite and that there are only a million ticktocks of it (a ticktock being a unit of hypertime as an hour is a unit of time). Unmolested by time-travelers, the block would grow some number of standard events per second and would hypergrow some number of seconds per ticktock, and there would be a certain happy historical agreement from the perspective of time and the perspective of hypertime—including agreement on statements such as ‘‘Caesar has died but once’’ and ‘‘Napoleon has met his Waterloo.’’ Bring a time-traveler into the story, however, and histories diverge. Suppose our time-traveler activates his annihilation-and-repositioning machine at 12:00a.m. on the first of January 2000, embarking for the Ides of March, 44 bc and hoping to witness a famous murder. Our hero pushes the button at midnight (and, as it happens, at ticktock 100). At hypertimes after but arbitrarily close to ticktock 100, the leading edge of the growing block is cut back to the Ides of March, 44 bc. You and I and the 7

On p.6 and again on p.15.

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roaring 20s and the Second World War are all annihilated, and surprisingly the phrase ‘the one and only murder of Caesar’ is about to have a satisfier for the second time on the hypertime stage. That is, our time-machine materializes, a little more hypertime passes, the block grows a little larger, and it is correct to report our temporal history with the claims that ‘‘Caesar has died but once’’ and ‘‘Napoleon has not yet met his Waterloo.’’ Those are not correct reports of hypertime history, however, which would instead require noting that ‘‘Caesar has died twice’’ and ‘‘Napoleon has met his Waterloo’’—it is just that whereas one of Caesar’s deaths and Napoleon’s defeat are both hyperpast, neither event is past. So—as advertised—time-travel and past-changing have been achieved.8 Note, though, that the repositioning of our time-traveler in the block wasn’t required to change the past. The annihilation machine was up to that task all on its own. Vary the story: The button is pushed by our would-be voyeur at midnight (and at ticktock 100), and as hypertime keeps slipping into the hyperfuture the growing block is cut to the first instant on the Ides of March, 44 bc as before. On this version, however, our protagonist and his machine are likewise annihilated, and that first instant on the Ides of March 44 bc contains exactly what it contained on the previous hypertime that the internal clocks in the leading edge of the growing block read that date—with one exception: some pebble on the floor of the Amazon rainforest has been annihilated, too. Accordingly, even though nothing has been repositioned in the block, and even though nothing has been added to an earlier slice of the block, histories diverge as soon as any hypertime goes by, for at ticktock 99 it was true that the first instant of the Ides of March, 44 bc contained that rock and at ticktock 101 it was not true that the first instant of the Ides of March, 44 bc contained that rock. And this, of course, is simply due to the facts that what is true of the past is fixed by features of the block and that the block looks very different at successive hypertimes—so that what is true of the past can change from hypertime to hypertime. But then it appears that not only is repositioning unnecessary, annihilation of the leading edge of the block is not required either. 8 Space constraints forbid reviewing all the details. We encourage the puzzled reader to review van Inwagen’s careful discussion of these sorts of scenarios before investigating their variants below.

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All we need is for there to be two hypertimes, for the growing block to have one of its temporal slices hyperpresent at both hypertimes, and for that slice to differ in its contents between the two hypertimes. Perhaps a malfunctioning machine responds to the press of its starter button at midnight (and at hypertime 100) by leaving the bewildered would-be time-traveler right where he is, letting the leading edge of the block continue to creep forward at its steady rate of n seconds per ticktock, and succeeding only in eliminating our Amazonian pebble from its place on the rainforest floor on the first moment of the day of Caesar’s murder. If this is the only hypertime such a machine is activated, then at hypertime 101 it is and will always be true that that pebble disappeared in the past on that date from its position on the rainforest floor. But at hypertime 99, it is true that the pebble did not disappear in the past on that date from its position on the rainforest floor. Once again, the past has been changed and no one had to go anywhen to do it. If it should turn out to be easier to build a creation machine (rather than an annihilation or repositioning machine) that would do the trick, too. And a machine that merely changes which color properties our pebble manifests would also do just as well (although one might think even this requires creating some individuals, too). But the general lesson is now clear. What is crucial is guaranteeing that somehow or other our block has different features in one of its timeslices at each of two hypertimes at which that slice is hyperpresent. (Moreover, in this scenario, no past or present objects had to be annihilated; so the first potential objection from the previous section is put to rest for good.) Just to be clear, it is not as if van Inwagen insists otherwise, but it is important for our purposes below that we clearly distinguish between what makes it true (if anything) that we have a case of time-travel and what makes it true (if anything) that we have a case of changing the past, for this distinction enables us to show just how the general lesson might be adapted to other theories of time. That task is the subject of the remaining two sections.

3. the eternalist block, time-travel, changing the past, and changing the future Consider the popular eternalist block model of time in which all temporal relations are B-relations, and in which our lives are—so

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to speak—events frozen in the block. No leading edge or puzzling growth to be explained here. For an event to occur in time is just for it to have a location somewhere or other in the block. Just as van Inwagen contends that hypertime is not necessary for but is nevertheless consistent with the growing-block theory, so too, we contend that (if hypertime is possible at all) it neither conflicts with nor provides some essential component of the eternalist block theory of time. It is simply an add-on with interesting consequences. In particular, it allows us to recapture the crucial component of van Inwagen’s story that was identified in the previous section. That is—an eternalist block can be present during each ticktock of hypertime, and for any instant of hypertime, there are facts about what is past and present (relative to any slice of the block) determined by the features of the block at that hypertime. But should the eternalist block happen to manifest different features at different hypertimes, we will be able to recast our story of time-travel and of past (and even of future) changing without any recourse to the growing block. Many of us have spent an afternoon or two daydreaming about different possible pasts that could have brought us to our present state and of different possible futures in which we could live out our days—knowing full well that those were not our pasts and will not be our futures. Still, as in the Myth of Er in Plato’s Republic, we might daydream about choosing the pattern of our lives, and we may fantasize about being embedded in one of those compossible, past–future pairs. With the tools furnished by hypertime, the present model shows how we could construct the Myth of (Hyp)-Er and how such dreams could be realized: Suppose that at ticktock 100 an eternalist block is hyperpresent and determines all the facts about the past and the future (relative to each of its time-slices). Further suppose that at ticktock 101, with the sole exception of some slice, S, a new eternalist block has replaced the old one (or else the old one has got some new filling sandwiching S—it does not much matter which). Of course, the items that characterize the eternalist block at ticktock 101 determine all the facts about the past and future (relative to each of its time-slices, as well). Finally, suppose that slice S, the common ingredient, was the slice in which you push the start button in your newly constructed time-changing machine. The bad news . . . at ticktock 99, it was true both that you

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did not have the fabulous past you would been dreaming about and you will not have the glorious future you had been hoping for, for at ticktock 99 it is true that the machine will not work. The good news . . . hyperwait for it . . . at ticktock 101, it was true both that you did have the fabulous past you’d been dreaming about and you will have the glorious future you’d been hoping for, for at ticktock 101 it is true that the machine did work.9 No need for such greedy, whole-scale change, though. Someone who is desirous of small changes in the past or of minor alterations in the future and who is willing to time-travel can MartyMcFly his way around the eternalist block fixing little things then and hence—provided that the relevant features of the eternalist block are hypertemporary and that they cooperate to yield the right hypersequence of past-and-future changes. Either way, the important point is that the hypertime account of time-travel can be combined with an eternalist theory of time. (Moreover, this combination provides for the possibility of changing the future in the same way that one might change the past, so the second potential objection identified earlier is no longer any problem at all.)

4. presentism, time-travel, changing the past, and changing the future Consider finally the presentist theory of time, according to which only present things and events exist. (For simplicity, we will continue to work with an eternalist picture of hypertime, according to which hyperpast, hyperpresent, and hyperfuture hypertimes all exist.) For the presentist, past and future truths are captured by means of present-tensed sentences prefixed by basic temporal operators like ‘It WAS the case that’ and ‘It WILL be the case that’ and metrical tense operators like ‘it WILL be the case one minute hence that’. For example, a hyperbeing hyperpresent at ticktock 100 might correctly report some of the temporal facts as follows: 9 For the record, we are not packing a great deal into the notion of ‘‘the machine’s working,’’ for like van Inwagen, we remain silent on how causal sequences are supposed to interact with and be restricted by time and hypertime. We do note, however, that even if the button pushings do not cause the eliminations, or redistributions, or alterations of content in the block, we are still left with perfectly good examples of time-travel and of the changing of past and future events.

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Hud Hudson and Ryan Wasserman • It is 12:00a.m. on 1 January 2000 ad. • A hopeful time traveler sits in his time machine with the dial set for 12:00am on 15 March 44 bc. • It WILL be the case one minute hence that (a hopeful time traveler pushes the start button). • It WILL be the case two minutes hence that (it is NOT the case that [it WAS the case over two thousand years ago that (a time-traveler witnesses a famous murder)]). • It WILL be the case two minutes hence that (a disappointed non-traveler sits in his machine).

More colloquially: someone will try to travel back to the ides of March, 44 bc, but will fail. He will thus be disappointed, for it will be the case that he did not travel back in time. In fact, the following report may even be true at ticktock 100: • It ALWAYS WILL be the case that (it is NOT the case that [it WAS the case that (there are time-travelers) ] )10 More colloquially: it will never be the case that there were timetravelers. Yet all of the reports made by our imagined hyperbeing at ticktock 100 are jointly consistent with the following reports made at ticktock 101: • It is 12:02a.m. on 1 January 2000 ad. • It is NOT the case that (a disappointed non-traveler sits in his machine). • It WAS the case two minutes ago that (a hopeful time-traveler sits in his time machine with the dial set for 12:00a.m., 15 March 44 bc). • It WAS the case one minute ago that (a hopeful time traveler pushes the start button). • It WAS the case one minute ago that (a hopeful time-traveler disappears). • It WAS the case over two thousand years ago that (a timetraveler witnesses a famous murder). • It WAS the case that (there are time-travelers). 10 ‘It ALWAYS WILL be the case that P’ is true iff for all n, [(it WILL be the case n units hence that P)].

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And, in fact, this is exactly the set of reports we should expect in a past-changing time-travel case according to the hypertime account, for these reports tell us that what was the case changes from one hypertime to another (and, we can add, that this change was due to the actions of the time-traveler and his machine). The presentist hypertime account is very similar to the eternalist hypertime account, but with one obvious difference. On the previous combination of views, there are past time-slices that exist relative to different hypertimes and exemplify different properties at those hypertimes. On the present combination of views, there are no past time-slices that exist relative to different hypertimes, for there are no past (or future) objects or events at all. Rather, there are simply different temporal facts, relative to the different hypertimes. The hypertime being in an eternalist world thus sees a changing block in which past slices, objects, or events alter their properties from one ticktock to the next. The hypertime being in a presentist world sees time-slices come and go from one ticktock to the next and also witness a change in what was or what will be. But in both cases, the hyperbeing observes time-travelers changing the past or future. We conclude that van Inwagen’s general picture can be adapted so as to avoid all of the potential objections outlined in section 1 above. Most importantly, the hypertime account of time-travel can be divorced from the growing-block theory of time, so that all hypertime realists can embrace the account, regardless of their preferred theory of time. Western Washington University

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PERSISTENCE THROUGH TIME

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4. Location and Perdurance Antony Eagle Two views dominate the contemporary discussion of persistence. Perdurance is the view that objects persist through time by having distinct parts (‘temporal parts’) at each moment at which they exist, so that persistence through time is just like ordinary extension through space. Endurance is the view that objects are wholly (not merely partially) present at each moment at which they exist, and objects thus persist through time by being present at different times. These different views give rise in turn to distinctive views on further metaphysical topics. Recently, Cody Gilmore (2007) has used some of these further consequences to develop an argument against perdurantism and in favor of endurantism. More specifically, he argues that perdurantism and endurantism involve different conceptions of what the location of a persisting object is, and that in certain cases, the perdurantistic conception of location seems to force the perdurantist to accept that there are coincident objects: distinct objects constituted by the same things and occupying the same location. Given that most philosophers reject the possibility of coincidence, this situation is a cost to the perdurantist.1 When one recalls that many take the supposed ability of perdurance theory to explain away cases of coincidence to which the endurantist appears committed to be the strongest argument in favor of perdurance, the cost is even more pronounced.2 But I am not convinced that Gilmore’s argument succeeds. In what follows I shall start by setting up the theory of locations and the various accounts of persistence, then describe Gilmore’s ingenious 1 This prohibition on coincident objects is nevertheless compatible with an object and its distinct constituting matter sharing a location (Fine, 2003), as long as the matter does not also constitute itself. 2 For example, Ted Sider argues that for problematic cases of coincidence, ‘if we believe in [perdurance], we can dissolve these and other puzzle cases; if we do not, we are left mired in contradiction and paradox’ (Sider, 2001: 10).

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cases and the conclusions he draws from them. I shall then argue (in Section 5) that the perdurantist has at least one natural and plausible response, which incurs no significant costs for the perdurantist: that is, the perdurantist can respond to Gilmore’s cases without accepting coincidence and without abandoning the advantages of perdurantism in dealing with other problem cases. I will sketch another perhaps more conciliatory response as well, in Section 6: this account may be of less interest to perdurantists, because it concedes much to endurantism, but perhaps for that reason will be more acceptable to such neutral parties as still exist in this debate. These responses have independent interest as they articulate and clarify aspects of the perdurantist picture which have not been sufficiently attended to. While I think Gilmore’s arguments ultimately fail, attending to them is of great importance for perdurantists, both to address a novel challenge and to further clarify what the view involves.

1. locations I assume that we have some background theory of spacetime regions, and some standard theory of mereology (which mereological theory to use will be discussed below, Section 5.2). It is worth noting, I think, that little in the present discussion depends on any particular theory of regions. One can safely think of regions as being just whatever those things are that an object can be found in. I will throughout the present chapter make the fairly standard assumption that regions are fundamentally made of points.3 Officially, regions will thus be mereological fusions of points.4 The following theory is also relativistically acceptable; for convenience I will mostly talk of temporally unextended regions, but this locution is intended to be neutral between the relativistic concept of a region all of whose subregions are spacelike separated from one another, and the pre-relativistic concept of a region of instantaneous duration (a ‘time slice’).5 3

Fairly standard, but not universal: see Tarski 1929 and Arntzenius 2003. Though, in line with standard approaches (Cartwright, 1987) I will sometimes model the mereological relations set theoretically, particularly in section 6. 5 As Gibson and Pooley (2006: 163) propose, a temporal part should ‘exactly occupy a region that is spacelike’; but I agree with their further claim that distinctively 4

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The theory of location I use is related to one recently developed by Josh Parsons (2007). The primitive notion I shall use is that of occupation of a region by an object. To fix ideas: an object occupies any spacetime region in which it can be found; relatedly, if indicating a region R would be a correct answer to the question ‘where is O?’, then O occupies R. So I occupy my office; I can be found in my office, and if someone asked where I was, ‘the office’ would be a correct answer. With occupation as the primitive, we can introduce some defined notions: Containment Filling Location

O is contained in R iff each part of O occupies a subregion of R. O fills R iff each subregion of R is occupied by O. A location of O is any region R that both contains O and is filled by O, as long as no proper subregion of R contains and is filled by O.

By way of illustration: I am contained in my office, because each of my parts occupies my office. If I was half out of my door, I could still be found in my office, but I would not be contained in my office because some of my parts would be outside that region. I do not fill my office, because some subregions of my office are free of me; I do fill the region of the interior of my head, as I have parts in each subregion. My location is a region I fill and am contained in; the region of my body is thus my location, as no proper subregion of my bodily region contains me. It should be noted that I have not assumed any constraints on which regions can be locations; I believe the present theory is compatible with both ‘liberal’ theories of receptacles (Hudson, 2002), as well as more moderate views (Uzquiano, 2006). Nothing in this theory commits me to the uniqueness of locations, and so this theory is compatible with the possibility of multiple location. This possibility arises because of the way that containment is defined; as it stands, an object can be contained in R if all of its parts are in R, whether or not those parts are also elsewhere. It seems perfectly intuitive to me that containment involves the object temporal parts of persisting objects are not of particular significance in the relativistic context. (Though a perdurantist view that retains a special role for distinctively temporal parts will be discussed in section 6.)

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being (to use Parson’s terminology) wholly within a region (every part of the object is in the region), whether or not it might also be entirely within that region (everywhere disjoint from the region is free of the object). By contrast, suppose we had defined: Containment*

O is contained* in R iff no part of O occupies any region not overlapping R.

If this was used in the definition of location, multiple location in the intuitive sense would not have been possible. But we can still define a location* in the obvious way, as a region filled by and containing* a given object, and we see that while objects may have many locations, they only have one location*—where the former is any region wholly filled by an object, and the latter is any region entirely filled by an object.6 The possibility of multiple location is why we need the additional clause in the definition of location. Consider instead the simpler definition: Location†

A location† of O is any region that both contains O and is filled by O.

If an object is multiply located in the intuitive sense, in that it fills one region R that contains it, and also fills another disjoint region S that contains it, then its locations are R and S, but its locations† are R, S, and their fusion R + S. But I do not think that the fusion R + S is intuitively among the locations of the object. For one thing, a non-scattered but multiply located object turns out to have a scattered location, severing the connection between the topology of an object and the topology of its location. For another, counting the fusion of the locations as a location seems to double count the locations. These counterintuitive results are avoided, and the spirit of the idea of multiple location preserved, if we restrict the locations of an object to those smallest minimal regions that it is contained in and fills, as my definition of location does. (But I will reconsider location and location† ; in the next section when discussing endurance.) 6 Parsons (2007) introduces the notion of ‘exact location’, which is my location*; he makes no use of my notion of location.

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2. theories of persistence and locations Before I proceed to describe Gilmore’s cases and argument, I need to be a bit more precise about perdurance and endurance and their relation to the theory of location. The perdurance theory I will discuss is the so-called ‘worm’ theory (Lewis, 1983).7 This view says that ordinary persisting objects have both spatial and temporal parts, so that each such object is spatially and temporally extended—a ‘worm’—with parts in each time and place where (when) the object exists. A continuant object occupies many regions, according to the perdurantist, but is contained only in temporally extended regions; the locations of a perduring object are thus temporally extended. A perduring object is wholly located in no temporally unextended region, though it has parts whose locations are temporally unextended—temporal parts. Perdurantists maintain that spatial and temporal extension are just species of the more general notion of extension, which in turn is analysed as having parts at different regions. This point provides the main contrast with endurance, which claims that persisting objects do so by existing entirely at each moment at which they exist. As it is sometimes put, an enduring object is ‘wholly present’ at each moment at which it exists (Haslanger, 2003: 317–18). The natural way to understand this locution is in terms of our notion of wholly located, so that an enduring object is contained within a temporally unextended region at each time at which it exists. An enduring object is such that if it exists at a time, it wholly exists at that time (it is contained in that time). This entails, of course, that enduring objects are temporally multiply located (Gibson and Pooley, 2006). Though some object to this (Barker and Dowe, 2005), it is straightforwardly compatible with the plausible theory of location articulated above. On this version of endurantism, persisting objects have zero temporal extent (they are genuinely three-dimensional), and persist by existing at—being located at—many different times. 7 Another perdurantist view is the ‘stage’ theory (Hawley, 2001; Sider, 2001), which shares with the worm theory a commitment to the existence of worms, but denies that terms for ordinary continuants refer to worms. Gilmore-style worries arise for the stage theory, but nothing specific to the stage theory needs to be said in response—the solution I propose for the worm view works equally well for the stage view, so I will not discuss the latter explicitly in what follows.

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One surprisingly common way to understand this temporal multiple location relies on the notion of an extended simple, which is an extended object with no proper parts. If such things are possible, then if they occupy a region, they are contained in it (by the definition of containment). An extended simple is then located† in every region it fills (Parsons, 2007: 212). However, the locations (according to the definition above) of an extended simple are the smallest regions it occupies, which are points in the present framework. If you believe in extended simples, and think that they have an extended location, that is one reason to prefer location† as the right conception of location in the intuitive sense. However, the counterintuitive results this produces in the most easily understood cases of multiple location make this quite unappealing in my view, particularly when combined with the controversial belief in extended simples.8 Accepting extended simples allows for a conception of endurance according to which objects persist by being extended temporal simples (though they will not in general be spatial simples): such things are therefore wholly located at each time at which they exist, in virtue of being extended across many times but lacking parts at those times. As Parsons shows, this version of multiple location based on extended simples can be used to define endurantism quite naturally, in much the same way as I did above. This version of endurantism maintains that persisting objects persist simply by filling a temporally extended region, but have no determinate temporal extent at all, since they are located† in regions that have no temporal extent, and are also located in regions that have temporal extent the same as the duration of the object’s persisting, and located† in regions of every extent between these extremes (van Inwagen, 1990: 251–2). 8 My view may have a decisive advantage over its rival when it comes to timetraveling objects, which play a crucial role in what follows. For if an object travels through time by making an instantaneous jump (perhaps living the first part of its career from 1997 to 2008, and the remainder from 2017 to 2020), it will be a disconnected object—but will also be (temporally) simple, if the second conception of endurance is right. But, unless it is supposed to be multiply located in my first sense, I find the idea of an extended scattered simple very puzzling, and the temporal version no easier to understand. The worry is naive: but how can something have one scattered location (as opposed to many non-scattered locations) except by having a part at each disconnected part of the location?

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Whatever the status of these debates over location versus location† , and over the existence of extended simples, both conceptions of multiple location will provide an adequate version of endurantism. Both views are united in their denial of temporal (proper) parts, the most significant marker of opposition to perdurantism. Crucially, both views agree that enduring objects are multiply located, and are located at every time at which they exist (though one may dispute whether these locations are intuitively accurate if the extended simple version of endurantism is adopted). If anything, the extended simples view is simply a way of spelling out the basic requirement of endurance, that persisting objects be multiply located. The differences between the views do not affect materially the substance of my argument, though it remains an interesting question which of these two conceptions is the right one for an endurantist to make use of. Let me now introduce some new terminology (adapted from Gilmore) to help capture these positions. Let an S-region of O be a maximal temporally unextended subregion of a location of O. S-regions correspond to what we intuitively regard as spatial locations of objects at particular moments in their career. It is fairly obvious that restricting an endurantistic location to a temporally unextended region will simply yield the same location back again, since endurantistic locations are temporally unextended to begin with. So it is natural for an endurantist to believe that the locations of an object are its S-regions. Of course, the way I have defined the concept, this is a trivial observation. But many people, including Gilmore, take themselves to have direct evidence as to what the S-regions of objects are, based on the idea that they are connected with spatial locations of objects at times. The idea is that an S-region is simply to be defined as a region we would ordinarily regard as the location of an object—it then follows, as a matter of fact rather than definition, that S-regions are maximally temporally unextended subregions of objects. Armed with direct opinions about S-regions, the trivial connection for the endurantist between S-regions and locations will allow one to discern the locations of an enduring object. It is equally obvious that no perdurantist could think that the location of a perduring object is an S-region, as no perduring object can be contained in a temporally unextended region. However, if

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we follow one standard definition of a temporal part, S-regions can be the locations of instantaneous temporal parts. According to Sider (2001: 60), x is an instantaneous temporal part of y at t iff (i) x is part of y; (ii) x exists wholly at t; and (iii) x overlaps every part of y that exists at t. Let R be the location of some object O; let Rt be the temporally unextended subregion of R at t. Then if Ot is O’s instantaneous temporal part at t, and Rt is the location of Ot , then Rt fulfils the conditions for being an S-region of O (Rt is maximal, and so contains every part of Ot ; Rt is a subregion of R and so is filled by Ot ).9 Even if S-regions cannot be locations of perduring objects, they may still be used to define locations. It is an obvious and natural principle that the location of an object is determined by the locations of its parts. One specific way—but not the only way—of fleshing out the natural principle is to claim that the location of an object is the fusion of the locations of its parts: that is, the location of an object is that region which has as subregions all and only the locations of the parts of the object. Every temporally unextended part of a perduring object is part of an instantaneous temporal part, and every temporally extended part of a perduring object is a fusion of temporally unextended parts, so in defining the location of a perduring object we need only consider the locations of its temporal parts. So the location of a perduring object is the fusion of its S-regions. Gilmore terms the fusion of the Sregions of an object its path. Using this terminology, the preceding discussion suggests that the location of a perduring object is its path. However, according to the view that locations are paths, multiple location of a perduring object is impossible. Even if an object’s temporal parts are multiply located, in the sense that one temporal part may be wholly located in disjoint S-regions, it will still be the case that there is just one path associated with a given set of S-regions (because any such set has a unique fusion), and thus just one location. Given that endurantism and the general theory 9 Note that it is possible that O’s temporal part at t may be identical to O’s temporal part at t’= t, if that temporal part is multiply located. We thus liberalize Sider’s conception, as he requires that a temporal part exist at one and only one time. We could also define non-instantaneous temporal parts, following Parsons (2007: 216, eqn. 10).

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of location are both compatible with multiple location, it must be seen as a cost associated with perdurantism to rule it out (though perhaps only a slight cost). Yet we can easily save the possibility of multiple location: define a sufficient collection of S-regions as any set C of S-regions for an object O such that every temporal part Ot of O has a location cOt ∈ C. Call a sufficient path the fusion of the members of a sufficient collection. The path of an object is a sufficient path, but the converse does not hold. If a temporal part of the object is multiply located, there will be more than one sufficient collection, and hence more than one sufficient path. Since the fusion of sufficient paths is also a sufficient path, we cannot simply say that a location of a perduring object is a sufficient path, for the reasons outlined in section 1. To bring the theory into line with the earlier definition of location, we must restrict our attention to those sufficient paths which do not have a sufficient path as a proper subregion. Calling such a path a minimal sufficient path, it follows that locations, for the perdurantist, are minimal sufficient paths.10 It seems inconceivable that a perduring object could be multiply located (rather than merely scattered) without having multiply located parts; thus if we adopt the notion of a minimal sufficient path as giving the perdurantistic concept of location we will at least have managed to give an account which is in agreement with the intuitive theory of location and is natural and attractive to the perdurantist. From now on I will take the thesis that locations are paths to mean that locations are minimal sufficient paths.11

10 Even here there is room for dispute. Consider O, which has two temporal parts, each of which is spatially bi-located. O therefore has four minimal sufficient paths. But, intuitively, it might seem that the bi-location of O’s parts is due to O being bi-located, not quad-located. We could get this outcome if we insist that a location should not overlap another location, and thus insist that O’s locations are the members of a maximal set of non-overlapping minimal sufficient paths. But there is more than one such set, which seems to introduce a distressing arbitrariness as to which set gives the ‘real’ locations of an object. I do not therefore proceed down this route, but stick with minimal sufficient paths as my analysis of perdurantist locations. 11 Note that the original notion of a path still has an important role: it corresponds to the exact location of an object (the region which entirely and exactly contains the object). In the case of an enduring object, it can be easily seen that its minimal sufficient paths are each of its S-regions, so the notion of a path is arguably more useful.

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3. coincidence problems for endurantists With these preliminaries in place, we can now introduce some famous problems for endurantists involving coincidence (distinct objects, with the same constituents, in the same location). Since we are supposing that such coincidence is impossible, such cases pose problems. In the cases we examine in this section, the apparent coincidence arises only under the assumption of endurance, which provides a reason to favor perdurance. A familiar case of this sort is the situation of Lumpl, a lump of clay, which is (at some time into its career) shaped into a statue, Goliath.12 Lumpl exists prior to Goliath’s creation, which is sufficient to make them distinct. Yet when Goliath is created, Lumpl is not destroyed, as the creation of Goliath involves simply reshaping Lumpl, and lumps of clay are not destroyed when reshaped. So Lumpl still exists when Goliath does, and though they are distinct they seem to be coincident and made of the same constituent clay after Goliath is created. Whether the appearance of coincidence is borne out depends on what the locations of Goliath and Lumpl turn out to be; and as we have seen, perdurantists and endurantists have differing views on what locations are. For the perdurantist who take Goliath and Lumpl each to persist by perduring, the location of Lumpl should include the locations of each of Lumpl’s parts, and since Lumpl exists prior to Goliath’s creation, Lumpl has parts that exist before any of Goliath’s parts. Therefore we can at least say that the location of Lumpl is distinct from the location of Goliath, as there are regions Lumpl occupies that Goliath does not. This follows straightforwardly from the thesis that a perduring object is partly located wherever its parts are located, which in turn follows from the thesis that perdurantistic locations are (minimal sufficient) paths. The threat of coincidence is avoided if the objects perdure. Locations are S-regions for the endurantist. Lumpl thus has some locations that Goliath lacks, in virtue of pre-existing Goliath, and thus Goliath and Lumpl are distinct. But there are apparently many S-regions and thus locations which Goliath and Lumpl share. In each of those locations, both Goliath and Lumpl are wholly present, and as they are distinct there are two distinct objects wholly 12

The case is due to Gibbard (1976); see also Sider (2001: ch. 5).

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present in the same location. Assuming as we have throughout that such coincidence is impossible, endurantism is committed to an impossibility. The problem arises because there are objects which share some but not all of their S-regions. The perdurantist proposes that these objects share some but not all of their (temporal) parts, and that the overlap of parts explains the partial overlap of location. But because endurantists regard S-regions as locations, sharing an S-region just is sharing a location, which quickly leads to coincidence problems. Considering this situation and others like it, it has been popular to conclude that the ease with which perdurantists can sustain the anti-coincidence principle, while the endurantist has difficulty, provides support for perdurance as an account of persistence. These cases are far from decisive, as becomes obvious when we consider further, more or less distantly related cases: What if Goliath and Lumpl are each created at the same time and thus share all their S-regions? What of the coincidence problems for the temporal parts of Goliath and Lumpl that exist after the creation of Goliath and which (it is tempting to think) must ground in their properties the modal differences between Goliath and Lumpl? Both perdurantists and endurantists have to say something about such cases in order to uphold anti-coincidence, and it is pretty clear that what the perdurantist says about the version of Goliath and Lumpl discussed above is not going to help with the variants just mentioned.13 But it should be (and generally is) admitted that the argument provides some presumptive support for perdurantism under the anti-coincidence assumption.

4. gilmore’s cases Analogous problems would arise for perdurantists if it were possible to have distinct objects which shared no S-regions but did have the same path. Such objects would not be coincident if they endured, but would be coincident if they perdured. Cody Gilmore has given putative cases of just this sort, by making an ingenious appeal to 13 In these other cases, the perdurantist’s best response is apparently to deny the distinctness of Goliath and Lumpl, usually by denying the existence of robust de re modal differences that would ground distinctness.

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the possibility of backwards time-travel. So before presenting his cases I will need to quickly discuss time travel and persisting objects. The standard story goes like this (Lewis, 1976). For many persisting objects, the conditions under which an object at one time is the same as an object at another time involve causal connections of specific sorts between those objects. For example, Locke (1976: §27) appears to claim that some things are the same person just in case the experiences of one object cause the other object to be in a position to veridically recall those experiences. In normal cases, these identity-constituting causal processes happen in a straightforward temporal order: the causes temporally, as well as causally, precede their effects. But if we can conceive of backwards causation, we can conceive of a case where the appropriate causal relationships hold between the object at one time, and the object at another time, but where the order of the times is out of step with the normal order for intrinsically similar causal relationships. So, if a person steps into a time-machine and travels back in time, they can be the same person because they remember various experiences of their causally prior selves, even though those selves may be many years in the future. Lewis introduces a distinction between ‘external’ time and ‘personal’ time, where the latter is a derivative temporal metric induced on a sequence of events by tracing the causal relations between them, and ordering them as surrounding intrinsic duplicate (or near-duplicate) causal processes are normally temporally ordered (if there is such a prevailing order). Since objects are unified by causal relations holding between the object at different times, each object has a personal time characterized by the normal temporal order for objects intrinsically like it causally. An object time-travels, says Lewis, if its personal time comes apart from external time.14 I can now introduce Gilmore’s two cases. Both have a similar structure; one is, he claims, ‘physically more plausible’, but both rely fairly heavily on backwards time-travel and are thus at most as plausible as that possibility.15 Nevertheless, backwards time-travel 14 It is sometimes objected that an endurantist cannot even make sense of timetravel; I will here assume that this objection is false, and point to Keller and Nelson (2001: sect. 9) for arguments. 15 The second case, of Adam and Abel, uses a more physically plausible version of Godelian time-travel, permitted in some models of general relativity (Earman, 1995: ¨ ch. 6). But there is no evidence that our world is one of those which permits such time-travel.

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is intuitively possible, and the easy way in which perdurantists can make sense of it has been used as an argument for perdurance. It is certainly fair to use it in setting up cases that are potentially problematic for the perdurantist, even if those cases are not nearly as obvious and familiar as cases like Goliath and Lumpl. Case 1 (Cell and Tubman). Suppose that some cell [‘Cell’] is originally created at the beginning of the year 2000 and that it jumps back in time over and over again, never venturing further back in time than the moment of its original creation, and never progressing beyond the end of the year 2002. The cell’s entire career is confined to this three-year interval. Suppose also that the cell never leaves the immediate vicinity of my bathtub. If this cell’s trips were structured properly, if it made enough of them, and if it underwent the right sorts of intrinsic changes along the way, the cell might compose some macroscopic object that sits in my bathtub for three years. Indeed, the cell might compose an object that by all appearances is a conscious, intelligent human being [‘Tubman’], one who exhibits the strange behavior of living in my bathtub, and whose constituent cells seem to pop into and out of existence, but who is otherwise quite normal. (Gilmore, 2007: 182)

The endurantistic analysis of this case appears straightforward. Had Cell not been a time-traveler, it would have been wholly located at each moment at which it existed, and thus temporally multiply located. By allowing Cell to time-travel in such a way as to overlap itself temporally, we turn temporal multiple location into spatial multiple location: Cell is wholly located in many places at each time at which it exists, so (by the endurantist definition of location) it has many S-regions at each time. Tubman has a part located wherever Cell is located, because Cell is part of Tubman, so Tubman’s Sregion at t is the fusion of the S-regions of its parts at t. That is, Tubman’s S-region at t is the fusion of all of Cell’s S-regions at t. Since a fusion of several things is not identical to any one of those things, Cell and Tubman have no S-regions (locations) in common. There is no threat of coincidence for these distinct objects. For perdurantists, who accept that a location of a persisting object is any of its minimal sufficient paths, the case is more problematic. Had Cell not been a time-traveler, it would have had a great many temporal parts stretched over a considerable period of time. When Cell time-travels in such a way as to overlap itself temporally, these parts have a more compact arrangement, and compose Tubman. There is a strong intuitive pull, however, toward regarding Cell’s Sregions as just the same as for the endurantist, though now they are

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all the locations of Cell’s temporal parts. Indeed, Gilmore suggests that ‘philosophers on both sides of the endurance v. perdurance dispute can all agree on . . . which regions are O’s S-regions’ (Gilmore, 2007: 179). As such, Cell’s S-regions differ from Tubman’s S-regions. But the fusion of Cell’s S-regions at each moment is Tubman’s S-region at that moment; so the fusion of Cell’s S-regions at every moment just is the fusion of Tubman’s S-regions at every moment. So the (minimal sufficient) paths of Tubman and Cell are the same, and for a perdurantist, Cell and Tubman are co-located. But they are also distinct. If the distinctness of their S-regions were not enough to convince us of distinctness, these further facts intuitively should: (i) Cell time-travels, while Tubman does not; (ii) Cell exhibits no conscious behavior, while Tubman does.16 As such, we have distinct but co-located objects, and hence coincidence. Assuming as we have throughout that such coincidence is impossible, the perdurantist is apparently committed to an impossibility. Before evaluating this case of Cell and Tubman, let me describe Gilmore’s other case: Case 2 (Adam and Abel). Consider . . . the career of a hydrogen atom, which we shall call ‘Adam.’ Adam is spatially bi-located throughout its two-billion-yearlong career. For any given moment of external time (or ‘global simultaneity slice’) t in the relevant universe, Adam is present at t ‘twice over:’ i.e., there are two different moments pt and pt* of Adam’s proper time such that, at pt, Adam is present at t, and at pt* Adam is present at t. Suppose that, at each moment of Adam’s proper time, Adam is chemically bonded to itself at a different moment of its proper time, thus forming a molecule of H2 , which we shall call ‘Abel.’ Abel is spatially mono-located throughout its career (which is only one billion years long). For any given external time t, Abel is present at t only once: i.e., there is only one moment of Abel’s proper time at which Abel is present at t. (Gilmore, 2007: 186–7)

As in the case of Cell and Tubman, the endurantist has an obvious analysis of the case of Adam and Abel: they share a path but none of their S-regions, so while they are distinct, they are not co-located. We can give additional grounds for the distinctness of Adam and Abel, as Adam, but not Abel, has a career extending over 2 billion years; and Abel weighs more than Adam over their entire careers. But again, if they are distinct, the perdurantist is committed to the impossible situation of distinct coincident objects. 16 I shall re-evaluate this supposed distinctness in Section 5.3; for now, I will accept it as intuitively plausible.

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4.1. Options for the Perdurantist There are two obvious ways for the perdurantist to resist the conclusions Gilmore draws about these cases. The first is to deny the distinctness of Cell and Tubman (and Adam and Abel); the second is to deny that they are co-located. As I will now go on to argue, both ways are independently plausible and could be attractive to the perdurantist. Indeed, views that reflect these ways are already part of the established literature on perdurance (though the first response, discussed in Section 5, is deservedly more popular). The responses I discuss are alternatives: each of them individually suffices to defuse Gilmore’s cases, but it is not plausible that a single perdurantist should appeal to both. Whichever alternative is chosen, however, we can conclude that no significant threat is posed to perdurance by Gilmore’s ingenious cases. The value of the cases remains however; they do show that erroneous ways of thinking about perduring objects are very easy to slip into, and that caution must be exercised to remain clear on what the view does, and does not, involve. In what follows I shall first explore the prospects for denying the distinctness of Cell and Tubman (Section 5), then explore the prospects for denying their co-location (Section 6).

5. denying distinctness 5.1. S-regions revisited Part of Gilmore’s argument from his two cases is that, if perdurantists and endurantists agree on the S-regions, they will agree on the distinctness of the objects in question. For, the reasoning goes, how could objects that were not distinct manage to have different instantaneous spatial locations? It is in fact fairly obvious that the perdurantist has no reason to agree with the endurantist on the S-regions of the objects in question. Recall that an S-region is a maximal temporally unextended subregion of a location (Section 2). That captures precisely the intuitive concept of an instantaneous spatial location: namely, it is the region which is the overlap (intersection) of a temporally extended location with a particular time (itself conceived of as a maximal spacelike fusion of points). It follows immediately from

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this that, if an object has just one location, it has just one S-region at a time, because intersection is a function. So if Gilmore is right in contending that there is just one path for Cell and Tubman, and paths are locations, then it follows immediately that the S-regions of Cell and Tubman must be the same as well. So it is not the case, as Gilmore uncritically supposes, that ‘philosophers on both sides of the endurance v. perdurance dispute can all agree on . . . which regions are O’s S-regions’: if Gilmore is right that the perdurantist is committed to Cell and Tubman having the same location, the perdurantist should be equally committed to Cell and Tubman having all the same S-regions. Exactly the same goes for the case of Adam and Abel.17 This outcome, notice, is precisely what perdurantists have accepted all along. For consider what the perdurantist says about the familiar kind of time-travel case, in which an older stage of a person visits a younger stage (perhaps to pass on some sage advice). In these cases, both stages are parts of a person, but they are not temporal parts, because they are not maximal (neither overlaps every part of the person existing at that time, because neither overlaps the other). The object that is the temporal part of the person at that time is a scattered object, which has both stages as its only parts. And it is the location of this temporally unextended scattered object which is the person’s S-region at that time. Nothing different from this case goes on in the Cell/Tubman case or the Adam/Abel case: and in each of those cases, the S-regions of the two purportedly distinct objects should be maximal objects also (though, given the description of the case, they will not be scattered).18 The sameness of S-regions derived from the sameness of perdurantistic locations significantly undermines the argument for the

17

It is, of course, possible to insist that Cell and Tubman do have distinct S-regions (see p. 70); but on the present conception of S-regions, that will entail that Cell and Tubman have different locations (since the S-region which is Cell’s but not Tubman’s, say, will be part of Cell’s location but not Tubman’s), which would obviously undermine Gilmore’s argument. 18 Of course this has the consequence that a person’s temporal part at a time need not itself be a person stage, and relations of personal identity do not necessarily hold between a person’s temporal parts, but sometimes hold between proper parts of those parts. But this is hardly inconsistent with intuition or commonsense, nor does it give rise to any great difficulties for perdurantism generally, except one needs to be careful when stating conditions of identity over time.

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distinctness of Cell and Tubman. Gilmore is careful to conclude that while he takes the distinctness of S-regions to be sufficient for distinctness in his two cases, he does not think it is the only argument; I shall consider his other arguments below (Section 5.3). For now, however, we can conclude that the perdurantist does not yet have a good reason to accept distinctness in those cases: not only are the overall locations of the pairs of objects the same, but so are the locations of their temporal parts. Something odd remains, however: the present conception makes it so obvious that the S-regions of Cell and Tubman are the same that Gilmore must be using a different conception of S-regions. The most plausible thought, and something that I mentioned above, is that Gilmore thinks that S-regions are not derivative from locations, but are instead themselves fundamental (or at least, they are closely associated with something fundamental, i.e., endurantistic locations). Gilmore’s informal glosses on the concept of an S-region strongly suggest this interpretation: he argues that an S-region ‘corresponds to what we would ordinarily think of as a spatial location of O at some instant in O’s career’ (179), and on this conception one can see how an endurantist would regard S-regions as independently specifiable. But the account of S-regions given in Section 2 makes perfect sense of this gloss: for in all ordinary and common cases, not involving this kind of backwards time-travel, the intersection of a location and a time is precisely what we ordinarily think of as a spatial location at an instant, and, moreover, this intersection is the location of a temporal part of an object. The perdurantist therefore is perfectly at liberty to accept that, in ordinary cases, the concept of an S-region is useful because it tracks the theoretically important concept of the location of a temporal part. But the perdurantist thereby incurs no commitment to regarding the notion of an S-region as a theoretically significant notion in its own right in all cases; and the perdurantist will take the lesson of Gilmore’s cases to be just that there are situations where S-regions do not pick out any fundamental feature of a persisting object.19 19 I think part of the intuitive appeal of Gilmore’s account of his cases rests on the following principle, which can look plausible on first acquaintance:

Correspondence

There is a one–one correspondence between S-regions of objects which are intrinsic duplicates.

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Importantly, the perdurantist has no need to accept the endurantistic thesis that, when an object has two contemporaneous stages, the object is multiply located at that time and thus has multiple S-regions. The much more natural thing to say, for the perdurantist, is that the object has two distinct parts, in different locations, and is merely scattered. But, as is quite clear, scattered objects do not thereby have multiple S-regions. Below, when I consider co-locationdenying responses to Gilmore’s argument, I shall return to the question of whether the perdurantist can adopt an alternative conception of S-regions according to which it is possible for a scattered object to have multiple S-regions at one time (Section 6). For now, I conclude, the perdurantist has no good reason to accept distinctness of Cell/Tubman, or Adam/Abel, on the basis of distinct S-regions. Even so, the endurantist could dig in at this point, and while defining an S-region as ‘what we ordinarily judge an object’s spatial location at a time to be’, also insist that, having read the Cell/Tubman scenario, we ordinarily and pre-theoretically want to say that Cell has numerous small cell-shaped locations at t, and that Tubman has just one, medium-sized, man-shaped location at t. It will then follow from this insistence that Cell and Tubman have distinct S-regions (and it will follow that S-regions cannot be defined as maximal temporally unextended subregions).20 But does it then follow that Cell and Tubman are distinct? Only if our ordinary judgments about S-regions are true claims about instantaneous spatial locations. Yet it is the contention of the perdurantist that the best theoretical account of our ordinary judgments is that spatial locations at a time are maximal temporally unextended subregions of objects—a theoretical account that would make the ordinary judgment report the endurantist insists on come out false.21 But this Correspondence thesis breaks down in cases where an object time-travels in such a way as to overlap itself. Given that ‘number of S-regions’ is not invariant between intrinsic duplicates, it cannot be a fundamental feature of a persisting object. So even if we have some sense of what Cell’s S-regions would be if it had not timetravelled, that gives no indication of what Cell’s S-regions in fact are. By contrast, there is a one–one correspondence between the parts of intrinsic duplicates. 20

Thanks to Martin Thomson-Jones for pressing me on this point. If the perdurantist concedes that intuition supports the contention that Cell and Tubman have different spatial locations (and it is by no means clear that it does), the perdurantist must be committed to an error theory about the intuitive judgments. Our ordinary views are not infallible guides to what is true about an 21

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There remains the possibility that a multiply located object can have more than one S-region at a time, if its parts at that time are multiply located. Gilmore hints at this possibility when he says of the second case that ‘Adam is present at t ‘‘twice over’’ ’. If we take this hint literally, Adam may be multiply located in virtue of having multiply located temporal parts. But keeping the correct definition of a path as a minimal sufficient path firmly in mind, this possibility poses no problem to the perdurantist. Adam certainly will turn out to have (at least) two minimal sufficient paths, and hence (at least) two locations. If we admit that Adam can bond with its disjointlylocated self to form Abel (a big ‘if’: see the following subsection), Abel will intuitively fill and be able to be contained only in the fusion of Adam’s minimal sufficient paths, and will thus have just one location. In a sense, then, we can claim that Adam is longerlived than Abel: Adam has at least two billion-year-long disjoint minimal sufficient paths, while Abel has only one billion-year-long minimal sufficient path. But this is straightforwardly compatible with the perdurantistic theory of locations, and is not paradoxical at all: there are (at least) two different locations of Adam, and only one for Abel, and they all differ, so there is no threat of coincidence. In this case, indeed, the relations between the regions Adam occupies, and the region Abel occupies, exactly mimic the mereological and compositional relations between Adam and Abel themselves. Below I will consider other ways, not involving brute multiple location, for the perdurantist to avail themselves of this kind of ‘mereological harmony’ (Uzquiano, 2006: 441): see Section 6. 5.2. Mereological Issues While Tubman is apparently composed of many cells, it is crucial to Gilmore’s example that in fact those apparently distinct cells are just the many parts of the trajectory of a single time-traveling cell. object’s instantaneous spatial locations, because those locations are to be analysed as maximal temporally unextended subregions of the object. Since Cell is Tubman, and since intuition entails that Cell and Tubman have different locations, intuition is in error here. It is not in grave error; after all, our intuitive judgments about object’s spatial locations are correct in cases not involving time-travel. And the error is explicable; I think it arises from a pre-theoretical tendency to accept the false principle I called (in n. 19) Correspondence, on the basis that spatial locations are fundamental features. But it is an error nevertheless.

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Similarly, while Abel is apparently a molecule composed of two atoms, it is crucial to Gilmore’s example that in fact those apparently distinct atoms are just two parts of the trajectory of a single timetravelling atom. These observations should give us pause: how can it be that just one cell or atom, in virtue of time-traveling, can compose, just with itself, something distinct from itself? If we have some objects, the Xs, the fusion of the Xs is that object (if it exists) which has all of the Xs as parts, and no part distinct from each of the Xs. It is plausible to suppose that composite objects are fusions, so we may begin to address the question of whether Cell can compose Tubman by thinking about the nature of the relation between fusions (like Tubman) and their parts. This is the subject matter of mereology, the theory of parts and wholes, and our question will now be: can standard theories of mereology accommodate the mereological relations required in Gilmore’s cases? It turns out that the possibility of Gilmore’s cases requires a significant and otherwise unmotivated revision of the standard mereology accepted by perdurantists; as such, it is reasonable and dialectically appropriate for the perdurantist to reject Gilmore’s cases as impossible. A highly attractive theory of mereology is classical extensional mereology. This theory can be completely characterized by the following three claims about the nature of parthood and fusion (Lewis, 1991: 74):22 Transitivity Unrestricted Composition

Uniqueness of Composition

If x is part of some part of y, then x is part of y. Whenever there are some things, then there exists a fusion of those things. It never happens that the same things have two different fusions.

This theory is not uncontroversial, and objections have been raised to each of the axioms (Simons, 1987: ch. 3). But this strife over mereology has divided largely along party lines, with perdurantists overwhelmingly favoring classical extensional mereology. Lewis, paradigm perdurantist, maintains that classical extensional mereology is ‘perfectly understood, unproblematic, and certain’ (1991: 75), 22

See also Simons (1987: part I) and Varzi (2006).

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and other perdurantists have largely agreed.23 A notable aspect of the perdurantist conception of parthood is that it is atemporal (Sider, 2001: 55–7). That is, the fundamental relation of parthood is not relative to a time. This conception of parthood is somewhat familiar; it is the one we use when considering parthood relations amongst stretches of time (Sider’s example is ‘the 1960s are part of the 20th century’), or amongst allegedly atemporal items (like geometrical objects). It is certainly the natural relation for the perdurantist to accept as fundamental, and because the perdurantist can easily define a notion of temporary parthood to correspond to the more familiar everyday sense, it cannot be objected that this theory is not a theory of parts and wholes in the everyday sense. What are the parts of Tubman in Gilmore’s cases? According to the perdurantist, those parts at least include an instantaneous temporal part Tubt at each moment t of Tubman’s existence. And each Tubt also has parts: because they are made of Cell, they have the temporal parts of Cell as their parts. Indeed, each Tubt is just the fusion of those temporal parts of Cell that exist at t; and Tubman is the fusion of all the Tubt s. So, by transitivity, all the temporal parts of Cell are parts of Tubman; Tubman has no parts that are not parts of Cell, by construction. So by the definition of ‘fusion’, Tubman is a fusion of the temporal parts of Cell, and by uniqueness, Tubman is the fusion of the temporal parts of Cell. But Cell is obviously the fusion of the temporal parts of Cell; therefore, Cell is Tubman. A precisely similar argument will show that while Atom’s temporal parts fuse to form Atom, the very same parts are supposed to fuse to form Abel; by uniqueness of composition, Atom must be Abel. Fairly clearly, it is the principle of Uniqueness of Composition that is used in the preceding argument to show that Cell and Tubman must be the same. Gilmore’s cases must involve the failure of Uniqueness, but since no perdurantist accepts the failure of uniqueness, no perdurantist can accept that Gilmore’s cases are genuinely possible.24 23 For example, Sider (2001) and Hudson (2006: 5–6); Sider (2007) argues that classical extensional mereology is well motivated by considering it as an articulation of the powerful intuition that parthood is an especially intimate relation. 24 We have here another argument that instantaneous spatial locations of objects should be analysed along perdurantist lines, as maximal temporally unextended subregions. For since Cell and Tubman have all the same parts, they are (by classical

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It might be pressed that the perdurantist should reject uniqueness of composition. The literature certainly contains two prominent arguments to that effect: 1. If there are ‘structural’ universals (Armstrong, 1997; Forrest, 1986), they are composed of their constituent parts in a way that violates uniqueness of composition, because structural universals can be distinguished on the basis of ‘how many times’ a certain universal is part of the structural universal. For example, according to defenders of structural universals, the structural universal methane must contain the hydrogen universal four times over. But, Lewis (1986) objects, what can it mean for the same part to be part of the same fusion many times over? And it seems perfectly appropriate, in the face of this mystery, to deny that the supposed unmereological ‘composition’ involved in structural universals is composition at all—and thus reject the possibility of structural universals (Lewis, 1991: 79, n. 8). 2. Fine (1999) objects that a fusion of two slices of bread and a piece of ham is not always a ham sandwich: before the sandwich is assembled, the fusion may exist but the sandwich does not. So the ham sandwich is not the fusion, and is rather a structured aggregate. But the ham sandwich does not have any parts that differ from the fusion, so the mode of structured composition of the ham sandwich must not be classical extensional mereological fusion. Note firstly that there is a straightforward perdurantist response (Sider, 2001: ch. 5.3):25 just because the fusion of the ham and bread does not at all times have the property of being a ham sandwich does not somehow mean that when that fusion comes to have that property, some new object comes into existence. The fusion which will be a ham sandwich exists (as a widely scattered object) before it becomes a ham mereology) identical; and since the instantaneous spatial locations of Tubman are intuitively maximal temporally unextended subregions, so too must the spatial locations of Cell be. Only if we have the antecedent conviction that Cell is not Tubman—a claim which is supposed to be the conclusion of Gilmore’s argument—would we even be able to judge that there are two objects with different instantaneous spatial locations. 25

See also response (5) of Lewis (1991: 78).

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sandwich; when it becomes a ham sandwich it remains the same unique fusion, but, in virtue of non-mereological facts about the arrangement of its parts, that fusion comes to be a ham sandwich. So there are not two modes of composition; there is classical fusion, and then there are changes that can occur to those classical fusions. In no way does this picture commit the perdurantist to non-unique composition.26 Furthermore, this Finean argument is not acceptable in the present context. For if we conclude that the fusion and the ham sandwich have the same parts but are distinct, we seem committed to the idea that where there is a ham sandwich, there is also a distinct fusion: and this would be an instance of coincidence. The assumption running behind Gilmore’s argument is that coincidence is impossible, so it is not appropriate in this context to object to the classical mereological framework the perdurantist adopts on grounds that will lead to coincidence. Ruling out coincident objects rules out all the most plausible alternatives to uniqueness of composition. I conclude, therefore, that the perdurantist has no pressing reason to abandon uniqueness of composition (consequences of doing so anyway will be explored in Section 6). Given the obvious fact that, on Gilmore’s description, Cell and Tubman have the same fundamental parts, they must be the same fusion, not distinct objects at all. I admit that there remains some residual worry, and note that when first hearing about the cases the intuition is that Cell/Tubman and Adam/Abel may well be distinct.27 Explaining away this intuition, and Gilmore’s other arguments for distinctness, is the task of the following subsection. Before turning to that, it is worth thinking about the mereological principles that an endurantist must subscribe to in order 26 And it certainly appears to capture our intuitions about the case: if we accept the idea that a fusion is intimately related to the plurality of the parts, then since surely we would like to say that since the plurality ‘the parts of the sandwich’ existed before they were made into a ham sandwich, we should accept that the fusion exists too; ‘being made into a ham sandwich’ is an operation that modifies a pre-existing plurality, rather than creates a new thing. 27 In related cases, some perdurantists have wrongly agreed that Cell and Tubman are distinct: see Effingham and Robson 2007: section 5, where they argue that in a parallel case, a time-traveling brick that composes a wall is not identical to the wall. This is just a mistake on their part.

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to accept Gilmore’s cases. The endurantist thinks that Cell is wholly present in many locations at any time when Tubman exists; and Tubman is composed of these many ‘copies’ of Cell. Of course, there are not really many copies; there is just Cell distributed over many locations. Thus, we can conclude, Tubman is composed from Cell, and from nothing else. This conclusion is in tension with a principle that is extremely widely accepted, even by those who reject classical extensional mereology: Weak Supplementation Principle (WSP)

If x is a proper part of y, then there is some proper part of y that is disjoint from x. 28

Simons (1987: 116) (no great friend of classical mereology or perdurantism) even maintains that ‘WSP is indeed analytic—constitutive of the meaning of ‘‘proper part’’ ’ (see also ibid., p. 362). But Gilmore’s cases violate WSP: Cell is a proper part of Tubman, since it is part of Tubman and is distinct from Tubman, by construction. But Tubman has no part that is not identical to Cell. So WSP fails. Given the centrality of WSP to all plausible conceptions of parthood, even an endurantist should not wish to accept Gilmore’s cases as possible.29 Gilmore (2007: 191, n. 32) is aware of the problem; his response is to reject the two-place parthood relation in favor of a four-place relation relativized to personal time: ‘x at moment tx of x’s personal time is part of y at moment ty of y’s personal time’. Perhaps this relation will do the job; it is no great surprise that endurantists should prefer temporally relativized versions of relations that the perdurantist understands atemporally, and once we introduce the distinction between personal and external time, why not relativize to the more fine-grained distinction? But it must be admitted that this relativizing strategy is not very intuitive, and moreover not every endurantist is going to want to adopt this view. In any case,

28 29

A ‘proper part’ of y is a part of y that is not identical with y. Effingham and Robson (2007: section 3) make a similar observation.

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these mereological issues are clearly in need of some attention from endurantists.30 5.3. Explaining Apparently Distinguishing Properties That Cell and Tubman (and Adam and Abel) are not distinct is the upshot of the previous two subsections. Gilmore tries to undermine this conclusion, suggesting for example that Cell (Adam) travels in time, while Tubman (Abel) does not; that Tubman is conscious, while Cell is not; that Abel has a ‘mass history’ different from Abel’s. The perdurantist needs to explain these suggestions away. The general strategy is this, letting a and b stand for two objects claimed to be distinct because one is an A and the other a B, but such that a and b have the same fundamental parts (as Gilmore claims for both pairs of Cell/Tubman and Adam/Abel): first, recalling the perdurantist thesis that for a persisting entity to be an F is for it to have temporal parts connected by the same-F-as causal relation, point out that the same-A-as relation and the same-B-as relation both hold amongst the parts of a and b. Second, note that while if those identity-constituting causal relations conflict with each other, then a (and b) is at most one of an A or a B, they need not necessarily conflict. It is quite possible that (in virtue of the same-A-as relation holding among its parts in one way) a is an A; and (in virtue of the same-B-as relation holding amongst those same parts, but in a different way), a is also a B. Finally, note that many other properties—like being conscious, or having a certain mass history—are properties had in virtue of being a certain kind of thing (as the perdurantist says about other coincidence cases, 30 Gilmore (p.c.) has given an argument for his conception of the parthood relation. Starting from the idea that composition should be unique (something which should be common to all anti-coincidentalists), different parthood relations will give rise to different ways of expressing that uniqueness. The standard two-place parthood relation, and the common endurantist three-place variant (‘x is part of y at t’), both fail if we take Gilmore’s cases as he describes them. So one is forced, by one’s adherence to uniqueness of composition, to something like Gilmore’s four-place parthood relation. And this four-place relation fits better with endurance than it does with perdurance. This may provide a reason for endurantists who accept distinctness in Gilmore’s cases to accept his conception of parthood. But there is no reason for perdurantists to do so, and I also see no compelling independent reason for endurantists to accept distinctness either—endurantists could just as easily take the argument of this footnote as a reductio.

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Goliath may be valuable in virtue of being a statue, and not in virtue of being a lump of clay—but still, that does not mean that the lump of clay is not valuable). So if C is some property a has in virtue of being an A, but not virtue of being a B, that might explain initially why it might seem that a and b, though identical, are distinct with respect to C. This general strategy applies to both the cases of Cell/Tubman and Adam/Abel. In a bit more detail in the Cell/Tubman case, there is a set of parts whose fusion occupies the path of both Cell and Tubman. (The perdurantist maintains that this fusion is identical to both Cell and Tubman.) There is a way of taking those parts such that all of them can be combined into a cell: there is an exhaustive set of fusions C of those parts such that each fusion in C is an instantaneous cell-slice, and such that the same-cell-as causal relation holds pairwise between all of the members of C. There is also a way of taking those parts such that all of them can be combined into a person: there is an exhaustive set of fusions T of those parts such that each fusion in T is an instantaneous person-slice, and such that the same-person-as causal relation holds pairwise between all the members of T . These two facts about the parts that make up Cell are sufficient to show that Cell is a cell, and also that Cell is a person; similarly for Tubman. Cell is the fusion of the cell-fusions in C; Tubman is the fusion of the person-fusions in T. In standard mereology, the fusion of all the members of C (Cell) is the same as the fusion of all the parts of the members of C, and the fusion of the all the members of T (Tubman) is the same as the fusion of all the parts of the members of T. It follows that, since all the parts of the members of C are parts of the members of T, and vice versa, Cell is Tubman. There is something slightly strange about the way that the samecell-as relations hold amongst the members of C, because some of those relations hold between instantaneous cell-slices that are separated in time by three years, and they also hold backwards in time. But these odd causal relations between those fusions in C do not undermine the obtaining of other causal relations between the members of T; in fact, as Gilmore describes the case, the causal relations between the members of C are set up precisely so as to ensure that the same-person-as relation holds between the members of T. As long as both sets of causal relations can obtain without

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conflict, there seems no insuperable objection to saying that the thing made from these parts is both a cell and a person (i.e., has the property being a cell and the property being a person): this is a person with an additional set of causal relations between the cell-parts of which it is composed, such that those cell-parts are part of one time-traveling cell. In that case it is just false to say that Cell is not conscious, if Tubman is. Of course Cell is not essentially conscious, because had Cell not time-traveled, it would not have been conscious (since it would not have been arranged so as to constitute a person).31 But as Cell happened to time-travel in the precise way that it did, it managed to have a structure that was sufficiently complicated to allow it to constitute a person, while retaining (in virtue of timetravel) a structure sufficient to allow it to be a cell. If Tubman is conscious, there seems little reason to say that Cell is not conscious, as the arrangement of parts and same-person-as causal relations sufficient for consciousness just is Cell’s arrangement of parts and causal relations. Gilmore’s contention that Cell time-travels, and Tubman does not, can also be answered by drawing on these resources. To be a time-traveler, remember, is to be something such that the identityconstituting causal relations sometimes hold in a manner divergent from external time. But these identity-constituting causal relations are not given independently of what kind of thing is in question. (It is accepted by everyone, that the identity conditions for statues are not the same as the identity conditions for lumps of clay, even if all statues are lumps of clay.) For the perdurantist, time-travel for an F just is the holding of backward causal relations that ground same-F-as over time. So there is an ambiguity in the question, ‘Is that thing a time-traveler?’—we cannot answer it until we know what kind of thing is in question, and which identity-constituting relations we need to examine to see if they run in a way divergent from external time. Since Cell is both a cell and a person, we can ask 31 This fits entirely neatly with the perdurantist’s typical antipathy towards essential properties and robust (non-conventional) de re modal properties (Sider, 2001: 207): Cell is both a person and a cell, and in virtue of that fact Cell has multiple counterpart relations (or so say most perdurantists); it is an elementary error to conclude from the existence of multiple counterpart relations that there are incompatible modal properties and hence distinct objects.

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both. Is it a time-traveling cell? Is it a time travelling person? The answer to the latter question is ‘no’, as there are no same-person-as causal relations in this case that run backwards in time. But there are same-cell-as causal relations that run backwards in time. Whether something is a cell is independent of whether it is a person, as the argument of the previous paragraph showed. And the present argument shows that whether something is a time-traveling cell is similarly independent of whether it is a time-traveling person. There is no genuine conflict between being a time-traveling cell and failing to be a time-traveling person.32 In his discussion of Adam/Abel, Gilmore rests considerable weight on his claim that Adam and Abel differ in their mass histories. He claims that Adam satisfies the property M1 , being an object that has a rest mass of one unit throughout its two-billion-year-long career, while Abel satisfies the property M2 , being an object that has a rest mass of more than one unit throughout its one-billion-year-long career. If this claim is true, and these properties are incompatible, then Adam and Abel must differ. The mass of Adam and Abel is the same: each part of one is part of the other, and the total mass of those parts must be the same for each.33 Gilmore thus rests the argument for distinctness on the distribution of that mass over each object’s career. If they have different-length careers, that same mass must be distributed differently (else it would not add up to the same mass for each object). To evaluate this, let us first ask: what it is to have a career of a certain length? Existing for a certain duration of external time is arguably the fundamental physical quantity that can be possessed non-relationally; but there is no sense in which the careers of Abel and Adam differ in length in external time. The only sense 32 There is a conflict between being a non-time-traveling cell and being a timetravelling person, but that is because non-time-traveling cells cannot be persons at all, as they have no temporally unextended stages which are person stages, and hence nothing appropriate to be relata of the same-person-as relation. 33 There is something strange about this, as mass cannot be an intrinsic property if we take Gilmore’s hint that each temporal part of Adam is bi-located (‘‘present twice over’’). Then Adam’s mass at a time will have to be twice the mass of Adam intrinsically, since Adam in each location exerts the same gravitational force. While this is strange, I am more inclined to think this a curious feature of multiple location than a problem for perdurantists—particularly as multiply located enduring objects which change mass over time will also not have mass as intrinsic, or indeed much else (see the final paragraph of this section).

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in which Adam has a two-billion-year-long career is in terms of its atomic personal time. Since personal time depends on timetravel, what an object’s personal time is, is not a monadic property, but depends, as we saw just above, on what sameness-constituting causal relations ground the time-travel. In terms of Adam’s molecular personal time, which involves no same-molecule-as backwards causal relations, and is thus identical to external time, Adam has a one-billion-year long career. But there is no reason to think that having different length careers is an incompatible property when we are not measuring length in the same time! One and the same object can have multiple mass histories, relative to the different personal times it has in virtue of the different kinds of things it is. So Adam has a two-billion-year long atomic career, and a one-billionyear long molecular career; and we know that the mass of the atom and molecule are the same overall. The distribution of that mass over the different trajectories is relative to the different sortals that Adam/Abel satisfies, but is no fundamental feature of Adam/Abel. Gilmore (2007: 192–6) considers a similar ‘relativizing’ approach to mass histories. While he admits it can succeed, he thinks that if adopted by the perdurantist, that perdurantist cannot in good conscience reject a similar relativizing response on the part of the endurantist to the argument from temporary intrinsics. The cases are importantly disanalogous, however, because the perdurantist should not accept that these relativized mass histories are fundamental physical properties. The only mass history with a fundamental role is the distribution of mass through external time, and in external time the only mass history, shared by Adam and Abel, is M2 . The endurantist confronted with the problem of temporary intrinsics, by contrast, cannot appeal to a fundamental non-relativized notion of intrinsic property. The perdurantist criticisms of relativizing moves do not apply to the unrelativized mass history in external time, and that mass history is the only one the perdurantist should take as basic.

6. denying co-location: non-extensional mereology of objects and regions What I have said does not go radically beyond the standard, though perhaps implicit, position of most perdurantists. If Gilmore’s cases

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are supposed to count as objections to the standard perdurantist position, those cases do not succeed. Most perdurantists will be content with the response in Section 5, and rightly so: it comports most naturally with other assumptions about extensionality of parthood that almost all perdurantists accept. I expect, in fact, that almost all perdurantists will wish to avail themselves of the response of the previous section. I would strongly urge them to do so, and for what it is worth my own preferred solution is the one outlined above. Nevertheless, I want in this section to explore an alternative proposal. One reason this is of interest is that it brings to prominence concerns about the relationship between the mereology of locations and the mereology of objects; while the discussion in this case focuses on a weak mereology, the lessons generalize. A second reason for bothering to explore this direction is that many of the observations in Section 5 will seem controversial to an impartial observer. We might wish to explore some deviations from the standard perdurantist line that might mesh more closely with our immediate intuitions about Gilmore’s cases, recognizing that one can come across Gilmore’s cases without feeling much initial resistance to the distinctness of Cell and Tubman. This seems to indicate that there is something intuitively acceptable about them being distinct. The orthodox perdurantist simply takes this to be a result of insufficient reflection on just what the cases involve, but it may be that there is more to be said in favor of that initial reaction. We should thus consider whether perdurantists can answer Gilmore’s cases even under weaker assumptions than classical extensional mereology; and in particular, try to develop a perdurantist theory of locations to accompany such a weaker mereology. The parthood relation, minimally, should be taken to be reflexive, antisymmetric, and transitive. If we also accept that weak supplementation is central to the correct conception of part (Section 5.2), we arrive at the theory of minimal mereology, MM (Varzi, 2006). However, strengthening MM without committing to unique composition is more difficult. For example, we cannot strengthen WSP by means of the following principle (Simons, 1987: 26–9):

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If y is not part of x, then there is at least some part of y that does not overlap x.

In the presence of SSP, we get classical extensional mereology back from MM: assuming that the Xs fuse to form distinct y and z, since y is not part of z, at least some of the Xs do not overlap z, but that cannot be since z fuses the Xs. So non-unique composition is impossible in the presence of SSP. This theory must also reject this principle, which, in the presence of WSP, entails SSP (Simons, 1987: 30–1): Product Principle (PP)

If x and y overlap, there is a thing that is part of both and has all their other common parts as parts (i.e., there is a maximal common part).

Unrestricted composition entails PP (Varzi, 2006: section 4.2), so the perdurantist who wishes to sacrifice unique composition also must sacrifice unrestricted composition. This is already a serious cost to perdurantism, as several arguments in the literature depend on the existence of arbitrary fusions (Sider, 2001: section 4.9). The interaction between supplementation principles and unrestricted fusions makes doubtful the existence of reasonably strong systems that are not as strong as full classical extensional mereology. So for now I will provisionally adopt MM as a stable way for the perdurantist to liberalize their conception of parthood. A useful model of MM for our purposes can be given by considering ordered sequences. Assume that there are mereological atoms (objects with no parts).34 Then we can model an object as a nonempty ordered sequence of such atoms without repetition; take the set of all such sequences as the domain of our model. An atom will be called an element of the sequence. If we interpret the parthood relation ‘x is part of y’ by the relation ‘x is a substring of y’ (where a sequence T is a substring of a sequence S iff T is a sequence of consecutive elements of S retaining the same order as in S), we 34 This assumption is true in the model I am presenting, but is not entailed by the axioms of MM, which also have atomless models.

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generate a model for MM. We can quickly show that in this model, parthood is reflexive (since every sequence is a trivial substring of itself), transitive (since if U is a sequence of consecutive elements from T, and T in turn a sequence of consecutive elements from S, then U must also be a string of consecutive elements from S), and antisymmetric (T cannot be a substring of one of its own substrings S unless S = T). We can also show that WSP is satisfied, since if T is a proper substring of S, there must be at least one further substring T*, with no substring in common between them (since S is without repetition), that is also a substring of S—obviously, it will be that string or strings such that if they are concatenated with T in the right order we get S back again (if we want to guarantee that for any part x of O, there is exactly one other disjoint part y of O such that the fusion of x and y is O, we should consider ordered loops). But quite clearly SSP fails in this model: take the simple case of the sequences S = and T = . Neither S nor T are parts (substrings) of each other; but every part of S overlaps (has a substring in common with) every part of T —those parts are , , and , the former of which are parts of T, and the latter of which, S itself, has parts which are parts of T. PP also fails, in the same case: S and T overlap on and , but there is no maximal common part. So we clearly have a model of MM which is not also a model of classical extensional mereology. Another way to see this is to note that concatenation of sequences is the analog of fusion in this model, and concatenation of some things gives different results depending on the order in which we take those things, unlike the case in extensional fusion. There will be more than one fusion of a single set of parts. In the context of this model of MM, we can treat the case of Adam and Abel quite simply (Cell and Tubman can be treated in exactly the same way, though the case is more tedious because Cell’s temporal parts are not atomic). An object is any sequence of atoms, and to model Adam and Abel we can choose any sequences which fit the intuitions Gilmore marshals. For example (if moments of time can be enumerated), we can take Adam (the time-travelling atom) to be composed at each moment t of two spatial parts, At,1 and At,2 , and take Adam as a whole to be the fusion of these parts. Order matters in our present model, but there is a natural order to use: personal time, or causal order (since that is the order that

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allows us to tell that all of these parts are parts of the atom Adam). On this ordering, Adam is the fusion . Note that Adam’s temporal parts are naturally taken to be a fusion of those of its parts that are simultaneous in personal time.35 Since no pair of Adam’s parts is simultaneous in personal time, Adam’s temporal parts are each 1-element substring of Adam. Abel’s temporal parts are also a fusion of its simultaneous parts, but as it is not a time-traveler, its personal time is the same as external time. Abel’s temporal parts, then, are fusions of Adam’s parts at each moment of external time.36 Abel is a fusion of its temporal parts, which is, again ordered by personal time, . It is obvious that these two sequences are not identical, and we can conclude that Adam is distinct from Abel. Moreover, this model captures some of the intuitions we have about the case: while the two objects have the same parts, the fact that each of Abel’s parts are personally simultaneous with exactly one other part, while none of Adam’s parts are, means that Adam has twice as many temporal parts as Abel, which supports the intuition that Adam’s personal duration (career) is twice as long as Abel’s. Their corresponding mass histories differ too, with each temporal part of Abel being heavier than each temporal part of Abel. Taking S-regions to be the spatial locations of the temporal parts, it is fairly clear that the one-to-one correspondence between Sregions and temporal parts will mean that the S-regions of Adam and Abel differ too. Continuing with the assumption from Section 1 that regions are fusions of points, a region will be a fusion of points occupied by mereological atoms; an S-region will be the fusion of those points occupied by the elements of a temporal part. As a path is a fusion of the S-regions, then Adam’s paths are Abel’s paths, and we immediately generate a coincidence scenario—but only on the 35 If we had chosen to take temporal parts as ordered by external time, Adam and Abel would have the same parts at each time. We are trying here to capture the intuitions Gilmore musters, to accommodate those who share them, and one of these intuitions is that personal time is fundamental in matters of composition. Gilmore, as mentioned above (p. 14) proposes to account for this by parameterizing the parthood relation by personal time; the perdurantist solution here retains the atemporal notion of parthood, and implements a significant role for personal time in fusion. 36 We must choose one of the two possible fusions of Adam’s parts at t to be Abel’s part at t—there seems no objective way to prefer one over the other, so I simply stipulate that the fusion is Abel’s part. I return to this issue below.

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assumption that the mereology of regions is classical extensional mereology. Yet having given up extensional mereology for objects, there seems no reason to retain it for regions. It seems more plausible instead to accept that the mereological relations amongst occupied regions are isomorphic to the mereological relations amongst the occupants of those regions. So the same minimal mereology MM should account for the parthood relations amongst regions. The path of Adam will be a fusion of the locations of its S-regions: , while the path of Abel will be a fusion of its S-regions: . If these paths are the locations of the objects, we avoid coincidence problems by having distinct locations for distinct objects. Having outlined one formal treatment of a non-extensional mereology that can be combined with perdurantism in a way that satisfies Gilmore’s intuitions, it is clear that such a perdurantism can avoid coincidence problems. Yet many philosophical issues remain, and to these I now turn. The first couple are technical worries about the framework. Having chosen sequences as the domain of our model of MM, it follows that any fusion can have only enumerably many parts (a sequence can be defined as a function from the natural numbers into some domain). This in turn means that we must treat time as having enumerably many instants, in order to allow objects to be fusions of temporal parts. This is an artifact of our simple model, and could be avoided if we had chosen a different model for MM. It is a simplification, obviously, to treat time as discrete, and I would want a more general solution if I were seriously defending this version of perdurantism (for example, take the domain of the model to be functions from the reals into some domain). But I wanted here mostly to illustrate the idea, and ordered sequences provide an easily visualized non-extensional way of fusing some set of things. Another technical worry is that the domain of the model is sequences without repetition, and this cannot account for the case of Adam and Abel if Adam is genuinely bi-located at each time, since we would need each temporal part of Adam to appear twice in the sequence. In fact, we cannot give a model of MM on arbitrary sequences, as WSP fails in such a model (counterexample:

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has as a proper part, but has no other proper part). I confess I’m not really sure how to accommodate this intuition of Gilmore’s, that Abel can be composed entirely of Adam and yet distinct; perhaps an even more non-standard mereology (maybe involving Gilmore’s variant 4-place parthood relation) is required for the perdurantist to make sense of this. My more serious worries about the present framework are two: one about spacetime, the other (which I flagged earlier) about overgeneration of fusions. The Spacetime Worry The spacetime worry is simply this: if the motivation for the anticoincidence principle is the worry that coincident objects would ‘crowd each other out’ of the one location, those who endorse anti-coincidence are unlikely to be content with a non-extensional mereology of regions, as those regions too seem to crowd one another out. For there are many regions associated with one set of spacetime points; and while it is all very well to say that extensional mereology provides a notion of fusion that is much too crude for material objects, it is extremely difficult to see what motivation internal to considerations about spacetime would move one towards a non-extensional fusion operation on sets of points. Spacetime is made of points; those points are intrinsic duplicates of one another; so setting aside purely haecceitistic differences, there seems no reason to think that the different sequences of points generated from the same underlying set of points can play any different role in being the locations of material objects. If this is right, spacetime should have an extensional mereology—and we get coincidence problems again. (Indeed, we get many more of them, as MM lacks unique composition, so whenever a region contains more than one atom we have more than one object fusing those same atoms in that region.) The best way to respond to this worry, it seems to me, is to reject the ‘spacetime first’ view of regions which seems to motivate it. If spacetime is viewed as a substantial receptacle, existing before there are entities to populate it, then it is natural to worry about whether the independently given structure of spacetime can be plausibly non-extensional. But this is not the only option. One alternative is relationism, the idea that spacetime is not real, but is at best a useful fiction to adopt when representing genuine facts about the relations

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between material objects. Since it is facts about material objects which are primary on this view, it would not be at all surprising if the best mereology of the locations of objects mirrored the best mereology for objects. Another very different view of spacetime, but with the same upshot, is supersubstantivalism—the view that material objects are to be identified with the regions of substantial spacetime that are their locations (Schaffer, 2009). Since the location relation is identity, the mereological relations on locations will even more clearly be the same as the mereological relations on objects. Both of these views, and others besides, end up supporting the following principle: Mereological Harmony

‘Mereological relations on material objects are perfectly aligned with mereological relations on the regions of space they exactly occupy [are located at]’ (Uzquiano, 2006: 441).

This principle is intuitively quite plausible. Nevertheless, it is not unproblematic: if there are such things as extended simples, Mereological Harmony appears to fail, for the location of an extended simple will have parts while the simple does not. Things are not as straightforward as this, since as the discussion of Section 2 made clear, there is a worry about whether extended simples have an extended location, or whether they have unextended locations but an extended location† (region they wholly and entirely fill)—if the latter, which is the view I prefer, Harmony can be maintained. Another difficulty for the principle is that it creates problems in the context of moderate views of which kinds of regions can be occupied (Uzquiano, 2006). So while it would help the view of the present section, and be independently interesting for a wide variety of mereological theories, Mereological Harmony is not entirely uncontroversial. A different argument that the mereology of regions is that of continuants comes from the following intuitively plausible principle:37 (SL)

The shape of an object is the same as the shape of its location. 37

But see Hudson 2006: 111–3.

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SL, as it stands, is too general for our purposes, because it has both temporary and non-temporary readings. The temporary reading claims that the shape of an object at a given time is the same as the shape of the location of that object at that time; both endurantists and perdurantists will think, for different reasons, that the temporary shape of an object at t is the same as the shape of the S-region that the object occupies at t (either by the object being located there, for the endurantist, or by the worm of the object’s temporal part being wholly located in the S-region at t).38 Of course, when spelled out this way the problem of temporary intrinsics immediately threatens the endurantist, so endurantists perhaps would not accept even this version of SL.39 No matter; our concern here is with the perdurantist. Perdurantists should also recognize another sense of SL. Just as each temporal part has a certain shape, so too it seems plausible that their fusion—the worm—might have a certain shape. Some worms are very long and thin (perhaps the worm of a fundamental particle is like this). Some are shorter and fatter (perhaps the worm of the Melbourne Cricket Ground is like this). The shapes that worms can have are surely different from the shapes that temporal parts of worms can have. Moreover, the shape of a worm surely depends on the shape and arrangement of its temporal parts, so one might suspect that worm shapes, and non-temporary shape in general, should be a derivative rather than a fundamental matter. Nevertheless, neither of these considerations should lead us to believe that there are no such things as the shapes of worms. If we accept that worms can have shape properties, the natural assumption is that these shape properties obey SL, so that the shape of the location of the worm is the same as the shape of the worm. The worm of a fundamental particle is long and thin; so too is its location. The worm of the MCG is shorter and fatter; so too is its location. The same intuitions support the thought that, if their distinctness is accepted, Abel’s worm is shorter and wider than Adam’s worm, and the mereological theory discussed in the present section permits that. If we accept SL, the location of Abel’s worm should be shorter and wider than the shape of Adam’s worm. Purely extensional 38 Something like SL, notice, lies behind the intuition exploited in Section 1 that the locations of a multiply located object should not include the fusion of its locations—since that fusion will generally have a different shape to its proper parts. 39 If shape is intrinsic: see Skow 2007.

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fusion of S-regions will not allow these distinctions, but the nonextensional conception of locations developed in this section does allow these distinctions. So if we agree that the four-dimensional shapes of distinct objects should be reflected in the shapes of the locations they occupy, and we accept the different shapes of Abel and Adam, then a non-extensional fusion of the distinct S-regions of these objects is required. This argument does not support an exact mirroring of the mereological structure of objects and regions; it concludes that when we have fairly robust intuitions about the structure of an object, the location of that object should have a structure rich enough to capture those intuitions in line with SL. This response to the spacetime worry could be turned around. The principles of Mereological Harmony and SL are not dependent on MM, and if one is worried (as I am) by the excessive multiplication of spacetime regions MM induces, one could use these principles to argue that an extensional mereology must also be the correct mereology for continuants—as argued in Section 5.2. The Overgeneration Worry The overgeneration worry is equally simple. Because there are many sequences that can be formed from the same elements, there will be intuitively too many fusions of a single collection of parts. We were able in the case of Adam to select a ‘natural’ fusion, the sequence ordered by personal time. But all the other possible sequences are fusions of those parts too, and they equally exist. Perhaps there is in this case only one plausible candidate fusion to be the continuant Adam, and we can appeal to pragmatic factors to explain why we ignore all the other relatively unnatural fusions. (In much the same way as believers in unrestricted composition defend that doctrine from the objection that it commits us to the existence of too many things, like Lewis 1991: 79–81.) But the case of Abel shows that this does not eliminate all the problems, since there will be many equally good fusions of simultaneous parts, none of which is more natural than any other (in the absence of any natural ‘order’ on points of space). The natural way to try and cut down on these additional spurious entities is to try to strengthen MM; but, as I mentioned earlier, many natural strengthenings collapse into extensional mereology. We could mark a distinction between temporal and spatial parts,

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perhaps treating spatial fusions extensionally and temporal fusions non-extensionally, but this undermines one strong motivation for perdurantism from perceived analogies between spatial and temporal features of objects. Finally, we might try to impose additional non-mereological constraints, just as we did when we chose the ‘natural’ ordering by personal time for Adam. Even if this move works in the case of Adam, I have no idea what kind of genuinely objective constraint would prefer one of the two alternative sequences and made from the parts a and b. But there is also considerable reason for worry in the case of Adam. The relevant non-mereological principle there is something like this: F-Fusion

There exists a fusion of some things just in case those things are an F or can be unified by same-F-as relation over personal time.

In this case, we get Adam, because those parts stand in the sameatom-as relation over personal time. But since the notion of personal time makes no sense for spacetime regions, which cannot timetravel, the F-Fusion principle will not allow us to form a fusion of the locations of Adam’s temporal parts that orders them other than in external time—that is, the only fusion that exists that is at all a candidate for Adam’s location is Abel’s location, and we get coincidence again. Cutting down on the number of continuants by appeal to non-mereological principles will very likely rule out almost all of the locations needed to avoid coincidence. Nothing in the present section has made any real headway against the arguments of Section 5 against the distinctness of Adam and Abel. So all those reasons to explain away the intuitive pull of distinctness need to be undermined, if we are to take the radical step of adopting a non-extensional mereology. Perhaps non-extensional mereologies are appropriate for endurantists, but there is a strong case to be made that perdurantism most naturally combines with classical extensional mereology—particularly because, for example, in some cases anti-endurantist arguments rely on there being a unique location for every spacetime region.40 That said, the aim of 40 For example, an endurantist could respond to the case of Deon and Theon by multiplying locations corresponding to the spacetime region where both are to be found.

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the present section was merely to indicate that, even if a perdurantist was so swayed by the intuition of distinctness that they accept that Abel is not Adam, there are somewhat plausible responses that can be made which avoid Gilmore’s conclusions.

7. conclusion While Gilmore’s cases are ingenious and interesting, once we pay careful attention to the theory of location and their mereological structure, there is no reason for an orthodox perdurantist, who accepts classical extensional mereology and the view that paths are locations, to be worried by Gilmore’s cases. Even a non-standard perdurantist, who believes in the distinctness of the objects concerned and hence accepts non-standard mereology, can respond to Gilmore’s cases by adopting that perfectly reasonable view that the mereological relations on locations mimic those of the objects which have those locations. Neither of these options undermines all common anti-endurantist arguments (though the view of Section 6 does worse on this score). The first does not even involve any modification to the express published views of standard perdurantists. Either way it is clear that there remains an asymmetry here: while endurantists fail to respond to apparent coincidence in some orthodox cases of objects with overlapping stages, the perdurantist is able to, and, orthodox or not, faces no related problem from Gilmore’s more recherch´e cases. Orthodox perdurantism moreover remains the best unified account of the paradoxes of coincidence.41 Exeter College, University of Oxford

references Armstrong, D. M. (1997), A World of States of Affairs (Cambridge: Cambridge University Press). 41 For discussion and comments, thanks to audiences at a meeting of the BSPS and at the Universities of Notre Dame, Nottingham, and Oxford, and to Cody Gilmore, Ben Caplan, Bill Child, Carrie Jenkins, Boris Kment, Daniel Nolan, Josh Parsons, Jeff Speaks, Martin Thomson-Jones, Gabriel Uzquiano, and Fritz Warfield. This research was partly supported by grant AH/E003184/1 under the AHRC Research Leave Scheme.

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Arntzenius, Frank (2003), ‘‘Is Quantum Mechanics Pointless?’’ Philosophy of Science, vol. 70: pp. 1447–57. Barker, Stephen and Dowe, Phil (2005), ‘‘Endurance is Paradoxical’’. Analysis, vol. 65: pp. 69–75. Cartwright, Richard (1987), ‘‘Scattered Objects’’. In Philosophical Essays (Cambridge, MA: MIT Press), pp. 171–86. Earman, John (1995), Bangs, Crunches, Whimpers and Shrieks: Singularities and Acausalities in Relativistic Spacetimes (Oxford: Oxford University Press). Effingham, Nikk and Robson, Jon (2007), ‘‘A Mereological Challenge to Endurantism’’. Australasian Journal of Philosophy, vol. 85: pp. 633–40. Fine, Kit (1999), ‘‘Things and Their Parts’’. Midwest Studies in Philosophy, vol. 23: pp. 61–74. (2003), ‘‘The Non-Identity of a Material Thing and Its Matter’’. Mind, vol. 112: pp. 195–234. Forrest, Peter (1986), ‘‘Neither Magic Nor Mereology: A Reply to Lewis’’. Australasian Journal of Philosophy, vol. 64: pp. 89–91. Gibbard, Alan (1976), ‘‘Contingent Identity’’. Journal of Philosophical Logic, vol. 4: pp. 187–221. Gibson, Ian and Pooley, Oliver (2006), ‘‘Relativistic Persistence’’. Philosophical Perspectives, vol. 20: pp. 157–98. Gilmore, Cody (2007), ‘‘Time Travel, Coinciding Objects, and Persistence’’. Oxford Studies in Metaphysics, vol. 3 (Oxford: Oxford University Press) pp. 177–98. Haslanger, Sally (2003), ‘‘Persistence Through Time’’. In Michael J. Loux and Dean W. Zimmerman (eds.), Oxford Handbook of Metaphysics (Oxford: Oxford University Press), pp. 315–54. Hawley, Katherine (2001), How Things Persist (Oxford: Oxford University Press). Hudson, Hud (2002), ‘‘The Liberal View of Receptacles’’. Australasian Journal of Philosophy, vol. 80: pp. 432–9. (2006), The Metaphysics of Hyperspace (Oxford: Oxford University Press). Keller, Simon and Nelson, Michael (2001), ‘‘Presentists Should Believe in Time Travel’’. Australasian Journal of Philosophy, vol. 79: pp. 333–45. Lewis, David (1976), ‘‘The Paradoxes of Time Travel’’. In Philosophical Papers, vol. 2 (Oxford: Oxford University Press), pp. 67–80. (1983), ‘‘Survival and Identity’’. In Philosophical Papers, vol. 1 (New York: Oxford University Press), pp. 55–77. (1986), ‘‘Against Structural Universals’’. Australasian Journal of Philosophy, vol. 64: pp. 25–46. (1991), Parts of Classes (Oxford: Blackwell).

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Locke, John (1976), An Essay Concerning Human Understanding. edit. P. H. Nidditch. (Oxford: Oxford University Press). Parsons, Josh (2007), ‘‘Theories of Location’’. Oxford Studies in Metaphysics, vol. 3 (Oxford: Oxford University Press) pp. 201–32. Schaffer, Jonathan (2009), ‘‘Spacetime the One Substance’’, Philosophical Studies vol. 145: pp. 131–48. Sider, Theodore (2001), Four-Dimensionalism: An Ontology of Persistence and Time (Oxford: Oxford University Press). (2007), ‘‘Parthood’’. Philosophical Review, vol. 116: pp. 51–91. Simons, Peter M. (1987), Parts: A Study in Ontology (Oxford: Oxford University Press). Skow, Bradford (2007), ‘‘Are Shapes Intrinsic?’’ Philosophical Studies, vol. 133: pp. 111–30. Tarski, Alfred (1929), ‘‘Foundations of the Geometry of Solids’’. In Logic, Semantics and Meta-Mathematics, Indianapolis: Hackett, pp. 24–9. Uzquiano, Gabriel (2006), ‘‘Receptacles’’. Philosophical Perspectives, vol. 20: pp. 427–51. van Inwagen, Peter (1990), ‘‘Four-Dimensional Objects’’. Noûs, vol. 24: pp. 245–55. Varzi, Achille (2006), ‘‘Mereology’’. In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Winter 2006 edn.), http://plato. stanford.edu/archives/win2006/entries/mereology/.

5. Coinciding Objects and Duration Properties: Reply to Eagle Cody Gilmore 1. introduction In my (2007) I presented a new puzzle for perdurantism1 (the ‘‘typeC puzzle’’), and I noted that perdurantists could solve it in various ways. I then defended the main conclusion of the paper: MC

Any perdurantist solution to the type-C puzzle would significantly weaken at least one familiar argument against endurantism.

Antony Eagle develops two perdurantist solutions to the puzzle. One of these, he agrees, poses no threat to MC. He takes the other, however, to be a counterexample to that conclusion. In what follows, I defend MC against Eagle’s challenge.

2. the type-c puzzle To state the puzzle, I introduced some technical terminology, which I briefly review. (1) I assumed that parties on both sides of the endurantism v. perdurantism dispute could grasp the two-place predicate ‘exactly occupies’, and that it would turn out to be obvious that a thing O exactly occupies a spacetime region R iff O has (or has-at-R) the same size, shape, and position as O, but that it would not turn out to be obviously impossible for a thing to exactly occupy each of several non-intersecting regions but not their union or any of their proper subregions. (2) I defined a thing’s path as the union 1 Roughly, the view that material objects persist by being temporally extended and having different temporal parts at different times. Endurantism, roughly, is the view that material objects persist without being temporally extended or having temporal parts, but rather by being wholly present at each moment of their careers.

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of the (region or) regions that the thing exactly occupies. (3) I (very informally) defined an S-region of an object as an instantaneous spacetime region that corresponds to what we ordinarily think of as a spatial location of that object at some instant in its career; I then claimed that according to endurantism, material objects exactly occupy just their S-regions, whereas according to perdurantism, material objects exactly occupy just their paths. (4) I said that things x and y coincide iff there is a region that they both exactly occupy.2 (5) Finally, I said that x and y are involved in a type-C situation iff x and y are numerically distinct material objects that have the same path but do not have any of the same S-regions. I described two main cases that appeared to count as type-C situations. I will focus on just one of them here: Adam and Abel. A hydrogen atom, Adam, has a path that follows a closed timelike curve.3 This closed curve, however, is not a ‘simple loop’; instead it is ‘doubled up’ like the edge of a m¨obius strip. This allows for Adam, at each moment of its career (or in each of its S-regions), to be chemically bonded to itself, at a different moment of its career (or in a different S-region), thus forming a molecule of H2 , Abel. Adam and Abel apparently have the same path but none of the same S-regions. Prima facie, Adam’s S-regions are atom-shaped, whereas Abel’s are all larger and shaped like molecules of H2 .

This case generates a puzzle for perdurantism that can be solved by shifting to endurantism. In light of the various apparent differences between them (which I discuss below), Adam and Abel are plausibly taken to be numerically non-identical. Moreover, if these objects perdure then, since they have the same path, and since perduring objects exactly occupy their paths, the objects coincide (at their shared path), in violation of the ‘anti-coincidence’ principle.4 But if 2 For simplicity, I am working with a spatiotemporal notion of coincidence when, to be strictly faithful to the views of most anti-coincidentalists, I would need to work with a mereological notion. See n. 4 below and my (2007: 178, n. 4). 3 A timelike curve is, roughly, a continuous one-dimensional spacetime region that could be the path of a spatially unextended particle that has mass. It need not by ‘curvy’; it can be straight. It is closed if it forms a loop. 4 For convenience, we can pretend (as I did in my (2007: 178, n. 4)) that this principle is just a ban on spatiotemporal co-location—i.e., that it is the view that it is impossible for there to be a spacetime region that is exactly occupied by two different material objects. In fact, a better approximation of the principle is this: it is impossible for two different material objects to exactly occupy the same spacetime

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they endure, then since they share none of their S-regions, and since an enduring object exactly occupies only its S-regions, the objects do not coincide, and the anti-coincidence principle is preserved. Eagle argues that perdurantists can solve the puzzle either by (i) maintaining that, despite appearances, Adam and Abel are in fact identical or by (ii) conceding that they are distinct, but using a non-extensional mereology for spacetime regions to claim that their paths, though both entirely composed of exactly the same spacetime points, are also distinct, in which case Adam and Abel could perdure without strictly coinciding, as I defined that term above. The second solution deserves more attention than I can give it here. As Eagle notes, however, it does undermine at least one wellknown style of argument against endurantism and so it poses no threat to my intended conclusion, MC. (See his note 40.) The remainder of the chapter, therefore, will focus on Eagle’s first solution.

3. identifying adam and abel: a PRIMA FACIE cost Eagle’s preferred response to the type-C puzzle is to maintain that Adam and Abel are numerically identical. Moreover, he claims—against MC—that this response does nothing to weaken any of the standard arguments against endurantism. This is a claim that I want to resist. One of my arguments for the non-identity of Adam and Abel appealed to differences in their ‘mass histories’: Adam has a rest mass of one unit throughout its two-billion-year-long career, but Abel does not. (Abel’s career is just one billion years long, and it has a rest mass of more than one unit throughout that career; this mass history is incompatible with Adam’s.) As far as I can tell, the only viable strategy for resisting this argument is to adopt a ‘relativizing’ treatment of mass histories. One can hold that my case involves just a single thing whose career region and be composed of the same things in that region. (Many self-described anti-coincidentalists are happy to allow for the possibility of worlds governed by unfamiliar laws of nature in which non-identical material objects exactly occupy the same region, so long as these objects are not composed of the same things.) Since my case obviously does not involve this sort of co-location without co-composition, the pretence above makes no difference.

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can be divided up into temporal parts in different ways. Relative to one such partition (the atomic partition), the thing is a long-lived, not-so-massive hydrogen atom; relative to a different partition (the molecule partition), it is shorter-lived, more-massive molecule of H2 . To adopt this view is to hold that while the relevant mass histories may appear to be incompatible, intrinsic, monadic properties, they are in fact ‘disguised relations’ that things can bear to partitions.5 The advocate of this view will reject the following principle: (L∗ )

If a small hydrogen atom with a 2 billion-year-long career and a constant rest mass of 1 unit completely composes a larger hydrogen molecule with a 1 billion-year-long career and constant rest mass of more than 1 unit (in the manner illustrated by my case), then: (i) there is a thing that just plain has the monadic, intrinsic, non-indexed property being an object that has a rest mass of 1 unit throughout its 2-billion-year-long career [M1 for short], and (ii) there is a thing that just plain has the monadic, intrinsic, non-indexed property being an object that has a rest mass of more than 1 unit throughout its 1-billion-year-long career [M2 for short], and (iii) necessarily: for any x and y, if x just plain has the monadic, intrinsic non-indexed property being an object that has a rest mass of 1 unit throughout its 2-billion-year-long career and y just plain has the monadic, intrinsic, non-indexed property being an object that has a rest mass of more than 1 unit throughout its 1-billion-year-long career, then x = y. (2007: 195)

So if one is willing to deny (L*) and be a relativizer about mass histories, then (so far as the current argument is concerned) one can identify Adam and Abel. 5 The relativizing strategy, like relativizing approaches to shapes and other apparently temporary properties, can be implemented in other ways as well: e.g., by positing an extra argument place in the instantiation relation rather than in the mass histories (or shapes). See Haslanger (2003). I assume that these alternatives do not require separate discussion.

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But (L*) is intuitively plausible, at least initially. Why is denying it any better than solving Lewis’s problem of temporary intrinsics (1986a: 202–4) by being a ‘relativizer’ about shapes and thus denying the following? (L)

For any material object O, if O changes from being bent to being straight, then: (i) there is a thing that just plain has the monadic, intrinsic, non-indexed property being bent, and (ii) there is a thing that just plain has the monadic, intrinsic, non-indexed property being straight, and (iii) necessarily: for any x and y, if x just plain has the monadic, intrinsic, non-indexed property being bent, and y just plain has the monadic intrinsic, nonindexed property being straight, then x = y. (2007: 195)

If the perdurantist identifies Adam and Abel and denies (L*), then he must reject our intuitions about the nature of the relevant mass histories. And if he does this, then he should concede that the endurantist can, at a comparable price, reject our intuitions about the nature of the relevant shapes. So, in the absence of some reason for treating these apparently similar cases differently, we can conclude that Eagle’s preferred solution does significantly weaken Lewis’s argument from temporary intrinsics, and that MC stands.

4. eagle’s attempt to find a significant disanalogy between the cases Is there anything especially bad about relativizing treatments of shapes (which deny (L)), or some problem for those views that does not apply equally to relativizing treatments of mass histories (which deny (L*))? This is the crux of the dispute between Eagle and myself, and he addresses it in the following passage: What is it to have a career of a certain length? Existing for a certain duration of external time is arguably the fundamental physical quantity that can be possessed non-relationally; but there is no sense in which the careers of Abel and Adam differ in length in external time. The only sense in which Adam has a two-billion-year-long career is in terms of its atomic personal time. Since personal time depends on time-travel, what an object’s personal time is is not a monadic property but depends, as we saw above,

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on what sameness-constituting causal relations ground the time-travel. In terms of Adam’s molecular personal time, which involves no same-molecule backwards causal relations, and is thus identical to external time, Adam has a one-billion-year-long career. But there is no reason to think that having different length careers is an incompatible property when we are not measuring length in the same time! One and the same object can have multiple mass histories, relative to the different personal times it has in virtue of the different kinds of things it is. So Adam has a two billion-yearlong atomic career, and a one-billion-year long molecular career; we know that the mass of the atom and molecule are the same. Gilmore considers a similar ‘relativizing’ approach to mass histories. While he admits that it can succeed, he thinks that if adopted by the perdurantist, that perdurantist cannot in good conscience reject a similar relativizing response on the part of the endurantist to the argument from temporary intrinsics. The cases are importantly disanalogous, however, because the perdurantist should not accept that these relativized mass histories are fundamental physical properties. The only mass history with a fundamental role is the distribution of mass through external time, and in external time the only mass history, shared by Adam and Abel, is M2 . The endurantist confronted with the problem of temporary intrinsics, by contrast, cannot appeal to a fundamental non-relativized notion of an intrinsic property. The perdurantist criticisms of relativizing moves do not apply to the unrelativized mass history in external time, and that mass history is the only one the perdurantist should take as basic. (2009)

Why is denying (L), and being a relativizer about shapes, so much worse than denying (L*), and being a relativizer about mass histories (or, more simply, career-lengths)? The reason, according to Eagle, is that while external time lengths, like shapes, are fundamental properties (which, so the thought goes, puts us under a special obligation not to be relativizers about them), personal time lengths are not fundamental; and while Abel does have a one-billion-yearlong career in external time, the only sense in which Adam’s career has the superficially incompatible length of two billion years is with respect to its atomic personal time. So, in light of the non-fundamentality of personal time (Eagle’s suggestion continues), we are free to be relativizers about lengths in personal time, without thereby undermining the argument from temporary intrinsics; and merely by relativizing in this innocuous way, we can resist the argument for the non-identity of Adam and Abel. For we can then say: while it is true that Abel’s career has the property having a length of one billion years, it is not

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true that Adam’s career has the incompatible property having a length of two billion years; rather, Adam’s career merely has the nonfundamental property having an ‘atomic personal time’ length of two billion years. And that is not enough to establish the non-identity of Adam and Abel, since there is clearly no incompatibility between that non-fundamental property and having a length of one billion years. To sum up, Eagle’s position is apparently this. If Adam’s career and Abel’s career plausibly had incompatible fundamental temporal lengths, then resisting the argument for their non-identity by appeal to some relativizing treatment of those lengths would undermine the argument from temporary intrinsics. But they do not plausibly have incompatible fundamental lengths. So the argument for their non-identity can be resisted without undermining the argument from temporary intrinsics.

5. reply One might wonder whether facts about fundamentality are relevant to the argument from temporary intrinsics in the way that Eagle apparently takes them to be. He suggests that what makes relativizing treatments of shapes so much worse than relativizing treatments of personal time lengths is that the former are so much more fundamental than the latter. This might be doubted. Is it any worse to hold that shapes are relations to times than to hold that apparently intrinsic aesthetic ‘properties’ such as beauty are relations to times?6 For the sake of argument, however, I will adopt Eagle’s suggestion: fundamentality matters. In a nutshell, my response to Eagle is this. It is quite plausible that the relativistic proper time length (not merely the Lewisian personal time length) of Adam’s career is two billion years, and likewise it is quite plausible that the relativistic proper time length of Abel’s career is one billion years. Since proper time lengths are the most fundamental temporal length properties, the fact that Adam’s career and Abel’s career plausibly have incompatible proper time lengths 6 Of course, if you think that beauty is highly fundamental, or if you deny that it is even prima facie monadic and intrinsic, then you should try to find a different example.

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is very significant, for it shows that if one resists the argument for the non-identity of Adam and Abel by adopting a relativizing treatment of the lengths with respect to which their careers appear to differ, then, by Eagle’s lights at least, one does undermine the argument from temporary intrinsics! 5.1. External Time Lengths v. Personal Time Lengths Lewis distinguishes between ‘‘time itself, external time as I shall call it’’ and ‘‘the personal time of a particular time-traveler,’’ and he sketches a reductive definition of the latter in terms of the former, together with certain notions concerning change and causation (1986b: 69). This distinction generates a more specific distinction between two families of temporal (or quasi-temporal) lengths: external time lengths and personal time lengths. Since these two families are directly relevant to my dispute with Eagle, I need to say something about each of them. Begin with external time lengths. In the context of a typical prerelativistic spacetime, there does not seem to be any question as to which properties count as the external time lengths. One such property is the one that would be expressed by the predicate in the following sentence, if spacetime in our world were Newtonian: (i) Bruce’s life has a length of 95 years. In relativistic spacetimes, however, there are several candidates for being the external time lengths: the proper time lengths, the inertial-frame-relative temporal lengths, and the so-called ‘cosmic time’ lengths. Proper Time Lengths Timelike curves and the careers of persisting objects have proper time lengths. Roughly, the proper time length of an object’s career is the property that would be measured by a clock that was carried along with the relevant object from the beginning of its career to the end. One famous feature of relativistic spacetimes is that different timelike curves or careers linking the same two points in such a spacetime will not in general have the same proper time lengths. (This gives rise to the ‘twins paradox’.) Crucially, the proper time

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length of a given curve or career is an invariant, not a frame-relative, matter; facts involving these properties are, so to speak, built into the metrical structure of the spacetimes in question. One proper time length property is expressed by the predicate in (ii) Bruce’s life has a proper time length of 95 years. Inertial-Frame-Relative Temporal Lengths The concept of an inertial reference frame is often invoked in presentations of special relativity, and associated with it is the concept of an inertial-frame-relative temporal length. Roughly put, where f is an inertial reference frame, the length-in-f of a given continuous timelike curve or career p is the temporal distance between the beginning of p and the end of p as measured by an observer at rest in f. In other words, it is the proper time length of a timelike curve p* at rest in f, where p* runs from the hyperplane of simultaneity-in-f that intersects the beginning of p to the hyperplane of simultaneity-in-f that intersects the end of p. Some inertial-frame-relative length properties are expressed by the predicates in the following sentences: (iii) Bruce’s life has a length of 95 years with respect to inertial frame f1 . (iv) Bruce’s life has a length of 50 years with respect to inertial frame f2 . ‘Cosmic Time’ Temporal Lengths Typical relativistic spacetimes admit of many different foliations, where a foliation is a partition of the spacetime into a set of non-overlapping ‘global time-slices’, or maximal spacelike hypersurfaces. But some spacetimes allowed by general relativity, while admitting of many foliations, have exactly one ‘preferred’ foliation that stands out from the rest by virtue of its geometrical properties. The rough idea is described by Michael Lockwood in the following passage: [The] fundamental observers . . . are observers whose state of motion coincides with average motion of matter in their own local regions of the universe, a region sufficiently large for the motion within it to be dominated by the recession of the local galaxies, in accordance with the overall expansion of the universe. It then follows that the local proper times of all these fundamental observers can be fused together to form a single coordinate time for the universe as a whole, known as cosmic time. (2005: 116)

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Associated with cosmic time will be a family of temporal length properties; call them the cosmic time lengths. On the assumption that cosmic time is definable in our spacetime, one such property is expressed by the predicate in (v) Bruce’s life has a length of 95 years with respect to cosmic time. Roughly, the cosmic time length of a given continuous timelike curve or career will be the temporal distance between its beginning and end, as measured by one of the so-called ‘fundamental observers’ that Lockwood mentions. The standard view among those who take relativistic spacetime seriously (a group that includes Eagle, I suspect) is that proper time lengths are more fundamental than frame-relative temporal lengths or cosmic time temporal lengths, on the grounds that the second and third families are defined in terms of the first. Relatedly, there are relativistic spacetimes (e.g., the Godel spacetime7 ) in which proper ¨ time lengths are instantiated but frame-relative and cosmic time temporal lengths are not (due to the absence of inertial reference frames and cosmic time); but there are no relativistic spacetimes in which the opposite holds: you can have proper time lengths without either of the other two families, but not vice versa. Insofar as external time has any claim to fundamentality, then, it seems to me that the best candidates (in a relativistic spacetime) for occupying the role of the external time lengths are the proper time lengths. Now we can turn from external time lengths to personal time lengths. Personal time, for Lewis, is non-fundamental: it is reducible to facts about external time, causation, and change. Here is Theodore Sider’s helpful gloss: Personal time is time experienced by the time-traveler, whereas external time is time simpliciter, time according to the public ordering of events. . . personal time, as construed by Lewis anyway, is not an additional fundamental physical element of the world, but is rather a defined quantity. Roughly, experiencing one minute of personal time is defined as undergoing the amount of change that would normally occur to a person during one minute of external time. (2001: 106)

It is worth adding that, at least in the case of ordinary objects such as persons, the change in question needs to be underpinned 7

For discussion, see (e.g.) Godel (1949), Earman (1995), and Lockwood (2005). ¨

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by causation. Lewis writes that ‘‘the properties of each stage depend causally on those of the stages just before in personal time, the dependence being such as tends to keep things the same’’ (1986b: 72). At first, some may be tempted to identify personal time with proper time and, more specifically, to identify personal time length properties with proper time length properties. After all, Lewis’s initial characterization of a time-traveler’s personal time is ‘‘roughly, that which is measured by his wristwatch’’ (1986b: 69), and very similar language is typically used in informal characterizations of relativistic proper time.8 Such an identification is clearly mistaken. First, proper time lengths are highly fundamental; they are the most fundamental temporal length properties. Personal time lengths are much less fundamental. Second, the possibility of Lewisian time travel would generate cases that block any proposed identification of a property from one of those families with a property from the other. Consider the property having a personal time length of 25 years. This property cannot be identified with having a proper time length of 25 years—not, anyway, if Lewisian time travel is possible. Given that possibility, a person’s personal time could fall out of step with the proper time elapsed along his path (and hence out of step with external time). This would happen if, e.g., a person’s behavior and all of his life-processes were ‘slowed down’ relative to his immediate surroundings, so that it took him (and his organs and cells, etc.) two hours to do what a normal person would do in one hour. Suppose that such a person is born in the year 2000, dies in the year 2050 (with the appearance of a normal 25-year-old), and never undergoes any unusual accelerations. Then his career has the property having a personal time length of 25 years, but it does not have the property having a proper time length of 25 years. (Its proper time length is roughly 50 years.) So the properties in question cannot be identified. Generalizing the argument in an obvious way, we can conclude that if Lewisian time-travel is possible, then no 8 For example, in their standard introductory textbook on special relativity, Edwin Taylor and John Archibald Wheeler write that ‘‘the length of a worldline between an initial and a final event is the elapsed time measured on a clock carried along the worldline between the two events. This is called the proper time, wristwatch time, or aging along this worldline’’ (1992: 162).

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personal time length property is the same as any proper time length property. 5.2. The Careers of Adam and Abel have Incompatible Fundamental Temporal Lengths I claimed that Abel’s career is one billion years long ‘in the fundamental way’, and Eagle agrees. What we disagree about is my claim that Adam’s career is two billion years long in that same ‘fundamental way’. By considering two series of cases, we should now be able to see that Eagle is right about Abel but wrong about Adam. We can begin with the Atom Series. Atom Case 1 A lone hydrogen atom drifts in deep interstellar space; its path has a beginning and an end, and the proper time elapsed along that path is two billion years. The spacetime the atom inhabits is relativistic but cosmic time is definable in it. The atom’s proper time never falls out of step with cosmic time. Atom Case 2 This is like the previous case, but the atom inhabits a Godel ¨ spacetime. As such, the spacetime contains no maximal spacelike hypersurfaces (it cannot be foliated into ‘global time-slices’) and, a fortiori, cosmic time cannot be defined in it. Nevertheless, the atom’s two-billion-year-long path has a beginning and an end. Intrinsically, the atom’s career is very much as it was in the previous case. Atom Case 3 This is like the previous case, but the atom does not merely ‘drift’. It accelerates in such a way that its path forms an almost closed timelike curve. The atom is created in a laboratory and kept there for three years before being sent on its journey. Near the end of the journey, the (quite old) atom returns to the laboratory, no worse for the wear, just as the physicists are creating its ‘younger self’. It spends the final three years of its career in the same room with its younger self, but the ‘two’ never chemically bond. Eventually the atom is destroyed.

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As in the previous cases, its path has a beginning and an end, and the proper time elapsed along that path is two billion years. Atom Case 4 This is like the previous case, but in this case the atom’s path forms a genuinely closed timelike curve, and the atom is never created or destroyed: its career has neither a beginning nor an end. Moreover, the atom never in any intuitive sense ‘coexists with another version of itself’: unlike in Atom Case 3, there are no smallish, locally spacelike regions that the atom’s path intersects in two different places. Rather, the path is a just a loop with a fairly simple shape. Again, the proper time elapsed along the path is two billion years. Atom Case 5 This is like the previous case, but the atom’s path forms a loop with a more complicated shape: it is doubled up like the edge of a mobius ¨ strip. But in this case (unlike the Adam–Abel Case), the atom is never chemically bonded to itself; instead, it is always at least 100 yards away from itself (as measured by it). Again, the proper time elapsed along its path is two billion years. Atom Case 6 This is like the previous case, but the atom is always chemically bonded to itself, forming a molecule of H2 , as in the Adam and Abel Case. But whereas the Adam and Abel Case occurs in a cylindrical spacetime, which can be carved up into global time-slices (and which perhaps even allows for cosmic time), Atom Case 6 occurs in a Godel spacetime. Again, the proper time elapsed along the atom’s ¨ path is two billion years. Adam Case 7 This case = the Adam and Abel Case. Atom Case 1 involves a hydrogen atom whose career has the property having a proper time length of two billion years, a property that is as fundamental as temporal length properties get. But for each pair of adjacent cases in the series, if the earlier case involves

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such an atom, then so does the later case. None of the differences between the adjacent cases are significant enough to block this ‘inductive step’. (I leave it to the reader to confirm this to his or her own satisfaction.) So the Adam and Abel Case contains such an atom—Adam, presumably. Adam’s career, then, has the property having a (relativistic) proper time length of 2 billion years. Since, as I noted earlier, this property is not the same property as having a (Lewisian) personal time length of 2 billion years, Eagle is just wrong when he says that ‘‘the only sense in which Adam has a two-billion-year-long career is in terms of its atomic personal time’’. Indeed, the proper time length property that I am attributing to Adam’s career belongs to the most fundamental family of temporal length properties. Eagle and I agree that Abel’s career is just one billion years long ‘in the fundamental way.’ But in case anyone is tempted to reject this claim, it can be supported by an argument parallel to the one just given. Consider the Molecule Series. Molecule Case 1 Two hydrogen atoms drift together in deep interstellar space. Each of them has its own continuous path with a beginning and an end, and the proper time elapsed along each of the two paths is one billion years. The atoms are chemically bonded to each other throughout their respective careers, thus forming a molecule of H2 . The spacetime they inhabit is relativistic, but cosmic time is definable in it. Neither of the atoms’ proper time ever falls out of step with cosmic time. Molecule Case 2 This is like the previous case, but the atoms inhabit a Godel ¨ spacetime. Again, the path of each atom has a beginning and an end and the proper time elapsed along it is one billion years. Molecule Case 3 This is like the previous case, but the molecule of H2 does not merely ‘drift’; it accelerates and follows an almost closed timelike curve, so that it manages to spend some time in the same room with its

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younger self before it (along with its constituent atoms) pops out of existence. But as before, each of the atoms has a path with a beginning and an end and a proper time length of one billion years. Molecule Case 4 This is like the previous case, but each of the two hydrogen atoms traces out its own closed timelike curve with a proper time length of one billion years. Neither atom is ever created or destroyed. Each of them has a career without a beginning or an end. Thus the molecule’s path consists of two ‘parallel’ or ‘side-by-side’ loops that represent the paths of the molecule’s constituent atoms. Molecule Case 5 This is like the previous case, but instead of having two atoms each tracing out a simple loop of one billion years in proper time length, we have just one atom tracing out an edge-of-a-mobius-strip-like ¨ loop of two billion years in proper time length. By being bonded to ‘another version of itself’ throughout its career, this atom forms a molecule of H2 . It would seem that the main difference between this molecule and the one described in the previous case is that this one performs one additional (or one fewer) 180-degree rotation over the course of its (apparently one-billion-year-long) career. This case, like the previous case but unlike the Adam and Abel Case, occurs in a Godel spacetime. ¨ Molecule Case 6 This case = the Adam and Abel Case. The argument now proceeds as before. Molecule Case 1 involves a molecule of H2 whose career has the property having a proper time length of one billion years,9 a property that is as fundamental as temporal length properties get. And for each pair of adjacent 9 One might deny this by appeal to the claim that, strictly speaking, the only entities that have (non-zero) proper time lengths are certain literally one-dimensional spacetime regions—namely, timelike curves. Of course, since Adam’s career is not strictly one-dimensional either (hydrogen atoms are not spatially pointlike), this would also disqualify Adam’s career from having a proper time length. (1) If the careers of spatially extended objects do not have proper time lengths, this would

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cases in the series, if the earlier case involves such a molecule, then so does the later case. Again, we don’t seem to have any differences between the cases that are significant enough to block this. So the Adam and Abel Case involves such a molecule—Abel, presumably. Just as Adam’s career has the property having a proper time length of two billion years, Abel’s career has the property having a proper time length of one billion years. These properties are as fundamental as any temporal length properties, and they are incompatible with each other. So, contrary to Eagle, Adam’s career and Abel’s career do quite plausibly have incompatible fundamental temporal lengths. By Eagle’s lights, then, we cannot resist the argument for the nonidentity of Adam and Abel without undermining the argument from temporary intrinsics.

6. conclusion I argued that if one identifies Adam and Abel (as a way of solving the type-C puzzle), then one must adopt a relativizing treatment of temporal lengths, and that, at least initially, this seems no better than adopting a relativizing treatment of shapes (as a way of resisting the argument from temporary intrinsics). Eagle tried to show that there is a significant disanalogy between the two cases: for the shapes that would need to be relativized are fundamental, whereas the only temporal lengths that would need to be relativized (personal time lengths) are highly non-fundamental. leave it a mystery as to why (e.g.) synchronized spatially extended clocks fall out of step when one of them accelerates rapidly back and forth while the other one drifts inertially. True, it may be harder to assign a precise, determinate proper time length to a spatially extended career than to a one-dimensional timelike curve, but this hardly shows that the former entities do not have proper time lengths at all or that such entities never determinately differ with respect to their proper time lengths. On the contrary, sometimes they clearly do so differ, and indeed, if the arguments given in this section show anything, they show that Adam and Abel plausibly differ in just this way. (2) Moreover, even if one is willing to concede (which I am not) that only timelike curves have the most fundamental temporal length properties, there is still plenty of room to argue that there is a another family of temporal length properties, the career-proper-time lengths, that can be possessed by spatially extended careers and that are sufficiently fundamental and sufficiently unlike mere personal time lengths as to put us under a fairly strong prima facie duty not to give a relativizing treatment of them.

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Perhaps there really is a significant disanalogy between the two cases lurking out there somewhere. I have not shown that there is not one. What I do take myself to have shown is that Eagle’s attempt to find such a disanalogy does not succeed. (Personal time lengths are not the only temporal lengths that would need to be relativized.) So far as his arguments are concerned, then, MC still holds.10 University of California, Davis

references Eagle, A. ‘Location and Perdurance’, in Oxford Studies in Metaphysics, this volume. Earman, J. 1995. Bangs, Crunches, Whimpers, and Shrieks. (Oxford: Oxford University Press). Godel. K. 1949. ‘A Remark on the Relationship between Relativity The¨ ory and Idealistic Philosophy’, in P. A. Schilpp, ed. Albert Einstein: Philosopher-Scientist (La Salle, IL: Open Court), pp. 557–62 Gilmore, C. 2007. ‘Time Travel, Coinciding Objects, and Persistence’, in Oxford Studies in Metaphysics vol. 3, pp. 175–98. Haslanger, S. 2003. ‘Persistence through Time’, in M. Loux and D. Zimmerman (eds.), The Oxford Handbook of Metaphysics (Oxford: Oxford University Press), pp. 315–54. Lewis, D. 1986. ‘‘The Paradoxes of Time Travel’’, in David Lewis, Philosophical Papers, Vol. 2, (Cambridge: Cambridge University Press), pp. 67–80. Lewis. D. 1986. On the Plurality of Worlds. (Oxford: Blackwell). Lockwood, M. 2005. The Labyrinth of Time (Oxford: Oxford University Press). Sider, T. 2001. Four Dimensionalism: An Ontology of Persistence and Time (Oxford: Oxford University Press). Taylor, E. and J. A. Wheeler. 1992. Spacetime Physics, 2nd edn. (New York: Freeman). 10 Thanks to Ben Caplan, Brad Morris, and Martin Thomson-Jones for helpful discussion.

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6. Duration in Relativistic Spacetime Antony Eagle In ‘Location and Perdurance’ (in this volume), I argue that there are no compelling mereological or sortal grounds requiring the perdurantist to distinguish the molecule Abel from the atom Abel in Gilmore’s original case (2007). The remaining issue Gilmore originally raised concerned the ‘mass history’ of Adam and Abel, the distribution of ‘their’ mass over spacetime. My response to this issue was to admit that mass histories needed to be relativized to a way of partitioning the location of Adam/Abel, but that did not amount to relativizing any fundamental natural intrinsic properties—the latter are all had unrelativized, and (so most perdurantists would say) the distribution of instances of these properties suffice to fix all the facts about Adam and Abel. No threat to perdurantism, or the argument from temporary intrinsics, comes from this direction. My response was too hasty, as it relied naively on the thought that proper time in the Adam/Abel case could be explained away in terms of external time. In his reply (in this volume), Gilmore points out, entirely correctly, that this cannot be done. Proper time along a trajectory is the only unrelativized (reference frame invariant) measure of duration. My original discussion at the end of section 5 was at best misleading. However, my basic mistake was to think that the case of Adam/Abel required a different treatment than the case of Cell/ Tubman. But in fact the two cases can be treated in a completely parallel way. In the Cell/Tubman case, there is no preferred duration that it ‘really’ has. There is, unrelativizedly, the volume of spacetime it occupies. There are two distinctive patterns of causal relations instantiated through that volume—patterns that, were they instantiated by intrinsic duplicate parts differently distributed through spacetime, would in one case constitute a three-year-long person, and in the other a 3n-year-long cell (where n is the number of cells in any timeslice of Tubman). Of course, given the existence of an

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external time in the classical spacetime I used in the discussion of Cell/Tubman, there is a strong temptation to say that the volume of spacetime occupied by Cell/Tubman has an unrelativized duration of three years, but I see no real significance as to whether perdurantists succumb to this temptation. In the Adam/Abel case, for some reason, I succumbed to the analogous temptation, and identified the duration of Adam/Abel with the apparent duration of Abel. This was an error, which I now retract—there is no requirement to pick a preferred duration in the classical case, and even less reason to do so in a relativistic spacetime. But the account then is just as for Cell/Tubman. Adam/Abel occupies a single volume (collection of points) in spacetime. The perfectly natural intrinsic properties instantiated at these points fix the intrinsic character of Adam/Abel. These properties suffice to establish that Adam/Abel is a molecule and that it is an atom—we now see, thanks to Gilmore’s cases, that these are not incompatible properties, and that a persisting object can have both, in virtue of the causal relations that its parts stand in. The particular way that Adam/Abel’s parts are distributed over spacetime causes some confusion when it comes to evaluating temporal duration. As mentioned above, the proper time is duration along a trajectory. What a trajectory is may be disputed; I will assume with Gilmore that they are not simply regions of spacetime, but rather trace the history of a persisting kind of object, an atom or molecule in this case, by tracing the causal connections among its stages.1 Because Adam/Abel satisfies both the properties of being a molecule and an atom, there are two ways of tracing continuous identity-constituting causal chains (trajectories) through the volume of space it occupies—the atomic way, giving a sequence of those parts of Adam/Abel which are atom-parts linked by the sameatom-as relation, and the molecular way, giving a sequence of molecular-parts linked by the same-molecule-as relation.2 Thus 1 Compare, e.g., ‘The world-line [trajectory] provides us with the information of where the particle was (and is, and will be) at every possible time . . . ’ (Geroch, 1978: 18). 2 One complication Gilmore mentions is that, strictly speaking, only onedimensional trajectories will have proper times associated with them, so the molecular trajectory will be a ‘center of mass’ trajectory, and may not fall entirely within the region filled by Adam/Abel. I will ignore this complication.

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there will be two proper time durations assignable to the volume of spacetime occupied by Adam/Abel, 1 billion years and 2 billion years. It is again true that intrinsic duplicates of Adam/Abel’s parts might be distributed through spacetime in such a way as to make, e.g., a molecule that is not an atom, and which has a 1-billionyear-long proper time along its molecular trajectory, and no other candidate proper time. In that case we would be justified in saying that the molecule had a duration of 1 billion years, and that may tempt us erroneously into saying that Abel has that duration. We can and should resist that temptation. This is, as Gilmore points out, a ‘relativizing’ response to this problem. But as before, no fundamental natural property is relativized. The situation thus remains quite different to what must happen in the case of endurantist intrinsic change, which does involve relativizing such properties. Since perdurantists can treat all facts in relativistic spacetimes—including all facts about the location and identity of persisting objects—as supervening on the distribution of perfectly natural intrinsic properties over spacetime, no fact in the supervenience base is relativized. (That is why fundamentality matters.) I see no weakening of the argument from temporary intrinsics in making this move. One potential concern might come from the thought that because proper times are Lorentz invariant, they are fundamental, and so not the kind of thing that a (non-hypocritical) perdurantist should relativize. But this thought is mistaken. Consider the purely existential claim that there exists some region of spacetime that is occupied. The truth-value of this claim is Lorentz invariant, yet the facts of unspecific occupation are not fundamental, depending as they do straightforwardly on the occupation of specific particular regions. Fundamental properties are invariant, but not vice versa. Finally, one might consider an analogous case in purely spatial terms, without the confounding persistence issues, invented by Oliver Pooley. Consider a physical 3-torus made of three-stranded rope. It turns out that the three ‘strands’ are in fact a single loop of cord wrapped around itself (see in Figure 6.1).3 So while the torus 3 In the figure, the single cord is black at one ‘end’ and white at the other. Incidentally, this variation in color poses a problem of spatial intrinsics for those who think a Pooley torus might be an extended simple, which must be solved, unlike the problem of length below, by relativizing intrinsic properties at points.

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Figure 6.1. A Pooley torus, with the rope threads shown separated (the (11, 3)-torus knot).

in cross-section has three strands, and a single cut would yield three small pieces of cord, the torus is actually composed only of one long twisted cord.4 This ‘Pooley torus’ occupies a given volume of 3-space, unrelativizedly. All the properties of the Pooley torus supervene on the distribution of properties throughout that spatial region. On this both endurantists and perdurantists agree. But what is the length of the Pooley torus? There seem two natural candidates: 1. The length of the rope, i.e., the diameter of the torus. A cut of the rope would yield a piece this long, so the Pooley torus, measured rope-wise, is this long. 2. The length of the cord, something somewhat larger than three times the diameter of the torus. A single cut of one strand 4 Mathematically speaking, this is therefore a torus knot (or a closed braid), a knot on the surface of a 3-torus obtained by wrapping a cord through the hole of a 3-torus p times, and looping the cord q times around the torus before joining it up, where p and q are relatively prime; this gives the (p, q) torus knot. Such a torus knot is hard to physically make, but it is clear which points of space it would fill, so simply consider the Pooley torus to be a filled region consisting of just those points.

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would yield a cord this long; so the Pooley torus, measured cord-wise, is this long. One conclusion might be that the cord is not the rope on the basis of these different length properties; anti-coincidence prohibits this conclusion for both endurantists and perdurantists. To preserve our judgments, we should accept that the cord is the rope, and relativize length to sortals, so we have rope-length and cord-length. This relativizing move seems natural and plausible in the spatial case; all I have argued is that it can be consistently and coherently extended to the temporal case by perdurantists, without undermining other perdurantist arguments.5 Exeter College, University of Oxford

references Eagle, Antony, ‘Location and Perdurance’, this volume. Geroch, Robert (1978), General Relativity from A to B (Chicago: University of Chicago Press). Gilmore, Cody (2007), ‘Time Travel, Coinciding Objects, and Persistence’. Oxford Studies in Metaphysics (Oxford: Oxford University Press), 3, 177–98. Gilmore, Cody, ‘Coinciding Objects and Duration Properties: Reply to Eagle’. Oxford Studies in Metaphysics, this volume. 5

Thanks to Oliver Pooley for discussion.

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7. Strange Kinds, Familiar Kinds, and the Charge of Arbitrariness∗ Daniel Z. Korman 1. prelude A snowdiscall is something made of snow that has any shape between being round and being disc-shaped and which has the following strange persistence conditions: it can survive taking on all and only shapes in that range. So a round snowdiscall can survive being flattened into a disc but cannot survive being packed into the shape of a brick. Ernest Sosa observes that one can avoid commitment to snowdiscalls, and a plenitude of other strange kinds, by embracing either some form of eliminativism on which there are neither snowballs nor snowdiscalls or else some form of relativism on which material objects do not exist simpliciter but only relative to some conceptual scheme or other.1 Curiously, the natural view that there are no snowdiscalls, that there are snowballs, and that snowballs exist simpliciter is not among the options that Sosa considers.

2. particularism Particularism about a given domain of inquiry is the view that our intuitive judgments about cases in the domain are largely correct and that, when intuitive judgments about cases conflict with compelling general principles, the cases should in general ∗

I am grateful to Derek Ball, John Bengson, Reid Blackman, Josh Dever, Kenny Easwaran, Adam Elga, John Hawthorne, Eli Hirsch, Cory Juhl, Shieva Kleinschmidt, Dave Liebesman, Dan Lopez de Sa, Marc Moffett, Bryan Pickel, Raul Saucedo, Peter ´ Simons, Ernest Sosa, Jason Turner, Chris Tillman, Michael Tye, audiences in Laramie and Urbana, and especially to George Bealer, Chad Carmichael, and Trenton Merricks for valuable discussion. 1

See his (1987, 178–9), (1993, 620–2), or (1999, 133–4).

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be treated as counterexamples to those principles.2 The distinction between particularists and nonparticularists cuts little ice in most domains. For instance, apart from skeptics, virtually all parties to the debates about empirical knowledge and justification are particularists—reliabilists and evidentialists, foundationalists and coherentists, internalists and externalists, contextualists and invariantists—never straying far from the bulk of our intuitive judgments about cases, even if they cannot accommodate all of them. The distinction does, however, cut ice in material-object metaphysics, in which many of the dominant views flout wide swathes of our intuitive judgments about cases. Here I especially have in mind views on which there are far more or far fewer things than we intuitively judge there to be: universalist views on which composition is unrestricted at a time or even across time, plenitudinous views on which familiar objects exactly coincide with countless other objects with slightly or wildly different modal profiles, and eliminativist views on which virtually none of the things that we intuitively judge to exist in fact exist.3 This constitutes a dramatic departure from standard philosophical methodology. What accounts for this departure? One possible explanation is that metaphysicians have become convinced—for instance, by familiar strategies for reconciling revisionary ontologies with ordinary discourse—that the relevant intuitive judgments are based on intuitions whose contents do not support those judgments and do not entail the falsity of their revisionary ontological theses.4 But these strategies have little prima facie plausibility, and it is difficult to believe that anyone who was not antecedently convinced that 2 By ‘intuitive judgments’, I mean the judgments that one is inclined to make on the basis of one’s intuitions together with either perceived details of actual cases or stipulated details of counterfactual cases when theoretical qualms are set to the side. See Bealer (2004, 14–15) on the primacy of judgments about cases. 3 See, among a great many others, Cartwright 1975, Unger 1979, Lewis 1986, van Cleve 1986, Yablo 1987, Heller 1990, van Inwagen 1990, Rea 1998, Sosa 1999, Horgan and Potr 2000, Hudson 2001, Merricks 2001, Sider 2001, Rosen and Dorr 2002, Hawthorne 2006, and Thomasson 2007. 4 Here I have in mind, e.g., the contention that the true content of apparently anti-eliminativist intuitions is only that there are mereological simples arranged thusand-so (`a la van Inwagen 1990) or that the contents of apparently anti-universalist or anti-plenitude intuitions are suitably—and perhaps inscrutably—restricted in such a way as to exclude strange kinds (`a la Lewis 1986).

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the intuitive judgments were mistaken would be moved by the suggestion that these intuitions are being misreported.5 The explanation is rather that they have been convinced by some argument against particularism in material-object metaphysics. These arguments fall into two broad categories: rebutting arguments and undercutting arguments. Rebutting arguments are arguments for conclusions that directly contradict some specific range of intuitive judgments about cases, the most prominent being the argument from vagueness, causal exclusion arguments, and arguments from the impossibility of distinct coincident items. Undercutting arguments are arguments for the conclusion that our intuitive judgments about cases are (probably) unreliable, but that do not purport to demonstrate the falsity of any specific range of intuitive judgments. Although the rebutting arguments are by far the more widely discussed of the two, it is difficult to believe that these arguments are primarily responsible for the widespread aversion to particularism in material-object metaphysics. After all, every philosophical domain has its share of powerful rebutting arguments, yet it is only in material-object metaphysics that such arguments do not typically inspire a Moorean confidence that at least one of the principles that drives the argument must be false. I suspect that it is rather the undercutting arguments that lie at the root of the aversion to particularism in material-object metaphysics. In what follows, I address one sort of undercutting argument, which turns on the claim that the particularist’s differential treatment of strange and familiar kinds is intolerably arbitrary. The literature is now replete with examples of such kinds—apceans, bligers, bonangles, carples, cdogs, cpeople, cupcups, dwods, gollyswoggles, incars, klables, monewments, shmees, shmrees, trables, troutturkeys, wakers—and the charge of arbitrariness has been leveled (in one form or another) by numerous authors.6 But, despite how influential the charge has been, there has been virtually no discussion of how particularists might respond to the charge. 5 See Merricks (2001, 162–70), Hirsch (2002a, 109–12), and my (2008 and forthcoming) for critical discussion of these reconciliatory strategies. 6 See, e.g., Cartwright (1975, 158), Quine (1981, 13), van Cleve (1986, 145), Yablo (1987, 307), van Inwagen (1990, 126), Hirsch (1993, 690), Hudson (2001, 108–11), Sider (2001, 156–7 and 165), Sidelle (2002, 119–20), Hawthorne (2006, 109), Johnston (2006, 696–8), and Schaffer (forthcoming, §2.1).

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There are at least two other sorts of undercutting arguments worth mentioning (setting aside those that generalize to intuitions in all domains). First, there are a variety of arguments having to do with the subject matter of material-object metaphysics; for instance, that questions in material-object metaphysics concern substantive facts about the world and therefore cannot be settled by (anything like) conceptual analysis.7 Second, there are a variety of arguments having to do with the apparent impossibility of subsuming our intuitive judgments about cases under interesting general principles.8 My ambitions in this chapter are modest in one respect, immodest in another. The modesty lies in its scope. I do not argue for particularism or against revisionary ontologies. I argue only that particularists have the resources to resist the argument from arbitrariness, and I have done my best to disentangle this argument from the others. The immodesty lies in the background metaontology. I will show that the argument from arbitrariness can be resisted without retreating to any sort of deflationary view of ontology. Particularists need not embrace any form of relativism about ordinary material objects, nor need they accept the deflationary doctrine of quantifier variance according to which there are counterparts of our quantifiers that are on a par with ours and that range over things that do not exist (but rather, e.g., shmexist).9 Let us then distinguish between deflationary particularists, who couple their particularist ontology with a deflationary metaontology, and robust particularists, who opt for a nondeflationary metaontology. I suspect that the argument from arbitrariness owes at least some of its influence to the presumption (implicit in Sosa’s trilemma) that robust particularism is a nonstarter and that the only viable alternative to a revisionary ontology is some form of deflationary particularism—or, in the words of John Hawthorne, ‘‘a kind of anti-realism that none of us should tolerate.’’10 For ease of exposition, I sometimes refer to ‘‘what particularists will say’’ about a given case. But no less than in other domains, particularism in material-object metaphysics is a matter of degree 7 See Hirsch (2002a, 107) for a statement of one version of this argument, and see Rodriguez-Pereyra (2002, 217) on appeals to intuition in metaphysics generally. 8 See, e.g., van Inwagen (1990, 66−8), Horgan (1993, 695), and Hudson (2001, 109). 9 10 See Hirsch (2002b). Hawthorne (2006, 109).

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and can come in endless varieties. Some particularists in epistemology and philosophy of language are willing to bite bullets in at least some cases (fake barn cases, seemingly informative identities, etc.), and particularists in material-object metaphysics may do the same. There also is endless room for disagreement among particularists about which kinds of things there are, about the persistence conditions of various familiar kinds, about whether it is at least possible for various strange kinds to have instances, about what it would take for various strange kinds to have instances, and so forth. Although particularism possibly deserves the label ‘folk ontology’ or ‘common-sense ontology’, I hesitate to use these labels for two reasons. First, particularists can be expected to reject highly intuitive general principles (e.g., about material coincidence) which the folk will assent to and which seem equally deserving of the label ‘common sense’. Second, the label may be misleading in the following respect. I wish to understand ‘intuitive judgments about cases’ not in terms of how the folk respond to philosophical interrogation or surveys, but rather in terms of how things seem to philosophers, who are alert to relevant distinctions and who know the difference between reporting their intuitions and reporting their considered judgments. However important the folk’s intuitions may or may not be, they are too likely to misreport or misrepresent their intuitions for their responses to be of much use to philosophers.11

3. the argument from arbitrariness The argument from arbitrariness turns on the claim that there is no difference between certain of the familiar kinds that we intuitively judge to exist and certain of the strange kinds that we intuitively judge not to exist that could account for the former’s but not the latter’s having instances. In short, there is no ontologically significant difference between the relevant strange and familiar kinds. Arguments from arbitrariness will have the following form: 11 Cf. Williamson (2007, 191): ‘‘Although the philosophically innocent may be free of various forms of theoretical bias, just as the scientifically innocent are, that is not enough to confer special authority on innocent judgment, given its characteristic sloppiness.’’

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There is no ontologically significant difference between Ks and K s.

(P2)

If there is no ontologically significant difference between Ks and K s, then it is objectionably arbitrary to countenance things of kind K but not things of kind K .

(C)

So it is objectionably arbitrary to countenance things of kind K but not things of kind K .

Deflationary particularists will typically deny P2. Countenancing familiar kinds but not strange kinds is objectionably arbitrary only if one thereby privileges the familiar kinds. But deflationary particularists will deny that existent kinds enjoy a privileged status. According to relativists, snowballs exist and snowdiscalls do not exist—relative to our scheme, that is—but, relative to other schemes, snowdiscalls exist and snowballs do not exist.12 According to quantifier variantists, snowballs exist but do not shmexist, and snowdiscalls shmexist but do not exist. So at bottom there is a uniform treatment of strange and familiar kinds. This sort of strategy is available only to deflationary particularists and, as indicated above, my goal is to show how robust particularists can resist the charge of arbitrariness. I know of no way for robust particularists to address all instances of the argument en masse; we will have to take them case by case. Before turning to the cases, however, let me make three preliminary remarks. First, in what follows I will identify what seem to be ontologically significant differences between various strange and familiar kinds without taking the further step of attempting to establish that the differences are indeed ontologically significant. I do not consider this a shortcoming of my response to the argument from arbitrariness. Consider this analogy. In explaining why a certain justified true belief counts as knowledge in one case but not in another, one might appeal to some feature F (e.g., having a defeater) that is present in the one case and absent in the other. There is an interesting question—which may or may not have an answer—of why F is epistemically significant, but it would be a mistake to insist that 12 Then again, relativists may be better understood as denying P1: the difference between snowballs and snowdiscalls that explains why the former but not the latter exist—relative to our scheme, that is—is that the concept snowball is part of our conceptual scheme and the concept snowdiscall is not.

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answering this question is a prerequisite to explaining why there is knowledge in one case but not the other. Furthermore, having an account of F’s epistemic significance is not required for having a reason to believe that F is epistemically significant: the reason would simply be that the presence of F seems to be making a difference both in the case at hand and across a wide range of cases. Analogously, there can be good reason to accept that a certain feature marks an ontologically significant difference between two kinds—that it explains why one kind has instances while another does not—even in the absence of an account of what makes that feature ontologically significant.13 I will therefore take myself to have defended the particularist against the charge of arbitrariness if I can achieve the more modest goal of identifying differences between strange and familiar kinds that do at least seem ontologically significant and that do not simply amount to the former’s being unfamiliar, or uninteresting, or intuitively nonexistent, or failing to fall under any of our sortals. Second, uncovering the metaphysics of familiar kinds is often quite complicated, and I suspect that part of the force of the charge of arbitrariness comes, illegitimately, from the intricacy of these issues and an impatience for long digressions into the metaphysics of snowballs, statues, solar systems, and so forth. If one finds one’s intuitions about familiar kinds unmanageable at times, one should bear in mind that this may be because metaphysics is difficult, not because the questions or our intuitions are somehow defective.14 Third, although I follow anti-particularists in characterizing these as cases of arbitrariness, this characterization is highly tendentious. Arbitrary judgments are those based on random choice or personal whim. The characterization is apt in other familiar charges of 13 Of course, the mere appearance of ontological significance will not be enough to convince some committed anti-particularists, but in that case their aversion to particularism presumably does not rest primarily on the absence of plausible candidates for ontologically significant differences and, therefore, lies beyond the scope of this chapter. 14 Moreover, difficulty in specifying a relevant distinction between two cases is not obviously evidence that there is no relevant distinction between them. Cf. Sider: ‘‘There are, one must admit, analogies between these cases [of genuine causes and epiphenomena], and it is no trivial philosophical enterprise to say exactly what distinguishes them. But setbacks or even failure at this task in philosophical analysis should not persuade us that there is no distinction to be made, since failure at philosophical analysis should never persuade anyone, on its own anyway, that there is no distinction to be made’’ (2003, 772). The italics are his.

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arbitrariness in metaphysics, for instance, that it is arbitrary to identify the number two with {{Ø}} rather than {Ø,{Ø}}.15 Here, not only can we find no difference between the two sets that could account for the one but not the other’s being the number two, there also is not even prima facie reason to believe that the one but not the other is the number two. By contrast, we do have at least prima facie reason for taking there to be snowballs but no snowdiscalls, for this view has strong intuitive support. Our reasons for the differential treatment of snowballs and snowdiscalls are therefore no different in kind from our reasons for the differential treatment of Gettier cases and paradigm cases of knowledge, and are hardly a matter of whim or random choice.16

4. toddlers and toddlescents A toddlescent is a material object that comes into existence whenever a child reaches the age of two, ceases to exist when the child reaches the age of fourteen, and is exactly co-located with the child at all times in between. Particularists will deny that there are toddlescents. Yet particularists will accept that there are toddlers. Is this differential treatment of toddlers and toddlescents arbitrary? Not at all, for there is an important difference between toddlers and toddlescents. Unlike toddlers, toddlescents would have to be things that cease to exist without any of their constitutive matter undergoing any intrinsic change. Toddlers, by contrast, do not cease to exist when they grow up; they merely cease to be toddlers. A separate question is whether there are toddlescent*s, where a toddlescent* is a child between the ages of two and fourteen. Particularists will of course agree that there are toddlescent*s. Some of them are toddlers, others are adolescents—things that we intuitively judge to exist. Toddlescents, however, cannot be either of these things on account of their strange persistence conditions.17 15 See Benacerraf (1965), as well as Armstrong (1986, 87) on ordered pairs, Bealer (1998, 6–7) on propositions, and Merricks (2003, 532–6) on counterpart theory. 16 Furthermore, we plausibly have reason to treat Gettier cases and paradigm cases of knowledge differently even before we manage to pin down the epistemically significant difference between the cases. 17 Here and elsewhere, I assume that individuals that can survive a given change cannot be identified with individuals that cannot survive that change.

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One must therefore take care to distinguish between strange phased kinds, like toddlescent*s, and strange individuative kinds, like toddlescents and snowdiscalls, when consulting one’s intuitions about strange kinds.18 Phased kinds are kinds whose instances can cease to belong to that kind without ceasing to exist. Individuative kinds are kinds whose instances cannot cease to belong without ceasing to exist: they are of that kind as a matter of de re necessity. Things belonging to strange phased kinds are often perfectly familiar things with perfectly ordinary persistence conditions; it is the things belonging to strange individuative kinds to which particularists take exception. I have found that those who cannot even see the pretheoretical reason for refusing to countenance the strange kinds discussed in the literature are often conflating phased and individuative kinds.

5. islands and incars A full-sized incar is like a car in nearly all respects. The main difference is that, unlike a car, it is metaphysically impossible for an incar to leave a garage. As the incar inches toward the great outdoors, it begins to shrink at the threshold of the garage, at which time an outcar springs into existence and begins growing. What it looks like for an incar to shrink and gradually be replaced by an outcar is exactly the same as what it looks like for a car to leave a garage. But an incar is not a car (or even a part of a car) that is inside a garage, for a (part of a) car that is inside a garage can later be outside the garage. Hawthorne maintains that ‘‘none but the most insular metaphysician should countenance islands while repudiating incars.’’19 The suggestion, I take it, is that there is no ontologically significant difference between islands and incars. Hawthorne evidently believes that islands shrink and ultimately cease to exist as their constitutive matter comes to be fully submerged, just as incars shrink and ultimately cease to exist as their constitutive matter leaves the garage. Particularists should reject this characterization of islands. Suppose that an island is entirely submerged every day at high tide. 18

Cf. Wiggins (2001, 29–33).

19

Hawthorne (2006, vii).

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Intuitively, it is still there at high tide—under the water—and when it re-emerges at low tide it has not suffered interrupted existence.20 Incars, by contrast, cease to exist when their constitutive matter leaves the garage, and without any of their constitutive matter undergoing any intrinsic change. This would seem to be an ontologically significant difference between islands and incars.21 It may be that those who were initially moved by Hawthorne’s objection were confusing incars with incar*s, where an incar* is a car that is inside a garage. There is no ontologically significant difference between islands and incar*s, but particularists will not deny that there are incar*s. Alternatively, it may be that they were confusing the question of whether the island ceases to exist when entirely submerged with the separate (and less pressing) question of whether an island ceases to be an island when it is entirely submerged. Some will be inclined to say that nothing that is entirely submerged, even momentarily, is at that time an island; others will say that islands continue to be islands when entirely submerged. I am inclined to say that an island ceases to be an island only when permanently submerged, or else when the waters recede and it comes to be part of a peninsula. Nothing hangs on this question of classification. For however one answers it, one can agree that all islands have perfectly ordinary persistence conditions and, in particular, that they do not cease to exist when submerged.

6. pages and monewments A monewment is like a monument insofar as it is a material object that has the function of commemorating a certain person or event. 20

Nor, for that matter, do islands shrink when the water levels rise. Islands are like icebergs: part of the island is above water and the rest of the island is underwater. (Submarines sometimes crash into islands.) And, like icebergs, they shrink by eroding. To the extent that we are ever inclined to say that the island is shrinking when the water levels rise, it seems plausible on reflection that all that is shrinking is the part of the island that is above water, not the island itself. And even this is evidently a fac¸on de parler. Nothing really becomes smaller when the part of the island that is above water gets smaller any more than something really becomes longer as the part of this sentence that you have read thus far gets longer. 21 I am grateful to Chad Carmichael here. E. J. Lowe raises similar points about islands in his (2007).

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But monewments have more permissive persistence conditions than monuments: if the constitutive matter of the monewment is annihilated, and a qualitatively identical material object is erected at the location of the original monewment, that material object is numerically identical to the original monewment. Particularists will deny that there are such things as monewments. Carl Ginet contends that we already countenance material objects of just this sort; for instance, pages of a typescript: Suppose that this typescript’s 18th page were now constituted of wholly different matter from that which constituted it yesterday, because I spilled coffee over it and had to retype it. The 18th page of this typescript (this page, I might say, holding it up) ceased to exist altogether for a while but now it exists again in a new embodiment.22

The suggestion is that pages, like monewments, are material objects that can survive undergoing a complete change of matter in a nonpiecemeal fashion. If so, then it may seem that there is no ontologically significant difference between the two. Particularists should deny that there is a single material object answering to ‘the 18th page’ that once had coffee spilled on it and is now in Ginet’s hand. Obviously, the mere fact that ‘the 18th page’ once referred to the coffee-stained page and now refers to the page in Ginet’s hand does not suffice to show that there is a single material object that was the 18th page at both times, any more than the fact that ‘the president’ once referred to Clinton and now refers to Obama suffices to show that there is a single individual who was the president at both times. There presumably is a type answering to ‘the 18th page’ which, once the typescript goes to press, will have multiple tokens; and perhaps this is a thing that ceases to exist and comes back into existence in the case that Ginet describes.23 But this is an abstract object, not a material object. There is a sheet of paper in Ginet’s hand, but that sheet never had coffee spilled on it; when the coffee was spilled, that sheet was across the room on top of a stack of other blank sheets. And even those who take the page to be a material object that is distinct from the sheet will insist 22

Ginet (1985, 220–1). Though, far more plausibly, this abstractum does not cease to exist when the original copy of the 18th page is destroyed, any more than there ceases to be an 18th letter of the alphabet when all of the tokens of that letter are destroyed. 23

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that the page in Ginet’s hand is something that began to exist—not something that came back into existence—when the words were retyped on the new sheet.24 So, unlike a monewment, neither the 18th page nor any of its tokens is a material object that can survive a complete and nonpiecemeal replacement of its constitutive matter. This would seem to be an ontologically significant difference between monewments and manuscript pages.

7. statues and gollyswoggles You have absent-mindedly kneaded a piece of clay into an unusual shape. Let us say that anything with exactly that shape is gollyswoggle-shaped. Something is a gollyswoggle just in case it is essentially gollyswoggle-shaped. Particularists will agree that there are statues but deny that there are gollyswoggles: some things are gollyswoggle-shaped, but nothing is essentially gollyswoggleshaped. Van Inwagen find this unacceptable: ‘‘I should think that if our sculptor brought a statue into existence, then you brought a gollyswoggle into existence.’’25 Van Inwagen evidently thinks that there is no ontologically significant difference between statues and gollyswoggles, including the presence of creative intentions in the one case and their absence in the other: ‘‘our sculptor intended to produce something statue-shaped while you, presumably, did not intend to produce anything gollyswoggle-shaped. But these facts would seem to be irrelevant to any questions about the existence of the thing produced.’’26 Yet our intuitive judgments about cases suggest that creative intentions are indeed relevant to what kinds of things there are. Suppose that a meteoroid, as a result of random collisions with space junk, temporarily comes to be a qualitative duplicate of some actual statue. Intuitively, nothing new comes into existence which, unlike the meteoroid, cannot survive further collisions that deprive 24 This is perhaps easiest to see if one imagines that this page is only one of several back-up copies of the 18th page that were produced after the spill. 25 Van Inwagen (1990, 126). 26 Ibid.

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the meteoroid of its statuesque form.27 Likewise, unintentionally and momentarily kneading some clay into the shape of a gollyswoggle does not suffice for the creation of something that has that shape essentially. When a piece of clay comes to be, and moments later ceases to be, gollyswoggle-shaped, this does not involve the generation of new objects, any more than a two-year-old’s becoming a three-year-old involves the generation of any new object. The particularist should therefore contend that the fact that many have set out to make statues, while no one has ever set out to make a gollyswoggle, is an ontologically significant difference between statues and gollyswoggles, in which case the differential treatment is not arbitrary.28 Does this view of artifacts as mind-dependent constitute a departure from the full-blooded ontological realism promised at the outset? Perhaps. But it is important to appreciate just how benign the needed degree of mind-dependence is. The artifacts cannot have begun to exist without us but, once created, they do not depend on us for their continued existence. Moreover, once created, their modal features remain entirely independent of how we later come to use them or conceive of them. This opens the door for community-wide error, for instance, of unearthing ancient cooking utensils and mistaking them for religious relics, or finding the statue-shaped meteoroid and mistakenly taking it to be an artifact and to have its form essentially. This is about as realist as one can get about artifacts.29 27 Cf. Baker (2008, 211). I leave it open whether this meteoroid is a statue. If so, then it turns out that, while most statues are essentially statues, others are only contingently statues, are identical with pieces of stuff, and share the persistence conditions of the piece of stuff. What matters for our purposes is that nothing with the persistence conditions normally associated with statues (and thus distinct from the meteoroid) comes into existence in the absence of the relevant creative intentions. Thanks to Reid Blackman and Josh Dever for helpful discussion here. 28 Particularists may hold that gollyswoggles’ inability to survive even minimal changes in shape is yet another an ontologically significant difference between statues and gollyswoggles. 29 Cf. Thomasson (2003). Some may object even to this minimal degree of minddependence and insist that the existence of a certain sort of object is always independent of human intentions and desires. See, e.g., van Cleve (1986, 149), Rea (1998, 353–4), Olson (2001, 347), Sider (2001, 157), and, for a dissenting voice, Baker (2008, 46–7). It is precisely their willingness to reject such intuitive principles in the face of what look to be clear counterexamples that distinguishes particularists from revisionary ontologists.

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There are a number of other questions that one might be tempted to ask at this point about the metaphysics of artifacts. Can one bring a new object into existence simply by placing a piece of driftwood in one’s living room and using it as a coffee table? Or by signing one’s name on a urinal and placing it in a museum? Or by pointing at some stuff, specifying some persistence conditions (however strange), and declaring that that stuff constitutes something with those persistence conditions? Is it possible to intentionally make a gollyswoggle (or incar, or snowdiscall)? If not, what are the constraints on our creative powers? I do not deny that these are difficult questions. There presumably are constraints on the creation of artifacts, and the nature of those constraints has been studied in some detail.30 But even those particularists who hold that creative powers are radically unconstrained may still insist that there are statues but no gollyswoggles and cite the absence of the relevant creative intentions as an ontologically significant difference between the two. In any event, answering the argument from arbitrariness is one thing, supplying a theory of artifacts is another, and I am here concerned only with the former.

8. snowballs and snowdiscalls Particularists will insist that the presence or absence of the relevant creative intentions is an ontologically significant difference between statues and gollyswoggles. It is open to particularists to account for the difference between snowballs and Sosa’s snowdiscalls along similar lines: clumps of snow sometimes constitute snowballs but never snowdiscalls because people have intended to make snowballs but (to my knowledge) no one has ever intended to make a snowdiscall. 30 See, e.g., Thomasson (2003, §3) and Baker (2008, 43–66). Some particularists are more liberal than others. Baker (2008, 53) allows that a wine rack can be brought into existence by brushing off a piece of unaltered driftwood and using it as a wine rack, so long as appropriate conventions and practices are in place. I am inclined to agree with Dean Zimmerman (2002, 335) that ‘‘changes in our ways of talking about things, even coupled with simple changes in some of our nonverbal reactions to things, could [not] by themselves bring any concrete physical object into existence’’ and to accept a more conservative view on which at least some alteration is required in order to bring a wine rack into existence (which is not yet to deny that the piece of driftwood is a wine rack; see n. 27).

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For what it is worth, I suspect that (pace Sosa) snowballs are not an individuative kind at all but rather a phased kind. Snowballs are identical to round clumps of snow, and they cease to be snowballs when flattened but do not cease to exist. That snowballs are just clumps of snow, while snowdiscalls are meant to be constituted by (and modally different from) clumps of snow, is itself an ontologically significant difference between snowballs and snowdiscalls. And, as should by now be clear, particularists will have no objection to instances of the associated phased kind, snowdiscall*, where a snowdiscall* is a clump of snow that has any shape between being round and being disc-shaped.

9. scattered objects It is often alleged that there is no ontologically significant difference between the scattered objects that we do countenance and those that we do not.31 In some cases, there are obvious differences: for instance, the disjoint microscopic parts of the table together exhibit a kind of unity, continuity, and causal covariance that is altogether lacking in the case of the alleged fusion of my nose and the Eiffel Tower.32 In other cases, the grounds for differential treatment are less obvious. I will discuss various strategies available to particularists for explaining away the apparent arbitrariness in such cases. As we have already seen, creative intentions do seem relevant to the existence of artifacts, and scattered artifacts are no exception. Whether a steel ball and steel rod arranged letter-‘i’-wise compose something will depend upon whether they came to be so arranged by accident or as a result of someone intending to make a lower-case letter ‘i’. Likewise, the ontologically significant difference between a work of art consisting of several disconnected parts and the alleged fusion of my nose and the Eiffel Tower is the presence of relevant creative intentions in the one case and their absence in the other. This account may also be extended to scattered institutional entities, like the Supreme Court, and scattered geopolitical entities, like the 31 See, e.g., Cartwright (1975, 158), Quine (1981, 13), van Cleve (1986, 145), and Hudson (2001, 108–12). 32 There is then the further question of how (and whether) such factors combine to yield necessary and sufficient conditions for composition, a question which lies outside the scope of this chapter.

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state of Michigan.33 In those cases, something is created without the sort of hands-on labor that is usually involved in bringing an artifact into existence. This is not to say that one can stipulate things into existence willy-nilly; as with ordinary artifacts, there presumably are constraints on the creation of such entities.34 Some may still feel that if this sort of ‘‘spooky action at a distance’’ is what is needed to vindicate our intuitive judgments about cases, then it is not worth the cost.35 Such is the difference between them and particularists. The charge of arbitrariness should move only those who already embrace a certain stringent view of what sort of factors are relevant to composition. Now let us now turn to scattered nonartifacts, taking the solar system as a representative example. Despite being a scattered object, the solar system exhibits a degree of unity altogether lacking in the universalist’s strange fusions. The solar system has boundaries demarcated by natural properties: the objects in the solar system are the smallest collection of objects containing the sun, each of whose primary gravitational influences are only the others in the collection. Furthermore, the solar system, not unlike an organism, is self-sustaining: it retains its form by means of forces internal to the system. So there do look to be ontologically significant differences between solar systems and the universalist’s strange fusions; though, as indicated in §3, the task of supplying an argument that these differences are ontologically significant lies outside the scope of this chapter.36 This, however, is not the only response available to particularists. The particularist could simply concede that there is no 33 The state of Michigan is not identical to the land that it now occupies. The land is a quantity of matter, and particularists need have no objection to arbitrary scattered quantities. There is some land some of which is on one side of Lake Michigan and some of which is on the other. There is even some flesh and metal, some of which is in Paris and some of which is on my face. And what, according to particularists, is the ontologically significant difference between this scattered quantity and the alleged individual whose parts are my nose and the Eiffel Tower? Their ontological category: one is some stuff, the other is an individual. 34 See, e.g., Thomasson (2003, §2). And even were our creative powers radically unconstrained, the fact that no one has directed such creative intentions at my nose and the Eiffel Tower would be ontologically significant, from the perspective of particularists, to their not composing anything. 35 See, e.g., van Inwagen (1990, 12–13), Rea (1998, 352), and Hudson (2001, 111). 36 I am grateful here to Kenny Easwaran.

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ontologically significant difference between the solar system and the strange fusion and to admit that the latter exists. Perhaps the strange fusion is not so strange after all: it too is a system (whose parts exert certain forces on one another) and, when we intuitively judged there to be nothing whose parts are my nose and the Eiffel Tower, it was because we had neglected to consider the system whose parts are my nose and the Eiffel Tower.37 Particularists who takes this line will still deny that there are such modally strange things as incars, monewments, gollyswoggles, and snowdiscalls. But they evidently must (on pain of arbitrariness) admit that composition is unrestricted, at least when it comes to items that exert some force on one another, thereby comprising a system. This response may seem to be in tension with particularism. But in a way, this would simply be a case of particularism in action: our concrete-case intuitive judgment that there is a system that has them as parts takes precedence over our intuitive judgment about the generalization that there is nothing that has them as parts. Alternatively, particularists might contend that the ontologically significant difference between the fusion of my nose and the Eiffel Tower and the solar system is that the former, but not the latter, is a single individual.38 The solar system is not a single individual; it is many individuals. ‘The solar system’ may be syntactically singular but, on the present account, it is nevertheless semantically plural: it refers, not to a set of heavenly bodies or to a fusion of heavenly bodies, but to some heavenly bodies.39 One problem with this account is that solar systems do not seem to have the right sort of modal profile to be pluralities. Pluralities presumably have exactly the parts that they do essentially, whereas solar systems can 37 It is precisely their immunity to this sort of error—having overlooked nonobvious exceptions—that makes concrete-case intuitions a more secure starting point. 38 Some may insist that strange fusions are mere pluralities and, therefore, ‘‘ontologically innocent.’’ If indeed the fusion of my nose and the Eiffel Tower just is my nose and the Eiffel Tower, then universalists and particularists have no disagreement, for they agree that my nose and the Eiffel Tower exist. That said, it is controversial (even among universalists) whether strange fusions are ontologically innocent in this way. 39 See Simons (1987, 142–3) for a related discussion. For what it is worth, the most common dictionary definitions of ‘the solar system’ are something along the lines of: the sun and the various heavenly bodies that orbit it.

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survive gaining and losing parts.40 However, particularists might take a page from the revisionary ontologist’s playbook here and insist that ‘solar systems can survive gaining and losing parts’ is true only in a loose and misleading sense.41 Just as no one thing actually becomes longer as the part of this sentence that you have read thus far gets longer, no one thing needs to get bigger or change parts in order for the solar system to grow or change parts; it is sufficient that one plurality of heavenly bodies is larger than a suitably related, earlier plurality of heavenly bodies. The things that are now (identical to) the solar system may be distinct from the things that had previously been the solar system.42

10. disassembled objects Let us turn now to a somewhat different way in which our treatment of cases may appear arbitrary. Thus far, we have been considering pairs of cases that allegedly do not differ in any ontologically significant respects. Now let us consider a single case that seems to admit of multiple permissible, but mutually incompatible descriptions. Suppose that a watch is disassembled and later reassembled. It seems equally permissible to describe the watch as coming back into existence upon reassembly as it is to describe the watch as having been scattered across the workbench prior to reassembly. But these descriptions are incompatible: the watch either did or did not exist after disassembly and prior to reassembly. So if the descriptions are equally permissible, it may seem that (on pain of arbitrarily favoring one over the other) one must either take them to be true of temporarily coincident things—only one of which 40 Furthermore, the solar system, unlike a mere plurality, would plausibly cease to exist if its parts were scattered across the universe. 41 An alternative would be to contend that some pluralities are mereologically flexible. For instance, Peter Simons (1997, 91–2) maintains that an orchestra is an ‘‘empirical collective,’’ which like a mere plurality is many things, not one thing, but unlike a mere plurality can survive gaining and losing parts. Whatever the merits of this view, it cannot (by itself) defuse the charge of arbitrariness, for one would still have to identify an ontologically significant difference between the solar system—understood as an empirical collective—and various strange empirical collectives of my nose and the Eiffel Tower. 42 I am grateful to Derek Ball, John Hawthorne, Dave Liebesman, and Peter Simons for valuable discussions of the points in this section.

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survived disassembly—or else true only relative to some convention or context.43 There is, however, more than one way of being permissible, and being true is only one of them. Another way of being permissible is by conveying something true despite being literally false. The police are looking for Carl, find a heap of bone and meat by the wood chipper, and one says to the other, ‘‘I think this is Carl.’’ She certainly does not think that this stuff is (the same thing as) Carl or that Carl still exists. She is speaking loosely. What she meant is that this heap of bone and meat is Carl’s remains. She takes herself to have been deliberately misunderstood when her partner replies: ‘‘How can this be Carl? Carl could not have survived that!’’ We react similarly when someone points to the disassembled parts and says, ‘‘You really think this is a watch? It is not shaped like a watch!’’ So it is plausible that we are likewise speaking loosely when we refer to the scattered parts as ‘a watch’. This could then serve as nonarbitrary grounds for favoring the description of the watch as coming back into existence upon reassembly.44 Yet another way of being permissible is by being penumbral. One mark of something’s being a borderline case that either verdict is permissible.45 It may well be that our ambivalence toward the two descriptions of the watch, and the permissibility of each description, is the result of its being vague whether the watch exists after disassembly. In that case, no more machinery is needed to account for the permissibility of these two descriptions of the watch than is needed to account for the permissibility of describing a borderline bald man as bald or as nonbald.46 43 See Hawthorne and Cortens (1995, 158–60) or Hawthorne (2006, 53–4) for discussion of a related case. 44 Some contend that all ordinary talk about nonliving composites (van Inwagen 1990) or mereologically flexible entities (Chisholm 1976) is loose talk about no such things and may go on to insist that there is no principled reason to take apparent reference to some but not other nonliving composites at face value. But there are principled reasons for the differential treatment. For the current appeal to loose talk is plausible and can be (and has just been) independently motivated, whereas these other appeals to loose talk are not plausible and cannot be independently motivated. Cf. Merricks (2001, 164–7), Hirsch (2002a, 109–11), and my (2007, §3). 45 See Sainsbury (1996, 259), Shapiro (2003, 43–4), and Wright (2003, 92–4). 46 It may well be that more needs to be said in accounting for the vagueness in the present case, but that is another matter; we are here concerned with the argument from arbitrariness, and the argument from vagueness will have to wait its turn.

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11. strange communities I have thus far made no mention of strange linguistic communities, though it is common for discussions of strange kinds to be carried out in terms of such communities. So, before concluding, I will make some remarks about communities that employ strange conceptual schemes and their relevance to particularism. One might think that the mere possibility of communities who make different intuitive judgments about strange and familiar kinds is enough by itself to cause trouble for the particularist. John Hawthorne seems to be suggesting something along these lines when he says: Barring a kind of anti-realism that none of us should tolerate, would it not be remarkable if the lines of reality matched the lines that we have words for? The simplest exercises of sociological imagination ought to convince us that the assumption of such a harmony is altogether untoward, since such exercises convince us that it is something of a biological and/or cultural accident that we draw the lines that we do.47

Hawthorne seems have in mind an argument along the following lines: we cannot expect intuitive judgments about which kinds of things there are to be correct because (1) we cannot expect intuitive judgments that are largely the result of biocultural accidents to be correct and (2) the intuitive judgments that lead us to draw the lines that we do are largely the result of biocultural accidents. We are meant to be convinced of the second premise by ‘‘the simplest exercises of sociological imagination.’’ Certainly Hawthorne is not suggesting that the mere fact that we are able to imagine strange communities is reason enough to accept (2). After all, just as easily as we can imagine perfectly functional communities who (say) take there to be snowdiscalls but no snowballs, we can imagine perfectly functional communities, no worse off than our own at satisfying their various needs and desires, with different intuitive judgments about the multiple-realizability of mental properties, the moral impermissibility of torturing babies for fun, the supervenience of moral facts on natural facts, the indiscernibility of identicals, the premises of the revisionary ontologists’ 47 Hawthorne (2006, 109). Cf. Hudson (2001, 107), Sider (2001, 156–7), and Rea (2002, ch. 8).

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favorite anti-particularist arguments, and so on for virtually all other intuitive judgments. Whatever reasons there may be for global skepticism about intuitive judgments, the mere imaginability of communities with different intuitive judgments is not one of them. Perhaps what Hawthorne has in mind is only that, when we imagine such communities, we see that there is no ontologically significant difference between the relevant strange and familiar kinds. But this would render the argument from strange imaginary communities parasitic on P1 of the argument from arbitrariness (§3), and the detour into strange communities superfluous. Alternatively, perhaps what Hawthorne has in mind is that we can easily imagine the sorts of circumstances that might have led us to draw the lines differently and that, since these circumstances could easily have obtained, we could easily have come to draw the lines differently. To a certain extent, this is right but poses no threat to the particularist. For instance, in other (easily imaginable) circumstances, we would have had reason to make different kinds of artifacts and, accordingly, would have taken there to be different kinds of artifacts. To that extent it is indeed a biocultural accident that we countenance the kinds of artifacts that we do. But given the way in which the kinds of artifacts there are depends upon the kinds of artifacts people have intended to make, our ability to judge correctly which kinds of artifacts there are is no more remarkable than our ability to know what kinds of artifacts people have intended to make. We can likewise easily imagine conditions under which we would have found it convenient to employ strange phased-kind concepts, like toddlescent* or incar*. But this would not be a case in which we would have made different intuitive judgments, for (as indicated in §§4–5) we do intuitively judge there to be such things as toddlescent*s and incar*s. So while it is almost certainly a biocultural accident that we employ the phased-kind concepts that we do, this is no indication that it is a biocultural accident that we intuitively judge there to be things answering to the relevant phased kinds. The real problem cases would be those involving strange nonartifactual individuative kinds, like toddlescents. However, I find it difficult to believe that there could easily have been communities that intuitively judged there to be toddlescents—just as I find it

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difficult to believe that there could easily have been communities with different intuitive judgments about the indiscernibility of identicals, multiple realizability, moral supervenience, and so forth. Why this could not easily have happened is a difficult question, but one which lies beyond the scope of this chapter.48 What would be more worrisome is if there turned out to be actual communities whose intuitive judgments about which kinds of things there are differed from ours, for one could hardly ask for better evidence that this could easily have happened than that it has happened. Suppose, for instance, that anthropologists or experimental philosophers discover an actual community that apparently takes there to be toddlescents. What then? Given that this is an actual case, whose details must be discovered (not stipulated), it will be open to debate whether they indeed intuitively judge there to be toddlescents. There will, in such cases, be at least two alternative explanations of whatever linguistic behavior it is that leads one to suspect that they intuitively judge there to be toddlescents. The first and most straightforward is that they have been misinterpreted: they do not take there to be toddlescents but, rather, toddlescent*s. Particularists do not deny that there are toddlescent*s, nor do they deny that there are things answering to countless other strange phased kinds. Consequently, the extensive anthropological literature on strange ways of categorizing has little if any bearing on particularism about material-object metaphysics. A further possibility is that they do judge there to be toddlescents but do not intuitively judge there to be toddlescents. Communities may come to form strange judgments about kinds or persistence conditions for reasons having nothing at all to do with their intuitions. Eli Hirsch discusses a case in which the Rabbis came to the counterintuitive conclusion that a sandal cannot survive the replacement of its straps, more than anything out of a practical need for a manageable criterion of persistence.49 Even if this judgment came to 48

Perhaps the case could be made that the correct explanation of why we could not easily have had different intuitive judgments about the indiscernibility of identicals, multiple realizability, moral supervenience (etcetera etcetera) does not carry over to our intuitive judgments about strange and familiar kinds. But I do not see how it would go. 49 Hirsch (1999). Objects that had become impure were not allowed into the temple, so the Rabbis needed principled ways of deciding whether a given object was the same object that at an earlier time had acquired the impurity.

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be shared by the entire religious community, this is not necessarily any indication of a difference in their intuitions about persistence, for they may have come to believe this on authority and despite finding it counterintuitive.50 Suppose, however, that (for one reason or another) these alternative explanations are untenable. What, according to robust particularists, should we do in that case? The same thing that we would do if we discovered an actual community whose scientists were running the same experiments but consistently obtaining radically different data. We would not throw out our equipment and burn all of our data, nor would we glibly ignore this other community. Rather, we would investigate, looking for possible sources of error on both sides. Likewise, for moral disagreement. Upon encountering communities with different ethical beliefs, we (realists, anyway) do not throw up our hands and conclude that there are no ethical truths or that we are both right relative to our respective standards. Rather we look for potential sources of error or bias. (Perhaps centuries of tyrannical rule have distorted their moral sense; perhaps centuries of overemphasis of the value of autonomy has distorted ours.) I see no reason to think that crosscultural ontological disagreements need be or should be treated any differently.51

12. conclusion I have examined numerous strategies for explaining away the apparent arbitrariness of our treatment of various cases: distinguishing between phased and individuative kinds, loose and strict talk, vagueness and arbitrariness, types and tokens, masses and individuals, intentional and unintentional activities, syntactic and semantic singularity, and simply thinking more carefully about the metaphysics of familiar kinds. I have not addressed every sort of case that has been, or might be, claimed to be indicative of arbitrariness (some will think I have gone on long enough already). But our 50 Similarly, differences in judgments about unobservables (e.g., ‘‘tree spirits’’) presumably have nothing at all to do with differences in intuitions. See Bealer (2004, 12–13) on the difference between intuition and belief. 51 I am grateful to George Bealer, Adam Elga, Marc Moffett, Bryan Pickel, and Chris Tillman for helpful discussion of points in this section.

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success above is at least some grounds for optimism that there will be a way of handling new problems as they arise. This has been only a partial defense of robust particularism. Consequently, I do not expect the foregoing to have won many converts. For one, I have not supplied any argument that the apparent ontologically significant differences (in particular, the presence of relevant creative intentions) are indeed ontologically significant. Nor have I shown that, all told, the costs of deeply revisionary ontologies are greater than the costs of robust particularism. This would require (among other things) examining strategies for—and costs associated with—blocking the rebutting arguments and the other undercutting arguments mentioned in §2, as well as assessing various strategies for mitigating the apparent costs of revisionary ontological theories. Nevertheless, I do hope to have emboldened fence-sitters and closet particularists, and to have shown that the particularist’s treatment of strange and familiar kinds is not as intolerably arbitrary as it is so often taken to be. University of Illinois, Urbana-Champaign

references Armstrong, D. M. (1986), ‘In Defence of Structural Universals’, Australasian Journal of Philosophy 64: 85–8. Baker, Lynne Rudder (2008), The Metaphysics of Everyday Life (Cambridge: Cambridge University Press). Bealer, George (1998), ‘Propositions’, Mind 107: 1–32. (2004), ‘The Origins of Modal Error’, Dialectica 58: 11–42. Benacerraf, Paul (1965), ‘What Numbers Could Not Be’, The Philosophical Review 74: 47–73. Cartwright, Richard (1975), ‘Scattered Objects’, in Keith Lehrer (ed.), Analysis and Metaphysics (Boston: Reidel Publishing Company). Chisholm, Roderick M. (1976), Person and Object, (London, George Allen & Unwin Ltd). Comesana. ˜ Juan (2008), ‘Could There Be Exactly Two Things?’, Synthese 162: 31–5. Ginet, Carl (1985), ‘Plantinga and the Philosophy of Mind’, in James E. Tomberlin and Peter van Inwagen (eds.), Alvin Plantinga (Dordrecht: D. Reidel), pp. 199–223. Hawthorne, John (2006), Metaphysical Essays (Oxford: Oxford University Press).

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and Cortens, Andrew (1995), ‘Towards Ontological Nihilism’, Philosophical Studies 79: 143–65. Heller, Mark (1990), The Ontology of Physical Objects: Four-Dimensional Hunks of Matter (New York: Cambridge University Press). Hirsch, Eli (1993), ‘Peter van Inwagen’s Material Beings’, Philosophy and Phenomenological Research 53: 687–91. (1999), ‘Identity in the Talmud’, Midwest Studies in Philosophy 23: 166–80. (2002a), ‘Against Revisionary Ontology’, Philosophical Topics 30: 103–27. (2002b), ‘Quantifier Variance and Realism’, Philosophical Issues 12: 51–73. Horgan, Terence (1993), ‘On What There Isn’t’, Philosophy and Phenomenological Research 53: 693–700. and Potrˇc, Matjaˇz (2000), ‘Blobjectivism and Indirect Correspondence’, Facta Philosophica 2: 249–70. Hudson, Hud (2001), A Materialist Metaphysics of the Human Person (Ithica: Cornell University Press). Johnston, Mark (2006), ‘Hylomorphism’, The Journal of Philosophy 103: 652–98. Korman, Daniel Z. (2008), ‘Unrestricted Composition and Restricted Quantification’, Philosophical Studies 140: 319–34. (2009), ‘Eliminativism and the Challenge from Folk Belief’, Noûs 43: 242–64. Lewis, David (1986), On the Plurality of Worlds (Malden: Blackwell). Lowe, E. J. (2007), ‘Metaphysical Essays’, Notre Dame Philosophical Reviews. Merricks, Trenton (2001), Objects and Persons, (New York: Oxford). (2003), ‘The End of Counterpart Theory’, The Journal of Philosophy 100: 521–49. Olson, Eric T. (2001), ‘Material Coincidence and the Indiscernibility Problem’, The Philosophical Quarterly 51: 337–55. Quine, W.V.O. (1981), Theories and Things (Cambridge: Harvard University Press). Rea, Michael C. (1998), ‘In Defense of Mereological Universalism’, Philosophy and Phenomenological Research 58: 347–60. (2002), World Without Design, (Oxford: Oxford University Press). Rodriguez-Pereyra, Gonzalo (2002), Resemblance Nominalism: A Solution to the Problem of Universals, (Oxford: Oxford University Press). Rosen, Gideon and Dorr, Cian (2002), ‘Composition as Fiction’, in Richard M. Gale (ed.), The Blackwell Guide to Metaphysics, (Oxford: Blackwell), pp. 151–74. Sainsbury, R. M. (1997), ‘Concepts Without Boundaries’ in Keefe and Smith (eds.), Vagueness: A Reader (Cambridge: MIT Press), pp. 251–64.

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Schaffer, Jonathan (forthcoming), ‘Monism: The Priority of the Whole’, The Philosophical Review. Shapiro, Stewart (2003), ‘Vagueness and Conversation’, in J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox, (Oxford: Oxford University Press), pp. 39–72. Sidelle, Alan (2002), ‘Is There a True Metaphysics of Material Objects?’, Philosophical Issues 12: 118–45. Sider, Theodore (2001), Four-Dimensionalism (Oxford: Clarendon). (2003), ‘What’s So Bad About Overdetermination’, Philosophy and Phenomenological Research 67: 719–26. Simons, Peter (1987), Parts: A Study in Ontology, (New York: Oxford University Press). (1997), ‘Bolzano on Collections’, in Künne, Siebel, and Textor (eds.), Bolzano and Analytic Philosophy (Amsterdam: Rodopi), pp. 87–108. Sosa, Ernest (1987), ‘Subjects Among Other Things’, Philosophical Perspectives, 1: 155–87. (1993), ‘Putnam’s Pragmatic Realism’, The Journal of Philosophy 90: 605–26. (1999), ‘Existential Relativity’, Midwest Studies in Philosophy 23: 132–43. Thomasson, Amie (2003), ‘Realism and Human Kinds’, Philosophy and Phenomenological Research 57: 580–609. (2007), Ordinary Objects (Oxford: Oxford University Press). Unger, Peter (1979), ‘There Are No Ordinary Things’, Synthese 41: 117–54. van Cleve, James (1986), ‘Mereological Essentialism, Mereological Conjunctivism, and Identity Through Time’, Midwest Studies in Philosophy 11: 141–56. van Inwagen, Peter (1990), Material Beings (Ithica: Cornell). Wiggins, David (2001), Sameness and Substance Renewed (New York: Cambridge University Press). Williamson, Timothy (2007), The Philosophy of Philosophy, (Malden: Blackwell). Wright, Crispin (2003), ‘Vagueness: A Fifth Column Approach’, in J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox, (Oxford: Oxford University Press), pp. 84–105. Yablo, Stephen (1987), ‘Identity, Essence, and Indiscernibility’, The Journal of Philosophy 84: 293–314. Zimmerman, Dean W. (2002), ‘The Constitution of Persons by Bodies’, Philosophical Topics 30: 295–338.

8. Many as One Thomas Sattig The problem of the many challenges our way of counting ordinary objects. We say that on an open plain on the northern boundary of Tanzania stands one mountain, Kilimanjaro. Yet there are many distinct, overlapping, mountain-shaped aggregates of rock, each of which is an equally good candidate to be this mountain. How can it be true, then, that there is one mountain on the plain, as opposed to many? Some reply that the many distinct aggregates are one and the same mountain. This slogan leaves our practice of counting mountains unscathed, but comes with a catch: orthodox, absolute identity is insufficient to sustain it. What is needed for the ‘double-count’ of aggregates and mountains is a notion of sortal-relative identity, which is standardly taken to replace the absolute notion. Irreducible relative identity, however, is plagued with problems. ‘Do not mess with identity’, critics urge, leaving the relative-identity solution to the problem of the many with few adherents. This chapter is a re-evaluation of the slogan ‘The many aggregates are one mountain’ as a response to the problem of the many. In order to sustain our ordinary conception of mountains in the face of this challenge, I shall develop and defend a metaphysically innocent picture of sortal-relative de re predication. This picture will be shown to avoid substantial problems for the traditional account of sortal-relative statements of identity, and to afford a solution to the problem of the many that is superior to competing solutions that also promise a metaphysically conservative vindication of ordinary mountain-talk.

1. the problem of the many Focus on Kilimanjaro, a mountain on an open plain. The mountain is alone, unaccompanied by any other mountains in the immediate vicinity. Seen from the distance, the mountain’s boundary appears

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precise. However, mountains are composed of rocks; and for many rocks on the mountain’s border there is no determinate answer to the question whether or not they are part of the mountain; it is not clear where the mountain ends and the surrounding countryside begins. So our mountain lacks a precise boundary; there is not just one way of drawing the mountain’s boundary, there are many ways. To each boundary we can draw corresponds an aggregate of rocks—assuming that for each set of rocks there is an object composed of the rocks in the set. Each of these aggregates is a candidate to be the mountain. If among many candidates a single one is a mountain, then there must be a fact of the matter singling out one candidate. Since each candidate has everything it takes to be a mountain, each of them is an equally good candidate to be the mountain, and hence there is no fact of the matter singling out one candidate. It follows that there are either many mountains or none where we thought there was just one. What holds for mountains holds for other macroscopic material objects. These are all composed of particles and have imprecise boundaries, with particles on the surface being neither clearly part nor clearly not part of the object. Accordingly, there are many ways of drawing the object’s boundary and many corresponding aggregates of particles, each a candidate equally suited to be the object. But without the means of selecting one, there are many or there is none. This is Peter Unger’s problem of the many.1 Among recent treatments of the problem, some surrender to the conclusion that there are many mountains or none, thereby revising our ordinary conception of mountains.2 Others save the latter conception by departing from a moderate metaphysics of material objects and their parts. An instance of this strategy is to deny that there are many aggregates, on the grounds that, mysteriously, among many largely intersecting sets of rocks only one such set has a fusion.3 A third approach to the problem is constituted by the hope of reconciling our ordinary conception of mountains with a moderate metaphysics of material objects. An instance of this strategy is to allow the many distinct aggregates to be one and the 1

See Unger (1980). The similar ‘problem of 1,001 cats’ appears in Geach (1962). See Unger (1980). 3 See Markosian (1998). For another way of twisting mereological orthodoxy, see Hudson (2001). 2

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same mountain. In what follows, I shall attempt a defense of this approach.4

2. the many are one Observing that in ordinary statements of identity of the form ‘a is the same K as b’ the identity predicate is relativized to a sortal term K, Peter Geach claimed that a may be the same K as b but a different K* than b, where K and K* are terms for different sorts. Suppose that the identity over time of persons is a matter of psychological continuity and that the identity over time of human beings is a matter of biological continuity. In a case of cerebrum-transplant from person/human being a to person/human being b with the result of a’s being psychologically but not biologically continuous with b, a is the same person as b but a different human being than b. Let us call this phenomenon ‘sortal variation’. Given the close relationship between the concept of identity and the concept of number, if statements of identity are sortal-relative, then so are statements of cardinality, statements about the number of things. If asked to count Ks, we collect things under the relation ‘is the same K as’. If asked to count K*s, we collect things under the relation ‘is the same K* as’. Taken by itself, the sortal relativity of cardinality is just as commonplace as the sortal relativity of identity. It is only when charged with sortal variation that the sortal relativity of cardinality receives philosophers’ attention. For then there may be two Ks while there is only one K*, as in the mentioned case there are two human beings but only one person.5 If sortal variation obtains, statements of the form ‘a is the same K as b’ cannot be analysed as ‘a is a K, b is a K, and a is identical with b’. For then the conjunction of ‘a is the same K as b’ with ‘a is a different K* than b’ yields a contradiction. How, then, are relativeidentity predicates of the form ‘is the same K as’ related to the absolute-identity predicate ‘is identical with’, or ‘is the same thing as’? Geach proposed to abandon the absolute predicate in favor of 4 For overviews of existing solutions and references, see Hudson (2001: ch. 1) and Weatherson (2003b). For a discussion of two further instances of the strategy of reconciliation, the almost-one solution and the standard supervaluationist solution, see Section 7. 5 See Geach (1962) and (1967).

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sortal-relative predicates: relative-identity predicates are primitive, or irreducible, and the absolute-identity predicate is meaningless. To construe relative-identity predicates as primitive is to hold that there are no more basic truths about a and b that make it the case that a is the same K as b. As a corollary, ‘is a K’ is understood in terms of the primitive ‘is the same K as’, by reading ‘a is a K’ as ‘a is the same K as some thing’.6 The problem of the many is the challenge to explain how it can be true that there is one mountain on the plain, while there are many overlapping, mountain-shaped aggregates of rocks, a1 , a2 , . . . , an , each of which is an equally good candidate to be this mountain. In response, Geach proposes to treat any massively overlapping, mountain-shaped aggregates ai and aj as distinct aggregates but as the same mountain. Given that K-relative-identity predicates are the basis for counting Ks, the aggregates are many but the mountains are one, just as expected. Since there is no absolute identity, there is no absolute count of how many mountain-shaped things there are on the plain. Facts of identity and cardinality are irreducibly sortal-relative, and hence metaphysically ultimate. The notion of relative identity at the heart of Geach’s solution to the problem of the many faces a number of substantial problems. I shall focus on the following two. The first problem concerns the thesis that relative identity replaces absolute identity. Can we do without absolute identity in ordinary thought and talk? Suppose that I believe that my friend, a person, is able to transform into a cat. Upon visiting her house I find a cat, and ask: Is the person I saw yesterday (identical with, the same thing as) the cat in front of me now? Of course, my background belief is false. Given this belief, however, my question is a pre-theoretically sensible one. The problem for Geach is that my question makes no sense if there is no absolute identity. I have in mind neither the question whether the person I saw yesterday is the same person as the cat in front of me now, nor the question whether the person I saw yesterday is the same cat as the cat in front of me now. For both of these sortal-relative versions of my original question have a trivially negative answer, whereas my original question does not. The point is that my question is a sensible question concerning 6

This is Geach’s derelativization thesis; see Geach (1962).

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inter-sortal identity over time. But inter-sortal identity over time is absolute identity. Hence my question cannot be asked. Setting aside the role of absolute identity in ordinary discourse, can we do mathematics and logic without absolute identity? It is hard to see how we can. Set theory provides a case in point. Our concept of a set is built upon the axiom of extensionality: a set x is identical to a set y iff x and y have the same members. This axiom employs the notion of absolute identity. If absolute identity is rejected, then it is unclear how the concept of a set is to be understood, and a significant portion of logico-mathematic orthodoxy is threatened.7 The second problem concerns the qualitative profiles of mountains, and is meant to be independent of the status of absolute identity. Since the aggregates of rocks on the plain differ in their qualitative profile, and since they are the mountain on the plain, the mountain has an inconsistent qualitative profile. For example, it may be true that the mountain on the plain both has and lacks a certain part at t, assuming that ‘The mountain on the plain has/lacks o as a part at t’ is read as ‘There is a mountain on the plain, all mountains on the plain are the same mountain as it, and it has/lacks o as a part at t ’.8 Are these inconsistencies a threat to our ordinary conception of mountains? One might deny that they are, on the grounds that the qualitative differences between aggregates that count as the same mountain are small, and that we ignore these small differences in ordinary contexts. This claim, however, is incorrect; we do not ignore these small differences in all ordinary contexts. We do perhaps ignore them when describing a mountain from a distance, judging naïvely that its boundary is clear-cut. But suppose that upon moving closer I point to a rock at the foot of the mountain—a rock that is a part of some but not all overlapping mountain candidates on the plain, and ask ‘Is the rock a part of the mountain?’. This question draws attention to a ‘small difference’ that appears firmly 7

For further endangered concepts from classical logic and semantics, see Hawthorne (2003). 8 Alternatively, ‘The mountain on the plain has/lacks o as a part at t ’ may be read as ‘Something is the same mountain as all mountains on the plain, and every mountain on the plain has/lacks o as a part at t ’. On this reading, the statements ‘The mountain on the plain has o as a part at t ’ and ‘The mountain on the plain lacks o as a part at t ’ may both be false, which is just as troubling as the possibility for both statements to be true.

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on the radar of ordinary intuition. In the capacity of ordinary speakers we would certainly not consider it a sensible response to the question that the rock both is and is not a part of the mountain. Yet this is the correct response if distinct aggregates with varying mereological profiles all count as the mountain on the plain. What we would respond to the question is that the status of the rock as a part of the mountain is indeterminate; it is neither clearly a part nor clearly not a part of the mountain. So, we ascribe these mountains a qualitative profile that is free of contradiction and sensitive to small differences between mountain candidates. If mountains are individuated by mountain-relative identity, however, we cannot ascribe mountains a qualitative profile that is free of contradiction and sensitive to small differences. Hence, a significant portion of our ordinary conception of mountains cannot be captured. For a case involving diachronic as opposed to synchronic sortalrelative identity, suppose that as a consequence of a cerebrumtransplant there are absolutely distinct human beings, a and b, such that a exists at t but b does not, and that a is psychologically continuous with b. If personal identity is a matter of psychological continuity, and if identity may be relative to personhood, then a is the same person as b. Does this person have a certain mass, shape, and size at t? Since a exists at t while b does not, the answer is ‘yes and no’. Hence, the entire qualitative profile that we would ordinarily ascribe to a person at a time may be in danger if persons are individuated by person-relative identity.9 My aim is to develop and defend a sortal-relativity solution to the problem of the many, captured by the slogan ‘The many aggregates are one mountain’, which avoids the massive costs of Geach’s view listed above. The key to this new solution is an account of sortal-relative statements of identity in terms of sortal representation. 9 In Geach’s framework, inconsistent profiles are worrying not only because they threaten our ordinary conception of material objects, but also because they point to a failure of Leibniz’s Law for sortal-relative identity. That is, the following inference schema is invalid: • α is F at t. • α is the same K as β. • Therefore, β is F at t.

In the absence of absolute identity, Geach is left without any version of Leibniz’s Law to characterize identity.

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3. the many are represented as one How can it be true that there is one mountain on the plain, while there are many overlapping, mountain-shaped aggregates of rocks, each of which is an equally good candidate to be this mountain? The answer I propose goes roughly as follows. There is a multitude of absolutely distinct, mountain-shaped aggregates of rocks—for short, mountain candidates—on the plain. When we count the mountains on the plain, we are not counting mountain candidates. What are we counting instead? We are counting mountain-representations. A mountain-representation is something that groups together mountain candidates in virtue of having these candidates as subjects. Counting mountains on the plain is counting mountain-representations with mountain candidates on the plain as subjects—in other words, counting mountains on the plain is counting mountain-representations that are ‘realized’ on the plain. If there are two mountains on the plain, one on the left, the other on the right, then there is a group of absolutely distinct, massively overlapping mountain candidates on the left, and a group of absolutely distinct, massively overlapping mountain candidates on the right. There is, further, a mountain-representation that has each candidate on the left but no candidate on the right as subject, while there is a distinct mountain-representation that has each candidate on the right but no candidate on the left as subject. If, as in the case of Kilimanjaro, there is only one mountain on the plain, then all of the absolutely distinct mountain candidates on the plain are subjects of the same mountain-representation. The purpose of the present section is to specify the details of this picture. I shall begin by developing the notion of a K-representation. An individual concept, as I shall use the term, is a partial function whose domain is a set of pairs of instants, or times, and spatial regions, or places, which assigns to each time t and place p in its domain a material object x that exists at t, and that exactly occupies place p at t.10 To an ordinary sortal term K, such as ‘mountain’, corresponds a certain class of individual concepts, namely the class of individual concepts that are K-unified. An individual concept 10 See Carnap (1947). Letting the domain of the function be a set of triples of times, places, and possible worlds is desirable for certain applications of individual concepts, but not for their application to the problem of the many.

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is K-unified if its values are maximally interrelated by the unity relation for Ks. Since we are, as far as the problem of the many is concerned, only interested in unity relations between objects existing at the same time, we may blank out cross-temporal unity relations, and restrict our attention to K-unification at a time t: (K)

An individual concept i is K-unified at t iff the set of i’s values at t is the maximal set of objects a1 , a2 , . . . , an , such that a1 , a2 , . . . , an are all K-shaped at t, and overlap extensively at t.

Finally, a K-unified individual concept is a K-representation of each of its material values. While a K-representation has distinct material objects as values, I shall assume that a material object is a value of at most one K-representation. By (K), distinct objects are subjects of the same K-representation if they are K-shaped and overlap extensively. As Geach pointed out, ordinary statements of identity of the form ‘a is the same K as b’ are relativized to a sortal term K. In order to regiment the sortal relativity of predications of identity, I shall introduce a sortal modifier ‘qua K’ (or ‘as a K’), whose syntactic function is to combine with a singular noun phrase to form a sortal-relative noun phrase. The ordinary sentence ‘a is the same mountain as b’ is to be read as ‘a qua mountain is identical with b qua mountain’. I shall further assign the operator ‘simpliciter’ the function of indicating the absence of sortal relativization by way of ‘qua K’. Thus, ‘a is identical with b, simpliciter’ is a sortally unrelativized predication of identity. Given the close relationship between the concept of identity and the concept of number, if statements of identity are sortal-relative, then so are statements of cardinality, statements about the number of things. The semantic function of ‘qua K’ in terms of the form ‘a qua K’ is to trigger a shift from a term designating a material object to a sortal-relative term designating the K-representation of that material object. Sortal-relative statements of identity may then be given the following truth conditions: for any material objects x and y, (T1)

x qua K is identical with y qua K iff the K-representation of x is identical with the K-representation of y.

For illustration of (T1), consider a mountain-representation i. If function i returns an object a for a time t and a place p as arguments,

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and if the same function i returns an object b for time t and a place p’ as arguments, then i represents b as the same mountain as a. Further, if i is the only function that has one or more objects with attribute φ as values, then it is true that there is one mountain with attribute φ. If, however, there is a mountain-representation i distinct from i, and both i and i have one or more objects with attribute φ as values, then it is true that there are at least two mountains with attribute φ. When counting mountains, we are not counting material objects; we are rather counting mountain-representations of material objects. Now back to the problem of the many. Given the representational function of sortal terms in predication, we must distinguish between descriptions of a situation as it really is and descriptions of a situation as it is represented under a sort. I shall assume that the following sortally unrelativized facts constitute the real basis of our case: there are various mountain-shaped aggregates of rocks that overlap extensively, and that are distinct simpliciter.11 Our aim is to sustain a description of this case ‘at the level of mountains’, according to which the absolutely distinct aggregates are represented as the same mountain. By principle (T1), ‘x qua mountain is identical with y qua mountain’ is true in virtue of x and y being subjects of the same mountain-representation. Principle (K) tells us what makes x and y subjects of the same mountain-representation: x and y are mountain-shaped and overlap extensively. Given the real basis of our case, as specified above, it follows by (T1) and (K) that our aggregates are identical, qua mountain; many aggregates are one mountain. Note that the representation of distinct objects as one mountain raises an issue concerning proper names. Suppose that the proper name ‘Kilimanjaro’ is introduced to designate the mountain at coordinates 03◦ 04’33’’S and 37◦ 21’12’’E. According to the representational account of sortal-relative statements of identity, there is a range of absolutely distinct aggregates of rocks with the mentioned coordinates, such that each aggregate in the range is the same mountain as any other aggregate in the range; in short, each aggregate 11 So talk of distinct aggregates is to be understood as talk of objects that are distinct simpliciter (though not disjoint, since distinct aggregates may overlap). It is, of course, possible to treat ‘aggregate’ as a sortal term and mean by ‘distinct aggregates’ objects that are distinct qua aggregate. Such relativized aggregate-talk, however, would be of little use for present purposes.

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is the mountain at the mentioned coordinates. Then which of these aggregates does the proper name Kilimanjaro designate? I will not address this issue in any detail but mention a natural view to take in response. On this view, the proper name Kilimanjaro is a vague term in virtue of lacking a precise referent. What the term has is a range of suitable candidate referents, the set of candidates being the maximal set of distinct, mountain-shaped aggregates of rocks with the mentioned coordinates. Statements containing the name may then be assigned supervaluational truth-conditions (see Section 6). Given the generality of the problem of the many, this view has the consequence that most ordinary proper names are vague terms.12 I have offered the core of a solution to the problem of the many in terms of sortal-relative statements of identity in which the predicate of identity is itself unrelativized. Since sortal-relative statements of identity ascribe the relation of absolute identity, there are no brute facts of relative identity. Sortal-relative facts of identity are facts about how reality is represented under a sort. Sortal relativity is metaphysically modest. Owing to this modesty, the present picture avoids the indispensability problem for Geach’s framework. For illustration, consider again the issue of inter-sortal identity over time. We may ask whether a person existing at one time is the same person as a person existing at another time. We may further ask whether a person existing at one time is the same thing as a cat existing at another time. Both questions are pre-theoretically sensible. Within the present framework, the first question is a matter of whether material objects are subjects of a common personrepresentation, whether they are represented as the same, whereas the second question is a matter of whether material objects really are the same. The present framework, unlike Geach’s, thus renders the second type of question just as meaningful as the first.13 12 See Hawthorne (2003). Another suggestion is to treat ‘Kilimanjaro’ as an instantial term: the name is introduced by existential instantiation and designates an arbitrary member of the maximal set of distinct aggregates of rocks with the mentioned coordinates; see Deutsch (2002). Geach’s distinction between a name for a mountain and a name of a mountain must also be mentioned in this context. For compact discussions, see Hawthorne (2003) and Noonan (1997: 641–2). 13 The account of sortal-relative statements of identity in terms of sortal representation belongs to the same family as the semantic theories developed in Lewis (1971), Gibbard (1975), and Gupta (1980). Space does not permit a discussion of how and

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The aim of this chapter is to develop a solution to the problem of the many that respects our ordinary conception of mountains; a solution that ‘saves the appearances’. The picture so far captures our everyday cardinality judgments on the basis of a moderate metaphysics of material objects. But there is more to saving the appearances than saving compelling cardinality claims. For the problem of inconsistent profiles encountered in Section 2 arises for any version of the relative-identity solution to the problem of the many, whether or not absolute identity is admitted, and hence the problem still arises within the present framework.

4. sortal-relative multiple location and variation The problem of inconsistent profiles is the following. We ordinarily ascribe mountains a qualitative profile that is free of contradiction and sensitive to small differences between mountain candidates. However, if mountains are individuated under mountain-representations, then it seems that we cannot ascribe mountains a qualitative profile that is free of contradiction and sensitive to small differences. The mountain-shaped aggregates of rocks on the plain differ in their qualitative profile; since they are the mountain on the plain, the mountain has an inconsistent qualitative profile. Hence a significant portion of our ordinary conception of mountains is lost. In this section I shall sketch a two-step extension of the framework of sortal representation, which guarantees that every material object has a consistent qualitative profile when individuated under a K-representation. First step: multiple spatial location. Consider distinct aggregates on the plain, a1 and a2 : a1 exactly occupies place p at t, and a2 does not exactly occupy place p at t, where a1 and a2 are mountain-shaped and overlap extensively.14 It follows that the mountain on the plain exactly occupies p at t and does not exactly occupy p at t, assuming why my picture differs from these. Note, however, that neither Lewis nor Gibbard nor Gupta employ their theories in response to the problem of the many. 14 A word on the notion of exact occupation is in order. An object exactly occupies a spatial region at a time if it fits into the region perfectly, without leaving any gaps. This is a gloss on exact occupation, not a definition, since the notion will be taken as primitive. See Sattig (2006: 48) for more details on this notion of occupation.

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that ‘The mountain on the plain exactly occupies p at t’ is read as ‘There is a mountain on the plain, all mountains on the plain are identical with it, qua mountain, and it exactly occupies p at t’. How can locational inconsistency be avoided? K-representations not only identify, but also qualify. A material object may be represented as being identical with an object from which it is really distinct. Likewise, a material object may be represented as having a certain qualitative profile that it really lacks. Correspondingly, sortal relativity, understood as invoking Krepresentation, is not confined to statements of identity. Ordinary de re temporal predications of the form ‘x is F at t’ may be sortal-relative in virtue of containing implicit sortal modifiers of the form ‘qua K’ with the syntactic and semantic function specified in Section 3.15 Let us focus on spatial location. If x is thought of as a mountain, then ‘x exactly occupies p at t’ is elliptical for ‘x qua mountain exactly occupies p at t’. Given that ‘qua mountain’ invokes the mountain-representation of x, x qua mountain exactly occupies p at t just in case x exactly occupies p at t according to its mountainrepresentation. How does a mountain-representation represent location? A mountain-representation is a function from pairs of times and places to objects. Such a function represents an object x as exactly occupying p at t iff x is a value of the function, and the function is defined at the pair of t and p—recall that if the function is defined at the pair of t and p, its value at this pair exactly occupies p at t, which value may or may not be identical with x. Sortal-relative locational statements thus get the following truth conditions: for any material object x, (T2)

15

x qua K exactly occupies p at t iff the K-representation of x is defined at .

There are a number of ways in which a relativizing sortal term in a temporal predication may be specified. Typically, when a predication contains a subject term that is governed by a sortal term K, then the predication is implicitly relativized by K as well. So ‘The K is F at t ’ has the default reading ‘The K qua K is F at t ’. However, a relativizing sortal need not be specified by a noun phrase in subject position. There are cases in which non-linguistic context determines a relativizing sortal that trumps the sortal in the subject term. And there are cases in which non-linguistic context determines a relativizing sortal, while no sortal governs the subject term. For a recent discussion of sortal relativity that questions its viability as a hypothesis about ordinary language, see Fine (2003). For responses, see Frances (2006), King (2006), and Sattig (2006: sect. 5.6).

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Returning to our initial example, aggregate a1 exactly occupies place p1 but not p2 at t, simpliciter, and aggregate a2 exactly occupies place p2 but not p1 at t, simpliciter. Since a1 and a2 are subjects of the same mountain-representation, it follows by (T2) that both a1 and a2 exactly occupy p1 and p2 at t, qua mountain. In the framework of K-representation, objects that are really uniquely spatially located at a time are represented as multiply spatially located. Hence, it is false that the mountain on the plain qua mountain exactly occupies p1 at t, and fails to occupy p1 at t. To sum up, locational inconsistency is avoided if statements of location about mountains are read as sortal-relative and sortal relativity is understood in terms of K-representation. Second step: spatial variation. Consider again our distinct aggregates on the plain, a1 and a2 : a1 has rock o as a part at t, whereas a2 lacks o as a part at t. Since a1 and a2 are the same mountain, it seems to follow that the mountain on the plain has o as a part at t, and lacks o as a part at t. How can mereological inconsistency be avoided? The answer is spatial variation. Consider the temporal case first. Ordinary objects exist at different times; and as persisting things they may change through time, possessing different, incompatible attributes at different times. If, analogously, ordinary objects exactly occupy different places at the same time, then, as spatially persisting things, they may change through space, possessing different, incompatible attributes at different places, at the same time. Now add sortal relativity to the mix. If a material object a qua mountain exactly occupies multiple places at the same time, then a’s attributes, qua mountain, need not only be relativized to times but also to places, so that a qua mountain may vary in attributes not only relative to different times but also relative to different places it occupies. In short, there is no sortal relativity without spatial relativity. The thesis of sortal relativity is that many ordinary predications contain implicit sortal modifiers. The thesis of spatial relativity is that sortal-relative predications contain implicit spatial modifiers of the form ‘at p’, where p is a spatial singular term designating a spatial region, which allow objects to vary in attributes across space, according to a K-representation. The sentential operator ‘at p’ attaches to ‘a qua K is F’ to yield ‘a qua K is F, at p’. The operator ‘at t ’ then attaches to ‘a qua K is F, at p’ to yield ‘a qua K is F, at p, at t ’. Let us call these spatial modifiers ‘spatial variation-modifiers’, since they

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are designed to allow the attribution of incompatible attributes to the same object at the same time. And let us assume for the moment that the spatial singular term p in ‘at p’ designates a spatial region determinately (we shall have reason to refine this picture shortly). Notice that none of our ordinary forms of spatial modification can perform the function assigned to variation modifiers. One familiar kind of spatial modifier is present in the statement ‘Alex is sitting in her car (at t)’. Here the spatial modifier ‘in her car’ is detachable, in the sense that the statement implies that Alex is sitting (at t). Our modifier ‘at p’, on the other hand, is non-detachable, for otherwise ‘a has mass m, at p, at t, and a does not have mass m, at p*, at t ’ would collapse into a contradiction. Another familiar kind of spatial modifier is present in the statement ‘The road is bumpy in the mountains (at t)’. Here the spatial modifier ‘in the mountains’ has the function of shifting the subject of predication from the road to a spatial part of the road, so that the sentence may be read as ‘The road has a spatial part in the mountains that is bumpy (at t)’. Our modifier ‘at p’, however, is no such shifter, since in ‘a has mass m, at p, at t, and a does not have mass m, at p*, at t ’ a enjoys multiple exact spatial locations at t, and varies in its mass across space at t, as opposed to just having different spatial parts at t, each possessing a different mass. Given that ordinary spatial modifiers cannot perform the function assigned to variation modifiers, there is likely to be no linguistic evidence available for spatial relativity. Is this a serious defect of the proposal? The thesis of spatial relativity is forced upon us by a gap, a mismatch, between how we represent the world in thought and talk, and how the world really is. Implicit spatial variationmodifiers are posited in order to close this gap, in order to save the appearances. The thesis of spatial relativity is thus not an empirical hypothesis. It is driven by metaphysical considerations, and therefore rests on a firm foundation even if no linguistic evidence for the presence of spatial variation-modifiers is available. For a precedent, compare the status of temporal relations in special relativity. In relativistic time no temporal relation is instantiated absolutely; it is not meaningful to ask whether an event is simultaneous with or earlier than another event. Instead, all temporal relations are relativized to frames of reference. Prima facie, ordinary thought and talk presupposes absolute temporal relations, and hence is out of synchronization with relativistic reality. Ordinary temporal talk may be

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saved, however, its mismatch with reality repaired, by positing a relativization to frames of reference that is hidden from ordinary speaker.16 Having recognized implicit spatial variation-modifiers, truth conditions of spatially as well as sortally modified de re predications may be specified as follows: for any material object x, (T3)

x qua K is F, at p, at t iff the K-representation of x, i, is such that i(t, p) is F at t.

For illustration of (T3), suppose that aggregate a1 , which has rock o as a part at t, simpliciter, exactly occupies p1 at t, and that aggregate a2 , which lacks o as a part at t, simpliciter, exactly occupies p2 at t. Since a1 and a2 are subjects of the same mountain-representation, it follows by (T3) that a1 qua mountain has o as a part, at p1 , at t, and lacks o as a part, at p2 , at t. Likewise for a2 . Hence, the mountain on the plain qua mountain has o as a part, at p1 , at t, and lacks o as a part, at p2 , at t. In the framework of K-representation, objects that are really uniquely spatially located at a time and have their attributes in a spatially insensitive way are represented as multiply spatially located at the same time and as varying in their attributes across these locations. The problem of mereological inconsistency was this: when objects are individuated under a mountain-representation they seem to end up with inconsistent mereological profiles. This consequence is avoided if mountain-representations are construed as triggering spatial variation; a mountain-representation represents a material object as having slightly different mereological profiles relative to different places at the same time. Accordingly, a mereologically consistent profile for mountains may be secured by reading mereological statements about mountains as both sortal-relative and spatial-relative. By reading ‘The mountain on the plain has o as a part at t and lacks o as a part at t ’ as ‘The mountain on the plain qua mountain has o as a part, at p1 , at t and lacks o as a part, at p2 , at t ’ the threat of inconsistency is banned. This is how sortal relativity plus spatial relativity guarantees that every material object has a consistent qualitative profile when individuated under a K-representation. 16

See Sattig (2006: 41–2).

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Now that we are able, within the framework of sortal representation, to ascribe mountains a consistent profile, the question arises whether we can ascribe mountains a profile that matches the one we ordinarily ascribe. This question will be addressed in the following two sections.

5. sortal-abstract unique location Prima facie, the thesis of sortal-relative multiple spatial location—the thesis that mountains and other ordinary objects exactly occupy multiple spatial regions at the same time—is unacceptably counterintuitive. For the thesis, together with the innocuous assumption that mountains are macroscopic material objects, seems to violate the platitude of common sense that macroscopic material objects exactly occupy a unique spatial region at a time.17 Why do we find this uniqueness principle so plausible? We are not committed to the principle because it derives from the specific ways in which we think about mountains and other kinds of material object. The impression that a mountain cannot be multiply located is independent of the geological and social features that make it a mountain, just as the impression that a person cannot be multiply located is independent of the psychological and biological features that make it a person; likewise for other sorts of material object. I suggest that we find the uniqueness principle so compelling because it partly constitutes our conception of macroscopic material objects in abstraction from the sorts to which they belong. Multiple exact location is a conceptual impossibility for distinct mountains, persons and plants simpliciter (recall that simpliciter is used to indicate abstraction from K-representation). In short, the uniqueness principle is a sortal-abstract principle, and may be stated perspicuously as follows: (U)

Macroscopic material objects exactly occupy a unique spatial region at a time, simpliciter.

17 One might add the qualification that multiple exact spatial location is a nomological impossibility for macroscopic material objects as long as we ignore the possibility of time-travel, and hence the possibility of an object meeting its younger self.

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Now reconsider the troubling inference we started with, this time premised on (U): • Macroscopic material objects exactly occupy a unique spatial region at a time, simpliciter. • Mountains are macroscopic material objects. • Therefore, mountains exactly occupy a unique spatial region at a time, simpliciter. The conclusion of this valid inference poses no threat to the thesis of sortal-relative multiple spatial location, since occupying multiple regions at the same time, according to a K-representation, is compatible with occupying a unique region at a time, simpliciter. The point is that if the platitude of common sense is understood as the sortal-abstract principle (U), then the platitude is compatible with the thesis of sortal-relative multiple spatial location. In other words, the thesis of sortal-relative multiple spatial location stands in no conflict with the platitude of common sense that macroscopic material objects occupy a unique place at a time, because the thesis and the platitude manifest different but compatible perspectives on the material world. The thesis manifests a sortally sensitive perspective on the material world, whereas the platitude manifests a sortally insensitive perspective, a perspective that abstracts from sortal input. The compatibility of these perspectives is afforded by the metaphysical innocence of sortal-relative discourse about material objects: K-representations do not always mirror reality; Krepresentations are often misrepresentations of reality. I conclude that the present picture of location is far less radical than it may have appeared at first. The friend of the representational account of sortal relativity may appeal to sortal-relative multiple spatial location in order to secure consistent qualitative profiles for mountains without violating our intuitions about the spatial profile of material objects on the whole.18 18

Hud Hudson (2001: ch. 2) builds his relative-identity free approach to the problem of the many on a rejection of the uniqueness principle, claiming that macroscopic material objects exactly occupy multiple spatial regions at a time, simpliciter. While this is the end of metaphysical innocence, Hudson claims that the price is right.

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6. mereological indeterminacy When setting up the problem of the many, we started with the intuition that a mountain can be alone, unaccompanied by other mountains in a given area. We then turned our attention to a further intuition: mountains lack precise mereological boundaries; for each mountain, there are things that are neither determinately part nor determinately not part of that mountain. Mereological indeterminacy of this type led us to a multitude of aggregates, corresponding to each suitable boundary that can be drawn for any mountain, thereby casting doubt on the initial intuition that a mountain can be alone in a given area. The primary task posed by the problem of the many is to save the first intuition concerning the number of mountains. The second intuition concerning the fuzzy boundaries of mountains, however, constitutes a serious constraint on any attempt to save the first: whatever guarantees that mountains can be unaccompanied by other mountains must allow that mountains have questionable parts.19 This constraint poses a challenge to the sortal-representation approach as it stands. According to the latter, each mountain is an aggregate of rocks. None of these aggregates has any questionable parts; for any aggregate a and any rock o, it is a determinate matter whether a has o as a part at any time. Moreover, when the aggregates are represented as a single mountain, this mountain inherits not only the aggregates’ spatial locations but also their determinate mereological boundaries, which the mountain possesses relative to different places. How, then, is the sortal-representation approach to capture the full story about mountains? Is there room for an account of a mountain’s imprecise mereological boundaries? In this section I shall propose such an account within the framework of supervaluationism.20 According to the theory of vagueness known as supervaluationism, vagueness is a linguistic phenomenon; there are no vague 19

With mereological indeterminacy come other forms of indeterminacy, such as indeterminacy of mass; it is unclear whether the indeterminate part’s mass should be counted towards the mountain’s mass. For simplicity, I shall focus on indeterminacy of parthood, mereological indeterminacy. The account of mereological indeterminacy given below may be straightforwardly extended to account for related forms of indeterminacy, such as indeterminacy of mass, as well. Indeterminacy of location, however, is a special case to which I shall return at the end of this section. 20 The standard way of putting to work supervaluationism in response to the problem of the many will be discussed in the final section.

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properties, objects or states of affairs, just vague linguistic expressions. To the supervaluationist, an expression is vague when its meaning can be extended, can be made more precise in different ways, each consistent with the expression’s intuitive behavior determined by its original content. A classical example of a vague expression is ‘heap’. The meaning of ‘heap’ and non-linguistic facts about piles of sand determine that some piles are heaps, some are not heaps, and others occupy a grey area, to the effect that the question whether they are heaps lacks an answer. The first cases are the clear cases, the second are the clear non-cases, and the third are the borderline cases. While the expression ‘heap’ leaves a grey area of application, there are many ways of extending its meaning. Each of these extensions of meaning is a precisification. A precisification of ‘heap’ is admissible iff it respects our intuitive assignments of truth and falsity to statements in English—that is, iff it makes ‘heap’ true of the clear cases, false of the clear non-cases, and either true or false of the borderline cases. As regards the borderline cases, an admissible precisification must further respect penumbral constraints arising from the expression’s original meaning.21 For example, if piles of sand a and b are borderline cases of ‘heap’ that differ from each other only in that b contains one grain of sand more than a, then the statement ‘If a is a heap, then b is a heap’ is intuitively true. This intuition yields the constraint that no precisification is admissible that makes a a heap but not b. That is, no matter where we draw the line we cannot turn a heap into a non-heap by adding a grain of sand. With the notion of an admissible precisification at her disposal, the supervaluationist introduces the notion of truth on an admissible precisification, and defines super-truth and super-falsity in terms of the latter: it is super-true that x is a heap iff it is true on all admissible precisifications that x is a heap; and it is super-false that x is a heap iff it is false on all admissible precisifications that x is a heap. Truth in the vague object-language is super-truth; and falsity in the object-language is super-falsity. Accordingly, if x is a borderline case of ‘heap’, then it is neither true nor false that x is a heap. Super-truth may be expressed in the object-language by means of a new operator ‘determinately’. ‘Determinately s ’ is true 21

The notion of a penumbral constraint is introduced in Fine (1975).

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iff s is super-true.22 Then x’s being a borderline case of ‘heap’ can be expressed by saying that x is neither determinately a heap nor determinately not a heap.23 Among the types of expression supervaluationism recognizes as vague are singular terms. Central to present purposes is the notion of a vague spatial singular term. Consider the sentence ‘There is where we first danced’, and suppose that its utterances are accompanied by a gesture of pointing in a certain direction.24 The expression ‘there’ in this sentence is a spatial singular term, in that it purports to designate a particular spatial region. The expression is vague, in that there is no determinate answer, no fact of the matter, as to which spatial region it picks out. We may say that associated with ‘there’ is a descriptive condition that is sensitive to the non-linguistic context of an utterance of the sentence; roughly, a condition along the lines of being a place in the direction of the pointing and the vicinity of the speaker. Several distinct places are natural satisfiers of this descriptive condition in each context in which ‘There is where we first danced’ is uttered; and each of these places is an admissible precisification of ‘there’. In supervaluational manner, an utterance of ‘There is where we first danced’ is supertrue iff it is true on all admissible precisifications I of ‘there’, that we first danced in I(there); an utterance of the sentence is super-false iff it is false on all admissible precisifications I of ‘there’, that we first danced in I(there); and an utterance is neither super-true nor super-false iff it is true only on some admissible precisifications I of ‘there’, that we first danced in I(there). Now recall the thesis of spatial relativity (Section 5), according to which ordinary, sortal-relative de re predications are in need of spatial relativization, which allows objects to vary in attributes across space, according to a K-representation. In the previous section I sketched a straightforward way of spatially relativizing sortal predications by incorporating spatial variation-modifiers of the form ‘at 22 More precisely, since standard supervaluational model theory only permits ‘super-truth at a model m’ and ‘truth at an admissible point i, at a model m’, ‘determinately’ should rather be introduced as follows: for all models m, for all admissible points i in m: ‘Determinately s ’ is true at i in m iff s is super-true in m. 23 For detailed introductions to supervaluationism, see Keefe (2000: ch. 7) and Williamson (1994: ch. 5). 24 This is an adaptation from an example in Schiffer (2000).

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p’, where p is a spatial singular term. Now we need to be more careful, and ask not only how sortal predications can be spatially relativized, but also how ordinary sortal predications are in fact spatially relativized. The picture I suggest is roughly the following. While an ordinary sortal predication about a material object a that is a K requires relativization to a place exactly occupied by a qua K at a given time t, no such predication is relativized to a determinate place, since a qua K exactly occupies multiple, minutely differing places at t, and neither the intentions of speakers nor the contexts in which the predication is uttered manage to select one place from this range of candidates. Note, then, that ordinary sortal predications are never relativized to any particular place in the way ordinary predications are typically relativized to a particular time (now; 6 July 2007, and so on). While ordinary sortal predications are not in fact relativized to a determinate place, such predications may in principle be so relativized. If it makes no difference to which place, out of a range of admissible places, the predication is relativized, then the predication is true. If, on the other hand, it does make a difference, then the predication lacks a truth-value, which is the mark of indeterminacy. The core details of this proposal, as applied to predications of parthood, may be filled in as follows. Ordinary object-language predications of parthood are temporally, sortally and spatially modified. The spatial modifier in such predications contains a spatial singular term. This term is vague similarly to ‘there’ in the example ‘There is where we first danced’. Ordinary sortal-relative predications of parthood thus have the form ‘a qua K has b as a part, at p, at t’, where the bold-face p is a vague spatial singular term, reserving the italic p for precise spatial singular terms, which are confined to the meta-language.25 The expression p is a spatial singular term, in that it purports to designate a particular place. The expression is vague, in that there is no determinate answer as to which place it picks out. Associated with p is a descriptive condition that is sensitive to the linguistic context—that is, sensitive to structure and components of the predication in which p occurs. 25 It is for reasons of simplicity that predications of parthood are here taken to contain only a single implicit sortal modifier; a but not b is modified.

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Roughly, associated with p is the descriptive condition of being a spatial region exactly occupied by the subject of the predication at the relativizing time, qua the relativizing sort. Thus, in ‘a qua K has b as a part, at p, at t’, the vague spatial singular term p purports to designate the spatial region exactly occupied by a qua K at t. The subject of an ordinary de re predication, such as the predication of parthood under consideration, is a macroscopic material object.26 By the thesis of sortal-relative multiple spatial location, each macroscopic material object that is a K exactly occupies multiple spatial regions at any time of its existence, qua K. Consequently, multiple distinct spatial regions satisfy the descriptive condition associated with p in ‘a qua K has b as a part, at p, at t ’—assuming that the singular term a designates a material object that is a K—and each of these spatial regions is an admissible precisification of p. The referent of p in ‘a qua K has b as a part, at p, at t ’ is thus constrained but not fully determined by a, t, and K. Sortal-relative predications of parthood containing variation-modifiers with vague spatial singular terms may be given the following supervaluational truth conditions: (P)

An utterance of ‘a qua K has b as a part, at p, at t ’ is supertrue (super-false) iff it is true (false) on all admissible precisifications I of the object-language, that I(a) qua K has I(b) as a part, at I(p), at t.27

Accordingly, an utterance of ‘a qua K has b as a part, at p, at t’ is neither super-true nor super-false iff it is true only on some admissible precisifications I of the object-language that I(a) qua K has I(b) as a part, at I(p), at t. Mountains have questionable parts. For each mountain there are rocks existing at a time t, such that it is not clear whether they are parts of the mountain at t. Consequently, there are distinct sets of rocks, where each rock exists at t, such that for each set, it is not clear whether its member-rocks compose the mountain at t. This is the mereological indeterminacy intuition at the heart of the 26

Ordinary de re predications may have a vague singular term—a term, such as ‘Kilimanjaro’, which lacks a determinate material object as referent—in subject position. I shall ignore this feature in my exposition but factor it into truth conditions (P) below. 27 (P) allows a and b to be vague singular terms of material objects.

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problem of the many. Any solution to the problem must be able to explain this type of indeterminacy. The vague spatial-modifier view of predications of parthood delivers such an explanation within the framework of sortal representation. Focus on the mountain on the plain at time t and on a rock o existing at the same time. Given the thesis of the sortal-relative spatial variation of parthood, we may assume that the mountain on the plain qua mountain exactly occupies multiple spatial regions at t, that the mountain on the plain qua mountain has o as a part at some regions it occupies at t, and that it fails to have o as a part at other regions it occupies at t.28 Note that these assumptions are stated in a meta-language with precise spatial variation-modifiers. Now consider the object-language predication of parthood ‘The mountain on the plain qua mountain has o as a part at t ’. On the vague spatial-modifier view, this sentence is elliptical for • The mountain on the plain qua mountain has o as a part, at p, at t, where ‘at p’ is a variation modifier containing a vague spatial singular term p. Given the supervaluational truth conditions stated in (P), each utterance of this sentence is neither super-true nor super-false, since it is true on some admissible precisifications I of p—each I(p) being a region exactly occupied by Kilimanjaro at t, qua mountain—that the mountain on the plain qua mountain has o as a part, at I(p), at t, but false on other admissible precisifications of p.29 What holds for the mountain on the plain, holds for every mountain; every mountain has questionable parts. More perspicuously, 28 The definite description ‘The mountain (currently) on the plain’ is understood as having a Russellian analysis involving identity qua mountain. As pointed out earlier, several absolutely distinct aggregates of rocks satisfy this definite description. 29 I suggested earlier that the proper name Kilimanjaro is vague. This vagueness is not needed to secure the indeterminacy of ‘Kilimanjaro qua mountain has o as a part, at p, at t ’. Kilimanjaro has a range of massively overlapping, mountain-shaped aggregates of rocks, a1 , a2 , . . . , an , as admissible candidate referents. For each ai from this range, it is neither super-true nor super-false that ai qua mountain has o as a part, at p, at t, since each ai qua mountain has multiple exact locations. Hence, each admissible precisification of ‘Kilimanjaro’ has an indeterminate mereological boundary. Indeterminacy arising from spatial relativization is independent of whether or not proper names are vague.

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• For all objects x, if x is a mountain, then there is an object y and a time t, such that it is indeterminate whether x has y as a part at t. On the sortal-representation account, this universal claim of mereological indeterminacy is elliptical for the following sortally and spatially relativized one: • For all objects x, if x is a mountain, then there is an object y and a time t, such that it is indeterminate whether x qua mountain has y as a part, at p, at t. Assuming the supervaluational truth conditions (P) of sortalrelative predications of parthood containing variation-modifiers with vague spatial singular terms, this claim of mereological indeterminacy comes out true.30 In summary, the source of the mereological indeterminacy of mountains is the following. By thinking of a material object as a mountain, we represent it as exactly occupying a multitude of massively overlapping spatial regions at a time and as having its parts relative to these regions at that time. Ordinary attributions of parthood to a mountain, while sensitive to spatial relativization, fail to single out a determinate relativizing place. Such attributions are not, but can in principle be relativized to a particular place. If it makes no difference to which place, out of a range of admissible places, of places exactly occupied by the mountain, the attribution is relativized, then the attribution is true. Since it does make a difference to which admissible place mereological statements about mountains are relativized, such statements are indeterminate.31 30 To be precise, (P) requires a slight modification to handle the universal claim. Assuming that the variables x and y range over material objects and are precise, an utterance of ‘x qua K has y as a part, at p, at t ’ is super-true (super-false) iff it is true (false) on all admissible precisifications I of p that x qua K has y as a part, at I(p), at t. 31 The present picture explains how an object that is determinately a mountain can have indeterminate parts. So the picture does not locate the source of mereological indeterminacy in the vagueness of the sortal term ‘mountain’; mereological indeterminacy arises whether or not ‘mountain’ is vague. (For a view that does locate the source of mereological indeterminacy in the vagueness of ‘mountain’, see Section 7). Yet the vagueness of ‘mountain’ must be recognized. An object x is a mountain just in case x is a value of a mountain-representation. By (K), x is a value of a mountain-representation just in case x is mountain-shaped. This is, of

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One loose end remains. We commonly believe that mountains lack a determinate decomposition, that they have a fuzzy mereological boundary. But that is not all. We also commonly believe that mountains lack a determinate location, that they have a fuzzy spatial boundary. The mereological belief can be captured within the framework of sortal representation, but the spatial belief is lost, since a mountain’s exact location is determinate; the mountain exactly occupies multiple spatial regions at a time. This is a tolerable cost of the present picture. While the picture does not render the spatial belief true, it does capture the source of this belief. Spatial regions are empirically inaccessible. So where does our false belief in the fuzzy spatial boundary of the mountain on the plain come from? I suggest that this belief has its source in a true belief about the mountain’s parts, which are empirically accessible; namely the belief that the mountain has a fuzzy mereological boundary: it is indeterminate where the mountain is located, because it is indeterminate what the mountain’s parts are. The spatial belief is a mere shadow of the mereological belief. Losing the former is tolerable as long as the latter is captured.32

7. alternatives The sortal-representation solution to the problem of the many is metaphysically conservative. In line with metaphysical orthodoxy, there are, in our example involving Kilimanjaro, absolutely distinct, massively overlapping, mountain-shaped aggregates of rocks on a plain, each occupying a unique place at a time, and each having a slightly different qualitative profile from the rest. At the same time, the sortal-representation solution respects our ordinary conception of mountains. In line with this conception, there is exactly one mountain on the plain. I shall close with a brief discussion of two alternative solutions that also promise a metaphysically course, a rough approximation of the application conditions of the sortal ‘mountain’ and of the predicate ‘is a mountain’. But even if those conditions are stated more cautiously, taking into account all sorts of geological and social factors that determine mountainhood, there will be objects that are neither clearly mountains nor clearly not mountains. 32 Of course, not all intuitions about location are shadows of intuitions about parthood. See the discussion of principle (U) in Sect. 5.

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conservative vindication of ordinary mountain-talk. My aim is to adduce some considerations on the topic of the previous section, mereological indeterminacy, which show that these alternatives are inferior to the sortal-representation solution. 7.1. The many are almost one Each of the mountain-shaped aggregates in our example is a mountain. If we count by identity, we get the result that there are many mountains on the plain instead of one. The aggregates, however, overlap extensively, and hence are almost (or partially) identical. In everyday contexts we do not count by identity, but rather by the weaker relation of almost-identity. That is, in ordinary contexts, the cardinality statement ‘There is one mountain on the plain’ receives the reading ‘There is a mountain on the plain, and all mountains on the plain are almost identical with it’. Since this reading is true in our case, we obtain the desired count of one mountain.33 While the almost-one solution captures our everyday cardinality judgments, it does not save the appearances completely. When mountains are individuated by almost-identity, then these mountains have inconsistent qualitative profiles. For example, it may be true that the mountain on the plain both has and lacks a certain part at t, assuming that ‘The mountain on the plain has/lacks o as a part at t’ is read as ‘There is a mountain on the plain, all mountains on the plain are almost identical with it, and it has/lacks o as a part at t ’.34 With the aim of alleviating the threat that these inconsistencies pose to our ordinary conception of mountains, one might emphasize that the qualitative differences between almost identical aggregates are small, and claim that we ignore these small differences in ordinary contexts. As pointed out in Section 2, the latter claim is incorrect; we do not ignore these small differences in all ordinary contexts. Perhaps we do ignore them when looking at a mountain from a distance. But suppose, again, that upon moving closer I point to a 33

The almost-one solution is proposed in Lewis (1993). Alternatively, ‘The mountain on the plain has/lacks o as a part at t ’ may be read as ‘Something is almost identical with all mountains on the plain, and every mountain on the plain has/lacks o as a part at t ’. On this reading, the statements ‘The mountain on the plain has o as a part at t ’ and ‘The mountain on the plain lacks o as a part at t ’ may both be false, which is just as worrying as the possibility for both statements to be true. 34

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rock at the foot of the mountain, a rock that is a part of some but not all overlapping mountain candidates on the plain, and ask ‘Is the rock a part of the mountain?’. This question draws attention to a ‘small difference’ to which we, in the capacity of ordinary speakers, are not blind. Surely we would deny that the rock both is and is not a part of the mountain. Yet this is the correct answer according to the almost-one approach. What we would respond to the question is that the status of the rock as a part of the mountain is indeterminate; it is neither clearly a part nor clearly not a part of the mountain. So, we do in ordinary contexts ascribe mountains a qualitative profile that is free of contradiction and sensitive to small differences between mountain candidates, which differences appear in our conception of mountains as indeterminacies. Since the almost-one solution to the problem of the many, in the form stated above, secures a consistent profile of mountains, as individuated by almost-identity, only if small differences are ignored, the almost-one solution does not allow taking seriously the mentioned intuitive indeterminacies, and hence fails to capture a significant portion of our ordinary conception of mountains. In light of the considerations of previous sections I conclude that the almost-one solution is inferior to the sortal-representation solution.35 7.2. The Many and the Super-One Vagueness is a linguistic phenomenon as explained by supervaluationism. The sortal term ‘mountain’ is vague in this sense. It has many clear non-cases, but no clear cases. There is a set of massively overlapping, mountain-shaped aggregates of rocks, a1 , a2 , . . . , an , each of which forms a candidate to be in the extension of ‘mountain’—in short, there is a set of massively overlapping mountain candidates. (I shall assume that ‘the set of mountain candidates’ is precise, and thereby ignore issues of higher-order vagueness.) For each of the mountain candidates, a1 , a2 , . . . , an , there are no linguistic or non-linguistic facts that settle the question as to whether it falls in the extension of ‘mountain’. So each candidate is neither 35 In response to this type of problem, Lewis proposes to combine the almostone solution with the standard supervaluationist solution; see Lewis (1993: 181–2). The latter will be criticized below, in a way that is independent of whether or not almost-identity is in the picture.

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determinately a mountain nor determinately not a mountain, and hence any precisification of ‘mountain’ can make each statement attributing mountainhood to one of the candidates either true or false. Given the vagueness of ‘mountain’, the standard supervaluationist captures the intuition that there is exactly one mountain on the plain in the following way. While each mountain candidate is neither determinately a mountain nor determinately not a mountain, it is true that there is exactly one mountain on the plain, since supervaluationism allows existential statements to be determinately true without any instance being determinately true. The trick is to require that on each admissible precisification of ‘mountain’, at most one of a range of massively overlapping candidates be in the extension of ‘mountain’. Since for any mountain candidate on the plain, ai , there is an admissible precisification of ‘mountain’ that puts ai but none of the other candidates in the extension of ‘mountain’, it is true on each admissible precisification, and hence super-true, that there is exactly one mountain on the plain.36 In securing the intuition that no mountain massively overlaps with other mountains, which grounds the desired count of mountains on the plain, the supervaluationist does not offer independent specifications of admissibility, but rather construes this intuition as a penumbral constraint on which precisifications of ‘mountain’ count as admissible. The supervaluationist claims that this strategy is innocuous, since the problem of the many does not pose the task of explaining uniqueness, but rather the mere task of sustaining uniqueness, in the sense of requiring a model in which the claim that there is one mountain on the plain is true.37 Whatever guarantees that mountains can be unaccompanied by other mountains must allow that mountains have questionable parts. This was earlier formulated as a constraint on any solution to the problem of the many. How does the standard supervaluationist solution fare with respect to mereological indeterminacy? 36 Put in terms of the scope of ‘determinately’—the -operator—the reading of ‘There is exactly one mountain on the plain’ intended by the supervaluationist is the wide-scope reading ∃!xMx, as opposed to the narrow-scope reading ∃!xMx, where M stands for ‘is a mountain on the plain’. 37 The standard supervaluationist solution appears most prominently in Lewis (1993) and McGee and McLaughlin (2000); see also Heller (1990) and Lowe (1995).

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Let us begin with the following singular claim of mereological indeterminacy: • There is an object x and a time t, such that it is indeterminate whether the mountain on the plain has x as a part at t. This is a perspicuous statement of the claim that the mountain on the plain has questionable parts. The standard supervaluationist is able to render this claim true (super-true) by holding that different admissible precisifications of ‘mountain’ put different candidates, of a range of massively overlapping candidates, into the extension of ‘mountain’. If rock o is a part of one mountain candidate on the plain at t but fails to be a part of another, massively overlapping candidate at t, then o makes our singular claim of mereological indeterminacy true. So far, so good. A major problem lies ahead, though. The mountain on the plain is not the only mountain with questionable parts. Surely, every mountain has questionable parts. This universal claim of mereological indeterminacy has the following more perspicuous form: • For all objects x, if x is a mountain, then there is an object y and a time t, such that it is indeterminate whether x has y as a part at t. On standard supervaluationism, this claim is false (super-false), since on each admissible precisification of ‘mountain’, the aggregates that are mountains have all their parts at any time of their existence determinately. The universal claim of mereological indeterminacy is therefore out of reach for standard supervaluationism; a significant shortcoming. Compare how the sortal-representation account handles this case. On the latter account, the universal claim of mereological indeterminacy is elliptical for the following sortally and spatially relativized one: • For all objects x, if x is a mountain, then there is an object y and a time t, such that it is indeterminate whether x qua mountain has y as a part, at p, at t. The representational account of sortal-relative predications and the supervaluational truth conditions (P) of sortal-relative predications

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of parthood containing variation-modifiers with vague spatial singular terms (see Section 6) have the consequence that the universal claim of mereological indeterminacy comes out true. By recognizing spatial modifiers as a source of indeterminacy in predications of parthood, the sortal-representation account has the significant advantage over the standard supervaluationist account of capturing the mereological-indeterminacy intuition both in its singular and in its universal form. I conclude that from the trio of solutions to the problem of the many promising a metaphysically conservative vindication of ordinary thought and talk, the sortal-representation solution is the most powerful.38 Washington University in St Louis

references Carnap, Rudolf (1947) Meaning and Necessity (Chicago: University of Chicago Press). Deutsch, Harry (2002) ‘Relative Identity’, Stanford Encyclopedia of Philosophy; http://plato.stanford.edu/entries/identity-relative. Fine, Kit (2003) ‘The Non-Identity of a Material Thing and its Matter’, Mind, 112: 195–234. (1975) ‘Vagueness, Truth and Logic’, Synthese, 30: 265–300. Frances, Bryan (2006) ‘The New Leibniz’s Law Argument for Pluralism’, Mind, 115: 1007–22. Geach, P. T. (1967) ‘Identity’, Review of Metaphysics, 21: 3–12. (1962) Reference and Generality, 3rd edn. (1980) (Ithaca: Cornell University Press). Gibbard, A. (1975) ‘Contingent Identity’, Journal of Philosophical Logic, 4: 187–221. Gupta, A. (1980) The Logic of Common Nouns (New Haven: Yale University Press). Hawthorne, John (2003) ‘Identity’, in Oxford Handbook of Metaphysics (Oxford: Oxford University Press). Heller, Mark (1990) The Ontology of Physical Objects: Four-Dimensional Hunks of Matter (Cambridge: Cambridge University Press). Hudson, Hud (2001) A Materialist Metaphysics of the Human Person (Ithaca: Cornell University Press). 38 Thanks to Richard Dietz, John Hawthorne, Hud Hudson, Trenton Merricks, and especially Jason Turner and Dean Zimmerman for helpful comments.

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Keefe, Rosanna (2000) Theories of Vagueness (Cambridge: Cambridge University Press). King, Jeffrey C. (2006) ‘Semantics for Monists’, Mind, 115: 1023–58. Lewis, David (1993) ‘Many, but Almost One’, in Ontology, Causality and Mind: Essays in Honour of D. M. Armstrong, Bacon, John ed. (New York: Cambridge University Press), 23–42. Reprinted in his Papers in Metaphysics and Epistemology (1999) (Cambridge University Press). Lewis, David (1971) ‘Counterparts of Persons and Their Bodies’, Journal of Philosophy, 68: 203–11. Lowe, E. J. (1995) ‘The Problem of the Many and the Vagueness of Constitution’, Analysis, 55: 179–82. McGee, Vann and McLaughlin, Brian 2000: ‘The Lessons of the Many’, Philosophical Topics, 28: 129–51. Markosian, Ned (1998) ‘Brutal Composition’, Philosophical Studies, 92: 211–49. Noonan, Harold (1997) ‘Relative Identity’, in A Companion to the Philosophy of Language, eds. Bob Hale and Crispin Wright (Oxford: Blackwell), 634–52. Sattig, Thomas (2006) The Language and Reality of Time (Oxford: Oxford University Press). Schiffer, Stephen (2000) ‘Replies’, Philosophical Issues, 10: 320–43. Unger, Peter (1980) ‘The Problem of the Many’, Midwest Studies in Philosophy, 5: 411–67. Weatherson, Brian (2003a) ‘Many Many Problems’, Philosophical Quarterly, 53: 481–501. Weatherson, Brian (2003b) ‘The Problem of the Many’, Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/problem-of-many. Williamson, Timothy (1994) Vagueness (London: Routledge).

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TIME, SPACE, AND LOCATION

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9. Extrinsic Temporal Metrics Bradford Skow 1. introduction When distinguishing absolute, true, and mathematical time from relative, apparent, and common time, Newton wrote: ‘‘Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly’’ [Newton 2004b: 64]. Newton thought that the temporal metric is intrinsic. Many philosophers have argued—for empiricist reasons or otherwise—that Newton was wrong about the nature of time. They think that the flow of time does involve ‘‘reference to something external.’’ They think that the temporal metric is extrinsic. Among others, Mach, Poincar´e, and Grünbaum seem to accept this view.1 And these are not the only two views available. Perhaps both Newton and his opponents are wrong and there is no temporal metric at all. Who is right? On the standard ways of understanding general relativity, quantum mechanics, special relativity, and Newtonian mechanics, these theories all postulate an intrinsic temporal (or spatiotemporal) metric. So, although we cannot know what future theories will look like, the evidence favors an intrinsic temporal metric. There are dissenters, though; Julian Barbour does not think there is an intrinsic temporal metric, and has developed alternative physical theories that do without one.2 I will not say anything here to settle this debate. Instead, my goal in this chapter is a conceptual one. I want to clarify the relationship 1

[Mach 1960: 272–3], [Poincar´e 2001], [Grünbaum 1968]. The claim that some other bit of geometrical structure is extrinsic also shows up in many places. Reichenbach [1957: 14–37] seems to believe that the spatial metric is extrinsic, defined in terms of the behavior of rigid rods, though his writing is difficult to interpret. And some interpreters attribute to Mach the claim that the affine structure of spacetime is extrinsic, defined in terms of the large-scale behavior of matter (see, for example, Friedman [1983: 67]). 2 [Barbour 1999] contains an elementary summary of Barbour’s theory.

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between the claim that the temporal metric is extrinsic and conventionalism about time. According to conventionalism, some appeal to our conventions must be made to explain how there could be an extrinsic temporal metric. I will argue that conventionalism is false. Extrinsic temporal metrics are as non-conventional and objective as the intrinsic temporal metric that Newton believed in. Section 5 contains my argument, and a presentation of some alternatives to conventionalism. Sections 6 and 7 consider objections to my view. Sections 2 to 4 are devoted to preliminaries. I say more about the difference between intrinsic and extrinsic metrics, and propose a definition of ‘‘conventionalism about the temporal metric.’’

2. intrinsic metrics, extrinsic metrics We can represent a temporal metric, mathematically, in many ways: as a function from pairs of times to real numbers, for example (where the number gives how many seconds elapse between those two times); or as a function from points of spacetime to functions from tangent vectors to real numbers (that is, as a one-form). In this chapter I take a temporal metric to be a relation like x and y have the same temporal length, or the same amount of time passes during x as during y—a two-place congruence relation on time intervals that (together with the betweenness relation on instants of time) satisfies some standard set of geometrical axioms, such as Hilbert’s axioms for neutral geometry. (I assume that the other mathematical representations can be recovered from the facts about how this relation is instantiated using a representation theorem. I also assume that we do not live in a relativistic world and that substantivalism is true, so that there are such things as times and time intervals to instantiate these relations. But I make these assumptions only to simplify the discussion. I believe that much of what I say remains true even if they are dropped. Of course, in the context of relativity theory, I would be discussing extrinsic spatiotemporal metrics.) What, then, does it mean to say that the temporal metric is intrinsic, or that it is extrinsic? There are standard intuitive explanations of what it is for a property or relation to be intrinsic or extrinsic. An intrinsic property is one that characterizes something that has it as it is in itself. What intrinsic properties something has does not

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depend on what else there is, what those other things are like, or how it is related to them. Similarly, an intrinsic relation is one that characterizes some things as they are in themselves. (For example, it is plausible that the relation x has the same mass as y is intrinsic: the existence of things other than a and b (and their parts) plays no role in determining whether a and b are equally massive.3 ) Extrinsic properties and relations are those that are not intrinsic. This explanation is not entirely clear: what does ‘‘in itself’’ mean, and what sort of dependence is at work? Many philosophers have attempted to give more precise definitions of ‘‘intrinsic.’’4 For this chapter I will not need the precision that these definitions provide, so I will be content to use the intuitive explanation. Figuring out whether a temporal metric is intrinsic or extrinsic is just a matter of applying this general characterization to the relevant temporal congruence relation. Indeed, we then end up back at the characterization Newton uses: extrinsic metrics do, and intrinsic metrics do not, involve ‘‘reference to something external’’—something other than times. Let us look at an example. Suppose that for two temporal intervals to have the same length is for the Earth to rotate through the same number of degrees during each of them. Then the temporal metric is extrinsic. Congruence of temporal intervals is analyzed in terms of the concrete physical processes that occur during those intervals—in this case, the rotation of the earth.5 In my example the temporal metric is analyzed in terms of the rotation of the Earth. But of course it may be that the temporal metric is extrinsic but congruence of temporal intervals is analyzed in terms of some other physical process. It might, for example, be 3 This relation is also usually thought to be internal: it supervenes on the intrinsic properties of its relata. But it is not necessary that intrinsic relations are internal. 4 See, for example, papers by Weatherson, Lewis, and others in the September, 2001 issue of Philosophy and Phenomenological Research. Horwich [1975] and Glymour [1972] discuss how to define ‘‘intrinsic’’ in the context of the debate about whether the metric is intrinsic. 5 One interesting feature of this analysis is that it presupposes that there is a spatial metric. The analysis makes sense only if there are facts about whether the Earth has rotated through the same distance during two intervals. (For all that has been said, though, the analysis does not presuppose that the spatial metric is intrinsic.) Is it possible to give an extrinsic analysis of the temporal metric (or the spatial metric, for that matter) without presupposing that there is any other metric around? I do not know.

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analyzed in terms of the period of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Or it might have some more complicated analysis. But I will continue to use the Earth’s rotation as my canonical example. Is any relation that has an analysis relevantly similar to the two examples given so far a candidate extrinsic temporal metric? What about the relation my watch ticks the same number of times during x as during y? Call this relation R. If the temporal metric is extrinsic, could it turn out that R is the temporal metric? It depends on how my watch behaves, but the likely answer is ‘‘no.’’ Suppose, as is likely, that my watch stops ticking some time in the next fifty years. I assume that there is not a last moment of time, so there are infinitely many times after the last tick of my watch. So every time interval located after the last tick of my watch bears R to every other such interval. But then R does not behave the way a congruence relation must behave. For example, it follows from the axioms of neutral geometry that each interval has a unique midpoint. But if t1 and t3 are times after my watch stops ticking, there is no unique time t2 between them such that the interval from t1 to t2 bears R to the interval from t2 to t3 . (These considerations probably also bar the Earth rotates through the same number of degrees during x as during y from being the temporal metric. But I do not think they bar every extrinsic relation on temporal intervals from being the temporal metric. To keep things simple, I will stick with the Earth rotates through the same number of degrees during x as during y as my main example.) Now that we have seen what an extrinsic temporal metric might look like, I want to make two points about such metrics. First, to say that the temporal metric is extrinsic is not merely to say that the time between two events depends, as a matter of physical law, on what physical processes happen between those two events. Something like this is true in general relativity; but in general relativity (standardly interpreted) the (space)time metric is not extrinsic. Second, to say that the temporal metric is extrinsic is not to say that the Earth turns out to be a perfect instrument for measuring independent facts about the lengths of temporal intervals. If the temporal metric is extrinsic, then the Earth’s rotation does not measure the time; it sets the time.

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So much for extrinsic temporal metrics. What about intrinsic temporal metrics? Unlike extrinsic metrics, intrinsic temporal metrics cannot be analyzed in terms of periodic physical processes. But it is hard to give a positive characterization of them. Most who believe that the temporal metric is intrinsic also deny that anything informative can be said about what makes two temporal intervals the same length: this is just a fundamental fact about those time intervals. Under certain assumptions about the topology of time, there may be something informative to be said about an intrinsic temporal metric. Suppose that time has the order-type of the integers—each instant of time has an immediate successor and an immediate predecessor. Then one might say that what makes two temporal intervals the same length is that they contain the same number of instants. A temporal metric defined in this way using the temporal topology would be intrinsic. But in our world time is densely ordered; there are infinitely many instants between any two. So if there is an intrinsic temporal metric in our world, it cannot be analyzed in this way.6 Let me make two more points about intrinsic temporal metrics, to make sure we have grasped the concept. First, even if the temporal metric is intrinsic it is possible that the Earth’s rotation is a perfect instrument for measuring time. That is, it is possible that two temporal intervals have the same length just in case the Earth rotates through the same number of degrees during each of them. But in this situation it is not the Earth’s rotation that makes it the case that the two temporal intervals have the same length. Second, that we refer to particular physical processes when we define the unit we use to measure time does not entail that the temporal metric is not intrinsic. In fact if we define our unit by saying something like, ‘‘temporal interval x is one day long iff as much time passes during x as during the temporal interval occupied by the last complete rotation of the Earth,’’ then we presuppose that there are already facts about which temporal intervals have the same length. Typically, people who disagree about whether the temporal metric is intrinsic or extrinsic also disagree about some associated 6 Grünbaum [1968: 12–13] seems to believe that if there is an intrinsic temporal metric, then it has an analysis in topological terms. Sklar [1974: 109–12] and Friedman [1983: 301–9] dispute Grünbaum’s claim.

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modal claims. Those who believe that the metric is intrinsic typically believe that time could pass in an otherwise empty universe; those who think that the metric is extrinsic typically believe that there can be no time without change. It is easy to see why these modal claims are typically associated with the corresponding view about the nature of the temporal metric. If metaphysical possibility is governed by some kind of combinatorial principle, then that the metric is intrinsic entails that time could pass in an otherwise empty universe. And the standard examples of extrinsic metrics have it that the amount of time that passes between two instants is defined in terms of the physical processes that occur between those two instants. For metrics like that there can be no time without change. (The definition of ‘‘extrinsic metric’’ leaves it open, though, that the amount of time that passes between those two instants is defined in terms of events and processes that do not occur between them.)7 There is another important modal difference typically associated with the difference between intrinsic metrics and (some) extrinsic metrics. If the metric is intrinsic, then every physical process could have happened faster than it actually happens. Indeed, if the metric is intrinsic, then the entire universe could have evolved in time faster than it actually does: it could have passed through the same sequence of instantaneous states that it actually does, but at a faster rate.8 (If it did so, some laws of physics would be violated; this possibility is not a physical possibility.) But if the metric is defined in terms of the Earth’s rotation then this is not possible. Since, in that case, how fast some physical process happens involves reference to another physical process (say, the Earth’s rotation), it could not happen that all processes happen faster than they actually do.9 7

Thanks to Frank Arntzenius for pointing this out. Cross-world comparisons of speed are tricky, especially if you think that the fundamental facts about temporal distance are facts about which temporal intervals are congruent to which other temporal intervals. But there is a similar modal difference between intrinsic and extrinsic temporal metrics that can be stated without making cross-world comparisons of speed. If the temporal metric is intrinsic, then there is a possible world w in which the universe passes through all the same instantaneous states that it actually passes through; but while in the actual world it takes the same amount of time to go from (say) state S1 to state S2 as it takes to go from state S3 to state S4 , in w it takes longer to go from S1 to S2 than it takes to go from S3 to S4 . 9 Three comments. First, I have said that if an extrinsic relation like the Earth rotates through the same number of degrees during x as during y is the temporal metric, then it 8

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Proponents and opponents of intrinsic metrics sometimes exploit this modal difference to argue for their view. In the scholium, Newton argues: any process could have happened faster than it actually does; so the temporal metric is intrinsic. Some of Newton’s opponents argue: it is not possible that every process could have happened faster than it actually does; so there is no intrinsic metric (See, for example, [Barbour and Bertotti 1982: 296]). Of course, if the temporal metric is extrinsic, some processes could happen faster than they actually do. It could have happened that the Earth rotated fewer times while Jupiter completed one orbit than it actually does. In a possible world where it does (and the temporal metric is defined in terms of the Earths’s rotation), Jupiter is orbiting faster than it actually does. But the Earth could not have rotated on its axis faster than it actually does.10

3. a puzzle about extrinsic metrics I said that there is a third possible view: that there is no temporal metric. (Even if there is no temporal metric there is still, I assume, a temporal topology.) It looks easy to say what the temporal facts are, according to this view. If there is no temporal metric, then while the question, ‘‘are those two particles getting closer together?’’ has an answer, the question, ‘‘how fast are they getting closer together?’’ is impossible for the entire universe to evolve in time faster than it actually does. But this may not be true of all possible extrinsic temporal metrics. In no. 16 below I give an example of an extrinsic metric for which it may be false. Second, as I said earlier, whether one recognizes these modal differences depends on one’s background modal assumptions. If you have strange enough views about modality, then you will not see a modal difference between intrinsic and extrinsic metrics. For example, if you think that all truths are necessary (an insane but consistent view), then you will think that even if the metric is intrinsic, it is impossible that every process happen faster than it actually does. Third, I want to make a note on the strength of the modality at work in this paragraph. Since a relation’s analysis is essential to it, ‘‘could have’’ and ‘‘possible’’ here express metaphysical possibility. So if the temporal metric is extrinsic, it could not have been intrinsic; those who disagree about whether the temporal metric is intrinsic or extrinsic disagree about a necessary truth. 10 Let T be the interval occupied by the most recent complete rotation of the Earth. It is possible that that rotation of the Earth occupy time interval T’ instead, where T’ is a proper part of T. But this does not entail that the Earth could have rotated faster than it actually does. Instead, (given that the temporal metric is extrinsic,) T would have been a longer temporal interval than it actually is.

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does not. That is very different from what we say if there is a temporal metric, either intrinsic or extrinsic; if there is either kind of temporal metric, then the second question does have an answer. In more abstract terms, we can put the difference like this: to say that there is a temporal metric (intrinsic or extrinsic) is to say that time has a geometrical structure. To say that there is no temporal metric, on the other hand, is to say that time lacks a geometrical structure. And there is certainly a difference between having and lacking a certain kind of structure. But there is a puzzle here. From another point of view it is difficult to say just what distinguishes the claim that there is a temporal metric from the claim that there is not. The problem is distinguishing between the claim that there is an extrinsic temporal metric and the claim that there is no temporal metric at all. Suppose that the temporal metric is extrinsic, and that it has the canonical analysis I have been using: for two temporal intervals to be the same length is for the Earth to rotate through the same number of degrees during each of them. An opponent denies that there is a temporal metric at all. But he still accepts that there is such a relation as the earth rotates through the same number of degrees during interval x as during interval y. Since he denies that there is a temporal metric, he denies that this relation is the temporal metric. It is not identical to the relation temporal interval x is the same length as temporal interval y. But what does this disagreement amount to? What does he think this relation is failing to do that the believer in an extrinsic temporal metric thinks it is succeeding in doing? What does an extrinsic relation on times have to do in order to be the relation that is responsible for time’s geometrical structure? We can also put the puzzle as a challenge to those who believe that the temporal metric is extrinsic. Suppose that, as before, the temporal metric is extrinsic, and that for two temporal intervals to be the same length is for the Earth to rotate through the same number of degrees during each of them. I have been speaking as if, in this circumstance, there is a unique extrinsic temporal metric. But why are there not lots of them? The relation Venus rotates through the same number of degrees during interval x as during interval y appears to be as good a candidate as the Earth rotates through the same number of degrees during interval x as during interval y for the job of temporal metric. Both are extrinsic relations on temporal intervals that satisfy

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the relevant set of geometrical axioms. What is the latter relation doing, then, that the former is not, that explains why it gets the job? One might think that these questions have a false presupposition: that the idea of an extrinsic temporal metric makes sense to begin with. Perhaps it is part of the concept of geometrical structure that something’s geometrical structure is intrinsic. If so, then there are really only two positions in logical space—either there is a temporal metric (in which case it is intrinsic), or there is not a temporal metric. To the extent that we are puzzled about the difference between the claim that the temporal metric is extrinsic and the claim that there is no temporal metric at all, it is because we are trying to make sense of an inconsistent view. But I do not think that it is part of the concept of geometrical structure that something’s geometrical structure is intrinsic. In fact, several people have defended views according to which some kind of geometrical structure is extrinsic. One example is the view that material bodies inherit their geometrical properties (including their shapes) from the regions of space they occupy. On this view, the shapes of material bodies are extrinsic.11 Another example is the platonist view that the fundamental spatial relation is the distance between x and y is r—a relation between points of space and numbers. This is a view according to which the geometry of space is extrinsic.12 When I reflect on these views, they seem like coherent views that we should take seriously, not views that are analytically false. So to ask what distinguishes the claim that the temporal metric is extrinsic from the claim that there is no metric at all is not to ask a question with a false presupposition. What, then, is the answer?

4. the conventionalist answer The puzzles in the previous section are generated by the fact that many extrinsic relations on temporal intervals are formally 11

I discuss this view in my [Skow 2007]; [McDaniel 2007] defends it. Field [1989] calls this view ‘‘heavy duty platonism.’’ I do not know of many explicit defenses of this view, but I suspect that many philosophers believe it. Mundy [1983] defends a view of this kind (though on his view the fundamental spatial relation is not the distance between x and y is r, but is some other relation to numbers). In my [Skow 2007] I defend the claim that on this view geometry is extrinsic, but I do not defend the view itself. 12

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eligible to be temporal metrics—they satisfy the relevant geometrical axioms—but nevertheless fail to be temporal metrics. So while satisfying those formal requirements is necessary, it is not sufficient, for being a temporal metric. What other conditions are required? In this section I will discuss an answer to this question that I will call ‘‘the conventionalist answer.’’ The claims that comprise the conventionalist answer resemble claims made by historical conventionalists like Poincar´e and Grünbaum. But I am not sure they would recognize the way that I have articulated the puzzle that conventionalism is supposed to solve. In any regard, my interest is in the conventionalist answer’s merits as an answer to my question, not its merits as an interpretation of those philosophers. According to the conventionalist answer, the Earth rotates the same number of degrees during x as during y gets to be the temporal metric because we have established linguistic conventions according to which it, and none of its competitors, is the semantic value of our expression ‘‘temporal interval x is the same length as temporal interval y.’’ (These conventions may not be explicit; they may be implicit in the way we use our words.) But conventionalists do not think that we established such conventions because this relation was more intrinsically worthy than its competitors. Either our decision was arbitrary, or it was based on other criteria. Here is my official statement of conventionalism about the temporal metric. It comprises two claims: (1)

Necessarily, all the extrinsic relations that are candidate meanings for ‘‘the same amount of time passes during x as during y’’ are equally qualified.

(2)

Necessarily, there is an extrinsic temporal metric just in case (there is no intrinsic temporal metric and) our linguistic conventions select one of the candidate extrinsic relations as the semantic value of ‘‘the same amount of time passes during x as during y.’’ The selected candidate is the temporal metric.

Some of the terms that appear in this statement need more explanation. First, ‘‘candidate meaning’’: by ‘‘candidate meaning’’ I just mean a relation that is formally eligible to be a temporal metric—it obeys the right geometrical axioms to be a congruence relation on times. Next, conventionalism says that the candidate meanings are ‘‘equally qualified’’ or ‘‘on a par.’’ What do I mean by these terms?

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They are best explained by looking at an example in which (the analog of) the conventionalist answer is clearly correct. Think about planets. What is a planet? Once upon a time, the answer was obvious: the planets were the relatively large bodies orbiting the sun; and we identified nine of them. But then our telescopes got better and we got into trouble. We discovered bodies orbiting the sun that are larger than Pluto; but we balked at calling them ‘‘planets,’’ even though ‘‘If something orbits the sun and is larger than some planet, then it is itself a planet’’ sounds analytic. Astronomers found this situation intolerable, so they got together to legislate a new meaning for ‘‘planet.’’ (Or, if you like, to precisify ‘‘planet’’—maybe you do not think precisifying a word gives it a new meaning.) It took them a long time to do this and some of them got very angry. Why? Because, given the recent astronomical discoveries, there is no natural boundary dividing the set of bodies that orbit the earth into the larger bodies and the smaller bodies.13 All the candidate meanings for ‘‘planet’’ are on a par. If there were a natural boundary, the obvious thing for the astronomers to do would be to identify that boundary with the boundary between the planets and the non-planets. Without a natural boundary the astronomers had no guidance. Forced to give ‘‘planet’’ some new meaning, they ‘‘made up’’ a boundary of their own. The difference between planets and non-planets, on the new meaning of ‘‘planet,’’ is for this reason merely conventional. Alien linguists learning our language for the first time might find themselves wondering what one of the smaller planets is doing that one of the larger asteroids is not, that explained why only one of them is a planet. The conventionalist answer (applied to planets) would dissolve their puzzlement. Conventionalists about the temporal metric think we are (or were at some point) in the same position as the astronomers. For practical reasons we could not just leave the predicate ‘‘the same amount of time passes during x as during y’’ meaningless, or completely indeterminate in meaning. We needed to precisify it, but no one precisification was better than any other. So we made an arbitrary decision. 13 Or, if there remains some natural boundary we can imagine that things turned out even worse. Suppose that we had discovered that there were bodies orbiting the sun of almost every size between the size of the Earth and the size of some tiny asteroid.

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When I say that the decision was ‘‘arbitrary,’’ I do not mean that (according to conventionalism) we had no reason at all to choose one meaning rather than another. Perhaps choosing one of the eligible relations rather than the others made for easier mathematical calculations when we did astronomy, or rocket science. Or perhaps choosing one of them made it easier to measure time. (Choosing the rotation of Pluto as our standard, for example, would make it very hard to measure the time.) As I will understand conventionalism, it contains the claim that pragmatic reasons like these are the only kinds of reasons we could have to choose one of the candidate meanings. Before I turn to evaluating the conventionalist answer (applied to temporal metrics), I want to say a word about a famous complaint about conventionalist views. Putnam [1975a], following Eddington, claimed that conventionalism about the temporal metric is just a consequence of Trivial Semantic Conventionalism. Trivial Semantic Conventionalism is the view, roughly, that our words mean what they do because of the linguistic conventions we have established. Putnam claims that Trivial Semantic Conventionalism is obviously true and that everyone knows that it is. If conventionalism about the temporal metric follows from it, he argued, then it is also obviously true and philosophically uninteresting. It should be clear that what I am calling ‘‘conventionalism about the temporal metric’’ does not follow from Trivial Semantic Conventionalism. Conventionalism about the temporal metric contains a claim about the nature of the candidate meanings for ‘‘temporal interval x is the same length as temporal interval y.’’ It is the claim that all these meanings are on a par—that none stands out from the others (claim (1) above). The denial of this claim is compatible with Trivial Semantic Conventionalism. Even if the candidate meanings are not all on a par, we could still get ‘‘temporal interval x is the same length as temporal interval y’’ to mean any one of them by making the right sorts of stipulations. We could even get it to mean some relation that is not a candidate at all—a relation that is not a congruence relation on times. We could get it to mean what we actually mean by ‘‘x is at least three feet away from y,’’ if the way we used the predicate departed radically from the way we actually use it. Since Trivial Semantic Conventionalism is compatible with the denial of conventionalism about the temporal metric, it does not entail it.

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If the conventionalist answer is correct, then it helps explain why it is hard to see just what someone who thinks the temporal metric is extrinsic and what someone who denies there is a temporal metric at all are arguing about. When I look at their dispute, it looks like a dispute about the answer to a metaphysical question: whether time has a certain kind of structure. But if conventionalism is correct, then it is not. (Maybe some conventionalists did think that the geometrical structure of time was somehow something we had power over—that time’s structure depended on what conventions we established. But this is no part of conventionalism as I am understanding it.) According to the conventionalist answer, the parties to this dispute agree about what the world is like. They have just established different linguistic conventions.

5. robust extrinsic metrics Conventionalism stands or falls with the truth of claim (1). (2) is only plausible if (1) is true; someone who accepts (2) but not (1) could fairly be accused of endorsing Trivial Semantic Conventionalism. I think the conventionalist answer is wrong because I think (1) is false. It is not a necessary truth that all candidate meanings for ‘‘the same amount of time passes during x as during y’’ are on a par. Why would anyone believe (1) in the first place? Perhaps for the following reason. It is obvious that obeying certain geometrical laws is necessary for being the temporal metric. But all of the candidate relations satisfy those laws; so this feature does not pick out one from the rest. There are, of course, other features that do discriminate between candidates. An alternative to conventionalism, a theory according to which (1) is false, will have to identify one (or more) of these features as a feature that is relevant to determining whether a candidate relation is the temporal metric. But it is hard to see how any of those features could be relevant. For example, the Earth rotates through the same number of degrees during x as during y makes reference to a terrestrial planet, while Jupiter rotates through the same number of degrees during x as during y makes reference to a gas giant. But this hardly makes the first relation more qualified to be the temporal metric than the second relation. While I agree about this particular example, I deny that, in general, there are no features in addition to formal eligibility that are relevant to determining whether a candidate relation is the

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temporal metric. Consider the following historical example, which I will use as an analogy. Descartes was a relationalist about motion. He thought that all motion is the relative motion of material bodies. But he did not think that all relative motions were on a par. One kind of relative motion was special. According to Descartes, something undergoes true, philosophical motion just in case it is separating from the bodies immediately contiguous to it, considered as at rest [Descartes 1991: part II]. And what made this state of relative motion special, what separated it from the others, is that it was this state of motion that the laws of motion (allegedly) governed. Descartes’ principle of inertia says that a body at rest will remain at rest, and a body in motion will continue moving in a straight line at constant speed, unless acted on by an outside force (which for him meant: unless struck by another material body). So for Descartes something that is not currently separating from the bodies immediately contiguous to it, and is not being struck by some other material body, will continue not to separate from the bodies immediately contiguous to it. But his laws say nothing about the conditions under which a body’s other relative motions will change.14 Just as the laws may distinguish one kind of relative motion from the others, the laws may distinguish one candidate meaning for ‘‘the same amount of time passes during interval x as during interval y’’ from the others. Suppose that there is no intrinsic temporal metric. It could turn out that the Earth rotates through the same number of degrees during x as during y plays a role in the dynamical laws that no other candidate relation plays. Maybe it is something’s speed relative to the Earth’s rotation that determines, in accordance with the laws, what it will do next. This example suggests the following alternative to conventionalism (call it ‘‘the simple view’’). In addition to formal eligibility, playing the ‘‘temporal metric role’’ in the laws is a feature relevant to determining which candidate is the temporal metric.15 That is, 14 Of course, as Newton [2004a] argued, Descartes’ laws do not really fit all that well with his definition of ‘‘true, philosophical motion.’’ But that should not be a problem for the analogy I wish to draw. 15 I will not give a precise definition of ‘‘the temporal metric role.’’ Different versions of the simple view might define this term differently. None of my arguments turn on how this term is defined.

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whichever candidate (actually) plays that role is the temporal metric. If we discover that that there is no such role to be played in the laws, then we will have discovered that there is no temporal metric. At this point the conventionalist will ask: why does playing the temporal metric role in the laws earn one of the candidate extrinsic relations the right to be the temporal metric? I will answer: because by playing that role it is doing the sort of thing that temporal metrics do. Do not misunderstand the simple view. On this view, neither the laws nor the role that the temporal metric plays in the laws appears in the temporal metric’s analysis. The analysis makes reference only to the earth’s rotation. The fact that the earth rotates through the same number of degrees during x as during y plays the temporal metric role in the laws serves only to justify the claim that this analysis is correct. If you believe that the temporal metric is extrinsic, and you adopt this simple view, you are going to face some tough questions about the modal consequences of your view. Suppose you claim that extrinsic relation R is the temporal metric. And to justify this claim you cite the fact that R plays the temporal metric role in the laws. Do you think that it is necessary that R plays that role in the laws? You might be in trouble either way. If you say ‘‘yes’’, some will say that your view places implausible restrictions on what the laws might be. If you say ‘‘no’’, then others will say that your view leads to skepticism. (They will say: what about those poor people in possible worlds at which some extrinsic relation other than R plays the temporal metric role in the laws? Those people will end up with false beliefs about which relation is the temporal metric. We are lucky we are not in one of those worlds. But if it is just a matter of luck that our belief that R is the temporal metric is true, then that belief is not justified.) If you do not like either answer but you do not want to be a conventionalist, there are alternatives to the simple view. Here is one: do not identify the temporal metric with the first-order relation that plays the temporal metric role in the laws. Instead, identify it with a second-order relation: x and y instantiate the unique relation that plays the temporal metric role in the laws.16 16 If the temporal metric is some kind of second-order relation defined in terms of the laws, then it may be possible for the entire universe to evolve in time faster than it actually does. This is possible if there is a possible world in which the universe passes

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I do not think the simple view is in that much trouble, though. If I were to defend the simple view I would deny that it is essential to the temporal metric that it play the temporal metric role in the laws. But I would also deny that this leads to skepticism. After all, R is the temporal metric. Possible worlds with strange laws that do not relate in the usual way to the temporal metric are going to be very far away from the actual world. If far-off worlds where I am a brain in a vat do not prevent me from knowing I have hands, these far-off worlds should not prevent me from knowing that R is the temporal metric. But my goal in this chapter is not to defend any particular alternative to conventionalism. I only claim that some alternative is viable. (I have only discussed alternatives to conventionalism according to which the feature playing the temporal metric role in the laws is the feature that makes the relation that has it into the temporal metric. But there may be defensible alternatives to conventionalism—analogs of either the simple view or the second-order alternative to the simple view—that pick some other feature.17 )

6. two objections In my argument against conventionalism I gave an example that (I said) shows that some candidate extrinsic relation could stand out from the rest by playing a special role in the laws. Doubts might be raised, though, about whether the scenario in the example is really possible, and about whether it achieves its purpose. Laws in which an extrinsic relation (like, for example, one that makes reference to my watch) plays the temporal metric role are through all the instantaneous states that it actually does, but the laws are different, and these different laws select as special a first-order extrinsic relation other than the one the actual laws select. Whether there is such a possible world is open to dispute. (Humeans about laws (for example, [Lewis 1983]) will deny it.) 17 I do want to briefly mention one alternative, since it looks at first more appealing than the alternative I use in the body of the chapter. Why not invoke the distinction between fundamental and non-fundamental properties, and say that (for example) the Earth rotates the same number of degrees during x as during y is the temporal metric and Venus rotates the same number of degrees during x as during y is not because only the former is a fundamental relation? This will not work, because in fact neither relation is fundamental. (Since these relations are extrinsic, they have analyses; and any relation with an analysis is non-fundamental.) Nor will it work to say that the former relation is, at least, more fundamental than the latter. In so far as I can judge such things, these two relations look equally fundamental.

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strange. If laws like that are actual, then how something in the Andromeda galaxy—a long way away—behaves depends on how fast it is moving relative to my watch. Laws like that are non-local, for one thing, and they are non-qualitative, they ‘‘make reference’’ to a particular thing, for another. One might object to what I have said, claiming that such laws are impossible. In Section 7 I will discuss this objection and dispute the claim that if an extrinsic relation plays the temporal metric role in the laws, then the laws must be non-qualitative. There is another objection to my example I want to discuss first. One might claim that it is impossible for some laws to distinguish one candidate extrinsic relation from the others, and defend this claim like this: Suppose we have written down an accurate statement of the laws that refer to your watch in their definition of ‘‘velocity’’ and other relevant terms. This does not show that your watch plays a special role in the laws that mine does not. That is because we could also write down a statement of the laws that refer to my watch in these definitions. If we make appropriate changes elsewhere, the new law-statements will be notational variants of the old ones: they will express the same propositions in a different language. So it is not right that only one of our watches plays a special role in the laws. Once again, our watches are on a par. So you have not shown us a property that distinguishes only one of the candidate relations.18

Let us become clearer on how this objection works. The key claim being defended is (3)

It is impossible for one candidate extrinsic relation to play a special role in the laws.

The rationale we are offered for (3) is (4)

For any possible set of law-statements that makes reference to one physical process, there is another set of law-statements that makes reference to some other physical process, and which is a notational variant of the first set of law-statements.

So, on this objection, the scenario I described in the previous section merely appears to be one in which the Earth plays a special role in 18 This ploy may look familiar. It closely resembles arguments given by Poincar´e [2001: part V], Grünbaum [1968: 59–70], and Reichenbach [1957: 30–37].

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the laws. Failure to recognize that (4) is true generates this illusion of possibility. Let us evaluate this objection. What reason is there to accept (4)? Time was you could argue for (4) by establishing that the two sets of law-statements are observationally equivalent, and then using the following premise: if the new law-statements and the old law-statements have all the same observational consequences, then the two sets of law-statements are notational variants.19 If one is a verificationist then one will accept this premise; but I am no verificationist. And, verificationism aside, I do not think this premise has much going for it. Two sets of law-statements may agree on their observational consequences but differ in their theoretical apparatus. They may disagree, for example, on whether some material body is suffering a net force. In that case, they are not notational variants. So far I have just wondered whether there is any good reason to believe (4). I will now turn to positive arguments against (4). I have two of them. The first argument is short. The first set of law-statements in the indented passage entails that if anything moves, then my watch exists. The second does not; it entails that if anything moves, then your watch exists. Since the two sets of law-statements differ in their consequences, they are not notational variants. This first argument, if it is sound, establishes that every possible set of law-statements that makes reference to my watch fails to be a notational variant of a set of law-statements that makes reference to your watch. But this is a stronger conclusion than I need. I only need to defend the claim that there is some possible set of law-statements that makes reference to my watch that fails to be a notational variant of a set of law-statements that makes reference to your watch. My second argument, to which I turn next, is a defense of this weaker claim. My second argument requires some stage-setting. Let us look at a procedure one might apply to laws that mention my watch to produce new laws that mention your watch and are notational variants of the old laws. I assume that our watches do not run at the same rate; sometimes my watch runs faster than yours and 19 This version of the argument is the standard interpretation of [Reichenbach 1957]; see [Putnam 1975b] and [Friedman 1983: 296].

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sometimes slower.20 Let us assume that one of the old laws is a version of the law of inertia: free bodies move in straight lines at a constant speed, relative to my watch. Now, if we use your watch as our standard for time, then free bodies will not move in straight lines at constant speed. Instead, they will move in straight lines but they will speed up and slow down. So we could write down new laws that refer to your watch and contain a complicated alternative to the law of inertia. The law will say how a free body’s speed varies with time. (Depending on the pattern of disagreement between our watches, this new law may be incredibly complicated.) I claim that, in this example, the new laws are not notational variants of the old laws. The two sets of laws are not true in all the same possible worlds. Here is the argument: it is compatible with the old laws that our watches run at the same rate. In a possible world in which they do, free bodies move at constant speed relative to both my watch and your watch. That means that the new laws are false in that world. So there is a world in which the old laws are true and the new laws are false. Since two sets of laws are notational variants only if they are true in all the same possible worlds, it follows that the old laws and the new laws are not notational variants. Of course, I only looked at one procedure for trying to generate news laws that are notational variants of the old laws. What about other possible procedures? Could they succeed where this one fails? Any procedure will fall prey to the same kind of argument. For example, consider the following alternative procedure. Suppose that our watches are very old and each keeps a complete record of how much time has passed, according to it. So at the moment of Jesus’s birth they both read (say) ‘‘0,’’ and each has counted continuously the number of times its second hand has ticked since then in arabic numerals on its display. Since my watch runs faster than yours and sometimes slower, they do not now display the 20 Normally when I say ‘‘my watch is fast’’ I mean only that it is set ahead of local time, not that each tick of its second hand is shorter than a second. But here when I say ‘‘my watch runs faster than yours’’ I do mean that each tick of my second hand is shorter than each tick of yours. (‘‘My watch ticks faster than yours’’ makes sense even if there is no temporal metric. This is easiest to see if we suppose our watches have continuously moving second hands. Then my watch ticks faster than yours during some interval of time just in case my watch’s second hand moves farther during that interval than yours.)

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same number simultaneously. But there is some function f such that when your watch reads t my watch reads f(t). Rewrite the laws to refer to your watch instead of mine using the following procedure: first, reinterpret the variable t to refer to the number displayed on your watch; then replace all occurrences of it with f(t). These two sets of laws will then agree on how much time has elapsed between any two instants, and so will agree that free bodies move at constant speed. This procedure fails to generate new laws that are notational variants of the old laws for the same reason as the first procedure. It is physically contingent that the numbers displayed on our watches are related by f . They could, instead, be related by identity. In a possible world in which they are, our watches run at the same rate. As before, the old laws are true at that world but the new laws are false. That concludes my presentation of two arguments against the claim that it is impossible for the laws to distinguish one candidate extrinsic relation from the others. I now return to the objection I mentioned at the beginning of this section.

7. qualitative extrinsic metrics I claimed that the laws might distinguish one candidate meaning for ‘‘the same amount of time passes during x as during y’’ from the others. But I also noted a problem with this claim. All the candidates I have mentioned are extrinsic relations; but they are also non-qualitative relations. They make reference to a particular individual. (My favorite example makes reference to the Earth.) But many philosophers claim that the laws of nature are purely qualitative.21 If they are right, then for any relation R, if R plays the temporal metric role in the laws, then R is a qualitative relation. So: all my examples of extrinsic temporal metrics are non-qualitative relations. Perhaps it is necessary that an extrinsic temporal metric is not a qualitative relation. Then it follows that, necessarily, no extrinsic temporal metric plays the temporal metric role in the laws. 21 The influence of this doctrine has waned as the influence of logical positivism has waned. So among others, Lange [2000: 34–9] and Lewis [1983] deny that the laws must be purely qualitative.

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We have here an argument that apparently shows that it is impossible for the laws to distinguish one candidate relation from the others. If the argument is a good one, then I do not have an example that refutes conventionalism. That my example extrinsic temporal metric figures in the laws has related absurd consequences. If it plays the temporal metric role in the laws, then it is physically necessary that the Earth exists. But that seems hard to believe. I do not think these arguments work. I deny the premise that necessarily, an extrinsic temporal metric is not a qualitative relation. There can be qualitative extrinsic metrics. For example, suppose that for two temporal intervals to have the same length is for the center of mass of the universe to move the same distance during each of them. If this metric plays the temporal metric role in the laws, there is no particular thing such that it is physically necessary that it exists. The laws are purely qualitative. This is merely an example. I do not mean to suggest that this is the only possible qualitative extrinsic metric; others are possible. Still, I want to explore a little bit what the world would be like if that were the temporal metric. First, what is the (intrinsic) spacetime geometry? Obviously, it has no intrinsic temporal metric. But it appears that the spacetime geometry must allow us to make sense of the distance between two non-simultaneous points of spacetime. The reason is this. In order for the extrinsic temporal metric, as defined, to make sense, there must be facts about whether the center of mass of the universe has moved the same distance during two time intervals. For only then can we define two time intervals to be the same length just in case the center of mass of the universe has moved the same distance during each of them. One way for there to be facts of this sort is for our spacetime geometry to provide a notion of absolute space: for there to be facts about the spatial distance between any two points of spacetime, including non-simultaneous points. Earman [1989: 27–36] provides a list of classical (non-relativistic) spacetime geometries, ordered from those with less structure to those with more structure. The spacetime geometry I have just described is not on the list. That is because no philosopher has taken it seriously as a possible spacetime geometry. Philosophers

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who deny that there is an intrinsic temporal metric are also hostile to absolute space. But it does not follow that that spacetime geometry is impossible. It does not lead me to think that there is something incoherent in the scenario I described.22 Now for my second point. If the laws and the temporal metric are as I said, then there we have a kind of holism. How fast some material body is moving depends on how fast it is moving relative to the motion of the center of mass of the universe. But the way the center of mass of the universe moves depends on how each of the material bodies in the universe is moving. So even if some material body were located at just the places it actually is at each time, it might still have moved at a different speed, had some other (perhaps very distant) material bodies moved differently, or not existed at all. Again, I do not think this fact suggests that there is anything incoherent in the scenario I described. I think it is just what we should expect from a world with an extrinsic temporal metric. And here I can claim that this is similar to other things some believers in extrinsic temporal metrics already accept. Consider the principle of inertia: a body unacted on by any force will not accelerate. Mach, for example, does not accept this principle as Newton understands it, because Newton understands ‘‘accelerate’’ as ‘‘accelerate relative to absolute space (and absolute time).’’ Mach proposes a replacement: a body unacted on by any force will not accelerate relative to the center of mass of the universe [Mach 1960: 286–8]. That is, its distance from the center of mass of the universe will change at a constant rate.23 So Mach, who denied that there is an intrinsic temporal metric, is willing to refer to a body’s changing distance from the center of mass of the universe in his statement of the laws. This leads to the same kind of holism I found in the possible world 22 Why not use Galilean spacetime in my example, and say that two intervals of time are the same length just in case the center of mass of the universe moves the same distance during each of them, in any inertial frame of reference? The example will not work with Galilean spacetime, because that spacetime comes with an intrinsic temporal metric. 23 This replacement does away with reference to absolute space but not to absolute time. For this and other reasons, it is not adequate for Mach’s purposes. (As Huggett [1999: 187] points out and as Mach was aware, something that orbited the center of mass of the universe at a constant speed would not be accelerating, according to this definition.)

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I described, in which a body’s speed relative to the motion of the center of mass of the universe appears in the laws.

8. conclusion I began this chapter by exploring the distinction between intrinsic temporal metrics and extrinsic temporal metrics. I then discussed a puzzle: from a certain point of view it is hard to see how the claim that there is an extrinsic temporal metric differs from the claim that there is no temporal metric. What does an extrinsic relation have to do to become the relation that is responsible for time’s geometrical structure? I argued that the conventionalist solution to the puzzle is wrong. I suggested, instead, that an extrinsic relation can become responsible for time’s geometrical structure by playing some special role in the laws of nature.24 Massachusetts Institute of Technology

references Barbour, Julian (1999). The End of Time (New York: Oxford University Press). and B. Bertotti (1982). ‘‘Mach’s Principle and the Structure of Dynamical Theories.’’ Proceedings of the Royal Society of London 382: 295–306. Descartes, Ren´e (1991). Principles of Philosophy (Boston: Kluwer Academic Publishers). Earman, John (1989). World Enough and Space-Time (Cambridge, MA: MIT Press). Field, Hartry (1989). ‘‘Can We Dispense With Space-time?’’ In Realism, Mathematics and Modality (New York: Basil Blackwell), ch. 6. Friedman, Michael (1983). Foundations of Space-Time Theories (Princeton: Princeton University Press). Glymour, Clark (1972). ‘‘Physics by Convention.’’ Philosophy of Science 39(3): 322–40. Grünbaum, Adolf (1968). Geometry and Chronometry in Philosophical Perspective (Minneapolis: University of Minnesota Press). Horwich, Paul (1975). ‘‘Grunbaum on the Metric of Space and Time.’’ British Journal for the Philosophy of Science 26(3): 199–211. 24 Thanks to Frank Arntzenius, Gordon Belot, Cian Dorr, Phillip Bricker, and Liz Harman.

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Huggett, Nick (1999). Space from Zeno to Einstein: classical readings with a contemporary commentary (Cambridge, Mass.: MIT Press). Lange, Marc (2000). Natural Laws in Scientific Practice (New York: Oxford University Press). Lewis, David (1983). ‘‘New Work for a Theory of Universals.’’ Australasian Journal of Philosophy 61: 343–77. (2001). ‘‘Redefining ‘Intrinsic’.’’ Philosophy and Phenomenological Research 63(2): 381–98. Mach, Ernst (1960). The Science of Mechanics. 6th edn. (LaSalle, IL: Open Court Publishing Company). McDaniel, Kris (2007). ‘‘Extended Simples.’’ Philosophical Studies 133: 131–41. Mundy, Brent (1983). ‘‘Relational Theories of Euclidean Space and Minkowski Space-time.’’ Philosophy of Science 50: 205–26. Newton, Isaac (2004a). ‘‘De Gravitatione.’’ In Andrew Janiak (ed.), Newton: Philosophical Writings (New York: Cambridge University Press). (2004b). ‘‘The Principia [excerpts].’’ In Andrew Janiak (ed.), Newton: Philosophical Writings (New York: Cambridge University Press). Poincar´e, Henri (2001). ‘‘The Measure of Time.’’ In The Value of Science: Essential Writings of Henri Poincar´e, (New York: The Modern Library) 210–23. Putnam, Hilary (1975a). ‘‘An Examination of Grunbaum’s Philosophy of Geometry.’’ In Mathematics, Matter, and Method: Philosophical Papers Volume 1 (New York: Cambridge University Press). (1975b). ‘‘The Meaning of ‘Meaning’.’’ In Mind, Language, and Reality: Philosophical Papers Volume 2 (New York: Cambridge University Press). Reichenbach, Hans (1957). The Philosophy of Space and Time (New York: Dover Publications). Sklar, Lawrence (1974). Space, Time, and Spacetime (Berkeley: University of California Press). Skow, Bradford (2007). ‘‘Are Shapes Intrinsic?’’ Philosophical Studies 133: 111–30. Weatherson, Brian (2001). ‘‘Intrinsic Properties and Combinatorial Principles.’’ Philosophy and Phenomenological Research 63: 365–80.

10. Parthood and Multi-Location Maureen Donnelly 1. introduction An object is exactly located at (or, more briefly, located at) a spacetime region just in case the object has (at a time) the same shape, size, and position as the region.1 In particular, I assume that an object has the same dimension as any spacetime region at which it is located. A number of philosophers have claimed that material objects are exactly located at more than one region of spacetime. See for example [van Inwagen, 1990], [Hudson, 2001], [Sattig, 2006], and [Gibson and Pooley, 2006]. Philosophers who, though perhaps not advocating multi-location, at least treat it as a viable option include [Sider, 2001, 3.4], [Beebee and Rush, 2003], [Bittner and Donnelly, 2004], [McDaniel, 2004], [Crisp and Smith, 2005], [Gilmore, 2006], [Balashov, 2008], and [Hawthorne, 2008].2 While some philosophers ( [Barker and Dowe, 2003, 2005], [Parsons, 2007] ) claim not to understand multi-location, I take it that enough has been said to address their concerns elsewhere (particularly in [Beebee and Rush, 2003], [McDaniel, 2003], and [Sattig, 2006]) and proceed under the assumption that spatiotemporal multi-location counts at least as a serious possibility. 1 In this chapter, I use ‘is (exactly) located at’ to mean what [Sattig, 2006] means by ‘occupies’, what [Hudson, 2001] and [Gilmore, 2006] mean by ‘exactly occupies’, and what [Hawthorne, 2008, 275–6] means by ‘is wholly located at’. I have borrowed my gloss on ‘exact location’ from Gilmore’s explanation of ‘exact occupation’: ‘‘This relation . . . is said to hold between a thing and a region just in case . . . the thing exactly fits into the region, where this is meant to guarantee that the thing and the region have precisely the same shape, size, and position’’[2006, 200]. 2 Most of the philosophers listed here treat multi-location as the default position for three-dimensionalists (or, endurantists) who are spacetime substantivalists. Given spacetime substantivalism, material objects must be located at spatiotemporal regions. If objects are three-dimensional, their locations must be three-dimensional. But then a persisting three-dimensional object has multiple locations in spacetime, since, taken together, its locations cut a four-dimensional path across multiple times.

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Multi-location raises the question of how location interacts with parthood. Let us say that an object is ‘uniquely located’ just in case it has exactly one spatiotemporal location (and thus is not multiply located). Over suitably delineated domains of uniquely located material objects3 , the logical link between location and parthood should be straightforward: object x is part of object y if and only if x’s unique location is included in y’s unique location. But if material objects are multiply located, things are more complicated. For then some, but not all, of x’s locations may be included in some, but not all, of y’s locations. We would in this case appear to have multiple possibilities for aligning a binary parthood relation with the location relation. In fact, though, the most common move among proponents of multi-location is to abandon a simple binary parthood relation in favor of a more complex parthood relation that somehow narrows down the range of locations under consideration. [Hudson, 2001], [Bittner and Donnelly, 2004], [McDaniel, 2004], [Crisp and Smith, 2005], [Gilmore, 2006], and [Balashov, 2008] all make use of ternary region-relative parthood relations. These relations are intended to capture a sense in which x may be part of y at some spacetime regions but not at other spacetime regions. However, although most authors leave the details of their region-relative parthood relations underspecified, it is clear enough from what is said that 3 The ‘suitably delineated’ qualification is intended to accommodate philosophers such as [Doepke, 1982] or [Lowe, 2003] who think that material objects x and y may be such that at least one of x’s locations is included in one of y’s locations even though x is never (or, at no region) a part of y. For example [Lowe, 2003] claims that a statue and its structural parts (its arms, legs, etc) are never parts of the lump of bronze constituting the statue even though the lump of bronze and the statue are temporarily co-located. [Doepke, 1982] makes similar claims concerning a ship and the wood constituting it. I have a hard time making sense of this view. But I take it that even philosophers of the same mind as Doepke or Lowe would agree that on certain restricted object domains—e.g., domains limited to just artifacts or to just lumps of stuff—parthood is completely determined by inclusion relations among objects’ locations. See also [Saucedo, forthcoming] which assumes that parthood and location are wholly distinct relations and argues that it is possible that i) there are material objects x and y such that x’s location is included in y’s location but x is not (ever) part of y; and ii) there are material objects x and y such that x is part of y but x’s location is not included in y’s location. I have a hard time seeing why anyone would think that parthood and location are wholly distinct relations. (In particular, I cannot imagine why anyone would think that (ii) is possible.) But even Saucedo concedes that some possible worlds (including perhaps the actual world) are such that: for any objects x and y, x is part of y if and only if x’s location is included in y’s location.

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the different authors assume different region-relative parthood relations. For example, one obvious source of variation among these relations is in different restrictions on the regions to which parthood is relativized. [Balashov, 2008] restricts the region argument of his parthood relation to achronal regions—regions which exclude spacetime points that are absolutely temporally separated. By contrast, [McDaniel, 2004] suggests that the region argument of his parthood relation should range only over maximally continuous three-dimensional slices of spacetime.4 And neither [Hudson, 2001] nor [Crisp and Smith, 2005] place any global restrictions at all on the range of the region argument for their parthood relations. This chapter is an attempt to work out in detail some of the more promising options for interpreting and axiomatizing regionrelative parthood relations. A particular objective of this chapter is to evaluate different region-relative parthood relations in terms of, on the one hand, classical mereological principles and, on the other hand, ordinary assumptions about parthood. Here, we should not expect an exact match on either count. Whereas classical mereology assumes a binary parthood relation, all region-relative parthood relations considered in this chapter are ternary. Still, we will see that the region-relative parthood relations considered below satisfy ternary counterparts of some classical principles. Also, although parthood is never explicitly relativized to spacetime regions in ordinary discourse, we do often relativize parthood to times. It is not unreasonable to expect that the ordinary time-indexed approach to parthood can be cashed out in the spacetime context in terms of a relation that links parthood to special regions—frame-relative time-slices or other achronal regions—that roughly correspond to ordinary times. My line of approach in this chapter is to first introduce specific parthood relations over classes of mathematical models and to then assess the properties of these relations, using classical mereology as a reference point. Because of the complexity of this task, I will, perhaps somewhat arbitrarily, narrow the scope of this study in the following ways. First, I assume that parthood relations among 4 Note that whereas an achronal region could include just one point, a threedimensional slice of spacetime must include a three-dimensional space. On the other hand, there are maximally continuous three-dimensional slices of spacetime that include absolutely temporally separated spacetime points and thus are not achronal.

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material objects are, in one way or another, entirely determined by inclusion relations among their locations. Without this assumption—and in the absence of any general principle detailing what exactly, besides some form of location inclusion, is required for parthood relations to hold—we could at best attribute only extremely weak logical properties to the relations introduced here.5, 6 Second, for the most part, I will say little about summation (or, fusion) relations in this chapter. In particular, I will not consider any version of a universal summation principle guaranteeing the existence of arbitrary sums of objects. I limit my discussion of summation relations because, though important, they bring along with them cumbersome baggage that would require too much digression from the main thread of this chapter.7 (However, I do note below some cases in which the introduction of an appropriate summation relation presents special difficulties for the parthood relation under consideration.) Third, although I allow (but do not require) that objects have multiple spacetime locations, I will throughout this chapter place certain other restrictions on the way objects may be located in spacetime. Most controversially, except where noted otherwise, I assume that no more than one object is exactly located at any spacetime region. This and two other (more modest) restrictions are designed to lead to a fairly close match between the regionrelative parthood relations and ternary counterparts of classical 5

This point is easy to illustrate in the simple case of unique location. Even if all objects had unique locations, we still could not infer, e.g., that parthood is transitive from just the assumption that, whenever x is part of y, x’s unique location is included in y’s unique location. If, on the other hand, we assume that locationinclusion is necessary and sufficient for parthood, then the transitivity of parthood follows immediately from the transitivity of the inclusion relation over the domain of spacetime regions. 6 As stated in n. 3, I assume that even philosophers like Doepke and Lowe, who think that something more than location-inclusion is required for parthood, will allow that on suitably restricted domains of objects, parthood is entirely determined by inclusion relations among locations. If this is right, then these philosophers can take the results of this chapter as valid at least on these proper sub-domains of material objects. (And [Saucedo, forthcoming] can take my results as valid at least in some possible worlds.) Even with these restrictions, the considerations raised in this chapter still establish that there are important differences between the various region-relative relations proposed in recent literature. 7 Baggage in the form of: i) additional machinery in the formal mereology for quantifying over sets, plurals, or something else along these lines and ii) controversy over universal summation principles.

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mereological principles. Though there is no reason to assume that any acceptable parthood relation must satisfy all classical mereological principles, many philosophers do assume that parthood behaves something like the classical mereological relation. It is thus worthwhile to see how close the region-relative parthood relations can come to supporting a classical mereological structure. But it is easy to see how, for those who want it, material coincidence can be accommodated (by discarding the prohibition on co-location) with only a localized adjustment in the formal axiomatization. I will discuss this alternative briefly in Section 4. The remainder of this chapter proceeds as follows. In Section 2, I introduce a general class of location models (L Models) as well as the four classical mereological principles that I use as a starting point for assessing the parthood relations. In Section 3, I consider possibilities for binary parthood relations on L Models. In Section 4, I suggest two alternatives for a (time-)slice-relative parthood relation. In Section 5, I focus on the very different region-relative parthood relation proposed in [Hudson, 2001].

2. location, regional inclusion, and classical mereological principles I initially assume little more than that spacetime is a non-empty set of points, that regions are the non-empty subsets of spacetime, that every object is (exactly) located at some—but possibly at more than one—region, and that there is no region at which two objects are located. These assumptions are represented in the most general class of models introduced in this chapter—Location (L) Models. L Models are ordered quadruples ST, R, OB, L where8 8 I adopt the following notational conventions throughout this chapter. I use BOLD-UPPERCASE PALATINO for the names of distinguished classes in the models (including relations—these are represented as sets of ordered pairs). I use lower case times italics for variables ranging over members of these classes. The mereological principles considered in this chapter are presented in standard first-order predicate logic with identity. In the formal language, I use BOLD UPPERCASE HELVETICA for predicates and lower-case Helvetica for variables. The reader is strongly advised not to confuse the predicates of the formal language with the relations of the models—I will consider different model theoretic relations as alternative interpretations for the same predicate in the formal language.

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1. ST (spacetime) is any non-empty set of points; 2. R (the region domain) is the set of non-empty subsets of ST (i.e. R = ℘(ST)\{Ø})9 ; 3. OB (the object domain) is any non-empty set disjoint from R; 4. L (the location relation) is any set of ordered pairs x, r with x ∈ OB and r ∈ R such that i) for each x ∈ OB, x, r ∈ L for at least one r ∈ R (i.e., OB is the domain of L); ii) if x, r, y, r ∈ L, then x = y; iii) if x, rx , y, ry  ∈ L and rx \ry = Ø, then there is some z, rz  ∈ L such that rz ⊆ rx \ry ; iv) if x, rx , y, ry  ∈ L and rx ∩ ry = Ø, then there is some

z, rz  ∈ L such that rz ⊆ rx ∩ ry . I will say that region r ‘is a location’ if and only if some object is located at r. Thus the class of all locations in an L Model

ST, R, OB, L is the subclass of R which is the range of L.10 Conditions (4.iii) and (4.iv) ensure that there are enough locations to ‘represent’ non-empty differences between locations and nonempty intersections of locations. Condition (4.iii) tells us that if the difference of location ry in location rx is non-empty, then there is some location rz lying within this difference. Condition (4.iv) requires that if the intersection of location rx and location ry is non-empty, then some location rz lies within this intersection. Conditions (4.iii) and (4.iv) will, for suitably defined parthood relations, guarantee that the mereological structure of the object domain at least weakly corresponds to the mereological structure of the region domain—objects need not have so many parts as their locations have subregions, but an object must have some part wherever its location is divided by intersecting locations.11 9 For any set X, ℘(X) is the power set of X; i.e., ℘(X) = {Z : Z ⊆ X}. For any sets X and Y, X\Y is the difference of Y in X; i.e., X\Y = {x : x ∈ X and x ∈ / Y}. Thus, the region domain,℘(ST)\{Ø}, is the set of all subsets of ST except, Ø (the sole member of {Ø}). 10 Warning: I am using ‘location’ as a convenient technical term for a certain class of spatiotemporal regions—those spatiotemporal regions at which some object is exactly located. This usage is not intended to match the more common use of the term ‘location’ to pick out certain kinds of spatial regions through which objects might move. 11 Weaker or stronger options for regulating the distribution of locations are possible. Philosophers who think that objects are located only at three- (or four-) dimensional regions may opt for weaker versions of (4.iii)–(4.iv), requiring only

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Notice that L Models are fairly general in that they leave open not only the specific geometric structure of spacetime, but also several important questions concerning the ways in which objects are distributed to locations in spacetime. As intended, L Models allow, but do not require, that objects are multi-located. In other words, an object x may stand in the location relation L to multiple regions. Or, x may stand in L to exactly one region. Further, in cases where an object has multiple locations, L Models place no restrictions on the spatiotemporal relations holding between these locations. For example, there is no requirement that all of an object’s locations overlap as [Hudson, 2001] assumes.12 Neither do L Models require that an object’s locations are pairwise discrete, as they would be if they were distributed to discrete time-slices as is assumed in [Sattig, 2006, 2.1]. Also, there is no requirement that an object’s locations are additive in the sense that x is located at both r and r* only if x is also located at r ∪ r*.13 Following [Gilmore, 2006] and [Balashov, 2008], I use the location relation to define the path of an object x as that region which is the union of x’s locations. PATH(x) = ∪ x,r∈L (r). Notice that since locations are not required to be additive, PATH(x) need not be one of x’s locations (or, for that matter, the location of any object). In the following sections I introduce different parthood relations into L Models or extensions of L Models and use formal that some object is located within all three- (or four-) dimensional intersections or differences of locations. At the other extreme, we might require that every subregion of a location is a location. Such a requirement would entail that all non-empty intersections of locations and all non-empty differences between locations are themselves locations (as are all of their proper subregions). For the most part, the particular choice of weaker or stronger versions of (4.iii) and (4.iv) is not relevant to our concerns. 12

Regions r1 and r2 overlap if and only if r1 and r2 share some subregion. (Equivalently, r1 and r2 overlap if and only if r1 ∩ r2 = Ø.) Regions r1 and r2 are discrete if and only if r1 and r2 do not overlap. (And r1 and r2 fail to overlap if and only if r1 ∩ r2 = Ø.) 13 The additivity assumption plays a key role in [Barker and Dowe, 2003], where it is used to derive a contradiction from the claim that objects persist by having multiple three-dimensional locations in spacetime. The additivity assumption is explicitly rejected in [Sattig, 2006], [Gilmore, 2006], and [Gilmore, 2007].

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mereological principles as one way of highlighting important distinctions between these relations. The most familiar mereological principles are those used to axiomatize classical mereology. Though we will need more complex principles to evaluate the ternary regionrelative relations, I take the following four formal principles for the binary parthood predicate P as a starting point from which to arrive at plausible ternary principles.14 (CM1) Pxx (every individual is part of itself) (CM2) Pxy & Pyz → Pxz (if x is part of y and y is part of z, then x is part of z) (CM3) ∼Pxy → ∃z (Pzx & ∼∃w(Pwz & Pwy)) (if x is not part of y, then x has a part that shares no parts with y) (CM4) Pxy & Pyx → x = y (if x is part of y and y is part of x, then x and y are identical) I will call the theory axiomatized by (CM1)–(CM4) in standard first-order predicate logic ‘CM’. Classical mereologies generally introduce several different defined mereological predicates. Within CM, I introduce, besides the primitive P, only the overlap predicate O. Definition (DO ) is standard. (DO )Oxy(x overlaps y) =def ∃z(Pzx & Pzy) (some individual is part of both x and y) Notice that supplementation axiom (CM3) is equivalent to the following more compact formula: ∼ Pxy → ∃z(Pzx & ∼ Ozy) (if x is not part of y, then x has a part that does not overlap y). 14 In fact, by a ‘classical mereology’ most philosophers mean a rather stronger theory which, in addition to counterparts of (CM1)–(CM4), also includes a universal summation axiom requiring that every collection of individuals has a mereological sum (or, fusion). (See [Simons, 1987] for a detailed comparison of different mereologies and a discussion of universal summation axioms.) As stated in the introduction, I do not consider universal summation principles in this chapter. Also, I do not introduce proper parthood, differences, or intersections into CM. For these reasons, CM should be considered a weak version of classical mereology. It is strong enough, however, to lead us to interesting comparisons between the parthood relations considered in this chapter.

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We already have one suitable interpretation for CM’s primitive P. It is easy to verify that when P is interpreted as the set inclusion relation ⊆ on the region domain R (i.e., as the set of ordered pairs ⊆R ={ r, r* : r, r* ∈ R and r ⊆ r*}), each of (CM1)–(CM4) is satisfied.15 On this interpretation of P, the overlap predicate O is interpreted as the relation that holds between r, r* ∈ R if and only if r and r* overlap. More precisely, O is interpreted as OR ={ r, r*: r, r* ∈ R and r ∩ r* = Ø}. It should come as no surprise that the inclusion relation on R satisfies the axioms of classical mereology. Classical mereologies were designed with the set theoretic inclusion relation (or, what for our purposes amounts to the same thing, the partial ordering of a Boolean algebra) as an intended model theoretic interpretation for the parthood predicate (see, e.g., [Simons, 1987], [Tarski, 1956]). But unless objects can be put in correspondence with regions (or, at least, with the subclass of regions that includes all locations) in a way that exactly aligns a candidate parthood relation on the object domain with the inclusion relation on locations, it is not obvious that there is an appropriate parthood relation for objects that satisfies (CM1)–(CM4). And multi-location makes any such a correspondence between objects and their locations particularly unlikely. For, given multi-location, the way in which objects are arranged in spacetime must be radically different from the way in which regions are arranged in spacetime—regions definitely do not have multiple positions in spacetime. Thus, given multilocation, we cannot assume that there is a binary parthood relation on objects which ‘mirrors’ the inclusion relation on locations. We will see in the next section that none of most reasonable potential candidates for a binary parthood relation on the object domains of L Models satisfies (CM1)–(CM4). So multi-location seems to require some kind of revision of classical mereological principles.16 The 15 The only one of CM’s axioms whose verification on this interpretation is not entirely trivial is (CM3). To see that (CM3) is satisfied, suppose that r, r*∈ R and r ⊆ r*. Then there is some s ∈ ST such that s ∈ r and s ∈ / r*. {s} ⊆ r and (since Ø is not a member of R) there is no region which is included in both {s} and r* (i.e., {s} and r* do not overlap). 16 A caveat: The fact that the binary relations considered in the next section fail to satisfy (CM1)–(CM4) over L Models depends on the restrictions i) –iv) on the location relation. As we will see at the end of the next section, one obvious way of forcing the binary relations to satisfy (CM1)–(CM4) is to eliminate multi-location by requiring

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ternary region-relative parthood relations that are the focus of this chapter retain much of the strength of a classical binary relation by restricting parthood ascriptions to regions.

3. binary parthood relations for object domains Just what is it about multi-location that drives us away from a more traditional binary parthood relation and motivates us to look for appropriate ways of relativizing parthood to times or to spacetime regions? In fact, even if we assume that objects have multiple locations in spacetime, there are a number of different ways we might reasonably attempt to introduce binary parthood relations over object domains. Here are the four most promising possibilities. (Occasional Parthood) x, y ∈ POC if and only if there are regions rx and ry such that x, rx , y, ry  ∈ L and rx ⊆ ry . (Bound Parthood) x, y ∈ PBD if and only if, for any region rx such that x, rx  ∈ L, there is some region ry such that y, ry  ∈ L and rx ⊆ ry . (Constant Parthood) x, y ∈ PCT if and only if, for any region ry such that y, ry  ∈ L, there is some region rx such that x, rx  ∈ L and rx ⊆ ry . (Path Parthood) x, y ∈ PPT if and only if PATH(x) ⊆ PATH(y). The differences between these four binary parthood relations are most easily illustrated if we assume for the moment that spacetime that each object stand in L to no more than one region. But, instead of prohibiting multi-location, we could reformulate conditions ii)–iv) so that that they are stated in terms of objects’ paths instead of their locations. I think such reformulated conditions seem less natural, but are not obviously untenable. With this change in the structure of L Models, the fourth of the binary relations considered in the next section, PPT (path parthood), would satisfy each of (CM1)–(CM4) even on models in which objects have multiple locations. But, even so, the path parthood relation would suffer the same inadequacy as the standard four-dimensionalist binary parthood relation—it does not preserve common-sense assumptions about which parts objects have. Thus, even if by introducing more complex restrictions on the way objects are located in spacetime, we can come up with a binary relation for multiply located domains which satisfies classical principles, we still have reasons (the same reasons four-dimensionalists have) for finding an alternative parthood relation that better fits common sense.

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is partitioned by a unique collection of instantaneous time-slices and that each ordinary object—each person, table, car, and so on—has a unique location within each time-slice through which it persists.17 On this view, for example, Jane Austen (JA) is located only at subregions of three-dimensional time-slices and has a unique threedimensional location within each instantaneous time-slice between 1775 and 1817. An occasional part of JA is any object having at least one location included within one or more of JA’s locations. For example, Jane’s teeth, hair, cells, head, and hands are all occasional parts of JA. To be not just an occasional part of JA, but also a bound part of JA, an object cannot have any location that is not included in one of JA’s locations. This condition is not satisfied by objects, such as JA’s head, hands, nose, and eyes, that we ordinarily think of as JA’s most salient parts. These objects all, however briefly, survive JA’s death. But some cells are bound parts of JA. For example, any red blood cell that remains within JA throughout its short life is a bound part of JA. (However, cells that survive removal from JA are not bound parts of JA.) On the other hand, JA’s head, hands, nose, and eyes are constant parts of JA. A constant part of JA has at least one location within each of JA’s locations and JA (fortunately) retained her head, hands, nose, and eyes throughout her life. By contrast, most cells did not remain with JA for all of her 41 years and thus are not constant parts of JA. Similarly, none of JA’s teeth or hairs is a constant part of JA. Nor are any of the molecules and atoms that passed in and out of JA during her lifetime. A path part of JA is any object whose path is included in JA’s path. It is easy to see that any bound part of JA must also be a path part of JA. Moreover, if ordinary objects such as hands, cells, and so on, are located only at disjoint three-dimensional subregions of time-slices (as we for the moment assume they are), then any one of these objects will be a path part of JA if and only 17 Something like this view of objects’ locations in spacetime is presented in [van Inwagen, 1990] and in [Sattig, 2006, 2.1]. But whereas [van Inwagen, 1990] leaves open (but does not advocate) the possibility that an object is located both at subregions of instantaneous time-slices and at its four-dimensional path, I assume in these examples that ordinary objects are located only at three-dimensional subregions of instantaneous time-slices.

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if it is a bound part of JA (since in this case its path is included in JA’s path if and only if, within each time-slice through which it persists, its location is included in JA’s location). To see that path parthood need not always imply bound parthood (and thus that path parthood is strictly weaker than bound parthood), imagine that, besides objects located at multiple three-dimensional regions, there are other objects—call them ‘events’—each of which is located at a unique four-dimensional region. Then an event—a particular clenching of one of JA’s hands or a blinking of her eyes—may be a path part of JA even though it cannot be a bound part of JA, or even an occasional part of JA, since its four-dimensional location is not included in any of JA’s three-dimensional locations. One immediate dilemma in attempting to introduce a binary parthood relation over the object domains of L Models is that no one of these four binary relations clearly distinguishes itself as the parthood relation. There is no obvious reason for, say, preferring constant parthood over the three other relations. Further, taken individually, each of POC , PBD , PCT , and PPT has shortcomings that should motivate us to look for an alternative treatment of parthood (even if we concede, as I think we should, that each of the four binary relations is useful on its own terms). The most obvious problem with PBD , PCT , and PPT is that, at least on the three-dimensionalist interpretation adopted for the Jane Austen example, none of these relations preserve uncontroversial commonsense assumptions about the parts of objects like JA. We saw this above—Jane’s head does not stand in PBD or in PPT to JA, most of Jane’s cells do not stand in PCT to JA, and so on. Less obviously, POC also fails to match ordinary usage in that it provides no mechanism for specifying the different times at which objects have specific parts. Objects that never comprise JA at the same time—her baby teeth and a mole that appears in her fortieth year—all count in the same way as occasional parts of JA. In ordinary usage, we can distinguish these objects as parts of JA at different times. A different sort of issue with POC , PBD , PCT , and PPT is that each of these relations is relatively weak, with POC (perhaps the least problematic from the point of view of common sense) being the weakest of the bunch. I take it that this is a separate concern from the failure of these relations to preserve intuitive assumptions about what parts

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objects have. In non-philosophical contexts, parthood relations are not usually put to work in complicated reasoning or organizational tasks. And I cannot see that non-philosophers have intuitions about what logical properties a parthood relation is supposed to have. But many philosophers do use parthood relations to introduce structure into object domains. Temporal parts, mereological sums (or fusions), and other key players in current philosophical debates are typically defined in terms of a parthood relation.18 Moreover, philosophers do often use the assumption that mereological relations have certain fairly strong logical properties—in particular, that parthood is transitive and satisfies some version of a supplementation principle—to support controversial claims.19 It would be an unfortunate thing for these philosophers if the only available parthood relation turned out to be much too weak to support interesting reasoning about the way objects are structured. I will try to make this point clearer by using the weakest of our binary relations as an example. Of CM’s four axioms, POC satisfies only (CM1) over all L Models—the occasional parthood relation is reflexive and not much else. In particular, POC is not transitive. For example (recurring to our three-dimensionalist picture of JA), a certain protein molecule may be an occasional part of a blood cell and the blood cell may, in turn, be an occasional part of JA, even though the protein molecule is not an occasional part of JA. This would be so if the blood cell acquired the protein molecule only after leaving JA (while it was, say, lying in a vial in her doctor’s laboratory). To see that POC does not generally satisfy the supplementation axiom, (CM3), imagine that a certain almond is never located within JA. But, after the almond is destroyed by being smashed into a marzipan paste, all of its component molecules are incorporated into JA’s body.20 In this case, the almond is not an 18 Temporal parthood relations are defined in terms of mereological relations in, e.g., [Sider, 2001, ch 3]. [Sider, 2001] also includes mereological definitions of summation relations (as does, e.g., [Simons, 1987] ). 19 Recent examples of assumptions about the logical properties of (binary or ternary) parthood at work in an argument are found in: [Sider, 2001, 65], [Crisp and Smith, 2005, 335], and [Olson, 2006, 742]. In each of these cases it is a variant of the classical supplementation principle, (CM3), which is used as a premise in an argument. 20 Let us avoid complications by assuming that the almond never loses molecules or other microscopic parts. If necessary, it can be a very short-lived almond.

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occasional part of JA, but all of its occasional parts share occasional parts with JA (so, substituting into (CM3), we get a conditional with a true antecedent and a false consequent). Because it is itself so very weak, the occasional parthood relation yields an extremely flimsy overlap relation. For x and y to overlap in the POC way, it is sufficient for some object z to be located within one of x’s locations and also within one of y’s locations. But z does not have to be located in the intersection of any location of x with a location of y. Thus, x and y may overlap in the POC sense even if all of their locations are disjoint. In practical terms, JA may POC -overlap with a man she has never met if just one atom is at one time located in her body and later incorporated into his body. Another weakness in POC -overlap is that it is not transferred from parts to wholes as is the overlap relation of classical mereology. More precisely, z may POC -overlap an occasional part, x, of y even though z does not POC -overlap y. For example, JA’s red blood cell in the laboratory vial POC -overlaps a protein molecule which (we may presume) does not POC -overlap JA. What kind of a summation relation would we have in such a murky mereology? Using plural quantification, the standard definition of (atemporal) summation runs something like this: x is a sum of the ys =def each of the ys is part of x and any object that is part of x overlaps at least one of the ys. Plugging the POC relations into this definition, we end up with sums that need not extend over the entirety of their summands. In particular, x may be a POC -sum of the ys even though some (or even all) of the ys have POC -parts that are not also POC -parts of x. For example, JA is a POC -sum of all of the molecules that ever make up her body. But any of these molecules may have POC -parts that are not POC -parts of JA—this could be so if, say, a particular water molecule is at one time incorporated in JA and acquires a new electron after leaving JA. Also, not only may one plurality have two or more POC -sums (so that, in general, there is no such thing as the POC -sum of a given plurality), but the different POC -sums of a fixed plurality need not even coincide. In fact, a given plurality may have POC -sums whose spacetime paths are completely disjoint. Suppose, for example, I build a shed from a bunch of sticks and later dismantle the shed to build a fence from the same sticks. Then

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the shed is a POC -sum of the sticks and the fence is also a POC -sum of the sticks, even though the shed and the fence are never in this world at the same time. And there is worse still. Nothing in our very weak mereology prevents a whole from being a POC -sum of just one of its proper POC -parts, where a proper POC -part of x is any occasional part of x other than x itself. Suppose there is some organism x which is so constructed that every microscopic particle ever entering its body is incorporated at some time into a special organ designed to arrange particles into a suitable form. (I have no idea whether there actually is such an organism. But surely it is possible.) Let z be x’s ‘rearrangement’ organ. Then x is a POC -sum of the plurality consisting of just z—z is an occasional part of x and z POC -overlaps every occasional part of x.21 But this is really odd. Summands are supposed to, in some sense, make up all of the object to which they sum. But z is just one organ within x. We may assume that x includes other organs which are spatially separated from z. Insofar as z does not (ever) extend over these other organs, z does not seem to make up all of x. I hope the discussion above suffices to convince readers that, if such a weak relation as POC were what philosophers refer to when they make claims about parthood, overlap, summation, and so on, then much of what they say would be false or confused. We should hope that there is a reasonable parthood relation which is stronger than POC . Now, each of PBD , PCT , and PPT is slightly stronger than POC . But not much stronger. Given our restrictions on L Models, each of PBD , PCT , and PPT is (unlike POC ) transitive. But besides being reflexive and transitive, I cannot see that PBD , PCT , and PPT have any other useful properties. In particular, none of these relations satisfies anything like CM’s supplementation principle, (CM3). And none satisfies the antisymmetry principle (CM4).22 21 To see that z POC -overlaps every occasional part of x, suppose that y is an object which is at some time included within x. Then, while within x, y is made up of particles within x. By assumption, each of these particles is an occasional part of z (since each is located within z at some time). But then z and y POC -overlap—each of these particles is an occasional part of both z and y. 22 Given suitable ontological assumptions, plausible additional restrictions on how objects are located in spacetime might render one (or all) of PBD , PCT , or PPT antisymmetric. But I cannot think of any reasonable additional restrictions that would force PBD or PCT to satisfy a supplementation principle. And the supplementation

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Also, importantly, like POC , each of PBD , PCT , and PPT generates odd overlap and summation relations. For example, given that a certain stick is a constant part of both my shed and the fence that I built after dismantling the shed, the fence and the shed would PCT -overlap even though these two objects are never in the world at the same time. But even if PBD , PCT , or PPT had nicer logical properties, they would still conflict with important intuitive assumptions about parthood (that JA’s head and hands are part of JA) in the ways noted above. It is thus worthwhile investigating region-relative parthood relations in the hopes that some such alternative to the binary parthood relations will do better at preserving intuitive assumptions about parthood while maintaining a structural power comparable to that of classical mereology. Before proceeding to the investigation of region-relative parthood relations, it is worth noting that two of the major obstacles in the way of binary parthood—that no one binary relation distinguishes itself as the parthood relation and that the most obvious candidate relations have weak logical properties—disappear if multi-location is eliminated. I think that this helps account for the fact that nearly all four-dimensionalists (at least among those who deny multi-location) assume that parthood is fundamentally a binary relation. We can see how unique location makes a crucial difference in some of the issues raised above if we consider, not all L Models, but only those in which there is no multi-location. A Unique Location (UL) Model is an L Model in which every object has a unique location. In other words, the UL Models are L Models in which the principle is really the more important of CM’s axioms. Philosophers who endorse material coincidence may want to reject antisymmetry anyway (as does as, e.g., Simons in [1987, 177–87]). But philosophers of many different persuasions have assumed that a parthood relation must satisfy some version of a supplementation principle (see n. 19 above). To see that, e.g., PCT does not satisfy (CM3), suppose that, at different times, the same small particles make up two oxygen atoms, O1 and O2 . Suppose further that O1 and O2 have each of the particles as constant parts—they are each made of the same particles throughout their lives. But their lives are confined to separate times. So neither of O1 or O2 is a constant part of the other. But every constant part of O1 shares a constant part (one of the particles) with O2 . (And, likewise, each of O2 ’s constant parts PCT -overlaps O1 ). Plugging into (CM3), we have a conditional with a true antecedent and a false consequence.

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relation L is a function from OB into R. In UL Models, each object is located only at its path. POC , PBD , PCT , and PPT collapse into a single relation on UL Models. This is because, for any objects x and y in a UL Model, the following are equivalent: i) one of x’s locations is included in one y’s locations; ii) all of x’s locations are included in all of y’s locations; iii) x’s path is included in y’s path. Let PUL be the binary relation defined on UL Models as follows: (Unique Location Parthood) x, y ∈ PUL if and only if PATH(x) ⊆ PATH(y).23 It is easy to verify that PUL satisfies (CM1), (CM2), (CM3), and (CM4) over all UL Models.24 Thus, the only obvious candidate for a binary parthood relation on UL Models turns out to have nice logical properties (or, in any case, the logical properties that philosophers whose thinking about parthood has been guided by classical mereology would expect it to have). Moreover, PUL serves as the basis for a plausible overlap relation. Plugging into CM’s definition (DO ), we get the relation OUL that holds between objects x and y in UL Models just in case some object stands in PUL to both x and y. It follows from Condition (4.iv) on L Models that x, y ∈ OUL if and only if x and y are located at overlapping spatiotemporal regions (i.e., if and only if PATH(x) ∩ PATH(y) = Ø). Thus, we avoid the ugly cases of ‘overlapping’ objects located in disjoint regions of spacetime (like JA and the man 23 Notice that x, y ∈ PUL if and only if: i) for some x, rx , y, ry  ∈ L and rx ⊆ ry ; ii) for any x, rx  ∈ L, there is some y, ry  ∈ L such that rx ⊆ ry ; or iii) for any y, ry  ∈ L, there is some x, rx  ∈ L such that rx ⊆ ry . Thus, if instead of introducing PUL along the lines of PPT we had followed the format for the definitions of POC , PBD , or PCT , we would have ended up with the same relation. 24 To see that PUL satisfies the supplementation principle (CM3), suppose that x and y are objects in a UL Model and that x, y ∈ / PUL . Then PATH(x) is not included in PATH(y). Since PATH(x) and PATH(y) are locations (the unique locations of x and y), it follows from condition (4.iii) on L Models that some object z is located within the difference PATH(x)/PATH(y). Since z’s unique location (its path) lies within PATH(x), z, x ∈ PUL . Also, since no object can have a path that is included in both PATH(x)/PATH(y) and PATH(y), no object can stand in PUL to both z and y. Thus, x, y ∈ / PUL implies that some object standing in PUL to x shares no PUL -parts with y.

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who inherited her atom) that plague the POC and PCT versions of overlap.25 However, PUL suffers from the same sorts of clashes with intuition as do PBD and PPT . For, if ordinary objects have unique locations in spacetime, then each ordinary object must be located at a fourdimensional region which extends exactly as long as that object persists. On this view, Jane Austen is located not at multiple threedimensional regions (as assumed earlier), but at a single fourdimensional region which extends from December 1775 to July 1817. But in this four-dimensionalist picture, JA’s hands, head, teeth, and so on, do not stand in the PUL relation to JA, since their unique locations all extend somewhat beyond July 1817. 26 Also, many artifacts—bicycles, computers, tables—would, on this view, lack what we ordinarily take to be their most salient parts. For example, the wheels, frame, and gears of my bicycle all pre-date my bicycle. Thus, on the four-dimensionalist account, none of these objects have locations that are included in my bicycle’s location and, as a result, none stands in PUL to my bicycle. This mismatch between the four-dimensionalist’s binary parthood relation and our ordinary assumptions about parthood is noted elsewhere (in, e.g., [Thomson, 1983], [Sider, 2001], [Sattig, 2006] ). In response, four-dimensionalists have introduced supplementary ternary parthood relations—usually linking parthood to times through temporal parts—that are much closer to the ordinary notion of parthood. Thus, even though an advocate of unique location may adopt a binary parthood relation with useful logical properties, he still has an interest in finding an alternative parthood relation that better fits ordinary usage. At the end of the next section, I suggest one region-relative parthood relation—a generalization of the time-relative parthood relation of [Sider, 2001, 53]—which, under appropriate conditions, can serve this purpose for four-dimensionalists. 25 More good news: It is easy to see that a reasonable summation relation can be introduced in terms of PUL and that a universal summation principle is satisfied over the subclass of UL Models in which the union of any collection of locations is also a location. Thus PUL satisfies all of the axioms of the strongest version of classical mereology on the expected subclass of UL Models. 26 Notice that the ordinary objects which are PUL -parts of JA on this fourdimensionalist account are just those objects which are PBD -parts and PPT -parts of JA on the three-dimensionalist multi-location account assumed at the beginning of this section.

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4. slice-relative parthood relations There are two general types of strategies for introducing regionrelative parthood relations. The first sort of approach—the focus of the current section—is to restrict the region argument of the ternary parthood relation to the members of a special subclass of regions. Examples of this approach are found in [Bittner and Donnelly, 2004], which restricts the region argument to absolute time-slices27 ; [McDaniel, 2004], which (tentatively) restricts the region argument to maximal three-dimensional slices of spacetime; and [Balashov, 2008], which restricts the region argument to achronal regions. The second strategy, which will be considered in Section 5, places no global restrictions on the sorts of regions at which objects may have parts. Examples of the second approach to region-relative parthood are found in [Hudson, 2001] and [Crisp and Smith, 2006]. The first strategy assumes that there are special kinds of regions—I will call them ‘slices’—within which objects have a limited number of locations. As we will see, ideally this should be no more than one location per object per slice. In the ideal case, relativizing parthood to a slice amounts to limiting the scope of parthood claims to regions of spacetime in which objects have unique locations. Here, we should expect that at a fixed slice the logical properties of slice-relative parthood relations more or less match those of the unique location parthood relation PUL . And the primary slice-relative parthood relation introduced below does indeed satisfy natural ternary counterparts of CM’s axioms. The primary slice-relative parthood relation considered in this section, PS-3D , assumes that all locations are included in some slice. If we take slices to be three-dimensional regions roughly corresponding to times (e.g., frame-relative hyperplanes of simultaneity or other achronal regions), then PS-3D is viable only on the assumption that all objects are located at regions of no more than three dimensions. But I will also briefly consider an alternative slice-relative parthood relation, PS-4D , which is compatible with some versions 27 In [Bittner and Donnelly, 2004], it is assumed that spacetime is the sum of a unique set of non-overlapping (instantaneous) time-slices. As it stands, this approach does not accommodate relativistic treatments of spacetime. But it could easily be generalized to allow that any region consisting of those points which are simultaneous relative to any frame counts as a time-slice. This requires only that we drop the assumption that time-slices are pairwise discrete.

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of four-dimensionalism or with mixed ontologies that include both three- and four-dimensional objects. A Slice3D Model (S3D Model)28 is an ordered quintuple ST, R, OB, L, S where ST, R, OB, L is an L Model (satisfying conditions 1—4 on L Models) and where, in addition, the following condition is satisfied: 5. S (the set of slices) is a subset of R such that i) for any x, r ∈ L, there is some s ∈ S such that r ⊆ s; ii) for any x ∈ OB and s ∈ S, if

x, r, x, r* ∈ L and r, r* ⊆ s, then r = r*. Condition 5 tells us that i) every location is included in some slice and ii) no object has more than one location within any given slice. The ternary slice-relative parthood relation PS-3D is defined over S3D Models as follows:

x, y, s ∈ PS-3D if and only if s ∈ S and for some rx , ry ∈ R,

x, rx , y, ry  ∈ L and rx ⊆ ry ⊆ s. In other words, x is part of y at slice s if and only if both x and y have locations within s and x’s unique location in s is included in y’s unique location in s. We can introduce additional slice-relative relations on S3D Models. The exists at relation ES-3D holds between an object and a slice just in case the object has a location within the slice.

x, s ∈ ES-3D if and only if s ∈ S and for some r ∈ R,

x, r ∈ L and r ⊆ s. 28 The indices S-3D and S-4D distinguish between the slice-relative parthood relation which is intended for a three-dimensionalist ontology and the slice-relative parthood relation which is intended for mixed or four-dimensionalist ontologies. The corresponding indices (3D and 4D) distinguish between the different classes of models into which these parthood relations are introduced. However, nothing in the abstract conditions on S3D Models or S4D Models requires that slices are three-dimensional regions. (This should be obvious, since I do not in this chapter introduce any device for distinguishing the dimension of a region. Within the models, I do not even assume that regions are the sorts of things that can have dimensions.) But, as I will emphasize below, in the sorts of S3D Models or S4D Models which offer plausible counterparts of ordinary temporalized parthood, the slices are roughly three-dimensional subregions of a four-dimensional spacetime.

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The overlaps relation OS-3D holds between objects x and y at slice s if and only if both x and y exist at s and x’s unique location in s overlaps y’s unique location in s.

x, y, s ∈ OS-3D if and only if s ∈ S and for some rx , ry ∈ R,

x, rx , y, ry  ∈ L, rx , ry ⊆ s, and rx ∩ ry = ∅. Though less common than formal treatments of binary parthood, axiomatizations of ternary parthood predicates have been proposed in several works including [Thomson, 1983], [Sider, 2001], and [Bittner and Donnelly, 2004].29 SM (Slice Mereology) is similar to these mereologies, but is somewhat stronger in that it assumes a ternary version of CM’s antisymmetry axiom. We will see that PS-3D satisfies this principle over all S3D Models. This is not necessarily a good thing. Three-dimensionalists who endorse material coincidence (including [Thomson, 1983] and [Simons, 1987]) tend to reject the ternary version of the antisymmetry axiom and will find PS-3D too strong. I will say something later about how S3D Models can easily be modified to accommodate material coincidence. But it is worthwhile first seeing how close we can come to preserving the full power of CM in a slice-relative setting. SM is developed in a standard sorted first-order predicate logic. Its domain of quantification is partitioned into two sorts: objects (over which the variables w, x, y, z range) and regions (over which the variables s, r range). All quantification in SM is restricted to a single sort. In the presentation below, quantifier restrictions are conveyed implicitly through the conventions on variable usage. SM’s only non-logical primitive is the ternary predicate P (parthood) which takes two object variables and one region variable as its arguments.30 Two additional predicates are defined in terms of P. (DSE ) Exs (x exists at s) =def Pxxs (x is part of itself at s) 29

See also the Mereology of Continuants in [Simons, 1987, 5.2]. Instead of using a ternary parthood predicate, Simons introduces a multitude of binary parthood predicates, each of which is indexed to a distinct time. 30 Of course SM’s ternary parthood predicate is not to be confused with CM’s binary parthood predicate. I might have introduced a new symbol for the ternary parthood predicate (and yet another symbol for SM’s ternary overlap predicate), but I do not see that this extra bit of notational complication is necessary because in what follows we will work within only one mereological theory at a time.

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(DSO ) Oxys (x overlaps y at s) =def ∃z (Pzxs & Pzys) (some object is part of both x and y at s) The following five axioms govern SM’s mereological predicates. (SM0) ∃s Exs (every object exists at some region) (SM1) Pxys → Exs & Eys (if x is part of y at s, then both x and y exist at s) (SM2) Pxys & Pyzs → Pxzs (if x is part of y at s and y is part of z at s, then x is part of z at s) (SM3) Exs & ∼Pxys → ∃z(Pzxs & ∼Ozys) (if x exists at s and is not part of y at s, then some object is part of x at s but does not overlap y at s) (SM4) Pxys & Pyxs → x = y (if x is part of y at s and y is part of x at s, then x and y are identical) When the ternary predicate P is interpreted in S3D Models as the ternary relation PS-3D , the defined predicate E is interpreted as ES-3D 31 , the defined predicate O is interpreted as OS-3D 32 , and SM’s axioms (SM0)-(SM4) are satisfied over all S3D Models33 . The principles (SM0)–(SM4) are, taken together, natural ternary counterparts of the axioms of CM in that any parthood relation satisfying these axioms behaves as a classical binary parthood relation within a fixed slice. More precisely, let OB, S, P be any model for SM, where OB is the object domain, S is a subset of the To see this: (⇒) If x, s ∈ ES-3D , then for some r ∈ R, x, r ∈ L and r ⊆ s. In this case, x, r, x, r ∈ L, r ⊆ r ⊆ s, and x, x, s ∈ PS-3D . (⇐) Conversely, if x, x, s ∈ PS-3D , then for some r, r* ∈ R, x, r, x, r* ∈ L, and r* ⊆ r ⊆ s. It follows immediately that

x, s ∈ ES-3D . (⇔) Thus, x, s ∈ ES-3D if and only if x, x, s ∈ PS-3D . 32 To see this: (⇒) Suppose x, y, s ∈ OS-3D . Then x and y have, respectively, locations rx and ry in slice s where rx ∩ ry = Ø. By Condition (4.iv) on L Models, there is a location r ⊆ rx ∩ ry . Let z be the object that is located at r. Since r ⊆ rx ⊆ s and r ⊆ ry ⊆ s, z, x, s ∈ PS-3D and z, y, s ∈ PS-3D . (⇐) Suppose z, x, s ∈ PS-3D and z, y, s ∈ PS-3D . Let r, rx , and ry be, respectively, z’s, x’s, and y’s unique locations within s. Then since r ⊆ rx and r ⊆ ry , rx ∩ ry = Ø. (⇔) Thus, x, y, s ∈ OS-3D if and only if there is some object z such that z, x, s ∈ PS-3D and z, y, s ∈ PS-3D . 33 To see that PS-3D satisfies (SM3), suppose that x, s ∈ ES-3D and x, y, s ∈ / PS-3D . Let rx be x’s unique location in s. (Case 1) y does not have a location within s. Then y does not overlap any object at s. Thus, x, x, s ∈ PS-3D and x, y, s ∈ / OS-3D . (Case 2) y / PS-3D , rx ⊆ ry and rx \ry = Ø. By Condition (4.iii), has a location ry ⊆ s. Since x, y, s ∈ some location r is included in rx \ry . Let z be the object located at r. Then since r ⊆ rx ⊆ s, z, x, s ∈ PS-3D . Since r ∩ ry = Ø, z, y, s ∈ / OS-3D . In both cases, the supposition that x, s ∈ ES-3D and x, y, s ∈ / PS-3D , implies that some part of x at s does not overlap y at s. 31

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region domain, and P is an interpretation of P over OB × OB × S satisfying (SM0)–(SM4). For example, given any S3D Model ST, R, OB, L, S, OB, S, P may be OB, S, PS-3D . Now let s be any member of S and define the binary relation Ps as:

x, y ∈ Ps if and only if x, y, s ∈ P. Let OBs = {x ∈ OB: x, x, s ∈ P} (i.e., OBs is the subset of OB consisting of those objects existing at s). Then OBs , Ps  is a model of CM. In particular, for any slice s in an S3D Model ST, R, OB, L, S, the binary slice-indexed parthood relation (PS-3D )s satisfies each of (CM1)–(CM4) over OBs .34 To this extent, we should be satisfied that, relative to a fixed slice, PS-3D replicates the structure of classical mereology. But, as mentioned above, three-dimensionalists who favor material coincidence will think that SM is too strong. For, it is a theorem of SM that no two objects can be composed of the same parts at the same slice. Taking slices as time-slices, this would prohibit a statue from temporarily coinciding with, while remaining distinct from, the lump of clay from which it is formed. But pro-coincidence threedimensionalists claim that the statue and lump are distinct objects which share a location and parts (at least their molecular parts) for as long as the lump constitutes the statue. The issue here is, at bottom, with the structure of the models. Condition (4.ii) on L Models and S3D Models prohibits distinct objects from standing in the exact location relation L to the same spacetime region. But proponents of material coincidence (at least those who endorse spacetime substantivalism) hold that distinct objects may be exactly located at the same spatiotemporal region. These philosophers will think that S3D Models misrepresent the way objects are located in spacetime since the models prohibit spatiotemporal coincidence. But the models can be easily modified to accommodate coincidence. Let us introduce the term ‘S3D * Model’ for any ordered quintuple ST, R, OB, L, S which satisfies all of the requirements of an S3D Model except Condition (4.ii). Let SM* 34 Furthermore, (SM1)–(SM4) are clearly also necessary for establishing that each of the reduced ‘single-slice’ domains is a model of CM. The additional axiom (SM0) just ensures that every member of the original object domain shows up in at least one of the single-slice domains.

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be that mereology which is just like SM except that SM* omits axiom (SM4). SM* is equivalent to the time-relative mereology introduced at [Sider, 2001, 58] and is quite similar to the timerelative mereologies of [Thomson, 1983] and [Simons, 1987, 5.2].35 It is trivial to verify that the S3D * counterpart of PS-3D satisfies all axioms of SM*, but does not satisfy SM’s (SM4). Whether we make room for coincident objects or not, P3D-S is fairly robust, at least from a logical point of view. Relative to a slice, it is reflexive and transitive and satisfies a supplementation principle. Thus, P3D-S supports much of the reasoning philosophers would like to do with a parthood relation, as long as they are willing to relativize parthood claims appropriately. Moreover, whether or not we leave room for coincident objects, P3D-S generates a plausible overlap relation—objects x and y share a P3D-S -part at slice s if and only if x’s unique location within s overlaps y’s unique location within s. With P3D-S -overlap, we cannot have ‘overlapping’ objects that are separated in space and time (as are JA and the POC -overlapping man who inherits her atom). Also, P3D-S serves as the basis for a reasonable summation relation. We can introduce slice-relative sums as x is a sum of the ys at s = def each of the ys is part of x at s and any object that is part of x at s overlaps at least one of the ys at s.36 It then turns out that P3D-S -sums must extend exactly so far as their summands within the slice in question. More precisely, if x is a sum of the ys at s, then x’s unique location within s is the union of the locations of the ys within s. With P3D-S -summation in a fixed slice, we cannot have summands that trail off past their sum (as the molecules that POC -sum to JA extend past JA) or sums that bulge out beyond their summands (as our possible organism from Section 3 extends beyond the single organ of which it is a POC -sum). 35

Thomson’s time-relative mereology is somewhat stronger than SM* in that it prohibits distinct objects from being parts of one another at all times at which at least one of them exists (see her axiom (CCL1 ) [1983, 216]). Besides being formulated in a free logic (instead of standard predicate logic as is SM*), Simons’ time-relative mereology uses multiple time-indexed binary parthood relations and a weaker supplementation principle than (SM3) (see his CTA10 [1987, 179]). 36 Except for its use of plurals instead of sets of summands, this matches the time-relative summation relation introduced at [Sider, 2001, 58].

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In these formal and structural respects, P3D-S works out better than the binary relations PBD , PCT , PPT , and (especially) POC . But what about our other important criterion—how well can a relation like P3D-S preserve intuitive assumptions about the parts of ordinary objects like JA? Obviously we do not normally think of ourselves as relativizing parthood to regions of spacetime. But we do explicitly link parthood to time (a certain cell is part of JA at some times but not others and the wheel that is now part of my bike was not part of it last year). I think P3D- S can be made to fit common sense thinking about parthood quite well if (and only if) there is a slice set consisting of regions that correspond roughly to times—slices that extend over only those spacetime points that might (at least from an appropriate frame-relative perspective) count as simultaneous. For such time-slices, the set of all objects existing at a slice includes all objects existing at the corresponding (frame-relative) time and the parthood relation holds between objects x and y at a slice just in case x is included in y at the corresponding time. So, for example, there will be some slice at which all of my current cells are part of me, another slice at which all of my cells from five years ago are part of me, another slice at which all of my cells from ten years ago are part of me, and so on. But there can be S3D Models in which the slices do not look anything at all like time-slices—models in which the slices zig-zag randomly all over spacetime but happen to pick up no more than one location per object as they do so. Although the slice-relative parthood relation for this sort of model would still enjoy SM’s or SM*’s nice logical properties, it clearly would not fit ordinary usage. There is no ordinary sense in which, say, all of my current cells are part of me at the same time (or region, or anything else) at which all of your cells from ten years ago are part of you. I will assume that the ideal spatiotemporal correlates of times are maximal achronal regions—spacetime regions that are achronal (i.e. include no absolutely temporally separated spacetime points) and are not properly included in larger achronal regions.37 For 37 See [Gilmore, 2006] and [Balashov, 2008] for more detailed discussions of absolute temporal separation and maximal achronal regions. Notice that we need not assume that every maximal achronal region is a natural spacetime correlate of some time instant. For example, [Balashov, 2008] argues that, in the context of special relativity, only flat frame-relative time-slices (and not curved maximal achronal

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example, assuming special relativity, for each inertial frame F, there is a partition of spacetime consisting of the equivalence classes of spacetime points which are simultaneous relative to F. Each of these frame-relative time-slices is a maximal achronal region. When wondering whether PS-3D parthood corresponds appropriately to ordinary temporalized parthood, the main question we should be asking then is: (*) Is any set S of maximal achronal regions such that i) every object is located only at subregions of the members of S and ii) no object has more than one location within any member of S? It is easy to name circumstances in which the answer to (*) is definitely ‘no’. This will be so if some (or all) objects are located at four-dimensional regions or if objects travel through time in a way that locates at least one object in multiple positions within an achronal region. But even if we assume that objects have only three-dimensional achronal locations and cannot travel backwards in time, it is not clear that the answer to (*) is ‘yes’. It might be, but then again it might not. It all depends on what exactly the three-dimensionalist can say about where objects are located in a relativistic spacetime. In a Galilean spacetime, there is an absolute simultaneity relation on spacetime points and the only maximal achronal regions are absolute time-slices (where each absolute time-slice consists of all points absolutely simultaneous with a given point). Importantly, absolute time-slices are pair-wise discrete. On this picture of spacetime, it makes sense for three-dimensionalists who eschew time-travel to hold that an object x is located at region r if and only if r is the intersection of x’s path with an absolute time-slice.38, 39 Since regions) play this role. Also, I think that spatiotemporal regions that are only roughly as spatially long and temporally short as maximal achronal regions could also play the role of time-slices. But the points made below about maximal achronal regions apply equally well to regions that are almost, but not quite, maximal achronal regions. 38 This is the characterization of location in spacetime which I initially assumed for the extended Jane Austen example in Section 3. Again, see [van Inwagen, 1990] and [Sattig, 2006, 2.1] for this sort of take on location in spacetime. 39 The three-dimensionalist who allows for time-travel will presumably want to say something different. Where x’s path crosses slice s twice, x has two separate locations within s and is not located at the intersection of its path with s (i.e., at the union of its two locations in s).

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absolute time-slices are pairwise discrete, no absolute time-slice could include more than one location per object. Since, in addition, each location is included in some absolute time-slice, the set of all absolute time-slices clearly satisfies the S3D criteria for slice sets. But in a relativistic spacetime, there are no absolute time-slices. There are at best only frame-relative time-slices. Importantly, regions that are time-slices relative to different inertial frames may have non-empty intersections. More generally, distinct maximal achronal regions in relativistic spacetimes may overlap. And, as Gilmore describes in his ‘corner slice’ example [Gilmore, 2006, 212–13], there may be an object x and overlapping frame-relative time-slices s1 and s2 such that PATH(x) ∩ s1 is properly included in PATH(x) ∩ s2 . Here, s1 cuts through a small end ‘corner’ of x’s path while s2 cuts through a larger swatch of x’s path—one that includes both PATH(x) ∩ s1 and PATH(x) ∩ s2 . If, following the example of non-relativistic three-dimensionalism, we assume that every object is located at any non-empty intersection of its path with a maximal achronal region, we must conclude that both PATH(x) ∩ s1 and PATH(x) ∩ s2 are locations of x. But then, since any region which includes PATH(x) ∩ s2 also includes PATH(x) ∩ s1 , no set of regions would satisfy the S3D criteria for slice sets. Of course, one could always argue that in transferring threedimensionalism to a relativistic framework, we should expect more complicated rules for locating objects within their paths. On behalf of the three-dimensionalist, Gilmore suggests that objects might be located only at all maximal achronal subregions of their paths (achronal subregions of their paths which are not properly included in larger achronal subregions of their paths) [Gilmore, 2006, 212–13]. On this modified location principle, maximal achronal regions cannot include more than one location per object. But it is not clear to me what reasons the three-dimensionalist has for thinking that objects are located only at maximal achronal subregions of their paths.40 Significantly, Gilmore himself ends up rejecting three-dimensionalism because he does not believe that the threedimensionalist can provide any general criteria for determining which subregions of an object’s path are its locations. While I do 40 See [Gibson and Pooley, 2006, 186] for doubts over the proposal that objects are located only at maximal achronal subregions of their paths.

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not see the difficulty in formulating a general location principle as a reason for rejecting three-dimensionalism, I do agree that it is not obvious what the three-dimensionalist who takes relativity seriously should say about where objects are located in spacetime.41 In particular, I do not think that the three-dimensionalist can simply assume that there must be a set of regions satisfying the S3D criteria for slice sets whose members correspond appropriately to ordinary time instants. How important is the requirement that the slices of S3D Models include no more than one location per object? Taking our cue from Gilmore’s corner slice example, might we not weaken Condition (5.ii) of S3D models so that it requires only that, if x has any location in slice s, then x has a maximal location in slice s? This modified condition would allow the object x in the corner slice example to have both PATH(x) ∩ s1 and PATH(x) ∩ s2 as its locations, since both of these regions (as well as any other location x might have in s2 ) are included in PATH(x) ∩ s2 . The corresponding slice-relative parthood relation would hold between objects x and y at slice s if and only if i) both x and y have locations in s and ii) x’s maximal location in s is included in y’s maximal location in s. However, unlike PS-3D , this modified slice-indexed parthood relation need not satisfy the supplementation principle (SM3).42 More generally, as far as I can tell, any weakening of (5.ii) results in a slice-relative parthood relation that does not satisfy all of SM*’s axioms and is thus significantly weaker than PS-3D . This sort of weakness does not necessarily disqualify a relation from serving as a parthood relation. But, as we saw in Section 3, it can make trouble for the kind of work philosophers have tried to do with mereological relations. I close this section by briefly sketching a different sort of variation on S3D Models and PS-3D parthood. Like S3D Models, S4D Models 41 See also [Gibson and Pooley, 2006] where it is argued, against Gilmore and in favor of three-dimensionalism, that there is no reason to expect that there is a general principle telling us where any object is located within its path. 42 To see this, suppose that for some slice s, x is located at both rx1 and rx2 where rx1 ⊂ rx2 ⊆ s and rx2 is x’s maximal location in s. Suppose further that i) no object is located within any proper subregion of rx1 and ii) there is some object y such that rx2 /rx1 is y’s maximal location in s. Then, for the proposed parthood relation, x would exist at s and would not be a part of y at s, but would have no part at s that fails to overlap y at s.

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assume a distinguished set of slices and require that no object has more than one location in any slice. However, unlike S3D Models, S4D Models do not require that every location is included in some slice. Thus, even if slices are three-dimensional subregions of spacetime, some of the objects in S4D Models may have four-dimensional locations. The parthood relation PS-4D is a spacetime counterpart of the time-indexed parthood relation that four-dimensionalists have introduced in terms of temporal parts (see, e.g., [Sider, 2001, ch. 3]).43 Most four-dimensionalists assume that objects are not multiply located and that there is a binary parthood relation along the lines of PUL satisfying the axioms of classical mereology over the object domain. But, as we noted in our examination of PUL , this sort of binary parthood relation does not preserve ordinary assumptions about parthood (e.g., that JA’s head is part of JA). The four-dimensionalist’s time-indexed parthood relation is introduced as a secondary parthood relation which is supposed to match ordinary temporalized parthood better than the binary parthood relation. If slices are spacetime regions that roughly correspond to time instants, then PS-4D should also fit ordinary assumptions better than the binary relation PUL . A Slice4D Model (S4D Model) is an ordered quintuple ST, R, OB, L, S where ST, R, OB, L is an L Model (satisfying conditions 1–4 on L Models) and where, in addition, the following condition is satisfied: 5*. S (the set of slices) is a subset of R such that i) if r ∈ R and there is some x, r* ∈ L such that r ⊆ r*, then there is some s ∈ S with r ∩ s = Ø; ii) for any x ∈ OB and any s ∈ S, if PATH(x) ∩ s = Ø, then there is some z ∈ OB such that z, PATH(x) ∩ s ∈ L; iii) for any x, r ∈ L and any s ∈ S, if r ⊆ s, then r = PATH(x) ∩ s. Condition (5*.i) requires that, taken together, slices cover every subregion of every location. Notice that (5*.i) is automatically satisfied if the slices cover all of spacetime (i.e., if ∪S = ST). (5*.i) replaces the stronger requirement in S3D Models that every location is included in some slice. Condition (5*.ii) requires that there is some object located at any region that is the intersection of a slice and an object’s 43 See also [Balashov, 2008] for a slightly different spacetime adaptation of the four-dimensionalist’s time-relative parthood relation.

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path. If slices correspond to times, the object which is located at the intersection of x’s path with a slice is, I will presume, a temporal part of x.44 Condition (5*.iii) requires that if object x has any location within slice s, that location must be the intersection of s and x’s path. It is an immediate consequence of (5*.iii) that no object has more than one location within any slice. But notice that for an arbitrary object x ∈ OB, none of (5*.i-iii) require that x has a location within any slice. The slice-relative parthood relation PS-4D is defined over S4D Models as follows:

x, y, s ∈ PS−4D if and only if s ∈ S and ∅ = PATH (x) ∩ s ⊆ PATH (y) ∩ s. In other words, x is part of y at slice s if and only if the intersection of x’s path with s is non-empty and is included in the intersection of y’s path with s. The supplementary exists at (ES-4D ) and overlaps (OS-4D ) relations are defined over S4D Models as follows:

x, s ∈ ES-4D if and only if s ∈ S and ∅ = PATH (x) ∩ s;

x, y, s ∈ OS-4D if and only if s ∈ S and PATH(x) ∩ PATH(y) ∩ s = ∅. Object x exists at slice s just in case x’s path overlaps s. Object x overlaps object y at slice s just in case x’s path overlaps y’s path within s. When the ternary predicate P is interpreted over S4D Models as PS-4D , the defined predicates E and O are interpreted as, respectively, ES-4D and OS-4D and all of SM*’s axioms are satisfied.45 Notice, 44 Whether this is so or not depends on exactly how temporal parts are defined. I do not wish to digress from the discussion of region-relative parthood in order to compare different ways of introducing temporal parts. But, given an appropriate set of time-slices, I cannot see that we would run into any trouble in introducing temporal parts either in terms of location, as is done in [Heller, 1984], or in terms of a binary parthood relation as is done in [Sider, 2001]. 45 To see that PS-4D satisfies (SM3) over all S4D Models, suppose PATH(x) ∩ s = Ø and PATH(x) ∩ s ⊆ PATH(y) ∩ s. Case 1: PATH(y) ∩ s = Ø. Then PATH(x) ∩ PATH(y) ∩ s = Ø and Ø = PATH(x) ∩ s ⊆ PATH(x) ∩ s. Case 2: PATH(y) ∩ s = Ø.

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though, that SM’s antisymmetry axiom (SM4) is not satisfied. Even when we retain the original requirement that no more than one object is exactly located at any region, distinct objects may still have paths that cross a slice at exactly the same place. More precisely, for x = y, there may be some slice s such that Ø = PATH(x) ∩ s = PATH(y) ∩ s. In this case, x and y would each be PS-4D -parts of one another at s and the object located at PATH(x) ∩ s would be a shared PUL -part of x and y. But this fits the standard four-dimensionalist treatment of temporalized parthood, which allows that objects like a statue and the lump of clay from which it is formed may each be part of the other at time t in the sense that they share a temporal part at t (see, e.g., [Heller, 1984] or [Sider, 2001, ch. 5] ). We noted above that, lacking a three-dimensionalist account of location in relativistic spacetimes, it is not obvious that PS-3D works out for slices that could be considered time-slices (at least, not if the slices over which PS-3D ranges are required to satisfy Conditions (5.i) and (5.ii) of S3D Models). Given four-dimensionalism and a relativistic spacetime, is the case for PS-4D parthood equally inconclusive? Not if the four-dimensionalist assumes (a) that each object has a unique location in spacetime and (b) that any nonempty intersection of a location and a maximal achronal region is a location. (Note that (b) is entailed by the stronger, but not unreasonable, assumption that every subregion of a location is a location.) Suppose that (a) and (b) hold and let MAX be the collection of all maximal achronal regions in the actual spacetime, ST. Since every point in ST is included in some maximal achronal region, ∪MAX = ST and MAX satisfies condition (5*.i) on S4D slice sets. It follows from assumption (b) that MAX also satisfies condition (5*.ii). To see that MAX satisfies the final condition, (5*.iii), suppose that object x is located within the maximal achronal region s. Then since x is uniquely located, PATH(x) ⊆ s and x’s location within s is just PATH(x) ∩ s = PATH(x).46 By (5*.ii), both PATH(x) ∩ s and PATH(y) ∩ s are locations. By (4.iii), since (PATH(x) ∩ s)/ (PATH(y) ∩ s) = Ø, there is some z, rz  ∈ L such that rz ⊆ (PATH(x) ∩ s)/ (PATH(y) ∩ s). Since rz ⊆ s, by (5*.iii), rz = PATH(z) ∩ s. Thus, Ø = PATH(z) ∩ s ⊆ PATH(x) ∩ s and PATH(z) ∩ PATH(y) ∩ s = Ø. 46 Given unique location, we obtain analogous results with the weaker assumption that, for some subset MAX* of MAX, ∪MAX* = ST and any non-empty intersection of an object’s path with a member of MAX* is a location. In the context of special

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5. other region-relative parthood relations In the previous section I proposed two general strategies for introducing ternary parthood relations that relativize parthood to special regions (slices). But some philosophers have made use of parthoodat-a-region relations whose third terms are not restricted to the members of a distinguished subclass of regions. The best-developed example of this approach is found in [Hudson, 2001, ch. 2]. The main task of this section is to examine a version of Hudson’s relation in the context of L Models. I will also briefly consider the rather different region-relative parthood relation of [Crisp and Smith, 2005]. Hudson holds that ordinary objects such as people, chairs, and tables are located at multiple overlapping four-dimensional regions. However, Hudson allows that there are three-dimensional objects (instantaneous temporal parts of ordinary objects) that have multiple locations within fixed achronal regions.47 Thus, neither of the slice-relative parthood relations considered in the previous section works out for Hudson’s ontology if we take slices to be something along the lines of maximal achronal regions. Hudson’s own regionrelative parthood relation is, as we shall see shortly, quite different from either PS-3D or PS-4D . As was the case for the slice-relative parthood relations, we will want to introduce Hudson’s parthood relation on a class of L Models that satisfies special criteria. For Hudson Models (H Models), we do not need a class of slices as in S3D and S4D Models. But for the Hudson relation to behave nicely, we do need to strengthen the original restrictions on the location relation. As I indicate in the notes below, the stronger restrictions are assumptions that Hudson endorses in [2001].48 relativity, the slice set MAX* might be restricted to frame-relative time-slices as is suggested in [Balashov, 2008, 28–37]. 47

In fact, when Hudson first provisionally attempts to develop his position in a three-dimensionalist ontology, he assumes that an ordinary object may occupy distinct spatial regions at a fixed time [Hudson, 2001, 52–3]. 48 This is not to say, however, that the H Models capture all of Hudson’s assumptions about location. Most notably, nothing in the H Models requires that any object is located at a four-dimensional region or that ordinary objects are located at multiple overlapping regions. I do not attempt to capture the former assumption because mereology has nothing to say about dimension. I do not attempt to capture the latter

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H Models are L Models in which the location relation satisfies the following additional restrictions: 4. v) if x, r ∈ L and Ø = r* ⊆ r, there is some y ∈ OB such that y, r* ∈ L; vi) if x, r ∈ L and r* ⊂ r, x, r* ∈ / L.49 (4.v) stipulates that every subregion of a location is a location, while (4.vi) prohibits an object from being located at two regions, one of which is a proper subregion of the other.50 The Hudson parthood relation PH is defined on H Models as follows:

x, y, r ∈ PH if and only if for some rx , ry ∈ R,

x, rx , y, ry  ∈ L, and rx ⊆ r ⊆ ry .51 Object x is part of object y at region r just in case x and y have locations, rx and ry , that ‘flank’ r in the sense that r includes rx and is included in ry . When we plug PH into the SM definitions (DSE ) and (DSO ), we get exists at and overlaps relations that are quite different from their counterparts in S3D and S4D Models. The exists at relation defined in terms of PH via (DSE ) turns out to be just the location relation L. This is because x, x, r ∈ PH if and only if x, r1 , x, r2  ∈ L and r1 ⊆ r ⊆ r2 . But it follows from condition (4.vi) that x, r1 , x, r2  ∈ L and r1 ⊆ r ⊆ r2 if and only if r1 = r = r2 and x, r ∈ L. (Notice how different this interpretation of E is from its interpretation over S3D Models. An object x in an S3D Model exists at slice s just in case x has a location assumption because I cannot see that it makes for an interesting difference in the logical properties of the parthood relation. 49 Notice that (4.v) entails (4.iii) and (4.iv). Thus, taken together, (4.1)–(4.vi) are redundant. 50 I take Hudson’s principles (SD) and (SDP) [2001, 65] as evidence that he endorses (4.v). (Note that in these principles, and throughout [2001, ch. 2], Hudson uses the term exactly occupies as I use is located at. His binary relation EO corresponds to my L.) As evidence that Hudson endorses (4.vi), see Hudson’s assumption that if an object x were located at regions r* and r where r* ⊂ r, then x would be of proper part of itself at r [2001, 68–9]. But this is impossible, since it would imply that x = x. 51 Though Hudson lists several principles governing his parthood relation, he never gives an intended model theoretic interpretation for it. However, he does explain the relation in terms of some examples (especially at [2001, 66–70]). In proposing this model theoretic treatment of Hudson’s relation, I am guided primarily by his explanation of how the relation is supposed to apply in his examples.

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that is included in s—x need not have a location that is identical to s. Over S4D Models, E has an even broader interpretation. An object x in an S4D Model exists at slice s just in case x has a location that overlaps s—again, x need not have any location that is identical to s.) The overlaps relation for PH is the relation OH such that

x, y, r ∈ OH if and only if there are x, rx ,

y, ry  ∈ L with ∅ = r ⊆ rx , ry . Objects x and y OH -overlap at region r just in case both x and y have locations that include r. (By contrast, OS-3D holds between x and y at slice s just in case x and y have overlapping locations that are included in s.) To see that OH is the result of plugging PH into (DSO ), suppose z, x, r ∈ PH and z, y, r ∈ PH . Then there are regions rz1 , rz2 , rx , and ry such that z, rz1 , z, rz2 , x, rx , y, ry  ∈ L, rz1 ⊆ r ⊆ rx , and rz2 ⊆ r ⊆ ry . It follows that r ⊆ rx , ry . Conversely, suppose that x, rx , y, ry  ∈ L and there is some region r which is included in both rx and ry . By (4.v), some object z is located at r. Since r ⊆ r ⊆ rx and r ⊆ r ⊆ ry , z, x, r ∈ PH and z, y, r ∈ PH . How do the logical properties of PH compare to those of the slice-relative relations P3D-S and P4D-S ? PH satisfies SM’s (SM0), (SM2), (SM3), and (SM4) over all H Models.52 But PH does not satisfy the existence axiom (SM1) on any H Model which includes objects located at extended regions. To see why, suppose that y is an object in an H Model, that y is located at region ry , and that region ry has a proper subregion rx . By condition (4.v), some object x is located at rx . Since x, rx , y, ry  ∈ L and rx ⊆ ry ⊆ ry , x, y, ry  ∈ PH . But it follows from Condition (4.vi) that, since x is located at rx , / L. (Similarly, x, y, rx  ∈ PH and x is not located at ry . Thus, x, rx  ∈ / L.) Since E is interpreted over H Models as L, PH does not

y, rx  ∈ satisfy (SM1) on this model. In its failure to satisfy (SM1), PH is slightly weaker than the slicerelative parthood relations of the previous section. But the most 52 PH serves as an important example for philosophers who think that specifying a relation requires little more than listing some of that relation’s logical properties. PH and P3D-S both satisfy the standard ternary mereological principles (SM0), (SM2), (SM3) and (SM4), but are clearly very different proposals for a region-relative parthood relation.

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remarkable difference between PH and the slice-relative relations is that PH satisfies some rather strong mereological principles that neither PS-3D nor PS-4D satisfies. For example, PH satisfies the following principle on all H Models. (HM1) Pxyr & Pwzr → Pxzr It is an immediate consequence of (HM1) that if y and z each has a part at r, then y and z have exactly the same parts at r. It is easy enough to see that neither PS-3D nor PS-4D generally satisfies (HM1). If x and y are objects in either an S3D Model or an S4D Model and x and y are located at disjoint subregions of slice s, then both x and y have parts at s, but x and y do not share any parts at s. To see that PH does indeed satisfy (HM1) over all H Models, suppose that x, y, r, w, z, r ∈ PH . Then for some rx , ry, rw , rz ∈ R, x, rx , y, ry , w, rw , z, rz  ∈ L, rx ⊆ r ⊆ ry , and rw ⊆ r ⊆ rz . It follows immediately that rx ⊆ r ⊆ rz and, consequently, x, z, r ∈ PH . I use the name ‘HM’ for the sorted first-order theory axiomatized by the following three formulas, where the predicates E and O are defined as in (DE ) and (DO ).53 (HM0) ∃r Exr (every object exists at some region) (HM1) Pxyr & Pwzr → Pxzr (if x is part of y at r and w is part of z at r, then x is part of z at r) (HM2) Pxyr & Pyxr → x = y (if x is part of y at r and y is part of x at r, then x and y are identical) PH satisfies HM’s three axioms over all H Models. Theorems of HM include counterparts of SM’s (SM2) and (SM3). In addition, each of following formulas is a theorem of HM which is satisfied by neither PS-3D nor PS-4D . 53 In his own presentation of the logical properties of his parthood relation, [Hudson, 2001] uses a region inclusion predicate. I think that this is definitely the right approach to take in a comprehensive treatment of Hudson’s relation. (If nothing else, we can only describe the distinctive ‘flanking’ of objects’ locations around the regions at which they stand in parthood relations if we can say something about region inclusion.) However, since my interest here is only in general differences between Hudson’s relation and the slice-relative parthood relations, I will not go into so much detail in my formal analysis of Hudson’s relation.

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(HT1) Exr & Eyr → x = y (if x exists at r and y exists at r, then x and y are identical) (HT2) Oxyr & Owzr → Oyzr (if x overlaps y at r and w overlaps z at r, then y overlaps z at r) (HT3) Oxyr & Oyzr → Oxzr (if x overlaps y at r and y overlaps z at r, then x overlaps z at r) (HT4) Exr & Oxyr → Pxyr (if x exists at r and x overlaps y at r, then x is part of y at r) (HT5) Pxyr & Pzwr → Oywr (if x is part of y at r and z is part of w at r, then y overlaps w at r) How suitable is PH for philosophical tasks? On the one hand, PH certainly does satisfy (ternary versions of) principles that philosophers commonly invoke to describe parthood relations. In particular, PH satisfies SM’s transitivity axiom, (SM2), and SM’s supplementation principle, (SM3). But, on the other hand, there is a sense in which PH ’s strong non-classical properties trivialize the familiar properties. For example, with only the weak assumption that y and z both have parts at r, (HM1) lets us transfer all of y’s parts at r to z (and vice versa). For PH , we do not need the extra assumption that y is part of z at r which is required by the antecedent of the transitivity principle (SM2). Also, it follows from (HT4) that if x exists at r, then x overlaps y at r if and only if x is part of y at r. Thus, if x exists at r and does not stand in PH to y at r, then x itself does not OH -overlap y at r—we do not need to invoke the supplementation principle to support the weaker conclusion that x has some PH -part at r which fails to OH -overlap y at r. I think that in its strength PH does not, after all, fit traditional philosophical thinking about parthood relations—why would philosophers have bothered with a transitivity principle for parthood if so strong a principle as (HM1) were on offer? Although philosophers do not generally explicitly deny (HM1), the assumption must have been all along that parthood is not that strong. Note also that (HT3) tells us that the overlap relation OH satisfies a ternary transitivity principle. Again, although most philosophers do not explicitly deny that overlap is transitive, transitivity is usually explicitly attributed only to parthood and not to the overlap relation.

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But, unlike the very weak binary parthood relation POC of Section 3, PH does have useful formal properties and can serve as a basis for interesting relations among objects. It is only important that care be taken in recognizing that these relations may not behave as expected. We have already seen some surprising properties of PH and OH . It is also worth noting that Hudson adopts a more complicated definition of summation than the standard definition used in Section 4 above (see [2001, 65] for Hudson’s definition). Hudson’s own summation relations works out nicely. But it is easy to verify that, plugging PH and OH into the standard definition of summation, we end up with an ugly relation that makes any object the ‘sum’ of any one of its atomic PH -parts at the (point-sized) region where the atomic part is located. Even if we can come to terms with its unexpected logical properties, PH is clearly at odds with ordinary thinking about parthood. PH holds between x and y at r only if the region r is included in one of y’s locations. Thus, if y is an ordinary object like a cat or a bicycle, the indexing region r must have a quite constricted spatiotemporal extent—it must be small enough to ‘fit inside’ of y. But we do not in ordinary discourse link parthood to anything like these small spacetime regions. (There is no time that corresponds in a natural way to any spatiotemporal region that fits inside my cat.) Moreover, we ordinarily assume that objects with widely separated locations—objects, like the Empire State Building and the Taj Mahal, which never occupy overlapping regions and never share parts—may have parts at the same time. By contrast, objects x and y can have PH -parts at the same region only if x and y have overlapping locations. In these respects, PH ’s regionrelativization of parthood takes a form quite different from the ordinary time-relativization of parthood. In addition, under Hudson’s assumptions concerning the dimension of objects’ locations in spacetime, PH would not preserve many of what we take to be the most obvious parthood linkages among objects. As stated earlier, Hudson claims that ordinary objects are located at multiple temporally-extended regions. Presumably then an object like Jane Austen is located at multiple regions, all of which extend from some time in 1775 to some time on 18 July, 1817. And Jane’s head (as well as Jane’s hands, Jane’s legs, and so on) occupy multiple regions, all of which extend somewhat past 18 July,

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1817. In this four-dimensionalist picture, there is no region r which includes one of Jane’s head’s locations and is included in one of JA’s locations. In other words, there is no region r at which Jane’s head is a PH -part of JA. Similarly, there is no region at which Jane’s hands, legs, liver, and so on, are PH -parts of JA. This is the same sort of divergence from ordinary parthood ascriptions which we have already noted in the binary relation PUL . Another region-relative parthood relation plays a central role in [Crisp and Smith, 2005]. Like PH (and unlike PS-3D and PS-4D ), Crisp and Smith’s ternary parthood relation—call it PCS —does not limit its third argument to the regions of any special collection such as a slice set. Although Crisp and Smith do not fill in many details concerning PCS , their explicit assumptions make it clear that this parthood relation is different from PH . It is only in this difference that I am interested here. Crisp and Smith stipulate that PCS must satisfy three principles [2005, 332–3]. I shall consider only the second of these principles.54 It is stated as follows: (*) ‘‘ . . .if x is wholly present at R and x is a part of y at R, then x is a part of y at every superregion and every subregion of R’’. [2005, 333] I take it that by ‘is wholly present at R’, Crisp and Smith mean roughly what I do by ‘is located at R’.55 On this assumption PH , unlike PCS , does not generally satisfy (*). From condition (4.vi) on H Models, it follows that if x is a PH -part of itself at region r, then x is not a PH -part of itself at any proper subregion or proper superregion of r. Let ST, R, OB, L be any H Model in which ST includes more than one spacetime point. In any such model, every region has a 54

For the record, the other two assumptions are — in case PCS is ‘‘analyzable in terms of parthood simpliciter, the analysis is given by . . .: x is part of y at R =def (i) x is a part simpliciter of y, and ii) x overlaps R’’ [2005, 332]; — ‘‘ . . . if y is wholly present at R and x is part of y at R, then x is wholly present at a subregion of R’’ [2005: 333].

55

This assumption has been confirmed by Crisp in e-mail correspondence.

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proper subregion or a proper superregion. It follows that for any object x ∈ OB, there is some region r at which x is wholly-present (i.e., located) and a PH -part of itself, but where there is at least one subregion or superregion of r at which x is not at PH -part of itself. Thus, (*) fails for PH on all H Models in which there are at least two spacetime points.

6. concluding remarks In this chapter, I have used mathematical models to represent different types of binary and ternary parthood relations. I have shown that these model theoretic relations have significantly different logical properties and that some of these relations can capture ordinary assumptions about parthood better than others. My primary conclusions are as follows. • Philosophers have proposed several different region-relative parthood relations for domains of multiply-located objects. These relations have significantly different logical properties and rely on different kinds of assumptions about how objects are located in spacetime. • On a relativistic account of spacetime, it is not obvious whether there is any relation that both fits ordinary thinking about parthood and can play the central role in an analysis of relations among objects that some philosophers have tried to assign parthood. Given multi-location, I think that PS-3D is the best shot at a relation that can satisfy both criteria. But, as we have seen, it is not obvious that there is an appropriate slice set for the PS-3D relation. Of course I do not think that the sort of model theoretic investigation pursued in this chapter can cover all issues relevant to parthood and location. There may be factors involved in parthood relations (e.g., relations of functional interdependence) that cannot be represented in a natural or illuminating way in terms of mathematical models. Also, there are important aspects of spatiotemporal location (the dimension of a location, the continuity of an object’s path) that can only be represented in more complex models than those used in this chapter.

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But I do think that model theoretic representations are a good starting point for a discussion of parthood relations, especially if we leave open the possibility of multiply located objects. As we have seen, over multiply located domains, the only plausible binary parthood relations are either extremely weak or fail to preserve important common sense intuitions. A ternary region-relative parthood relation may be a more appropriate choice for these domains but different types of region-relative parthood relations are possible. A model theoretic representation can help determine which relation is intended, what logical properties this relation is supposed to have, and what general assumptions are made about the interaction between location and parthood. Too little is said about these important issues in philosophical work that makes use of region-relative (or time-relative) parthood relations. As a result, the reader is sometimes given little indication of how he is to understand the author’s claims about region-relative parthood.56 University at Buffalo, SUNY

references Balashov, Y. (2008). ‘‘Persistence and Multilocation in Spacetime’’. In D. Dieks (ed.), The Ontology of Spacetime. Philosophy and Foundations of Physics Series, vol. 2, (Amsterdam: Elsevier), 59–81. Barker, S. and Dowe, P. (2003). ‘‘Paradoxes of Multi-Location’’ Analysis 63: 106–14. and Dowe, P. (2005). ‘‘Endurance is Paradoxical’’. Analysis 65: 69–74. Beebee, H. and Rush, M. (2003) ‘‘Non-Paradoxical Multi-Location’’. Analysis 63: 311–17. Bittner, T. and Donnelly, M. (2004) ‘‘The Mereology of Stages and Persistent Entities’’. In R. Lopez de Mantaras and L. Saitta (eds.), Proceedings of the European Conference on Artificial Intelligence (Amsterdam: IOS Press), 283–7. Crisp, T. and Smith, D. (2005) ‘‘ ‘Wholly Present’ Defined’’. Philosophy and Phenomenological Research 71: 318–44. Doepke, F. (1982) ‘‘Spatially Coinciding Objects’’. Ratio 24: 45–60. Gibson, I. and Pooley, O. (2006) ‘‘Relativistic Persistence’’. In J. Hawthorne (ed.), Philosophical Perspectives, vol. 20, Metaphysics (Oxford: Blackwell), 157–98. 56 I am grateful for the helpful comments of Yuri Balashov, Thomas Crisp, Cody Gilmore, Hud Hudson, and Dean Zimmerman on the work presented in this chapter.

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Gilmore, C. (2006) ‘‘Where in the Relativistic World are We?’’. In J. Hawthorne (ed.), Philosophical Perspectives, vol. 20, Metaphysics (Oxford: Blackwell), 199–236. (2007) ‘‘Time Travel, Coinciding Objects, and Persistence’’. In D. Zimmerman (ed.), Oxford Studies in Metaphysics (Oxford: Oxford University Press), 177–98. Hawthorne, J. (2008) ‘‘Three-Dimensionalism vs. Four-Dimensionalism’’. In T, Sider, J. Hawthorne, and D, Zimmerman (eds.), Contemporary Debates in Metaphysics (Oxford: Blackwell), 263–82. Heller, M. (1984) ‘‘Temporal Parts of Four Dimensional Objects’’. Philosophical Studies 46: 323–34. Hudson, H. (2001) A Materialistic Metaphysics of the Human Person (Ithaca: Cornell University Press). McDaniel, K. (2003) ‘‘No Paradox of Multi-Location’’. Analysis 63: 309–11. (2004) ‘‘Modal Realism with Overlap’’. Australasian Journal of Philosophy 82: 137–52. Lowe, E. J. (2003) ‘‘Substantial Change and Spatiotemporal Coincidence’’. Ratio 16: 140–60. Olson, E. T. (2006) ‘‘Temporal Parts and Timeless Parthood’’. Noûs 40: 738–52. Parsons, J. (2007) ‘‘Theories of Location’’. In D. Zimmerman (ed.), Oxford Studies in Metaphysics (Oxford: Oxford University Press), 201–32. Sattig, T. (2006) The Language and Reality of Time (Oxford: Clarendon Press). Saucedo, R. (forthcoming) ‘‘Parthood and Location’’. In D. Zimmerman (ed.), Oxford Studies in Metaphysics (Oxford: Oxford University Press). Sider, T. (2001) Four-Dimensionalism: An Ontology of Persistence and Time (Oxford: Clarendon Press). Simons, P. (1987) Parts: A Study in Ontology (Oxford: Oxford University Press). Tarski, A. (1956) ‘‘On the Foundations of Boolean Algebra’’. In Logic, Semantics, Metamathematics, trans. J. H. Woodger (Oxford: Clarendon). Thomson, J. J. (1983) ‘‘Parthood and Identity over Time’’. Journal of Philosophy 80: 201–20. van Inwagen, P. (1990) ‘‘Four-Dimensional Objects’’. Noûs 24: 245–55.

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THE METAPHYSICS OF SOUNDS

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11. Constructing a Theory of Sounds Casey O’Callaghan 1. sounds and vision Vision has dominated philosophical thinking about perceptual experience and the nature of its objects. Color has long been the focus of debates about the metaphysics of sensible qualities, and philosophers have struggled to articulate the conditions on the visual experience of mind-independent objects. With few notable exceptions, ’’visuocentrism’’ has shaped our understanding of the nature and functions of perception, and of our conception of its objects. The predominant line of thought from the early modern era to the present is that, in the philosophically interesting respects, as things are with vision, so they are with hearing, touch, olfaction, and the rest. A closely related line of thought has been particularly strong in the case of the secondary qualities. The more or less implicit assumption is that as things are with colors, so they are with sounds, tastes, and smells. This chapter is predicated on skepticism about this kind of claim. I suggest that we put to rest the traditional lines of thought because hearing and the world of sounds are rich with raw material that presents both novel philosophical problems and telling new instances of old ones. The case of sounds and audition demonstrates that attention to modalities other than vision enriches our understanding of perception and the natures of its objects. This chapter presents the framework for a philosophical account of sounds that I develop and defend in Sounds: A Philosophical Theory (2007). In particular, contrary to the traditional philosophical understanding of sounds as secondary qualities, and contrary to the commonplace scientific view that sounds are waves in a medium, I argue that sounds are events located in the environment near their sources.1 This proposal is designed, first and foremost, to capture 1 Casati and Dokic (1994) propose a related view in La Philosophie du Son. As the following account illustrates, my proposal differs both in the arguments that motivate it and in the events it identifies as the sounds.

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the essentially temporal nature of sounds. Furthermore, it aims to explain the features of auditory perceptual experience in a way that avoids attributing widespread, systematic illusion. According to this account, sounds are particular individuals that bear audible qualities, persist, and travel only if their sources do. Perceivers hear publicly available, distally located sounds thanks to the waves that bear and transmit information about those sounds. I am a realist about sounds. Sounds are individuals in the world that possess many of the features we hear them to have. The proposal that sounds are events of this sort has consequences for theorizing more broadly about perception. This theory of sounds discloses greater variety among the objects of perception than the traditional lines of thought imagine, and forces us to reconsider our visuocentric understanding of perception.

2. what kind of thing is a sound? Sounds are public objects of auditory perception, I maintain until convinced otherwise. You might hallucinate a sound, but in that case you fail to hear a sound—you just think that you do. In principle, others might hear any sound you hear. Tinnitus sufferers suffer hallucinations. Furthermore, I will assume that if you successfully hear anything at all, you hear a sound. Whatever else you hear, such as an object or a happening in your environment, you hear it by way of or in virtue of hearing the sounds it makes. Sounds are, in this innocuous sense, the immediate objects of auditory perception. This sense is innocuous because it is neutral on the question concerning whether you are, in another sense, immediately aware only of auditory sense data. Finally, sounds are frequently characterized by pitch, timbre, and loudness. This tells us very little about what kind of thing a sound is—what ontological category it belongs to.

3. sounds as properties The traditional philosophical outlook has grouped sounds with the colors, tastes, smells, and other sensible attributes or secondary qualities. Popular analyses of such qualities then imply that sounds are either dispositions to cause auditory experiences in

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suitably equipped perceivers under the right sorts of circumstances: categorical bases of such dispositions; physical properties; simple, primitive, or manifest properties; or mere projections of qualities of experiences. The options are familiar from the literature on color. Locke, for one, held that sounds are secondary qualities: powers, grounded in the primary qualities of bodies, to produce auditory experiences (1975: II, viii, 10). But to which bodies did Locke mean to attribute these powers? On a natural reading of the Essay, he meant to attribute them to sounding objects so that sounds, like colors, are dispositions ordinary objects have to affect perceivers’ experiences (see, in particular, II, viii, 9–14). However, Locke may have spoken loosely and meant instead to attribute sounds to the medium that intervenes between object and perceiver so that sounds are dispositions of the medium itself, considered as a body, to produce auditory experiences.2 Depending which Locke meant, we get two views that differ on where sounds are located. Robert Pasnau (1999) takes a stand on this issue concerning the locations of sounds. Pasnau introduces a view according to which sounds are properties of sounding objects, not of the medium. Sources themselves have or possess sounds on this view. For Pasnau, sounds either are identical with or supervene upon the vibrations of the objects we ordinarily count as sound sources, so sounds are properties that depend upon the categorical bases of Lockean powers. Pasnau and Locke thus both reflect the traditional understanding of sounds as secondary qualities or sensible attributes. We can classify views developed in the spirit of the traditional model of sounds as sensible qualities according to their stance on two questions. (1) What is the correct account of the sensible qualities, in general? That is, are they dispositional properties, physical properties, or primitive properties with which perceptual experience acquaints us? (2) Are sounds properties of the medium or of the objects? A matrix of property views of sound results. 2

In a passage from the later Elements of Natural Philosophy, Locke (1823) says:

That which is conveyed into the brain by the ear is called sound; though, in truth, till it come to reach and affect the perceptive part, it be nothing but motion. The motion, which produces in us the perception of sound, is a vibration of the air, caused by an exceeding short, but quick, tremulous motion of the body from which it is propagated; and therefore we consider and denominate them as bodies sounding.

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However, independent of providing the details of a philosophical account of sounds as sensible attributes, we need to ask whether this model is the right approach to sounds in the first place. I want in what follows to suggest that it is not. Both of the questions that yield the above matrix depend upon a misguided supposition. The suggestion that sounds themselves are sensible qualities is attractive only if we are in a mood that overemphasizes similarities with color and entices us to provide an account that subsumes sounds with colors under a single metaphysical category. This should be resisted. Sounds themselves are not properties or qualities at all. Sounds are best understood as particular individuals that possess the audible qualities of pitch, timbre, and loudness, perhaps along with other audible and inaudible properties. Sounds bear similarity and difference relations to each other that are based upon the complexes of audible qualities they instantiate. Sound sources, among which we count ordinary objects and events, such as bells, whistles, and collisions, make or produce sounds, but are not at intervals simply qualified by their sounds in the way that walls are qualified by colors. Several kinds of consideration support this suggestion. First, sounds survive changes to their properties and qualities. A sound that begins high-pitched and loud may continue to exist though it changes to being low-pitched and soft. An object does not lose its sound and gain a new one when it goes from being high-pitched to being low-pitched, as with an emergency siren’s wail. The sound of a spoken word begins with certain audible characteristics and ends with others, but a pitch shift is not the end of a sound. Determinate perceptible or sensible qualities, however, do not survive change in this way. The red color of the fence does not survive the whitewashing. The dank smell of the dog does not survive the perfuming. Particular individuals, such as the fence and the dog, however, survive changes to their qualities. In addition, the identities of many recognizable sounds are tied to the pattern of audible qualities they exhibit over time. To be the sound of a duck’s quack, or the sound of a spoken syllable, requires a certain complex pattern of changes in pitch, timbre, and loudness over time. The sound of the spoken word ’treatise’ differs from the sound of the spoken word ’treason’ precisely because each exhibits a different pattern of change in audible qualities over time.

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Since sounds survive changes to their properties across time, sounds last through time. In particular, sounds have durations or lifetimes. Sounds have beginnings, middles, and endings. A sound can have a low-pitched part and a high-pitched part, and this is not just a matter of some object’s being low-pitched at one time and high-pitched at another. This intuitive philosophical picture of sounds as particulars, not properties or qualities, finds empirical support from research on audition. According to our best understanding of the central task of auditory perceiving, sounds are the individuals that ground the grouping and binding of audible qualities. Perceiving sounds requires discerning coherent and significant streams of auditory information from an intertwined set of signals bound up with irrelevant ’’noise’’. Albert Bregman (1990) likens this problem, which he calls auditory scene analysis, to determining the number, size, and location of pebbles thrown into a lake by observing just the motions of a pair of handkerchiefs moved by the waves that travel up two narrow channels dug at the lake’s edge. Hearing, as we experience it, is made possible in information-rich environments by the auditory system’s ability to sort through the complex information available at the ears and extract cues about significant items the environment contains. The experienced result is a set of distinct, temporally extended sounds heard as generated in the surrounding space. Audition accomplishes this by grouping or bundling audible qualities into distinct auditory perceptual ’’objects’’ or ’’streams’’. A set of assumptions and grouping principles for auditory perceptual items (auditory objects or streams) enables us to associate correctly the low pitch with the soft volume and faraway location, and at the same time to group correctly the high pitch with the loud volume and nearby location, without mixing things up into a garbled ’’sound soup’’ of high pitch, nearness, soft volume, low pitch, loud volume, and distance. Our ability to group correctly the qualities of auditory perceptual objects or streams grounds our ability to discern complex individual sounds in the environment on the basis of information arriving at the ears. Auditory scene analysis amounts to sound perception precisely because the auditory system invokes principles founded upon assumptions that capture genuine regularities in the world of sounds.

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The auditory system solves the problem of auditory scene analysis by segregating the auditory scene into separate sound objects or streams characterized by complexes of pitch, timbre, loudness, and location. This answer, in effect, turns on the auditory system’s treating the auditory objects or streams in question as particular individuals. First, auditory objects or streams bear pitch, timbre, and loudness and thus serve as the locus for property binding. Second, discrete auditory objects may be represented as distinct both at a single time and across time. That is, distinct sounds can be heard as simultaneous, and successive but qualitatively similar sounds need not be identified. Third, as the term ’’stream’’ indicates, they last through time and persist by having duration, and may be represented to persist even through masking noise. Finally, auditory perceptual objects or streams regularly survive changes to their properties though time, as the sound of a spoken word or waning siren demonstrates. These considerations strongly indicate that auditory objects or streams are particulars that ground audible property grouping and binding, auditory attention, and figure-ground distinctions. Awareness of an auditory object or stream constitutes awareness of a sound, an audible particular. Finally, sounds have sources. Although we commonly experience sounds as sounds of something—we hear the sound of a car, a bell, or a dog—that does not imply, in the first instance, that their sources bear or possess the sounds. We might experience a sound without experiencing its source, and sounds might appear to outlast their sources. Sounds, it seems, are produced or generated by their sources. Ordinary objects and happenings cause sounds. Properties and qualities, on the other hand, are not commonly understood as standing in causal relationships to their bearers. These arguments show that we do not regard sounds merely as repeatables that account for the dimensions of similarity among other items. Rather, sounds are distinct particulars that bear similarity and difference relations to each other based on the complexes of audible qualities—the properties of pitch, timbre, and loudness—to which their identities are tied. Sounds have identity, individuation, and persistence conditions that require us to distinguish them from properties or qualities of the objects and happenings that produce sounds.

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Identifying sounds with properties has a defect that in my view cannot satisfactorily be repaired. The defect is a failure to account for the essential temporal characteristics of sounds. Property bearers may instantiate and persist through the loss and gain of properties and qualities, while properties, qualities, and their instances exhibit quite different temporal characteristics. This serves as an important indication that sounds are not just properties things gain and lose. The way sounds persist and have duration distinguishes them most sharply from the traditional secondary quality understanding implicit in much philosophical work on sensation and perception. Once appreciated, the temporal characteristics of sounds present the greatest theoretical obstacle to a perceptually tractable and phenomenologically plausible account of sounds along the contours of the property model. All of this is not to say that no account of properties could make sense of the particularity and temporal character of sounds in a way that dealt with auditory grouping and binding through time. A trope theorist, for example, might capture the particularity of sounds by understanding sounds as particularized complexes of particularized pitch, timbre, and loudness complexes bearing particularized temporal relations to each other. The success of the theory of sounds, however, should not rest on such a controversial theory about the metaphysics of properties. My claim is that given the apparent particularity of sounds, which is required to capture certain aspects of how we perceptually individuate sounds, and given the temporal characteristics of sounds, including duration and change, the property model assumed by both the traditional secondary quality view of sounds and Pasnau’s more recent account is ill suited as a perceptually realistic account of the metaphysics of sounds. Abandoning that model frees us from a host of cumbersome and weakly motivated metaphysical commitments. This points the way to a richer and more nuanced understanding of auditory perception and its objects.

4. sounds as waves The standard philosophical understanding of sounds, of which I have been critical, has not gained broad popularity. The science of acoustics has taught that sounds are waves. We learn early on that sounds are longitudinal pressure waves that travel from a source to

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our ears and that these waves are the proximal causes of auditory experiences. The sound just is the wave train leading from source to subject. Just what the customary wave view of sounds amounts to metaphysically is somewhat obscure. One way to characterize the wave is as a pattern of pressures at each point in the surrounding medium over time. This interpretation makes the wave a complex property of the medium that evolves through time. On the version of the secondary quality view that ascribes sounds to the medium, pressure patterns are candidates for the categorical bases of dispositions to produce auditory experiences. This proposal, however, is a version of the property understanding of sounds, and faces just the same problems that stem from treating sounds as repeatable properties instead of particular individuals. As an account of the metaphysics of sounds it makes little headway. There are, however, other promising ways to develop the view that sounds are waves. If the wave view is plausible as a view about what sounds are, then the wave in question is a particular that persists and travels through the medium. First, waves stand in causal relations. Waves are produced or generated by their sources. Sound waves are the causal byproducts of the activities of objects and interacting bodies and have among their effects the motions of resonating bodies and the auditory experiences of hearers. Second, the wave bundle responsible for the experience also has spatial boundaries. It is characterized by a wavefront that propagates outward from the source, and its spatial extent depends on when the wave-generating activity ceases and the last pressure disturbance brings up the rear. Even when the waves rebound from a reflecting surface, spatial boundaries may remain intact, though altered. Furthermore, these spatial boundaries are perceptually significant. For example, the onset of periodic pressure differences at one ear is assumed to share a cause with their onset at the other ear, despite a delay. The spatial boundary responsible for differential onset is critical for auditory localization. Third, the waves propagate or travel at a speed determined by the density and elasticity of the medium. In 20 degrees Celsius air at sea level, we say that the speed of sound waves is 344 meters per second (1497 m/s in water; 6420 m/s in aluminum).

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Finally, waves are capable of surviving changes to their shape and to other properties and qualities. A wave’s form and amplitude may change as it propagates, resulting in different heard attributes, but the wave persists throughout. Such spatially bounded, traveling particulars are in certain respects surprisingly object-like. They can be created; they have reasonably defined spatial boundaries, but persist through deformation; they survive changes to their locations and other properties; and they are publicly perceptible. To be sure, they make peculiar sorts of objects: their capacity to overlap and pass through themselves makes them stranger than most everyday objects. Though this may be a mereologically interesting problem, it seems to pose no fundamental obstacle to viewing wave bundles as in some, perhaps minimal, sense object-like. Another important qualification to this object-like nature is that waves are dependent particulars. Sound waves depend for their existence on a medium. Their survival conditions differ from those of the medium, and they depend on different bits of the medium at different times, but without an elastic medium no sound waves exist. It is likely that lots of other things are dependent particulars, too, like tables and chairs and anything else not identical with its constituting matter. This seems to pose no obstacle to viewing the waves as object-like. The dependence of waves on a medium is significant for a different reason. In light of the awkward fit of understanding waves as object-like particulars, the dependence points to an alternative take on the wave bundle altogether. The wave is in an important sense something that happens to the medium. The wave is not just a parasitic item passing through the medium; it constitutes a dynamic occurrence that takes place within the medium. The existence, propagation, and boundaries of the wave depend on processes that occur within and essentially involve a medium, so to highlight the medium dependence of the wave and its attributes is to highlight the wave’s event-like characteristics. It is more plausible to think of the waves the wave conception of sound identifies as the particular sounds not as the object-like bundle, but instead as a variety of event that takes place and evolves in the medium through time. Whether or not the wave view of sounds can accommodate it, the event-like construal is far more plausible as an account of sounds

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than the object-like construal. Features central to how we conceive of object-like particulars, in contrast to time-taking particulars like happenings and events, make for poor characterizations of sounds. One telling point already played a key role in rejecting the property understanding and delivers a central desideratum in theorizing about sounds. An account of sounds should capture the fact that the qualitative profile of a sound over time is crucial to its being the sound that it is, as we recognize in the difference between the sounds ’protect’ and ’protean’. But it is an intuitive feature of the way we perceive and perceptually understand objects that they persist by enduring through time, as opposed to perduring by having numerically distinct temporal parts at different times. That is, we intuitively think of objects, as opposed to time-taking particulars, as being wholly present at each time at which they exist. This is what led Thomson to say of perdurantism, ’’It seems to me a crazy metaphysic—obviously false’’ (1983: 210). And that is why the perdurantist must motivate the view with philosophical considerations. This fact about the way that objects appear to persist does not apply to events and other time-taking particulars, which intuitively have parts that exist and take place at different times. In particular, it does not apply to sounds as we perceptually individuate them, since sounds simply are not candidates for being entirely present at a given moment. Sounds, instead, are things that occur over time. Now, if objects do perdure, in contrast to the intuitive way we perceive and understand them, then the difference between events and time-taking particulars and objects may be just a matter of degree. If so, sounds are quite a distance from the end of the continuum occupied by tables, chairs, and even persons. In any case, I do not want my account of the metaphysics of sounds to hinge essentially on a discussion of how objects persist. What is clear is that sounds differ in important respects from ordinary objects in their ways of extending through time. My goal has been to point out that the widely accepted wave view is not completely clear either from a metaphysical standpoint or as a theory of sounds. The understanding of waves as eventlike particulars is the most promising way to develop the view that sounds are longitudinal compression waves. That work seems worthwhile because the view that waves are dependent, spatially bounded, event-like particulars that persist and travel from their

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sources outward through the surrounding medium captures many of our commonly held beliefs about sounds. But the model of sounds as waves, like the traditional philosophical model of sounds as properties, has important shortcomings that make it unsuitable for a philosophical theory of sounds. It is a strength of the wave view that it counts sounds as particulars that persist. But a theory of sounds should identify not only the ontological kind to which sounds belong, but also just where in space and time sounds exist. The wave account’s problems stem primarily from its implication that such particulars exist or occur in different parts of the medium over time. The claim that sounds travel, however, turns out to be an unnecessary and, indeed, undesirable commitment for a theory of sounds.

5. the locations of sounds Any realist account of sounds should say just where in space and time sounds exist. As with property accounts, other theories may differ in where they locate the sounds. If sounds are waves, and waves are events, sounds are located throughout the medium and travel in the sense that their position changes from one time to another. At one time the waves are there but not here; at another time the waves are here but not there. But hearing, like vision and probably unlike olfaction, is a locational modality. Hearing furnishes information about the locations of objects and events in the surrounding environment. We learn on the basis of hearing not just that a plate has broken, but also something about where to look for the mess. Though hearing lacks the fine spatial resolution of vision, audition presents information about the relative locations of audible events and objects. Hearing furnishes information about the locations of objects and events in the surrounding environment by presenting sounds themselves as located. Sounds seem to be located not only in a particular direction, but also at some distance. Auditory researchers refer to this phenomenon as extracranial localization. One of the most active areas of research into locational hearing seeks to explain the mechanisms that ground the experiential sense that sounds occur at particular locations around us and do not just seem to be located, for example, at the ears.

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Why say that sounds themselves seem to be located? First, the data of psychological research supports this claim. In Spatial Hearing, Blauert (1997) says: Research has shown that the region of most precise spatial hearing lies in, or close to, the forward direction and that, within this region, a lateral displacement of the sound source most easily leads to a change in the position of the auditory event . . . . The spatial resolution limit of the auditory system [about 1 degree of arc] is, then, about two orders of magnitude less than that of the visual system, which is capable of distinguishing changes of angle of less than one minute of arc. (38–9)

The spatial information conveyed in audition, however, is not just directional. Concerning what he calls distance hearing, Blauert reports: For familiar signals such as human speech at its normal loudness, the distance of the auditory event corresponds quite well to that of the sound source. (45)

Blauert notes that although distance localization is much less accurate for unfamiliar sounds, including ’’unusual types of speech,’’ even in such cases, ’’The auditory event is, to be sure, precisely spatially located’’ (45–6). This is representative of the intuitive and widely accepted view among auditory researchers that hearing informs subjects about the locations of sounds in egocentric space. This view also is apparent when we compare it to alternative phenomenological descriptions. Sounds do not ordinarily seem in auditory experience to travel. Imagine hearing a sound that seemed to be generated across the room, and that subsequently seemed to move toward your head like the auditory analog of a missile. You probably would try to duck out of the way. The experience of such a traveling auditory particular would be quite unlike your ordinary experience of sounds, which seem to be located at a distance in some direction. We often, however, describe sounds as coming from their sources, and not as being at or near their sources. My auditory missile example illustrates that sounds do not auditorily seem to travel toward us from their sources. Sounds also do not seem to be nearby (at the ear) but to have come from somewhere else, as a breeze is felt on the face as having come from the left. Headphone listening illustrates the contrast. Ordinary headphones noticeably lack distance

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or externalization cues, though they support directional hearing.3 Sounds therefore do not seem to come from their sources in any spatial sense of coming from. The sense in which it is correct to say that sounds seem to come from their sources must be a causal sense. Sounds seem produced or generated by their sources. The claim that sounds are phenomenologically located in the environment, at a distance in some direction, grounds an important fact about locational hearing. It is clear that we gain information about the locations of items and happenings around us by means of audition. Furthermore, this locational information is perceptually available to us in audition—we can act upon and form beliefs about the locations of things in the environment just on the strength of auditory experience. Since sounds are the immediate objects of auditory awareness, awareness of a sound and its audible qualities must furnish or bear locational information about sound sources. But sounds do not seem to come from their sources in a sense that includes travel from those sources, and sounds do not seem to come from their sources in the sense that they seem to be nearby but to have come from the source. Sounds seem to come from their sources only in the sense of being produced or generated by those sources.4 So, hearing sounds themselves as located makes possible one’s audition-based awareness of the arrangement of everyday things and happenings, and it grounds perceptual beliefs about their locations. I have argued that sounds seem located and that sounds seem to travel only if their sources do. Sounds in this way mediate auditory perceptual access to the locations of things and events in the environment. Unless we are subject to a systematic illusion of spatial location in audition, a theory of sounds must locate the immediate objects of hearing at a distance from perceivers, in the neighborhood of their sources. Not only do we sometimes get the locations of sounds wrong in hearing if sounds are not distally located and relatively stationary, we almost never perceive a sound to occupy its true location. If the phenomenological claim is correct, and if 3 Expensive headphones that retain the cues required for externalization exist. Such headphones require custom measurements to determine the effects of the pinnae on incoming sound waves to calculate the individual head related transfer functions (HRTFs) that proper externalization requires. See, e.g., Carlile (1996). 4 Nudds (2001) discusses at length the perception of the generation of sounds by sources.

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auditory experience is not systematically illusory with respect to the locations of its objects, then sounds do not travel through the surrounding medium, and the wave model fails.5

6. duration Locational hearing is not all that is mistaken if sounds propagate through the medium. The illusions multiply. The wave-based understanding of sounds is unable satisfactorily to account for a critical dimension of sounds and auditory experiences. It, too, fails to capture the temporal characteristics of sounds. Perceiving the durations of sounds is clearly an important part of auditory perception. Sounds inform us about happenings in and states of our environment, and part of what they inform us about is how long those happenings last. I learn through hearing when the coin stops spinning, when the fridge starts up and shuts down, and how long the car idles in the driveway. I experience how long the nine-year-old who lives next door practices violin each afternoon—I sometimes wish the sessions had shorter durations. If sounds are spatially bounded particulars that travel through the medium, what in fact I experience when I take the sound to have duration, however, is not the duration of a sound at all. Rather, my encounter with a spatial boundary of a sound leads to my enjoying an auditory experience while the sound passes. On later encountering the far boundary of the sound, I experience the sound to end. Whether the wave is an object-like particular that passes by, or an event-like particular that unfolds at different places in the medium over time, domino-wise, my experience of the sound is caused by the spatial parts of the sound wave bundle as it passes. I do not experience the lifetime of an object-like entity or the duration of an event other than my own sensing. Apparent duration perception results from encounters with the spatial boundaries of sounds, according to the wave view. This means that each time I hear a sound, I mistake an experience of the spatial boundaries of a sound for an experience of the duration and temporal boundaries of that sound. The experienced duration of a sound is therefore nothing 5 Pasnau (1999) argues that locational hearing is incompatible with a wave account of sounds that does not attribute widespread spatial illusion to audition.

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more than a form of crude projective error: I mistake the duration of an experience alone for the duration of the thing I am experiencing. Duration perception, too, is a wholesale illusion if sounds are waves. Perhaps you are willing to live with the illusion to preserve the common scientific view. So suppose the experience of a sound’s duration is an illusion. Since experiencing a sound mediates our awareness of sound-producing events, and, in particular, since experiencing the duration of a sound mediates awareness of the duration of a sound-producing event, your awareness of the durations of sound-producing events is mediated by illusory awareness of the durations of sounds. We have no reason, however, to doubt that awareness of the durations of sound-producing events is veridical. Such awareness regularly grounds true perceptually-based beliefs. It follows that this case constitutes an instance of veridical mediated awareness that is mediated by an illusion.6 This complication strikes me as the most important negative consequence of a commitment to illusory sound duration perception, since it is arguably among the primary functions of auditory perception to inform us about the temporal characteristics, including the durations and patterns of change, of happenings in our environment. The account of sounds as waves entails that we do not hear the durations of sounds and that our justification for believing that the violin practice lasted 45 minutes cannot come just from hearing because what we experience is an illusion. These consequences result from the claim that sounds construed as waves travel through the medium. The other important consequence is that our ways of perceptually individuating and tracking sounds through time are wildly misguided. If sounds persist and travel in the manner of the waves, then our perceptually-based estimates of the lifetimes and survival conditions of sounds all are incorrect because the waves may continue to exist long after the sound has seemed to cease. It is simply a mistake according to the wave account to state that Time Is On My Side by the Rolling Stones is three minutes and two seconds long if the song is the sounds. 6 It is important here to keep in mind that the mediatedness in question is of a sort to which the subject has access. It is not, for instance, the kind of mediatedness in question when we say that hearing is mediated by activity in the cochlea or auditory nerve.

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I have claimed that the traveling wave view of sounds runs into problems with duration perception, since it makes the perceived duration of a sound a systematic illusion. The problem lies with saying that the sound—what you most immediately hear—is the bundle of waves that passes. Suppose we omit the claim that sounds travel. Because it is a central fact about pressure waves that they travel through a medium, we must then abandon the suggestion that sounds are waves. I contend that the illusions of location and duration warrant doing just this. Sounds, I claim, are located roughly where we hear them to be: at or near their sources. The sound does not travel as do the waves. The waves, however, are causally intermediate between the sounds and the auditory experiences of perceivers. The waves bear or transmit information about sounds through the medium, and thus furnish the materials for auditory experience. Sounds are stationary relative to their sources. If sounds are stationary events, then the auditory experience of location does not involve a systematic and pervasive illusion, and audition-based beliefs about the durations of sounds are for the most part true.

7. sounds as events The wave understanding of sounds gets several things right. According to the best version of the account, individual sounds are particulars that can be counted and quantified over, and possess a range of attributes and qualities. Sounds need not be repeatables or properties ascribed either to ordinary objects or to the medium. It recognizes that sounds are temporally extended occurrents with temporal parts and durations and counts sounds as persisting particulars capable of surviving change. Under its best interpretation, sounds are event-like particulars. A wave-based understanding, however, is unable to capture correctly the temporal characteristics of sounds and the nature of our perceptual acquaintance with sounds that extend through time. In short, it mistakes the lifetime of a train of sound waves in an environment for the duration of a sound. But the claim that sounds are particular events captures important truths about sounds and meets defining desiderata for a theory of sounds and the objects of auditory perception. Sounds, intuitively, are happenings in one’s environment. We speak of sounds, like

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lectures but unlike colors, shapes, and tubas, as occurring, taking place, or lasting. Sounds also stand in causal relations. They are caused by ordinary events like collisions and vibrations, and give rise to reverberant vibration, auditory experiences, and recordings. According to the standard account of causation, causal relata are events. Sounds have straightforward temporal boundaries that circumscribe durations, but, like events and unlike objects, their spatial boundaries are less obvious. Sounds, in addition, appear to tolerate co-location or overlap with other sorts of things and events. A sound might occupy part of the same region as a fiddle or a bowing. Sounds, that is, appear to relate to space and time in ways characteristic to events. Understanding sounds as events of a certain sort amounts to a powerful framework for a satisfactory account of both the metaphysics of sound and the objects of auditory experience. There is one caveat. The critical features of the theory of sounds should not turn on some one account of the metaphysics of events. I would like the theory of sounds to be reasonably neutral on the nature of events and viable no matter what events turn out to be.7 One might even hold it against a theory of events if it fails to capture facts about sounds. So, within reason, whatever events turn out to be, sounds should count as events. I think there is good chance for this, though once we get down to the detailed theory of sounds, some decisions will turn on just what is the right account of events. I want for now to operate with an intuitive conception of events as potentially time-taking individuals—happenings that may or may not essentially involve change. Events as I wish to understand them are immanent or concrete individuals located in space and time. Sounds, among the events, are akin to processes or activities. Sounds are not instantaneous events, but require time to unfold. Some sounds—such as spoken words, bird calls, or an eighth-note at Csharp—may lend themselves to treatment as performances or accomplishments with a certain natural trajectory toward completion. So, sounds are events located at a distance from their perceivers. They occur at or near their sources, and travel only if their sources travel. Sounds have durations and are capable of surviving changes to their properties and qualities across time. Sounds stand in causal 7 Candidates include, for instance, theories stemming from Davidson (1970), Kim (1973), Galton (1984), Lewis (1986), and Bennett (1988), among others.

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relations to the activities of objects and events that are sound sources, and they fulfil the causal requirement on any account of their veridical perception. Sounds thus occupy distinctive causal roles. Which distal events are the sounds? Consider the case of a tuning fork struck in air. The striking of the fork makes or causes a sound in virtue of the oscillating arms of the fork disturbing the surrounding air and creating regular compressions and rarefactions. However, since sounds do not travel through the medium, but remain stationary relative to their sources, the sound does not travel as do the waves. Since sound waves that reach the eardrums cause auditory experiences, sounds must be causally intermediate between ordinary, everyday events and traveling sound waves. Since waves bear and transmit information about sounds, sounds cause waves. And since sounds indicate something about the events and happenings that occur in an environment, ordinary objects and happenings cause sounds. Recall that what you perceive as the duration of a sound is in fact the duration of the process of sound wave production. Since the event in which sound waves are produced occupies a role causally intermediate between ordinary collisions or strummings and subsequent sound waves propagating throughout the medium, this event plays a centrally important part in developing the theory of sounds. My claim is that such events are strong candidates for the particular events that are the sounds. Consider the tuning fork. The sound, I propose, is the event of the tuning fork’s disturbing the medium. According to this way of articulating the proposal that sounds are events, particular sounds are events of oscillating or interacting bodies disturbing or setting a surrounding medium into wave motion. This event occupies the appropriate causally intermediate role between the everyday events that cause sounds and the compression waves that travel through the medium bearing the marks of sounds and producing experiences. If a sound just is an object or interacting bodies’ disturbing the surrounding medium in a wave-like or periodic manner, then sounds do not travel through the medium, but remain stationary relative to their sources. A sound unfolds over time at a location determined by the sound source. Though it does not travel through the medium, however, it necessarily involves a medium. If sounds are the immediate objects of hearing,

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such disturbing events are the best candidates for the sounds. Its creating the disturbance constitutes the tuning fork’s sounding.8 According to this account, sounds are particular events of a certain kind. They are events in which a moving object disturbs a surrounding medium and sets it moving. The strikings and crashings are not the sounds, but are the causes of sounds. The waves in the medium are not the sounds themselves, but are the effects of sounds. Sounds so conceived possess the properties we hear sounds as possessing: pitch, timbre, loudness, duration, and spatial location. This distal event understanding of sounds counts among its greatest strengths the resources to capture convincingly the conditions under which sounds are identified and individuated. The disturbance event account individuates sounds primarily in terms of their causal sources and their spatio-temporal boundaries. A given sound particular has a unified causal source and must be spatially and temporally continuous throughout its entire history. A change in causal source, or a spatial or temporal discontinuity, suffices for numerically distinct sound particulars. Qualitative resemblance, however, is neither necessary nor sufficient for numerical identity of a sound. A temporally seamless transition from one instrument’s playing a C-sharp to another instrument’s playing a Csharp involves numerically distinct sounds of the same sound type, since it involves different disturbance events. Qualitatively similar sounds with numerically distinct sources are the same sound in nothing stronger than a qualitative sense. Temporally discontinuous soundings from the same source likewise are at most qualitatively identical since they involve different medium disturbance events. But when a single instrument seamlessly shifts from playing Csharp to playing B, only its state of sounding changes. There is still a single sound event of which each note instance is a part, and so each note instance is part of a single continuous sound. A sound can extend over considerable time and might change a great deal qualitatively. It may at times be loud and low-pitched; at times it may be soft and high-pitched. If the causal source remains the same and the 8 My account thus differs from the located event theory of sound proposed by Casati and Dokic (1994, 2005) in holding that a medium is a necessary condition on the existence of a sound, rather than just a condition on the perceptibility of a sound. See O’Callagham (2007: ch. 4).

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disturbing is spatio-temporally continuous, it may remain a single sound.9 Difficult cases for the spatial and temporal criteria, such as a tele-transported or time-traveling trumpeter, may of course arise. These cases should be decided by appeal to whether the causal source criterion is satisfied. When the causal source is numerically identical, spatial and temporal continuity from the point of view of the source may obtain and resolve the question in favor of identity. None of this rules out that there might be complex sounds comprised of distinct sounds from a number of sources arranged either across space, over time, or both, as when an orchestra plays. Complex sounds might even include periods of silence. Consider the sound of a song or of a spoken sentence. Complex sounds, however, are complex events constituted by many distinct sounds or disturbance events, and some principle of unity must exist. There may be many different justifiable ways of counting sounds in these kinds of cases, but ways of counting complex sounds are intelligible because they invoke complex event types or complex sound universals. The ways of counting or individuating sounds may differ depending on one’s purpose. Understanding the metaphysics of music or of speech sounds differs from developing a metaphysical account of environmental sounds because the kinds and complexity of the sound events of interest to each enterprise differ. It is striking, however, that disputes over individuation principles for sounds and disagreements about the number of sounds one has heard mirror disputes about individuating or counting events themselves—disputes that are notoriously difficult to resolve. This makes it a virtue of the event model of sounds that it leaves room for disputes concerning how many sounds have occurred, since it inherits that feature from questions and uncertainty about counting and individuating events.

8. sound-related phenomena: interference, echoes, and doppler effects The discussion so far leaves unresolved a host of questions about pervasive sound-related phenomena. The familiar wave model is 9 A significant and sharp qualitative change may suffice for distinct sound particulars in absence of a temporal gap when it diagnoses a different medium disturbance.

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fantastically successful at explaining the experiential impact of effects such as constructive and destructive interference; transmission through interfaces and barriers; echoes and reflected sounds; and the Doppler effects. Divorcing the sounds from the waves traveling in a medium means the distal event proposal, or any account that locates sounds at or near their sources, owes equally explanatory accounts of these phenomena and the related contents of perception. The event model surpasses the wave view’s success at convincingly accounting for such phenomena. The distal event account claims that sound waves transmit information about the sounds. It therefore can explain interference, transmission, echoes, and the Doppler effects as wave phenomena that have little to do with sounds themselves. For example, the situation in which destructive interference among the waves from two different sounds creates a ’’silent spot’’ at a node where the summed amplitudes of waves cancel is not just a place where there is no sound. Instead, wave interference creates places from which one cannot hear the two sounds that exist in the surrounding space. Because information about sounds is transmitted through a medium by pressure waves, and because waves behave as they do, from such a ’’silent’’ node it is as if there are no sounds around. The situation from that place mimics locally the situation, with respect to waves, in which no sounds exist to be heard in one’s surroundings. What about echoes? According to theories on which sounds are waves, an echo is a sound that travels through space and rebounds from reflective surfaces. No such story is available if sounds do not move with their waves. According to the event account I have proposed, the experience of an echo is not a second encounter with a traveling sound at a later stage in its career. An echo experience instead is a second, illusory, experience of the original primary sound. One enjoys a second experience of the original sound event thanks to the way sound waves travel and rebound from reflecting surfaces. The second experience, however, includes illusions of space and time. The echo experience presents the sound as located where it is not (at the reflecting surface), and though the sound heard is past, the echo experience presents it as occurring now. This temporal illusion, however, is no more troublesome than the minimal temporal illusion in ordinary hearing, or for that matter

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in vision. Sounds heard and objects seen are heard and seen as they were due to the delayed arrival of information about them. The delay in hearing is greater than that in vision, as we easily confirm at a fireworks display. Explaining the Doppler effect is one of the event account’s strong suits. According to the wave account there actually are two Doppler effects. When a source travels toward a stationary subject, individual wave peaks compress to yield a higher frequency and higher perceived pitch. Since the frequency of the wave is higher than if the source were stationary, and since pitch is tied to frequency, the pitch of the sound itself is higher than when the same source is stationary. If, however, a subject travels toward a stationary source, the subject encounters more wave peaks per unit of time and falls prey to an illusion of increased pitch. Understanding sounds as events located near their sources, however, yields a unified explanation of source-motion and subject-motion Doppler effects. Both source motion and subject motion produce illusions of altered pitch thanks to how waves transmit information about sounds and excite auditory experiences. In neither case do the qualities of a sound change due to relative motion of source and subject. Rather, a sound merely seems to have altered its pitch thanks to such relative motion. The event view thus captures the way experienced pitch depends upon the motions of subjects and sound sources. As with interference and echoes, Doppler effects are perceptual effects that result from our encounters with waves; none involves the sounds themselves.

9. concluding remarks The foregoing discussion illustrates that the event view furnishes the materials for an explanatorily robust understanding of sounds and their perception. The key is that sound waves transmit information about sounds but are not identical with the sounds. Waves are the proximal stimulus to audition but are not themselves the objects of auditory perceptual experience. This account relies on a model of auditory perception that differs in important respects from the received understanding of hearing as involving awareness of sounds constituted by perceptible patterns of pressure difference throughout a medium. The medium according to the event account

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is a necessary condition on the perceptibility of a sound, but the medium cannot satisfy the constraints that must be met by the proper objects of auditory perception. What consequences does this account have for theorizing more generally about perception? Sounds, I have argued, are not among the traditional secondary or sensible qualities because sounds are particular individuals. Pitch, timbre, and loudness, however, provide auditory analogs of color and other sensible qualities. But sounds are not ordinary objects, and sounds are not even objectlike particulars. The event view therefore challenges the simple understanding according to which perception reveals just ordinary objects and their attributes. The event view of sounds thus entails that more variety exists among the immediate objects of perception than many modern views acknowledge. If sounds are the immediate objects of auditory awareness, and sounds are events, then audition involves unmediated awareness of events. That is, in the sense discussed earlier, awareness of a sound is awareness of an event that is not mediated by prior awareness of some object and its states or changes. In fact, any auditory awareness of the activities of ordinary objects is mediated by awareness of sounds. Events figure into the immediate contents of audition according to this account. I have aimed to demonstrate how thinking about perception and the natures of its objects is made richer by attention to audition. Such attention contributes, more broadly, to understanding how to reconcile the manifest and scientific images of the world. The guiding suggestion has been that the tyranny of the visual undermines a complete understanding of perception and the things we perceive. We are likely to miss the most interesting and distinctive features of sounds and audition if we remain bound to the model of vision. Just as we miss what is most striking about vision and its objects if we neglect spatial features, appreciating audition and the nature of its objects requires taking seriously their temporal characteristics. Just as ordinary visual objects are essentially spatial, sounds are essentially temporal. Traditional visuocentric ways of understanding the objects of experience simply fail to capture what is most interesting about sounds. I have argued that taking the temporal features of sounds seriously shows that sounds are neither secondary qualities as the traditional philosophical outlook has it nor waves as the common

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scientifically grounded view suggests. Sounds are bearers of audible qualities, and waves, which facilitate audition, are the causal byproducts of sounds. Everyday events like collisions and vibrations cause sounds whose locations remain stationary relative to their sources. Sounds do not travel. Sounds are particular events whose locations and durations are, when things go well in hearing, as they seem to be. Rice University

references Bennett, J. (1988). Events and Their Names. (Oxford: Clarendon Press). Blauert, J. (1997). Spatial Hearing: The Psychophysics of Human Sound Localization. MIT Press. Bregman, A. S. (1990). Auditory Scene Analysis: The Perceptual Organization of Sound. (MIT Press.) Carlile, S. (ed.) (1996). Virtual Auditory Space: Generation and Applications. (Austin, TX: R. G. Landes). Casati, R. and Dekie, J. (1994). La Philosophie du Son (Nimes: Chambon). (2005). ‘‘Sounds’’. Standford Encyclopedia of Philosophy. E. Zalta, ed. Davidson, D. (1970). ’’Events as Particulars.’’ In Essays on Actions and Events. (1980) (Oxford: Clarendon Press). Galton, A. (1984). The Logic of Aspect. (Oxford: Clarendon Press). Kim, J. (1973). ’’Causation, Nomic Subsumption, and the Concept of Event.’’ Journal of Philosophy 70: 217–36. Lewis, D. (1986). ’’Events.’’ In Philosophical Papers, Volume II. (Oxford: Oxford University Press). Locke, J. (1975). An Essay Concerning Human Understanding. P. Nidditch, ed., (Oxford: Clarendon Press). Locke, J. (1823). Elements of Natural Philosophy. In The Works of John Locke, volume III. Printed for Thomas Tegg, London. Nudds, M. (2001). ’’Experiencing the Production of Sounds.’’ European Journal of Philosophy 9: 210–29. O’Callaghan, C. (2007). Sounds: A Philosophical Theory. (Oxford: Oxford University Press). Pasnau, R. (1999). ’’What is Sound?’’ Philosophical Quarterly 49: 309–24. Thomson, J. J. (1983). ’’Parthood and Identity Across Time.’’ The Journal of Philosophy 80: 201–20.

12. Hearing Sounds1 Roger Scruton I entirely endorse Casey O’Callaghan’s contention that sounds are not secondary qualities. And I agree with him that they are events, that they are publicly audible and that they are situated (though often not precisely situated) in space. However, I am not convinced that sounds are the kind of physical event that O’Callaghan says they are. His arguments against the view that sounds are secondary properties are persuasive, to me, largely because they are arguments against the view that sounds are properties. I agree that sounds are objects, that they belong to that amorphous class of objects that we refer to as events, that they have temporal duration and that they are the bearers of properties. But I wish to argue that they are secondary objects, in something like the way that red is a secondary property. I also argue elsewhere that they are pure events—events which do not happen to anything else, but just happen.2 About this second thesis I will not have anything to say, though I think that it is of the first importance for the metaphysics of music. Another way of putting the point that I wish to make is to say that sounds are audibilia—they are essentially things heard, and are absent from the world of the deaf person in the way that colors are absent from the world of the blind. Underlying O’Callaghan’s argument is a kind of scientific realism, not to say physicalism, which regards as real only those things that are referred to (quantified over) in the true theory of the world. Sounds, on this view, are the events that explain our auditory experiences, and are to be identified in terms of changes and transformations in the primary qualities of material things. In opposition to that, I want to argue for a richer ontology, 1 I am very grateful to Dean Zimmerman and Casey O’Callaghan for comments on an earlier version of this chapter. 2 See The Aesthetics of Music, Oxford: Oxford University Press, 1997, ch. 1., and ‘Sounds as Secondary Objects and Pure Events’, in M. Nudds and Casey O’Callaghan, Sounds and Perception: New Philosophical Essays. Oxford: Oxford University Press, 2009.

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and one that includes objects and properties that do not figure in the scientific theory that explains our experience of them. It is best to begin from examples of secondary objects that are sensibilia of some other sense modality than hearing. Consider smells. These are objects of the sense of smell, and are absent from the world of the person who does not have that sense. Like sounds, they are located in space, but not precisely located. The bale of hay at my feet emits a smell, and this smell hovers in the vicinity of the hay. But it is not a property of the hay, since it (this very smell) can and does exist without the hay, and lingers in the stable after the hay has been taken away. I am not sure that the smell can be described as an event, and indeed I am at a loss to know quite what ontological category, other than the broadest category of object, to which to assign it. But I am sure that its existence is in a deep sense dependent on our capacity to observe it. For something to be red, it has been plausibly argued, is for it to look red to a normal observer: there is an ontological dependence of the property on the capacity of competent observers to experience it. Likewise, I suggest, for there to be a smell of hay at this place in the stable is for normal observers, situated at or near this place, to sense just such a smell. This means that there can be illusory smells. It is one of the distressing features of certain schizophrenic states that they involve powerful and often humiliating smell-illusions. Smells can be described and compared, and there can be experts at discerning them. People like Robert Parker can insure their noses for a million dollars, on the grounds that their ability to discern and distinguish smells is both vital to their career and also an objective guide to the real character of the olfactory world. But what about the smells that normal people cannot smell, because their noses fall short of them? Dogs and horses recognize each other by their smell; deer smell the approaching stalker from a mile away downwind; a pack of hounds follows a scent that was left by the quarry half an hour earlier, and which no normal person can smell; bees are attracted by their sense of smell to flowers that have no scent to our noses. In such cases I think we should say that the concept of normality that we are invoking is itself relative to the smell to be discerned. The normal observer is the one equipped to distinguish smells of the relevant kind. The scent of a fox exists along a certain trail if creatures equipped to discern smells of that

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kind will discern it when following that trail. No matter that we do not discern it; it is there for the foxhound, and therefore it is there. You might seriously wonder why we should reify, in this way, the object of smell. Why not simply refer to the experience of smelling things, and leave it to science to discover the causes of that experience—the interaction between molecules in the air and receptor cells located in the nose and mouth, for example. That way the ‘secondary object’ drops out of the picture, leaving only ordinary objects and their primary qualities as the ultimate and explanatory truth. My response is to say that, indeed, there is no great metaphysical loss involved in the refusal to reify smells. The loss would occur only if there were some practice in which we collectively participate, which has an importance for us, and in which smells, construed as objects of perception, play an indispensable part. Most candidates for such a practice—including the sniffing and snorting of the wine taster—do not build upon some primitive act of reification. We do not have to think of the smell as an object lingering in the glass in order to appreciate the distinction between one bouquet and another. It would be enough to recognize that the experiences of smelling the wines are sufficiently distinct, and may be capable of being identified through their normal causes. To smell this wine is something like smelling hay; to smell that one something like smelling sour milk; and so on. All the—often painfully—lyrical descriptions of the wine critics could be recast as descriptions of individual experiences without loss of sense. When it comes to secondary objects in the visual world, however, the move towards reification seems rather more firmly grounded. Rainbows provide a particularly vivid example. That there is a rainbow visible over the hill beyond my window follows from the fact that a normal observer, located here, would have just such a visual experience when looking towards the hill. This does not mean that rainbows have only secondary qualities: on the contrary, rainbows have many primary qualities, such as shape, size, and duration. But their having these qualities depends upon a counterfactual about experience. Rainbows are located, but not precisely. There is no pot of gold at the end of the rainbow because there is no place which is the end of the rainbow; nor is there a stretch of sky that the rainbow occupies. In this case location too is experience-dependent. To say

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that there is a rainbow visible over the hill is to say that a person located in a certain place and looking towards the hill would see the arch of a rainbow lying over it. Rainbows do not take up space, and do not exclude other objects from the spaces where they appear. Nevertheless there is a distinction between the places where rainbows are and the places where they are not. Rainbows are real and objective. Someone who claims to see a rainbow where the normal observer could not see one either is under an illusion or has made a mistake. Such a person is wrong about the way the world is. Rainbows can be other than they seem, and seem other than they are. There were rainbows in the world before there were creatures to observe them, for the truth about rainbows consists in the truth of a counterfactual, concerning what normal observers would see were their eyes to be turned in a certain direction. The rainbow, like the photon, is an ungrounded disposition, and it illustrates the way in which ungrounded dispositions can be part of the fabric of reality. None of that implies that we could not give a full explanation of rainbows in terms of the primary-quality structure and changes of normal primary objects. The explanation is indeed familiar to us, and invokes light waves, their refraction by water droplets in the air, and their subsequent impact on the retina, to be processed by the brain. This explanation of rainbows is very similar to the explanation of the experience of smells in terms of the interaction of molecules in the air with receptor cells in the nose. But it will not mention any particular object that is identical with the rainbow, in the way that O’Callaghan urges us to consider the disturbing of the medium around the source as identical with the sound. Hence it will leave us free to locate the rainbow in the area where it appears, and not at some place chosen for its prominent role in the theory of rainbows (for instance the patch of water drops in the air, the sun, the eye of the beholder). Of course, the rainbow appears to be in place X only from place Y: someone standing at X might not see it. But this illustrates once again the peculiar relation of rainbows to the space in which they are situated: a rainbow is visible at a place from a place, and that is the last word as to where it is. The reification of rainbows is, nevertheless, better grounded than the reification of smells. Rainbows summon our attention in just the way that other spectacular visual objects summon it. They

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have a place in our visual experience, which is comparable to the place occupied by many ordinary objects: stars, for example. Hence rainbows feature in story telling and myth in ways that assume their existence as independent objects. God gave the rainbow as a covenant to Noah, that He would never again flood the world. In Das Rheingold the gods enter Valhalla across the rainbow bridge; and the goddess Iris spreads out the rainbow as a sign of her peace. Nor is it only in myths and religions that the rainbow appears as a secondary object. We point to rainbows, describe them to each other, count them, attribute properties to them, and reidentify them at successive times. There are ways of perceiving rainbows and incorporating them into our plans and projects that are shared and rewarding, and which would be damaged were we to cease to identify rainbows as objects, so as to refer instead to our visual experiences and their hypothetical cause. The need to reify is even more apparent in the case of sounds. And it is by considering them that I believe we can most clearly discern the ontological impoverishment that is foisted on us by the scientific realist view of the world—the view which condemns as unreal, all those objects and properties that play no role ˆ in the ultimate explanatory picture. By treating sounds as objects, we can develop a shared language of sound-descriptions, which attributes publicly recognizable properties to those objects: pitch, loudness, timbre, and so on. Having done this, we can then rise to another level of intentionality, so to speak, in which we do not merely hear those objects, and hear them as objects, but also hear them as the peculiar entities which I call tones—objects organized and animated by the forces that govern the world of music. Tones exist only for those with musical imagination; to hear them we must endow sounds, in our perception, with properties that sounds do not have—positions in an auditory space; movement through that space; tension and release. Tones belong to melodies and harmonies; they have a rhythmic pulse and a magnetic effect on their neighbors and successors. Tones are bound together by a virtual causality that we hear but which no animal—not even the most sonorous songbird—has ever heard. Spelling all that out in philosophically acceptable terms is not an easy task, but it is surely true that, while we are dealing here primarily with the way things are heard, that way of hearing would not exist, were it not for our

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disposition to identify sounds as public objects, whose identity and properties are fixed by the way they sound. Someone who took O’Callaghan’s approach would not recognize, in what I have said, anything that undermines the physicalist view of sounds. What sounds really are, he would say, is the local events that cause sound-waves to spread through the surrounding medium to the ear. All I have added, he would suggest, is a complex phenomenology—an account of the intentional objects of auditory experiences. Talk of secondary objects if you will; but the facts can be as well dealt with by regarding sounds as physical events with audible secondary properties. It is these secondary properties that constitute the audibilia to which you draw attention; and nothing is added, other than metaphysical puzzles, by attributing those properties to another kind of object than the one responsible for the physical disturbance. Is there a dispute here? I think there is. For O’Callaghan there is no reason in principle why a deaf person should not perceive all the sounds that I perceive. With the aid of a sophisticated vibromator, such a person could discover the place, the pitch, the loudness and even the timbre of the sounds in his surroundings. He might become better at some auditory judgments than a person who hears. But still, I want to say, he has no knowledge of sounds, just as a blind person has no knowledge of colors. To have knowledge of sounds you must be acquainted with them, so as to know what it is like to hear them. That kind of knowledge is not knowledge that a deaf person has. One who is acquainted with sounds also confronts them as publicly identifiable objects observable to all others with normal hearing. And when we arrange those objects in musical space, as we do when we hear them as tones, their nature as audibilia is the foundation on which we build. Sounds are objects all of whose properties are audible properties, and they stand in audible relations to other objects of their kind. It is for this reason that the art of sound is possible; and the art of sound is music. Now the deaf person who explored the world of sounds with a cunning vibromator will draw many conclusions from his investigations. But one thing he would not know, since it can be known only by hearing it, is that this sound begins the melody, which continues through this sound and ends in this one. He could not

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know that the process which began five minutes ago is, after evermounting tension, now resolving itself in a cadence. And so on. The art in which sound achieves its metaphysical apogee, so to speak, as a system of intricately related and separately identifiable objects, sustaining a drama in which all the characters have only auditory personalities, would be forever unavailable to him. There are qualifications to be added to that picture. There are deaf people who understand and create music: one of the greatest of composers was one of them. However, Beethoven heard what he wrote—it is just that, because of his unfortunate ailment, he could hear only in imagination. There are even people who have been deaf from birth who enjoy music. That is to say they experience pleasure in the concert hall, discriminate harmony from discord in their surroundings, and follow the gestures of conductor and orchestra with real interest and pleasure. English National Opera used to have (before the advent of surtitles) performances accompanied by sign language, so that deaf members of the audience could follow the plot. About such cases I would say only that we are confronted with another sense modality here—the sensory experience of vibrations—and not with the experience of sounds. If my deaf companion, Angela, were actually to be acquainted with the musical movement, and the virtual causality that generates tone from tone in the musical line, then we should have to say that she hears after all. That is, she would be experiencing the sounds as audibilia, under an auditory description. As it is, we should expect Angela, asked to describe the sounds that she imagines, to resemble Locke’s blind man, who imagined red to be like the sound of a trumpet. If Angela were to say that she imagines the sound of the trumpet to be like a red flag waving, then we would understand her. If, however, she were to say that the sound of the trumpet is higher, more shrill, more penetrating than that of the French horn, that it stands out above the chorus of horns and blends beautifully with the oboe, that it loses color as it descends below the upper alto range, and so on, we should be puzzled. Has she been fooling us in saying she was deaf? Why do these ontological disputes matter? It seems to me that they matter because philosophy is not, and ought not to be, the handmaiden of science. Philosophy should not be in the business of clearing the world of all merely intentional objects, and leaving

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the field to theory and explanation. There is another and more important task for it to perform—a task that only philosophy can perform. This is the task of describing the world as it is presented to us in consciousness. That task got a bad name on account of the inspissated technicalese of phenomenology, and badly needs to be rescued from Husserl and his followers. But it can be so rescued, I believe. Philosophy can validate the ways in which we achieve a shared understanding of the world, and show that there are objectively decideable questions concerning entities that would not be referred to in any explanatory science. Why is that? To put the answer as briefly as it can be put: because understanding is one thing, explanation another. Institute for Psychological Sciences

13. What Sounds Are Matthew Nudds Everyone agrees about the facts of the matter. Sounds are produced when objects collide, scrape, or knock together, or are otherwise caused to vibrate. Their vibration disturbs the surrounding air (or other medium), producing a pressure wave which propagates throughout the medium.1 This pressure wave, when it reaches our ears, is focused on to eardrums, which vibrate: some small bones move, sensitive hairs are disturbed, something psychological happens, and we experience sounds. Together these facts constitute the ‘‘sound-producing process’’. Not everyone agrees about what sounds are. Although there is no part of the sound-producing process that sounds have not been claimed to be, sounds have most often been claimed to be identical with a property instantiated by the sound source—a property of the object that produced the sound—or with an event involving the sound source,2 or to be identical with a property instantiated by the pressure wave produced by the sound source. How do we decide which is correct? That is, how do we answer the question of what part, if any, of the sound-producing process should be identified with sounds? Sounds are objects of our auditory experience. So whatever, according to our account, sounds are, they must be such that it is plausible that those things are objects of our auditory experience. The properties of the sounds we hear determine (at least in part) the character of our auditory experience. When we hear a sound we experience it as being some way. So sounds are those things whose properties determine whether sounds are as we experience 1 Pressure waves can occur in the absence of objects (something I discuss in more detail below); the result is the same. 2 See Casati and Dokic (1994), Pasnau (1999), and O’Callaghan (2005, 2007).

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them to be;3 they are whatever part of the sound-producing process determines the veridicality of our experience of sounds, the things on which the veridicality of our experience depends. Any account of sounds that claims to identify sounds with things that do not determine the veridicality of our experience will have misidentified sounds with something that is merely part of the causal process involved in sound production. How do we decide which part of the sound-producing process determines the veridicality of our experience of sounds? To answer that we need to know what are the veridicality conditions of our experience of sounds. We might attempt to discover the veridicality conditions of our auditory experience by introspective reflection: by attending to how sounds seem or appear in our experience of them, we can discover how our experience presents sounds to be. For auditory experience to be veridical, sounds must be as they are presented by that experience to be. Some writers claim that sounds seem to have spatial location and temporal duration. O’Callaghan, for example, claims that we can hear where things are, and can do so because sounds seem to have spatial properties; in particular sounds seem to be spatially located. Hearing ‘‘consciously present[s] sounds as located in the surrounding environment . . . experience presents the sound as occurring at some distance and in a particular direction’’ (O’Callaghan 2007: 32). He claims, too, that sounds seem to have a determinate duration—they have a beginning, middle, and end: a time-course characteristic of the sound in question. If our experience presents sounds as having spatial location and temporal duration, then our experience of sounds will be veridical only if sounds actually have the spatial location and temporal duration they are presented as having. Any account of sounds that claims to identify sounds with things that do not have that spatial location or temporal duration will be vulnerable to the objection that it has misidentified sounds with something that is merely part of the causal process involved in sound production. According to the account that identifies sounds with a property instantiated by the pressure wave produced by sound sources, 3 Compare this with vision: when we see an object, it looks a certain way; the object we see is whichever object it is whose character determines whether things are as they look to be.

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sounds are not spatially located only at their sources; they are located wherever the pressure wave that instantiates them is located. The location of the pressure wave that instantiates a sound changes as it propagates from the sound source, and therefore the location of the sound changes: sounds, like sound waves, propagate from their sources. In the same way, if sounds are identical with a property of the pressure wave produced by sound sources, then their temporal duration is the same as that of the pressure wave: a sound exists for as long as it is instantiated by the pressure wave produced by its source. So, according to this account, sounds do not have the properties they are presented by our experience as having. It follows that if sounds are the objects of auditory experience then, of the three accounts described above, they must either be identical with a property of the sound source—a property of the object that produced the sound—or with an event involving the sound source; either way, they cannot be identical with a property instantiated by the pressure wave produced by the sound source. That, anyway, is the line of thought that has led to the rejection of the wave view of sounds by a number of writers. The problem with this line of thought is that introspective reflection is a poor guide to the veridicality conditions of our perceptual experience in general and of our auditory experience in particular.4 Consider, for example, the claim that our auditory experience presents sounds as spatially located. The claim is based on introspection: it introspectively seems that sounds are presented as spatially located. But although introspection reveals auditory experience to have spatial content, there is more than one way to interpret that spatial content, and introspection alone is not sufficient to decide which is correct. According to one interpretation—the one described above—sounds themselves seem to have a spatial location; the veridicality conditions of our auditory experience therefore require the sounds we hear to be spatially located. According to an alternative interpretation, sounds themselves do not seem to have a spatial location: ‘‘While we have the auditory experience of hearing that a sound comes from p, we do not have any experience that it 4 For example, it is not possible to tell, simply on the basis of introspection, what properties our visual experiences represent objects as having, hence the debate about whether experience represent objects as belonging to natural kinds, as causally related, and so on.

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is here where it now sounds . . . And this is so for a very interesting reason: namely, that we absolutely never immediately perceive sounds to be at any place’’ (O’Shaughnessy, 2000: 446); the veridicality conditions of our auditory experience therefore do not require the sounds we hear to be spatially located. The first interpretation claims that sounds seem to be located, the second claims that sounds seem to have a source at, or to originate from, a location but do not themselves seem to be located. Since it is not clear what difference it would make to how our auditory experience seems for sounds to be presented as coming from a location rather than for them to be presented as having a location, it not clear how introspection alone could decide between these two interpretations.5 A different problem arises when we consider what we should say about experiences of (apparent) sounds when those experiences have been produced in abnormal ways. Given the nature of the sound-producing process it is possible, for any naturally produced sound that we experience, to bring about an experience of an exactly similar sound—a sound with exactly the same character—by bringing about a pressure wave at the ears which is the same as that which in fact produced the experience of the naturally produced sound. There are many abnormal ways of producing such a pressure wave. Perhaps it is possible to do so by directly causing the air to move.6 Is an experience produced in this way veridical or not? That is, in having such an experience do we hear a sound, albeit one that was produced in an abnormal way, or do we merely seem to hear a sound? The answer matters to what account we give of the nature of sounds. If we hear a sound then, since the sound we hear is not instantiated by any source, those accounts that identify sounds with properties instantiated by their sources, or with events involving their sources, must be mistaken. Conversely, if we do not hear a sound then those accounts that identify sounds with properties instantiated by pressure waves must be mistaken. What account we give of the nature of sounds will, therefore, vary according to whether or not experiences of this kind are veridical, 5

For more on this, see Nudds (2009). It is possible, for example, to produce sounds by directly heating the air: the crackle of a spark and the rumble of thunder are both produced in this way. So called ‘plasma’ loudspeakers produce sounds by modulating a spark-like corona which heats the air and directly produces a pressure wave. 6

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but introspection alone cannot tell us: all introspection can tell us is that such experiences seem to involve the presentation of a sound, not whether they really do so. Introspection, then, is both an unreliable guide to what properties sounds are presented as having, and cannot tell us when an auditory experience is veridical: whether an experience presents a sound, or merely seems to. The argument developed above was that sounds are objects of our auditory experience—they are that part of the sound-producing process that determines the veridicality of our experience of sounds—so if we know the veridicality conditions of our auditory experience we can determine what part of the process satisfies those conditions. Since it is not possible to discover the veridicality conditions of our auditory experience by introspective reflection alone, introspective reflection is an unreliable guide to the nature of sounds; in particular, introspective reflection will not decide between the three accounts of sounds described above. We need some other way to determine which part of the sound-producing process determines the veridicality of our auditory experience. When we hear a sound we can attend either to whatever it was that apparently produced the sound or to the sound produced: to the glass breaking or to the sound of the glass breaking. When we are asked to describe what we hear (in psychoacoustics experiments, for example) we are often encouraged to attend to it in the second way: to describe the sensory attributes of the sounds we hear—their pitch and loudness, say—in abstraction from whatever it was that produced them.7 We may be helped by being played harmonically simple sounds produced by a tone generator, sounds which do not develop over time and which have little or no ecological significance. There is little to describe about an experience of such sounds over and above the sensory qualities of the sounds experienced. Most of the sounds we hear—‘ecological’ sounds produced by events occurring in our environment—are far richer. When we are asked to describe such ecological sounds, we do not describe the qualities of sounds that we hear, we describe the apparent sources of the sounds—the events that produced them. We are typically very 7 When we do this we adopt what Gaver (1993a) calls a ‘musical’ and Scruton (1987, pp. 2 ff.) an ‘acousmatic’ attitude to what we hear.

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good at identifying both the kind of event—footsteps, a door opening—and the kind of object—a person, a door—that produced the sounds we hear, and we are often able to perceive features of that object or event—the hardness of the object, the force with which it was struck, its location, whether it was in an enclosed space, and even its approximate size and shape.8 Most everyday hearing is of this kind: we attend to the apparent sources of the sounds we hear and listen to the things going on around us—to the objects and events that produce sounds (‘sound sources’ for short). In most everyday listening we are more concerned with the sound-producing events and the environment in which they occur, than with qualities of the sounds they produce. It is uncontroversial to suggest that auditory perception tells us about the sources of sounds as well as about sounds. The suggestion that I am going to develop is that the function of auditory perception is to tell us about the sources of sounds—that perceiving the sources of sounds is what auditory perception is for and that what sounds we hear we hear as a consequence of the particular way auditory perception functions to tell us about the sources of sounds. My argument has two parts: first I explain how auditory perception could function to tell us about the source of sounds; second, I argue that its so functioning best explains our experience of sounds. Suppose we ask how auditory perception tells us about the sources of the sounds we hear. In explaining the acquisition of perceptual beliefs, we can make a distinction between what we come to believe on the basis of perception and the strictly perceptual basis of those beliefs. It might be suggested that in auditory perception all we strictly perceive are sounds and their auditory qualities, and that we form beliefs about the sources of sounds on the basis of perceiving these purely auditory qualities. Such a suggestion is not generally plausible. For the most part, we do not experience sounds as having auditory qualities that could be the basis of our beliefs about their sources—qualities the recognition of which could make it reasonable for us to form beliefs about the features of the things that produced those sounds. For example, two qualitatively 8 Research has tended not to focus on these aspects of auditory perception, but for a survey of evidence that supports the claims in the text see Carello et al. (2005). See also Wildes and Richards (1988), Freed (1990), McAdams (1993), Lakatos et al. (1997), Kunkler-Peck and Turvey (2000).

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identical sounds may differ in that one may appear to be coming from something on my right and the other from something on my left, but there is nothing in the qualities of the two sounds which could rationalize the different beliefs about the location of their sources that I form on the basis of my experience of them. In this example, changes in a feature of the source—its location—are not reflected in changes in the auditory qualities of the sound it produces. Even when changes in features of the source are reflected in changes in the auditory qualities of the sound it produces, it is not necessarily the case that we form beliefs about the source on the basis of perceiving qualities of the sounds. The experience we have of the sound made by striking a coin gently on the desk is different to the experience we have of the sound made by striking a coin forcefully on the desk; in the first case we can tell that the desk was struck gently, in the second that it was struck forcefully. But there is no way to characterize the experience on the basis of which we can tell that other than in terms of the beliefs formed: the first sound appears to have been produced by a gentle strike, the second by a forceful strike, and in forming the beliefs I am merely accepting that things are as they appear to be.9 Appearing to have been produced by a gentle or a forceful strike is part of the way the sound is presented in experience to be—it is part of the content of the experience of the sound. What these examples suggest is that auditory perception tells us about the sources of sounds not by presenting sounds as having auditory qualities on the basis of which we form beliefs about what produced them, but by presenting sounds as having been produced by sources of certain kinds or by sources having certain features or properties. How could features of the source of a sound be part of the content of an auditory experience? Sounds are produced when objects are caused to vibrate. The vibration of an object is complex—composed of many different frequency components—and varies over time; the particular structure and time-course of the vibration is determined in a law-like way by the character of the event that caused the vibration and the properties of the vibrating object. Both the 9 It would be a mistake to think the difference was a difference in the loudness of the sounds. If we adjusted the sounds to be equally loud we could still hear the difference.

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spectral composition and amplitude of the resultant vibration, for example, varies according to the force with which the object was struck; the size of the object determines the harmonic structure and the frequency of the lowest frequency component of the vibration; the material composition of the object determines the way the vibration changes over time; and so on.10 The structure and timecourse of an object’s vibration therefore carries or embodies a great deal of information about the vibrating object and about the event that caused it to vibrate. When a vibrating object is immersed in a suitable medium, such as air, its vibration produces a pressure wave in that medium. This pressure wave interacts with other objects in the environment and is differentially reflected by surrounding surfaces. These interactions alter the pressure wave in characteristic ways. The structure of the vibration that is transmitted to the ears by the pressure wave therefore carries or embodies information, not only about the objects and events that produced the pressure wave, but also about the environment within which the events that produced it occurred. Sound waves, then, carry information about objects and events in our environment that it would be possible, in principle, for a perceptual system to recover. In this respect sound waves are no different from light: light may carry more information than sound waves, but the objective import of the information is the same in both cases. The recovery of this information is hindered by the fact that the single pressure wave that reaches our ears instantiates a vibration that is the result of pressure waves produced by distinct sound sources, and their reflections, being superimposed on each other. When pressure waves are superimposed, the individual frequency components they instantiate combine additively to produce a single vibration. To extract information about individual sound sources the auditory system must interpret the complex vibration instantiated by the pressure wave that reaches our ears; in what follows I describe how our experience of sounds is determined by this process of interpretation. The auditory system detects the pattern of frequency components that constitute the complex vibration instantiated by the pressure wave that reaches the ears. It constructs a representation of individual sound sources in the environment by extracting the 10 For a useful introduction to some of the ways objects vibrate see Fletcher et al. (1998).

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information about those sound sources that this pattern embodies. Information about an individual sound source is embodied in the structure and time-course of the frequency components produced by that source. To extract this information, the auditory system must first work out which of the frequency components it detects have been produced by the same source, and which have been produced by different sources. It must do this both synchronically—to determine how many sound producing events are occurring at any one time—and diachronically—to determine whether or not an event occurring at one time is a continuation of an event occurring at an earlier time.11 There is nothing intrinsic to a particular frequency component that marks it as having been produced by one source rather than another, or as having been produced simultaneously with any other components, that could enable it to do this, but there are relationships between frequency components that have been produced by the same source that are unlikely to exist by chance. It is on the basis of these relationships that the pressure wave can be interpreted. For example, an object’s vibration involves frequency components that are harmonics of a fundamental frequency. Therefore, the frequency components of the pressure wave produced by the vibration of a single object will be harmonically related; such harmonic relationships are unlikely to exist between components produced by distinct sources since it is unlikely that two simultaneously occurring natural events produce overlapping sets of harmonics. Therefore, if the auditory system detects a number of frequency components that are harmonically related they are likely to have been produced by a single source. Similarly, the pressure wave produced when an object is struck will have frequency components that share temporal properties: they will all begin at the same time and follow a similar time-course. It is unlikely that frequency components produced by distinct events will begin at precisely the same time, and unlikely that they will all follow the same time-course. Therefore, if the auditory system detects frequency components with the same temporal properties then they are likely to have been produced by a single source. 11 My discussion here and in what follows draws on Bregman (1990), especially ch. 3, to which the reader should refer for details of the empirical support for the claims in the text.

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There are other relationships between frequency components that are likely to exist only when those components have been produced by a single source, and unlikely to exist otherwise, and which, therefore, are evidence of their having been produced by a single source. In interpreting a sound wave the auditory system exploits these relationships: when frequency components are detected that are related in these ways they are treated by the auditory system as having been produced by a single source; only then can information about that source be extracted. This account of how auditory perception can recover information about the things in our environment that produce sounds lends empirical support to the suggestion that auditory perception functions to tell us about the sources of sounds. What is the connection between this account and our auditory experience? What sounds we experience, and how we experience them to be, is determined by the way that auditory system interprets the complex vibration instantiated by the pressure wave that reaches the ears. That vibration is made up of many different frequency components: if the auditory system treats all of those frequency components as having been produced by a single source then we experience a single sound; if it treats them as having been produced by two sources then we experience two sounds; and so on. The character of each individual sound is determined by the particular combination of frequency components that the auditory system treats as having the same source.12 We can think of the relationships between frequency components detected by the auditory system as evidence that supports an interpretation of the sound wave as having been produced by certain sources. When the circumstances are ideal, or nearly so, this evidence will support only a single interpretation. Circumstances are sometimes not ideal: sound waves suffer interference during transmission and frequency components may become distorted or attenuated; in noisy environments some components may be masked by others; damage to or deterioration of the ears may mean that some components are not detected. In such circumstances the 12 A simple demonstration of this can be made with a piano. If you play middle C you hear a single tone; play middle C and the C an octave above and strike the keys exactly simultaneously and you still hear a single tone which has a slightly different character; play middle C and B an octave above and you hear two distinct tones.

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evidence may support more than one interpretation, and the auditory system must make best sense of the sound wave it detects. Making best sense of the sound wave will often produce correct results and determine an experience of sounds that correspond to the sources that produced them; sometimes however, the auditory system interprets the sound wave incorrectly and treats components that were in fact produced by distinct sources as having been produced by a single source, or treats components that were produced by the same source as having been produced by distinct sources. The result of an incorrect interpretation is an experience of a sound that does not correspond to any source. Experiences of this kind have become commonplace with the advent of recorded and electronically produced sounds: a loudspeaker can produce sound waves with frequency components that could never be naturally produced by a single source. We can experience the sounds produced by a loudspeaker as having any number of distinct sources; and we can experience the sound waves produced by two (or more) distinct loudspeakers—two distinct sources—as a sound apparently produced by a single source. Both kinds of experience are a result of the way the auditory system interprets the sound wave. Our experience of sounds is determined by the way the auditory system interprets the sound wave that reaches the ears. The auditory system functions to interpret the sound wave in such a way that we can hear the sound producing events in our environment. In normal circumstances, components that are produced by the same source are treated as having been produced by the same source and information about that source—embodied in the structure of components it produced—can be extracted. The consequence of this interpretation is that, normally, the sounds we experience correspond to events in our environment that produced them. However, if the sound wave has been produced in an abnormal way, or has been altered or distorted, the sounds we experience may not correspond to events in the environment that produced them; when that happens we experience sounds that would correspond to the events that, in normal circumstances, would have produced the sound wave detected by the ears.13 In both cases we experience the sounds we do because those sounds would correspond to events 13 Our auditory system evolved in an auditory environment that was in many ways quite different from the one we now inhabit. In our present environment

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that, in normal circumstances, would have produced the sound wave detected by the ears. So the best explanation of our experience of sounds—of why we experience the sounds we do—is that it results from the way that auditory system interprets the sound wave in order to extract information about what produced it; in terms, that is, of a process whose function is to tell us about the sources of those sounds. I described above how, in hearing sounds, we can attend to their apparent sources and listen to the things going on around us, and I argued that we can do this in virtue of our auditory experience presenting sounds as having been produced by sources of certain kinds or by sources having certain features. The account I have given of how the auditory system functions gives us some understanding of how our auditory experience could present sounds as having been produced by their sources.14 It also suggests an answer to the question of what sounds are. Each sound we experience is determined by a structure or pattern of frequency components instantiated by the pressure wave that reaches the ears and treated by the auditory system as having been produced by the same source; the qualitative character of each of these sounds is determined by the particular combination of frequency components that make up that structure or pattern. I think we can best understand sounds as structures or patterns of frequency components: those structures or patterns of frequency components, instantiated by sound waves, that would normally be interpreted by the auditory system as having been produced by a single source event. Since the way our auditory system interprets the sound wave determines our experience of sounds, we can also say that a sound is a structure or pattern of frequency components, instantiated by a sound wave, that would normally be experienced as a single sound. It follows that an experience of a sound is veridical only if it is produced by the instantiation of a structure or pattern of frequency components that would normally produce an experience many of the sounds we hear are produced in ways that are, from that evolutionary perspective, abnormal: many of the sounds we hear are produced electronically, or by musical instruments, or are recorded sounds. In such cases, the connection between the sound wave and the thing that produced it is abnormal. The resulting experiences almost invariably mislead us about the sources of the sounds we hear. 14 There is a more to be said about the content of auditory experience and how sounds are presented. I go into more detail elsewhere.

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of that sound. It is not veridical if it is not produced by any such structure or pattern, or if it is produced by a structure or pattern that would not normally produce an experience of that sound. Hallucinatory experiences of sounds are non-veridical in the first way; experiences that are non-veridical in the second way are most likely to occur as a result of damage to the auditory system. Any account of the nature of sounds ought to be consistent with our intuitions about the identity conditions of sounds; that is, with our everyday ways of counting them. Sounds are structures or patterns of frequency components instantiated by the sound wave that reaches the ears, but the same pattern or structure of frequency components can be instantiated in more than one place and time. Exactly this happens, for example, when I tap a pen in the same way twice on my desk. Each tap produces a pressure wave which instantiates the same pattern or structure of frequency components, but each tap produces a distinct sound. We cannot, therefore, identify sounds with a type of pattern or structure of frequency components. That suggests we should identify sounds with instances of the pattern or structure of frequency components. When are instances of a pattern or structure of frequency components instances of the same sound? The answer appears to be independent of the qualitative similarity of the sounds determined by the pattern:15 there are circumstances in which we would say that you and I both hear the same particular sound even when we each hear the sound as having a different character. Suppose we both hear the sound of a gunshot, but I am standing next to the gun and you are much further away. I experience the sound as loud and sharp, you experience it as quiet and muffled. The structure or pattern of frequency components which determines your experience will be different in various respects from the pattern which determines mine. Given these differences, what makes it the case that both of our experiences are determined by the same instance of the pattern or structure? We cannot answer this question other than by asking what makes it the case that we both hear the same sound, to which the most plausible answer is that we hear the same sound because the sound we hear was produced by the 15 Though that may not always be the case. In some contexts, when our interest is in the character of the sound, it may be that we do regard qualitatively different sounds as distinct.

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same source event and so, in hearing the sound, we both hear the same event: we both hear the gunshot. If the sounds were in fact produced by different source events then we would not say that we heard the same sound even if the sounds we experienced were qualitatively indistinguishable. That suggests that it is a necessary condition for two experiences to be experiences of the same sound that the sounds experienced were produced by the same event; it follows that, normally, two experiences are of the same sound only if, in virtue of hearing the sound, we hear the same source event. But there are difficult cases. An array of loudspeakers may be arranged in such a way that when a subject stands in a particular ‘sweet spot’ they hear a sound that is otherwise inaudible—someone not in the sweet spot will simply hear the sounds produced by each of the individual loudspeakers. In virtue of what do two people standing in the sweet spot hear the same sound?16 We are not going to get answers to these difficult cases by appealing to the nature of sounds as instantiated structures or patterns. Answers to these questions must depend on our everyday ways of counting sounds and will inherit whatever vagueness is present in those ways of counting. What’s important is that the account of sounds as structures or patterns instantiated by sound waves is consistent with our everyday ways of counting sounds and, as far as I can tell, it is. I have arrived at an account of the veridicality conditions of auditory experiences and of the nature of sounds, and have done so not simply by appealing to introspective descriptions of our auditory experience, but by asking what sounds must be according to the best explanation of certain features of our auditory experience, in particular our experience of sounds.17 Since the argument is in terms of what best explains our experience of sounds the account satisfies the condition that it be an account of the objects of our auditory experience. According to my account, sounds are properties of the pressure waves produced by sound sources; they are not properties of the 16 What, in general, should we say about recorded sounds? What should we say about echoes? In hearing a sound and its echo do we hear the same sound again, or do we hear two sounds? 17 This is a methodology which, I would argue, can be applied generally to answer questions about the correct attribution of content to perceptual experiences (see my ‘Visual content attribution’ MS).

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sound source or of an event involving the sound source. It might be objected that, for all I have shown, sounds could be properties of their sources. After all, when everything goes as it should, the frequency components that are treated by the auditory system as having been produced by a single source will in fact have been produced by a single source. When that happens the structure or pattern of frequency components that determines our experience of a particular sound will be the same as that instantiated by the source of that sound. If that is right then the account I have given of the function of auditory perception would seem to be consistent with the accounts of sounds that identify them with a property of their sources: when everything goes as it should, our experience of sounds is determined by that pattern or structure of frequency components instantiated by sound sources. So why not identify the sound with the structure or pattern of frequency components instantiated by the source of the sound rather than with that instantiated by the sound wave that reaches our ears? There are two reasons why not. First, even when things do go as they should, the pattern of frequency components that determines our experience of a sound may not be identical to that instantiated by the source of the sound. The character and composition of the pattern of frequency components often changes during transmission. Some frequency components may be masked or have become attenuated or the whole structure may shift in frequency so that the sound, when we hear it, might sound quite different to the way it would have sounded when it was first produced. Such changes may not greatly affect the information embodied by the pattern, but will alter the qualitative character of the sound we experience. A sound may be very loud when it was first produced but—if I am a long way from its source—quite quiet by the time I hear it, or it might—having been transmitted, say, through the walls of a building and so lacking any high-frequency components—have a muted character quite unlike that that it had when it was first produced. So even when things go as they should, the pattern or frequency components that determine my experience may be quite unlike the pattern instantiated by the sound source. To maintain that the sound we hear in these circumstances is the structure or pattern of frequency components instantiated by the source of the sound would be to claim that we misperceive the sound—that we

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experience the sound as having an auditory character that it does not really have. If we experience a quiet sound that was loud when it was first produced then our experience involves a misperception of the loudness of the sound. But such a claim is implausible. We can distinguish the loudness of a sound from the apparent force or violence of the event which produced it. When we shut the door to keep out the sounds made by someone hammering nails next door, the sounds we hear get quieter, but the hammering itself does not appear to change—shutting the door does not make it appear that the nails are struck with any less force. That is because information about the force or violence of an event that produced a sound is not embodied in the amplitude of the sound wave (which determines how loud it sounds), but in an aspect of the pattern of the frequency components it produces, and shutting the door does not disrupt that aspect of the pattern. That means that we can hear a quiet sound that seems to have been produced by a forceful or violent event: a quiet sound may seem to have been produced by an event that we can tell produced a loud sound, so we can correctly say: a quiet sound seems to have been produced by loud hammering. Since there need be nothing misleading about the experience, there are no grounds for claiming that an experience of a sound that was loud when first produced but that we experience as quiet involves a misperception. Second, even if it were true that when everything goes as it should the structure or pattern of frequency components that determines our experience of a particular sound will be the same as that instantiated by the source of that sound, it would only follow that sounds are instantiated by the source of a sound if the following is true: we only hear sounds when the frequency components that are treated by the auditory system as having been produced by a single source have in fact been produced by a single source. I have already described cases in which we at least seem to hear sounds when this is not true—cases, for example, in which two distinct sources produce frequency components that are treated as having been produced by the same source; or cases in which we hear a sound as a result of a pressure wave that was not produced by any source. In these cases there is nothing—apart from the pressure wave—that instantiates the pattern or structure of frequency components treated by the auditory system as having

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been produced by a single source, so if I do hear a sound I do not hear a sound that is instantiated by its source. Are there any grounds for claiming that although in these cases I seem to hear a sound I do not really do so? I have argued that our experience presents sounds as having been produced by their sources. Given the way the auditory system functions we normally experience sounds that correspond to their sources; such sounds are presented as having been produced by sources that did in fact produce them. When we experience a sound that was produced by more than one source our experience presents the sound as having been produced by a single source; that is, we experience a sound that seems to have been produced by a something that did not produce it. The possibility of this kind of error makes sense given that the auditory system functions to tell us about the sources of sounds. In carrying out this function the auditory system relies on the connection between the sound wave and the things that produced it. When that connection breaks down the resulting experience will mislead us about what produced the sounds we hear, but it does not follow that it misleads us about the sounds themselves.18 If that is right then we hear a sound even when the frequency components that are treated by the auditory system as having been produced by a single source have not in fact been produced by a single source; we hear a sound even though the sound seems to have been produced by a source that did not produced it. There are two ways, therefore, in which auditory experience may mislead us. It may mislead us about the existence or the character of a sound—we may seem to hear a sound when there is none, or hear a sound be to other than it is—or it may mislead us about the source of a sound that we hear—we may hear a sound that seems to have been produced by something that did not produce it. This pattern of possible error means that auditory perception is indirect in a way that visual perception is not. We can hear sounds and their sources; we sometimes hear sounds without hearing their sources, 18 It would be a mistake to think that the auditory system functions to tell us about the way the object vibrates. It uses the way the sound wave that reaches the ears vibrates to tell us about objects, but it does not attempt to recover information about how the object vibrates. It is not possible to recover such information and it is anyway not necessary to do so.

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but cannot hear sources other than by hearing the sounds they produce; therefore we hear sources ‘indirectly’ in virtue of hearing the sounds they make. Hearing informs us about the length of time events in our environment last. We can hear the durations of the events that produce sounds: for how long someone plays the violin, for example, or for how long a spinning coin spins on the desk. But if we can hear the duration of these events then that must surely be because we can hear the duration of the sounds they produce. If we do not hear the duration of the sounds they produce then how can we hear the duration of the events that produce them? According to the account of sounds that I have been defending, sounds exist for as long as the pressure wave that instantiates them exists. That is, a sound that I hear typically exists—unheard by me—before and after I hear it and, therefore, I do not hear its duration. It has been objected that this implies that our auditory experience is illusory. What in fact I experience when I take a sound to have duration is not duration of a sound at all. Rather, my encounter with a spatial boundary of a sound leads to my enjoying an auditory experience while the sound passes. . . . This means that each time I hear a sound, I mistake an experience of the spatial boundaries of a sound for an experience of the duration and temporal boundaries of that sound. (O’Callaghan 2007, 43–4)

If my experience of the duration of a sound is illusory then how, in hearing that sound, can I hear the duration of the event that produced it? The account of sounds as waves entails that we do not hear the temporal features of sounds things make as those sounds unfold over time. It entails that we do not hear the durations of sounds and that our justification for believing that the violin practice lasted forty-five minutes cannot just come from hearing because what we experience in hearing is an illusion. (Ibid, 45)

There are two kinds of error that might be involved here. It might be claimed that experience presents sounds as having a duration. If sounds do not have a duration then experience misrepresents them. Even if experience does not present sounds as having a duration, it may be that we take them to have a duration on the basis of the way our experience does present them—as having features

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which lead us to judge that they have a duration. In that case our experience does not misrepresent sounds, but it misleads us into making perceptual judgments that are false: we take ourselves to perceive that sounds have a duration that they do not in fact have. Does the wave view of sounds commit us to either of these kinds of error? What does it mean to say that we experience something as having a duration? Something exists for a duration if and only if there is a period of time during which it exists, and an earlier and later time during which it does not exist; an object instantiates a property for a duration if and only if it instantiates the property at some time and there is an earlier and later period of time when it does not instantiate that property. We need to be careful to distinguish experiencing something for a duration and experiencing something as having a duration. We experience something for a duration if there is a period of time during which it is experienced. For example, if I open my eyes, look at the cup on my desk, and then close my eyes, then there is a period of time during which I am aware of the cup; or if I see a train passing at the end of a narrow gap between buildings then there is a period of time during which I see the train; or if I run my hand over the surface of my desk and feel that it is smooth, then I am aware of the smoothness of the desk for a duration. Although in these cases my experience has a duration, in none of them do I experience something as having a duration. We experience something as having a duration if we experience it as existing or lasting for a period of time. If it is possible to have such experiences then the experiences of seeing a light flashing on and off, or of seeing someone blow a soap bubble that floats for a moment and then bursts are surely examples: I am aware of the duration of the illumination of the light, and I am aware of the duration of the bubble’s existence. We have, then, a contrast between two kinds of experience: the experience of something for a duration and the experience of something as having a duration. What is the difference between them? When we experience something for a duration, it is our having the experience which has duration and the experience itself is consistent with the object experienced not having a duration. When we experience something as having a duration our experience is not consistent with the object experienced not having a duration.

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For example, in seeing the light flash, we experience the light as first illuminated and then not illuminated—we experience the object at one time as being a way that is inconsistent with the way we experience it as being at a later time. If our experience is veridical then the object must have changed. In seeing the bubble burst we see the bubble disappear without moving out of our visual field or becoming occluded by another object. At one time we experience the bubble, at a later time (and without any other relevant changes) we do not experience the bubble. If our experience is veridical then (given the circumstances) the bubble must have ceased to exist. In both cases, the pattern of change in the course of our experience is not consistent with there being no change in the object.19 When we merely experience something for a duration the change in the course of our experience is consistent with there being no change in the object experienced. That suggests a necessary condition for experiencing something as having a duration is that our experience represents that thing as being in two incompatible states; represents it, that is, as having changed. Do we experience the events that produce sounds—the violin playing or the spinning of the coin—as having a duration? Hearing the coin spin may seem parallel to seeing a light flash on and off: just as we can see the light on and then off, we hear the coin producing a sound and then no longer producing a sound. But there is a difference: although (for a range of properties) we can see that something does not have that property, we cannot have an auditory experience of something not producing a sound: a buzzing object may cease buzzing or may be placed in a soundproof box; in both cases we no longer hear the object as buzzing, but in neither case do we hear the object as not buzzing (or as being some way that is incompatible with its continued buzzing). If we cannot hear the coin not producing a sound, then we cannot hear the coin as changed from spinning to not spinning; it seems, therefore, that although we can hear the coin spinning for a duration, we cannot hear the spinning as having a duration. 19 Although our experience requires there to be a change in the object, it is a further question whether it represents that change as a change: whether it represents the light as flashing or the bubble as bursting. I do think that we can experience changes as changes—and doubt, for example, that we can give an explanation of visual change blindness without appealing to such experiences—but the discussion here does not presuppose an answer to the question.

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In fact there is a difference between the time-course of a vibration that decays and that of a vibration that is interrupted. Our auditory system may be sensitive to such differences and so in many cases enable us to tell the difference between an object that has stopped producing a sound and an object that continues to produce a sound but ceases to be audible.20 But even if we cannot hear such differences, we can hear something make a high-pitched sound and then a low-pitched sound and so we can experience something as producing one sound and then producing a different, incompatible, sound and so as having changed. To hear the duration of an event that produced a sound is a matter of hearing for how long something produces a sound. To hear that we need to do no more than hear the object start and stop producing a sound, or to hear the character of the sound it produces change, and to be able to judge or experience the time taken for such changes to occur. This is not inconsistent with the wave view of sounds since it does not depend on hearing how long a sound lasted and judging on that basis how long the coin was spinning. We simply hear changes in the object in virtue of hearing the sound it makes and we can judge the duration of the sound-producing event on the basis of when we hear those changes occur in the object. That is sufficient to explain how we are able to hear how long the coin spins, how long the performance lasts, and so on. We can hear the duration of sound-producing events whether or not we can hear the duration of the sounds they produce, but do we hear the durations of sounds they produce? Suppose that I hear a brief melody part of which consists of a low-pitched, then a high-pitched, and then a low-pitched note. In hearing this melody, do I experience the high-pitched note as having a certain duration? I suggested that to experience something as having a duration requires that our experience represent it as being in two incompatible states, and so as having changed. When we hear a sound as first high-pitched and then low-pitched do we experience it as having incompatible properties? That depends on whether a sound can be simultaneously high- and low-pitched. We cannot hear a single sound as simultaneously high- and low-pitched, but 20 These would be auditory cues that are analogous to the low-level cues to visual occlusion used by the visual system to determine the difference between the edge of a surface and a boundary formed by an occluding surface.

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whether a sound itself can be both depends on the nature of sounds. If sounds are properties instantiated by the object that produced them, or by the event of their production, then being high-pitched would be incompatible with being low-pitched. A single object or event cannot instantiate a sound that is simultaneously high- and low-pitched. On the other hand, if sounds are waves then being high-pitched is not incompatible with being low-pitched. One part of the wave can instantiate a high-pitched sound at the same time as a different part of the wave instantiates a low-pitched sound. The experience of a sound as first high- and then low-pitched is exactly what you would expect if the sound were a wave experienced for a duration, with parts experienced at different times instantiating different patterns or structures of frequency components. Whether hearing a sound as high- and then low-pitched implies that we experience it as changed depends, therefore, on the nature of sounds. It follows that whether we experience sounds as having a duration, rather than merely for a duration, also depends on the nature of sounds. Our experience itself is neutral and does not favor one account of sounds over another. Even if we do not experience sounds as having a duration, are we nonetheless misled by our experience: do we judge or take sounds to have a duration and so mistake what are in fact the spatial boundaries of a sound for temporal boundaries? If we do, the wave view implies those judgments are mistaken but since, in hearing things, most of our interest is in the properties—including the temporal properties—of the events that produce sounds such mistakes would not be very serious or consequential. Suppose that the church bells are rung, and we judge that the ringing lasted a long time, then we are making a judgment about the event that produced the sound. But what if we ask about the sound of the ringing—how long did that last? In most contexts it would be appropriate to answer that the sound lasted for as long as the ringing did. Such a judgment would simply reflect the fact that the sound of the ringing was audible for as long as the ringing was audible: in a echoing church where the sound was audible after the ringing had stopped, we might say that the sound of the ringing lasted longer than the ringing did. If our judgments about how long a sound lasts are usually judgments about the length of time for which the sound was audible, then they are not inconsistent with any particular

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account of sounds. If we sometimes make judgments—based on the time for which a sound was audible—about how long a sound exists, then such judgments are likely to be inconsistent with what the wave theory tells us about how long sounds exist. But such judgments are mistaken, and they give us no reason to think that the view of ‘sounds as waves’ is false. University of Edinburgh

references Bregman, Albert S. (1990) Auditory scene analysis: the perceptual organization of sound (Cambridge, Ma.: MIT Press). Carello, C., J. B. Wagman, and M. T. Turvey (2005) ‘Acoustic specification of object properties’. In Moving image theory: Ecological Considerations, ed. by J. D. Anderson and B. Fisher (Carbondale, IL: Southern Illinois University Press). Casati, Roberto, and J´erome Dokic. 1994. La philosophie du son (Nimes: ˆ Editions Jacqueline Chambon). Fletcher, Neville H., and Thomas D. Rossing. 1998. The Physics of Musical Instruments, 2nd edn. (New York: Springer). Fowler, C. A. (1991) ‘Auditory perception is not special: We see the world, we feel the world, we hear the world’, Journal of the Acoustical Society of America 89:2910–15. Freed, D. J. (1990) ‘Auditory correlates of perceived mallet hardness for a set of recorded percussive events’, Journal of the Acoustical Society of America 87:311–22. Gaver, W. W. (1993a) ‘How do we hear in the world? Explorations in ecological acoustics’, Ecological Psychology 5:285–313. (1993b) ‘What in the world do we hear? An ecological approach to auditory event perception’, Ecological Psychology 5:1–29. Kunkler-Peck, A., and M. T. Turvey (2000) ‘Hearing shape’, Journal of Experimental Psychology: Human Perception and Performance 1:279–94. Lakatos, S., S. McAdams, and R. Causs´e (1997) ‘The representation of auditory source characteristics: Simple geometric form’, Perception and Psychophysics 59:1180–90. Li, X., R. J. Logan, and R. E. Pastore (1991) ‘Perception of acoustic source characteristics: Walking sounds’, Journal of the Acoustical Society of America 90:3036–49. McAdams, Stephen (1993) ‘Recognition of sound sources and events’. In Thinking in Sound, eds. S. McAdams and E. Bigand (Oxford: Oxford University Press).

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McAdams, Stephen and Emmanuel Bigand, eds. (1993) Thinking in Sound (Oxford: Oxford University Press). Neuhoff, John (2004) ‘Auditory motion and localisation’, In Ecological Acoustics, ed. J. Neuhoff (London: Academic Press). Nudds, Matthew (2009) ‘Sounds and Space’. In Nudds and O’Callaghan, eds., Sounds and Perception (Oxford: Oxford University Press). O’Callaghan, Casey (2007) Sounds: A Philosophical Theory (Oxford: Oxford University Press). (2009) ‘Sounds and Events’. In Nudds and O’Callaghan, eds. Sounds and Perception (Oxford: Oxford University Press). O’Shaughnessy, Brian (1957a) ‘An impossible auditory experience’. Proceedings of the Aristotelian Society. 57:53–82. (1957b) ‘The location of sound’, Mind 66:471–90. (1971–1972) ‘Processes’. Proceedings of the Aristotelian Society. 72. Pasnau, Robert (1999) ‘What is Sound’. Philosophical Quarterly 49:309–24. (2000) ‘Sensible Qualities: The Case of Sound’, Journal of the History of Philosophy 38:27–40. Peretz, Isabelle (1993) ‘Auditory agnosia: a functional analysis’. In Thinking in Sound, eds. S. McAdams and E. Bigand (Oxford: Oxford University Press). Scruton, Roger (1997) The Aesthetics of Music (Oxford: Oxford University Press). Wildes, R., and W. Richards (1988) ‘Recovering material properties from sound’. In Natural computation, ed. W. Richards (Cambridge, MA: MIT Press).

14. Sounds and Temporality Jonathan Cohen Space and time are the forms of combination in intuition and serve for applying the categories in concreto. (Kant, Notes on Metaphysics, AA 18 §5934, 1783–1784)

What is the relationship between sounds and time? More specifically, is there something essentially or distinctively temporal about sounds that distinguishes them from, say, colors, shapes, odors, tastes, or other sensible qualities? And just what might this distinctive relation to time consist in? Apart from their independent interest, these issues have a number of important philosophical repercussions. First, if sounds are temporal in a way that other sensible qualities are not, then this would mean that standard lists of paradigm secondary qualities offered by Locke, Galileo, and other modern philosophers—lists which include colors, odors and sounds without any significant distinctions—overlook significant metaphysical differences. This, in turn, would threaten to undermine the coherence of the modern understanding of secondary qualities itself. Moreover, a number of authors have recently urged that the essential temporality of sounds makes it impossible to understand sounds as properties (except on a trope theory of properties; see note 3). If true, and given the more or less universal view that colors are properties, this last conclusion would make potentially inapplicable to sounds much of the comparatively well-developed philosophical taxonomy and apparatus that has arisen in philosophical disputes over the status of colors (for presentations of this taxonomy and apparatus see, for example, Byrne and Hilbert (2003); Cohen (2009)).1 Therefore, the conclusion that sounds are distinctively temporal would be a serious blow to hopes for a theoretically unified treatment of the 1

Nothing I say here will takes sides in those disputes about color.

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sensory qualities.2 For all these reasons, quite a lot seems to hang on the question of the temporality of sounds. The thesis of this chapter is that the contemplated form of sound exceptionalism based on the allegedly distinctive temporal features of sound is unjustified. I allow, of course, that sounds occur in time, that they have temporal durations, and perhaps bear many other interesting relations to time; but I will suggest that, claims to the contrary in the literature notwithstanding, there is no metaphysically significant disanalogy between sounds and colors on this score. I shall begin by reviewing critically a series of arguments intended to show that sounds are unlike other sensible qualities in that the former must be understood as concrete individuals (§1). Next I shall consider whether the temporal features of sounds—in particular, their having temporal locations and durations—distinguishes them from other sensible qualities (§2). Finally, I shall assess the claim that sounds can be distinguished from other sensible qualities by their capacity to survive qualitative change (§3). I shall conclude that none of these considerations reveal a distinctively temporal aspect to sounds.

1. sounds as individuals The idea that sounds are distinctively and essentially temporal is largely tied up with the contention—pressed especially by recent writers who argue that sounds should be understood as events—that sounds are (unlike colors) concrete particular individuals rather than abstracta such as properties (O’Callaghan, 2007, 17–19).3 And this has been thought to lead to the conclusion that 2 Defenders of the essential temporality of sounds may welcome these consequences as overdue correctives to a history of unreflective, ‘‘visuocentric’’ attempts to force sounds into terms taken from debates about color (which has, of course, dominated discussion in philosophy of perception for centuries). Whether the charge of visuocentrism is justified (and whether visuocentrism is a bad thing) depends, of course, on whether sounds really are distinct in the way proposed. But it strikes me as good methodology to start by attempting to leverage tools that have worked in the past; if so, then visuocentrism might make for a useful working methodology even if it turns out to be untenable at the end of the day. 3 Of course, if properties are understood as tropes, as suggested by Williams (1953); Mertz (1996), then the worries discussed below vanish; for properties would then be nothing other than (certain kinds of) concrete individual particulars. In what

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sounds are temporal in a way that distinguishes them from nonsound sensible qualities such as color because it is generally thought that concrete individuals can, while abstracta (such as properties, and, in particular, color properties) cannot, have temporal locations and extents. I shall return to the question of the temporal features of properties in §2. For now, however, I want to ask what reasons there are for supposing that sounds are concrete individuals rather than abstract properties. A first reason for taking sounds to be concrete individuals involves noting the ways in which we count and quantify over them. As the church bell chimes noon, we hear one sound (the first chime), then another (the second chime), and so on. After the chimes end, we say that we heard every one of them, or that none of them occurred before noontime, and so on. In these respects, then, sounds are like individuals: we count one chair, then a second chair, and so on, and quantify over all the chairs in the room. Unfortunately, however, this consideration does not appear to discriminate between individuals and properties. For while we can indeed count and quantify over individuals in these ways, it appears that the same is true of properties. Of course, this follows immediately if we understand properties as tropes (see note 3). But it seems true of properties construed as universals as well. To see this, suppose that the church bell has a size, shape, and color, that these are among its properties (nothing hangs on this; the reader who disagrees is invited to substitute alternative properties of the bell she recognizes), and that these properties are universals rather than tropes. Then the size, shape, and color of the bell are properties, and they are different properties (for they come apart in extension). Moreover, they can be counted (I have mentioned exactly three) and quantified over (the properties I have mentioned). I conclude that the present argument does not rule out a property construal of sounds. A second piece of support for taking sounds to be individuals rather than properties is that we do not ordinarily think of sounds as being exemplified by individuals. If the sound of the bell’s chiming were a property, and assuming we have perceptual access to this follows, however, I shall be assuming a non-trope understanding of properties both for the sake of sustaining interest in this class of arguments and so as to avoid making property views of sound hostage to such a tendentious theory of properties.

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sound, it must be exemplified by something. The best candidates for the role of property bearer are the source of the sound (the bell) or the medium of transmission between source and perceiver (in this case, air). Yet it sounds odd to say that the church bell has the sound; instead we say that the bell makes or produces the sound. And if anything it sounds worse to say that the medium has the sound; instead we say that the medium transmits or carries the sound. Thus, this argument concludes, since sounds are not exemplified by individuals, they are inapt for being understood as properties. Now, as Casati and Dokic (2005) point out, this motivation, which turns on the evidence of ordinary English idioms, will be unpersuasive to anyone who thinks that metaphysics can fail to be revealed by ordinary linguistic usage. Even putting aside this concern about the argument, there are other candidates for the role of sound bearer that may be available to the proponent of sounds as properties. To my ear, at least, it does not sound so bad to ascribe sounds to spatio-temporal regions that are occupied by (and so not identical to) portions of the sound-transmitting medium. In particular, the property theorist might suggest that the sound is exemplified by the spatio-temporal region occupied by its source.4 If so, then the property theorist will be able to meet the concern that she find a bearer for her properties. A third piece of support for conceiving of sounds as individuals is that sounds themselves seem to exemplify qualities. In particular, sounds exemplify the so-called ‘‘auditory qualities’’ of pitch, timbre, and loudness. Indeed, sounds stand in a network of similarity relations to one another on the basis of the auditory qualities they exemplify. But if, as some have thought, the things that bear properties are all individuals, then this shows that sounds must be individuals too. Once again, this argument fails because the criterion on which it turns is not unique to individuals. This can be seen clearly from consideration of colors (everyone’s canonical example of properties). Quite apart from the current controversy, it is plausible that there are higher-order properties—namely, qualities or properties 4 This proposal is compatible with the possibility that one spatio-temporal region might exemplify multiple sounds; consequently, the occurrence of multiple sounds in one region is no obstacle to the view.

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of properties. And in particular, it is often said that colors exemplify (e.g.) hue, saturation, and lightness/brightness qualities. Admittedly, this may be a secondary, derivative use: it is plausible that x s color exemplifies a hue quality H (say) only if x exemplifies H. Whether or not they are derivative, however, ascriptions of hue, saturation, and lightness/brightness qualities to colors are commonplace and apparently can be true. Additionally, it is a standard view that colors can be organized in a network of similarity and difference relations on the basis of the hue, saturation, and lightness qualities they bear.5 But if colors—archetypal properties for all parties to the dispute—can themselves exemplify qualities and be organized into a network of similarity relations on that basis, then it is no argument against the property view of sounds that the same is true of them. The arguments we have considered, then, fail to preclude a property understanding of sounds, and therefore fail to show a significant disanalogy in the way that sounds and color properties are related to time.

2. temporal location and temporal extent We have been reviewing, and finding fault with, arguments for the conclusion that sounds are concrete individuals rather than abstract properties. And we have been interested in the latter conclusion because it is generally held that concrete individuals can, but abstract properties cannot, occupy temporal locations and extents, so this conclusion would provide one way of securing the needed conclusion that sounds are more closely bound up with time than other sensible qualities. However, whatever one makes of the arguments so far, I take it to be clear (even without the aid of subtle metaphysical arguments) that sounds do have temporal locations and extensions. We say that one sound occurs before another (as it might be, that the one commences at 2:00 and the other at 3:00), and that a first lasts longer than a second (as it might be, that the first extends for an hour and 5 It is also true that colors can be organized into other similarity and difference networks on the basis of other similarity metrics defined over others of the qualities they bear; see Kuehni (2003).

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the second extends for only a minute).6 Whether these observations show that sounds are in some way distinctively temporal, of course, turns crucially on whether non-sound sensible qualities (e.g., color properties) are without temporal locations and extensions. Given that non-sound sensible qualities are generally taken to be properties, there is an obvious reason for thinking that they cannot bear temporal locations and extensions. Namely, barring a fairly heterodox tropist understanding of properties (whose adoption would limit the attractions of whatever we can say for non-sound sensible qualities—see note 3), we will presumably want to say that these qualities are universals or some other sort of abstracta, hence outside—namely, not located in—space and time. But, on reflection, this contemplated reason for denying temporal locations and durations to non-sound sensible qualities seems to show far too much. In particular, it would seem to apply indiscriminately to all properties, and to show that all of them are without both temporal locations and extents. Indeed, it would also seem to show that all properties are without spatial locations and extents, in so far as abstracta are outside both time and space. But surely it is common ground in the present dispute that some properties—colors, as it might be—can have spatial locations and extents. Or, more cautiously, if it is not granted that colors and other properties literally have spatial locations and extents, it will be allowed that we ordinarily ascribe spatial locations and extents to such properties. And now an obvious worry arises: that is, that whatever makes it true (or at least permissible) to say that colors and other properties have spatial locations/extents will apply, mutatis mutandis, as an explanation of what makes it true (or at least permissible) to say that sounds and other properties have temporal locations/ extents. In particular, and although there are many variations on this theme, the most obvious universalist-friendly account of our ascriptions of spatial locations/extents to properties would involve the idea that those locations/extents apply in the first instance to the particular concrete instances of properties, and only in a derivative (possibly pragmatically governed) sense to the properties 6 Part of the motivation for insisting that sounds have temporal extent comes from the view that they survive change over time; more on the latter in §3.

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themselves.7 By way of analogy, then, a property universalist might treat sounds as properties and then account for our ascription of temporal locations/extents to sounds as derivative from the temporal locations/extents of the instances of sounds.8 Now, one possible objection to the line of thought just sketched is that, if successful, it would not explain why colors (say) lack temporal locations/extensions, or why sounds lack spatial locations/extensions. For, since the strategy seems equally applicable to any property with concrete instances—and equally applicable to any such property in both spatial and temporal forms, it could not provide for any asymmetry in the spatial and temporal features of the properties to which it is applied. However, I suggest that the alleged asymmetries in the spatial and temporal features of the properties at issue are exaggerated. Continuing with the paradigmatic properties used up to now, it seems to me that colors can permisibly (perhaps even truly) be ascribed temporal locations/durations. For one salient example, we complain about the excessive duration of redness when stopped at a traffic light. (On my proposal, this is probably best understood as a complaint about the duration of a particular instance of redness.) Likewise, I join the theoretical consensus in thinking that we can quite naturally ascribe spatial locations/durations to sounds (Pasnau, 1999; Casati and Dokic, 2005; O’Callaghan, 2007; Sorenson, 2008). To be fair, I think the ways we talk about sounds and colors is asymmetric: we more typically ascribe temporal locations/extents to sounds than colors, and we more typically ascribe spatial locations/extents to colors than sounds. And this asymmetry in our talk about sensible qualities is prima facie in conflict with my proposed explanation that treats the sensible qualities symmetrically with respect to their spatial and temporal features. But I think this conflict is only apparent, and can be explained without taking the

7 This pragmatic strategy for rescuing the ascriptions of temporal locations/extents to sounds mirrors in some ways the pragmatic strategy offered by Sorenson (2008) for rescuing the ascriptions of spatial locations/extents to sounds on behalf of the wave theory. 8 Of course this leaves unanswered the question of what are the instances of sounds; but presumably this is a matter a property theory would have to address in any case.

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asymmetry in our linguistic practices as reflecting a metaphysical difference between the sensory qualities themselves. In particular, I think we can explain the asymmetry in our linguistic as a result of our differential sensitivity to spatial and temporal inhomogeneities in color, on the one hand, and to spatial and temporal inhomogeneities in sound, on the other. On the visual side, the evidence suggests strongly that our perceptual systems are more sensitive to spatial than to temporal inhomogeneities in color. For example, we seem to be far better at discerning color differences in patches presented simultaneously at different regions of the visual field than color differences in patches presented successively in the same region of the visual field—this is why our performance in simultaneous color matching tasks is superior to our performance in successive color matching tasks (Newhall et al., 1957; P´erez-Carpinell et al., 1998). In contrast, we seem to have the opposite bias with respect to auditory discrimination: we are more sensitive to temporal than to spatial auditory inhomogeneities. One demonstration of this difference comes from the discrepancy between our abilities to discriminate melodies on the one hand, and our abilities to discriminate chords on the other. Even musically untrained subjects are very good at distinguishing one four note melody—namely, temporal discontinuity in sound—from another when these are played by a single trumpet (as it might be) from a single location in the space around the subject. In contrast, distinguishing corresponding spatial discontinuities in sound is much more difficult. If four trumpet players stand arrayed before the subject and each play simultaneously a single note, they will thereby produce a chord that constitutes a sonic spatial discontinuity. As any music student who has suffered through this sort of ear training will attest, discriminating one such spatial discontinuity from another takes considerable effort and practice. Thus, we have reason for thinking that our perceptual systems are more sensitive to spatial than to temporal inhomogeneities in color, but more sensitive to temporal than to spatial inhomogeneities in sound. Given this (presumably contingent) fact about our perceptual endowment, it is considerably easier, and so more useful, for creatures like us to notice, think about, and talk about, the spatial distribution of colors and the temporal distribution of sounds rather

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than vice versa. But if this is so, then we should not take the asymmetry in our ascriptions of spatial and temporal features to sounds and colors to reveal an underlying metaphysical difference between these two sorts of perceptual qualities. Rather, this asymmetry is plausibly a result of our contingent perceptual endowment (and what, given that endowment, it is useful and interesting for us to talk about). Returning to the main thread, it seems to me that the abstractness of properties in general, and color properties in particular, does not prevent them from sharing the types of temporal features (namely, temporal locations and durations) that sounds have. In short, then, temporal locations and durations do not make for a distinctively temporal aspect of sounds of the sort we were seeking.

3. survival through qualitative change A further reason some have given for thinking sounds are distinctively temporal is based on the allegation that sounds survive qualitative changes in their auditory qualities (e.g., their pitch, loudness, or volume). Thus, O’Callaghan writes that, . . . sounds survive changes to their properties and qualities. A sound that begins high-pitched and loud may continue to exist though it changes to being low-pitched and soft. An object does not lose its sound and gain a new one when it goes from being high-pitched to low-pitched, as with an emergency siren’s wail. The sound of a word begins with certain audible characteristics and ends with others, but a pitch shift is not the end of a sound. Determinate perceptible or sensible qualities, however, do not survive change in this way. The red color of the fence does not survive the whitewashing. The dank smell of the dog does not survive the perfuming. Particulars, such as the fence and the dog, however, survive changes to their qualities. (O’Callaghan (2009, 250); cf. O’Callaghan (2007, 22))

The charge that sounds survive qualitative changes is not only offered as a way in which sounds are distinctively temporal, but also as yet another reason for resisting a property theory of sounds of the sort held by Pasnau (1999, 2000). Indeed, O’Callaghan goes on, following the quoted passage, to claim not only that sounds survive qualitative change, but that their temporally evolving pattern of qualitative features is just what individuates one sound from another.

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I do not believe these considerations about survival are decisive. The first point to make is that, even if it were true that sounds survive qualitative change, this would not distinguish sounds from colors given a standard (though controversial) description of the phenomenon of color constancy.9 On this standard description, instances of color constancy involve one color that differs in its qualitative presentation—e.g., the one color has different chromaticities (as revealed by color matching tasks) when presented under each of two different illuminants. The presentations can be simultaneous (as in cases of simultaneous color constancy), or not simultaneous (as in cases of successive color constancy). If this is the right description of the phenomenon, then in instances of color constancy there is a single color that survives qualitative change (at different spatial regions and a single time, in simultaneous cases, or at one spatial region and different times, in successive cases). Consequently, the claim that sounds survive qualitative change would fail to mark out a distinctively temporal dimension of sounds among the sensory qualities. However, and even more importantly, I am not convinced that sounds survive qualitative change. Rather, it seems to me, the evidence is equivocal between survivalism and non-survivalism, so these should be treated as something like alternative and equally acceptable methods of bookkeeping; of course, this would also mean that it is inappropriate to use a criterion of survival to rule out metaphysical theories of sound that are incompatible with it. To see why, consider one of O’Callaghan’s examples, the wail of the siren, that occurs over an extended temporal interval T. Focus on two instants in T —call them t0 and t1 —such that what you hear at t0 differs in respect of pitch from what you hear at t1 . Grant, as seems plausible, that you hear a sound at t0 and that you hear a sound at t1 . (This leaves open that there may be other things that you hear at these instants, including possibly other things that you hear in virtue of hearing these sounds). Is it true that the sound you hear at t0 is the same as (numerically identical to) the sound you hear at t1 ? 9 In particular, I myself reject this description of color constancy for reasons alluded to in n. 16 and discussed in much greater detail elsewhere (Cohen, 2008). But the dialectical point I am making stands: if one accepts the standard description of color constancy, then the alleged survival of sound across qualitative change fails to distinguish sounds from colors.

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Here are two ways of answering this question that strike me as equally acceptable. • First answer: survivalism. You hear a single, temporally extended sound over the entire interval T —that is, you hear a single extended sound at each moment that comprises T. A fortiori, you hear that single sound at both t0 and t1 . Now, it is a feature of the hypothesized case that the pitch exemplified at t0 is distinct from the pitch exemplified at t1 , so it follows from the present answer that the qualities of the one sound heard—in particular, its pitch qualities—change over time. And this means that a sound can survive changes to its pitch qualities. Indeed, analogous responses to slight variants of our set-up that involve variations in timbre and loudness suggest that a sound can survive changes to these auditory qualities as well. In addition to the single sound we hear over T, there are, of course, many temporal parts of the sound. While standards for temporal part individuation can vary, one potentially fruitful strategy would be to individuate the temporal parts of the sound heard over T by their auditory qualities. It would then follow that the temporal parts of the sound, unlike the sound itself, do not survive changes to their auditory qualities. And, of course, the duration of these temporal parts is less than T itself. But the one, changing sound we hear extends in duration over the entire interval T. • Second answer: non-survivalism. You hear a number of distinct sounds over interval T. These sounds can be distinguished by differences in their pitch (or other auditory) qualities. It follows from this that a sound does not survive changes in its auditory qualities. Since the pitch quality exemplified by what you hear at t0 is distinct from the pitch quality exemplified by what you hear at t1 , the sound you hear at t0 is numerically distinct from the sound you hear at t1 .10 In addition to the multiple individual sounds we hear over T, there is, of course, the single, temporally extended stream of which these sounds are temporal parts. It is natural to say that the stream can bear 10 In order to set aside further issues about the metaphysics of time and change, I assume here that the sounds recognized by this answer have non-zero temporal durations less than the duration of T.

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Now, it must be admitted that, as O’Callaghan points out in the quoted passage, some aspects of our ordinary usage favor the survivalist option. We talk about the sound (singular) of the siren, not the sounds (plural) of the siren, despite the fact that the pitch we hear when we hear the siren at t0 is distinct from the pitch we hear when we hear the siren at t1 . Similarly, if you instruct someone to reproduce the sound of the siren she will ordinarily produce a temporally extended sound stream that varies in its pitch qualities between t0 and t1 , thereby answering your request for one sound by producing something—presumably a sound, if your interlocutor is cooperative—that varies in (hence, survives changes in) pitch. But ordinary usage points the other way as well. We might describe an extended stream produced by a siren in terms of its being made up of one sound (with one stable set of auditory qualities) at t0 , then another (with another stable set of auditory qualities) at t1 , and so on. (Breaking down a stream into these isolated and qualitatively unchanging parts is one natural mode of instructing novices—e.g., children—in the oral reproduction of sound streams of interest.) Similarly, we are inclined to describe the auditory effect of footsteps as a sequence of qualitatively uniform sounds (the individual steps) rather than a single repetitive sound that alternates between distinct qualities (a step, then a silence, then a step, etc.). These ordinary ways of talking about sound, it seems to me, fit better with non-survivalism. It appears, then, that ordinary usage pulls in both directions, and so does not settle the issue. One might, therefore, look to semantics as a further possible way of choosing between survivalism and non-survivalism. In particular, one might notice that sound can be

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used as a mass noun, and infer from this that what the word picks out is a collection of individuals (e.g., a surviving stream, as per survivalism) rather than an individual. Evidence that sound functions as a mass noun consists in the observations that, in environments such as sound of a siren, it cannot be directly modified by a number without specifying or at least implying a unit of measurement (1a–1c), and is more acceptable with a definite than an indefinite article (2a–2c). In these respects, sound in this environment patterns with mass nouns such as furniture rather than count nouns such as chair: 1a Mary hears the sound/?two sounds of a siren. 1b Mary sees the chair/two chairs. 1c Mary sees the furniture/* two furnitures. 2a Mary hears the sound/?a sound of a siren. 2b Mary sees the chair/a chair. 2c Mary sees the furniture/*a furniture. But these reflections seem unconvincing as well. For one thing, the very same criteria suggest that sound functions as a count noun in other environments (e.g., hear a sound), as shown by (1d) and (2d). 1d Mary hears the sound/two sounds. 2d Mary hears the sound/a sound. More fundamentally, there is no reason to think that the semanticist’s notions of mass and count correspond to the metaphysical categories of qualitatively-unstable individual and qualitativelyunstable stream. Consequently, the evidence we have been considering seems, once again, unable to choose between survivalism and non-survivalism. Finally, one might hope to support survivalism by pointing to empirical research on the way the auditory system segments or parses the occurrent flux of auditory information into temporally extended streams—a process Bregman (1990) calls ‘‘auditory scene analysis.’’ It seems clear that human perceptual systems do perform this sort of analysis: we distinguish auditorily the wail of the siren from the temporally overlapping roar of the passing automobile engine. Moreover, it is clear that the streams that are so segregated survive changes to their auditory qualities; indeed, a main focus of

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this area of empirical research is to discern the exact dimensions of variation and amount of variation along these dimensions that a single stream can have before it is parsed as more than one stream.11 There is a further reason for being interested in the streams picked out by auditory scene analysis: they are plausibly the loci for the binding of auditory qualities.12 After all, it is not that audition merely represents loudness at one time, high pitch at another, and so on; were this the case, we would be unable to discriminate a scenario with one loud, high tone and a second soft, low tone, on the one hand, from one with a loud, low tone and a soft, high tone. The best explanation of our capacity to make such discriminations is that we represent auditory qualities not by themselves, but as applying to individuals. For if so, we could describe the first scenario as one in which loudness and highness are represented as applying to one individual while softness and lowness to a second individual; this would then differ from the second scenario, under which loudness and lowness are represented as applying to a first individual while softness and highness are represented as applying to a second. This idea connects with auditory streams in so far as the latter are plausible fillers of the role of auditory individuals needed for the success of the proposed explanation of the binding of auditory qualities.13 Psychological considerations about auditory scene analysis and auditory binding, then, give us further reason (in case more was needed) for thinking that there are sound streams that survive qualitative change, and that any acceptable philosophical account of sound must recognize them. But what is not clear, and what is not settled by the empirical research, is whether sounds should be identified with such streams (as per survivalism) or not (as per nonsurvivalism). After all, non-survivalists, too, accept the existence and importance of sound streams. And non-survivalists have at hand a gloss on what auditory scene analysis amounts to; they will 11 For example, it turns out that streams fail to survive sufficiently large variation in pitch (more accurately: frequency), spatial location, timbre, loudness, and temporal location (Bregman, 1990, ch. 2). 12 The problem under discussion here is an auditory version of the manyproperties problem discussed by Jackson (1977), and later by Clark (2000). For discussion and comparison between auditory and visual instances of perceptual binding, see O’Callaghan (2008). 13 For persuasive arguments against the alternative view that auditory properties might serve as the needed individuals, see O’Callaghan (2007, 19ff).

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characterize auditory scene analysis as a process of grouping or integrating distinct sounds that fail to survive qualitative change, rather than as individuating one qualitatively changing sound from another.14 Yet again, it seems that the phenomena at issue are neutral between a survivalist and non-survivalist metaphysics of sound. More generally speaking, I doubt that there are perceptual phenomena—as opposed to theoretically laden descriptions of perceptual phenomena—that are better accounted for by one or the other of these views. This is because survivalists and nonsurvivalists agree about what there is: they both recognize the existence of temporally extended entities that survive change in auditory qualiti´es (the survivalist treats these entities as sounds, the non-survivalist does not). And they both recognize the existence of temporal parts of such streams, which do not survive change in auditory qualities (the non-survivalist treats these as sounds, the survivalist does not). Finally, they both agree that we have perceptual contact with the temporal parts that make up the streams, and, derivatively, with the streams themselves, so both are in a position to say that we succeed in hearing sounds.15 This suggests that the two views will not differ in the treatments of perceptual phenomena 14 For what it is worth, Bregman (1990, ch. 1) himself repeatedly characterizes auditory scene analysis as a process of grouping auditory components. 15 That we can make auditory contact with the temporally extended sound stream despite at every instant only being auditorily exposed to one of its temporal parts is a special case of the so-called puzzle of ‘‘presence in absence’’ discussed by O’Regan and No¨e (2002); No¨e (2004, 2006b, 2007); No¨e finds it generally puzzling that we manage to sense at one time and place (and know that we sense at one time and place) whole objects—a whole tomato, as it might be—given that we seem to make direct perceptual contact with only a small subset of spatialtemporal parts of the whole—just the facing surface of the tomato, as it might be. His solution to the puzzle is to hold that our perceptual contact (alternatively, sometimes, just our ‘‘sense’’ of our perceptual contact) at t with the whole is constituted in part by our implicit grasp of ‘‘sensorimotor contingencies’’—counterfactuals about what we would sense were we to change our spatiotemporal relation to the whole. Thus, for No¨e, perceptual contact with wholes requires that the parts be perceptually accessible, not that they be perceptually accessed. A prima facie objection is that this treatment is inapplicable to sound stream perception: when you hear a siren wail W (a temporally extended sound stream) at t, past temporal parts of W are not accessible (indeed, depending on how the counterfactuals are understood, perhaps necessarily not accessible) in any of the ways you, as an ordinary perceiver, have at your disposal (cf. Clark, 2006). But No¨e (2006a) is unbothered by this worry: he holds that there is no ‘‘presence in absence’’ phenomenology in the auditory case of the sort that motivated his account of the visual case.

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they make available, but only in their descriptions of the perceptual phenomena—in their understandings of how the explanatory burdens of accounting for such phenomena are distributed across their shared apparatus.16 I take the foregoing to show that we can think about and talk about both streams and temporal parts of streams, that both answer to at least some parts of the best job description for sounds we I find this response unsatisfactory. Perhaps I am just a phenomenological boor, but I confess to not recognizing in either the visual or the auditory case the alleged phenomenology of ‘‘presence in absence’’—although of course I recognize that we have a ‘‘sense’’ (namely a belief) that we make perceptual contact with wholes. If the intended scope of the account is limited to phenomenology, then, I do not see that there is anything to be explained. In contrast, if we take No¨e’s view as an account of how we make perceptual contact with wholes, then there is something to be explained, but, as have seen, No¨e’s account is inapplicable to cases of auditory perception of sound streams, and this should make us doubt it in other (e.g., visual) cases as well. A more plausible and more general view on this matter is that, just as S can make non-perceptual contact with x by achieving non-perceptual contact with only a subset of x’s proper parts, so too S can achieve perceptual contact with x at t by virtue of achieving perceptual contact with only some proper parts of x at t (or even merely in some neighborhood of t, in order to avoid the problems about bottom-up atomism discussed by Matthen (2008)). This condition on perceptual contact is, of course, compatible with the perceiver’s failing to bear counterfactual relations to currently unsensed spatial or temporal parts of x. Finally, if the question is why we believe that we achieve perceptual contact with wholes despite occurrently contacting only proper parts, then surely the best answer is that we believe this because we do make perceptual contact with wholes. Once again, there is no need for perceivers to bear counterfactual relations to currently unsensed spatial or temporal parts. 16 There is an instructive parallel here with certain issues concerning color constancy. Philosophers (following perceptual psychologists) have traditionally described the phenomenon of color constancy as a kind of invariance in apparent color across changes in illumination, and have then appealed to the phenomenon as the most important empirical motivation for views that identify colors with illuminationindependent properties of surfaces. However, as I have pointed out elsewhere (Cohen, 2008), this description of the phenomenon is either empirically inadequate or theoretically biased. A more neutral, and more empirically adequate, description is that apparent colors exhibit variance (in one sense) and invariance (in some other sense) across changes in illumination. Given this more adequate characterization, it is then up for grabs whether what is variant across such changes or what is invariant across such changes should be understood as color; and therefore it is also up for grabs whether or not the phenomenon of color constancy (so-called) lends support to illumination-independent accounts of color. Analogously, it begs the question in favor of sound survivalism to say that it is the sound rather than (as the non-survivalist maintains) the sound stream that persists across changes in auditory qualities.

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know how to formulate, and that no total theory of sound and sound phenomena can do without either. In particular, then, it seems to me that what is known about the (neutrally described) facts about sound survival is agnostic between survival and nonsurvival views. (I am not asserting that there is no fact of the matter about which is right, but only that if there is a fact of the matter, the sources of evidence I have discussed fail to disclose what it is.) Once again, therefore, considerations about survival through qualitative change fail to provide for a temporal aspect that would distinguish sounds from other sensory qualities.

4. conclusion Sounds occur at particular times and have temporal durations. In those senses, and perhaps others, they are creatures of time. But I do not see that the case has been made that there are distinctively temporal aspects of sounds—aspects that distinguish them in metaphysically significant ways from other sensory qualities. While it may be that metaphysical differences between sounds on the one hand and colors, tastes, and odors on the other, ultimately undermine the prospects for a uniform treatment of sensory qualities, it seems to me that what we know about the temporal features of sounds does not license this conclusion at this time.17 University of California, San Diego

references Bregman,A. S. (1990). Auditory Scene Analysis. Cambridge, Mass: MIT Press. Byrne, A. and Hilbert, D. R. (2003). ‘Color realism and color science’, Behavioral and Brain Sciences, 26(1), 3–64. Casati, R. and Dokic, J. (2005). ‘Sounds’. In E. N. Zalta, ed., The Stanford Encyclopedia of Philosophy. Clark, Andy (2006). ‘That lonesome whistle: A puzzle for the sensorimotor model of perceptual experience’, Analysis, 66(1), 22–5. 17 I am grateful to Kalle Aho, Craig Callender, Casey O’Callaghan, and Chris Wüthrich for helpful discussion of these matters.

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Clark, Austen (2000). A Theory of Sentience. New York: Oxford University Press. Cohen, J. (2008). ‘Color constancy as counterfactual’, Australaian Journal of Philosophy, 86(1), 61–92. (2009). The Red and The Real: An Essay on Color Ontology. Oxford: Oxford University Press. Jackson, F. (1977). Perception: A Representative Theory. New York: Cambridge University Press. Kuehni, R. G. (2003). Color Space and Its Divisions: Color Order from Antiquity to the present. New York: Wiley. Matthen, M. (2008). ‘On the diversity of auditory objects’, European Review of Philosophy, 7. Mertz, D. W. (1996). Moderate Realism and its Logic. New Haven: Yale. Newhall, S. M., Burnham, R. W., and Clark, J. R. (1957). ‘Comparison of successive with simultaneous color matching’, Journal of the Optical Society of America, 47(1), 43. No¨e, A. (2004). Action in Perception. Cambridge, Mass.: MIT Press. (2006a). ‘Experience of the world in time’, Analysis, 66(1), 26–32. (2006b). ‘Experience without the head’. In T. S. Gendler and J. Hawthorne, eds., Perceptual Experience, pages 411–33. New York: Oxford University Press. (2007). ‘Real presence’, Philosophical Topics, 33(1), 235–64. O’Callaghan, C. (2007). Sounds. Oxford: Oxford University Press. (2008). ‘Object perception: Vision and audition’, Philosophy Compass. (2009). ‘Constructing a theory of sounds’, Oxford Studies in Metaphysics (Oxford: Oxford University Press). O’Regan, J. K. and No¨e, A. (2002). ‘A sensorimotor account of vision and visual consciousness’, Behavioral and Brain Sciences, 24(5), 883–975. Pasnau, R. (1999). ‘What is sound?’, Philosophical Quarterly, 49(196), 309–24. (2000). ‘Sensible qualities: The case of sound’, Journal of the History of Philosophy, 38, 27–40. P´erez-Carpinell, J., Baldoví, R., de Fez, M. D., and Castro, J. (1998). ‘Color memory matching: Time effect and other factors’, Color Research & Application, 23(4), 234–47. Sorenson, R. (2008). ‘Hearing silence: the perception and introspection of absences’. In M. Nudds and C. O’Callaghan, eds., Sounds and Perception: New Philosophical Essays. Oxford: Oxford University Press. Williams, D. C. (1953). ‘The elements of being’. Review of Metaphysics, 7, 3–18, 171–92.

INDEX Armstrong, D. 74 Arntzenius, F. 184 n. 7

Earman, J. 64 n. 15, 199 Effingham, N. 75 n. 27

Baker, L. R. 131 n.27, 132 n. Balashov, Y. 205, 209, 221, 227 n. Barbour, J. 179, 185 Barker, S. 57, 209 n. 13 Bertotti, B. 185 Bittner, T. 221, 221 n. Blauert, J. 258 Bregman, A. 287 n., 316 n. 11 auditory scene analysis 251, 315, 317 n. 14 Broad, C. 7

Field, H. 187 n. 12 Fine, K. 53 n., 163 n. structured aggregates 74 Forrest, P. 38 n., 74 Friedman, M. 196 n.

Carnap, R. 151 n. Cartwright, R. 54 n. 4 Casati, R. 306 Chisholm, R. 8, 137 n. 44 Clark, A. 317 n. 15 Cohen, J. 312 n. 9 color constancy 318 n. 16 Crisp, T. 205, 215 n. 19, 221 region-relative parthood relation 240 Descartes, R. 192 Deutsch, H. 154 Doepke, F. 204 n. 3 Dokic, J. 306 Donnelly, M. 221, 221 n. Dowe, P. 57 additivity assumption 209 n. 13 Eagle, A. 95, 97, 99–102, 104, 106, 108, 110–11

Gaver, W. 283 n. Geach, P. 147–8, 150, 152, 154, 154 n. 2 Geroch , R. 114 n. 1 Gibbard, A. 62 n. Gibson, I. 54 n. 5, 57 Gilmore, C. 53–4, 57, 59, 60, 63–92, 113–15, 209 corner slice example 229–30 Ginet, C. 129–30 Grunbaum, A. 183 n. ¨ Haslanger, S. 57, 98 n. Hawley, K. 57 n. Hawthorne, J. 122, 154 n. 12 anti-realism and language 138–9 incars 127–128 Hirsch, E. 122 n. 9, 140 Howard-Snyder, F. 42 n. 2 Hudson, H. 55, 73 n. 24, 205, 209, 221 parthood-at-a-region relation 234–5, 237 n., 239 uniqueness principle 161 n. Huggett, N. 200 n. 23

322

Index

Kant, I. 303 Kuehni, R. 307 n. Lange, M. 198 n. Lewis, D. 4 n.3, 64, 72, 74, 99, 170 n. 32, 171 n., 198 n. anti-universalist intuitions 120 n. 4 external time vs. personal time 101–102, 104, 105 time travel 64 unrestricted composition 90 worm theory 57 Locke, J. identity of persons 64 sounds 249 Lockwood, M. 103–4 Lowe, E. J. 204 n. 3 McCall, S. 31 McDaniel, K. 205, 221 Mach, E. 200 Markosian, N. 42 n. 2, 146 n. 3 Mundy, B. 187 Newhall, S. M. 310 Newton, I. 179–81, 185, 192 n. 14, 200 No¨e, A. 317 n. 15 O’Callaghan, C. 269, 271, 274, 276, 280, 296, 304, 311, 312, 314 Olson, E. T. 215 n. 19 Parsons, J. 55, 56, 56 n., 58, 60 n. Pasnau, R. 253, 260 n. 5, 311 sound location 249 Perez-Carpinell, J. 310 Pooley, O. 54 n. 5, 57, 115 n. Pooley torus 115–16

Price, H. 35–7 Prior, A. 30 Putnam, H. 190 interpretation of Reichenbach 196 n. Reichenbach, H. 37, 179 n., 196 n. Robson, J. 75 n. 27 Saucedo, R. 204 n. 3, 206 n. 6 Schaffer, J. 88 Schiffer, T. 164 n. 23 Scruton, R. 271, 283 n. Sider, T. 12 n. 15, 53 n. 2, 73 n. 23, 79 n., 125 n.14, 215 n. 18, 215 n. 19, 220, 226 external time vs. personal time 104 fusions 74, 83 instantaneous temporal parthood 60, 60 n. stage theory 57 n. Simons, P. 72, 217 n. 22, 223 n. 29 empirical collectives 136 n. 41 Strong Supplementation Principle 83 Product Principle 83 Weak Supplementation Principle 76, 82 Skow, B. 89 n. 39 Smart, J. 31 Smith, D. 205, 215 n. 19, 221 region-relative parthood relation 240 Sosa, E. 119 Taylor, E. 105 n. Thomasson, A. 131 n. 29 Thomson, J. 226 n. 35, 256

Index Unger, P. 146 Uzquiano, G. 88 van Inwagen, P. 46, 137 n. 44 anti-eliminativist intuitions 120 n. 4 extended temporal simples 58 objects’ locations 213 n.

323

statues vs. gollyswoggles 130 theory of time travel 29, 41–3 Varzi, A. 82–3 Wiggins, D. 127 n. 18 Williamson, T. 123 n. Zimmerman, D. 132 n.