Modal Matters: Essays in Metaphysics 0199676569, 9780199676569

Table of contents :
Dedication
Contents
Preface
Acknowledgments
Part 1. Reality
1. Introduction: A Sketch of Reality (2018)
2. Realism without Parochialism (1992)
Postscript to “Realism without Parochialism” (2018)
Part 2. Possible Worlds Realism and Actuality
3. Concrete Possible Worlds (2007)
4. Island Universes and the Analysis of Modality (2001)
5. Absolute Actuality and the Plurality of Worlds (2006)
6. Isolation and Unification: The Realist Analysis of Possible Worlds (1996)
7. Reducing Possible Worlds to Language (1987)
8. Quantified Modal Logic and the Plural De Re (1989)
Part 3. Modal Plenitude
9. Principles of Plenitude (1986)
Postscript to “Principles of Plenitude” (2016)
10. Plenitude of Possible Structures (1991)
Postscript to “Plenitude of Possible Structures” (2016)
11. All Worlds in One: Reassessing the Forrest-Armstrong Argument (2011/2018)
12. On Living Forever (1985)
Part 4. Humean Perspectives on Truthmaking, Mereology, Spacetime, and Quantities
13. Truthmaking: With and without Counterpart Theory (2015)
14. The Relation between General and Particular: Entailment vs. Supervenience (2006)
15. Composition as a Kind of Identity (2016)
16. Composition as Identity, Leibniz’s Law, and Slice-Sensitive Emergent Properties (2019)
17. The Fabric of Space: Intrinsic vs. Extrinsic Distance Relations (1993)
18. Is There a Humean Account of Quantities? (2017)
Bibliography
Index

Citation preview

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Modal Matters

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Modal Matters Essays in Metaphysics

Phillip Bricker

1

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Great Clarendon Street, Oxford,  , United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Phillip Bricker  The moral rights of the author have been asserted First Edition published in  Impression:  All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press  Madison Avenue, New York, NY , United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number:  ISBN –––– Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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To Margi, Nora, and Adam, for all their love and support.

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Contents Preface Acknowledgments

ix xiii

Part . Reality . Introduction: A Sketch of Reality () . Realism without Parochialism () Postscript ()

  

Part . Possible Worlds Realism and Actuality . Concrete Possible Worlds ()



. Island Universes and the Analysis of Modality ()



. Absolute Actuality and the Plurality of Worlds ()



. Isolation and Unification: The Realist Analysis of Possible Worlds ()



. Reducing Possible Worlds to Language ()



. Quantified Modal Logic and the Plural De Re ()



Part . Modal Plenitude . Principles of Plenitude () Postscript () . Plenitude of Possible Structures () Postscript ()

   

. All Worlds in One: Reassessing the Forrest-Armstrong Argument (/)



. On Living Forever ()



Part . Humean Perspectives on Truthmaking, Mereology, Spacetime, and Quantities . Truthmaking: With and without Counterpart Theory ()



. The Relation between General and Particular: Entailment vs. Supervenience ()



. Composition as a Kind of Identity ()



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. Composition as Identity, Leibniz’s Law, and Slice-Sensitive Emergent Properties ()



. The Fabric of Space: Intrinsic vs. Extrinsic Distance Relations ()



. Is There a Humean Account of Quantities? ()



Bibliography Index

 

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Preface This volume consists of thirteen papers that have been previously published (one only in part) and five papers that have not been published before. I have made only minor changes to previously published papers and the unpublished papers that had been available on my website. Footnotes that were added for this volume are in square brackets. For three of these papers I have added substantial postscripts, either to account for relevant literature written after my paper, or to note how my views have developed. David Lewis wrote, in the introduction to his first volume of papers: “I would have liked to be a piecemeal, unsystematic philosopher, offering independent proposals on a variety of topics” (Lewis a: ix). It’s different with me: all of my work in metaphysics, from my days as a graduate student at Princeton, has been loosely governed by a certain underlying picture of reality, and certain fundamental principles that I take to hold sway. But owing to some combination of timidity, distraction, and just plain laziness, I never attempted to lay out in detail “my system of the world.” Most of the papers in this volume (Chapter  excepted) are but fragments of that system. In the introductory Chapter , “A Sketch of Reality,” I provide the backdrop that ties the fragments together. No doubt, among philosophers, my views in the metaphysics of modality are closely associated with Lewisian modal realism. And, indeed, I do believe in a plurality of concrete possible worlds, on one natural way of understanding ‘concrete’. (See Chapter .) But two caveats are in order. First, my belief in mathematical entities and structures has always been much firmer than my belief in possible worlds. When I was a graduate student, there was talk of Kreisel’s dictum, as paraphrased by Dummett: “The problem is not the existence of mathematical objects but the objectivity of mathematical statements” (Dummett : xxxviii). That stuck in my craw. How can there be objective statements with nothing for those statements to be about, truths ungrounded in reality? (See Chapters  and .) I suspect I have been a mathematical Platonist since, as a young child, I took the numbers to be my special friends, with distinct personalities, and with whom I competed in all manner of sports (though they needed my assistance to swing the bat, or throw the ball). My Platonism has evolved since then, but I am no less convinced that my thoughts about mathematical entities reach their target. The objectivity of mathematics demands that there be a subject matter for mathematics, and there is no reason to think that the physical world can play that role. What makes my belief in possible worlds less secure is just that it is less clear that modal statements are objectively true, or that only concrete possible worlds can supply the required subject matter. The first caveat, then, is that my realism about possible worlds is part of a much wider realism. The second caveat is that my realism about possible worlds is not Lewisian modal realism because I do not find it coherent to hold that the actual and the merely possible have the same ontological status. As a graduate student, I worried

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that this rejection of Lewisian modal realism would require taking modality seriously, something I was loath to do. But I soon came to realize that it was only actuality that had to be taken seriously, that Lewis’s account of actuality as merely relative had to be replaced by an account of absolute actuality, and that absolute actuality did not require any sort of primitive modality. My realism about possible worlds, then, is in some ways closer to “abstractionist” views than to Lewis’s, in that I accept that merely possible worlds do not have the same ontological status as the actual world of which I am a part. But, with respect to how worlds represent what is true at them, my view is much closer to Lewisian modal realism than to any abstractionist view; like Lewis, I reject representation by magic. A theme that runs through these papers is my commitment to Humeanism, a commitment that has only grown stronger over the years. I reject all forms of primitive modality, be it metaphysical or nomic or causal, that cannot be reduced to logic and the relations between concepts. That’s not to say that I don’t struggle with my own non-Humean intuitions, especially with respect to physical laws. Why not, then, go with the flow and give in to temptation? A simple reason: positing nonHumean whatnots to ground laws, or causation, or metaphysical modality, does nothing to provide any genuine explanation, no more than positing a deity does anything to explain why the universe exists. It is all too easy to be seduced by such pseudo-explanations. Throw away these crutches, I say, and try to walk on your own! Another theme that runs through these papers is conspicuous by its absence: I abjure the orthodox Quinean methodology for deciding what metaphysical theories to believe. That concrete possible worlds are “serviceable” for systematic philosophy (as Lewis argues), that mathematical entities are “indispensable” for our scientific theorizing about the world (as Quine and Putnam argue), these claims, even if true, provide no good reason to believe that possible worlds or mathematical entities exist. That one metaphysical theory is simpler or more economical than another, either by positing fewer ontological kinds or having fewer ideological primitives, gives us no good reason to think that theory more likely to be true. How could it, unless we have good reason to think that reality itself is simple? And it seems just obvious to me that we do not. Those philosophers who engage in the Quinean methodology, seeking the simplest theory that can account for our manifest world, seem to me to be playing an elaborate game. It may be a fun and challenging exercise, like when I assign to my logic students the task of finding the shortest proof for some theorem. But it has nothing to do with seeking truth. I hold instead that all of our a priori knowledge about reality comes from rational insight. I’m optimistic that this can provide an adequate foundation for our knowledge of reality. But to whatever extent our rational insight falls short, there must we withhold our belief. Better to be agnostic than to believe without good reason. Writing papers, for me, is a solitary endeavor, with little or no input from others. That does not mean, however, that I do not owe an enormous debt to other philosophers: to my students, my colleagues, and my former teachers. For it is only through critically engaging with their work that I have been able to develop my own views. This has been true of all of my doctoral students; but those whose work has been most relevant to the ideas in this volume are (chronologically) Ted Sider,

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David Denby, Stephan Torre, Kris McDaniel, Sam Cowling, Barak Krakauer, and Cameron Gibbs. Among colleagues, I will mention Jonathan Schaffer and Brad Skow, both of whom moved on from UMass far too soon. But I thank all of my colleagues at UMass for providing a supportive environment for practicing philosophy. My most influential teachers at Princeton were Paul Benacerraf, Saul Kripke, and especially my thesis advisor, David Lewis. David was a magnanimous mentor and a true friend. Perhaps I am not alone in saying that, even now, eighteen years after his passing, he is the imagined audience for all that I write. Finally, I would have accomplished nothing without the love and support of my wife and two children. It is to them that this volume is dedicated.

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Acknowledgments Thirteen of the chapters in this volume were previously published, in whole or in part. The place of first publication is listed below. I thank the publishers for permission to republish. “Concrete Possible Worlds,” in T. Sider, J. Hawthorne, and D. Zimmerman (eds.), Contemporary Debates in Metaphysics (Blackwell, ), –. Reprinted with permission of John Wiley & Sons, Inc. “Island Universes and the Analysis of Modality,” in G. Preyer and F. Siebelt (eds.), Reality and Humean Supervenience: Essays on the Philosophy of David Lewis (Rowman and Littlefield Publishing Group, ), –. “Absolute Actuality and the Plurality of Worlds,” in J. Hawthorne (ed.), Philosophical Perspectives, vol.  (Blackwell, ), –. Reprinted with permission of John Wiley & Sons, Inc. “Isolation and Unification: The Realist Analysis of Possible Worlds,” Philosophical Studies  (), –. “Reducing Possible Worlds to Language,” Philosophical Studies  (), –. “Quantified Modal Logic and the Plural De Re,” Midwest Studies in Philosophy  (), –. Reprinted with permission of John Wiley & Sons, Inc. “Plenitude of Possible Structures,” Journal of Philosophy  (), –. “Truthmaking: With and without Counterpart Theory,” in B. Loewer and J. Schaffer (eds.), A Companion to David Lewis (Blackwell, ), –. Reprinted with permission of John Wiley & Sons, Inc. “The Relation between General and Particular: Entailment vs. Supervenience,” in D. Zimmerman (ed.), Oxford Papers in Metaphysics, vol.  (Oxford University Press, ), –. “Composition as a Kind of Identity,” Inquiry: An Interdisciplinary Journal of Philosophy  (), –. Reprinted by permission of Taylor and Francis Group, LLC, a division of Informa plc. “Composition as Identity, Leibniz’s Law, and Slice-Sensitive Emergent Properties,” Synthese (), –. https://doi.org/./s---y. “The Fabric of Space: Intrinsic vs. Extrinsic Distance Relations,” in Midwest Studies in Philosophy  (), –. Reprinted with permission of John Wiley & Sons, Inc. “Is There a Humean Account of Quantities?,” Philosophical Issue  (), –. Reprinted with permission of John Wiley & Sons, Inc.

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PART 

Reality

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 Introduction: A Sketch of Reality () . Preamble Every account of reality needs a creation myth. Here is mine, in three stages: In the beginning, reality was without form, and darkness was upon the face of the deep. And then—wham!—there was logic. And the laws of logic gave structure to reality, not only by proscribing the ways that reality could not be, but by demanding that, for any way that reality could be structured, some portion of reality was structured that way. And then— splash!—fundamental properties, and perhaps relations, were hurled across reality in every combination. And various portions of reality became colored, and massy, and conscious, and acquired all manner of qualitative character. And then—buzz!—a portion of reality of which I am a part was lit up with actuality from within, acquiring an elevated ontological status. And that is all.

Most of the papers included in this volume concern issues that arise at one or more of the three stages in the above myth. But they leave a lot unsaid, sometimes because I lack arguments to back up my beliefs, sometimes because I simply don’t know what to believe. In this introduction to the volume, I fill in some of the gaps. My hope is that a reader who knows, even in rough outline, the picture of reality that I had in the back of my mind as I wrote these papers will better understand the motivations for the views presented, and how those views fit together. I don’t have space here to fully support the claims that I make; indeed, in many cases, I am raising what for me are open questions. My goal is to scope out the terrain in a useful way.

. Reality What is reality? As a first pass, I want to say: reality consists entirely of things having fundamental properties and standing in fundamental relations.¹ Included in this This chapter has not been previously published. Parts of Section  were presented at MIT in November . Thanks to the audience for their helpful comments. And thanks especially to Sam Cowling, Cameron Gibbs, Kris McDaniel, Jonathan Schaffer, and Jonathan Vogel for comments on earlier drafts. ¹ Two terminological points. First, elsewhere in this volume I frequently use Lewis’s term ‘perfectly natural’ instead of ‘fundamental’. I take these terms to pick out the same properties and relations. Both are multiply ambiguous; but the ambiguities sway together. I say more about what I take ‘fundamental’ to include below in Section .. Second, I use ‘property’ and ‘relation’ in accord with an abundant conception of properties and relations. (See Lewis (a: –) on the distinction between abundant and sparse conceptions.) Thus, for any things, there is a property had by all and only those things. Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

:     ()

characterization is that reality is ruled by Hume’s Dictum, the denial of necessary connections between distinct existents;² for it is constitutive of my use of the term ‘thing’ that distinct things stand in no necessary connections. (More on what that means below in Section ..) Given my acceptance of unrestricted composition, reality can be identified with a single thing, the fusion of all things. Less contentiously, it can be identified with the many things plurally. Either way, reality is the totality of things, not of facts (or states of affairs). For facts, unlike things, do not abide by Hume’s Dictum. (See Chapters  and .) For any thing or things—any portion of reality³—one can ask three questions, one corresponding to each of the three stages of creation. First, what is its structure? Second, what is its content, or qualitative character? And, third, is it actual? The answers to these questions, I think, are grounded in different ways. Roughly, I am a nominalist about structure, a trope theorist about qualitative character, and a sort of mysterian about the property of actuality: there is a fundamental distinction of ontological category between the actual and everything else that we have good reason to accept but about which there is very little positive to say. (On how we can know that we are actual, we can say a bit more; see Chapter .) Not only are the three stages differently grounded, they (presumably) apply to progressively smaller parts of reality. All of reality has structure. Reality then divides into a region of pure structure, where mathematical systems reside, and a region where there is both structure and qualitative character, where the possible worlds reside. And then there is the region where, in addition to structure and qualitative character, there is actuality, where we and what we are externally related to (and perhaps more) reside. Reality is thus threefold, with content heaped on structure, and actuality heaped on structure and content. There is no room in reality for sets (or classes, or other set-like entities), not if sets are taken to be entities composed in some non-mereological way from their members. As an uncompromising Humean, I reject non-mereological modes of composition and the mysterious necessary connections that come with them. (See especially Chapters  and ; see also Lewis : –.) I do allow that there is in reality a mathematical system in which Zermelo-Frankel set theory is true, indeed, many such systems since ZFC, even second-order ZFC, is not categorical. (See Chapter .) And I have no objection to calling the things that inhabit such systems “sets.” But these “sets” are not the composite entities that most friends of sets believe in. In the mathematical theory of sets, and the mathematical structure determined by the “membership” relation, it is irrelevant whether the “sets” have internal structure.

² In Hume’s words: “There is no object, which implies the existence of any other if we consider these objects in themselves . . . .” Hume (: ). Calling contemporary versions of this “Hume’s Dictum” goes back (at least) to Goodman (), and has recently been made popular by Wilson (). ³ The expression ‘portion of reality’ is a term of art that, although syntactically singular, is semantically neutral with respect to the plural/singular distinction; ‘part of reality’ and ‘region of reality’, on the other hand, I take to be both syntactically and semantically singular. Talk of “portions of reality” could be regimented within a version of plural logic that has only a single style of variable rather than separate singular and plural variables. But I prefer to understand it as schematic, to be replaced by either a singular or plural variable. See Section ..

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

But although I reject sets as composite entities, I do not reject our ordinary or philosophical talk of sets (or sets of sets, or . . . ). And I help myself to such talk in many of the chapters of this volume. It is just that such talk, I am now convinced, must be interpreted in a way that does not introduce ontological commitment to anything beyond the “individuals,” the entities on the ground floor. There are two main methods on the market for accomplishing this. There is the structuralism about sets developed (but not yet endorsed) by Lewis (: –, –). (See also Nolan (: –) for a variation on this method.) I do not accept Lewis’s structuralism, however: it does not give a plausible account of what our set talk means. Instead of giving an interpretation of the membership relation, it generalizes over interpretations in a way that reduces the content of set theory to a claim about the size of reality. I prefer instead to understand talk of sets as talk of pluralities, and pluralities of pluralities, and so on into the transfinite. That provides a more straightforward interpretation of our ordinary and philosophical talk of sets; and an easy translation from the axioms of set theory into a higher-order plural logic. Moreover, it allows our talk of sets to inherit the ontological innocence of plural logic. Please don’t misunderstand. My rejection of sets is not motivated by a desire to resolve the paradoxes associated with sets: whatever pressure there was to say that the universe of sets is “indefinitely extensible” re-emerges as pressure to say that the hierarchy of pluralities is “open-ended.” But resolving the paradoxes was not the goal. Rather, sets had to be excised from reality in service of a thoroughgoing Humeanism. (For a bit more on Humeanism and sets, see the postscript to Chapter .) A defense of this approach to discourse about sets would require, not just a defense of pluralism—that is, the legitimacy and primacy of plural logic—but also a defense of higher-order plural logic: pluralities of pluralities, and on up the ladder. (Whether English has such higher-order plural terms is beside the point.) I cannot provide such a defense here.⁴ It would include my conviction that the singularist bias (according to which plural terms and quantifiers must be construed as singular terms and quantifiers) is more prevalent even than most pluralists realize. It has afflicted not only the development of formal logic, but our understanding of ordinary language as well. For on my view, although English terms referring to classes, collections, groups, and so on are grammatically singular, a metaphysical semantics of English would take them to be plural terms, and adjust their logic accordingly. Here, as in so many other places, our ordinary language, being built for convenience, fails to reflect the underlying reality. I called my initial characterization of reality a “first pass” for at least four separate reasons. First, it is too limiting to say that reality consists of things instantiating allor-nothing properties and relations: that may not allow for an adequate account of quantities. I put this issue to one side in this introduction, but see Chapter . Second, it is also too limiting to say that reality consists of things instantiating ⁴ Some defense, along with precise formulations and discussion of what is philosophically at stake, can be found in Rayo () and Linnebo and Rayo (). In the terminology of the latter work, I am a “liberalist” about sets who accepts an open-ended “ideological hierarchy.” But that is compatible with also holding, as I do, that reality is fixed once and for all, that there is no open-ended “ontological hierarchy.” For more on the “size of reality,” see Chapter .

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

:     ()

fundamental properties and relations. I also allow for higher-order fundamental properties. For example, there could be an emergent fundamental property instantiated by a plurality of things, but not by any one thing. I also put this issue to one side here, but see Chapter . Third, we can eliminate the apparent duality of things and their properties if, as I think, the instantiation of fundamental properties and relations by things is a matter just of the existence of certain particulars, such as tropes, out of which all things are composed. (See Section . below.) Fourth, in addition to the reality of things described above, there are also representations of reality. I have in mind here not mental or linguistic tokens, which can easily be located within the reality of things. Rather, I speak of mind- and languageindependent propositions that represent, truly or falsely, the reality of things and its parts. And, I speak of the properties and relations and higher-order “intensional entities,” such as propositional operators and quantifiers, that provide the constituents of propositions on a structured conception.⁵ Perhaps God (were there such a creature), being infinite and omnipresent, could think about reality in some intimate way, communing with the things that comprise reality without having to represent them. But for us, being finite and occupying some small portion of reality, thinking about reality goes primarily by way of our representations of it. And now we come to what is perhaps the most fundamental question of metaphysics: do these representations belong to some realm separate and independent of the reality of things, a Fregean third realm? Or can these representations, these propositions and other “intensional entities,” be reduced to the reality of things? If the latter, then in this respect at least our first pass at what reality is will also be our last pass. David Lewis, of course, was committed to reducing representations of reality to the reality of things. Primitive representational properties could not be understood by us, he claimed, and a primitive external relation of representing (or corresponding to, or being true of ), in addition to not being understandable by us, would violate Humean strictures against necessary connections. (See Lewis’s argument against what he calls “magical ersatzism” in Lewis a: –.) No problem, Lewis thought: by extending reality to include a plurality of possible worlds, he believed that (with the aid of set theory) he could find room for all manner of intensional entity, abundant propositions and properties, structured and unstructured. (See Lewis a: –.) Lewis’s account faces a problem, I think, with respect to the content of mathematical propositions (see Bricker : –); but my inclusion of mathematical systems within reality provides the wherewithal to set this right. The reduction of the third realm, however, faces a stiffer challenge from logic: logical truths differ from one another, not just in structure, but in content (on at least some conceptions of content). For example, I take the truths of mereology to be truths of logic. (See Chapter  and Section . below.) But, surely, the truth that every plurality has a fusion differs in content from the truth that everything is self-identical. ⁵ See Lewis (a: –) on structured vs. unstructured conceptions of propositions and properties. It won’t matter for present purposes whether propositions are structured like sentences in some language of mathematical logic, or sentences in some language with a categorial grammar, or in some other way, as long as the principle of compositionality is satisfied.

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Should we expand reality to include logically impossible “worlds”? No, here I draw the line; allowing reality to contain logically impossible “worlds” would be self-undermining for the realist.⁶ We will need another way to account for the content of logical truths, and the theory of understanding that accompanies it. (See Section . below.) A working hypothesis that underlies many of the chapters of this volume (this introduction included) is that a reduction of representations of reality to the reality of things can be carried out. I sometimes speak in the chapters of this volume, perhaps too glibly, of the space of concepts, and of some concepts being fundamental; but I am under no illusion that it is easy to square this talk with my Humean conviction that all there is is the reality of things. I am especially concerned that the relations of fundamentality among logical concepts will not be derivable from the relations of fundamentality that hold among things and their properties and relations. (See Chapter  and Section . below.) And I am unsure whether what I call “perspectival concepts” can be reduced to the reality of things. (See Chapter  and Section . below.) If the project of reduction were to fail, then reality would consist not only of the reality of things but also a Fregean third realm of entities that are representational by their very nature; and the Humean prohibition against necessary connections would apply only to the reality of things, not to reality as a whole. Presumably there would then be a hierarchy of such entities—representations of representations, and so on up the ladder—and the impoverished budget of solutions that are on offer to ward off paradox. A major disappointment to be sure; but one that I am prepared to swallow, if need be.

. Logic What is logic? First off, set aside conceptions of logic based on the mathematics of formal systems or the psychology of human reasoning. They are not much relevant to our current endeavor. But there are two rather different, and independent, conceptions of logic that are relevant, two different faces of logic. One face looks outward from the realm of representations to the reality of things; the other looks inward, toward the structure of the realm of representation itself. It is the first conception I have generally had in mind when I use the term ‘logic’ in the chapters of this volume. Call it, for easy reference, the external conception. On this conception, roughly, logic is a body of very general truths about reality. But let me start with a few words about the second conception according to which logic is internal to the realm of representation. Call it the internal conception. On the internal conception, logic is often characterized as the study of logical consequence, where the consequence relation is taken to be formal, to hold between propositions in virtue of their logical form. But the consequence relation I take to be fundamental to logic—call it (logical) implication—is material, not formal: whether one proposition implies another depends only on the content of those propositions, ⁶ Lewis (a: ) gives an argument that there are no (concrete) worlds in which contradictions are true. See Stalnaker () for interesting discussion.

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:     ()

not on their form.⁷ Intuitively, one proposition logically implies another just in case it has at least as much (informational) “content” as the other: it says everything that the other says, and perhaps more. Propositions with the same content mutually imply one another, and are (logically) equivalent. This notion of content does not presuppose any particular reduction of propositions to the reality of things, or even that any reduction is possible. It is clear enough, I believe, to stand on its own. But, of course, the reduction that I have in the back of my mind identifies the content of a proposition with the portions of reality of which it is true. What do the theses of logic look like on the internal conception? They tell us how logical implication relates to the various logical operations and relations that apply to our representations of reality, the (abundant) propositions, properties, and relations. They include (to give a couple of random examples): for any propositions p and q, the conjunction of p and q implies p; and, for any propositions p and q, the negation of the conjunction of p and q mutually implies the disjunction of the negations of p and q. And they include (on an unstructured conception of propositions): whenever two propositions imply one another, they are identical. A succinct formulation of a part of logic that is concerned only with the “logic of (unstructured) propositions” would be: the propositions under implication form a complete Boolean algebra.⁸ But there will be theses to reflect how all of the logical notions including quantifiers relate to one another and to logical implication. When I use the term ‘logic’, however, it is generally the external conception that I have in mind according to which the laws of logic are general truths about reality. This conception of logic is grounded in tradition—or so I think—but it is out of step with contemporary ways of thinking about logic. Moreover, the scope of logic is broader for me than for most of my contemporaries; for example, I take logical notions to include not only the Boolean operators and singular and plural quantifiers, but also notions of mereology. And, even more controversially, I take logic to be productive, to give us substantial knowledge of what exists.⁹ The laws of logic express how our representations of reality relate to the reality of things. For each propositional operator, there will be a law expressing “truth conditions” for proposition-denoting terms involving that operator. For example, one law will be: a disjunction of p and q is true of a portion of reality just in case p is true of that portion or q is true of that portion. Another law will tie the relation of logical ⁷ If indeed they have form. On an unstructured conception of propositions, the structure of a sentence that expresses a proposition does not correspond to any internal structure of the proposition expressed. Of course, saying that implication is not formal is compatible with holding that our recognition that implication holds between propositions often hinges on the forms of the sentences we use to express those propositions, or the terms we use to denote them. ⁸ Structured propositions form a Boolean algebra only after equivalent propositions have been “identified.” For more discussion of the relevant notion of propositional content, and how it relates to the “logic of propositions,” see Bricker (). ⁹ Frege () took logic to be a body of truths about reality, some of which make substantive claims about what exists, so-called “logical objects”; otherwise his combination of logicism and platonism would be incoherent. Frege () also called the laws of logic “laws of truth.” I can accept that as well, as long as it is understood to say only that the logical laws are naturally expressed in terms of truth, not that they are about some primitive notion of truth. I am a deflationist about truth as it applies to propositions (given the hypothesis that the realm of representation reduces to the reality of things).

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implication to reality: p logically implies q just in case every portion of reality of which p is true is a portion of reality of which q is true. Some logical laws, however, do not involve any particular logical notion, but express general truths about how propositions relate to reality. For example, there is a law of logic that expresses the abundance of propositions: for any distinct portions of reality r and s, some proposition is true of r but not of s. This law guarantees that whatever the size and character of reality there are representations enough to describe it.¹⁰ It is sometimes thought mistakenly that the two conceptions of logic can be reduced one to the other. For, starting with the internal conception of logic and the notion of logical implication, we can define a “logical truth” as a proposition that is implied by any proposition and then we can take the logical laws to be those “logical truths” that are appropriately general. It seems we thus have no need for the external conception. But this proposed reduction rests on a confusion between two senses of ‘logical truth’: the notion defined in terms of the relation of logical implication is better called “logical validity.” Without the laws of logic relating the realm of representation to reality, there is no guarantee that these logically valid propositions will be logically true in the desired sense: true of all portions of reality. To make that case, we need to use the laws of logic given above that were provided by the external conception. Then we can argue as follows. Let p be a logical validity, and thus implied by any proposition. Let r be any portion of reality. Let q be a proposition that is true of r. Then, since q implies p, p is true of r as well. Do we need a third conception of logic to capture the way in which logic is normative? No. Again following Frege () I do not think the laws of logic differ, say, from the laws of physics or the laws of biology with respect to the source of their normativity. Given that we have a desire to believe what is true, the laws of physics put constraints on what beliefs we should have about the physical world, and on what inferences we should make from our beliefs. For example, given Newton’s third law and a belief that a impresses a force on b, I should believe that b impresses an equal and opposite force on a. Logic is different only in being more general: given our desire to believe what is true, it constrains how we should form our beliefs about any portion of reality. This approach to the normativity of logic also provides a response to Lewis Carroll’s () point that the rule of inference modus ponens cannot be captured by any statement of logic, such as the conditional ‘for any p and q, if p and p implies q, then q’. Rather, it is the conditional statement, together with our desire to believe what is true, that supports modus ponens as a normative rule of inference.¹¹ I have said that the laws of logic are distinguished from other truths by their generality. But how to characterize the relevant notion of generality is a delicate affair. Certainly, we should require that a logical law refer to no particular part of the reality of things; it generalizes over all portions of reality. But that won’t be enough: we should also require that it refer to no particular property or relation, even if we ¹⁰ See Bricker () for a detailed account of these laws as applied just to the Boolean notions. Note that I now take the laws to quantify over all portions of reality, not just possible worlds. ¹¹ For some discussion of how the laws of logic for Frege can be descriptive and yet have normative implications, see MacFarlane (: –).

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 :     () take these to be abundant and not constituents of the reality of things. But we cannot say that a logical law refers to no particular entity in the realm of representation, no particular (abundant) property or relation or proposition. For laws of logic require for their expression particular logical notions such as ‘and’, ‘not’, and ‘all’; and these notions need not be generalized over. It seems, then, that we need some independent way of characterizing the logical notions so that we can exclude them when defining the appropriate sort of generality: logical truths generalize over all non-logical notions and entities. Various ideas have been floated for how to characterize the logical notions, the socalled “logical constants.” But they all seem susceptible to the same problem: what results one gets out depends on the assumptions one puts in; and thus by varying the assumptions one can pretty much get whatever result one wants. To illustrate, consider what may be the most developed idea according to which the logical notions are characterized in terms of permutation invariance: the logical notions are just those that are invariant over all permutations of objects in the domain.¹² The “domain” in our case is all of reality and the “objects of the domain” are the things that are parts of reality. To simplify, I will consider only which properties and relations over the domain are logical, and I will ignore logical notions of higher type. Now, as is often noted, the disputed question whether or not the identity relation is logical is decided on this proposal: identity is preserved over all permutations in virtue of permutations being mappings that are single-valued and one-one. But—wait—why assume that the mappings over which we require invariance should be all and only permutations of objects in the domain? Suppose instead we require invariance, not just over permutations, but over all mappings, whether or not they are single-valued or one-one. Then the identity relation no longer counts as logical, though the property being a thing still does. Or suppose, in the other direction, we require invariance not over all permutations, but only over permutations that preserve “intrinsic structure.” If we take intrinsic structure to include mereological structure, then the generalized identity relation, being the same portion of reality, comes out logical, as do all the mereological relations, such as parthood and fusion, that, I have argued, are definable in terms of it. (See Chapter ; here I am supposing the mappings may take plural arguments.) The axioms of mereology will now be among the laws of logic, a result that I endorse. Or we can go further and take the “intrinsic structure” of a thing to include whatever structure results from the external relations among its parts. Then the relation, being externally related to, is a logical notion, again a result that I endorse. And it will be a matter of logic whether reality divides into isolated regions, as I think, or is instead a single interrelated whole. (A region is isolated iff no part of the region is externally related to any part of the rest of reality. For a defense of isolated regions of reality, see Chapter .) Or, we can go even further and take the “intrinsic structure” of a thing to include its intrinsic qualitative nature, thereby making the relation, is a duplicate of, a logical relation. In any case, it is clear that the decision what to count as the relevant “intrinsic structure,” and thus the mappings over which the logical notions are required to be

¹² This was the suggestion in Tarski (). See also Sher ().

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invariant, is what is determining the results. One might better just call the relevant intrinsic structure “logical structure” to make the circularity apparent.¹³ On my view, the feature that best captures what sets the logical notions apart is their universal applicability: logical notions, and only logical notions, apply whatever the subject matter.¹⁴ Thus, propositions can be conjoined and disjoined whatever they are about. For entities from any part of reality, one can ask whether they are identical to one another, or (say I) related as part to whole. On the other hand, there is a clear sense in which, say, the relation being seventeen feet from applies only to worlds with space and the relation being a successor of applies only to mathematical systems with numbers. This notion of “applicability” is, inevitably, a technical notion, but one that is intuitively based. Although it is used to identify logicality, it needn’t be incorporated into the logic itself; in particular, it doesn’t demand a three-valued logic. Making the notion more precise would involve assigning to each basic notion, of whatever type, a domain of application. The domains of derived notions will then be assigned domains recursively in terms of the domains of the basic notions from which they are derived. The simplest rule would be that the domain of a derived notion is the intersection of the domains of the notions from which it is derived. What matters is that a derived notion is logical—has universal application—if and only if all of its basic notions are logical. In particular, structured propositions, which are themselves derived from basic notions, will be classified as logical if and only if all of the basic notions from which they are derived are logical.¹⁵ Logical propositions have the sought-after “generality.” We can now say that the laws of logic are the logical propositions that are true of reality as a whole, which is to say, true simpliciter. Two questions remain. First, are all true logical propositions laws of logic? What about those that involve “gruesome” logical notions that are not at all fundamental? No problem: we can say that the fundamental laws of logic involve only fundamental logical notions. I am not here concerned with the project of singling out a few fundamental laws as axioms from which all the laws can be derived as theorems. But, second, don’t we still need to restrict the laws of logic—whether fundamental or derived—to those that belong to a “best system” according to a best system analysis of laws? No. In the case of natural laws, we need to somehow demarcate the laws from accidental generalizations, and the best system analysis provides a plausible way of doing that. But there are no “accidental” generalizations among the general truths of logic. The best system analysis loses its main point when applied to laws that govern a non-contingent domain.

¹³ The previous paragraph owes much to the insightful discussion in MacFarlane (, ch. ). But we have very different views as to what might plausibly be counted as relevant “intrinsic structure.” ¹⁴ Frege famously used the universal applicability of the concept of number—applying to “the widest domain of all . . . not only the actual, not only the intuitable, but everything thinkable”—as a mark of the logical in his defense of logicism about arithmetic. See Frege (, section ). For discussion, see Dummett (: –). Sometimes the view I espouse is characterized by saying that logic is topic-neutral. If that is taken to include the claim that logic does not have its own subject matter, however, then I do not endorse it: applying to all subject matters is not the same as not having its own subject matter. ¹⁵ I am here taking propositions, and derived notions generally, to be structured and allowing propositions with the same content to differ in their logicality. If propositions are unstructured, we should say instead that a proposition is logical just in case some logical structured proposition has the same content.

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 :     () For the universal applicability approach to logicality, just as with the permutation invariance approach, what one gets out depends on what one puts in, in this case, on the domain assignments to the basic notions. Not all philosophers will agree with the domain assignments that I make. Indeed, some philosophers even deny that identity can be applied to regions of reality made up of “stuff ”; and many philosophers deny that the part-whole relation of mereology can be applied to regions of reality that are not, in some sense, “concrete.” Thus, I do not suppose that the idea that logical notions are universally applicable, even if accepted, will do much to resolve disputes over logicality. Moreover, some philosophers would challenge the idea itself. Most, I think, would allow that universal applicability is necessary for being a logical notion. But is it also sufficient? Certainly, those with an impoverished conception of reality—for example, those who take reality to consist only of our physical universe—may reject sufficiency: even fundamental physical properties will be universally applicable without being logical. But those who, like me, take the expanse of reality to be governed by Humean principles of plenitude will have a strong case for sufficiency. (See Chapter , in particular, what I call the Principle of Interchangeable Parts.) Consider any fundamental property (or relation) that makes for qualitative character, and consider the region of reality where it is applicable. That region will not be all of reality; for reality will also contain a distinct region whose fundamental properties are all alien to the chosen property. It follows that no fundamental property that makes for qualitative character will be universally applicable. But could there be a fundamental property that neither makes for qualitative character nor is logical, and yet is universally applicable? The most plausible counterexamples to sufficiency involve notions of intentionality. Consider, for example, the property being a possible object of thought, that is, being an object of thought for some possible thinker. Isn’t that property universally applicable? Can’t thought range indiscriminately over all portions of reality? Here I am happy to embrace the consequence that such general intentional notions are logical. My conception of reality is intimately tied to intentionality, to what can be an object of thought. (See Section ..) Such general intentional notions, then, are indistinguishable in content from the notion of being something, and share in that notion’s logicality. I said above that the relation of logical implication is material, not formal. It may seem, then, that my account is far removed from the orthodox conception of logical consequence embodied in Tarski’s () model-theoretic account. Not so: my account is ideally suited to the Tarskian conception. First, since I accept an objective delineation of the “logical constants,” I have an objective notion of the “logical form” of a (structured) proposition, or a pair of (structured) propositions. That allows me to say that the implication relation, though not formal, is formalizable in this sense: whenever p implies q, there exists a p0 equivalent with p and a q0 equivalent with q such that, for every p00 and every q00 such that the pair < p00 , q00 > has the same logical form as the pair < p0 , q0 >, p00 implies q00 . Second, both my account and Tarski’s account are reductive accounts of logical modality. Third, because I accept a plenitudinous reality, I can interpret Tarski’s reduction of logical modality as quantification over portions of reality without being susceptible to the familiar critique that the account makes logically contingent claims about the size of

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reality come out logically true.¹⁶ Far from rejecting Tarski’s account of logical consequence, I am in a position to fully endorse it. Among all the propositions, some are singled out as contradictory: the propositions that imply every proposition. (Only one, on an unstructured conception of propositions.) It is a law of logic, a version of the law of non-contradiction, that contradictory propositions are true of no portion of reality. That is how logic proscribes the ways that reality could not be: there are no logically impossible objects or worlds. The law of non-contradiction is controversial, to be sure.¹⁷ But it is not nearly as controversial as its converse, which I also take to be a law of logic. Say that a proposition is consistent if it is not identical with any contradictory proposition. Then I claim: every consistent proposition is true of some portion of reality.¹⁸ Call this the law of plenitude. This law has far-reaching implications for the extent and structure of reality. Consider, for example, second-order Peano arithmetic which I suppose is not contradictory. The proposition that is the conjunction of all its theorems is, according to our law, true of some portion of reality. So there is a portion of reality that divides into an infinite succession of things. If we add to the theory that these things have no (substantial) qualitative character, and that they stand in no fundamental relation other than “is a successor of ” to one another or to any other objects, we get that there is a portion of reality where this augmented theory is true, an isolated portion of reality satisfying Peano arithmetic, what I call a “mathematical system.” But there was nothing special about Peano arithmetic here. We could start with any logically consistent mathematical theory positing any sort of structure, and the law demands that there exist, as a part of reality, an isolated mathematical system where that theory is true. We have the “plenitudinous platonism” with respect to mathematics that I defend in Chapter  and its postscript.¹⁹ Plenitudinous platonism is properly classified as a version of mathematical structuralism. For each structure posited by the mathematical structuralist, I say there is a system composed of objects occupying the places of that structure, objects that have no (substantial) qualitative character. But it is the system of objects that is primary, not the structure. Facts about the structure and its places reduce to facts about the system and the relations between its objects; and those facts in turn derive just from the objects themselves, taken plurally. Neither the structures, nor the relations that ground the structures, are parts of reality.²⁰ This allows for a uniform Humean treatment of the mathematical and qualitative realms. Both realms can be characterized simply as objects (or things)

¹⁶ See Etchemendy (: –) on the reductive commitments of Tarski’s account, and Etchemendy (: –) for the main critique. ¹⁷ Dialetheists, such as Priest (), hold that there are true contradictions. They will not want to accept my claim that all contradictory propositions are logically equivalent and have the same content. ¹⁸ See Bricker (, section ) for discussion. An important derived law will be: distinct propositions are true of different portions of reality. See Bricker (, section ). Note that the restriction ‘of some portion of reality’ is needed to avoid the “bad company objection.” See the postscript to Chapter . ¹⁹ I first defended a version of plenitudinous platonism in my doctoral thesis, Bricker (, section ). Versions have been independently developed by Balaguer () under the designation “full-blooded platonism” and by Eklund () under the designation “maximalism.” ²⁰ For arguments against positing structures, see Lewis (b).

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 :     () standing in relations; or, equivalently, as objects occupying places in a structure. The difference between the realms has to do with whether or not the objects have (substantial) qualitative character. (More on “qualitative character” in Section . below.) Thus far I have applied the law of plenitude only to mathematical theories. But laws of logic are fully general, and apply whatever the subject matter. If we start instead with a physical theory, whether physical laws or particular facts or some combination, then, as long as the theory is not logically contradictory, the law of plenitude demands that there exist some portion of reality in which the theory is true: such portions when unified and isolated I call “possible worlds.” There will be Aristotelian worlds and Newtonian worlds; indeed, there will be Newtonian worlds in which our solar system has a planet Vulcan orbiting inside the orbit of Mercury. And there will be worlds to match even the wildest fictions with flying pigs and talking donkeys. There will be a plenitudinous plurality of worlds among the portions of reality, indeed, a plurality that surpasses in scope even the worlds of Lewis’s modal realism.²¹ And all this, as I see it, will be demanded by the laws of logic. I am well aware that my conception of logic as productive is opposed to the currently fashionable idea that logic is neutral with respect to what exists, that logic is ontologically innocent.²² My conception harks back to the time when comprehension principles were considered a part of logic. For example, the naive comprehension principle for extensions—that for any concept, there exists an extension whose members are all and only those things that fall under the concept—followed from Basic Law V of Frege’s logic. Indeed, the law of plenitude is a comprehension principle applied to consistent propositions (or theories).²³ But it differs from the naive comprehension principle for extensions in a number of ways, some significant, some not. First, the fact that the law of plenitude applies to propositions rather than concepts or properties is not significant. Propositions themselves are properties of maximally connected, isolated portions of reality—what I call islands of reality—such as possible worlds or mathematical systems. And in any case I also endorse a more general law of plenitude which applies to properties: for any consistent property, there exists a portion of reality of which the property is true. Second, and more significantly, the law of plenitude, unlike the naive comprehension principle, does not posit the existence of entities of some special ontological category, entities that the theory, intuitively, is not about: no extensions, or sets, or courses of value. In that way, it is more like the comprehension principle for plurals: for any concept, if

²¹ See Chapters  and  on the individuation and character of possible worlds. Unlike Lewis, I do not suppose that worlds must be spatiotemporal. ²² See, for example, Yablo (). He takes it to be “out of the question” to accept the rationalist view according to which existence can be established a priori from “truths of reason.” Quineanism, he thinks, has won the day: one can establish the existence of mathematical or modal entities only by means of a “holistic a posteriori indispensability argument.” ²³ In Chapter , I say any coherent theory is true of some portion of reality so as to allow that there may be some constraints on theories that do not come from logic itself. Whether that is needed will depend on how certain issues about the scope of logic are decided. If theories are identified with unstructured propositions and all principles of the framework are deemed part of logic, then coherence and logical consistency will coincide.

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something falls under the concept, then there exist some things that are all and only the things that fall under the concept. But, third, unlike the comprehension principle for plurals, the law of plenitude is not conditional on any thing or things existing; it posits existence ab nihilo. It is conditional only on the proposition (or property) being consistent. But, fourth, although naive comprehension is not likewise conditional on the concept being consistent, that just reflects that, whereas even inconsistent concepts have an extension—the null class—there is no corresponding “null” part of reality. Fifth and most significantly, the law of plenitude, unlike naive comprehension, leads to no contradiction, not under my working hypothesis that the realm of representation reduces to the reality of things. For then propositions and properties are true only of portions of the reality of things, not of propositions or properties or other representational entities. There is no way for them to feed on themselves to generate Russellian contradictions. I hear a chorus of naysayers: “Your ontological profligacy offends egregiously against Occam’s Razor.” Well, I reply, that depends on what it means to posit entities beyond necessity; I think I posit exactly those entities that it is necessary to posit according to the general metaphysical or logical principles that I accept.²⁴ The chorus responds: “But, surely, positing such a bloated reality is not indispensable to our theorizing about the world.” Well, I concede that the claim that reality is plenitudinous is not indispensable to our theorizing about the physical world. But I can’t help but think it is indispensable to our theorizing about reality as a whole: after all, I think it is true! If the charge is that I fail to take various pragmatic features of a theory, such as its simplicity or usefulness, to count as evidence for the truth of the theory, then I am guilty as charged, and proudly so. The Quinean orthodoxy that takes pragmatic features to be evidence of the truth about reality is absurd on its face. Those who endorse it are false friends of ontological realism. They are instead what I call parochialists: they hold that reality is made in our image, that it somehow conforms to our desire for simple and useful theories. To be sure, pragmatic features have a role to play in our metaphysical theorizing, but not as evidence of truth. I lay out that role in Chapter . When I read the work of self-proclaimed Quineans, I find that they often help themselves to rationalist intuitions no less than I do. Is that not blatant hypocrisy? No; for they can always respond that these intuitions are just more a posteriori grist to feed into their holistic pragmatic mill. In that way, they get to invoke rationalist intuitions without the burden of having to provide a priori justification for them. I can never win that argument, never convince them that they are dissembling ²⁴ See Chapter , where I argue that one can support a plenitudinous reality either by invoking a truthmaker principle or an intentionality principle. These two approaches may be more similar than appears at first blush. The truthmaker principle that I accept holds that every proposition has a subject matter; propositions are always about portions of reality. (See Chapter , and the discussion of the “Subject Matter Principle.”) The intentionality principle that I accept holds that intentional thought is genuinely relational; thoughts are always about portions of reality. But propositions and thoughts (in the relevant sense) are entities in the realm of representation, and all such entities are “relational” in that they require the existence of those portions of reality that they are about. Thus, the demands on reality made by truthmaker principles and intentionality principles are at bottom one and the same.

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 :     () rationalists. But it is not about winning arguments. It is about holding fast to realist principles. I would rather admit that my view of reality is based on a shaky rationalist foundation than endorse a Quinean methodology which provides no foundation at all.

. Modality Propositions, I have supposed, have truth values relative to isolated portions of reality. Isolated portions of reality include not only possible worlds and mathematical systems, but also pluralities of possible worlds and mathematical systems.²⁵ The broadest alethic modality, what I will call logical or absolute modality, quantifies unrestrictedly over this domain. A proposition is logically necessary just in case it is true of every isolated portion of reality; a proposition is logically possible just in case it is true of some isolated portion. (These analyses can be extended to apply to properties by dropping the restrictor ‘isolated’; but I here focus on the modal features of propositions.) Note that the analyses are not very discriminating when applied to unstructured propositions, since there is only one unstructured proposition that is logically necessary and only one that is not logically possible. I will suppose, then, just as with the laws of logic, that we are dealing with structured propositions. One might wonder whether the logically necessary propositions are just the laws of logic. But that identification fails in both directions. First, not every law of logic is logically necessary. For a logical proposition to be a law of logic, it is sufficient for it to be true of reality as a whole; but many logical propositions are true of reality as a whole without being true of smaller portions of reality. For example, it is a law of logic, on my broad conception of logic, that everything has a distinct duplicate; but there are possible worlds, certainly, such that no part of the world has a distinct duplicate as part of that same world. Purely universal laws of logic, such as that everything is self-identical, will all be logically necessary; but that need not be so for laws of logic that involve existential quantification. Second, considering the other direction, not every logically necessary proposition is a law of logic. The laws of logic are fully general; but it is natural, if not inevitable, to understand the logical notions in a way that allows propositions that refer to particular objects or properties to be logically necessary. I will illustrate with two test cases. Consider the proposition that Trump is identical with Trump; call it Ti. And consider the proposition that Trump exists; call it Te. Are either or both of these propositions logically necessary? It seems we have three options. () Ti is logically necessary and Ti implies Te. In that case, Te is logically necessary as well—not a result that I like. () Neither Ti nor Te is logically necessary: both are true only at portions of reality that contain Trump as a part. Or () Ti is logically necessary but Te is not, in which case Ti does not imply Te and the resulting logic is a version of free logic. Only ²⁵ In Chapter  I argue that propositions have truth values relative to pluralities of possible worlds to allow for the possibility of island universes. And I argue that one can introduce a “null plurality” to allow for the possibility of nothing. If one chooses to do that, then the null plurality will count as a “portion of reality” in the definitions that follow. (But nota bene: talk of a “null plurality” or “null portion of reality” is a terminological convenience, not an ontological posit.)

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this third option, which I prefer, allows us to hold in accordance with custom that identity is (strictly) necessary whereas existence is contingent. It requires that in our assignment of truth conditions to a proposition the existential quantifier, but not the singular component, is restricted to parts of the portion of reality where the proposition is being evaluated. For a second test case, consider a possible world w in which grass is purple and consider the “world-indexed” proposition that grass is purple in w. Is that proposition logically necessary? With our free logic in hand, we can say “yes” without having to say that the world w exists in every world: what exists in each world is just that world and its parts. By choosing to understand existential propositions and certain singular propositions as I have while granting that I could choose to understand them in other ways, do I thereby endorse the conventionality of logic? No. It is a matter of convention that I have chosen to interpret our ordinary existential and singular claims as involving the logical notions that I have, with the truth conditions I have given them. The alternative interpretations are not alternative logics; they simply invoke alternative logical notions I have chosen to ignore. Thus, there is an alternative logical notion of “existence” that is always unrestricted: using that notion, it is true to say that Trump “exists” even in worlds in which neither he (nor any counterpart of him) is a part.²⁶ There are logical notions of identity and truth-in according to which that Trump is identical with Trump, or that snow is purple in w, are true in some worlds and not in others. One can be a pluralist about logical notions without being a pluralist about logic. It is no different here than with propositional logic: the same logic can be formulated using different combinations of Boolean operators. There is one true logic, I say, and reality makes it so. There could only be a plurality of logics, on my understanding of logic, if there were a plurality of realities. And talk of a plurality of realities is plainly incoherent.²⁷ As a Humean, I reject all primitive modality, be it logical, natural, metaphysical, or epistemic. So far, I have dealt only with logical modality: I have analyzed it in terms of quantification over portions of reality. But even here there may seem to be a problem. How do I square this analysis with my realist account of logic, and my acceptance of logical notions such as consistency as fundamental? I said in the preamble that logic gives structure to reality. But the myth of logic impressing form on a featureless reality is just that: a myth. And my calling logic “productive” is not to be construed as bestowing on logic some mysterious metaphysical power. (Compare how the platonist about sets should understand the metaphors embodied in the iterative conception.) The way that reality is is not ²⁶ Indeed, it is true to say that necessarily everything is necessarily something, the thesis that Williamson () takes to define what he calls “necessitism.” Using the unrestricted notion of existence, as Williamson notes with respect to modal realism, necessitism is trivially true. I can’t object to that: ‘necessitism’ is Williamson’s coinage, and he can define the term as he wishes. But I can insist that we speak truly in ordinary contexts when we say that things exist contingently. For I am within my rights to interpret ‘exists’ in the way that best accords with our ordinary usage as it applies to reality as I take it to be. See Bricker (a). ²⁷ Not that that is a deterrent to philosophical invention. See, for example, the “fragmentalism” considered by Fine ().

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 :     () grounded in logic. On the contrary, the truths of logic are grounded in the way reality is. But that is compatible with also holding that logical notions are epistemically (or conceptually) fundamental, that is, fundamental at the level of our representations. I take a notion to be epistemically fundamental, roughly, if it belongs to our basic ideology for theorizing about reality. (For a bit more on this, see Chapter .) Logic is one of the pillars on which our access to reality depends. But the logical notions are not metaphysically fundamental. How to analyze the various other modalities is a long story that I can only touch on here. For the varieties of natural modality associated with laws, causation, dispositions, and objective chance, suffice it to say that they must all be reduced, one way or another, to the Humean mosaic. Note that the reduction of natural modality is independent of the acceptance of a plenitudinous reality. The laws, causes, and so forth in our world reduce to the Humean mosaic in our world. The Humean reduction of metaphysical modality proceeds along different lines. In my view, the notion of metaphysical modality as it is understood by most of my contemporaries is based on conventional and/or contextual features of our modal discourse, not on the nature of reality itself. This is most obviously true, I claim, with respect to metaphysical modality de re. That water is necessarily composed of hydrogen and oxygen or that I couldn’t have existed without my parents existing are true, today, when asserted in the philosophy room (owing to the influence of Kripke and others); but there are ordinary contexts where these claims are false, contexts that are in no way defective. Counterpart theory, of course, is the semantic tool that allows for such contextual variability. Perhaps there are conventional limits on this variability; perhaps, for example, there is no context where it can be truly asserted that I might have been a poached egg, or a prime number (though I doubt it, see Section .. for discussion). But in any case, what counts as an essential property of a thing will be highly language dependent. There are no deep metaphysical facts about the essences of things. Metaphysical modality de dicto, as commonly understood by philosophers, I take to be a conventional restriction of logical modality.²⁸ As I interpret their usage, it quantifies only over some of reality—the realm of “possible worlds”—while ignoring the rest of reality, including the realm of mathematical systems. And it in effect expands the domain of each possible world uniformly so that it includes the entire mathematical realm, thus making the truth value of mathematical statements invariant from world to world. Perhaps this expansion is plausible for the platonist who reduces all of mathematics to a single mathematical system, say, some version of set theory. But once one allows that reality is composed of a plurality of mathematical systems no less than a plurality of possible worlds, and that the truth value of mathematical propositions varies from system to system in a way analogous to the

²⁸ Sider (, ch. ) also takes metaphysical modality to involve convention, and not to carve reality at its joints. He is a fellow Humean in his rejection of all primitive modality. But there the similarities between our accounts end. Because he does not endorse a plenitudinous reality, he does not take the relevant conventions to provide, as I do, a restriction on the space of logical possibilities. Rather, he takes the relevant conventions to specify a list of types of truth that are to count as metaphysically necessary, one type being the logically necessary propositions.

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way the truth value of ordinary contingent propositions about the natural world vary from world to world, there is no reason not to allow for mathematical contingency. That every number has a square root is contingent, true in some mathematical systems and not others, just as that every house has a square window is contingent, true in some worlds and not others. Of course, we can say, if we choose, that the relational claim, that every number has a square root is true in the complex number system is a necessary truth; but we can say the same about that every house has a square window is true in w, where w is a world where it is true. Differences in how we have chosen to apply modal talk to mathematics often do not reflect genuine differences in reality itself.²⁹ I said there is no reason not to allow for mathematical contingency. But when asked whether mathematics is contingent or necessary, I hem and haw and say “it depends what you mean by ‘mathematics’.” When focusing on the content of mathematical statements, I find it natural to say that those statements are contingent, true at some systems but not others. For example, ‘+=’ is true in mod  arithmetic, but not in Peano arithmetic. But when focusing on our a priori knowledge of mathematical truths, I find it more natural to take those truths to generalize over mathematical systems, thus making them necessarily true—indeed, making them truths of logic (in my broad sense). The statement ‘+=’, made in the context of doing mod  arithmetic, would be interpreted to say that every system satisfying the axioms of mod  arithmetic has the relevant feature, and there are such systems. Taking mathematics to be a body of statements interpreted in the former way makes mathematics contingent; taking mathematics to be a body of statements interpreted in the latter way makes mathematics necessary. As I said, “it depends what you mean by ‘mathematics’.”³⁰ Logical modality, I have said, is the broadest alethic modality. But it is also, as I see it, a form of epistemic modality. For consider an ideally rational thinker. The logically necessary propositions are the sum total of what such a thinker knows a priori in virtue of being ideally rational. That is not a claim that can be argued: it is constitutive of ‘ideally rational thinker’ that it be so. It is then straightforward to analyze more restrictive epistemic modalities, modalities that apply to particular thinkers with varying amounts of a posteriori knowledge, as restrictions on logical modality. A posteriori knowledge serves to rule out some portion of reality. What the thinker ²⁹ There is however at least one genuine difference between the mathematical and the modal realms that will affect our modal discourse. Because mathematical systems lack qualitative content, all counterpart relations underlying modality de re will be based on similarity of structure in the mathematical case. A second ostensible difference is this. The distinction between being actual and being merely possible appears to apply only to the modal realm, not the mathematical realm (except in the trivial semantic way in which the entire mathematical realm can be taken to be “actual by courtesy”; see Lewis a: –.) If that is so, there is no notion of truth simpliciter for mathematical statements: all mathematical truth is system relative. For scientific statements, on the other hand, we can ask not only whether they are true at this or that possible world, but whether they are true simpliciter, true in actuality. (But see Section . where I question whether this second difference is genuine.) ³⁰ This sort of bifurcation in how to understand mathematical statements is familiar, of course, and has been introduced, with variations, to serve different philosophical ends: nominalism, fictionalism, logicism, and structuralism.

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 :     () knows is that she is located somewhere within the portion of reality that hasn’t been ruled out. Because the knowledge gained from experience is de se, the portion of reality that remains as a possibility for the thinker need not be an isolated portion of reality. Her knowledge, then, is best captured by taking its objects to be properties rather than propositions. (See Lewis .) In any case, what is epistemically possible for such a thinker is a restriction of what is logically possible, and so easily analyzable as such. All this is standard, of course, and goes through smoothly as long as one holds to the assumption that the thinker is ideally rational and knows the laws of logic. But what if we drop that assumption, and consider instead what is compatible with the knowledge of a less than ideally rational thinker? Consider some such non-ideal thinker that fails to know a law of logic. It seems that there are ways for reality to be according to the epistemic state of that thinker that are not ways that reality could possibly be. To capture the content of the thinker’s thought, it seems, we need to expand reality to include portions where the logical law in question is false, logically impossible portions of reality. And thus a popular response to the problem of the less than ideally rational thinker has been to introduce impossible worlds (and even impossible mathematical systems) so as to capture the content of such a thinker’s thought. I do not recommend that response. It’s not just that I find “concrete” impossible worlds and objects incoherent. Even if one instead takes impossible worlds to be abstract constructions of some sort, they do not provide a stable solution to the problem at hand. For suppose we add “impossible worlds” where some law of logic is false to capture the epistemic state of our non-ideal thinker. Those impossible worlds, like all portions of reality, could be represented by a thinker in multiple ways. An ideally rational thinker would know that these representations are equivalent and have the same content. But a non-ideal thinker need not know this. And so those impossible worlds will not be able to serve as the content, relative to that thinker, of the different representations. To capture the content of the thinker’s thought, we would need to expand reality once again. And we would be embarking on an infinite regress. That regress will be vicious if one holds, as I do, that reality is definite, not indefinitely extensible. How bad would it be to stonewall, and simply refuse to assign content to the thought of a less than ideally rational thinker? Here my broad notion of logic exacerbates the problem. I really don’t much care if there is no way to capture the content of thought of someone who claims to reject the law of non-contradiction. But I include principles of plenitude, of truthmaking, and of mereology as part of logic, indeed, all the principles of metaphysics that I take to be a priori. I do not want to say I cannot understand philosophers when they deny these principles. For I am able to accurately predict what other principles they hold or deny. I can engage them in critical discussion. I don’t treat what they say as gibberish, or in a purely syntactic way. There must be some way, in virtue of my understanding these denials, that I attribute genuine content to them, and not just the content of a logical contradiction. Consider, for example, the thesis of unrestricted composition: for any things, there exists a fusion of those things. I take this to be a law of logic, absolutely necessary. When some philosopher asserts its denial, I can only suppose that they do not mean what I do by ‘fusion’. For if I take them to mean what I mean, what they assert is

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logically incoherent, equivalent to the contradictory proposition. (See Chapter  for my take on unrestricted composition.) But that is not the end of the matter: it does not mean that I cannot make any sense of what they say. I can make (some) sense of what they say by finding an interpretation of their words into my own words that allows me to engage with their thought, for example, by allowing me to successfully predict what else they will say. I may know full well that this interpretation does not capture all that they mean. But I choose the interpretation that makes the totality of what they say as coherent, by my lights, as possible. In the case of the denial of unrestricted composition, I will interpret them to be speaking about some portion of reality whose structure matches the structure that they take to hold of the parthood relation. Thus, I interpret their words ‘is a part of ’ not as the parthood relation, but as a structurally similar relation that is instantiated in some limited portion of reality. (This assumes a strong principle of plenitude for structures; see Chapter  and its postscript.) I then can predict what mereological claims they will endorse or deny by asking what is true of that portion of reality. Thus, I interpret their mereological claims to be contingent truths (whether or not they agree that mereology is contingent). And in this way, I am able to “make sense” of what they say. Sometimes philosophers argue as follows. Surely, I can make sense of what my opponent says when she denies unrestricted composition. But then I must allow that the denial of unrestricted composition is at least coherent, and has non-trivial content. Moreover, because I take a proposition to have non-trivial content just when it is true of some portions of reality and not others, I must allow that the denial of unrestricted composition is contingent.³¹ But I hope that what I said above makes plain that this argument conflates the proposition that I express by the denial of unrestricted composition and the proposition that I interpret my opponent as expressing when she appears to assert that denial. Let S be a sentence in my language and call the sentence not-S the linguistic denial of S. Let E be the function from my sentences to the propositions that those sentences express in my language, and let I be the function from your sentences to the propositions that give my interpretation of your sentences. For any proposition p, let ~p be the propositional denial of p. Then it may well be that I(not-S) 6¼ ~E(S). And that is how I can make sense of your linguistic denial of what I say even when the propositional denial of what I say is logically incoherent, and lacks any non-trivial content. I conclude that the consideration of less than ideally rational thinkers does not give good reason to hold that there are epistemically possible propositions that are not logically possible.

. Qualitative Character Shape, color, mass, charge, pain, pleasure: all these contribute to the qualitative character of things. Qualitative character, as I understand it, supervenes on the distribution of fundamental properties and relations: two portions of reality have the same (intrinsic) qualitative character—are qualitative duplicates—just in case ³¹ Cameron () seems to endorse this line of thought when he claims that, since the Composition as Identity theorist can “make sense” of things not composing, she should accept that it is conceptually possible that things not compose.

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 :     () there is a one-one correspondence between their parts that preserves all fundamental properties and relations.³² But what this amounts to won’t be clear until more is said about the kinds of fundamental properties and relations, and how they differ. After the first stage of our mythical creation, reality was all structure. There was the structure induced by logical relations such as is-identical-to, is-a-fusion-of, and is-one-of. And there was the structure induced by the fundamental relations that unite portions of reality into mathematical systems. Call these relations mathematical.³³ The mathematical relations are purely structural in this sense: their nature is determined entirely by their pattern of instantiation. Distinct but isomorphic mathematical systems (if any there be—see Chapter ) are each unified by the same mathematical relations. Non-isomorphic systems are each unified by different relations. For example, the “successor” relation that unites the natural numbers and the “successor” relation that unites the integers are the same in name only. Now, because after the first stage of creation the fundamental properties and relations were all logical or mathematical, the qualitative character of things at that stage was determined by logic and structure alone. The simples that composed the mathematical systems had no qualitative character beyond their bare simplicity. Could that be all there is to qualitative character? I think not. Something more is needed, I think, to get the colors and masses, a second stage of creation. Something must be added to reality in virtue of which things acquire properties, and perhaps relations, that do not derive from structure alone; content must be added to structure. These newly acquired, non-structural fundamental properties have “suchnesses” or quiddities. They have substantial intrinsic natures in virtue of which they may be cross-identified between distinct portions of reality. The qualitative character of a portion of reality thus has three separate components, arising from three different kinds of fundamental properties and relations: logical, mathematical, and quiddistic. The qualitative properties and relations, then, are just those that supervene on the distribution of these three kinds of fundamental property and relation.³⁴ Not all properties are qualitative. The non-qualitative properties derive from two different sources. First, there are “thisnesses” or haecceities: properties of being identical with some given entity or thing. That these properties are sometimes non-qualitative follows from my rejection of (non-trivial) principles of the identity of indiscernibles; indeed, I hold that every part of a system or world has somewhere in reality a distinct qualitative duplicate. Moreover, since distinct portions of reality have no things in common, no haecceity instantiated in one is instantiated in the ³² A complication: because I allow that fundamental properties may apply to pluralities (see below), the one-one correspondence has to be extended in the natural way to apply to all subpluralities of the two portions of reality. See Chapter . ³³ Mathematical systems may also involve fundamental properties, functions, and constants. My use of “mathematical relation” is meant to encompass all these, but whether it is by way of reduction or terminological stipulation I need not decide here. Nor need I decide here whether there are mathematical systems with properties only, and no unifying relations. Such a system would be unified by a fundamental plural property that applies to the elements of the system taken together. ³⁴ On a narrower notion of “qualitative,” only the quiddistic properties are qualitative, not the logical or mathematical properties. I stick with the broader notion of “qualitative” from here on out, and use ‘quiddistic’ for the narrower notion. But I often use the narrower notion of “qualitative” in other chapters of this volume (and earlier in this introductory chapter). Context should decide.

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other. There is no cross-identification in virtue of haecceities.³⁵ Second, there is the property of actuality. That actuality is non-qualitative follows from my belief that actual things have qualitative duplicates that are merely possible.³⁶ This property only arises at the third stage of creation; see Section . below. Because I take the qualitative to supervene on the fundamental, and haecceities and actuality to be non-qualitative, it follows that I do not take haecceities and actuality to be fundamental properties. That is terminologically awkward, perhaps, especially with respect to actuality since I say also that actuality marks a fundamental ontological distinction, a distinction of kind. One could instead take haecceities and actuality to be a fourth and fifth kind of fundamental property, and define the qualitative to supervene on just the first three kinds.³⁷ But the terminology I have here introduced best coheres with how I speak in other chapters of this volume. It is enough to bring the awkwardness to the surface. Quidditism, as I understand it, is the view that structurally indiscernible portions of reality can differ qualitatively. In that case, I will say that there is a quiddistic difference between those portions. Structurally indiscernible portions of reality agree on all their logical and mathematical properties and relations. And they agree on the pattern of instantiation of fundamental properties and relations with quiddities (if any), what I call instantial structure. If they differ qualitatively, they differ with respect to the distribution of fundamental properties or relations with quiddities. They differ either because some of the quiddistic properties or relations instantiated in one of the portions of reality have been switched in the other portion of reality, or because some have been replaced in the other portion of reality by quiddistic properties or relations alien to the first portion of reality, or because of some combination of switching and replacing. Note that we can speak of these properties having been switched or replaced because they have substantial intrinsic natures, or quiddities, in virtue of which they have been cross-identified.³⁸ I have presupposed quidditism throughout my writing on modality. (See especially Chapter , where I use quidditism together with the possibility of alien fundamental properties to argue against (what Lewis later called) linguistic ersatzism.) But why be a quidditist? Why not hold instead that the qualitative character of composite things

³⁵ If a haecceity is taken more broadly to be a property had by a thing and all of its counterparts, then things in distinct portions of reality can share haecceities, and qua counterparts be “cross-identified.” Whether haecceities in this broad sense are qualitative then reduces to whether the counterpart relation is qualitative. For some discussion, see Section .. Note, however, that haecceities in this broad sense won’t be able to play the traditional role of “thissnesses” if the counterpart relation is not an equivalence relation. ³⁶ See Chapter . But note that, on what I there call the “transformation account” of absolute actuality, the property of actuality would be trivially qualitative under the hypothesis of universal actualization. ³⁷ Cowling () defends the supervenience account of qualitative character that I have relied on in many of the following chapters. He also argues that some non-qualitative properties should be deemed fundamental. ³⁸ Lewis () uses switching and replacing to illustrate quiddistic differences. But note that Lewis (and many others engaging in this debate) have a stronger notion of structural indiscernibility in mind, and that potentially makes quiddistic differences harder to come by. On this stronger notion, causal and nomological structure, whether or not fundamental, is relevant to whether portions of reality are structurally indiscernible, not just logical, mathematical, and instantial structure. I also endorse quidditism in Lewis’s sense.

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 :     () emerges as structure becomes sufficiently complex? In that case, no superaddition of content is needed to get a world with mass or color or pain. There are many ways to argue for quidditism that I endorse. But some of them rest heavily on modal intuitions as to the individuation of possibilities, intuitions that an opponent might simply reject.³⁹ Here I sketch two sorts of argument that are less dependent on brute modal intuition. According to the first, support for quidditism comes directly from the phenomenology of our experience, for example, our experience of color qualia. Consider the visual experience of seeing a solitary blue square. Now consider the experience that results from switching the color to red while holding the shape fixed. This switch in color doesn’t alter the internal structure of the visual impression, or its relations to anything external in the visual experience. I claim that experiences of this sort give us a clear and distinct idea of how portions of reality can differ qualitatively without differing in structure. There are two steps to the argument. First, the differing phenomenal content of these two experiences is given by two different portions of reality. This follows from my more general view that all contentful thought and experience is relational (see Section .) together with the law of plenitude discussed above. Second, the portions of reality that give the phenomenal content have no structure beyond what is apparent in the visual experience. Phenomenal content, when basic, is entirely manifest.⁴⁰ Given these admittedly controversial assumptions, quidditism follows as a thesis about reality as a whole. That is, somewhere in reality there are portions that are structurally, but not qualitatively, indiscernible. But, of course, the argument does nothing to support that quidditism holds as a thesis restricted to our neighborhood of reality, the sphere of worlds that have the same fundamental properties as our world. Indeed, I incline towards materialism. I therefore doubt that color qualia are fundamental in our world. A second way of arguing for quidditism bases quiddistic differences, not on our experience, but on our scientific theorizing. First off, it is natural to take our scientific theorizing to give insight into what structures are possible. (See Chapter  and the principle (BS).) According to classical physical theories, the structure of a classical world is given by the pattern of instantiation of fundamental properties instantiated by point particles occupying a uniform spacetime. Is that all the fundamental structure in these worlds (excepting logical structure)? As a Humean, I say “yes.” In particular, I reject fundamental causal or nomological structure. But the argument that follows does not depend on that. Second, it is natural to take our scientific theorizing to give insight into what fundamental properties are possible and how they can be arranged. According to (a simplified) classical physical theory, the fundamental (non-kinematical) properties are the determinates of mass and charge, and any assignment of mass and charge over finitely many point particles in any spatiotemporal arrangement in accordance with Newton’s law of gravitation and Coulomb’s

³⁹ Moreover, intuitions favoring quiddistic differences can be accommodated by a “cheap substitute” for quidditism that parallels Lewis’s (a: –) “cheap substitute” for haecceistism. Hawthorne () recommends this strategy to the causal structuralist. ⁴⁰ Lewis is a quidditist. But he would reject this argument for quidditism at its second step. It rests on something akin to what Lewis calls the Identification Thesis: anyone acquainted with a quale knows just which property it is. See Lewis (: ).

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law represents a possible world. Now consider the following two simple worlds. Each world has two point particles, and the masses of the particles are the same between the two worlds. The charges of the particles, however, differ between the two worlds, but in such a way that the trajectories of the two point particles do not differ between the worlds. At the first world, the two particles revolve around one another as determined by their masses and charges. At the second world, the two particles follow the exact same trajectories as in the first world, but the forces between them depend on a different combination of charges.⁴¹ These two worlds are structurally indiscernible: they both have the same pattern of instantiation of fundamental properties over the same uniform spacetime. But they differ qualitatively because the fundamental properties at one world are different from the fundamental properties at the other. Classical physics, at least as standardly interpreted, supports quidditism.⁴² Again, as with the argument from experience, we can conclude that quidditism holds as a thesis about reality as a whole, but not that it holds when restricted to our neighborhood of reality. After all, we no longer believe that there is a uniform spacetime whose structure is independent of the configuration of mass and charge; so it may be that the different configurations correspond to different spatiotemporal structures. That would indeed be the case if geometrodynamics were true, the view summed up by the physicist John Wheeler as follows: “There is nothing in the world except empty curved space. Matter, charge, electromagnetism, and other fields are only manifestations of the bending of space. Physics is geometry.” (Wheeler : .) That program was soon abandoned even by its most ardent proponents. But what grounds are there for thinking that no mathematical structure, not even one more complex than the spatiotemporal structure posited by Einstein’s General Relativity, could deliver the goods? If there were such a structure, we could conclude: Physics is mathematics.⁴³ David Lewis once wrote: “I cannot believe (though I know not why not) that our world is a purely mathematical entity.” (Lewis : .) Few would dispute this sentiment, or bother asking why we hold it. But all of our beliefs, I think, call out for justification. Whether this belief can be justified depends, I think, on what we take the mathematical entities to be. In Chapter , I give an account of how we could know that we are individuals and not sets (if, contrary to what I think, sets exist and belong to a different fundamental ontological category than individuals). Roughly, I take the concepts of individual and set to have indexical components: it is part of the meaning of ‘individual’ and ‘set’ that I am an individual and not a set. But that account (and the similar indexical account I give of how we can know that we are actual) rests on ⁴¹ Let the charges of the particles in the first world be c₁ and c₂ and the charges in the second world be c₃ and c₄. Then it suffices (in our simplified theory that ignores magnetic forces) to set c₃  c₄ = c₁  c₂. ⁴² In Chapter , I argue that a Humean should take determinables, not determinates, to be fundamental; determinates arise from the instantiation of determinables in an enhanced world structure. On that view, the simple worlds described do not support quidditism: the worlds differ in instantial structure. The argument from scientific theorizing can still be made, but it requires a more complex theory. In Chapter , I argue, based more on hypothetical than actual scientific theorizing, that there are quiddistic differences between worlds with different kinds of homogeneous matter. I would classify that argument as an argument from modal intuition, not an argument from scientific theorizing. ⁴³ The cosmologist Max Tegmark believes that our universe is part of a multiverse that ultimately reduces to pure mathematics. According to his Mathematical Universe Hypothesis, “our external physical reality is a mathematical structure” (Tegmark : ).

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 :     () the assumption that individuals and sets (or actualia and mere possibilia) belong to different fundamental ontological categories. I don’t think, however, that mathematical and physical entities belong to different ontological categories in a way that would allow for an indexical solution. (See Section . below on how I understand ‘ontological category’.) Is there some other way to justify my belief that I am not a mathematical entity, that I have a quiddistic nature? A conception of reality different from the one I have been touting might be in a better position to justify the belief that physics does not reduce to mathematics. I have supposed thus far that reality divides into separate realms: the purely structural or mathematical, and the quiddistic realm. (There is also a “mixed” realm, by recombination; see below.) And that opened up the troubling question: which realm do I inhabit? Suppose instead we take all of reality to be infused with quiddities. Mathematical entities, then, are not portions of reality, but something abstracted from those portions; not systems, but structures (and constituents of structures). And since mathematical entities are now universals, they belong to a different ontological category then the physical portions of reality, and an indexical solution to how I know I am not a mathematical entity can be applied. But I reject this alternative conception of reality, not just on the mathematics side (owing to my rejection of structural universals), but also (more tentatively) on the physics side. For, on this alternative conception, it is natural (though not inevitable) to take the fundamental relations, not just the fundamental properties, to have quiddities. For example, structurally indiscernible worlds can differ qualitatively merely because their fundamental spatiotemporal relations have different quiddities. And that raises the important question: are there relational quiddities? It might seem that the arguments given above for quiddistic properties can be used to support quiddistic relations as well. But I am doubtful. The argument from scientific theorizing is less convincing because, at least with respect to classical theories, the non-spatiotemporal fundamental properties and relations can all be reduced to fundamental scalar and vector quantities that hold at points of spacetime; and those quantities, I argue in Chapter , can be construed as properties instantiated within an enhanced world structure. There is no need to posit fundamental relations that are not purely structural.⁴⁴ The argument from perceptual experience may seem more promising. Indeed, Frege (, section ) famously argued that the principle of duality in projective geometry supported the Kantian view that space has intuitive content that goes beyond what can be captured by the axioms and theorems. This argument could be adapted to support quiddistic relations as follows. Call the relation that holds between a point and a line that the point lies on the incidence relation. Suppose we take incidence to be a fundamental geometric relation. In that case, we do not need to take the property of being a point or of being a line to be fundamental geometric properties. We can define a point as anything that bears the incidence relation to something and a line as anything to which the incidence relation is borne by something. Now, according to duality, for any configuration of points and lines, there is a dual configuration that results from switching the incidence relation for its ⁴⁴ I believe this is also true of the fundamental non-spatiotemporal relations that may be needed for quantum mechanics. But this is not a topic that I am prepared to pronounce on.

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converse and thereby switching points and lines.⁴⁵ The resulting configuration is structurally isomorphic to the original configuration. But the two configurations look completely different from one another! For example, a single line containing many points becomes a single point through which many lines pass. So, again making the controversial assumptions needed by the argument from perceptual experience, there are structurally isomorphic portions of reality that differ quiddistically. The incidence relation that is instantiated in the portions of reality that give content to our visual experience of points and lines is a quiddistic relation. I do not think, however, that this argument for quiddistic relations has merit. The incidence relation in geometry is a mereological relation, the relation of being a (simple) part of. Portions of reality are structurally isomorphic only if there is a oneone mapping between them that preserves logical relations, which include (I say) the parthood relation, and so maps points to points and lines to lines. When we switch points with lines and the incidence relation with its converse, we do not thereby move to a part of reality that is structurally isomorphic; and so the differing content of the switched experience does not require the positing of quiddities. I conclude that this Frege-inspired argument for relational quiddities does not succeed. Indeed, I know of no argument for relational quiddities that I find convincing.⁴⁶ Are there arguments against? To answer this we need to know more about the nature of quiddities and their ultimate grounds. Different accounts of quiddities may differ as to whether to allow for relational quiddities. What grounds the third component of qualitative character? When portions of reality differ quiddistically, what grounds the difference? Don’t say: the quiddistic properties. I have been using ‘property’ in an abundant sense. The quiddistic properties are not themselves parts of reality; they belong rather to the realm of representation. The question is: when a thing instantiates a quiddistic property, what in reality makes it so? The nominalist answers: just the thing itself. Nothing but things are needed to serve as truthmakers for all the fundamental truths. Moreover, distinct fundamental truths may have a single truthmaker; for example, that a is P and that a is Q, for fundamental properties P and Q, are both made true by the thing a. The truthmakers are not very discriminating. (See Section . and Section . for an account of things as truthmakers.) The realist answers: the instantiation of a quiddistic property is grounded in some part or constituent of the thing, a universal or trope. It is then natural to say that it is the universal or the ⁴⁵ One needs the full projective plane, with its points at infinity, to have a full duality of theorems of geometry: that every theorem has a dual theorem gotten by replacing ‘point’ with ‘line’ and ‘lie on’ with ‘contains’. But the argument from experience need not assume that the geometry of our visual experience is projective. It need only draw on the duality of configurations, that for every configuration of points and lines, there is a dual configuration gotten by switching points with lines and the relation of lying on with its converse. For, presumably, our perception of a finite part of space does not depend on what goes on at infinity. ⁴⁶ When I presented this material at MIT, Stephen Yablo suggested that perhaps a stronger argument from perceptual experience could be based on our perception of temporal order. Perhaps. Consider an experience involving a blue flash followed immediately by a red flash. The experience that instead has a red flash followed by a blue flash differs in phenomenal content from the first. But here the difference can be attributed to a switching of quiddistic properties. In any case, how to understand the content of our experience of temporal order is not a simple matter.

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 :     () trope that has a quiddity, a substantial intrinsic nature; the (abundant) properties have quiddities, when they do, in a derived sense. Throughout most of the chapters included in this volume, I have tried to remain neutral between nominalist and realist accounts of qualitative character. But in fact I strongly favor a realist theory with tropes. The basic tenets of trope theory are well known, and I will be brief.⁴⁷ Tropes are particulars, which is just to say that they have qualitative duplicates. Tropes are mereologically simple. Tropes have non-qualitative haecceities, which is just to say that, as with all simple particulars, their identity and distinctness is primitive and not analyzable. And tropes have quiddities in virtue of which distinct tropes may or may not be qualitative duplicates of one another. Let me illustrate with respect to a possible world whose structure is spatiotemporal, and whose (quiddistic) fundamental properties apply only to point-sized things. According to the trope theory I endorse, this world, and all the things that exist in this world, are composed entirely of tropes. Because tropes are particulars, the trope theorist will agree with the nominalist that whatever exists in the world is particular. But unlike the nominalist the trope theorist holds that point-sized things may be composite, having tropes as non-spatiotemporal parts. When these point-sized things instantiate the same fundamental property, it is in virtue of having parts that are duplicate tropes. The (quiddistic) fundamental properties correspond one-one with maximal sums of duplicate tropes, summing not just across this world but across all of reality. Questions remain. First, how do the tropes occupy places in a structure? There are two options. Return to our myth. After the first stage of creation, there were systems exhibiting pure structure. Each place in the structure was occupied by an object having only logical and purely structural character. Call such objects bare particulars.⁴⁸ Now, at the second stage of creation, the stage at which reality takes on quiddistic character, should we think of the tropes as replacing the bare particulars that make up mathematical systems so as to create fully qualitative systems, including the possible worlds? Or should we instead think of the tropes as being added to the bare particulars of mathematical systems, so that the qualitative systems have a dualism of bare particulars co-located with tropes? In that case, contrary to what was said above, the spatiotemporal world would not be composed only of tropes; it would be composed of tropes together with points of spacetime. When I spoke in the preamble of “hurling” the tropes across some of reality, that suggested the second view. But in fact I find the first, monistic view more credible. (Requiring that tropes always coexist with bare particulars would violate a reasonable Humean principle of recombination; see Chapter .) That leaves us with the following picture of reality after the second stage. First, there is still a mathematical realm, a realm of pure ⁴⁷ Loci classici for trope theories include Williams () and Campbell (). But there have been many defenders of tropes, from ancient times to the present. ⁴⁸ Use of the term ‘bare particular’ is sometimes restricted to substance-attribute metaphysics but I use it broadly to refer to particulars none of whose parts instantiate a quiddistic property. Trope theorists may have need for bare particulars in their ontology even if they are not wanted to serve as substances that tropes instantiate. They may be needed to serve as mathematical objects if the trope theorist endorses a structuralist version of platonism (as I do); or to serve as spacetime points if the trope theorist is a spacetime substantivalist. Bare particulars have sometimes gotten a bad rap in contemporary metaphysics. For some arguments in their defense, see Sider ().

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structure left untouched by the hurling of tropes. In this realm, systems are composed entirely of bare particulars with one or more bare particulars occupying each of the places in the structure of that system. Bare particulars, I surmise, are all simples— mereological atoms—or sums of simples; and simple bare particulars are all qualitative duplicates of one another. (See the postscript to Chapter  on why my (tentative) rejection of gunk does not offend against principles of plenitude.) Second, there is now a quiddistic realm, a realm of structure and content. The unified and isolated portions of reality in this realm—the “islands of reality”—are composed entirely of tropes, with one or more tropes occupying each of the places in the structure of that island. There are no bare particulars in the quiddistic realm. I would have liked to say that every island of reality in the quiddistic realm is a possible world; but such a broad notion of “possible world” would gratuitously conflict with my fellow metaphysicians (see Section .). I therefore say that only those islands of the quiddistic realm that are sufficiently world-like are called “possible worlds.” Finally, there will be a mixed realm where the hurled tropes occupy some but not all of the places in the structure of an island; in this mixed realm, tropes and bare particulars reside side by side. The existence of a mixed realm will follow from any reasonable principle of recombination; again, see Chapter . At first blush, the islands of this mixed realm might seem to be mere oddities. But if physical theories are best interpreted as positing a structure with places unoccupied by tropes—as I argue in Chapter  may be the case on a Humean account of quantities—then this mixed realm may well be the realm we inhabit. For purposes of this introduction, however, I set the mixed realm aside and focus on portions of reality composed entirely of bare particulars or entirely of tropes. Another question is this. If reality is composed entirely of tropes and bare particulars, what happened to the things that, on my first pass as to what reality is, I took to instantiate the fundamental properties and relations? In the mathematical realm, the things can be identified with bare particulars, and the sums of bare particulars, belonging to a single system. In the quiddistic realm, the things can be identified with maximal sums of tropes occupying a single place in a structure, and the sums of such maximal sums belonging to a single “world.” (For more on how I understand the co-location of tropes, see Section ..) That identification will not allow distinct point-sized things to occupy the same place in a spatiotemporal world. Causal relations, however, may be called on to get a finer individuation of things. I here set this finer grained notion of thing to one side: it belongs more to contingent physics than to metaphysics. We are now in a position to correct our first pass at what reality is by saying that reality consists of bare particulars and tropes (and sums of such) instantiating fundamental properties and relations. The pattern of instantiation of fundamental properties and relations determines what fundamental structures are exhibited throughout reality. So we can also say that reality consists of bare particulars and tropes occupying places in fundamental structures.⁴⁹ ⁴⁹ Whether these two characterizations of reality are equivalent will depend on subtle issues having to do with the individuation of structures. In this introductory sketch, I have set those issues aside and used these two modes of speaking—fundamental-properties-and-relations talk vs. fundamental-structures talk—interchangeably.

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 :     () Another question pertains to the extent and variety of the quiddistic realm. According to the myth in the preamble, the tropes were hurled across portions of reality in every combination. It is the business of Chapter  to make this more precise, to develop and defend particular Humean principles of plenitude. In rough outline the picture is this. First, the laws of logic, spearheaded by the law of plenitude, provide a principle of plenitude for structures: they determine what the logically possible structures are, that is, what structures are instantiated throughout reality.⁵⁰ Then, we Humeans endorse the following principle of recombination: for any logically possible structure, and any way of arranging tropes within that structure, there is a portion of reality that arranges duplicates of those tropes in that way. This too follows from the law of plenitude on a Humean account of possibility.⁵¹ Finally, to get the full range of Humean possibilities, we need a principle that guarantees an inexhaustible supply of qualitatively distinct tropes, what I call a principle of plenitude for contents: any trope in any island of reality can be replaced, without altering the structure, by an alien trope, by a trope that has no duplicates in that island. (See the Principle of Interchangeable Parts, and the principle (B) in Chapter .) Without such a principle, there would be a kind of necessary connection between distinct tropes. With these three sorts of principle of plenitude—plenitude of structures, of contents, and of recombination—the extent and variety of the quiddistic realm is fully settled. I can now, finally, return to the question: are there relational quiddities? Given the trope-theoretic account of quiddities adumbrated above, that reduces to the question: are there relational tropes? For example, when a fundamental relation holds between two point-sized things, is that ever made true by a relational trope that somehow spans the location of those two things? Or instead are the fundamental relations that hold in a physical world—for example, the spatiotemporal relations—no different from the purely structural relations that hold between points of a mathematical spacetime? For ease of exposition, let me restrict attention to fundamental dyadic relations. Prima facie, there are two cases to consider: fundamental symmetric relations and fundamental non-symmetric relations. The non-symmetric case raises problems for relational tropes that go beyond the problems raised by the symmetric case.⁵² Here I focus on the symmetric case which raises problems enough. The question, then, is whether relational tropes are coherent given a Humean conception of possibility. The first thing to note is that relational tropes, if there are ⁵⁰ In Chapter , I develop more narrowly a principle of plenitude for metaphysically possible structures, that is, structures instantiated within some “possible world.” Not all logically possible structures are metaphysically possible in this sense. ⁵¹ In Chapter , following Lewis, I applied the principle of recombination to things (what I there call “individuals”), rather than to tropes. But only a minor adjustment to the principle is required: in an arrangement of tropes within a structure, allow multiple tropes to be located in the same place. Whether distinct duplicate tropes, however, are ever co-located is a delicate problem. For more on how co-location can be accommodated within an account of plenitude, see Section .. ⁵² See Fine () for some of the problems with positing worldly non-symmetric relations, and possible responses.

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such, are extended simples. A dyadic trope occupies two places, and is thus extended, without having parts that separately occupy those places. For if it did divide into two parts—one in one place, one in the other—the relationality would be lost. Relational tropes, then, violate a general principle of mereological harmony according to which the mereological structure of an occupant of spacetime matches the mereological structure of the places that it occupies.⁵³ Moreover, if things are just sums of tropes, mereological harmony for things must be abandoned as well whenever things have relational tropes among their parts. That might make trouble for perdurantism, the view that things persist in virtue of dividing into temporal parts: perdurantism will have to be restricted to worlds where things are not composed in part of relational tropes that span distinct times. And then it will have to be argued on contingent grounds that the things of our world divide into temporal parts, which may be none too easy. But these consequences of admitting relational tropes, however unfortunate, do not amount to incoherence. Extended simples are sometimes thought to be incoherent in virtue of their having primitive distributional properties.⁵⁴ For example, an extended simple could be red and blue striped without having parts that are red and blue. Indeed, I am doubtful that primitive distributional properties are coherent, but we can set that issue aside here; for there is no pressure to deny that relational tropes are homogeneous. When an extended thing composed of relational tropes has a distributional property, that reduces to a matter of the co-location of various monadic tropes with the relational tropes. No incoherence there. But there is a problem for the stout Humean. Relations require relata.⁵⁵ That is an analytic truth if ever there was one. But then grounding fundamental relations in relational tropes will introduce unacceptable necessary connections. It will be impossible to have a relational trope existing in isolation, all by itself. Whenever a relational trope exists, there would have to be monadic tropes co-located with the relational trope to serve as the relata. Note that this problem is specific to relations and does not afflict properties. It is also an analytic truth, I say, that properties require bearers; but that is no obstacle to a property trope existing in isolation all by itself. On a “bundle theory” according to which the bearers of properties are sums of tropes, a lonely trope, in effect, does double duty: it grounds a fundamental property and is also that property’s bearer.⁵⁶ Now, it may seem that the Humean has a way out of this problem: she can hold that the relationality of relational tropes is an extrinsic feature, that when a relational trope exists all by itself, its relationality is lost. But then how

⁵³ I endorsed mereological harmony in Chapter . But if one makes the amendment to the principle of recombination recommended above, allowing multiple entities to occupy the same place, mereological harmony is already lost. ⁵⁴ See Parsons (), however, for a defense of primitive distributional properties. ⁵⁵ This has been denied by “ontic structural realists” (see e.g. French and Ladyman ) who take structures to be primitive; but their inevitable talk of “places” in the structure, it seems to me, belie this denial. Note, however, that I do not suppose that the relata must have a substantial intrinsic nature; they may all be bare particulars. ⁵⁶ On this fundamental difference between relational tropes and property tropes, I am in agreement with Campbell (: ).

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 :     () does a relational trope differ from a property trope? If the distinction isn’t structural, which it can’t be given that tropes are simple, then it seems to be utterly mysterious. Thus, I do not think a Humean trope theorist should admit different categories of trope, relational tropes in addition to property tropes. It is worth noting, however, that property tropes can sometimes serve as substitutes for relational tropes; property tropes can ground the instantiation of symmetric relations. What I have in mind is this. There are worlds in which fundamental properties are instantiated by pluralities of things. (See Chapter ; when a fundamental property is instantiated by a plurality, it is emergent.) And when these properties have quiddities, their instantiation is grounded in the existence of extended tropes. Consider for example a world where property dualism is true, where fundamental mental properties, such as experiencing a red quale, are instantiated by brains. I ground a given instantiation of this experiential property in a trope co-located with the brain, call it RED. RED is an extended simple in the world in question. But there is nothing in the nature of RED that requires that it be co-located with a plurality of things. Indeed, a duplicate of RED could exist all by itself, in which case RED would not be an extended simple. Being extended is an extrinsic feature of tropes. Now consider a putatively fundamental symmetric relation, R. R is associated with a unique plural property P: R holds between a and b if and only if P holds collectively of the plurality with members a and b. It would not be proper to call P a relation, or the tropes that ground instantiations of P relational tropes. But the instantiation of R is now grounded in the existence of property tropes, allowing the instantiation of R to make a quiddistic difference. Strictly speaking, however, it is the plural property P, not the relation R, that is fundamental. Note that there is no hope of using property tropes in this way to ground non-symmetric relations. So whereas the trope theorist has the option to take putatively fundamental symmetric relations to have quiddities, that option is closed for non-symmetric relations. In rejecting relational tropes, I made no mention of what is sometimes taken to be the main objection: Bradley’s relation regress.⁵⁷ According to the regress argument, if one posits a worldly relation, say a relational trope, in order to ground that two things stand in some fundamental relation, then one will be led also to posit a worldly relation to ground that the relation applies to those things, a relational trope of relation application. And so on, leading to a vicious infinity of grounds. But the regress can be curtailed at the start. It rests on the faulty assumption that if a relation is unanalyzable in the realm of representation—is a primitive notion in our ideology—then it must have a worldly correlate. But a relation may be fundamental to our thought without being fundamental to reality. (See the Chapter  for more on this distinction.) Identity, I claim, is not analyzable; but no relation of identity need be posited to ground facts of identity. The identity of a thing with itself is grounded in the existence of that very thing; the non-identity of two things is grounded in the existence of those two things. Similarly, the relation of instantiation that holds between a relation and what it relates is unanalyzable, but no worldly instantiation relation is needed to ground facts of instantiation: whatever grounds the fact that

⁵⁷ See MacBride () for discussion of this objection.

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a Rs b also grounds the fact that a and b instantiate R. Even the realist about relations should agree with the nominalist that some relations fundamental to our thought require no worldly correlates to serve as grounds or truthmakers. Everyone’s a little bit nominalist. But I am more than just a little bit nominalist. Not only do I hold that there are fundamental logical relations, such as identity and predication, that have no worldly correlates, no “relational” parts of reality that determine when they apply, I hold the same for all fundamental relations, and all fundamental structure, be it mathematical or physical. The structure of a portion of reality is fully grounded in the things—more generally, the tropes and bare particulars—that compose that portion of reality. Consider, for example, the natural numbers: countably many bare particulars united by the successor relation in accordance with Peano’s axioms. In reality, there are just the bare particulars. The successor relation, and all the properties, relations, and functions defined in terms of it, belong to the realm of representation, as do the truths of number theory. But those truths need nothing but that plurality of bare particulars to make them true. Other pluralities of bare particulars instantiate other mathematical structures, and make other mathematical theories true—the integers, the rationals. But in each case, there is nothing but the bare particulars to do the work of truthmaking, or grounding. I expect resistance to this nominalist mode of thinking. If both the natural numbers and the rationals are just countable pluralities of bare particulars, an opponent might ask, where does the difference in structure come from? Well, these particulars get together one way to make up the natural numbers, those particulars get together another way to make up the rational numbers. They need no help from worldly relations to get together in these different ways. To suppose that they do would, for no good reason, be to embark on Bradley’s regress. Banishing structures to the realm of representation helps solve a problem with the individuation of possibilities. If structures are immanent, there will be an artificial proliferation of portions of reality, distinctions without a difference. To take the simplest case: consider a portion of reality composed of two nonduplicate simples standing in a symmetric relation. This portion of reality instantiates a structure with two places. Now, there are two ways of arranging those simples within the structure: there is a second arrangement where the simples switch places. Then, by the principle of recombination (see the principle (LPR) in Chapter ), for each of these arrangements, there is a portion of reality where duplicates of the simples are arranged in that way. But do these two arrangements correspond to two distinct portions of reality, two distinct possibilities? The principle of recombination does not say; it says that both arrangements are possible, but not whether the possibilities are distinct. But if structures are immanent, the answer will be “yes.” (This assumes that the “places” in structures can be cross-identified between different instantiations; but I take that to be part of a realist account of structures.) It seems, however, that the answer should be “no.” For example, if a red and blue simple are ten meters apart (and there is no substantival space), one does not get a distinct possibility where the simples have switched places. Removing the structures from reality and placing them in the

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 :     () realm of representation allows us to say what we should: the two different arrangements are two different representations of the same portion of reality, not two different portions of reality.⁵⁸ Let me conclude this long section by returning to a question earlier left hanging: Might all of reality be infused with quiddities? In that case, there would be no separate mathematical realm. Instead mathematics would somehow be abstracted from or reduced to the quiddistic realm which would be all of reality. I do not think my rejection of relational tropes answers this question. After all, a nominalist about structure believes, no less than the realist, that there are truths about structure. No doubt in some sense one does not need a separate mathematical realm to ground the truths of mathematics. One can choose to reinterpret the theories of mathematics as generalizations over the quiddistic realm by ramsifying mathematical theories.⁵⁹ Given the plenitude of structures instantiated in the quiddistic realm, no mathematical truths would be lost. But why should one reinterpret mathematical theories if they are perfectly coherent as they are written, not as generalizations over possible worlds, but as statements about entities and systems that lack quiddities? The question isn’t whether we need the mathematical realm, but whether logic demands that it exist. And, as I argued in Section , the law of plenitude will demand it. I therefore cannot reject the mathematical realm. And that leaves me in the rather awkward position of having to say that I do not know how to justify my belief that I am not an inhabitant of the mathematical realm, an entity with structure but no quiddity. For, verbal tricks aside, nothing that I know a priori—much less a posteriori—rules that out. Only a dualist about phenomenal consciousness, it seems, could have good reason to deny (with Lewis) that “our world is a purely mathematical entity.”

. Actuality I am actual. So is the racehorse Secretariat. But Mr. Ed, the talking horse, is not; he is merely possible.⁶⁰ In what does this difference consist? According to Lewis, the difference in question is a matter of how these two horses relate to me: Secretariat is spatiotemporally related to me, and is thereby truly said by me to be “actual”; Mr. Ed is spatiotemporally isolated from me, and is thereby truly said by me to be “merely possible.” It is not a difference in ontological status. Secretariat and Mr. Ed are both flesh-and-blood horses, and both concrete; indeed, they belong to all the same fundamental ontological kinds. It is just that Secretariat is here, in my little ⁵⁸ Compare the critique of positionalism in Fine (). Fine’s solution to the problem allows relations to be immanent, but it is not a solution a Humean can accept. ⁵⁹ As Lewis (, section .) does for set theory. But note that Lewis’s reinterpretation does not take structure seriously, although he has the means to do so by quantifying over natural properties and relations. For discussion, see Chapter . ⁶⁰ I take fictional entities to be merely possible. But fictional names, such as ‘Mr. Ed’, do not uniquely refer. They have descriptive meanings that pick out a suitable class of counterparts, each member of which is indeterminately referred to by the fictional name. Statements about fictional entities are evaluated using supervaluations.

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portion of reality, and Mr. Ed is out there, in portions of reality disconnected from mine. So says Lewis. Like many others, I find Lewis’s relational account of actuality incoherent. (For more on this, see Chapters , , and .) Actuality, I claim, is categorial and absolute. The actual things comprise a fundamental ontological category. In virtue of belonging to that category, they have a different ontological status than merely possible things. And the ontological status conferred by the property of actuality is had absolutely: it does not depend on relations to me, or to anything else. Because I take actuality to be categorial and absolute, I cannot help but interpret Lewis as believing in a bloated actuality, chock-full of actual universes; for according to Lewis, all the worlds are ontologically on a par with the actual world. And it is because I take actuality to be categorial and absolute that I can safely and literally say, in agreement with ordinary language, that Mr. Ed is a talking horse. I don’t need to take this back in the philosophy room, saying instead that Mr. Ed is some abstract whatnot that magically represents something being a talking horse, in order to accept the philosophical truism: Mr. Ed is not ontologically on a par with me. Return one last time to my little creation myth. At the end of the second stage, reality is infused with all manner of qualitative character: protons, pigs, people in pain. But I am nowhere to be found. There are qualitative duplicates of me scattered throughout reality, indeed, qualitative duplicates of the entire cosmos of which I am a miniscule part. But nothing as yet has been made actual. A third stage of creation is needed, I claim, to confer that special ontological status in virtue of which actual things differ from their merely possible counterparts. I know a priori that at least one world has this special status: the world I inhabit. Perhaps other worlds have it as well, in which case the realm of the actual is composed of island universes, disconnected cosmoi (see Chapter ); perhaps I can even have indirect evidence of such. But I know nothing a priori as to how or whether actuality extends beyond the world I inhabit. For all I know a priori, actuality could infuse the entire quiddisitc realm, or even all of reality. The property of actuality is mysterious, to be sure. I differ in some fundamental way from my merely possible qualitative duplicates.⁶¹ But how? It can’t be that there is some background buzz that I hear and my duplicate does not, or that my experience of the world is more vivid or vivacious than his; for these are qualitative differences, and there are none between me and my duplicate. Nor can there be “actuality tropes” that ground the property of actuality by virtue of having some special quiddity; for, again, that would be to posit a qualitative difference where none exists. Attempting to give some defining characteristic of actuality is a futile endeavor. But that is only to be expected given that actuality is a fundamental ontological distinction that cuts across all other fundamental properties.

⁶¹ Is ‘qualitative’ redundant in ‘qualitative duplicate’? That depends on whether duplicates are characterized as “sharing all intrinsic qualitative properties” or “sharing all intrinsic properties.” On the latter characterization, since the property of actuality is naturally taken to be intrinsic, I have no merely possible duplicates. But in the chapters that follow, I sometimes invoke the former characterization and take ‘duplicate’ and ‘qualitative duplicate’ to be synonymous.

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 :     () Still, something must be said to motivate the need for both a second and third stage of creation, for a distinction between the quiddistic and the actual. Why not simply collapse the distinction, either by being Lewisians or “actualists”? Lewisians agree with me that the quiddistic realm is plenitudinous, but take actuality to have a much greater extent than we ordinarily take it to have (although Lewis, of course, would not describe his view this way). Actualists restrict the quiddistic realm to what we ordinarily take to be actual, and posit non-qualitative substitutes for my quiddistic realm. Endorsing either of these views would avoid having to posit an ideological primitive of absolute actuality. I could simply identify the actual with the quiddistic, calling them both “concrete.” I have no decisive argument that a separate ontological category of the actual must be posited. But there are two considerations I take to be weighty against Lewisians. The first is metaphysical. I take the supposition that actuality consists of island universes—spatiotemporally disconnected regions—to be coherent. Moreover, as I argue in Chapter , the possibility of island universes follows from plausible Humean principles of plenitude, indeed, principles that Lewis himself accepts. But on Lewis’s relational analysis of actuality, it is analytically false that the actual realm is composed of island universes. If actuality is absolute, on the other hand, a believer in concrete worlds can allow that the actual realm is composed of island universes simply by allowing that more than one world be actual. The second consideration is epistemic. I claim that we have fundamentally different modes of access to reality, modes of forming beliefs about what exists. We access what we take to be actual by acquaintance, and by the causal and statistical inferences founded on such acquaintance. Knowledge of reality that we acquire in this way is a posteriori. But, I have claimed, we also access portions of reality by description, positing the existence of whatever portions of reality are needed to serve as the content of our thought. Knowledge of reality that we acquire in this way is a priori, and based purely on logic (in my broad sense). Those portions of reality that we can access only a posteriori, it seems to me, have a special ontological status. We learn something about those portions of reality in virtue of having accessed them in this way, founded on acquaintance: this mode of access to a thing is indicative of the thing’s ontological nature. Now, it is certainly traditional to hold that the objects of a posteriori disciplines such as physics and a priori disciplines such as mathematics differ ontologically in some fundamental way. But I have no argument to persuade a Lewisian who claimed instead that we have here different modes of accessing things that are ontologically on a par. A challenge for any account that holds that actuality is absolute is to say how one can know that one is actual. I have given my response at length in Chapter , so here I will be brief. The key move is to distinguish the concept of actuality, which is indexical, and the property of actuality expressed by uses of the concept, which is absolute. If the indexical component of the concept is something like, belongs to the same fundamental ontological category as me (or categories, if I belong to more than one), then I know trivially that I am actual; there is no need to make the property I thereby know relational, as Lewis does. This is not the end of the matter, however; for one might also ask what my merely possible qualitative duplicates believe when they think to themselves that they are actual. I argue in Chapter  that, unless there

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are certain symmetries between the structure of the actual and merely possible realms, the concept of actuality is what I call a “perspectival concept.” Whether or not one can grasp such a concept, and express thoughts by means of it, depends on having the right perspective on reality. I have it in virtue of being actual; my merely possible duplicates do not. The concept of actuality is defective for them, and designates no property in their mouths. This requires a special exception to my otherwise internalist views about knowledge. Two subjects can have the same evidence, the same concepts, the same powers of reasoning, and yet differ in what they know, not because of differing causal relations to their environment, but merely because of differing perspectives on reality. One but not the other, in virtue of being actual, has the right perspective, and thereby is in a position to know. A final question has to do with whether and how my account of actuality counts as a version of “ontological pluralism.”⁶² That depends, of course, on how ontological pluralism is characterized. If to be an ontological pluralist is just to believe in more than one fundamental ontological category where the ontological categories are just the most general kinds of thing, then anyone who believes in, say, sets and individuals or particulars and universals counts as an ontological pluralist. (I was using ‘fundamental ontological category’ in this weak sense in the chapters of this volume.) My belief in the distinction between actual and merely possible things would count as well. But if fundamental ontological kinds differ from, say, kinds of dog or kinds of ice cream just by being more general, then ontological pluralism doesn’t have much punch. It’s just a matter of extending the classification of things by genus and species all the way up the ladder. What more might be meant by “ontological pluralism”? Sometimes it is said that the ontological pluralist believes there are different “ways of being,” or that members of one category are “more real” than members of some other. I have no objection to saying that actual things are “more real” than merely possible things. But such talk is metaphorical at best, and lacks the substance to hang a philosophical theory on. How, then, should a strong version of ontological pluralism be characterized? The characterization of ontological pluralism that is currently fashionable is in terms of quantification: ontological pluralism is the view that there are multiple (singular) existential quantifiers, each of which is fundamental and “carves at the joints.”⁶³ But this seems to me to put the “pluralism” in the wrong place. There is only one fundamental (singular) existential quantifier, say I, so I prefer to locate the multiplicity in what the quantifier ranges over, in different domains of the quantifier.⁶⁴ Ontological pluralism, then, is the view that there are multiple fundamental domains of quantification, but no more inclusive fundamental domain. But that still won’t do. For I think that some fundamental domains, such as the actual and the merely possible, support ontological pluralism, whereas other fundamental domains, such as the domain of atomic things and the domain of composite things, do not. What distinguishes the former sort of case from the latter? McDaniel (: –) ⁶² It is used as one of a number of illustrations of ontological pluralism in McDaniel (: –). ⁶³ See especially Turner () and McDaniel (). ⁶⁴ If an argument is wanted, see Schaffer (manuscript). He argues (in sections . and .) on semantic grounds that “quantifier variance is best understood as domain variance.”

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 :     () and Turner () argue that when different domains support ontological pluralism, different logical principles will apply to the different domains. But I do not think the logic that governs actual things is any different from the logic that governs merely possible things. There is only one logic. I do, however, think that there is a way in which my acceptance of a fundamental distinction between the actual and the merely possible makes me an ontological pluralist in a robust sense; and if I accepted a distinction between sets and individuals, or universals and particulars, or present and past or future, those views too would count as ontologically pluralist. The distinction between ontological categories cannot, however, be made out in terms of fundamental properties. It has to do, rather, with the concepts by means of which we pick out the fundamental properties and thereby gain a priori knowledge of reality, and our place in reality. I say: a fundamental property corresponds to an ontological category (in a robust sense) just in case there is a perspectival concept by means of which thinkers gain access to that property, and may thereby know that they have the property. This account will not fit what all philosophers have meant by “ontological category”; but it is broad enough to include any candidate that I would take seriously. It identifies ontological categories with privileged perspectives on reality. But note that the “privilege” in question has an epistemic source. Although any fundamental property creates a divide across reality, with each side of that divide providing a perspective on reality, it takes more for a perspective to be privileged in the sense that makes for an ontological category. There must be a perspectival concept, the grasping of which provides me with knowledge de se, knowledge of which side of the divide I inhabit. Ontological pluralism thus becomes a thesis, not just about reality’s structure, but about our epistemic access to reality, qua thinker.

. Conclusion So concludes this whirlwind tour of reality as I see it. I expect my fair share of “incredulous stares.” The incredulity should not stem, as it did with Lewis, from the positing of flying pigs and talking donkeys ontologically on a par with the ordinary pigs and donkeys of my acquaintance; I do not believe in those. But I do believe in merely possible flying pigs and talking donkeys. Although they exist as objects of my thought, I believe they are no less parts of reality for that. And, yes, they are made out of flesh and blood. Some philosophers, so-called naturalists, hold to a creation myth much simpler than mine. There is only a single stage of creation at which the physical cosmos is brought into being. And that is all. All of reality is actual and concrete. There can be worthwhile philosophical debate, they allow, over the constituents of the cosmos— facts or things? universals or tropes? atoms or gunk?—but the extent of reality is a matter for cosmologists to determine (if it can be determined at all), not for philosophers. There is no a priori knowledge of some greater reality, no knowledge that transcends the knowledge bequeathed to us by science. My response, simply put, is not to take their denials at face value. Their discourse is subject to the same truthmaking principles, and harbors the same intentional content, as my own.

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Their thoughts about flying pigs do not differ in content from mine. I cannot consistently take my thoughts to be ontologically committing but not theirs. Other philosophers, sometimes called abstractionists, allow that reality must be extended to include truthmakers for our modal claims, and contents for our thoughts, but disagree with me over the nature of those entities. Perhaps there are possible flying pigs, but they are not pigs and not concrete (as Williamson holds). Or perhaps there are just actually existing properties that are possibly instantiated (as Stalnaker holds). These are legitimate disputes. As are disputes over the truthmakers for mathematics: perhaps they are sets, or ante rem structures, and not the mathematical systems of bare particulars that I believe in. For I would not go so far as to claim that it is illegitimate to challenge the Humeanism that underwrites many of my arguments as to the nature of reality. I once wrote (in the paper that is Chapter ), after presenting my view of concrete possible worlds with absolute actuality, that “I wouldn’t stake my life on [the account] being true—or even my next paycheck.” If forced to assign a degree of belief to the account I have here put forth in this introduction, that number would not be very high. (Although at least I can say I put more credence in my own view than in any of the alternatives!) I have reasons and arguments for my views; but the very subject matter ensures that much of what I affirm is speculative in the extreme. Sometimes I wish I had chosen to pursue a field where certainty, or near certainty, is not so far out of reach: mathematics, or physics, or some less speculative area of philosophy. But out of reach or not, my work in metaphysics has ever only been about truth, about getting it right. Truth may be elusive in matters of fundamental metaphysics. But it remains the overriding goal.

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 Realism without Parochialism () . Introduction I am a realist of a metaphysical stripe. I believe in an immense realm of “mathematical” and “modal” entities, entities that are neither part of, nor stand in any causal or external relation to, the actual, concrete world. For starters: I believe in mathematical objects and structures; in (concrete) possible worlds and individuals; in propositions, properties, and relations (both abundantly and sparsely conceived); and in sets (or classes) of whatever I believe in.¹ Call these sorts of entity, and the reality they comprise, metaphysical. In contrast, call the actual, concrete entities, and the reality they comprise, physical.² Physical and metaphysical reality together comprise all that there is. In this chapter, it is not my aim to defend realism about any particular metaphysical sort of entity. Rather, I ask quite generally whether and how any brand of realism about metaphysical sorts of entity can be justified.³ Belief in metaphysical sorts of entity does not rest on acquaintance, or anything analogous to perception; by definition, we bear no causal relations to them. If we have beliefs about such entities at all, it is by way of description, through theories that postulate their existence. Thus, the question of belief in metaphysical sorts of entity may be shifted to the question of belief in metaphysical theories. I believe in metaphysical sorts of entity because I believe theories postulating their existence to be true, to provide an accurate description of what there is.⁴

This chapter has not been previously published. It was presented as a symposium paper at the Pacific APA meetings in March, . Because it has been available on-line for many years, and cited in the literature, I have made only minor revisions, and added a postscript. Additions to the original paper are in square brackets. ¹ [See Chapter  for an updated account of the entities I believe in; and see the postscript to this chapter for relevant discussion. In particular, with respect to abundant properties and relations, and sets or classes, while I am committed to our discourse being true, the interpretation of such discourse need not make them out to be entities in their own right.] ² [The distinction I make here between the “physical” and “metaphysical” realms is essentially the distinction between the “actual” and the “merely possible” that figures prominently in Chapters  and . Neither terminology is ideal. Allowing unreduced mental entities and properties to be classified as “physical,” if any there be, sounds odd. But allowing mathematical entities to be classified as “merely possible,” as I do elsewhere in this volume, is also not in accord with usual ways of speaking.] ³ I mean nothing fancy by ‘realism’: to be a realist about some sort of entity is just to believe that entities of that sort exist. ⁴ I use ‘metaphysical theory’ broadly to include any theory whose quantifiers are interpreted to range over metaphysical sorts of entity. [See the postscript for discussion.] Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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



What criteria do I use in deciding which metaphysical theories to believe? Of course, if a theory is incoherent, it can be rejected out of hand. One way for a theory to be incoherent is for it to be logically inconsistent, but I suppose there are other ways.⁵ Moreover, if a theory is unfaithful to the notions it aims to elucidate, be they notions of ordinary or of scientific thought, that too is a form of incoherence; it too can be rejected out of hand. Unfortunately, however, criteria of coherence appear to leave the choice of metaphysical theories vastly underdetermined. What to do? Enter here the broadly pragmatic criteria. According to conventional wisdom, we should believe theories that are, on balance, more fruitful, simple, elegant, unified, or economical than their rivals. We should believe pragmatically virtuous theories. It won’t matter, for this chapter, what a complete list of the pragmatic virtues would look like, or how the virtues are to be weighed one against another. The problem I want to discuss would remain even if only one pragmatic virtue played a role in deciding which metaphysical theories to believe. The problem is this. It is one thing for a theory to be pragmatically virtuous, to meet certain of our needs and desires; it seems quite another thing for the theory to be true. On what grounds are the pragmatic virtues taken to be a mark of the true? It is easy to see why we would desire our theories to be pragmatically virtuous: the virtues make for theories that are useful, productive, easy to comprehend and apply. But why think that metaphysical reality conforms to our desire for simplicity, unity, and the other pragmatic virtues? Moreover, standards for simplicity, unity, and the like have been notoriously difficult to pin down objectively; it seems such standards may differ from culture to culture, era to era, galaxy to galaxy. Why think that metaphysical reality, even if simple and unified by some standards, conforms to our standards for simplicity and unity? Believing a metaphysical theory true because it is pragmatically virtuous leads to parochialism, and seems scarcely more justified then, say, believing Ptolemaic astronomy true because it conforms to our desire to be located at the center of the universe. Here I take my stand as a realist. I deny categorically that the pragmatic virtues of metaphysical theories are a mark of the true. Do I then abjure the use of pragmatic criteria in metaphysics? Not at all. The pragmatic virtues, I maintain, serve as criteria of acceptance, without serving as criteria of truth, or of reasonable belief. What I mean by ‘acceptance’ is this.⁶ Suppose I want to write the book on metaphysics, to develop a grand unified theory of what there is. Of course, I want the book to be true. I also want the book to be systematic and comprehensive. It should include precise explications of all the fundamental concepts of our ordinary and scientific thought; it should be a complete articulation of our conceptual scheme. But I also want the book to be succinct. Metaphysics, after all, is a human endeavor. There is no expectation that all true metaphysical theories will earn a place therein. In particular, when different theories cover more or less the same ground, all but one may be omitted ⁵ [But see Chapter  for my current, expansive notion of logical consistency applied to theories understood as collections of language-independent propositions. Thus understood, consistency and coherence will coincide. See also the postscript to this chapter.] ⁶ My use of ‘acceptance’ needs to be distinguished from that of van Fraassen (). Unlike van Fraassen, I take acceptance to entail belief.

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    () without sacrificing comprehensiveness. The metaphysical theories I accept are just those I would include in the book, in a grand unified theory of metaphysics. I thus distinguish between acceptance and belief. I accept a metaphysical theory in part because I believe it true and in part because I would include it in the best comprehensive, succinct systematization of our fundamental ordinary and scientific beliefs. Pragmatic criteria of theory choice are relevant only to the goal of systematization, not to the goal of truth. I recognize no presumption that the more fruitful, simple, elegant, unified, or economical theory is more likely to be true; no presumption that reality is made in our image. Call the view that I endorse absolute realism, or absolutism. It holds, first, that the notions of truth and reality are absolute: the metaphysical theories that are true for us are true for the Alpha Centaurians, true for all actual and possible thinkers. And it holds, second, that the notion of reasonable or rational belief is absolute: the epistemic principles that are correct for us, are correct for the Alpha Centaurians, correct for all actual and possible thinkers; and the correct application of these correct principles would lead all thinkers to belief in the same metaphysical theories (or to none at all).⁷ Call any realist view that opposes either or both of these claims parochial realism, or parochialism. One of the opponents I have in mind is David Lewis. Lewis squarely rests his defense of realism about possible worlds on pragmatic grounds. At the beginning of On the Plurality of Worlds, he writes: “Why believe in a plurality of worlds?—Because the hypothesis is serviceable, and that is a reason to think that it is true.” Lewis (a: ). According to Lewis—and I concur—possible worlds and individuals have proven enormously fruitful in diverse areas of philosophy. They provide “the wherewithal to reduce the diversity of notions we must accept as primitive, and thereby to improve the unity and economy of . . . total theory. . . . ” Lewis (a: ). And, for Lewis, such theoretical benefits provide good (though not conclusive) reason for believing that possible worlds and individuals exist. Lewis has not, so far as I know, acknowledged that the use of pragmatic criteria leads to parochialism (in my sense); but I do not see how it could plausibly be denied.⁸ In this chapter, I examine the prospects for absolute realism, for realism without parochialism. My aims are extremely modest. I do not expect to sway a content parochialist, much less an ardent renouncer of metaphysical sorts of entity. There is no thought of proving my basic conviction, that a parochial foundation for belief is no foundation at all. Nor will I attempt to provide an alternative foundation. My chief concern will be to show how absolutism can be reconciled with the free and inevitable use of pragmatic criteria of theory choice. In particular, I ask: what must one presuppose about metaphysical reality to ensure that the use of pragmatic criteria will not lead one to accept false theories. I argue that an absolutist must posit a

⁷ Of course, I do not assume that all thinkers are capable of discovering the correct principles, or of correctly applying them, even if discovered. ⁸ Other opponents—for example, Putnam and Quine—openly embrace parochialism. Quine (: ) writes, for example: “The very notion of object, or of one and many, is indeed as parochially human as the parts of speech; to ask what reality is really like, however, apart from human categories, is selfstultifying. . . . Positivists were right in branding such metaphysics as meaningless.”

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   



vastly greater metaphysical reality then the parochialist ever would or could accept. Pragmatic criteria must be seen as selecting from, rather than determining, what is metaphysically real.⁹ I conclude the chapter by discussing briefly the views that I oppose. The problem of pragmatic criteria in theory choice has been more often discussed in relation to scientific theories, than in relation to metaphysical theories.¹⁰ I shall have little to say here about scientific realism. In my view, the cases are substantially different. The choice of scientific theories is based in large part on inductive and causal-explanatory criteria that play no role in the metaphysical case.¹¹ These criteria are not, in my view, essentially pragmatic; that is to say, although theories that satisfy these criteria tend to be pragmatically more virtuous than those that do not, the epistemic ground of these criteria is independent of their pragmatic consequences. Thus, pragmatic virtues are selected, not for their own sake, but because they ride piggyback on inductive and causal-explanatory virtues. Those pragmatic virtues that are systematically selected in this way may indeed be a mark of the true. But they are no more a ground of truth, I would argue, in science than in metaphysics.

. Realism about Mathematical Theories I reject the use of pragmatic criteria as grounds for truth or reasonable belief, not as grounds for acceptance. When the parochialist makes use of pragmatic criteria in a decision to accept some metaphysical theory, I want to be able to do so as well. The dispute between absolutism and parochialism, as I see it, need have little effect on metaphysical practice; it is a dispute over the best interpretation of that practice.¹² Suppose, then, that a parochialist uses pragmatic criteria to choose one among a class of competing theories. An absolutist who wants to match that choice has two basic strategies at her disposal. She can find epistemically correct, non-parochial criteria that dictate the same choice, thus showing that the pragmatic virtues of the chosen theory ride piggyback on non-pragmatic virtues. Unfortunately, this strategy has limited use when dealing with metaphysical theories. Or, as a second strategy, the absolutist can argue that the theories in question should all be believed true—or, at least, believed true to the same high degree—in which case the pragmatic criteria serve only as grounds for acceptance, not as grounds for belief. That is the strategy I want to pursue in what follows. First, I will consider mathematical theories in some ⁹ [I first argued for this view in my doctoral dissertation, Bricker (, section ).] ¹⁰ See, for example, van Fraassen (: –) and Boyd (). ¹¹ We do say that metaphysical, as well as scientific, theories have “explanatory power.” In the case of metaphysical theories, I take it this is an amalgamation of pragmatic features involving fruitfulness, unification, and perhaps others. By causal-explanatory criteria, I have in mind principles that support inference to the existence of unobserved, and even unobservable, causes. [Those contemporary metaphysicians who accept a heavy-duty, non-pragmatic notion of metaphysical explanation, perhaps based on a metaphysical grounding relation analogous to causation, may have a way of rejecting parochialism without being plenitudinous realists. It will depend crucially on the epistemology that accompanies their metaphysics. I have in mind, especially, Fine () and Schaffer ().] ¹² Reconciliation has its limits. The dispute over the interpretation of metaphysical theories, we shall see, carries with it a dispute over the extent of metaphysical reality; and that dispute is genuine, not verbal.

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    () detail, where the strategy is familiar and more widely accepted; then I will briefly consider other metaphysical theories, and propose a parallel treatment. There is no doubt that pragmatic criteria play a dominant role with respect to theory choice in mathematics. Theories that are fruitful and elegant earn a place within the body of mathematics; theories that are sterile or clumsy may earn a Ph.D., but are quickly forgotten. The use of pragmatic criteria in mathematics, however, is no threat to absolutism. The discarded theories, if consistent, are thought no less true for being sterile or clumsy. Pragmatic criteria determine which theories are worth pursuing and worth preserving for posterity, not which theories are true, or reasonably believed to be true.¹³ What if the choice is between logically incompatible theories, as frequently appears to be the case? The strategy has the onus of providing a plausible interpretation of the theories under which the prima facie incompatibility disappears. Otherwise, joint belief in all the theories would lead to belief in a logical contradiction. Consider the strategy’s most famous application: the case of non-Euclidean geometry. On the face of it, Euclidean and the non-Euclidean geometries (say, of three dimensions) are logically incompatible theories: where Euclidean geometry asserts that through a point not on a line there is exactly one parallel to the given line, non-Euclidean geometries assert that there is no parallel, or more than one. If the geometrical terms, such as ‘point’, ‘line’, and ‘intersect’, have the same meanings throughout the different theories, then logic dictates that at most one of the theories be true. But modern mathematics treats all of these theories on a par: all are true, if any are. The solution, of course, assuming a realist interpretation, is to hold that some or all of the terms are equivocal between the different theories. Once the equivocation is set right, the theories are seen not to be logically incompatible, and there is no logical obstacle to believing all of them true.¹⁴ The equivocation will be differently diagnosed on different methods of interpreting geometrical theories. It will be worth our while to consider the interpretation of geometrical, and, more generally, mathematical theories in some detail as preparation for the discussion of other metaphysical theories. Let us say, as usual, that an interpretation of a theory consists of a domain of entities, and an extension over the domain (of appropriate type) for each primitive, non-logical term of the theory.¹⁵ The simplest diagnosis of the equivocation would be this: the Euclidean theory and the non-Euclidean theories are each fully interpreted theories, that is, each has a unique “intended” interpretation; but the “intended” domains of the theories are mutually ¹³ This is controversial with respect to set theory. Gödel (), for example, held that the pragmatic consequences of accepting, say, the continuum hypothesis, were relevant to its truth or falsity. ¹⁴ I pursue only realist interpretations of theories in this chapter. Another familiar reaction to nonEuclidean geometry, endorsed by formalists and logical empiricists, treats geometrical theories—if mathematical, rather than physical—as wholly uninterpreted, and thus as lacking in truth value or ontological commitment. See, for example, Hempel (). ¹⁵ If our metalinguistic framework includes set theory, then domains and extensions of predicates can be identified with sets (or classes) in the usual way. But when the interpretation of set theory is itself at issue, talk of interpretations must be reconstrued within a framework admitting plural quantification and quantification over relations. (Relations need not be taken to be primitive if the framework includes mereology. See Lewis (), especially the appendix by Burgess, Hazen, and Lewis. [But note that I now prefer a different reduction of relations, and set theory generally; see Section ..])

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   



disjoint. Thus, the Euclidean theory makes assertions about Euclidean points and lines, the various non-Euclidean theories about various non-Euclidean points and lines, and no logical incompatibility can arise. But just what is a Euclidean or non-Euclidean point or line? The theories themselves do not tell us; nor do the geometers who present the theories. The view that geometrical theories are fully interpreted, each with a unique intended interpretation, does not accord with modern mathematical practice. On the modern approach, a geometrical theory serves to characterize a geometrical structure (or, in more abstract branches of geometry, a class of geometrical structures), but without singling out a domain of entities instantiating that structure. There are many equally intended interpretations; the terms of the theory are thus partially, not fully, interpreted.¹⁶ Which interpretations count as intended? The theory itself tells us something about the relations among points and lines; only interpretations that satisfy the theory count as intended. (An interpretation satisfies a theory iff all assertions of the theory are true in the interpretation, using the standard model-theoretic account of truth.) What about denumerable, “Skolemized” interpretations that satisfy firstorder formulations of the theory? Such interpretations are clearly unintended. I suppose that the geometrical theories are not formulated in a first-order way, so that the “Skolemized” interpretations do not satisfy the theories. That allows geometrical theories to be categorical, to determine their interpretation “up to isomorphism.” Moreover, geometers sometimes—though not always—tell us more than the theory itself: points are simple and have no proper parts; lines are composite and have points as their simple parts; points are intrinsic duplicates of one another; and, perhaps, points and lines are “mathematical,” not “physical,” entities. I suppose that the intended interpretations satisfy these extra-theoretical constraints. But nothing we are told, explicitly or implicitly, fully interprets the terms of a geometrical theory. Ordinary geometrical assertions about points and lines are now seen to be doubly equivocal. One equivocation is set right by relativizing to theory, for example, by replacing ‘point’ and ‘line’ by ‘Euclidean point’ and ‘Euclidean line’. The other equivocation we let stand, but without thereby forfeiting talk of truth. A geometrical assertion about Euclidean points and lines, though equivocal, is true if true in all intended interpretations of Euclidean theory, and if there are some.¹⁷ The Euclidean theory itself is then true just in case some intended interpretation exists. And, similarly, for the other geometrical theories. (Of course, for axiomatized theories, one can speak more simply of intended interpretations of the axioms.) Again, there is no logical obstacle to believing both Euclidean and non-Euclidean theories true. Geometers rarely explicitly affirm belief in the truth of their theories; but such belief, I take it, is presupposed by the acceptance of the theory into the body of mathematics.¹⁸ ¹⁶ I do not mean to suggest that there is not some perfectly good sense of ‘meaning’ according to which the meanings of the geometrical terms are fully determined, say, by their theoretical or conceptual role. But on my usage, full meaning or interpretation requires determinate reference. ¹⁷ Here I suppose it natural to adapt the method of supervaluations: an assertion is false if false in all intended interpretations, or if there are none; neither true nor false if true in some intended interpretations and false in others. ¹⁸ One might prefer to view geometrical theories, not as partially interpreted, but as “Ramsified”: what appear to be predicates are instead second-order variables bound to existential quantifiers prefixed to the

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    () I have said that a theory is true if true in some intended interpretation. In other words, the theory is true if the theory and intentions, however formulated, together are true in some interpretation; that is, if the theory and intentions together are consistent. Is consistency, then, the way to truth? No, even supposing there is no problem about formulating the intentions, consistency is no easier (or harder) to find than truth itself, for two well-known Gödelian reasons. First, given that mathematical theories are not first-order, there is no proof procedure in terms of which consistency can syntactically or formally be defined; consistency is thus itself a semantic notion, and cannot provide a non-semantic criterion of truth. Second, even if we were (mistakenly) to interpret theories as first-order, and define consistency as formal consistency, we would have no general method for establishing the consistency of a theory without simply assuming the consistency of some stronger theory.¹⁹ Tying truth to consistency may serve to suggest the scope of truth; but it does nothing to provide a foundation. I said there is no logical obstacle to believing all the geometrical theories true? What about ontological obstacles? Does believing all geometrical theories true step up demands on ontology? Not on the above characterization of intended interpretation. If each of the theories has some intended interpretation, then each has an intended interpretation over one and the same domain. That is because, on the usual modeltheoretic account, nothing about the domain other than its size contributes to the satisfaction of the theory. And the extra-theoretical assertions, being the same for all of the theories, can all be satisfied by a single domain. Indeed, if we do not require that the domain consist of “mathematical” entities, then actual physical points (assuming there are continuum many), and fusions of physical points, will serve as an intended domain for both Euclidean and non-Euclidean theories. Of course, at most one such theory will have ‘point’ and ‘line’ interpreted, respectively, as the class of physical points and physical lines; but the other theories come out true under some non-physical interpretation. If, on the other hand, we require that the domain consist of “mathematical” entities, then a moderate belief in “pure” sets will meet the ontological demands of any ordinary geometry positing continuum many points; and to meet the demands of one is to meet the demands of all.²⁰ As theories have thus far been construed, one can multiply belief without ontological cost because the intended interpretations are ontologically indiscriminate. That is a false victory for ontological parsimony. The present model-theoretic construal of truth for partially interpreted theories, though standard, does not seem to me plausible. Geometrical theories, I have said, posit a structure, and make theory as a whole (or, better, the conjunction of its axioms). Logically speaking, the differences are small: an equivocal predicate behaves logically just like a second-order variable (with appropriate range); and the presupposition of existence commits one to belief no less than its explicit assertion. But to the extent that the differences are genuine, they favor the partial interpretation view as being closer to actual practice. For a discussion of Ramsification as applied to set theory, see Lewis (: –, –). ¹⁹ We have no formal method, according to Gödel’s second incompleteness theorem; and I know no reason to think there is some non-formal, accessible method. ²⁰ The extra-theoretical mereological demands may be satisfied too, if we accept the thesis, defended in Lewis (), that the parts of a (non-empty) set are its (non-empty) subsets. Points will then be singletons; lines will be unions (fusions), rather than sets, of points.

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   

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assertions about whatever entities instantiate that structure. (I assume for simplicity we are considering only categorical geometrical theories.) Surely, whether or not some entities instantiate a posited structure is not solely a matter of their number; if it were, then geometrical theories would tell us nothing about which entities are points and lines. That is too much inscrutability. I think geometrical theories tell us that the points and lines, whatever they may be, instantiate the posited structure in virtue of their genuine, or natural, properties and relations. Call this genuine, or natural, instantiation: a domain of entities naturally instantiates the structure posited by a theory iff the theory comes out true under some natural interpretation over the domain, that is, some interpretation that assigns only natural properties and relations over the domain to the primitive, non-logical terms of the theory. I hold that, for any domain, irrespective of the nature of the entities, it makes sense to ask what natural properties and relations the entities stand in, and thus what structures the entities naturally instantiate. In general, only an infinitesimal minority of the classes of entities, and of the classes of n-tuples of entities, will be (or correspond to) natural properties and relations.²¹ The distinction between natural and unnatural properties and relations is indispensable to the task of describing what there is, be it physical or metaphysical reality. It belongs to the universal framework of all theories, and as such is no less a part of logic than the existential quantifier or identity.²² The truth of geometrical theories, then, requires natural instantiation. Natural instantiation should be compared with model-theoretic instantiation, according to which, for any two domains of the same size, either both or neither instantiate the structure posited by a theory, and with elementary instantiation, according to which, for any two infinite domains, of whatever size, either both or neither instantiate the structure posited by a theory. Equating truth of a geometrical theory with there being some model-theoretic instantiation of the posited structure seems to me hardly more plausible than equating truth with there being some elementary instantiation. Neither, I think, captures the intentions of geometers. Let us say, then, that an interpretation of a geometrical theory counts as intended only if its domain naturally ²¹ On the need for a sparse conception of properties and relations that distinguishes between the natural and the unnatural, see Lewis (a: –). Lewis’s discussion focuses, however, on natural physical properties and relations. Some of what he says does not apply, I think, to natural mathematical properties and relations. In particular, I do not hold that all natural properties and relations are qualitative: mathematical entities and domains have no qualitative character; their nature is determined by structure alone. Lewis (: ) suggests it would be “overbold” to think there are natural mathematical properties and relations other than, perhaps, a single primitive of set theory (for Lewis, the singleton relation). I am so emboldened. [Note that both ‘natural’ and ‘qualitative’ have narrow and wide construals. On the narrow construal, fundamental logical or purely structural (i.e. mathematical) properties and relations do not count as “natural” or “qualitative.” Here, I am using ‘qualitative’ narrowly but ‘natural’ broadly. In Chapters  and , my usage differs: I there take ‘natural’ more narrowly so that it excludes fundamental logical relations.] ²² In a section of the Aufbau, Carnap (: section ) endorses a structuralist interpretation of theories that quantifies over “natural relations,” in effect endorsing what I call “natural instantiation.” And he claims that the notion of a natural relation belongs to logic. (He calls natural relations “founded.”) [Thanks to David Lewis for this citation when he sent me comments on a draft of this chapter. Lewis writes (letter of April , ): “[This little-known passage] is part of why I admire Carnap much more as a metaphysician than as an anti-metaphysician.” Lewis’s letter, which contains an extensive response to some of the issues raised in this chapter, will be published in Beebee and Fisher (forthcoming).]

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    () instantiates (henceforth, just instantiates) the structure posited by the theory; that is, only if the interpretation both satisfies the theory and is natural.²³ Is the strategy of believing all geometrical theories true now ontologically demanding? That depends. In the case where we drop the requirement that geometrical entities be “mathematical,” we can no longer expect the points of physical space to provide an intended domain for more than one geometrical theory, since the physical points presumably do not instantiate more than one geometrical structure in virtue of their natural metric relations. A moderate belief in a plenitude of possible worlds, however, would provide all the theories with intended domains, assuming the structure of space is contingent. In the case where we keep the requirement that geometrical entities must be “mathematical,” the mere existence of “pure” sets no longer suffices for truth. But the “pure” sets are rich in structure; they stand in myriad natural relations definable in terms of the membership relation. Familiar ways of interpreting Euclidean and non-Euclidean geometry within set theory—for example, by identifying points with tuples of real numbers, and real numbers with . . . —show how natural set-theoretical properties and relations may be assigned to primitive geometrical terms in such a way that the theory comes out true.²⁴ Thus “pure” sets, assuming they exist, provide intended domains for any ordinary geometry; and the strategy of multiplying belief need not yet result in multiplying entities if one already believes in sets. Are all mathematical theories partially interpreted? The chief evidence for partial interpretation is this. Mathematicians are generally aware of the reducibility of mathematics to set theory, but they don’t much care whether the basic entities of which they speak are taken sui generis, or are identified with pure sets. Either option

²³ It might appear that in more abstract areas of mathematics—abstract algebra and algebraic approaches to geometry—the model-theoretic approach to interpretation has taken over: the “points” of an abstract “space,” it is explicitly asserted, may be any set, relations between “points” may be any set of ordered pairs of “points,” and so on. I dispute this. The arbitrary, non-natural interpretations are themselves objects posited by the theory; they are not used to interpret the theory. The theories of abstract mathematics, nowadays, are couched entirely in set-theoretic terms, and the question how to interpret them is just the question how to interpret set theory. And for set theory, I suppose, only natural interpretations are intended. ²⁴ I cannot discuss here three important questions: () Which set-theoretical properties and relations are natural? First-order definability in terms of the membership relation (and identity) is both too broad and too narrow: too broad, because even single disjunctions of natural properties need not themselves be natural; too narrow, because, for example, the ancestral of a natural relation is itself natural. In any case, I suppose that the set-theoretical properties and relations that come into the standard reduction of mathematics to set theory are all natural. (Note that a definition can be arbitrary, in the sense that there are others that are just as adequate, without the defined property or relation being unnatural.) () Is naturalness all or nothing, or a matter of degree? If naturalness is a matter of degree, does instantiation of structures depend only on the perfectly natural properties and relations, or is it itself a matter of degree? If instantiation is a matter of degree, is the truth of partially interpreted theories a matter of degree as well? Probably all are a matter of degree; but I will continue to speak as if they are all or nothing. () Is the distinction between natural and unnatural relations compatible with the joint identification of relations with sets of ordered pairs, and of ordered pairs with sets, given that there are many natural ways of identifying ordered pairs with sets? If not, either relations or ordered pairs will have to be taken as primitive entities, sui generis. For discussion, see Sider ().

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   



is considered perfectly satisfactory.²⁵ When pressed about the existence of the entities, some turn formalist or structuralist; put them to one side.²⁶ Others maintain realism about the entities, and are aware that the two options are incompatible; but they nonetheless refrain from choosing between them. The choice is not considered to be a mathematical choice because it would make no difference to the truth of any mathematical theorem. Leaving the choice unmade is tantamount to leaving their theories partially interpreted. This evidence obviously does not apply to set theory itself. But there are reasons for taking set theory—even second-order set theory—to be partially interpreted. For one thing, there seems to be no end to the discovery of undecidable “large cardinal axioms” that extend the height of the set-theoretic hierarchy; neither the axioms of Zermelo-Frankel set theory nor the iterative conception that underlies it determines the interpretation of the membership relation “up to isomorphism.” Thus, many structures are compatible with both the theory and the iterative conception. If only one such structure were instantiated, I suppose we could single out the membership relation as the relation of this instantiated structure. But what reason could there be for thinking that just one such structure is instantiated? Perhaps pragmatic reasons of ontological parsimony would support unique instantiation; thus a parochialist need not deny that set theory is fully interpreted on account of the undecidable “axioms.” But an absolutist cannot support full interpretation in this way. Second, even if somehow our conception of set did manage to single out a unique relation between entities and the sets formed from those entities, the universe of “pure” sets would not yet be determined. For that depends on singling out some entity to be the null set, and nothing set theorists say seems to come close to accomplishing that. Set theory tells us that the null set is the only set that has no members; and set theorists may add that it is a “mathematical” entity, not an ordinary individual. But since set theorists, as just noted, typically allow other “mathematical” entities that are memberless (but are members of sets), such as sui generis numbers or geometrical points, it follows that they do not intend to uniquely specify the domain of pure sets. I conclude, then, that mathematicians leave all their theories only partially interpreted.²⁷ The task of fully interpreting them, of fixing references for mathematical terms, if it can be done at all, is left to the philosophers.

²⁵ For example—and examples could be multiplied—Enderton (: ) begins the chapter on the natural numbers: “There are, in general, two ways of introducing new objects for mathematical study: the axiomatic approach and the constructive approach.” Either approach, he says, may be used to introduce the natural numbers. ²⁶ We can bring them back as a last resort, but only if more straightforward interpretations of mathematical theories are known to fail. By “structuralism,” I mean the view that mathematical theories are committed to the existence of structures only, not objects that instantiate the structures. See, for example, Resnik (). Another use of ‘structuralism’ is almost the opposite: one keeps the objects, but denies that they naturally instantiate structures, thus forcing a model-theoretic construal of theories. See Lewis (: –). ²⁷ Mathematicians do speak of the null set, the number three, the Euclidean plane. But this, of course, is compatible with partial interpretation; such talk requires unique reference within each intended domain, not across intended domains. Similarly, our use of ‘the cloud’ in ordinary English does not give evidence that we believe in a single object with an indeterminate border, a so-called “vague object.”

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    ()

. The Problem of Uniqueness Can it be done at all? There are two problems that need to be separated: uniqueness and existence. Consider first uniqueness. Start with a categorical theory, such as (second-order) Peano arithmetic. If I am right that this theory is partially interpreted by number theorists, then, relative to a context in which number theory is being done, there is no fact as to whether natural numbers are sui generis entities, or are von Neumann “numbers,” or Zermelo “numbers.” One may stipulate, say, that ‘natural number’ means Zermelo “number”; that is, that ‘zero’ denotes the null set, that ‘successor’ denotes the singleton function, and that ‘number’ denotes the intersection of all classes containing the null set and closed under taking singletons. One thereby creates a context in which the ordinary meanings of the arithmetic terms have been changed by narrowing the range of “intended” interpretations. Relative to a context in which the stipulation is made, the following sentences are true: ‘natural numbers are sets’, ‘two is the singleton of the singleton of the null set’, and ‘two is a member of three’. If instead one stipulates that ‘natural number’ means von Neumann “number,” one creates a context in which the first and third sentences above are true, but the second sentence is false. The von Neumann stipulation has now become standard, and so stands as an established technical meaning for ‘number’ alongside its more ordinary meaning; thus today, in contexts where set theory is being done, the stipulation is presupposed unless explicitly denied. Still, relative to any ordinary context, the three sentences above are neither true nor false.²⁸ Does the von Neumann or Zermelo stipulation fully interpret the arithmetic terms? That depends, of course, on whether the set-theoretic terms are themselves fully interpreted. One may also stipulate that natural numbers are sui generis. What does that mean? I take it that sui generis natural numbers—if any there be—have no superfluous structure among themselves, no structure that they are not required to have by the Peano axioms (or the extra-theoretical axioms, if any). I also suppose that the sui generis natural numbers stand in arithmetic relations only to one another. I will say that a domain of entities matches a structure iff () it instantiates that structure, and no more inclusive structure; and () the instantiating natural relations never hold between entities inside and outside the domain. We have, then, the following: a domain of entities is sui generis relative to a mathematical theory iff the domain matches some structure posited by the theory.²⁹, ³⁰ ²⁸ Compare vague words of ordinary language (such as ‘adult’) which similarly may have one or more precise (or more precise) established meanings that are selected in certain contexts; and may for the nonce be given a precise (or more precise) meaning to serve some purpose at hand. ²⁹ The second condition on matching may not be part of the meaning of ‘sui generis’, as that phrase is commonly understood. Requiring sui generis domains to be thus isolated from one another does not allow, for example, the sui generis natural numbers to be included among the sui generis integers; or the sui generis points of two-dimensional Euclidean geometry to be included among the sui generis points of threedimensional Euclidean geometry. If the second condition is left out, however, unique reference demonstrably fails. Infinitely many subdomains of Euclidean -space match the structure of the Euclidean plane. ³⁰ [The definition of a sui generis domain in the original paper has been simplified by incorporating isolation into the definition of ‘matching’. When Michael Jubien commented on this paper at the APA meetings, he complained that no purely structural account of sui generis could capture what philosophers ordinarily mean by the notion. He claimed, for example, that sui generis numbers are entities that “in and

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   



For any mathematical theory, one may stipulate that the entities posited by the theory are sui generis (still waiving the problem of existence). Does the stipulation fully interpret the theory, and thus uniquely fix the reference of its terms? Not if the theory fails to be categorical; at best it would narrow the range of intended interpretations to one for each structure compatible with the theory. Thus, I doubt the stipulation that sets are sui generis uniquely fixes the reference of ‘set’. What about categorical theories, such as Peano arithmetic? Does the stipulation that natural numbers are sui generis uniquely fix the reference of ‘natural number’? That would require a non-trivial version of the identity of indiscernibles. Consider: Domains that match the same structure are identical. No; that is too strong. I am a realist about possible worlds, and I suppose that domains of worlds may match the same structure without the domains being identical; they may differ in their purely qualitative features.³¹ That suggests we restrict our attention to “mathematical” domains. Intuitively, what characterizes a domain as “mathematical” is that its entities, as well as all fusions of its entities, lack any intrinsic qualitative character. (I say an entity lacks intrinsic qualitative character if its intrinsic nature is entirely determined by the number of its parts, and the pattern of instantiation of natural properties and relations among its parts.) Would the stipulation that the domain is both sui generis and mathematical fully interpret a mathematical theory? One still needs an indiscernibility principle: mathematical domains that match the same structure are identical. I know of no reason to disbelieve the principle; but no reason to believe it either.³², ³³

. The Problem of Existence It may be, then, that uniqueness of reference is impossible to achieve when dealing with metaphysical sorts of entity. But uniqueness is a side issue. On the partial interpretation approach, uniqueness is not required for truth. Existence is another matter. I hold that every coherent mathematical theory is true, with or without the stipulation that the posited entities are sui generis.³⁴ That is ontologically quite demanding. It multiplies the number of basic kinds of entity well beyond what is needed for the truth of mathematics. It is time I said something to defend it. And what I say had better be compatible with absolutism: I do not want to replace a

of themselves are numbers.” I find that confused. But if he is right that that is what philosophers ordinarily mean by sui generis, then I should be understood to be offering a useful replacement for that notion.] ³¹ [I thus endorse quidditism. See Section ..] ³² It wouldn’t help to stipulate that a domain is mathematical only if it satisfies the principle; that would merely shift the problem from uniqueness to existence. ³³ [I am now inclined to disbelieve it. The argument in Chapter  against the identity of qualitatively indiscernible worlds can be applied to mathematical systems if one allows that it is contingent whether mathematical systems are actual.] ³⁴ [In saying that every coherent mathematical theory is true, I was implicitly understanding this as “true of some domain, some portion of reality”: the quantifiers of the theory are restricted to entities in the domain. This, of course, is the currently standard practice for interpreting mathematical theories. For the importance of this restriction, see the postscript.]

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    () parochial desire for ontological uniformity with a no less parochial desire for ontological variety. I ask first: why believe that every coherent mathematical theory is true? As noted above, a belief in pure sets would suffice, assuming the reducibility of mathematics to set theory; but my belief does not rest on a belief in sets. For one thing, there is nothing special about my belief in sets; the reasons I have for belief in sets apply, mutatis mutandis, to my belief, say, in natural numbers or in geometrical objects. For another thing, if I somehow discovered that set theory was incoherent and I had to retract my belief in sets, I would not also retract my belief in natural numbers or geometrical objects. My belief in sets, then, cannot be the sole support for those other beliefs. How, then, do I support my belief in the truth of mathematical theories? Here is the barest sketch of an argument. I do not present it to convince: its premises are no less controversial than its conclusion. Consider the case of natural numbers. I understand Peano arithmetic. Moreover, I understand it as it is written, with existential quantifiers over entities called “numbers,” not in some devious way. (Partial interpretation, I insist, is not devious; if it were, then the standard understanding of vague language, and so of most of ordinary language, would be devious as well.) Understanding is a relation between a thinker and what is thought about, in this case, between me and domains that instantiate the Peano structure. If no such domains existed, I could not stand in this or any relation to them; relations hold only between what exists and what exists (in the broadest sense of ‘exists’). Therefore, my understanding of Peano arithmetic entails the existence of domains that instantiate the Peano structure, that is, the truth of Peano arithmetic. Of course, there are mathematical theories that I will never understand; and mathematical theories no human being will ever understand. But if the theory is capable of being understood by some actual or possible thinker, if it is in the broadest sense intelligible or coherent, then the above argument will apply. Or so says the absolutist: epistemic arguments must be the same for all actual and possible thinkers, and lead to belief in the same mathematical theories. I conclude that the coherence of a mathematical theory is sufficient grounds for its truth. What about the added stipulation that the entities posited by the theory are sui generis? The same argument applies. The theory plus the stipulation, if coherent, is true. For example, I understand Peano arithmetic with the stipulation that the numbers are sui generis. Or at least I think I do. And if I do, then, by the above argument, sui generis numbers exist. Of course, my claim to understand a theory is fallible, to varying degrees. For one thing, understanding requires logical consistency, and even the best of us can be wrong about that (as witness Frege and naive set theory). Moreover, the framework I have used for interpreting theories may itself be incoherent, with its mathematical domains, and its natural mathematical properties and relations. In that case, the whole notion of sui generis mathematical entities may be incoherent as well. But the rejection of such entities on grounds of incoherence is compatible with the view that coherence suffices for truth and existence. And it is compatible with absolutism: such entities would not be rejected on grounds of ontological superfluity, or pragmatic undesirability.

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   



But I do not believe my framework is incoherent. Thus, I believe in a vast universe of sui generis mathematical entities: for any structure, a mathematical domain that matches that structure. There are sui generis natural numbers, rational numbers, real numbers; sui generis Euclidean and non-Euclidean points and lines; and for each structure compatible with set theory, there are sui generis sets that instantiate the structure, as well as sui generis ordered pairs, sequences, ordinal and cardinal numbers. And on and on and on.³⁵ Some would say my beliefs are extravagant. They would say: “since mathematics is reducible to set theory, all of these basic kinds but one are theoretically dispensable in science, mathematics, and (at least most of) philosophy; they are indefensible on pragmatic grounds.” That moves me not. They may also say: “the vastness of your ontology flies in the face of common opinion.” That would have some force if it were true: it would cause me to question my own beliefs. But I doubt that common opinion has much definite to say about the nature or extent of metaphysical reality. There is an offhand reluctance to admit the existence of any metaphysical sort of entity. When that is overcome, there remains a reluctance to identify kinds of entity that have been introduced into language or thought as distinct: the identification of, say, numbers with sets receives no support from the man on the street. As to the vastness of metaphysical reality: common opinion makes no distinction between orders of infinity; belief in the iterative hierarchy of sets has already left common opinion far behind. My liberality does not extend to physical reality, to the actual, concrete world. With respect to an arbitrary physical theory positing some physical kind of entity, there is, if anything, a presumption against (physical) existence. To defeat that presumption, to justify belief in that physical kind, one needs evidence of causal interaction, direct or indirect, with entities of that kind. The more physical kinds one believes in, the more justification one needs. In general, with physical reality, believing in more is harder to justify than believing in less.³⁶ Just the opposite is true, on my account, with respect to metaphysical reality. Our apparent understanding of a metaphysical theory carries with it a presumption in favor of existence. We need a reason to defeat that presumption, a reason for thinking our understanding is not genuine. Believing in less than all we think we understand is what requires justification. In general, with metaphysical reality, believing in less is harder to justify than believing in more.

³⁵ I do not say that distinct mathematical theories always have distinct sui generis domains. Distinct theories may posit the same structure, and thus be associated with one and the same basic kind. Identifications of this sort are discovered, not stipulated. But such identifications do little to limit the abundance of basic kinds. (A standard example: Boolean structures characterized in terms of the operations of “meet” and “join” are identical with corresponding Boolean structures characterized in terms of the “less than” relation.) [Note: what I here call simply “mathematical domains” I now call “mathematical systems”; see Sections . and ..] ³⁶ [I would now restrict this claim to the entities that make up our physical universe. As to whether other possible worlds, isolated from our physical universe, are absolutely actual I am wholly agnostic.]

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    ()

. The Role of Pragmatic Criteria Let us return, at long last, to the debate between absolutism and parochialism, and the absolutist strategy of reconciliation. Suppose the absolutist takes my broadminded approach to metaphysical reality. Then she can freely make use of pragmatic criteria in deciding which theories to accept, even when ontological reduction is at issue. Suppose we are deciding whether to accept a metaphysical theory that identifies all mathematical entities with pure sets on grounds of ontological parsimony and theoretical unification. For the parochialist, this is a decision as to what to believe about the extent of metaphysical reality: to accept the theory is to decide that sets exist, and that mathematical entities other than sets do not. For the absolutist, this is a decision whether to narrow, by linguistic stipulation, the class of intended interpretations of mathematical theories to interpretations whose domains consist entirely of sets. It is a decision what to talk about. And the sui generis numbers, and other nonsets, are thought no less real for the decision not to talk about them. Now, there is no reason why an absolutist cannot allow the decision what to talk about to be based on its pragmatic consequences. All sides agree that doing mathematics entirely within set theory has pragmatic advantages: it unifies and simplifies the vast array of mathematical notions; it facilitates the cross-fertilization of mathematical theories; and so on. Thus, an absolutist, no less than a parochialist, can accept on pragmatic grounds the metaphysical theory that identifies all mathematical entities with pure sets. For the absolutist, however, the pragmatic criteria serve to select some portion of metaphysical reality to be the universe of discourse for our mathematical theories; they do not determine what is metaphysically real. Moreover, an absolutist, no less than a parochialist, can reject on pragmatic grounds any metaphysical theory that posits sui generis mathematical entities other than sets. For the absolutist, however, theories rejected on pragmatic grounds are not thereby false. They are merely useless because essentially redundant; we can say all that we care to say without referring to the reality of which they speak. Conversely, an absolutist who does not take my broad-minded approach to metaphysical reality cannot freely make use of pragmatic criteria in deciding which metaphysical theories to accept. For suppose the absolutist is considering a class of competing metaphysical theories, all of which are thought coherent, and thought equally likely to be true. And suppose she narrow-mindedly takes the theories to be genuine alternatives: one of them at most is true. Finally, suppose she nonetheless accepts one of the theories on pragmatic grounds. Then she accepts a theory she does not believe to be true, since, as an absolutist, the pragmatic grounds do nothing to boost her degree of belief. Perhaps that is not so bad. Perhaps the goal of metaphysical theorizing, rightly understood, has only to do with systematization, and nothing to do with truth. Perhaps, the way to uphold an absolutist epistemology and a realist interpretation of metaphysical theories is to be wholly agnostic about the truth of metaphysical theories. But to embrace agnosticism is to abandon absolutism as herein characterized; it is to abandon absolute realism. Agnosticism is an alternative to realism, not a species thereof.

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   



. Realism about Metaphysical Theories I turn now, all too briefly, to the consideration of non-mathematical metaphysical theories, such as theories of possible worlds, or propositions, or impure classes. You will not be surprised to find that I hold: every coherent metaphysical theory is true.³⁷ If that is to be at all plausible, coherence must go well beyond logical consistency, as ordinarily conceived. In particular, any theory that conflicts with principles of the framework by which I interpret and understand metaphysical theories will be rejected as incoherent. That will include numerous traditional theories of truth and existence, including parochialism itself. If all coherent metaphysical theories are true, then much of what was said about the interpretation of mathematical theories will apply, mutatis mutandis, to other metaphysical theories; and again the absolutist can freely make use of pragmatic criteria in choosing among coherent theories. Consider, for example, theories of propositions. On their face, they are often incompatible: some take propositions to be “structured,” some “unstructured”; some take propositions to be “intensional,” some “hyperintensional.” When properly interpreted, however, the incompatibilities disappear. Each theory serves to explicate a different conception of proposition; and the different conceptions are in peaceful coexistence. I do not say that every conception of propositions is coherent; some, for example, founder on the Liar paradox, and its kin. But if a conception of propositions is coherent, then the theory articulating that conception is true, and the entities posited by the theory exist. An absolutist who accepts only the most fruitful of coherent conceptions need have no fear of accepting false theories. My approach to ontological reduction, too, is the same for metaphysical theories generally as for mathematical theories. For many philosophical purposes, the propositional theories are left partially interpreted; no attempt is made to provide each theory with a unique domain. For purposes of “ontology,” however, the theorist must decide whether the entities posited by a propositional theory are to be taken as basic, and if not, how they may be identified with the entities posited by some other theory. On my view, these “ontological” decisions are not, in general, matters for discovery;³⁸ they are matters for linguistic stipulation. Perhaps there are coherent metaphysical theories whose posited entities may not coherently be taken as basic; I address this question below. But when different “ontological” decisions are equally coherent, metaphysical reality accommodates them one and all; and again the absolutist can choose between them on pragmatic grounds. Although I have used mathematical theories to motivate my approach to metaphysical theories in general, I do not claim that the case of mathematics is in all ³⁷ [Again, it is important to understand this as saying ‘true of some portion of reality’. Moreover, I should have been more careful distinguishing two types of “metaphysical” theory. There are theories that are true of reality as a whole, such as theories of possible worlds or propositions; these theories may themselves deserve to be included among the principles of the framework. And there are “metaphysical” theories that are true only of some portion of reality, such as contingent theories describing the structure, or laws, or qualitative character of some possible worlds, or class of possible worlds. See the postscript.] ³⁸ There are exceptions. As in the mathematical case, I allow for non-trivial discoveries that theories are analytically equivalent, and thus posit the same entities. For example, I suspect (with many others) that on some conceptions of properties, and some conceptions of classes, properties and classes coincide.

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    () respects representative. There are ways that metaphysical theories may be incoherent that have little or no application in the mathematical case. For one thing, nonmathematical metaphysical theories typically serve to explicate notions of ordinary language and thought; if such a theory veers too far from ordinary usage, it is incoherent because analytically false. Mathematical theories, in contrast, have broken away from their ordinary origins, and cannot be charged with incoherence on these grounds. Mathematical “rings” cannot be faulted for not being round. There is another, related way that mathematical theories may be protected from charges of incoherence. The mathematical entities of which they speak are isolated from the physical entities, not just causally, but with respect to all natural relations; and the mathematical and physical entities have nothing but purely structural natural properties in common. Being thus isolated and dissimilar from the physical realm, the mathematical realm runs little risk of conflicting with fundamental principles of ordinary thought; for ordinary thought is directed first and foremost towards the physical. Non-mathematical metaphysical theories posit entities more closely tied to the physical realm; that makes judgments as to their coherence inevitably less secure. Let me illustrate. Begin with the mathematical theory of Newtonian spacetime. Add that the spacetime points match the Newtonian structure. Now consider two purely qualitative natural properties that hold of point-sized objects (assuming there are such); call them ‘red’ and ‘blue’. Add to the theory that everything is “blue” up to and including some time, and then “red” thereafter. Add that nothing has any other qualitative property. I think I understand the resulting theory, modulo an understanding of ‘red’ and ‘blue’. I therefore think the theory is coherent (if qualitative natural properties of point-sized objects are), and that there exists a domain of entities of which the theory is true. I call (the fusion of) any such domain, naturally enough, a possible world. (Of course, the example generalizes. Start with any spatiotemporal structure, with any class of purely qualitative natural properties and relations, and with any distribution of just those properties and relations over the domain of the structure; one thereby describes a possible world.³⁹ The details needn’t concern us here. What I have to say about coherence applies whether one posits one world or many.) Are the worlds that I believe in the worlds of David Lewis? No, it is a principle of my framework that the distinction between physical and metaphysical reality—between the actual, concrete world and everything else—is a fundamental ontological distinction; whatever is of the same ontological kind as a part of physical reality is itself a part of physical reality. Since Lewis’s worlds are of a kind with physical reality without themselves being physical, I judge them incoherent relative to my framework. And Lewis, I suspect, would return the favor. What I call worlds Lewis might call “pictorial ersatz worlds.”⁴⁰ On my account, parts of possible worlds and parts of physical reality can share qualitative character—indeed, can be qualitative ³⁹ What if one starts with an arbitrary mathematical structure, say, a four-element group? The resulting theory is no less coherent, and the domains that satisfy the theory are no less real; but they would not, in general, properly be called “possible worlds.” [See Chapter  and its postscript.] ⁴⁰ Except that I include the actual, concrete world among the possible worlds, and deny that there must be some “ersatz” world that represents the actual, concrete world. See Lewis (a: –), for Lewis’s discussion and dismissal of pictorial ersatzism. [I now think the view I espouse is better classified, not as pictorial ersatzism, but as realism with absolute actuality; see Chapters  and .]

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duplicates—even though they are of fundamentally different ontological kinds. Is that coherent? I think it is, but I won’t try to defend that here. My point has been made. Metaphysical theories positing entities that share qualitative character with physical entities face challenges to their coherence that do not arise in the mathematical case. Mathematical domains and unactualized possible worlds have this in common: they are all isolated from the physical realm. Other metaphysical sorts of entity are less aloof. Consider again some theory of propositions. Of the propositions that purport to describe physical reality, some succeed and some do not; the successful ones we call true. Any adequate theory of propositions, I suppose, will be in part a theory of this relation between the true propositions and the physical reality they purport to describe. The theory thus posits a structure with a “mixed” domain: part physical, part metaphysical. And the two parts are interrelated, I suppose, by some natural external relation. Now, within the physical realm, an object’s intrinsic qualitative character does not determine the natural physical relations that it bears to other objects: this pen is touching a piece of paper, but nothing about the pen’s intrinsic qualitative character necessitates that that be so. Within the mathematical realm, the same principle vacuously applies: since a mathematical entity has no intrinsic qualitative character, nothing is necessitated by its intrinsic qualitative character. What about a mixed realm, containing physical and metaphysical entities standing in natural external relations to one another? The principle is violated. Somehow, the intrinsic qualitative character of this pen makes it absolutely necessary that the pen stand in the truthmaking relation to some propositions, but not others (and, for good measure, in the instantiation relation to some properties, but not others, and in the membership relation to some classes, but not others). A fundamental modal principle that holds for all natural physical relations fails to hold for all natural relations. Is that coherent?⁴¹ I think it is.⁴² I do not see why the modal principle in question must apply generally to all of reality. One must beware of false projection. But I won’t try here to defend theories of propositions (or properties, or impure classes). My point is this: theories positing metaphysical entities that stand in external relations to the physical realm face serious challenges that do not arise in the mathematical case. The working mathematician’s attitude towards the objects of which she speaks, as summarized by David Lewis, is this: “No worries, it’s all abstract!” I have more or less supported that attitude towards mathematical entities, naive though it be. Worry seems out of place in mathematics, formalists and intuitionists aside. Lewis considers metaphysicians who defend controversial ontology by mimicking the mathematician’s response: “No worries, it’s all abstract!” And when asked what that means, they say: “You know, abstract the way mathematical entities are abstract.”⁴³ I have tried to distance myself a little ways from that response. ⁴¹ The above is adapted from Lewis’s (a: –) argument against magical ersatz worlds. Lewis would not quite say that the (unreduced) propositions, or properties, or classes, are incoherent; only that if we do somehow understand them, we know not how we do it. ⁴² [I am less sanguine today about the coherence of positing an irreducible realm of propositions, or other “intensional entities.” Any restriction of Humean principles to some parts of reality will be hard for the Humean to justify. See Section . for some discussion of whether the “realm of representation” reduces to the “reality of things.”] ⁴³ Lewis (a: ). Lewis is addressing the various ersatz modal realists.

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    () First, I have refrained from calling all the parts of metaphysical reality “abstract.” What makes the mathematical entities abstract (in one sense of ‘abstract’) is their purely structural character. “Modal” entities, such as possible worlds, are not abstract in that way, because they have qualitative character. “Intensional” entities, such as propositions, are not abstract in that way, because their character depends in part on relations to the physical. Labeling all these kinds of entity ‘abstract’ would serve only to cover up their fundamentally different natures. Second, worry does not seem out of place when considering “modal” or “intensional” entities. The carefree existence of mathematical entities need not transfer to other metaphysical kinds. I worry that my belief in, say, possible worlds or propositions, may be wrong. But worry alone cannot defeat the presumption of existence. Until an argument convinces me that I could not understand what I think I understand, I will continue to believe.

. Frege vs. Hilbert on Truth and Existence The approach that I take to truth and existence is often associated with David Hilbert and the formalist philosophy of mathematics.⁴⁴ Hilbert wrote, in a well-known letter to Gottlob Frege: “If the arbitrarily posited axioms together with all their consequences do not contradict one another, then they are true and the things defined by the axioms exist. For me, this is the criterion of truth and existence.”⁴⁵ Frege responded with two objections. The first was epistemological. Since one can only know that a theory is consistent if one knows that there exist objects of which the theory is true, consistency cannot provide a foundation for truth and existence. Frege’s objection pushed Hilbert down the path of formalism. I do not follow Hilbert down that path. For one thing, it leads to Gödel’s cul-de-sac: not even formal consistency is epistemologically secure. But more important, to even begin down that path is to forfeit any claim to be giving a criterion for truth and existence. On a formalist interpretation, mathematical theories are not true, and have no existential import. Formalism does not provide an account of truth and existence in mathematics, but a discounting of these notions altogether. How, then, can a realist respond to Frege’s objection? Concede the point: the criterion does not provide a foundation. Frege’s second objection to Hilbert’s criterion was posed as a question: Let us suppose that we know that the propositions: . A is an intelligent being. . A is omnipresent. . A is omnipotent. together with all their consequences did not contradict one another. Could we infer from this that there exists an omnipotent, omnipresent, intelligent being? (Frege : )

⁴⁴ [More recently, it has been discussed by Balaguer () under the name “full-blooded platonism” and Eklund () under the name “maximalism.” It also has affinities with Thomasson’s () “easy ontology” and Hale and Wright’s () “neo-Fregeanism.”] ⁴⁵ Reprinted in Frege (: –).

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It appears that Hilbert’s criterion applies, and that the answer to Frege’s question is “yes.” But that, I take it Frege would say, is objectionable. It shouldn’t be that easy to prove the existence of a deity. Hilbert promised Frege a letter in response; to Frege’s chagrin, it never came. But I don’t much care what Hilbert’s formalist response might have been. I offer a realist response in its place. First, Frege’s question needs disambiguating. There are two distinct realms of existence: the physical and the metaphysical. Typically, when we use ‘exist’ and its cognates, we speak only of some or all of physical reality. Is Frege asking whether the criterion leads to the existence of a being that is, among other things, physically located and physically active? If that is his question, then the answer is “no”—at least as I construe the criterion. The criterion has no implications for the existence of physical beings. The criterion leads only to a priori knowledge. All knowledge of physical reality is a posteriori. On the other hand, Frege may be asking whether the criterion leads to the existence of an intelligent, omnipresent, omnipotent being in the widest sense of ‘exist’, as a part of either physical or metaphysical reality. Indeed, I think it does, assuming Frege’s mini-theory is coherent. But the existence of such a being as a part of metaphysical reality, presumably off in some possible world, does not seem to me objectionable. Such a being is causally isolated from the physical realm. An easy proof of its existence provides small comfort for the deist.

. Conclusion I conclude the chapter by discussing briefly the views that I oppose: first the parochialist, then the skeptic. Suppose we have before us a metaphysical theory that is universally agreed to be fruitful and elegant. I ask: why is fruitfulness or elegance a reason to believe the theory true? I hold that fruitfulness and elegance are good reasons to develop and use a theory believed true on other grounds; the parochialist holds that fruitfulness and elegance are good reasons (though, I suppose, not sufficient reasons) to believe the theory true. I distinguish two such parochialists, depending on whether they deny the absoluteness of truth, or of epistemic rationality. The first parochialist is a traditional pragmatist, and endorses a pragmatic theory of truth: having appropriate pragmatic features is part of what constitutes a theory’s being true, part of the meaning of the word ‘true’. This seems both wrong and unhelpful: wrong as a claim about ordinary language; unhelpful, because it evades rather than answers the question I intended to ask. Suppose I give the pragmatist the word ‘true’. I can rephrase my question thus: why believe that the theory in question agrees with reality, or describes what there is? Well, you know how the dialectic goes. The pragmatist won’t be content with the word ‘true’; he’ll claim that pragmatic criteria in part determine the meaning of ‘reality’, ‘existence’, our whole vocabulary for talking about what there is (including ‘is’). But this is still unhelpful. Suppose I give the pragmatist this whole vocabulary. The question I intended to ask still remains, inexpressible though it be in the appropriated language. Pragmatic reinterpretation does not make it go away. The other parochialist holds to a realist conception of truth, and a realist interpretation of theories. She holds that the pragmatic criteria of theory choice are

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    () constitutive, not of truth, but of reasonable belief. Perhaps she argues thus. The standards by which our community in fact designates beliefs as reasonable, she claims, determine the very meaning of ‘reasonable’, and these standards are pragmatic through and through. I deny both conjuncts. Moreover, even if I granted that pragmatic criteria were part of what we mean by ‘reasonable belief ’, it would do nothing towards answering the question I intended to ask. First off, I doubt that the standards by which we in fact adjudge beliefs reasonable are pragmatic through and through. Often, we deem it reasonable to believe the simpler or more fruitful hypothesis, not on pragmatic grounds, but because of the nature of the subject matter; for example, reasonable hypotheses about human behavior tend to be simple and fruitful because human beings are simple creatures, who typically do things for a reason. The subject matter of all of physical science cannot be supposed simple in this way; but here, I think, the use of pragmatic criteria has been much overestimated. Scientist’s affirmations of belief are normally comparative judgments among alternative theories; and scientists rarely, if ever, consider alternatives differing only on pragmatic features. Thus, the case for pragmatic criteria as grounds for belief in science is virtually impossible to make.⁴⁶ In any case, our concern here is with metaphysical, not scientific, theories; and our standards of reasonableness in the two cases need not coincide. In mathematics, I have said, theories that are unfruitful or inelegant are not thereby deemed false. In philosophy, the use of pragmatic criteria as grounds for belief has waxed and waned over the years. I doubt that philosophical practice, historically viewed, manifests any consensus on standards of reasonable belief in metaphysics. Even supposing that our standards of reasonable belief are pragmatic, I deny that such standards constitute the meaning of ‘reasonable belief ’, any more than, say, our standards for measuring distance constitute the meaning of ‘distance’; neither term is “operationally” defined. Our standards for measuring distance may be wrong, whether or not we could ever discover that they were wrong; and similarly for our standards of reasonable belief. What is constitutive of reasonable belief is that it be formed according to standards that are reliable, though perhaps fallible, guides to the truth. If I somehow discovered that one of the standards we use was not reliable in this way, I would retract the designation ‘reasonable’ from beliefs formed in accordance with that standard without the meaning of ‘reasonable’ having thereby changed. Finally, even were I to grant that pragmatic criteria are constitutive of reasonable belief, that would not help answer the question I intended to ask. I could rephrase it thus: Why have reasonable beliefs? Why think this will further our pursuit of the truth? For the realist, at least, our practice of forming beliefs has a goal external to itself—the goal of truth—and so the question inevitably arises, whether the practice is justified in the sense of being suited to attain that goal. It is no justification to simply point out that it is our practice, or that, on reflection, we are satisfied with it, or that we know of no better. Nor, for that matter, is it justification to point out (wrongly, I think) that, perhaps for evolutionary reasons, we are (biologically) incapable of

⁴⁶ Recall what was said above: inductive and causal-explanatory criteria of theory choice are not, on my view, pragmatic.

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conforming to any other.⁴⁷ At most, that would justify—that is, make blameless—our use of the practice; it would not justify the practice in the relevant sense of showing that it leads reliably to the truth. Please don’t misunderstand. I am not demanding that the parochialist justify her use of pragmatic criteria; I point out that she has not done so only in order to make room for my own view. The demand for a justification of all epistemic principles is surely illegitimate. Indeed, I myself have no intention of foregoing the use of inductive criteria pending a solution to the problem of induction. Nonetheless, the cases of inductive and pragmatic criteria are not, for me, alike. The very conceivability of justification for pragmatic criteria is ruled out by their parochial nature; no criteria directly tied to specifically human needs, interests, or desires could conceivably be linked to truth and reality, absolutely understood. Although it has been argued that inductive criteria, too, are parochial, it can plausibly be denied; and I deny it. I turn now briefly to my opponent on the other side: the skeptic about metaphysical sorts of entity. The skeptic and I agree in rejecting pragmatic criteria as grounds for belief. But the skeptic is not much impressed by my argument that coherence suffices for truth. The skeptic allows that one may reasonably choose to contemplate a metaphysical theory, or to develop it, or to examine its consequences; but one may not reasonably believe the theory true. The only reasonable position, says the skeptic, is to withhold belief in metaphysical sorts of entity. Here the realist may protest: metaphysical theories have their “data,” no less than scientific theories; there are fundamental mathematical and modal intuitions that any account of reasonable belief must respect. The skeptic has a familiar argument in response. Metaphysical sorts of entity, by their nature, do not causally affect us in any way. Thus, all of our intuitions, our beliefs, indeed, all of our psychological states would be the same whether such entities existed or not.⁴⁸ It follows that our intuitions, beliefs, and states cannot provide evidence that these entities exist. So argues the skeptic. But the argument has no force against the realist. How are we to understand this subjunctive conditional: if metaphysical sorts of entity did not exist, our psychological states would be just as they are? As such conditionals are standardly understood, it is the barest triviality.⁴⁹ If the antecedent is true, the conditional is true merely in virtue of the truth of its consequent. If the antecedent is false, it is impossible, and anything follows from an impossible supposition: our psychological ⁴⁷ Lycan (), for example, pursues an evolutionary response. I am the “snooty epistemologist” of his scenario. ⁴⁸ This assumes a “narrow” individuation of psychological states; but I have no problem granting that assumption. ⁴⁹ [Since this was written, there have been various proposals for using impossible worlds to provide a non-trivial semantics for counterpossibles; see especially Nolan (). But the distinctions of content that these proposals aim to capture are elusive, and, in any case, won’t do much to bolster this argument. See Section . for my take on the content of impossible propositions. Note, however, that because I take mathematical theories to be contingent—true at some mathematical systems, false at others—I allow that some counter-mathematicals are determinately false on the orthodox semantics. For example: “If the real number – had had a square root, it would have had only one square root” is determinately false: in the closest mathematical systems in which it has a square root, it has two.]

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    () states would be just as they are, they would be different, they would be whatever you please. To the extent that the conditional seems non-trivial, it only reiterates the assumption that the metaphysical entities are causally independent of our states; and to suppose causal dependence is a prerequisite of knowledge, or reasonable belief, is to suppose just what the realist is at pains to deny. The skeptic’s argument cannot compel the realist. Nor, I think, can the realist’s appeal to fundamental intuitions as incontrovertible “data” compel the skeptic. For what are fundamental intuitions if not fundamental beliefs, and such beliefs cannot serve as grounds for themselves. Nor will the skeptic be moved by Cartesian appeals to the clarity and distinctness of our fundamental intuitions or beliefs: illusions need not lack for clarity and distinctness. No, the realist should not expect to counter the skeptical challenge by argument; the only counter to a coherent skepticism is belief itself. Thus, I reject the skeptic’s demand that I give good reasons for my belief in metaphysical sorts of entity. I have my reasons, to be sure, but they are not reasons the skeptic will accept; and if asked to give reasons for these reasons, my reason giving soon comes to an end. I am not against believing without good reasons; for that is just to allow that some beliefs are basic, and cannot be supported by reasons at all. What I am against is believing for bad reasons. Better, I say, to reject the skeptic’s demand for reasons altogether, then to put forth parochial reasons as grounds for belief.

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Postscript to “Realism without Parochialism” () The linchpin of my project to renounce pragmatic criteria of truth without sacrificing realism is my acceptance of the principle: every coherent metaphysical theory is true. By “metaphysical theory” I meant, not “theory promulgated by metaphysicians,” but “theory interpreted to be about (what I call) metaphysical sorts of entity,” such as mathematical or modal entities, entities that are not part of the concrete, physical realm. In this postscript, I consider two familiar objections to this principle that I think are easily disarmed. I then consider what I take to be the largest challenge: determining the proper scope of the principle. But first, I want to say three things by way of clarifying the principle itself. I apply the principle first, in Section ., to mathematical theories. This application has a historical pedigree, as I note, going back at least to Hilbert.¹ But it has also been endorsed by many contemporary philosophers who accept versions of mathematical structuralism. If one takes mathematical theories to be true, not of mathematical structures, but of mathematical systems (as I do in Chapter ), the resulting view may be called “plenitudinous platonism.”² In Section ., I apply the principle also to physical (and other scientific) theories, theories that describe alternative ways that concrete, physical reality might be. In this case, the principle leads to the positing of concrete possible worlds and possibilia: false physical theories, if coherent, are true in non-actual possible worlds. This application of the principle is well known but less well accepted by philosophers when the posited worlds are concrete, and goes by the name “modal realism.” (See Chapter .) Somewhat surprisingly, philosophers who accept plenitudinous platonism rarely accept modal realism, and the philosopher most known for modal realism—namely, David Lewis—was not a plenitudinous platonist. I find this surprising because what I take to be the chief motivation for either of these views is that coherent theories need content—entities that the theories are about—and this motivation applies with equal justice and in similar ways to mathematical and physical theories. Indeed, by applying the principle to all coherent theories, I endorse a unified conception of the mathematical and modal realms. (Again, see Chapter .) But wait: didn’t I restrict ¹ It has also been attributed to Poincaré. He wrote, for example: “a mathematical entity exists provided there is no contradiction in its definition.” Poincaré (: ). But Poincaré was no platonist; he had a deflationary notion of mathematical existence. ² See especially the “full-blooded platonism” of Balaguer (). Structuralist accounts of mathematics are developed in Shapiro () and Resnik (). Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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   “  ” () the principle to coherent metaphysical theories? As noted above, that is not a restriction on theories but on how the theories are to be interpreted. For any theory, one can ask whether it is made true by the physical realm or whether it is made true by the metaphysical realm. The principle I endorse applies to all coherent theories when interpreted metaphysically. I call this view “plenitudinous realism.” A second clarification is this. The notion of truth involved in the principle, as I hope the examples in the chapter made clear, is not truth simpliciter, but relative truth: every coherent theory is true of some portion of reality. Coherent mathematical theories are true of mathematical domains; coherent physical theories are true in possible worlds. In determining whether a theory is true of some portion of reality, one must restrict all the quantifiers of the theory so that they range only over the entities that make up that portion. And if the theory refers to an individual entity (or natural kind) by name, one must consider counterparts of that entity (or kind) within the relevant portion of reality. I do not reject truth simpliciter, truth about reality as a whole. Indeed, if the restriction to the quantifiers and reference to counterparts is made explicit, the resulting theory is true simpliciter. But the principle has us get at what is true simpliciter in a piecemeal way, by determining first what is true in this portion of reality or that. The third point of clarification answers this question: why ‘coherent’ instead of ‘logically consistent’? In what ways could a consistent theory fail to be coherent, and so fail to be true of some portion of reality? When I wrote this chapter, I had two main reasons for formulating the principle using ‘coherent’. First, I was taking theories to consist of sentences, not propositions. A sentence may be logically consistent in virtue of its form but analytically or conceptually impossible and so not coherent, at least as these terms are ordinary applied to sentences. For example, ‘some bachelor is married’ is logically consistent, but not coherent. But if instead we take theories to consist of propositions—as I now prefer—whether structured or unstructured—and we take the consistency of propositions to depend only on content, not on form (as I do in Chapter ), then logical consistency and conceptual possibility coincide. This first reason for using ‘coherence’ no longer applies. Second, I was understanding ‘coherent’ to mean: consistent with the principles of the framework. Thus, unless all principles of the framework are logically necessary, consistency and coherence will not coincide. For example, I take the mereological thesis of unrestricted composition to be a principle of the framework. On the usual understanding of logical notions, this thesis is not logically necessary, and so the denial of the thesis is consistent; but if unrestricted composition is a principle of the framework, its denial is not coherent. But I now prefer to understand logic in a much broader way, indeed, in a way that makes all the principles of the framework, even the principle of plenitude now under discussion, logically necessary; see Chapter . And on that broad conception of logic, consistency and coherence coincide. To some extent, this is just a terminological decision; and it is not a decision most philosophers make. But, in any case, it means that my second reason for using ‘coherent’ instead of ‘consistent’ also no longer applies on my current understanding of these terms. I thus use ‘coherent’ and ‘consistent’ interchangeably in what follows. With these three clarifications in place, I can state the principle more simply. Rather than speaking of theories, I will speak of the single proposition that is the

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  “  ” ()



conjunction of all the members of the theory. (There is no problem here taking infinitary conjunctions if need be, since the Boolean algebra of propositions is complete; see Bricker : –). Then the principle I endorse is equivalent to what I call in Chapter  the “law of plenitude”: every consistent proposition is true of some portion of reality. The notion of consistency, and so this law, applies in the first instance to unstructured propositions. But since every structured proposition corresponds to at most one unstructured proposition, consistency, and this law, apply derivatively to structured propositions as well, however we choose to delineate structure. I am now in a position to make short shrift of the two objections to plenitudinous realism I most commonly hear. These objections may apply to close cousins of my view, but they do not apply to the view as I understand it. One of these close cousins is the view that Eklund (: ) calls “maximalism,” which he introduces as follows: “For a given sortal F, Fs exist just in case (a) the hypothesis that Fs exist is consistent, and (b) Fs do not fail to exist, simply as a matter of contingent empirical fact.”³ Without clause (b), he claims, the maximalist would be committed to the existence of entities—such as yetis—that “we have empirical reasons not to believe in.” He concedes that there are “significant problems” concerning the formulation of (b), but sets those problems aside and relies on “an informal understanding.” Indeed, clause (b) is problematic. If “matters of contingent fact” include some or all facts of existence, it is hard to see how the maximalist criterion of existence avoids circularity; it needs a characterization of “what exists as a matter of contingent fact” that is independent of “what exists simpliciter.” Call this first problem for maximalism the problem of empirical conflict.⁴ Fortunately, it is not a problem for the plenitudinous realist’s criterion of existence, the law of plenitude, because as propositions are being interpreted there is no need for a clause (b): the extent of the physical realm—what I elsewhere call the realm of absolute actuality—is irrelevant to what the law claims. To see this, consider the following familiar hypothesis, once believed by the scientific community: the laws of physics are Newtonian, and there is a planet, Vulcan, located between Mercury and the sun that is responsible for the anomalous precession of Mercury’s perihelion. This hypothesis, of course, is now known to be false—false, that is, of the physical universe that we inhabit. But the hypothesis is consistent, and so by the law of plenitude, there is a portion or reality of which is it true. But wouldn’t that portion of reality have to contain Mercury and the sun, and thus require that Vulcan exist in the physical universe after all, contradicting well-established empirical fact? Of course not. According to plenitudinous realism, because the hypothesis is empirically false, the portion of reality of which the proposition is true is isolated from the portion of reality we inhabit. It contains counterparts of Mercury and the sun, not Mercury and the sun themselves. Because the hypothesis is interpreted “metaphysically,” so as to ³ I do not know how broadly Eklund understands ‘sortal’, or even whether he takes sortals to be linguistic expressions or non-linguistic concepts or properties. It won’t matter for what I say. And I will follow Eklund in being loose with use and mention. ⁴ Restall () argues that Balaguer’s plenitudinous platonism falls victim to this problem.

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   “  ” () allow its quantifiers to range over the metaphysical, or non-actual, realm, the facts of the physical, actual realm are irrelevant to whether or not the law of plenitude is satisfied. And because the hypothesis is interpreted in terms of counterparts, it makes no claims about the gravitational interactions of our Mercury and sun, no claims that can come into conflict with the contingent empirical facts. The second problem for maximalism—the problem Eklund takes to be more serious—is a version of the bad company objection that has been leveled against the neo-Fregean’s use of abstraction principles. There are abstraction principles, such as Hume’s Principle, that can only be satisfied in an infinite domain; and there are abstraction principles, such as Boolos’s Parity Principle, that can only be satisfied in a finite domain. (See Boolos .) Both principles are consistent. So maximalism requires that reality be both infinite and finite. Not good. But, of course, plenitudinous realism skirts this problem by invoking a relative notion of truth. The law of plenitude says only that each of these principles is true of some portion of reality, not reality as a whole. And there is no contradiction in holding that some portions of reality are finite while other portions are infinite. (Compare the treatment of nonEuclidean geometry in Section ..) Eklund goes on to consider examples involving what he calls incompatible objects; and these examples might, at first glance, appear to be more worrying for the plenitudinous realist. Suppose that it is consistent that F’s exist and it is consistent that G’s exist, but it is not consistent that F’s and G’s co-exist. Then the F’s and the G’s are what Eklund calls “incompatible objects.” Incompatible objects are, indeed, a problem for maximalism. But perhaps plenitudinous realism is threatened by them as well, even though it invokes a relative notion of truth. For, according to the law of plenitude, the F’s exist in some portion of reality, R, and the G’s exist in some portion of reality, S. Now consider the portion of reality that is their fusion, R+S. Don’t the F’s and G’s coexist in this portion of reality, thereby contradicting the supposed incompatibility of F’s and G’s? To solve this conundrum, we need to note an ambiguity in the characterization of “incompatible objects.” When we say, an F cannot coexist with a G, do we give F and G wide scope or narrow scope? If we give them wide scope, we get: for any x that is an F and any y that is a G, necessarily, x and y do not coexist. Then any F and any G are indeed incompatible objects. But if we give F and G narrow scope, we get instead: necessarily, there does not exist both an F and a G. In this case we might better say that F and G are incompatible sortals. Let us take these two readings in turn. The plenitudinous realist simply denies that there are incompatible objects. Let a be an F that is part of R and b be a G that is part of S. Then, a and b are also a part of R+S, and so co-exist in R+S (as well as in reality as a whole). a may not be an F in R+S, nor b a G. But in asking whether a and b are incompatible objects, we are concerned with the objects themselves, independently of how they are picked out.⁵ On the other hand, the plenitudinous realist will allow that there are incompatible sortals, where ⁵ Alternatively, we might understand “incompatible” in terms of (intrinsic) duplicates, as with Humean recombination principles; see Chapter  and below. The plenitudinous realist will also deny that there are incompatible objects in this sense. For any F and any G, there is a world where a duplicate of that F co-exists with a duplicate of that G.

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  “  ” ()



sortals F and G are incompatible just in case there is no portion of reality of which it is true to say both that there is an F and that there is a G. In particular, if F or G contains an explicit or implicit universal component, then F and G may be incompatible. For although the truth of purely existential propositions is preserved under expansions, the truth of purely or partly universal propositions is not. Consider the following illustration. Let F apply to objects that are spatiotemporally related to everything and let G apply to objects that are spatiotemporally related to nothing. That there is an F and that there is a G are both consistent propositions, the former true in spatiotemporal worlds, the latter true in worlds with no spacetime. But there is no world or portion of reality in which both these propositions are true. Such incompatible sortals, however, make no trouble for the plenitudinous realist. They just provide an illustration of something we knew all along, namely, that the consistency of propositions isn’t preserved under conjunction. The main challenge for plenitudinous realism is to say what the principles of the framework are, and then to justify that choice. The latter project is tantamount to providing an epistemology for the principles I take to be a priori. As an absolutist, I am prohibited from invoking pragmatic criteria in justifying these principles. For, applying my strategy for avoiding parochialism, I would then have to allow that there are alternative coherent frameworks, each with its own principles true of its own portion of a bigger reality. And because I would need a meta-framework to make sense of the alternative frameworks, I would be off on a vicious regress. So the project of justifying the principles of the framework is urgent, to be sure. But here I set it aside and just say something about which principles I accept. That will allow me to say something about which metaphysical theories I take to be absolutely necessary— built into the framework—and which metaphysical theories I take to be contingent— true of some portions of reality but not others. For the plenitudinous realist, different choices of principles for the framework lead to substantially different commitments as to the nature of reality. At one extreme, we might leave out some or all of the principles of classical logic, thus allowing that alternative logics are somehow coherent, each holding in some portion of reality. Towards the other extreme, we might take all of mathematics to follow from principles of the framework, thus rejecting plenitudinous platonism. Or, going farther, we might take our laws of nature to be principles of the framework, allowing only for a very limited version of modal realism. I do not find these extreme positions plausible, or well motivated. But there is a lot of middle ground over which plenitudinous realists may tussle. More liberal philosophers include less in the framework, thereby allowing more to be coherent, or in some sense contingent. More conservative philosophers include more in the framework, thereby taking more to be incoherent. As a Humean, I find myself towards the conservative end of this scale. In what follows, I say a few words about the principles I include in the framework, and then illustrate where I stand with respect to some controversial theories. The framework I accept starts with what Lewis (b) calls “megethology,” which adds to the logical apparatus of truth functions, identity, and singular quantification the apparatus of plural quantification and mereology. But, unlike Lewis, I also include higher-order plural quantification: quantification over pluralities of pluralities, and

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   “  ” () on up the hierarchy. (See Chapter  and below.) And, because I accept (a moderate version of) composition as identity, I take a generalized identity relation—being the same portion of reality as—as my single mereological primitive. (See Chapter .) In stating the principles of logic, I help myself to quantification over (abundant) propositions, properties, and relations, and to primitive logical notions that apply to these entities. Although these logical notions are ideological primitives of the framework, I take it as a working hypothesis that “intensional entities” and the logical notions that apply to them can be ontologically reduced to the particulars that make up reality. What is ideologically basic to our representation of reality need not be ontologically basic. (Again, see Chapter , and below.) The principles of logic and mereology that I include in the framework won’t be stated here. It suffices to note that their implications are classical through and through. Next I add to the framework a primitive notion of being (perfectly) natural, or being fundamental, that applies to properties and relations.⁶ As noted in the chapter, I apply naturalness to mathematical properties and relations no less than physical properties and relations; for it is the natural properties and relations that make mathematical and physical reality intrinsically structured.⁷ The structure of a portion of reality, be it a mathematical system or a possible world, is determined by the distribution of natural properties and relations over that portion of reality; it is the pattern of natural properties and relations. So, with a primitive notion of naturalness in hand, we can define the notion of structure.⁸ We can also define the notion of being an intrinsic duplicate, and the notions of internal and external relations, notions that will be needed to state the principles of the framework. (For the standard definitions, see Lewis a: –.) Two points of clarification are important. First, although I apply ‘natural’ and ‘fundamental’ only to properties and relations, I don’t thereby mean to exclude “intensional” entities of other or higher type; it is just that my preferred framework is a categorial grammar according to which all components of (structured) propositions are either individuals or properties/relations. But I do mean to exclude individuals: I do not think there is any distinction among individuals of being more or less natural, or more or less fundamental, that is not derivative from the naturalness ⁶ I hold, as a second working hypothesis, that a notion of relative naturalness, or fundamentality, can then be defined using logic. Should this project fail, relative naturalness would need to be taken as primitive. ⁷ On Lewis’s (, b) structuralist interpretation of mathematics, the truth of mathematics depends only on the size of reality: how or whether reality is (intrinsically) structured is irrelevant. This might aptly be called “structuralism without structure.” As I said in the chapter, I do not think that it gives a plausible interpretation of mathematical theories. ⁸ Two portions of reality r₁ and r₂ have the same structure iff there is a one-one mapping φ between their parts and a one-one mapping ψ between the perfectly natural properties and relations instantiated in r₁ and the perfectly natural properties and relations instantiated in r₂ such that (to take the case of a dyadic relation) for any relation R instantiated in r₁ and any parts x and y of r₁, x bears R to y iff ϕx bears ψR to φy. Note that we will need a principle of the framework that tells us which defined properties and relations are perfectly natural. Example: I take it that for any structure picked out by the earlier-than relation, the same structure could instead by picked out by the later-than relation. To get that result, we need to know that whenever a relation is perfectly natural, so is its converse. (Here, I am rejecting Lewis’s claim that the natural properties are a minimal supervenience base; see Lewis a: ). Intuitively, if the definition is in no way disjunctive, then the perfect naturalness of the components of the definiens is transferred to the definiendum. But it is no trivial matter to state the required principle rigorously, and in full generality.

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  “  ” ()



of the properties/relations they instantiate. In particular, I do not take parts to be more fundamental than the wholes that they compose simply in virtue of being parts. Second, it is important to distinguish different ways that properties or relations may be fundamental. They may be ideologically basic, or ontologically basic. The notions that are needed to express the principles of the framework count as ideologically basic. They are fundamental to our representation of reality, but need not be fundamental to reality itself. These two ways of being fundamental cut across one another. For example, I would say that the truth functions and quantifiers are ideologically basic, but not ontologically basic; “fundamental” physical properties and relations (mass, distance) and mathematical relations (successor) are ontologically basic, but not ideologically basic (because the principles of the framework are general, and quantify over these properties and relations); the primitive relations of mereology and plural logic—is the same portion of reality as and is one of—are both ideologically and ontologically basic. I do not think that these different ways of being fundamental amount to different conceptions of fundamentality, or naturalness. But it is important to keep the distinction in mind when asking how what we take to be fundamental is relevant to the structure and scope of reality itself. (See Chapter .) Among the principles of the framework that go beyond megethology are Humean principles of plenitude. I need principles strong enough to undergird all the various applications of Hume’s Dictum, that there are no necessary connections between distinct existents. I divide these principles into three sorts. (See Chapter  for more precise formulations; here they are generalized to apply to reality as a whole, not just the realm of possibilia, and to apply to any sort of mathematical structure, not just spatiotemporal structure.) In stating these principles, I make use of the notion of an island of reality: a maximal unified portion of reality. Islands may be unified by any external relation, and each island is absolutely isolated from every other: no inhabitant of an island stands in any external relation to anything not on the island. Possible worlds and mathematical systems are, I claim, the prime examples of islands of reality. (See Chapter  for arguments, and precise definitions.) Now for the principles. First, we need a principle of recombination such as the following: (R)

For any things, however scattered throughout reality, and any way those things could be arranged, there is an island of reality inhabited by duplicates of those things arranged in that way.

Second, we need a principle of plenitude guaranteeing the existence of aliens, what I call a principle of plenitude for contents. (In Chapter , I call this the principle of alien individuals.) (A)

Consider any thing and the island of reality it inhabits. There is another island of reality exactly like the first island except that the thing has been replaced by something alien to the first island.

Third, we need a principle of plenitude for structures, a principle that tells us what structures are instantiated by islands of reality. In Chapter  and its Postscript, I consider principles that tell us what structures are instantiated by possible worlds. But the principle I accept is simpler if we can ignore whether the island of reality is properly called a “world.” I accept:

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   “  ” () (S)

For any structure compatible with the framework,⁹ there is an island of reality that has (exactly) that structure.

Principle (S) is behind my claim in the chapter that, not only is any coherent theory true of some portion of reality, but true of some portion of reality that matches the theory. These three principles of plenitude work together to ensure that Hume’s Dictum will be in full force. I need also to mention a (meta-)principle that constrains what primitives can be added to the framework. Goodman (: ) expressed it this way: “There cannot be difference of entities without difference of content.” Understanding the “content” of an entity to be the entities that it, in some sense, is composed of, we can take Goodman’s principle to be this: (G)

For any mode of composition, numerically distinct entities are never composed of the same elements.

The instance of (G) that applies to mereological composition is already a principle (or theorem) of the framework: mereological composition is unique. But (G) tells us more; it tells us that the framework can brook no non-mereological mode of composition that fails to be unique. Strictly speaking, (G) does not rule out a nonmereological mode of composition if it satisfies uniqueness. But, even if there were a plausible example of such,¹⁰ I do not think it could be included together with mereological composition as part of the framework. For the disjunction of two modes of composition would itself be a mode of composition that violates uniqueness. Thus, as I understand (G), it rules out adding any non-mereological compositional primitives to the framework. Mereology is the only mode of composition. I consider (G) to be part of my Humeanism. Following Lewis (a: ), I take (G) and the Humean dictum to be inextricably linked. (See Section . for some discussion.) The necessary connections that come with non-mereological modes of composition do not get a free pass by labeling them “constitutional”; they are not thereby made intelligible. But Humeanism is a big tent; and many philosophers who are Humean in other respects will not follow me in accepting (G). It conflicts, or at least appears to conflict, with the acceptance of entities they hold dear: maybe classes, or structural universals, or states of affairs. (G) is sometimes associated more with “nominalism” than with Humeanism, perhaps because of its link to Goodman. But that isn’t quite right either: a philosopher who accepts simple universals while rejecting structural universals and states of affairs need not run afoul of (G). In any case, however classified, (G) is a powerful principle with significant consequences for the nature of reality.¹¹

⁹ How the framework is relevant to what structures are possible comes out in the final section below. ¹⁰ The operation that forms classes from their members might be taken to be a non-mereological mode of composition that satisfies uniqueness. But if the relation is composed of must be transitive, as I am inclined to say, then classes are composed, not of their members, but of their urelements; and the operation that generates the classes is the transitive closure of the membership relation. ¹¹ Goodman’s principle, it seems to me, is more often implicitly relied on than explicitly endorsed; see, for example, the literature rejecting coincident entities. Lewis (c: ) is explicit: “there is only one mode of composition; and it is such that, for given parts, only one whole is composed of them.” Lewis

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  “  ” ()



I have provided only a partial characterization of the framework and its principles.¹² But enough has been put in place to take up some familiar metaphysical controversies, for example, over the existence of “intensional entities,” of classes, of universals, and of emergent properties. I begin with theories of “intensional entities.” As I noted in the chapter, I am a pluralist: I accept abundant and sparse conceptions; structured and unstructured conceptions; extensional, intensional, and hyperintensional conceptions. (See Bricker .) For the pluralist, these different conceptions are not in conflict with one another. But in this case the avoidance of conflict does not derive from an application of the law of plenitude. It is not that the different conceptions apply to different portions of reality. Rather, all the conceptions apply to all portions of reality. For any portion, we can ask what abundant or sparse properties are instantiated there, and we can take the instantiated abundant properties to be either structured or unstructured, intensional or hyperintensional. In this case, we avoid conflict the old-fashioned way: we disambiguate. The conceptions do not conflict because they do not mean the same thing by ‘property’. In the chapter to which this is a postscript, I should have been more careful to distinguish between two types of coherent metaphysical theory. In both cases, in virtue of being coherent the theory is true of some portion of reality. But some coherent theories are true of the whole of reality, such as general theories of propositions or possible worlds; these theories may themselves be taken to be, or follow from, principles of the framework. Other coherent metaphysical theories are true only of some smaller portion of reality, such as contingent theories describing the structure, or laws, or qualitative character of some world or plurality of worlds. Now, because quantification over propositions and other “intensional entities” will be needed to state the principles of logic (see Section .), it follows that a theory of “intensional entities” will need to follow from the principles of the framework. That raises the question: what constraints should be imposed on the interpretation of this theory? When I include some theory as part of the framework, I mean to commit myself to a realist interpretation of that theory. Take the case of propositions. It is not enough to make our discourse about propositions true by means, say, of a fictionalist paraphrase that shirks ontological commitment. The quantifiers over propositions must be taken to be objectual, and objectual quantification, I hold, is ontologically committing. That leaves open whether the propositions, under a given conception, should be taken as basic or ontologically reduced to other entities. But the former avenue is closed off: taking propositions to be basic would conflict with Humean principles of plenitude, as Lewis (a: –) convincingly argued in his critique of magical ersatzism. The relation that holds between a proposition and what makes that proposition true holds with absolute necessity. But if this argument against taking intensional entities to be basic is accepted, then the judgment that the theory is () also explicitly endorses the principle, but emphasizes that it doesn’t rule out his mereological account of classes according to which the singleton relation is primitive: a singleton is nowise composed of its member. Van Cleve () also explicitly endorses the principle. ¹² In Chapter  I endorse additional principles of the framework having to do with the size of reality and the size of islands of reality.

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   “  ” () coherent is held hostage to there being an acceptable ontological reduction. As I note in Chapter , I take it as a working hypothesis that there is some such reduction of what I call the realm of representation to the reality of things. For example, abundant unstructured propositions may be taken to be classes of portions of reality, and abundant structured propositions may be taken to be more complex class-theoretic constructions in familiar ways. But that brings us straightway to the problem of the existence of classes. For without classes (or other class-like entities), it is hard to see how the reduction could succeed. It is common indeed to include a theory of classes (or sets) among the principles of the framework. But I do not think that theories of classes, as ordinarily understood, are coherent. Classes, no less than propositions, make trouble for Humeans. They are up to their neck in illicit necessary connections. For, surely, as classes are ordinarily understood, I cannot exist without my singleton existing, nor can my singleton exist without me. Nonetheless, a great many Humeans unreflectively accept classes. And even those Humeans who renounce classes do so on grounds of their being abstract, or causally isolated from the concrete realm, not on explicitly Humean grounds. How do they justify excluding classes from the Humean denial of necessary connections? Humean apologists for classes take various lines. Most common, I suppose, is to hold that classes are composed in some non-mereological way from their urelements, thereby rejecting (G). On this approach, a class and its members are not, in the relevant sense, “distinct.” Hume’s Dictum, they then claim, with ‘distinct’ properly understood, is no threat to classes; the necessary connections are harmless, indeed, to be expected.¹³ This defense of classes seems rather strained when applied to singletons: how does it make sense to say that a singleton is composed, even nonmereologically, of its sole member? But let me set this problem aside, stipulating that ‘mode of composition’ as it occurs in (G) ranges over purported modes that allow one thing to be the sole component of another thing. With that stipulation in place, I cannot fault this reply if made by a self-proclaimed Humean who rejects (G). I can only repeat what I claimed above: a full-blooded Humean should accept (G). Lewis (: –) takes a different line. He concedes that singletons are mysterious, that if we understand the singleton relation, we know not how we do it. But he nonetheless accepts the singleton relation as primitive, and uses it to formulate principles of the framework. He also accepts (G), and therefore understands Hume’s Dictum to say: there are no necessary connections between mereologically distinct existents. Moreover, singletons for Lewis are mereological atoms, and are mereologically distinct from their sole member. Why then does he not take the necessary connections that accompany the singleton relation to be Humeanly objectionable? Lewis’s strategy is this. Humean principles of plenitude are violated only if a singleton is necessarily connected to the intrinsic nature of its member, whereas a singleton is instead necessarily connected to the member itself. The necessity involved is thus necessity de re, and for Lewis necessity de re is not objectionable ¹³ See Wilson (: –) who in effect takes it to be a requirement on an adequate interpretation of “Hume’s Dictum” that ‘distinct’ be understood in a way that does not rule out classes, or other entities that involve “constitutional necessities.”

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  “  ” ()



when interpreted in terms of counterpart theory. For example, the Humean who accepts counterpart theory can allow that I could not exist without my parents by holding that any world that contains a counterpart of me also contains counterparts of my parents; that does not violate any Humean principle of recombination. Similarly, Lewis claimed, the Humean can allow that I cannot exist without my singleton by holding that any world that contains a counterpart of me also contains a counterpart of my singleton. The flexibility of counterpart theory will guarantee that counterparts can be assigned in a way that makes that true (or, if preferred, true in some contexts and false in others). In short: Humean principles rule out necessary connections between the intrinsic natures of things not the things themselves. That is why, Lewis thought, the necessary connections accompanying the singleton relation did not violate Hume’s Dictum.¹⁴ I do not think, however, that this defense of singletons succeeds. It is not enough to hold that there is some interpretation of the necessity involved under which the necessary connections are Humeanly acceptable. One must show that there are no violations of Humean principles however the necessity is interpreted. Granted, one cannot evaluate whether violations occur without making some assumptions as to the intrinsic nature of singletons. But if there are violations on all plausible assumptions, then singletons—and classes generally—are in trouble. Consider first (A), the principle of plenitude that guarantees aliens. Because the singleton relation is external, a thing and its singleton inhabit the same island of reality. Applying (A) to the singleton results in a different island that contains a duplicate of the thing but no duplicate of the singleton. And this conflicts with the following plausible assumption: singletons of duplicates are themselves duplicates. This assumption will hold if a singleton inherits the intrinsic character of its member, or if a singleton has no substantial intrinsic character at all. The only way the principle could fail, it seems, is if the intrinsic nature of a singleton depends on its particular member; and that would violate the Lewisian understanding that intrinsic natures are purely qualitative. It seems, then, that classes run into Humean trouble that no resort to counterpart theory can obviate. Consider next (S), the principle of plenitude for structures. Start with any island of reality that contains a thing and its singleton. Consider the structure that applies to just the individuals—that is, the non-classes—that inhabit the island. Applying (S), we have an island of reality that has exactly that structure, and therefore where the individuals have no singletons at all. This conflicts with the assumption, made by Lewis () and all standard class theorists, that every individual has a singleton. Again, Humean principles of plenitude make trouble for theories of classes that no resort to counterpart theory can obviate. None of this, however, is a problem for purely mathematical theories of classes, not if one accepts the structuralist claim, as I do, that mathematics is concerned only with structure. As I noted in the chapter, there will be islands of reality—mathematical systems—where second-order Zermelo-Fraenkel set theory is true, islands that differ ¹⁴ There was still the problem of how we could possibly understand the singleton relation. Not long after publishing Parts of Classes, Lewis inclined more towards a structuralist account of singletons. See Lewis (b).

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   “  ” () in size. Moreover, for any cardinal number κ, there will be islands of reality where Zermelo-Fraenkel set theory with κ urelements is true. If we let κ be the number of parts of our physical universe, we can use principles of plenitude to get an island of reality, one part of which is a duplicate of our physical universe, the rest of which is a “superstructure” of “classes” satisfying Zermelo-Fraenkel set theory with κ urelements.¹⁵ But no purely structural account of classes can do the work required of philosophical theories of classes. Take, for example, the identification of abundant properties with classes. Abundant properties are instantiated by things inhabiting different islands, different worlds. But no purely structural account can allow classes to have members from different islands. The Humean who wants to invoke classes for purposes of ontological reduction will need a different approach. There is a different approach, one that allows the Humean to accept our ordinary conception of classes and the necessary connections that come with them. As I mentioned above, I take the framework of megethology to include higher-order plural quantification. I propose, then, to understand classes as pluralities of individuals, and pluralities of pluralities of individuals, and so on up the hierarchy all the way into the transfinite. On this account, there is no problem saying that classes have members taken from different islands of reality, for “being a member” is just the logical relation “is one of,” not a mathematical or structural relation. And there is no longer any mystery over the necessary connections. My singleton is not distinct from me. The difference between me and my singleton, so to speak, is a difference in modes of referring to a single entity. Wherever I go, so goes my singleton because wherever I go, so goes the possibility of referring to me using either singular or plural modes of reference. Since I am identical with my singleton, there is obviously no objectionable necessary connection, no violation of Hume’s Dictum. Note that what I am proposing is a thoroughly realist interpretation of theories of classes; it is not some sort of eliminativist paraphrase. Plural quantifiers, no less than singular quantifiers, are ontologically committing. But plural quantification over individuals—even of higher order—is not committed to anything beyond the individuals themselves.¹⁶ Classes exist, sure enough, but they are not controversial abstract entities, not if one rejects the singularist dogma that plural quantification is just quantification over classes, or class-like entities.¹⁷ With classes resurrected, I move now to our third topic: the coherence of universals. In Chapter , I plumped for an account of reality that grounded the qualitative character of things in tropes, not universals; everything that exists is particular. But if theories of universals are coherent no less than theories of tropes, the law of ¹⁵ How do we know whether or not we inhabit a physical universe with such a superstructure of “classes”? We don’t. The superstructure makes no difference to any observation we could make; it is causally and explanatorily inert. But ignorance of this sort is a feature, not a bug. It is only to be expected once one embraces plenitudinous realism. ¹⁶ Though it may be committed to the individuals in different ways; see Rayo (: –) on what he calls “plethological commitment.” ¹⁷ See Section ., for a bit more discussion. Rayo () gives one way of developing this sort of approach. I hope to present my own account elsewhere. There are land mines that need to be carefully avoided.

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  “  ” ()



plenitude demands that there be portions of reality where things instantiate universals. Indeed, reality will divide into regions with universals but no tropes, regions with tropes but no universals, and regions with both. One could choose to accept a theory of tropes on pragmatic grounds. But, for the plenitudinous realist, that would be a decision to ignore universals, not a reason to excise universals form reality. But I do not think that theories of universals are coherent. One problem afflicts theories that posit structural universals. Structural universals involve some nonmereological mode of composition; I deem them incoherent in virtue of violating (G). (See Lewis b.) But the incoherence goes deeper, afflicting the simple universals themselves; and the violations involve the core Humean principles of plenitude, not the principle (G) that many Humeans do not accept. The problem comes from the very idea that universals are immanent, that they are part of reality. That requires characterizing them either as “multiply located” or “wholly present” in their instances. Either way, the universals theorist must be an ontological dualist: either she must posit, in addition to the ontological category of universals, an ontological category of locations (whether spatiotemporal or something more general), allowing distinct locations to be “occupied” by the same universal; or she must posit an ontological category of particulars, allowing distinct particulars to be “tied” by instantiation to the same universal.¹⁸ That leads to a rather flatfooted argument against universals based on the principle of recombination (R): there would have to be portions of reality, for example, where particulars occupy universals, or where universals instantiate particulars. But that argument, I think, is rather weak. It is only to be expected that a Humean that was an ontological dualist would need to restrict (R) to “category-preserving” arrangements. It is enough that the Humean allows all particulars to freely recombine with one another and all (monadic) universals to freely recombine with one another, without also allowing particulars and universals to freely recombine. Indeed, I am an ontological dualist: I accept a categorial distinction between the actual and the merely possible. (See Chapter .) But I take the distribution of actuality over reality to be brute; it is not determined by applying a principle of recombination. Recombination, I hold, applies only to qualitative properties, and the property of actuality is not qualitative. The incoherence of universals on my account has a different source: it arises because in applying (R) to universals, it is the universals themselves that are recombined at other worlds, not their duplicates. Indeed, it is constitutive of universals that they have no qualitative duplicates. That leads to a conflict with the principle of plenitude for structures (S). For according to (S), reality divides into absolutely isolated regions—islands of reality—that do not overlap. Different structures are instantiated by distinct islands. (See Chapter .) Consider one such island and a universal U located there. By (R), U will also be located in other islands. But then the islands overlap one another, and so are not islands after all, contradicting (S).

¹⁸ It would take me too far afield to say why I do not think a “bundle theory” of universals is compatible with the idea that universals are “multiply located.” In any case, retreating to a bundle theory would only escape the argument from recombination below, not the argument from isolation.

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   “  ” () I conclude that the Humean account of plenitude that I accept as part of the framework is incompatible with immanent universals. I close with a case that swings the other way, a case where I am quite liberal with respect to the framework. First, note that all of the theories considered in the chapter involved singular predicates, that is, predicates all of whose argument places are singular. When these theories are naturally instantiated by some domain—some portion of reality—the instantiating natural properties and relations are singular as well. The entities that instantiate these properties and relations, of course, may be composite. Thus, the law of plenitude already requires that there be emergent natural properties instantiated in reality, where a natural property is emergent iff its instantiation by a whole does not depend on the instantiation of natural properties and relations among its proper parts. There will be coherent theories that posit emergent natural properties. But so far there is nothing surprising: emergent natural properties are widely taken to be metaphysically possible, indeed, even actual on many interpretations of quantum mechanics. When an emergent property is singular, the different ways that a whole can be decomposed into proper parts is irrelevant to the instantiation of the property by the whole. But now consider theories that involve plural predicates, predicates some of whose argument places are plural. It is a consequence of my acceptance of plural logic in the framework that there are such theories. When these theories are naturally instantiated by some portion of reality, some of the instantiating natural properties and relations are irreducibly plural. The law of plenitude will require that there be emergent natural properties instantiated in reality that are slice-sensitive, that apply to one plurality and not another, even though these pluralities have the same fusion, are the same portion of reality. There will be coherent theories that posit slice-sensitive fundamental properties, and that, I think, has far-reaching consequences. For example, I argue in Chapter  that the possibility of slicesensitive fundamental properties (or relations) is a reason why strong versions of composition as identity should be rejected. For such properties require that the distinction between singular and plural is not just a matter of how we represent reality, but a feature of reality itself. The four examples presented here have, I hope, given a taste of the power of the Humean framework, both to rule out widely accepted theories as incoherent, and to allow for possibilities that go beyond what we ordinarily take to be possible. For the plenitudinous realist, these are not mere possibilities in some abstract sense. They are concretely realized in the vast expanse of reality.

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PART 

Possible Worlds Realism and Actuality

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 Concrete Possible Worlds () . Introduction Open a book or article of contemporary analytic philosophy, and you are likely to find talk of possible worlds therein. This applies not only to analytic metaphysics, but to areas as diverse as philosophy of language, philosophy of science, epistemology, and ethics. Philosophers agree, for the most part, that possible worlds talk is extremely useful for explicating concepts and formulating theories. They disagree, however, over its proper interpretation. In this chapter, I discuss the view, championed by David Lewis, that philosophers’ talk of possible worlds is the literal truth.¹ There exists a plurality of worlds. One of these is our world, the actual world, the physical universe that contains us and all our surroundings. The others are merely possible worlds containing merely possible beings, such as flying pigs and talking donkeys. But the other worlds are no less real or concrete for being merely possible. Fantastic? Yes! What could motivate a philosopher to believe such a tale? I start, as is customary, with modality.² Truths about the world divide into two sorts: categorical and modal. Categorical truths describe how things are, what is actually the case. Modal truths describe how things could or must be, what is possibly or necessarily so. Consider, for example, the table at which I am writing. The table has numerous categorical properties: its color, perhaps, and its material composition. To say that the table is brown or that it is made of wood is to express a categorical truth about the world. The table also has numerous modal properties. The table could have been red (had it, for example, been painted red at the factory), but it could not, it seems, have been made of glass, not this very table; it is essentially made of wood. Just where to draw the line between the categorical and the modal is often disputed. But surely (I say) there is some level—perhaps fundamental physics—at which the world can be described categorically, with no admixture of modality. Now, suppose one knew the actual truth or falsity of every categorical statement. One might nonetheless not know which truths are necessary nor which falsehoods are possible. One might be lacking, that is, in modal knowledge. In some sense, then, the modal transcends the categorical. First published in T. Sider, J. Hawthorne, and D. W. Zimmerman (eds.), Contemporary Debates in Metaphysics (Blackwell, ), –. Reprinted with permission of John Wiley & Sons, Inc. It was written with an audience of non-specialists in mind: advanced undergraduates and beginning graduate students. Thanks to Ted Sider for comments on an earlier draft. ¹ The fullest statement of Lewis’s theory of possible worlds is contained in his magnum opus, Lewis (a), On the Plurality of Worlds. Lewis’s view is sometimes called “modal realism.” ² Historically, it was the attempt to provide semantics for modal logic that catapulted possible worlds to the forefront of analytic philosophy. The locus classicus is Kripke (). Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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    () And that’s trouble; for modal statements are problematic in a way that categorical statements are not. I see that the table is brown, but I do not see that it is possibly red. I can do empirical tests to determine that it is made of wood, but no empirical test tells me that it is essentially made of wood. By observation, I discover only the categorical properties of objects, not their modal properties. That makes special trouble for the empiricist, who holds that all knowledge of the world must be based on observation. But empiricist or not, modal properties are mysterious: they do not seem to be among the basic or fundamental ingredients that make up our world. What to do? Should we turn eliminativist about modality, holding that modal statements are unintelligible, or, at any rate, that their communicative purpose is not descriptive?³ That would be implausible. We assign truth and falsity to modal statements in principled ways; we reason with modal statements according to their own peculiar logic. No, we must hold that modal statements are descriptively meaningful, but not fundamental. Thus begins the search for an analysis of modal statements, the attempt to provide illuminating truth conditions for modal statements without just invoking more modality. Consider this. Modal statements can be naturally paraphrased in terms of possible worlds. For example, instead of saying “it is possible that there be blue swans,” say “in some possible world there are blue swans.” Instead of saying “it is necessary that all swans be birds,” say “in every possible world all swans are birds.” When paraphrased in this way, the modal operators ‘it is possible that’ and ‘it is necessary that’ become quantifiers over possible worlds. Intuitively, these paraphrases are merely a façon de parler, and not to be taken with ontological seriousness. But perhaps the ease with which we can produce and understand these possible worlds paraphrases suggests something different: the paraphrases provide the sought after analyses of the modal statements; they are what our modal talk has been (implicitly) about all along.⁴ And if that is so, then we have a reduction of the modal to the categorical after all: although the modal properties of this world transcend the categorical properties of this world, they are determined by the categorical properties of this world, and other possible worlds. But the introduction of possible worlds may seem to raise more questions than it answers. What are these so-called worlds? What is their nature, and how are they related to our world? Philosophers who believe in possible worlds divide over whether worlds are abstract or concrete. I use the terms ‘abstract’ and ‘concrete’ advisedly: there are (at least) four different ways of characterizing the abstract/concrete distinction, making the terms ‘abstract’ and ‘concrete’ (at least) four ways ambiguous.⁵ Fortunately, however, on the Lewisian approach to modality I am considering, the worlds turn out (with some minor qualifications) to be “concrete” on all four ways of drawing the distinction.

³ The locus classicus of the eliminative approach to modality is “Reference and Modality” in Quine (). ⁴ Of course, there is much more to modality than statements of (metaphysical) possibility and necessity. But the project of paraphrasing more complex modal locutions in terms of possible worlds has also met with considerable success. ⁵ For a discussion of the four ways, see Lewis (a: –).

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



(C)

Worlds (typically) have parts that are paradigmatically concrete, such as donkeys, and protons, and stars.

(C)

Worlds are particulars, not universals; they are individuals, not sets.

(C)

Worlds (typically) have parts that stand in spatiotemporal and causal relations to one another.

(C)

Worlds are fully determinate;⁶ they are not abstractions from anything else.

It is convenient, then, and harmless, to call worlds “concrete” if they satisfy all four conditions listed above. The concrete worlds taken altogether I call the modal realm, or, following custom, logical space.⁷ The denizens of logical space—the worlds and their concrete parts—are called possibilia. In one sense, concrete possible worlds are like big planets within the actual world: two concrete worlds do not have any (concrete) objects in common; they do not overlap. Thus, you do not literally exist both in the actual world and in other merely possible worlds, any more than you literally exist both on Earth and on other planets in our galaxy. Instead, you have counterparts in other possible worlds, people qualitatively similar to you, and who play a role in their world similar to the role you play in the actual world. This constrains the analysis of modality de re—statements ascribing modal properties to objects. When we ask, for example, whether you could have been a plumber, we are not asking whether there is a possible world in which you are a plumber—that is trivially false (supposing you are not in fact a plumber), since you inhabit only the actual world. Rather, we are asking whether there is a possible world in which a counterpart of you is a plumber.⁸ (More on this in Section . below.) In another sense, however, concrete worlds are quite unlike big planets within the actual world. Each possible world is spatiotemporally and causally isolated from every other world: one cannot travel between possible worlds in a spaceship; one cannot view one world from another with a telescope. But although this makes other possible worlds empirically inaccessible to us in the actual world, it does not make them cognitively inaccessible: we access other worlds through our linguistic and mental representations of ways things might have been, through the descriptions we formulate that the worlds satisfy. I say that merely possible concrete worlds, no less than planets within the actual world, exist. I do not thereby attribute any special ontological status to the worlds (or planets): whatever has any sort of being “exists,” as I use the term; ‘existence’ is coextensive with ‘being’. Of course, merely possible concrete worlds do not actually exist. For the realist about concrete worlds, existence and actual existence do not coincide. In most ordinary contexts, no doubt, the term ‘exists’ is implicitly restricted

⁶ An object is fully determinate if and only if, for any property, either the property or its negation holds of the object. In the case of worlds, this is equivalent to: for any proposition, either the proposition or its negation is true at the world. ⁷ Because logical relations between propositions can be represented by relations between classes of worlds; for example, one proposition logically implies another just in case the class of worlds at which the one is true is included in the class of worlds at which the other is true. But see n. . ⁸ The analysis of modality de re in terms of counterpart relations was first introduced in Lewis ().

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    () to actual things—as, for that matter, is the term ‘is’. The phenomenon of implicit contextual restriction allows us to truly say, in an ordinary context, “flying pigs do not exist” or “there are no flying pigs,” without thereby denying the existence of concrete non-actual worlds teeming with flying pigs. It is the same phenomenon that allows us to truly say, when there is no beer in our fridge, “there is no beer,” without denying that there are other fridges in the world packed with beer. Ordinary assertions of nonexistence, then, do not count against realism about concrete worlds. Call approaches to modality that analyze modality in terms of concrete possible worlds and their parts Lewisian approaches. I take the following four theses to be characteristic of Lewisian approaches to modality. () () () ()

There is no primitive modality. There exists a plurality of concrete possible worlds. Actuality is an indexical concept.⁹ Modality de re is to be analyzed in terms of counterparts, not transworld identity.

In what follows, I devote one section to each of these theses. I write as an advocate for Lewisian approaches, and feel under no obligation to give opposing views equal time. For each thesis, I take Lewis’s interpretation and defense as my starting point. I then consider and endorse alternative ways of accepting the thesis, some of which disagree substantially with Lewis’s interpretation or defense. There is more than one way to be a Lewisian about modality.

. No Primitive Modality The rejection of primitive modality is a central tenet of Lewisian approaches. It motivates the introduction of possible worlds, the most promising avenue of analysis. And it motivates taking possible worlds to be concrete: Lewis’s most persistent complaint against accounts of worlds as abstract is that they must invoke primitive modality in one form or another. But for all the talk of rejecting primitive modality by Lewis and others, there is no clear agreement as to just what this means. Indeed, I think three different and independent theses have been taken to fall under the “no primitive modality” banner. Although Lewis accepts all three theses, only one of them is truly central to the Lewisian approach. Before turning to discuss that central thesis, I will briefly mention the other two. First off, it is natural to interpret “no primitive modality” as a supervenience thesis: the modal supervenes on the categorical. To say that modal statements supervene on categorical statements is to say that there can be no difference as to how things are modally without some difference as to how they are categorically. Or, cashed out in terms of possible worlds: whenever two possible worlds differ in their modal features, they differ in their categorical features as well. Thus, for example, no two possible worlds differ just with respect to brute dispositional properties, or primitive causal ⁹ What it means for the concept of actuality to be indexical, and how that relates to whether the property of actuality is relative or absolute, is discussed in Section .. [See also Chapter .]

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  



relations. This supervenience thesis is central to Lewis’s (broadly) Humean analyses of laws, causation, and counterfactuals, of the physical and causal modalities. But the thesis is not directed at the metaphysical (or logical) modality that is the target of our current analytic endeavor. I therefore set it aside. A second way one might understand the “no primitive modality” thesis is as what Lewis calls a principle of recombination. His initial formulation of the principle is: “anything can coexist with anything else, and anything can fail to coexist with anything else.” Lewis (a: ). The first half, when spelled out, says that any two (or more) things, possibly from different worlds, can be patched together in a single world. To illustrate: if there could be a unicorn, and there could be a dragon, then there could be a unicorn and a dragon side by side. Since worlds do not overlap, a unicorn from one world and a dragon from another cannot themselves exist side by side. So the principle is to be interpreted in terms of intrinsic duplicates: at some world, a duplicate of the unicorn and a duplicate of the dragon exist side by side. The second half of the principle of recombination, spelled out in terms of worlds and duplicates, says this: whenever two (non-overlapping) things coexist at a world, neither of which is a duplicate of a part of the other, there is another world at which a duplicate of one exists without a duplicate of the other. To illustrate: since a talking head exists contiguous to a living human body, there could exist an unattached talking head, separate from any living body. More precisely: there is a world at which a duplicate of the talking head exists but at which no duplicate of the rest of the living body exists.¹⁰ According to Lewis, the principle of recombination expresses “the Humean denial of necessary connections between distinct existents.” Lewis (a: ). But two caveats are needed. First, only the second half, strictly speaking, embodies a denial of necessary connections; the first half embodies instead a denial of necessary exclusions. And, second, the thesis that the modal supervenes on the categorical is an alternative way to capture the denial of necessary connections, one that may be closer to Hume’s intent; it denies that there are necessary connections in the world, ontological interlopers somehow existing over and above the mere succession of events, and somehow serving as ground for modal truths about powers or laws or causation. Lewis sometimes charges those who reject principles of recombination with being committed to primitive modality.¹¹ But I do not think that a principle of recombination is the right way to capture the “no primitive modality” thesis. Violations of recombination impose a modal structure on logical space, allowing that the existence of some possibilia necessarily entail or exclude the existence of other possibilia; but the structure imposed need not be primitive modal structure. It may be that the violations can all be accounted for in non-modal terms, that necessary connections and exclusions only occur when some non-modal condition is satisfied. Thus, contra what Lewis suggests, violations of recombination are not—or at least not by themselves—primitive modality.

¹⁰ [See Chapter  for a detailed exposition of principles that generalize Lewis’s recombination principle, and make it more precise.] ¹¹ See Lewis’s discussion of “magical ersatzism” in Lewis (a: –).

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    () The first two interpretations of the “no primitive modality” thesis each put constraints on logical space, though in opposite ways. The supervenience thesis demands that there not be too many worlds, that there never be two worlds that differ modally without differing non-modally. Principles of recombination demand that there not be too few worlds, that there always be enough worlds to represent all the different ways of recombining. The third interpretation—the one we’re after—is different: it puts constraints on our theorizing about logical space, not (directly) on logical space itself. It demands that our total theory, our best account of the whole of reality, can be stated without recourse to modal notions, that the (primitive) ideology of our total theory be non-modal.¹² Can the Lewisian meet this demand? First, we need to know what terms of our total theory count as non-modal. Here I will suppose this includes the Boolean connectives (‘and’, ‘or’, ‘not’) and unrestricted quantifiers (‘every’, ‘some’) of logic, the ‘is a part of ’ relation of mereology, the ‘is a member of ’ relation of set theory, the ‘is an instance of ’ relation of property theory, and a (second-order) predicate applying to just those properties and relations that are fundamental, or perfectly natural.¹³ I will also suppose that the notion of a spatiotemporal relation is nonmodal; perhaps it can be given a structural analysis in terms of the above.¹⁴ Now, let us suppose that the Lewisian has succeeded in providing possible worlds paraphrases for the vast panoply of modal locutions. That still leaves the notion of a possible world itself, an ostensibly modal notion. It won’t do simply to take this notion as primitive. At best, that would reduce the number of modal notions to one. If primitive modality is to be eliminated, the Lewisian must provide an analysis of ‘possible world’ in non-modal terms. It might appear that analyzing ‘possible world’ involves two separate tasks: first, analyzing the notion of world; then, distinguishing those worlds that are possible from those that are not. This second task, however, would appear to land the Lewisian in vicious circularity: possibility is to be analyzed in terms of possible worlds, which in turn is to be analyzed in terms of possibility, and round and round and round.¹⁵ But the threat of circularity is bogus because there is no second task to perform. On a conception of possible worlds as concrete, there are no impossible worlds. For suppose there were a concrete world at which both p and not-p, for some proposition p. Then there would be a property corresponding to p such that the world both had and didn’t have the property. Contradictions could not be confined to impossible worlds; they would infect what is true simpliciter, thereby ¹² I use ‘ideology’ roughly in Quine’s sense; see Quine (: –). But I do not suppose our total theory is couched in first-order predicate logic. Thus, all primitive terms of the language contribute to the ideology, not just the primitive predicates. ¹³ The perfectly natural properties make for intrinsic qualitative character; the perfectly natural relations are the fundamental ties that bind together the parts of worlds. (See also n. .) For discussion of the notion of a perfectly natural property, and a defense of its legitimacy, see Lewis (a: –). [For ways in which the primitive ideology that I accept differs from Lewis’s, see especially the postscript to Chapter .] ¹⁴ Lewis sketches such an analysis in Lewis (a: –). (He calls them the “analogical spatiotemporal relations,” but I drop the ‘analogical’.) ¹⁵ For (a version of ) the argument that Lewisian analyses of modality are circular, see McGinn (: –). For a response, see Bricker ().

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making our total theory contradictory.¹⁶ The law of non-contradiction, then, demands that the Lewisian reject impossible worlds. But if there are no impossible worlds, the ‘possible’ in ‘possible world’ is redundant, and no separate analysis is needed to pick out the worlds that are possible from the rest. Let us, then, focus on the one and only task: analyzing the notion of world. To accomplish this task, it suffices to provide necessary and sufficient conditions for when two individuals are worldmates, are part of one and the same world. Lewis’s proposal is this: individuals are worldmates if and only if they are spatiotemporally related to one another, that is, if and only if every part of one stands in some distance (or interval) relation—spatial, temporal, spatiotemporal—to every part of the other. (Lewis a: ) This leads immediately to the following analysis of the notion of world: a world is any maximal spatiotemporally interrelated individual—an individual all of whose parts are spatiotemporally related to one another, and that is not included in a larger individual all of whose parts are spatiotemporally related to one another. On this account, a world is unified by the spatiotemporal relations among its parts. If one further assumes with Lewis that being spatiotemporally related is an equivalence relation (reflexive, symmetric, and transitive), it follows that each individual belongs to exactly one world: the sum (or aggregate) of all those individuals that are spatiotemporally related to it. Now, the point to emphasize for present purposes is this: if Lewis’s analysis is accepted, the notion of world has been characterized in non-modal terms, and the claim to eschew primitive modality has been vindicated. The acceptability of Lewis’s analysis of world hinges on the acceptability of his analysis of the worldmate relation. One direction of the analysis (sufficiency) is uncontroversial. Whatever stands at some spatiotemporal distance from us is part of our world; or, contrapositively, non-actual individuals stand at no spatiotemporal distance from us, or from anything actual. In general: every world is spatiotemporally isolated from every other world. According to the other direction of the analysis (necessity), worlds are unified only by spatiotemporal relations; every part of a world is spatiotemporally related to every other part of that world. This direction is more problematic, for at least three reasons. Although I believe Lewis’s account needs to be modified to solve these problems, in each case the modification I would suggest does not require introducing primitive modality. First, couldn’t there be worlds that are unified by relations that are not spatiotemporal? Indeed, it is controversial, even with respect to the actual world, whether entities in the quantum domain stand in anything like spatiotemporal relations to one another; the classic account of spacetime may simply break down. I defend in Chapter  a solution that Lewis considered but (tentatively) rejected: individuals are worldmates if and only if they are externally related to one another, that is, if and only if there is a chain of perfectly natural relations (of any sort) extending from any part of one to any part of the other. This analysis quantifies over all perfectly natural relations rather than just the spatiotemporal relations, but it is none the worse for that with respect to primitive modality. ¹⁶ Lewis’s brief argument for this occurs in a footnote in Lewis (a: ). For further discussion, see Stalnaker (). Of course, the argument presupposes classical logic, that our total theory satisfies the law of non-contradiction. For a non-classical approach to contradictions, see Priest ().

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    () Second, isn’t it possible for there to be disconnected spacetimes, so-called “island universes”? Couldn’t there be a part of actuality spatiotemporally and causally isolated from the part we inhabit? Lewis must answer “no.” When Lewis’s analysis of world is combined with the standard analysis of possibility as truth at some world, island universes turn out to be impossible: at no world are there two disconnected spacetimes. But although Lewis rejects the possibility of island universes, he is uneasy, and for good reason: principles of recombination that Lewis seems to accept entail that island universes are possible after all.¹⁷ Of course, if one took the worldmate relation to be primitive, one could simply posit as part of the theory that some worlds are composed of disconnected spacetimes. But that’s no good. A primitive worldmate relation is primitive modality: what is possible—for example, the possibility of island universes—depends on how the worldmate relation is laid out in logical space. Is there a way to allow for the possibility of island universes without invoking primitive modality? This time, I think, the best strategy is to amend the analysis of modality instead of the analysis of world. The modal operators should be taken to quantify over worlds and pluralities of worlds. For example, to be possible is to be true at some world, or some plurality of worlds. What is true at a plurality of worlds is, intuitively, what would be true if all the worlds in the plurality were actualized. If a plurality of two or more worlds were actualized, then actuality would include two or more disconnected parts; and so, on the amended analysis, island universes turn out to be possible.¹⁸ A third problem afflicts Lewisians who are also platonists of a certain kind, namely, those who believe that in addition to the modal realm consisting of the isolated concrete worlds there is a mathematical realm consisting of isolated mathematical systems of abstract entities. Most platonists (although not Lewis) believe in at least one such system: the pure sets, externally related to one another in virtue of the structure imposed by the membership relation. But I, for one, believe also in sui generis numbers, externally related to one another in virtue of the structure imposed by the successor relation, in sui generis Euclidean space, and much, much more. But, then, on the (amended) analysis of world being considered, the pure sets comprise a world, the natural numbers comprise a world, and so on for all the mathematical systems one believes in. That makes metaphysical possibility, which is analyzed as a quantifier over worlds, depend on what mathematical systems there are, and that seems wrong.¹⁹ The Lewisian, then, needs to find a way to demarcate modal reality from mathematical reality, a way that does not invoke primitive modality. ¹⁷ For example, according to one such principle, for any individual, simple or composite, it is possible that a duplicate of that individual exist all by itself. Applying this principle to a disconnected sum leads to the possibility of island universes. See Chapter  for detailed argument. Lewis (a) does not explicitly accept any principle this strong; but Langton and Lewis (: ) accept such a principle, and claim that it is part of the combinatorial theory put forth in Lewis (a). ¹⁸ This view is presented and defended in Chapter . Note that, on this approach, if island universes actually exist, then there is more than one actual world. But I will continue to speak of the actual world for ease of presentation. ¹⁹ Note that on a view that accepts mathematical systems alongside the concrete worlds [see Chapters  and ], it is natural to distinguish between logical, metaphysical, and mathematical modality. Logical modality is absolute modality: it quantifies over concrete worlds and abstract systems both. Metaphysical and mathematical modality are both restricted modality, quantifying over just the concrete worlds or just

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It might seem that the conditions (C)–(C) used to characterize what makes a world concrete could do the job. For the mathematical systems, one might hold, are abstract in virtue of violating (C): they are not fully determinate; it is neither true nor false of numbers, for example, that they are red, or weigh ten grams. But that is not how I see it. Indeed, the pure sets, or sui generis numbers, or Euclidean points have no intrinsic qualitative nature, but not because they are somehow abstractions from something else.²⁰ Rather, they determinately fail to instantiate every qualitative property.²¹ Their nature is purely relational, but it is no less determinate for that. But perhaps there is a simple fix. I think there is a fifth way in which entities can be said to be concrete: concrete entities have an intrinsic qualitative nature in virtue of instantiating, or having parts that instantiate, perfectly natural properties. This provides a fifth condition to be satisfied by the concrete worlds: (C)

A world is a sum of individuals, each of which instantiates at least one perfectly natural property.²²

Incorporating (C) into the analysis of world provides the needed distinction between the modal and the mathematical realm. And since the notion of a perfectly natural property is non-modal, we have not had to invoke primitive modality to do the job. The elimination of primitive modality is a central goal of Lewisian approaches. To achieve this goal, the Lewisian must make a case for the following controversial claims. First, there is the claim that there exists a plurality of concrete worlds; that claim will be the focus of Section .. Second, there is the claim that the analysis of modality in terms of concrete worlds is materially and conceptually adequate; that claim will be put to the test in Section . with respect to the analysis of actuality, and in Section . with respect to the analysis of modality de re. Finally, there is the claim, noted above, that there are no impossible concrete worlds. That claim rests ultimately on a defense of classical logic, a topic too far afield to pursue further here.

the mathematical systems, respectively. The term ‘logical space’ is now best used to refer to mathematical and modal reality together, not just modal reality. ²⁰ I do not accept that there are any entities that are abstract in virtue of violating (C). There is a mental operation of abstraction, which involves ignoring some features and attending to others; and we can represent the results of this mental procedure, if we want, by using set-theoretic constructions in ways made familiar by mathematicians. But, in my view, there are no “indeterminate objects.” ²¹ A property is qualitative (in the narrow sense) if its instantiation does not depend on the existence of any particular object, but does depend on the instantiation of some perfectly natural property. (The broad sense drops the second clause.) Every object a instantiates the non-qualitative property: being identical with a. Every natural number other than zero instantiates the non-qualitative (structural) property: being the successor of something. I suppose that the perfectly natural properties are all qualitative, and that the qualitative properties supervene on the perfectly natural properties and relations. All qualitative properties are categorical, but not conversely. [See Chapter  for more on the (narrow) notion of a qualitative property, and its relation to the doctrine of quidditism.] ²² This “simple fix” presupposes that material objects can be identified with regions of spacetime. That is, it presupposes that worlds do not divide into two distinct domains: an immaterial spacetime, and material objects that occupy regions of spacetime. On the “dualist” view, the immaterial spacetime regions, arguably, would not instantiate any perfectly natural properties.

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. Concrete Worlds Exist Why believe in concrete worlds other than the actual world? Throughout his career, Lewis held to a broadly Quinean methodology for deciding questions of existence. Roughly, we are committed to the existence of those entities that are quantified over by the statements we take to be true.²³ And we should take those statements to be true that belong to the best total theory, where being “best” is in part a matter of being fruitful, simple, unified, economical, and of serving the needs of common sense, science, and systematic philosophy itself. What we should take to exist, then, is determined by criteria both holistic and pragmatic. Early on, Lewis applied the Quinean methodology directly to statements we accept in ordinary language. (Lewis : .) We accept, for example, not only that “things might be otherwise than they are,” but also that “there are many ways things could have been besides the way they actually are.” This latter statement quantifies explicitly over entities called “ways things could have been,” entities that Lewis identifies with concrete possible worlds. But that identification is far from innocent. It was soon pointed out (in Stalnaker ()) that the phrase ‘ways things could have been’ seems to refer, if at all, to abstract entities—perhaps uninstantiated properties—not to concrete worlds. Indeed, it is doubtful that any statements we ordinarily accept quantify explicitly over concrete worlds. In On the Plurality of Worlds, Lewis abandoned any attempt to apply the Quinean methodology directly to ordinary language, and applied it instead to systematic philosophy. Concrete worlds, if accepted, improve the unity and economy of philosophical theories by reducing the number of notions that must be taken as primitive. Moreover, concrete worlds provide, according to Lewis, a “paradise for philosophers” analogous to the way that sets have been said to provide a paradise for mathematicians (because, given the realm of sets, one has the wherewithal to provide true and adequate interpretations for all mathematical theories). Here Lewis has in mind not just the use of possible worlds to analyze modality, but also their use in constructing entities to play various theoretical roles, for example, the meanings of words and sentences in semantics and the contents of thought in cognitive psychology.²⁴ So, when asked—why believe in a plurality of worlds?—Lewis responds: “because the hypothesis is serviceable, and that is a reason to think that it is true.” Lewis (a: ). Lewis does not claim, of course, that usefulness by itself is a decisive reason to believe: there may be hidden costs to accepting concrete worlds; there may be alternatives to concrete worlds that provide the same benefits without the costs. Lewis’s defense of realism about concrete worlds, therefore, involves an extensive cost-benefit analysis. His conclusion is that, on balance, his realism defeats its rivals: rival theories that can provide the same benefits all have more serious costs. I will not

²³ Or, in Quine’s slogan: “To be is to be the value of a variable.” That is to say, we are ontologically committed to those entities that belong to the domain over which the variables of our quantifiers range. See Quine (: ). ²⁴ Lewis (a: –) surveys some of the uses to which possible worlds have been put in systematic philosophy. Lewis’s oeuvre taken altogether provides a monumental testament to the fruitfulness of possible worlds.

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attempt here to summarize Lewis’s lengthy and intricate discussion.²⁵ But I will say something about the general idea that belief in concrete worlds can be given a pragmatic foundation, and I will ask whether an alternative foundation is feasible. Lewis’s argument for belief in concrete worlds depends on the assumption that we should believe pragmatically virtuous theories, theories that, on balance, are more fruitful, simple, unified, or economical than their rivals. Although this assumption is orthodoxy among contemporary analytic philosophers, I find it no less troubling for that. It is one thing for a theory to be pragmatically virtuous, to meet certain of our needs and desires; it seems quite another thing for the theory to be true. On what grounds are the pragmatic virtues taken to be a mark of the true? It is easy to see why we would desire our theories to be pragmatically virtuous: the virtues make for theories that are useful, productive, easy to comprehend and apply. But why think that reality conforms to our desire for simplicity, unity, and the other pragmatic virtues? Believing a theory true because it is pragmatically virtuous leads to parochialism, and seems scarcely more justified than, say, believing Ptolemaic astronomy true because it conforms to our desire to be located at the center of the universe. Such wishful thinking is no more rational in metaphysics than in science or everyday life.²⁶ But if we reject a pragmatic foundation for belief in concrete worlds, what is there to put in its place? My hope is that there are general metaphysical principles that support the existence of concrete worlds, and that we can just see, on reflection, that these principles are true. This “seeing” is done not with our eyes, of course, but with our mind, with a Cartesian faculty of rational insight. This faculty is fallible, to be sure, as are all human faculties. (Contra Descartes, I do not take the faculty to be invested with the imprimatur of an almighty deity.) But fallible or not, some such faculty is needed lest our claim to have a priori knowledge be bankrupt. Now, what general principles could play a foundational role for belief in a plurality of concrete worlds? I will consider, briefly, two lines of argument. One way to argue for controversial ontology is to invoke a truthmaker principle: for every (positive) truth, there exists something that makes it true, some entity whose existence entails that truth.²⁷ Truths don’t float free above the ontological fray. They must be grounded in some portion of reality. For example, that Fido is a dog, if true, has Fido himself as a truthmaker. (Assuming, as is customary, that an animal belongs to its species essentially.) That some animals are dogs has multiple truthmakers: each and every animal that is a dog. (On the other hand, a negative truth, such as that no dog is a bird, is made true by the lack of falsemakers, by the non-existence of any dog that is a bird.) A more controversial case: that there are infinitely many prime

²⁵ Chapter  of Lewis (a) argues that the cost of accepting concrete worlds is manageable by responding to eight objections from the literature. Chapter  argues that rival views that take worlds to be abstract entities all have serious objections. For summaries of some of these arguments, see Bricker (). ²⁶ [See Chapter  for how I think a realist can endorse the use of pragmatic criteria without being committed to parochialism.] ²⁷ For an introduction to truthmaking, see Armstrong (). For reasons to accept only a weak truthmaker principle that applies to positive truths, see Lewis (a). [For more on my own take on truthmaking, see Chapters  and .]

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    () numbers, I claim, is made true by the existence of the system of natural numbers. Mathematical truths have mathematical entities as truthmakers. Consider now a (positive) modal truth such as it is possible that there be unicorns. What could be a truthmaker for this truth? Not any actual unicorn: there aren’t any. Not actual ideas of unicorns, or other actual mental entities; for the possibility of unicorns doesn’t depend on whether any mind has ever conceived of unicorns, or even whether any mind has ever existed. What then? To find the truthmakers for a statement, it helps to ask what the statement is about. That unicorns are possible appears to be about unicorns, if about anything at all. And, by the truthmaker principle, it is about something. But since there are no actual unicorns, that leaves only merely possible unicorns for it to be about: it has each and every possible unicorn as a truthmaker. Thus baldly presented, the truthmaker argument for concrete possibilia may fail to convince. Indeed, the argument would need to be supplemented in at least two ways. First, even if one grants that modal truths require possibilia for truthmakers, why hold that the possibilia in question must be concrete? Perhaps abstract possibilia can meet the demand for truthmakers. Filling this gap in the argument, it seems, must wait on a decisive critique of all abstract accounts of possibilia—a tall order. Second, the truthmaker principle is often restricted to contingent truths, and for good reason. A truthmaker for a statement is an entity whose existence entails that statement. As entailment is ordinarily understood in terms of possible worlds, one statement entails a second just in case every world at which the first is true is a world at which the second is true. Thus understood, any statement entails a necessary truth, and so truthmaking for necessary truths becomes a trivial affair, devoid of ontological consequence. But the thesis that concrete worlds exist (with ‘exists’ unrestricted) is a necessary truth. If the truthmaker principle is to apply to this thesis, truthmaking must be based on a more discriminating notion of entailment. It won’t do to take this discriminating notion of entailment as primitive, lest the “no primitive modality” thesis be violated. So, some non-standard account of truthmaking in terms of worlds will need to be developed—no easy task. Given these difficulties with the truthmaker argument for concrete possibilia, I find a different line of argument more promising, one that focuses on the nature of intentionality. Intentionality, in the relevant philosophical sense, refers to a feature of certain mental states such as belief and desire: these states are always “directed” towards some object or objects; one doesn’t just believe or desire, one always believes or desires something.²⁸ That some of our mental states have this feature is something we know a priori by introspection. We know, that is, a general principle to the effect that mental states with this feature—“intentional states”—are genuinely relational. The second line of argument, then, is that this general principle can serve as foundation for belief in concrete possible objects and possible worlds. Concrete possibilia are needed to provide the objects of our intentional states, to provide an ontological framework for the content of our thought.

²⁸ For an introduction to the logical and metaphysical issues raised by this notion of intentionality, see Priest (). [I say a bit more about it in Section ..]

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To illustrate how the argument might go, consider the intentional state of thinking about some object or objects. Suppose, for example, that I am now thinking about a dodecahedron made of solid gold. I can do this, of course, whether or not any such object actually exists. If there is, unbeknownst to me, an actual gold dodecahedron, then I am related to it in virtue of being in my current intentional state; it is an object of my thought. But what if there is no actual gold dodecahedron? Does that somehow prevent me from thinking about one? Of course not. In either case, I claim, thinking about is relational, and relations require relata. In the latter case, only merely possible gold dodecahedrons are available to be objects of my thought; I am related to possible but non-actual objects. However, if the relationality of intentional states is to serve as a foundation for a Lewisian account of worlds, at least three further claims require support. First, the objects of thought, even when merely possible, must instantiate the same qualitative properties as actual objects of thought. Suppose again that a merely possible gold dodecahedron is an object of my thought. Does this object instantiate the property of being gold? Or is it an abstract object that somehow represents the property of being gold? I say the former. It is one thing to think about a gold dodecahedron, another thing to think about some abstract simulacrum thereof. If I am thinking about a gold dodecahedron and thinking about is genuinely relational, then there is a gold dodecahedron that I am thinking about. That it is made of gold and shaped like a dodecahedron is independent of whether it is actual or merely possible. Indeed, nothing prevents actual and merely possible objects from being perfect qualitative duplicates of one another. But, second, more is needed if the objects of my thought are to count as concrete: they must not only instantiate qualitative properties, they must be fully determinate in all qualitative respects. How can that be? Intentional states such as thinking about do not seem to be determinate with respect to their objects. In thinking about a gold dodecahedron, I wasn’t thinking about a gold dodecahedron of any particular size. Should I say, then, that I was related by my thought to an object that has no definite size? No. It is one thing to think indeterminately about a gold dodecahedron, another thing to think about an indeterminate object. The indeterminacy is in the thinking, not the object of thought. I am related by my thought to a multitude of possible gold dodecahedrons with a multitude of different, but fully determinate, sizes.²⁹ But still more is needed if the argument is to support a Lewisian account: each concrete object of thought must be part of a fully determinate concrete world. In thinking about a gold dodecahedron, I wasn’t thinking about how it is situated with respect to other objects. But that is just another aspect of the indeterminacy of my thought. Each possible gold dodecahedron has a determinate extrinsic nature; my thought doesn’t discriminate between differently situated gold dodecahedra, and it therefore ranges indeterminately over them all. Perhaps, as I believe, there exists in logical space a solitary gold dodecahedron that is a world all by itself. But then distinguish: it is one thing to think about a solitary gold dodecahedron, another thing

²⁹ We can call on the method of supervaluations to explain why I can truly say that there is one thing that I am thinking about: a dodecahedron made of solid gold. See Lewis (a).

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    () to think about a gold dodecahedron without considering how it relates to other objects. In the former case, what I am thinking about stands in no spatial or temporal relations to other objects; in the latter case, it is indeterminate whether what I am thinking about stands in spatial or temporal relations to other objects. In either case, the possible gold dodecahedra that are objects of my thought belong to fully determinate concrete worlds. That, in barest outline, is how the relationality of thought could serve as foundation for a Lewisian account of concrete worlds. A thoroughgoing Quinean pragmatist would say, of course, that the thesis of the relationality of thought—or the truthmaker principle of the preceding argument, or any fundamental metaphysical principle— can support belief in concrete worlds only to the extent that its acceptance confers benefits on our total theory. But if I am right that the pragmatic virtues are never, in and of themselves, a mark of the true, then a “pragmatic foundation” is not to be had; indeed, it is a contradiction in terms. Contra what Lewis claims, that a belief is “serviceable” for the project of systematic philosophy provides no reason at all to hold it. Founding belief in concrete worlds instead on a (fallible) faculty of rational insight into matters metaphysical is controversial, to be sure, and in need of much development. But better a shaky foundation, I say, than no foundation at all.

. Actuality Is Indexical Thus far, I have said very little about the notion of actuality. But some of the commitments of a Lewisian account are already clear. Since the Lewisian believes that merely possible worlds exist, she rejects the thesis that whatever exists is actual; that is to say, the Lewisian is a possibilist, not an actualist. Moreover, since the Lewisian believes that merely possible concrete worlds exist, she rejects any identification of the actual with the concrete. Furthermore, since the Lewisian holds that actual things have qualitative duplicates in merely possible worlds, actuality cannot itself be any sort of qualitative property. What, then, is it? In virtue of what do actual things differ from their merely possible counterparts? The Lewisian needs a positive account of actuality. Lewis responds by proposing a deflationary account of actuality. The actual world and the merely possible worlds are ontologically all on a par; there is no fundamental, absolute property that actual things have and merely possible things lack. Nonetheless, I speak truly when I call my world and my worldmates “actual” because ‘actual’ just means ‘thisworldly’, or ‘is part of my world’. For Lewis, ‘actual’ is an indexical term, like ‘I’ or ‘here’ or ‘now’. What ‘actual’ applies to on a given use depends on features of the context of utterance, in particular, on the speaker, and the speaker’s world. When I say of something that it is “actual,” I say simply that it is part of my world; when my otherworldly counterpart says of something that it is “actual,” he says simply (if he is speaking English) that it is part of his world. I call my worldmates “actual” and my otherworldly counterparts “merely possible”; my counterpart calls his worldmates “actual” and me “merely possible.” And we all speak truly, just as many people in different locations all speak truly when each says, “I am here.” For Lewis, being actual or merely possible does not mark any ontological

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

distinction between me and my counterparts because—as with being here or being there—there is no ontological distinction to be marked. What sort of property is expressed by a given use of ‘actual’ on Lewis’s account? When Lewis’s indexical analysis of actuality is combined with his analysis of world, we get that, in any context, ‘actual’ expresses the property of being spatiotemporally related to the speaker in that context. Thus, in any context, the property expressed by ‘actual’ is a relative property, a property things have in virtue of their relations to things, not in virtue of how they are in themselves. That makes actuality, on Lewis’s account, doubly relative: what property is expressed by a given use of ‘actual’ is relative to the speaker; and the property thus expressed is itself a relative property. Lewis’s indexical account conflicts rather severely with our ordinary way of thinking about actuality. As Robert Adams vividly put it: “We do not think the difference in respect of actuality between Henry Kissinger and the Wizard of Oz is just a difference in their relations to us.” Adams (: ). According to Lewis, however, a believer in concrete worlds has no choice but to accept an indexical analysis of actuality according to which actuality is doubly relative. For, Lewis argues, if my use of ‘actual’ instead expressed an absolute property that I have and my otherworldly counterparts lack, then no account could be given of how I know that I am actual. I have counterparts in other worlds that are epistemically situated exactly as I am; whatever evidence I have for believing that I have the supposed absolute property of actuality, they have exactly similar evidence for believing that they have the property. But if no evidence distinguishes my predicament from theirs, then I don’t really know that I am not in their predicament: for all I know, I am a merely possible person falsely believing myself to be absolutely actual. Thus, Lewis concludes, accepting concrete worlds together with absolute actuality leads to skepticism about whether one is actual. Since such skepticism is absurd, a believer in concrete worlds should reject absolute actuality.³⁰ Lewis’s indexical account makes short work of the skeptical problem, and that is an argument in its favor. On Lewis’s analysis, ‘I am actual’ is a trivial analytic truth analogous to ‘I am here’. Just as it makes no sense for me to wonder whether I am here (because ‘here’ just means ‘the place I am at’), so it makes no sense for me to wonder whether I am actual (because ‘actual’ just means, according to Lewis, ‘part of the world I am part of ’). Moreover, my otherworldly counterparts have no more trouble knowing whether they are actual than I do. When one of my counterparts says, “I am actual,” he speaks truly (if he is speaking English), and he knows this simply in virtue of knowing that he is part of the world he is part of. Thus, Lewis can explain why it strikes us as absurd for someone to wonder whether or not she is actual. Is Lewis correct, however, that a believer in concrete worlds has no choice but to reject absolute actuality? I hope not. I, for one, could not endorse the thesis of a plurality of concrete worlds if I did not hold that there was a fundamental ontological distinction between the actual and the merely possible. A Lewisian approach to modality that rejects absolute actuality does not seem to me to be tenable.

³⁰ Lewis first introduced his indexical theory of actuality and invoked the skeptical argument to support it in Lewis (). See also Lewis (a: –).

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    () Actuality, I claim, is a categorial notion: whatever belongs to the same fundamental ontological category as something actual is itself actual. When Lewis insists, then, that all worlds are ontologically on a par, this can only be understood as saying that all worlds are equally actual—his denials notwithstanding. But that undercuts Lewis’s defense of concrete worlds: an analysis of modality as quantification over concrete parts of actuality, no matter how extensive actuality may be, is surely mistaken.³¹ A way to test whether actuality is absolute or merely relative is to ask whether it is coherent to suppose that actuality is composed of island universes: parts that stand in no spatiotemporal (or other external) relations to one another. If actuality is absolute, the hypothesis of island universes is coherent: something could be actual even though entirely disconnected from the part of actuality we inhabit. But if actuality is merely relative, as Lewis supposes, then the hypothesis of island universes is analytically false—and that seems wrong. Nor would it help for Lewis to switch to the amended analysis of possibility suggested in Section .: that analysis would make the hypothesis of island universes metaphysically possible—true at some plurality of worlds; but the hypothesis would remain analytically false of (what Lewis calls) actuality. Accepting that combination compounds the problem, rather than solving it. Fortunately, I think a Lewisian can accept absolute actuality without falling victim to the skeptical problem. To see how, we first need to distinguish, for any predicate, the concept associated with the predicate from the property expressed by a given use of that predicate. The concept associated with a predicate is naturally identified with its meaning. It embodies a rule that determines, for each context of use, what property is expressed by the predicate in that context. A predicate is indexical if it expresses different properties relative to different contexts of use; in that case, the associated concept can also be called indexical. Indexicality is one kind of relativity: relativity to features of context. But that sort of relativity must be distinguished from the relativity of the property expressed. Lewis’s argument against absolute actuality presupposes that these two sorts of relativity must go together, that any indexical analysis of actuality will be doubly relative. But that assumption, I think, is mistaken. Indexical concepts can be associated with predicates that express either relative or absolute properties. For example, consider the indexical predicate ‘is a neighbor’. On different occasions of use, it expresses different properties. When I use the predicate, it expresses the property of being one of my neighbors; when you use the predicate, it expresses the property of being one of your neighbors. On any use, the property expressed is a relative property: whether a person has the property expressed depends on that person’s relations to the speaker. Other indexical predicates, however, express absolute properties on each occasion of use. For example, the indexical predicate ‘is nutritious’ expresses different properties relative to different speakers (depending on age, or state of health). But, on each use, the property expressed is absolute, not relative: something is nutritious (for the speaker) in virtue of its chemical nature, not in virtue of its relative properties; if two things are chemical duplicates of one another, then either both or neither are nutritious (for the speaker).

³¹ As Lewis himself concedes: “if the other worlds would be just parts of actuality, modal realism [Lewis’s brand of realism about possible worlds] is kaput.” Lewis (a: ).

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

Now, on Lewis’s analysis of actuality, the concept associated with the predicate ‘is actual’ is indexical, and the property expressed by the predicate, on each context of use, is relative. It is the indexicality of the concept that allows for a solution to the skeptical problem. It is the relativity of the property that leads, I have claimed, to an untenable position. Is there a way of analyzing actuality so that the concept is indexical but the property is absolute? Consider this: ‘is actual’ in my mouth expresses the property, “belonging to the same fundamental ontological category as me.” That builds the categorial nature of actuality directly into the analysis. But, thanks to the indexical component, it makes short work of the skeptical problem. On this analysis, I know that I am actual simply in virtue of knowing that I belong to the same ontological category as myself. Knowledge that I am actual is just as trivial as on Lewis’s analysis of actuality, as it should be, but the property of actuality remains ontologically robust.³² Lewis would no doubt object that a theory of concrete worlds with absolute actuality is less parsimonious than his own. Granted. What matters, however, is which theory gets it right. If actuality is a categorial notion, as I believe, then Lewis’s indexical theory must be rejected. Lewis would also, I suspect, object that the notion of absolute actuality is mysterious. Perhaps. (It is not, however, the mystery of primitive modality: absolute actuality is no more primitive modality than is absolute truth.) It does not help our understanding, for example, to say that merely possible entities are “less real” than actual entities: both merely possible and actual entities exist, and I do not understand how existence could be a matter of degree. Nor does it help to say that actual and merely possible entities exist in different ways, that there are two modes of existence: if that means anything at all, it just means that there are two fundamental ontological categories. The best inroad to understanding the distinction between the actual and the merely possible, I would say, comes from considering how we and our surroundings differ from what exists merely as an object of our thought. Lewis himself allows that there are entities of distinct ontological categories: individuals and sets. The distinction between an individual and its singleton is arguably no less mysterious than the distinction between an actual thing and its merely possible qualitative duplicates. In the case of sets, Lewis embraces the mystery.³³ Why not also, in the case of possibilia, embrace the mystery of absolute actuality? The answer turns on whether concrete talking donkeys and flying pigs are any easier to believe in if they belong to a different ontological category than actual donkeys and pigs. I leave that to the reader to ponder.

. Modality De Re and Counterparts It is traditional to divide modal statements into two sorts: de dicto and de re. A modal statement is de dicto, it is sometimes said, if the modal operator applies to a proposition (Latin: dictum); it is de re if the modal operator applies to a property to form a modal property, which is then attributed to some thing (Latin: res). ³² For a detailed attempt to develop an alternative indexical analysis of actuality on which actuality is absolute, see Chapter . ³³ See Lewis (: –) on “mysterious singletons.”

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    () Thus, ‘necessarily, all birds are feathered’ is de dicto; ‘Polly is necessarily feathered’ is de re. The traditional characterization, however, is defective in a number of ways. For one thing, equivalent statements are not always classified alike. Indeed, any de dicto modal statement is equivalent to a statement attributing a modal property to the (actual) world. Thus, ‘necessarily, all birds are feathered’ is equivalent to ‘the world is necessarily such that all birds are feathered’, which the above criterion classifies as de re. Moreover, a de re modal statement such as, ‘Polly is necessarily feathered’ is equivalent to ‘necessarily, Polly is feathered’ (or, perhaps, ‘necessarily, if Polly exists, Polly is feathered’), which the above criterion classifies as de dicto. Another defect of the traditional characterization is that it fails to provide an exhaustive classification of modal statements: modal statements with a complicated structure will not be classified either as de dicto or de re. A better way to characterize the de dicto/de re distinction looks to the content of modal statements, rather than their form. Here is the rough idea, neutrally expressed so as to apply to Lewisians and non-Lewisians alike. All possible worlds theorists will have to provide an account of how truth at a world is to be determined, how a world represents that things are one way or another. Part of any such account will involve providing for each world a domain of entities—the entities that in some primary sense “inhabit” the world—and saying, for each entity in a world’s domain, what properties it has at the world. Any such account will also have to say how it is determined, when an entity is picked out as belonging to the domain of one world, whether the entity exists at some other world, and what properties it has at this other world. Call this “crossworld representation de re.” Now, what makes a modal statement de re is that in the course of evaluating its truth or falsity, one must have recourse to facts about crossworld representation de re. A modal statement is de dicto, on the other hand, if no recourse to crossworld representation de re is needed to evaluate its truth or falsity. To illustrate with a standard example: compare the de re ‘everything is necessarily material’ with the de dicto ‘necessarily, everything is material’. The former statement depends on crossworld representation de re: it says that every entity in the domain of the actual world is material, not only at the actual world, but at every possible world (better: at every possible world at which it exists). The latter statement does not depend on crossworld representation de re: it says that at every possible world, everything in the domain of that world is material. These two statements are not equivalent. The de dicto statement is made false by a possible world whose domain contains a non-material object; but if that possible world doesn’t represent de re concerning any actual object that it is non-material, then the de re statement may still be true. How is crossworld representation de re determined? The simplest answer, of course, would be this: an entity picked out as belonging to the domain of one world exists at some other world just in case it also belongs to the domain of that other world. On this account, the domains of different worlds overlap, and all facts about what properties an entity has at a world are given directly by how that world represents that entity to be. To exist at a world is just to belong to the domain of that world. I will say, following standard though somewhat misleading usage, that such an account endorses “transworld identity.” To illustrate, consider again George W. Bush. On the transworld identity theory, Bush belongs not only to the actual

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domain, but also to the domain of many merely possible worlds. Some of these worlds represent him as having properties he doesn’t actually have, such as losing the presidential election in , or being a plumber. The Lewisian, however, needs a different account of crossworld representation de re. For the Lewisian, each world has as its domain just the entities that are part of the world. Bush is part of the actual world, and is in the actual domain. But since worlds do not overlap, Bush is not in the domain of any merely possible world. How, then, does a merely possible world represent Bush as existing and having properties he doesn’t actually have? Lewis responds: by having in its domain a counterpart of Bush. A merely possible world represents de re concerning Bush that he exists and, say, is a plumber by containing a counterpart of Bush that is a plumber. So, for the Lewisian, existing at a world must be distinguished from being in the domain of a world: Bush exists at many worlds, although he is in the domain of—is part of—only one. Because Bush exists at many worlds, the Lewisian can be said to accept “transworld identity” in a weak, uncontroversial sense; but not in the stronger sense that requires overlapping domains. What makes an entity in one world a counterpart of an entity in another? According to Lewis, the counterpart relation is a relation of qualitative similarity. He writes: Something has for counterparts at a given world those things existing there that resemble it closely enough in important respects of intrinsic quality and extrinsic relations, and that resemble it no less closely than do other things existing there. Ordinarily something will have one counterpart or none at a world, but ties in similarity may give it multiple counterparts. (Lewis : )

With a counterpart relation in place, de re modal statements can be analyzed in terms of concrete worlds and their parts. For example, to consider the simplest cases: ‘Bush is possibly a plumber’ is true just in case at some world some counterpart of Bush is a plumber; ‘Bush is necessarily (or essentially) human’ is true just in case at every world every counterpart of Bush is human.³⁴ For the Lewisian, we have a simple way of distinguishing de re from de dicto: de re modal statements depend for their evaluation on the counterpart relation; de dicto modal statements do not. Whether one is a Lewisian or not, there are good reasons to prefer counterpart relations to transworld identity as an account of representation de re. I have space here to consider just one such reason, namely, that only counterpart relations can allow for essences that are moderately, without being excessively, tolerant.³⁵ Let me

³⁴ Not all de re modal attribution follows this pattern. For example, ‘Bush necessarily exists’ should be analyzed as the falsehood, ‘at every world there is some counterpart of Bush’, not the trivial truth, ‘at every world every counterpart of Bush exists’. For discussion, see Lewis (a: –). ³⁵ Two other important arguments supporting counterpart theory are the following. () Contingent Identity Statements. Only counterpart theory allows one to hold, for example, that a statue is identical with the lump of clay from which it is made, even though one can truly say: “that statue might have existed and not been identical with that lump of clay.” () Inconstancy of Representation De Re. Only counterpart theory allows one to hold, in accordance with ordinary practice, that attributions of essential properties may vary from context to context. In both of these cases, the counterpart theorist introduces multiple counterpart relations to achieve the desired result. See Lewis (a: –).

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    () explain. Modality de re is the realm of essence and accident. A property had by an individual is essential to the individual if that individual couldn’t exist without the property; it is accidental if it is not essential. An individual’s essence is the sum of all its essential properties. These notions will be translated into the framework of the counterpart theorist and the transworld identity theorist in different ways. For the counterpart theorist, an essential property of an individual is a property had by the individual and all of its counterparts. For the transworld identity theorist, an essential property of an individual is a property had by the individual itself at every world at which it exists. These two accounts come apart if the counterpart relation does not have the logical properties of identity, such as transitivity.³⁶ When they come apart, the counterpart theorist enjoys a flexibility that the transworld identity theorist cannot match. In particular, only a counterpart theorist can allow that an individual could have been somewhat different, say, with respect to its material composition or its origins, but could not have been wildly different. The transworld identity theorist will have to hold that essential properties are either not tolerant at all (with respect to material composition, or origins) or excessively tolerant; moderation will have to be abandoned. And that, in many cases, will lead to the wrong truth conditions for de re modal statements. To illustrate the problem of moderately tolerant essences, consider the following simple, though somewhat implausible, example. Suppose that it is essential to any person to have at least one of the (biological) parents he or she in fact has, but that it is not essential to have both. Thus, I could have had a different mother, and I could have had a different father, but I couldn’t have had both a different mother and a different father. My essence, then, is moderately tolerant with respect to my origins. Now, if representation de re works by transworld identity, an essence such as this leads to contradiction. Call my mother m and my father f. Since my essence is (moderately) tolerant, I could have had a different father and the same mother. So there is a world w and a person p existing at w such that p = me and p has father f 0 (6¼ f ) and p has mother m. But now, since p’s essence is (moderately) tolerant, p could have had a different mother and the same father. So there is a world w 0 and a person p0 existing at w 0 such that p0 = p and p0 has mother m0 (6¼ m) and p0 has father f 0 . But if p0 = p and p = me, then p0 = me (by the transitivity of identity). So, at w0 I exist and my father is f 0 and my mother is m0 . I could have had both a different father and a different mother after all, which contradicts the supposition that my essence is moderately tolerant. What to do? It would not be plausible, I think, to deny that individual essences can be moderately tolerant. A better solution is to switch to counterpart theory. If representation de re works by counterpart relations instead of transworld identity, then moderately tolerant essences are unassailable. A counterpart relation is based (at least in part) on qualitative similarity, and relations of similarity are not in general transitive. For the example at hand, the counterpart theorist will simply say that although p is a counterpart of me and p0 is a counterpart of p, p0 is not a counterpart of

³⁶ A relation is transitive if and only if whenever one thing bears the relation to a second, and the second bears the relation to a third, then the first bears the relation to the third.

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me. Thus, the counterpart theorist is not driven to assert that I could have had both a different mother and a different father, and no contradiction can be derived.³⁷ So much in favor of counterpart theory; now for something on the other side. Many philosophers have argued, the theoretical benefits of counterpart theory notwithstanding, that counterpart theory provides unacceptable truth conditions for de re modal statements. If Lewisian realism is committed to counterpart theory, they say, so much the worse for Lewisian realism. I will briefly consider two of the most common lines of attack, both of which can be traced to Saul Kripke’s influential discussion of counterpart theory in Naming and Necessity.³⁸ For the first line of attack, consider Kripke’s complaint that according to counterpart theory: . . . if we say “Humphrey might have won the election (if only he had done such-and-such),” we are not talking about something that might have happened to Humphrey, but to someone else, a “counterpart”. Probably, however, Humphrey could not care less whether someone else, no matter how much resembling him, would have been victorious in another possible world. (Kripke : )

Kripke’s objection naturally falls into two parts. The first part is that, on the analysis of modality de re provided by counterpart theory, the modal property, might have won the election, is attributed to Humphrey’s counterpart rather than to Humphrey himself. But surely, the objection continues, when we say that “Humphrey might have won,” we mean to say something about Humphrey. This part of the objection, however, is easily answered. According to counterpart theory, Humphrey himself has the modal property, might have won the election, in virtue of his counterpart having the (non-modal) property, won the election. Moreover, that Humphrey has a winning counterpart is a matter of the qualitative character of Humphrey and his surroundings; so on the counterpart theoretic analysis, the modal statement is indeed a claim about Humphrey. The second part of Kripke’s objection is more troublesome. We have a strong intuition, not only that the modal statement, “Humphrey might have won the election,” is about Humphrey, but that it is only about Humphrey (and his surroundings). On counterpart theory, however, the modal statement is also about a merely possible person in some merely possible world; and that, Kripke might say, is simply not what we take the modal statement to mean. The first thing to say in response is that the charge of unintuitiveness would apply equally to any theory that uses abstract worlds to provide truth conditions for modal statements; for our intuitive understanding of modal statements such as “Humphrey might have won the election” does not seem to invoke abstract worlds any more than counterparts of Humphrey. The objection, then, if it is good, would seem to cut equally against all possible worlds approaches to modality; if anything, it would favor a non-realist approach that rejects possible worlds, concrete or abstract. But is the objection good? Should our pre-theoretic intuitions as to what our statements are and are not about carry much, or even any, weight? I think not. A philosophical analysis of our ordinary ³⁷ The problem of moderately tolerant essences was first introduced in Chisholm (). The counterpart-theoretic solution is discussed in Lewis (a: –). ³⁸ See Kripke (: –). For other well-known objections to counterpart theory, see Plantinga ().

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

   ()

modal statements must assign the right truth values and validate the right inferences; moreover, it must be able to withstand mature philosophical reflection. But requiring that philosophical analyses match all our pre-theoretic intuitions would make systematic philosophy all but impossible. The second line of attack on counterpart theory I want to consider also has its origin in Kripke’s Naming and Necessity; but I will present the argument in the way I find most effective, even though it may not coincide with Kripke’s intentions. The argument begins with the observation that we often simply stipulate that we are considering a possibility for some given actual individual. For example, we can simply say: consider a possible world at which Bush lost the election in . In doing so, we consider a possible world that represents de re concerning Bush that he lost. Of course, such stipulation cannot run afoul of Bush’s essential properties: we cannot stipulate that a world represents de re concerning Bush that he is a poached egg if Bush is essentially human. The point, rather, is that such stipulation may be legitimate even if no loser at the world in question can be singled out as qualitatively most similar to Bush. But then, the argument concludes, representation de re cannot be based (entirely) on relations of qualitative similarity. If counterpart relations are relations of qualitative similarity, as Lewis asserts, then counterpart theory must be rejected. To illustrate the argument, consider the possibility that I have an identical twin. It seems coherent to suppose that, in the possibility being considered, neither my twin nor I is qualitatively more similar to the way I actually am than is the other. Nonetheless, in the possibility being considered, I am one of the twins and not the other; indeed, we can stipulate that I am the first-born twin. How can the Lewisian account for such a possibility? It seems that the Lewisian has to hold that there is a possible world that represents de re concerning me that I am the first-born twin without representing de re concerning me that I am the second-born twin. But if representation de re works by counterpart relations, that would seem to be impossible. Both twins are equally good candidates to be my counterpart if the counterpart relation is a relation of qualitative similarity. A Lewisian has the following perfectly adequate response. If counterpart relations are relations of qualitative similarity, then indeed each twin is a counterpart of me at the world in question. Given that, the world can only represent de re concerning me that I am the first-born twin if it also represents de re concerning me that I am the second-born twin, lest representation de re not be determined by counterpart relations. But the Lewisian can allow that the one world represents two distinct possibilities for me: it represents de re concerning me that I am the first born of two identical twins in virtue of containing a counterpart of me that is a first-born identical twin; but it also represents de re concerning me that I am the second born of two identical twins in virtue of containing a counterpart of me that is the second born. In this way, all the facts of crossworld representation de re still depend only on the one qualitative counterpart relation; but when there are multiple counterparts at a world, multiple possibilities are represented within a single world.³⁹

³⁹ Lewis presents and defends this response in Lewis (a: –).

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



There is another sort of example involving stipulation de re, however, that the above response does nothing to accommodate. Not only can we stipulate that we are considering a possibility that involves a given individual, we can also stipulate that we are considering a possibility that does not involve a given individual. And, with this sort of stipulation, there do not seem to be any qualitative constraints. Indeed, it is perfectly legitimate to say: consider a possibility qualitatively indiscernible from actuality but in which I do not exist. In the possibility envisaged, I have a doppelganger, a person exactly like me in every qualitative respect, intrinsic and extrinsic; but that person isn’t me. I find this intuition compelling, and think that any account of modality de re must find a way to accommodate it. But now the counterpart theorist is in trouble if counterpart relations are relations of qualitative similarity. If the possibility in question is represented by some non-actual world qualitatively indiscernible from the actual world, then an inhabitant of that nonactual world is qualitatively indiscernible from me without being my counterpart. If the possibility in question is instead somehow represented by the actual world, then there would have to be a counterpart relation under which I am not a counterpart of myself. Either way, the Lewisian would have to reject the idea that counterpart relations are (always) relations of qualitative similarity. So be it. That is a retreat from Lewis’s original understanding of counterpart theory, but it is by no means a defeat for the Lewisian. Counterpart theory, first and foremost, is a semantic theory for providing truth conditions for de re modal statements. As such, it should adapt to those de re modal statements we take to be true. As long as this is accomplished in a way that doesn’t compromise the metaphysics of Lewisian realism, nothing of value is lost.⁴⁰

. Conclusion When Lewis first began advocating the thesis that there exists a plurality of concrete worlds, he received in response mostly “incredulous stares.” That soon changed. Over the ensuing years, arguments for and against Lewisian realism have filled philosophical books and journals. Lewisians have had to develop and revise their position in the light of powerful criticism; non-Lewisian alternatives have sprouted like weeds in the philosophical landscape. The debate goes on; as with other metaphysical debates, a decisive outcome is not to be expected. And through it all, the incredulous stares remain: Lewisian realism does disagree sharply, as Lewis himself concedes (Lewis a: ), with common sense opinion as to what there is. There seems to be a fundamental rift—unbridgeable by argument—between ontologically conservative philosophers who have what Bertrand Russell called ⁴⁰ Anti-haecceitism—the view that representation de re supervenes on qualitative features of worlds (in the broad sense)—is arguably an essential component of the metaphysics of Lewisian realism. But the Lewisian need not abandon anti-haecceitism to accommodate the possibility that things could be qualitatively the same as they actually are and yet nothing actual exist. The Lewisian can say that the actual world represents this possibility with respect to a counterpart relation under which nothing is a counterpart of anything. Although, contra Lewis, this counterpart relation is not a relation of qualitative similarity, it is nonetheless qualitative (in the broad sense): it does not distinguish between qualitative indiscernibles. For Lewis’s characterization and defense of anti-haecceitism, see Lewis (a: –).

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   ()

“a robust sense of reality,” and ontologically liberal philosophers who respond, echoing Hamlet: “there is more on heaven and earth than is dreamt of in your philosophy.” No doubt, the Lewisian approach to modality will always be a minority view. But the power and elegance of the Lewisian approach has been widely appreciated by philosophers of all stripes. The bar is set high for the assessment of alternative views.

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 Island Universes and the Analysis of Modality () . Introduction Is physical reality a single, unified whole—say, a spacetime of four, or ten, or twenty-six dimensions? Or does physical reality divide into parts that are causally and spatiotemporally isolated from one another, so-called “island universes”?¹ I doubt we could ever have good reason to answer this question one way or the other. In this chapter, I focus instead on a prior question, one more amenable to philosophical analysis: Is the supposition that island universes exist even coherent? Is it metaphysically or logically possible? I will argue that island universes are metaphysically possible, and then consider the substantial impact this has on realist theories of possible worlds and the realist analysis of modality. The question whether island universes are possible has been raised intermittently through the history of philosophy, and most often answered in the negative. Kant maintained the necessary unity of space and of time: necessarily, everything spatial is spatially related to everything else spatial; mutatis mutandis for time.² Bradley disagreed about the unity of space and of time, but held that reality (the Absolute) is necessarily unified in some non-spatial, non-temporal way.³ In this century, logical positivists held that the supposition that island universes exist is meaningless—and so, a fortiori, not genuinely possible—on grounds of the verifiability criterion of meaning: the existence of a part of physical reality causally and spatiotemporally isolated from us would be unverifiable (by us), even in principle.⁴ I have little First published in G. Preyer and F. Siebelt (eds.), Reality and Humean Supervenience: Essays on the Philosophy of David Lewis (Rowman and Littlefield Publishing Group, ), –. Reprinted with the permission of Rowman and Littlefield Publishing Group. I began the essay with this note: “It is a privilege and a pleasure to contribute to a volume on the philosophy of David Lewis. His work has been a fountain of philosophical inspiration—and good sense—wherein I continually replenish myself.” Portions of this chapter were presented at Princeton University in March, . Thanks to David Lewis and Ted Sider for helpful discussion over a number of years. ¹ In astronomical usage, galaxies have been called “island universes” owing to their relative isolation. In metaphysical usage, the isolation of “island universes” is more complete. The phrase ‘island universe’, in its metaphysical sense, occurs in Armstrong (a). (See Section .. below for a precise definition.) ² Kant (), A, B. For critical discussion, see Strawson (). ³ Bradley (). For arguments against the unity of space and of time, see pp. –, –; for arguments that reality (the Absolute) is necessarily unified, see pp. –. ⁴ In arguing against logical positivism, Russell ponders whether the supposition that “there is a cosmos which has no spatiotemporal relation to the one in which we live” is meaningful, even though there could be no evidence for or against it. See Russell (: ). Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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       ()

sympathy with verificationist or idealist arguments against the possibility of island universes, but nothing new to say against them. I mention them to set them aside. More recently, the possibility of island universes has been challenged by David Lewis.⁵ His argument is indirect. Island universes make trouble for his realist theory of possible worlds; to the extent that he has good and sufficient reason to accept the theory, he has good and sufficient reason to reject the possibility of island universes. Trouble comes when one tries to say, as any realist must, how the possible worlds are demarcated one from another. According to Lewis, possible individuals are part of one and the same possible world if, and only if, they are spatiotemporally related.⁶ It follows immediately that no possible world is composed of island universes, of spatiotemporally isolated parts. Given the standard analysis of possibility as truth at some possible world, island universes, then, are impossible. I agree that island universes make trouble for Lewis’s realism about possible worlds, perhaps more than Lewis realizes. But Lewis’s realism is not the only realism in town. On Lewis’s brand of realism, the actual world and the merely possible worlds are ontologically on a par: there is no absolute ontological distinction between the actual and the merely possible. If, instead, the realist endorses some form of absolute actuality, then, I will argue, there is a simple and natural solution to the problem of island universes. Simple and natural, but in one sense radical, because it involves an emendation in the standard analysis of possibility as truth at some possible world, and in the standard method of giving truth conditions relative to possible worlds. On the emendation, as I will develop it, modal operators are analyzed as plural, rather than individual, quantifiers over possible worlds. When the amended analysis of possibility is combined with absolute actuality, a number of problems faced by Lewis’s realism are neatly solved. () The possibility of island universes, which I defend in Sections .. and .., can easily be accommodated. () The possibility of nothing, which I defend in Section .., can also easily be accommodated (if so desired). () A version of Lewis’s “principle of recombination”—roughly, that anything can coexist with anything—can be accepted without qualification; it is invulnerable to the Forrest-Armstrong argument (Section ..). () The principle of the identity of qualitatively indiscernible worlds, mysteriously undecidable on Lewis’s theory, can be decisively refuted (Section ..).⁷ All this is good news for those of us who are favorably inclined towards realism about possible worlds, and the project of analyzing modality in terms of them. Those who reject realism about possible worlds on grounds of crazy or bloated ontology, and who have no misgivings about primitive modality, will not, of course, be moved.

⁵ See Lewis (a: –). ⁶ Actually, Lewis holds that possible individuals are part of the same world if and only if they are spatiotemporally related, or analogically spatiotemporally related. See Lewis (a: –). Since this complication won’t matter for what follows, I will simply use ‘spatiotemporal’ broadly so as to include what Lewis calls “analogically spatiotemporal.” ⁷ All four difficulties with Lewis’s view are discussed in Armstrong (a); the second difficulty, according to Armstrong, is not genuine.

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. Realist Theories .. Realism What is realism about possible worlds? It won’t do simply to say that, according to the realist, possible worlds exist: given the range of disagreement over what sort of thing ‘possible world’ refers to, those who assert that possible worlds exist cannot be taken to share a single view. Nor will it do to say that, according to the realist, possible worlds exist and are concrete: there is too much disagreement (and unclarity) over what to call “concrete.” (See Lewis a: –.) The following characterization, however, should be sufficiently definite for our purposes. Call a theory of possible worlds realist if it holds that () worlds exist;⁸ () worlds are individuals rather than classes, or functions, or mathematical models; () worlds are particulars rather than properties or universals; () (most) worlds are complex rather than simple—for example, many worlds have parts that stand in spatiotemporal relations to one another. The problems I raise in this chapter involving island universes are problems for realism, so characterized. Thus, I address my argument and proposed solution primarily to realists. But non-realists who have as their goal an ontologically “innocent” reinterpretation of realist theory—various fictionalists and ersatzers—will also want to take heed.

.. Actuality According to David Lewis’s brand of realism about possible worlds, actuality is “indexical” and relational: the actual world is this world, the world we inhabit; to be actual is to be appropriately related to us. And that is all. Being actual confers no special ontological status; our world and the other possible worlds do not differ in ontological kind. The difference, so to speak, is that we are here and not there.⁹ The alternative for the realist is to hold that actuality is absolute, that there is an ontological distinction of kind between the actual and the merely possible. In my opinion, this is the only viable option for the realist. Our conceptual scheme demands that actuality be categorial: whatever is of the same ontological kind as something actual is itself actual. To hold, then, as Lewis does, that the actual world and the merely possible worlds do not differ in kind is simply incoherent.¹⁰ Our conceptual scheme also demands, I believe, a partial alignment between ontological and epistemological distinctions. In particular, whatever is of the same ontological kind as (and external to) an actual thinking subject cannot be known a priori by that subject ⁸ I use ‘world’ and ‘possible world’ interchangeably. I use ‘exist’ here to range over absolutely everything, the broadest category of being. In what follows, it will be clear from context, I hope, whether or not ‘exist’ is to be restricted to actual things. ⁹ See Lewis (: –); Lewis (a: –, –). ¹⁰ One must be careful not to conflate two related objections: the objection that actuality must be categorial; and the objection that actuality must be universal, applying to whatever has being, in the broadest sense of ‘being’. Lewis (a) discusses only the latter—I think, less plausible—objection. Note that either objection could be behind the tendency of philosophers to interpret Lewis—his denials notwithstanding—as believing in an actuality “much bigger and more fragmented than we ordinarily think.” See Lewis (a: –). [In the previously published version of this chapter, I wrote ‘categorical’ instead of ‘categorial’. But the term ‘categorical’ is best reserved for the notion that contrasts with modal. What I have in mind here is that actuality is a fundamental ontological category.]

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       ()

to exist. To hold, then, as Lewis does, both that the actual world and the other possible worlds do not differ in kind and that we know a priori that other possible worlds exist is, again, simply incoherent.¹¹ These are not arguments of course. But, if I am right, no arguments are needed: the alleged incoherence is right on the surface. A coherent realism, then, must hold that the actual and the merely possible differ in kind. I would develop the view, within a broader context, as follows. There are two fundamental categories of being. There is physical reality, which contains us as a part, and which we have epistemic access to, when we do, through causal interaction. And there is metaphysical reality, which contains the possible worlds (and, I think, mathematical entities and structures) as parts, and which we have epistemic access to, when we do, through our powers of thought, through our mental representations. The distinction I have in mind between physical and metaphysical does not coincide, on current usage, either with the distinction between actual and non-actual, or with the distinction between concrete and abstract. The mathematical part of what I call metaphysical reality—if not the whole of it—is typically called “actual” (by realists and non-realists alike); it would be mistaken, therefore, to identify physical reality with “actuality.” The modal part of metaphysical reality, as I conceive it, may properly be called “concrete,” since it is qualitatively no less determinate than physical reality; it would be misleading, therefore, to identify physical reality with the “concrete.”¹² On the brand of realism I am propounding, there is an absolute fact as to which among all the possible worlds has been actualized; call it realism with absolute actualization. Absolute actualization comes in two versions depending on whether the whole of physical reality is itself one of the possible worlds, or instead has a representative among the possible worlds. To make the different versions vivid, consider Leibniz’s God surveying the possible worlds prior to actualization.¹³ Did actualization consist in God conferring a special ontological status on the best possible world? Strictly speaking, this is transformation, not creation. Or did actualization consist in God creating ab nihilo a new “world,” qualitatively indistinguishable from the best possible world, but differing from it in ontological status? I think the second version is superior, but I won’t argue that here; the difference won’t much matter for what follows.¹⁴

¹¹ This objection was raised early on in Skyrms (: ). ¹² My distinction between physical and metaphysical is closer to the outmoded distinction between existence and subsistence. Note that I do not exclude irreducibly mental entities—if any there be—from physical reality. ¹³ Here and throughout, I call on Leibniz’s God to illustrate absolute actualization. But there are two implications of the image that I want to explicitly reject: I think actualization is primitive as well as absolute; and I deny that God could actualize a world without being a part of the world he actualized. ¹⁴ On this second version, what we call “the actual world”—that is, physical reality—is not, technically speaking, a world. But use of ‘the actual world’ is so ingrained that I won’t go out of my way to avoid it. The second version of realism with absolute actualization falls under what Lewis calls “pictorial ersatzism” (Lewis a: –). But its realist credentials are not thereby impugned. Indeed, Lewis’s chief criticism of pictorial ersatzism is that it offers no ontological gain for those who find his own account of worlds incredible. Agreed. The gain lies elsewhere: in giving an adequate account of actuality, and in solving the problems addressed in this chapter. [In Chapter , I reverse my judgment that the “creation” version should be preferred to the “transformation” version, and give some reasons. In that chapter, I call realism with absolute actualization “Leibnizian realism.”]

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No doubt I will be asked to explain further this distinction between physical and metaphysical reality. I can say something the distinction is not: it is not a difference in some quality. For any quality, there would have to be worlds at which some things have, and some things lack, the quality. But at every possible world everything is metaphysical, and at the actual “world” everything is physical. I can say something about how the distinction relates to us. What we think about, in one sense, is not limited by what actually exists—as when we think about golden mountains, or perfect cubes. Metaphysical reality is what we can think about, in that sense; it is the realm of “content” for all actual and possible thought. Physical reality, crudely put, is what we can bump into. I cannot, however, say anything about how physical and metaphysical reality differ in and of themselves. That is only to be expected: the distinction is fundamental and unanalyzable. The most serious objection to realism with absolute actualization, as Lewis has emphasized, is that it seems to allow for a coherent skepticism about one’s own actuality, whereas such skepticism is absurd. (See Lewis ; Lewis a: .) Granted, such skepticism is absurd. When asked—how do I know that I am actual?— I can give only one response: I just know it.¹⁵ I think that response is acceptable when dealing with a fundamental ontological category; talk of “evidence” here is beside the point. Moreover, anyone who accepts more than one fundamental ontological category, be it individuals and classes, or particulars and universals, must face the same sort of question, and, I claim, give the same answer. How do I know that I am an individual and not a class? I just know it. How do I know that I am a particular and not a universal? I just do. If this is right, then all ontological pluralists are in the same boat. And ontological monism, while attractively simple, faces formidable obstacles. I do not say that realism about possible worlds with absolute actualization is without difficulties. I wouldn’t stake my life on its being true—or even my next paycheck. But I can say with confidence that it is a serious contender in the metaphysical arena, worth seriously pursuing.

.. Isolation and Unification: The Realist Analysis of World Realists, I have noted, must provide a criterion of demarcation for worlds. Moreover, this must be done without relying on modal notions lest the analysis of modality in terms of worlds run in a circle.¹⁶ The basic strategy is this. Let logical space be that part of metaphysical reality over which (alethic) modal operators range; in other words, the aggregate sum of possibilia.¹⁷ Some regions of logical space are unified; the maximal such unified regions are the worlds. On this basic strategy, I am in agreement with Lewis. But I disagree with Lewis over two substantial issues having to do with the manner of unification. First, for Lewis, all worlds are globally unified ¹⁵ [In Chapter , I say more. I claim that although the concept of actuality is indexical, which is needed to make my knowledge that I am actual trivial, the property of actuality is absolute. Roughly, it follows from our concept that something is actual just in case it belongs to the same ontological category as me.] ¹⁶ This section draws on Chapter , which was written earlier. ¹⁷ If logical space is a proper part of metaphysical reality, as I think, it is also imperative that the extent of logical space be characterized without relying on modal notions. I believe that can be done, but I won’t discuss it here.

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       ()

(or almost globally unified): at any world, (almost) every part is directly linked to (almost) every other part.¹⁸ I hold instead that some worlds are locally unified: at some worlds, parts are directly linked only to “neighboring” parts. Second, for Lewis, each world is spatiotemporally unified; every world is spatiotemporally isolated from every other. I hold instead: a world may be unified by non-spatiotemporal relations; every world is absolutely isolated from every other. If I am right with respect to (either or both) of these issues, then Lewis’s conception of logical space is impoverished: some possible worlds are missing. I have argued for my views elsewhere; here I present the requisite definitions, and my analysis of world, without defense.¹⁹ First, I need to introduce some familiar metaphysical machinery.²⁰ I start with an abundant conception of property according to which, for every class of possibilia, there is (at least) one property had by all and only the members of that class. Then I need the distinction between the fundamental, or (perfectly) natural, properties and relations, and the rest. The natural properties and relations are those that correspond to immanent universals or tropes, if there are universals or tropes. They make for qualitative similarity: if two things instantiate the same natural property, or each divides into parts that stand in the same natural relation, then the things are objectively similar in some qualitative respect. Moreover, the qualitative supervenes on the natural: fixing the natural properties and relations suffices to fix all the qualitative properties and relations. In terms of naturalness, a number of indispensable metaphysical notions can be defined. I will be brief.²¹ Things are (intrinsic, qualitative) duplicates just in case there is a similarity map from one to the other: a one-one correspondence between their parts that preserves all natural properties and relations (and the part-whole relation). An intrinsic nature is a property had by all and only the duplicates of some thing. An intrinsic property is one that never differs between duplicates; a property is extrinsic just in case it is not intrinsic. An internal relation is a relation that supervenes on the intrinsic natures of its relata. Having-the-same-mass-as is an example of an internal relation, assuming the mass properties are intrinsic. An external relation is one that, although it fails to supervene on the intrinsic natures of its relata, does supervene on the intrinsic natures of its relata and of the fusion of its relata.²² Being-adjacent-to is an example of an external relation: whether two things are adjacent to one another is

¹⁸ This follows from Lewis’s claim that the unifying relations are “pervasive.” See Lewis (a: ). ¹⁹ In Chapter , I argue that if Einsteinian relativity is true (on its most natural interpretation), then we live in a locally unified world; such worlds had better be possible! In Chapter , I defend the view that worlds are absolutely isolated from one another. (Note, however, that for a realist who accepts universals— unlike myself—that view will have to be qualified.) ²⁰ In this and the following paragraph, I more or less follow Lewis (a: –). ²¹ Quantifiers range over all parts of physical and metaphysical reality unless explicitly restricted. I assume familiarity with mereology, the theory of part and whole. In particular, I assume unrestricted mereological composition: for any things, there is a (mereological) sum, or fusion, of those things. ²² More precisely, say that an ordered pair and an ordered pair are internal duplicates iff a is a duplicate of c and b is a duplicate of d; external duplicates iff, in addition, the composite of any similarity map from a to c and any similarity map from b to d induces a similarity map from the fusion of a and b to the fusion of c and d. Then, an internal (dyadic) relation is one, the holding of which never differs between pairs that are internal duplicates; an external (dyadic) relation is one that is not internal, but the holding of which never differs between pairs that are external duplicates. (Analogously for relations of three or more places.)

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not determined by their intrinsic natures, taken separately, but it is determined if one also takes into account the intrinsic nature of their fusion. A relation that is either internal or external is intrinsic; all others are extrinsic. Note that it is built into the definitions that all natural properties and relations are intrinsic. I am ready, finally, to define unification, and then in terms of unification to analyze the notion of world. Unification, I take it, is to be characterized in terms of the holding or failing to hold of natural external relations. But which relations? Different choices yield different notions of unification. I will give the definitions simultaneously with respect to two choices: the natural spatiotemporal relations, and all natural external relations. Two parts of logical space are spatiotemporally separated (externally separated) if and only if they are non-overlapping and no part of one stands in any natural spatiotemporal (external) relation to any part of the other. A part of logical space is spatiotemporally unified (externally unified) if and only if it is not the sum of two spatiotemporally separated (externally separated) parts. Two parts of logical space are spatiotemporally related (externally related) if and only if some spatiotemporally unified (externally unified) part of logical space includes them both; otherwise they are spatiotemporally isolated (externally, or absolutely, isolated).²³ Now we can analyze the notion of world in terms of unification. For Lewis, were he to accept the above definitions, the analysis would be this: a world is any maximal spatiotemporally unified part of logical space, that is, a spatiotemporally unified part not properly included in any other spatiotemporally unified part. As already noted, I reject this analysis as too narrow. (Indeed, for all we know, not even the actual “world” is spatiotemporally unified; perhaps, as physicists have pondered, spatiotemporal relations do not apply at the “submicroscopic” level.) I accept instead: a world is any maximal externally unified part of logical space. From this it follows that worlds are absolutely isolated from one another, and, in particular, that no two worlds overlap. Finally, parts of logical space are worldmates if and only if they are part of the same world, if and only if they are externally related. In what follows, I will assume that worlds are demarcated by external interrelatedness, although all of my main arguments could be modified to apply to Lewis’s notion of world. I conclude this section with an example that serves to illustrate the definitions, and that will play a role in the argument of Section ... Worlds may be unified to a greater or lesser degree. At one end of the spectrum, we have globally unified worlds at which no part is externally separated from any other part. At a globally unified world, points of spacetime (if such exist and are atomic) are directly linked to one another by some natural external relation, presumably, by some external relation of spatiotemporal distance (interval). At the other end of the spectrum, we have locally unified worlds at which the only parts that are not externally separated are overlapping or adjacent parts.²⁴ (The separated parts are nonetheless externally related in

²³ In Chapter , which was written earlier, I (unwisely) used ‘isolation’ both for what I here call “separation” and (in informal discussion) for what I here call “isolation.” When worlds are not globally unified (see Section .. below), the difference matters. ²⁴ Topologically speaking, two regions are adjacent iff they are non-overlapping, but one contains a boundary point of the other. (For example, on the real line, the open interval (, ) is adjacent to the closed interval [, ], but not to the open interval (, ).) Only worlds with topological structure can be locally unified.

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

       ()

virtue of belonging to a single externally unified region of logical space.) At a locally unified world with continuous spacetime, distinct points of spacetime are separated (being non-adjacent), and so are not directly linked by any natural external relation; relations of spatiotemporal distance are extrinsic, rather than external, because the distance between points depends on the intervening spacetime, on the lengths of paths from one point to the other.²⁵

. Are Island Universes Possible? .. Island Universes: Strong and Weak Senses We now have the resources to define the notion of island universe, and to consider whether or not island universes are metaphysically possible. But we must be careful to distinguish an absolute, or strong, sense of ‘island universe’ from various nonabsolute, or weak, senses, depending on whether island universes are required to be absolutely isolated, or only isolated in one or another respect. Thus, island universes in the strong sense exist if and only if physical reality is not externally unified; the island universes are the maximal externally unified parts of physical reality. Island universes in a (prominent) weak sense exist if and only if physical reality is not spatiotemporally unified; the island universes are the maximal spatiotemporally unified parts. Other weak senses of ‘island universe’ can be had by replacing ‘spatiotemporally’ with ‘spatially’ or ‘temporally’ or ‘causally’ or any combination of these.²⁶ If worlds are demarcated by external interrelatedness, as I claim, then the possibility of island universes in a weak sense is not problematic; for example, a world might well have spatiotemporally isolated parts that are externally related by some non-spatiotemporal relation. The possibility of island universes in the strong sense, however, is blatantly contradictory (if possibility is truth at some world): worlds, being externally unified, cannot have absolutely isolated parts. In Sections .. and .. I give my arguments for the possibility of island universes in the strong sense; in Sections .. through .., I consider how the contradiction is to be avoided.

.. Lewis on the Possibility of Island Universes If worlds are demarcated by spatiotemporal interrelatedness, as Lewis holds, then even the possibility of spatiotemporally isolated island universes is contradictory (if possibility is truth at some world): no world can both be spatiotemporally unified and have spatiotemporally isolated parts. Lewis therefore rejects the possibility of spatiotemporally isolated island universes (what he calls, “disconnected spacetimes”). He writes: I would rather not [reject the possibility]; I admit some inclination to agree with it. But it seems to me that it is no central part of our modal thinking, and not a consequence of any interesting general principle about what is possible. So it is negotiable. (Lewis a: ) ²⁵ In Chapter , I argue that distance relations are extrinsic, rather than external, at (some) worlds with continuous spacetime. ²⁶ For Lewis, adding causal isolation to spatiotemporal isolation would have been redundant. See Lewis (a: –). Not so for those who take causation to be a primitive external relation between events.

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

To make the rejection of island universes more palatable, Lewis offers up an assortment of substitutes. Perhaps, he suggests, when we think we conceive of the possibility of island universes, what we really have in mind is a big world, spatiotemporally unified, but with many causally isolated world-like parts. For example: The spacetime of the big world might have an extra dimension. The world-like parts might then be spread out along this extra dimension, like a stack of flatlands in three-space. (Lewis a: )

And he provides three other ways that a big world might have causally isolated world-like parts. I have no objection to Lewis’s substitutes: they are genuinely possible, one and all. But they do nothing to counter our inclination to believe in the possibility of island universes. If the notion of island universe were obscure, or very complex, it might be otherwise; we might not know what possibility we had in mind. But since the notion of island universe, once disambiguated, is simple and clear, Lewis’s substitutes are plainly beside the point. (Compare the question whether there could be a world at which space is “curved.” To point out the possibility of a world at which a “curved” space is embedded within a higher-dimensional “flat” space is unresponsive; it merely changes the subject.) An “inclination to believe,” however, by itself, does not count for much. Lewis’s rejection of the possibility of spatiotemporally isolated island universes would be unassailable if, as he says, “[it is] not a consequence of any interesting general principle about what is possible.” I will argue below, on the contrary, that the possibility of island universes does follow from general principles about what is possible, from (strong versions) of the Humean denial of necessary connections. These principles, I think, will be hard for an arch-Humean such as Lewis to resist!

.. Are Island Universes Physically Possible? Before turning to metaphysical arguments for the possibility of island universes in the strong sense, I want to briefly consider some arguments loosely based on contemporary physics. I grant, of course, that if island universes are physically possible, then, a fortiori, they are metaphysically possible. But arguments for the physical possibility of island universes either fail outright, or, in the best case, rest on a controversial interpretation of objective chance. Thus, I rate the case for physical possibility, at best, as inconclusive. To begin, consider the following proposals, all made by reputable physicists. () Perhaps our “universe” is but one of many “universes” in a series of “cosmic oscillations”: big bang, followed by big crunch, followed by big bang, followed by big crunch, and so on. The different “universes” may even differ in their physical laws. But, surely, if talk of “oscillations” is to be appropriate, there must be (spatio) temporal (and causal) relations between the many “universes.” () Perhaps our “universe” is the result of a quantum vacuum fluctuation, one of those things, according to quantum electrodynamics, “that happens from time to time.” And then why not more than once, resulting in many “universes”? But, surely, talk of “vacuum fluctuations” requires a pre-existing “vacuum,” and the many “universes” (for all the theory says) are spatiotemporally (and causally) related to one another via

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

       ()

this “vacuum.” () Perhaps on the correct interpretation of quantum mechanics our “universe” is but one of many branching “universes” splitting off from one another with every measurement-like interaction. But since any two branching “universes” have a common history, events from different “universes” are spatiotemporally (and causally) related to one another via events that occurred before the “universes” split off. None of these proposals, then, provides an argument for the possibility even of spatiotemporally (and causally) isolated island universes, let alone island universes in the strong sense.²⁷ Perhaps General Relativity can do better. It won’t do, however, to simply note that Einstein’s field equation admits solutions with disconnected spacetimes because sums of solutions are solutions. Physicists do not automatically suppose that mathematical solutions to their equations have any genuine physical significance; further argument is needed. Try this. There is a well-known solution to Einstein’s field equation in which two regions of spacetime are connected only by a single, shortlived “wormhole”: the “wormhole” evolves spontaneously between two spatially disconnected regions, it gradually grows to a maximum diameter, then it shrinks until, pinching off, it leaves behind two spatially disconnected regions as before. Thus far, we have only spatial, not spatiotemporal, isolation.²⁸ But now consider physical indeterminism. Presumably, there was some non-zero objective chance that the “wormhole” would never have evolved, that is, some non-zero objective chance that the two regions would have been spatiotemporally isolated island universes. And, whatever has a non-zero objective chance of happening is physically possible.²⁹ I don’t really know whether our best physical theory allows General Relativity and indeterminism to be combined in the way required for this argument. In any case, I suspect the argument supports only the possibility of spatiotemporally isolated island universes, not the possibility of island universes in the strong sense. Here’s why. In the possibility being envisaged, the spatiotemporally isolated island universes are related in virtue of their being some objective chance that a “wormhole” connect one with the other. This cannot be an internal relation, lest there also be some objective chance that a “wormhole” connect the one island with each and every duplicate of the other island, spread out through logical space! It must then be an

²⁷ See Leslie (: ch. ), for a discussion (with references) of these and other physical mechanisms for generating many “universes.” If the many universes are needed only to solve the “fine-tuning problem”— the problem of rendering unsurprising the fact that our “universe” is “fine-tuned” for the existence of life— the “universes” needn’t be spatiotemporally (or causally) isolated; there need only be enough variety. ²⁸ Weingard (: ) uses this solution—due to Martin Kruskal—to argue that “it is physically possible for space (relative to a frame of reference) to be, at some time, in disconnected pieces.” ²⁹ The idea of using objective chance at this point in the argument I get from Bigelow and Pargetter (). A related argument goes like this. Presumably, whether and how the “wormhole” evolved might depend on features of the two spacetime regions, so that, had the regions been appropriately different, no “wormhole” would have evolved; the regions would have been spatiotemporally isolated. But, here, an appeal must be made to some principle of recombination to ensure that the regions could have been appropriately different; one may no longer be within the realm of physical possibility. In Bigelow and Pargetter (), the counterfactual argument is not distinguished from the argument from objective chance. Incidentally, Lewis () gives the counterfactual argument, and claims that “the intuitive case that island universes are possible has been much strengthened” thereby. But he does not suggest any realist response.

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   ?



external relation, either itself natural, or grounded in other natural external relations between the islands. The two islands, then, are externally related; they are not island universes in the strong sense.

.. Island Universes and Plenitude: The Principle of Solitude (PS) So much for arguments based on speculative physics. I turn now to arguments based on speculative metaphysics. As already noted, I think the possibility of island universes (in the strong sense) follows from general principles of the plenitude of possibilities, in particular, from strong versions of the Humean denial of necessary connections. To start, let us consider what I call the principle of solitude, roughly: anything can exist all by itself.³⁰ This principle captures the idea that actualization is unconditional: whether or not one thing is actualized is not conditional on whether or not any other thing is actualized. Unconditional actualization, it seems to me, is part and parcel of the conception of intrinsic nature presented above (in Section ..): each thing has an intrinsic nature in virtue of which it can be conceived of apart from anything else; and, if it can be conceived apart from anything else, then, possibly, it exists apart from anything else, that is, all by itself. To illustrate the principle of solitude, consider a discriminating God. While surveying the possible worlds prior to actualization, he comes on the world corresponding to our actual “world.” He is not pleased with everything he sees. Perhaps only one thing pleases him—say, Leibniz. Then, according to the principle of solitude, God could choose to actualize Leibniz, and nothing else. Or perhaps only one thing displeases him—say, Hume. Then, according to the principle of solitude, God could choose to actualize the world minus only Hume, leaving a hole where Hume would have been. In deciding what to actualize, God does not have to take a world all or nothing: he can exercise a line item veto. How should the principle of solitude be expressed within a realist framework? First, there is no reason to restrict the quantifiers to actual individuals; we can quantify universally over individuals from all the worlds. It is, however, restricted to worldbound individuals, individuals that are part of some world.³¹ Second, since the principle is not a claim about the essences, or potentialities, of things, it is to be interpreted in terms of duplicates, rather than counterparts. The principle requires that a duplicate of Leibniz could exist all by itself; it does not say that the duplicate represents anything de re of Leibniz, or would properly be called “Leibniz.” (Perhaps some of Leibniz’s essential properties are extrinsic, having to do with his origins, or what have you.) Thus, the principle can be formulated as follows: (PS)

Principle of Solitude.For any worldbound individual, possibly, a duplicate of that individual exists all by itself.

Three brief comments may help to forestall misunderstanding. First, to say that the duplicate “exists all by itself,” of course, is to say that nothing wholly distinct from the ³⁰ [On how the principle of solitude relates to more general principles of plenitude, see Chapters  and , especially the postscript to Chapter .] ³¹ By ‘individual’, I always mean “thick” particular. See Section .., which is the only place the distinction matters.

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

       ()

duplicate exists: nothing exists but the duplicate and its parts. Second, (PS) does not rule out the necessary existence of “mathematical” entities: necessary beings are compatible with (PS) as long as they are not parts of worlds, and so not within the range of the quantifiers. Third, existing all by oneself is not the same as existing surrounded by empty spacetime. In the possibility posited by (PS), spacetime, if it exists at all, has whatever shape the duplicate has; in the case of lonely Leibniz, spacetime has the shape of a spacetime worm. Now we are ready for the argument that (PS) leads to the possibility of island universes (in the strong sense). I need two modest assumptions. First, I suppose there are worlds at which the only natural external relations are spatiotemporal relations; worlds at which spacetime is empty will do. Second, I suppose that some such worlds are locally unified: individuals at the world that are neither adjacent nor overlapping are spatiotemporally separated; they (and their parts) are not related by any natural spatiotemporal relation (see Section ..). Now consider the sum of any two spatiotemporally separated individuals at any such locally unified world. Apply (PS) to that sum. In the possibility that results, the duplicate of the sum is composed of two individuals that are not only spatiotemporally separated, but spatiotemporally isolated as well; the intervening spacetime that unified them in the original world has been removed. By assumption, there are no non-spatiotemporal, natural external relations to unify them. So, they are absolutely isolated individuals: island universes in the strong sense.³²

.. Lewis and (PS); Strong vs. Weak Denials of Necessary Connections The principle of solitude, as I see it, encapsulates an especially strong form of the Humean denial of necessary connections between distinct existents. Lewis cannot accept (PS)—not if I am right that it leads to the possibility of island universes. When Lewis champions the denial of necessary connections, then, it must be something weaker he has in mind.³³ The difference between the strong and weak denial can be seen to arise from an ambiguity of quantifier scope. Let us say there are necessary connections between distinct existents if: there exists some x such that, necessarily, x coexists with some y distinct from x. This is ambiguous: the quantifier ‘some y’ can be given wide scope or narrow scope. If given wide scope, we get: (NC1)

There exists some x and some y distinct from x such that, necessarily, x coexists with y.

When Lewis explicates the Humean denial of necessary connections as (in part) “anything can fail to coexist with anything else,” it is apparently the denial of (NC) that he has in mind. (Lewis a: ). Indeed, that denial is all one needs to support the standard Humean arguments about laws and causation. If the quantifier ‘some y’ is given narrow scope, we get (the equivalent of ): ³² I present this example, to a slightly different end, in Chapter . ³³ [Maybe not. After this chapter was published, Lewis explicitly endorsed (PS). See Langton and Lewis (: ). In the terminology of that paper, (PS) is the assumption that “everything has a lonely duplicate.” They write: “that assumption is part of an attractive combinatorial conception of possibility such as that advanced on pp. – of On the Plurality of Worlds.”]

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

(NC2) There exists some x such that, necessarily, x does not exist by itself. (PS) is the denial of (NC). I do not think a Humean can rest content with the denial of (NC): that denial is too weak to capture the full plenitude of possibilities. To illustrate, consider a red ball and a green cube. (Pretend that colors and shapes are natural properties.) According to the denial of (NC), it is possible for (a duplicate of ) the red ball to exist without (any duplicate of ) the green cube existing. But, surely, there is more. It is also possible for (a duplicate of ) the red ball to exist without anything green existing. Yet this does not follow from the denial of (NC). To allow for this possibility, we need a stronger principle of plenitude. But there is more. It is also possible for (a duplicate of ) the red ball to exist without anything (else) colored, without anything (else) colored or extended. We have started down a slippery slope. (PS)—the denial of (NC)—is waiting at the bottom of that slope: it allows for all these possibilities in one fell swoop. There is no stopping short.³⁴

.. Generalizations of (PS) There are two ways to generalize (PS) corresponding to two ways to unrestrict the quantifiers over worldbound individuals. We can quantify instead over all individuals, transworld individuals included (where a transworld individual is any sum of worldbound individuals from two or more worlds); and we can quantify, not only over individuals, or thick particulars, but over thinned-down particulars that have had some or all of their universals or tropes stripped away (if there are universals or tropes). Either generalization leads directly to the possibility of island universes (in the strong sense), without taking a detour through locally unified worlds. Generalizing the first way gives this: (GPS)

Generalized Principle of Solitude. For any worldbound or transworld individual, possibly, a duplicate of that individual exists all by itself.

The possibility of island universes follows immediately from (GPS) by instantiating the quantifier to any transworld individual. What motivates (GPS) is that there should be no restrictions on what can be actualized. On the realist position I espouse, actualization is primitive and absolute. What justification could there be for restricting actualizability to some parts of logical space—the worldbound individuals—while excluding it from others—the transworld individuals. Or look at it from God’s perspective. To restrict actualizability to worldbound individuals would be to put a limitation on God’s power to choose, one not grounded in any logical necessity. Suppose that, in surveying the worlds prior to actualization, God found that two or more worlds were tied for best. Why must God choose between actualizing one world, or the other? He’s all-powerful! He can simply say: “Actualize those!” ³⁴ There is another way to allow for the possibilities mentioned. One can instead invoke principles of plenitude for alien possibilities: instead of “removing” the world surrounding the red ball, one can “replace” it by something alien to actuality. [I formulate and defend such principles in Chapter .] But one would still be stuck with necessities involving the disjunction of all natural properties, actual and alien. On “alien” possibilities, see Lewis (a: –).

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

       ()

The second way of generalizing (PS) comes into play only for theories that accept universals or tropes. Any such theory will distinguish between “thick” particulars— what I have been calling “individuals”—and “thinned-down” particulars.³⁵ Thick particulars are maximal “bundles” of co-instantiated universals or tropes, perhaps together with a substratum, or “thin” particular. If one selectively strips off some universals or tropes from a thick particular, or from any of its thick particular parts, what is left is a thinned-down particular; if one strips off all these universals or tropes, and anything is left, what is left is a thin particular. Now, the generalization we seek simply applies unconditional actualization to all particulars: (GPSP) Generalized Principle of Solitude for Particulars. For any particular (thick, thinned-down, or thin), possibly, a duplicate of that particular exists all by itself. To illustrate, consider some actual individual—say, an electron. Among its natural properties, let us suppose, are having unit negative charge, having spin one-half, and having a mass of . MeV. Then, according to (GPSP), possibly, there exists a particle (an ordinary individual) just like an electron but with no charge property; or with no charge or spin property; or (on a substratum view) with no properties at all— a “bare” particular. To get from (GPSP) to the possibility of island universes, one can start with any world containing two or more individuals and strip off relations, that is, polyadic universals or tropes. One way: apply (GPSP) to the world minus all polyadic universals or tropes. In the possibility that results, distinct individuals are absolutely isolated island universes. Another way is more selective, but works only for tropes: divide the world into two distinct individuals and apply (GPSP) to the world minus all polyadic tropes connecting the two individuals. In the possibility that results, the duplicates of the two individuals are absolutely isolated island universes. That concludes my case for the possibility of island universes in the strong sense. My belief in this possibility is not an offhand modal opinion. It follows, in many ways, from general principles of plenitude applied to ordinary individuals and worlds, principles that are hard to deny for anyone with a broadly Humean approach to modality. It is not negotiable.³⁶

. Realist Responses .. Primitive Worldmate Time for the realist to face the music. Island universes in the strong sense, I have argued, are metaphysically possible. That leaves the realist with two options: somehow revise the analysis of world so as to allow a world to have absolutely isolated ³⁵ For example, see Armstrong (b: –). ³⁶ According to the classification of principles of plenitude in Chapter , (PS) and its generalizations count as principles of plenitude for structures: they have implications as to which structures are possible (that is, possibly instantiated or actualized). (PS) implies that possibility is preserved under the operation of taking substructures; (GPS) implies, in addition, that possibility is preserved under the operation of taking sums of structures.

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 



parts; or somehow revise the standard analysis of modality so that island universes can be possible, without there being a world at which island universes exist. I begin with the first option. According to Lewis, the alternative to rejecting the possibility of island universes is to posit a primitive worldmate relation.³⁷ A world, then, would be a region of logical space such that every part stands in this worldmate relation to every part, and to nothing else. But there is a major problem. A primitive worldmate relation is primitive modality, in at least two ways. First, the realist will need to posit necessary connections between the worldmate relation and other relations: necessarily, things are worldmates if they are spatiotemporally (or externally) related. Second, general facts of modality (facts expressible without reference to specific properties, relations, or things) will be made to depend on how the worldmate relation is laid out in logical space; in particular, the very possibility under discussion, the possibility of island universes, will so depend.³⁸ To resort to a primitive worldmate relation, then, would spell defeat for the realist project of analyzing modality. Even leaving the question of primitive modality to one side, it is far from clear how a primitive worldmate relation would solve the problem at hand. For what sort of relation is primitive worldmate? Not internal, of course: a duplicate of one of my worldmates need not be my worldmate. So it is external (since, presumably, being primitive, it is not extrinsic). But if it is a natural external relation, then it is no help at all! For then worldmates are externally connected, and no world has absolutely isolated parts, island universes in the strong sense. Could primitive worldmate be some non-natural external relation? In that case, it would not genuinely unify, or tie together, its relata. (Compare non-identity, which is also, I claim, external but non-natural.) It would be best thought of, I think, not as a primitive relation, but as a plethora of primitive, non-natural properties, one per world; the one that applies to the actual world could do much of the work done by primitive actualization on my own theory.³⁹ I cannot complain that these primitive, non-natural (and so non-qualitative) properties are mysterious, since I have helped myself to one for my own realist theory (though I can wonder at their abundance). But realism with primitive actualization has stark advantages over realism with primitive worldmate. It avoids positing primitive modality, and it upholds the ontological bifurcation, without which I claim realism is untenable, between the “actual” and the “merely possible.” Fortunately, given primitive actualization, we don’t need primitive worldmate to respond to the realist dilemma. There is a better way.

³⁷ Lewis (a: –). For Lewis, rejecting island universes is “more credible” than positing primitive worldmate, though he doesn’t give his objections to the latter. ³⁸ Note that primitive actualization is not in the same boat: modal facts do not depend on which region of logical space has been actualized; only truth so depends. Actuality is no more a modal notion than is truth. ³⁹ I do not know whether Lewis would allow that worldmate could be both primitive and non-natural. In any case, since Lewis only considers primitive worldmate as a means to accommodate spatiotemporally isolated island universes, he only needs primitive worldmate to be non-spatiotemporal; it needn’t also be non-natural.

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

       ()

.. The Amended Analysis of Modality: Class Version The realist has a second option: amend the standard analysis of possibility. It should be obvious by now how the emendation will go. If possibilities are represented not just by single worlds, but also by pluralities of worlds, then our problem is instantly solved: the possibility of island universes will be represented throughout logical space, by every plurality of worlds.⁴⁰ The solution comes in different versions, however, depending on how “pluralities” are understood. If pluralities are understood as classes, we have: Amended Analysis (Class Version). A proposition is (metaphysically) possible if and only if it is true at some (non-empty)⁴¹ class of worlds. The analysis requires that we make sense of truth at a class of worlds. The idea, of course, is to imagine that all the worlds in the class are actualized, and then to ask what would be true. More exactly, starting from a standard interpretation of a language in possible worlds semantics, we extend the interpretation as follows. The domain of a class of worlds is the union of the domains of the worlds in the class. An existentially or universally quantified sentence, when evaluated relative to a class of worlds, will have its lead quantifier restricted to the domain of that class. The extension of a predicate relative to a class of worlds is just the union of its extensions at the worlds in the class. That suffices to assign truth values to sentences of modal predicate logic. It might seem rash to tinker with the standard analysis of possibility: it is a cornerstone of modern modal metaphysics, not only for realists, but for all who traffic in the language of possible worlds. Note, however, that the proposed emendation is conservative in its consequences. For one thing, although possibilities are added, no possibilities are taken away. Whatever was possible under the standard analysis—true at some world—remains possible under the amended analysis: true at the corresponding singleton world. For another thing, the added possibilities are extremely limited in scope. One new possibility, of course, is the proposition that there exist absolutely isolated individuals: it is true at any class of two or more worlds. And every other new possibility entails this proposition. So only propositions having to do with island universes change their possibility status under the amended analysis. Possibility does not stand alone. There will have to be corresponding emendations in the analyses of other modal operators. Metaphysical necessity, of course, becomes truth at all classes of worlds. Restricted modal operators—nomological, doxastic, deontic, and so on—will have to be analyzed in terms of accessibility relations taking classes of worlds as their relata. For example, an agent’s (de dicto) beliefs will be represented, not by a class of doxastically accessible worlds, but by a class of classes of

⁴⁰ Warning: Lewis has also argued that possibilities need not be represented by possible worlds, but on entirely different grounds. Lewis claims that possibilities for an individual—possibilities de re—are represented, not by whole possible worlds, but by possible individuals, that is, parts of worlds. See Lewis (a: –). We are concerned here, however, only with possibilities de dicto. ⁴¹ On whether ‘non-empty’ should be dropped, see Section ...

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 



doxastically accessible worlds. Suppose an agent believes that island universes exist; then only classes of worlds containing two or more members are doxastically accessible (for that agent). Truth conditions for counterfactuals, too, will have to change: the closeness relation will take classes of worlds as its relata. To illustrate, consider again a world that contains a single “wormhole” connecting two otherwise disconnected regions of spacetime. (This time make it deterministic, with a static spacetime.) Now consider the counterfactual: had there been no “wormhole,” there would have been absolutely isolated regions of spacetime. I take it this counterfactual is true. (Should one say instead that there would have been some other “wormhole”? That the antecedent is impossible?) What makes the counterfactual true is this: from the standpoint of the “wormhole” world, the closest class of worlds at which the antecedent is true is a doubleton, rather than a singleton; and the consequent is true at that closest class. It would be onerous, to be sure, to have to rewrite the textbooks on possible worlds semantics, giving truth conditions relative to classes of worlds, instead of worlds. I recommend against it! Once one becomes convinced that it can be done, one need only indulge on those rare occasions—the present included—where the possibility of island universes comes into play.

.. The Amended Analysis with and without Absolute Actualization: Semantical Considerations Could Lewis accept the amended analysis so as to allow for the possibility of island universes? I think not. The amended analysis and Lewis’s “indexical” theory of actuality do not mix well. The problem comes, not with possibility, but with truth. Suppose I assert: “Island universes exist.” On the semantical framework that underlies the amended analysis, the truth or falsity of my utterance is to be evaluated relative to a class of worlds. But which class? The world at which my utterance occurs belongs to many classes of worlds, and without absolute actualization there is nothing to choose between them. I consider three options. () Stay as close as possible to the old method according to which the truth or falsity of an utterance is evaluated relative to the world at which the utterance occurs. On the new semantical framework, this becomes: my utterance, “island universes exist,” is true, simpliciter, if and only if it is true at the singleton whose sole member is the world at which my utterance occurs; otherwise, false, simpliciter. But, then, on semantical grounds alone, my utterance is false, simpliciter, since it is false at any singleton world. So, if we combine the Amended Analysis with option (), we have it that my utterance is both contingently possible and analytically false. Not a happy combination. () Invoke supervaluations. Call any class of worlds containing the world at which my utterance occurs a relevant class. Then, my utterance, “island universes exist,” is true, simpliciter, if and only if it is true at all relevant classes; it is false, simpliciter, if and only if it is false (not true) at all relevant classes; otherwise, it is neither true nor false, simpliciter. But, then, on semantical grounds alone, my utterance is neither true not false, simpliciter, since it is false at one relevant class (the singleton) and true at all the others. So, if we combine the Amended Analysis with option (), we have it that my utterance is both contingently possible and analytically neither true nor false.

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

       ()

Still not a happy combination. () Deny that utterances (at any rate, utterances that express contingent propositions) are ever true or false, simpliciter: there is only relative truth, truth relative to this or that class of worlds. But now I’ve lost my grip on the semantical enterprise. One job of an adequate semantics is to provide the machinery for moving from actual utterances to truth values. The assignment of truth conditions to utterances takes us only partway; there must also be a path from truth conditions to truth or falsity. Indeed, if there is no truth or falsity, simpliciter, in what sense are truth conditions conditions of truth? Option (), being semantically inadequate, is a non-starter. And that’s it, I think, for plausible options. The amended analysis, then, and the semantical framework that underlies it, are not for Lewis: without absolute actualization, the alternative to rejecting the possibility of island universes, as Lewis said, is to posit primitive worldmate.⁴² When the amended analysis is combined with absolute actualization, my assertion, “island universes exist,” acquires a definite truth value. One or more worlds has been actualized, and the truth or falsity of my assertion is evaluated relative to the class of actualized worlds. Of course, the assertion, “island universes exist,” is rather extraordinary. But it does no harm also to evaluate ordinary assertions relative to the class of actualized worlds. For most ordinary assertions, the quantifiers are implicitly restricted to the world at which the assertion occurs, or some part thereof; island universes, then, should they exist, would be irrelevant to the truth values of ordinary assertions. For some assertions, however, it is unclear whether or not island universes would be relevant. For example, when a physicist says, “nothing travels faster than light,” should her quantifiers be restricted to the world she inhabits, so that superluminal particles in other island universes would be irrelevant? Or should her quantifiers extend to other actualized worlds, so that her assertion would be falsified by an island universe at which light is not a first signal? I doubt there is any linguistic fact of the matter. There is no reason why the physicist should have bothered to decide which she means. Nor, then, should we decide. It is enough to note that we can allow for either interpretation. So much for actual utterances. Should we also assign truth values to merely possible utterances? We can if we want. Typically when we ask whether a possible utterance is true, we are engaged in counterfactual thinking: would the utterance have been true, had it been made. In that case, we carry over information about actuality to the counterfactual situation. In particular, we carry over whether island universes do or do not exist, since, presumably, the existence of island universes is counterfactually independent of whether or not an utterance is made. A possible utterance of “island universes exist,” thus counterfactually considered, has the same truth value as an actual utterance of the same. We might mean something else, however, when we ask for the truth value of a possible utterance. We might be asking whether the possible utterance is true or false, simpliciter, true or false from its own perspective. In that case, the absolute facts about actuality are irrelevant. We are back to the three options ⁴² I have focused on the problem of evaluating the proposition expressed by an utterance. There is a parallel problem—with three parallel options—having to do with determining what proposition an utterance expresses in case the proposition expressed depends on the world in which the utterance occurs (as happens, for example, with restricted modalities and counterfactuals).

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 



considered above. (But this time, we needn’t worry that the options will be semantically inadequate, either by making truth values semantically determined where they should not be (options () and ()), or by denying such truth values exist (option ()); the assignment of truth values to possible utterances is not essential to the semantical enterprise, and can be introduced or jettisoned with impunity.) I think option (), the method of supervaluations, gives the intuitively right results. When a merely possible person (speaking our language) says, “island universes exist,” what she says is neither true nor false, simpliciter. Although she intends her assertion to depend on the facts of absolute actualization, there are no such facts (from her perspective), and so nothing to make her assertion either true or false. But when the merely possible person says “I exist,” she speaks truly: her utterance is true at all relevant classes of worlds.

.. The Amended Analysis: Plural Quantifier and Aggregate Versions On the amended analysis of modality, possibility is truth at some world, or some plurality of worlds. I began with a class version of the analysis—interpreting pluralities as classes—not because I favor that version, but because among contemporary metaphysicians classes are familiar and, for the most part, accepted tools of the trade. In this section, I consider two further versions, and state my preferences. Because the philosophical arguments that favor one version over another largely cut across the issues of this chapter, I will be brief. All three versions allow equally for the possibility of island universes. The version I favor analyzes modality as plural quantification over worlds, and plural quantification, I have been convinced, is not to be reduced to singular quantification over classes (or class-like entities).⁴³ Consider first an ordinary language example. I am deciding which books to put on a shelf. You warn: “Some books will bring down the shelf.” You deny, however, that any single book will bring it down. Then you have quantified plurally over books. In longwinded paraphrase: there are some books such that they will bring down the shelf. You have not thereby quantified over anything other than books; in particular, you have not unwittingly, surreptitiously also quantified over, or somehow trafficked in, classes (or class-like entities). Just as the predicate ‘will bring down the shelf ’ may be either singular or plural, I propose to interpret the relational predicate ‘is true at’ as either singular or plural in its second argument place, allowing it to take either singular or plural quantifiers over worlds. Thus, a proposition may be true at some world, or true at some worlds, where the latter does not entail the former. Recasting the amended analysis in terms of plural quantification, we have: Amended Analysis (Plural Quantifier Version). A proposition is (metaphysically) possible if and only if it is true at some world, or some worlds. Of course, a proposition is true at some worlds just in case it is true at the class containing those worlds as members according to the account given in Section ... So the class version and the plural quantifier version do not differ as to what ⁴³ For convincing arguments, see Boolos (), Boolos (), and Lewis (). For interesting dissent, see Resnik () and Hazen ().

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

       ()

propositions are possible; in particular, on both versions, it is possible that island universes exist. Why, if the versions are extensionally equivalent, do I prefer the plural quantifier version? Because, first, on the class version, one cannot say that a proposition is possible without being ontologically committed to classes; and that is plainly wrong. A (Harvard) nominalist who refused to countenance classes could nonetheless consistently aver that island universes are possible. Classes have nothing to do with it. Second, the plural quantifier version deviates less from the standard analysis, and so is less suspect. What is central to the standard analysis is that modal operators are quantifiers over worlds. The plural quantifier version of the amended analysis doesn’t reject that. It just adds: they may be plural, as well as singular, quantifiers. A final reason I postpone until Section ...⁴⁴ A third version of the amended analysis interprets truth at a plurality of worlds, not in terms of classes or in terms of plural quantification, but in terms of aggregates (that is, mereological sums). Thus, Amended Analysis (Aggregate Version). A proposition is (metaphysically) possible if and only if it is true at some aggregate of worlds. Since worlds do not overlap, aggregates of worlds and (non-empty) classes of worlds are in one-one correspondence. That guarantees that the aggregate version is extensionally equivalent to the others. The aggregate version and the plural quantifier version share a common advantage: assertions of possibility do not carry ontological commitment to classes. Of course, the aggregate version is committed to aggregates of worlds. But, appearances notwithstanding, that is no disadvantage: aggregates, I have been convinced, are an ontological free lunch; if one is committed to some things, then one is committed to the aggregate of those things, willy-nilly. (See Lewis .) Why, then, do I prefer the plural quantifier version to the aggregate version? I have three reasons. First, as before, the plural quantifier version deviates less from the standard analysis by analyzing possibility as a quantifier over worlds. Second, if one accepts the aggregate version, one is tempted to generalize the analysis, so that a proposition is possible also if it is true at some part of some world. But that would be wrong. It would make possible the proposition: something exists that is spatiotemporally related to something that doesn’t exist. And then we might wonder whether, if we went far enough out in spacetime, we would encounter the merely possible! That’s absurd. The third reason I postpone until Section ...

.. Aggregates of Worlds are Worlds: A Third Realist Response? The realist, I have said, can allow for the possibility of island universes in either of two ways: revise the criterion of demarcation for worlds, or amend the standard analysis of modality. With a mere shift of terminology, however, a solution of the ⁴⁴ Note that one may reject the class version but still choose to formulate possible worlds semantics in terms of classes of worlds (if, that is, one believes in classes). The ontological commitments of the metalanguage within which we do semantics for natural language may transcend the ontological commitments of natural language.

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

second sort—the aggregate version of the amended analysis—can be transformed into a solution of the first sort. Say that a world, under a revised criterion, is any aggregate of worlds, under the original criterion. Then, the standard analysis applied to worlds in the revised sense is identical with the aggregate version of the amended analysis applied to worlds in the original sense; the two approaches are essentially equivalent. Since the difference is only terminological, there can be no substantive reason to prefer one approach to the other. We do not have a third realist solution. The question can only be: which way of speaking deviates less from established usage? First off, the word ‘world’ as it occurs in modal metaphysics is a philosophical term of art, not a part of ordinary language. Moreover, since the standard analysis of modality is couched in terms of worlds, it can be no more a part of ordinary language than is the notion of world itself. To answer our question, then, we must look to established philosophical usage. For realists from Leibniz through Lewis, worlds have been essentially unified, and they have either not overlapped, or not overlapped extensively. The revised criterion of demarcation would be a radical departure from that tradition.⁴⁵ On the other hand, the amended analysis of modality—at least, the plural quantifier version—preserves the central core of the standard analysis: modal operators are quantifiers over possible worlds. It counts as a minor modification. Thus, I recommend on terminological grounds amending the standard analysis.⁴⁶

. Further Applications Thus far, I have dealt with one problem for Lewis’s modal realism, the problem of island universes. I have argued that, if a realist accepts the amended analysis of modality (and absolute actualization), the problem is easily resolved. Three further problems for Lewis’s modal realism are likewise easily resolved: () an unqualified principle of compossibility can be accepted, thereby allowing for the possibility of universal actualization; () the possibility of nothing can be endorsed, if desired, with an appropriate modification of the amended analysis; and () the principle of the identity of indiscernible worlds, undecidable on Lewis’s theory, can be decisively refuted. I treat these three problems in turn, followed by a brief conclusion.

.. Universal Actualization and Lewis’s Principle of Recombination It is natural to think that only part of logical space has been actualized: flying pigs, planets of pure gold, these are merely possible beings existing nowhere in actuality. But is it not at least conceivable, and metaphysically possible, that all of logical space ⁴⁵ However, non-realists who gloss ‘possible world’ as ‘counterfactual situation’—such as Kripke ()—might find (the non-realist analogue of ) the revised criterion compatible with their usage. ⁴⁶ Ted Sider suggested in conversation (in ) the idea that every possible individual is a world, transworld individuals included; that aggregates of worlds are worlds is a special case. It wasn’t until I worked out the ideas of this chapter the following year that I came to appreciate the insight behind his suggestion. When this work was presented at Princeton in March, , I was shown an unpublished manuscript by Richard B. Miller, “Actuality, Island Universes and Schrodinger’s Cat,” in which the idea that aggregates of worlds are worlds is adopted to allow for the possibility of island universes.

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

       ()

has been actualized, that everything possible exists? It follows from principles of plenitude that I accept—for example, from (GPS)—that the answer is “yes.” Indeed, many philosophers, from ancient times to modern, have defended a principle of plenitude for actuality according to which whatever can exist, does.⁴⁷ Their reasoning, if set within the present Humean approach to possible worlds, would go like this: God, being perfectly good, will choose to actualize the best; but more is always better than less; so, God will choose to actualize, not this world or that, not these worlds or those, but all the worlds in logical space. I won’t vouch for the theology; but the possibility of universal actualization seems perfectly coherent. The amended analysis allows for the possibility of universal actualization: ‘everything possible exists’ is true at the class (aggregate, plurality) of all possible worlds. On the standard analysis, however, universal actualization, literally interpreted, is out of the question: ‘everything possible exists’ is true at no world. Is some non-literal interpretation of universal actualization compatible with the standard analysis? Lewis accepts a “principle of recombination” which, if left unqualified, entails that anything can coexist with anything, or, more generally, that, for any things, those things can coexist. Since the “things” may be in different worlds, possible coexistence is to be understood in terms of duplicates: for any things, some world contains distinct duplicates of those things. The unqualified principle of recombination, then, would allow for the possibility of universal actualization in an attenuated sense: possibly, every possible intrinsic nature is (distinctly) instantiated. But a well-known argument due to Forrest and Armstrong, if appropriately beefed up, shows that the unqualified principle of recombination leads to contradiction: the big world that contains distinct duplicates of all the worlds would, in a sense that can be made precise, have to be bigger than itself.⁴⁸ To avoid the contradiction, Lewis adds a qualifying proviso: for any things, those things can coexist size and shape of possible spacetimes permitting. When it comes to all things, no possible spacetime will be big enough. That avoids the contradiction, but at a substantial cost: it rejects as impossible what a great many philosophers throughout history have thought possible, even actual. When we switch to the amended analysis, the Forrest-Armstrong argument loses its bite. Although no one world mirrors all the worlds in logical space, that no longer rules out the possibility of universal actualization. (GPS) can serve as a pure, unqualified Humean principle of plenitude for compossibility: contra Leibniz, all things are compossible. Precisely what recombination principle to adopt in addition to (GPS) is a matter for another occasion.

⁴⁷ See Lovejoy () for a detailed account of the history of the “principle of plenitude” (for actuality), which holds in part “ . . . that no genuine potentiality of being can remain unfulfilled, that the extent and abundance of the creation must be as great as the possibility of existence . . . ” (p. ). ⁴⁸ The argument is in Forrest and Armstrong (), and reformulated by Lewis in Lewis (a). As pointed out in Nolan (), the contradiction does not follow from any premises explicitly presented either by Forrest and Armstrong or by Lewis. The gap can be filled, however, by appealing to either a principle of plenitude for possible structures, or a principle of plenitude for alien natural properties. [See Chapter  for the details.]

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 



.. Nothing I believe that there might have been nothing at all. Nothing physical, that is; not even empty spacetime. Logical space would have existed as unrealized potentiality, waiting on an absent creator. This belief is controversial, to be sure—more so, I think, than the possibility of island universes or universal actualization. I defend it, as best I can, at the end of this section. Realism about possible worlds when combined with the standard analysis of modality cannot accommodate the possibility of nothing. Any part of any world exists at that world, and any world has itself as a part; so there is no world at which nothing exists. Then, given the standard analysis, it is impossible that nothing exist. (See Lewis a: –.) Nor does switching to the amended analysis, as it stands, help. To accommodate the possibility of nothing, the amended analysis must be modified so as to include, in effect, a “null plurality” of worlds. But now it matters which version of the amended analysis the realist accepts; for not all versions can incorporate the modification in a natural way. Consider first the aggregate version. Here we would have to allow a “null aggregate” of worlds, and say: possibility is truth at some aggregate of worlds, the null aggregate included. But there is no such thing as a null aggregate! On this version, the possibility of nothing is really an ad hoc special case; it does not follow from the analysis of possibility as a quantifier over aggregates of worlds. Consider next the class version. Here the modification seems to be better off: possibility is truth at some class of worlds, the null class included. But is the null class ontologically more respectable than the null aggregate? I doubt it.⁴⁹ In which case, the possibility of nothing is just as ad hoc on the modified class version, as on the modified aggregate version. Consider, finally, the plural quantifier version. Here the modification seems to be in trouble: plural quantifiers in English do not range over things “in the null way” that would be required for the modification. But the trouble is one of ordinary language, not logic or metaphysics, one of expression, not understanding. For we understand second-order logic with the second-order monadic quantifiers ranging over all subclasses of the domain, the null class included. And we understand, I have claimed, how to interpret quantification over non-empty classes as ontologically innocent plural quantification. To hold that ontological commitment to non-empty classes can be eliminated in this way, but not ontological commitment to the null class, would be absurd! The fact that the quantifiers of second-order logic do not match up neatly with the plural quantifiers of ordinary language is a mere technicality—of no more importance to logic or metaphysics than the fact that the quantifiers of first-order logic do not match up neatly with the singular quantifiers of ordinary language.

⁴⁹ On Lewis’s mereological theory of classes—classes are aggregates of singletons—the null class and the null aggregate are in exactly the same boat (though one might choose to introduce something arbitrarily to play the theoretical role attributed to the null class, say, in mathematics or possible-world semantics). See Lewis (: –).

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

       ()

That leaves a small problem of expression. If we want to express the modified amended analysis in something resembling English, we will need to coin a phrase. Say that a proposition is true at nothing just in case, intuitively, had no world been actualized, the proposition would have been true. To be more precise, we can let some developed version of free logic—logic over an empty domain—be our guide. The plural quantifier version of the amended analysis then becomes: possibility is truth at some world, or at some worlds, or at nothing. Since the proposition that nothing exists is true at nothing, it comes out possible, as desired, even though it is true at no world. Note that being true at nothing is not the same as being true at no world: a contradiction, for example, though true at no world, is not true at nothing; if it were, contradictions would be possible. On the surface, the modified amended analysis looks like an ad hoc collection of clauses. But deeper down, its content is seamless. When asked which worlds might be actualized, we answer: all, or some, or none. We cover the full range of quantifiers. What would be arbitrary would be to leave off the ‘none’. The possibility of nothing does not follow from the Humean principles of plenitude accepted in Sections .. and ... They need to be strengthened. That is easily done. The relevant question is then: do the arguments that served to motivate the original principles also serve to motivate the strengthened versions? Consider first (PS). Say that a worldbound individual x at a world w is strongly contingent if and only if, possibly, x fails to exist without anything taking its place; that is to say, possibly, all and only duplicates of the parts of w that do not overlap x exist. Then (PS) is equivalent to: any worldbound individual other than a whole world is strongly contingent. To strengthen (PS), we simply omit the italicized clause: any worldbound individual is strongly contingent. The possibility of nothing now follows by applying the strengthened (PS) to whole worlds. My defense of (PS) rested on the claim that actualization is unconditional: whether or not some thing can be actualized does not depend on whether or not anything else is actualized along with it. To defend strengthened (PS), we need instead a form of unconditional de-actualization: whether or not some thing can be de-actualized—can fail to exist without anything taking its place—does not depend on whether or not anything else exists at its world. For consider some worldbound individual x that is not a whole world. By ordinary (PS), there is a world v that is a duplicate of x. If we do not strengthen (PS), then x is strongly contingent (can be de-actualized) but v is not (cannot), even though they differ only extrinsically: v, but not x, exists all by itself. I accept unconditional de-actualization; but I do not think it should be allotted a fundamental role in a theory of plenitude alongside unconditional actualization. A case for the possibility of nothing based on unconditional de-actualization is weak. A stronger case can be made by considering the argument behind (GPS). To strengthen (GPS), we simply move from an aggregate to a plural quantifier formulation, allowing plural quantifiers, as above, to range over “nothing”: for any things, possibly, distinct duplicates of those things exist all by themselves. My defense of original (GPS) rested on the claim that there should be no restrictions on what can be actualized. The argument applies no less to strengthened (GPS). Consider a donothing God, content to contemplate the eternal verities. To hold that he must actualize some world, or some worlds, is to restrict his power to choose: being

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 



all-powerful, he can choose to actualize nothing. And, if rejecting the possibility of nothing would be a restriction on God’s power to choose, so too would it be a restriction on primitive actualization, a restriction on what can be actualized.

.. Indiscernible Worlds Any realist account of possible worlds must face the question whether distinct worlds are ever qualitative duplicates of one another, or, equivalently, given the absolute isolation of worlds, the question whether there are counterexamples to the identity of qualitatively indiscernible worlds.⁵⁰ For if worlds are particulars—as opposed, say, to properties or universals—then duplication is not ruled out categorically as incoherent. David Lewis writes: “My modal realism does not say whether or not there are indiscernible worlds; and I can think of no very weighty reason in favor of one answer or the other.” Lewis (a: ). Indeed, as long as we hold to the standard analysis according to which possibility is truth at a single world, modal intuitions are powerless to decide the issue: no judgment of possibility would be affected by the presence or absence of indiscernible worlds. There is a mismatch between the framework of possible worlds and the judgments of possibility that the framework serves to interpret. A feature of the framework, the existence or non-existence of indiscernible worlds, appears arbitrary and artificial—an ontological dangler, if you will. Something, I think, is amiss. The amended analysis of modality sets this right. It allows familiar arguments against the identity of indiscernible worldmates to be straightforwardly applied to the worlds themselves. For, surely, if island universes are possible, then it is possible for the islands to be qualitatively very similar; and, if very similar, why not exactly alike?⁵¹ But the possibility of duplicate island universes, on the amended analysis, requires the existence of distinct but indiscernible worlds. The identity of indiscernible worlds, then, is false. Is this argument conclusive? It does assume that the possibility of indiscernible island universes is to be analyzed in a way analogous to the possibility of almost indiscernible island universes: since the latter possibility is made true by a pair of distinct worlds, so is the former. But this assumption cannot plausibly be challenged by a realist. For on what grounds would the possibilities be treated differently? Granted, the possibility of indiscernible island universes is made true by a single type of world, instantiated twice-over, whereas the possibility of almost indiscernible island universes is made true by two distinct types. But to analyze possibility in terms of types of world is to move away from realism, as here characterized, and to identify worlds instead with uninstantiated properties or universals.⁵² As long as worlds are taken to be particulars, the argument, I think, is conclusive.

⁵⁰ On duplicates vs. indiscernibles, see Lewis (a: –). ⁵¹ This is an adaptation of Robert Adams’s “argument from the possibility of almost indiscernible twins.” See Adams (: –). ⁵² As is done, for example, in Forrest () and Stalnaker ().

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

       ()

. Conclusion The standard analysis of modal operators as individual quantifiers over worlds is well entrenched, but not sacrosanct. Analyzing modal operators instead as plural quantifiers over worlds (or individual quantifiers over classes, or aggregates, of worlds) has a lot to recommend it for the realist. No ordinary possibility judgments are affected by the shift, and the newly added possibilities are theoretically very satisfying: they allow the realist to accept the plenitude of possibilities to its fullest extent. That includes the possibility of island universes (in the strong sense), even the ultimate possibility of universal actualization. It includes (on the plural quantifier version) the possibility of nothing. And the amended analysis resolves the otherwise mysteriously aloof identity of indiscernible worlds. The price for all this is absolute actualization. But, if I’m right, that price is not so great as is often supposed.

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 Absolute Actuality and the Plurality of Worlds () . Introduction Let’s fix some terminology at the start. A world (or possible world—for me, the ‘possible’ is redundant) is, first, an individual, not a set or class; second, a particular, not a property or universal; third, concrete in this sense: it is fully determinate in all qualitative respects; and, fourth, a maximal interrelated whole: each world is internally unified, and isolated from every other world.¹ There is at least one world, the world we are part of. It is an actual world, the actual world if there are no “island universes.”² Worlds that are not actual (if any) are merely possible. A realist about possible worlds believes that there is a plenitudinous plurality of worlds: whenever something is possible—for example, that donkeys talk, or that pigs fly—there is a world in which it is true. There is more than one way to be a realist about possible worlds. Realists divide into two camps depending on their account of actuality. According to David Lewis, the worlds are ontologically all on a par; the actual and the merely possible differ, not absolutely, but in how they are related to us. Call this Lewisian realism.³ Most philosophers grant that Lewisian realism, if true, would bring substantial theoretical benefits to systematic philosophy. Nonetheless, few philosophers have been willing or able to believe it. Often the obstacle to belief is the supposed ontological extravagance that accompanies any full-blown realism about possible worlds: belief in talking donkeys and flying pigs—even if they are spatiotemporally and causally isolated First published in J. Hawthorne (ed.), Philosophical Perspectives, Metaphysics, vol.  (Blackwell Publishing) : –. Reprinted with the permission of John Wiley and Sons, Inc. An earlier version of this chapter was presented at Tufts University and at the Eastern APA Meetings in . Thanks to the audiences on those occasions, and especially my APA commentator, John Divers. ¹ For precise definitions of unification and isolation, see Chapter . I argue that one needs to allow not only spatiotemporal relations, but any and all external relations, to serve as unifying relations for worlds. ² In Chapter  I argue that the claim that there are island universes—disconnected regions of actuality— is best understood (by a realist) as the claim that more than one world is actual rather than the claim that the one actual world has disconnected regions; but the issue is largely terminological. It may grate on the ears at first to hear ‘an actual world’ instead of ‘the actual world’, but you’ll get used to it. When I want to refer to the unique world we are part of, I will use ‘our world’. ³ A.k.a. “modal realism.” But ‘modal realism’ is sometimes used broadly to include any view that accepts a plurality of concrete worlds, sometimes narrowly to include only views that accept specific Lewisian theses having to do with plenitude or isolation; I think it best to avoid it. Lewis presents and defends his realism about possible worlds in Lewis (a). Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

       ()

from us—is deemed simply preposterous. But that objection is based on prejudice, not argument; and it is not a prejudice I share. Objections to Lewis’s account of actuality, however, are another matter. I take it to be conceptually evident that actuality is absolute, not relative, and that, moreover, the distinction between the actual and the merely possible is a distinction in ontological status: whatever is ontologically of the same fundamental kind as something actual is itself actual. When Lewis insists, then, that all worlds are ontologically on a par, I can only understand this—his protests notwithstanding—as saying that all worlds are equally actual. But that makes Lewis’s defense of a plurality of worlds incoherent. For there could be no good a priori reasons for believing in a plurality of actual concrete worlds. And an analysis of modal operators as quantifiers over concrete parts of actuality, no matter how extensive actuality may be, is surely mistaken. I thus reject Lewisian realism.⁴ Can one preserve the theoretical benefits of realism about possible worlds by combining it with an account of actuality as an absolute property that marks a distinction in ontological status? Call this combination Leibnizian realism.⁵ Lewis thought not. He writes: “given my acceptance of a plurality of worlds, the relativity [of actuality] is unavoidable.”⁶ Indeed, by the time Lewis wrote On the Plurality of Worlds, he thought it sufficient to devote less than one page containing two brief arguments to the refutation of Leibnizian realism, in part because the two arguments were already well known⁷, in part, no doubt, because he took the arguments to be decisive.⁸ True, Lewis did spend eight pages presenting and objecting to a view that in some ways resembles Leibnizian realism, a view he calls “pictorial ersatzism”—an odd, hybrid view that, I suspect, no one has or ever will hold. But in the end his chief objection to pictorial ersatzism is that, when fully and properly developed, it collapses into (a version of ) Leibnizian realism, in which case it loses the supposed advantage of having a safer and saner ontology than Lewisian realism, and takes on the distinct disadvantage of having already been refuted. What are the two arguments that, supposedly, refute Leibnizian realism? One is that the Leibnizian realist cannot account for the contingency of actuality. But this problem, I will argue, admits of an easy and natural solution as soon as the Leibnizian distinguishes between what is true of a world and what is true at a world—a

⁴ Lewis considers the objection that he is committed to holding that all worlds are actual in Lewis (a: –). He concedes that “if the other worlds would be just parts of actuality, modal realism is kaput.” Lewis (a: ). But his defense assumes that the objector holds that ‘actual’ is a blanket term (it is analytic that everything is actual), not, as I would urge, that ‘actual’ is a categorial term (it is analytic that actualia comprise a fundamental ontological category). ⁵ I use ‘Leibnizian’ primarily for marketing purposes. It seems plausible to me that Leibniz’s account of possible worlds and actuality, when stripped of his theology—that possible worlds are ideas in the mind of God, that our world derives its actuality from a free choice of God—has as remainder what I call Leibnizian realism. But this comes with the usual disclaimers to appease the historians. If one adds to Leibnizian realism the thesis that the property of actuality is unanalyzable, one gets what Adams calls the simple property theory of actuality. See Adams (). ⁶ Lewis (a: ). See also Adams’s dilemma in Adams (: –). ⁷ Lewis had presented versions of the arguments before, one in Lewis (), the other in Lewis (), as had Adams in Adams (). ⁸ Fifteen years later, when John Divers () published his extensive survey of approaches to possible worlds, Leibnizian realism as an alternative to Lewisian realism received nary a word.

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



distinction the Lewisian needs in any case for the analysis of de re modality. A property is true of a world when the world has that property; a property is true at a world when the world represents itself as having that property. With respect to actuality, these two notions do not coincide: every world represents itself as being actual. By interpreting possibility and necessity in terms of what is true at a world rather than (just) what is true of a world, the contingency of actuality can be secured by the Leibnizian. There is indeed a residual question whether this notion of truth at a world requires primitive modality. I shall argue below that it does not. The second argument against Leibnizian realism is more challenging. Surely, Lewis says, we know that we are actual; skepticism about our own actuality is absurd. With this I agree. But, the argument continues, if Leibnizian realism were true, we could not have this knowledge. For the Leibnizian must allow that there are concrete merely possible people who are epistemically situated exactly as we are: there is no evidence that can distinguish our predicament from theirs. But then we can’t rule out the possibility that we are the merely possible people inhabiting a merely possible world. We don’t know that we’re actual after all. The first thing to say in response is that this is not just the Leibnizian realist’s problem. Suppose you believe in sets. I ask: how do you know you are an individual, and not a set? Suppose you believe in universals. I ask: how do you know you are a particular, and not a universal? These questions, I argue below, do not have the answers one might at first expect. The problem of skepticism about actuality for the Leibnizian realist is in crucial ways analogous to the problem of skepticism about individuality, or particularity, for the realist about sets, or universals. So if the Leibnizian realist is in trouble, he has a lot of company. Lewis is a realist about sets.⁹ Most prominent “actualists” believe in sets or universals or both, and make essential use of them in their accounts of modality.¹⁰ The Leibnizian realist, then, when confronted with the skeptical problem, can justifiably respond: tu quoque! Unfortunately, tu quoque arguments, even when sound, are never fully satisfying: knowing that (almost) everyone sinks or floats together does nothing to help one float. The main thrust of this chapter, once preliminaries are out of the way, will be to develop and defend a positive Leibnizian solution to the problem of skepticism about actuality.¹¹ Knowledge of one’s actuality, I will argue, is fully compatible with actuality being an absolute property that some things have and other things lack. As with other cases of contingent, a priori knowledge, what matters is that the property in question is picked out indexically; the property thereby picked out may be as robust and absolute as you please. In this way, the Leibnizian rejects what is objectionable in Lewis’s account—that the property of actuality is merely relative— while preserving what is needed to solve the skeptical problem—that the concept of actuality is indexical. ⁹ See Lewis (). Note, however, that on a version of the structuralist approach that does not include the postulate that no individual is a set, the answer to the question “how do you know you are an individual and not a set?” is: “I don’t know.” ¹⁰ Adams (), Stalnaker (), Plantinga (), Armstrong (a). ¹¹ In Chapter  I suggested the tu quoque, but I did not defend it there, or provide a positive solution to the skeptical problem. This is not the first time that Lewis’s disparate treatment of worlds and of sets has led to a charge of tu quoque; see van Inwagen (: –).

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

       ()

Explaining how we know that we are actual is the easy part of the problem. Explaining just what we know in virtue of knowing that we are actual is more problematic. Is knowledge that I am actual trivial and uninformative, like knowledge that I am here? Or is it substantial knowledge de se, knowledge of where I am located in logical space? The answer, surprisingly, depends on the particular version of Leibnizian realism that one accepts. The Leibnizian who holds that there is substantial knowledge de se should endorse, I claim, a form of (ontological) perspectivalism: our perspective on reality is an integral part of that reality; a perspectiveless account of reality is inevitably incomplete. Moreover, this Leibnizian has the additional burden of explaining how we can have such knowledge even though our merely possible counterparts do not. This Leibnizian, it turns out, can only be an ontic egalitarian—allowing, for example, that merely possible people exist and have the same sort of qualitative properties, physical and mental, that we do—by being an epistemic chauvinist. We are epistemically privileged simply by virtue of being actual; our perspective on logical space cannot fail to be the right perspective for acquiring knowledge. At first blush, resorting to epistemic chauvinism might appear to be an ad hoc stratagem for saving (this version of) the Leibnizian theory from refutation. Not so. Epistemic chauvinism, in my view, will need to feature in any adequate account of a priori knowledge. To reject it would be to embrace, not just skepticism about actuality, but a general skepticism about matters a priori.

. Motivations for Realism about Possible Worlds I should say something about the motivation for realism about possible worlds and possibilia, as I see it. Unlike Lewis (a: –), I do not take the theoretical benefits of belief in possibilia to be a reason to think the belief true: wishful thinking is no more rational in metaphysics than in everyday life. Rather, I believe in possibilia because I take the existence of possibilia to be a prerequisite, not just for modal thought, but for thought in general. Thus, I take the primary motivation for belief in possibilia to arise from the nature of intentionality: possibilia provide the requisite objects that our intentional states are about (in one sense of ‘about’), and, more generally, the framework for the content of all language and thought. But the nature of intentionality, by itself, takes us only partway. We also need to consider the nature of modality: possibilia provide the requisite domain over which modal operators range, and, more generally, the subject matter for all modal statements. Considerations of intentionality are primary, establishing the existence of possibilia; considerations of modality are needed in addition to establish their abundance. I present the line of thought that leads me to realism about possible worlds in a series of six theses. My commentary on these theses will be brief, contentious, and largely dogmatic; my primary goal in this chapter is to defend Leibnizian realism against Lewis’s charges, not to expound the theory from scratch. First, a terminological warning. Although I make heavy use of intentionality in motivating my belief in possibilia, I speak like a possibilist, not a Meinongian.¹² What I mean by ‘is actual’

¹² See, for example, Parsons (); Routley ().

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     



is, more or less, what the Meinongian means by ‘exists’. What I mean by ‘exists’ is, more or less, what the Meinongian means by ‘is’ or ‘has being’. I say: whatever is, exists. The Meinongian says: there are some things—for example, merely possible objects—which do not exist. It is an interesting question which mode of speaking is more in accord with ordinary language; I think a case can be made either way. Neither can be in very close accord with ordinary language because both abstract from the heavy context-dependence of words like ‘is’ and ‘exists’. In any case, this difference with the Meinongian is superficial. More substantial differences come out in the third and fourth theses below. First Thesis: Narrowly psychological intentional states are genuinely relational. Intentional states can be given either a wide or a narrow interpretation. When interpreted narrowly, the content of the state is independent of whatever relations of acquaintance the subject has to objects in her environment. I am interested here only in narrow intentional states. There are two sorts of narrow intentional state, neither reducible to the other: those states, such as belief, that relate subjects to propositions (or perhaps properties), and those states, such as fear (in one sense), that relate subjects directly to objects. The path to possibilia is more direct with the latter sort of state, so let us focus on one of them: the state of thinking about some object or objects. Suppose, for example, that I am now thinking about a dodecahedron made of solid gold. I can do this, of course, whether or not any such object actually exists. Thinking about, I claim, is relational, and relations require relata. So, if there are no actual gold dodecahedra, only merely possible gold dodecahedra are available to be objects of my thought; I am related to possible but non-actual objects. Second Thesis. The qualitative nature of objects of thought is independent of their status as actual or merely possible. Suppose again that a merely possible gold dodecahedron is an object of my thought. I ask: does this object of my thought instantiate the properties, being gold and being a dodecahedron? Or does it instead somehow represent, or encode, or stand in some pseudo-instantiation relation to these properties? I say the former. It is one thing to think about a gold dodecahedron, another thing to think about some abstract simulacrum thereof. If I am thinking about a gold dodecahedron and thinking about is genuinely relational, then there is (in some part of reality) a gold dodecahedron that I am thinking about. That it is made of gold and shaped like a dodecahedron is independent of whether it is actual or merely possible. Indeed, nothing prevents actual and merely possible objects from being perfect qualitative duplicates of one another.¹³ Third Thesis. Objects of thought are fully determinate. Intentional states such as thinking about are typically indeterminate with respect to their objects. In thinking about a gold dodecahedron, I was not thinking about a gold dodecahedron of any particular size. Should I say, then, that I was related by my thought to an object that has no definite size? No. It is one thing to think indeterminately about a gold dodecahedron, another thing to think about an indeterminate object. The indeterminacy was in the thinking, not the object of thought. I was related by my thought to

¹³ As the Meinongians might say: Sosein is independent from Sein. But I would substitute actual/merely possible for Sein/Nichtsein.

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

       ()

a multitude of possible gold dodecahedra with a multitude of different, but fully determinate, sizes. We can call on the method of supervaluations to explain why I can truly say that there is one thing that I was thinking about: a dodecahedron made of solid gold.¹⁴ Fourth Thesis. There are no impossible objects of thought. Prima facie, I can think about impossible no less than possible objects, round squares no less than flying pigs. Why doesn’t the relational character of intentionality apply with equal force in both cases? The reason, in a word, is logic; and logic trumps appearances. To follow appearances by positing impossible objects would be to endorse contradictions and thereby sink into incoherence. Better to concede that, not being logically omniscient, we are fallible when we judge that a thought has reached its target. Frege believed that there were entities satisfying his set-theoretic axioms, entities that he had thought long and hard about. There weren’t, and he hadn’t.¹⁵ Fifth Thesis. Every object of thought is part of a fully determinate possible world. When I think about a gold dodecahedron, I need not think about how it is situated with respect to other objects. But that is just another aspect of the indeterminacy of my thought. Each possible gold dodecahedron has a determinate extrinsic nature; my thought doesn’t discriminate between differently situated gold dodecahedra, and it therefore ranges indeterminately over them all. Perhaps, as I believe, there is a world consisting of a solitary gold dodecahedron, and nothing else. But then distinguish: it is one thing to think about a solitary gold dodecahedron, another thing to think about a gold dodecahedron without considering how it relates to other objects. In the former case, what I am thinking about stands in no spatial or temporal relations to other objects; in the latter case, using the method of supervaluations, it is neither true nor false that what I am thinking about stands in spatial or temporal relations to other objects. In either case, the possible gold dodecahedra that are objects of my thought belong to fully determinate possible worlds. Sixth Thesis. There is a plenitude of possible worlds. Human intentionality takes us only so far. Our intentional states are limited in their discriminatory powers; a somewhat impoverished space of possibilia could account for the relationality of our thought. But it would be deficient as a subject matter for modality. We need in addition to require that the space of possibilia satisfy principles of plenitude, principles stating that if such-and-such is possible, then such-and-so is possible as well. The specific formulation of such principles won’t matter for this chapter.¹⁶ I will simply assume that there is a plenitude of possible worlds sufficient to ground the fundamental link between possibility and possibilia: whatever is possible is true in ¹⁴ As follows: each admissible interpretation of ‘thinking about’ relates me (at the time in question) to a single possible gold dodecahedron; the sentence, ‘there is one thing that I was thinking about’, is then true in all admissible interpretations, or super-true; and, in ordinary contexts, we count what is super-true as true. See Lewis (a). ¹⁵ I have no objection to abstract substitutes for impossible objects: sets of incompatible properties, or whatever. Such substitutes may even play a role in providing a semantics for intentional attributions. And I have no objection to non-classical logics when used to model psychological or semantic phenomena. But only classical logic delineates the ultimate nature of what there is. ¹⁶ For Lewis’s account of plenitude, which is based primarily on a Humean Principle of Recombination, see Lewis (a: –). But principles of recombination are only part of the story; see Chapter .

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 .  



some possible world. With plenitude in place, we can take the realm of possibilia to be the realm of thought for all actual and possible thinkers.¹⁷

. Lewisian vs. Leibnizian Realism Both a Lewisian and a Leibnizian can accept the six theses adumbrated above. They can agree on the concrete nature and the extent of possibilia. They disagree, however, on the nature of actuality. According to the Lewisian, it makes no sense to ask whether something is actual simpliciter, only whether something is actual relative to something else. In short: Lewisian Thesis.

Actuality is relative.

For the Lewisian, what is fundamental is not the property, being actual, but the relation, being co-actual with. When we say simply that something x is actual, that is to be understood as saying that x is co-actual with some y, where y is supplied by the context of utterance, perhaps the speaker, perhaps the speaker’s world. Co-actuality is an equivalence relation, and it therefore partitions the space of possibilia—logical space—into distinct regions. The inhabitants of any one region are all actual relative to one another, but not actual relative to the inhabitants of any other region. We inhabit one of these regions. We have perfect qualitative duplicates that inhabit others of these regions. According to the Lewisian, we do not differ from these duplicates ontologically in any way: we all belong to the fundamental ontological kind, concrete individual. A Lewisian might take the relation of co-actuality to be primitive and unanalyzable. That has the advantage of allowing for the possibility of island universes: worlds that are spatiotemporally (and otherwise) isolated from one another may nevertheless be co-actual; and two or more co-actual worlds may be taken to represent the possibility of island universes. But it also has a severe disadvantage: primitive coactuality is primitive modality. The truth or falsity of some general modal claims, such as that island universes are possible, depends on how the co-actuality relation is laid out through logical space. Since one of the chief merits of realism about possible worlds is the avoidance of primitive modality, the Lewisian may well choose to sacrifice the intuitive possibility of island universes and opt instead for an analysis of co-actuality. That is what Lewis himself does. With the possibility of island universes off the table, the co-actuality relation reduces for Lewis to the worldmate relation: possible objects are co-actual iff they are worldmates. And, then, for Lewis, possible objects are worldmates iff they are spatiotemporally related.¹⁸

¹⁷ If the “realm of possibilia” is the “realm of thought,” are possibilia somehow mind-dependent? There is no dependence on individual minds, nor any dependence on the human mind. But if asked whether possibilia are independent of mind in general, I can only respond with Frege: “To answer that would be as much as to judge without judging, or to wash the fur without wetting it.” Frege (: ). The interdependence between thought and the intentional objects of thought is conceptual. ¹⁸ Or, more strictly, iff they are related by “analogical spatiotemporal relations.” See Lewis (a: –). For an alternative analysis of the worldmate relation, see Chapter .

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

       ()

The Leibnizian will have none of this. According to the Leibnizian, it is incoherent to suppose that actual objects and merely possible objects do not differ in ontological status. The Leibnizian holds that actuality is a property that objects have or fail to have simpliciter, not relative to something else. In short: Leibnizian Thesis.

Actuality is absolute.

For the Leibnizian, the property, being actual, is more basic than the relation, being co-actual with: two objects are co-actual just in case both are actual. Is the property of actuality analyzable? There have, of course, been historically important attempts at analysis—most memorably, to be actual is to belong to the best of all possible worlds—but none of these analyses, I think, would be considered at all plausible today. I will therefore assume that the Leibnizian takes the property of actuality to be primitive.¹⁹ Is that a cost to the theory? Not because primitive actuality is primitive modality: as we shall see in the next section, what is possible—including what is possibly actual—is independent of the extent of actuality. Perhaps there is a cost whenever one accepts more than one fundamental ontological kind. But if the Leibnizian is right that the actual and the merely possible differ in ontological status, then that is a cost that will have to be borne. Taking actuality as primitive doesn’t mean that anything goes: even primitives have their constraints. In particular, the Leibnizian will surely want to accept the harmless half of Lewis’s relational analysis: anything spatiotemporally (or, I would say, externally) related to something actual is itself actual. But since distinct actual objects need not be spatiotemporally (or externally) related to one another, the possibility of island universes is not ruled out. Score one for the Leibnizian. Traditional Leibnizian realism also adds the thesis that exactly one world is actual. As noted at the start, I prefer instead to leave it open how many worlds are actual to accommodate the possibility of island universes. Even the two extremes—that every world is actual, that no world is actual—are not ruled out. The latter extreme, however, I shall argue, is not epistemically possible: we know that our world is actual. Leibnizian realism, whether traditional or non-traditional, comes in two main versions. According to one version, the property of actuality applies to a realm of entities entirely separate from the realm of possibilia postulated by the six theses. Thanks to plenitude, however, our world will have an exact qualitative duplicate among the merely possible worlds. That duplicate is actualized, not actual.²⁰ (More generally: a merely possible world is actualized if and only if some actual world is a qualitative duplicate of it.) Call this the creation version of Leibnizian realism because, on the myth of God surveying the possible worlds and then making a world actual, this version requires that God create a not yet existing actual world to match the possible world he likes best. According to the other version of Leibnizian realism, the property of actuality applies directly to the possibilia postulated by the six theses, dividing them into two ¹⁹ As we shall see in Section ., taking the property of actuality to be primitive is not the same as taking the concept of actuality to be primitive. ²⁰ Lewis makes use of this distinction between being actual and being actualized in his presentation of ersatzist views; see Lewis (a: –).

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 .  



classes: actual and merely possible. On this version, the actual world need not have a merely possible qualitative duplicate. Call this the transformation version of Leibnizian realism because, on the God myth, God makes a possible world actual, not be creating it anew, but by transforming it into an actual world. Once this talk of God is excised, however, the distinction between being actual and being actualized drops out on this version: a possible world is actual just in case it itself is actualized. The difference between the creation and transformation versions is starkest if the Leibnizian accepts the identity of qualitatively indiscernible worlds: only on the creation version will each actual world have a qualitative duplicate among the merely possible worlds. But if the identity of qualitatively indiscernible worlds is uniformly rejected,²¹ and, say, there is only one actual world, then the creation and transformation versions of Leibnizian realism will not differ in their depiction of logical space: the actual world will have a merely possible duplicate on either view. But for the nontraditional Leibnizian the two versions can still be clearly distinguished, for example, by considering the possibility of universal actualization, the possibility that every possible world is actualized.²² On the creation version, this possibility is realized when, for every possible world, there is a duplicate world that is actual: there are still two distinct realms, the realm of the merely possible and the realm of the actual. On the transformation version, this possibility is realized when every possible world is itself actual, when there are no merely possible worlds.²³ Which version of Leibnizian realism is to be preferred? Suppose we understand intentional relations, such as thinking about, on the model of grasping: an actual subject mentally reaches out and grasps merely possible objects occupying an ontological realm distinct from that of the subject; the subject is directly related by this mental grasping only to merely possible objects, although the subject can be indirectly related to an actual object by grasping a merely possible duplicate of it; moreover, these grasping relations are genuine, external relations that connect one ontological realm with another. This model of intentionality leads straight to the creation version of Leibnizian realism according to which an actual world is required to have a merely possible duplicate. For if an actual world in its entirety lacked a merely possible duplicate, it would be unthinkable (in its entirety), even in principle. I reject the grasping model of intentionality. For one thing, if I allowed that thinking about was an external relation, I would be hard-pressed to explain why the objects of our thought are not themselves actual; for I hold that whatever we are externally connected to, whether it be by spatiotemporal, or causal, or mental grasping relations, is thereby a part of actuality. (See Chapter .) We are related by ²¹ As it is in Section ... ²² In Section .. I claim that some arguments for the possibility of island universes naturally carry over to the possibility of universal actualization. But since the possibility of universal actualization is more controversial than the possibility of island universes, I will consider below Leibnizian realism both with it and without it. ²³ Which version one plumps for also affects the interpretation of recombination principles such as: anything can co-exist with anything else. On the creation version, this must be understood, as Lewis does, in terms of duplicates. On the transformation version, this is made literally true by considering the possibility of universal actualization. (The flip side, that anything can fail to co-exist with anything else, must still be interpreted in terms of duplicates.)

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

       ()

our thought to mere possibilia by description, not by acquaintance. For another thing, I find the supernatural relation of grasping utterly mysterious. I take thinking to be a natural phenomenon: if we have the capacity to think about merely possible objects, we have that capacity in virtue of the natural properties and relations instantiated by us, our parts, and perhaps the objects in our environment. Intentional relations, such as thinking about, supervene on the intrinsic natures of the thinker (and perhaps the thinker’s world) and the objects the thinker thinks about. I cannot think about one but not the other of two indiscernible merely possible objects; nor can an indiscernible merely possible counterpart of me be thinking about anything that I am not. Intentional relations take their relata—the subjects and objects of thought—indiscriminately from either realm: there are merely possible thinkers; there are actual (direct) objects of thought. When we switch from the supernatural to the natural model of intentionality, we can give up the unnecessary duality of actual worlds and their merely possible copies. There is simply the realm of possibilia, and the absolute distinction that divides the realm into actual and merely possible. Thus, I favor the transformation version of Leibnizian realism. But in what follows, I will mostly put this preference to one side.²⁴ There is one final issue over how to understand Leibnizian realism that will play an important role in what follows. I have spoken of an absolute distinction between the actual and the merely possible, and of an absolute property of actuality that grounds this distinction. This property of actuality, of course, is not a property merely “in the abundant sense,” according to which any old collection of things can be said to share a property. Actuality must be a property “in the sparse sense”: when two things are actual, they genuinely have something in common.²⁵ What they have in common, however, cannot be a qualitative feature; rather, it is that they belong to the same fundamental ontological kind. How can one distinguish those properties (in the sparse sense) that are qualitative from those that are not? The fundamental nonqualitative properties and relations are needed to provide the underlying framework for logical space. Here I include, in addition to sameness-of-ontological-kind properties, identity, part-whole, instantiation, and perhaps spatiotemporal and other external relations. Fundamental qualitative properties and relations, on the other hand, can be distinguished as those that are subject to principles of recombination: they are distributed over logical space every which way. These partial explanations are not intended as analyses, of course; they are merely hints to the reader, aids to understanding. Now, the issue that will be needed below can be raised by asking the following questions: Is there, in addition to the property of actuality, also a property (in the sparse sense) of being merely possible, a property shared by all and only the mere possibilia? Or, is something merely possible simply in virtue of lacking the property of actuality? In the latter case, I will speak of one-property Leibnizian realism: actual things are distinguished by having a special property of actuality; the actual and the ²⁴ I have wavered somewhat over the years as to which version of realism I prefer. In Section .., I reported a preference for the creation version; I have now switched sides, I hope for good. ²⁵ On the difference between conceptions of properties as abundant and conceptions of properties as sparse, see Lewis (a: –).

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   



merely possible are ontologically distinct, but the merely possible do not form a genuine ontological category.²⁶ In the former case, I will speak of two-property Leibnizian realism: there is an ontological symmetry between the actual and the merely possible; there are two distinct ontological categories, the members of each lack a property had by the members of the other. One-property Leibnizian realism seems most naturally aligned with the transformation version: actual things have a special ingredient that distinguishes them, ontologically, from their merely possible qualitative duplicates. I do not think that there is much more that can be said about the nature of this property: even characterizing it as making the actual “more real” than the merely possible seems to be lacking in genuine content. Two-property Leibnizian realism seems better suited to the creation version: there are two separate realms of being, two fundamental ontological categories, neither ontologically inferior to the other. But I suppose the creation version could be combined with a one-property view, or the transformation version with a two-property view. So, although I tend to favor the one-property view, I will remain officially neutral, and when it matters, I will consider all available options.

. The Contingency of Actuality I am actual. But it didn’t have to be that way. Had a massive asteroid plowed into the Earth ten million years ago wiping out all life, I would not have been actual. Flying pigs, on the other hand, are merely possible (let us suppose). Again, it didn’t have to be that way. Had a mutation arisen that gave pigs (powerful!) wings, some of the merely possible flying pigs would have been actual. Individual things are contingently actual or non-actual (excepting necessary existents, if any). So too for entire worlds. Any actual world might not have been actual; indeed, I think that there might have been no actual world at all. And any merely possible world might have been actual; indeed, I think that every world might have been actual together.²⁷ Can the contingency of actuality be squared with the absoluteness of actuality, as the Leibnizian is committed to believe? Lewis thought not. Here is his entire argument as it appeared in On the Plurality of Worlds: Surely it is a contingent matter which world is actual. A contingent matter is one that varies from world to world. At one world, the contingent matter goes one way; at another, another. How can this be absolute actuality?—The relativity is manifest! (Lewis a: , his emphasis)

²⁶ Compare Lewis’s presentation of “pictorial ersatzism”: “There is a special ingredient of the concrete world—vim, I shall call it—which is entirely absent from all the ersatz worlds and their parts.” Lewis (a: –). ²⁷ The cognoscenti are aware that ‘actual’ and its cognates are ambiguous in modal contexts between rigidified and non-rigidified readings. (Ditto for ‘merely possible’.) The rule to follow here is: disambiguate in such a way that what I say comes out true! Generally, in sentences expressing the contingency of actuality, predications of ‘actual’ should be taken to be non-rigid. But, for example, the first ‘actual’ in ‘possibly, some actual world is not actual’ must be taken to be rigid. On the need for two readings, see Lewis (a: ).

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

       ()

We need to flesh this out a bit. Here is one way of developing the argument.²⁸ Suppose, for reductio against the Leibnizian, that there is an absolute property of actuality. Let w be a world that is (absolutely) not actual. Since actuality is contingent, there is a world at which the proposition, w is actual, is true—presumably, w itself. But—and here is the crucial move—if w is actual is true at w and actuality is absolute, then w is actual is true simpliciter, contradicting the initial assumption. The contradiction can be avoided, it seems, only by making actuality relative to worlds. What justifies the crucial move from “the proposition w is actual is true at w” to “w is actual is true simpliciter”? Lewis is supposing that all realists about possible worlds must accept something like his own account of truth at a world: For me, truth at a world is a simple matter. It is just truth, with world dependent linguistic elements evaluated at the world in question. For instance—and this alone will take us a lot of the way—quantifiers must (usually) be restricted to inhabitants of the proper world. Thus it is true at a certain world that some pigs fly iff some inhabitants of that world are pigs that fly. (Lewis a: –)

The argument can now be filled in as follows: If actuality is absolute, then the sentence ‘w is actual’ contains no “world-dependent linguistic terms.” Therefore, the ‘at w’ drops out as superfluous. That is to say, switching harmlessly from sentences to propositions: w is actual is true at w if and only if w is actual is true. Which it isn’t. Farewell contingency. In response, the Leibnizian must deny that Lewis’s account of truth at a world is fully general. Let us say, for any property F, that the proposition w is F is true of w if and only if w is F. Then, what the Leibnizian must deny, at least for the case of actuality, is that what is true at a world—how the world represents itself to be— coincides with what is true of the world—how the world really is. Whenever F is a qualitative property, the two notions coincide for the Leibnizian: if a world represents itself, say, as having seventeen dimensions, or having flying pigs as parts, then the world does have seventeen dimensions, or flying pigs as parts. This contrasts sharply with the ersatzers, linguistic and magical, for whom what is true at a world is rarely if ever what is true of the world. But, in the case of actuality, every world represents itself as being actual, according to the Leibnizian, whether it is actual or not. Since truth conditions for modal statements, for realists and ersatzers alike, are given in terms of what is true at worlds, what worlds represent to be the case, the Leibnizian can have her cake and eat it too: the contingency of actuality without sacrificing its absoluteness. Incidentally, it is not just the problem of the contingency of actuality that pushes the Leibnizian down this road. When we contemplate possible worlds in the course of making decisions or forming beliefs, we always consider possible worlds under the supposition that they are actual. Suppose I ask whether I would have been better off had I done something other than I did. It would be absurd for you to reply: “Of course not. You would have been much worse off; you would have been merely possible instead of actual!” Thus, the Leibnizian agrees with the Lewisian that, with ²⁸ What follows is more or less how Lewis presents the argument in Postscript B to Lewis (). See Lewis (a: –).

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   



respect to the role that possible worlds play in our mental life, the worlds are all on a par: when we consider any world, we consider it as actual. But for the Leibnizian, there is a sober metaphysical fact as to whether a world is actual or not, no matter how irrelevant that fact is to the world’s ordinary role. Could a Lewisian realist charge that, by detaching true at a world from true of a world, the Leibnizian forgoes her realist credentials? I think not. All realists need to invoke the distinction between what is true at a world and what is true of a world in order to provide adequate truth conditions for de re modality. Consider the standard example. Humphrey might have won the election in ; there is a world w, then, at which Humphrey wins.²⁹ According to the realist, this is to be understood in terms of Humphrey’s counterparts: Humphrey wins at w in virtue of w having among its parts a winning counterpart of Humphrey, not Humphrey himself. Now consider the property, having Humphrey among its parts. The proposition w has Humphrey among its parts is true at w, but not of w. The Lewisian thus distinguishes between what individuals a world represents de re as existing, and what individuals in fact exist there. This distinction is motivated, of course, by a desire to get the truth conditions for de re modal statements right while holding to a metaphysics of worlds believed on independent grounds, in particular, that worlds don’t overlap. If the Lewisian can invoke this distinction to account for de re modality, why can’t the Leibnizian do the same to account for the contingency of actuality, to get the truth conditions for modal statements about actuality right? The Lewisian may legitimately insist, however, that whenever what is true at a world differs from what is true of a world, there be an account of how representation by worlds works that does not rely on primitive modality. This demand can be met in the case of representation de re so long as counterpart relations are based on qualitative similarity. Can this demand be met by the Leibnizian in the case of the representation by worlds of their own actuality? It won’t do, of course, for the Leibnizian simply to call on the general formula: what is true at a world w is what would have been true had w been actual. That leads to the desired result, that every world is actual at itself, but it invokes modality to do it. The formula should follow from the Leibnizian’s account of representation by worlds, not constitute it. What saves the Leibnizian is that actuality is a special case: no general analysis of representation is needed to account for it; a one-off analysis will do. The Leibnizian can simply accept whatever account a Lewisian provides for representation by worlds, and then add a separate clause to deal with the property of actuality. For the traditional Leibnizian, the added clause is: world w is actual is true at v if and only if w is identical with v.³⁰ The contingency of actuality then follows immediately: for any world w, possibly, w is the one and only actual world. For the non-traditional Leibnizian realism that I endorse, the added clause is less familiar because truth conditions for propositions are given relative to classes (or aggregates, or pluralities) of worlds, rather than relative to single worlds. (See Section ...) Intuitively, a ²⁹ This presupposes what Lewis calls the “simple account” of modal operators. See Lewis (a: –). ³⁰ More generally: individual x is actual is true at v if and only if some counterpart of x is a part of v. I here ignore abstract individuals (if any there be) that one may want to count as “actual by courtesy” so as to include them in the domain of our actualist quantifiers. See Lewis (a: –).

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

       ()

proposition is true at a class of worlds if it would have been true had all and only the worlds in that class been actual. The clause, then, that should be added to the Lewisian account is: for any world w and any class of worlds V, world w is actual is true at V if and only if w is a member of V. The analyses of possibility and necessity must now be amended as well: a proposition is possible if and only if it is true at some class of worlds; necessary if and only if it is true at every class of worlds. To illustrate: the proposition exactly two worlds are actual is true at any pair of worlds, and is thus possible. The proposition every world is actual is true at the class of all worlds, and is thus possible; but it is not necessary since it is not true at any proper subclass of the class of all worlds. Now, given the added clause and the amended analysis of possibility, we have the contingency of actuality in the following form: for any class of worlds W, possibly, all and only the members of W are actual. With the contingency of actuality secured in this way, the Leibnizian can provide a sense in which logical space as a whole (the aggregate of all worlds) might have been different with respect to the distribution of actuality without having to conjure up a disheartening regress of ever more inclusive logical spaces to account for it. Whether one opts for traditional or non-traditional Leibnizian realism, there is no way that the extra clauses providing truth conditions for world w is actual can conflict with the Lewisian account to which they are added. That is because, on the Lewisian account, what worlds represent to be the case depends entirely on the qualitative nature of those worlds, whereas, for the Leibnizian, the property of actuality is independent of all qualitative features. There is thus no risk that in accepting the extra clause, a world may end up representing itself both as p and as not p, for some proposition p; no need for primitive modality to ensure that worlds are genuinely possible. Of course, the oneoff analysis would be modal if actuality itself were a modal property. But, as already noted in Section ., it is not: what worlds are possible is prior to, and independent of, what world or worlds are actual. I conclude that the Leibnizian has an adequate response to the first of Lewis’s two arguments.

. Skepticism about Actuality I know that I am actual and not merely possible—no doubt about that. The question is: how do I know it? According to Lewis’s second argument against Leibnizian realism, the Leibnizian is in a poor position to answer this question. Here is how Lewis presents the argument in On the Plurality of Worlds: . . . one world alone is ours, is this one, is the one we are part of. What a remarkable bit of luck for us if the very world we are part of is the one that is absolutely actual! Out of all the people there are in all the worlds, the great majority are doomed to live in worlds that lack absolute actuality, but we are the select few. What reason could we ever have to think it was so? How could we ever know? Unactualized dollars buy no less unactualized bread, and so forth. And yet we do know for certain that the world we are part of is the actual world . . . . (Lewis a: )

I would object to Lewis’s assumption that, for the Leibnizian, only one world is absolutely actual, so that “the great majority” of people lack absolute actuality. For all I know, it may be the other way around. But that hardly helps. The argument doesn’t

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  



get its force from probabilistic considerations. If there are merely possible people epistemically situated exactly as I am—no matter how many—then I can’t rule out my being in their predicament. I can’t know that I am actual. Let me lay out the skeptical argument in a series of numbered steps for future reference, and then add some brief commentary. Say that two subjects of thought are epistemic counterparts iff they have exactly matching perceptual experience (and other evidence, if there is any). Now, the Leibnizian realist is committed to: () I have epistemic counterparts inhabiting other possible worlds; and () actuality is an absolute property that I have and (some of ) my epistemic counterparts lack. It follows that () none of my evidence rules out the possibility that I am merely possible rather than actual. Therefore, the argument concludes, () I do not know that I am actual.³¹ Two brief comments are in order. First, the argument is presented from an “internalist” perspective, as is befitting Lewis’s own views about knowledge (Lewis: ). But by choosing epistemic counterparts that are qualitatively indiscernible from me—or very nearly so, if one rejects qualitatively indiscernible worlds—the argument will apply to a wide range of “internalist” and “externalist” accounts of knowledge. Indeed, the only restriction on what counts as “evidence” is that evidence be qualitatively characterizable; I do not need to suppose it is “internally accessible.” Secondly, premise (), as stated, is incompatible with universal actualization. This could be remedied by prefixing each premise with the epistemic operator, ‘for all I know’; but I will ignore this complication. I will return to this argument in due course. I want to begin, however, by briefly dismissing two Leibnizian responses that are not, I think, satisfactory. According to the first response, the Leibnizian should embrace skepticism about actuality, at least when in the philosophy room. After all, it is not uncommon for metaphysicians to introduce new skeptical possibilities in the course of developing their theories. To reject a metaphysical theory because it leads to skepticism—as Lewis does— would be wrongly to hold metaphysics hostage to epistemology. If we have good reason to accept Leibnizian realism and Leibnizian realism leads to skepticism about our actuality, then we have good reason to be skeptical about our actuality. So goes the first response.³² This response fails, however, because skepticism about actuality is not at all analogous to familiar brands of skepticism.³³ Skepticism about the external world, or the past, or other minds, is palatable in the philosophy room only because we have a satisfactory account of how in ordinary contexts we speak truly when we say that we know. For example, if I say in an ordinary context, “I know that my arm is broken,” I speak truly (we may suppose) even though I cannot rule out being a brain in a vat with simulated experiences that match my own. In ordinary contexts, alternative ³¹ In Lewis (: ), the conclusion of the argument is disjunctive: “Either we know in some utterly mysterious way that we are actual; or we do not know it at all.” This is a rhetorical flourish: Lewis certainly rejected the first disjunct, and expected the reader to apply disjunctive syllogism. Whether Lewis would have classified the solution I give below as “utterly mysterious” I am not sure. ³² See Parsons (: –). ³³ In Section ., I give an account of ‘actual’ according to which we know a priori that we are actual simply in virtue of understanding the concept of actuality. Familiar skepticisms cannot be defeated in this way, though not for lack of trying.

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

       ()

possibilities in which there are such bodiless brains in a vat are irrelevant, and “properly ignored.”³⁴ But skepticism about actuality cannot be made palatable in this way. To save my knowledge of actuality in ordinary contexts, all mere possibilities containing epistemic counterparts of me would have to be deemed irrelevant; and the only principled way to do that would be to deem all mere possibilities irrelevant. That gives the wrong result for knowledge of other matters of fact. If only one world is actual, knowledge of actuality would be saved in ordinary contexts by making knowledge of other matters too easy: our true beliefs would amount to knowledge in ordinary contexts, at least with respect to contingent matters of fact. In any case, however many worlds are actual, knowledge of actuality would be saved in ordinary contexts by making knowledge of other matters depend on an epistemically irrelevant factor: the extent of actuality. But if skepticism about actuality can’t be confined to the philosophy room, then embracing skepticism about actuality just in the philosophy room isn’t an option for the Leibnizian. A second response invokes again the distinction between what is true at and what is true of a world. I know that I am part of some world; and that any part of a world is actual at that world. (The latter results, as seen above, from my understanding that actuality is contingent.) Thus, I know that I am actual at my world; I know, in other words, that the proposition, I am actual, is true at my world. But, truth at my world is truth simpliciter. Therefore, I know that I am actual. So goes the second response. Clearly, this response evades the problem rather than solving it. Granted, given the distinction between what is true at and what is true of a world, we can distinguish two questions: how do I know that I am actual is true at my world? And, how do I know that I am actual is true of my world, that is, how do I know that my world is (absolutely) actual? Perhaps the question—“How do I know that I am actual?”—is ambiguous within the Leibnizian framework, and can be interpreted either way. Interpreted the first way, the question is indeed easily answered by reflecting on the contingency of actuality, and the requirement that every world represents itself as being actual. But answering this first question does not make the second question go away. No doubt, when Lewis asks—“How could we ever know?”—it is the second question that he has in mind, and nothing has yet been done to resolve the skeptical problem that it raises. It may seem that my approach to the problem of skepticism is at variance with my approach to the problem of contingency. For the claim, every world is contingently actual, can also be taken to be ambiguous within the Leibnizian framework; when interpreted in terms of truth of rather than truth at, it comes out false, not true. Why do I allow the contingency claim to come out false on one of its interpretations, while requiring the knowledge claim to come out true on both its interpretations? The difference is this. The problem of contingency, as I see it, is essentially a semantical problem. To solve it, the Leibnizian needs to provide plausible semantical analyses of the modal statements in question within her framework of possible worlds. Success is measured by getting the truth conditions right. The existence of other plausible analyses that get the truth conditions wrong is beside the point. The problem of

³⁴ For a contextualist account of knowledge along these lines, see Lewis ().

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 



skepticism, on the other hand, is an epistemological problem. It isn’t enough to show that the proposition, I know that I am actual, comes out true on some plausible analysis. Every plausible analysis potentially introduces a different skeptical problem. The Leibnizian, to be successful, must be able to respond to them all. With these first two responses off the table, the Leibnizian appears to be in a bind. If there were a priori grounds for holding that everything is actual, as actualists believe, then skepticism could be defeated; but there aren’t. If actuality were relative rather than absolute, as Lewis believes, then knowing that one is actual at one’s own world is all there is to know; but it isn’t. Has the Leibnizian, by taking actuality to be an absolute property that some things have and other things lack, put knowledge of actuality beyond reach? Perhaps it will help, before answering, to consider a different but related skeptical problem.

. Tu Quoque I know that I am an individual, not a set—no doubt about that. The question is: how do I know? The correct answer, I think, will surprise many realists about sets.³⁵ For purposes of this discussion, I will suppose that sets exist and satisfy the standard axioms for iterative set theory,³⁶ and that sets and individuals are two distinct fundamental ontological kinds of entity that together exhaust all of reality.³⁷ Now, I suspect that most realists about sets would accuse anyone who raises the question—how do I know I am not a set?—of being (intentionally) obtuse. Surely, they will say, there are numerous properties that I know I have and that I know no set has. It then follows, by an application of Leibniz’s Law, that I know I am not a set. But it is no simple matter to come up with the required discerning properties. I will not challenge our claim to know, for many ordinary properties, whether we have the property. The problem, rather, is with our claim to know that sets do not have the property. We know less than we tend to think we know about the properties of sets. Consider the following attempts to argue that I am discernible from any set. First Try. I am concrete. Sets are abstract, not concrete. Therefore, I am not a set. That gets us nowhere, even if we allow that the abstract/concrete distinction is well understood. The skeptic’s question can just be rephrased: how do I know I am concrete and not abstract? Of course, we do know that we are concrete; but the

³⁵ Similarly, I would ask the realist about universals: how do you know you are a particular, and not a universal? Since the dialectic is relevantly similar to the case of sets, I will here discuss only the latter. ³⁶ That is, Zermelo-Fraenkel set theory with Choice (ZFC) and extensionality adapted to allow for individuals. What I call “individuals” are sometimes called “urelements,” or “atoms.” ³⁷ Two kinds exhaust reality if every part of reality belongs to one or the other or is a fusion of things belonging to one or the other. Note that saying that the two kinds are distinct rules out Quine’s set theory according to which an individual is its own singleton. Note also that, for Lewis, the fundamental ontological distinction is between individuals and classes: proper classes are not sets, and the null set, by stipulation, is an individual, not a class. See Lewis (). But, since what I have to say can be said equally well within the framework of sets or of classes, I will simply assume in what follows that there are no proper classes, or other “set-like” entities.

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

       ()

question is how. The answer to the rephrased question is no easier to come by than the answer to the original question. Indeed, if one holds that all and only individuals are concrete and that all and only sets are abstract,³⁸ then the rephrased question is equivalent to the original. Second Try. I am spatially located. No set is spatially located. Therefore, I am not a set. The common assumption that sets are not spatially located is nothing but unsupported dogma. Perhaps pure sets lack spatial location—although that may depend on what one takes the null set to be—but that does nothing to support the claim that impure sets lack spatial location as well. Indeed, if we ever refer to impure sets with our ordinary talk of collections, groups, and so forth, then it is far more natural to suppose that sets are spatially located: they are located where their members are. A baseball team, for example the New York Yankees, is plausibly identified with a set. The question—where is the team now?—does not seem to be deviant in any way. The team is where its members are: in New York, if all the team members are in New York; partly in New York, partly in Boston, if some of the team members are in New York and some are in Boston; and so on. Similarly, it is natural to say that my singleton is located exactly where I am. Knowing that I am spatially located, then, does nothing to rule out my being a singleton. The same goes, mutatis mutandis, for knowing that I am temporally located or spatiotemporally located. Third Try. But I do know that I am not a singleton. I am mereologically complex. Singletons are mereologically simple; they are atoms. Therefore, I am not a singleton. I hold, with Lewis (: –), that mereology applies to sets: the parts of a set are all and only its non-empty subsets; singletons are mereological atoms. I therefore grant that the argument is sound: I know that I am not a singleton. But consider instead the set whose members are all and only my atomic parts. That set is mereologically complex, and its complexity exactly matches my own.³⁹ Knowledge of my mereological complexity, then, does nothing to rule out my being that set. Fourth Try. I have qualitative properties that no set has. For example: I have a mass of  kilograms. No set has a mass of  kilograms, or any other mass. Therefore, I am not a set. More unsupported dogma. Why not say instead that a singleton inherits the qualitative properties of its sole member, and, more generally, that a set inherits the qualitative properties, including structural qualitative properties, of the aggregate of its members? More precisely, consider the following qualitative inheritance principles for sets: (Q)

Whenever x has a qualitative property, {x} also has that property, for any individual or set x. (Q) Whenever x₁, . . . xn stand in an n-ary qualitative relation, {x₁}, . . . {xn} also stand in that relation, for any individuals or sets x₁, . . . xn.

³⁸ As Lewis does, except that he would say ‘class’ where I say ‘set’. See Lewis (). ³⁹ Unless I am composed partly or wholly of atomless gunk. But I, for one, do not think atomless gunk is metaphysically possible. Be that as it may, all I need for the above argument is that we do not know that we are partly or wholly composed of atomless gunk, which I think is a safe assumption. On gunk, see Lewis (: –).

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 



For all I know, (Q₁) and (Q₂) give the truth about sets.⁴⁰ But, then, on the plausible assumption that the qualitative properties of composites are wholly determined by the qualitative properties and relations among atomic individuals,⁴¹ it follows that, for all I know, I am qualitatively indiscernible from the set of my atomic parts (and, indeed, from infinitely many other sets higher up the iterative hierarchy). Knowledge of my qualitative nature, then, does nothing to rule out my being a set; in particular, nothing to rule out my being a set having a mass of  kilograms. Fifth Try. Even if sets share qualitative properties with individuals, they do not belong to the same kinds. I am a person, a thinker, an agent. No set is a person, or a thinker, or an agent. Therefore, I am not a set. But being of the kind person, or thinker, or agent supervenes on the having of qualitative properties and the standing in qualitative relations. Once one grants that, for all one knows, every individual is qualitatively indiscernible from some set, one cannot consistently deny that sets and individuals fall under the same kinds. For all I know, the set of my atomic parts is a person that thinks and acts and wonders whether it is a set. Knowledge that I am a person or a thinker or an agent, then, does nothing to rule out my being a set. Sixth Try. Individuals can be distinguished from sets by their modal properties. I could exist without some of my parts. No set could exist without some of its parts. Therefore, I am not a set. Here, again, the philosophers’ dogma about sets—that sets have their members essentially—clashes with ordinary applications of sets. After all, a team can lose a member without thereby ceasing to exist. But even if one grants the two premises of the argument, the conclusion doesn’t follow by Leibniz’s Law. Not, in any case, for a counterpart theorist. There is overwhelming evidence that ordinary attributions of de re modality are context-dependent. This context-dependence, I would argue, is best explained by holding that different counterpart relations are evoked in different contexts. Which counterpart relation is evoked in a context determines which modal property is expressed by a given modal predicate in that context. Now, talk of persons ordinarily evokes a counterpart relation—the personcounterpart relation—according to which not all parts are essential; talk of sets ordinarily evokes a different counterpart relation—the set-counterpart relation— according to which all parts are essential. But then the modal property attributed in the first premise—having a person-counterpart lacking (counterparts of) some of my parts—need not be the modal property that is denied in the second—having a setcounterpart lacking (counterparts of) some of my parts. The application of Leibniz’s Law is spurious.⁴² Six tries, six failures. At this point, I hope the problem of skepticism about one’s individuality is looking oddly analogous to the problem of skepticism about

⁴⁰ Please don’t misunderstand. I am not saying that, for all we know, sets are spatially located or have mass in some extended Pickwickian sense; we could just stipulate that. I am saying that, for all we know, sets have spatial location and have mass in exactly the same sense that we do. ⁴¹ Alternatively, if one accepts gunk or emergent properties and relations, one can generalize (Q) and (Q) by switching to plural variables. Thus, (Q) becomes: whenever X has a qualitative property, {x: x is one of the Xs} also has that property, for any individuals or sets X. And (Q) can be similarly generalized. ⁴² Lewis first used multiple counterpart relations to provide for “contingent identity” in Lewis (). He defends the context dependence of de re modal attributions in Lewis (a: –).

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

       ()

one’s actuality. In both cases, the problem is that, for all we know, there are thinkers qualitatively indiscernible from us who are deceived about their ontological status: in the one case, merely possible thinkers deceived about their actuality; in the other case, thinkers that are sets(!) deceived about their individuality. Thus, when a realist about sets claims that the problem of skepticism refutes Leibnizian realism, the Leibnizian can respond: tu quoque! This response is especially powerful, ad hominem, against Lewis. For example, with respect to whether sets are located in space and time, Lewis writes: I don’t say the classes are in space and time. I don’t say they aren’t. I say we’re in the sad fix that we haven’t a clue whether they are or whether they aren’t. We go much too fast from not knowing whether they are to thinking we know they are not, just as the conjurer’s dupes go too fast from not seeing the stooge’s head to thinking they see that the stooge is headless. (Lewis : )

And, directly after this passage, he affirms our ignorance as to whether sets have qualitative properties. Sets are mysterious, according to Lewis, and we know much less about them than we often take ourselves to know. He would have to conclude, at least if only qualitative properties are being considered, that we have no way to rule out that we ourselves are sets. I suspect many realists about sets will still accuse me (and Lewis) of being obtuse. Of course, they will say, if one has to somehow discover the properties of sets, one will not succeed; there is no faculty of intuition, analogous to perception, by which we can “see” whether sets are spatially located, or massive, or what have you. The properties of sets are stipulated, not discovered. In particular, it is stipulated that sets have only the properties they are required to have to satisfy the standard axioms of (impure) set theory.⁴³ It follows from the stipulation that sets share no qualitative properties with us. End of skeptical worries. I am sympathetic with the stipulational approach. But it merely shifts the skeptical problem rather than eliminating it. Just where the bulge in the carpet reappears depends on one’s underlying ontological views. If one’s ontology is sparse, then stipulation is risky. I have supposed that there are some entities that jointly satisfy the axioms of impure set theory; on a sparse ontology, no other entities jointly satisfy the axioms. To stipulate anything further about the nature of these entities, whether positive or negative, risks getting it wrong: only one such stipulation gets it right. What if the stipulation that gets it right is the one that attributes the qualitative inheritance principles, (Q) and (Q), to sets? We have shifted the uncertainty, but not done away with it. Given the proposed stipulation, I now know that, if sets exist, they have no qualitative properties. Thus, I now know that I am not a set, but only at the cost of no longer knowing whether sets exist. Realism about sets has been undermined. If instead one’s ontology is expansive, then stipulations are fairly safe. The stipulations must meet minimal standards of logical consistency (and, I think, somewhat more), but we can be fairly sure that there exist entities corresponding to our stipulations. Suppose it is stipulated that sets have no qualitative properties. ⁴³ Perhaps incorporating the mereological theses, à la Lewis, with the singleton relation taken as primitive, rather than the membership relation. And perhaps with the ontological axiom that individuals and sets belong to disjoint ontological kinds.

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       



Then I can be (fairly) sure that sets, so stipulated, exist, and that I am not one of them. I know that I am an individual and not a set. But now consider an alternative stipulation according to which the entities stipulated—call them “schmets”—satisfy the axioms and the inheritance principles, (Q) and (Q). This stipulation, it seems to me, is internally consistent. If one’s ontology is sufficiently expansive, then there are entities satisfying the alternative stipulation: schmets exist. I now ask: how do I know that I am an individual and not a schmet? Skepticism is back.

. Indexicality and the Concepts of Individual and Set But, of course, it is absurd for me to wonder whether or not I am a set. I know I am not a set as soon as I understand the concept of set. I think there is only one way that could be: the concept of set is an indexical concept. Ditto for the concept of individual. That I am an individual is built into the concept of individual. That sets and individuals belong to distinct ontological kinds, and thereby that I am not a set, is built into the concept of set. This indexicality, and this alone, can explain how I know that I am not a set. Before turning to the specific analyses, it will be useful first to introduce a distinction, familiar under various terminological guises, between concepts and properties.⁴⁴ To illustrate, consider the timeworn example: water. The concept that is associated with the predicate ‘is water’ is something like: being the substance of my acquaintance which descends from clouds in rain, which forms rivers, lakes, seas, etc., and which is used for drinking, cooking, bathing, etc.⁴⁵ This concept is what I grasp in virtue of understanding the predicate ‘is water’, what would most naturally be identified with its meaning. The property designated by a use of the predicate ‘is water’ is determined jointly by the associated concept and the context of use. (A property for a realist about possible worlds can be identified with the class of its actual and possible instances.) When I use the predicate ‘is water’, the property designated is something like: being composed of molecules of H₂O. Now, the concept expressed by ‘is water’ is an indexical concept: the property it designates on a given use depends on the context of use. For example, Twin Earthlings have the same concept of water that we Earthlings do, but when they use the predicate ‘is water’—so the oft-told story goes—it designates the property of being composed of XYZ. Indexical concepts can be taken to be functions (perhaps partial) from contexts to properties. However, since in this chapter the only relevant feature of context is the thinking subject (or speaker), I will simplify and take indexical concepts to be functions (perhaps partial) from subjects to properties.

⁴⁴ In the framework of Kaplan (), it is the distinction between character and content applied to monadic predicates. See also the two-dimensional semantical frameworks in Stalnaker (), Chalmers (), and Jackson (). ⁴⁵ Taken almost verbatim from Webster’s New International Dictionary, except that I add “of my acquaintance” to allow for the off chance that a different substance might meet these conditions elsewhere in the universe, and I substitute ‘substance’ for ‘liquid’ so that I can say, simply, that whatever is composed of H₂O is water.

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

       ()

Indexicality is a kind of relativity: when an indexical concept is associated with a predicate, which property (if any) is designated by the predicate on a given use is relative to the subject using the predicate. Another kind of relativity has to do with the nature of the property designated: the designated property is itself a relative property, a property derived from an underlying relation. These two kinds of relativity are independent of one another. A predicate associated with an indexical concept can designate, in every context, an absolute property; the predicate ‘is water’ is an example of such. On the other hand, the predicate associated with an indexical concept can designate, in every context, a relative property; the predicate ‘is a neighbor’ has this dual relativity. I claim that the concepts of individual and set are, in this respect, like the concept of water, not the concept of neighbor: although the concepts are indexical, the properties designated are absolute. But the concepts of individual and set are in other respects quite unlike ordinary indexical concepts. Start with the concept of individual. I distinguish three independent components. First, there is the standard mathematical characterization in terms of the membership relation: x is an individual if and only if x is memberless and distinct from the null set. Typically, this (or something equivalent) is taken to be all there is to being an individual. But, although it provides a necessary and sufficient condition for being an individual, it does not exhaust the concept. There is also an ontological component: anything ontologically like an individual—anything belonging to the same fundamental ontological kind as an individual—is itself an individual. For it is part of the concept of individual that the individuals constitute a fundamental ontological kind, that being an individual is a categorial property; and categorial properties, I assume, are absolute. Finally, there is the crucial indexical component: I am an individual. This provides an independent sufficient condition: x is an individual if x is identical with me. Putting the ontological and the indexical components together, we have the stronger sufficient condition: x is an individual if x is ontologically like me.⁴⁶ These three components, I claim, are all analytic: no one could be said to understand the concept of individual if they failed to grasp any of the three.⁴⁷ The concept of set can similarly be divided into three components. Mathematical: x is a set if and only if x has members or is identical with the null set.⁴⁸ Ontological: anything ontologically like a set is a set. Indexical: sets and individuals are distinct. Now, if asked how I know I am an individual and not a set, I can answer with the ⁴⁶ If everything belongs to exactly one fundamental ontological category, then one also has: anything ontologically unlike an individual is not an individual. In this case, the ontological and indexical components together provide a second necessary and sufficient condition: x is an individual if and only if x is ontologically like me. But I will not include this so as to accommodate ontological frameworks that allow fundamental ontological categories to cut across one another, as would be the case if one accepts four fundamental ontological categories: individual, set, actual, merely possible. ⁴⁷ Well, it may be problematic just how to divide the analyticity between the concepts of individual and of set; all I need is that what I claim to hold a priori for individuals, or for sets, follows jointly from the concepts of individual and set (and any related concepts, such as membership). I can allow that the analysis of the separate concepts is somewhat indeterminate. ⁴⁸ Of course, the concepts of individual and set will be indeterminate to whatever extent the concept of member is indeterminate; they will be loaded (for example, with content from ZFC) to whatever extent the concept of member is loaded. But let that pass.

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       



greatest of ease. That I am an individual is built into the concept of individual. That I am not a set follows from my being an individual and what is built into the concept of set. These answers, which depend only on the indexical components of the concepts, are trivial and analytic, just as they should be. Not only is skepticism about individuality defeated; it is defeated in a way that explains why such skepticism strikes us as absurd. No dissertation on the metaphysical nature of individuals or sets is needed, or relevant.⁴⁹ The mathematical components of the analyses of set and individual (together with the axioms of set theory) provide me with a priori knowledge de dicto of the iterative hierarchy: the individuals all occupy the bottom level; the sets appear successively at levels higher up. But that’s not all. The mathematical and indexical components, when combined, provide me with substantial a priori knowledge de se, knowledge of where in the hierarchy I am located: I know that I am at the bottom. No amount of knowledge de dicto could provide me with this knowledge de se: one cannot pull knowledge de se out of a purely de dicto hat.⁵⁰ The indexicality of the concepts is crucial; it provides me with a perspective on the hierarchy. I am not, as it were, outside the hierarchy looking in, wondering where I am. The hierarchy is built on top of me; I cannot be anywhere but at the bottom. My place in the hierarchy is in part constitutive of what the hierarchy is.⁵¹ That there can be substantial knowledge de se based solely on the understanding of concepts is a surprising—and, I suspect, controversial—feature of my account. It arises because the concepts of individual and set are indexical concepts of a special sort. The indexical and non-indexical components of the concepts are not conjoined to form a single necessary and sufficient condition. For example, if to be an individual were to be both ontologically like me and memberless (and distinct from the null set), then I could not know that I was an individual without first knowing that I was memberless (and distinct from the null set). But since instead the indexical and non-indexical components of the concept of individual provide independent, complementary necessary and/or sufficient conditions, then I can first, via the indexical component, conclude that I am an individual, and then, via the non-indexical component, conclude that I am memberless (and distinct from the null set), that I am located at the bottom of the hierarchy. Say that an indexical concept is perspectival if it meets the following four conditions. First, if the concept picks out a property relative to a subject, then the subject has that property. Second, one and the same (absolute) property is picked out by the

⁴⁹ How to respond to the skeptical argument will be discussed in Section .. ⁵⁰ On the irreducibility of knowledge de se to knowledge de dicto, see Lewis (). ⁵¹ A personal note. When developing these views in –, my son Adam, then age , asked a question that helped crystallize my thought. We were reading Eric Carle’s The Very Hungry Caterpillar and I turned to the page showing the caterpillar inside its cocoon. Adam asked: but how did the caterpillar get inside? His puzzlement betrayed a failure to conceive of all possible solutions. The skeptic, I claim, makes essentially the same mistake when asking how I know I am at the bottom of the hierarchy. The answer, in the caterpillar case, is that the caterpillar didn’t have to get inside because he built the cocoon around him. The answer to the skeptic is essentially the same, except that the sense of ‘built’ is conceptual, rather than physical.

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

       ()

concept relative to all subjects for whom it picks out a property. Third, relative to some subjects, no property at all is picked out: the concept is defective relative to those subjects. And, fourth, if the concept is not defective relative to a thinking subject that grasps the concept, the subject knows a priori it is not defective. The concept of individual, I claim, is a perspectival concept. Relative to me, and any subject ontologically like me, it picks out an absolute property had by all the memberless entities (except for the null set). Thus it meets the first two conditions. Relative to a set as subject, it is defective and picks out no property at all.⁵² Thus it meets the third condition. Finally, I know the concept is not defective for me simply in virtue of grasping the concept. (How? More on this in Section ..) Thus it meets the fourth condition. When a concept is perspectival, we have indexicality without the usual variability. Perhaps that is why it is easy to miss the indexical component of these concepts. But the indexicality nonetheless plays a crucial role: it provides a perspective on reality, and the conceptual knowledge de se that goes with it. Call the view perspectivalism that holds that perspectival concepts are needed to give a complete account of reality. They are needed because not all perspectives on reality are equally valid: mine is; yours is, if you are ontologically like me; but yours is not if you are ontologically alien—a set, or a merely possible being.⁵³ I will return to perspectivalism after considering the concept of actuality in the next section. But, first, there is an urgent matter to attend to.⁵⁴ Perspectival concepts, if abundant, threaten to make knowledge de se too easy, even knowledge of contingent matters. Suppose I have entered a lottery. The winning ticket has just been drawn and, unbeknownst to me, I am the winner. How can I find this out? Easy. Concoct a perspectival concept whose non-indexical component is: something satisfies the concept if and only if it won the lottery in question. Now, reason as follows. By the indexical component that is part of any perspectival concept, I know that I satisfy the concept. Then, by the non-indexical component, I conclude that I won the lottery. Good news! And I never had to leave my armchair to get it.

⁵² To see this, suppose for reductio that a property were picked out. According to the indexical component, that property would have to be satisfied by a set, the subject; but according to the mathematical component, it can only be satisfied by entities that are memberless (and distinct from the null set). Contradiction. Therefore no property is picked out by the concept of individual relative to a set as subject. ⁵³ Ever since reading Stalnaker () as a graduate student, I have been intrigued by his apparent endorsement of perspectivalism. He asks (in Stalnaker (: –), a revision of Stalnaker ()) whether, on his version of “moderate realism” about possible worlds “ . . . from an objective, absolute standpoint, merely possible people and their surroundings are just as real as we and ours?” He replies: “Only if one identifies the objective or absolute standpoint with a neutral standpoint outside of all possible worlds. But there is no such standpoint. The objective, absolute point of view is the view from within the actual world, and it is part of the concept of actuality that this should be so.” Stalnaker (: ). And, indeed, I think the “moderate realist,” (or “magical ersatzer,” as Lewis would say), no less than the Leibnizian realist, has reason to endorse perspectivalism. But later in the paper, Stalnaker endorses “analytic actualism,” writing: “the thesis that the actual world alone is real has content only if ‘the actual world’ means something other than the totality of everything there is, and I do not believe that it does.” Stalnaker (: ). I find this puzzling because analytic actualism would seem to make perspectivalism with respect to actuality otiose. ⁵⁴ Here I am indebted to Ted Sider.

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     



I respond by denying that there are such concocted perspectival concepts. There are only as many perspectival concepts as there are genuine ontological perspectives, perspectives built into the nature of reality. Genuine ontological perspectives can’t be donned and doffed at will, like an old suit of clothes, any more than objects can be willed in and out of existence. Granted, there is an algebra of concepts that allows concepts to be introduced by the arbitrary combination of conditions, but only when those conditions are individually necessary and jointly sufficient. Thus, I could concoct a concept that is satisfied by anything that is identical with me and won the lottery. But that concept does not provide a path to easy knowledge. As best I can tell, the only perspectival concepts are those that correspond to firstperson ontological kinds: individual for the realist about sets, particular for the realist about universals, actual for the Leibnizian realist, present for the A-theorist about time. Such concepts provide us with ontological orientation, with genuine knowledge de se. They are a path to easy knowledge, all right, but only in cases where the knowledge should be easy: knowledge of what fundamental ontological kind or kinds I belong to.

. Indexicality and the Concept of Actuality Our way is now clear to return to Leibnizian realism and the problem of skepticism about actuality. The solution I propose will hardly come as a surprise. The concept of actuality, like the concept of individuality, is an indexical concept; that explains how I know—and how I know trivially—that I am actual. The property of actuality—the property designated by my use of the predicate ‘is actual’—is nonetheless an absolute property that holds of everything ontologically like me, without regard to the holding of any external or spatiotemporal relations. Again, as with the contingency of actuality, the Leibnizian can have her cake and eat it.⁵⁵ Of course, the solution requires, in both the mathematical and the modal cases, that it is meaningful to speak of things differing in ontological status, or belonging to distinct ontological kinds; and it requires that the ontological status of a thing, or the ontological kind to which it belongs, is a matter of how it is in itself, not how it is related to other things.⁵⁶ But this commitment is shared by all ontological pluralists. Only a strict nominalist, one who accepted only “concrete particulars,” could ⁵⁵ I find it somewhat surprising that the idea that a realist about possible worlds could combine indexicality of the concept of actuality with absoluteness of the property of actuality, which seems so natural to me, has received no discussion in the literature. An exception is Parsons (unpublished). He introduces and rejects what he calls a “hybrid view” that combines an indexical semantics for ‘is actual’ with absolute actuality. But Parsons’s hybrid view introduces separate actuality properties for each world, rather than the one absolute actuality property accepted by the Leibnizian realist. And he does not notice what I think is most interesting and problematic about the approach, namely, that it appears to require what I call perspectivalism. ⁵⁶ Intuitively, the properties corresponding to fundamental kinds are intrinsic. But one must be careful here, because the standard Lewisian account of an intrinsic property—a property is intrinsic just in case it can never differ between two duplicates—only gives the expected results when restricted to the qualitative properties. The property of actuality, for the Leibnizian, fails to be intrinsic on this account, since actual entities have merely possible (qualitative) duplicates. (A similar problem arises with respect to haecceities— properties of being identical with a given object.)

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

       ()

accuse the Leibnizian of trafficking in obscure notions without herself being subject to a tu quoque. There are, however, at least two ways in which the mathematical and modal cases differ. First, in the case of sets and individuals, there is a well-established, fully developed theory with precise axioms: Zermelo-Frankel set theory with individuals. In the case of the actual and the merely possible, the relevant theory is less developed and more contentious: the theory of Leibnizian realism. Indeed, as discussed in Section ., there are competing formulations of Leibnizian realism: traditional (exactly one world is actual) vs. non-traditional; creation vs. transformation; oneproperty vs. two-property. The analyses of the concepts of actual and merely possible will vary depending on the formulation of Leibnizian realism that is accepted. Second, sets and individuals can be distinguished from one another structurally, by their standing vis-à-vis the membership relation. The actual and the merely possible, on the other hand, are structurally isolated from one another; they must be distinguished from one another internally, if they are to be distinguished at all. But whether or not they can be distinguished internally depends on which formulation of Leibnizian realism is accepted. If they can’t be distinguished internally, then the concept of actuality will be indexical, but not perspectival. More on this to follow. I turn now to the analysis of the concept of actuality. It will be easiest to suppose we have before us a formulation of the theory of Leibnizian realism, non-indexically presented. Since the creation and transformation versions of Leibnizian realism lead to somewhat different conclusions, I will give them separate treatment. I will combine the creation version with two-property Leibnizian realism, and the transformation version with one-property Leibnizian realism. Other combinations are possible; but I will leave what to say in those cases as an easy exercise for the reader. I list only theses directly relevant to the distinction between the actual and the merely possible. Leibnizian Realism (Two-Property Creation Version). There are two fundamental ontological kinds of entity, the A-kind and the P-kind. These two kinds are mutually exclusive, and together exhaust all of reality. The A-kind is copied in the P-kind; that is, the P-region of logical space contains an exact qualitative duplicate of the A-region. The P-kind satisfies the principles of plenitude. On the transformation version, there is less to distinguish the A-kind from the Pkind: the A-kind need not be copied in the P-kind; and it is reality as a whole, not the P-kind, that is required to be plenitudinous. But the one-property transformation version introduces a different asymmetry: Leibnizian Realism (One-Property Transformation Version). There is a fundamental ontological kind of entity, the A-kind, all and only the members of which share a primitive, non-qualitative property. (There is no P-kind; no primitive, nonqualitative property shared by all and only the entities that are not members of the A-kind.) Now, with respect to either version, the analysis of the concept of actuality is straightforward. I present it as a schema to cover both versions at once. As with the concept of individual, there are three components. The indexical and ontological components are just as before; but now the “mathematical” component is replaced

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     



with a “modal” component. Indexical: x is actual if x is identical with me. Ontological: if x is actual, and y is ontologically like x, then y is actual. Modal: x is actual if and only if x belongs to the A-kind. Thanks again to the indexical component of the concept, I know, and know trivially, that I am actual. That explains how I know that I am actual; but it doesn’t tell me what I know. Can the indexical and non-indexical components be combined, as in the case of knowledge that I am an individual, to provide substantial knowledge de se, knowledge as to where I am located in logical space? That depends on which version one accepts. Consider first the two-property creation version of Leibnizian realism. As formulated above, there are no constraints on the extent of actuality; the theory is compatible with exactly one world being actualized, or every world being actualized, or anything in between. If exactly one world is actualized, then we have substantial knowledge de se based solely on our understanding of the concepts. Logical space then divides into two regions—the A-region and the P-region—that are structurally distinguishable: the A-region is unified, and contains a single world; the P-region is disunified, and contains many, many worlds. We know we are located in the A-region. This knowledge de se is analogous to the knowledge we had in the case of the iterative hierarchy, knowledge that we are located at the bottom. Thus, for the traditional Leibnizian, one who adds to the above formulation that exactly one world belongs to the A-kind, the concept of actuality is perspectival. But now suppose instead that every world is actualized. In that case, the two regions of logical space are mirror images of one another—both regions satisfy the principles of plenitude—and we have no grounds for locating ourselves on one side rather than the other. Our understanding of the concept of actuality does not provide us with a perspective on logical space, with any substantial knowledge de se. Thus, for the non-traditional Leibnizian who doesn’t rule out universal actualization, the concept of actuality is not guaranteed to be perspectival. That could be remedied by giving up the possibility of universal actualization. Certainly, it is ordinarily supposed that the realm of the actual is dwarfed in size by the realm of the possible. We could add to the creation version of Leibnizian realism the thesis that the A-region is smaller than the P-region, measuring the size of a region by the number of worlds it contains.⁵⁷ That addition still allows more than one world to be actualized, and so supports the possibility of island universes. It also guarantees us a perspective on logical space: of the two ontologically distinct regions of logical space, we are located in the smaller region. I am committed, however, to the possibility of universal actualization. Were I to accept the two-property creation version of Leibnizian realism, I would accept that the concept of actuality is not guaranteed to provide us with a perspective on logical space. It is still the case that we know in a trivial analytic way that we are actual: the indexical component is enough to secure that. But in this case the concept of actuality, unlike the concept of individuality, is not a perspectival concept. Relative to a merely possible subject, the predicate ‘is actual’ designates the (absolute) ⁵⁷ I hold that it follows from principles of plenitude that there are “too many” possible worlds for them to form a set. See Chapters  and . The best way to implement this strategy, then, would be to add the thesis: there is a set of all and only the A-worlds.

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

       ()

property of belonging to the P-kind. When she thinks to herself, “I am actual,” what she thinks is true: she is nowise deceived. The concept of actuality is not defective for her any more than it is for us. Now consider the one-property transformation version of Leibnizian realism, the version I recommend. Here the concept of actuality is a perspectival concept as long as not every world is actual. A special property has been conferred on the actual entities that make them different in kind from the merely possible entities, and that is enough to provide us with a perspective on logical space, and genuine knowledge de se. In knowing that I am actual, I know that I am located in that ontologically distinct region of logical space, composed of entire worlds, throughout which everything shares a genuine property. However, when a merely possible person thinks to herself, “I am actual,” she is deceived. The concept of actuality is defective for her, because the entities ontologically like her do not share a genuine property, and so do not belong to the A-kind. No property is designated by her use of ‘is actual’. Finally, what about the case of universal actualization, which, on the transformation version, amounts to every world being actual? In this case, there is only one fundamental ontological kind, and knowing that I belong to that kind affords no genuine knowledge de se. As with the two-property creation version, the concept of actuality fails to be perspectival on the supposition that every world is actualized.

. Epistemic Chauvinism Time to face the music. I have claimed that, if the concept of actuality is indexical and the associated property is absolute, I can know trivially that I have the absolute property; I can know that I am actual. But I have not yet diagnosed where the skeptical argument from Section . goes wrong. There are two cases to consider. If the concept of actuality is indexical without being perspectival, the problem is easy to locate. Line () of the argument—“None of my evidence rules out the possibility that I am merely possible rather than actual”—is correct; my merely possible epistemic counterparts represent epistemic possibilities for me, and, by definition, they have the same evidence as I do. But the inference from () to the conclusion ()—“I do not know that I am actual”—fails. For (), at least on its most natural interpretation, is equivalent to: “I do not know that I have the property designated by my use of ‘actual’ ” (with the property description having narrow scope). And so interpreted, () is false. My epistemic counterparts all have the property designated by their use of ‘actual’, which is just to say that there is no epistemic possibility according to which the property designated by my use of ‘actual’ fails to apply to me. It is exactly analogous to how I can know that I am here, even though ‘here’ may have different denotations relative to me and my epistemic counterparts, actual or merely possible. If the concept of actuality is perspectival, however, as I believe, then the above diagnosis does not apply. For then I have epistemic counterparts for whom the concept of actuality is defective, and fails to designate any property. When they think to themselves “I am actual,” they fail to express any proposition with those

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 



words; they are deceived in thinking they know they are actual. If these epistemic counterparts are genuine possibilities for me, then for all I know I am in their predicament. For all I know, the concept of actuality is defective for me, in which case, I am deceived in thinking I know I am actual. The skeptical argument has not been defeated. It is tempting to respond: “Who cares what my epistemic counterparts think; they’re merely possible!” I think this is the right response for the case at hand; but it comes too quick at this stage of the argument. In other cases, the predicament of my epistemic counterparts tells me what I am in a position to know. If I have an epistemic counterpart that lacks a property, my claim to know that I have the property is thereby undermined. For example, I hold that because I have epistemic counterparts that are brains in a vat, I do not know (in the strictest sense) that I am not a brain in a vat. I accept the conclusion of the skeptical argument in this case, at least in the confines of the philosophy room. I need, then, to explain why the case of actuality is special. How can I hold that my brain-in-a-vat epistemic counterparts show that being a brain in a vat is a genuine epistemic possibility for me, whereas my merely possible epistemic counterparts do not show that being merely possible is an epistemic possibility for me? Indeed, since brains in vat are (presumably) merely possible, isn’t this on its face incoherent? To untangle this web, I need to return to the idea (mentioned in Section .) that, ordinarily, when we consider a possibility for ourselves, we consider that possibility as actual. When I think—“for all I know I am a brain in a vat having experiences qualitatively identical with my actual experiences”—I am considering that brain in a vat as actual. A merely possible brain in a vat represents a possibility for me of being an actual brain in a vat. If the genuine epistemic possibilities for me are given by what properties my epistemic counterparts represent, not by what properties they have, then the problem is immediately solved. Even though my epistemic counterparts are merely possible, being merely possible is not a genuine epistemic possibility for me: the possibilities represented by my epistemic counterparts are all possibilities in which I am actual. But I dare not rest with this response. The skeptical argument receives its force from the idea that all my experience and evidence is compatible with my being merely possible. That compatibility is established by the existence of merely possible epistemic counterparts. That these counterparts are actual at their own world, that they represent possibilities in which I am actual, fails to respond to the argument. Absolutely considered, they are merely possible. So, we are back to the question: why don’t my merely possible epistemic counterparts, when considered as merely possible rather than actual, count as genuine epistemic possibilities for me? How can the Leibnizian rule them out? Perhaps all my merely possible counterparts are zombies: they lack conscious experience. More generally, we can add to the analysis of the concept of actuality that only actual individuals have conscious experience. Then the skeptical problem dissolves. Line () of the skeptical argument is false: my having conscious experience rules out my having any merely possible epistemic counterparts; and, so, being merely possible is not an epistemic possibility for me. Of course, we still

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

       ()

want to say that merely possible people are conscious at their worlds so that modal claims about consciousness will come out right. But absolutely speaking, they are zombies.⁵⁸ This seems to me to be a desperate response. For materialists, of course, it is unintelligible: it violates the supervenience of the mental on the physical, even if the supervenience claim is restricted to worlds appropriately like ours. But dualists, I think, should also find the response unintelligible. Why should a dualist, any more than a materialist, deny that there are merely possible worlds differing from our world in some minute physical feature—perhaps an electron added to or removed from a galaxy lacking consciousness—without differing in the presence of consciousness? Allowing that some merely possible worlds that are physically similar to our world are populated with zombies gives no support to the view that all merely possible worlds physically similar to our world are populated with zombies. The Leibnizian needs a different response. Perhaps the skeptical argument, with its emphasis on experience and evidence, is simply not suited to drawing conclusions about a priori knowledge. A priori knowledge, at least when basic, isn’t about having evidence, but about grasping concepts. If two subjects have the same evidence without grasping the same concepts, it is only to be expected that one may have a priori knowledge that the other lacks. The inference from () to () loses its backing. But, for the case at hand, this doesn’t help. I have been supposing that my epistemic counterparts have the same concepts of actuality and mere possibility that I do, not just the same experience and evidence. They cannot be ruled out as genuine epistemic possibilities for me on grounds of being conceptually deficient. Yet, the very concepts that we share give rise to knowledge for me and not for them. Wherefore this asymmetry? The only response possible, I think, is to say that knowledge sometimes depends not only on having the right evidence, the right concepts, the right reasoning, but also on having the right perspective. I have it; my merely possible counterparts do not. Having the right perspective, moreover, is accessible to whoever has it. Knowing that I have the right perspective, I know that the concept of actuality is not defective for me. My merely possible counterparts can’t know this, not because they lack evidence or have different concepts, but just because they are in no position to know it. Thus, the Leibnizian who holds that knowledge of our actuality is perspectival will be led to a form of epistemic chauvinism, at least with respect to a priori knowledge. An epistemic chauvinist, as I use the term, holds that her beliefs may constitute knowledge even though another subject, actual or possible, with the same evidence, the same concepts, and the same powers of reasoning holds beliefs contrary to hers. In that case, this other subject is deceived, even though no rational argument could persuade her to change her view. Disagreement, even disagreement at the end of the day when all the evidence and arguments are in, need not undermine knowledge. ⁵⁸ Forrest (: –) suggests that accepting a “zombie hypothesis” according to which all our counterparts lack consciousness may provide the best way for a realist about possible worlds (which Forrest is not) to develop her view. In particular, he writes: “If we accept the zombie hypothesis we no longer need Lewis’s token-reflexive account of actuality. For we could hypothesize that to be actual is to be in a world with consciousness in it.”

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



I am an unabashed epistemic chauvinist. I claim to know, for example, that the law of non-contradiction holds universally: there is no proposition such that both it and its denial are true. I do not think this knowledge is undermined by the existence of actual or possible dialetheists, even if I grant that a dialetheist may share my concepts (of proposition, negation, and truth), and reason in an internally consistent fashion. I am in a position to know the logical truths; the dialetheist, I therefore conclude, is not. Or consider: I claim to know that possibilia exist in spite of the preponderance of benighted philosophers who disagree. It’s not that I think there are any nonquestion-begging arguments that will force them, by the light of reason, to see the error of their ways. The light of reason, I simply conclude, shines on me and not on them. Now, I am not claiming that my perspectival knowledge that I am actual leads to quite the same brand of epistemic chauvinism as the two examples above. I am saying only that, since I am committed to epistemic chauvinism already on other grounds, I do not take the fact that Leibnizian realism leads to epistemic chauvinism to count against it. A final point to ward off misunderstanding. I am not saying that perspectivalism with respect to actuality explains how we know that we are actual. That knowledge is explained by pointing to the indexical component of the concept of actuality, an explanation available to realists and non-realists alike. I am saying that perspectivalism is the necessary consequence—the fallout, as it were—of taking on the Leibnizian’s fundamental a priori commitments: that possibilia exist and are plenitudinous, that the actual and the possible differ in ontological status, and that I know that I am actual. Perspectivalism with respect to actuality is not an independently supported view that is introduced to explain those commitments. I expect that you, dear reader, probably lack those commitments, and will happily turn my modus ponens into a modus tollens. And there is nothing, rationally, that I can say to change your mind. (But, still, I can know that you’re wrong.)⁵⁹

. Conclusion Consider a map located in some part of reality with a point highlighted to show the map’s location. For someone viewing the map, the point says: “you are here!” We don’t take the highlighting of the point to be part of the depiction of reality: a map located in some other part of reality with a different point highlighted may depict the same reality; a map with no point highlighted need not thereby be incomplete. Now consider a map where the highlighted point has a different interpretation: the highlighted point on the map depicts a point of reality with a special property; ⁵⁹ [The question naturally arises: to what extent is the perspectivalism I endorse ontological as opposed to epistemic. In Chapter , I give an entirely third-person description of reality, and characterized the problem of knowing that we are actual as a problem of epistemic access to a feature of reality that some parts of reality have and others lack. In this chapter, I say that any third-person description of reality is “inevitably incomplete,” and that our perspective on reality is “built into” reality itself. But I don’t think these alternative ways of talking are genuinely in conflict. For I hold that questions of epistemic access— more generally, of what we can know a priori—are objective, and that whatever is objective is somehow grounded in the nature of reality. The epistemic and the ontological are thus blended together, with no firm line between them.]

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

       ()

perhaps the point of reality is the center of its world and all objects revolve around it. In this case, the highlighting of the point is part of the depiction of reality: a map with some other point highlighted would be an incorrect depiction; a map with no point highlighted would be incomplete. Now combine these two sorts of maps. Consider a map whose highlighted point plays both roles simultaneously. It says to the viewer of the map: “you are here, at this ontologically special point.” The map’s correctness as a depiction of reality depends both on where the map is located and on which point is highlighted. A map that highlighted a different point would be an incorrect depiction of reality by misidentifying the ontologically special point of reality; a map that was located at another point of reality would be an incorrect depiction of reality by misidentifying the ontologically special perspective; a map with no point highlighted would be twice-over incomplete, first, for leaving out the ontologically special point of reality, and, second, for leaving out the ontologically special perspective. Perspectives, in addition to objects with their properties and relations, are needed to provide a complete account of reality. This third sort of map, of course, is analogous to the perspectival concepts that, I have argued, the Leibnizian should endorse, except that instead of a highlighted point, there is a highlighted region of reality: a region containing all and only the members of a fundamental, first-person ontological kind. That it should take perspectival maps, or perspectival concepts, to provide a complete description of reality may seem strange: how different it is from what we have been taught! But I think it is strange because it is unfamiliar, not because it is incoherent. If I am right that any philosopher who divides reality into two (or more) fundamental ontological kinds—actual/merely possible, individual/set, particular/ universal—and who claims to know which kind is ours must accept a modest perspectivalism, it is time to begin making the strange familiar. It will take some getting used to.

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 Isolation and Unification: The Realist Analysis of Possible Worlds () . Introduction Realism about possible worlds bears analytical fruit. The prize plum, perhaps, is the analysis of modality, in particular, the analysis of modal operators as quantifiers over possible worlds. But if the goal is the elimination, and not just the systematization, of primitive modality, it won’t do for the realist simply to “take possible worlds as primitive.” For, what sort of primitive is “possible world”? It wears its modality right on its face! Nor will it do for the realist to “take the worldmate relation as primitive,” the relation that holds between things inhabiting one and the same possible world. On a realist account, ‘possible world’ and ‘worldmate’ are trivially interdefinable. No, the realist must provide an analysis of possible world (and worldmate)—or forfeit the prize. The modern champion of realism about possible worlds, of course, is David Lewis; but a realist need not accept all of Lewis’s “modal realism.”¹ The core of realism about possible worlds, I think, is captured by the following five claims. () Worlds exist.² () Worlds are individuals rather than classes, or functions, or mathematical structures. () Worlds are particulars rather than properties or universals. () Worlds are “concrete” in this sense: they are fully determinate in all qualitative respects. () Worlds are (for the most part) mereologically complex rather than simple—for example, many worlds have parts that stand in spatiotemporal relations to one another.³ The mereological complexity of worlds suggests that worlds can, and should, be analyzed.

First published in Philosophical Studies  (): –. Reprinted with the permission of Springer. Portions of this chapter were presented in talks at Princeton University and at the University of Massachusetts, Amherst. Thanks especially to David Lewis for helpful comments. ¹ On Lewis’s brand of realism, there is no absolute actuality: the actual world and the merely possible worlds are ontologically all on a par. I find that implausible, perhaps even incoherent. I would argue, contra Lewis, that realism with absolute actuality is a viable alternative. [See Chapter .] The fullest presentation and defense of Lewis’s realism is in Lewis (a), passim. ² I use ‘world’ and ‘possible world’ interchangeably; for the realist (excepting a few deviants), there are no “impossible worlds.” I use ‘exist’ without restriction to cover everything “real,” with any sort of “being.” ³ I assume familiarity with mereology, the theory of part and whole. In particular, I assume unrestricted mereological composition: for any things whatsoever, there is a (mereological) sum, or fusion, of those things, the least inclusive thing that includes each of those things as a part. Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

      ()

How should the realist analysis of world proceed? Let reality be the sum of whatever exists. Let logical space be that portion of reality over which (alethic) modal operators range, in other words, the sum of possibilia. There are two tasks for the realist, two distinct problems of demarcation. First, there is the problem of demarcation for logical space: Where does logical space end and the rest of reality begin? What is the criterion for determining which parts of reality are parts of logical space, and so ingredients for making the worlds? (Of course, if logical space is the whole of reality—if, for example, there are no “abstract” mathematical entities, no transcendental universals—then this problem doesn’t arise.) Second, there is the problem of demarcation for the worlds themselves: Where does one world end and another begin? What is the criterion for determining when two (or more) parts of logical space are parts of one and the same world? I believe that both problems can be satisfactorily solved. In particular, criteria of demarcation can be found that do not rely on anything modal—a prerequisite for the elimination of primitive modality. In this chapter, I discuss only the demarcation of worlds.⁴ David Lewis has tackled the problem of demarcation for worlds.⁵ The basic strategy for a solution is this. Some regions of logical space are unified; the maximal such unified regions are the worlds. On this basic strategy, Lewis and I agree. But I disagree with Lewis over two substantial issues having to do with the manner of unification. First, for Lewis, all worlds are globally unified (or almost globally unified): at any world, (almost) every part is directly linked to (almost) every other part.⁶ I hold instead that some worlds are locally unified: at some worlds, parts are directly linked only to “neighboring” parts. I have discussed local unification elsewhere (with respect to spatial and spatiotemporal relations); here I cover the issue only in passing.⁷ Second, for Lewis, each world is spatiotemporally unified; every world is spatiotemporally isolated from every other.⁸ I hold instead: a world may be unified by non-spatiotemporal relations; every world is absolutely isolated from every other. If I am right with respect to (either or both) of these issues, then Lewis’s conception of logical space is impoverished: perfectly respectable worlds are missing. I will proceed as follows. First, in order to properly frame these disputes, I need to introduce some basic metaphysical machinery. Second, I develop notions of isolation and unification, and the analysis of world in terms of them, with sufficient generality to allow for locally unified worlds. Third, I give my argument against Lewis’s requirement that worlds be spatiotemporally unified. Fourth, I present an alternative analysis which allows for non-spatiotemporal unification of worlds, and I defend it in detail, first against an objection raised by Lewis, then against five further objections. ⁴ [See Chapter  for my current thinking on the problem of demarcation for logical space, what I there call “the quiddistic realm.”] ⁵ In the section entitled “Isolation” in Lewis (a: –). ⁶ This follows from Lewis’s claim that the unifying relations are “pervasive.” See Lewis (a: ). ⁷ See Chapter . I argue that, if Einsteinian relativity is true (on its most natural interpretation), then we live in a locally unified world. Such worlds had better be possible! ⁸ Actually, Lewis holds that worlds may also be unified by what he calls analogically spatiotemporal relations—relations appropriately analogous to the actual spatiotemporal relations. Since this complication won’t matter for what follows, I will simply use ‘spatiotemporal’ broadly so as to include what Lewis calls “analogically spatiotemporal.” See Lewis (a: –).

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   :  .  



Fifth and finally, I briefly discuss how the problem of demarcation for worlds is related to the problem of “island universes.”

. The Analysis of Worlds: Global vs. Local Unification To start, I need the distinction between the fundamental, or (perfectly) natural properties and relations, and the rest.⁹ The natural properties and relations are those that correspond to immanent universals or tropes, if there are universals or tropes. They make for qualitative similarity: if two things instantiate the same natural property, or each divides into parts that stand in the same natural relation, then the things are objectively similar in some qualitative respect. Moreover, the qualitative supervenes on the natural: fixing the natural properties and relations suffices to fix all the qualitative properties and relations. In terms of naturalness, a number of indispensable metaphysical notions can be defined. I will be brief. Things are (intrinsic, qualitative) duplicates just in case there is a similarity map from one to the other: a one-one correspondence between their parts that preserves all natural properties and relations (and the part-whole relation). An intrinsic nature is a property had by all and only the duplicates of some thing. An intrinsic property is one that never differs between duplicates; a property is extrinsic just in case it is not intrinsic. An internal relation is a relation that supervenes on the intrinsic natures of its relata. Having-the-same-mass-as is an example of an internal relation, assuming the mass properties are intrinsic.¹⁰ An external relation is one that, although it fails to supervene on the intrinsic natures of its relata, does supervene on the intrinsic natures of its relata, and of the fusion of its relata.¹¹ Being-adjacent-to is an example of an external relation: whether two things are adjacent to one another is not determined by their intrinsic natures, taken separately, but it is determined if one also takes into account the intrinsic nature of their fusion. A relation that is either internal or external is intrinsic; all others are extrinsic. Note that it is built into the definitions that all natural properties and relations are intrinsic. Now we are ready to begin our task proper: to define isolation, and in terms of isolation to analyze the notion of world. Isolation, I take it, is to be characterized in terms of the holding or failing to hold of certain natural external relations. But which relations? Different choices yield different notions of isolation. For the sake of generality, I first present the definitions in the form of a schema. Let ℱ be a family of natural external relations. Two parts of logical space are ℱ-isolated if and only if they are non-overlapping, and no part of one stands in any relation from ℱ to any ⁹ In this paragraph and the next, I more or less follow Lewis (a: –). ¹⁰ [I no longer hold that determinate quantities are intrinsic; see Chapter .] ¹¹ More precisely, say that a(n) (ordered) pair and a(n) (ordered) pair are internal duplicates iff a is a duplicate of c and b is a duplicate of d; external duplicates iff, in addition, the composite of any similarity map from a to c and any similarity map from b to d induces a similarity map from the fusion of a and b to the fusion of c and d. Then, an internal (dyadic) relation is one, the holding of which never differs between pairs that are internal duplicates; an external (dyadic) relation is one that is not internal, but the holding of which never differs between pairs that are external duplicates. (Analogously for relations of three or more places.)

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

      ()

part of the other. A part of logical space is ℱ-connected (or ℱ-unified) if and only if it is not the sum of two ℱ-isolated parts; otherwise it is ℱ-disconnected (or ℱ-disunified). Two parts of logical space are ℱ-related if and only if some ℱ-connected part of logical space includes them both. Note that parts of logical space can be ℱ-related even though they are ℱ-isolated, and so not related by any member of ℱ. (An illustration follows shortly.)¹² Worlds can now be analyzed schematically in terms of ℱ-connectedness: a different analysis results for each choice of ℱ. A world is any maximal ℱ-connected part of logical space; that is, an ℱ-connected part not properly included in any other ℱ-connected part. It follows from the analysis that any two worlds are ℱ-isolated from one another, and, in particular, that no two worlds overlap.¹³ Finally, parts of worlds are worldmates if and only if they are part of the same world, if and only if they are ℱ-related. Since worlds do not overlap, the worldmate relation is an equivalence relation over parts of worlds. Worlds may be unified to a greater or lesser degree. At one end of the spectrum, we have globally ℱ-unified worlds at which no part is ℱ-isolated from any other part. At a globally ℱ-unified world, points of spacetime (if such exist and are mereologically atomic) are directly linked to one another by some natural relation in ℱ, presumably, by some external relation of spatiotemporal distance (interval). At the other end of the spectrum, we have locally ℱ-unified worlds at which the only parts that are not ℱ-isolated are overlapping or adjacent parts.¹⁴ (The ℱ-isolated parts are nonetheless ℱ-related in virtue of belonging to a single ℱ-connected region of logical space.) At a locally ℱ-unified world (with continuous spacetime), distinct points of spacetime are ℱ-isolated (being non-adjacent), and so are not directly linked by any natural relation in ℱ; relations of spatiotemporal distance are extrinsic, rather than external, because the distance between points depends on the intervening spacetime, on the lengths of paths from one point to the other.¹⁵ Lewis does not allow for locally unified worlds. Let me recast his account within my framework for purposes of comparison. For Lewis, worlds are maximal ℱinterrelated regions of logical space, where ℱ-interrelatedness is defined narrowly as follows: a part of logical space is ℱ-interrelated if and only if every part stands in ¹² [In Chapter , I say “ℱ-separated” where I here say “ℱ-isolated.” I then say that parts of logical space are ℱ-isolated iff they are not ℱ-related. That terminology is better because when dealing with ℱ-locally unified worlds (defined below), the informal notion of isolation corresponds with “not being ℱ-related,” rather than with what I here call “ℱ-isolated.”] ¹³ Proof. Let W and V be two worlds, two maximal ℱ-connected parts of logical space; and suppose, for a reductio, that W is not ℱ-isolated from V. Claim: W + V (the sum of W and V) is ℱ-connected, violating the maximality of W or of V. For consider any Z and Y such that Z + Y = W + V. There are two cases (by mereology). () W and/or V overlaps both Z and Y; then Z is not ℱ-isolated from Y, owing to the ℱ-connectedness of W and/or V. () W = Z and V = Y, or W = Y and V = Z; then Z is not ℱ-isolated from Y since, by assumption, W is not ℱ-isolated from V. Therefore, in all cases, Z is not ℱ-isolated from Y, and W + V is not the sum of two ℱ-isolated parts; that is, W + V is ℱ-connected, as was to be shown. ¹⁴ Topologically speaking, two regions are adjacent iff they are non-overlapping, but one contains a boundary point of the other. (For example, on the real line, the open interval (, ) is adjacent to the closed interval [, ], but not to the open interval (, ).) Only worlds with topological structure can be locally unified. ¹⁵ In Chapter , I argue that distance relations are extrinsic, rather than external, at (some) worlds with continuous spacetime.

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       



some relation from ℱ to every other (non-overlapping) part.¹⁶ For Lewis, then, interrelatedness is the only kind of unification for worlds: all worlds are globally unified. That worlds are ℱ-isolated from one another, and never overlap, cannot be proven on Lewis’s account; it is a substantial extra postulate.

. Lewis’s Analysis: Spatiotemporal Isolation What should we take ℱ to be in the analysis-schema for world? According to Lewis, ℱ is the family of natural spatiotemporal relations: worlds are maximal spatiotemporally unified regions of logical space; all and only worldmates are spatiotemporally related.¹⁷ But this proposal, I think, is open to a decisive objection. Physicists have often speculated, in trying to make sense of quantum mechanical mysteries such as wave-particle duality, that spacetime is not physically fundamental, that the spatiotemporal relations holding at the “macroscopic” level are reducible to more fundamental properties and relations holding only at the “microscopic” level, in the way that, say, relations of chemical bonding are reducible to more fundamental physical properties and relations. Moreover, it may be that none of the fundamental, (perfectly) natural relations are even structurally analogous to the spatiotemporal relations. Suppose this speculation is true. Then, had the fundamental physical laws or the “initial” conditions been otherwise than they are, there might have been no spacetime at all, just as had the physical laws or “initial” conditions been different, atoms might never have clumped into molecules. If logical space is to make room for these possibilities, there must be worlds that are unified by non-spatiotemporal relations. Indeed, if actual spatiotemporal relations hold only at the “macroscopic” level, then not even the actual world is spatiotemporally unified: the “microscopic” parts of actuality stand in no spatiotemporal relations to anything, just as quarks stand in no relations of chemical bonding. Lewis’s analysis, by requiring that worlds be spatiotemporally unified, in effect rules out the physicist’s speculation a priori. That’s not right. Any possibility for actuality must find a place in logical space. Lewis’s conception of logical space, then, is too narrow.¹⁸

. Absolute Isolation and the Objection from Relational Charge In the analysis of world, the family ℱ must contain natural external relations that are not spatiotemporal. But which ones? It would be arbitrary, I think, to include some while excluding others. I propose, then, that we take ℱ to contain all natural external relations: worlds are maximal externally unified regions of logical space; all and only ¹⁶ See Lewis (a: ). On p. , Lewis allows that some exceptional parts of an ℱ-interrelated world may be indirectly linked by a chain of relations from ℱ, rather than directly linked by a single relation from ℱ. Worlds, then, may be almost globally ℱ-unified, but still not locally ℱ-unified. ¹⁷ Lewis (a: ). I henceforth speak only of unification; the difference between interrelatedness and unification plays no role in what follows. ¹⁸ For an early discussion of the possibility that space and time are not fundamental, and further references, see Smart (: –).

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

      ()

worldmates are externally related; every world is externally—that is, absolutely— isolated from every other. Lewis considers this proposal, but then rejects it (tentatively) for a reason I find unconvincing. In this section, I present and respond to Lewis’s objection. In the next section, I present and respond, more briefly, to five other objections that naturally come to mind. Lewis bases his objection on the case of relational charge. Suppose, as can be argued on physical grounds, that the charge properties, such as having-unit-positivecharge, and having-unit-negative-charge, are not intrinsic.¹⁹ Suppose instead that there are natural external relations of like-chargedness and opposite-chargedness (and, perhaps, of all ratios of chargedness); all the facts about charge supervene on the holding or failing to hold of these relations. If this is the case, a charged particle and its anti-matter twin—say, an electron and a positron—may be intrinsically exactly alike; they differ relationally, however, in that whenever one is like-charged a given particle, the other is opposite-charged that same particle. Now, the case of relational charge, Lewis thinks, makes trouble for the absolute isolation of worlds. He writes: “Could two particles in different worlds stand in these external relations of like- and opposite-chargedness? So it seems, offhand; and if so, then the [proposal] fails.”²⁰ But this offhand judgment can, and should, be resisted. It is a holdover, I suspect, from the (more customary) view that charge is intrinsic. On that view, of course, transworld comparisons of charge are always meaningful. But on the view that charge is relational, it is unnecessary and gratuitous to suppose that transworld comparisons of charge are always meaningful, that the fundamental charge relations ever link world to world. The actual distribution of charge, and the laws it obeys, are determined by intraworld relations of charge at the actual world; the possible distributions of charge, and the possible laws, are determined by intraworld relations at other possible worlds. Wherefore the supposition of transworld relations of like- and opposite-chargedness? Might transworld relations of charge be needed to individuate possibilities, to avoid conflating possibilities that, intuitively, are distinct? Consider these possibilities. It is possible that the world be just as it is except for the addition of a single electron; or, it is possible that the world be just as it is except for the addition of a single positron. Surely, these possibilities are distinct. But, the objector asks, on the relational account of charge, what could distinguish them other than the fact that, according to the first possibility, there is an extra particle like-charged actual electrons, whereas, according to the second possibility, there is an extra particle oppositecharged actual electrons? The possibilities are distinct, all right, but no transworld external relations of charge are needed to distinguish them. At worlds where the possibilities are realized, there are particles that are counterparts to our actual electrons, particles that play the same role vis-à-vis their world as the electrons play vis-à-vis ours. This determination of counterparts rests on global comparisons of similarity; it requires only the holding ¹⁹ Symmetries at the actual world suggest that charge and handedness are coordinate: either both or neither are intrinsic. But handedness is not intrinsic. For example, a right-handed glove can be superimposed on a left-handed glove by taking a trip through higher-dimensional space. ²⁰ Lewis (a: –). Lewis says no more: the amplifications considered below are not his.

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   



of internal, not external, relations between worlds. The two possibilities can then be distinguished as follows: one is realized at worlds containing some particle that is not a counterpart of any actual particle, and that is like-charged its worldmates that are counterparts of actual electrons; the other is realized at worlds containing some particle that is not a counterpart of any actual particle, and is opposite-charged its worldmates that are counterparts of actual electrons. Only intraworld relations of like- and opposite-chargedness ever come into it. But, the objector persists, consider these possibilities. It is possible that nothing exists but a single electron; or, it is possible that nothing exists but a single positron. If these possibilities are distinct, the objector continues, they can only be distinguished by transworld external relations of charge. But the relational account of charge, as I understand it, can and should deny that these possibilities are distinct. After all, a world realizing the first possibility and a world realizing the second are exact qualitative duplicates. What independent grounds could there be for distinguishing them? I can think of one other way of arguing for transworld external charge relations. Consider the following plausible principle: if some natural relation, or family of natural relations, has some general structural feature at every world, then it has that feature simpliciter. For example, if some natural relation is necessarily symmetric, is always symmetric between worldmates, and if (contrary to my view) it holds also between non-worldmates, then it is always symmetric between non-worldmates. For, if the necessary symmetry does not (somehow) come from the nature of the relation, whence does it come? To deny the principle would be to impose arbitrary necessities not grounded in the natures of things. Now, the relevant structural feature of the family of external charge relations is this. At any world, the charge relations are universal over their field of application: if a stands in some charge relation to something, and b stands in some charge relation to something, then a stands in some charge relation to b. But then, the argument goes, taking a and b to be charged particles in different worlds with relational charge, the principle demands that there be some transworld external charge relation between a and b. I do not dispute the principle, but I reject the argument. I deny that, at every world, the family of external charge relations is universal over its field of application. Perhaps that holds at the actual world. But then it holds contingently, as a matter of physical law. I accept a principle of recombination for relations: any natural relation, or family of natural relations, can be instantiated in any pattern whatsoever.²¹ Thus, at some worlds, two things stand in charge relations to other things, but not to one another. The argument never gets off the ground.

. Five Objections and Replies I know of no other argument that supports transworld external relations of charge. They can be rejected, I think, with impunity. But there are other objections to the

²¹ Roughly speaking. For some discussion and supporting argument, see Armstrong (a: –). [Today, I would be careful to restrict recombination to properties and relations with quiddities; and on my preferred ontology, only properties, not relations, have quiddities. See Chapter .]

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

      ()

view that worlds are absolutely isolated. I will briefly consider five. The first three are easily dealt with; the last two are potentially more serious. Objection. According to realists, we are related to other worlds by our thoughts, for example, whenever we contemplate what might have been. Then, with us as intermediaries, these other worlds are linked by a unifying chain. They are not absolutely isolated. Reply. Our thoughts relate us to other worlds, all right, but not by way of any natural external relation. Rather, our mental representations, which are part of the actual world, stand in relations of similarity to the worlds they represent, and these similarity relations, being internal, are no threat to absolute [isolation]. In short, we are related to other worlds, not by any sort of acquaintance, but only by description. Objection. There is no such thing as absolute isolation: any two things stand in the external relation of non-identity. Every plurality, then, is unified by external relations.²² Reply. Indeed, non-identity comes out external according to the definition.²³ But non-identity is not a natural relation; it does not make for qualitative similarity.²⁴ If it did, then every composite individual would be qualitatively similar to every other composite individual in virtue of being composed of non-identical parts, which is absurd. Moreover, for any natural relation, there are worlds at which distinct things fail to stand in that relation, which is absurd when applied to non-identity. Finally, for any natural property or relation, the negation of that property or relation is not natural: its instances are too miscellaneous to make for qualitative similarity. But identity and non-identity would have an equal claim on being natural, which disqualifies them both. Identity and non-identity are properly classified as logical relations; they are not qualitative, much less (perfectly) natural.²⁵ Objection. There is no such thing as absolute isolation if mereological composition is unrestricted. For then parts of different worlds are always unified by the transworld fusions that include them. Reply. This objection can be handled along the same lines as the preceding. The relation of part to whole, though external, is not natural. Mereological relations, no less than identity and non-identity, are properly classified as logical. As such, they have no power to unify. In particular, a transworld fusion cannot serve as a unifying

²² Lewis mentions this objection, but does not endorse it. See Lewis (a: ). ²³ Let a and b be two duplicates. Then, the ordered pairs and are internal duplicates, but only the latter is a non-identity pair. So, non-identity is not internal. Let the ordered pairs and be external duplicates. Then there is a one-one correspondence taking c to e and d to f, which is impossible unless both pairs, or neither, are non-identity pairs. So, non-identity is external. ²⁴ Warning: there is a broader sense of ‘natural’ afoot that is not tied to qualitative character; it applies also to “fundamental” logical (or mathematical) properties and relations. The broader notion is needed to help resolve indeterminacy of the content of thought. See Lewis (b: –). [See Chapter , where I distinguish between notions that are fundamental to our thought (ideologically basic) and those that are fundamental to reality (ontologically basic); and I distinguish three kinds of property and relations that are fundamental to reality: logical, mathematical, and quiddistic.] ²⁵ The failure to distinguish between qualitative relations, which can unify their instances, and nonqualitative relations like non-identity, which cannot, appears to be behind the idealist doctrine that there is unity in every plurality. See, for example, Bradley (: –) on the unity of the Absolute.

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   



intermediate link between its worldbound parts: any such link would have to be wholly distinct from the parts that it unifies, which the fusion is not.²⁶ Objection. There can be no absolute isolation if class formation is unrestricted. For then parts of different worlds are always unified by the transworld classes of which they are members. Reply. I grant that classes, as ordinarily conceived, make trouble for the absolute isolation of worlds. Classes are wholly distinct from their members. Transworld classes, then, unlike transworld sums, seem able to serve as intermediate links in a unifying chain. Can we say that the membership relation, though external, is not natural? Indeed, membership imposes necessary connections that violate principles of recombination for natural relations. But if membership is not natural, what is it? Unlike part-whole, it does not seem to be properly classified as logical. There doesn’t seem to be anything coherent for it to be! I see two options. The more radical option is to reject classes outright. This option carries with it the substantial burden of showing how essential uses of classes—for example, in semantics—can be accomplished by other means.²⁷ A more conservative option is to invoke Lewis’s mereological theory of classes: a class is the fusion of the singletons of its members; the singleton relation, not membership, is (perfectly) natural.²⁸ On this option, the objection involving transworld classes can be assimilated to the objection involving transworld sums. Parts of different worlds are, indeed, linked to their singletons; no problem there, the singletons are confined each to their own world. Are the singletons linked to one another? No; the singletons are no more linked to one another by way of the transworld set that includes them, then the parts are linked to one another by way of their transworld sum. There is no unifying chain from world to world. But there is a catch. For this reply to work, the transworld sets cannot themselves have singletons. I don’t think such singletons will be much missed. Within each world, the entire set-theoretic hierarchy can be constructed. Objection. Universals unify their instances. If the instantiation relation is the relation of whole to part, then universals unify by overlap. If the instantiation relation is some non-mereological external relation, then universals are an intermediate link in a unifying chain. Either way, parts of different worlds are not absolutely isolated when they instantiate the same universal. Reply. I grant that universals make trouble for the absolute isolation of worlds. But, I would argue, universals are to be rejected on independent grounds. I lean, instead, towards a theory of tropes. When parts of different worlds instantiate the same natural property, each world has as a part its own particular trope. These tropes are duplicates of one another; they are internally related. There is no threat to absolute isolation.

²⁶ [For more on this deflationary notion of composition, see Chapter .] ²⁷ What about uses of classes in mathematics? As I see it, the “pure” sets needed for mathematics are not as problematic as the “impure” classes, classes with parts of worlds as their members (or members’ members, etc.). Necessary connections are at home in the realm of mathematics. Perhaps only the “impure” classes need to go. [For discussion, see Chapters  and .] ²⁸ See Lewis ().

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

      ()

. Conclusion The view that worlds may be unified by non-spatiotemporal relations, I have argued, is needed to allow for the possibility that spacetime isn’t fundamental. There appears to be a second advantage to the view. Consider the following question. Is it possible that physical reality divides into two or more spatiotemporally isolated parts, into socalled “island universes”? I join a chorus of others in answering “yes.”²⁹ Perhaps we could never have good reason to believe we inhabited one such island among many; but it is possible nonetheless. Lewis’s criterion of demarcation, however, leads him to reject the possibility: all worlds are spatiotemporally unified; so no world divides into spatiotemporally isolated parts; assuming the standard analysis of possibility as truth at some world, it follows that spatiotemporally isolated island universes are impossible. The criterion of demarcation I am defending, on the other hand, allows the possibility to be easily accommodated. If worlds may be unified by nonspatiotemporal relations, there is nothing to exclude a world that divides into parts, each spatiotemporally unified, but each spatiotemporally isolated from the others, a world with island universes. This advantage for my view, however, is more apparent then real. Arguments that support the possibility of spatiotemporally isolated parts often support with equal force the possibility of absolutely isolated parts. The view that worlds are externally unified does no better at accommodating the latter possibility than the view that worlds are spatiotemporally unified does at accommodating the former. The problem of absolutely isolated island universes is still with us. Should we, then, seek some further broadening of the criterion of demarcation for worlds, one that gives up on the idea that worlds are, in any sense, unified? I think not. The best solution to the problem of island universes lies elsewhere.³⁰

²⁹ I give my arguments in Chapter . See also Armstrong (a). ³⁰ In Chapter , I argue that the problem of island universes, and a number of others, can best be solved by emending the standard analysis of modality: modal operators are plural, rather than individual, quantifiers over possible worlds; to be possible is to be true at some world, or some worlds.

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 Reducing Possible Worlds to Language () . Introduction Can possible worlds be reduced to language? That is, can talk about possible worlds be reconstrued as talk about respectable linguistic entities, thus allaying ontological worries as to what strange manner of entity a possible world might be? I will argue in the final section of this chapter that no such reduction can succeed. But if my argument is to be conclusive, it will have to be directed against the strongest possible case for reduction, not against the proposals most commonly heard. The commonly heard proposals succumb to a simple cardinality argument: on quite modest assumptions, it can be shown that there are more possible worlds than there are linguistic entities provided by the proposal; it follows straightway that the linguistic entities cannot be the possible worlds. One might be tempted to think that some version of the cardinality argument could be used quite generally to show that any attempt at reducing possible worlds to language must fail. This, however, is not the case. In this chapter I will show how the standard proposals for reduction can be generalized in a natural way so as to make better use of the resources available to them, and thereby circumvent the cardinality argument. Once we see just what the limitations are on these more general proposals for reduction, we will be able to see more clearly where the real difficulty lies with any attempt to reduce possible worlds to language. Roughly, the difficulty is this: no actual language could have the descriptive resources needed to represent all the ways things might have been. I will conclude by arguing that this same difficulty spells doom for any nominalist or conceptualist proposal for reducing possible worlds. In order to set the stage for what will follow, let me briefly recapitulate some of the ontological positions that might be held with respect to possible worlds. The realist with respect to possible worlds takes possible worlds to be primitive to her ontology.¹ She means her talk about possible worlds to be taken literally, not as disguised talk about some other kind of entity. Non-realists can take either of two stances towards

First published in Philosophical Studies  (): –. Reprinted with the permission of Springer. It was taken from my doctoral dissertation, Bricker (: –). Versions were presented at Princeton University in  and Rutgers University in . David Lewis made helpful comments at various stages of its development. ¹ [Today I would not say that a realist about possible worlds must “take possible worlds to be primitive.” See, for example, Chapter  for the realist analysis of possible world that I accept.] Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

     ()

possible worlds: they can simply reject talk of possible worlds outright, or they can try to reinterpret talk of possible worlds in a way they find ontologically acceptable. The first sort of non-realist, if consistent, must also reject the philosophical analyses that have been based on the notion of a possible world, or, at any rate, take them to have at most heuristic value and to lack any serious philosophical import. On the other hand, the reductionist with respect to possible worlds often finds such analyses to be fruitful, illuminating, and philosophically important. The problem is not that she finds talk of possible worlds meaningless or incomprehensible, but that her philosophical conscience will not permit her to take such talk at face value. Possible worlds are nowhere to be found in her ontology. In this chapter, I will be concerned only with ontological reduction: an ontological reduction of possible worlds leaves talk of possible worlds syntactically unchanged, but reinterprets the values of the variables by identifying possible worlds with entities that the reductionist accepts.² Moreover, I will assume that the reductionist is an actualist and a nominalist (of sorts): her ontology consists only of actual, concrete entities together with what can be constructed out of such entities by means of set theory.³ Some philosophical consciences, it is true, have even balked at an ontology of sets; but, as will become clear, any attempt at reduction would be crippled at the outset if the reductionist did not at least have set theory at her disposal. Now, what the reductionist must do if she is to succeed in appeasing her philosophical conscience is to show that all talk about possible worlds that she wants to preserve can be interpreted as disguised talk about some kind of entity that she does accept. It is natural at this point for the reductionist to turn to language. For among the entities that she does accept are linguistic entities: finite sequences of types of concrete marks or sounds (where sequences and types are given their standard interpretations in terms of sets), and set-theoretic constructions out of such sequences. Moreover, we do commonly gain access to the notion of a possible world through language, through what purport to be descriptions of possible worlds. So the reductionist might hope that talk about possible worlds could be made respectable by identifying possible worlds with their purported linguistic descriptions, or, at any rate, constructions out of such descriptions.⁴ To what language shall we assume that these descriptions belong? If we allow the reductionist to take the notion of a language too broadly, the resulting reduction is likely to be circular. For example, if she can merely stipulate that the language contains a name for every possible world, then the existence of the language (with this interpretation) is at least as dubious as the existence of the possible worlds themselves. The surest safeguard against such circularity is to require that possible ² I am skeptical that other types of reduction—for instance, a translation of talk of possible worlds into a language containing modal operators and higher-order quantifiers—can succeed in eliminating an ontological commitment to possible worlds; but I will not attempt to argue this here. ³ This is the ontology accepted by Quine throughout most of his career. See, for example Quine (a: –). Possible exceptions are an early paper with Goodman (Goodman and Quine ) in which he rejected the abstract entities; and a late paper (Quine ) in which he seems to reject the concrete ones. [This admittedly odd use of the term ‘nominalist’ occurs only in this chapter.] ⁴ Of course, when I speak of language, it will always be an interpreted language that I have in mind; uninterpreted linguistic entities do not purport to describe possible worlds, or anything else.

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    



worlds be reduced to a natural language, perhaps enriched by the languages of science, mathematics, and formal logic. Unfortunately, many sentences of a natural language are unsuitable for use in constructing the possible worlds. I will thus assume that the reductionist has extracted from the enriched natural language a sublanguage satisfying all of the following conditions: () () () ()

All sentences are declarative sentences. Truth values of sentences are independent of contexts of inscription or utterance. All sentences are unambiguous. There is no vagueness in the truth conditions for sentences, let alone indeterminacy of a more radical sort. () Sentences can be uniquely parsed so as to exhibit their truth-functional and quantificational form. Any sublanguage of a natural language satisfying these conditions will be called a reasonable language—reasonable in the sense that it is appropriate for the project of reducing the possible worlds.⁵ One characteristic of the original natural language will be shared by all of its sublanguages: the expressions of the language are finite sequences over a finite alphabet, and thus the language has at most a countably infinite number of distinct expressions. One of the chief contentions of this chapter, however, is that this limitation is not as severe as might first appear.

. Three Necessary Conditions for Reduction What conditions must be met by a successful reduction of possible worlds to language? Let us suppose that the reductionist has fixed on a particular theory of possible worlds, that is, the set of those sentences about possible worlds whose truth she wishes to preserve. Let us call this the possible worlds theory. Just which statements about possible worlds will be included in this theory is largely up to the reductionist. But I assume that at the very least she will want the theory to be strong enough to support the standard analysis of the alethic modalities as quantifiers over possible worlds. Of course, the reductionist cannot hope to preserve the truth of everything that a realist would assert about possible worlds. For example, a realist would assert that possible worlds are not linguistic entities, whereas this cannot be preserved by any reduction of possible worlds to language. Similarly, other statements directly or indirectly about the ontological status of possible worlds must remain a source of disagreement between the realist and the reductionist. What the reductionist must do is to fix on a theory that is strong enough to support the possible worlds analyses that she wants to accept, but not so strong that the very possibility of reduction is excluded by the theory itself. Now, a minimal condition that the reduction must satisfy is that it provide a translation of sentences about possible worlds into sentences about linguistic entities that maps truths of the possible worlds theory into truths, and falsehoods into falsehoods. ⁵ Some would say that no natural language has a significant sublanguage satisfying all of the above conditions, even if the natural language is enriched by the languages of science, mathematics, and formal logic. This claim is certainly controversial; but there is no space to discuss it here.

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

     ()

But the sort of reductionist here considered not only wants the translation to preserve the truth of the possible worlds theory, she wants it to preserve the structure of the theory as well. For example, sentences that make existential claims about possible worlds are to be translated into sentences that make existential claims about linguistic entities. The reduction is not intended to provide a restructuring of logical form, only a switch of the underlying ontology. If a reduction provides a truth- and structure-preserving translation of the possible worlds theory, then I will call the translation faithful, and say that the reduction satisfies the faithfulness condition. There is a routine method for providing such a faithful translation, at least if we assume that the possible worlds theory is couched within an extensional language. To each possible world one assigns a linguistic entity to serve as that world’s representative—in effect, to be the possible world. Distinct possible worlds are assigned distinct linguistic entities if they are discernible⁶, but in general the assignment need not be one-one. Any such assignment uniquely determines a faithful translation of the possible worlds theory into a theory of the linguistic entities. For example, given any unary predicate expressing some property of possible worlds, a corresponding predicate expressing some property of the linguistic entities is introduced as follows: the new predicate is true of just those linguistic entities that were assigned to some possible world of which the old predicate was true. More generally, the entire possible worlds ideology (to use Quine’s term), as represented by the predicates and function symbols of the language of the possible worlds theory, is transferred by way of the assignment into a corresponding ideology of the linguistic entities, as represented by corresponding new predicates and function symbols. Every sentence about possible worlds can then be translated into a sentence about linguistic entities simply by replacing old predicates and function symbols by corresponding new ones. The translation clearly preserves logical structure. Moreover, by wending one’s way up Tarski’s inductive definition of truth, it can easily be shown that the translation is truth-preserving as well. So, any such assignment of linguistic entities to possible worlds can be used to ensure that the faithfulness condition be satisfied. But this whole procedure has an air of circularity about it. The reductionist wishes to show that she need not admit a primitive notion of possible world to her ontology. According to the above procedure, she does this by showing that, for each possible world, a corresponding linguistic entity can be found. But if she thus invokes the possible worlds in selecting the linguistic entities, how can she claim to have eliminated the possible worlds? I do not think that she can unless the reduction meets the following non-circularity condition, which then shows that the circular way of describing the reduction can be avoided: the possible worlds, and all the possible worlds ideology, must be constructed out of the actualist ontology and ideology that is

⁶ Here and throughout, what ‘discernible’ means is relative to the possible worlds theory chosen by the reductionist. Strictly speaking, it is relative discernibility with respect to the predicates of the possible worlds theory (excluding identity). But if we assume that the possible worlds theory has a binary predicate ‘is discernible from’, then relative discernibility with respect to the predicates of the theory will correspond to a notion of metaphysical discernibility (although, as we shall see, it will not correspond to a notion of qualitative discernibility if the theory is haecceitist). On the distinction between relative and absolute discernibility, see Quine (a: ).

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 



already available to the reductionist. In other words, all the primitive predicates and function symbols of the possible worlds theory must be translated by predicates and function symbols that are definable within the reductionist’s theory of the language in question. Thus, the reduction will be circular if the possible worlds are eliminated by stepping up the ideology of the reductionist’s theory of language. As we shall see, the non-circularity condition is difficult to apply in practice if it is unclear just what the reductionist should be allowed to include within her ideology. My general policy will be to allow the reductionist whatever property of linguistic entities she thinks she can make sense of; she will need all the help she can get.⁷ There is a further condition that, I believe, must be satisfied by any successful reduction of possible worlds to linguistic entities. As yet I have said nothing to require that the linguistic entities correspond in a natural way with the possible worlds that they are to replace. It might be that the entire syntactic structure of the possible worlds theory was duplicated, sentence for sentence, by the syntactic structure of some wholly unrelated actualist theory of linguistic entities. But the existence of such a duplication of structure cannot by itself provide grounds for concluding that the linguistic entities are fit to play the role of the possible worlds. We need to require that the linguistic entities be naturally linked to the possible worlds that they are to replace. Now, the obvious place to look for such a natural correlation is to the semantics of the language: the linguistic entities represent possible worlds in virtue of what they mean. More exactly, I will say that the reduction satisfies the naturalness condition as long as each possible world is replaced by a linguistic construction that can serve as a complete description of that possible world, where a linguistic construction completely describes a possible world if it is true at that world, and perhaps at worlds indiscernible from that world, but at no others. Again I have taken the realist’s perspective and spoken as if there were possible worlds existing over and above the linguistic entities. Only the realist can speak of a correlation between possible worlds and linguistic entities as being more or less natural. But I think that the naturalness condition can be restated in a way that the reductionist can accept. For although it makes no sense to the reductionist to speak of a linguistic construction as providing a complete description of a possible world, it does make sense to speak of a linguistic construction as purporting to provide such a description, even if she doesn’t believe that anything exists for the description to describe. So the reductionist too can recognize which linguistic entities are naturally suited to play the role of the possible worlds, although, as we shall see, modal notions may be needed for this purpose.

. First Proposal With these three conditions in hand—faithfulness, non-circularity, and naturalness—let us turn to the evaluation of specific proposals for reducing possible worlds to language. Suppose that the reductionist has fixed on some particular ⁷ The two conditions required thus far are in rough agreement with the conditions given by Quine (a: –).

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

     ()

language L satisfying the five requirements listed in the introduction to this chapter. If two worlds are such that there is a sentence of L true at one of the worlds but not the other, then I will say that the worlds are linguistically discernible with respect to L, or, L-discernible, for short.⁸ The first proposal to be considered is based on the following idea: a class of possible worlds (in particular, the class of all possible worlds) is reducible to L as long as any two discernible worlds from that class are also L-discernible. For if discernible worlds are always L-discernible, then the reductionist can succeed in assigning distinct linguistic entities to discernible worlds by assigning to each possible world the set of those sentences of L that are true at the world. Note that the reductionist is not required to assign distinct linguistic entities to indiscernible possible worlds (if any there be); I assume that the principle of the identity of indiscernible worlds is compatible with the possible worlds theory. Which sets of sentences of L will be identified with possible worlds under this proposal? Such a set of sentences must be consistent, that is, there must be a possible world at which all of the sentences of the set are true. Moreover, let us assume that it follows from the possible worlds theory that possible worlds are fully determinate, and thus that for any sentence of L and for any possible world, either that sentence or its negation is true at that world. Then the set of sentences true at a world will be a maximal consistent set, containing for any sentence of L either that sentence or its negation. Thus the proposal for reduction that we are considering can be formulated as follows: Proposal 1.

Possible worlds are maximal consistent sets of sentences of L.

Note that the notion of consistency used in Proposal  is a modal notion; it cannot be taken to be narrowly logical consistency, where this is defined, for example, as truth under some reinterpretation of the non-logical vocabulary, lest there turn out to be possible worlds in which bachelors are married. Proposal  is the proposal made by Richard Jeffrey (: –), who called such maximal consistent sets of sentences novels. It has been defended by Frank Jackson (: ), and, more extensively, by Andrew Roper ().⁹ It is related to Carnap’s well-known proposal to identify possible worlds with state descriptions, where a state description is defined as a set of sentences (of some given language) which contains for every atomic sentence either that sentence or its negation, but not both, and no other sentences.¹⁰ One of the ways in which Carnap’s proposal differs from Proposal  is that it places additional restrictions on the language in question.¹¹ Thus, Carnap must assume that the language contains names (or there would be no atomic sentences at all) and that distinct names denote distinct individuals. More significantly, Carnap must assume that the primitive predicates of the language have been chosen (and so can be chosen) in such a way as to guarantee that all the state ⁸ Note that, to use Quine’s terminology, L-discernibility is, in effect, absolute discernibility with respect to the predicates of L. ⁹ Roper’s paper came to my attention after this chapter was completed. Roper has noticed (p. ) that Proposal  can be generalized so as to meet the cardinality objection, although he provides few details. He says nothing to defend the proposal against the objection from descriptive impoverishment. Perhaps he is not requiring that the possible worlds be reduced to what I call a reasonable language. That would help to explain his puzzling conflation of reductions to sentences with reductions to propositions. See n.  below. ¹⁰ In Carnap (: ). ¹¹ Cf. the discussion in Carnap (: –).

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  



descriptions are consistent (in the modal sense). This second assumption, which is quite strong, can be avoided by changing the proposal to read that possible worlds are to be identified, not with state descriptions, but with maximal consistent sets of basic sentences, where a basic sentence is either an atomic sentence or the negation of an atomic sentence. Then the proposal can be compared with Proposal  by assuming that the language in question is a reasonable language, and that it contains a name for every individual that can be uniquely picked out by some formula of the language. But this revised Carnapian proposal is still not equivalent to Proposal . In the “possible worlds” it provides, only individuals that can be given names exist. Thus the proposal cannot account for the fact that other individuals might have existed, and it thereby misrepresents the facts of modality. In what follows I will say no more about Carnap’s proposal, and focus on the more plausible Proposal  and its generalizations.¹²

. The Circularity Objection It might seem that Proposal  violates the non-circularity condition because it makes use of the notion of a consistent set of sentences, and consistency has simply been characterized in terms of truth at a possible world. The reductionist, if she is to succeed, must be able to define consistency in a way that makes reference only to entities and notions that she already accepts. Here, I think, the reductionist must turn to empirical linguistics. Recall that L is assumed to be a sublanguage of an actual language used by actual people. For the reductionist, whatever facts there are about the consistency of sets of sentences of L will be grounded in actual usage. Now, empirical data of actual usage is available to the reductionist in constructing the possible worlds. Contained within the data will be direct or indirect reports of modal intuitions that the reductionist can cash out in terms of the consistency of sets of sentences. Indeed, whenever the realist uses her modal intuitions to decide the truth or falsity of some sentence about possible worlds, the reductionist can use reports of those modal intuitions in constructing a notion of possible world that preserves the truth value of the sentence. Presumably, there will not be enough data to answer all questions about the consistency of sets of sentences. But all that matters for the reduction is that there be enough data from which a notion of possible world can be constructed that provides a faithful reinterpretation of the possible worlds theory. Of course, where modal intuitions give out and are unable to decide the truth value of some sentence about possible worlds, the realist and the reductionist disagree as to what to say. The realist maintains that there is a fact of the matter, but that the fact is unknown and perhaps unknowable. The reductionist maintains that there is no fact, and that the truth value of the sentence is to be conventionally decided one way or the other. But this difference need not hinder the reductionist in her attempt to preserve the possible worlds theory. For although the realist thinks that the reductionist is bound to get many of the unknown facts of modality wrong, she cannot object in this way without merely begging the question whether or not there are any such facts. ¹² For further discussion of Carnap’s proposal, and a more detailed comparison with Proposal , see Bricker (: –).

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

     ()

These brief remarks suggest how I think Proposal  could be vindicated with respect to the charge of circularity. Much more would have to be said as to what methods the reductionist should be permitted, and, in particular, as to what can count as admissible data. But I want to bracket the problem of circularity for the rest of this discussion and turn instead to the problem that results from considerations of the cardinality of the class of possible worlds. This will provide a decisive refutation of Proposal .

. The Cardinality Objection We have numerous beliefs about what is possible, and about which possibilities exclude which other possibilities—that is, about compossibility, or what I have called consistency. If we accept the thesis that whatever is possible is true at some (fully determinate) possible world, then our beliefs about possibility and compossibility will lead us to beliefs about the number of possible worlds: we should believe that there are at least as many possible worlds as are needed to support our beliefs about possibility and compossibility. In this section, I shall use this method to set a lower bound on the cardinality of the class of possible worlds.¹³ Here are some modal beliefs that I take to be fairly uncontroversial. I believe that the world might have consisted of nothing but (a single kind of) uniformly dense matter distributed throughout a Euclidean space and time. Moreover, the world might have contained nothing but a single solid cube of such matter, persisting without change throughout all eternity. Let us fix our attention on one such world containing one such cube. I believe that that world might have had less matter than it had. For example, the cube might have been missing one of its corners, or it might have had holes through it like a Swiss cheese. Indeed, all of the cube’s matter might have been missing except for that of a single point. In general, for any collection of points of matter of the original cube, the world might have had the matter of just those points, spatially arranged in just that way. If these beliefs about what is possible are correct, how many possible worlds must there be? Let the worlds that result from the elimination of some of the matter of the original cube be called the cube worlds. Different metaphysical positions with respect to transworld identity will lead one to count the cube worlds in different ways, although all methods of counting ultimately give the same result. Let us first consider how a haecceitist with respect to matter would count the cube worlds.¹⁴ A haecceitist is someone who believes that there are primitive facts as to whether individuals inhabiting different worlds are numerically identical or not, and that therefore two worlds might be just alike with respect to all their qualitative properties, but nevertheless differ with respect to which individuals inhabit them. A haecceitist with respect to matter, then, believes that there are primitive facts as to whether individuals ¹³ The argument of this section is derived from a similar argument in Lewis (: ). I have presented the argument in such a way as to make it more neutral with respect to controversial metaphysical issues about identity over time, and identity across possible worlds. ¹⁴ I use the terms ‘haecceitism’ and ‘anti-haecceitism’ roughly in accordance with the usage of Kaplan (). But there is no general agreement as to exactly how to use these terms.

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  



inhabiting different worlds are composed of the numerically same matter. Now, the haecceitist with respect to matter calculates the number of cube worlds as follows. Sets of points of the original cube are in one-one correspondence with cube worlds, each set of points corresponding with the unique cube world in which the cube has retained just the matter occupying the points of that set, in just that spatial configuration. Since space is assumed to be Euclidean, there are continuum many, or ‫ב‬a₁, points of the original cube, and thus there are power-set-of-the-continuum many, or ‫ב‬a₂, sets of such points. It follows that there are ‫ב‬a₂ cube worlds. Since every cube world is a possible world, there are then at least ‫ב‬a₂ possible worlds. So calculates the haecceitist. According to the anti-haecceitist, however, distinct worlds must differ in some qualitative feature. In particular, distinct cube worlds must differ in the size or shape of their respective aggregates of matter, for there are no other qualitative properties that could distinguish them. (Note that qualitative does not exclude quantitative on this usage.) But then the anti-haecceitist will find the above calculation guilty of double counting: where the haecceitist sees many worlds, the anti-haecceitist often sees only one. For example, imagine the original cube to be divided into two equal halves. Call the cube world in which just the matter in one half is retained w₁, and the cube world in which just the matter in the other half is retained w₂. The haecceitist claims that w₁ and w₂ are distinct worlds because they differ as to which half of the original cube, and thus as to which matter, they contain. But the anti-haecceitist claims that w₁ and w₂ are one and the same world, namely, that cube world that can be completely described in qualitative terms as the Euclidean world consisting of nothing but a solid, rectangular block of a certain kind of matter, of a certain shape and size, persisting unchanged throughout all time. So according to the antihaecceitist, the calculation done above counted the same world twice.¹⁵ More generally, wherever the haecceitist sees distinct cube worlds whose respective aggregates of matter have the same shape and size, the anti-haecceitist sees but a single world. This can be made precise as follows. Two aggregates of matter contained within the original cube, a and b, have the same shape and size just in case one can be superimposed on the other by some combination of translations, rotations, and reflections; that is, just in case one can be superimposed on the other by a Euclidean transformation. (Note that this definition also covers wildly scattered and discontinuous aggregates of points of matter.) Let wa and wb be the cube worlds that result from removing all the points of matter not contained within a and b respectively. Then whereas the haecceitist holds that wa is identical with wb if and only if a is identical with b, the anti-haecceitist holds that wa is identical with wb if and only if a and b have the same shape and size.¹⁶ Thus, we must distinguish between the ¹⁵ Of course, I am speaking of an anti-haecceitist not only with respect to matter, but also with respect to points of space. Otherwise, she counts the cube worlds like the haecceitist with respect to matter: w₁ and w₂ are distinct cube worlds because their respective blocks of matter occupy different points of space. ¹⁶ Nothing compels the anti-haecceitist to use same shape and size as her criterion for individuating the cube worlds. She could use same shape alone; or she could use something even weaker such as same topological structure. It depends on whether or not she believes that there are absolute notions of shape and size that can support transworld comparisons. A discussion of this issue is beyond the scope of the present chapter.

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

     ()

haecceitist cube worlds on the one hand, and the anti-haecceitist cube worlds on the other. Since the relation having-the-same-shape-and-size-as is an equivalence relation over the set of all aggregates of matter contained within the original cube, it induces an equivalence relation over the set of haecceitist cube worlds, and thus partitions this set of worlds into equivalence classes. The anti-haecceitist cube worlds, then, are in one-one correspondence with these equivalence classes. It is now a simple matter to count the number of anti-haecceitist cube worlds by counting the number of equivalence classes. Recall that there are ‫ב‬a₂ haecceitist cube worlds in all. Each equivalence class contains at most of ‫ב‬a₁ members because there are only ‫ב‬a₁ Euclidean transformations of a Euclidean space onto itself. But it follows from the arithmetic of infinite cardinals that if a class of ‫ב‬a₂ members is partitioned into subclasses each containing at most ‫ב‬a₁ members, then there are ‫ב‬a₂ such subclasses. Therefore, there are ‫ב‬a₂ anti-haecceistist cube worlds. Although the haecceitist and the anti-haecceitist may disagree as to how to interpret the various modal beliefs listed at the beginning of this section, they can both agree that those modal beliefs commit them to there being at least ‫ב‬a₂ possible worlds. How does all this bear on Proposal ? Let us make the modest assumption that the possible worlds theory will be sufficiently strong to guarantee that there are cube worlds as described above, and that every cube world is a possible world. Then, whether the theory is haecceitist or not, it will follow from the theory that there are at least ‫ב‬a₂ possible worlds. If Proposal  is to provide a faithful translation of the possible worlds theory, it must provide at least ‫ב‬a₂ linguistic entities to serve as substitutes for the possible worlds. But it can’t. Since there are only a countable number of sentences of L, there are at most ‫ב‬a₁ maximal consistent sets of sentences of L. That will not be enough linguistic entities to provide a faithful translation of the possible worlds theory.

. Second Proposal The reductionist needs a proposal for reduction that can provide more linguistic entities than are provided by Proposal . The problem is not that there is a lack of linguistic entities in her ontology: by taking sets of sets of expressions of L, ‫ב‬a₂ linguistic entities can be made available. The problem is that, if the proposal is to satisfy the naturalness condition, these entities will have to be able to serve as complete descriptions of possible worlds. But I will show that Proposal  can be generalized in a natural way so as to provide ‫ב‬a₂ linguistic entities, and thus circumvent the cardinality argument. In order to motivate such a generalization, it will be helpful to look more closely at how Proposal  satisfies the naturalness condition. The naturalness condition requires that possible worlds be identified with constructions out of L that can serve as their complete descriptions, but it does not require that any single sentence of L be a complete description of a possible world. For example, Proposal  may identify a possible world with a set of sentences none of whose members is a complete description of that world. It is rather the set of sentences as a whole that is to be taken as the complete description of a world; the set of sentences is to be thought of as describing a world at which all of the sentences of the set are true. Thus, Proposal  satisfies the naturalness condition because we can

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 



think of a maximal consistent set of sentences as playing the role, semantically speaking, of the infinite conjunction of all its members. This shows that Proposal , in effect, allows the reductionist to make use of the expressive power of a certain infinitary expansion of L; an infinitary language that adds to the logical apparatus of L an infinitary connective of sentential conjunction. This suggests a way of generalizing Proposal . An infinitary language that only permits infinite conjunctions of complete sentences is quite weak as far as infinitary languages go. Why not allow the reductionist to make use of an infinitary logical expansion of L that permits infinite conjunctions of open formulas as well? Let L* be the infinitary logical expansion of L that permits infinite conjunctions over sets of less than ‫ב‬a₂ open (or closed) formulas; infinitely long sentences of L* are formed by attaching finite strings of quantifiers to such infinitely long formulas. (Thus, L* is the infinitary language L‫ב‬a₂‫ב‬a₀, to use a standard nomenclature.) The semantics for the language L* is developed in the obvious way. In particular, the appropriate clause in the definition of truth and satisfaction reads: an infinite conjunction of formulas is satisfied by an assignment of objects to its free variables if and only if every conjunct is satisfied by that assignment. In general, the language L* will be richer in expressive power than the original language L. For example, let L contain a predicate for the greater-than relation between real numbers and a name for every (standard) natural number, but no other non-logical constants. Although the Archimedean property of the reals, that every real number is exceeded by some (standard) natural number, cannot be expressed in L, it can be expressed in L* by: for all reals r, it is not the case that r is greater than  and greater than  and greater than  and . . . . Such examples as this suggest—rightly, as we shall see—that a stronger proposal for reduction will result if L is replaced by L* in Proposal : Proposal 2.

Possible worlds are maximal consistent sets of sentences of L*.

How does Proposal  fare with respect to the three conditions for reduction laid down earlier in this chapter? It fares at least as well as Proposal  with respect to the non-circularity condition. Both proposals are faced with the problem of defining consistency. True, questions about the consistency of sets of sentences of L* go beyond questions about the consistency of sets of sentences of L in that they require, in effect, judgments about the compossibility of infinite sets of open formulas for their answers. But such judgments seem no more problematical than the judgments about the compossibility of finite sets of open formulas already required by Proposal . Moreover, the constructions out of linguistic entities that are to replace the possible worlds, the maximal consistent sets of sentences of L*, are all non-circularly available to the reductionist. For the expressions of L are all assumed to belong to the reductionist’s ontology, and the sentences of L* can be defined as set-theoretic constructions out of the expressions of L (perhaps together with some symbols for the infinitary connectives). Indeed, the work of constructing the sentences of L* has already been done for the reductionist by the logicians who developed the syntax for infinitary languages.¹⁷

¹⁷ For example, see Karp ().

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

     ()

Proposal  also satisfies that part of the naturalness condition that requires that possible worlds be replaced by linguistic entities that can serve as their descriptions. (Whether or not the linguistic entities provided by Proposal  can serve as complete descriptions of possible worlds (for a reasonable language L) will be discussed below.) The sentences of L* all have a definite meaning as long as the sentences of L do. Indeed, truth conditions for sentences of L* are completely determined by the semantics for L together with the semantical rules for the infinitary connectives. These truth conditions naturally correlate the maximal consistent sets of sentences of L* with the possible worlds that they purport to describe. With respect to the faithfulness condition, Proposal  is a definite improvement over Proposal ; for Proposal  can provide enough linguistic entities to faithfully translate the sentence: ‘There are at least ‫ב‬a₂ possible worlds’. This will be shown with respect to the cube worlds introduced above. It will suffice to show that, for some appropriate choice of L, distinct cube worlds are L*-discernible; for then the set of sentences of L* true at one of the worlds and the set of sentences of L* true at the other will be distinct maximal consistent sets of sentences of L*. Since there are ‫ב‬a₂ cube worlds, it follows that Proposal  provides at least ‫ב‬a₂ maximal consistent sets of sentences. First I will show that, if L contains some rudimentary mathematical language, then L* will contain, in effect, a name for every real number.¹⁸ Thus, let us assume that L contains at least a predicate for the greater-than relation between real numbers, function symbols for the arithmetical operations, and names for the numbers  and . Then every rational number is designated by some term of L. We can use the fact that every real number is the least upper bound of some set of rational numbers to show that every real number uniquely satisfies some open formula of L*. For consider any real number r, and let {qi} be a set of rational numbers (indexed by ωa) that has r as its least upper bound. (The set of all rationals less than r will do.) There is a formula of L* that is satisfied by a real number just in case it is the least upper bound of the qi; that is, just in case it is greater than q₁ and greater than q₂ and . . . and such that any other real number that is greater than q₁ and greater than q₂ and . . . is greater than it. Such a formula uniquely picks out the real number r. The names of real numbers contained in L* can be used in describing the cube worlds if we assume that L has the means to speak of a Euclidean assignment of spatial coordinates to points of matter. For example, L might contain an -place predicate that holds between four points of matter and four triples of real numbers just in case the assignment of those four triples to those four points determines a Euclidean coordinatization of space, that is, an assignment of triples of real numbers to all points of matter that is in conformity with the Euclidean structure of space. And L might contain a -argument function symbol representing the assignment of coordinates (triples of real numbers) to points of matter relative to an initial assignment of coordinates to four points of matter.

¹⁸ That is, L* will have the expressive power of a language that contains a name for every real number. Such names can be provided by introducing into L, and thus into L*, a description operator that is contextually defined à la Russell.

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 



Consider now the haecceitist version of the cube worlds. Here we must further assume that L contains names for four points of matter of the original cube, and that these points have been assigned coordinates in such a way as to determine a Euclidean coordinatization of the entire cube. On these assumptions, it is a simple matter to formulate, for any two cube worlds, a sentence of L* that is true at one of the worlds but not at the other, that is, a sentence of L* that discerns the two worlds. This is because, since L* contains a name for every real number, and so (with a modicum of set theory) a name for every triple of real numbers, L* also contains a name for every point of matter of the original cube: every such point can be picked out by reference to its spatial coordinates. Now, consider any two haecceitist cube worlds. Since the worlds are distinct, there must be some point of matter of the original cube that exists at one of the worlds but not at the other. But then any sentence of L* that asserts that that point of matter exists is true at one of the worlds but not at the other. Since any two haecceitist cube worlds are thus L*‐discernible, and since there are ‫ב‬a₂ cube worlds in all, it follows that there must be a least ‫ב‬a₂ maximal consistent sets of sentences of L*. Let us now turn to the anti-haecceitist version of the cube worlds. Here it does no good to use the names for real numbers to introduce names for the individual points of matter. Rather, the anti-haecceitist uses the names for real numbers to formulate qualitative descriptions of the cube worlds, descriptions of the overall shape and size of the worlds’ aggregates of matter. Thus, let T be the set of triples of real numbers that are assigned to points of matter of the original cube under some arbitrary Euclidean coordinatization; and let S be an arbitrary subset of T, and wS the corresponding (anti-haecceitist) cube world. The world wS can be described by a sentence of L* that asserts the following: On some Euclidean coordinatization of space, all and only the coordinates in S are assigned to points of matter existing in the world. For each (anti-haecceitist) cube world, there will be such a sentence of L* that is true at that world but at none of the others. So here again we see that Proposal  can provide ‫ב‬a₂ maximal consistent sets of sentences, and thus enough linguistic entities to undermine the cardinality argument. Proposal  can succeed where Proposal  failed because Proposal  is based on a weaker requirement for reducibility to L. Recall that Proposal  was based on the idea that a class of possible worlds is reducible to L if any two discernible worlds from that class are L‐discernible. But this condition for reducibility to L, although sufficient, is not necessary. Proposal  makes use of a weaker, but still sufficient, condition for reducibility: a class of possible worlds is reducible to L if any two discernible worlds from that class are L*-discernible. Since, as we have seen, there may be worlds that are discerned by a sentence of L* but not by a sentence of L, Proposal  is essentially more powerful than Proposal , and can reduce wider classes of possible worlds.

. Further Generalizations Proposal  can itself be generalized, in at least two directions. First, recall that L* went infinitary with respect to conjunction, but not with respect to quantification. L* can be further expanded by introducing a stock of ‫ב‬a₂ individual variables to be used in the construction of formulas, and permitting universal quantification with respect to sets

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

     ()

of such variables of cardinality less than ‫ב‬a₂. (This gives the infinitary language L‫ב‬a₂, to use a standard nomenclature.) Secondly, the bound on the number of formulas that can be conjoined, and the number of variables that can be quantified over, can be extended to arbitrarily high cardinality. For each infinite cardinal κa, there is an infinitary expansion of L, Lκa, which permits conjunctions over any set of less than κaformulas, and universal quantification with respect to any set of less than κa individual variables. These infinitary languages provide, for each infinite cardinal κa, a distinct proposal for reduction: Proposal 3κa. Possible worlds are maximal consistent sets of sentences of Lκa. Each of the Proposals κa is based on the idea that Lκa‐discernibility provides a sufficient condition for reducibility to L. Taken together, the series of Proposals κa suggests the following necessary and sufficient condition for reducibility to L: a class of possible worlds is reducible to L if and only if any two discernible worlds from that class are Lκa-discernible, for some infinite cardinal κa.Corresponding to this necessary and sufficient condition, there is a maximally general proposal for reduction. It can be most easily formulated by introducing the infinitary language L∞, which is defined as the union of the languages Lκa, for all infinite cardinals κa: Proposal 4.

Possible worlds are maximal consistent classes of sentences of L∞.

Note, however, that the maximal consistent classes of sentences of L∞ are proper classes—they are “too large” to be sets. Thus, only a reductionist who admits proper classes as well as sets into her ontology can feel free to make use of Proposal . Whether or not the reductionist will need to make use of the full power of Proposal  will depend on which theses about possible worlds she has included in the possible worlds theory. For example, if the reductionist believes that any possible world can be described by giving the distribution of no more than ‫ב‬a₁ qualitative properties over a space-time of no more than ‫ב‬a₁ points, then the cardinality of the set of all possible worlds will be no more than ‫ב‬a₂, and Proposal  will succeed if any proposal will. But if the reductionist believes that, for any ordinal number, there is a possible world in which time is composed of a succession of instants having the order type of that ordinal, then there will be no set of all possible worlds, and the full generality of Proposal  will be needed in any attempt to reduce the class of possible worlds to language.¹⁹ But the question now arises: what are the limitations on even this most general proposal for reducing possible worlds?

. The Objection from Descriptive Impoverishment Initially, one might have thought that there were two sorts of limitation that must be overcome by any proposal for reducing possible worlds to a language L. Let us assume, as is customary, that the vocabulary of L has been divided into two parts: a logical part and a non-logical, or descriptive, part. Then, one might have thought ¹⁹ [For more on “worlds with ordinal time” and whether there are “too many” worlds to form a set, see Chapters  and .]

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    



that the prospects for a successful reduction would have been limited, on the one hand, by any impoverishment of the logical apparatus of L, and, on the other hand, by any impoverishment of the descriptive apparatus. What I hope that the generalized proposals of the preceding section have shown is that an impoverishment of the logical apparatus does not in fact limit the prospects for a reduction at all. As long as the reductionist has set theory (and perhaps class theory) at her disposal, she can always cook up set-theoretic constructions out of the expressions of L that can do all the work that the expressions of any infinitary expansion of L can do. Indeed, the standard proposals for reduction such as Proposal  already make use of this idea in an implicit way. I have suggested making the idea explicit by simply using sentences of an infinitary expansion of L in forming the required set-theoretic constructions. The move is perfectly legitimate; it merely allows the reductionist to make full use of the descriptive resources of the original language L. The problem of an impoverishment of the descriptive vocabulary, however, cannot be dealt with as easily. I claim that any language L that is appropriate to the task of reducing the possible worlds will have its descriptive vocabulary impoverished in such a way as to present insurmountable difficulties for the reductionist. In brief, the problem is this. If the language L is to be able to provide a non-circular reduction of possible worlds to an actualist ontology, then the descriptive resources of L will have to be, in a sense to be illustrated, imprisoned within the actual world. But then only possible worlds that are, in some broad sense, rearrangements of the actual world can be constructed (in a natural way) out of the linguistic entities of L. That will not be all of the possible worlds. Indeed, not even the cube worlds, it seems to me, can be taken to be rearrangements of the actual world, and thus constructible out of the expressions of a reasonable language. So if the cube worlds are possible worlds, as I have claimed, then no attempt at reducing possible worlds to language can succeed. Let us first see where the problem lies with respect to the haecceitist version of the cube worlds. When I argued above that any two haecceitist cube worlds were L*‐discernible, and thus that the class of such worlds was reducible to L, I had to assume that L contained names for points of matter of the original cube. But if L is required to be a reasonable language, in particular, an actual language used by actual people, then that assumption is unacceptable. Presumably, there does not exist a perfect cube of uniform matter anywhere in the actual world; nor do there exist dimensionless points of matter out of which such a cube might be composed. But providing names for nonactual points of matter of a non-actual cube is beyond the reach of the descriptive apparatus of an actual language. Such points cannot be named by ostension; nor can they be distinguished one from the other by their qualitative properties, or by their qualitative relations to actual existents. It follows that there will be cube worlds that are discernible (according to the haecceitist), but not linguistically discernible with respect to any infinitary logical expansion of L, for any reasonable language L: just take two cube worlds whose aggregates of matter have the same shape and size. But then no linguistic construction out of L can be naturally correlated with one of the two worlds but not the other; no linguistic construction can provide a complete description of either of the two worlds. In short: the haecceitist cube worlds are not reducible to L. The generalized proposals for reduction of the preceding

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

     ()

section cannot help the reductionist here because the problem arises from an impoverishment of the descriptive vocabulary, not from an impoverishment of the logical vocabulary. The reductionist with anti-haecceitist leanings runs into difficulties of a somewhat different sort in attempting to reduce the cube worlds. I argued above that, for any anti-haecceitist system of cube worlds, distinct worlds of that system are L*‐discernible. Moreover, no assumptions on L were needed that would make the reduction circular, as was the case with the haecceitist cube worlds. Nevertheless, showing that any two worlds from some one system of cube worlds are L*‐discernible shows only that the class consisting of worlds from that one system is reducible to L; it does not show that the class consisting of worlds from all the different systems of cube worlds is reducible to L. I claim that there are numerous different systems of cube worlds, differing with respect to the kind of matter that their worlds contain. I argue thus. Consider a world that contains two kinds of uniformly dense matter, and suppose that the two kinds of matter have all of their qualitative properties in common. However, they are distinguished relative to one another by the fact that matter of different kinds mutually attracts, matter of the same kind mutually repels. Surely, such a world is possible. Call one of the kinds of matter p‐matter and the other kind n‐matter. Consider the possibility that there exists nothing but a single cube of p‐matter, and the possibility that there exists nothing but a single cube of n‐matter. I claim that these are distinct possibilities, and thus that the system of cube worlds composed of p‐matter is distinct from the system of cube worlds composed of n‐matter. This claim does not admit of demonstration; but to deny it would be to hold not only that we cannot specify other worlds by stipulating what individuals they contain, but that we cannot specify other worlds by stipulating what kinds they contain. Such an extreme form of anti-haecceitism is strongly at variance with modal intuitions, and no philosopher to my knowledge has endorsed it.²⁰ Now, unless L has the descriptive resources to single out one of these two kinds of matter, there will be distinct worlds from different systems of cube worlds that are not L*‐discernible, and so that cannot be assigned distinct linguistic constructions out of L: just take a cube world composed of p‐matter and a cube world composed of n‐ matter whose aggregates of matter have the same shape and size. Could an actual language produce an expression that applied to only one of these kinds of matter? I think not. According to modern science, nothing that exists in the actual world is the stuff of which a cube world is made; so we cannot fix on either p‐matter or n‐matter by means of ostension. Moreover, by the symmetry built into the case, any purely qualitative description that applies to p‐matter applies to n‐matter, and vice versa. So, for the anti-haecceitist as well as the haecceitist, the cube worlds resist

²⁰ [This “extreme form of anti-haecceistism” would today be called “anti-quidditism” or “structuralism.” Since this chapter was first drafted (in ), it has been endorsed by many philosophers, first and foremost by Shoemaker (). Note that today I would not say that p-matter and n-matter “have all their qualitative properties in common.” I distinguish quidditism from the weaker haecceitism about properties. The quidditist holds that the system of cube worlds composed of p-matter and the system of cube worlds composed of n-matter are not just distinct, but qualitatively discernible from one another. See Section . for arguments in support of quidditism.]

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    



reduction. The haecceitist’s problem with respect to distinguishing individual points of matter rearises for the anti-haecceitist with respect to distinguishing kinds of matter. By arguing along lines similar to these, I think it can be shown that much of our talk about possible worlds cannot be successfully reinterpreted as disguised talk about linguistic entities. But might the reductionist have done better to have chosen other entities from her nominalist ontology to be the possible worlds, rather than the linguistic entities? That would have been of no avail. The proposals presented in this chapter will succeed in reducing possible worlds to a nominalist ontology if any proposal will. For assume that some proposal for reduction makes use of a nonlinguistic, actual, concrete entity in constructing the possible worlds. That entity can be uniquely described by some open formula of some infinite expansion of some reasonable language L. At any rate, this is true if, as I suppose, every actual, concrete entity can be singled out by means of its spatiotemporal and causal relations to entities with which we are familiar.²¹ So the reductionist can just as well use the open formula in constructing the possible worlds as use the concrete entity that the open formula describes. In general, whatever can be reduced to actual, concrete entities can equally be reduced to their descriptions. I have supposed that the reductionist has a nominalist ontology and ideology. Would it help to provide her in addition with a conceptualist ontology and ideology? It would not. The proposals of this chapter have already, in effect, made use of such conceptualist resources. Whatever is conceivable by actual people is describable within a reasonable language, an actual language used by actual people.²² So whatever is conceivable can be represented by some linguistic construction. Moreover, since the reductionist’s ideology has already been allowed to include modal notions such as consistency, there seems to be nothing left for a conceptualist ideology to offer. So, adding a conceptualist ontology or ideology would not improve the reductionist’s chances of success. The proposals of this chapter already make full use of the combined resources of the nominalist and the conceptualist. The failure of these proposals, then, marks the failure of any nominalist or conceptualist proposal for ontologically reducing possible worlds. These proposals fail, I conclude, because the possible outruns the actual not in number, but in kind.²³

²¹ The most plausible exception would be a concrete part of the actual world that was spatiotemporally and causally disconnected from the part we inhabit. [For arguments that such “island universes” are possible, see Chapter .] ²² Although, on some views, the public natural language would have to be supplemented with private languages, one for each creature capable of conception. ²³ What about proposals to reduce possible worlds to maximal consistent sets of propositions, rather than sentences (as, for example, in Adams )? If the proposal does not require that the propositions all be expressible within (an infinitary expansion of) a reasonable language, then my objections do not apply. I have argued against such proposals along different lines in Bricker (: –).

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 Quantified Modal Logic and the Plural De Re () . Introduction Quantified modal logic has proven itself a useful tool for the formalization of modal discourse. It has its limitations to be sure: many ordinary modal idioms must be artificially restructured if they are to be expressed within a language whose only modal operators are the box and the diamond; other modal idioms cannot be expressed within such a language at all. Nonetheless, quantified modal logic has enjoyed considerable success in uncovering and explaining ambiguities in modal sentences and fallacies in modal reasoning. A prime example of this success is the now standard analysis of the distinction between modality de dicto and modality de re. The analysis has been applied first and foremost to modal sentences containing definite descriptions. Such sentences are often ambiguous between an interpretation de dicto, according to which a modal property is attributed to a proposition (or, on some views, a sentence), and an interpretation de re, according to which a modal property is attributed to an individual. When these sentences are translated into the language of quantified modal logic, the de dicto/de re ambiguity turns out to involve an ambiguity of scope. If the definite description is within the scope of the modal operator, then the operator attaches to a complete sentence, and the resulting sentence is de dicto. If the definite description is outside the scope of the modal operator, then the operator attaches to a predicate to form a modal predicate, and the resulting sentence is de re. Quantified modal logic has the resources to clarify and disambiguate English modal sentences containing definite descriptions. In this chapter, I explore to what extent the analysis in terms of scope can be applied to modal sentences containing denoting phrases other than definite descriptions, phrases such as ‘some F ’ and ‘every F ’.¹ I will focus on categorical modal sentences of the following two forms:

First published in Midwest Studies in Philosophy  (): –. Reprinted with the permission of John Wiley and Sons, Inc. Versions of this chapter were presented at Rutgers University, the University of California Santa Barbara, and the University of Massachusetts Amherst. David Lewis made helpful suggestions on an earlier draft. ¹ I use ‘F ’ and ‘G’ as schematic letters replaceable by simple or compound English general terms; single quotes should be read as quasi-quotes, where appropriate. I use ‘denoting phrase’ without prejudice towards any theory as to how, or whether, such phrases denote. Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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      



ð◊AÞ Every F might be G: ð□IÞ Some F must be G: These sentences, I will argue, have a threefold ambiguity. In addition to the familiar readings de dicto and de re, there is a third reading on which they are examples of the plural de re: they attribute a modal property to the Fs plurally in a way that cannot in general be reduced to an attribution of modal properties to the individual Fs. The plural de re readings of (◊A) and (□I) cannot be captured simply by varying the scope of an individual quantifier. Indeed, there is an ambiguity associated with the general term ‘F ’ that cannot be analyzed at all within standard quantified modal logic.² I will consider three basic strategies for extending standard quantified modal logic so as to provide analyses for the sentences in question. On the first strategy, all denoting phrases have a rigid/non-rigid ambiguity paralleling the ambiguity some have proposed for definite descriptions and formalized using Kaplan’s ‘dthat’ operator. I will argue that, although there is some plausibility to the ambiguity posited, the first strategy fails to provide a general solution because it cannot provide adequate translations for sentences involving iterated modality. On the second and third strategies, the ambiguity associated with the denoting phrase is again a matter of scope: in this case, the scope of the general term ‘F ’. The second strategy introduces new operators that serve to represent the scope of a general term by indexing it, implicitly or explicitly, to distant modal operators; the third strategy represents scope by appropriately relocating the general term, and then introduces either quantifiers over sets or Boolos’s plural quantifiers to solve a resulting problem of cross-reference. I will argue that only the third strategy with plural quantifiers can provide an adequate formalization of modal discourse within the framework of quantified modal logic.

. Criteria for Evaluating Formalizations of Modal Discourse I will make use of two principles in evaluating proposals for formalizing modal discourse. Let S be an English sentence to be formalized, and let T(S) be its translation into the formal language. The first principle requires that T(S), when interpreted, provide a correct semantic analysis of S in at least the following minimal sense: For any possible context of utterance, if S has a determinate truth value in that context, then T(S) has the same truth value as S in that context.³

² For definiteness, by ‘standard quantified modal logic’ I will mean the language and semantical treatment in Kripke (). I assume that quantifiers range only over individuals (concrete or abstract), not over sets. ³ Of course, one generally requires also that T(S) in some sense capture the logical form of S; but it will not be necessary to appeal to such a requirement in what follows.

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

        ()

The first principle applies to formalization in general. What further requirements should be imposed will depend on the goals of the particular project of formalization at hand. Such goals might include, for example, any of the following: () exploring the expressive power of a particular logical framework; () developing a perspicuous logical regimentation of English; () showing that English is free of certain unwanted ontological commitments; () modelling the psychological processes by which a language user comprehends English. The first- and second-mentioned goals are relevant to the present project of formalization; especially, exploring the expressive power of the framework of quantified modal logic. It is essential to this framework that the concepts of possibility and necessity be expressed by means of propositional operators that do not use the full resources of quantification over possible worlds. Thus, for the project at hand, there is a second principle that proposals must satisfy: The formal language must not contain the equivalent of full variable-binding operators ranging over possible worlds. A full variable-binding operator has the power to bind a variable occurring at any position syntactically within its scope. Although the notion of equivalence in question is difficult to make precise, standard quantified modal logic itself clearly satisfies the principle. When sentences of quantified modal logic are translated in the usual way into first-order world theory, the box and the diamond become quantifiers that are constrained by the rule: world variables must be bound by the nearest possible quantifier (unless they occur in an argument-place of the accessibility predicate). For this reason, a box or a diamond, unlike a full variablebinding operator, always has its influence disrupted by the presence of another box or diamond within its scope. The principle does restrict, however, the ways in which standard quantified modal logic can be extended for purposes of formalizing English modal sentences. What about the ontological goal of showing that English modal discourse lacks a realist commitment to possible worlds and possibilia? Formalization within quantified modal logic has less to offer the non-realist, I think, than has sometimes been supposed. I will touch on this question briefly at the end of the chapter.

. Scope Ambiguity in Modal Discourse I turn now to the formalization of particular English sentences. It will be useful to begin by illustrating a method for applying the analysis in terms of scope to modal sentences containing definite descriptions. Consider the following familiar example: ()

The President is necessarily a U.S. citizen.

The source of ambiguity in () is immediately apparent if one applies Russell’s analysis of definite descriptions. On Russell’s analysis, there are two ways of eliminating the definite description in (): the description can be taken to have either narrow scope or wide scope.⁴ As a result, there are the following two possible translations into quantified modal logic (using the obvious abbreviations): ⁴ For Russell’s theory, see Russell (: –). Russell speaks of primary and secondary occurrences of a description instead of wide and narrow scope, respectively. Russell’s theory is applied to the modal case in Smullyan ().

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     ðÞ

□ð9xÞððyÞðPy

ðÞ

ð9xÞððyÞðPy $ y ¼ xÞ & □CxÞ:



$ y ¼ xÞ & CxÞ:

In (), the box attaches to a complete sentence; () is therefore de dicto. So interpreted () is presumably true: it asserts that at every accessible possible world the President at that world is a U.S. citizen, and this will be true as long as only worlds that conform to the U.S. constitution are considered accessible. In (), the box attaches only to the predicate ‘Cx’; () is therefore de re. So interpreted () is presumably false: it asserts of the person who is in fact President, Ronald Reagan, that he has the modal property of being necessarily a U.S. citizen. Reagan lacks that property because his parents might have renounced their citizenship and left the country before he was born. Thus, sentences of quantified modal logic can be provided that succeed in capturing the two possible readings of (), and that show the difference in readings to be a matter of scope.⁵ The explanation of ambiguity in terms of scope has also been applied to modal sentences containing denoting phrases other than definite descriptions. For example, as has often been noted, the difference between uses of ‘any’ and ‘every’ can sometimes be explained by the rule that the former takes the wider of two available scopes whereas the latter takes the narrower scope.⁶ Thus, suppose that a lottery is to take place in which various numbers are to be chosen, and compare () with (): ()

Any number less than a hundred might be chosen.

()

Every number less than a hundred might be chosen.

() asserts of each number less than a hundred that it has a certain modal property: the property of possibly being chosen. The quantifier is outside the scope of the modal operator and the sentence is de re:⁷ ()

(x)(Nx ! ◊Cx). (Nx = x is a number less than a hundred.)

(), on the other hand, is ambiguous. On one reading it is equivalent to () and analyzed as (). On another reading, it asserts that it is possible for a certain proposition to be true: the proposition that every number less than a hundred is chosen. The quantifier is within the scope of the modal operator and the sentence is de dicto: ðÞ

◊ðxÞðNx

! CxÞ:

⁵ An alternative analysis takes () to involve a primitive description operator. See, for example, Hintikka (). Thomason and Stalnaker () use a description operator and a device for forming complex predicates to analyze (). All these methods agree in attributing the ambiguity in () to a distinction of scope. ⁶ There is a discussion of various non-modal examples in Quine (a: –). Thomason and Stalnaker (: ) explicitly apply the rule to a modal example. ⁷ On this extended (though now standard) use of de re, the individual or individuals to whom the modal property is attributed need not be individually named or described. This allows the classification of sentences as de dicto or de re to be exhaustive for standard quantified modal logic. For a precise explication of an exhaustive de dicto/de re distinction, both syntactic and model theoretic, see Fine ().

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

        ()

In this example, then, the difference between ‘any’ in () and ‘every’ in () can be accounted for in terms of quantifier scope in sentences of quantified modal logic.⁸ Distinctions of quantifier scope can also be used to resolve ambiguities involving the denoting phrase ‘some F ’. Thus, ()

Some number less than a hundred must be chosen.

is ambiguous between the de re assertion ðÞ

ð9xÞðNx & □CxÞ;

which would be true if the lottery were rigged to ensure that, say, the number seventeen be the chosen number, and the de dicto assertion ðÞ

□ð9xÞðNx & CxÞ;

which would be true if only ninety-nine tickets were sold, numbered consecutively from one. The explanation of the ambiguity involving ‘some F ’ in terms of quantifier scope exactly parallels the explanation of the ambiguity involving ‘every F ’. In the case of ‘some F’, however, English provides no alternative denoting phrase that serves to force either the narrow scope or the wide scope interpretation.⁹

. The Plural De Re So far, so good. But when one considers sentences of the form (◊A) and (□I) that have readings that are plurally de re, the standard analysis in terms of scope breaks down. A modal proposition is plurally, as opposed to individually, de re if it involves the assertion or denial of a joint possibility for two or more individuals. A plurally de re proposition is not in general reducible to a combination of individually de re propositions; for example, given the possibility that a is F and the possibility that b is F, nothing in general follows about the joint possibility that both a and b are F.¹⁰ I turn now to an example where the difference between ‘any F ’ and ‘every F ’ cannot be attributed simply to the scope of an individual quantifier. Suppose that a drawing for prizes is about to occur. Three of the people who entered the drawing—

⁸ It should be noted, however, that sometimes the role of ‘every’ in ‘every F ’ is to signal that the Fs are to be taken collectively rather than distributively, and the main predicate interpreted accordingly. Thus, although the sentence ‘I can say any English word in less than a minute’ can be formalized by giving a universal quantifier wide scope, the sentence ‘I can say every English word in less than a minute’, on the reading that makes it false, cannot be formalized by giving a universal quantifier narrow scope because the predicate ‘is said in less than a minute’ is to be applied, not to individual English words, but to English words taken altogether. In the examples discussed below, all predicates are to be taken distributively. ⁹ Russell (: –) distinguishes between ‘some F ’ and ‘a F ’, giving the former wide scope and the latter narrow scope. Perhaps English exhibits some tendency in this direction; but, in contrast to ‘any F ’, each of these denoting phrases can take either scope. ¹⁰ An exact analysis of the plural de re is beyond the scope of this chapter. It requires the problematic— though, I think, genuine—distinction between qualitative and non-qualitative properties. Thus, even ‘◊Ga’ is plurally de re if ‘Gx’ is equivalent to the non-qualitative ‘Fx & Fb’. The plural de re has been discussed in connection with counterpart theory in Hazen () and Lewis (a: –). A full account requires the consideration of ordered pluralities, that is, sequences of individuals.

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   



Tom, Dick, and Harry—are gathered together in a room awaiting the results. Compare the following two assertions: ()

Any person in the room might win a prize.

()

Every person in the room might win a prize.

() asserts that each person in the room has the modal property, possibly wins a prize. It can be translated by the standard de re: ()

(x)(Rx ! ◊Wx). (Rx = x is a person in the room.)

(), on the other hand, is ambiguous. Like (), the standard de re and de dicto formulas provide possible readings. Unlike (), however, () has a third—plurally de re—reading which, I will argue, is not equivalent to either of the other two. This reading can be expressed in a preliminary way as follows: () asserts of the people who are actually in the room—in this case, Tom, Dick, and Harry—that it might be the case that all of them win a prize. Assume throughout what follows that () is to be interpreted according to the plurally de re reading just given. There is no problem finding a plurally de re sentence of modal logic (enhanced with proper names) that is guaranteed to have the same truth value as () for all contexts of utterance in which Tom, Dick, and Harry are the people in the room: ðÞ

◊ðWt & Wd & WhÞ:

But (), of course, fails to provide an analysis of (); it does not have the same truth value as () for every context of utterance.¹¹ How, then, can () be analyzed as a sentence of quantified modal logic? I argue first that () cannot be analyzed as the standard de re (). For suppose that when () is uttered Tom, Dick, and Harry are in the room, and suppose that according to the rules of the drawing only one person can win a prize. Then (), and so (), is false, since there is no accessible world at which all three of them win a prize (allowing only worlds that satisfy the rules of the drawing to be accessible). But () is true. In the context in question, () has the same truth value as the individually de re ðÞ

◊Wt & ◊Wd & ◊Wh;

and () is true as long as Tom, Dick, and Harry each have a chance to win. So () cannot provide an analysis of (). If the difference between () and () were simply a matter of the scope of the universal quantifier, then () could be translated by the de dicto: ðÞ

◊ðxÞðRx

! WxÞ:

¹¹ In standard quantified modal logic without names, the sentence ‘(9x)(9y)(9z) (x6¼y & y6¼z & Rx & Ry & Rz & ◊(Wx & Wy & Wz))’ has the same truth value as () for all contexts in which there are three people in the room; but, again, this does not provide an analysis of ().

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

        ()

But () fares no better than () as an analysis of (). Since in () the predicate ‘Rx’ is within the scope of the diamond, the truth value of (), unlike (), will depend on who is in the room at worlds other than the actual world. This allows there to be cases where () and () diverge in truth value. Suppose again that according to the rules of the drawing only one person can win a prize, thus making () false. But () is true. () asserts that the proposition every person in the room wins a prize is a possible proposition, and so true at some possible world. Consider a world at which Tom is the only person in the room, and at which Tom wins the prize. Such a world is possible assuming only that being in the room is a contingent property of Dick and Harry, and that Tom has a chance to win a prize (irrespective of who is in the room). Moreover, the proposition every person in the room wins a prize is true at this world. So (), unlike (), is true, and () cannot provide an analysis of (). I conclude, then, that the de dicto/de re, narrow scope/wide scope distinction is unable by itself to capture all the possible readings of sentences of the form (◊A) ‘Every F might be G’.¹² The difficulty in formalizing () within quantified modal logic afflicts other modal constructions involving other denoting phrases. Thus, suppose that the following sentence of the form (□I) ‘Some F must be G’ is uttered in the same circumstances as () above: ()

Some person in the room must win a prize.

Both the de re () and the de dicto () provide possible readings of (): ðÞ

ð9xÞðRx & □WxÞ:

ðÞ

□ð9xÞðRx & WxÞ:

But () also has a plural de re reading that is captured neither by () nor by (). On this reading, () asserts of the people actually in the room—in this case, Tom, Dick, and Harry—that it must be the case that at least one of them wins a prize. To see that neither () nor () can capture this reading, suppose that the drawing has been rigged by removing all tickets belonging to entrants other than Tom, Dick, or Harry. In this case, () is true, but () and () are false (under the natural accessibility assignment). () is false because there is no particular person who is guaranteed to win a prize: it could be either Tom, Dick, or Harry. () is false because being in the room is, I suppose, a contingent property of Tom, Dick, and Harry, and irrelevant to the selection of a winner.¹³ How widespread is the plural de re phenomenon exhibited by () and ()? For one thing, it is not restricted to the logician’s favorite denoting phrases: ‘every F ’ and

¹² It should now be apparent why (), unlike (), does not possess a reading that cannot be captured by the standard de dicto and de re formulas. The property being a number less than a hundred, unlike the property being in the room, applies necessarily to whatever has it. ¹³ Note, in contrast, that for sentences of the form (□A) ‘Every F must be G’ and (◊I) ‘Some F might be G’, the plurally de re reading is equivalent to the ordinary de re reading. This is due in essence to the distributivity of the box over conjunction and the diamond over disjunction.

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  ;  



‘some F ’. The threefold ambiguity in the following examples should now be readily apparent to the reader: ()

Most students from out-of-state must live off-campus.

()

Exactly five students in my class can win a fellowship.

Moreover, the phenomenon occurs not only in connection with modal operators, but in connection with temporal operators as well. The ambiguity in () gives rise to the same difficulty in formalization as do the modal examples: ()

Every book in the store was on sale.¹⁴

Indeed, the phenomenon can also be recognized in connection with propositional attitude constructions such as: ()

Ralph believes someone in the house committed the murder.¹⁵

For each of these examples, the usual de dicto/de re, narrow scope/wide scope distinction can be used to analyze two possible readings, but a third, plural de re reading remains unanalyzed.

. Formulating the Plural De Re: The Actuality Operator; Actuality Quantifiers I turn now from illustration to diagnosis. In what follows, I will focus on the two schemas (◊A) and (□I) interpreted in the plural de re way illustrated above. Why were the ordinary de re and de dicto analyses unable to provide translations for (◊A) and (□I)? Consider (□I): ‘Some F must be G’. On the de re analysis, the existential quantifier is outside the scope of the box, and the box attaches to the predicate ‘Gx’. On this analysis, (□I) would assert that one and the same individual has the property expressed by ‘G’ at every possible world. This, we have seen, misconstrues the plurally de re (□I) (unless there is only one F), because (□I) is compatible with the property expressed by ‘G’ being had by different individuals at different worlds. On the de dicto analysis, the quantifier occurs within the scope of the box. This forces the predicate ‘Fx’ also to occur within the scope of the box, for ‘Fx’ must occur within the scope of the quantifier that binds its free variable. Since ‘Fx’ occurs within the scope of the box, which individuals are F at non-actual worlds is relevant to the truth value of the de dicto analysis. And that, we have seen, also misconstrues the plurally de re (□I), since only which individuals are F at the actual world is relevant to its truth value. A correct analysis of (□I), it seems, must have the predicate ‘Fx’ governed by the

¹⁴ Temporal examples of the plural de re were noticed by Hans Kamp and Frank Vlach; but neither provides an adequate general solution to the problem of formalization. On Vlach’s solution, see n.  and n.  below. ¹⁵ Belief sentences such as () give rise to further ambiguities having to do with the issue of actual vs. imaginary objects of thought. For this reason, and others, I think it best to use modal (or temporal) examples to isolate the phenomenon here in question. But what I say about the modal case, it should be apparent, can be applied to the propositional attitude case as well.

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

        ()

existential quantifier, but not governed by the box that governs the existential quantifier. No sentence of standard quantified modal logic can do that.¹⁶ In what follows, I will consider three basic strategies for extending quantified modal logic so as to provide formalizations for (◊A) and (□I). The first strategy focuses on the fact that only the actual Fs, not the otherworldly Fs, are relevant to the truth value of (◊A) and (□I). According to this strategy, the analyses need to have an actuality operator prefixed to the predicate ‘Fx’ in order to ensure that only the individuals that are actually F will be considered, even when ‘Fx’ occurs within the scope of a modal operator. As a first step, then, let us add to the two modal operators of standard quantified modal logic an actuality operator, ‘A’, to be interpreted as follows (‘φ‘ stands for a formula of the object language that may or may not contain free variables; f is an assignment of individuals, actual or possible, to the variables of the object language): ‘Aφ’ is true at world w on assignment f if and only if ‘φ’ is true at the actual world on assignment f.¹⁷ With the actuality operator at hand, (□I) can be formalized by: ðÞ

□ð9xÞðAFx & GxÞ:

It is instructive to compare the truth conditions of () with the truth conditions of the failed de dicto analysis ðÞ

□ð9xÞðFx & GxÞ:

() is true just in case, at all worlds w, there exists an individual at w that is F at w and G at w; () is true just in case, at all worlds w, there exists an individual at w that is F at the actual world and G at w. () can accomplish what () could not because the actuality operator provides the means by which the predicate ‘Fx’ can be syntactically within the scope of a modal operator, but semantically unaffected by its presence. Simply prefixing the actuality operator to the predicate ‘Fx’, however, cannot be trusted by itself to give a correct analysis of sentences of the form (◊A) or (□I) unless one makes the implausible assumption that the same individuals exist at every possible world. Let us first consider the problem with respect to (◊A) ‘Every F might be G’, whose translation using the actuality operator alone would be: ðÞ

◊ðxÞðAFx

! GxÞ:

¹⁶ That neither (□I) nor (◊A) can be expressed by any sentence of standard quantified modal logic follows (for S) from Theorem  in Hodes (). A more general result can be derived from the proof of Theorem  in Kamp (). ¹⁷ Only sentences uttered at the actual world are here considered; otherwise ‘the actual world’ should be replaced by ‘the world of the utterance’ making the actuality operator overtly indexical like ‘now’. An indexical actuality operator was introduced in Lewis (); see also Hazen (). Kamp () uses the ‘now’-operator to formalize a temporal example of the plural de re.

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  ;  



Suppose that (◊A) is false, that is, that there is no possible world at which all the actual Fs are G.¹⁸ () might nonetheless be true. For suppose further that at some world one of the actual Fs is not G because it fails to exist at the world, although all the other actual Fs exist and are G at the world. At this world, all the individuals that exist at the world and that are F at the actual world are G. So () is true, and () does not adequately translate (◊A). The problem with () is easily diagnosed. The predicate ‘Fx’ has been freed from the tyranny of the diamond, but the universal quantifier remains enslaved. On the standard interpretation of the quantifiers—the inner interpretation—the quantifier in () ranges only over the individuals that exist at the world at which the quantification is being evaluated. If one of the actual Fs does not exist at this world, then the quantifier will not range widely enough to capture the sense of (◊A). This suggests that we try adding to the inner quantifiers of standard quantified modal logic outer quantifiers: quantifiers that range over the entire universe of possibilia. Using ‘’ and ‘’ for the outer quantifiers, (◊A) can be translated by: ðÞ

◊ðAFx

! GxÞ:

For the case considered above, (), unlike (), will be false as required: the subformula ‘(AFx ! Gx)’ is false at a world at which an actual F fails to be G by failing to exist at the world.¹⁹ But if inner quantifiers sometimes fail to range widely enough, outer quantifiers sometimes have the opposite defect of ranging too widely. To see this, consider the translation of (□I) that results from the joint use of the outer quantifier and the actuality operator: ðÞ

□ðAFx & GxÞ:

() need not correctly capture (□I) in cases where other worlds contain individuals that do not exist at the actual world. Thus, suppose that (□I) is false, that is, that there are worlds at which none of the actual Fs are G. () might nonetheless be true. For suppose further that at every such world there exists a G that is F at the actual world without existing at the actual world. (‘F ’ might be a compound, negative general term, such as ‘person not in the room’.) Such individuals are irrelevant to the truth value of (□I); but since they lie within the range of the quantifier in (), they satisfy the subformula ‘AFx & Gx’, and so make () true. In an extreme case, () could be true even though there were no Fs existing at the actual world. But surely (□I), as it would ordinarily be understood, has existential import and implies the sentence ‘(9x)Fx’. It follows that () does not provide a correct analysis of (□I).

¹⁸ By ‘the actual Fs’, I mean the individuals that are F at the actual world and exist at the actual world. The second clause is not redundant: the Kripkean semantics here presupposed allows that an individual be F at the actual world without existing at the actual world (both for simple and complex ‘F ’). Taking the alternative approach, however, would affect what follows only in detail. ¹⁹ For other examples of the expressive power conferred by the joint use of outer quantifiers and an actuality operator, see Hazen (). The inner and outer quantifiers are often called actualist and possibilist quantifiers, respectively, but I prefer to reserve the term ‘actualist’ for the quantifiers to be introduced below.

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

        ()

If the first strategy is to succeed, then, it needs to introduce, not outer quantifiers, but rather what might be called actuality quantifiers: quantifiers that range over all and only the individuals that exist at the actual world even when occurring within the scope of a modal operator.²⁰ Using ‘[9x]’ and ‘[x]’ as the actuality quantifiers, the first strategy provides as the final translations for (◊A) and (□I): ðÞ

◊½xðAFx

! GxÞ:

ðÞ

□½9xðAFx & GxÞ:

The problems associated with earlier attempts to translate (◊A) and (□I) no longer arise; in particular, () has existential import like (□I) and unlike (). Moreover, () and (), like (◊A) and (□I), are not equivalent to either of the standard de dicto or de re sentences that can be formulated in standard quantified modal logic: the additional apparatus plays an essential role. This strategy for handling the denoting phrases ‘every F ’ and ‘some F ’ can be generalized in a natural way beyond (◊A) and (□I). Consider any sentence having one of the forms: ()

O(every F is G).

()

O(some F is G).

where ‘O’ stands for a simple or compound modal propositional operator. For each such sentence, the strategy posits an ambiguity in the denoting phrase ‘every F ’ or ‘some F ’. On one reading, the denoting phrase is contextually analyzed by way of an ordinary inner quantifier; on the other reading, by way of an actuality quantifier with an actuality operator. Note, for comparison, that the strategy posits a similar ambiguity in the denoting phrase ‘the F’. Thus () can be analyzed (using Russell’s theory) as either () or (): ðÞ

Oðthe F is GÞ:

ðÞ

Oð9xÞððyÞðFy $ x ¼ yÞ & GxÞ:

ðÞ

O½9xð½yðAFy $ x ¼ yÞ & GxÞ:

When analyzed as (), the denoting phrase ‘the F ’ functions as if it had Kaplan’s dthat-operator prefixed to it, at least in cases where there is one and only one F. Indeed, the ambiguity here posited for ‘every F ’ and ‘some F ’ can be seen as a natural generalization of the purported ambiguity in ‘the F ’ captured by ‘dthat the F ’.²¹ On the strategy being considered, one would expect an ambiguity to be present even in the simple categorical sentences ‘Every F is G’ and ‘Some F is G’, that is, even in the case where the operator ‘O’ in () and () has been dropped. For, in this case, the two readings of () may diverge in truth value at other possible worlds, as may the two readings of (); and on most accounts, this is sufficient for divergence in

²⁰ Alternatively, an actuality predicate can be introduced, and the actuality quantifier defined as a restricted outer quantifier. ²¹ For the logic of ‘dthat’, see Kaplan (; b).

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  ;  



meaning. But since the readings cannot diverge in truth value at the actual world, it is difficult to find evidence for or against the presence of an ambiguity in English. Both readings have, in effect, been put forward. The possible-worlds literature standardly uses inner quantifiers to give the truth conditions of simple categorical sentences. But writers of logic texts over the years have frequently opted, perhaps unwittingly, for an actuality interpretation of the quantifiers. Whenever it is said that ‘Every F is G’ is equivalent in meaning to a (perhaps infinitary) conjunction and that ‘Some F is G’ is equivalent to a (perhaps infinitary) disjunction, and that in a (finite) universe in which all individuals had names, the quantifiers would be dispensable (except as a convenient abbreviation), the actuality interpretation is tacitly being endorsed.²² That the strategy being considered allows for a possible ambiguity in simple categorical sentences counts, if anything, in its favor. But when one turns to cases involving iterated modality, the strategy being considered is inadequate to the task at hand. For example, consider the sentence that results from prefixing a possibility operator to (): ()

It might have been the case that some person in the room had to win.

Interpret () as asserting that the plurally de re () might have been true. Suppose again that Tom, Dick, and Harry were actually in the room. Suppose further that Heckle and Jeckle have entered the drawing, that the lottery might have been rigged so as to ensure that either Heckle or Jeckle win a prize by removing all tickets belonging to entrants other than Heckle or Jeckle, and that this is the only way the drawing might have been rigged. Finally, suppose that Heckle and Jeckle might have been in the room instead of Tom, Dick, and Harry, but that being in the room has nothing to do with whether or not the drawing is rigged. () is true in the situation just described, but all of the available translations of () are false, and so fail to capture the intended interpretation. Let me quickly run through the options. Suppose first that () is taken to be of the form () with ‘O’ standing in for ‘◊□’.²³ Then there are two translations available: ðÞ

◊□ð9xÞðRx & WxÞ:

ðÞ

◊□½9xðARx & WxÞ:

According to (), it might have been the case that the drawing was rigged so as to guarantee that a room-dweller win a prize. By assumption, this is false, since the only way the drawing might have been rigged was so as to guarantee that either Heckle or Jeckle win; and guaranteeing that either Heckle or Jeckle win does not guarantee that a room-dweller win because Heckle and Jeckle might not have been in the room. According to (), it might have been the case that the drawing was rigged so as to ²² For example, see Quine (: ). But note that Quine (and others) take ‘Some F is G’ to be equivalent to ‘(Fa & Ga) v (Fb & Gb) v . . . ’ (where ‘a’, ‘b’, . . . are all the actual individuals), which is equivalent to using the actuality quantifier without prefixing the actuality operator to ‘Fx’—a most implausible hybrid. To get the second reading above, ‘Some F is G’ should be taken to be equivalent to ‘Ga v Gb v . . . ’ (where ‘a’, ‘b’, . . . are all the actual Fs). ²³ For the case at hand, the diamond and the box may be tied to different accessibility relations. This should be made notationally evident, but I will not bother since it affects nothing that follows.

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

        ()

guarantee that Tom, Dick, or Harry win a prize, which, by assumption, is false. Nor can the reading under which () is true be captured by giving the existential quantifier wide or intermediate scope. Varying the scope of the actuality quantifier in () has no effect on truth conditions. Varying the scope of the quantifier in () results in the following two readings: ðÞ

ð9xÞðRx & ◊□WxÞ:

ðÞ

◊ð9xÞðRx & □WxÞ:

Neither () nor () captures the sense in which () involves the plural de re. The wide scope reading, (), asserts that either Tom, Dick, or Harry is such that the drawing might have been rigged so as to guarantee that he win. The intermediate scope reading, (), asserts that there might have been some person in the room such that the drawing was rigged so as to guarantee that that person win. Both of these are false because, by assumption, it was not possible to rig the drawing so as to guarantee that any one person win, only that of two people, one of them win. In sum, () is true when interpreted as saying that the plurally de re proposition () might have been the case; but no sentence of quantified modal logic, even when enhanced with actuality quantifiers and an actuality operator, can capture that interpretation. It should now be clear why the first strategy cannot handle sentences involving iterated modality. In evaluating the truth value of a sentence like () there is a double shift away from the actual world, one shift for each modal operator. We have seen that if a translation of () is to capture the sense in which it involves the plural de re, the existential quantifier—and so the predicate ‘Rx’—must be within the scope of the box. In standard quantified modal logic, if the predicate ‘Rx’ is within the scope of the box (as in ()), then the people in the room at doubly shifted worlds will be relevant to the evaluation of truth value. In the extended modal logic with actuality quantifiers and an actuality operator, it is possible to have the predicate ‘Rx’ within the box (as in ()) and yet to have the people in the room at the actual world be relevant to the evaluation of truth value. But since the actuality apparatus always takes us all the way back to the actual world, we still lack the means to construct a sentence of quantified modal logic that would make the people in the room at singly shifted worlds relevant to the evaluation of truth value, and so we are unable to provide a translation for the reading of () on which it is true. The first strategy fails to provide a general solution to the problem of analyzing the use of denoting phrases to express the plural de re; for a general solution must be able to handle not only simple examples of the plural de re such as (◊A) and (□I), but also an example such as () in which the plural de re is embedded within a modal context.

. Formulating the Plural De Re: The Method of Indexing We need a fresh diagnosis of the failure of standard quantified modal logic to capture the plural de re, one that will generalize to cases involving iterated modality. Consider again the denoting phrase ‘some F’ or ‘every F ’ as it occurs within a (perhaps iterated) modal context. As we have seen, there are often two sorts of ambiguity

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

    :    



associated with the denoting phrase, only one of which can be analyzed in terms of quantifier scope. The second sort of ambiguity was analyzed in the previous section as, in effect, a rigid/non-rigid ambiguity: when interpreted rigidly, the denoting phrase serves to pick out the actual Fs, and rigidly refers to them at all worlds throughout the process of evaluation; when interpreted non-rigidly, the denoting phrase refers to whatever is F at the world at which the evaluation is taking place. Positing a two-way, rigid/non-rigid ambiguity, however, could not handle the multiple ambiguity associated with assertions of iterated modality. On the strategy now to be considered, the second sort of ambiguity involves, like the first, an ambiguity in scope: the scope of the general term ‘F ’. To illustrate what is meant by the “scope” of the general term, return once more to (). We have seen that () has three distinct readings depending on whether the people in the room at the actual world, at singly shifted worlds, or at doubly shifted worlds are relevant to the evaluation of truth value. For these three readings, I say that the term ‘person in the room’ has, respectively, wide scope, intermediate scope, or narrow scope. Normally, the way to give the term ‘person in the room’ the appropriate scope is to place the predicate ‘Rx’, respectively, outside the diamond, between the diamond and the box, or within the box. But in standard quantified modal logic, the appropriate placement cannot always be had because the predicate ‘Rx’ cannot take wider scope than the quantifier that binds its variable. Thus, on the new diagnosis, the crucial limitation of standard quantified modal logic is that it does not provide the means by which the scope of a predicate can vary independently of the scope of the quantifier that governs it. How might standard quantified modal logic be extended so as to provide for such independence? As a first method, we might try introducing two new operators, # and ", which can be used in tandem to give a selected predicate any available scope.²⁴ If the predicate is governed by the #-operator, it need not have narrow scope, but may instead have whatever scope is indicated by the placement of the "-operator. To illustrate, consider again (). Standard quantified modal logic was constrained to give the predicate ‘Rx’ narrow scope if the quantifier was given narrow scope, resulting in the mistranslation: ðÞ

◊□ð9xÞðRx & WxÞ:

Our two new operators allow the predicate ‘Rx’ to be syntactically within the scope of the box, although semantically tied to the diamond: ðÞ

◊ " □ð9xÞð# Rx & WxÞ:

By placing a ‘#’ in front of ‘Rx’ and a ‘"’ after the ‘x’, the predicate ‘Rx’ is given intermediate scope, thus making the people in the room at singly shifted worlds relevant, as desired. However, () fails as a translation of () because the quantifier wrongly ranges over the individuals inhabiting doubly shifted worlds. On a correct translation of ²⁴ Frank Vlach used the #- and "-operators to formalize an example of the plural de re involving iterated tenses. A semantics for these operators can be given by the method of “double indexing,” that is, by assigning truth values relative to ordered pairs of worlds rather than single worlds. See Vlach () and Lewis (: –).

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi



        ()

(), the domain of the quantifier has intermediate, not narrow, scope. This suggests relocating the ‘#’ as follows: ðÞ

◊ " □ # ð9xÞðRx & WxÞ:

Out of the frying pan and into the fire! In (), the predicate ‘Wx’ is wrongly given intermediate, instead of narrow, scope. Using the operators # and " by themselves cannot succeed in capturing the sense of ().²⁵ Even supposing that the problem of assigning scope to quantifier domains could be separately solved (for example, by partial use of the method of indexing introduced below), there is a more general objection. The #- and "-operators provide some freedom in representing the scope of predicates, but not enough. Although any given predicate can be assigned any available scope, it is not the case that any two or more given predicates can independently be assigned any available combination of scopes. Suppose we have a sentence with n predicates, each of which has m available scopes. Then there are mn possible assignments of scope, each of which corresponds to a distinct proposition with distinct truth conditions. But only a fraction of these propositions can be expressed using the #- and "-operators. Thus, countless sentences will have readings involving the plural de re that cannot be formalized within the extension of quantified modal logic that adds only the operators # and ".²⁶ We need to extend quantified modal logic in a way that provides for complete independence in the assignment of scope to predicates (and quantifier domains). This suggests a second method, what I call the method of indexing. We can index predicates directly to modal operators to indicate the desired scope. Let us use the letters ‘w’ and ‘v’ (with or without subscripts) as indices, placing them as superscripts after an operator and as subscripts after a predicate. Then, the predicate ‘Rx’ in () can be given intermediate scope as follows: ðÞ

◊w □ð9xÞðRw x & WxÞ:

To capture the sense of (), however, we must index the quantifier ‘(9x)’ to the diamond as well so that it will appropriately range over the individuals inhabiting singly shifted worlds: ðÞ

◊w □ð9xÞ

w ðR w x & WxÞ:

In (), although the quantifier has narrow scope, the domain of the quantifier has intermediate scope. This notion of scope can be applied not only to predicates and quantifier domains, but also to modal operators: the box in () has a suppressed index binding it to the diamond, since we are interested in who wins at worlds ²⁵ Vlach focuses on an example that involves an analog of (◊A) rather than (□I), and formalizes it by using # and " together with outer quantifiers. But as we saw in connection with () above, outer quantifiers range too widely to capture sentences like () that involve (□I). ²⁶ If an example is wanted, consider “It might have been the case that someone in the room who lost had to win” uttered in the same circumstances as (). (Assume also that Heckle and Jeckle actually lost.) For the reading on which this is true, the predicates ‘person in the room’, ‘person who lost’, and ‘person who wins’ are inside the ‘□’ and have intermediate, wide, and narrow scope, respectively—a combination that cannot be had using only # and ".

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

    :    



accessible to singly shifted worlds, not at worlds accessible to the actual world—and these may differ if the logic is not S. In general, predicates, quantifier domains, and modal operators that are to have the narrowest possible scope can be seen as being indexed to the operator immediately governing them, but with their indices suppressed. Restoring the suppressed indices in () gives: ðÞ

◊w □v ð9xÞ w

w ðRw x & Wv xÞ:

Finally, for the case where a predicate, quantifier domain, or modal operator is to be given wide scope, the symbol ‘α’ for the actual world can be used as the index. Thus, () is equivalent to: ðÞ

◊w □v ð9xÞ α

w

w ðRw x & Wv xÞ:

Applying the method of indexing to the plurally de re readings of (◊A) and (□I), we have, with all suppressed indices restored, respectively: ðÞ

◊w ðxÞ

ðÞ

□ w ð9xÞ

α

α

α ðFα x

! Gw xÞ:

α ðFα x & Gw xÞ:

27

The method of indexing, it is clear, provides the resources to formalize the plurally de re readings of modal sentences, even those involving multiply iterated modality.²⁸ The method of indexing is powerful. But if the goal is to formalize English modal discourse within the framework of quantified modal logic, then the method is a cheat. The “indices” are nothing but variables ranging over possible worlds; the “indexed” modal operators are full variable-binding operators—namely, (variably) restricted quantifiers over possible worlds. To see this, note that sentences () through () can be transformed into sentences of first-order world theory by making the following notational substitutions (where vRw iff v is accessible from w, and xIw iff x exists at w): ‘Fwx’

‘Fw x’

for

‘ðvÞðvRw ! ̲ ̲ ̲Þ’

for

‘□vw ’

‘ð9vÞðvRw & ̲ ̲ ̲Þ’

for

‘ ◊w ’

v

‘ðxÞðxIw ! ̲ ̲ ̲Þ’

for

‘ðxÞw ̲ ̲ ̲’

‘ð9xÞðxIw & ̲ ̲ ̲Þ’

for

‘ð9xÞw ̲ ̲ ̲’

²⁷ If ‘F ’ or ‘G’ is complex rather than atomic, then each atomic predicate, quantifier, and modal operator within ‘F ’ or ‘G’ is to be appropriately subscripted. ²⁸ The method of indexing used here is similar to that introduced in Peacocke () but with the following difference: Peacocke introduces indexed operators, ‘Ai’, to tie the evaluation of predicates, quantifiers, and operators within their scope to a previous modal operator with index ‘i’; I directly index atomic predicates, quantifiers, and operators to previous modal operators. The two methods are equivalent if nesting of the indexed operators ‘Ai’ is permitted (with precedence given to the innermost competing operator). Indexed operators (with nesting) are used extensively in Forbes ().

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

        ()

Applying these substitutions to () and () results in the following formalizations of (◊A) and (□I) within first-order world theory: ðÞ

ð9wÞðwRα & ðxÞðxIα ! ðFαx ! GwxÞÞÞ:

ðÞ

ðwÞðwRα ! ð9xÞðxIα & ðFαx & GwxÞÞÞ:

The method of indexing is a notationally deviant way of formalizing modal sentences within first-order world theory. As such, it does nothing to help accomplish the present goal, explicated in Section ., of formalizing modal sentences within the framework of quantified modal logic.²⁹

. Formulating the Plural De Re: Quantifying over sets; Plural Quantifiers Let us turn, then, to a third strategy for formalizing modal examples of the plural de re. As seen above, we need to extend quantified modal logic so as to allow the scope of a predicate to vary independently of the scope of the quantifier that governs it. In the previous section, we considered extensions to quantified modal logic that represented the scope of a predicate not, as is customary, by its syntactic placement, but by means of exotic modal operators. In this section, I consider the other tack: representing the scope of a predicate by syntactic relocation. Thus, consider the plurally de re (◊A). It can be paraphrased in a way that locates the general term ‘F ’ outside the scope of the modal operator: The Fs are such that, possibly, all of them are G. But translation of this paraphrase into quantified modal logic is blocked because there is no way to capture the pronoun ‘them’ as it refers back plurally to the Fs. This suggests that the deficiency in standard quantified modal logic resides not in its modal apparatus, but in its apparatus for expressing plurality. One solution, familiar from other contexts, is to systematically replace plural reference by singular reference to sets. That will turn, for example, the above paraphrase of (◊A) into: The set of Fs is such that, possibly, all of its members are G. And this can straightforwardly be formalized if we add to standard quantified modal logic first-order quantifiers ranging over sets. Before turning to particular formalizations, however, we need to say how modal sentences with quantifiers over sets are to be interpreted. The interpretation of such sentences will depend on our choice of modal set theory, that is, on decisions about the modal properties of sets.³⁰ Different decisions are possible, but the most natural, I think, are these:

²⁹ I here disagree with Forbes (: –) when he denies that the indexed operators “are really nothing but devices for disguised quantification over worlds” on the grounds that “each successive step in introducing the operators was motivated by the production of an English sentence which required . . . the operator introduced at that step.” But, first, Forbes has not shown that the English sentences he gives require the use of indexed operators; indeed, some of them can be handled by the method to be introduced below. And, second, if some English sentences do require indexed operators, why not conclude, in light of the above equivalences, that some English sentences involve disguised quantification over worlds? ³⁰ For developments of modal set theory, see Fine () and Forbes (: –).

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

  ;   (A) (A)

(A)



Contingency of set existence. A set exists at a world if and only if all of its members exist at the world. Necessity of set membership. If an entity is a member of a set at some world, then it is a member of that set at every world—including worlds at which either the entity or the set doesn’t exist. Transworld criterion of identity. A set existing at one world is identical with a set existing at another world if and only if they have the same members.

We are now in a position to provide translations of the plurally de re readings of (◊A) and (□I). Suppose we add to quantified modal logic first-order variables ‘s’ and ‘t’ (with or without subscripts) ranging over sets. Consider the following translation of (◊A): ðÞ

ð9sÞððyÞðy 2 s $ FyÞ & ◊ ðx 2 s ! GxÞÞ:

It is clear that () is de re, although the res in question is a set rather than an individual: it asserts of the set of Fs that all of its members might be G. It is perhaps not surprising that one way to express the plural de re is to make assertions that are de re sets. Note that () contains an outer quantifier ‘’ for reasons similar to those given in discussing () above; for if there is a world at which some actual Fs fail to exist and the rest of the actual Fs are G, then the version of () with an inner quantifier comes out true, although (◊A) as intended may be false.³¹ Applying the same technique to (□I) results in the translation ðÞ

ð9sÞððyÞðy 2 s $ FyÞ & □ðx 2 s & GxÞÞ:

() asserts of the set of Fs that at least one of its members must be G. Note that there is no danger in (), as there was with (), that the outer quantifier will range too widely: the use of an inner quantifier ‘(y)’ together with the restriction on the outer quantifier given by ‘x2s’ ensure that only the actual Fs will be relevant to the evaluation of truth value. Note further that () has existential import as required. Finally, note that the assertion that (◊A) or (□I) is possible or necessary, such as the once problematic (), can be translated, as would be expected, simply by prefixing a diamond or a box to () or (). Can the addition of quantifiers over sets (together with outer quantifiers) match the expressive power of the method of indexing without introducing the equivalent of variable-binding operators ranging over possible worlds? Indexed predicates can always be eliminated using quantified set variables in accordance with the following schema (where ‘O’ is ‘□’ or ‘◊’ or absent if the index is ‘α’): Ow ð:::Fw x::: Þ ¼ Ow ðððy 2 s $ FyÞ & ð::: x 2 s :::ÞÞÞ:

³¹ Michael Jubien suggested ‘(9s)((y)(y2s $ Fy) & ◊((9t)t=s & (x)(x2s ! Gx)))’ as a translation of (◊A). This avoids outer quantifiers and captures an intuition that worlds at which the set of actual Fs fails to exist are to be ignored. But such worlds cannot always be ignored, as is seen by considering the plurally de re sense of ‘Everyone in the room might fail to exist’. I prefer to provide uniform translations for all sentences of the form (◊A), and to ignore worlds, when appropriate, by restricting the accessibility relation.

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi



        ()

(Note that outer quantifiers are needed for the general case to ensure that the set picked out contains all the possible individuals that are F at the world in question, whether or not they exist at the world.) Indexed inner quantifiers can always be replaced by outer quantifiers restricted by an indexed existence predicate; and then the indexed existence predicate can be eliminated as above. This results in the following schema for replacing the indexed universal quantifier: Ow ð:::ðxÞw ð ̲ ̲ ̲Þ::: Þ ¼ Ow ðððy 2 s $ EyÞ & ð:::ðx 2 s ! ð ̲ ̲ ̲ÞÞ:::ÞÞÞ; and similarly for the indexed existential quantifier. (The existence predicate, ‘Ey’, is definable by ‘(9x)x=y’.) Superscripts may be dropped from modal operators as soon as all subscripts have been dropped to which they were previously tied. Applying the above schemata to () and () results in sentences longer than, but logically equivalent to, () and (). If the underlying modal logic is S, then sentences with subscripted modal operators are equivalent to their unsubscripted counterparts interpreted as having their subscripts suppressed; so subscripts on modal operators can simply be dropped. But if the underlying logic is not S, subscripts on modal operators cannot always be eliminated. For example, quantification over sets can do nothing to help formalize ‘Necessarily, something exists that might not have existed’ when it is given the following reading using indexed operators: □wα (9x)w ◊vα~Evx. Thus, the addition of quantifiers over sets cannot quite match the power of indexing—that is, of full quantification over possible worlds. But the cases in which it falls short have nothing to do with the plural de re. The use of sets to formalize the plural de re is on the right track, I think; but a serious problem remains. There are sentences of the form (◊A) and (□I) for which the formalizations given above, () and (), fail even to get the truth value right. Whenever there are “too many” Fs for them to form a set, () and () come out false; but the corresponding English sentences might well be true. For example, consider the plurally de re reading of ()

Every impure set might fail to exist,

where a set is impure if one of its members, or its member’s members, . . . , is not a set. Assuming that there is a world at which all the actually existing non-sets fail to exist, () is true. But the formalization of () given by () is false, since there exists no set whose members are all and only the impure sets. In general, quantified modal logic with quantifiers over sets cannot be trusted to translate a plurally de re sentence containing the general term ‘F ’ unless the Fs form a set at every world.³²

³² One might be tempted to replace quantifiers over sets with quantifiers over classes, thus allowing the initial quantifiers in () and () to range over proper classes as well as sets. But that would be a mistake. For one thing, embarrassing questions will arise when the Fs are themselves proper classes, and so do not even form a class. More importantly, proper classes are dubious entities, and I for one do not believe in them. Yet, clearly, I can believe that () is true without inconsistency. [Proper classes were made more respectable by David Lewis if you buy into his mereological account of classes; see Lewis (). But today I would say that sets, no less than proper classes, are dubious entities; see Chapter  and the postscript to Chapter .]

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

  ;  



Although () and () give the wrong truth value only in the special case where there is no set of Fs, this failure is symptomatic, I think, of a more general problem. Even when the Fs do form a set, the plurally de re (◊A) and (□I) make no reference to this set. For it is clear that (), which is of the form (◊A), makes no reference to a set of all impure sets. Assuming that all plurally de re sentences of the form (◊A) are to be translated alike, it follows that no translation of (◊A) should make reference to a set of Fs, whether or not such a set exists. We need a means for referring plurally to the Fs that makes no mention of the set of Fs. To find such a means, we need look no further than our native language: English. Return to the paraphrase of (◊A) given above by: The Fs are such that, possibly, all of them are G. This in turn can be paraphrased: There are some things such that each of them is F and each F is one of them and, possibly, all of them are G. English already contains just the device we need for referring plurally to the Fs without mentioning the set of Fs: the plural quantifier ‘there are some things such that . . . they (them) . . . ’.³³ I thus propose that we add the plural quantifier to quantified modal logic, and use plural quantification instead of quantification over sets to formalize the plural de re. If the plural quantifier is represented by means of a second-order existential quantifier, ‘(9X)’, then (◊A) and (□I) can be formalized, respectively, by: ðÞ

ð9XÞððyÞðXy $ FyÞ & ◊ðXx ! GxÞÞ:

ðÞ

ð9XÞððyÞðXy $ FyÞ & □ðXx ! GxÞÞ:

(More generally, in the translation schema above, ‘’ can be replaced throughout by ‘’, and ‘x2s’ by ‘Xx’.) It is important to realize that the quantifier ‘(9X)’ in () and () ranges neither over sets, nor classes, nor properties; it ranges in an irreducibly plural way over the Fs themselves. Truth conditions for sentences of quantified modal logic with plural quantifiers can be given within a metalanguage that itself partakes of plural quantifiers; and that is enough, since plural quantification, being part of English, is antecedently understood.³⁵ To demand that such truth conditions be given without using plural quantifiers is no more legitimate here than it would be to demand that truth conditions for individual quantifiers be given without

³³ George Boolos has championed the use of plural quantifiers in a series of recent articles. See especially Boolos (). ³⁴ In the case where there are no Fs, we want () and () to agree in truth value with () and (); so in this case the second-order quantifier ‘(9X)’ cannot be read as ‘there are some things such that’. This minor mismatch between the formal language and English is no more problematic here, however, than it is with the individual quantifiers. An exact scheme for translating second-order sentences, such as () and (), into English can be found in Boolos (). [Today I would formulate plural quantification using special plural variables (‘xx’, ‘yy’) and a predicate for the “is one of” relation (‘κa-diverse. The second principle we need is: Substructure Principle (SP). Any (spatiotemporally connected)²² substructure of a possible world structure is a possible world structure. These two principles allow us to reformulate the Forrest-Armstrong argument in the following way. After (F), add: (F.)D (F.)D (F.)D

Assume: Principle of Structural Diversity (PSD) and Substructure Principle (SP). The world structure of Giganto has size κa(by (F)). Therefore by (PSD): There are more than κa non-isomorphic world structures. Let WD be a set of more than κaworlds no two members of which have isomorphic world structures. Let GD be a set of duplicates of these worlds in Giganto. For each individual d in GD, let Ad be the atoms in d, and let ℬd be the arrangement of Ad in Giganto restricted to the substructure determined by Ad.

Then the argument continues as before, substituting a version of (F)** in place of (F). ((F) is no longer relevant.) That results in the following: (F)**D (F)D

Therefore by (SP) and Strong (LPR): For each subset Ad of A, there is a variant of Giganto that recombines members of Ad according to ℬd but without the members of A – Ad. There are more than κasuch variants of Giganto.

(F) through (F) are unchanged except that ‘more than κa’ is substituted for ‘κa– ’. (F)D follows from (F)**D for the following two reasons: first, if d 6¼ e, for d and e in ²² I would say instead “externally connected” because I do not think that all worlds are unified by spatiotemporal relations; see Chapter . Some qualification is needed because no world divides into spatiotemporally disconnected parts (on Lewis’s account), or externally disconnected parts (on my account). For more on how I understand the Substructure Principle, see the postscript to Chapter .

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

  :    



GD, then Ad does not overlap Ae and so the variant based on Ad is different from the variant based on Ae (because these variants have non-isomorphic world structures); and, second, the cardinality of GD is greater than κa. Appealing to (PSD) and (SP) has successfully bridged the gap.²³ It remains to ask: why accept (PSD) and (SP)? As I said above, they follow from the account of plenitude of structures that I accept. I defended that account in Chapter  and its postscript, so here I will be brief. Roughly, I hold that a structure is possible— instantiated in some possible world—if either (S) it plays, or has played, an explanatory role in our theorizing about the actual world, or (S) it belongs to a natural (mathematical) generalization of structures that are possible in virtue of (S). (Compare the principles (B) and (PPS) in Chapter .) For example, a Euclidean space of three dimensions is clearly possible by (S); and therefore so are Euclidean spaces of one and two dimensions (since they are instantiated whenever Euclidean space of three dimensions is instantiated). Euclidean spaces of any finite dimension are possible by (S), being a natural generalization of Euclidean spaces of dimension one, two, or three. I also hold (S) that any (spatiotemporally connected) possible structure is a possible world structure—the complete underlying structure of some possible world. (Compare the principle (PW) in the postscript to Chapter .) Returning to our example: it follows from (S) that, for each finite n, there is an (entirely spatial) world with n-dimensional Euclidean space. The Substructure Principle (SP) follows immediately from this account of the plenitude of structures. Any (spatiotemporally connected) substructure of a possible world structure is automatically a possible structure, and so, by (S), a possible world structure. The Substructure Principle serves to ground a Humean principle of plenitude that I, and many others, accept, a principle I call the Principle of Solitude: for any (spatiotemporally connected) possible individual, there is a world containing a duplicate of that individual and nothing that isn’t a part of that duplicate (not even empty spacetime). (For the derivation of this principle, see the postscript to Chapter ; for an application of the principle, see Chapter .) The principle of solitude provides one way—though not the only way (see Chapter )—of supporting the Humean denial of necessary connections (DNC). The principle of structural diversity (PSD) also follows from my account of plenitude. A simple way to illustrate this invokes what I have elsewhere called “worlds with ordinal time.” (See Section . for a more detailed version of the argument below; see also Chapter  for more discussion of worlds with transfinite duration.) First, it follows from (S) that temporal structures with discrete time are possible, including finite structures with n instants of time, for any (non-zero) natural number n. For, surely, both discrete time and finitism have been taken seriously in our theorizing about the actual world. It then follows from (S) that,

²³ Have we strayed from the core of the original Forrest-Armstrong argument by no longer appealing to a Cantorian diagonalization argument? The argument is still a Cantorian argument that there can be no largest size. Or we could more closely follow the original Forrest-Armstrong argument by switching from worlds with ordinal time (introduced below) to worlds with linear time, and make use instead of the . This theorem is proved by theorem that there are κanon-isomorphic linear orderings of size k, for infinite κa a straightforward Cantorian diagonalization argument. See Rosenstein (: ).

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

  -  (/)

for any ordinal number αa, temporal structures with discrete time where the instants are temporally well-ordered like the ordinal αaare possible structures.²⁴ For, surely, the ordinal numbers are a natural generalization of the natural numbers. Finally, it follows from (S) that, for each ordinal αa, there is a world structure, and so an (entirely temporal) world, with αainstants of time.²⁵ We have then: (OT)

For any ordinal αa, there is a world structure isomorphic to αa.

We can use (OT) to argue for (PSD) as follows. Consider any cardinal κa. There are more than κaordinals of size at most κa.²⁶ Different ordinals correspond to nonisomorphic world structures of worlds with ordinal time. Therefore, there are more than κanon-isomorphic world structures of size κa, which is to say that modal reality is >κa-diverse. This shows that, in the Forrest-Armstrong argument fortified with (PSD) and (SP), just focusing on the worlds with ordinal time and their duplicates in Giganto is enough to generate the contradiction. Again, we have successfully bridged the gap.

. Worlds Whose Atoms are Many? Nothing stronger than (PSD) was needed to bridge the gap in the Forrest-Armstrong argument. But in supporting (PSD) with (OT), we have invoked a much stronger principle. For it follows immediately from (OT) that for any κa, modal reality is >κa-diverse. It might seem that this stronger principle undermines, rather than supports, the Forrest-Armstrong argument. Here’s why. The principle (OT) leads directly to the conclusion that the worlds are many, that there is no set of worlds. Accepting that conclusion, however, suggests a way to protect (UPR) from the fortified Forrest-Armstrong argument. Thus far, we have treated the ForrestArmstrong argument as a reductio with (UPR) as its target. But (UPR) wasn’t the only assumption of the argument (in addition to the other principles of plenitude). There was also (F), the assumption that the atoms in any world are few, and form a set. Given that the worlds are many, (UPR) is in direct conflict with (F): the world Giganto guaranteed to exist by (UPR) will have non-overlapping duplicates corresponding to all the worlds, and so will itself have many atoms. That suggests two things. First, the Forrest-Armstrong argument wasn’t needed to attack (UPR): (F) and the fact that the worlds are many can do the job all by themselves. But, second, if we instead want to hold on to (UPR), we can use it and the fact that the worlds are many to reject (F). In that case, we simply take the target of the fortified ForrestArmstrong argument to be (F) rather than (UPR). (UPR), it appears, is immune even to fortified versions of the Forrest-Armstrong argument. ²⁴ As is usual, I understand an ordinal αato be a well-ordered set every member of which is the set of its predecessors. An ordinal, then, is well-ordered by 2. ²⁵ One might wonder what distinguishes a temporal structure with αainstants from a one-dimensional spatial structure with αapoints. I would say: they do not differ intrinsically; they differ, when they do, in virtue of the pattern of instantiation of qualitative features. But note that the present argument can be run without supposing the structure is intrinsically temporal or spatial. ²⁶ That is how the next cardinal number after κais defined when cardinals are identified with initial ordinals. See, for example, Hartogs’s Theorem and its proof in Enderton (: –).

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    ?



But rejecting (F) cannot provide a defense of (UPR). (F) was not essential to the Forrest-Armstrong argument. It was merely a convenient posit that allowed the argument to be given a set-theoretic formulation. Even if we were to allow that worlds may have many atoms, we could still prove the contradiction that Giganto is bigger than itself. The proof can be carried out in various frameworks that invoke proper classes, such as Bernays-Gödel class theory with urelements.²⁷ Or it can be carried out in Lewis’s “megethology,” without ever mentioning proper classes (or sets, for that matter). I will here just sketch informally how such a proof would go. Let us say that some things are barely many iff they correspond one-one with the cardinals of pure set theory.²⁸ (F) and (F) are replaced with the weaker: (F)+ (F)+

Assume: For any world, the atoms in that world are few or barely many. Then: The atoms in Giganto are few or barely many. Call the atoms in Giganto aa.

We then complete the argument as a simple dilemma. One fork, assuming that aa are few is unchanged. The other fork, assuming that aa are barely many, continues as follows: (F)+ (F)+ (F)+

There are many, but not barely many, fusions of atoms among aa. For each fusion of atoms among aa, there is a variant of Giganto in which just the atoms in that fusion remain and the rest have been deleted. There are many, but not barely many, such variants of Giganto.

(F) through (F) are unchanged except that ‘many, but not barely many’ is substituted for ‘κa – ’. (F)+ follows immediately from (F)+ for the haecceitist version of the argument. The anti-haecceitist can fortify the argument by invoking principles of plenitude stronger than (PAD) or (PSD). Thus, we can strengthen (PAD) by adding: (PAD)+ If some world contains barely many atoms, then some world contains barely many atoms, no two of which are duplicates of one another. (PAD)+ together with (UPR) will guarantee that, if Giganto contains barely many atoms, then Giganto contains barely many kinds of atom. And then, by an argument paralleling the argument from (PAD) in Section ., it will follow that there are many, but not barely many, variants of Giganto. Invoking (PAD)+ bridges the gap on the “barely many” fork of the dilemma. Alternatively, we can bridge the gap using the plenitude of world structures. We can strengthen (PSD) by adding: (PSD)+

If some world contains barely many atoms, then there are many, but not barely many, non-isomorphic world structures.

²⁷ See Uzquiano (: ) for the coding trick that will be needed to carry out the reasoning in BernaysGödel class theory. Hawthorne and Russell () give a detailed version of the Forrest-Armstrong argument where (F) (what they call “Marble Set”) is rejected, and worlds are allowed to be large. ²⁸ Lewis (b) has instead this definition: xx are barely many iff they correspond one-one with all the atoms in reality. On the significance of this difference, see Section . below.

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

  -  (/)

We can support (PSD)+ in the same way that we supported (PSD) by considering worlds whose instants of time are well-ordered. If we were to allow worlds whose atoms are barely many, there could be no good reason not to allow a world whose instants of time are ordered like all the ordinals, one instant for each ordinal number. But we couldn’t stop there. There would be a world with one more instant, and another, and so on. To turn this into a proper argument, we could introduce, on top of the ordinal numbers, the super-ordinal numbers. For every way of well-ordering barely many elements, let there be a super-ordinal number that represents that wellordering.²⁹ Just as the ordinal numbers are a natural generalization of the natural numbers, the super-ordinal numbers are a natural generalization of the ordinal numbers. Given the account of plenitude of world structures I endorsed above, we will have: (OT)+

For any ordinal or super-ordinal αa, there is a world structure isomorphic to αa.

(OT)+ will entail (PSD)+ by the same reasoning that (OT) entails (PSD). And then, by an argument paralleling the argument from (PSD) in Section ., it will follow that there are many, but not barely many, variants of Giganto. Invoking (PSD)+ gives a second way of bridging the gap on the “barely many” fork of the dilemma. I conclude that (F) was not needed for the Forrest-Armstrong argument. A weaker assumption, (F)+, will do. But wait: why not take the +-version of the Forrest-Armstrong argument to be a reductio of (F)+ and allow that the atoms of some world may be many, but not barely many? Well, you know how the dialectic will go. Whatever imaginary über-size we allow worlds to have, the über-sized version of the Forrest-Armstrong argument will show that either (UPR) must be rejected, or we must allow worlds to have some über-über-size. If I thought that modal reality was indefinitely extensible, I suppose that this would allow me to hold on to a modalized version of (UPR). But if, as I have been assuming, modal reality has some determinate size, whatever it may be, then some version of (F) that gives the size of worlds must hold, and the Forrest-Armstrong reductio will be aimed squarely at (UPR). There is no escape.

. Modal Reality and Limitation of Size The upshot of the somewhat intricate argumentation of the last three sections is this. The Forrest-Armstrong argument, when fortified in various ways by plausible principles of modal plenitude, is alive and kicking. And so the unrestricted principle of recombination (UPR) must go. It must be replaced by a restricted principle such as ²⁹ These super-ordinals should not be thought of as set-like entities. They are just well-ordered pluralities. Hazen (, ) would claim that it is not even conceptually possible to posit “superordinals,” or, indeed, a world whose atoms are many. Any attempt to do so just extends the domain of sets so as to include the posited entities. It is constitutive of the distinction between sets and proper classes that proper classes always remain beyond the horizon of our speculations about reality. Hazen’s view raises deep questions. In short, I would say that it rests on a conception of reality as indefinitely extensible that I reject.

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     



(LPR). (Or strong (LPR). But if one follows me in accepting (PIP), strong (LPR) can be derived from (LPR) and (PIP); see Chapter .) According to (LPR), some things can be recombined only if there is a possible arrangement of those things. And what arrangements are possible will depend in part on what world structures there are within which to do the arranging. The question—what is the appropriate restriction on recombination?—is reduced, at least in part, to the question: what world structures are possible? For the purposes of this chapter, what matters most is just one aspect of that question: what are the possible sizes of world structures? The principle (OT) gives part of the answer. For every cardinal κa, there is a world, and so a world structure, of size κa. There is no cardinal bound on the size of worlds, or world structures. That tells us that the worlds are many, but it leaves open how many. To get purchase on that question, we need to make some supposition as to what the possible sizes are that exceed every cardinal. I propose, in line with many others, that we endorse some version of the doctrine of Limitation of Size. By Limitation of Size, I mean more than the claim that some pluralities are “too big” to form a set. What I take to be essential to the doctrine is, first, that there is some maximal size, the size of all the parts of reality, and, second, that the cardinal numbers of ZFC determine this maximal size. That there is some maximal size follows from my view that reality is definite, not indefinitely extensible; and this follows from my rejection of primitive modality. In saying that the cardinal numbers determine this maximal size, I do not say that there is only one size that exceeds all the alephs. There are more fusions of cardinals than cardinals (if, as I think, the pure sets are atoms and therefore do not overlap). So there are at least two sizes that exceed all the alephs. Moreover, for all I know, there may be sizes between the size of the cardinals and the size of the fusions of the cardinals. (To deny that would be to endorse, for no known reason, an even more general General Continuum Hypothesis.) So, in saying that the cardinals determine the limit on size, I am saying that the maximal size is given by the size of the fusions of the cardinals, not by the size of the cardinals.³⁰ The possible sizes divide into three categories—few; barely many; and many, but not barely many— where the third category includes a maximum size. If a plurality has this maximal size, we can say that it is maximally many. With Limitation of Size in place, we can give definite answers to the two questions about the size of modal reality raised in Section .. First, we have: Size of Reality.

The worlds in modal reality are barely many.

³⁰ If one accepted proper classes as parts of reality, then one could instead take the classes of cardinals to give the maximal size. But if, like me, one takes talk of proper classes to be best interpreted in terms of plural terms and plural quantifiers, then this would amount to taking the pluralities of cardinals to give the maximal size, and I would reject that because I do not apply Limitation of Size to pluralities. Indeed, since there are more pluralities of fusions of cardinals than there are fusions of cardinals, the size of the fusions of cardinals would not be the maximal size that a plurality could have. Could we instead take the number of pluralities of fusions of cardinals to give the maximal size? I think not. I speculatively endorse (in Chapter ) an “ideological hierarchy” that allows the taking, for any plurality, a plurality of its subpluralities, and so on without end. See Linnebo and Rayo () on the distinction between ontological and ideological hierarchies.

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

  -  (/)

For the worlds in modal reality are many, by (OT), and if the worlds were many, but not barely many, there would be more worlds than cardinals, and so more fusions of worlds than fusions of cardinals (since worlds do not overlap), contradicting Limitation of Size. Since worlds do not overlap, it follows from Size of Reality that the atoms in modal reality are many, and that modal reality is large. Second, returning to our question—what are the possible sizes of world structures?—we have what was earlier labelled ‘(F)’: Size of Worlds. For every world, the atoms in that world are few; in other words, every world is small. Moreover, for every cardinal κa, some world has exactly κa atoms. The second half, as noted above, follows from (OT). To see why the first half holds, suppose there was a world with barely many atoms. Then by (PSD)+ (which I defended in the last section on the assumption that some world has barely many atoms), there would be many, but not barely many, non-isomorphic world structures, and so many, but not barely many worlds, contradicting Size of Reality and Limitation of Size. Size of Reality and Size of Worlds together entail that there are barely many atoms in modal reality. It then follows that, although there is more than one way for a plurality to be many, there is only one way for a part of reality to be large: a part of reality is large if and only if it contains barely many atoms. This neat little package respects the fortified versions of the Forrest-Armstrong argument as well as all of the principles of plenitude that I have endorsed throughout the course of this chapter.³¹ It ensures that there will be a restriction on recombination sufficient to avoid the paradox. For according to Size of Worlds, every world is small, and so no world is big enough to recombine all the many worlds. There is no one world such as Giganto that reflects all of modal reality, and so (UPR) can be safely rejected. Is there any cost at all to rejecting (UPR)? One might think that (UPR) is needed to support an intuition that universal actualization is possible. How, without Giganto, do we get the possibility that the goings-on of every possible world could be actualized together? But (UPR) is not needed, nor wanted, to capture this possibility. In Chapter , I argued that we need to divorce what is possible from what is true in some possible world. Rather, possibility should be analyzed in terms of plural, not singular, quantification over worlds. To be possible is to be true in some world or some worlds. The possibility of (absolutely isolated) island universes, for example, is the possibility that more than one world is actual. The possibility of universal actualization, then, is just the extreme case of the possibility of island universes: the possibility that every world is actual. So, if one rejects the orthodox (and Lewisian) analysis of possibility as I do, there is no problem accommodating universal actualization without (UPR). And that is to the good, because the intuition supporting this possibility has nothing to do with recombination. Even if there were such a world as Giganto, it would be a poor substitute for the possibility of universal actualization. For the situation in which Giganto is actualized and other worlds are not only

³¹ It is a package I have been promoting since the mid-s. See Chapters , , and .

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     



indirectly represents the situation in which all of modal reality is actualized. Worse, it imputes additional structure—the world structure of Giganto—that was no part of the possibility in question. Recombination is needed to capture many of our beliefs about the plenitude of possibilities. But our belief in the possibility of universal actualization is not one of them. Is there a cost to founding an account of modal reality on a doctrine of Limitation of Size? Limitation of Size is a fundamental assumption that does not follow from more fundamental assumptions; but if that is a cost, it is a cost borne by all theories. Still, even fundamental assumptions need reasons to believe them. My version of Limitation of Size holds that the mathematical part of reality provides an absolute measure of size for all of reality. In particular, the size of modal reality is constrained by the size of the cardinals. There can be too many atoms in modal reality for them to form a set, but not too many for them to be in one-one correspondence with the cardinals. This must be included as a substantial “principle of the framework.” (On “principles of the framework,” see the postscript to Chapter .) I confess I do not know how to defend the truth of this principle. No doubt there are pragmatic arguments based on the relative simplicity of the resulting account of reality, or based on some doctrine of philosophical conservatism. But I do not take such arguments to bear directly on truth. (See Chapter .) Limitation of Size is a reasonable supposition to make: it attributes to reality the minimal size that leaves room for all the principles of plenitude I have reason to believe; and selecting any other size for reality would be arbitrary. But I am doubtful that that provides much of an argument for its truth. Lewis () also endorses a version of Limitation of Size according to which there are at least two sizes beyond the alephs. On his account of classes, classes are fusions of singletons. It follows that there are more classes than there are singletons and, since the singletons are many, at least two ways for a plurality to be many. Lewis’s account differs from the account I have given above, however, by defining ‘many’ and ‘barely many’ relative to reality as a whole. He does not take the cardinals of pure set theory to provide an absolute measure of size. Lewis asserts, as I do, that the atoms in reality are barely many, where for him the atoms include the singletons. But for him this is the definition of ‘barely many’, and so not a substantial assumption. Does that make his account preferable in virtue of carrying less baggage? I think not. Lewis’s account leaves it open whether the cardinals of pure set theory are barely many, or some lesser size. For all that Lewis explicitly says, there may be fewer cardinals than there are atoms in modal reality, atomic urelements.³² No doubt Lewis did not think there were more atomic urelements than cardinals. But his account does not seem to rule this out. He has two options. He can add the assumption that the cardinals are barely many, thus equating the size of the cardinals with the size of the atoms in modal reality. But then, definitions aside, the account would carry the same baggage as my account, and be no easier to defend. Or, he can decline to make the ³² By ‘urelement’, I mean any entity that is not a set, or class, or other set-like entity. Lewis’s definition of ‘urelement’ is more restrictive, requiring that urelements be individuals, that is, members of some class. Mixed fusions of classes and individuals are not members of classes for Lewis, and so not urelements. But note that the usages agree on what counts as an atomic urelement.

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

  -  (/)

assumption, keeping all measures of size beyond the alephs relative, and not absolute. He can plead ignorance, leaving it entirely open how many atomic urelements there are in any absolute sense. But then I do not see how the account deserves to be called “Limitation of Size.” No limit is being placed on the size of reality. It is whatever size it is, and all measures of size are determined relative to that. Might it be possible to do away with Limitation of Size and rest solely on the iterative conception? We can still say, based on the iterative conception, that some pluralities are “too big” to form a set. But we put no limits on how big these pluralities can be. Indeed, Size of Worlds can be based on an iterative conception of world structures if we take world structures to be represented by sets (as I do). Let us say that a mathematical structure is an ordered pair whose first element is a pure set, the “domain” of the structure, and whose second element is a set of relations over that domain, where a relation over the domain is a set of ordered pairs of members of the domain, and ordered pairs are taken to be sets in the usual way. Then every mathematical structure is itself a set that shows up in the iterative hierarchy five levels beyond the level where the domain shows up. It follows that there are barely many mathematical structures. Now, I am not suggesting that we identify world structures with mathematical structures. For one thing, world structures have coarser individuation conditions. For another, I have not been supposing in this chapter that every mathematical structure corresponds to a world structure, only that the spacetime mathematical structures do.³³ But it is enough to say that each world structure is represented by some mathematical structure, where the (atomic) places in the world structure correspond one-one to the members of the domain. There is thus a mapping from some of the mathematical structures onto the world structures. It follows that there are barely many world structures, and that the (atomic) places of any world structure form a set. An iterative conception of mathematical structures thus leads to an iterative conception of world structures. And an iterative conception of world structures is all that is needed to get Size of Worlds. (Remember, we are still assuming that there is no co-location; I will revisit that assumption in the final section.) But an iterative conception of world structures does nothing to support Size of Reality. We need Limitation of Size to get any definite claim about the size of modal reality as a whole. For if any atom, or any world, has many, but not barely many, duplicates in modal reality, then Size of Reality is false. The iterative conception plausibly places a limit on the plenitude of world structures, but not on the plenitude of world contents. We need a limit on both to get a definite size for modal reality. We need Limitation of Size. Of course, one can object in familiar ways to the iterative conception of mathematical structure. Consider the ordinals. On the usual construction of the ordinals as sets, the ordinals are well-ordered by 2, the membership relation. Doesn’t that show that the ordinals form a mathematical structure, a structure beyond all the levels of

³³ In fact, I hold to a very expansive principle of plenitude for structures: every mathematical structure is instantiated by some maximal isolated portion of reality; see the postscript to Chapter . And I hold that mathematical structures may be “higher order,” and thus go beyond the relational structures mentioned in the text; see Chapter . But all that is compatible with an iterative conception of structure. See Hawthorne and Russell () for ways of regimenting principles of this sort within model theory.

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     



the iterative hierarchy? But then shouldn’t there also be a corresponding world structure, and a world with ordinal time whose instants of time are unbounded by any ordinal number? I think not. The pressure to accept large structures is really no different from the pressure to accept large set-like entities beyond all the sets, such as proper classes. I have nothing new to add to the debate. To accept proper classes, or large structures with proper class domains, where these classes are somehow forbidden from being members of other classes, is unacceptably ad hoc. To allow these large set-like domains to be members of other set-like domains gains nothing, and lands us back in the same dilemma. The best response is to reject the large set-like entities and structures. The intuition that whenever there are some things, there is a set-like entity with those things as members, or that whenever there are some things related in some way, there is a structure with a set-like domain with those things as members, is faulty, and has to go. It is a vestige of a general conflation in thought and language between pluralities and single entities. When there are some birds flying overhead, I may say that there is a flock of birds. I may add that the flock has the shape of a vee. But I don’t thereby refer to a set or set-like entity. What I say can be fully captured within a language with plural terms and plural quantifiers. It is no different with the ordinals, or other pluralities of many things. We can say everything we want to say about the ordinals plurally, including how they are well-ordered, using plural terms and irreducibly plural quantification. We can even compare the sizes of pluralities of many things by quantifying over relations between those pluralities where this is reduced to plural quantification over ordered pairs and ordered pairs are construed non-set-theoretically. (For example, we can quantify plurally over things that encode the ordered pairs mereologically; for one way of doing this, see Lewis b: –.) I do not see any way that large set-like entities or structures will be missed.³⁴ One might try to argue directly for a large world, and so a large world structure, whose instants of time correspond to the ordinal numbers. For although I deny that whenever there are some things, there is a set-like entity with those things as members, I accept that whenever there are some things, there is fusion with those things as parts. If there were worlds with ordinal time nested one within the next, with one world for each ordinal, then indeed taking the fusion of these worlds would lead to a large world with as many instants of time as all the ordinals. But here is where the thesis of structural pluralism introduced in Section . comes into play. The worlds with ordinal time are all isolated from one another, and do not overlap. They have a fusion all right; but their fusion is an aggregate of worlds, not a world. I conclude that there are no good reasons why an account that accepts the worlds with ordinal time, all of which are small, must also accept a large world whose instants of time match the entire sequence of ordinal numbers.

³⁴ Hawthorne and Russell (: ) introduce what they call “plural structures”—certain indexed families of pluralities—whose domains may be too big to form sets. If we allowed such “plural structures” to be instantiated at worlds, then Size of Worlds, of course, would be false. I have no objection to speaking of these “plural structures,” but I do not count them as possible structures in the relevant sense: they are not candidates for providing the underlying or instantial structure at worlds. Worlds I take it are essentially unified. Instantiating a “plural structure” does not bestow the requisite unification.

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

  -  (/)

. Hawthorne and Uzquiano: Arguments against Modal Realism Hawthorne and Uzquiano () reject the neat little package expressed by Size of Reality and Size of Worlds. They have two main arguments, one based on the Urelement Set Axiom, the other based on a version of Limitation of Size stronger than what I endorsed. They conclude that a “modal realist” will be forced to hold that there is some cardinal bound on the size of possible worlds, the view endorsed by Lewis (a). I argued against there being any such cardinal bound in Chapters  and . It would require that, for some ordinal αaand beyond, there are no worlds with ordinal time with αainstants of time. In this section, I say why I do not find their arguments persuasive. In short: one can reject with impunity the main assumption of each of their arguments. They also have an independent argument ad hominem against Lewis’s theory of classes. I do not accept Lewis’s theory (see Section .); but in any case, there is a natural fix that avoids the problem. An independent third argument based on the possibility of co-location can also be gleaned from their discussion. I postpone my response until the next section. I am still under the working assumption that all parts of worlds are spatiotemporal parts. The thesis Size of Worlds is captured by what Hawthorne and Uzquiano call “Indefinite Extensibility,” which they express as follows: Indefinite Extensibility. There could not be so many angels as to exceed each and every aleph, but for each αa, there could be exactly ℵαa-many angels in existence. Before discussing the arguments they give against Indefinite Extensibility, let me say three things with respect to how it compares with the thesis Size of Worlds that I endorse. First, a terminological point. I prefer to restrict “indefinite extensibility” to the view that some primitive modality is needed to characterize the extent of reality, a view I reject. They are not supposing that the modality in terms of which Indefinite Extensibility is expressed is primitive. Indeed, it is interpreted in terms of quantification over possible worlds when they present their arguments against the modal realist. Second, although the discussion around their introduction of Indefinite Extensibility suggests that they are concerned with how many angels can be colocated at a single point of spacetime, their two main arguments apply no less to how many angels can exist in a world, whether co-located or not. I consider the special problem raised by co-location in the next section. Third, nothing in their arguments depends on any theological assumptions about angels. I set their “transcendental theology” aside here, and consider only how many (mereological) atoms can exist in a world. Wherever they say ‘angels’ in their arguments, I say ‘atoms’. I turn now to their two main arguments, which I will take to be directed against Size of Worlds. Their first argument is just that Size of Worlds is incompatible with the Urelement Set Axiom, that the urelements form a set; for the second half of Size of Worlds entails that the atoms throughout modal reality are many. I do not contest that the Urelement Set Axiom is part of orthodoxy; it has been presupposed, if not explicitly stated, by the majority of those who accept an iterative conception of impure sets. (Zermelo (), however, is an important exception.) But this is only, I think, because prior to serious reflection on principles of modal plenitude, the idea that

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  :    



the urelements fail to form a set was given little serious consideration. In any case, accepting the Urelement Set Axiom is out of the question if, like me, one accepts unrestricted composition and holds that the pure sets (or even just the singleton pure sets) are mereological atoms. For any pure set, there is a distinct fusion of that pure set with the Eiffel Tower; so there are as many such fusions as there are pure sets, too many urelements for them to form a set. Orthodox or not, I do not know of any good reason to accept the Urelement Set Axiom. As noted in Section ., the cardinals of pure set theory are all that is needed to determine which pluralities of urelements form a set: a plurality of urelements form a set just in case they are in one-one correspondence with the members of some cardinal. Thus, if a plurality of urelements is few, they form a set; if many, they do not form a set. Given that we have the means to say which pluralities of urelements form sets and which do not, what reason could there be for requiring that the plurality of all urelements form a set? I am here disputing Lewis’s (a: ) claim that we “have no notion what could stop any class of individuals—in particular, the class of all worlds—from comprising a set.” In support of this, he writes: “the obstacle to sethood is that the members of the class are not yet all present at any rank of the iterative hierarchy. But all the individuals, no matter how many there may be, get in already on the ground floor.” I do not see why using the cardinals of pure set theory as an absolute measure of size for pluralities of urelements does not suffice to give us a notion of why some pluralities of urelements do not form a set. Thus there is no need to define an iterative hierarchy over the urelements in order to say which pluralities form a set. But if we choose, we can apply the iterative conception directly to the urelements and get the same result. Go ahead and put all the urelements (along with the null set) on the ground floor, at stage . Then use the hierarchy of pure sets to determine which impure sets to add at subsequent stages. In particular, in forming the next stage of the hierarchy based on the urelements, put in sets whose members are pluralities of sets and urelements from previous levels just when the cardinality of that plurality does not exceed all the cardinalities of the pure sets at that stage of the pure hierarchy. Eventually, all and only those pluralities of sets and urelements that are not too big to form a set will find a place in the hierarchy. Letting all the urelements “get in already on the ground floor” does not require that one ever introduce a set of all the urelements.³⁵ The main reason Hawthorne and Uzquiano give for accepting the Urelement Set Axiom is that rejecting it “threatens the universality of mathematics, which is supposed to investigate structures presented by the other sciences”: there may be “structures constituted by non-sets” that cannot be mathematically represented ³⁵ This is, in essentials, the method of Zermelo (), what he calls the “canonical development” of the cumulative hierarchy. A different response to Lewis’s argument comes from Menzel (). If one is willing to reject Limitation of Size and weaken the Replacement Axioms, then one can allow that the urelements are greater in size than any aleph and still form a set—a so-called “wide set.” This allows one to accept Size of Worlds and the Urelement Set Axiom. And it allows one, as is customary, to take that next stage of the iterative hierarchy to contain, for any plurality of sets and urelements formed at previous stages, a set with that plurality as its members. I object that, without Limitation of Size, we have no way of saying how wide the set of urelements is. But such agnosticism might instead be taken to be a positive feature, not a bug.

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

  -  (/)

(Hawthorne and Uzquiano : –). But in my view the structures instantiated in worlds are mathematically represented by (pure) mathematical systems that share those structures, by isomorphism. No sets based on urelements are needed for mathematical representation. Granted, such mathematical systems can only represent structures with small domains. But given my structural pluralism—the absolute isolation of worlds—there is no (non-logical) structure that encompasses (large) aggregates of worlds. I conclude that Size of Worlds, and the rejection of the Urelement Set Axiom, is no threat to the universal applicability of mathematics. Hawthorne and Uzquiano concede that the case for the Urelement Set Axiom is not overwhelming. They put more weight on a second argument to which I now turn. First, they characterize the thesis of Limitation of Size as follows: “a plurality forms a set if and only if they are not in one-one correspondence with the entire universe of all objects.” Call this the “axiom” of Limitation of Size. It follows immediately from this that the only way for a plurality to fail to form a set is for them to be in one-one correspondence with the universe of all objects, and that therefore there is only one size that pluralities that do not form sets can have. But this conflicts with the principle of unrestricted composition (if the atoms in modal reality are barely many), a principle that I am committed to. Their argument is this. First, for any plurality greater than one, there are more subpluralities than members of that plurality.³⁶ Now consider the plurality of atoms in modal reality. On my view they are barely many, the same size as the cardinals. But every subplurality of the plurality of atoms has a fusion, and different subpluralities have different fusions. So there are more fusions of atoms than there are atoms. But that contradicts the axiom of Limitation of Size, which only allows one way for a plurality to be many. It would not be fruitful to enter into a dispute over what to call “Limitation of Size.” Weaker and stronger versions of the doctrine have been put forward from Cantor to the present day.³⁷ The strong version that Hawthorne and Uzquiano focus on comes from von Neumann. It takes a plurality’s failure to be in one-one correspondence with the universe of objects to be a sufficient condition for the plurality forming a set. The weaker version of Limitation of Size that I endorse must reject this sufficient condition, since there are pluralities that neither form a set nor are in oneone correspondence with the universe. Rather, the sufficient condition for forming a set is being in one-one correspondence with some cardinal. The question that matters is: what benefit does the stronger version provide that the weaker version does not? According to Hawthorne and Uzquiano, a significant consequence of von Neumann’s axiom of Limitation of Size is this: “The scale of alephs . . . form a proper foundation for the metaphysics of size by forming a kind of universal ruler, in the sense that the size of a plurality is determined by its relation to the ruler” (Hawthorne and Uzquiano : ). Their ruler, they go on to say, has a notch for each aleph: if the plurality corresponds with an aleph, that aleph gives its size; if not, then one can deduce that it has the size of the universe. (Here I suppose they intended the ruler to have a notch not just for each aleph, but for each cardinal number, including the ³⁶ They sketch a proof of this claim, which they call “Remark ,” on pp. –. ³⁷ See Hallett () for a discussion of weaker and stronger versions of the doctrine of Limitation of Size.

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  :    



natural numbers.) But it seems to me the weaker version of Limitation of Size also provides a foundation for the metaphysics of size. Extend the ruler to add a notch for the size of the cardinals and another notch for the size of the fusions of cardinals. (The space between these two notches can be left as a blur.) We can speak of a plurality corresponding to these notches beyond all the alephs no less than to the notches given by the alephs themselves. If a plurality corresponds to the former added notch, it is barely many; if it corresponds to the latter added notch, it is maximally many. We can still say, it seems to me, that “the scale of alephs [cardinals] forms a proper foundation for the metaphysics of size.” If there is a cost to endorsing the weaker version of Limitation of Size, it presumably comes from this: on the weaker version, Global Choice does not follow from Limitation of Size and would need to be posited as an additional axiom. (Global Choice is the principle that there is a function that selects from every plurality exactly one member of that plurality; how to represent the “function” in question will depend on the framework.) That may be a cost in systematization. But it is not a cost in our ability to know the truth. Global Choice, I say, is obviously true. (Indeed, I never met a choice principle applying to sets, or set-like entities, or pluralities that I didn’t like.) But perhaps the problem is this. Global Choice implies a global wellordering of the universe by the usual argument. But if the universe is greater in size than the ordinals, there will be well-orderings with no standard ordinal numbers to represent them. There is a temptation, then, to introduce super-ordinals (as set-like entities) to measure these large well-orderings. But the introduction of super-ordinals would lead to an argument for super world-structures, say, to be the structure of worlds with super-ordinal time. (See Section . above.) And we would be off to the races once again. But, as already noted, such super-ordinals can and must be resisted. We can allow that some pluralities can be well-ordered without introducing set-like entities in reality to serve as measures of those well-orderings.³⁸ It is worth noting that Hawthorne and Uzquiano are committed to allowing, in a sense, that there is more than one size beyond the alephs. For they accept that, for any plurality more numerous than one, there are more subpluralities than members. And this suggests a clear sense in which it is legitimate to ask how many subpluralities of alephs there are, and to answer: a size greater than the size of all the alephs. There are at least two absolutely infinite sizes. A similar dialectic then ensues. Global Choice should somehow be extended to apply not only to pluralities, but also to pluralities of pluralities; and it should follow that any plurality of pluralities can be well-ordered. There is the same pressure to introduce super-ordinals, and the same reason to resist that pressure. To deny that we can meaningfully assign sizes to these pluralities of pluralities, whether directly or by coding, would seem to belie the obvious. One must beware, however, that “seeming obvious” has a checkered history when it comes to sets and classes and pluralities; paradox always lies in wait just beyond the corner. Hawthorne and Uzquiano (: –) consider the weaker version of Limitation of Size that I endorse only in connection with Lewis’s theory of classes. They rightly

³⁸ See Uzquinao (: –) who argues on somewhat different grounds that allowing for more than one size beyond the alephs will push one to introduce what I call “super ordinals.”

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

  -  (/)

notice that Lewis’s theory runs into trouble if he rejects the Urelement Set Axiom, and holds that the urelements are many. Their argument is this. According to Lewis, the atoms in reality set the standard of size: there are barely many atoms. Suppose Lewis holds that there are barely many urelements among the atoms (as I do). Then, since Lewis accepts unrestricted composition, there are more fusions of urelement atoms than atoms; that is, the fusions of urelement atoms are many, but not barely many. But Lewis also accepts a fundamental principle called Domain that gives the domain of the singleton function, part of which says, any part of the null set has a singleton. (The null set is the sum of all the urelements.) Then, since any fusion of urelement atoms is a part of the null set, it follows that any fusion of urelement atoms has a singleton. So there are many, but not barely many, singletons. But the singletons are mereological atoms on Lewis’s theory. So, there are many, but not barely many atoms. Contradiction. The first thing to say is that this problem is peculiar to Lewis’s theory of classes. Since I reject that theory, I do not take the problem to bear on my acceptance of Size of Reality and Size of Worlds. In particular, I endorse no principle like Domain that maps the urelements one-one into the atoms. The second thing to say is that the problem does not afflict Lewis prior to  when Parts of Classes was written. At that time, he still believed, as he explicitly endorses in On the Plurality of Worlds, that the urelements are few, not many. But by , Lewis was willing to allow that the worlds are many, and that there is no cardinal bound on the size of worlds. (See Chapter , n. .) That change in view would require some modification to his theory of classes. The most natural fix would be to accept Domain in a weaker version that claims only: any small part of the null set has a singleton. That restriction is mysterious, I think, but no more mysterious, and along similar lines, as the restriction on singleton formation that Lewis is already committed to, namely, that only sets, not proper classes, have singletons. With the restriction in place, it no longer follows that there are many, but not barely many singletons, and the contradiction is blocked.³⁹

. Discharging Assumptions and Conclusion Prior to plunging into the details of the Forrest-Armstrong argument and its numerous variations, I made two simplifying assumptions. I assumed that no worlds have non-spatiotemporal parts (thus ruling out spatiotemporal co-location), and that no worlds contain atomless gunk. In fact, I disbelieve the first assumption, and am (officially) agnostic about the second. (For how I think the possibility of gunk relates to principles of plenitude, see the postscript to Chapter .) In this final section, I check whether these assumptions might have affected any of my conclusions. ³⁹ When I first read a draft of Parts of Classes in , I sent Lewis comments, one of which was aimed at the problem currently under discussion. I wrote: “For Domain, do you want: any small part of the null set has a singleton? Or do you want another hypothesis: the null set is small? (Am I the only one who worries about this?!)” (Letter of March , ). Replying to this comment, Lewis wrote: “I don’t know whether I want to add either of these things. Do you see any trouble from leaving them off? If not, I want to be guided by the pursuit of orthodoxy. However I’m not clear whether orthodox set theory rules out that there might be so many individuals that the class of them is a proper class. Part of the problem is that axiomatic set theory is so often pure set theory” (Letter of June , ).

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   



The most important thing to note in this regard is that each of these assumptions, if false, had the effect of restricting our attention to only some of the worlds in modal reality. But adding more worlds does nothing to undermine the force of the ForrestArmstrong argument. For if Unrestricted Recombination requires that there be a world Giganto that leads to contradiction, it matters not whether Giganto recombines some but not all of the worlds. All that matters is that Unrestricted Recombination when combined with plausible principles of plenitude entails that some world exists that cannot possibly exist. Thus the main conclusion, that the Forrest-Armstrong argument or one of its variations is sound, in no way depended on those two assumptions. But perhaps rejecting one or both of these assumptions makes trouble in another way, by undermining my defense of Size of Worlds. That defense rested on my account of plenitude of structures, and in particular, an iterative conception of structure according to which world structures are represented by sets. But perhaps once co-location or gunk is allowed, an iterative conception of structure will not be sufficient to defend Size of Worlds. Consider first co-location. In fact, I favor a trope theory that allows multiple tropes to be co-located at the same point of spacetime. (See Section . and Chapter . What I say here about co-location of tropes applies also to other putative cases of co-location involving, say, bosons or angels, whether or not the bosons or angels are themselves, as I think, composed of tropes.) I also allow that throughout modal reality the atomic tropes are many, indeed, the kinds of atomic tropes are many, where atomic tropes are of the same kind iff they are duplicates of one another. In virtue of what do I hold that the atomic tropes colocated at a single spacetime point must be few? Even if an iterative conception of structures is accepted, so that the places in any world structure are few, how does that entail that the atoms at any world are few in accordance with Size of Worlds? Why can’t a world be large, not in virtue of having a large spacetime, but in virtue of having many tropes (or bosons or angels) co-located at some point of spacetime? Call this hyper-co-location. It appears that a separate principle would have to be introduced to prohibit hyper-co-location, a restriction on what recombinations of tropes are possible. But that goes against my claim that the only restriction on recombination comes from the plenitude of structures, from what structures are possible. A separate restriction on recombination, say, restricting how many duplicates can be co-located, might seem unmotivated and ad hoc.⁴⁰ I respond as follows. No separate restriction on recombination is needed because co-location is a structural feature; worlds with and without co-location differ structurally. There are two ways, however, in which co-location may figure into the structure of a world, and it is important to distinguish them. One way takes colocation to be reflected in the world’s “underlying structure”; the other takes co-location to be reflected in the world’s “instantial structure.” Recall from ⁴⁰ This argument is adapted from Hawthorne and Uzquiano (: ), though they direct it explicitly against Lewis’s (a) view that there is a cardinal bound on the size of worlds. They write: “a restriction on the possible size and shapes of concrete universes [does] not help us much in a context where angels can be packed into a single point.” See also Pruss () who had earlier argued that Lewis’s proviso “size and shape permitting” will be ineffective if co-location is possible.

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

  -  (/)

Section . that the underlying structure, which we have been supposing is spatiotemporal, is the structure over which the fundamental elements are distributed. Suppose, for purposes of illustration, that these fundamental elements are atomic tropes. The instantial structure then arises from the pattern of instantiation, or occupation, of the atomic tropes, where the pattern reflects which places in the underlying structure are occupied by duplicate tropes (and so instantiate the same fundamental property). Consider first how spatiotemporal co-location may be reflected in the underlying spacetime structure. When atomic tropes are co-located, each of the tropes separately stands in spatiotemporal distance relations to some or all of the other tropes; the co-located tropes themselves are distance zero from one another. On this way of understanding spacetime structure, the co-located tropes each occupies its own place in the underlying structure; we only get a standard spacetime structure (Minkowskian, Euclidean) when we identify the places occupied by the co-located tropes, and take the quotient structure. (If the standard spacetime is a metric space, the underlying structure is a pseudo-metric space, allowing distinct elements to be zero distance apart.) Now, the iterative conception of structure requires that the places in the underlying pseudo-spacetime structure are few. And so if spatiotemporal co-location is understood in this way, as being reflected in the underlying structure, it follows that the co-located atomic tropes must be few. No separate restriction on recombination is needed. But I suppose we might instead understand spatiotemporal co-location to be reflected in the instantial structure. On this approach, the underlying structure is the same, whether there is co-location or not; but the places in the underlying structure take plural arguments. If the distribution of tropes over the underlying structure assigns a plurality of more than one trope to a given place in the structure, then tropes are co-located at that place; if a single trope (a “plurality of one”) is assigned to the given place, then there is no co-location at that place. The facts of colocation are determined by how the tropes are distributed across spacetime. The possibilities of co-location, then, are determined by how tropes can be recombined in a possible spacetime. Since hyper-co-location can occur in an ordinary spacetime, is a special restriction on recombination needed to rule out hyper-co-location? I think not. Spatiotemporal co-location is no longer reflected in the underlying spacetime structure, but it still a structural feature; it has been shifted to instantial structure. For once we allow plural occupation, we must include the size of the plurality occupying a given place as part of the instantial structure, the pattern of instantiation. Thus the only restriction needed comes from considerations of what structures are possible. To see how this works, consider (LPR), now extended to allow for the recombination of tropes: for any tropes and any non-overlapping spatiotemporal arrangement of those tropes, there is a world that recombines those tropes according to that arrangement. As noted in Section ., (LPR) is a restricted, or qualified, principle of recombination: some tropes can be recombined only if there is a possible arrangement that recombines them. What arrangements are possible depends in part on what underlying structures are possible, and so is covered by plenitude of structures. But what arrangements are possible also depends on what instantial structures are possible. And that too is covered by plenitude of structures, though not plenitude of world (i.e. underlying) structures. It is no less governed by the iterative conception of

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   



structure, which is the ultimate determiner of what structures are possible, that is, instantiated in some world. In any case, no adjustment to (LPR) is needed, in particular, no separate restriction on how tropes can be recombined. I conclude, then, that allowing spatiotemporal co-location does not undermine the motivation for accepting Size of Worlds. I turn now to atomless gunk. The assumption that no world contains gunk was made for convenience. It allows one to measure the size of a world—or a part of a world, or modal reality as a whole—by how many atoms it contains. But we can do just as well in a gunky setting by making use of “relative atoms.”⁴¹ To measure the size of a world, be it atomic, gunky, or partly gunky, consider all the ways of partitioning the world into non-overlapping parts. Each such partition has associated with it some number: the number of non-overlapping parts. Let the size of the world be the least upper bound of all those associated numbers. If the world is atomic, this method gives the same result as measuring by the number of atoms. If the world is gunky, this method makes the world smaller than a corresponding expanded atomic world which has atoms added, so to speak, at infinity. For example, a world with a standard spacetime continuum (and no co-location) has size ‫ב‬a₁. A world with a gunky spacetime continuum, where each part of spacetime corresponds to a regular open set of the standard continuum, has size ‫ב‬a₀ (= ℵ₀). All partitions of that gunky spacetime are countable. Allowing gunky worlds raises a question analogous to the question raised by allowing co-location. And it gets an analogous answer. The question is: does hypergunk exist in any world?⁴² An object is hypergunk if and only if it is gunk and its parts are many. If some world contains hypergunk, then Size of Worlds is false: hypergunk can be partitioned into κanon-overlapping parts, for any cardinal κa, and so a world with hypergunk is large. But hypergunk, no less than hyper-co-location, is ruled out on an iterative conception of structure. For the mereological structure of a world is part of the world’s underlying structure. And no world’s underlying structure, on the iterative conception, can accommodate hypergunk. Hypergunk, then, is no threat to Size of Worlds. Note, however, that there is this difference between the case of hyperco-location and hypergunk. Since hyper-co-location occurs in no world, it is impossible. But if one allows, as I do, that it is possible for more than one world to be actual, indeed, even for every world to be actual, then hypergunk is possible (if gunk is) even though it exists at no world. For suppose that, for any cardinal κa, gunk of size κaor larger exists at some world. It follows that the fusion of all the gunk in all the worlds will be hypergunk. (Hazen  calls this “the Big Blob.”) If it is possible for every world to be actual, then hypergunk is possible, even though it exists at no world. But perhaps when Nolan () claims that hypergunk is possible, he has in mind what Hazen () calls strong hypergunk. Strong hypergunk is gunk such that the parts of any of its parts are many. If strong hypergunk were to exist in modal reality, then it would exist in any world that it overlaps, violating Size of Worlds. Strong hypergunk, then, is impossible, even if we allow for the possibility of universal actualization. ⁴¹ Armstrong (a: ) introduced relative atoms to extend his combinatorial account to possibilities (perhaps doxastic only) containing individuals or properties that were not composed of simples. ⁴² On hypergunk, see Nolan () and Hazen ().

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

  -  (/)

It is impossible not because it is formally inconsistent, but because it conflicts with the principles of the framework. So much for hyper-co-location and hypergunk. Having discharged assumptions, I can now briefly conclude. The Forrest-Armstrong argument, once fortified, demands that the Unrestricted Principle of Recombination (UPR) be rejected. But the restriction comes entirely from the plenitude of structures, from independently motivated considerations as to what structures are instantiated in possible worlds. No additional restriction on recombination is required. When a modest version of Limitation of Size is added as a principle of the framework, a simple and attractive picture of modal reality emerges, as captured by Size of Reality and Size of Worlds. In sum: the worlds are many, but the parts of any world are few. Or, in other words: modal reality is large, but each of the worlds included in modal reality is small. That, I think, is a reasonable way for modal reality to be. Indeed, it may just be the only reasonable way for modal reality to be.

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 On Living Forever () . Introduction I want to live forever; but just what is it that I thereby want? Prior to  (or thereabouts) my want would have seemed quite clear: I would have wanted to live for an unending sequence of years. But our horizon has since been expanded by the teachings of Georg Cantor.¹ The natural numbers all together amount only to the smallest order of infinity, ℵ₀. There are countless greater infinities that dwarf ℵ₀ as surely as ℵ₀ dwarfs our customarily allotted three score and ten. Why settle for a piddling ℵ₀ years if there are limit cardinals out there to vault over, inaccessible cardinals waiting to be surpassed? This chapter divides into three parts. In the first part, I argue that trans-ω longevity is (conceptually) possible: there are possible worlds that endure beyond a single ω-sequence of years, and a person can survive in these worlds from one ω-sequence to another. In the second part, I discuss two reasons why one might want to live for more than a single ω-sequence, one having to do with the pursuit of mathematical knowledge, the other with the maximization of pleasurable experience. Finally, in the third part, I provide an analysis of wanting to live forever, in the sense of wanting to live as long as possible. Is it wanting to inhabit a world in which one’s life has the maximal possible duration? I argue that there are no such worlds, and then provide an alternative analysis. I want trans-ω longevity, but not at any cost. Wanting to live beyond a single ωsequence of years is, for me, a conditional want, as is wanting to live to be . Both wants are conditional, at the very least, on my still having my wits about me, and on there still being a fair balance of pleasure over pain. In claiming that trans-ω longevity is desirable, I claim only that there is some possible world, even if quite remote from our own, in which I have trans-ω existence and the above conditions are satisfied. Some, it is true, have argued that such conditions could never be satisfied even for ordinary immortality because a life too long inevitably leads to perpetual boredom.² I suspect that those who argue in this way either lack imagination or This chapter has not been previously published. It was presented at Queens College, CUNY in December, , and at the APA Pacific Division Meeting in March, . Because it has been available on-line for many years, and cited in the literature, I have made only cosmetic revisions. I should note, however, that the claims relating to modal plenitude have been superseded by the more precise account given in Chapters , , and . Additions to the original paper are in square brackets. ¹ In  Cantor first published a proof that the real numbers are not equinumerous with the natural numbers. See Cantor (). ² See, for example, Williams ().

Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

   ()

become too quickly jaded with the good things in life; at any rate, their arguments do not, so far as I can introspect, apply to me. I will not attempt to respond directly to their arguments here. Instead, I will assume that immortality of some sort or another is desirable, and ask what kind of immortality I would choose, if I had the choice.

. Worlds and Lives with Transfinite Duration Are Possible One objection to the claim that I want to live for more than ℵ₀ years needs to be dismissed at the outset. What if the actual world has at most ℵ₀ years to give? Would it follow that I could not want to live longer than that? No, the life span of the actual world sets no limit on what I can want. I could want to live ℵ₀ years even if the universe were destined to end in the Big Crunch some few billions of years down the line. Similarly, I can want to live for more than ℵ₀ years whether or not the universe actually has those years to give. My wants are not circumscribed by what is actual, only by what is possible.³ Still, one might wonder whether it is even possible for a world to endure for more than ℵ₀ years. In answering this question, I will focus on those worlds that can be decomposed into a temporally ordered aggregate⁴ of instantaneous world-stages, and that support a (possibly transfinite) metric that measures time intervals between stages in years. Moreover, I will assume that each world-stage is contained within an ω-sequence of years, where an ω-sequence of years is itself an aggregate of worldstages that is isomorphic to the half real line. Finally, the world as a whole is a wellordered aggregate of these ω-sequences.⁵ A world with only one ω-sequence of years I will call an ordinary world. I will say, somewhat metaphorically, that stages from neighboring ω-sequences are separated by a gap. More exactly, two stages are separated by a gap if and only if the interval between them as measured by the temporal metric is infinite. Cantor demonstrated the abstract possibility of temporal orderings that allow world-stages to sum to worlds that endure for more than ℵ₀ years. But there are two sticking points: there is a problem of quantity, of there being enough world-stages to make such a world; and there is a problem of unification, of uniting the world-stages into a single world. I will take up the second problem first. A world that endures for more than ℵ₀ years would have to have an ω-sequence of years followed by more years. On what grounds would the further years be accounted part of the very same world? Wouldn’t a mysterious world-building glue have to be posited to hold such a world together across the gap?

³ Are my wants even circumscribed by what is possible? I claim that one cannot genuinely want what is impossible; but this claim is controversial, and I cannot here discuss how I would deal with the various prima facie counterexamples. Very little will depend on this claim in what follows. ⁴ [By ‘aggregate’ I mean just (mereological) sum or fusion, a use of the word that is now largely out of favor.] ⁵ It is for convenience only that I restrict my attention to this class of worlds. I do not mean to suggest that there are no worlds outside of this class: worlds with no time at all, or time but no temporal metric; relativistic worlds, or other worlds that cannot be decomposed into temporal parts; worlds with no beginning, or with a beginning and an end; worlds whose ω-sequences are not well-ordered; and so on.

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       



Mysterious, perhaps; but no more mysterious than what is already needed to keep an ordinary world from falling apart at its Dedekind seams. One way of arguing for this invokes a modal principle of the elasticity of time: Elasticity Principle. Any sequence of events might have occurred faster or slower without affecting any non-temporal relations between the world-stages comprising the world. Consider the sequence of events that will occur between : p.m. and midnight on the eve of the year . There is a possible world in which the sequence from : to : takes one year to occur, in which the sequence from : to : also takes one year to occur, and so on ad infinitum. In this elongated world, the events prior to the year  are stretched out over an ω-sequence of years; the events of midnight occur not at the beginning of the third millennium, but at the beginning of a second ω-sequence. Whatever (non-temporal) relations serve to hold together the worldstages before and after midnight in the ordinary world are still around to keep the elongated world from falling to pieces. So the unification of worlds composed of successive ω-sequences of years is in general no more problematic than the unification of an ordinary world. The argument assumes, of course, that the temporal relations do not themselves supply the glue; else the world might come unglued when the temporal relations are tampered with. This assumption is unproblematic for those (like myself) who accept the irreducibility of temporal relations. How do things stand for the reductionist with respect to time? The reductionist believes: No time without a timekeeper! The timekeeper might take many forms: it might be a transcendent being that establishes temporal relations between stages by its subjective perception of them; it might be a clock ticking away in some corner of an otherwise uneventful universe; normally it will be a convergent pattern of cyclical, lawful processes from which a temporal metric can be defined. The reductionist can accept the Elasticity Principle in the amended form: Amended Elasticity Principle. Any sequence of events might have occurred faster or slower without affecting any relations between world-stages other than relations involving the timekeeper. The argument still goes through if the reductionist believes there is a world in which the timekeeper is sufficiently localized or isolated that it is implausible to take the timekeeper to be what holds the world together; the Amended Elasticity Principle can then be applied to this world. Reductionists with an essentially sticky conception of time may remain unconvinced by the argument; but I suspect there are other paths to the same conclusion. If two ω-sequences of years can be brought together to form a single world, I see no reason why the same would not hold for a larger number of ω-sequences, even infinitely many. That brings us to the problem of quantity. Repeated application of the Elasticity Principle leads to more and more complicated temporal orderings, but never turns an ordinary world into a world with more than ℵ₀ years. To ensure the existence of such worlds, we can use the following modal principle of plenitude: Duplication Principle. quantity whatsoever.

Whatever can be duplicated can be duplicated in any

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

   ()

This principle follows from the more general modal maxim: possibility respects no arbitrary bounds. There may be in the actual world no more than seventeen duplicates of some object, say, some kind of quark; but that couldn’t be a necessary truth. The same goes for ℵ₁₇. Not everything that exists can be duplicated. Consider the number three. It is as much an absurdity to speak of many threes as to speak of a round square.⁶ The distinction between those objects that cannot by their nature be duplicated and those objects that can be duplicated corresponds roughly with the traditional distinction between abstract and concrete; indeed, it provides one way of explicating that distinction. Among the entities that can be duplicated are world-stages; otherwise there could be no world at which time passes without anything happening.⁷ So the Duplication Principle ensures that there are enough world-stages around to make a world with more than ℵ₀ years, indeed, with as many years as one pleases. Such worlds might be exceedingly dull, for all the principle tells us, but they are possible nonetheless. I have argued that a single world can encompass any number of ω-sequences of years, and can hold together across the gaps between them. But I need more. I need to argue that a person inhabiting such a world can survive the gaps. What relations must hold between person-stages on either side of a gap for them to be stages of one and the same person? I answer, with many others, that they need to belong to an aggregate of person-stages that exhibits the right sort of psychological continuity and connectedness.⁸ Now, the elongated world introduced above already demonstrates the possibility of surviving the gap: whatever relations of continuity and connectedness hold between the stages of an ordinary person whose life straddles the year  in the ordinary world also hold between the stages of his elongated counterpart whose life straddles the gap in the elongated world. But this is longevity without the benefits. The life and accomplishments of the elongated person are quite ordinary in all non-temporal respects. Is there a more interesting way to survive the gap? First, let us consider the problem of continuity. The problem is that complete continuity at the limit point initiating a second ω-sequence of years would require that all changes in one’s psychological makeup became vanishingly small as the preceding ω-sequence of years progressed. And that seems to undercut the rationale for wanting to live more than one ω-sequence of years because it leaves no room for personal growth. The problem evaporates, however, as soon as one realizes that continuity need not be absolute for identity to be preserved. Say that my interest in trans-ω longevity stems from an interest in the transfinite accumulation of knowledge. My surviving the gap would no more be threatened by a constant increase in knowledge than would my ⁶ [Today I would not use numbers as examples of entities that cannot be duplicated. Due to my structuralist leanings, I hold that there are many mathematical systems whose domain contains an entity that, in virtue of its place in the system, is properly called “the number three” (when the domain of discourse is restricted to that system). It is still true to say: “there is only one number three” when interpreted using supervaluations. (Compare: there are many sums of molecules that are properly called “Mt. Everest”; but it is still true to say “there is only one Mt. Everest.”) For more on this, see Chapters  and .] ⁷ If an argument is wanted for the possibility of time without change, see Shoemaker (). ⁸ Locke is generally held to be the father of this view. Modern exponents include Parfit (), Perry (), and Lewis ().

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   ?



ordinary survival be threatened by the ingestion of a Britannica pill, a pill that furnishes instant encyclopedic knowledge. Survival only requires that there be continuity in enough of the psychological traits that matter. And, certainly, in wanting to live beyond a single ω-sequence of years, I do want most of my important character traits and values to stay with me. So, on a reasonable criterion of psychological continuity, my wants are not incompatible with surviving the gap. What about psychological connectedness? The relation of psychological connectedness generally held to be most important to personal survival is the relation that holds between two person-stages whenever one contains, or under appropriate conditions would contain, a memory of an experience had by the other. I see no obstacle to remembering experiences from a previous ω-sequence. True, facts of brain physiology severely restrict the time periods over which direct connections of memory can persist in the actual world. But these facts are contingent, and thus do not preclude the possibility of surviving for longer periods, even ω and beyond. They only show that in wanting to live for ω years or beyond, I also want there to be a more reliable mechanism for encoding memories. But there is another problem. It is arguably essential to us as persons that our minds be finite, and, in particular, that at any moment we can remember at most a finite number of distinct experiences from our past. Thus, at the onset of a new ωsequence of years, all but finitely many of the previous years will be lost to oblivion. This might indeed sour some to the prospects of trans-ω living, namely, those who cherish their experiences more in the remembering than at the time. But I don’t think it is an obstacle to surviving the gap between ω-sequences. It is still possible for any two person-stages to be connected by the transitive closure of the memory relation. That is all we ask for in ordinary cases of survival; why demand more when trans-ω survival is at stake?⁹

. Why Want Transfinite Longevity? I have argued that trans-ω existence is within the realm of possibility, and thus a legitimate object of desire. But I can hear the reader impatiently demand: “What could be the point of it? What could one do or experience in more than one ωsequence of years that could not already be done or experienced by an ordinary immortal living for one ω-sequence?” I will consider two replies. The first reply invokes lofty intellectual pursuits, but is somewhat inconclusive. The second reply invokes more creaturely pleasures and would equally be available to a slug. I think it succeeds, however, in giving me reason to want trans-ω longevity. Consider first my interest in mathematics. For example, I have always wanted to know whether Fermat actually possessed “a truly marvelous proof that would not fit in the margin.” Is Fermat’s Last Theorem true?¹⁰ Trans-ω longevity would give me the means to find out. And I wouldn’t even need a Ph.D. in mathematics; just the ⁹ The above problem is like the familiar case of the senile general, but with a vengeance. For the solution in the ordinary case, see Perry (: ). ¹⁰ [Since this chapter was written, Fermat’s Last Theorem has been proved by Andrew Wiles. One can substitute Goldbach’s Conjecture in what follows.]

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

   ()

patience to perform trivial, though increasingly tedious calculations.¹¹ I could simply begin a systematic search for counterexamples, deciding in advance to write the word “no” in some designated place when and only when a counterexample was found. Then, assuming a world in which inscriptions survive the gap, the answer will be waiting for me at the beginning of the next ω-sequence: if I find the word “no,” the conjecture is false; otherwise, true. There might appear to be a catch in the above procedure in light of what was said about the essential finiteness of memory. I do not have in mind the fact that I would be able to recollect performing at most finitely many of the calculations necessary to a verification of the conjecture; the rest is mercifully forgotten. But the procedure does require that I have some way of knowing when the next ω-sequence has arrived so that I can know whether to interpret the lack of a “no” as a “yes.” Can I know that a new ω-sequence has arrived without remembering the intervening years? I think so. The fact that I no longer intend to perform another tedious calculation gives evidence of such. Moreover, the world might contain an ideal clock so designed that its hands move half the distance to twelve o’clock with each passing year; the chimes at midnight then signal the arrival of the new epoch. Memory is not essential to marking the passage of time, either in a world of ordinary or of trans-ω duration. Why, then, do I find this first reply inconclusive? Because there are other sure ways of determining the truth or falsity of Fermat’s Last Theorem that would not involve so much waiting around. My interest in Fermat’s Last Theorem doesn’t so much give me reason to want trans-ω longevity as reason to want to inhabit a world in which infinity machines exist, machines that can perform an ω-sequence of tasks in a finite amount of time.¹² If my interest in acquiring mathematical knowledge is to give me a reason to want trans-ω longevity, there will have to be mathematical problems that are solvable given enough time, but not solvable by an infinity machine in an ordinary world. Here one naturally turns to undecidable problems in set theory. Solutions to some of these problems would be worth waiting around for no matter how long the wait. Unfortunately, there seems to be no systematic way to inspect the universe of sets, one set at a time, analogous to the way it is possible to inspect the universe of numbers. The problem has to do with the non-constructive character of the cumulative hierarchy, and is not at all affected by merely adding more time. Consider the case of the Continuum Hypothesis. Its truth or falsity is already decided by the time the cumulative hierarchy has been carried out two levels past level ω. If one could inspect all the sets at this level, one could decide the Continuum Hypothesis simply by checking to see whether for every set of reals, there exists (a set that is) a one-to-one correspondence between it and either the set of all the reals or an enumerable set of reals. But it is not that simple. Ignore for now the problem that the requisite bookkeeping could not be done within the confines of an ordinary

¹¹ Of course, the enterprise need not monopolize all of my time: I could arrange it so that the time spent calculating took up an increasingly smaller proportion of my total time; or a trustworthy computer could be employed. ¹² For more on infinity machines, see the essays by Black, Thompson, Benacerraf, and Grünbaum in Salmon ().

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   ?



Euclidean space; perhaps we could expand our spatial horizon just as we have expanded our temporal horizon. The insurmountable problem is that we would have no way of knowing that we had inspected all the sets of reals, or all the oneto-one correspondences between them. It seems doubtful that transfinte living would do much for the pursuit of mathematical knowledge that could not also be done by other means.¹³ But there is another sort of reply to the question: what point could there be to living for more than a single ω-sequence of years? And if it is right, it equally gives reason for wanting to live for any number of years, no matter how large the number. It begins by noting that there are pleasures that we never grow tired of, of which we can say: “the more, the better.”¹⁴ Take, for example, my unflagging desire for Thai food: I want to eat it this week, and next week, and the week after, and so on. Do I thus desire more than ℵ₀ experiences of Thai food? There is a problem owing to an ambiguity in the phrase ‘and so on’. Do I really want my Thai experiences to be iterated far into the transfinite? Or do I want only that every Thai experience be followed by another? The latter want, of course, could be satisfied within the lifetime of an ordinary immortal. To the extent that it is unclear which of these wants I have, it is also unclear whether my enjoyment of Thai food gives me reason to want to live for longer than an ordinary immortal. And indeed, although I am quite sure that I never want to be eating my last Thai dinner, I am less sure what attitude I have towards the prospect of future ω-sequences without Thai food. The ambiguity occurring in ‘and so on’ occurs in exactly the same way in ‘forever’, and that takes us back to where we began. In wanting to live forever, do I want only that the sequence of years comprising my life has the ordinal property of having no last member? Or do I also want that the number of years of my life have the cardinal property of being as large as possible? Only in post-Cantorian times has it become clear that the former property fails to entail the latter. There is no doubt that I have the ordinal want. It comes from viewing my life from within and realizing that I never want to be at the end of my life, that I always want to have more to look forward to. But here, in contrast to the Thai food example, I feel quite sure that I also have the cardinal want.¹⁵ It comes from viewing my life from without and realizing that the longer life is the better life provided only that the added years are themselves worth living. And, I claim, there always is a longer life: for any of my possible lives in any possible world, there is another possible world in which that life is (cardinally) extended by adding years that I deem worth living.¹⁶ That there

¹³ But I here leave open the question whether any interesting mathematical proposition could be decided in this way. Perhaps each level of the constructible universe L could be exhaustively inspected, one set at a time, so as to decide sentences undecidable in the theory ZF + V = L. Perhaps attempts to produce infinitary proofs would improve on what infinity machines can prove in an ordinary world. ¹⁴ [Fischer (: ) uses pleasures we never grow tired of—what he calls “repeatable pleasures”—to defend the view that (ordinary) immortality is not so bad.] ¹⁵ Here, of course, I can only speak for myself. I do not claim that one who lacks this want is irrational. ¹⁶ [Segal (: ) contests my claim that the longer life is better—where a person’s life is composed of a well-ordered sequence of (non-overlapping) person-stages, each person-stage lasting for some finite duration. He claims that every such life will have a “Zenoian alternative” that is confined to a finite interval of time and that is equally good. (The existence of these Zenoian alternatives is grounded in something

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

   ()

should always be such worlds follows from the Duplication Principle, and the fact that I am easy to please. For one of the pleasures that I will never grow tired of is life itself: at any rate, life spent in a thinking, active mode, free of pain.

. Analysis of Wanting to Live Forever One doubt may remain. I have claimed that in wanting to live forever, I want the number of years of my life to be as large as possible. But how large is that? Since the cardinal numbers increase without bound there is no particular number of years that I want to live. It appears, rather, that I want the following to be the case: for any particular number of years, I live longer than that. But, assuming that I must live for some particular number of years, the above want isn’t satisfiable in any world; in wanting to live forever, I would be wanting the impossible. One solution would be to give up the assumption that I must live for some particular number of years. One could posit the existence of worlds that endure for longer than any cardinal number of years, worlds, for example, whose ω-sequences are ordered like the class of all ordinal numbers. Such worlds might be called ageless because no numerical age can be assigned to them. On this view, wanting to live forever is wanting to be ageless in an ageless world. I prefer not to posit ageless worlds, although I have no decisive objection to them.¹⁷ I cannot object on the grounds that they bring with them a prima facie commitment to proper classes. True, the domain of individuals existing at an ageless world is “too large” to form a set, since there are as many world-stages as there are ordinal numbers, and each world-stage is itself an individual existing at the world. But I am already committed to the totality of worlds being “too large” to form a set because, I have claimed, for every cardinal number there is a world that endures for exactly that many years. I find proper classes obscure and expect they will find no place in a final inventory of what there is. But I know of no reason to think that positing ageless worlds is any worse than positing arbitrarily long-lasting worlds on the score of proper classes.¹⁸ In either case, I believe, a Zermelo-style solution could be brought to bear to eliminate the problematic classes. Still, I think there is reason enough to reject ageless worlds. They conflict with ordinary modal intuitions about time and persistence. The idea that something might endure, but not endure for any particular length of time, sounds, on the face of it, absurd. And further modal reflection does nothing to overturn the initial impression. In particular, principles of plenitude such as the Duplication Principle do not require the existence of ageless worlds, at least not for a non-believer in proper classes. The Duplication Principle requires that, for any object that can be duplicated and any cardinal number, there is a world containing just that number of such duplicates. similar to my Elasticity Principle.) But he seems to miss that the life I claim is better is extended not just ordinally, but cardinally. If the sequence of temporal parts is uncountable, there is no Zenoian alternative.] ¹⁷ [I would not say that today. See Chapter  where I argue that the Forrest-Armstrong argument gives a realist about possible worlds compelling reason to reject ageless worlds, or any worlds with proper-class many individuals.] ¹⁸ Terrence Parsons has influenced my thinking on this point.

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     



It thus requires that the totality of worlds be “too large” to form a set. But it does not require that the domain of individuals at a single world ever be “too large” to form a set. Transfinite longevity notwithstanding, our days are necessarily numbered. Nor are ageless worlds required to provide an analysis of wanting to live forever. To see this, let us first consider a structurally analogous case that admits of a similar solution. I once knew a person who, I would say, had a desire to memorize as many digits in the decimal expansion of pi as possible. How might this desire be analyzed? Did she want the following to be the case: for any (finite) number, she memorized more than that many digits in the decimal expansion of pi? If we assume what was said above about the essential finiteness of memory, that would be to want the impossible. Is there some way to explain what she wanted without attributing to her a confused desire for the impossible? Indeed, what she had was an infinite sequence of distinct wants: for each (finite) number, she wanted to memorize more than that many digits in the decimal expansion of pi. Each want in this infinite sequence is satisfied in some possible world. What is impossible is only that all of these wants be simultaneously satisfied.¹⁹ This solution transfers easily to the case at hand: providing an analysis of wanting to live forever, in the sense of wanting to live as long as possible. Although there is no world in which I could be said to live forever, it is possible to understand what I assert when I say: I want to live forever. There is no unitary want that I have, but instead a transfinite sequence of wants, each satisfiable: for any cardinal number of years, I want to live longer than that. Or, even better, think in terms of a personal utility function defined over the worlds: in my case, worlds in which I live longer in general score higher, but without there being a highest-scoring world.²⁰ This shows that, by my lights, there is no best of all possible worlds: they get better and better as I live longer and longer. Pace Leibniz, this world barely rates at all.

¹⁹ Of course, this solution only works if one need not want the conjunction of what one separately wants, lest one be back to wanting the impossible. But this is familiar behavior of wants-that as a propositional operator. ²⁰ It cannot be a real-valued utility function. There aren’t enough real numbers to make the necessary discriminations.

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PART 

Humean Perspectives on Truthmaking, Mereology, Spacetime, and Quantities

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 Truthmaking: With and without Counterpart Theory () . Introduction According to the Truthmaker Principle: every truth has a truthmaker. Spelled out a bit more exactly: for every true proposition, there is some entity whose existence entails, or necessitates, the truth of the proposition. The demand for truthmakers has been championed most vigorously by David Armstrong, who, following C. B. Martin, touts it as a way to keep philosophers honest by requiring that they pay the full ontological cost of their theories.¹ Numerous philosophical views, including versions of phenomenalism, behaviorism, and presentism, would seem to run afoul of the Truthmaker Principle. Moreover, according to Armstrong, the demand for truthmakers provides an alternative to the orthodox Quinean approach to ontological commitment, which, due to its narrow focus on the quantifiers, tends to underestimate the ontological cost of theories.² As David Lewis writes: “We can scarcely exaggerate the importance of the demand for truthmakers throughout Armstrong’s writings” (Lewis : ). Attempts to come to grips with the Truthmaker Principle played a prominent role in Lewis’s metaphysical writings over the last fifteen years of his career. He first grappled with the principle in “A Comment on Forrest and Armstrong.”³ Then he launched a critique of the principle in reviews of Armstrong’s books, A Combinatorial Theory of Possibility and A World of States of Affairs.⁴ Although Lewis agreed that the truth of propositions must somehow be ontologically grounded, the Truthmaker Principle was too strong: it conflicted with two of Lewis’s most fundamental metaphysical assumptions, the uniqueness of composition and the Humean denial of necessary connections. Lewis endorsed instead a weaker principle according to which truth supervenes on being: any true proposition is made true by the pattern of instantiation of fundamental, or perfectly natural, properties

First published in B. Loewer and J. Schaffer (eds.), A Companion to David Lewis (Blackwell, ), –. Reprinted with permission of John Wiley & Sons, Inc. Thanks to Jonathan Schaffer for helpful comments. ¹ See especially Armstrong (a), (), and (). ² I critically discuss this aspect of truthmaking theory in Bricker (b). ³ Lewis (c). This was a reply to comments delivered by Peter Forrest and David Armstrong on Lewis (b). ⁴ Lewis () and Lewis (), respectively. Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

:      ()

and relations among particular things.⁵ Then, in a series of three papers, among his final writings in metaphysics, he further developed and crystallized his view. In Lewis (b) he argued that the Truthmaker Principle, even if accepted, does not support a correspondence theory of truth. In Lewis (a), he sharpened his critique of the Truthmaker Principle by showing that it is equivalent to an implausible principle as to how possible worlds must differ. Finally, in Lewis () and Lewis and Rosen (), he changed course, noting that his critique of the Truthmaker Principle rested on essentialist assumptions that he, as a counterpart theorist, does not accept. Once freed from those assumptions, a counterpart theorist can accept the Truthmaker Principle after all without buying into unmereological composition and mysterious necessary connections. Ironically, it is Lewis, not Armstrong, who can accept the Truthmaker Principle without paying a hefty metaphysical cost. The divide over the Truthmaker Principle corresponds to a divide over fundamental ontology. For Armstrong, the demand for truthmakers leads to an ontology of facts, or states of affairs, thus supporting Wittgenstein’s famous Tractarian saying: “the world is the totality of facts, not of things.” (I will stick with Armstrong’s preferred term ‘state of affairs’ in what follows.) States of affairs are (somehow) composed of particulars and immanent universals, and are thus themselves immanent entities that go toward making up the world. How abundant are the states of affairs? I will suppose at a minimum that all states-of-affairs theorists accept a full slate of atomic states of affairs: for any atomic proposition Ra₁a₂ . . . , there is an atomic state of affairs S such that, necessarily, S exists (obtains) if and only if Ra₁a₂ . . . is true.⁶ For Lewis, states of affairs are trouble. Lewis is a thing theorist: the world is a thing, the biggest thing, and every spatiotemporal part of the world, no matter its shape or size, is also a thing. Perhaps in addition to things there are nonspatiotemporal components of things—immanent universals or tropes. Lewis remained agnostic between a nominalist and a realist version of thing theory. But a thing theorist does not admit states of affairs. Both Lewis and Armstrong hold that truth is ontologically grounded. But whereas Armstrong thinks that states of affairs are needed to ontologically ground truth, Lewis holds that an ontology of things (perhaps together with immanent properties and relations) provides ground enough. Here, in brief outline, is the plan of the following chapter. In Section ., I introduce the idea of truthmaking, and consider how much truthmaking can be done by things without making controversial assumptions in modal metaphysics. In Section ., I present Armstrong’s account of truthmaking, and the states of affairs he thinks are needed to play the role of truthmakers. In Section ., I present Lewis’s view that the Truthmaker Principle does not make any claim about the notion of truth, and a fortiori should not be taken to be a version of the correspondence theory

⁵ See also Lewis (a) and Lewis (b) for statements and endorsement of the idea that truth supervenes on being. Lewis uses ‘fundamental’ and ‘perfectly natural’ interchangeably; see Lewis () for the terminological pros and cons. In this chapter I stick with ‘fundamental’. ⁶ Here a₁, a₂, . . . are (one or more) particulars and R is a (monadic or polyadic) universal. Since the universals are sparse on Armstrong’s theory, so are the atomic propositions, and the atomic states of affairs. On the distinction between sparse and abundant conceptions of properties and propositions, see Lewis (a: –).

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   



of truth. In Section ., I present Lewis’s critique of the Truthmaker Principle, and the ontology of states of affairs that (typically) goes with it. In Section ., I present Lewis’s claim that, as a counterpart theorist, he can accept the Truthmaker Principle after all. Here, however, I think Lewis’s position needs to be amended. Finally, in Section ., I present and discuss Lewis’s alternative to the Truthmaker Principle, the idea that “truth supervenes on being.”

. The Theory of Truthmaking Before considering Lewis’s critique of the Truthmaker Principle, we need to bring the principle into sharper focus, and view the principle through Lewis’s eyes. Lewis’s discussion of the principle (prior to Lewis ) was meant to be neutral with respect to controversial metaphysical assumptions, such as his own modal realism, or his counterpart theory. He supposed only that talk of possible worlds made sense, whether worlds were taken to be concrete, or abstract, or fictional; and that entities could be said to exist in multiple possible worlds, whether existing-in-a-world was to be understood in terms of “transworld identity” or counterpart relations. Moreover, as should go without saying, Lewis interpreted the principle in terms that he found intelligible, so without recourse to a primitive grounding, or dependence, or in-virtue-of relation. First, I ask: how does Lewis understand the truthmaking relation? For Lewis, truthmaking is a modal relation, and is to be understood in terms of strict implication. When an entity makes a proposition true, it is metaphysically impossible for that entity to exist without the proposition being true. Thus, as an initial formulation of the Truthmaker Principle, we have: (TM)

For every true proposition P, there exists some entity T such that, necessarily, if T exists, then P is true.

Three things should be noted at the start. First, on this account, the truthmaking relation—a relation between entities and propositions—is analyzed in terms of strict implication—a relation between propositions and propositions.⁷ The analysis requires the assumption that, for any entity, there is a proposition asserting that that entity exists. This assumption, however, is not problematic on an abundant, non-linguistic conception of propositions of the sort that Lewis is supposing (see below). The analysis has the consequence that it is a trivial analytic truth that any (existent) entity is a truthmaker for the proposition that that entity exists. But that seems exactly right. Second, on this account, the Truthmaker Principle is an assertion of modality de re: it involves quantifying into a modal context. It thus presupposes that modality de re is coherent (pace Quine ). In particular, it presupposes that it makes sense to say that an entity exists in multiple worlds, and that an entity’s essential properties are those properties that it has in every world in which it exists (pace Fine ). For Lewis, since possible worlds are concrete and do not overlap, ⁷ This contrasts with Armstrong’s account, according to which truthmaking is a primitive, crosscategorial relation (Armstrong : ). Armstrong argues against reducing the truthmaker relation to entailment (p. ). But his argument depends on taking entailment to be a more fine-grained relation between propositions than is strict implication, and so would not apply to Lewis’s account.

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

:      ()

modality de re must be interpreted using counterpart theory (see Section .). But Lewis’s discussion of (TM) (prior to Lewis ) does not presuppose any particular account of possible worlds or of de re representation. Third, (TM) does not by itself assert that all truths are ontologically grounded because the truthmaking relation, as Lewis understands it, does not require truthmakers to be in any way fundamental. Consider the following example. On standard assumptions, I cannot exist without my singleton, nor can my singleton exist without me. Thus, according to (TM), not only am I a truthmaker for the proposition that my singleton exists, but my singleton is a truthmaker for the proposition that I exist; for my singleton is such that, necessarily, if it exists, then I exist. But, surely, the truth that I exist is not grounded in the existence of my singleton. If a truthmaker theorist wants to capture the idea that truth is ontologically grounded, she has a choice. She can enhance the notion of a truthmaker, requiring that a proposition P is made true by an entity T only if both P is strictly implied by the existence of T and P holds in virtue of T (cf. Armstrong : ). (Depending on how the in-virtue-of relation is understood, the first conjunct may be redundant.) Or, she can keep the truthmaking relation as it is, realizing that something more will have to be added to (TM) to capture the idea that all truths are ontologically grounded. I will follow Lewis in taking the latter course. Lewis’s critique of (TM) is not affected by this choice. Arguably, the intuitive notion of making a proposition true includes the notion of grounding, but the terminological decision not to so include it won’t lead to trouble if it’s made clear at the start. Next I ask: in taking the bearers of truth to be propositions, what conception of proposition does Lewis intend to invoke? For Lewis, there are many equally legitimate conceptions of proposition; which conception is appropriate to a given task depends on the role that the propositions are required to play, be it semantic, or epistemic, or metaphysical (see Lewis a: –). In the context of seeking truthmakers, what matters is only the content, irrespective of how that content is represented, or whether that content is in any way fundamental. When content alone matters, Lewis identifies propositions with classes of possible worlds. But for his critique of the Truthmaker Principle, Lewis does not want to impose his own view, and assumes only the following. First, propositions are not linguistic entities, nor in any way tied to language; the metaphysical demand for truthmakers should not be limited by the expressive capabilities of actual languages, or even humanly possible languages. Second, propositions are abundant to at least this extent: for any possible world, there is a proposition asserting that that world alone is actualized. Beyond this, Lewis’s assumptions on propositions are standard: the propositions form a Boolean algebra, and thus are closed under the standard Boolean operations of conjunction, disjunction, and negation. By formulating the Truthmaker Principle as (TM), Lewis endorses what Armstrong calls “Truthmaker Necessitarianism”: if T is a truthmaker for P, then there is no possible circumstance in which T exists but P fails to be true. For to allow that there might be such a circumstance is just to concede that the existence of T, by itself, is not enough to make P true, that something more is needed (see Armstrong : –). Truthmaking, then, unlike for example causation, isn’t a merely contingent affair.

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   



Formulating the Truthmaker Principle as (TM) commits Lewis also to the converse of Truthmaker Necessitarianism: if the existence of T strictly implies P, then T is a truthmaker for P. This consequence of (TM) appears to be problematic. For example, supposing arithmetical truths to be necessary, it makes me a truthmaker for the proposition that + = . More generally, it makes any (existent) entity whatsoever a truthmaker for any necessary truth. Truthmaking for necessary truths is thus trivialized. Lewis has little to say about the problem of truthmaking for necessary truths. I take it that there are three main lines of response. First, one could simply accept that necessary truths lack non-trivial content: they apply indiscriminately to everything, and so everything is on a par with respect to making them true. This response might be acceptable for logical truths, and perhaps even for analytic truths; but it seems especially implausible for truths of mathematics or metaphysics. Second, one could replace strict implication with a more discriminating relevant entailment relation: my existence does not relevantly entail that + = .⁸ Third, one could introduce alongside the metaphysically possible worlds mathematically possible worlds wherein the mathematical entities reside. (This is the response that I would prefer; see Chapter .) Mathematical truths are, strictly speaking, true only in mathematical worlds, and so not, strictly speaking, necessary. Since I do not exist in any of the mathematically possible worlds, I am not a truthmaker for any (purely) mathematical truth. Only mathematical entities, on this approach, can be truthmakers for mathematical truths. Nor are we forced to say that any mathematical entity is a truthmaker for any mathematical truth. The number , for example, is arguably not a truthmaker for + =  because there are mathematical worlds in which mod  arithmetic holds, and so in which the number  exists but the number  does not, and so + 6¼ . Although Lewis suggests briefly that he would favor some version of the second (relevant entailment) response, he chooses instead in all of his discussions of truthmaking to sidestep the issue by considering only a restricted version of (TM) according to which only contingent propositions are required to have truthmakers. For the remainder of this chapter, I will suppose that (TM) (and any variant introduced below) has been restricted in this way, and I will say no more about the problem of finding truthmakers for necessary truths.⁹ ⁸ For suggestions along these lines, see Restall (). ⁹ Restricting the scope of (TM) to contingent propositions, however, may not succeed in sidestepping the problem if there are distinct contingent propositions that are necessarily equivalent; any two such propositions could not differ with respect to their truthmakers. (On Lewis’s own account of propositions, according to which propositions are classes of possible worlds, necessarily equivalent propositions are identical; but Lewis’s professed neutrality prohibits him from presupposing this account.) The conjunction of any necessary proposition with a contingent proposition is contingent. Once we allow that there are distinct necessary propositions that differ with respect to their truthmakers, it is hard to see how the conjunctions of these necessary propositions with a contingent proposition would not also be distinct propositions that differ with respect to their truthmakers. So if there was a problem to begin with, simply restricting (TM) to contingent propositions won’t solve it. The right way to sidestep the problem is this: first, form a quotient algebra of the Boolean algebra of propositions by identifying necessarily equivalent propositions; then, second, restrict the scope of (TM) to the resulting equivalence classes of contingent propositions, what might be called the thoroughly contingent propositions. All quantification over propositions in what follows is tacitly restricted to the thoroughly contingent propositions.

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

:      ()

Next, I turn to the task of finding truthmakers for various sorts of truth. I start with cases of truthmaking that are undisputed common ground, acceptable to both a thing theorist and a states-of-affairs theorist. Consider first essential predications. Suppose, for example, that the property of being a dog is essential to whatever has it, and that Fido is some actual dog. Then, Fido is a truthmaker for the proposition that Fido is a dog. For Fido is such that, necessarily, if he exists then he is a dog. Similarly, Fido is a truthmaker for the existential proposition that dogs exist. For Fido is such that, necessarily, if he exists then dogs exist. Now consider any proposition that is strictly implied by Fido is a dog, for example, Fido is a mammal. Clearly, Fido is also a truthmaker for this weaker proposition. More generally, we have as a consequence of (TM) what Armstrong (: ) calls the Entailment Principle, but with strict implication standing in for entailment: (EP)

Whenever T is a truthmaker for a proposition P, and P strictly implies Q, then T is also a truthmaker for Q.

It follows from (EP) that the truthmaking relation is one-many: a single entity is a truthmaker for many—infinitely many—true propositions. Now, let’s consider truthmakers for disjunctions and conjunctions. Suppose that the property of being a cat is essential to whatever has it, and that Tabby is some actual cat. Consider the disjunctive proposition that Fido is a dog or Tabby is a cat. Since a disjunction is strictly implied by each of its disjuncts, it follows immediately from (EP) that both Fido and Tabby individually are truthmakers for the proposition that Fido is a dog or Tabby is a cat. The truthmaking relation, then, is many-one as well as one-many. In particular, disjunctive entities needn’t be introduced to make disjunctions true. Now, what about the conjunctive proposition that Fido is a dog and Tabby is a cat? Neither Fido nor Tabby alone can make the conjunction true, since there are worlds where Fido exists without Tabby, and vice versa. The most natural thing to say, surely, is that they together make it true, that in this case the truthmaking relation holds between two things and a proposition, and that, in general, the truthmaking relation takes a plural argument in its first place. This suggests that we consider a plural version of the Truthmaker Principle: (TMP)

For every true proposition P, there exist some one or more entities T₁, T₂, . . . such that, necessarily, if all of the Ts exist, then P is true. (Cf. Lewis a: .)

Is (TMP) a weaker demand for truthmakers than (TM)? That depends on whether whenever some entities plurally make a proposition true, there is always a single entity that makes the proposition true on its own. There are two prominent candidates for bridging the gap between (TMP) and (TM): classes and mereological sums. Suppose first that one is a realist about classes. Then classes, it seems, can bridge the gap, since, first, class formation is universal—whenever there are some things, there is a class having as members all and only those things—and, second, classes have their members essentially. It follows that whenever T₁, T₂, . . . are plurally truthmakers for P, the class of T₁, T₂, . . . is a single

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   



truthmaker for P.¹⁰ But realism about classes is controversial; and if instead one accepts a fictionalist or structuralist account of classes, classes lack the bona fides to perform the task of truthmaking. I suggest, then, that we look elsewhere for bridging the gap. So suppose now that one is a realist about the mereological sums of classical mereology. In particular, assume (with Lewis and Armstrong) universal composition: whenever there are some things, there is a mereological sum of those things. Assume further a version of mereological essentialism according to which wholes have their parts essentially: for any thing and any part of that thing, necessarily, if the thing exists, then the part exists as well. Then sums can bridge the gap between (TMP) and (TM). Whenever there are Ts such that, necessarily, if all of the Ts exist, then P is true, then the sum of the Ts is such that, necessarily, if the sum exists, then P is true. Mereological essentialism, of course, is controversial in ordinary contexts: my left hand is part of my body, but my body, we ordinarily think, could have existed though it never had a left hand as a part. A counterpart theorist such as Lewis, however, will allow that there are also contexts that evoke a counterpart relation that makes mereological essentialism true. If truthmaking contexts evoke such a counterpart relation, then we have as much of mereological essentialism as we need. (See Section . for further discussion of mereological essentialism.) In any case, since nothing that follows hangs on the difference between (TMP) and (TM), I will forgo neutrality by assuming universal composition and mereological essentialism (in truthmaking contexts), and focus henceforth only on (TM). Taking sums of truthmakers to be truthmakers allows that truthmakers need not be ordinary things; for example, a truthmaker for the conjunction, Fido is a dog and Tabby is a cat, is the sum of Fido and Tabby. But, ordinary or not, nothing beyond the category of things has had to be introduced to serve as truthmakers for the propositions thus far considered. Once we accept universal composition and mereological essentialism, we can establish a flipside to the Entailment Principle that I will call the Parthood Principle: (PP)

If T is a truthmaker for P, and T is a part of T 0 , then T 0 is a truthmaker for P.

Call the mereological sum of all (actual, existing) entities the world. It follows immediately from (PP) that, if a proposition has any truthmakers at all, then the world is a truthmaker for the proposition. Say that an entity discerns one proposition from another just in case it is a truthmaker for the one but not the other. Then, the world is the least discerning truthmaker, and, as such, not a very interesting truthmaker.¹¹ If, as Armstrong thinks, the search for truthmakers is a rival to Quine’s method for uncovering the ontological commitments of propositions, we should be searching for more discerning truthmakers; it is no news, after all, that the world exists. Two ways of supplementing the Truthmaker Principle so as to require more

¹⁰ As Lewis (b: ) notes, this requires that none of the Ts is itself a proper class, since proper classes are not members of proper classes. Lewis provides a way around the problem. But note that the problem only arises if there are worlds with proper class many individuals. In Chapter , I argue that such worlds should be rejected. ¹¹ Unless, perhaps, one is a priority monist. See Schaffer ().

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

:      ()

discerning truthmakers come readily to mind. First, atomic truths (and intrinsic predications generally) should have discerning truthmakers: (SUP₁)

For any atomic truth Ra₁a₂ . . . , there exists a truthmaker involving at most a₁, a₂, . . . and R.

For the states-of-affairs theorist, of course, the corresponding atomic state of affairs is the required discerning truthmaker. Second, although in general the truthmaking relation is many-many, for the case of atomic propositions, there are truthmakers with respect to which the relation is one-one: (SUP₂)

Distinct atomic truths have distinct truthmakers.

Again, this supplement to the Truthmaker Principle is automatic for a states-ofaffairs theorist: atomic states of affairs are the required discerning truthmakers. The extent to which a thing theorist can accept these supplements to the Truthmaker Principle will be discussed in Section .. Let’s take stock. Thus far, we have found truthmakers for essential (monadic) predications, for singular existential propositions, for general existential propositions that generalize over essential properties, and for propositions that can be generated from these by taking disjunctions and conjunctions. And these truthmakers have all been compatible with a thing ontology, where mereological sums of things are taken also to be things. But what about all the propositions not yet included? Don’t they too have to be ontologically grounded? The acceptance of (TM), what Armstrong calls Truthmaker Maximalism, demands that every truth have a truthmaker. The Truthmaker Maximalist needs, then, to find truthmakers for inessential predications, both monadic and polyadic, for general propositions, and for negative propositions. According to Armstrong, a thing ontology will not be able to meet this demand. Finding appropriate truthmakers for these propositions leads to an ontology of states of affairs.

. Truthmaking and States of Affairs I turn now to Armstrong’s theory of truthmaking and the introduction of states of affairs. I start with the simplest case: a predication where the property predicated is intrinsic but not essential to the subject. For example, consider the (singular) proposition I express when, looking at a red ball, I say: “the ball is red.” Is the ball a truthmaker for this proposition? Although it may seem as though, in some sense, the ball does indeed make this proposition true (see later in this section, and also Section .), taking the ball to be a truthmaker is incompatible with (TM) given our ordinary attributions of modality de re. For the ball is only contingently red; it could have existed and been blue, or some other color. So, it is false that the ball is such that, necessarily, if the ball exists, then the ball is red; the ball fails as a truthmaker. Somehow, a truthmaker for the proposition that the ball is red must carry the redness with it wherever it goes. Suppose, then, that there is an immanent universal of redness that the ball instantiates; and consider the mereological sum of the ball and this universal. Could this sum be a truthmaker for the proposition that the ball is red? Although the sum necessitates that the ball exists and that redness exists (is

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somewhere instantiated), it does not necessitate that the ball is red; the sum could exist if the ball were blue and something else were red. (The class having as members the ball and the universal does no better, for a similar reason.) A truthmaker for the proposition that the ball is red must somehow unify the ball and redness, and the unification must itself be necessary. The “obvious candidate” for such a unified truthmaker, Armstrong claims (: ), is the state of affairs of the ball’s being red. For this state of affairs is such that, whenever it exists, not only do the ball and redness both exist, but the ball instantiates redness. States of affairs, if such entities exist, are made for the role of truthmaking. Accepting the demand for truthmakers, then, gives good and sufficient reason for believing that states of affairs exist. Armstrong calls the above argument for states of affairs the truthmaker argument, and says (in Armstrong : ) that it is “perhaps the fundamental argument of this book.” If the argument is sound, then the demand for truthmakers has substantial ontological consequences. Three caveats, however, are in order. First, positing a state of affairs to serve as truthmaker for an intrinsic predication is only justified, given Armstrong’s sparse account of states of affairs, when the property predicated is (or corresponds to) a universal. Without that restriction, the truthmaker argument could be used to support the existence of negative, or disjunctive, states of affairs. Second, as Armstrong concedes, states of affairs might not be the only entities that can fill the truthmaking role (Armstrong : ). Indeed, if we switch from an ontology of universals to an ontology of tropes (particularized properties), and we allow that tropes are non-transferrable—that a trope cannot be instantiated by anything other than what actually instantiates it—then tropes can take the place of states of affairs as truthmakers for intrinsic predications. But, for lack of space, I will say no more in this chapter about tropes as truthmakers. Problems for taking states of affairs to be truthmakers tend to have parallel problems for non-transferrable tropes; and, in any case, Lewis’s critique of (TM) was directed predominantly at Armstrong’s account in terms of states of affairs. Third, Armstrong identifies what he calls “thick” particulars with certain conjunctive states of affairs, which has the effect of allowing thick particulars to be truthmakers for intrinsic predications. Consider a state of affairs of a’s being F. This state of affairs has two constituents: the “thin” particular, which I will call a–, and the universal F; the relation of instantiation holds between a– and F. The thin particular, according to Armstrong, is “the particular in abstraction from its properties” (Armstrong : ). The thick particular, which I will call a+, on the other hand, is “the particular taken along with all and only the particular’s non-relational properties” (Armstrong : ).¹² Let N (for nature) be the conjunction of all the universals instantiated by a–. Then, the thick particular, a+, according to Armstrong, is the state of affairs of a’s being N. Now, back to our red ball. The proposition that the ball is red predicates a property of a thin particular. In the truthmaker argument, it is this thin particular that fails to be a truthmaker for the proposition. But if instead we consider the ball as a thick particular, a conjunctive state of affairs, then the ball is a ¹² We need not concern ourselves, when we say ‘a is F ’, with whether ‘a’ refers to the thick particular a+ or the thin particular a–; either way, the meaning of the predicate ‘is F’ can be adjusted to make the truth conditions of the whole sentence come out right.

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

:      ()

truthmaker for the proposition after all; for the state of affairs of the ball’s being red is a conjunct (and so for Armstrong a part) of the thick particular, and must exist whenever the thick particular exists.¹³ This concession undermines the claim, from the perspective of a states-of-affairs theorist, that a thing ontology is inadequate to the task of providing truthmakers for intrinsic predications; for it is thick particulars, presumably, not thin particulars, that a states-of-affairs theorist should take to be the things of the thing theorist. (In Section ., we will see that the claim is also undermined from the perspective of a thing theorist who accepts counterpart theory.) At best, the states-of-affairs theorist can claim that things cannot provide sufficiently discerning truthmakers for intrinsic predications, since the ball is a truthmaker not only for the proposition that the ball is red, but also for other intrinsic predications, such as the proposition that the ball is round. But put this aside for now. Lewis (prior to Lewis ) accepts the truthmaker argument, that the demand for truthmakers leads to the postulation of states of affairs; his target, rather, is (TM) and the states of affairs themselves. In any case, perhaps the Truthmaker Principle makes a stronger case for states of affairs when applied to general or negative propositions. Consider, for example, the general proposition that all humans weigh less than a ton. (Pretend that being human and weighing less than a ton are universals.) This proposition, if true, is contingently true: it is not impossible for there to exist a supersized human weighing more than a ton. But no thing (or thick particular) could be a truthmaker for the proposition because, assuming even a weak Humean principle of recombination, no thing necessarily excludes the existence elsewhere of a supersized human. In general, no contingent general proposition is made true by a thing, because any thing is compatible with the existence of a distinct thing that is a counterexample to the general proposition.¹⁴ To meet the demand for truthmakers for general propositions, Armstrong introduces totality states of affairs. Say that a sum totals a property if and only if everything that has the property is a part of the sum. For Armstrong, the totaling relation is a dyadic universal, second-order in its second relatum. Whenever a sum stands in the totaling relation to a property, a (second-order) totality state of affairs exists.¹⁵ For example, there is a totality state of affairs asserting that the sum of all humans totals the property being human. Now return to the proposition that all humans weigh less than a ton. For each human, consider the state of affairs asserting that he or she is human, and the state of affairs asserting what that human weighs. Conjoin these states of affairs with the above-mentioned totality state of affairs. This conjunctive state of affairs is such that, necessarily, if it exists, then all humans weigh less than a ton. We have found a truthmaker for the general proposition. More ¹³ Here, however, I wonder what justifies Armstrong’s identification of a conjunction of atomic states of affairs with a state of affairs having a conjunctive universal as a constituent. ¹⁴ See Chapter  for a more nuanced discussion of the argument that things cannot be truthmakers for general propositions. See also Section . below. See Lewis (a) for the (weak) Humean principle of recombination that underlies the argument. ¹⁵ Two notes. First, Armstrong says “aggregate” rather than “sum,” but elsewhere identifies aggregates with (mereological) sums. Second, I will suppose that the second relatum of the totaling relation is always a universal; it is unclear how else to justify the existence of the corresponding totality state of affairs.

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

generally, consider the proposition that all Fs are Gs, where F and G are universals. Let a₁, a₂, . . . be all the Fs. Then, the state of affairs of a₁’s being F & a₁’s being G & a₂’s being F & a₂’s being G & . . . & the sum of a₁, a₂, . . . totaling F is the sought-after truthmaker. Further, if we take a₁, a₂, . . . to be thick rather than thin particulars, then all the conjuncts are included in the one totality state of affairs, and the totality state of affairs, by itself, is a truthmaker for the general proposition.¹⁶ One might hope that a solution to the problem of finding truthmakers for general propositions would immediately carry over to the problem of negative existentials, since every negative existential proposition is logically equivalent to a general proposition. But if the equivalent general proposition generalizes over a negative property, the solution won’t carry over unless we already have a solution to the problem of finding truthmakers for negative predications. Consider, for example, the negative existential that there are no purple cows. (Pretend that being a cow and being purple are universals.) This is equivalent to the general proposition that all cows are not purple. We have the totality state of affairs of the sum of c₁, c₂, . . . totaling the property being a cow, where c₁, c₂, . . . are all the cows. But we don’t yet have truthmakers for the negative predications c₁ is not purple, c₂ is not purple, and so on. Moreover, taking c₁, c₂, . . . to be thick particulars won’t guarantee that the totality state of affairs is a truthmaker for the negative existential unless we have a guarantee that being purple couldn’t possibly be added to the nature of any of the cs. Let us then consider the problem of finding truthmakers for negative predications such as a is not F, where F is a universal. We could, of course, simply postulate negative states of affairs to play the truthmaking role. But, if the states-of-affairs theorist cares about ontological economy, this should be a last resort. Sometimes, at least, positive atomic states of affairs can serve as truthmakers for negative predications. For example, returning to the red ball, the proposition that the ball is not blue is plausibly made true by the state of affairs of the ball’s being red. For, arguably, being blue and being red are necessarily incompatible properties. In general, if a thing a instantiates a determinate F from a determinable, then (arguably) the atomic state of affairs of a’s being F is a truthmaker for the negative predication that a is not G, where G is any other determinate from that determinable. But this solution (called the “incompatibility solution” by Armstrong) doesn’t generalize. For it does not seem necessary that, whenever a particular lacks a property, there must be some other property that it has that is incompatible with the property it lacks. Suppose, for example, that neutrinos can be massless in the sense of instantiating no mass property whatsoever, not even the property of having zero mass (if such there be). Consider the negative predication that some neutrino n doesn’t have mass  kg. In this case, all positive states of affairs involving n are compatible with n having a mass of  kg

¹⁶ Armstrong gives a somewhat different, but equivalent, account of how totality states of affairs provide truthmakers for general propositions in (Armstrong : ). In any case, I see a problem. If the Fs overlap in such a way that some F is mereologically composed of parts of other Fs, then it is possible for a sum of Fs that are G to total F while having a part that is F but not G, and the account fails. This problem could be solved by taking the first relatum of the totaling relation to be a class, rather than a sum; though this raises further issues about the ontological status of classes, and states of affairs involving classes.

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

:      ()

(given a plausible principle of recombination for properties), and so cannot serve as truthmakers for the negative predication. But perhaps totality states of affairs, which are needed in any case to serve as truthmakers for general propositions, can do double duty as truthmakers for negative predications. To see how, note that the proposition that a is not F (for universal F) is equivalent to a general proposition quantifying over universals: all universals instantiated by a are distinct from F. Suppose that G₁, G₂, . . . are all the universals instantiated by a, and that F is distinct from all of the Gs. Consider the totality state of affairs asserting that the sum of G₁, G₂, . . . totals the property of being a universal instantiated by a. This is a truthmaker for the proposition that a is not F. For this totality state of affairs is such that, necessarily, if it exists, then no universal other than G₁, G₂, . . . is instantiated by a, and so (since F is necessarily distinct from all of the Gs) F is not instantiated by a. Taking this totality state of affairs to provide the ontological ground for a simple negative predication might seem suspicious: unlike the second-order totality states of affairs postulated to ground general truths, this totality state of affairs is effectively third-order, relating a sum of properties to a property of properties. Such is the price for repudiating negative atomic states of affairs. By introducing totality states of affairs, Armstrong provides truthmakers for general and negative propositions, where the properties generalized or negated are universals. But what about the rest of the propositions, including the vast majority of propositions that we think or assert? If we drop any concern with finding discerning truthmakers, we can retreat to the idea that the world in toto is a truthmaker for all truths. What is the world for a states-of-affairs theorist? The world must be taken to include, in addition to all the atomic states of affairs, the grand totality state of affairs asserting that these are all the atomic states of affairs. If the world instead were taken to be a big thing, the cosmos, it would not be a truthmaker for general and negative truths, since it would not include the limits that these general and negative truths depend on. Totality states of affairs are needed—at least the one grand totality state of affairs—to provide these limits, to provide the ground for general and negative truths. Or so says Armstrong, who, following Russell, rejects as inadequate the Tractarian account according to which all states of affairs are atomic and first-order.

. Truthmaking and Theories of Truth Before turning to Lewis’s critique of Armstrong’s truthmaking theory, it is worth considering a side dispute over the relation between the Truthmaker Principle and theories of truth. Truthmaker theorists such as Armstrong often claim that the Truthmaker Principle is a stripped-down version of the correspondence theory of truth. “Stripped-down,” because the correspondence it invokes between truths and states of affairs (aka facts) is many-many, not one-one, and because it is neutral as to whether the internal structure of propositions, if any, “pictures” the internal structure of states of affairs. This allows the correspondence theorist to embrace a sparse account of states of affairs, paralleling a sparse account of the universals that are constituents of states of affairs. Just as there are no negative or disjunctive universals, so there are no negative or disjunctive states of affairs. Nonetheless, the essential

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core of a correspondence theory is upheld: a proposition is true if and only if it (appropriately) corresponds to some state of affairs, where the truthmaker relation, it is claimed, provides the appropriate correspondence. Lewis disagrees. In “Forget about the Correspondence Theory of Truth” (Lewis b), he argues that Armstrong’s truthmaker theory is not correctly described as a version of the correspondence theory of truth, or as a rival to the other standardly mentioned theories of truth such as the redundancy, coherence, or pragmatic theories. Indeed, Lewis thinks that nothing that has gone by the name “correspondence theory of truth” should be counted among the rival theories of truth. Talk of the correspondence theory of truth should be banished from philosophy. First, some stage setting. Lewis, himself, is a redundancy theorist, at least with respect to propositional truth.¹⁷ For each proposition, there is what he calls a redundancy biconditional. For example, for the proposition that cats purr, there is the biconditional: the proposition that cats purr is true iff cats purr. This, and any other, redundancy biconditional is “trivial, necessary, and knowable a priori” (Lewis a: ). That is the positive side of the redundancy theory. But there is also a negative side: the redundancy biconditionals are all there is to a theory of (propositional) truth. No substantial property of truth is needed to play any theoretical role. Rather, the truth predicate is needed in ordinary language only to play a practical role: it allows us to form generalizations that “make a long story short.” To use his example, the generalization “whatever the Party says is true” is equivalent to an infinite bundle of conditionals: “if the Party says that two and two make five, then two and two make five”; “if the Party says that we have always been at war with Eastasia, then we have always been at war with Eastasia”; and so on, with a conditional for every proposition. If we had world enough and time, perhaps we could assert each member of the bundle individually; but we don’t, and hence the need for a device, such as the truth predicate, which allows us to assert the entire bundle with a single, compact sentence. But the important point is this: since this bundle of conditionals has nothing especially to do with truth, it follows that the generalization also has nothing especially to do with truth. The truth predicate is here merely a syntactic device for increasing the expressive power of our finitary language. (Alternatively, we could have introduced propositional quantifiers and prosentences. But we didn’t, not in ordinary language.) How does the redundancy theory relate to the other standard theories of truth? Most of these theories, it seems, must reject the positive side of the redundancy theory. For example, a version of the pragmatic theory of truth holds that biconditionals such as the following are a priori: the proposition that cats purr is true iff it is useful to believe that cats purr. If the redundancy biconditionals were also a priori, it would follow that the biconditional—cats purr iff it is useful to believe that cats purr—was itself a priori, which, Lewis says, it manifestly is not. Putative correspondence theories of truth, on the other hand, can accept that the redundancy ¹⁷ In Lewis (a) he explicitly endorses what he there calls a “deflationary theory of truth.” And although his views changed somewhat between Lewis (a) and Lewis (b) (see the following note), I know of no reason to think that what is called “the redundancy theory” in Lewis (b) isn’t just the same theory renamed.

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

:      ()

biconditionals are a priori (and Lewis assumes throughout that they do). They conflict with the redundancy theory instead only by rejecting its negative side, by denying that the redundancy biconditionals are all there is to truth. Now, back to the main argument. For a putative theory of truth to be aptly termed a “correspondence theory” and be counted among the rival theories of truth, according to Lewis, it would have to meet at least the following four conditions: first, it would have to go beyond the redundancy theory, thus conflicting with its negative side; second, it would have to conflict with the positive doctrines of the other theories of truth, such as the coherence and pragmatic theories; third, it would have to be aptly summarized by the slogan “truth is correspondence to fact”; and fourth, it would have to be a theory of truth, not a bundle of claims having nothing especially to do with truth. Lewis conjectures that all putative correspondence theories will fall into one of two camps. Those in the first camp, in effect, identify “facts” with true propositions. These theories are vacuous: it is no news to be told that truths correspond to true propositions (if truths themselves are propositions, as we are supposing). These putative correspondence theories, then, fail to satisfy the first two conditions: they do not go beyond the redundancy theory, nor do they conflict with the coherence or pragmatic theories. “Away with them!” Lewis cries. The other sort of putative correspondence theory is represented by Armstrong’s Truthmaker Principle. On this sort of theory, facts are not identified with true propositions, but with “Tractarian” facts, Armstrong’s states of affairs. This theory is certainly not vacuous, and satisfies the first two conditions. But, according to Lewis, it fails to satisfy the third and fourth condition. It fails to satisfy the third condition, Lewis claims, because truthmakers need not be states of affairs, and so the theory isn’t aptly summarized by the slogan that “truth is correspondence to fact.” For example, as we saw in Section ., any entity is a truthmaker for the proposition that that entity exists, and that entities of its (essential) kind exist. This objection, however, is not in my view very serious. Armstrong presumably holds that any entity whatsoever, if not itself a state of affairs, is a constituent of some state of affairs. Moreover, any state of affairs is a truthmaker for the existence of any of its constituents. It follows from the transitivity of the truthmaking relation that, whenever an entity makes a proposition true, some state of affairs also makes that proposition true. Thus, all truths have states of affairs among their truthmakers, and the slogan seems to be captured well enough, even if some truthmakers are not states of affairs. The chief reason to deny that Armstrong’s theory should be called “a correspondence theory of truth” is that it fails to satisfy the fourth condition: it is a bundle of claims having nothing especially to do with truth, and therefore is not really a theory of truth at all. According to Lewis, the word ‘truth’ occurs in the Truthmaker Principle just for the purpose of making a long story short. The Truthmaker Principle is equivalent to a bundle of biconditionals, one member of which, for example, is the following: the proposition that cats purr is true iff there exists some entity T such that, necessarily, if T exists, then cats purr. But, given the redundancy biconditionals, this biconditional is equivalent to: cats purr iff there exists some entity T such that, necessarily, if T exists, then cats purr. This biconditional says nothing of truth; rather, it claims that the purring of cats is existentially grounded. Similarly, the other biconditionals of the bundle make claims about the existential grounding of all

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’   ()   -- 



manner of things—the flying of pigs, or what-have-you—but say nothing about the concept of truth. Lewis concludes, “if the Truthmaker Principle amounts to a bundle of claims that are not at all about truth, it should not be called a ‘theory of truth’ ” (Lewis b). Consequently, disputes over whether or not to accept the Truthmaker Principle should be sharply separated from disputes over what is the correct theory of truth.¹⁸

. Lewis’s Critique of (TM) and a States-of-Affairs Ontology I turn finally to Lewis’s critique of Armstrong’s truthmaker theory.¹⁹ Following Lewis, I focus in this section on just two types of proposition: negative existentials and inessential (monadic) predications. Offhand, Lewis claims, we do not expect such propositions to have truthmakers. Consider, for example, the proposition that there are no unicorns. Intuitively, this is true not because of anything that exists, but because of what fails to exist. Sure, we can say: the absence of unicorns makes the proposition true. But only someone beholden to a naive theory of reference would take this to imply that absences populate the world along with people, and planets, and protons. Consider now the proposition that the ball is red. (Pretend, as before, that being red is a universal.) Intuitively, this proposition is true not because of what things there are, but because of how things are. Sure, we can form a gerundial phrase, ‘the ball’s being red’; and using that phrase, we can say that such-and-such is true in virtue of the ball’s being red. But such use of the gerundial phrase, by itself, should not lead us to say that the world is populated by an entity, the ball’s being red, distinct from, but co-located with, the ball. To put flesh on these intuitive bones, Lewis recasts the Truthmaker Principle as a principle as to how possible worlds must differ. Before recasting, however, we first need to strengthen the principle. Truthmaker theorists typically hold that (TM) is necessarily true: contingently false propositions would have had truthmakers had they been true.²⁰ Appending ‘necessarily’ to the front of (TM) and regimenting the result in the language of possible worlds gives:

¹⁸ It is worth noting that Lewis’s discussion of theories of truth in (b) differs in substantial ways from his discussion in (a). (Although both papers were published in the same year, and neither refers to the other, I suppose that (b) contains his later, considered view. Lewis (a) was circulated in draft form in May, , and presented at AAP in July, ; Lewis (b) was published in Analysis, which typically has a quick turnover.) In Lewis (a), the correspondence theory of truth is included among “the grand theories of truth”; and it is claimed that all the grand theories are compatible with the redundancy biconditionals, and so none of the grand theories of truth are really about truth. In Lewis (b), as noted above, he argues that the pragmatic and coherence and other substantial theories of truth are incompatible with the redundancy biconditionals. It is because the correspondence theory (if there were such a thing) would presumably be taken in conjunction with the redundancy biconditionals that it, alone, turns out not really to be a theory of truth. ¹⁹ For an alternative, substantial discussion of Lewis’s critique of the Truthmaker Principle with different points of emphasis than what follows, see MacBride (). ²⁰ There is one sort of actualist, however—an actual world exceptionalist—who holds that at alien possible worlds, existential propositions may lack truthmakers.

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

:      ()

(TM+)

For any proposition P and any world W, if P is true in W, there exists some entity T in W such that, for any world V, if T exists in V, then P is true in V. (Lewis a: )

(TM+) implies: For any two worlds W and V and any proposition P, if P is true in W but not in V, then there exists some entity T in W that does not exist in V. For if T is a truthmaker for P in W, then T cannot exist in V lest P be true in V. But on the abundant conception of propositions being assumed, for any two worlds W and V, there is a proposition true in W but not in V: the proposition that W is actualized. The following difference-making principle therefore follows from (TM+): (DM)

For any two worlds W and V, there exists some entity T in W that does not exist in V. (Lewis a: )

(DM) is a two-way difference-making principle: for any two worlds, each world contains some entity not contained in the other.²¹ The case of negative existentials, however, suggests that the difference between world populations need not be two-way. There are no unicorns in our world, but there are unicorns in some other possible worlds (I suppose). In moving from our nounicorn world to a world populated with unicorns, why can’t we simply add unicorns to the population? Why must we also take something away? The case of (inessential) predications suggests that there need not even be a oneway difference in population. Consider a red ball and green bat in our world, and another world where that ball and bat have switched colors: the ball in the other world has the exact shade of green that the bat has in our world; the bat in the other world has the exact shade of red that the ball has in ours. Why must there be any difference in population between these two worlds? Why must a change in how things are bring with it a change in what things there are? These difference-making considerations against the Truthmaker Principle are suggestive, but they are not likely to have much force for a states-of-affairs theorist such as Armstrong. They presuppose that the right way to think of a world population is in terms of things, or things and universals, not in terms of states of affairs. To get a more decisive argument against (TM+), Lewis attacks directly the states of affairs that (he allows) would be needed to serve as truthmakers. States of affairs violate two principles that are fundamental to Lewis’s metaphysics: the uniqueness of composition, and the Humean denial of necessary connections. Because of these violations, Lewis concludes that an ontology of states of affairs is “bad news for systematic metaphysics” (a: ). Let’s start with the composition of states of affairs. Lewis holds to a twofold principle of uniqueness of composition: “there is only one mode of composition; ²¹ Note that (DM) demands that there be no indiscernible possible worlds, since indiscernible worlds do not differ with respect to what exists in them. If one wants to avoid this consequence, one can restrict the Truthmaker Principle to what Lewis (oddly) calls discerning propositions, where a proposition is discerning iff it is never true at one but not the other of two indiscernible worlds: only discerning propositions have truthmakers. (Correlatively, the initial quantifier in (DM) is ‘for any two discernible worlds W and V ’.) See Lewis (a: –). Note that for an anti-haecceitist, a proposition is discerning if and only if it is qualitative. For Lewis’s affirmation of anti-haecceitism, see Lewis (a: –).

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’   ()   -- 



and it is such that, for given parts, only one whole is composed of them” (Lewis c: ). For Lewis, the laws of this single mode of composition are given by classical mereology. Many philosophers reject uniqueness of composition because they accept sets into their ontology, and hold that sets are composed of the individuals that are their members, or their members’ members, or . . . . On this view, distinct sets, such as {a, {b}} and {{a}, b} are composed of the same elements. Lewis argues, however, that the formation of sets involves two operations: the forming of singletons and the forming of unions. Only the latter operation is a mode of composition, and the composition is mereological. Sets, then, on Lewis’s account, do not violate uniqueness of composition.²² Now, consider states of affairs a’s being F, b’s being G, and a’s having R to b, where a and b are particulars and F, G, and R are universals. Somehow, the state of affairs of a’s being F is supposed to be composed of the particular a and the universal F. But, as noted in Section ., the composition cannot be mereological because (on standard essentialist assumptions) the sum a+F can exist even though the state of affairs of a’s being F does not. Moreover, whatever sort of composition is involved, it is not unique. The dyadic states of affairs of a’s having R to b and b’s having R to a are distinct (at least if R is not necessarily symmetric), even though they are composed of the same particulars and universals. And, for good measure, the conjunctive states of affairs of a’s being F & b’s being G and b’s being F & a’s being G are distinct, even though again they do not differ in their (ultimate) components. For Lewis, the idea that states of affairs can be unmereologically composed in this way from particulars and universals makes them totally mysterious. Consider next how states of affairs violate the Humean denial of necessary connections. To capture the Humean prohibition, Lewis introduces a principle of recombination, initially formulated as follows: “anything can coexist with anything else, and anything can fail to coexist with anything else” (Lewis a: ). The first half, strictly speaking, is a prohibition against necessary exclusions. Lewis’s illustration: if there could be a unicorn, and there could be a dragon, then there could be a unicorn and a dragon side by side. How should this be interpreted in terms of worlds? Since worlds do not overlap for Lewis, a unicorn from one world and a dragon from another cannot themselves exist side by side. The principle is to be interpreted in terms of intrinsic duplicates: in some world, a duplicate of the unicorn and a duplicate of the dragon exist side by side. The second half of the principle of recombination is the prohibition against necessary connections. Spelled out in terms of worlds and duplicates, it says: whenever two distinct things coexist in a world, there is another world in which a duplicate of one exists without a duplicate of the other.²³ Lewis’s illustration: since a talking head exists contiguous to a living human body, there could exist an unattached talking head, separate from any living body.

²² See Lewis (). Other potential counterexamples to uniqueness of composition—for example, involving compositional change over time—are also discussed and dismissed (pp. –). ²³ ‘Distinct’, in this context, means non-overlapping, rather than non-identical; I trust that context successfully resolves this ambiguity throughout this chapter. Lewis’s statement of the principle of recombination is rough, and in need of qualification. For example, distinct duplicates cannot “fail to coexist,” as Lewis understands that phrase. For a detailed attempt to set all this right, see Chapter .

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

:      ()

More precisely: there is a world at which a duplicate of the talking head exists but at which no duplicate of the rest of the living body exists. Now, suppose that the atomic proposition that a is F is true for some particular a and universal F, and consider the atomic state of affairs, a’s being F, that according to the states-of-affairs theorist makes it true. As noted above, a’s being F is not mereologically composed of a and F, and thus a’s being F and a are mereologically distinct. Moreover, since a’s being F is a truthmaker for the proposition that a is F, a’s being F is necessarily such that if it exists, then a is F. But also, necessarily, if a is F, then a exists. Thus a’s being F cannot fail to coexist with a, and the prohibition against necessary connections is violated. Next, suppose that the negative existential proposition that there are no Fs is true, and consider a state of affairs S that according to the states-of-affairs theorist makes it true. (For Armstrong, S is, or includes, a totality state of affairs.) Since S is a truthmaker for the proposition that there are no Fs, S cannot possibly coexist with an F. Thus, Lewis concludes, the prohibition against necessary exclusions is violated. Lewis’s second argument against states of affairs, however, falls short of the mark, as can be seen by recasting it in terms of worlds and duplicates. S, we are supposing, exists in the actual world where there are no Fs. Let W be a world where Fs exist, and let a be an F that exists in W. And suppose as is usual that F, being a universal, is an intrinsic property of a. The most that Lewis’s argument establishes is that S itself cannot coexist with a duplicate of a: there is no world in which both S and a duplicate of a exist. But what needs to be shown is that there is no world in which a duplicate of S and a duplicate of a both exist. If the exclusionary power of S arises from its extrinsic nature—as one would naturally suppose—then the argument fails. The way in which S manages to necessarily exclude a may be mysterious—Lewis no doubt thinks that it is—but not because it is a violation of Lewis’s Principle of Recombination. The first argument against states of affairs also faces difficulties. Although it establishes a violation of the Principle of Recombination, as Lewis understands it, it has no force against a states-of-affairs theorist who thinks that states of affairs have an unmereological mode of composition. For such a theorist, the state of affairs of a’s being F and the particular a, though mereologically distinct, are not distinct simpliciter because a is an unmereological component of a’s being F. And since they are not distinct simpliciter, necessary connections between them are excusable, indeed, are to be expected. The Humean denial of necessary connections between “distinct” existences does not apply. So Lewis’s argument that states of affairs violate the Humean denial of necessary connections depends on his defense of the principle of Uniqueness of Composition. Lewis’s metaphysical views on composition are primary. Interestingly, Lewis (a: ) thinks it is the other way around, claiming that the complaint involving necessary connections “subsumes” the complaint involving unmereological composition. He says that unmereological composition can be defined in terms of necessary connections, so if he could understand the necessary connections, then he could understand unmereological composition. Although he doesn’t elaborate on this, the definitions he has in mind are, presumably, something like the following. First, some terminology. Let’s say that a is a part of b just when a is

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()   -- , 



a mereological component of b; and let’s say that a is a constituent of b just when a is an unmereological component of b. (‘Component’, then, is the inclusive, neutral term.) We suppose that ‘part’ (or ‘mereological component’) is already understood. We can define the neutral notion of being a component as follows: a has b as a component iff, necessarily, whenever a exists, b exists. We can then define the unmereological notion of a constituent like this: a has b as a constituent iff a has b as a component, but b is not a part of a. Then, unmereological composition can be defined in terms of ‘constituent’ just as mereological composition is defined in terms of ‘part’. These definitions show that if we could understand necessary connections between mereologically distinct entities, then we could understand unmereological composition. And that is why Lewis thinks the Humean objection to states of affairs subsumes the mereological objection. This puts pressure on Lewis to defend the Humean denial of necessary connections. For all the importance that it plays throughout Lewis’s metaphysics, his defense essentially comes down to this: “it is the Humean prohibition against necessary connections that gives us our best handle on the question what possibilities there are” (a: ). But that does not help with determining the exact scope of the principle. The Humean prohibition does not apply without restriction: entities that are not mereologically distinct, that have a part in common, are allowed to stand in necessary connections. Lewis needs a principled reason why these, and only these, entities are excluded from the scope of the principle. One good reason, I think, would be this. Parthood is partial identity; so it is as much to be expected that there are necessary connections between an entity and its parts as that there are necessary connections between an entity and itself; and, surely, no entity could fail to coexist with itself! But Lewis has backed away somewhat from the view that composition is identity—he says instead that composition is analogous to identity (Lewis : –)—and so it is unclear whether he can avail himself of this response. In any case, we see again that it is the account of composition that is primary, since one’s views on composition inevitably inform the interpretation and defense of the Humean denial of necessary connections. This turns Lewis’s claim on its head: if we could understand why mereological composition is, or is not, the only mode of composition, we could understand why the Humean prohibition always, or only sometimes, applies to mereologically distinct entities. The mereological objection subsumes the Humean objection.²⁴

. (TM) and a States-of-Affairs Ontology, Reconsidered Over the course of fifteen years and four papers (c, , , a), Lewis never wavered in his critique of the Truthmaker Principle and the states-of-affairs ontology that goes with it. Then, toward the end of his career, he takes it all back—or so it might seem. In Lewis (), he argues that intrinsic predications have ²⁴ For a more detailed discussion of the relation between the Humean prohibition and composition, see Chapter .

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

:      ()

truthmakers after all. In a postscript to that paper (Lewis and Rosen ), he argues that negative existentials have truthmakers as well. In a note written shortly thereafter (Lewis ) he withdraws his objections to a states-of-affairs ontology. This is certainly a new view; but is it a change in view? Lewis thought not. His critique had relied on standardly accepted attributions of essential properties; he did not want his own views in the metaphysics of modality, his own brand of essentialism, to prejudice the debate. When he stopped to consider what could be said from his own perspective, there was a radical shift. A counterpart-theoretic account of essential properties allows for a more flexible interpretation of the Truthmaker Principle, an interpretation under which it can be taken to be literally true without running afoul of Uniqueness of Composition or the Humean prohibition. For a truthmaker theorist who does not embrace counterpart theory and the inconstancy of de re modality, Lewis’s critique still stands.

.. Truthmaking and Counterpart Theory: Lewis’s Approach. First, we need some background on counterpart theory.²⁵ As noted in Section ., the Truthmaker Principle (for Lewis) is an assertion of modality de re. It depends for its interpretation on how a world represents de re of an object whether it exists in the world, and what properties it has in the world. For Lewis, since objects inhabit—are part of—only one world, this must be done by considering an object’s counterparts in other worlds. Thus, a world W represents de re of an actual object a that a exists in W just in case a has a counterpart that inhabits W. A world W represents de re of a that a has property F just in case some counterpart of a inhabits W and is F.²⁶ The counterpart relation can then be used to characterize which properties of a are contingent, and which essential: a has F contingently just in case a has F, but some counterpart of a in some world doesn’t have F; a has F essentially just in case every counterpart of a in every world has F.²⁷ Three conditions that must be met by any proposed counterpart relation are especially significant for what follows (see Lewis : ). First, the counterpart relation must be based on qualitative similarity. Second, the respects of similarity that count must be predominantly intrinsic. And, third, the respects of similarity that count must be “important,” where what counts as important can vary, within limits, from context to context. This leads to the inconstancy of de re modality, and to a multiplicity of admissible counterpart relations. Lewis writes: Today, thinking of Saul Kripke as essentially the occupant of a distinguished role in contemporary philosophy, I can truly say that he might have been brought by a stork. Tomorrow, thinking of him as essentially the man who came from whatever sperm and egg he actually

²⁵ Counterpart theory was introduced in Lewis (). It was expanded and modified in Lewis (), Lewis (), and Lewis (a). ²⁶ Note that if a has multiple counterparts inhabiting W, some of which are F and some of which are not F, then W represents de re of a both that a is F and that a is not-F, although not, of course, that a is F-and-not-F. ²⁷ See Lewis (a: –) for some limitations on these definitions. It is controversial, to be sure, whether essence should be analyzed in terms of modality; see Fine (). But here I follow Lewis in presupposing it.

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()   -- , 



came from, I can truly say that he might never have had a philosophical thought in his life. I would be right both times, but relative to different, equally admissible, counterpart relations. (Lewis : )

One of the chief benefits of counterpart theory with multiple counterpart relations is that it allows us to hold on to ordinary ways of counting and individuating objects. Consider the stock example from Gibbard (): suppose I peer into a room that is empty except for a statue of Goliath made of clay. We ordinarily think that there is one object in the room, an object that is both a statue and a lump of clay. But philosophers often disagree. They say that the statue—let’s call it Goliath—and the lump of clay—let’s call it Lumpl—cannot be one and the same object. For there is a property Lumpl has that Goliath lacks: Lumpl, but not Goliath, could have survived a squashing. By an application of Leibniz’s Law, Lumpl is not identical with Goliath. But if ‘could have survived a squashing’ expresses different modal properties when applied to ‘Lumpl’ and when applied to ‘Goliath’, the inference from Leibniz’s Law is invalid. The counterpart theorist diagnoses the situation as follows. There is one counterpart relation, counterpartS, under which all counterparts of the object are statues; with respect to the counterpartS relation, the object is essentially a statue, and could not have survived a squashing. There is a different counterpart relation, counterpartL, under which all counterparts of the object are lumps of clay, but need not be statues; with respect to the counterpartL relation, the object is not essentially a statue, and could have survived a squashing. When we use the name ‘Goliath’, we typically (though not invariably) evoke the counterpartS relation. So interpreted, ‘Goliath could have survived a squashing’ falsely attributes the property has a counterpartS that survives a squashing to the object in the room. When we use the name ‘Lumpl’, we typically evoke the counterpartL relation. So interpreted, ‘Lumpl could have survived a squashing’ truly attributes the property has a counterpartL that survives a squashing to the object. Sometimes both counterpart relations are needed for the interpretation of a single sentence, such as ‘Lumpl, but not Goliath, could have survived a squashing’. The best way of capturing this in a semantic or pragmatic theory need not detain us. What matters is that counterpart theory with multiple counterpart relations allows one to attribute different modal properties to an object depending on how the object is considered.²⁸ Now, in order to apply counterpart theory to truthmaking, we need to reformulate (TM) explicitly in terms of worlds and counterparts: (TMC)

For every true proposition P, there exists some entity T and some admissible counterpart relation such that, for every world W, if T has a counterpart (under that relation) in W, then P is true in W.²⁹

²⁸ Lewis introduced multiple counterpart relations to solve the problem of contingent identity in Lewis (). ²⁹ This deviates from what one would get if one slavishly applied the translation scheme from Lewis () to (TM) (even putting to one side the introduction of multiple counterpart relations). As discussed in Lewis (a: –), the translation scheme does not give the expected results when applied to sentences that are, or contain, existence propositions.

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

:      ()

(TMC) makes explicit that, for Lewis, the counterpart relation used to interpret truthmaking claims is not fixed once and for all; it may vary from context to context depending on which truthmaking claims are under discussion. I consider this further in Section ... To illustrate (TMC), let’s return to the problem of finding truthmakers for intrinsic predications such as ‘the ball is red’. As noted in Section ., there seems to be some sense in which the ball itself can serve as truthmaker for the proposition that the ball is red; but we were prohibited from saying this because being red is a contingent property of the ball, and so not a part of its essence. To serve as truthmaker, the ball would have to be essentially red. Then, and only then, would the ball be such that, necessarily, if it exists, then it is true that the ball is red. Counterpart theory to the rescue! A flexible essentialism can allow that there are contexts with respect to which being red is part of the ball’s essence. One way to conventionally evoke such a context is to introduce special ‘qua’-names of objects, such as ‘the ball qua red’. When we refer to the ball as ‘the ball qua red’, we evoke an unusual counterpart relation under which all counterparts of the ball are red; thus we can truly assert ‘the ball qua red is essentially red’. What are these peculiar qua-entities, and how do they fit into a thing ontology? The ball qua red, of course, is nothing other than the ball itself, just as the statue Goliath is nothing other than the lump of clay Lumpl. Once again, by multiplying counterpart relations, we avoid having to multiply entities. But is there an admissible counterpart relation under which all counterparts of the ball are red? Suppose we start with an ordinary counterpart relation and simply add the restriction that all counterparts of the ball are red. The new counterpart relation is based on predominantly intrinsic respects of qualitative similarity, since the ordinary counterpart relation is; and the new counterpart relation is still based on respects that are important in the context, since use of ‘the ball qua red’ made being red important by suggesting that that was how we were to consider the ball. The strategy clearly generalizes. For any intrinsic predication, a is F, we can say that a qua F is a truthmaker where a qua F is none other than a itself; for, under the counterpart relation evoked by ‘a qua F’, any world in which a exists (by having a counterpart) is a world in which a is F (by having a counterpart that is F). Consider next the problem of finding truthmakers for negative existentials such as the proposition that there are no unicorns. Perhaps the qua-names, and the unusual counterpart relations they evoke, can help here as well. For example, we could say that an ordinary thing, such as the Eiffel Tower, is a truthmaker for the proposition that there are no unicorns if we conceive of the Eiffel Tower as: the Eiffel Tower qua unaccompanied by unicorns. Does this qua-name evoke a counterpart relation under which all of the Eiffel Tower’s counterparts are in worlds uninhabited by unicorns? Unlike the use of qua-names to show that ordinary things are truthmakers for intrinsic predications, this, according to Lewis, is a cheap trick. The supposed counterpart relation evoked by use of this qua-name is based on respects of similarity that are almost entirely extrinsic, depending neither on the intrinsic nature of the Eiffel Tower nor the intrinsic nature of its immediate surroundings. Thus, it is not an admissible counterpart relation. Even a flexible essentialism, Lewis thinks, must have limits to its flexibility. Lewis () thus still denies that negative existentials have truthmakers.

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()   -- , 



But in the postscript (Lewis and Rosen ), Lewis reconsiders. By considering bigger truthmakers, the needed counterpart relations become less extrinsic. Although no proper part of the world can, in virtue of its intrinsic nature, make it true that there are no unicorns, perhaps the world in its entirety can do the job, where the world is just the biggest thing, the cosmos. Indeed, surely there is some sense under which the world as a whole makes it true that there are no unicorns. A truthmaker theorist would argue, however, that no thing can make negative existentials true, because any thing, even the cosmos, might have been a proper part of a bigger thing. (Consider, for example, a possible world containing a series of cosmic oscillations—big bang, big crunch, big bang, and so on—and suppose one of the cycles is a duplicate of our cosmos.) If our world might have been a proper part of a bigger world, then it might have existed in its entirety while there also existed somewhere outside of its bounds a unicorn. Thus the world, if taken to be a thing, is not a truthmaker for the proposition that there are no unicorns. We can, indeed, the truthmaker theorist continues, say that the world as a whole is a truthmaker for the proposition that there are no unicorns; but only because the world is a state of affairs, not a thing, and is composed in part of totality states of affairs. But the thing theorist who accepts the inconstancy of de re modality has a ready response. Although we can say that the world might have been part of a bigger world, we also can say, with no less propriety, that the world might have been bigger than it is. (For example, the world might have contained a series of cosmic oscillations.) Both of these claims are naturally and straightforwardly interpreted as modality de re; moreover, the very same possible world (say, with cosmic oscillations) can serve to validate both claims. To validate the first claim, we identify the world by intrinsic character alone, so that counterparts of the world must be duplicates of the world. To validate the second claim, we identify in part by extrinsic character, taking it to be essential to a world that it be a world, that is, the biggest thing there is; on this way of identifying, counterparts of the world must be worlds. Combining these two ways of identifying, we get a counterpart relation according to which counterparts of worlds are always duplicate worlds. This counterpart relation is admissible, being based on respects of similarity that are predominantly intrinsic. And the respect that is not intrinsic, being the biggest thing, and so unaccompanied, is clearly an important respect. To evoke this counterpart relation, we can use the qua-name: ‘the world qua unaccompanied and intrinsically just as it is’. When the world is thus considered, the world is a truthmaker for every truth. A thing theorist, then, can endorse (TM) and Truthmaker Maximalism. Thus far we have seen how the flexibility of counterpart theory allows things to be truthmakers for intrinsic predications and negative existentials; states of affairs aren’t needed to satisfy (TM). But the states-of-affairs theorist is not content. Suppose that there are immanent universals, and that some particular a instantiates universals F and G. Although the states-of-affairs theorist grants that some thing—the thick particular associated with a—makes true both the proposition that a is F and the proposition that a is G, she thinks that each of these propositions has its own, more discerning, truthmaker. The state of affairs of a’s being F makes true that a is F; the state of affairs of a’s being G makes true that a is G. To get sufficiently discerning truthmakers, one still needs states of affairs.

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

:      ()

As we saw in Section ., Lewis had objected to states of affairs on two grounds: their unmereological composition and their involvement in necessary connections. But in a note originally titled “States of Affairs Reconsidered” (Lewis ) written shortly after Lewis (), he withdraws these objections. With the flexibility of counterpart theory on board, states of affairs can be made innocent by identifying them with mereological sums of particulars and universals. The Truthmaker argument touted by Armstrong and endorsed by Lewis, that the state of affairs of a’s being F could not be identified with the mereological sum a+F, rested in part on mereological essentialism, the view that the whole cannot exist without its parts, nor can the parts exist with the whole. Formulated in counterpart theoretic terms, mereological essentialism amounts to this: (ME)

For all entities a and b, if a+b has a counterpart, (a+b)0 , in a world W, then both a and b have counterparts, a0 and b0 , in W, and a0 +b0 = (a+b)0 ; and if a and b have counterparts, a0 and b0 , in a world W, then a+b has a counterpart (a+b)0 in W, and a0 +b0 = (a+b)0 .

But a flexible essentialist will say that (ME) is true in some contexts but not in others. To illustrate, consider the question whether a chair is identical with the sum of its legs, seat, and back. A dogmatic mereological essentialist must answer “no” because we can truly say that the chair could have existed without one of its legs, and that the sum could have existed even though the legs, seat, and back were never assembled into a chair. In counterpart-theoretic terms: because there are counterparts of the chair that are not identical with any sum of the counterparts of the chair’s parts, by (ME), the chair cannot be identical with the sum. But, once again, flexible counterpart theory comes to the rescue. There is only one thing, alternately referred to as “the chair” or “the sum of the legs, seat, and back.” When we refer to it as “the chair,” we create a context that evokes a counterpart relation under which the parts are not essentially tied to the whole and mereological essentialism is false. When we refer to it as “the sum of the legs, seat, and back,” we create a context that evokes a counterpart relation under which the parts and the whole are essentially linked and mereological essentialism is true. One thing has different essences depending on how it is considered. The same treatment applies to sums and states of affairs. An enlightened states-ofaffairs theorist, one who accepts a flexible essentialism, can say that the state of affairs of a‘s being F is identical with the mereological sum a+F. (More exactly, a–+F, the sum of the thin particular and the universal.) But this one entity, which is both a state of affairs and a sum, has different essences depending on how it is considered. Considered as a state of affairs, it is essentially a state of affairs, and exists if and only if a is F. Considered as a sum, it is not essentially a state of affairs, and exists whenever a and F exist, whether or not a is F. Again, one multiplies essences (and counterpart relations) without multiplying entities. This same treatment applies to any state of affairs with particulars and universals as components: the state of affairs is identical with the mereological sum of those particulars and universals.³⁰ ³⁰ One might wonder whether a similar treatment could be applied to structural universals, thus taking them to be mereologically composed. Such a “reconsideration” of structural universals would undercut Lewis’s reason for rejecting universals in Lewis (b). But if there are, or could be, basic laws involving

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

Lewis’s earlier objections to a state-of-affairs ontology now dissolve. The first objection was that states of affairs violate Uniqueness of Composition. But if states of affairs are just mereological sums, then no two states of affairs have the same components because mereological composition is unique. An opponent might, however, try to turn this on its head: don’t the examples in Section . introduced to show that the composition of states of affairs is not unique tell against the identification of states of affairs with mereological sums? Suppose, for example, that the states of affairs of a’s having R to b and of b’s having R to a both exist, for some dyadic universal R. (Suppose also that R is not necessarily symmetric.) Aren’t these states of affairs distinct, contra Lewis’s proposal? There are two complementary replies. First, perhaps our offhand opinion that these states of affairs are distinct comes from thinking of states of affairs as proposition-like. On that conception of states of affairs, the opinion is correct, but irrelevant. Second, even when thinking of states of affairs as Tractarian facts that compose the world, we may still be in the grip of some Leibniz’s Law argument that lacks force for a flexible essentialist. In any case, notwithstanding our offhand opinions, perhaps overall theoretical considerations favor identifying these dyadic states of affairs with mereological sums, and thus with one another. Or so Lewis could claim (but see Section .. for a problem).³¹ The second objection was that states of affairs are involved in mysterious necessary connections. Whenever the state of affairs of a’s being F exists, necessarily, a exists and instantiates the universal F. Whence come these necessary connections? They arise from the counterpart relation evoked when speaking of states of affairs, and thus are no more mysterious than the workings of a flexible counterpart theory— something Lewis thinks we understand well enough. Thus, it is no mystery on this account how there could be something such that (in some contexts) it is true to say that, necessarily, it exists only if a is F. This entity, qua state of affairs, is a truthmaker for the proposition that a is F. This same entity, qua mereological sum, is not a truthmaker for the proposition that a is F. Once again, whether or not an entity is a truthmaker for a proposition depends on how that entity is considered.

.. Truthmaking and Counterpart Theory: An Alternative Approach Lewis’s strategy for rehabilitating the Truthmaker Principle, however, is open to a decisive objection.³² That strategy is this. For any true proposition, find some thing whose essence can be tailor-made, using flexible counterpart theory, so that the thing’s existence necessitates the proposition. For the ball is red, use ‘the ball qua red’ to evoke a counterpart relation under which the ball is essentially red; for there are no unicorns, use ‘the world qua unaccompanied by unicorns’ to evoke a counterpart relation under which the world cannot co-exist with a unicorn. And so on. What constrains this tailoring of essences to truths? Without constraints, we could structural universals, then such a reconsideration would not be compatible with either Lewis’s or Armstrong’s theory of laws. ³¹ A third, and I think better, reply is to hold that fundamental relations are necessarily symmetric; but I know of no reason to think Lewis would support it. Dorr () provides some arguments for this view. ³² Material from this section was presented in a lecture entitled “The World: Facts or Things?” at NYU in February, .

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

:      ()

apply the strategy according to a simple formula: for any true proposition P and any thing T, use ‘T qua inhabits a world where P’ to evoke a counterpart relation that makes T’s existence necessitate P. Lewis would reply: ‘T qua inhabits a world where P’ will often fail to evoke an admissible counterpart relation; P may be about matters that are too extrinsic, or unimportant, for P to be taken to be essential to T. I demur. I think a flexible essentialist should allow that for any T and P, some context could be concocted according to which the truth of P is essential to T. But put that worry aside. There is a bigger problem. Suppose, then, that the problem of placing limits on what counts as an admissible counterpart relation is satisfactorily solved. At best, Lewis’s strategy shows that, for each truth, there is a counterpart relation under which that truth has a truthmaker. What we need, I claim, is something stronger: there is a single counterpart relation under which every truth has a truthmaker. That is, we should replace (TMC) with the stronger: (TMC+)

There is an admissible counterpart relation, call it counterpartT, such that: for every true proposition P, there exists some entity T such that, for every world W, if T has a counterpartT in W, then P is true in W.

The counterpartT relation is evoked by “truthmaker contexts,” contexts in which we ask whether a proposition is made true by some thing, or is true in virtue of the existence of some thing. Without a single such counterpart relation, we do not have a uniform interpretation of the Truthmaker Principle under which it is true. We have only, for each instance of the Truthmaker Principle, an interpretation that makes that instance true. If we take the Truthmaker Principle to be a schema, an infinite bundle of assertions, one for each proposition, then a non-uniform interpretation might be good enough. But if we take the Truthmaker Principle to be a single assertion (as we have heretofore), the assertion that every truth has a truthmaker, then the variable ranging over potential truthmakers needs to be interpreted uniformly with respect to a single counterpart relation.³³ For comparison, suppose I walk into a room filled with statues made of clay and say: “some of these are essentially statues; others are essentially lumps of clay.” If there are no relevant features distinguishing some of the statues from the others, I find this scarcely intelligible. What conditions would a counterpart relation have to meet in order to validate the Truthmaker Principle? First, since any aspect of the intrinsic character of a thing may be relevant to what propositions the thing makes true, a counterpart of the thing must preserve its intrinsic character. We can guarantee this by requiring: ()

CounterpartsT are always intrinsic duplicates.

() ensures that a true intrinsic predication of a thing will have that thing as a truthmaker. Thus, if a has F, for intrinsic F, then a is such that, necessarily, whenever it exists it has all of its actual intrinsic properties, and so, in particular, it has F. ³³ Lewis (b: ) does say that the Truthmaker Principle “is equivalent” to an infinite bundle of biconditionals so as to emphasize the dispensability of the notion of truth; see Section . above. But elsewhere, he formulates the Truthmaker Principle as a single assertion by quantifying universally over propositions. The argumentation of Lewis (a) depends on it.

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()   -- , 



Second, on any view that accepts the Truthmaker Principle, the world should be the supreme truthmaker, making every truth true. But, as we saw in Section .., if the world is a thing—the biggest thing—then the world will fail to be a truthmaker for negative existentials if the world could have been a proper part of a bigger world. The thing theorist, then, needs to require, at least in truthmaking contexts, that the world is essentially the biggest thing, that the world is essentially a world. In terms of counterparts: ()

CounterpartsT of worlds are always worlds.

A counterpart relation satisfying () and () trivially validates the Truthmaker Principle. Under any such counterpart relation, the counterpart of a world is a duplicate world. But duplicate worlds are indiscernible;³⁴ they agree with respect to the truth or falsity of any (qualitative) proposition. Thus, for any true proposition P, the world (the cosmos) is such that, necessarily, if it exists, then P is true. The world is a truthmaker for every truth. If the goal were just to find a counterpart relation that makes the Truthmaker Principle true, we could stop here. But sometimes, at least, more discerning truthmakers can be had. We have seen this, so far, only for the case of intrinsic monadic predications. This is the only case that Lewis considers. But I think discerning truthmakers can be found as well for intrinsic polyadic predications, and for the negations of intrinsic predications. In each of these cases, the object or objects of predication can serve as truthmakers. Consider first the case of negated intrinsic monadic predications. Intuitively, when asking what makes it the case that a thing has, or fails to have, some intrinsic property, we need look no further than the thing itself. For an intrinsic property F, a is a truthmaker for a has F, if it is true that a has F, and a is a truthmaker for a doesn’t have F, if it is true that a doesn’t have F. In asking what constraints this puts on the counterpart relation in truthmaking contexts, there are two cases to consider: a nominalist thing theorist who rejects universals or tropes; and a realist thing theorist who accepts universals or tropes. For the nominalist thing theorist, () already suffices. Suppose a doesn’t have F is true. By (), any counterpartT of a at any world lacks F. Thus, a is such that, necessarily, if it exists, then it is not F. And that’s what it takes for a to be a truthmaker for the proposition that a is not F. But now consider a realist thing theorist who takes things to be thick particulars composed of thin particulars and universals. In seeking a truthmaker for a is not F, we need to keep track of the distinction between the thin particular, a–, and the thick particular, a+. It is the thick particular, a+, that the thing theorist takes to be a truthmaker for a is not F. But () does not suffice to guarantee that the thing a+ is a truthmaker. For consider a world W where a+ has a (unique) counterpartT b that is not itself a thick particular, but is instead a “middle-sized” particular properly included in a thick particular b+. And suppose that, although b does not include F (since by () it is a duplicate of a+), the thick particular b+ does include F. Now, I suppose that for any particular a—thin, thick, or middle-sized—it is true that a is F ³⁴ This requires two assumptions. First, that duplicate worlds are qualitatively indiscernible requires that worlds are externally isolated, that no part of any world is externally related to any part of any other world. I defend this in Chapter . Second, that qualitatively indiscernible worlds are indiscernible tout court requires anti-haecceitism. See Lewis (a).

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

:      ()

iff the thin particular component of a instantiates F. On those truth conditions, although a+’s counterpartT in W, b, does not include F, the proposition that a is F is nonetheless true in W. To defuse this counterexample, we need to require that a thick particular could not have failed to be a thick particular, that things are essentially things. In terms of counterparts: ()

CounterpartsT of things are always things.

() and () together ensure that a true negation of an intrinsic predication of a thing will have that thing as a truthmaker.³⁵ Finally, consider the problem of finding things to serve as truthmakers for intrinsic polyadic predications. Start with the dyadic case; the generalization to higher adicity is routine. Thus, suppose that a has R to b, where R is a fundamental relation, and thus intrinsic.³⁶ A flexible truthmaker theorist might hope that the sum a+b will be a truthmaker for the proposition that a has R to b. This will require (as we saw in Section .) that, in truthmaking contexts, mereological essentialism (ME) holds, at least with respect to things. Does () together with (ME) (for counterpartT) ensure that a+b is a truthmaker for a has R to b? Indeed, given () and (ME), every counterpartT of a+b is a duplicate of a+b, and so has a duplicate of a as a part that is R-related to a part that is a duplicate of b. But that may not be sufficient to make a +b a truthmaker for aRb. For a counterexample, consider this. Suppose that a and b are duplicates, and that R is an intrinsic relation such that aRb, but not bRa. Consider a world where a and b each have a single counterpartT, a0 and b0 , respectively, and such that b0 Ra0 . By (), a, b, a0 , and b0 are all duplicates of one another. By () and (ME), a0 +b0 is a duplicate of a+b, and so not a0 Rb0 . Then, although a+b has a counterpartT in the world, no counterpartT of a has R to any counterpartT of b. Thus, the existence of a+b does not necessitate that a has R to b. The solution, as in other cases involving essential relations, is to consider not only counterparts of individuals, but also counterparts of pairs (and more generally counterparts of sequences of arbitrary length).³⁷ Say that two pairs, and , are intrinsic isomorphs iff a and a0 are duplicates and b and b0 are duplicates, and, for any intrinsic relation R, aRb iff a0 Rb0 . To ensure that a+b will be a truthmaker for aRb we can, first, put the following constraint on the pair-counterpart relation: () If a and b have counterpartsT in W, then has a counterpartT pair in W; and counterpartT pairs are always intrinsic isomorphs.³⁸ ³⁵ Proof. Suppose a is not F, for intrinsic F; and let W be a world where a exists, that is, where a has counterpartsT. By () and (), all of a’s counterpartsT in W are thick particulars and duplicates of a+. Since they are thick particulars, they have a property F iff they include F. Since they are duplicates of a+, they do not include F. Therefore, no counterpartT of a+ has F, and the proposition that a is not F is true in W. Which goes to show: a is a truthmaker for a is not F. ³⁶ A relation is intrinsic if it is either internal or external, and thus supervenes on the intrinsic natures of its relata, taken together. See Lewis (a: ). Lewis supposes that fundamental properties and relations are intrinsic. (See Section ..) ³⁷ See Hazen () and Lewis (a: –). ³⁸ Lewis (a: –) calls the counterpart pair a “joint possibility” for . So () could be rephrased: if W contains individual possibilities for a and for b, then it contains a joint possibility for , and joint possibilities preserve intrinsic relations between a and b.

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()   -- , 



Then, second, we modify the counterpart-theoretic semantics for doubly de re modal assertions so as to quantify over counterpart pairs rather than counterparts.³⁹ Thus, a world represents de re of that a has R to b iff some counterpart pair of , , exists in W, and a0 has R to b0 in W. Given () through (), (ME), and the modified semantics, a+b is a truthmaker for aRb whenever aRb is true and R is intrinsic.⁴⁰ Call any counterpart relation that satisfies () through () and (ME) a truthmaking counterpart relation. (Counterpart relations are now expanded to include sequences among their relata.) Truthmaking counterpart relations have at least as much claim to legitimacy as the counterpart relations admitted by Lewis. Because of () and (), the respects of similarity that count are “predominantly intrinsic.” And the extrinsic respects of similarity that count according to () and ()—the property of being a world and the property of being a thing—are undeniably “important” in contexts where ontology is under discussion. Moreover, unlike Lewis’s piecemeal approach to finding truthmakers that appeals to multiple counterpart relations, fixing on a single truthmaking counterpart relation allows for a uniform interpretation of (TM) and its supplement (SUP₁), an interpretation that makes them both literally true. ((SUP₁), recall, was the thesis: for any atomic truth Ra₁a₂ . . . , there exists a truthmaker involving at most a₁, a₂ . . . and R.) The ‘qua’-names are not needed to evoke a truthmaking counterpart relation. Rather, in truthmaking contexts, contexts in which the search for truthmakers is explicit, a truthmaking counterpart relation is automatically evoked. That provides the best explanation, for a thing theorist, as to why (TM) and (SUP₁) have the ring of truth. In considering (TM) or (SUP₁), we create a truthmaking context; and in truthmaking contexts, they are true. There is one casualty, however, in the switch to a uniform interpretation of (TM): states of affairs, understood in Lewis’s way as mereological sums of universals and (thin) particulars, no longer provide sufficiently discerning truthmakers to validate (SUP₂), the other supplement to (TM). ((SUP₂), recall, was the thesis: distinct atomic truths have distinct truthmakers.) The problem arises with certain relational or complex states of affairs. Consider, for example, a dyadic universal R such that aRb and bRa. By (SUP₂), since aRb and bRa are distinct atomic propositions, they have distinct (i.e. non-identical) truthmakers. A thing theorist will simply reject (SUP₂) and say that there is a single truthmaker, a+b, for both aRb and bRa. A states-ofaffairs theorist, however, is committed to (SUP₂), and must find distinct truthmakers for aRb and bRa. But if states of affairs are mereological sums, how can the one sum, a+R+b, provide two distinct truthmakers? On Lewis’s piecemeal approach, this is easily done: by multiplying counterpart relations, one multiplies truthmakers. The ‘qua’-name ‘a+R+b qua a’s-having-R-to-b’ evokes one of these counterpart relations; ‘a+R+b qua b’s-having-R-to-a’ evokes the other. Although there is only

³⁹ See Hazen (: –). For the general case, Hazen quantifies over the “representative functions.” The modification in the semantics is needed in any case to solve the problem of “essential relations.” ⁴⁰ Proof. Suppose aRb, for intrinsic R. Consider any world W where a+b has a counterpartT. By (ME), a and b have counterpartsT in W. By (), has a counterpartT pair in W, where and are intrinsic isomorphs. Therefore, since aRb and R is intrinsic, a0 Rb0 . Thus, aRb is true in W, as was to be shown.

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

:      ()

one entity involved—a+R+b—one can allow that (in truthmaking contexts) the two ‘qua’-versions of this one entity count as two different truthmakers.⁴¹ But this way of validating (SUP₂) is ruled out if we want a uniform interpretation of the truthmaking relation. If there is only a single counterpart relation with respect to which we interpret (TM) and (SUP₂), then no one entity can provide two different truthmakers. I conclude, then, that Lewis’s irenic attempt to rehabilitate a states-of-affairs truthmaking theory falls short of the mark.⁴²

. Truth Supervenes on Being In the previous section, we saw that a thing theorist, armed with a flexible counterpart theory, can accept the Truthmaker Principle as literally true. That is not to say, however, that the thing theorist takes the Truthmaker Principle to be a fundamental principle of metaphysics. On the contrary, showing how to make the Truthmaker Principle true is a metaphysical sideshow, interesting only as a way of appeasing ordinary intuitions about truthmaking, and thereby placating truthmaker theorists overly enthralled by those intuitions. Here are three reasons why the Truthmaker Principle should be no part of fundamental metaphysics. First, the Truthmaker Principle, being an assertion of modality de re, depends for its interpretation on context; and no fundamental metaphysical principle should be context dependent. This objection, however, can be gotten around by formulating the Truthmaker Principle in the language of possible worlds, explicitly incorporating the truthmaking counterpart relation into the formulation. That would eliminate the context dependence. Second, all fundamental metaphysical principles are necessary, whereas the Truthmaker Principle, even if true, is not necessarily true. Or so I claim, because I take it to be metaphysically possible that nothing exists—or, at any rate, that no contingent entity exists.⁴³ But, the Truthmaker Principle is incompatible with such a possibility: if it is true that no (contingent) entity exists, then that very truth lacks a truthmaker, and the Truthmaker Principle is false. Admittedly, this objection lacks bite for those modal metaphysicians, including Lewis and Armstrong, who reject the possibility of nothing.⁴⁴ But there is worse to come. The third and most important reason why the Truthmaker Principle is no part of fundamental metaphysics is that it is motivated by a wrong account of ontological grounding, wrong on two counts. Let us take the relation of ontological grounding to ⁴¹ There are various ways in which a flexible counterpart theorist can make precise the notion that “one entity can count as two” in intensional contexts. And, of course, truthmaking contexts, on modal accounts of truthmaking, are intensional contexts. ⁴² Would it help to take states of affairs to be sets—sequences of universals and (thin) particulars— instead of mereological sums? No, on any states-of-affairs theory, the (concrete) world is composed of states of affairs; thus, if states of affairs are sets, the (concrete) world itself, implausibly, would have to be a set, or composed of sets. ⁴³ See Section .. on how a realist about possible worlds can accommodate the possibility of nothing. ⁴⁴ Lewis nonetheless transforms it into a meta-metaphysical objection. The Truthmaker Principle, he writes, provides “a swift reason why there must be something, and not rather nothing . . . Altogether too swift, say I” (Lewis a: ).

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   



be a relation between propositions and propositions, with both arguments plural. Wherever one might speak of entities doing the grounding, we will instead say that the corresponding existence propositions provide the ground. (This parallels what was said in Section . about the truthmaker relation.) When some propositions are fundamental, and sufficiently inclusive to ground all truths, call those propositions an ontological base; if there is an ontological base, say that truth is ontologically grounded. Now, although both Lewis and the truthmaker theorists agree that truth is ontologically grounded, the truthmaker theorist puts two conditions on ontological grounding that Lewis would reject. First, truthmaker theorists hold that for truth to be ontologically grounded there must be an ontological base consisting entirely of existence propositions. I will return to this below. Second, truthmaker theorists hold that entailment by propositions in the base is necessary for ontological grounding: truth is ontologically grounded only if every truth is entailed by some truths in the ontological base. Call this the entailment constraint. (Entailment may be strict implication, or may be something even stronger.) Lewis holds instead that only supervenience, not entailment, is necessary for ontological grounding: truth is ontologically grounded if and only if every truth supervenes on the ontological base. Call this the supervenience constraint. (Supervenience here is global supervenience applied to propositions.) If the ontological grounding of truth satisfies the entailment constraint, it also satisfies the supervenience constraint, but not vice versa. The supervenience constraint, then, is weaker than the entailment constraint, and demands less of the ontological base. Truthmaker theorists, because they accept the entailment constraint, must deny that the atomic propositions form an ontological base. For example, negated atomic truths and general truths need not be entailed by the true atomic propositions. Thus, truthmaker theorists must add non-atomic propositions to the ontological base to ground negated atomic and general truths; Armstrong, for example, adds propositions involving totality states of affairs. Lewis, because he accepts the supervenience constraint, disagrees: no propositions beyond the atomic propositions need be included in the ontological base. This is because all truths supervene on the atomic truths; two worlds that agree with respect to the truth value of all atomic propositions must agree with respect to the truth value of all negated atomic propositions, of all general propositions, indeed, of all (qualitative) propositions. Speaking picturesquely: once God has fixed the truth value of all the atomic propositions, the qualitative nature of the world is thereby fully determined.⁴⁵ The other condition that truthmaker theorists wrongly require for ontological grounding is that there be an ontological base consisting entirely of existence propositions. This condition is easily met by states-of-affairs theorists. Each fundamental proposition in the ontological base can be replaced by a necessarily equivalent existence proposition asserting that the corresponding state of affairs exists. For example, the atomic proposition that a is F can be replaced by the existence proposition that a’s being F exists; the totality proposition that a totals F can be ⁴⁵ The modifier ‘qualitative’ is needed because indiscernible worlds (if any) agree with respect to the truth value of all atomic propositions. Note that the atomic propositions must include all possible atomic propositions, including those (if any) involving alien universals.

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

:      ()

replaced by a’s totaling F exists; and so on. Lewis, however, feels no compulsion to whittle down the fundamental propositions in the ontological base to existence propositions.⁴⁶ One aspect of being involves what entities there are; but another aspect of being involves how those entities are, and how they are arranged. The atomic propositions by themselves give full expression to both of these aspects of being. Thus, if truth supervenes on the atomic propositions, then we can say that truth supervenes on being: (TSB)

For any proposition P and any worlds W and V, if the same entities exist in W and V, and those entities instantiate the same fundamental properties and relations in W and in V, then P is true in W iff P is true in V.⁴⁷

For an anti-haecceitist, the conditional simplifies to: if W and V have the same pattern of co-instantiation of fundamental properties and relations, then P is true in W iff P is true in V. According to Lewis, it is (TSB), not (TM), that is a fundamental principle of metaphysics.⁴⁸ Nothing more than (TSB) is needed to ensure that truth be ontologically grounded. A defense of (TSB) must show that, although weaker than the Truthmaker Principle, it is still strong enough to do some ontological heavy lifting. For truthmaker theorists, an important role of the Truthmaker Principle, is to “catch cheaters”: philosophers who, by positing truths without truthmakers, fail to own up to the ontological cost of their theories. Consider, for example, the phenomenalist who holds that propositions about physical objects can be analyzed in terms of sensedata.⁴⁹ To analyze propositions about unobserved objects, such as that a ball in an otherwise empty room is red, the phenomenalist typically calls on the sense-data that an observer would have had, had she been in a position to observe the ball. But, the truthmaker theorist asks, what are the truthmakers for these counterfactual truths? Not the unobserved ball (or states of affairs involving it) because these, according to the phenomenalist, do not exist. And not any actual sense-data (or states of affairs involving them) because these are all compatible with the ball not being red. The phenomenalist counterfactuals, it seems, are “brutely true” in violation of the Truthmaker Principle.

⁴⁶ No compulsion. But a thing theorist can do this without cost by interpreting existence propositions using the duplication relation as the counterpart relation: a thing exists in all and only those worlds that have a duplicate of that thing as a part. This leads to a stronger version of Truth Supervenes on Being that I call the Subject Matter Principle: every proposition has a subject matter, entities such that the truth or falsity of the proposition is determined by whether or not those entities exist. See Chapter . ⁴⁷ This is essentially the formulation in Lewis (a: ). See also Bigelow (a: ). (TSB) is sometimes called “Truthmaker” in the literature. Although (TSB) can be understood as characterizing a weaker truthmaker relation, I think it best to only speak of truthmaking if the truthmaker necessitates, or entails, the truth. ⁴⁸ Must one be a modal realist to take (TSB) to be a fundamental principle of metaphysics? Lewis thought not; he took the ontological dispute between (TM) and (TSB) to cut across disputes in the metaphysics of modality. But, certainly, some actualist reformulations of (TSB)—by linguistic ersatzists, for example—are not plausible candidates for fundamental metaphysical principles; and some actualists lack the modal means to provide any formulation of (TSB). ⁴⁹ Armstrong uses this example to introduce truthmaking in Armstrong ().

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   



Lewis is in rough agreement with the truthmaker theorists as to which philosophical views are guilty of cheating—except, of course, that he doesn’t think the thing theorist, whether or not a flexible counterpart theorist, counts among the cheaters. It is thus imperative for Lewis that (TSB) be strong enough to rule out views that posit brute counterfactuals (as well as brute dispositions, brute lawhood, and brute tensed properties). And so it is. For example, on any plausible version of phenomenalism, the phenomenalist counterfactuals don’t supervene on being: two worlds can agree on the truth value of all atomic propositions, propositions about (actual) sense-data, and yet disagree on the truth value of counterfactuals as to what sense-data would have existed, had unmade observations been made. (This is especially obvious if one grants that there are possible worlds without any observers, and so without any (actual) sense-data.) Thus Lewis, no less than the truthmaker theorists, can reject phenomenalism because it violates the dictum that truth be ontologically grounded. The philosopher who invokes brute counterfactuals has a ready response. “Every view is entitled to choose its own base of fundamental propositions. I choose to take some counterfactual propositions to be fundamental. Perhaps these brute counterfactuals can be understood to attribute dispositional properties to the entities that populate the world. But, if not, they can always be taken to attribute fundamental properties to the world as a whole. If it is part of my view that the brute counterfactuals are fundamental propositions belonging to the ontological base, then my view is compatible with (TSB).” And a similar speech can be made by the other supposed cheaters. Lewis would protest: “a philosopher is not entitled to take any proposition to be fundamental; the fundamental properties and relations involved in the fundamental propositions are all categorical; their essential nature is entirely intrinsic, given by their quiddities, not by their causal or nomological roles.” And what justifies this view of the fundamental properties and relations? “Humean recombination principles would fail if the fundamental properties and relations were not categorical.”⁵⁰ This would seem to take us in a circle, since Humean recombination principles are only plausible if one assumes that the fundamental properties and relations are categorical, that their natures are intrinsic. A better Lewisian response is that the whole package is justified holistically by the success of the Humean metaphysics, and especially by its perspicuity. Its metaphysical rivals are shrouded in mystery, engendering at best an illusion of understanding. It is now plain that the way in which Truth Supervenes on Being “catches cheaters,” and puts constraints on metaphysical theorizing, has little to do with the “supervenience” part, and everything to do with the “being” part, and the conception of fundamentality that informs it. Oft-heard complaints that supervenience by itself is not a dependence relation, or a relation of ontological priority, though true enough, are beside the point. It is supervenience on being, where being is characterized in terms of the pattern of instantiation of fundamental properties and relations, that ⁵⁰ See Lewis (a: –). Sider (: ) suggests that replacing (TSB) with “the correct fundamental ideology is that of predicate logic” would have “essentially the same upshots regarding cheaters.” But I don’t see how that would catch “cheaters” who gladly trade their fundamental propositional operators for fundamental properties of worlds.

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

:      ()

provides an ontological ground for truth. And it is the conception of fundamental properties and relations as categorical that gives (TSB) its ontological punch.⁵¹ Those seeking an informative characterization of this Humean conception of fundamental properties will be disappointed; Lewis offers little more than gestures and hints. But one could scarcely exaggerate its importance throughout Lewis’s writings. His entire Humean metaphysics is incomprehensible without it.

⁵¹ Lewis’s most extensive discussion of the nature of fundamental properties is in Lewis (). Lewis does not much use the word ‘categorical’; perhaps a better label would be ‘Humean’, since, for Lewis, only properties that satisfy Humean recombination principles are candidates for being fundamental.

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 The Relation between General and Particular: Entailment vs. Supervenience () . Introduction I say (with many others): the world is a thing, the biggest thing, the mereological sum (or aggregate) of all things. Truth is determined by the distribution of fundamental, or perfectly natural, properties and relations over the parts of this biggest thing. For want of a better name, call this the thing theory.¹ Some say instead: the world is a fact, the most inclusive fact, the conjunction (also mereological sum) of all the facts. Truth is determined by (some sort of) correspondence to the facts. Call this, following Armstrong, factualism.² The dispute between the thing theory and factualism can be traced, of course, to the opening lines of Wittgenstein’s Tractatus (Wittgenstein ), where it received second billing:  .

The world is all that is the case. The world is the totality of facts, not of things.

Could there be a starker division between fundamental ontological theories of the world? On closer inspection, however, the division appears less stark. A factualist need not reject things; nor need a thing theorist reject (all) facts. For each thing accepted by the thing theorist, there is an associated fact accepted by the factualist saying that the thing has the nature that it does. A factualist can identify things with their associated facts at no ontological cost: (some) facts are also things.³ In the other direction, a

First published in D. Zimmerman (ed.), Oxford Papers in Metaphysics, Volume  (Oxford University Press, ), –. This chapter was part of Metaphysical Mayhem VII at Syracuse University in August  and was presented to the Philosophy Department at Arizona State University in February . It expands on material presented to the Philosophy Department at New York University in February . Thanks especially to Daniel Nolan, Laurie Paul, and Ted Sider for their comments. ¹ David Lewis is a prominent thing theorist. See the opening paragraph of Lewis (a). ² Armstrong’s version of factualism is presented in Armstrong (). Armstrong prefers ‘state of affairs’ to ‘fact’. Although I will stick with the shorter term ‘fact’, be warned that ‘fact’ herein is a philosophical term of art, ontologically loaded unlike its natural language homonym. Facts are immanent: part of the furniture of the world. ³ Armstrong holds that things—what he calls “thick particulars”—are facts, in both cases taking intrinsic nature to be essential. See Armstrong (: –). Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

 .  ()

thing theorist can identify the fact associated with a thing with the thing itself: things are also facts. Some metaphysicians may protest: the thing and its associated fact differ in their modal existence conditions, and so cannot be identified; a red ball, for example, can exist without being red, whereas whenever the associated fact exists, the ball exists and is red. But an enlightened factualist or thing theorist, one who appreciates the inconstancy of de re modality, need no more reject the identity of things with facts on this basis than, say, the identity of statues with hunks of clay; in both cases, counterpart relations may be multiplied, not entities within the world.⁴ The factualist, however, is not yet satisfied: even if things are facts, they cannot be all of the facts. For there is a distinct fact for each (sparse) property had by a thing, the fact that the thing has that property; there is a multiplicity of facts for each thing. And, the factualist may continue, this is an immanent multiplicity, lest the world be impoverished in causes and effects.⁵ But whether a thing ontology is too coarse, say, to capture the causal structure of the world belongs to the traditional dispute over the reality and nature of properties; it is not what divides factualists from thing theorists. A trope theory, for example, can provide for an immanent multiplicity of causes and effects without positing anything but things and the tropes that are their parts; an enlightened trope theorist can then identify facts with tropes, and sums of tropes.⁶ As long as one considers only the case for (positive) particular facts, it seems, the dispute between factualists and thing theorists is either subsumed under a more general dispute over de re modality, or is transformed into a dispute over realism about properties. But when one considers instead the case for general facts, one comes up against a genuine disagreement. If a factualist can successfully argue that general facts exist as something “over and above” particular facts and particular things, then the thing theorist is in trouble. In this chapter, I focus on two contrasting arguments: Russell’s well-known argument, endorsed and bolstered by Armstrong, that general facts are needed in addition to particular facts because general truths are not entailed by particular truths; and an argument, endorsed by Lewis among others, that general facts are not needed in addition to particular facts because general truths (globally) supervene on particular truths.⁷ Needless to say, as a thing theorist, I reject the former argument and accept the latter. My goal, however—difficult to attain in matters of fundamental ontology—is to present my main arguments from a neutral perspective, so that they will have force even from within the factualist framework: a factualist, no less than a thing theorist, should dispense with general facts.⁸

⁴ See Lewis (a: –) for the general case. For an application of counterpart theory to the case at hand, see Lewis (). ⁵ See, for example, Armstrong (b: –). Lewis () provides a non-immanent multiplicity. ⁶ Armstrong (b: –) acknowledges that tropes can do much of what he requires of states of affairs. I do not agree, however, that tropes must be “non-transferable.” ⁷ See David Lewis’s discussion of the supervenience of truth on being in Lewis (: –). Brian Skyrms is a factualist who rejects general facts because they supervene on (first-order) particular facts; see Skyrms (: –). ⁸ The arguments for and against general facts apply, with some changes, to the case of negative particular facts. In this chapter, however, for reasons of space, I omit explicit discussion of negative particular facts.

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



In what follows I tacitly restrict attention to first-order particular facts and their first-order generalizations. Some factualists hold that second-order facts involving relations between universals are needed to ground the distinction between laws and accidental regularities. Their arguments, however, threaten only nominalist versions of thing theory, not thing theory generally. In any case, the issues raised by the problem of laws cut across the issues raised by the problem of generality; in this chapter, I set the former aside. I begin with some clarifications, and some basic assumptions. As a thing theorist, I may well be asked: what is a thing? Don’t expect a definition, however. Things are basic for the thing theory. I can give examples: people and puddles and protons are things. But I can’t say much of anything as to what makes people and puddles and protons things without taking sides on metaphysical disputes—three-dimensionalism vs. four-dimensionalism, absolutism vs. relationism about space and time, realism vs. non-realism about properties (universals or tropes)—with respect to which I intend the thing theory to be neutral. I will assume, however, that aggregates of things are things, and that spatiotemporal parts of things are things. But non-spatiotemporal parts of things (if any—for example, universals or tropes) are not things. Finally, I believe that any respectable thing theorist will follow Hume in denying necessary connections between (mereologically) distinct things; but since the Humean thesis is disputed, I will be careful to flag those arguments that depend on it. What, according to the factualist, are facts? The best way, I think, to get a handle on facts is by way of their relation to propositions. Propositions, I will suppose, are necessarily existing, non-linguistic entities.⁹ Truth conditions for propositions are classical: for any proposition and any possible world, the proposition is (definitely) true at the world, or its denial is (definitely) true, but not both. Propositions represent the way the world is from “without”; they do not belong to the basic inventory of the world, all of whose members contingently exist. Facts, on the other hand, are immanent; they are part of the world; they ground the truth and falsity of (contingent) propositions. To each fact there corresponds a unique (up to necessary equivalence) true proposition (truth, for short): necessarily, the fact exists if and only if the corresponding proposition is true. But, whereas the propositions are abundant, so that every sentence (with definite, classical truth conditions) expresses some proposition, the facts are sparse. Only a select minority of the truths correspond one-one with the facts. Which truths so correspond? Start with the (first-order) atomic propositions: all propositions that predicate a fundamental property of a basic particular, or a fundamental n-ary relation of an n-tuple of basic particulars.¹⁰ All factualists hold ⁹ Since in this chapter I am concerned only with contingent truth, propositions may be taken to be classes of possible worlds. But it wouldn’t matter for what follows if instead propositions were taken to be sentences (better: equivalence classes of necessarily equivalent sentences) of some sufficiently idealized language, say, with predicates for fundamental properties and relations, and names for basic particulars. And it wouldn’t matter for most of what follows if some propositions were taken to exist contingently, existing only when the entities they are about exist. ¹⁰ I assume that for our world, or any world, both the factualist and the thing theorist can speak of the particulars that exist at the world, and the fundamental (or perfectly natural) properties and relations

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

 .  ()

that, for each true atomic proposition, there is a unique atomic fact; and that the atomic facts are ontologically basic, part of any complete inventory of what there is. But are these atomic facts all the (first-order) facts? Define a (first-order) general proposition to be the result of applying a universal quantifier to a qualitative property; the general proposition is true (true at a world w) just in case every particular (every particular existing at w) instantiates the property.¹¹ A property is qualitative, roughly, if it can be defined using Boolean operators and first-order quantifiers from the fundamental properties and relations (including the part-whole relation).¹² For example, if F and G are fundamental properties and R is a fundamental relation, then being G if F, bearing R to something, and having a part that is F and bears R to a part that is G are all qualitative properties; and their generalizations, all Fs are Gs, everything bears R to something, and everything has a part that is F and that bears R to a part that is G are all general propositions. Now, the question to be addressed is this. Must the factualist posit general facts in addition to the atomic facts to serve as ontological ground for true general propositions? Or are the general truths determined by the atomic truths? Clearly, the issue will turn on what is meant by “determined.” Perhaps it helps somewhat to note that the determination relation in question is non-causal, and holds of necessity. But more must be said: a mere logical or functional determination is not to the point unless it carries with it ontological force. Thus, if the atomic truths determine the general truths, in the relevant sense, then the general propositions hold or fail to hold in virtue of, or because of, the holding or failing to hold of the atomic propositions. In this case (I argue below) general facts needn’t be added to the inventory; the atomic facts suffice. On the other hand, if the atomic truths fail to determine the general truths, in the relevant sense, then the atomic facts do not suffice; additional entities will need to be added to serve as ontological ground for the general truths—presumably, for the factualist, general facts. Call the relevant determination relation between propositions ontological determination. What has given the debate over general facts its longevity, I believe, is that there are two relations between propositions—entailment and supervenience—either of which, at first glance, might plausibly be taken to be necessary for ontological determination. Often it makes no difference which is taken to be necessary because instantiated by particulars at the world. I call a particular basic just in case it instantiates some fundamental property or relation; thus I do not suppose that basic particulars must be mereologically simple. Note that, for the factualist, facts are not constructed from particulars, properties, and relations. Rather, facts are basic, and particulars, properties, and relations are somehow abstracted from facts. Cf., for example, Armstrong (a: ). ¹¹ The restriction to qualitative properties is a convenient stipulation: it allows me to ignore questions of de re modality when considering truth conditions for the general propositions; but it doesn’t unduly limit the scope of my arguments, since the case for general facts applies with equal force to the qualitative and the non-qualitative propositions. ¹² A more formal treatment would take place within a framework of algebraic logic: the qualitative properties (and relations) are defined recursively by the application of Boolean, quantificational, and combinatorial operators to the fundamental properties and relations. For an illustration of one such framework, see Quine (b). (But the framework would need to be generalized to take advantage of the resources of infinitary logic, including infinite Boolean operators and, perhaps, infinite strings of quantifiers.)

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 -      



entailment and supervenience coincide. But for the case at hand, one must choose: taking entailment to be necessary and taking supervenience to be necessary lead to opposite results. In what follows, I will argue that when entailment and supervenience diverge, it is only supervenience that is necessary for ontological determination; failure of entailment carries no ontological force.

. The Non-Entailment of General Truths by Particular Truths Russell famously argued during his logical atomist stage for the existence of general facts. In The Philosophy of Logical Atomism Russell writes: “you cannot ever arrive at a general fact by inference from particular facts, however numerous” (: ). He later concludes from this: “you must admit general facts as distinct from and over and above particular facts” (: ). In short, Russell argues: the atomic truths do not entail the general truths; therefore, general facts are needed in addition to atomic facts.¹³ Call this the Non-Entailment Argument. The argument tacitly supposes that entailment, of some sort, is a necessary component of ontological determination. What is entailment? Although I suppose that Russell had in mind some notion of formal entailment (within an ideal language), I suggest instead that we take the relevant entailment relation to be strict implication defined in terms of possible worlds.¹⁴ Let A and B be classes of propositions.¹⁵ B entails A iff every world at which every member of B is true is a world at which every member of A is true. (B entails a single proposition Z iff B entails {Z}.) Note that, on this interpretation of ‘entails’, necessary general truths are trivially entailed by the atomic truths, and so should be excluded from the scope of the argument. The main premise of the Non-Entailment Argument, then, is the following Non-Entailment Thesis. Some contingent general truth is not entailed by the class of atomic truths (if there are any contingent general truths). As stated, the Non-Entailment Thesis is a claim about truths at the actual world: a local thesis. A stronger, global thesis, generalizing over all worlds, is the following Global Non-Entailment Thesis. At any world (at which there are contingent general truths), some contingent general truth is not entailed by the class of atomic truths.

¹³ Armstrong endorses Russell’s argument, and gives it a truthmaker twist. I discuss Armstrong’s variation in Section .. Frank Ramsey () objected to Russell’s argument for general facts along different lines. But Ramsey’s objection rests on a modal fallacy. For a diagnosis, see Hazen (). ¹⁴ Interpreting entailment as strict implication does not prejudice the case. If entailment is strict implication, it is more difficult to show that general truths are not entailed, but (presumably) easier to show that non-entailment has ontological import; and I will be challenging only the latter. ¹⁵ I adopt the following convenient policy. Formal definitions and proofs are given within a framework of classes allowing, if need be, for proper classes (e.g. “the class B entails the class A”); informal discussion is carried out within a framework of plurals (e.g. “the Bs entail the As”). This is a distinction of style, not substance. All such occurrences should be interpreted the same way; for purposes of this chapter, it won’t matter which.

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

 .  ()

In this section, I will examine arguments for versions of the Non-Entailment Thesis, ultimately defending a version even stronger than Global Non-Entailment. In the final section, I reject the Non-Entailment Argument. I claim that the NonEntailment Thesis does not give reason, even from within the factualist framework, for positing general facts as anything “over and above” the atomic facts. Arguments for all versions of the Non-Entailment Thesis start from a base world, perhaps the actual world, and then move to an expanded world at which the atomic truths at the base world still hold, but some general truth at the base world fails to hold. (I say a world w is an expansion of a world w0 just in case the atomic truths at w include all the atomic truths at w0 , and at least one more.) There is more than one way, however, to choose an expanded world, and the choice may matter. Let me illustrate with respect to a simple world, schematically described. Suppose that at the base world there are just four atomic truths: Fa, Ga, Hb, and Rab.¹⁶ Thus the world contains two basic particulars, three fundamental properties, and one fundamental relation. The general proposition, all Fs are Gs, is true at this world. Is it entailed by the atomic truths? No. First Method of Argument. Expand the base world vertically by adding a property to an already existing particular: in this case, add F to b so that Fb but not Gb is true at the expanded world. Then, the atomic propositions true at the base world are still true at the expanded world, but the general proposition, all Fs are Gs, is false. Second Method of Argument. Expand the base world horizontally by adding a new basic particular: in this case, add a new particular c standing in relation R to a and to b, and having just one fundamental property F, so that Fc but not Gc is true at the expanded world. Then, again, the atomic propositions true at the base world are still true at the expanded world, but the general proposition, all Fs are Gs, is false. Thus, on either method of expansion, it is shown that the atomic truths do not entail the (contingent) general truths. But this is much too quick. There are positions in the metaphysics of modality on which one or both of these methods of arguing may be blocked. I will briefly outline the difficulties faced by these methods. Then, I will show how, given the metaphysical assumptions that I accept, the argument for the Non-Entailment Thesis can be improved. The first method of argument might seem preferable to the second—expanding the base world vertically rather than horizontally—because it makes no assumptions about the possibility of alien (basic) particulars (or alien fundamental properties and relations).¹⁷ But the first method’s scope is limited, in two ways. First, the method works for some general truths, but not for others. Whether it works depends, for one thing, on the logical features of the property being generalized. Thus, if the property has the feature, is preserved under vertical expansion—whatever has the property continues to have it in any vertically expanded world—then the first method is blocked. For example, the proposition, everything is F, for a fundamental ¹⁶ If particulars can exist without instantiating any fundamental properties, then Hb can be omitted. If particulars can co-exist unconnected by fundamental relations, then Rab can be omitted as well. ¹⁷ Alien properties and individuals—and their cost—are discussed in Lewis (a: –, –). [See also Section ..]

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 -      



property F, cannot be falsified by a vertical expansion. For another thing, whether the first method works depends on contingent features of the base world. For example, suppose we add to the simple world considered above a fifth atomic truth, Gb. Then, all Fs are Gs is again true, but cannot be falsified by a vertical expansion: something which is G if F cannot be made to be not G and F by adding properties, because everything at the base world is G. The first method’s scope is limited in a second way. Thus far, I have spoken uncritically of atomic propositions, as if all theorists would agree as to what proposition Fa expresses when a basic particular a instantiates a fundamental property F. But, of course, different ways of “identifying” particulars across worlds, different accounts of individual essence, result in different propositions being labeled “the atomic propositions.”¹⁸ Now, if individual essences can exclude fundamental properties, if particulars can necessarily fail to have a fundamental property, then there is no guarantee that in vertically expanding the base world, some atomic truth at the base world won’t be falsified. For example, with respect to the simple world considered above, if b’s individual essence excludes F, there will be no expanded world at which Fb is true, and so the argument is blocked. More generally: say that a basic particular a at world w has a vertically exclusive essence iff in some vertical expansion of w, a does not exist. The first method of argument for the Non-Entailment Thesis cannot be counted on if vertically exclusive essences are allowed. But vertically exclusive essences would seem to be unexceptionable: an electron essentially has the fundamental properties it has—say, a particular mass property, charge property, and so on—and essentially has no more. The second method of arguing—expanding the base world horizontally—can be applied much more generally. It can be applied to any contingent general truth, irrespective of the property generalized or the contingent features of the base world. And the restriction on essences is much less severe. Say that a basic particular a at world w has a horizontally exclusive essence iff in some horizontal expansion of w, a does not exist. The second method cannot be counted on if horizontally exclusive essences, such as being the only F, are allowed. But I know of no modal theorists who identify atomic propositions by way of essences of this sort. The second method, then, appears far superior to the first as a way of supporting the Non-Entailment Thesis. But the second method isn’t for everyone because of its reliance on aliens, on there being basic particulars at the expanded world that do not exist at the base world. Of course, when the base world is a simple world, such as considered above, this is not a problem; plenty of actual particulars are alien to the simple world. But when the base world is the actual world, as it must be to argue for the Non-Entailment Thesis, there may be a problem for certain strict actualists: those who hold that any basic particular

¹⁸ I remain neutral throughout this chapter on issues of “transworld identification” except when considering my own modal view, in which case my counterpart theoretic bias comes to the fore. Note that a counterpart theorist should distinguish between atomic predications, which have truth values only at a single world, the world that includes the particular in question, and atomic propositions, which have truth values at every world, truth values that depend on the choice of a counterpart relation. The atomic predications are metaphysically basic for a counterpart theorist, not the atomic propositions.

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

 .  ()

that exists in any non-actual possible world also exists in the actual world. (Actualists who are anti-haecceitists face no problem, however, since for them horizontal expansions can add new, qualitatively different basic particulars without adding alien properties, and thus without violating actualist scruples.) The second method, however, has a serious limitation for anyone, actualist or not, who holds that some worlds are “maximal” and cannot be horizontally expanded. For, in that case, it is a contingent matter whether the actual world can be horizontally expanded as the argument requires, and the argument for the Non-Entailment Thesis, and thus for general facts, will depend on a contingent premise. That’s no way to argue for fundamental ontology. Certainly it would be better to establish the NonEntailment Thesis entirely from premises that are necessary and a priori. But enough meddling in other philosophers’ modal affairs. Since my strategy is to give the factualist the Non-Entailment Thesis, and reject the Non-Entailment Argument, it is enough if I explain why, on the modal metaphysics I accept, the Non-Entailment Thesis holds. A revision of the second method will allow me to do this without relying on premises that are contingent. First, I allow as a genuine metaphysical possibility what might be called universal actualization.¹⁹ The best way to illustrate the possibility I have in mind is from a realist perspective; ersatzists and fictionalists can translate into their framework in familiar ways. Thus, consider Leibniz’s God surveying the realm of possible worlds prior to actualization. I suppose that within each world the parts are all interconnected (by spatiotemporal relations, or other external relations), but that the parts of distinct worlds are wholly disconnected from one another.²⁰ On the standard assumption, Leibniz’s God must actualize exactly one world: necessarily, one and only one world is actual. But what prohibits Leibniz’s God from actualizing two (or more) worlds? Or, dropping Leibniz’s God from the picture: why exclude the possibility that two or more worlds together are actual? No good reason, I fear; only custom. If two (or more) worlds are actual, then actuality includes two disconnected parts: island universes. If all worlds are actual, then actuality includes every possibility: every possible individual is actual. If we allow the possibility that two (or more) worlds are actual, then there are more possibilities (for the whole of actuality) than there are worlds: every world is a possibility, but so is every plurality of worlds. Propositions, then, will have to be assigned truth values relative not just to single worlds, but to pluralities of worlds.²¹ For example, the proposition, island universes exist, is false at any single world, but true at any plurality of worlds. (Truth at a plurality of worlds is not, of course, to be identified with truth at each world in the plurality; rather, one is to suppose actuality is the aggregate of the worlds in the plurality, and then to ask what is true of that aggregate.) Then, the standard analyses of possibility and necessity must be adjusted to accommodate the change in truth conditions for propositions: a proposition is ¹⁹ The view set forth in this and the next paragraph is defended at length in Chapters  and . ²⁰ For defense, see Chapter . ²¹ What is a “plurality”? I use ‘plurality’ as a way of staying neutral between three ways of presenting the view: classes, aggregates, or a framework of plurals. I prefer the latter, but the differences won’t much matter for present purposes.

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 -      



possible iff it is true at some world, or at some plurality of worlds; a proposition is necessary iff it is true at all worlds, and at all pluralities of worlds.²² I also need a modest assumption about essences: necessarily, any particular that exists would still have existed, and would have had the same fundamental properties and stood in the same fundamental relations to other particulars, had the world been (qualitatively) exactly the same except for the addition of island universes.²³ It follows that, if a world or plurality of worlds is included in a larger plurality of worlds, then the atomic propositions true at the former are still true at the latter; atomic truths are not falsified by the addition of island universes. More generally, when a particular instantiates a qualitative property at a world or plurality of worlds, it does so also at any larger plurality of worlds. It is now a simple matter to establish the Non-Entailment Thesis without relying on a contingent premise by establishing the Global Non-Entailment Thesis. Indeed, I will establish an even stronger thesis, claiming that, necessarily, no contingent general truth is entailed by the atomic truths, with necessity understood as quantification over worlds, and pluralities of worlds. This will be seen to follow from the fact that, at the plurality of all worlds, all general truths—equivalently, all negative existential truths—are necessary truths; for example, if there aren’t any unicorns in the plurality of all worlds, then, necessarily, there aren’t any unicorns. Strong Global Non-Entailment Thesis. For any world or plurality of worlds W, any contingent general proposition true at W is not entailed by the class of atomic propositions true at W.²⁴ Proof. Consider any world or plurality of worlds W. Let P be any contingent general truth at W. (If there are none, then the non-entailment claim holds vacuously.) Let U be the plurality of all worlds. Given our assumption about essences, the atomic propositions true at W are all true at U since W is included in U. But P is false at U. For, being contingent, P is false at some V included in U, and, making use of our assumption once again, any counterexample to P in V is also a counterexample to P in U.²⁵ ²² This raises a terminological problem. Usually when I write ‘world’ in this chapter, what I really mean is ‘world, or plurality of worlds’. But since my own view rears its head only in this section, and in one paragraph of the final section, it does little harm that I speak with the vulgar. ²³ The assumption for relations requires that island universes be absolutely isolated: no part of one stands in any fundamental (external) relation to any part of another. Note also that, in interpreting modality de re on this view, counterpart relations must be taken to be relative to worlds and to pluralities of worlds (because pluralities of worlds overlap). In particular, where W and V are distinct worlds or pluralities of worlds, a is a particular existing at W, and b is a particular existing at V: b may be a counterpart of a at V, even though b, presumably, is not a counterpart of a at the plurality of worlds that includes V and W: a is its own counterpart at any plurality of worlds that includes W. ²⁴ Of course, the definition of entailment now reads: a class of propositions A entails proposition Z iff at every world or plurality of worlds at which all the members of A are true, Z is true. ²⁵ It might appear that the proposition, everything coexists with some F (where F is any contingently instantiated fundamental property), is a counterexample to Strong Global Non-Entailment; for this proposition is entailed by any atomic truth of the form a is F. But everything coexists with some F is not a general proposition as herein defined because the property, coexisting with some F, is not a qualitative property. It depends on the distinction between the actual and the merely possible, and, if island universes are possible, that distinction cannot be defined in qualitative terms. Being spatiotemporally related to some

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

 .  ()

. The Supervenience of General Truths on Particular Truths Standing in opposition to the Non-Entailment Argument is the Supervenience Argument: general truths supervene on atomic truths; therefore, general facts are not needed in addition to atomic facts. (A stronger version, to be discussed below, concludes: therefore, there are no general facts.) The argument tacitly supposes that supervenience, not entailment, has ontological import. I will defend that assumption in due course. First, I consider the supervenience claim. As with the Non-Entailment Thesis, it comes in both a local and a global version: Supervenience Thesis: If a world agrees with the actual world on all atomic truths, then it agrees also on all general truths. Global Supervenience Thesis: If any two worlds agree on all their atomic truths, then they agree also on all their general truths.²⁶ Since it is unlikely that anyone would accept the local version while rejecting the global version, I will focus entirely on the latter in what follows, what I call the Supervenience of the General on the Particular. For the thing theorist, the Supervenience of the General on the Particular is almost automatic. The actual world is a thing. Other worlds, then, are things as well. (For the realist about worlds. But it will do no harm to speak from a realist perspective because ersatzist or fictionalist thing theorists will represent other worlds as things, and the realist arguments will carry over.) The intrinsic qualitative nature of a thing, including a world, is determined by the distribution of fundamental properties and relations over its parts. The qualitative nature of a world, then, is determined by its atomic truths, and worlds that agree on atomic truths agree in their intrinsic qualitative nature: they are qualitative duplicates. Moreover, since worlds are biggest (interconnected) things—parts of distinct worlds stand in no fundamental relations to one another—worlds that agree on atomic truths are qualitatively indiscernible, agreeing in both their intrinsic and extrinsic qualitative nature.²⁷ Now, for the thing theorist, propositions are properties of worlds. In particular, a general proposition, everything is Q, is the property, having every part be Q. But having every part be Q is a qualitative property given that Q is qualitative. Therefore, worlds that agree on atomic truths, being qualitatively indiscernible, agree on any general truth, everything is Q.²⁸ F is a qualitative property; coexisting with some F is not. (This is the only argument in the chapter for which the restriction of general propositions to qualitative propositions plays an essential role.) ²⁶ Two asides. First, it is more common nowadays to define global supervenience as a relation between classes of properties, rather than classes of propositions. Couching the discussion in terms of properties is obviously inconvenient for the case at hand, though it could be done by turning propositions into properties of worlds. Second, the distinction I draw between global and local theses—applying to all worlds, or just to the actual world—is not the distinction that originally gave Global Supervenience its name. ²⁷ See Lewis (a: –) on the distinction between duplicates and indiscernibles. ²⁸ For the thing theorist, Supervenience of the General on the Particular is an instance of what Lewis has dubbed (following Bigelow) the Supervenience of Truth on Being: “truth is supervenient on what things there are and which perfectly natural properties and relations they instantiate” (Lewis : ). A stronger version of Supervenience of Truth on Being holds that truth supervenes on what things there

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       



The factualist will not be impressed by this argument. But I think the factualist has good reason, from within her own framework, to accept the Supervenience of the General on the Particular. To see why, ask what sort of world would have to exist in order for such supervenience to fail. Thus, suppose worlds w and w0 agree on atomic truths. Then they agree on all matters of particular fact: for any particular, basic or not, and any qualitative property, the particular instantiates the property at both worlds, or at neither.²⁹ Now suppose some general proposition, everything is Q, holds at w but not at w0 . Then, the existential proposition, something is not Q, holds at w0 . But there is no particular a at w0 such that a is not Q; for if there were, a is not Q would be true at w as well, which contradicts everything being Q at w. So, at w0 , a true existential proposition is unwitnessed by any particular. Rejecting the Supervenience of the General on the Particular requires embracing worlds at which existential truths lack witnesses. But what is objectionable about that? It is customary and proper, for thing theorist and factualist alike, to place the following two demands on a theory of possible worlds: worlds must be possible, in a broad metaphysical or logical sense; and worlds must be determinate, leaving nothing unsettled. These demands constrain which classes of propositions can be the class of truths at some world. Thus, according to the first demand, truth—that is, the class of true propositions—is consistent at any world (where consistency is a modal notion, perhaps primitive, perhaps reduced to possibilia). The second demand is sometimes expressed by saying that truth, in addition to being consistent, is maximal consistent at any world: no proposition could be added to the truths without falling into contradiction. Truth is maximal consistent just in case it is consistent and complete: for any proposition, either that proposition or its denial is true. But requiring only that truth be maximal consistent, or consistent and complete, at every world falls short of capturing the idea that worlds are determinate. Truth must be witnessed as well: every existential truth must be witnessed by some particular; that is, if an existential proposition is true at a world, the property being existentially generalized holds of some particular existing at the world. Let Completeness be the thesis that truth at any world is complete; Witnessing the thesis that truth at any world is witnessed. To clarify the relation between Completeness and Witnessing, it is useful to consider a “world” that everyone agrees is not in general determinate: the “world” of a work of fiction. Suppose, for example, there is a story, “Who Killed Peter Rabbit?” that has only four characters (among sundry objects): Flopsy, Mopsy, Cottontail, and Peter. Suppose it is true in the story that someone kills Peter, but it is never revealed who did it. (For the sake of the illustration, pretend that killing is a fundamental relation.) Then truth in the story violates Witnessing, because there is no character (or object) in the story that witnesses the existential proposition in question. Now consider two versions of the story. On one version, it is revealed that either Flopsy, Mopsy, or Cottontail is the killer, but not who. In this case, Completeness is violated as well as Witnessing: for example, neither the proposition that Flopsy killed Peter, nor its denial, is true in the story. This

are (full stop). That corresponds to the Subject Matter Principle to be introduced below. (To avoid confusion, the weaker version might better be called the Supervenience of Truth on Particular Truth.) ²⁹ Again, any proof of this must await a rigorous account of the qualitative properties.

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

 .  ()

is one way for the story to fail to be determinate. On the second version, it is revealed that neither Flopsy, Mopsy, nor Cottontail (nor Peter himself) is the killer. In this case, there need be no violation of Completeness: the story could be filled out so that, for every proposition, either it or its denial is true in the story. But, Witnessing is still violated: for any character in the story—indeed, any actual or possible individual—it is not true that he or she (or it) is the killer. Clearly, the “world” of the story fails to be determinate no less in the second version than in the first: someone killed Peter, but who the killer is remains undetermined. Once it is appreciated that Completeness and Witnessing are two independent demands on truth and that both are needed to capture the idea that worlds are determinate, it is hard to justify holding on to the former while rejecting the latter. If the factualist concedes, as most factualists will, that truth at any world is maximal consistent, and thus complete, it would be arbitrary to then deny that truth need be witnessed as well. But might a factualist choose to give up both Completeness and Witnessing? Could the idea that worlds are determinate be just a stale piece of thing propaganda? Perhaps the following argument for Completeness and Witnessing will carry some weight. I suppose that the factualist works within a framework of classical logic, and that the meaning of the Boolean operators and the quantifiers is given by the standard Tarskian truth conditions. But meaning has modal force: the Boolean operators and the quantifiers must have these truth conditions, not just at the actual world, but at all possible worlds. Now suppose we were to allow a world at which truth is not complete: some proposition p is such that neither it nor its denial, not p, is true at the world. But, presumably, the proposition, p or not p, is true at the world, since tautologies are necessary truths. And that violates the standard Tarskian truth conditions for disjunction: a disjunction is true iff at least one of its disjuncts is true. To reject Completeness, then, is to suppose that there is a world at which disjunctions do not have their standard truth conditions. That, I claim, is incoherent, and thus worlds at which truth is not complete should be rejected. The argument applies, mutatis mutandis, to Witnessing. For suppose we were to allow a world at which truth is not witnessed: there is a true existential proposition, something is Q, even though for every particular a existing at the world, it is not true that a is Q. That violates the standard Tarskian truth conditions for existential propositions. To hold that existential propositions behave in this way, I claim, is incoherent. Worlds at which truth is not witnessed, no less than worlds at which truth is not complete, should be rejected.

. Supervenience and Entailment Compared General truths, I have argued, supervene on, but are not entailed by, atomic truths. Thus supervenience and entailment do not always coincide. In this section, I consider just how these notions relate to one another. Before going any further, however, it would be wise to head off a potential terminological confusion. Armstrong, as already mentioned, is a staunch defender of the Non-Entailment Argument. But this might seem to conflict with his espousal of what he calls “the doctrine of the ontological free lunch” applied to supervenience: “what supervenes is no addition of being”; “the

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   



supervenient is ontologically nothing more than its base” (Armstrong : –). Thus Armstrong apparently accepts the premise of the Supervenience Argument that supervenience has ontological import. But wait: Armstrong also appears to accept the other premise of the Supervenience Argument, the Supervenience of the General on the Particular.³⁰ How, then, can he avoid the conclusion of the Supervenience Argument? Why aren’t general facts free for the eating? The mystery is solved as soon as one realizes that Armstrong has a non-standard notion of supervenience, at least as applied to propositions. He defines supervenience in terms of possible worlds as follows: “Q supervenes on P if and only if there are P-worlds and all P-worlds are Q-worlds.”³¹ In this definition, ‘Q’ and ‘P’ may range over all manner of entity. But when applied to (contingent) propositions, the definition amounts to equating ‘supervenience’ with ‘entailment’ (between single propositions), rather than what is standardly called ‘supervenience’—viz., global supervenience.³² Thus—returning now and henceforth to standard usage— Armstrong applies the doctrine of the ontological free lunch only to entailment, not to (global) supervenience; he would not use the doctrine to support the Supervenience Argument. On the contrary, according to Armstrong, because the General is not entailed by the Particular, one must pay with an ontology of general facts. I return now to the question: how exactly are supervenience and entailment related to one another? Are there conditions, say, on the base propositions—conditions not satisfied in the present case—under which these notions do coincide? In comparing supervenience and entailment it is important to compare like with like: local with local, global with global. For maximum generality, I will compare global with global. The definitions, for arbitrary classes of propositions A and B, are as follows. (Recall, the A-truths (B-truths) at a world are all and only those propositions in A (B) that are true at the world.) A globally supervenes on B iff for any worlds w and w 0 , if the B-truths at w coincide with the B-truths at w 0 , then the A-truths at w coincide with the A-truths at w 0 . A is globally entailed by B iff for any world w, the B-truths at w entail the A-truths at w. Or, equivalently, unpacking for easy comparison: A is globally entailed by B iff for any worlds w and w 0 , if every B-truth at w is a B-truth at w 0 , then every A-truth at w is an A-truth at w 0 . ³⁰ Armstrong (a: ). Specifically, Armstrong concedes that two worlds could not “be exactly alike in all lower-order states of affairs” and yet differ in their “totality states of affairs.” For Armstrong’s account of totality states of affairs—his version of general facts—see Armstrong (: –). ³¹ Armstrong (: ). Armstrong seems to be aware that his use of supervenience is, in some way, non-standard. He writes: “supervenience in my sense [my emphasis] amounts to entity P entailing [his emphasis] the entity Q, but with the entailment restricted to the cases where P is possible.” ³² Given the proliferation of supervenience notions, it is with some reluctance that I label any “nonstandard.” But the classic formulations of supervenience are discernibility theses—there can be no difference of one sort without a difference of some other sort—not entailment theses. (Two loci classici are Hare (: ) and Davidson (: )). Even Kim, who promulgated entailment formulations for “strong” and “weak” supervenience, called the discernibility formulations “canonical” (Kim : ). See also the following footnote.

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

 .  ()

Clearly, entailment is a stronger notion than supervenience: Claim. Whenever A is globally entailed by B, A globally supervenes on B as well. Proof. Suppose A is globally entailed by B. Consider any two worlds, w and w 0 , whose B-truths coincide. Then, every B-truth at w is a B-truth at w 0 , and every Btruth at w 0 is a B-truth at w. Since A is globally entailed by B, this gives: every A-truth at w is an A-truth at w 0 , and every A-truth at w 0 is an A-truth at w. In other words, the A-truths at w and at w 0 coincide, as was to be shown. For an example where both global entailment and global supervenience hold, let B be the class of atomic propositions, and A the closure of B under (finite or infinite) conjunction. (That is, A is the smallest class that contains B, and contains all conjunctions of whatever it contains.) We already have before us one example—the atomic propositions and the general propositions—where supervenience holds and entailment fails. For a second, more revealing, example, let B again be the class of atomic propositions, and let A be the class of all denials of atomic propositions. Supervenience holds because whenever worlds assign the same truth value to propositions, they assign the same truth value to their denials. But entailment fails to hold: any two worlds, one of which is a (vertical or horizontal) expansion of the other, provide a counterexample. Consideration of this example suggests a simple condition under which supervenience and entailment coincide: Claim. Suppose that B is closed under denial: the denial of any proposition in B is in B. Then: if A globally supervenes on B, A is globally entailed by B. Proof. Suppose B is closed under denial, and that A globally supervenes on B. Let w and w 0 be such that every B-truth at w is a B-truth at w 0 . Then, also, every B-falsehood at w is a B-falsehood at w 0 . For let Z be any B-falsehood at w. Not Z is a B-truth at w (because B is closed under denial), and so not Z is a B-truth at w 0 , making Z a B-falsehood at w 0 . Thus, the B-truths at w and w 0 coincide. Since A globally supervenes on B, the A-truths at w and at w 0 coincide as well, from which it follows that every A-truth at w is an A-truth at w 0 , as was to be shown. We are now in a position to see why supervenience and entailment often can be, and are, substituted for one another without ontological consequence. The majority of global supervenience theses on the market involve base propositions that are closed under denial; so, for ontological purposes, it wouldn’t matter if the corresponding global entailment thesis were considered instead.³³ Thus, one often takes the ³³ In Kim’s seminal work on formulations of supervenience, the base properties and supervenient properties were assumed to be closed under Boolean operations. That allowed Kim to prove the equivalence of the discernibility and entailment formulations for “strong” and “weak” supervenience. (Kim : ; Kim : –.) (What I have called “global entailment,” of course, is the entailment formulation that corresponds to global supervenience.) Others, however, have sometimes been less careful in their discussions of supervenience. For example, Chalmers (), in an influential discussion, introduces supervenience (applied to properties) as a discernibility thesis (p. ), but then claims that “in general, when B-properties supervene logically on A-properties, we can say that the A-facts entail the B-facts . . . ” (p. , his emphasis). Moreover, he explicitly does not take the A-properties and B-properties to be closed under denial: “supervenience theses [for capturing materialism] should apply only to positive facts and properties . . . ” (p. , his emphasis). Later, supervenience is redefined, in effect, as a (local) entailment thesis

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   



qualitative propositions as the base and asks whether the laws of nature, or counterfactuals, or propositions about causation, or about chance, globally supervene. Since the qualitative propositions are closed under denial, they supervene just in case they are entailed. Similarly, supervenience of the mental on the physical, or the evaluative on the descriptive, could just as well be presented in terms of entailment. But when the base propositions are the atomic propositions, as in the case at hand, supervenience and entailment come apart. Now it matters whether it is supervenience, or only entailment, that entitles one to an ontological free lunch.

. The Supervenience Criterion Defended: The Subject Matter Principle In this section, I defend the Supervenience Argument: general truths supervene on atomic truths; therefore, general facts need not, and should not, be added to the fundamental inventory of what there is. There are two familiar obstacles to arguments of this sort: one having to do with the relation between supervenience and determination (or dependence), the other having to do with the relation between supervenience and reduction. I will take them in turn. Ontological determination, I said earlier, supports such locutions as ‘in virtue of ’: the determined propositions hold or fail to hold in virtue of the holding or failing to hold of the determining propositions. It has often, and rightly, been pointed out that supervenience cannot by itself be taken to be such a relation of ontological determination.³⁴ Supervenience is necessary covariation: if the truth values of the supervenient propositions vary between two worlds, then the truth values of the base propositions must vary between them as well. Necessary covariation, however, is not asymmetric. There are classes of propositions such that each necessarily covaries with the other. But, certainly, we would not want to say that each ontologically determines the other. For example, the denials of atomic propositions supervene on the atomic propositions; but also vice versa. Yet, surely, it is the atomic propositions that are ontologically basic, not their denials. Thus, supervenience, characterized as necessary covariation, falls short of ontological determination. So far, so good. But here approaches to the problem diverge. The approach I favor asks: What must be added to supervenience to get ontological determination? How is ontological determination to be analyzed in terms of supervenience? The other approach rejects attempts at analysis. Supervenience, it is claimed, is a superficial relation that must be explained by some deeper, metaphysical relation of ontological because the original discernibility thesis is deemed inadequate to capture the ontological thesis of materialism. I suspect a typical reader will not be aware that the original discernibility notion of supervenience has been replaced by a non-equivalent entailment notion (also called “supervenience”). Another example: Horgan (: –) uses the discernibility and entailment formulations interchangeably without (explicitly) making any closure assumption that would justify such usage. For further discussion of how the discernibility and entailment notions differ, and why they should be kept apart, see McLaughlin (: –). ³⁴ See especially Kim (: –).

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

 .  ()

determination. On this approach, to try to analyze ontological determination in terms of supervenience would be to put things wrong way around.³⁵ I reject the second approach. Primitive ontological determination is dark and mysterious, and primitive modality, to boot. Why buy into the framework of possible worlds if not to rid the world of such creatures of darkness? Otherwise, one buys the dog and does the barking oneself. How, then, might ontological determination be analyzed? Although I have no detailed analysis to offer, I think the ingredients of a correct analysis are clear enough. We need here, as in so many other analytic endeavors, the notion of a fundamental, or perfectly natural, property or relation. But since we are concerned with supervenience of propositions, rather than properties and relations, we will need a derived notion of fundamental that applies to classes of propositions. Then, a plausible sufficient condition on ontological determination can be formulated: A is ontologically determined by B if B is fundamental, and A supervenes on B. This condition, of course, is quite limited in application; but it will be enough for moving on with. First, let us see how this partial analysis applies to our example with symmetric supervenience. We say: the denials of atomic propositions hold or fail to hold in virtue of the holding or failing to hold of the atomic propositions, not the reverse. Why? Surely, the class of atomic propositions is a fundamental class; it consists entirely of predications of fundamental properties and relations. Although supervenience goes in both directions—either class of propositions is logically definable in terms of the other using Boolean negation—only one direction corresponds with the true order of analysis. Fundamentalness of the base propositions supplies the direction, and turns supervenience into ontological determination. Nothing more is needed to justify use of the “in virtue of ” locution. The partial analysis applies to the case of general propositions in exactly the same way: the general propositions are ontologically determined by the atomic propositions because the atomic propositions are fundamental, and whatever supervenes on what is fundamental is ontologically determined by it. Note that the argument does not require as premise that the general propositions are not fundamental. Perhaps we start out unsure whether (some of) the general propositions are fundamental. After the supervenience thesis is established we conclude that they are not. For, a constraint on any correct analysis of ‘fundamental’ applied to classes of propositions is: if two (nonoverlapping) classes of propositions are fundamental, neither supervenes on the other. Some philosophers claim to find the distinction between properties and relations that are fundamental, and those that are not, mysterious, whether couched within a realist theory of universals or tropes, or a nominalist theory of natural properties and relations, or objective resemblance. An analysis of ontological determination in terms of fundamental properties and relations would leave them still in the dark. But, even to these philosophers I can say: better one mystery than two, and there is no hope of running an analysis in the other direction. Moreover, although these philosophers may endorse some primitive modal notions, and make use of them in constructing a framework of possible worlds, once that framework is available they will presumably

³⁵ This is more or less the view of Kim (: –).

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   



want to hold the line and analyze further modal notions, such as ontological determination, in terms of possible worlds and non-modal notions. The notion of fundamental, being non-modal, meets that standard. The second familiar problem involves the relation between supervenience and reduction. I have claimed that, under the right conditions, supervenience goes hand in hand with ontological determination. But ontological determination, as characterized, is a relation of conceptual or analytic priority; I have done nothing to justify my use of the adjective ‘ontological’. Let it be granted that the general propositions hold or fail to hold in virtue of the atomic propositions holding or failing to hold. Why should this have any ontological consequence as to the existence of general facts? Philosophers who make use of notions of supervenience in their theories and analyses appear to be divided by this issue into two camps: roughly half take supervenience to be a reductive relation (at least when it is assumed the base propositions are fundamental), roughly half take it to be non-reductive.³⁶ Indeed, for the former being reductive is essential to supervenience (when the base propositions are fundamental), for the latter being compatible with non-reduction is essential. Strange goings on! Sometimes, no doubt, the disagreement is superficial: different notions of reduction are being invoked, or restricted supervenience theses are at stake. Here, of course, only ontological reduction and unrestricted supervenience theses (quantifying over all worlds) are at issue. But even when the discussion is narrowed in this way, a fundamental disagreement sometimes remains. What might it be? What provides the link between supervenience and ontological reduction, I think, is (a version of) the Humean denial of necessary connections between distinct existents. Disagreement on the Humean principle, then, is a likely source of disagreement on the relation between supervenience and reduction. (It is no accident that the modern champion of the Humean denial, David Lewis, is also a chief proponent of the reductionist camp!) Whenever one class of propositions supervenes on another, there are necessary connections between the subject matters of the two classes. The only way to avoid such necessary connections is to maintain that the subject matter of the two classes is not distinct. Here is a way of making these ideas more precise. I will need the notion of a subject matter of a class of propositions: all the entities that the propositions are about, in one sense of ‘about’. Say that a (non-empty) class of actual or possible entities E is a subject matter for a (non-empty) class of propositions A iff the existing or failing to exist of members of E entails the truth or falsity of members of A. More exactly, where X is any (contingent) proposition saying for each member of E whether or not that member exists, and Z is any proposition in A: E is a subject matter for A iff either

³⁶ Lewis and Armstrong are chief proponents of the reductionist view. Lewis’s most complete statement is in Lewis (a: –); Armstrong states his view in Armstrong (a: –) and Armstrong (: –). As we have seen, however, Armstrong applies his “doctrine of the ontological free lunch” to entailment, not to (what I have been calling) supervenience when it differs from entailment. The widespread view that supervenience is compatible with non-reduction was promoted, in large part, by Davidson ().

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

 .  ()

X entails Z or X entails not Z.³⁷ (E is a subject matter for a single proposition P iff E is a subject matter for {P}.) Note that this is an intensional notion of subject matter. In one sense of ‘about’, if asked what the proposition, there are no unicorns, is about, I correctly answer (as a thing theorist): “nothing” (assuming there are no actual unicorns). In another sense, I should answer: “all actual and possible unicorns.” It is the latter sense that I have in mind. Note also that subject matters are not required to be minimal, much less unique.³⁸ Indeed, when E is a subject matter for A, every class that includes E is also a subject matter for A; the class of all actual and possible entities is a subject matter for any class of propositions (assuming it has a subject matter at all). It might be useful to present some elementary facts about subject matters that follow immediately from the definition. First, any subject matter for a proposition P is also a subject matter for its denial, not P, and vice versa. Second, if D is a subject matter for a proposition P and E is a subject matter for a proposition Q, then the union of D and E is a subject matter both for the conjunction, P and Q, and for the disjunction, P or Q. Third, any (non-empty) class is a subject matter for a necessary or an impossible proposition. Finally, if D is a subject matter for a class of propositions A, and E is a subject matter for a class of propositions B, then the union of D and E is a subject matter for the union of A and B. Now, I would like to put forward for consideration, by thing theorists and factualists alike, the following fundamental principle of ontology: Subject Matter Principle. matter.³⁹

Every (non-empty) class of propositions has a subject

This principle expresses in a direct way the (strong) Supervenience of Truth on Being.⁴⁰ Factualists should find this principle agreeable; if anything it is too weak— weaker, for example, than the Truthmaker Principle accepted by many factualists (see Section .). Thing theorists, however, will only find the principle plausible if ³⁷ My use of “subject matter” here, as the class of entities the propositions are about, should not be confused with David Lewis’s related but ontologically noncommittal use of “subject matter” according to which subject matters are partitions of logical space, roughly, all the ways a world could be with respect to the subject matter. See Lewis (). ³⁸ In special cases, a proposition will not have a minimal subject matter. Consider, for example, there exist infinitely many things. Any “co-finite” class of possibilia—that is, any class consisting of the entire universe of possibilia with finitely many entities removed—is a subject matter for this proposition. (The parallel observation that this proposition lacks a minimal class of truthmakers is due, I believe, to Greg Restall.) ³⁹ If the identity of indiscernible worlds is rejected—as I do in Chapter —then the Subject Matter Principle must be restricted to propositions that never differ in truth value between indiscernible worlds. (Qualitative propositions, for an anti-Haecceitist.) ⁴⁰ Not to be confused with the weaker version of the Supervenience of Truth on Being considered earlier which asserts only that Truth supervenes on Particular Truth. See n. . The Subject Matter Principle is roughly equivalent to Bigelow’s weakened Truthmaker Axiom: “If something is true, then it would not be possible for it to be false unless either certain things were to exist which don’t, or else certain things had not existed which do” (Bigelow b: ). It also turns out to be equivalent to a simple, (one-way) differencemaking principle: for any two (discernible) worlds w and w 0 , either something exists at w but not at w 0 , or something exists at w 0 but not at w. For an illuminating discussion of the relation between various truthmaking and difference-making principles, see Lewis (a).

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   



they are willing to deviate from ordinary ascriptions of essential properties: when determining which things make up the subject matter for a class of propositions, intrinsic nature must be taken to be essential; a counterpart of a thing is always a duplicate of the thing. Otherwise, the thing theorist will be hard-pressed to find a subject matter for predications of properties ordinarily not taken to be essential, such as color or shape applied to macroscopic objects: the existence of some particular red ball will not entail that the ball is red. This is a substantial commitment, but one that I happily accept. I thus endorse the Subject Matter Principle. How does the notion of subject matter apply to atomic and general propositions? Consider an atomic proposition, a is F. For a thing theorist, a itself provides a subject matter for a is F (assuming a is essentially F). For a factualist, a subject matter is provided by the atomic fact, a’s being F. (If the proposition a is F is false, then a’s being F is a merely possible fact.) Now consider a general proposition, everything is F, for a fundamental property F. For a thing theorist, its subject matter is the class of all actual and possible non-Fs (assuming that a non-F is essentially a non-F). What is its subject matter for a factualist? It follows immediately from the definitions that if A supervenes on B, and E is a subject matter for B, then E is a subject matter for A. Applying this to the case at hand: the general propositions supervene on the atomic propositions; the atomic facts, according to the factualist, are a subject matter for the atomic propositions; therefore, according to the factualist, the atomic facts are a subject matter for the general propositions. If we take the Subject Matter Principle to be our guide in the ontological enterprise—as I think we should—we arrive already at the weak conclusion of the Supervenience Argument, that general facts are not needed in addition to atomic facts. For, the notion of a subject matter provides a clear sense in which the atomic facts are an ontological ground for the general propositions: the truth or falsity of any general proposition is fixed by the existing or failing to exist of the atomic facts. That nothing more should be required of an ontological ground will be further argued in Section .. What of the strong conclusion of the Supervenience Argument, that general facts do not exist? The Subject Matter Principle, by itself, is compatible with there being a realm of general facts alongside the atomic facts—a second, separate subject matter for the general propositions. If ontological determination is to make the reduction of that which supervenes not merely permissible, but obligatory, we will need to invoke, in addition to the Subject Matter Principle, some version of the Humean denial of necessary connections. Say that two subject matters are distinct just in case no member of one is identical with, or overlaps, any member of the other. I propose calling on the Humean denial in the following convenient form: Non-Distinctness of Subject Matters. subject matters.

No contingent proposition has two distinct

The Non-Distinctness of Subject Matters follows from the Humean denial. For suppose some contingent proposition P has two distinct subject matters, D and E. Then, whether the entities in D exist or fail to exist is not independent of whether the

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

 .  ()

entities in E exist or fail to exist.⁴¹ That is to say, there are necessary connections between distinct existents. How does Non-Distinctness of Subject Matters apply to the Supervenience of the General on the Particular? As already noted, for the factualist, the atomic facts (actual and possible) are a subject matter for the general propositions. Now suppose there existed a general fact distinct from the atomic facts. Since, necessarily, a fact exists if and only if its corresponding proposition is true, it follows that this general fact would all by itself be a subject matter for the general proposition to which it corresponds. But then this general proposition would have two distinct subject matters—the general fact and the atomic facts—violating the Non-Distinctness of Subject Matters. It follows—at least for the factualist with strong Humean scruples— that there are no general facts. The factualist may well balk at this last step in the Supervenience Argument. Humean scruples come more easily to a thing theorist than a factualist. For one thing, there is the perennial problem of determinates and determinables. More relevant to the present case, there is the problem that general facts, were they to exist, would by their very nature stand in necessary connections with particular facts. So any argument against general facts that invokes the Humean denial can fairly be accused of begging the question. I would be content, however, in conversation with such a factualist, to retreat from the claim that general facts do not exist to the claim that general facts are not needed in addition to atomic facts. Atomic facts are subject matter enough for the general propositions.⁴²

. The Non-Entailment Criterion Rejected: The Truthmaker Principle In the previous section, I argued that even a factualist has reason to accept (at least the weak conclusion of ) the Supervenience Argument and deny the need for general facts. In this section, I examine the Non-Entailment Argument, and try to pinpoint where it goes wrong. The Non-Entailment Argument, recall, is: the general truths are not entailed by the atomic truths; therefore, general facts are needed in addition to atomic facts. A natural way to try to bridge the gap between failure of entailment and the need for general facts is to invoke a truthmaker principle—every truth has a truthmaker—and argue that general facts (or, at any rate, non-atomic facts) are needed to serve as truthmakers for general truths. The Non-Entailment Argument then becomes a species of the Truthmaker Argument frequently employed by Armstrong to support the existence of facts (“states of affairs”).⁴³ ⁴¹ Why? Let X be a proposition entailing P that says, for each entity in D, whether it exists or fails to exist; such a proposition exists because P is contingent and D is a subject matter for P. Let Y be a proposition entailing not P that says, for each entity in E, whether it exists or fails to exist; such a proposition exists because not P is contingent and E is a subject matter for P. X and Y are incompatible. ⁴² Perhaps the factualist will be swayed instead by Occam’s razor to abandon general facts. But I, for one, do not think Occam’s razor has any force in metaphysics. ⁴³ See especially Armstrong (: –, –, –). Armstrong’s rendition of the NonEntailment Argument is extremely brief, though it is clear that the Truthmaker Principle plays a crucial supporting role.

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 -  



To begin, we need a precise formulation of the Truthmaker Principle. I will begin with a “local” version that applies only to the actual world: actual truths have actually existing truthmakers. And I will consider the weaker, plural version which demands, not that there be a single entity that serves as truthmaker, but only that there be some plurality of entities that serve jointly as truthmakers.⁴⁴ A plurality of entities makes true a proposition when their joint existence entails the proposition. Restating in terms of worlds and classes gives: a (non-empty) class of actual entities E provides truthmakers for a (non-empty) class of truths A iff every world at which every member of E exists is a world at which every member of A is true. (E provides truthmakers for a single truth Z iff E provides truthmakers for {Z}.) Then, Truthmaker Principle.

Every (non-empty) class of truths has truthmakers.⁴⁵

For the factualist, the atomic facts are truthmakers for the atomic truths. In this case, the correspondence between truthmakers and truths is one-one. In general, however, the correspondence is one-many and many-one. For example: a truthmaker for a truth is also a truthmaker for the many disjunctions with that truth as one of its disjuncts (one-many); but also an existential truth, something is F, for fundamental F, has as truthmaker the many atomic facts Fa, Fb, and so on (many-one). The Truthmaker Principle thus supports a “correspondence theory of truth” according to which a proposition is true just in case there exists some fact (or facts) that bears the truthmaker relation to the proposition; but it is a stripped-down correspondence theory, because the facts are sparse, and the correspondence provided by the truthmaker relation is many-many, and not one-one. The Truthmaker Principle, if sound, can bridge the gap in the Non-Entailment Argument. For suppose, for reductio, that the atomic facts were all the facts. By the Truthmaker Principle, the general truths must have truthmakers. So, the atomic facts would be truthmakers for the general truths; that is, the existence of the atomic facts would entail the general truths. But, necessarily, an atomic fact exists just in case the corresponding atomic proposition is true. So, the atomic truths would entail the general truths, contradicting the Non-Entailment Thesis (assuming there are contingent general truths). Therefore, the atomic facts cannot be all the facts: the general truths must have truthmakers beyond the atomic facts—presumably, for the factualist, general facts. With the Non-Entailment Argument thus expanded, it is clear that the Truthmaker Principle goes hand in hand with the view that failure of entailment is what matters for ontology, not the holding of supervenience. At this point, one might expect that a proponent of the Supervenience Argument would simply reject the Truthmaker Principle. But it is not that simple. An enlightened thing theorist can, and should, accept the Truthmaker Principle, properly interpreted. Ironically, it is only the factualist, I shall argue, who should reject the demand for truthmakers. ⁴⁴ Whether or not this is equivalent to requiring a single truthmaker for every truth depends on one’s views on unrestricted mereological composition, and mereological essentialism. [See Section ..] ⁴⁵ Note that the truthmaking relation here defined is uninteresting for necessary truths: any (nonempty) class of entities is a truthmaker for a necessary proposition. But it does no harm to include this case. For discussion, see Restall (). [See also Section ..]

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

 .  ()

I begin by considering the Truthmaker Principle from the thing theorist’s perspective, asking: what thing (or things) could make a general truth true? Take, for example, the general proposition, all planets are less than ³⁰ kg in mass. Suppose for the sake of argument that this proposition is true. The planets, whether taken singly or jointly, seem deficient as truthmaker: all the planets might exist just as they are and yet an extra jumbo planet greater than ³⁰ kg exist as well. A truthmaker for this proposition must somehow include, not only the planets, but the spaces between the planets. But such a truthmaker is not far to seek. Although no proper part of the world will do as truthmaker for this or any other (contingent) general truth, the world as a whole does the job just fine, where the world is understood to be the biggest thing. The world as a whole makes it true that there are no planets greater than ³⁰ kg simply by having no such planets among its parts. Armstrong would disagree: no thing could be a truthmaker for a (contingent) general truth; since the world is indeed a truthmaker for all truths, the world is not a thing. Why couldn’t a thing be truthmaker for a general truth, according to Armstrong? Because any thing might have been a proper part of some bigger thing. If the world is a thing, then we will have to say that the world might have been a proper part of a bigger world. (Consider, for example, a possible world containing a series of cosmic oscillations—big bang, big crunch, big bang, and so on—and suppose one of the cycles is a duplicate of our world.) If our world might have been a proper part of a bigger world, then it might have existed in its entirety while there also existed outside its bounds a planet greater than ³⁰ kg in mass. So, the existence of the world does not entail that no such planet exists. To ensure truthmakers for general propositions, Armstrong is driven to introduce “totality facts,” higherorder facts that exist in addition to the atomic facts that constitute things. But the thing theorist who accepts the inconstancy of de re modality has a ready response. Though we say that the world might have been part of a bigger world, we also say, with no less propriety, that the world might have been bigger than it is. (For example, the world might have contained a series of cosmic oscillations.) Both of these claims are naturally and straightforwardly interpreted as modality de re; moreover, the very same possible world (say, with cosmic oscillations) can serve to establish both claims. To establish the first claim, we identify the world by intrinsic character alone, so that counterparts of the world must be duplicates of the world. To establish the second claim, we identify in part by extrinsic character, taking it to be essential to a world that it be a world, that is, the biggest thing there is; on this way of identifying, counterparts of worlds must be worlds. Don’t ask: which is the right way of identifying, the way that captures the real essence of the world? Rather, follow the lead of ordinary discourse and let attributions of essence vary with context. It is ordinary indeed, and unobjectionable, to say that general truths are made true by the world as a whole, all the while understanding by ‘world’ the aggregate of all things. In this “truthmaking” context, both intrinsic and extrinsic aspects of the world are taken to be essential: only duplicates of our world that are themselves worlds are counterparts of our world.⁴⁶ ⁴⁶ Again—as with the Subject Matter Principle—if the identity of indiscernible worlds is rejected, then the Truthmaker Principle should be restricted to propositions that are true at all worlds indiscernible from the actual world. (Of course, stipulating that identity is essential to worlds, that a world has only itself as

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 -  



If we interpret the Truthmaker Principle in a way that allows the extrinsic character of the truthmaker to be relevant, then the truthmaker relation is not an internal relation: it does not supervene on the intrinsic character of its relata. Armstrong objects that the truthmaker relation must be internal. Why? I can discern two arguments in his writing. In one, he suggests that if the truthmaker relation is not internal, then we will have to allow, a priori, that “anything may be truthmaker for any truth.”⁴⁷ But here Armstrong seems to have conflated “not internal” with “purely external.” A purely external relation, indeed, is not in any way constrained by the intrinsic character of its relata (taken separately). For example, if distance relations are purely external, then any two distinct things can stand in any distance relation to one another. But one can deny that a relation is internal without denying that it is constrained by the intrinsic character of its relata. From the fact that intrinsic character isn’t all that matters, one cannot conclude that intrinsic character doesn’t matter at all. When I say that the world, qua world, has as part of its essence the extrinsic property of being the biggest thing, that is compatible with saying that some or all of its intrinsic nature is also essential. Thus, one can deny that the truthmaker relation is internal without holding that “anything may be truthmaker for any truth.” A second argument that the truthmaker relation must be internal is hinted at in a number of places.⁴⁸ Suppose, the argument goes, that an entity may be a truthmaker for a proposition in virtue of its extrinsic character. Then the entity is not by itself sufficient as truthmaker, it is only sufficient in conjunction with something else; that something else must also be brought into the truthmaker. But here I suspect a conflation between “extrinsic” and “relational.”⁴⁹ Relational properties are always extrinsic; but extrinsic properties need not be relational. The property, being the biggest thing, is extrinsic (i.e. not intrinsic) because it can differ between duplicates. But it is not relational: an entity need not stand in any (external) relation to something else in order to have it. One cannot conclude from the fact that truthmaking depends on an extrinsic property of a thing—for example, the property of being the biggest thing—that there exists some entity distinct from the thing that must be “brought into the truthmaker.” I conclude that, so far as general truths are concerned, a thing theorist is free to accept the Truthmaker Principle by holding, at least in truthmaking contexts, that the world is essentially a world. Does this do anything to advance the conclusion of the Non-Entailment Argument, as recently expanded? Even if we suppose that the thing theorist accepts atomic facts, say, by identifying them with the tropes that things are composed of, one only gets so far as the conclusion that some truthmaker is needed in addition to the atomic facts. For the thing theorist, that extra truthmaker is just the world, the biggest thing. Note that the world may be an extra truthmaker even if it is

counterpart, would trivialize the Truthmaker Principle.) For an extended discussion of the essential properties of worlds, but one from a rigid essentialist perspective, see Bigelow (). [For a more general treatment of how a counterpart theorist may take things to be truthmakers for a wide array of propositions, see Section ...] ⁴⁷ Armstrong (: ). ⁴⁸ For example, Armstrong (: –). ⁴⁹ Armstrong uses “intrinsic” and “non-relational” interchangeably throughout his work. See, for example, Armstrong (: –).

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

 .  ()

identified with the aggregate of the atomic facts: the world qua aggregate can be allotted different truthmaking capabilities than the atomic facts taken plurally (or, equivalently, than the class of atomic facts).⁵⁰ Although I accept the Truthmaker Principle, I accord it no fundamental ontological importance. I say this not because taking the world to be truthmaker for every (contingent) truth, as I do, trivializes the principle. It does not. (For example, if someone insisted, per impossibile, that there were brute counterfactual or dispositional truths, I would not allow that the world made them true; for, on my view, the fundamental properties and relations that hold among parts of the world are all categorical, not modal.) The Truthmaker Principle lacks ontological importance because, although it is true, and even known a priori to be true, it is not necessarily true, whereas all fundamental ontological principles hold necessarily. And because it is not necessarily true, the connection it makes between failure of entailment and the existence of extra truthmakers is not necessary either. How can the Truthmaker Principle fail to be necessary? Here I need to return briefly to my view, introduced above, that not all possibilities for actuality are possible worlds. Consider again Leibniz’s God surveying the realm of possible worlds. I said that an industrious God might have actualized more than one world, even all the worlds. I say no less that a lazy God might have actualized no world at all, in which case nothing contingent would have existed: no world, and no truthmakers for the contingent truth, nothing exists. There is thus a possibility (though not a possible world) at which the Truthmaker Principle is false: the possibility of nothing.⁵¹ However, because the possibility of nothing is controversial—and, in particular, rejected by Armstrong⁵²—I will not suppose it in discussing the factualist view. I turn now to consider the Truthmaker Principle—and the Non-Entailment Argument—from the factualist perspective. The strategy of holding on to the Truthmaker Principle by allowing the world to be composed entirely of atomic facts while taking the world nonetheless to be a truthmaker for general truths is not available to the factualist. For, as noted in the introduction, for any fact, necessarily, the fact exists if and only if its corresponding proposition is true. I take this to be constitutive of what facts are vis-à-vis their role as truthmakers. But now suppose that the world is merely the aggregate—i.e. the conjunction—of all the atomic facts. Then, necessarily, if all the atomic facts exist, then the proposition that is the conjunction of all the atomic truths is true, and so the world exists as ⁵⁰ I suppose a thing theorist could introduce a limited multiplicity of truthmakers for general truths by taking the aggregate of all the Fs to be a truthmaker for the general truth, all Fs are Gs (for any G). This could be done by saying that, in truthmaking contexts, the extrinsic property, including all the Fs, is an essential property of the aggregate of all the Fs. Again, these extra truthmakers would be no addition to the thing theorist’s ontology. But here, I fear, the thing theorist can no longer claim any support from ordinary discourse. And an awkward problem arises in case all Fs are Gs and there is only one F. ⁵¹ For some defense of this view, see Section ... To accommodate the possibility of nothing, the analysis of possibility must be further expanded to read: a proposition is possible iff it is true at some world, or some plurality of worlds, or at nothing, where a proposition is true at nothing, intuitively, iff it would have been true had no world been actualized. All (contingent) existential propositions are false at nothing; all universal propositions are vacuously true. ⁵² See Armstrong (a: –).

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 -  



well—even if additional atomic facts exist. The world could not have any (horizontally or vertically) exclusive essential properties, such as the property: being the totality of atomic facts. And so the world could not be the “extra truthmaker” required by the Non-Entailment Argument. The factualist, then, must choose: accept the Truthmaker Principle and posit general facts, or reject the Truthmaker Principle. I have already argued, in effect, that a factualist with a Humean bent should reject the Truthmaker Principle. In the previous section, I invoked the Humean denial of necessary connections to defend the Supervenience Argument. If the Supervenience Argument is sound, the Non-Entailment Argument is not, and, for the factualist, the culprit must be the Truthmaker Principle. But the Non-Entailment Argument can be attacked directly without relying on Humean principles. According to the NonEntailment Argument, general facts are needed because the general truths are not entailed by the atomic truths. But when we look more closely at why such entailment fails—when we look to a proof of the Non-Entailment Thesis—we find nothing to support the need for general facts. Entailment fails because there are two worlds, w and w 0 , such that all the atomic truths at w are true at w 0 , but some general truth at w is false at w 0 . What grounds this difference in general truths between w and w 0 ? There are atomic propositions true at w 0 that are not true at w. Thus, for the factualist, there are entities that exist at w 0 —atomic facts—that do not exist at w. This difference in the existence of atomic facts is all that the proof needs to ground the difference in general truths, and establish the failure of entailment. No mention of general facts here: nothing beyond atomic facts need be posited to exist either at w or at w 0 . The argument generalizes. Consider any case in which a class of propositions A supervenes on, but is not (globally) entailed by, a class of propositions B. Then, by failure of (global) entailment, there are worlds w and w 0 such that every B-truth at w is a B-truth at w 0 , but some A-truth at w is not an A-truth at w 0 . What grounds the difference in A-truth at w and w 0 ? Supervenience ensures that the difference can be grounded entirely in the subject matter for B. For, by supervenience, since w and w 0 differ with respect to A-truth, they must differ with respect to B-truth. Whatever entities, then, ground the difference in B-truth also ground the difference in A-truth: no special subject matter for A is required. Thus, whenever (global) entailment fails and supervenience holds, there is a full and adequate explanation for the failure of entailment in terms of the difference in B-truth, and the subject matter of B. If the factualist rejects the Non-Entailment Argument, as I have argued she should, the Truthmaker Principle goes with it. I recommend that the factualist accept in its place the Subject Matter Principle of the previous section. To properly compare these two principles, we need to consider the “global” version of the Truthmaker Principle that applies, not just to the actual world, but to all possible worlds:⁵³ Global Truthmaker Principle. For every (non-empty) class of propositions A and every world w, the A-truths at w have truthmakers at w. (The truthmaking relation is now relativized to worlds: a (non-empty) class of (actual or possible) entities E provides truthmakers at w for a (non-empty) class of propositions

⁵³ On my view, it would be further generalized to apply to all possibilities for actuality.

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

 .  ()

A iff every member of E exists at w and, for any world v, if every member of E exists at v, then every member of A is true at v.) Claim.

The Global Truthmaker Principle implies the Subject Matter Principle.

Proof. Let A be any non-empty class of propositions, and Z any member of A. Let w be any world. By Global Truthmaker (and excluded middle), either Z has truthmakers at w or not Z has truthmakers at w. Let EZw be any class that provides truthmakers for Z at w or that provides truthmakers for not Z at w. Let EZ be UwEZw. We need to show that EZ is a subject matter for Z. Consider any proposition X saying for each member of EZ whether or not that member exists. If X is impossible, then it trivially entails Z (as well as not Z); so suppose that X is possible. X is compatible either with Z or with not Z. Suppose first that X is compatible with Z, and let v be a world at which both X and Z are true. Consider EZv. X entails that each member of EZv exists (since X is true at v, each member of EZv exists at v, and EZv is included in EZ), which in turn entails Z (because EZv provides truthmakers for Z). So, X entails Z. By a similar argument, if X is compatible with not Z, then X entails not Z. It follows that EZ is a subject matter for Z. Finally, let E be UZ2AEZ. E is a subject matter for A, as was to be shown. We have the following proportional analogy: the Subject Matter Principle is to supervenience as the (Global) Truthmaker Principle is to (global) entailment. If the factualist concedes that only failure of supervenience, not failure of (global) entailment, has ontological import, then she should accept only the Subject Matter Principle. In the opening paragraph of this chapter, I said—being intentionally vague—that, for the factualist, truth is determined by (some sort of ) correspondence to the facts. If the factualist trades in the Truthmaker Principle for the Subject Matter Principle and allows that the atomic facts are all the (first-order) facts, what remains of correspondence? Even less, certainly, than the stripped-down correspondence supported by the Truthmaker Principle. Correspondence becomes more holistic: it can no longer be said, except in special cases, that an individual truth corresponds with anything less than all the facts. But it can still be said that truth in its entirety corresponds with the facts in their entirety: for any world, the (first-order) truths at that world are determined by the atomic facts that exist at the world. Moreover, this plural correspondence between truths and facts is one-one: worlds that differ with respect to (first-order) truth differ with respect to the existence of atomic facts as well. The correspondence relation is now, of course, a relation of supervenience, not entailment. Whether such plural correspondence is enough, terminologically speaking, to count as a “correspondence theory of truth,” or a “factualist theory,” is an empirical matter of usage. Metaphysically speaking, it is correspondence enough.

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 Composition as a Kind of Identity () . Introduction The situation is untenable. Something that I take to be absolutely obvious is rejected by many, if not the majority, of my philosophical peers. No, I don’t mean my belief in possible worlds or mathematical entities. I understand full well how my peers can disagree with me about that. I speak rather of my belief in: Unrestricted Composition: For any things whatsoever, there is something that those things compose: a fusion of those things. Unrestricted Composition follows with perfect clarity from my understanding of the notion of composition. It baffles me to no end how this notion, which I grasp so clearly, can elude so many other philosophers.¹ Because no one who means what I do by ‘composition’ could coherently deny Unrestricted Composition, I am faced with a problem of interpretation. Either such a denier and I are talking past one another, or the denier is deeply confused. (To think that I might be the one who is deeply confused would be self-undermining, and is not an option.) Sometimes I think it is the former, for example, when the denier claims that composition only occurs when the components are “cohesive” or “causally integrated.”² Here, I suppose, the denier is trying to capture the contextually restricted application conditions for ordinary uses of the word ‘object’; and I can understand that project well enough. Sometimes I think it is the latter, for example, when the denier claims that whether or not composition occurs is a brute fact, or is a contingent matter.³ On my understanding of composition, this makes about as much First published in Inquiry: An Interdisciplinary Journal of Philosophy  (): –. Reprinted with the permission of Taylor and Francis. This chapter is an expansion of the David Lewis Lecture, delivered at Princeton University on May , . It was also presented at UC Irvine May , . I thank the audiences on those occasions for their helpful comments. Thanks also to Einar Bohn. ¹ Peter van Inwagen has often been praised for introducing the Special Composition Question, roughly: What has to happen in order for some things to compose another thing? But if the answer is, as I think— nothing—focusing on the Special Composition Question was a big mistake. With its suggestion that something does have to happen, it sent a horde of philosophers in search of an account where none is needed. Van Inwagen asks, and gives his own answer to, the Special Composition Question in van Inwagen (). ² Some such views are discussed and rejected in van Inwagen (). ³ Ned Markosian () defends brutal composition. Ross Cameron () defends the contingency of composition. More recently, Cameron (: , ) has claimed only that brute, contingent composition is at least coherent, or conceptually possible. But he seems to mean something weaker by ‘coherent’ Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

      ()

sense as claiming that it is a matter of brute contingency whether a thing is identical with itself; and I know of no other understanding of composition under which it makes better sense. Sometimes I am not sure what to think, as when I am confronted by the nihilist who rejects all composite objects.⁴ For in one respect the nihilist and I are in agreement: we both have a deflationary notion of “composite object.” But, unlike the nihilist, what I think is deflationary about “composite object” is the operation of composition, not the object composed; I do not recognize any interesting metaphysical sense in which composite objects are less “real,” or less “fundamental,” than their parts. Be all that as it may, I can’t help but wonder: is there something I can do to help these philosophers latch on to the deflationary notion of composition that I so clearly grasp? David Lewis (: ) famously wrote: “I myself take mereology to be perfectly understood, unproblematic, and certain.” As best I can tell, my own notion of composition is precisely the same as Lewis’s. And, like Lewis, when called on to elucidate this notion to those who don’t fully grasp it, I find it natural to call on some version of the doctrine of “composition as identity.” But composition as identity has not found many adherents: strong versions of the doctrine are rejected as being incoherent, weak versions as being too weak to be interesting. In what follows, I have a modest goal and a hope. The goal is to lay out clearly what I take the doctrine of composition as identity to be. I hold to a moderate version of the doctrine: although the many-one relation of composition, unlike the one-one relation of identity, does not satisfy a principle of the indiscernibility of identicals, it nonetheless is a kind of identity and not merely analogous to one-one identity. The hope is that, once laid out, the doctrine will force itself on you as being true. But I will settle for less: an understanding of the notion of generalized identity of which composition is a species. For those philosophers who claim still not to understand, I will consider in the final sections whether the notion can be elucidated in modal terms. That project faces serious obstacles, but perhaps something will be learned from the exercise.

. Composition as Identity: Informal Characterization The idea behind composition as identity is supposed to be simple: The whole “is nothing over and above” its parts. Talking about the whole and talking about the parts that compose it are just different ways of talking about “the same portion of reality.” If portions of reality are what our terms, both singular and plural, refer to, then it follows that ‘the whole’ and ‘its parts’ co-refer; and so the whole is identical to its parts.

than I do. I grant that, in some sense, I understand what it means to say that composition sometimes occurs and sometimes does not, just as, in some sense, I understand what it means to say that the operations of conjunction and disjunction sometimes apply to propositions and sometimes do not. What I don’t, and can’t, understand is how these things could be true. [See Section . for some discussion of this issue.] ⁴ Nihilism has been defended by Dorr and Rosen (), by Cameron (a), and, most recently, by Sider ().

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  :  



Perfectly clear, right? Maybe not. For I also grant that there is a sense in which the whole is something over and above its parts. How, then, do I get you to understand the intended sense in which it is identical to its parts? And to say that a plural term refers to a portion of reality, rather than to portions of reality, just begs the very question at issue: whether identity statements between plural and singular terms are ever meaningful and true. So if you didn’t understand my notion of composition as identity before I trotted out the above explanation, it seems unlikely you will understand it after. Perhaps I can get you to understand by example. Consider Baxter’s (a: ) six-pack: Someone with a six-pack of orange juice may reflect on how many items he has when entering a “six items or less” line in a grocery store. He may think he has one item, or six, but he would be astonished if the cashier said “Go to the next line please, you have seven items.” We ordinarily do not think of a six-pack as seven items, six parts plus one whole.

Counting is tied to identity: since we don’t count the six-pack as a seventh item, we must be taking the six-pack, the whole, to be identical to its parts—not with each of the parts individually, of course,⁵ but with the parts taken collectively. Perfectly clear, right? Maybe not. Our reluctance to say that there are seven items could more simply be explained by a quantifier domain restriction in ordinary contexts: include either the parts or the whole in the contextually determined domain, but not both. That makes room for extraordinary contexts where we do count the whole as something additional to the parts. How many squares are in the following diagram?

The answer “fourteen” is certainly permitted. Indeed, intelligence tests sometimes contain questions of this sort, and only the answer ‘fourteen’ would be marked as correct. Apparently, rejecting composition as identity is considered a sign of intelligence!

⁵ I say “of course.” But there are accounts of composition as identity that, by denying principles I take to be undeniable, hold that a whole is identical to each of its parts. See, most notably, Baxter (b: –).

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

      ()

. The Formal Theory: Adding Generalized Identity to Classical Mereology Let’s get serious. Perhaps I can better elicit an understanding of “composition as identity” by providing a formalized theory with precisely formulated axioms and definitions. Perhaps the generalized notion of identity of which composition is a species can then be understood implicitly by way of the axioms and theorems that involve it. In any case, it will be worth our while to consider the theory in some detail. I will couch the theory within a framework of plural logic. The framework I have in mind adds to first-order predicate logic (with identity): plural terms, both variables, ‘xx’, ‘yy’, etc., and constants, ‘aa’, ‘bb’, etc.; plural quantifiers ‘9xx’, ‘8xx’, etc.; a relation ‘≺’ for ‘is one of ’ that links singular and plural terms; and predicates and functors that may take plural arguments, interpreted to apply collectively rather than distributively.⁶ The most straightforward approach is to start with classical mereology, and then add theses that characterize the intended generalized identity relation. If we take the parthood relation, ‘P’, to be primitive, we can characterize Classical Mereology (or CM) by the following three theses (and accompanying definitions): Transitivity of Parthood (TP). (xPy & yPz) ! xPz.⁷ (Any part of a part of a thing, is a part of that thing.) Say that two things overlap iff some thing is a part of them both: xOy $def 9z(zPx & zPy). Say that a thing x is a fusion of yy (or yy compose x) iff each of yy is part of x and every part of x overlaps at least one of yy: xFyy $def 8z(z ≺ yy ! zPx) & 8w(wPx ! 9z(z ≺ yy & zOw)).⁸ The second thesis asserts that fusions are unique: Uniqueness of Composition (Unique). those things have at most one fusion.)

(xFzz & yFzz) ! x = y. (For any things,

The third thesis asserts, what was already introduced above, that any plurality of things has a fusion: Unrestricted Composition (Unrestricted). things have at least one fusion.)

8xx9y yFxx. (For any things, those

Given the second and third theses, a totally defined fusion operator can be introduced that can take either plural or singular arguments: fus xx = y $def yFxx; fus x = y $def x = y. : Now, add to this language a special symbol, ‘¼’, to be understood (putatively) as expressing a generalized identity relation that can take either singular or plural arguments. (I do not use ‘=’ to avoid the appearance of impropriety.) When both ⁶ I assume throughout that plural logic is legitimate and fundamental; in particular, it is not to be understood as singular logic plus set theory. For a presentation and sturdy defense of plural logic, see Oliver and Smiley (). ⁷ Outermost universal quantifiers taking wide scope will often be omitted for readability. ⁸ There are other less standard definitions of being a fusion; but nothing I want to say will depend on choosing this one. See Simons () for various alternative formulations of classical mereology.

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     



: arguments are singular, ‘¼’ expresses familiar one-one identity; when one or both : arguments are plural, ‘¼’ will be said to express many-one or many-many identity, respectively.⁹ It will be convenient, in presenting theses about generalized identity, to make use of schematic letters X, Y, etc. to be replaced uniformly either by singular variables x, y, etc. or by plural variables xx, yy, etc. Then, the fundamental relation between generalized identity and mereology is captured by: : Generalized Identity (GI). X ¼ Y $ fus X = fus Y. GI comprises four theses that, when simplified and combined, become: : Many-Many. xx ¼ yy $ fus xx = fus yy. : : Many-One. xx ¼ y $ fus xx = y; x ¼ yy $ x = fus yy. : One-One. x ¼ y $ x = y. : Needless to say, GI must not be taken to be a stipulative definition of ‘¼’. In that case, adding GI to classical mereology would just be classical mereology with a new : abbreviation. Rather, the idea is that ‘¼’ expresses a notion antecedently understood—what I have suggestively called “being the same portion of reality”— and that GI is a substantial claim about how that notion relates to mereology. Sometimes, it will be useful to have a neutral word to express generalized identity: : : when xx ¼ y, or xx ¼ yy, I will say that the x’s coincide with y, or the x’s coincide with the y’s. : It follows immediately from GI that ‘¼’ is an equivalence relation, where the relata may be singular or plural or mixed: Equivalence. : Reflexive. X ¼ X. : : Symmetric. X ¼ Y ! Y ¼ X. : : : Transitive. (X ¼ Y & Y ¼ Z) ! X ¼ Z. Note that Equivalence comprises fourteen theses in all: two under Reflexive, four under Symmetric, and eight under Transitive. (Some of these can be derived from others, but I am not interested in economy here.) Finally, it follows immediately from Many-One that: : Fusion: xx ¼ fus xx.¹⁰ : Let CM´ be the extension of classical mereology that adds ‘¼’ to the primitive of mereology, ‘P’, and adds GI to the axioms TP, Unique, and Unrestricted. ⁹ [Note that this characterization of “many-one” and “many-many” is syntactical, and applies to : formulae. Thus, ‘aa ¼ b’ is many-one even if ‘aa’ refers to a single thing as is permitted by the semantics of plural logic.] : ¹⁰ We also, of course, have Weak Fusion: xFyy ! x ¼ yy. Often Weak Fusion by itself is called “composition as identity” (e.g. Sider : ). But that nomenclature has the potential to mislead. As a number of philosophers have noted (see especially Cameron ), Unrestricted Composition does not follow logically from Weak Fusion. But Unrestricted Composition is an essential component of the doctrine of composition as identity as I see it; it flows from the underlying conception of reality. (More on this below.) I prefer therefore to apply the phrase ‘composition as identity’ to an entire theory, and not to any one thesis.

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

      ()

It will be useful to have before us a stock example to illustrate the theory, and to have around for future reference. Consider a deck of cards, d. The deck is composed of  cards, c₁, c₂, . . . c₅₂. Call this plurality of cards, cc. Then, d = fus cc, and so, by : Many-One, cc ¼ d. We can say, if not quite in ordinary English: “the cards are the deck”; “the deck is the cards.” (Or we can instead use the technical term introduced above and say “the cards coincide with the deck”; it matters not.) The deck is also composed, let us suppose, of some enormous number of molecules, mm. Then: : : d = fus mm and mm ¼ d. By Transitive: mm ¼ cc. That is, a many-many identity holds between the molecules and the cards: the cards are the molecules; the molecules are the cards. Many-many identity must not be confused with plural identity: the x’s are plurally identical with the y’s iff every one of the x’s is one of the y’s and every one of the y’s is one of the x’s. Plural identity entails many-many identity, but not vice versa. Plural identity and many-many identity are different but compatible ways to generalize the relation of one-one identity so that it applies to plural arguments. It is important, therefore, to speak of a generalized identity relation rather than the generalized identity relation.

. The Formal Theory: Composition as Identity Now that the theory has been provided in full, the doctrine of composition as identity is perfectly clear, right? Problems remain. The first is a problem of formulation: the “axioms” of the theory CM´ contain theses that are supposed to follow from the doctrine of composition as identity, not be assumed at the start. Thus we should not start with mereology and then expand it to arrive at the doctrine of composition as identity. We should start with the conception of reality that underlies composition as identity, and show how classical mereology follows from this conception under appropriate definitions. Only then can it be claimed that composition as identity explains classical mereology. : Let us, then, reverse direction. We take ¼ as our only primitive, putatively to be understood as a relation of generalized identity, and formulate axioms intended to characterize the composite structure of reality. It is this theory that I will call composition as identity, or CAI for short. The notions of parthood and fusion will be defined in terms of generalized identity, and classical mereology will then be : derived. To start, we want ¼ to be an equivalence relation; for only then can it capture a notion of “same portion of reality.” The first axiom, then, is Equivalence. : We also want ¼, when applied to single things, to reduce to one-one identity. So we take One-One to be the second axiom.¹¹ We then define the fusion relation as: : xFyy $def x ¼ yy. That is, when one thing and many things coincide—are the same portion of reality—the one thing fuses the many things. Finally, we define parthood : by: xPy $def 9xx(xx ¼ y & x ≺ xx). To prove that parthood, so defined, is transitive, we need an axiom guaranteeing that, in an appropriate sense, reality is transparent: ¹¹ If I were not supposing that one-one identity is a primitive in our logical system, I would instead take One-One to be a definition of one-one identity.

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  :    Transparency.



: : : (X,Y ¼ Z & W ¼ X) ! W,Y ¼ Z.¹²,¹³

Say that some things divide some thing or things if the former coincide with the latter, every one of the former is part of some one of the latter, and every one of the latter has one of the former as a part: : xxDY $def xx ¼ Y & (8x ≺ xx)(9y ≺ Y) xPy & (8y ≺ Y)(9x ≺ xx) xPy. Transparency ensures that whenever the x’s divide the y’s and the y’s divide the z’s, the x’s divide the z’s. It follows that the composite nature of a thing—that is, the ways of dividing a thing—depends in turn on the composite natures of the things that divide that thing. Transparency fails, for example, if reality has levels where things of one level can be divided into things of the next level down, but cannot be divided into things of a level lower than that. Uniqueness of Composition follows trivially from the axioms introduced: if xFzz : : and yFzz, then x ¼ zz and y ¼ zz, and so x = y by Equivalence and One-One. To derive Unrestricted Composition, we need the fundamental underlying idea that every many is also a one: every plurality of things coincides with some single thing. : E Pluribus Unum (EPU). 8xx9y xx ¼ y.¹⁴ Unrestricted Composition now follows immediately from EPU and the definition of fusion. Finally, we need to prove that the definition of fusion in terms of parthood given by classical mereology can be derived as a theorem. That is, we need to prove: xFyy $ ½8yðy ≺ yy ! yPxÞ & 8wðwPx ! 9yðy ≺ yy & yOwÞÞ: For the left-to-right direction, I propose we take as an axiom that reality has a particulate structure: whenever some things xx coincide with some things yy, there exist some things zz that divide both xx and yy. We have, then, a necessary condition on being the same portion of reality: : Particulate. xx ¼ yy ! 9zz(zzDxx & zzDyy). To get a sense of what Particulate demands of reality, consider how it might fail. Suppose some object divides into a top half and a bottom half, and into a right half and a left half, but has no other parts. Then reality would fail to have a particulate structure. Particulate (together with the other axioms) demands that the object also divide into a top-right corner, a top-left corner, a bottom-right corner, and a bottom-left corner. Note that, if reality has an atomic structure, then it has a particulate structure, but not necessarily vice versa. For example, if everything is composed of particles, and ¹² I use a comma to form compound plural terms in the obvious way: x ≺ Y,Z $def x ≺ Y v x ≺ Z, where ‘≺’ is ‘=’ when the second argument is singular. : ¹³ Proof of transitivity of parthood. Suppose xPy and yPz. That is to say: 9xx(xx ¼ y & x ≺ xx) and : : : : 9yy(yy ¼ z & y ≺ yy). But then () xx, yy ¼ z. For, since yy ¼ z and y ≺ yy, we have y, yy ¼ z. And then : () follows by Transparency and xx ¼ y. But we also have () x ≺ xx, yy, because x ≺ xx. Putting () and () together gives: xPz. ¹⁴ The phrase ‘e pluribus unum’ has instead been applied to the formation of sets from their elements rather than, as here, to the formation of wholes from their parts. See e.g. Burgess ().

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

      ()

each of those particles in turn is composed of more particles, and so on ad infinitum, then reality is particulate without being atomic. Although it is controversial whether or not we can know a priori that reality is atomic, that reality is particulate cannot, in my view, coherently be denied. If xx and yy have no common division, there is nothing to tie them to a single “portion of reality”; having a common division is in part constitutive of being the same “portion of reality.” For the right-to-left direction, we need a supplementation axiom, an axiom that guarantees the existence of differences when portions of reality do not coincide. When restricted to singular arguments, we have: if x 6¼ y, then either x has a part that does not overlap y, or y has a part that does not overlap x. Generalizing to plural arguments, we have a sufficient condition on being the same portion of reality: : Difference. ~xx ¼ yy ! 9z[(9x ≺ xx)zPx & (8y ≺ yy)~zOy] v 9z[(9y ≺ yy)zPy & (8x ≺ xx)~zOx]. Portions of reality cannot differ unless some portion of reality makes the difference. That portions of reality are, in this sense, extensional is again in part constitutive of “portion of reality” and cannot, in my view, coherently be denied. But, or so it seems, it has often been denied: states of affairs and structural universals are among the entities that, prima facie, would violate Difference were they to exist. : That completes the formulation of CAI that takes ¼ as primitive and includes as axioms Equivalence, One-One, Transparency, E Pluribus Unum, Particulate, and Difference.¹⁵ Now, finally, is composition as identity, and how it explains classical mereology, perfectly clear?

. Preliminary Skirmishing I expect the following complaint: The doctrine of composition as identity is supposed to embody a picture of the structure of reality that explains classical mereology. But CAI and CM0 are interderivable: when definitions are added to the axioms, they are broadly logically equivalent.¹⁶ It seems, then, that ¹⁵ Proof of: xFyy $ [8y(y ≺ yy ! yPx) & 8w(wPx ! 9y(y ≺ yy & yOw))]. : : Left-to-Right. Suppose xFyy; that is, x ¼ yy. To prove the first conjunct, consider any y ≺ yy. Since x ¼ yy, : : yPx. To prove the second conjunct, let wPx; that is, 9xx(xx ¼ x & w ≺ xx). By Equivalence, xx ¼ yy. By Particulate, 9zz (zzDxx & zzDyy). Since zzDxx, (8x ≺ xx)(9z ≺ zz) zPx. So, (9z ≺ zz) zPw. But then, since zzDyy, (8z ≺ zz)(9y ≺ yy) zPy. So, (9y ≺ yy) zPy, and thus (9y ≺ yy) yOw. : : Right-to-Left. Suppose ~xFyy; that is, ~x ¼ yy. Suppose also that (8y ≺ yy) yPx. By EPU, 9v v ¼ yy. v 6¼ x, so by Difference either 9z(zPx & ~zOv) or 9z(zPv & ~zOx). But the second disjunct is false. For consider : any zz ¼ yy, and consider any z ≺ zz. By Particulate, 9ww (wwDzz & wwDyy). So, (9w ≺ ww) wPz because wwDzz. Moreover, (9y ≺ yy) wPy because wwDyy. But then wPx by TP, and so zOx. Therefore, the first disjunct is true (changing variables): 9w(wPx & ~wOv). But since (8y ≺ yy) ~wOy follows from : ~wOv & v ¼ yy, we have 9w(wPx & (8y ≺ yy) ~wOy), which is just what we needed to prove. ¹⁶ I say theories are narrowly logically equivalent if the axioms of each logically imply the axioms of the other; theories with different primitives are never narrowly logically equivalent. I say theories are broadly logically equivalent if the axioms plus definitions of each logically implies the axioms plus definitions of the other.

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 



CAI is just the same theory as CM0 , reformulated; and, surely, a theory cannot explain itself! We might as well just take the familiar formulation of classical mereology as fundamental, and be done with it.

In response, I say, CM0 and CAI are different theories, both metaphysically and conceptually. Metaphysically, they attribute different fundamental structures to reality in virtue of having different primitives. According to CM0 , the parthood relation is fundamental to reality, and facts about generalized identity (if any) are grounded by facts about parthood. According to CAI, the generalized identity relation is fundamental, and facts about parthood are grounded by facts about generalized identity. The two theories conflict over what grounds what, reversing the order of explanation. Conceptually, CM0 and CAI occupy different neighborhoods in conceptual space, and thus have different explanatory potentials; for the theories differ as to how reality would be were the theory to be false. CM0 has as its closest neighbors in conceptual space theories that posit alternative “parthood” relations. For example, if Transitivity of Parthood fails, we may have a theory of sets or classes that takes membership to be a “parthood” relation. In contrast, CAI has as its closest neighbors in conceptual space theories that posit alternative “composite structure” to reality; the composition relation falls out of that composite structure. For example, if Transparency fails, the composite nature of a thing is not determined by the composite nature of the parts of that thing. If Particulate fails, then different ways of dividing a thing may have no common subdivision; reality has no grain. These conceptual differences between CM0 and CAI allow CAI to gain explanatory purchase on CM0 , notwithstanding that they are broadly logically equivalent.¹⁷ Of course, that is not to say that the explanation will be fully satisfying. In particular, doubts about Unrestricted Composition will, for many philosophers, just transfer over to doubts about E Pluribus Unum. Even if one grants that the truth of CAI could explain the truth of CM0 , that just invites the obvious question: why accept CAI? And, I do not see much point in seeking an even more fundamental theory to explain CAI. With CAI, we have hit rock bottom. : Does it help to point out that ¼ is to be understood as a generalized identity relation? Could we then say, based on our prior understanding of identity, that the axioms of CAI can be seen to be true? Perhaps. For example, Transparency and Particulate could then be seen as instances of the substitutivity of identicals; and E Pluribus Unum—arguably—could be seen as an instance of the universality of : identity. But, in any case, we first need a reason to think that ¼ is a generalized identity relation, and not just a generalization of identity—that is, not just a relation that has identity as a special case. Indeed, every reflexive relation—e.g. having the same mass as, being at least as massive as—could be said to have identity as a special case. But, surely, not every reflexive relation is a kind of identity, or a generalized identity relation. ¹⁷ What counts as an explanation is a disputed matter, but I take it that not all explanations in metaphysics are grounding explanations: an explanation may enhance our understanding by revealing objective relations between the concepts involved without reducing the explanandum to something more fundamental. See also n.  below.

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

      ()

: Might the interpretation of ¼ as generalized identity be forced on us by the formal system, by the axioms of CAI together with the interpretation of plural logic? It is easy to disabuse oneself of that idea. For even if one held that plural terms in some sense co-refer with singular terms, nothing but an aggregative bias could lead to the : conclusion that plural terms refer to the plurality’s fusion. Indeed, if we interpret ¼ to mean “have the same intersection” rather than “have the same fusion,” and similarly interpret the other notions by substituting their duals, then the axioms of CAI still come out true. (Well, almost; we would need either to posit a “null element,” an element that is a part of every element, for EPU to come out true, or substitute a qualified version of EPU.) It may seem counterintuitive to interpret plural terms as, in some sense, referring to their intersection rather than their fusion, but nothing about the framework of plural logic forces one interpretation over the other. CAI, considered as a formal system, no more supports the thesis that composition is identity than the thesis that intersection is identity. It should come as no surprise that CAI, when considered as a formal system, : cannot force an interpretation of ¼ as generalized identity. When considered as a formal system, CAI merely posits that its domain has a certain structure—the structure of a complete Boolean algebra with no minimal element—and there is more to interpretation than structure. But, still, it seems legitimate to place some : demands on CAI. My claim that ¼ in CAI is a generalized identity relation is only plausible if the axioms and theorems of CAI capture whatever features are essential to identity relations. If we take one-one identity to serve as our paradigm, those features are two: the universality of identity and the indiscernibility of identicals.¹⁸ The universality of identity is the thesis that everything is identical with something; the indiscernibility of identicals is the thesis that identicals have all of their properties in common. Let us say that the doctrine of strong composition as identity holds that these two theses, appropriately reconfigured, apply to generalized identity. Have these theses already been incorporated into CAI? Consider first the universality of identity. On its face, there are two ways to generalize the universality of one-one identity within the framework of CAI, and both have already been accounted for. The universality of many-many identity : follows immediately from an instance of Reflexive: from 8xx xx ¼ xx it follows : that 8xx 9yy xx ¼ yy. The universality of many-one identity, although it cannot be : derived from Reflexive, is given directly by EPU: 8xx 9y xx ¼ y.¹⁹ Neither generalization follows logically from the universality of one-one identity; both, however, are natural ways of generalizing to the plural framework. Any of these versions of

¹⁸ In the formal presentation of identity theory, the two features are often taken to be reflexivity and the indiscernibility of identicals. Reflexivity and universality are interderivable given the indiscernibility of identicals, so either could be taken as an axiom. But when generalizing identity so as to allow it to take either singular or plural arguments, universality is stronger than reflexivity. In any case, reflexivity is captured directly by CAI as part of Equivalence. : ¹⁹ [The universality of many-one identity also includes ‘8x 9yy x ¼ yy’. This follows trivially from the universality of one-one identity because ‘yy’ can be assigned the same value as ‘x’. It does not have as a consequence that everything is composed of gunk. But cf. Lechthaler () who argues that to whatever extent the universality of identity motivates unrestricted composition, it also motivates that everything is composed of gunk.]

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    



universality could, I suppose, be denied. The universality of one-one identity would fail if, per impossibile, some thing failed to “constitute a unit,” and thereby failed to be identical to anything, even itself. The universality of many-one identity would fail if, per impossibile, some things (collectively) failed to “constitute a unit,” and thereby failed to coincide with any single thing. On the deflationary notion of composition embodied in CAI, the second denial is no more coherent than the first.²⁰ In any case, with respect to the issue at hand, there can be no doubt that CAI incorporates generalizations of the universality of identity among its axioms and theorems; if anything, some might think, it incorporates too much universality. Consider next the indiscernibility of identicals. The usual way to incorporate the indiscernibility of identicals into a system such as CAI that lacks quantification over properties is as an inference schema: : Substitutivity of Identicals (Sub Id). X ¼ Y ‘ φ(X) $ φ(Y). where ‘φ’ is any open sentence in the language that has the appropriate sort of variable free. Since Sub Id is not generally valid in CAI, let us add it explicitly and call the resulting theory strong CAI. There is no doubt that Sub Id is a generalization of the substitutivity of identicals for one-one identity: x = y ⊦ φ(x) $ φ(y). But if Sub Id : is required to uphold the claim that ¼ is a generalized identity relation, then that claim is in serious trouble. Strong CAI should be rejected.

. Against Strong Composition as Identity Some predicates that take plural arguments can be added to the language of CAI without making trouble for Sub Id. If the deck of cards weighs  grams, then the cards that compose the deck (collectively) weigh  grams, and the molecules that compose the deck (collectively) weigh  grams.²¹ But such predicates are rather special. Many predicates, when added to the language of CAI, produce counterinstances to Sub Id, notably numerical predicates. The cards that compose the deck are  in number, but neither the deck itself, nor the molecules that compose the deck, are  in number. Call predicates that provide counter-instances to Sub Id slice-sensitive predicates. (Other terms in use are “count-sensitive” and “set-like”). Whether or not a slice-sensitive predicate applies to a portion of reality is relative to how the term referring to that portion of reality slices it up.²² ²⁰ The universality of one-one identity is captured by Quine’s slogan “no entity without identity.” The universality of many-one identity, “no entities without (collective) identity,” is no less compulsory on my picture of reality. I say a bit more in support of this below. ²¹ I skirt grammatical issues involving plural verbs by supposing that both ‘weigh  grams’ and ‘weighs  grams’ are represented by a single predicate in CAI that takes either singular or plural arguments. I do not, however, suppose that all predicates in CAI are thus polymorphous. ²² Thus, Lewis (: ) writes in rejecting a version of strong composition as identity: “Even though the many and the one are the same portion of Reality . . . we do not really have a generalized version of the indiscernibility of identicals. It does matter how you slice it—not to the character of what’s described, of course, but to the form of the description. What’s true of the many is not exactly what’s true of the one. After all they are many while it is one.” [But see Chapter  for an interpretation of this that is compatible with a version of strong composition as identity.]

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

      ()

One might hope to save strong CAI by simply banning the offending predicates from the language, but the problem goes deeper. Sub Id together with basic principles of mereology and plural logic lead to what Sider ominously calls: Collapse. yPfus xx $ y ≺ xx. (Something is part of the fusion of the x’s iff it is one of the x’s.) Collapse, of course, is unacceptable. For example, it would require that anything that is part of the fusion of the cards is itself one of the cards. But molecules are parts of the fusion of the cards without being cards. Collapse is to be avoided at all cost.²³ : Is there a way to reject Sub Id while holding on to the claim that ¼ satisfies the indiscernibility of identicals? That is, is there a way of rejecting the formal system strong CAI while holding on to the doctrine of strong composition as identity? Perhaps we can call on the familiar distinction between the substitutivity of identicals as a semantic, language-relative principle and the indiscernibility of identicals as a metaphysical, or logical, principle, since it is only the latter that, we are now considering, is an essential feature of identity relations. Perhaps, then, violations of Sub Id always involve predicates that fail to express genuine properties, or that express different properties with different occurrences. In that case, we could say that, although there are slice-sensitive predicates, there are no slice-sensitive properties. Violations of Sub Id would not carry over to violations of the indiscernibility of identicals. To illustrate the strategy, consider how the counterpart theorist rejects coincident entities without running afoul of the indiscernibility of identicals. We have before us (as per usual) a statue of Goliath made of clay. Call the statue ‘Goliath’, the lump of clay ‘Lumpl’, and suppose that Goliath and Lumpl came into existence and went out of existence together. Now, it is natural to think that there is a single object before us that is both the statue and the lump of clay; that is, that Goliath is identical to Lumpl. But how can that be? Aren’t there properties that Lumpl has and Goliath lacks? For example, Lumpl, but not Goliath, could have survived a squashing. It then follows from the indiscernibility of identicals that Lumpl and Goliath are not identical. But wait: if the predicate ‘could have survived a squashing’ expresses different modal properties when applied to ‘Lumpl’ and when applied to ‘Goliath’, the inference to non-identity is invalid. We have a violation of the Substitutivity of Identicals in English, to be sure: Goliath is identical to Lumpl; Lumpl could have survived a squashing; it is not the case that Goliath could have survived a squashing. But we have no violation of the indiscernibility of identicals, no property that Lumpl has but Goliath lacks. The counterpart theorist diagnoses the situation as follows.²⁴ A de re modal claim such as ‘a could have φ’ed’ is to be analyzed in terms of a counterpart relation: some counterpart of a φ’s, where the quantifier ‘some counterpart’ ranges over possibilia. But there are multiple counterpart relations; different counterpart relations are needed for the evaluation of de re modal claims in different contexts. In particular, there is a counterpart relation, call it the s-counterpart relation, under which counterparts of statues are always statues; with respect to this counterpart relation, the ²³ Sider () details a parade of absurd consequences of Collapse. But if the question is whether or not to accept strong CAI, the parade can be cut short: one absurdity is as bad as a thousand. ²⁴ See especially Lewis () and Lewis (a: section .).

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    



object before us is essentially a statue and could not have survived a squashing. There is a different counterpart relation, call it the l-counterpart relation, under which counterparts of lumps of clay are always lumps of clay (but need not be statues); with respect to this counterpart relation, the object before us is not essentially a statue and could have survived a squashing. When we use the name ‘Goliath’, we typically (though not invariably) evoke the s-counterpart relation. So interpreted, ‘Goliath could have survived a squashing’ falsely attributes the property has an s-counterpart that survives a squashing to the object before us. When we use the name ‘Lumpl’, we typically evoke the l-counterpart relation. So interpreted, ‘Lumpl could have survived a squashing’ truly attributes the property has an l-counterpart that survives a squashing to the object before us. Sometimes both counterpart relations are needed for the interpretation of a single sentence, such as ‘Lumpl, but not Goliath, could have survived a squashing’. The best way of capturing this in a semantic or pragmatic theory need not detain us. What matters is that we explain the violation of the substitutivity of identicals (in English) in a way that does not involve a violation of the indiscernibility of identicals. How we consider the subject of the predication determines in part what modal property is attributed to the subject by the modal predicate. Now, let’s see if this strategy can be applied to slice-sensitive predicates, such as : ‘are  in number’. We have a failure of Sub Id, to be sure: the cards ¼ the deck; the cards are  in number; but it is not the case that the deck is  in number. But if ‘are  in number’ does not express a single property when applied to ‘the deck’ and when applied to ‘the cards’, we have no violation of the indiscernibility of identicals. Let us say, then, that the property expressed by a numerical predicate depends in part on how the portion of reality referred to by the subject term is sliced. For any portion of reality, there are as many slicings as there are ways of dividing the portion into non: overlapping parts. Thus, we define: X are a slicing of Y iff fus X ¼ Y, and X do not pairwise overlap.²⁵ There are numerous different slicings of the deck. Which slicing is relevant to the application of a slice-sensitive predicate depends, typically though not invariably, on how the subject of the predication is referred to. Referred to as ‘the deck’, the relevant slicing is just the deck itself; referred to as ‘the cards’, the relevant slicing is the  cards. Once we make this relativity to slicings explicit, we see that there is no violation of the indiscernibility of identicals: the portion of reality referred to by ‘the deck’ and ‘the cards’ has the property being  in number relative to slicing c₁, . . . , c₅₂, and fails to have the property, being  in number relative to slicing d. Just as in the modal case considered above, the property expressed by a predicate depends in part on how the subject of the predication is being considered; but now the different ways of considering the subject are given by the different slicings rather than the different counterpart relations.²⁶ : Now, at long last, is the doctrine of composition as identity, with its claim that ¼ is a generalized identity relation, perfectly clear? No, we have taken a wrong turn. ²⁵ We might want to drop the second condition to allow “double counting” in special contexts, in which case the term ‘slicing’ should be replaced. ²⁶ Compare Frege’s (: section ) claim that “a statement of number contains an assertion about a concept.” For Frege, numerical predications hold or fail to hold relative to concepts where concepts provide the slicing.

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

      ()

The analogy between the modal case and the plural case is spurious in a crucial respect. In the modal case, the counterpart theorist supposes that there is a full reduction of the modal to the non-modal. When the underlying theory of modality is presented in a fundamental language that quantifies over possibilia, the indiscernibility of identicals will be reflected in that language by a valid rule of substitutivity of identicals. In the plural case, however, there is no reduction of the plural to the singular, no more fundamental language in which we can escape all violations of substitutivity. For the fundamental predicate of plural logic, ‘is one of ’, is itself a slicesensitive predicate that expresses a slice-sensitive relation. So there can be no escaping violations of the indiscernibility of identicals, or failures of substitutivity of identicals, in the fundamental language. If the doctrine of composition as identity, : with its claim that ¼ is a generalized identity relation, requires that a generalization of the indiscernibility of identicals hold, then composition as identity is dead. We can put the problem in the form of a trilemma. First horn. Leave ‘is one of ’ (and its ilk) out of the fundamental logical framework. But that is just to abandon plural logic. And without plural logic, the doctrine of composition as identity (as I understand it) cannot even be stated. So this is not an option. Second horn. Put ‘is one of ’ into the fundamental logical framework. But then ‘is one of ’ is a fundamental relation that is slice-sensitive in its second argument. Slice-sensitive properties that violate the indiscernibility of identicals are then close at hand. For example, the cards have the property, having the ace of spades as one of them, but the deck does not. Strong composition as identity has been abandoned. Third horn. Include ‘is one of ’ in the fundamental logical framework in relativized form, like other slice-sensitive predicates. That is, the fundamental relation of plural logic is represented by a three-place predicate: x is one of yy relative to slicing zz. But this predicate, although no longer slice-sensitive in its second argument, is slice-sensitive in its third argument. For example, although the ace of spades is one of the cards relative to the card slicing, it is not one of the cards relative to the deck slicing; and, again, violations of the indiscernibility of identicals are close at hand. Again, strong composition as identity has been abandoned. Did we perhaps go astray in the implementation of the strategy? Should we have taken slicings to be sets, rather than pluralities? Say that ‘mem x’ is a plural term that denotes the : members of x if x is a set. We can define: x is a slicing of Y iff x is a set, fus mem x ¼ Y, and mem x do not pairwise overlap. Then, indeed, the three-place relativized ‘is one of ’—x is one of yy relative to slicing z—is no longer slice-sensitive. But this effectively reduces ‘is one of ’ to ‘is a member of ’: x is one of yy relative to slicing z iff x is a member of z; the second argument drops out. It gets plural logic back only by letting sets and set membership do all the work that plural terms and ‘is one of ’ were called on to do. And that project cannot succeed. To give just one reason: we can refer collectively to all the sets, and predicate properties of them collectively (e.g. that they can be well-ordered), even though there is no set of all sets.²⁷ Strong composition as identity should be rejected.²⁸

²⁷ Nor would it help to introduce proper classes, or even more outré class-like entities. See Lewis (: –). ²⁸ My reason for rejecting strong composition as identity is essentially the same as that given by Sider (): that it is incompatible with the framework of plural logic. But I wanted to tell the story in my own way. (Sider no longer accepts this reason because he now rejects the framework of plural logic; see Sider : –.) (continued on next page . . . ).

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     



. Composition as Merely Analogous to Identity Now what? It is common at this point, supposedly following David Lewis (), to retreat to the claim that composition is analogous to identity. Lewis presents five aspects of the analogy. The first is ontological innocence: if one is ontologically committed to some thing, one is thereby also ontologically committed to anything identical to that thing; similarly, if one is ontologically committed to some things, one is thereby also ontologically committed to the fusion of those things. The second aspect is what I called “universality”: if some thing exists, then automatically something identical to that thing exists; similarly, if some things exist, then automatically their fusion exists. The third aspect is uniqueness: two different things are never both identical to some thing; similarly, two different things are never both the fusion of some things. The fourth and fifth aspects have to do with how the character and location of a thing, or things, determines the character and location of anything that is identical to that thing, or is a fusion of those things. Lewis concludes: “This completes the analogy that I take to give the meaning of Composition as Identity.”²⁹ It is natural at this point to wonder: is that all there is? Wasn’t composition as identity introduced to provide support for various controversial theses about mereology? But if composition is merely analogous to identity, how can it do any explanatory work? We have the following dilemma. Either the controversial theses of mereology—ontological innocence, unrestricted composition, uniqueness of composition—are included among the aspects of the analogy, or they are not. If they are included (as Lewis clearly does), then composition as identity as we are now understanding it presupposes those controversial theses, and does not support them. If they are not included, then we have only an “argument from analogy” to support the controversial theses; and such arguments are notoriously weak. (Indeed, in my view, they lack epistemic force altogether in a priori domains.) Either way, we have done little or nothing to explain why the controversial theses should be accepted.³⁰ Sider (: ) contrasts strong composition as identity, which he calls “fun and interesting,” with the view he attributes to Lewis that composition is analogous to identity, which he calls “wimpy, dreary, and boring.” I’m not sure how a view that leads to absurdity can be fun and interesting. (Perhaps the way jumping out of a plane without a parachute is fun and interesting?) But, in any case, if all there is to Philosophers who employ versions of this strategy in defense of strong composition as identity include Bohn (), Wallace (b), and Cotnoir (). Each of these views deserves a separate extended discussion, but here I can just say, summarily, that they all seem to me to be subject to the difficulty adumbrated above: when cashed out at the fundamental level, they must either reject basic principles of plural logic, or restrict the indiscernibility of identicals, or both. Baxter is often taken to hold strong composition as identity—and, indeed, his view is quite strong—but since he rejects the indiscernibility of identicals, his view is not strong composition as identity as herein characterized, and so is outside the scope of the current argument. See Baxter (). ²⁹ Lewis (: ). Sider (: ) adds three additional aspects to the analogy: identity and composition are both absolute, cross-categorial, and precise. ³⁰ Yi () argued early on that the view that composition is analogous to identity does nothing to support the ontological innocence of mereology. But, oddly, he interpreted Lewis as intending that the analogy provide such support, ignoring that Lewis explicitly included ontological innocence as an aspect of the analogy.

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

      ()

Lewis’s view is that composition is analogous to identity, then Sider’s assessment of that view as “wimpy, dreary, and boring” would seem to be hard to deny.

. Composition as a Kind of Identity But wait: there is more. Prior to specifying the aspects of the analogy that he takes to give the meaning of composition as identity, Lewis (: –) writes: The mereological relations are something special . . . . They are strikingly analogous to ordinary identity, the one-one relation that each thing bears to itself and to nothing else. So striking is this analogy that it is appropriate to mark it by speaking of mereological relations—the manyone relation of composition, the one-one relations of part to whole and of overlap—as kinds of identity. Ordinary identity is the special limiting case of identity in the broadened sense.

Although Lewis does not include the claim that composition is a kind of identity when he states what he “takes to give the meaning of Composition as Identity,” it is a substantial component of his view nonetheless. Certainly, he does not retract or qualify the claim when explicating the analogy between composition and identity; and the claim is repeated elsewhere in his body of work.³¹ Moreover, the claim that composition is merely analogous to one-one identity is fully compatible with taking composition to be (literally) a kind of identity if one-one identity is itself one of many kinds of identity. Let us take weak composition as identity to be the view that composition is merely analogous to one-one identity; it does not include the claim that composition is a kind of identity. Let moderate composition as identity be the view that composition is a kind of identity as well as being analogous to one-one identity. (Both weak and moderate composition as identity, I will suppose, endorse the formal system CAI; they differ with respect to the underlying picture that motivates and interprets CAI.) It is moderate composition as identity that Lewis held,³² and that I want to defend. For all that weak composition as identity says, the ³¹ See especially Lewis (a: –); see also Lewis (d: ). ³² See also Bohn (). Bohn and I are in agreement that Lewis () has often been wrongly understood as holding only weak composition as identity. Still, we are faced with an exegetical puzzle. Early in the section on “Composition as Identity” he writes: “The ‘are’ of composition [as in, ‘the parts are the whole’] is, so to speak, the plural form of the ‘is’ of identity. Call this the Thesis of Composition as Identity. It is in virtue of this thesis that mereology is ontologically innocent . . . ” (p. ). How can this be squared with his later claim that the ontological innocence of mereology is part of what “gives the meaning of Composition as Identity” (p. )? It is common to suppose that Lewis weakens what he takes to be the thesis of composition as identity between the beginning and end of the section, and then somehow missed that he can no longer take composition as identity to support ontological innocence. (For example Burgess (: ) writes: “It is difficult, also, to see how Lewis can be acquitted of question-begging when he argues that one respect in which there is analogy is in ontological innocence.”) But I find that interpretation wholly implausible. Rather, I think that when Lewis says that the analogy “gives the meaning of Composition as Identity,” he is not taking back his earlier characterization quoted above; he is not saying that the thesis of composition as identity is to be defined as: the five aspects of analogy hold. I am not entirely sure what he had in mind by “gives the meaning,” but all of the following are plausible alternative readings that are compatible with Lewis not having to retract his claim that mereological innocence holds in virtue of composition being a kind of identity: the analogy “gives significance to composition as identity”; the analogy “provides substantial support” for composition as identity; and (as he explicitly says) the analogy makes it “appropriate to speak of the mereological relations” as kinds of identity. Uses of “gives meaning to” that do not entail “provides the definition of ” are altogether common.

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     



analogy between composition and identity might be “accidental”; it might fail to reflect the natural order. But if composition is a kind of identity, and the kinds of identity themselves form a natural kind, then the analogies are not merely “accidental”; they have a deeper significance. How should Lewis’s “broadened sense” of identity be understood? There is no question what Lewis means by it: it is the mereological relation of overlap generalized to apply to plural arguments.³³ If we introduce the symbol ‘≑’ to express this relation, we have: X ≑ Y $def fus X O fus Y. Strict one-one identity has been broadened along two dimensions: singular to plural, and total overlap to partial overlap. What has hitherto been called “generalized identity” is a kind of identity that has been : broadened only along the former dimension: X ¼ Y entails X ≑ Y, but not vice versa. We can call generalized identity and composition total identity relations; broadened identity, overlap, and parthood are partial identity relations. One could, I suppose, say that the partial identity relations are not, properly speaking, identity relations, any more than a half brother is, properly speaking, a brother. I do think that generalized identity is the fundamental kind of identity in terms of which the family of identity relations, total and partial, is to be analyzed, a view reflected in my formal theory CAI. But whether the partial identity relations are, properly speaking, identity relations is a terminological decision of little consequence. But might not the entire difference between weak and moderate composition as identity be a terminological decision of little consequence? Sider (: ), for example, writes: “Whatever else one thinks about identity, Leibniz’s Law [my “Substitutivity of Identicals”] must play a central role.” Once Leibniz’s Law is seen to fail for composition, and strong composition as identity is thereby rejected, Sider never seriously considers the idea that composition is a kind of identity. What I have called “kinds of identity” he calls relations of intimacy, and he then goes on to consider how relations of intimacy such as parthood and composition are to be understood. If the only difference between us is that what I call “(partial or total) identity” he calls “intimacy,” then weak and moderate composition as identity are just terminological variants of one another. What’s in a name? Before answering that question, a brief recap is in order. I argued that the theory : CAI can be said to explain classical mereology because its primitive notion, ¼ (“same portion of reality as”), is more fundamental than the notion of parthood that mereology (typically) takes as its primitive. We are now asking: what is added to : this explanation by holding to the claim that ¼ is a kind of identity, a generalized identity relation, as opposed to just claiming that it is a special relation of intimacy that is in some ways analogous to (one-one) identity? What can moderate composition as identity provide that weak composition as identity cannot? And here, I think, it is generally supposed that, for a defender of moderate : composition as identity, the answer must be that taking ¼ to be a kind of identity ³³ Based on the above quoted passage, Megan Wallace seems to identify Lewis’s “broadened sense” of identity with Butler’s “identity in the loose and popular sense.” She writes: “Identity in this broadened sense is presumably how we understand personal identity over time, qualitative similarity or type-hood, and (as Lewis suggests) the composition relation” (Wallace a: ). But I know of no reason to think Lewis would include non-mereological relations that sometimes are confused with identity as kinds of identity.

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

      ()

does additional explanatory work; it can explain why the analogies hold, and in particular why universality holds, why every plurality coincides with some single thing. But, as already noted, with universality we have hit rock bottom: there is nothing more fundamental in terms of which it can be explained. If explanations in metaphysics must always provide grounds, then it is not about explanation. What, then, does moderate composition as identity have to offer? There is only one possible answer. It is not about explanation (in the sense of grounding); it is about understanding, about getting our mind right. That is to say, it is about getting the order and classification of our concepts to line up properly with the structure of reality.³⁴ Because taking composition to be a kind of identity is not introduced to ground the analogies with (one-one) identity, it is perfectly proper to hold both that composition as a kind of identity gives reason to accept the analogies and that the analogies give reason to take composition to be a kind of identity. These claims are not in conflict; indeed, they mutually reinforce one another. Such interlocking circles enhance understanding. Perhaps an analogy would be useful. Consider the belief, common in preCantorian days, that the following feature is essential to the notion of cardinal number as applied to sets (or pluralities): if an element not already a member of a set is added to that set, then the set increases in number. Whoever holds that belief is making a kind of mistake, one that is not merely terminological. They are taking a feature of the finite cardinal numbers to be a feature of cardinal numbers generally. Historically, this mistake hindered the development of the Cantorian theory of infinite cardinality. Similarly, though less momentously, taking the indiscernibility of identicals to be essential to identity relations blinds one to natural generalizations that should inform our metaphysical picture of reality. Indeed, calling composition a kind of identity is not metaphysically inert. It matters whether our terminology reflects the true order and classification of our concepts. Consider two examples. First, to see composition as a generalized identity relation is to see mereology as part of logic, a logic more general than either firstorder predicate logic or first-order plural logic.³⁵ And once we see mereology as logic, it is natural, even inevitable, to see it as necessary and a priori. Further, seeing composition as a logical operation naturally constrains how composition should be understood. For, say I, whether a logical operation applies should not be a matter of brute contingency, or a vague matter. So moderate answers to van Inwagen’s special composition question go by the board in favor of one of the two extremes.

³⁴ As mentioned in n. , I do not accept that all explanations in metaphysics are grounding explanations. There are other notions of explanation that are concerned with “unification” or “systematization” rather than “grounding,” notions that are commonly applied to mathematics but apply no less to metaphysics. On these notions of explanation, a global conceptual re-orientation can be “explanatory” without providing new grounds. Taking the various relations of mereology to be of a kind with one-one identity is explanatory in this sense. For accounts of explanation in mathematics, see Kitcher () and Lange (). ³⁵ Lewis (: ) writes, when introducing the framework of plural logic and mereology: “I would have liked to call it ‘logic’ . . . But that is not its name, and, with names, possession is nine points of the law.” Armstrong (: ) writes: “mereology may be thought of as an extended logic of identity, extended to deal with cases of partial identity.” [See Section . for my own broad conception of logic.]

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  :   



If composition occurs at all, then composition is unrestricted. For a second example, consider the debate between priority monists and pluralists. As often characterized, this is a debate over whether the parts are prior to the whole or the whole is prior to the parts. But if composition is a kind of (total) identity, and kinds of (total) identity are all symmetric, then the debate so characterized makes no sense; the mereological relations between objects, by themselves, are irrelevant to questions of priority. The only way in which it could make sense to speak of some objects being prior to others is derivatively from the properties and relations that they instantiate, and whether those properties and relations are fundamental. Now, I am not saying the thesis that composition is a kind of identity has these (and other) metaphysical views as consequences; for I have not attempted to give precise formulation to the thesis. Rather, I am saying that there is a substantial unified picture with the idea that composition is a kind of identity at its core, a picture that differs profoundly from what comes from weak composition as identity, and merely taking mereological relations to be relations of intimacy.³⁶ If I were doing philosophy on a desert island, I might be content to end my train of thought here. But I can’t help but want my philosophical peers to see the light as I have seen it, to get their minds right. And yet I suspect that, to this point, I have made few if any converts. Perhaps the doubters will grant that, if composition is a kind of identity, where these kinds of identity are all unified together along with oneone identity to form a natural kind, then the various analogies will have been shown not to be accidental, and to reinforce one another. But why believe the antecedent? What unifies these various kinds of identity into a single natural kind? Failing an answer to that question, I cannot hope to change many minds.³⁷

. Elucidating Generalized Identity: The Humean Strategy Try this. As a steadfast Humean, there is no principle more central to my metaphysical outlook than the prohibition against necessary connections. The prohibition, of course, is not absolute; after all, any thing is necessarily connected with itself. Perhaps our understanding of when necessary connections are and are not prohibited can inform our understanding of generalized identity relations. Thus, consider Hume’s Dictum as traditionally formulated: Hume’s Dictum. There are no necessary connections between distinct existences. ³⁶ Or so it seems. But it is hard to say, in the final accounting, just how much Sider’s () view, which is couched in terms of intimacy, differs from my own view, couched in terms of kinds of identity. Both of us believe that there is a single underlying picture that unifies the various analogies between composition and identity. In describing his picture, he says: “the intuitive picture of the intimacy of parthood demands that we adhere as closely as possible to strong composition as identity . . . ” (p. ). Will that picture lead to metaphysical views at odds with my own picture of the intimacy of parthood as a kind of identity? I’m not sure. ³⁷ Sider (: ) puts the challenge as follows: “Without strong composition as identity, it seems impossible to express what is special about parthood in a single, precise thesis.” I am asking whether that challenge can be met.

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

      ()

The notion of “distinct” that is relevant here, as Humeans know, is not the denial of one-one identity; for example, a whole and one of its parts do not count as distinct. Rather, the relevant notion of “distinct” is linked to the very notion that we are at pains to elucidate, the “broadened sense of identity.” And that suggests the following strategy. Although it is customary for the Humean to take the notion of distinct as given and use Hume’s Dictum to delimit the scope of metaphysical possibility and necessity, perhaps we can turn this around. That is, perhaps we can instead use our pre-theoretic understanding of metaphysical necessity to elucidate the relevant notion of distinct, and the broadened sense of identity. Call this the Humean strategy. Three things about this strategy should be emphasized at the outset. First, the strategy is useless to non-Humeans. If you tell me that it is absolutely necessary that you could not have existed without your parents, you do not thereby have an unusually broad notion of identity according to which you are your parents. Rather, you have exited the conversation. I have nothing more to offer you. Second, if like me you reject primitive modality, then you must see this as a project of elucidation, not analysis. But that has been the project from the start: the notion of generalized identity that is the primitive of CAI is fundamental, and so not analyzable in terms of necessity or anything else. Even fundamental notions need to be understood, and often we best understand them by invoking less fundamental notions. The order of understanding need not match the order of analysis. All that is demanded is that, once the notion of necessary connections is analyzed (say in terms of possibilia), the analyzed notion coheres with the intuitive notion with which we began. Third, I do not assume that the broadened sense of identity that is relevant to Hume’s Dictum is just mereological overlap generalized to apply to plural arguments. I believe that; but to suppose it at the start would risk alienating my already diminished audience. And, indeed, it is not an essential component of the view that composition is a kind of identity. Thus, I allow that Humeans may differ as to what necessary connections they claim to understand. For example, I allow that Humeans may hold that sets stand in necessary connections to their members, and so, in the broadened sense, are identical with their members, even though sets are not mereologically composed of their members. Hume’s Dictum, by itself, does not rule out unmereological modes of composition. The Humean strategy of using the notion of necessary connections to elucidate general relations of composition and identity is suggested indirectly by some remarks of David Lewis in a critique of Armstrong’s states-of-affairs ontology. I start with some stage setting.³⁸ In previous critiques, Lewis had made two complaints. The first complaint was that Armstrong’s states of affairs violate “a twofold principle of the uniqueness of composition: there is only one mode of composition; and it is such that, for given parts, only one whole is composed of them” (Lewis c: ). Armstrong holds instead that states of affairs have some unmereological mode of composition that allows different wholes to be composed of given parts. For example, if R is a relational universal that is not necessarily symmetric, and a and b are particulars such that both a has R to b and b has R to a, then there are two states

³⁸ See Chapter  for a fuller discussion of Lewis’s critique of Armstrong’s states-of-affairs ontology.

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  :   



of affairs composed unmereologically of a, R, and b. Lewis’s second complaint was that Armstrong’s states of affairs violate Hume’s prohibition against necessary connections, which Lewis (a: ) formulated roughly as: “Anything can coexist with anything else, and anything can fail to coexist with anything else.” (Here, Lewis understood ‘anything else’ as ‘anything mereologically distinct’.) For example, the state of affairs of a’s having F cannot exist without the particular a existing, even though the state of affairs and the particular are mereologically distinct. The problem, for Lewis, is that such necessary connections are unintelligible. But then, in a later critique, Lewis (a: ) writes: But now I think that the second complaint subsumes the first. For we can explain how [the states of affairs] are constructed out of their constituents, provided we define the “construction” simply in terms of the necessary connections themselves. Then indeed the requisite mode of “unmereological composition” has been explained—but in a way that does nothing at all toward excusing or explaining the necessary connections.

The idea, as I understand it, can be fleshed out as follows. Whenever xx compose y (or y is “constructed out of constituents” xx), there are necessary connections between xx and y. Since, for the Humean, composition relations are the only source of necessary connections, we can use the necessary connections to define a general notion of composition as follows: Def.

xx compose y iff, necessarily, every one of xx exists iff y exists.

We can then use this general notion of composition to define a general notion of being a component of: Def.

x is a component of y iff, for some xx, x is one of xx and xx compose y.

It then follows immediately that: x is a component of y iff, necessarily, y exists only if x exists.³⁹ And from this it follows that the relation, is a component of, is reflexive and transitive. Thus, the definitions together with the relevant facts of necessity lead to a very minimal general theory of composition.⁴⁰ Note that what might be called “compositional essentialism” is simply built into the theory: if some things compose a whole, then, necessarily, whenever the whole exists, so do all of those things, and whenever all of those things exist, so does the whole. But the theory leaves open whether there are multiple “modes of composition,” some of which fail to satisfy analogues of the mereological axioms Unique and Unrestricted. These, and analogues of the various supplementation axioms, are not justified by Hume’s Dictum alone. Indeed, the theory leaves open whether there is any such thing as mereological composition: if mereological essentialism is rejected,

³⁹ Proof. Left-to-right. Suppose some things, one of which is x, compose y. Then, necessarily, y exists iff each of those things exists. And so, necessarily, if y exists, then x, which is one of those things, exists. Rightto-left. Suppose, necessarily, if y exists, then x exists. We need to find some xx such that x is one of xx and xx compose y. Consider any yy that compose y. (There always is such yy, since everything composes itself.) Then add x to yy to get the desired xx. ⁴⁰ See Fine () for a very different and more developed and discriminating general theory of composition. Fine is not concerned, as I am here, with finding a single thesis to characterize when composition of any mode takes place.

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

      ()

then the fusion operation of mereology won’t count as a mode of composition, and all composition will be unmereological. In any case, our little theory, even in its undeveloped state, provides an explication of Lewis’s idea that, if we could understand the necessary connections between mereologically distinct things, then we could understand unmereological composition. And it shows why Lewis thinks the Humean objection to states of affairs subsumes the mereological objection. Put unmereological composition to one side. Our focus here is to generalize the general notion of composition defined above to total and partial identity relations that take either plural or singular arguments. It appears to be altogether straightforward. Def. X is totally identical with Y iff, necessarily, every one of X exists iff every one of Y exists. We can say that a relation R is a kind of total identity iff whenever X R Y, X is totally identical with Y. (We might want to add that R is a somewhat natural or fundamental relation to rule out gruesome strengthenings of total identity.) We have, then, that : any mode of composition is associated with a kind of total identity. Whether ¼, the basic notion of CAI, is a kind of total identity will depend, once again, on the status of mereological essentialism.⁴¹ If we allow that there are sets and that a set exists if and only if its members exist, then the relation, having the same urelements, will be a kind of total identity among the sets. Note that, in this case, because uniqueness of composition fails, we have things that are totally identical without being one-one identical; for example, {a, {b}} and {{a}, b}. Defining a generalized partial identity relation appears to be equally straightforward. First, we need to generalize the relation ‘is a component of ’ to take pluralities in its second argument place: x is a component of yy iff, for some xx, xx are totally identical with yy, and x is one of xx. (Note that, if there are multiple modes of composition, then one thing may be a component of another, or of others, due to a mixing of modes; for example, if forming wholes from parts and forming sets from members are both modes of composition, and a is a part of b and b is a member of c, then a is a component of c.) Second, we simply take partial identity to be overlap— having a component in common—generalized to apply to plural arguments: Def. X is partially identical with Y iff, for some z, z is a component of both X and Y. We can say that a relation R is a kind of partial identity relation iff whenever X R Y, X is partially identical with Y. (Again, we might want to add that R is a somewhat natural or fundamental relation.) Whether ≑, generalized mereological overlap, is a kind of partial identity will depend, again, on mereological essentialism. The difference between total and partial identity, on this view, corresponds to different senses of ‘necessary connection’. Say that X and Y are strongly necessarily connected iff X is totally identical with Y, weakly necessarily connected iff X is partially identical with Y but not totally identical with Y. ⁴¹ [Indeed, it requires the strong, bi-conditional form of mereological essentialism: not only must the parts exist whenever the whole exists, the whole must exist whenever all of its parts exist. For more on mereological essentialism, see Section ...]

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   



We now see how the generalized identity relations—the kinds of identity—can be characterized in a unified way that does not resort to an appeal to analogies with oneone identity. The analogies, those that still hold in this general setting, flow from the characterization. In particular, tying the generalized identity relations to necessity puts the ontological innocence of composition, supposedly the hallmark of composition as identity, in a new light. It does not rest on an analogy with one-one identity. Rather, it rests on what Armstrong (: –) calls “the doctrine of the ontological free lunch,” the idea that entities whose existence is necessarily entailed are no “addition to being.” But I, for one, think that the dispute over the ontological innocence of mereology has generated more heat than light. (That is why, unlike Lewis, I did not feature it centrally in my presentation of composition as identity.) Quinean methods of scoring ontological commitments, counting by one-one identity, are perfectly legitimate for some purposes. Lewisian methods of scoring, counting by generalized identity, are fit for other uses. One need not choose between them.⁴² But I can say this. If I accepted unmereological modes of composition, I would take them to be on a par with mereological composition with respect to ontological innocence. For example, if I believed in sets and held that sets were composed of their members, I would think that set formation and mereological composition were on a par with respect to innocence; a set and its members, no less than a whole and its parts, would comprise the same portion of reality. But I don’t believe in sets, or other entities unmereologically constructed. Although the necessary connections involved in such belief are compatible with the above account of kinds of identity, and with Hume’s Dictum broadly construed, they are not compatible with my strict standards of intelligibility. They violate my Humean scruples.⁴³ It appears, then, that I can only hope to communicate my understanding of kinds of identity if I can get you to accept Hume’s Dictum as I understand it, with my own understanding of necessity. And that thought leads to a more pressing issue. Is the above characterization of generalized identity in terms of necessary connections even compatible with my own modal metaphysics, according to which necessity is analyzed in terms of possible worlds? When recast in that framework, can it even meet the minimal condition of extensional adequacy without going around in a circle?

. Against the Humean Strategy I am a realist about possible worlds. Like Lewis, I hold that possible worlds are concrete and do not overlap. And, like Lewis, I analyze modality in terms of possible worlds and their parts.⁴⁴ How, in this framework, should the Humean prohibition ⁴² But which method of scoring, one might ask, is the right way to calculate the ontological costs of a theory? Don’t we need an answer if we are to give prescriptions for theory choice in metaphysics? I think not. Ontological economy, however characterized, should play no role in theory choice. See Chapter . ⁴³ [For more on this, see Section . and the discussion of the principle (G) in the postscript to Chapter .] ⁴⁴ My view differs from Lewis’s in various respects, most notably, in my acceptance of absolute actuality; but the differences will not matter for what follows. For Lewis’s account of possible worlds, see Lewis (a). For a summary of ways that my own view differs from Lewis’s, see my Chapter .

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

      ()

against necessary connections be understood? (I will focus on the strong necessary connections associated with total identity, since the weak necessary connections associated with partial identity can be defined in terms of the strong.) I do not say: entities are (strongly) necessarily connected just when any world that contains one contains the other. That would make it trivially true that any two actual entities are necessarily connected (since worlds do not overlap). I do not say: entities are (strongly) necessarily connected just when any world that contains a counterpart of one contains a counterpart of the other. Counterpart relations were introduced to capture ordinary claims of necessity de re, and I do not want to deny that, in ordinary contexts, we may truly claim that mereologically distinct entities are necessarily connected—for example, that you cannot exist without your parents. The notion of necessity de re that is relevant to Hume’s Dictum requires a special interpretation. Like Lewis, I understand the prohibition against necessary connections in terms of duplicates. I say, roughly: entities are (strongly) necessarily connected just when any world that contains a duplicate of one contains a duplicate of the other.⁴⁵ Unfortunately, attempting to make the analysis less rough uncovers an insurmountable obstacle with the Humean strategy.⁴⁶ When generalized to apply to plural as well as singular arguments, the most straightforward rendition is this: NC. X is strongly necessarily connected with Y iff, for every world w, duplicates of every one of X exist at w iff duplicates of every one of Y exist at w. There is a problem with NC, however, that does not arise from the generalizing; it occurs already for singular arguments. Thus suppose that a and b are duplicates but are not totally identical. Since a and b are not totally identical—not the same portion of reality—they should not be strongly necessarily connected according to the Humean. But NC gets this wrong: any world at which a duplicate of a exists is a world at which a duplicate of b exists because any duplicate of a is also a duplicate of b. The following fix immediately suggests itself: NC. X is strongly necessarily connected with Y iff, for every world w, duplicates of every one of X exist at w iff duplicates of every one of Y exist at w, and if both X and Y have duplicates existing at w, then some duplicates of X are distinct from some duplicates of Y. Out of frying pan and into the fire! Even ignoring the problem (for a moment) what ‘distinct’ can mean here without going in a circle, the analysis now gives the wrong result when a is (totally) identical with b. If there is a world with just one duplicate of a (or b), then it turns out according to NC that a (or b) is not strongly necessarily connected with itself. The problem is quite general: no analysis that refers only to duplicates of X and duplicates of Y on the right-hand side of the analysis can

⁴⁵ For Lewis’s interpretation of Hume’s Dictum as a principle of recombination in terms of duplicates, see Lewis (a: –). For a fuller treatment of principles of recombination, and plenitude more generally, see Chapter . Entities are duplicates, roughly, iff they have all of their (qualitative) intrinsic properties in common. See Lewis (a: –). ⁴⁶ [This paragraph and the next have been slightly altered to correct mistaken formulations in the published version.]

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 



successfully distinguish the cases where X are totally identical with Y, and where X are duplicates of Y but not totally identical with Y. When necessary connections are analyzed entirely in terms of duplicates, Hume’s Dictum becomes in effect a prohibition against necessary connections between the intrinsic natures of things, not between the things themselves. The phrase ‘distinct existences’ is interpreted to mean ‘things with distinct intrinsic natures’, not ‘things that are distinct’ in the sense of ‘distinct’ relevant to generalized identity. And on that interpretation, there is no longer any reason to move from NC to NC. But that is not an interpretation of Hume’s Dictum that allows it to be reverse engineered to provide an understanding of generalized identity. As best I can see, the only way to capture an interpretation of Hume’s Dictum that makes identicals, but not distinct duplicates, necessarily connected is to treat the case where X and Y are totally identical as a special case, resulting in a disjunctive analysis: NC. X is strongly necessarily connected with Y iff, either X is totally identical with Y, or, for every world w, duplicates of every one of X exist at w iff duplicates of every one of Y exist at w, and if both X and Y have duplicates existing at w, then some duplicates of X are not totally identical with some duplicates of Y. (Note that I have also replaced ‘distinct’ in NC with ‘not totally identical’ so as to capture the minimal condition needed for there to be necessary connections between duplicate pluralities.) NC, indeed, is the analysis of necessary connections that I endorse.⁴⁷ But on that analysis it is plain that the notion of necessary connection presupposes the very notion we were hoping it could elucidate, indeed, presupposes it twice-over, in both disjuncts. Moreover, the necessity of identity relations, and mereological essentialism, are just built into the analysis from the start; they do not amount to substantial claims about the space of possible worlds.⁴⁸ I conclude, then, that there is no way to use Hume’s Dictum to further an understanding of kinds of identity, or their kin, within my own modal framework.

. Concluding Remarks Where does that leave us? Return to Lewis’s claim that, if we could understand the necessary connections between mereologically distinct things, then we could ⁴⁷ It generalizes to plural arguments the notion of necessary connection that was incorporated in the principle (B) that I endorsed in Chapter . ⁴⁸ How can I say that I accept the necessity of identity while at the same time invoking counterpart theory to reject coincident entities? I say: “Goliath is identical with Lumpl, but not necessarily identical with Lumpl.” Isn’t that a rejection of the necessity of identity? No, not in any way that is relevant to fundamental metaphysics. It is speaking with the vulgar. It is providing a semantics for statements of necessity that captures ordinary, superficial ways of speaking. Counterpart theorists speak out of both sides of their mouth, but without losing track of which side is speaking metaphysics. The content of the claim that everything is necessarily identical with itself is given neither by counterparts nor duplicates; it amounts to no more than the claim that everything is identical with itself. The same goes for mereological essentialism, and the necessity of total and partial identity relations. I invoke counterpart theory to speak with the vulgar, to be able to say, for example, that you might have existed without your hands, or your heart. But when I say, doing metaphysics, that, necessarily, you exist if and only if your parts exist, what I say cannot be understood in terms of counterparts or duplicates; it merely reiterates that you are your parts.

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

      ()

understand unmereological composition. Perhaps. But it doesn’t follow that an understanding of mereological composition derives from an understanding of the necessary connections between a thing and its parts. On my modal metaphysics, such necessary connections are simply stipulated. For me, then, mereological composition and the associated kinds of identity are basic—not just in the order of analysis, but in the order of understanding. My understanding of Hume’s Dictum depends on a prior understanding of generalized identity to delimit the scope of the Humean prohibition. But I need not lament the failure of the Humean strategy. : I already understand the theory CAI with the relation ¼ interpreted as a kind of identity. But enough about me. This was supposed to be about you. As I said at the start: I want to find a way to get you to latch on to the deflationary notion of composition that I so clearly grasp. And if you do not share my realist metaphysics of possible worlds, you may still find merit in the Humean strategy of understanding composition in terms of necessary connections. Even so, however, I fear it may not help much with understanding what I mean by kinds of identity. The Humean strategy is a victim of its own generality: because it is compatible with there being multiple modes of composition, it can provide at most a partial understanding of the mereological mode of composition that is my target. If you failed to understand why Unrestricted Composition or Uniqueness of Composition should hold, even after enduring my presentation of CAI and moderate composition as identity, embracing the Humean strategy will make you none the wiser. This final attempt at fostering mutual understanding will end in failure.

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 Composition as Identity, Leibniz’s Law, and Slice-Sensitive Emergent Properties () . Introduction Composition as identity, I am convinced, expresses a deep truth about the structure of reality. But even among its adherents, there is little agreement as to just what that truth is. In Chapter , I developed and endorsed a “moderate” version of composition as identity according to which there is a generalized identity relation, being the same portion of reality as, of which composition and numerical identity are distinct species. Composition is a genuine kind of identity, I claimed, but it is not numerical identity; for unlike numerical identity, it fails to satisfy Leibniz’s Law. Prima facie counterexamples to composition satisfying Leibniz’s Law abound. For example, a composite whole and its parts differ with respect to their numerical properties: the whole is numerically one; the parts (collectively) are numerically many. That a genuine identity relation can fail to satisfy Leibniz’s Law is controversial, to be sure. Many philosophers and logicians take Leibniz’s Law to be constitutive of identity. There is a risk, then, that disagreements over composition as identity will devolve into terminological disputes. The moderate theorist, therefore, has the burden of saying why composition should be classified with numerical identity as a genuine identity relation, and not with relations that are “identity in a loose and popular sense,” such as so-called “qualitative identity.” Whether and how that can be done will be one of the topics of this chapter. Strong versions of composition as identity, in contrast, have no truck with kinds of identity that fail to satisfy Leibniz’s Law. The strong theorist holds: the whole is identical with the parts that compose it; therefore, the properties of the whole do not differ from the properties of the parts taken together. The strong theorist has the burden of somehow showing that the offending numerical properties do not provide counterexamples to Leibniz’s Law, perhaps because they are relations in disguise. In Section ., I argued that various strategies for defending the strong theory are incompatible with taking the framework of plural logic to be fundamental.¹ First published in Synthese (), –. It was presented at a conference on Mereology and Identity in Pisa, Italy, in July, . I thank the conference participants for their helpful comments. ¹ A similar argument was given in Sider (). But Sider (: –) now rejects the argument because he no longer takes the framework of plural logic to be fundamental. I claim below that there is a Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

  , ’ ,  -

As a second topic of this chapter, I consider whether there is a version of strong composition as identity that evades my earlier argument, a version that formulates composition as identity within the framework of plural logic, but claims nonetheless that a composite whole and its parts taken together have all of their genuine properties in common. I believe that there is such a view, and that it is the most formidable version of strong composition as identity. It holds that, at the fundamental level, portions of reality have mereological, but not plural, structure. Although there is a fact of the matter as to whether a portion of reality is composite or atomic, there is no fact of the matter as to whether it is plural or singular. This will need some explaining. It turns out that these two topics merge into one. The propped-up moderate theory is just the new version of the strong theory in disguise. But however one characterizes the view, I am inclined to reject it. The main issue, we shall see, has to do with whether slice-sensitive emergent properties are possible. I will argue that they are, making use both of specific examples and general principles of modal plenitude. I do not claim that my argument against this version of composition as identity is irresistible. But it cannot be evaded as easily as a related argument against strong composition as identity given by Kris McDaniel (). I take a detailed look at McDaniel’s argument to pave the way for my own argument.

. Leibniz’s Law Leibniz’s Law is a principle of the logic of identity. As traditionally understood, it applies to individuals: x is identical with y if and only if x and y have all of their properties in common. Here ‘property’ must be understood in an abundant sense: for any things, there is at least one property had by all and only those things.² Call this identity relation between individuals singular identity. Leibniz’s Law can be extended naturally to plural identity, where a plurality xx is identical with a plurality yy if and only if every one of xx is one of yy and every one of yy is one of xx.³ Here ‘xx’ and ‘yy’ are plural variables that range indifferently over individuals and pluralities; but to save words, I will count an individual as a plurality of one so that I can just say that plural variables range over pluralities. Properties of pluralities—plural properties—apply collectively, not distributively. And again,

crucial ambiguity in what it means to take the framework of plural logic to be “fundamental.” In any case, however, I do not see how composition as identity can even be formulated properly without the use of plural logic. Using schematic letters in place of plural variables weakens the theory, and introduces an irrelevant dependence on language. Using quantification over sets (or classes) in place of plural quantification, as was done in early presentations of classical mereology (Tarski ), also weakens the theory (since not all pluralities form sets), and introduces an irrelevant dependence on sets, something I find especially problematic. For I prefer the reverse reduction of sets to plural logic over the reduction of plural logic to set theory. See Chapter . ² On abundant vs. sparse conceptions of properties, see Lewis (a: –). ³ Pronounce ‘xx’ as “the x’s” if you like. Although ‘plurality’ is grammatically singular, it is important not to think that, on my usage, a plurality is a single set-like object.

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’ 



‘property’ must be understood in an abundant sense: for any pluralities of things, there is at least one property had by all and only those pluralities.⁴ I can now introduce an important distinction for what follows among plural properties: a plural property is slice-sensitive iff it holds of xx but not of yy, for some xx and yy that have the same fusion.⁵ Let us say that singular and plural identity are forms of numerical identity. Because plural variables range over both individuals and pluralities of individuals, plural identity subsumes singular identity, and Leibniz’s Law for plural identity entails Leibniz’s Law for singular identity. We could, then, just take Leibniz’s Law for plural identity as our basic principle of the logic of identity. We could even eliminate singular variables and quantifiers altogether. But it will be more convenient to stick with the standard formulation of plural logic that contains both singular and plural variables, and to formulate Leibniz’s Law in a way that allows it to apply to a wider range of purported identity relations. We can do this most easily by using schematic variables x and y that can be replaced by either singular or plural variables, and letting ‘is identical with’ take either singular or plural arguments. We can then express Leibniz’s Law as a schema: Leibniz’s Law. For any x and any y, x is identical with y if and only if x and y have all of their properties in common. Note that Leibniz’s Law, so formulated, includes four instances, two of which mix singular and plural variables. That requires that it be meaningful to ask whether one thing is identical with two or more things. But accepting the Leibniz’s Law schema does not prejudge whether one thing is ever identical with two. We can now make good sense of the claim that composition fails to satisfy Leibniz’s Law. When we put composition in place of identity, one of the instances is this: for any xx and any y, xx compose y if and only if xx and y have all of their properties in common. Composition fails to satisfy Leibniz’s Law if this instance is false. And prima facie it is false. Consider, for example, a deck of cards. The cards compose the deck, but although the cards are  in number, the deck is not  in number. But perhaps this “first appearance” is deceiving. In this chapter I will be concerned just with the left-to-right direction of Leibniz’s Law, the principle of the indiscernibility of identicals. (Indeed, some philosophers take Leibniz’s Law to be just the left-to-right direction.) The indiscernibility of identicals needs to be distinguished from the linguistic principle, the substitutivity of identicals, which can fail with respect to a language whose predicates do not express genuine properties, or express different properties in different linguistic contexts. For example, the substitutivity of identicals fails with respect to English. From “Shorty is so-called because of his size” and “Shorty is identical with Frankie,” one cannot conclude “Frankie is so-called because of his size.” But that failure does ⁴ Those who do not accept plurally plural quantification will have to approximate this using quantification over classes (taking properties of pluralities to apply to the corresponding classes): for any class of classes, there is a least one property had by all and only those classes. ⁵ I use ‘slice’ broadly: whenever xx composes y, xx give a way to “slice” y, whether or not xx overlap. I use ‘fusion’ and ‘compose’ to denote the same relation: y is the fusion of xx iff xx compose y.

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

  , ’ ,  -

not challenge the indiscernibility of identicals: the predicate ‘is so-called because of his size’ does not express a genuine property, but rather a relation. (Or, if ‘express’ is understood so that it does express a property, it expresses different properties in different linguistic contexts.) What the genuine properties are, and what exemplifies those properties, is a matter of the structure of reality; it is a matter of metaphysics, not language. Whether an application of Leibniz’s Law is legitimate, then, is also a matter of metaphysics, not language. Return to the example of the deck of cards, and the swift argument from an instance of Leibniz’s Law to the conclusion that the cards are not identical with the deck. That is only a legitimate application of Leibniz’s Law if ‘are  in number’ expresses a genuine property. And to determine whether or not that is so, we must look to the fundamental structure of reality, not the grammatical structure of language.

. Composition as Identity According to the doctrine of composition as identity, in some sense, a whole is identical with its parts taken together. But in what sense? Adding that the whole “is nothing over and above” its parts is suggestive, but not much more. Saying that a whole and its parts are “the same portion of reality,” ⁶ as I do, just raises the question whether being the same portion of reality entails identity. For in English we can say that things are the same F—the same color, the same age—without it following that those things are identical. The problem of how to characterize composition as identity is especially acute given that philosophers who espouse composition as identity often appear to have very different doctrines in mind. We can get a start on distinguishing different versions of composition as identity by asking two questions: first, is the “identity” in question a genuine identity relation? And, second, if it is a genuine identity relation, does it satisfy Leibniz’s Law? We can then formulate strong and weak versions as follows, roughly in accord with how these versions are characterized in the literature. Strong composition as identity holds that there is only one identity relation, that it satisfies Leibniz’s Law, and that whenever some things xx compose a thing y, xx are identical with y. Weak composition as identity also holds that there is only one identity relation and that it satisfies Leibniz’s Law, but it denies that composition literally is or entails identity. Rather, composition is merely analogous to identity in striking and important ways. The version of composition as identity I call “moderate” is not much discussed in the literature.⁷ As I use the term, moderate composition as identity holds that there are multiple kinds of identity which can be reduced to two basic kinds: numerical identity, which satisfies Leibniz’s Law, and generalized identity—same portion of reality—which does not. Whenever xx compose y, xx are identical with y according to this latter ⁶ The expression ‘portion of reality’, as I use it, is a term of art that, although syntactically singular, is semantically neutral with respect to the plural/singular distinction. Talk of “portions of reality” could be regimented within plural logic using the schematic variables introduced above. ⁷ But cf. Cotnoir (), who can be construed as introducing a generalized identity relation distinct from numerical identity. It is unclear, however, whether he insists that any identity relation satisfy Leibniz’s Law.

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      



kind of identity: xx are the same portion of reality as y. Composition, then, is literally a kind of identity.⁸ I call the view moderate because, without Leibniz’s Law, it lacks many of the consequences of the strong version some of which, within a framework of plural logic, are clearly false if not outright contradictory. For example, within the framework of plural logic one can derive from composition as identity and substitutivity of identicals the principle known as Collapse: Collapse.

x is one of yy if and only if x is part of the fusion of yy.⁹

But, surely, the right-to-left direction of Collapse is false: no molecule that is part of the fusion of the cards—that is, is part of the deck—is one of the cards. This gives powerful reason to prefer moderate over strong composition as identity. In other ways however, moderate composition as identity is not moderate at all. For one thing, it allows that a genuine identity relation can fail to satisfy Leibniz’s Law. In this way, it shares at least some of the “strangeness” of Don Baxter’s version of composition as identity.¹⁰ For another thing, it allows that xx be the same portion of reality as yy even though xx are not the same plurality as yy. In this way, it shares at least some of the “strangeness” of Peter Geach’s relative identity according to which x can be the same F as y, but not the same G, even though both x and y are Gs.¹¹ But on closer inspection, the association between Baxter’s or Geach’s views on identity and moderate composition as identity turns out to be superficial. The moderate theory must be evaluated on its own terms.

. A Defense of Moderate Composition as Identity Moderate composition as identity faces the following challenge: how in the absence of Leibniz’s Law can one characterize what counts as a genuine kind of identity? Unless this challenge can be met, one might wonder whether moderate composition as identity just collapses into weak composition as identity. In Chapter , I argued that moderate composition as identity is a distinct view, with the power to shape and inform our metaphysical conception of reality; but I despaired of providing an elucidation of the generalized notion of identity that would be of any assistance to someone who claimed not to understand it. In that chapter, I bypassed an obvious strategy: perhaps, although generalized identity fails to satisfy Leibniz’s Law, it satisfies a restricted version of Leibniz’s Law; and it is in virtue of satisfying the ⁸ I prefer to understand strong and moderate composition as identity broadly so that they include much more than what is stated here: they encompass a full theory of the composite nature of reality, including classical mereology. But the additional content will not be relevant here. See Chapter  for the details. ⁹ For arguments that strong composition as identity leads to Collapse, see Yi () and Sider (). For further unacceptable consequences of Collapse, see Sider (). Yi () considers a related kind of collapse that results when composition as identity is combined with plural logic: generalized identity collapses into plural identity (if both satisfy substitutivity). It then follows that if xx are the same portion of reality as y, then xx are one. And that trivializes strong composition as identity. ¹⁰ For Baxter’s groundbreaking approach to composition as identity, see Baxter (a) and (b). ¹¹ For Geach’s defense of relative identity, see Geach (). For discussion and critique, see Perry ().

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

  , ’ ,  -

restricted version that it is appropriately called a genuine identity relation. But what is the right restriction? It won’t do just to say that a genuine identity relation satisfies Leibniz’s Law restricted to properties that are not slice-sensitive; for plural properties are called “slice-sensitive” in virtue of failing to satisfy Leibniz’s Law, and we would be going in a circle. But perhaps there is a more informative way to capture the restriction, and thus to characterize what relations are genuine kinds of identity. To see what I have in mind, consider how David Lewis describes the failure of Leibniz’s Law. He writes: Even though the many and the one are the same portion of Reality, and the character of that portion is given once and for all whether we take it as many or take it as one, still we do not really have a generalized principle of the indiscernibility of identicals. It does matter how you slice it—not to the character of what’s described, of course, but to the form of the description. (Lewis : , my italics)

The italicized sentence suggests two things. First, it suggests that the sought-after restricted version of Leibniz’s Law quantifies only over properties that ascribe “character.” I take it that the character of a portion of reality supervenes on the fundamental properties and relations had by that portion of reality and its parts, taken singly or plurally. In particular, then, we have the following consequence. Suppose that xx and yy are the same portion of reality, but that they slice that portion of reality differently. Then: no fundamental property applies to xx but not to yy. For if it did, then the character of a portion of reality would depend on how you slice it. The second thing suggested by the passage from Lewis is this: failures of Leibniz’s Law are due to differences in our ways of describing or representing reality, not to differences in reality itself. In Chapter , I argued against strong composition as identity on the grounds that the framework of plural logic is fundamental, and so failures of Leibniz’s Law that result from logical properties definable from the fundamental relation of plural logic, ‘is one of ’, show that Leibniz’s Law fails at the fundamental level. Numerical properties taking plural arguments illustrate this failure. But I now think there are two different notions of “fundamental” at play that my argument conflates. We need to distinguish between what is fundamental at the level of our representations of reality from what is fundamental to reality itself, at the level of being.¹² Properties and relations that are fundamental at the level of our representations are ideologically basic, but need not be ontologically basic. This distinction will have broad application, but it applies first and foremost to logical notions. For example, our representations of reality involve content with Boolean and quantificational structure; and that structure is fundamental at the level of our representations. But, as I see it, there is no Boolean or quantificational structure at the level of being. At the level of being, there is only a pattern of instantiation of fundamental properties and relations. Boolean and quantificational propositions are made true by that pattern. To think that one must posit fundamental Boolean

¹² Compare Ismael () where this distinction is made in connection with Humean accounts of laws and chance.

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      



or quantificational structure in reality itself is to hold to a naive picture account of representation.¹³ The passage from Lewis suggests that something similar should be said about the distinction between plural and singular that is fundamental to plural logic. Even if we grant that the framework of plural logic is fundamental at the level of our representations, we need not grant that there is plural structure at the level of being. Propositions that we formulate within a framework of plural logic are made true, somehow, by what is fundamental at the level of being; but the plural structure inherent in those propositions is not mirrored by any corresponding plural structure in reality. Portions of reality, the fundamental existents, are either composite or atomic, but they are not in themselves either singular or plural. Let me illustrate the proposed view a bit further. Among the apparatus that we use to represent reality are structured propositions. Say that two structured propositions have the same content just in case they are true of the same portions of reality. (This notion of “same content” presupposes a plenitudinous reality such as a modal realist accepts; propositions that have the same content are “necessarily equivalent” according to some metaphysical modality.) Since the framework of plural logic is fundamental to our representations of reality, structured propositions can differ in that one involves plural predication, the other singular predication. Consider some (composite) portion of reality; and let F be the full intrinsic character of that portion. Then, on the view being considered, the structured proposition that some x exemplifies F and the structured proposition that some yy exemplify F are distinct structured propositions that have the same content. The propositions, we might say, differ at the level of representation but not at the level of being. Now, suppose the portion of reality in question is composed of seventeen atoms. Let ‘aa’ denote the plurality of atoms: each one of the atoms is one of aa. Then it is true to say: aa is seventeen in number. Let the fusion of aa be b. Then it is true to say: b is one in number. But there is simply no fact of the matter whether the portion of reality is seventeen in number or one in number. These numerical properties do not supervene on the intrinsic character of that portion of reality. If we accept this view that plural logic is not fundamental at the level of being, then we have a neat way of characterizing a moderate version of composition of identity that is clearly distinct from weak composition as identity. For we now have a way of saying when a relation is a genuine kind of identity and not just a relation analogous to identity. We have the following analysis: Identity. A relation taking plural or singular arguments is a kind of identity just in case it satisfies Leibniz’s Law restricted to properties that supervene on properties and relations that are fundamental at the level of being. We can still say that the composition relation fails to satisfy unrestricted Leibniz’s Law: the numerical properties are still around to serve as counterexamples. But because the numerical properties do not supervene on properties and relations that are fundamental at the level of being, they are no threat to composition being a kind

¹³ See Chapter  for how this plays out with respect to general propositions.

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

  , ’ ,  -

of identity. If all slice-sensitive properties similarly fail to supervene and so do not contribute to character, then composition is a kind of identity and moderate composition as identity is vindicated.

. From Moderate Composition as Identity to Strong Composition as Identity With a slight change in perspective, however, the moderate theory just presented looks like a version of strong composition as identity in disguise. Let me explain. We are now supposing there is a clear distinction between those features that are fundamental at the level of being, the features that give rise to character, and those features that arise from our representations of reality, including features that depend on our slicing reality in different ways. To get at fundamental ontology, the existents that are fundamental to reality, we need to “factor out” the slice-sensitive features and posit that the fundamental existents have only the features that give rise to character. Considering only these fundamental existents and their properties and relations, it seems that unrestricted Leibniz’s Law holds after all, at the level of being. For at the level of being the properties that are putative counterexamples to Leibniz’s Law simply do not exist, or at any rate, can be deemed not genuine in the relevant sense. So, on the view being considered, strong composition as identity holds with respect to fundamental reality. Although at the level of representation, there are multiple kinds of identity, at the level of being there is just the one relation of identity that every portion of reality bears to itself. What I called numerical identity turns out to itself belong to our representational apparatus, and so can be deemed non-genuine. This change in perspective turns the usual way of thinking about identity on its head: generalized identity, not numerical identity, is the genuine identity relation. I asked in the introduction: is there a version of strong composition as identity that is compatible with taking plural logic to be fundamental? We now see that that question is ambiguous, and that it is enough if the version of the strong theory is compatible with taking plural logic to be fundamental at the level of representation, fundamental to our theorizing about reality. That is all that is needed for strong composition as identity to be properly formulated. It may at first seem odd that strong composition as identity cannot be stated purely in terms that are fundamental to reality, without the aid of plural logic. Composition as identity, then, is a truth that can only be expressed using a representational apparatus that does not accurately mirror reality at the fundamental level. But if I am right that Boolean and quantificational structure are no more mirrored by reality than is plural structure, then all of our theorizing about reality will be in the same boat. If we had to write our theories in terms that are fundamental at the level of being, they would be reduced to infinite lists of particular facts. That’s no way to gain insight into the nature of reality. Call this moderate-cum-strong version of composition as identity Mods. For those sympathetic to the intuitions behind composition as identity, there is a lot to like about Mods. It seems to be a reasonable interpretation of Lewis’s remarks, even though Lewis is often taken, wrongly I think, to accept only weak composition

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      



as identity.¹⁴ Indeed, it is more or less the view that I myself held, somewhat confusedly, from the time I first read Lewis’s Parts of Classes up until about ten years ago. Moreover, it seems to be a natural development of the view of strong composition as identity theorists, such as Einar Bohn, who want to attribute all violations of Leibniz’s Law to our ways of representing reality, and not to reality itself. But on the proposed version, unlike Bohn (), there is no attempt to avoid violations of Leibniz’s Law by introducing an explicit relativization to concepts. That cannot succeed. For one thing, it inevitably mangles plural logic because the fundamental notion of plural logic, is-one-of, is itself slice-sensitive, and would have to be relativized to concepts. For another thing, it raises uncomfortable questions as to what these concepts are, and how they fit into reality. Rather than struggling to avoid violations of Leibniz’s Law, Mods accepts them, but relegates them to the realm of representation. Most importantly, Mods is not committed to Collapse or any of its nasty consequences because the principle of substitutivity that leads to Collapse does not hold in the plural framework within which the theory is expressed. Granted, in some sense there is a collapse at the level of being. Singular identity, plural identity, and generalized identity all collapse into one. But there is no collapse at the level of representation. So there is no need to endorse claims formulated in plural logic that are obviously false. Ordinary truths couched within plural logic come out as literally true on the proposed version of strong composition as identity. It’s just that differences in these truths sometimes reflect, not differences in fundamental reality, but differences in the representational apparatus we use to describe that reality. In spite of its many attractions, I reject Mods, whether it is taken to be a version of moderate or a version of strong composition as identity. It does not jibe with my conception of reality. The deflationary notion of composition that I accept is essentially intertwined with the notion of plurality. To hold that reality is composite at the fundamental level in the way that I do is just to hold that reality has plural structure through and through. That is not an argument, of course, and need have no force against a proponent of Mods who rejects my deflationary notion of composition. Can I do better? To see how one might formulate an argument against Mods, note that Mods has implications for emergent properties. Say that a property is emergent if and only if it is fundamental (at the level of being) and whether or not it holds of a composite portion of reality does not supervene on the fundamental properties and relations of that portion’s proper parts, taken singly or plurally.¹⁵ According to Mods, portions of reality are not plural or singular in themselves. Thus, according to Mods, emergent properties of portions of reality are not plural or singular in themselves, but only in how we designate them. An emergent property applies indifferently to a plurality or to the fusion of that plurality. But, then, no emergent property applies to only one of two different pluralities with the same fusion; there are no slice-sensitive emergent properties. This suggests that the best—and perhaps the only—way to ¹⁴ See Chapter  for more discussion of Lewis’s view. See also Bohn (), who similarly thinks Lewis has been wrongly understood. ¹⁵ Note that this is strong emergence, requiring the failure of logical supervenience. On strong vs. weak emergence, see Chalmers ().

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

  , ’ ,  -

argue against Mods is to argue that slice-sensitive emergent properties are possible. And that is something, I hope, that even those attracted to Mods will feel pressure to accept on independent grounds.

. McDaniel’s Argument against Strong Composition as Identity Before turning to my argument against Mods, I want to consider a related argument against strong composition as identity in McDaniel (). Like me, McDaniel thinks that the possibility of emergent properties can make trouble for strong composition as identity. But McDaniel does not consider—or even allow for—emergent properties that take plural arguments. And for that reason, his argument cannot succeed.¹⁶ McDaniel begins with some standard terminology. Say that w and z are duplicates iff there is a one-one correspondence between the parts of w and the parts of z that preserves all perfectly natural properties and relations.¹⁷ (One must add that it preserves the part-whole relation, if that relation is not perfectly natural.) This notion of duplicate can be extended to apply to pluralities in a natural way: xx and yy are plural duplicates iff there is a one-one correspondence between xx and yy that preserves all perfectly natural properties and relations, and such that corresponding elements are duplicates of one another. (If the xx and yy are atomic, the second clause can be dropped.) Now, before going any further I want to flag an important assumption that McDaniel appears to be implicitly making, an assumption that will be crucial in applying the above definitions. It is this: perfectly natural properties are exemplified by individual objects, not by pluralities of objects; perfectly natural relations hold between individual objects, not between pluralities of objects. That, of course, is an assumption I will be at pains to argue against below. The assumption plays a role in judgments as to whether xx and yy are plural duplicates. Suppose, contrary to the assumption, that there is a perfectly natural property F that is exemplified (collectively) by xx but is not exemplified by yy. Then no one-one correspondence between xx and yy preserves all perfectly natural properties: xx and yy are not plural duplicates. And that is as it should be for the strong theorist, since the portion of reality referred to by xx does not have the same qualitative character as the portion of reality referred to by yy. After introducing these definitions, McDaniel claims, as the first premise in his argument against strong composition as identity, that the strong theorist will be committed to what he calls the Plural Duplication Principle:

¹⁶ Sider () has argued, on different grounds, that McDaniel’s argument does not succeed against versions of strong composition as identity that accept Collapse. But Sider’s argument does not apply to a version such as Mods, which rejects Collapse. ¹⁷ Following McDaniel and Lewisian tradition, I use ‘perfectly natural’ in phrasing these definitions, instead of ‘fundamental’, But note that only properties and relations fundamental at the level of being should count as perfectly natural, and factor into the definition of duplicate.

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’      



(PDP) If the xx compose w, the yy compose z, and the xx are plural duplicates of the yy, then w is a duplicate of z.¹⁸ McDaniel’s argument that the strong theorist is committed to (PDP) invokes Leibniz’s Law, and is roughly this. Since, for the strong theorist, xx is identical with w and xx and yy are plural duplicates, w is a duplicate of any individual identical with yy, and so w and z are duplicates. It is important to note two things about this argument. First, the argument is only a legitimate application of Leibniz’s Law if the relation of duplication between individuals and the relation of duplication between pluralities is one and the same relation. But that is something the strong theorist is committed to. For the qualitative character of a portion of reality does not depend on whether it is represented as a single object, or as a plurality of objects; and so the relation, having the same qualitative character, should apply either way. The argument invoking Leibniz’s Law switches the form of the duplication relation, but it does not illegitimately switch the relation being said to hold. From now on, I will simply speak of the duplication relation, allowing it to take either singular or plural arguments.¹⁹ Second, the argument assumes that the duplication relation in question is not slice-sensitive. For if it were slice-sensitive, the application of Leibniz’s Law would not be legitimate. This second assumption is also not problematic for the strong theory under discussion; it also follows from the idea that the qualitative character of a portion of reality is the same, no matter how you slice it. Now, granting these two assumptions, the argument that the strong theorist is committed to (PDP) is unassailable. Indeed, (PDP) just expresses the strong theorist’s view that once we give the qualitative character of the xx, we have thereby given the qualitative character of w, the fusion of the xx. In Lewis’s words (quoted by McDaniel): “Describe Magpie and Possum fully—the character of each and also their interrelation—and thereby you fully describe their fusion” (Lewis : ). The second premise of McDaniel’s argument against strong composition as identity is the claim that (PDP) is incompatible with the possibility of emergent properties. His definition of emergent property is a bit different than the one I gave above, but not in any way that will matter. It is as follows (omitting the qualifier ‘strongly’ before ‘emergent’): A property F is emergent iff (i) F is perfectly natural, (ii) F can be exemplified by a composite material object, and (iii) F does not locally supervene on the perfectly natural properties and relations exemplified by only atomic material objects. (McDaniel : )

The argument for the second premise is this. (I quantify over possible objects, but the argument could instead be given using modal operators.) Suppose that emergent properties are possible. Let F be an emergent property that holds of some composite object w, and let xx be the atomic parts of w.²⁰ Then, by clause (iii), there will be some other object z that does not exemplify F whose atomic parts, yy, are plural duplicates ¹⁸ This formulation is from Sider (). It is more intuitive than McDaniel’s formulation, and does not differ in any way relevant to the argument. ¹⁹ McDaniel does not explicitly make this “one-relation” assumption. He is content to leave the argument at an informal level, one that does not invoke any specific formulation of Leibniz’s Law. ²⁰ As McDaniel notes, the mereological atomism presupposed by this argument is not essential. I am happy to go along with it.

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

  , ’ ,  -

of xx. It follows that w and z are not duplicates, contradicting (PDP). I will return to this argument shortly. The third and final premise is that emergent properties are indeed possible. He mentions two sorts of example: quantum properties of entangled systems and phenomenal properties of conscious subjects. McDaniel’s third premise is weaker than what I will be arguing for below, namely, that slice-sensitive emergent properties are possible. In any case, I do not contest the third premise. Putting this all together, we have the following argument against strong composition as identity. () () () ()

Strong composition as identity entails (PDP). (PDP) is incompatible with the possibility of emergent properties. Emergent properties are possible. Therefore, strong composition as identity is false.²¹

I do not think that this argument succeeds. The strong theorist can and should reject the second premise. McDaniel’s argument for that premise depended on his implicit assumption that perfectly natural properties and relations never take plural arguments. That assumption is needed to go from the failure of local supervenience to there being duplicate pluralities whose fusions are not duplicates. To see this, consider again the case of an emergent F exemplified by w with atomic parts xx. By clause (iii) of the definition of emergent property, there will be yy that are “just like” xx with respect to the perfectly natural properties and relations that are exemplified by atoms, but such that the fusion of yy, z, doesn’t exemplify F. Can we conclude that xx and yy are plural duplicates, so as to contradict (PDP)? No, for if we allow that there may be natural properties exemplified by some of the xx plurally, then the xx can be “just like” yy in the sense relevant to the failure of local supervenience even though xx are not plural duplicates of yy. For, unless F is slice-sensitive, the strong theorist can and should say that F is exemplified not only by w but also by xx. After all, w is identical with xx! For the strong theorist, there is just the one portion of reality that can be represented either singularly or plurally, and however we represent it, it exemplifies F. So the assumption that natural properties are exemplified by objects, and never by pluralities, should be rejected by the strong theorist. Now since z doesn’t exemplify F, neither does yy, and xx and yy are not plural duplicates after all. There is a natural property, F, that is not preserved by any one-one correspondence between xx and yy. And so emergent properties do not pose a challenge to (PDP).²² To get a sound argument against strong composition as identity, we need to replace the third premise with the stronger “slice-sensitive emergent properties are possible,” and accordingly replace the second premise with the weaker “(PDP) is incompatible with the possibility of slice-sensitive emergent properties.” That is an

²¹ Actually, what McDaniel asserts for the first premise is that “any reasonable formulation of [strong] composition as identity” will entail (PDP); but I take it that Mods would count as a “reasonable formulation,” and so McDaniel’s argument should apply to it. ²² See Bohn () for a similar response to McDaniel’s argument.

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-  :  



argument I can accept. But there is no need to give the details.²³ When arguing against Mods, there is no need to take a detour through (PDP) at all. The possibility of slice-sensitive emergent properties is directly incompatible with Mods.

. Slice-Sensitive Emergent Properties: Three Illustrations I turn now to consider three sorts of case that support, in somewhat different ways, the possibility of slice-sensitive emergent properties. In each case, I think, there is pressure to say, if emergent properties are possible at all, then so are slice-sensitive emergent properties. I hope that the scenario described in each case will seem intuitively possible on its face. But, for me, the judgments of possibility are supported by general principles of modal plenitude. In this section, I present the three cases; in the next section, I consider the general principles that support them. First case: infinite series. Perhaps the clearest cases of slice-sensitive emergent properties arise from considering infinite series. When infinite series have both positive and negative terms, the sum may depend on how the terms are grouped or ordered. That provides a model for how a possible world could have emergent properties of the whole world that depend in part on how the world’s atoms are sliced. Here is a very simple illustration. Consider a world with denumerably many atoms, a₁, a₂, a₃, . . . , each of which has positive or negative unit charge. Suppose the distribution of charge is as follow: + to a₁, – to a₂, + to a₃, and so on. Suppose there are two fundamental properties, C₁ and C₂, that each assign a “net charge” to the world as a whole. (We can concoct scenarios in which these properties figure in the world’s fundamental laws.) C₁ is the net charge relative to the slicing given by the plurality a₁+a₂, a₃+a₄, . . . . (I use ‘+’ for both mereological sum and arithmetic sum; context decides.) C₁ has the value ( – ) + ( – ) + . . . = . C₂ is the net charge relative to the slicing given by the plurality a₁, a₂+a₃, a₄+a₅, . . . . C₂ has the value  + (– + ) + (– + ) + . . . = . C₁ and C₂ are slice-sensitive emergent properties.²⁴ I expect the following objection: the net charge properties are not emergent properties, for they supervene on the charges of the individual atoms together with fundamental relations that give the groupings. In other words, there are fundamental relations among the atoms, R₁ and R₂, where a₁R₁a₂, a₃R₁a₄, . . . and where a₂R₂a₃, ²³ The argument for the revised second premise is roughly this. Suppose slice-sensitive emergent properties are possible. Suppose xx and yy are plural duplicates, where xx compose w and yy compose z, with w and z composite. Because xx and yy are plural duplicates, w and z have the same emergent properties on one way of slicing. But on other ways of slicing w and z need not have the same emergent properties. One way of slicing w and z is the trivial one-membered slicing given by w and z themselves. With an appeal to plenitude, we can suppose that w and z do not have the same emergent properties on this trivial slicing. But that is just to say that w and z are not duplicates, contradicting (PDP). ²⁴ One could question whether the values of C₁ and C₂ are legitimately “sums” of the infinite series, since these series do not converge; but I don’t think that matters to the force of the example. In any case, one could easily construct fancier examples involving conditional convergence. An infinite series is conditionally convergent iff it converges, but the series of absolute values of the terms diverges. The sum of a conditionally convergent series depends on the ordering of its terms; and different orderings corresponds to different slicings in the world where the infinite series is exemplified. For example, the sequence of atoms corresponds to the slicing given by the plurality: a₁, a₁+a₂, a₁+a₂+a₃, . . . .

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

  , ’ ,  -

a₄R₂a₅, . . . . (Maybe R₁ and R₂ can be thought of as fundamental relations of bonding between atoms.) If the world is correctly described by taking the relations R₁ and R₂ to be fundamental, not the properties C₁ and C₂, then the case for slice-sensitive emergent properties evaporates. But why claim that the world envisaged must have fundamental relations that group the atoms, rather than allowing that there are worlds of both sorts, some with fundamental relations, others with fundamental emergent properties? The objection only has force if it derives from a general strategy for eliminating all emergence, for redescribing any purported case of an emergent property by adding fundamental relations to the supervenience base. I reject the general strategy. Indeed, even if we allow that there is no fundamental plural structure, worlds with fundamental properties and with fundamental relations have distinct structures, and so are distinct worlds. But in any case, to apply the strategy only in cases of purported slice-sensitive emergent properties would be ad hoc at best. I conclude, then: if emergent properties are possible at all, then slice-sensitive emergent properties are possible as well. That is the lesson of the world with emergent properties of net charge. Second case: spatial and temporal continua. It is sometimes thought that even if time and space are both continua composed of dimensionless atoms, the instants of time are unified in a way that the points of space are not. Using Bergsonian terminology, we might say that the instants of time form a qualitative multiplicity, the points of space a mere quantitative multiplicity.²⁵ The instants of time are not merely juxtaposed, but somehow permeate one another; the points of space, in contrast, are separate and several. It won’t much matter for the example whether this makes much sense; it is enough if one grants that it is possible that there be an emergent property of instants of time—call it “Q”—that does not apply to points of space. Now, consider a four-dimensional Bergsonian world. I will suppose that its metrical structure is Newtonian; any two points of spacetime have an absolute temporal and an absolute spatial separation. But that leaves out the additional structure it has in virtue of the emergent property Q. Consider a four-dimensional block, either the whole world or a portion of the world, and ask whether that block has Q. It seems that the answer depends on how we slice it. If we slice it into stationary world lines, then it doesn’t have Q: the different world lines, each parallel to one another, form a quantitative multiplicity, like the points of space. If we slice it into simultaneity “planes,” then it does have Q: the planes, added together, form a temporal continuum, a qualitative multiplicity. On its face, then, Q is a slice-sensitive emergent property. We could object as before that the slicings are given by fundamental relations; and we can give the same reply. But now there is a more telling objection. If we take Q to be an emergent property of one-dimensional world lines, rather than fourdimensional blocks, we no longer have to allow that Q is slice-sensitive. The extension of Q to four-dimensional blocks will supervene on fundamental properties and relations of proper parts of the block, and there is no need to posit slice-sensitive

²⁵ See Bergson (). But I use Bergson merely as a foil; he would not accept the assumption that a continuum is composed of dimensionless points.

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-  



emergence. This objection is easily countered, however, if we allow that gunky spacetime is possible.²⁶ For in that case every proper part of a four-dimensional block of spacetime is itself four-dimensional. The slice-sensitive emergence of Q applies all the way down, to smaller and smaller regions of spacetime. It cannot be eliminated. Third case: directional properties. Some properties of objects are directional: they do not hold absolutely, but only relative to a chosen direction. Consider, for example, the three-dimensional block letters, made famous by the book Gödel, Escher, Bach, that present as three different letters when viewed from three different directions. For example, the letter on the cover of the book is a ‘G’, an ‘E’, or a ‘B’ depending on the direction from which it is viewed. A direction naturally corresponds to a slicing of the object, where slices are perpendicular to the direction. The block letter is a portion of reality, then, that can be said to have three different properties depending on how that portion is sliced: one slicing makes it a ‘G’, one an ‘E’, and one a ‘B’. For another example, consider the phenomenon of iridescence. The color of an iridescent object varies with the direction from which the object is viewed. Again, we can take the color to be relative to a slicing. Of course, there is no reason to think that these physical properties of objects are emergent at the actual world. Presumably, the directional shape properties can be reduced to fundamental geometric relations between the points that compose the block; the directional color properties can be reduced to non-directional fundamental properties and relations that hold at the microscopic level. But it seems to me to be possible that shape or color properties of extended objects be emergent. A world at which shape or color properties are emergent might have directional shape or color properties, and those directional shape or color properties would be emergent as well. If emergent directional properties are slice-sensitive, as I have claimed, it follows that slice-sensitive emergent properties are possible. The arguments for the possibility of slice-sensitive emergent properties embodied in these three cases rest heavily on modal intuition. That makes them vulnerable to an opponent who simply claims not to share the relevant intuition. Arguments from modal intuition are weak unless the modal intuitions are backed by general principles. I turn, then, to consider how general principles of modal plenitude might provide support for the possibility of slice-sensitive emergent properties.

. Slice-Sensitive Emergent Properties: Principles of Modal Plenitude In Chapters  and , I argued that principles of modal plenitude fall naturally into three sorts. First, there is a principle of recombination (narrowly construed). Roughly, for any way of arranging elements taken from sundry possible worlds, there is a possible world that arranges (duplicates of ) those elements in that way. ²⁶ Spacetime is gunky iff there are no atoms: every part of spacetime has a proper part. Actually, I am doubtful that gunk is possible. But since most philosophers seem to think otherwise, the example has dialectical force.

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

  , ’ ,  -

Second, there is a principle of plenitude for world contents. Roughly, for any possible world, any element of that world can be replaced by an element alien to the world. And third, there is a principle of plenitude for world structures. Roughly, any structure compatible with the framework is instantiated within some possible world (or, at any rate, some portion of reality, whether or not it is properly called a “world”). How might these principles be used to generate possible worlds with slicesensitive emergent properties? I do not see how a principle of recombination on its own will be of much help. Recombining the fundamental properties from worlds without emergence will not lead to worlds with emergence. But perhaps the latter two sorts of principle of plenitude can do the job. Start with the plenitude of world contents, and take the fundamental elements of a world to include its fundamental properties. Say that a fundamental property is alien to a world if it is not instantiated at the world, or, more simply, it is not among the world’s elements.²⁷ It is alien simpliciter if it is alien to the actual world. A world with alien properties is an alien world. It is widely accepted that alien worlds with alien fundamental properties are possible. If the laws of physics had been different, then the fundamental physical properties might have been different as well. What grounds this belief? Not, I think, some belief about the poverty of the actual world. I hold that for any possible world, there is a world alien to it. (See Chapter  for discussion.) Moreover, this belief in the possibility of alien worlds is independent of any beliefs about the possible structures of worlds: for any world, there is a world alien to, but having the same structure as, that world. We can switch out content, as it were, without changing structure. This leads me to accept the following as a fundamental principle of plenitude: Plenitude of Alien Fundamental Properties. For any world w and any fundamental property P instantiated at w, there is a world just like w except that P has been replaced by a fundamental property Q alien to w. Other than having Q in the place that was occupied by P, the worlds are exactly alike. But this principle of plenitude does no better than a principle of recombination in generating possible worlds with emergent properties. Starting from a world where all fundamental properties are instantiated by atoms, one gets a world with different fundamental properties instantiated by atoms. One doesn’t get a world with fundamental properties instantiated by composites, a world with emergence. To see what more we need, consider the modal intuition behind the third case involving directional properties. The idea there was that, wherever a structural property is instantiated, there could be an emergent property instantiated in that place. Given a world like ours where the color properties of a composite object (let us suppose) are structural properties of the object’s constituent atoms and molecules, there is an alien world where the color properties of that object are emergent. (Perhaps this alien world is something like how an especially “naive realist” takes the actual world to be.) ²⁷ I am supposing that conjunctive and structural properties are not fundamental. Also, I take determinables rather than determinates to be fundamental; see Chapter . If determinates are fundamental, one would want to add a clause saying that no determinate of the property’s determinable is instantiated at the world.

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-  



The structural properties are still there alongside the new emergent properties; but the latter are now properly called the “color properties” for they are the more eligible semantic values. That suggests that we consider the following extension of our principle of plenitude: Plenitude of Alien Emergent Properties. For any world w and any structural property P instantiated at w, there is a world just like w except that P is coinstantiated with a fundamental property Q alien to w. But note that this principle not only supports the possibility of emergent properties, it supports the possibility of slice-sensitive emergent properties as well. For the directional properties considered in the third case are also structural properties, where the different directions correspond to different properties. For a block letter to be a ‘G’ is for it to be composed of slices with certain shapes; for it be an ‘E’ is for it to be composed of other slices with other shapes. Applying the principle will give different emergent properties for the different directional properties, emergent properties that are slice-sensitive. Do we have, then, a more general and principled argument for the possibility of slice-sensitive emergent properties? There is a problem. The principle of plenitude of alien emergent properties is not a pure principle of world contents. It rests in part on the plenitude of world structures. If we could say that the entire structure of a world is given by its spatiotemporal structure, then we could say that the alien world with emergence has the same structure as the world without emergence. But that conception of a world’s structure fails to take into account the pattern of instantiation of fundamental properties and relations, which is also a part of the world’s structure. The emergence world does not have the same pattern of instantiation, for it has an additional fundamental property co-instantiated with structural properties; and that changes the structure. Indeed, it changes the structure in a radical way, from a pattern of instantiation that does not exhibit emergence to a pattern of instantiation that does. It seems, then, that any attempt to support the possibility of emergence will need to look to principles of plenitude for world structures. I accept an extremely liberal principle of plenitude for structures: Plenitude of Structures. Any structure compatible with the framework is instantiated in some portion of reality. Some comments are in order. First, this principle needs further elaboration if we are concerned only with the structures instantiated in possible worlds. A portion of reality is only properly called a “possible world” if it is unified and isolated from the rest of reality; and perhaps it must be unified in some special way to be a possible world, by spatiotemporal relations, or relations analogous to spatiotemporal relations.²⁸ But we can set that issue aside here. Second, to fully interpret the principle, we need to answer questions about what the structures are, and how they are individuated. Are we nominalist or realist about the instantiation of structure? If realist, are structures ante rem or in rebus? But those questions can also be set aside here. What is relevant

²⁸ As Lewis (a: –) held. See Chapter  for discussion.

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

  , ’ ,  -

to present concerns is what the framework is taken to be, and what it is for a structure to be compatible with the framework. I assume that the framework includes standard first-order logic with identity. Structures that are compatible with first-order logic can be represented, in the usual way, by set-theoretic models. Such a model consists of a domain together with a set of designated properties and relations over that domain, where a property is represented as a subset of the domain and an n-ary relation as a subset of n-tuples of elements of the domain. The designated properties and relations will correspond to the fundamental properties and relations in a world that instantiates the structure. It will be convenient to separate out the relational structure at a world, the structure determined by the relations alone, and the property structure, the pattern of instantiation of the properties over the relational structure. Let me illustrate what I have in mind. Consider a Newtonian world consisting of points of spacetime; and suppose it has fundamental properties of mass and charge instantiated directly by those spacetime points. The relational structure of the world, we may suppose, is determined by the fundamental spatiotemporal relations between the points. The property structure of the world is determined by the distribution of mass and charge over the points. A Newtonian world with different fundamental properties—say schmass and scharge—has the same property structure, the same pattern of instantiation of fundamental properties, if its distribution of schmass and scharge is isomorphic to the distribution of mass and charge in the first world. Thus far, there is nothing to support the possibility of emergent properties. But, in addition to first-order logic with identity, I hold that the framework includes classical mereology; indeed, I take mereology to be a part of logic, broadly construed. It follows that every structure compatible with the framework has mereological structure: the part-whole relation applies to elements of the domain no less than does the identity relation; and the domain of every structure is closed under the taking of fusions. Now, among the structures compatible with the framework will be structures whose designated properties are instantiated by composite elements of the domain, properties that are not definable in terms of other designated properties and relations. Worlds that instantiate such structures will be worlds with emergent properties. So the principle of plenitude for structures will demand that there be worlds with emergence. But not yet slice-sensitive emergence. For the principle of plenitude for structures to demand that slice-sensitive emergent properties be possible, we need the framework to include plural logic. In structures compatible with plural logic, the designated properties and relations may be plural, that is, they may have pluralities of elements of the domain as their arguments; and there will be structures whose designated properties apply to pluralities in every which way. Indeed, there will be structures with a designated property that applies to one but not the other of two pluralities that have the same fusion, but which property is not definable in terms of other designated properties and relations. A world that instantiated that structure would be a world with slice-sensitive emergent properties. Thus, if we include plural logic as part of the framework, the principle of plenitude for structures will demand that there be a world with slice-sensitive emergence.

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-  



But there is a problem. Mods takes plural logic to be fundamental at the level of our representations, not fundamental at the level of being. A proponent of Mods would be within her rights to say that the structures that are used to interpret plural languages are not “compatible with the framework” in the relevant sense; they are not compatible with the framework that is fundamental at the level of being. According to Mods, a structure with a designated slice-sensitive property would represent a world where a fundamental property both is instantiated and is not instantiated by a single portion of reality. That structure can be rejected as impossible with impunity. I must, then, be guarded in stating my conclusions. I believe that slice-sensitive emergent properties are possible, and therefore that Mods should be rejected. I hope that the three illustrative cases I presented have some force to persuade others of this. But I have to concede that whoever rejects the modal intuitions embodied in those cases will have no problem rejecting the principle of plenitude that, for me, grounds the intuitions. For that principle and those modal intuitions are inseparable from the belief that the distinction between singular and plural is fundamental at the level of being. If the framework of plural logic is taken to be fundamental at the level of our representations, but not at the level of being, then some of the structures I take to be possible will not be possible in the relevant sense: they will not be instantiated at the level of being. A stalemate ensues.

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 The Fabric of Space: Intrinsic vs. Extrinsic Distance Relations () . Introduction Start with ordinary Euclidean space. Be a realist about points, and about distances between points. I ask: how are the points interwoven to form the fabric of space? Are there direct ties only between “neighboring points,” so that points at a distance are connected only indirectly through series of such direct ties? Or are there also direct ties between distant points, so that the fabric is reinforced, as it were, by irreducibly global spatial relations? To fix ideas, roughly, try the following thought experiment. Take a scissors and cut along some plane in space, severing the points just to one side of the plane from the points just to the other. Does space thereby fall into two totally disconnected pieces, so that points on one side now stand at no spatial distance from points on the other? Or does space, being reinforced, retain its shape? If space is maximally reinforced by direct ties of distance, then a distance relation, such as being twenty feet from, is intrinsic to the points that stand in it. Whether or not the relation holds depends solely on the intrinsic nature of the two points, and of the composite of the points. On the other hand, if space is not maximally reinforced by direct ties of distance, then a distance relation will not in general be intrinsic to the points that stand in it, and its holding may depend in part on features of the surrounding space. What features? To fix ideas, roughly, try the following thought experiment. Start with two points, say, twenty feet apart, and remove some of the space directly between the two points. (I don’t mean just the matter or energy occupying the space; I mean the space itself.) I ask: now how far apart are the points? Are they still twenty feet apart, on the grounds that distance, being intrinsic, is indifferent to changes in the intervening space? Or are they now less than twenty feet apart, on the grounds that there is now less space between them? Or are they now more than twenty feet apart, on the grounds that the shortest (continuous) path from one to the other is now more than twenty feet long? We have three competing answers, each, I think, with some intuitive appeal. Which is correct? And how can we tell? The situation is familiar. We start with some notion from ordinary language or thought—in this case, the notion of distance (but compare the notions of person, cause, law, matter). We notice that there are different criteria associated with the notion, depending on the context of application or of thought. But given the First published in Midwest Studies in Philosophy () : –. Reprinted with the permission of John Wiley and Sons, Inc. Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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



presuppositions under which the ordinary notion has evolved—in this case, presuppositions about the Euclidean nature of space—the different criteria fit together as well as you please. Then, driven perhaps by science, by mathematics, or by analytic philosophy, we consider extraordinary physical or logical possibilities that violate the presuppositions—for example, space with a “hole”—and the different criteria are seen to come apart. We are left with a plurality of competing conceptions, typically none of which captures all that was thought to be essential to the original ordinary notion. The question then arises: which conception should we accept? It would be wrong in general to expect a univocal answer. Competing conceptions may be evaluated along at least three different dimensions. One can ask: which conception best corresponds to the ordinary notion with which we began? One can ask: which conception is mathematically, or philosophically, more fruitful, say, by leading to more interesting and powerful generalizations. Or one can ask: which conception has application at the actual world according to our best physical, or perhaps philosophical, theories? In this chapter, I evaluate various conceptions of distance. There are clear losers, but no clear winner, no conception that dominates the score on all dimensions of evaluation. I recommend pluralism: different conceptions can peacefully coexist as long as each holds sway over a distinct region of logical space. But when one asks which conception holds sway at the actual world, one conception stands out. It is the conception of distance embodied in differential geometry, the conception that underlies modern treatments of physical space (and spacetime) based on Einstein’s general relativity. On this conception, all facts about distance are analyzed in terms of “local” facts about distances between “neighboring points.” Putting quantum mysteries to one side, I would say that this “local” conception gives the best account of distance at the actual world.¹ But there is a problem: the “local” conception, notwithstanding its mathematical and physical credentials, appears metaphysically suspicious. In the final section, I try to give the “local” conception a sound metaphysical footing. A word of caution. My question whether distance relations are intrinsic to pairs of points should not be confused with the oft-discussed question whether “space has an intrinsic metric.” Reichenbach, Grünbaum, and many others held that facts about the congruence of intervals of space are imposed from the outside, as it were, by our conventions for interpreting the behavior of material rods or rays. Without these conventions, they held, there are no facts about congruence; with different conventions, there are different such facts. I simply reject this here. The question here isn’t whether to be a metrical realist or conventionalist, but rather, assuming realism, what sort of realist to be.²

¹ What if quantum mysteries are taken into account? Must I endorse some conception of space as foamy, or spongy, or stringy, or loopy? I haven’t a clue. ² For the conventionalist line, see Reichenbach () or Grünbaum (: chapter ). For a realist response, see Nerlich ().

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

 .    ()

. Preliminary Notions I need some preliminary notions and assumptions. I will speak of possible worlds. In this chapter, I restrict attention to worlds at which space exists, at which space is composed entirely of points, and at which there are determinate facts about the distance, say in feet, between pairs of points. For these worlds, both intraworld and transworld comparisons of distance are meaningful. For the most part, I also restrict attention to Newtonian worlds, worlds at which there are determinate facts about the “identity over time” of points of space, and at which distances between points do not change over time. Newtonian worlds need not have Euclidean space. I assume that a variety of spatial structures will be exhibited at Newtonian worlds: curved and flat, finite and infinite, with and without boundaries, continuous and discrete.³ Although I speak for simplicity primarily of spatial distance in Newtonian worlds, what I say applies more generally, mutatis mutandis, to temporal duration, and to intervals of spacetime in relativistic worlds. In Section ., the focus will switch from space to spacetime. It is a matter of indifference whether one speaks of a distance function assigning non-negative reals to pairs of points, or of a multitude of distance relations, one for each non-negative real; I will speak of distance relations. I assume that the distance relations satisfy, at each Newtonian world, the usual constraints. Write ‘Dr(p, q)’ for ‘p is r feet from q’. Let ‘r’, ‘s’, and ‘t’ range over non-negative reals.⁴ Then, for any world, for any points p, q, and r at the world: (D) (D) (D) (D)

Dr(p, q), for exactly one r. Dr(p, q) iff Dr(q, p), for all r. D₀(p, q) iff p = q. If Dr(p, q) and Ds(q, r) and Dt(p, r), then t < r+s, for all r, s, and t.

I will freely apply mereology to points of space. Whenever there are some points, there is a unique fusion of those points. In particular, any pair of points, p and q, has a unique fusion, p + q. Whenever X is a proper part of Y, there is a unique difference, Y – X, which is the fusion of the parts of Y that do not overlap X. Space at a Newtonian world is the fusion of all the points of space existing at the world. I stay neutral as to whether a Newtonian world contains, in addition to the parts of space, entities that occupy those parts. If not, then the properties and relations ordinarily attributed to the “occupants” of parts of space must be attributed directly to the parts of space themselves. I assume that the spaces of distinct worlds do not overlap, that no point inhabits more than one world. Modal assertions that are de re points or

³ For a discussion of which spatial structures are possible, that is, instantiated at some possible world, see Chapter . ⁴ For convenience, I include ∞ among the non-negative reals, and read ‘D∞(p, q)’ as ‘p stands in no distance relation to q’. As usual, ∞ is greater than any other non-negative real, and ∞ added to any nonnegative real is ∞. This artifice allows the distance axioms to hold at worlds with “island universes,” spatially disconnected parts. [I now prefer to say that, although island universes are possible, there is no world with island universes as parts. See Chapter  on how I square these two claims.]

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 

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regions of space must therefore be interpreted with respect to an appropriate counterpart relation.⁵ I assume that there are certain primitive or fundamental properties and relations, the holding or failing to hold of which suffices to determine, at any world, all the qualitative facts at that world. In particular, there are primitive or fundamental spatial properties and relations which suffice to determine, at any world, all the facts about distance. Whether or not the distance relations are themselves among the primitives is a question soon to be addressed. I will call the primitive or fundamental properties and relations perfectly natural properties and relations.⁶ I assume that the part-whole relation is perfectly natural. I will need to speak of worlds or parts of worlds being (intrinsic) duplicates of one another. I define ‘duplicate’ in terms of perfectly natural properties and relations:⁷ for all X and Y, X and Y are duplicates iff there is a one-one correspondence between the parts of X and the parts of Y that preserves all perfectly natural properties and relations. (Remember: everything is a part of itself.) I call any such correspondence establishing that X and Y are duplicates an (X,Y)-counterpart relation; to each part Z of X, it assigns a unique part W of Y to be its (X,Y)-counterpart. (I drop the prefix when context allows.) Note that, for any (X,Y)-counterpart relation, (X,Y)counterparts are duplicates of one another. However, duplicate parts of X and Y will not be (X,Y)-counterparts, for any (X,Y)-counterpart relation, unless they are similarly related to the other parts of X and Y. Note also that, since there will in general be more than one (X,Y)-counterpart relation, a part Z of X and a part W of Y may be (X,Y)-counterparts relative to some (X,Y)-counterpart relations, but not others. Nonetheless, in presenting examples I will leave the relation unspecified, and say simply that Z and W are (X,Y)-counterparts. That won’t lead to trouble because what I say will hold true for an arbitrarily chosen (X,Y)-counterpart relation, assuming, of course, that a single such relation is held fixed throughout the example. I turn now to the notion of an intrinsic property or relation. Intuitively, a property is intrinsic just in case whether it holds of an object depends only on the way the object is in itself. Let us take the way an object is in itself—its intrinsic nature—to be given by the disposition of perfectly natural properties and relations among the object and its parts. Then we have, in terms of duplicates: A property P is an intrinsic nature iff P is had by all and only the duplicates of X, for some X. And, since an intrinsic property of an object is one that depends only on the object’s intrinsic nature, we have: A property P is intrinsic iff, for all X and Y, if X and Y are duplicates, then X has P iff Y has P. Note that, on this notion of intrinsic, an object’s haecceity— the property of being that object—is not one of the object’s intrinsic properties, since it is not shared by the object’s duplicates.

⁵ Those who prefer genuine transworld identity may simply suppose that one of the appropriate counterpart relations is the relation of identity. No trouble arises because, for the counterpart relations introduced below, counterparts are always intrinsic duplicates. ⁶ Following David Lewis. For discussion, see Lewis (a: –). ⁷ Again following Lewis (a: –). The definitions below of ‘intrinsic’, ‘internal’, and ‘external’ are also adapted from Lewis, although he does not apply the word ‘intrinsic’ to relations. Note that quantifiers here and below range over all possibilia.

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 .    ()

The notion of intrinsic can be extended to relations in a natural way. Let us say that a (dyadic) relation is intrinsic just in case whether or not it holds of a pair depends only on the intrinsic natures of X₁, X₂, and the fusion X₁ + X₂. Then, we have in terms of duplicates: A (dyadic) relation R is intrinsic iff, for all X, X₁, X₂, Y, Y₁, Y₂, if X and Y are duplicates, X = X₁ + X₂, Y = Y₁ + Y₂, X₁ a counterpart of Y₁, and X₂ a counterpart of Y₂, then R holds of iff R holds of .⁸ (Similarly for relations of three or more places.) If a property or relation is not intrinsic, it is extrinsic. Note that it follows immediately from the definitions that the perfectly natural properties and relations are themselves intrinsic. No surprise: this assumption was built in to the definitions from the start.⁹ Finally, I will need two modest assumptions about the plenitude of possible worlds. One, a principle of recombination for points, I will introduce when I need it in Section .. The other is this: for any part of the space of a Newtonian world, there is a Newtonian world whose entire space is a duplicate of that part. Actually, I only apply this assumption to rather ordinary parts of a three-dimensional Euclidean space. To illustrate: start with an ordinary world satisfying the laws of Newtonian mechanics. The principle posits a world just like it except for a “hole” in space. At this world, conservation laws fail in the vicinity of the “hole.” Objects entering the “hole” simply vanish; objects emerging from the “hole” appear out of nowhere. Bizarre, indeed. But logically impossible? Contemplate that as you drift towards the black hole at the center of the Milky Way!

. Distance Exam I now present a multiple-choice exam. I invite the reader to try it. The questions involve precisely formulated variations on the thought experiments from the introduction. The original formulations were unsatisfactory. They were naturally understood to involve de re counterfactuals—e.g. if some of the space between two points were removed, those points would be closer together. The interpretation of such counterfactuals is not fixed once and for all: different contexts may favor different comparative similarity relations on worlds, and different counterpart relations on points.¹⁰ I had a particular interpretation in mind; only when so interpreted do responses to the thought experiments have the intended metaphysical consequences. Therefore, to rule out unintended interpretations, I bypass the counterfactual formulations of the thought experiments and speak directly in terms of possibilia. (Once the intended interpretation is well-established, I will allow myself to slip back into the counterfactual mode.) ⁸ As Lewis notes (a: ), on a theory of universals according to which universals are parts of the particulars that instantiate them, we must everywhere replace the fusion X + Y with an augmented fusion, which includes among its parts not only X and Y, but the dyadic universals that hold between X and Y (or between parts of X and parts of Y ), and the monadic universals that hold of the fusion X + Y (or of its parts). Mutatis mutandis for a theory of tropes. ⁹ The intrinsic relations can be further divided into internal and external. A (dyadic) relation is internal just in case whether it holds of a pair depends only on the intrinsic nature of X₁ and of X₂, not on the intrinsic nature of X₁ + X₂. In terms of duplicates: A (dyadic) relation R is internal iff, for all X₁, X₂, Y₁, Y₂, if X₁ and Y₁ are duplicates and X₂ and Y₂ are duplicates, then R holds of iff R holds of . R is external iff R is intrinsic but not internal. This distinction, however, will not be needed below. It is agreed on all sides that distance relations, if intrinsic, are external. ¹⁰ On the interpretation of de re counterfactuals, see Lewis (: –).

OUP CORRECTED PROOF – FINAL, 7/3/2020, SPi

 



DISTANCE EXAM Part I: Removing Space. Consider a world with a three-dimensional Euclidean space, E. Let p and q be points of E twenty feet apart. Let Xi (for i from  to ) be a part of E that includes p and q. Consider a world whose entire space Yi is a duplicate of Xi. Let p0 and q0 in Yi be counterparts of p and q in Xi, respectively. () X₁ is E – A, where A is an open¹¹ infinite slab bounded by two parallel planes, ten feet wide, centered on and perpendicular to the line segment connecting p and q. (See Fig. .; the shaded region represents space that has been removed.) How far apart are p0 and q0 in Y₁? (a) Twenty feet apart. (b) They stand in no distance relation. (c) Ten feet apart.

p′

5′

q′

5′

Fig. .

() X₂ is E – B, where B is an open sphere, ten feet in diameter, centered on the point midway between p and q. (See Fig. ..) How far apart are p0 and q0 in Y₂? (a) Twenty feet apart. pffiffiffi (b)   + π/ feet apart. (Greater than twenty feet!) (c) Ten feet apart. 5π/3′ 5√3′

5√3′

p′

q′ 5′

5′

Fig. .

¹¹ That is, the points of A form an open set in the usual topology. A part of space is open iff it excludes all of its boundary points.

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

 .    ()

() X₃ is E – C, where C is an open right circular cone with height and radius each ten feet, with vertex on the point midway between p and q, and with axis perpendicular to the line segment connecting p and q. (See Fig. ..) How far apart are p0 and q0 in Y₃? (a) Twenty feet apart. (b) Twenty feet apart. pffiffiffi (c)   feet apart. (Less than twenty feet!)

5√2′ p′

5√2′ 10′

10′

q′

Fig. .

() X₄ is the surface of a sphere with diameter twenty feet and with p and q at opposite poles. How far apart are p0 and q0 in Y₄? (a) Twenty feet apart. (b) π feet apart. (Half the sphere’s circumference.) (c) Zero feet apart. () X₅ is an infinite wavy plane whose hills and valleys are an alternating series of infinite half cylinders of diameter ten feet, with p and q on adjacent summits. (See Fig. ..) How far apart are p0 and q0 in Y₅? (a) Twenty feet apart. (b) π feet apart. (A quarter of the cylinder’s circumference, quadrupled.) (c) Zero feet apart. p′

20′

q′

10π′

Fig. .

Part II: Adding Space. Consider a world with a two-dimensional space, Xi (i =  or ). Let p and q be points of Xi twenty feet apart. Consider a second world with a threedimensional Euclidean space, E, and a part Yi of E that is a duplicate of Xi. Let p0 and q0 in Yi be counterparts of p and q in Xi, respectively.

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   



() X₆ is the surface of a sphere with circumference eighty feet. How far apart are p0 and q0 in Y₆? (a) Twenty feet apart. pffiffiffiffiffiffiffiffi (b)  =π feet apart. (Length of chord subtending a quarter of a great circle.) () X is a Euclidean plane. How far apart are p0 and q0 in Y₇? (a) Twenty feet apart. (b) Could be any distance d,  < d <  feet, depending on the nature of E and the choice of Y₇.

. Three Conceptions of Distance Now for the answers. Unfortunately, no one answer key will do. Different conceptions of distance answer the questions differently. Consider first the conception according to which distance relations are intrinsic to the pairs of points that stand in them. Call this the intrinsic conception of distance. (Presumably—though nothing rests on it—distance relations are not only intrinsic on this conception, but perfectly natural; for what other intrinsic features of the two points or their fusion could suffice to determine the distance between them?) On the intrinsic conception of distance, if points p and q are twenty feet apart, and p0 , q0 , and p0 + q0 are duplicates of p, q, and p + q, respectively, then p0 and q0 are twenty feet apart. Now, for all seven questions, p0 , q0 , and p0 + q0 are counterparts of p, q, and p + q, respectively; and counterparts are duplicates; so, on the intrinsic conception, the answer seven times over is: (a) twenty feet. “Additions to” and “removals from” the space surrounding two points are nowise relevant to the distance between them.¹² Central to the intrinsic conception is the notion of congruence, generalized to apply to parts of space perhaps from different worlds: X is congruent to Y iff there is a one-one correspondence between the points of X and the points of Y that preserves all distance relations. On the intrinsic conception, duplicate parts of space are congruent. I assume the intrinsic conception accepts a partial converse as well: congruent parts of space are spatial duplicates, that is, they agree with respect to all their intrinsic spatial properties.¹³ Thus, on the intrinsic conception, congruence serves to delimit the border between the intrinsic and the extrinsic. The mathematical embodiment of the intrinsic conception is the abstract structure of a metric space. A metric space consists of a non-empty universe of points together with a family of distance relations (or a single distance function—it matters not) satisfying the axioms for distance listed above. The distance relations are taken as primitive, and other features of space—e.g. topological features—are defined in terms of the distance relations. The notion of a metric space is mathematically simple, yet extremely general: it encompasses spaces that are curved, flat, continuous, discrete, ¹² The intrinsic conception of distance is endorsed by Lewis (a: ). Lewis does not consider alternative conceptions. ¹³ What about a right- and a left-handed glove that are mirror images of one another, and thus congruent? That is no counterexample. The gloves differ in orientation, and orientation is not intrinsic, as consideration of a Möbius strip (or its three-dimensional analog) should make clear.

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

 .    ()

and all manner of hybrids thereof.¹⁴ Since the distance axioms all quantify only universally over points, any part of a metric space, and so any duplicate of that part, is itself a metric space. That ensures that one may speak without impropriety of distances between points in the spaces Y₁ through Y₇. Now consider a second conception of distance: the distance between two points of a space is given by the length of the (or a) shortest continuous path through space from one of the points to the other. (More exactly: the greatest lower bound of the lengths of continuous paths from one to the other, since in general there need be no least length; but I henceforth ignore this complication.) Call this the Gaussian conception of distance. The paths through space are themselves parts of space, fusions of points.¹⁵ On the Gaussian conception, the assignment of lengths to paths is prior to the assignment of distances to pairs of points. The Gaussian characterization of distance may have an air of circularity about it; but the air is only apparent. One metrical notion—distance between points—is defined in terms of another metrical notion—length of paths. There is no attempt to analyze away all metrical notions. Later we shall ask how length of path can itself be analyzed; and then, of course, we shall have to be careful not to close the circle. On the Gaussian conception, I suppose, the length of a path through space is an intrinsic property of that path. However, the distance between two points turns out not to be an intrinsic relation of those points. If some of the space surrounding two points is “removed,” some or all of the paths connecting those points may no longer exist, and the length of the shortest remaining path—the new distance between the points—may be greater than it was, or not defined. If the space surrounding two points is embedded in a larger space, new paths connecting the points may come into existence, and the length of the shortest connecting path—the new distance between the points—may be less than it was. In short: the distance between two points does not depend solely on the intrinsic nature of the fusion of the two points. To illustrate, turn to the exam. The Gaussian conception answers (b) seven times over. In question (), none of the paths connecting p and q in E have counterparts in Y₁. Thus no path in Y₁ connects p0 with q0 , and the distance between them is undefined (equivalently, ∞, given our convention). Y₁ is composed of two “island universes” with p0 and q0 inhabiting different islands.¹⁶ In question (), the straightline path from p to q in E has no counterpart in Y₂. The shortest path from p0 to q0 in Y₂ is one that follows a tangent from p0 to the edge of the hole, hugs the hole for a sixth of a turn, and then follows a tangent back to q0 . (See Fig. ..) Hence, the

¹⁴ Of course, a further generalization is needed to account for the interval relations of Minkowski spacetime, since the interval squared may be positive, negative, or zero. ¹⁵ In what follows, unless otherwise noted, I assume that paths are continuous and “smooth”—that is, without corners or cusps. (Technically, I assume that paths can be given a parametrization that is differentiable with non-zero derivative at all points along the path.) In order that the notions of continuity and “smoothness” be applicable to parts of space, the Gaussian must assume that space is a manifold, that space has both topological and differential structure. ¹⁶ I assume island universes are possible. An upholder of the Gaussian conception who denied this should refuse to answer question () on the grounds that there is no world whose entire space is a duplicate of X₁. [Again, my view has changed, and I no longer think the possibility of island universes requires there be worlds with disconnected spatial parts. See Chapter  and the postscript to Chapter .]

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   



pffiffiffi answer:   + π/ feet apart. In question (3), the straight-line path from p to q in E has a counterpart in Y₃, so the distance between p0 and q0 in Y₃ is the same as the distance between p and q in E: twenty feet. In question (), the shortest path connecting p0 and q0 in Y₄ is half a great circle. In question (), the shortest path connecting p0 and q0 in Y₅ slides down a quarter circle hill, around a half circle valley, and then up another quarter circle hill. (See Fig. ..) In either case, the shortest path between p0 and q0 is a counterpart, not of the straight-line shortest path between p and q in E, but of a longer, more circuitous path between p and q whose length is given by the answer (b). I consider questions () and () below. Central to the Gaussian conception is the notion of an isometry between parts of space: X and Y are isometric iff there is a one-one correspondence between the points of X and the points of Y that (when extended to fusions of points) preserves lengths of paths. (More exactly, the image of a path with endpoints p and q is a path of the same length with endpoints the image of p and the image of q.) Since Gaussian distance is defined in terms of lengths of paths, isometries preserve Gaussian distance as well. Duplicate parts of space are isometric, since the length of a path is intrinsic. With that the intrinsic conception can agree. But the Gaussian conception accepts, whereas the intrinsic conception must deny, the partial converse: isometric parts of space are spatial duplicates, and thus agree with respect to all their intrinsic spatial properties. On the Gaussian conception, isometries delimit the border between the intrinsic and the extrinsic: spatial properties are intrinsic just in case they are preserved by isometries, just in case they are isometric invariants. To illustrate the difference between congruence and isometry, consider a “flat” plane F and a “wavy” plane W (such as X₅), each embedded in a three-dimensional Euclidean space E. F and W are not congruent, since no one-one correspondence between F and W can preserve the distances among four points of W not co-planar in E. Therefore, on the intrinsic conception, F and W are not spatial duplicates. However, F and W are isometric. Intuitively, this is because something flat, such as a piece of paper, can be made to wave without stretching or tearing, and thus without changing the lengths of any paths confined to its surface. Thus, on the Gaussian conception, F and W are spatial duplicates, and agree with respect to all their intrinsic spatial properties. Let us now see how the Gaussian conception answers questions () and (). For question (), the Gaussian reasons: X₆ and Y₆ are duplicates; therefore isometric. The only parts of E isometric to X₆ are themselves surfaces of spheres with circumference eighty feet.¹⁷ Therefore Y₆ is one such. Since the shortest path between two points on the surface of a sphere is part of a great circle, and p and q are twenty feet apart, p and q are connected by a quarter great circle in X₆. By the isometry, p0 and q0 are connected by a quarter great circle in Y₆. The distance, then, between p0 and q0 in Y₆, and so in E, is the length of a “wormhole” through the interior of the sphere; namely, the length of the chord that subtends the quarter great circle connecting p0ffi pffiffiffiffiffiffiffi and q0 . This length, which is less than twenty feet, is given by answer (b):  =π ¹⁷ This is because the surface of a sphere (unlike a flat plane) is rigid in E: any part of E that is isometric to the surface of a sphere in E is also congruent to it, and so itself the surface of a sphere of the same size. Intuitively, a surface in E is rigid if it cannot be deformed without stretching or tearing.

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

 .    ()

feet apart.¹⁸ For question (), the Gaussian reasons: X₇ and Y₇ are duplicates; therefore isometric. Since E contains both flat planes and wavy planes that are isometric to X₇, Y₇ may be either wavy or flat. If Y₇ is a wavy plane, then p0 and q0 are closer together in Y₇ (and E) than p and q are in X₇: p0 and q0 are connected by a (straight-line) “wormhole” in E. How close together? The distance between p0 and q0 is just the length of the (straight-line) “wormhole,” which may have any value in feet greater than zero and less than twenty, depending on the “wavelength” of Y₇. Thus, the Gaussian answers (b) to question (). The Gaussian conception of distance finds mathematical expression in the development of differential geometry. Here is how the conception is typically motivated.¹⁹ Start with a two-dimensional surface X embedded in a three-dimensional Euclidean space E. Ask: what geometrical features of the surface X could be ascertained by twodimensional geometers whose measurements were entirely confined to X? These features comprise X’s “intrinsic geometry.” The length of a path confined to X can be ascertained to any specified degree of accuracy by placing sufficiently small measuring rods end to end along the path; so lengths of paths are intrinsic to X. Moreover, features that depend only on lengths of paths—the isometric invariants—could all be ascertained by the geometers, and so are intrinsic to X; this includes, most famously, the Gaussian curvature at a point. But the true Euclidean distances between points of X are not intrinsic to X: they cannot be ascertained without “leaving the surface.” The geometers do, however, have an intrinsic substitute for distance between points: distance-within-X, that is, the length of a shortest path in X between the points. Indeed, if the geometers (wrongly) take X to be all of space, they will (wrongly!) take distance-within-X to be the true distance.²⁰ The next step is to do away with the embedding space E, to consider a surface Y intrinsically just like X, but not embedded in any larger space. For this surface Y, the intrinsic geometry is all there is to geometry, and distance-within-Y is all there is to distance. The final step generalizes the Gaussian conception to apply to surfaces not embeddable within Euclidean space, and to spaces of dimension greater than two. In particular, since Euclidean distances are identical with distances-within-E, the Gaussian conception applies to E as well. I will attempt to evaluate the motivation that underlies the Gaussian conception in Section .. First I want to consider a third conception of distance. It is not, in my view, a serious contender. But I dare not ignore it: the most popular answer by far to question () is neither (a) nor (b), but (c). Call it the naive conception. The leading idea is this: the distance between two points should be a measure of the amount of space between the points; but, unlike the Gaussian conception, the amount of space between two points need not be identified with the length of any continuous path. Indeed, although the naive conception agrees with the Gaussian conception that distance relations are not intrinsic, the dependence of distance on the surrounding

¹⁸ Note that, on the intrinsic conception, p0 and q0 are separated by more than a quarter great circle of Y₆, since p0 and q0 are twenty feet apart in Y₆, and Y₆ is the surface of a sphere of circumference eighty feet. ¹⁹ For an excellent introduction to differential geometry, including the standard motivational asides, see O’Neill (). ²⁰ Unfortunately for our purposes, distance-within-a-surface is often called “intrinsic distance,” since it is part of the surface’s intrinsic geometry. I will avoid that usage.

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   



space is reversed. On the naive conception, if space is “removed” from between two points, the points will then be closer together; if space is “added” between two points, the points will then be farther apart. How might the amount of space between two points be measured so as to capture these naive intuitions? Version . Many who answer (c) to question () have in mind closing up the gap left by the missing slab, and “stitching” the two remaining parts of Y back together. The amount of space between p0 and q0 in Y₁ is then the length of the straight-line path in the stitched-up space. But that won’t do. For one thing, on topological grounds the stitching cannot be seamless. One cannot avoid the seam by identifying boundary points on opposite sides of the gap; for that would violate the supposition that X₁ and Y₁ are duplicates (assuming the topological property being connected is intrinsic, and so must be shared by duplicates). But a seam composed of distinct, copresent boundary points would violate the distance axiom (D) requiring that distinct points be some positive distance apart. How bad is that? Perhaps violations of (D), if restricted to boundary points, could be tolerated on the grounds that boundary points may in effect be contiguous, and thus no distance apart.²¹ In any case, the stitching idea doesn’t generalize. Depending on the shape of the part of space removed, the stitching might be done in any number of ways, the choice among which is arbitrary; indeed, this is so for Y₂ through Y₅. Thus, version  cannot in general provide determinate answers to questions about distance. We need another idea. There are, however, ways to make the naive conception more precise; I will consider the two most promising. For each, I assume that the length of a path is an intrinsic property of that path; and I will speak of “disconnected paths,” whose lengths (when defined) are the sum of the lengths of their connected parts. Version . Determine the distance between p0 and q0 in Yi (i from  to ) as follows: start with the straight-line path in E connecting p and q; take the part of this path, perhaps disconnected, that overlaps Xi; then take as answer the length of this perhaps disconnected path (equivalently, of its counterpart in Yi). When applied to questions () and (), the result is answer (c): ten feet apart. But consider question (). On version , the answer would be twenty feet, in agreement with the intrinsic and Gaussian conceptions. But someone who holds the naive conception could say that p0 and q0 are closer together in Y₃ than p and q are in E. They could take the amount of space between p0 and q0 to be the amount of space between p0 and the cone-shaped hole plus the amount of space between the cone-shaped hole and q0 . (See Fig. ..) This illustrates version . Determine the distance between p0 and q0 in Yi as follows: start with all the (continuous) paths in E connecting p and q; for each such path, take the perhaps disconnected part of the path that overlaps Xi; then take as answer the least of the lengths of these perhaps disconnected paths (or, equivalently, of their counterparts in Yi). (More generally, take the greatest lower bound pffiffiffi of the lengths.) When applied to question (), version  gives the answer (c):   feet apart. In the cases considered, either version  or  may appear to give plausible answers. But trouble looms for both. First, note that on either version distinct boundary points

²¹ This notion of a boundary point of space requires differential (but not metrical) structure: a boundary point is one such that some path to the point cannot be “smoothly” extended.

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

 .    ()

of a space may be zero feet apart in violation of axiom (D). For example, two points on opposite “shores” in Y₁, or two points on the edge of the “hole” in Y₂. Perhaps, as noted above, that is tolerable. But now look more closely at how version  applies to Y₂. Since boundary points r0 and s0 in Y₂ are zero feet apart (see Fig. .), we have D₅ (p0 , r0 ), D₀(r0 , s0 ), and D√(p0 , s0 ) in violation of the triangle inequality (D). Not so tolerable. No family of relations that violates the triangle inequality deserves to be called a family of distance relations. That leaves version . Version  satisfies the triangle inequality, in much the same way as the Gaussian conception, by defining distance as the length of a shortest “path” (under an expanded notion of path). Since the shortest “path” in Y₂ from p0 to s0 goes by way of r0 , D₅(p0 , s0 ) on version . But when applied to Y₄ and Y₅, version , and version  to boot, fail to give plausible answers. Let’s focus on Y₄, the duplicate of the surface of a sphere. On both versions  and , not only p0 and q0 , but any two points of Y₄ are assigned the distance: zero feet apart. That violates axiom (D) in a big way. It obliterates all distinctions of distance, treating Y₄ in effect as a “space” with but a single point. Moreover, Y₄ is a space without boundaries; no path in Y₄ abruptly comes to an end. Even were one to tolerate violations of (D) for boundary points, I see no comparable grounds for leniency here. I conclude that neither version  nor version  gives acceptable answers to questions about distance between points of Y₄. Could some fourth version of the naive conception answer any differently? The distance between p0 and q0 in Y₄ must be less than or equal to twenty feet, on the naive conception, because space was “removed” from E; yet the distance must be a measure of the amount of space between p0 and q0 . How could any such distance, other than zero, be singled out over any other? I reject the naive conception.

. The Gaussian Conception: Must Shape Be Intrinsic? That leaves two contenders: the intrinsic and the Gaussian conception. In this section, I question the Gaussian demarcation between intrinsic and extrinsic spatial properties. First, I ask the Gaussian whether the shape of a thing is intrinsic to that thing. Suppose it is. Consider a wavy plane and a flat plane embedded in a threedimensional Euclidean space. They are isometric; therefore they are spatial duplicates, says the Gaussian; therefore, since shapes are intrinsic, they have the same shape. That is plainly wrong: one is curved and the other is flat! So the Gaussian must deny that shapes are intrinsic. That looks bad. We ordinarily take the shape properties to be the very paradigm of intrinsic properties, of properties that depend only on the way something is in itself. And we ordinarily would say that two things cannot be duplicates of one another unless they have the same shape. It appears that the Gaussian conception clashes with our ordinary ways of thinking about shape, and so, derivatively, about distance. The intrinsic conception, on the other hand, can uphold the intuition that shapes are intrinsic, since the wavy plane and the flat plane are not congruent with one another. I suppose a Gaussian might respond as follows. Ordinary intuition, rightly understood, does not conflict with the Gaussian conception. Our intuitions about shape

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  :    ?



apply only to the objects of our experience, not to mathematical abstractions therefrom; and the objects of our experience are three-dimensional and Euclidean. Now, for ordinary three-dimensional parts of a three-dimensional Euclidean space— spheres, cubes, even paper-thin sheets—the parts are isometric only if they have the same shape. That allows the Gaussian to hold that shapes are “intrinsic” in a restricted sense: for ordinary three-dimensional parts of a three-dimensional Euclidean space, duplicate parts always agree in shape. And ordinary intuition demands no more. Just as the domain of ordinary intuition is restricted to objects of experience, so should the sense in which ordinary intuition takes shapes to be intrinsic similarly be restricted. This response lacks conviction. The two-dimensional surfaces of threedimensional things, no matter how “abstract” or “ideal,” seem to be objects of our intuition no less than the three-dimensional things themselves; and intuition pronounces the shapes of the former intrinsic no less than the latter. Nor does it much help to note that many two-dimensional surfaces, such as that of a sphere or a cube, are rigid, and so are isometric only if they have the same shape; for rigidity plays no role in the relevant intuition. The Gaussian should concede the clash with ordinary intuition. It no more condemns the Gaussian conception than, say, the acceptance of continuous paths through space that are nowhere “smooth”—once thought monstrous by intuition—condemns the standard mathematical analysis of continuity. Ordinary intuitions about matters susceptible to mathematical precision have often been found in need of revision. That is a small price to pay for the power and generality conferred by mathematics; or for the explanatory and predictive success of scientific theories mathematically based. Adherence to ordinary intuition is to some extent necessary to keep our bearings; but when mathematics gives us clear vision above and beyond, we should not hesitate to change our course. Agreement with ordinary intuition, by itself, favors the intrinsic conception but little.

. The Gaussian Conception: No Phantom Embedding Space? In this section, the Gaussian takes the offensive. Suppose the geometers of a world have access to every part of their space. They have measured the length of every path, the area of every surface, the volume of every region. Then, says the Gaussian, they have the wherewithal to know all there is to know about the structure of their space; there are no spatial facts that are in principle inaccessible to geometric measurement. The intrinsic conception, we have seen, must disagree. Consider again a world with a two-dimensional space isometric to the Euclidean plane. On the intrinsic conception, the space may be “wavy” or “flat,” depending on facts purportedly about “distance”; but these “distance” facts are inaccessible to geometric measurement, even assuming the geometers have access to every part of their space. Now the Gaussian objects: these facts are mysterious. If the two-dimensional space were embedded in some inaccessible higher-dimensional space, the “distance” facts could be understood in terms of inaccessible facts about the nature of the embedding. But by assumption there is no such embedding space. Rather, the inaccessible

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

 .    ()

“distance” facts reflect the possibilities of embedding, what would have been the case, had the space been embedded in some higher-dimensional space. But to explain such facts in terms of the possibilities of embedding is to reverse the true order of things. What reason could there be, then, for taking these facts which are inaccessible to geometric measurement to be facts about distance, to be spatial facts? By maintaining that these inaccessible facts are facts about distance, the intrinsic conception posits a phantom embedding space, the ghost of a departed embedding. Note that this argument does not rest on a positivist premise. The Gaussian need not deny that there could be perfectly natural relations between points of space— even relations satisfying the distance axioms—knowledge of which is inaccessible to geometers who have access to all parts of space. The Gaussian need only deny that such relations could be the distance relations. Before attempting to evaluate the argument, I want to examine the assumption that lies at its heart. It is a supervenience thesis. When stripped of colorful talk of tiny geometers, it comes to this: if the spaces of two worlds are isometric, then the spaces are congruent as well. In short: distances supervene on lengths of paths.²² This assumption needs to be qualified in at least two ways. The alleged supervenience is contingent, not logical. First, consider worlds with discrete space. In a discrete space, there are no continuous paths between points, so no lengths of continuous paths. Therefore, any two discrete spaces with the same number of points agree vacuously on all lengths of continuous paths. But the spaces need not agree on all distances between points, say, by having each point be an island universe all to itself. That is one possibility, I suppose. But I also suppose the points of a discrete space may stand in various distance relations, as long as the relations satisfy the axioms for distance. Indeed, I suppose there could be physical evidence that actual space (or spacetime) is discrete, and thus that the actual distance relations are not Gaussian. Since the Gaussian analysis of distance cannot account for the variety of possible discrete— more generally, disconnected—spaces, the Gaussian assumption must be qualified: distances supervene on lengths of paths, for worlds with continuous space. Gaussian supervenience also fails, I think, at some worlds with continuous space. Consider worlds with “action at a distance”: worlds at which forces act directly from one point to another without being propagated along a continuous path connecting the points. At such worlds, distance relations need not be Gaussian. For example, consider again Y₂, the space with a “hole.” Suppose that away from the “hole,” Newtonian laws of motion and of universal gravitation have been well-confirmed: the force of gravity produces an acceleration that varies inversely with the square of the distance. Then, a measurement of the acceleration of objects located at p0 and q0 could give evidence against the Gaussian conception. Moreover, the “action at a distance” need not be instantaneous. Suppose that away from the “hole” it is wellconfirmed that all causal signals travel no faster than the speed of light. Then, a measurement of the time it takes signals to travel from p0 to q0 could give evidence

²² Globally supervene, that is, since the assumption only applies to entire worlds. On the distinction between various notions of supervenience, see Teller ().

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    



against the Gaussian conception. The most the Gaussian is entitled to claim is this: there could be no evidence against the Gaussian conception at worlds where it has been established that all action is local, that is, propagated locally along continuous paths. For only in such worlds must evidence for distance relations be, ipso facto, evidence for lengths of paths. The Gaussian supervenience thesis thus applies at most to local-action worlds, to worlds with continuous space and no action at a distance. The Gaussian argument against the intrinsic conception must similarly be limited in scope. (The argument erred, in particular, by focusing too narrowly on what would be accessible to geometers, rather than what would be accessible to physicists more generally.) It follows that the intrinsic conception of distance cannot be jettisoned from logical space. The Gaussian must accede to pluralism: at some worlds, distance relations are intrinsic, and presumably perfectly natural; at other worlds, distance relations are extrinsic, and subject to the Gaussian analysis. Under pluralism, worlds with Euclidean space may have either intrinsic or extrinsic distance relations. That points up a flaw in my distance exam. The embedding space E was underspecified. Interpreted one way, the answer is (a) throughout; interpreted the other way the answer is (b). I shall have to give a lot of ‘A’s. Pluralism is the best the Gaussian can hope for. Unfortunately, the Gaussian isn’t yet in a position to demand her share of logical space. Limiting the scope of the Gaussian argument undermines its force altogether. Any world at which Gaussian supervenience fails—be it a world with discrete space or with action at a distance—is a world with a phantom embedding space, no less than a world whose space is a “wavy plane.” The argument began as a general indictment of worlds with a phantom embedding space. What remains is a specific indictment—based none too clearly on considerations of physical evidence—of local-action worlds with a phantom embedding space. But the intrinsic conception is not committed to such worlds. If one takes a local-action world with intrinsic distance relations, and one “removes,” say, a sphere from its middle, one gets a world with a phantom embedding space all right, but the world is no longer a local-action world. I conclude that the Gaussian is left without any argument against the intrinsic conception. For the sake of uniformity, why not take the intrinsic conception to apply to all worlds with distance relations?

. General Relativity and Humean Supervenience Here’s why. Our best physical theory of space and time, Einstein’s general relativity, is based on differential geometry, and is Gaussian through and through. I suppose general relativity is logically possible, that is, true at some possible worlds. At these worlds, distance relations are Gaussian, not intrinsic. Moreover, to whatever extent we believe that general relativity is true at the actual world, to that extent we should believe that actual distance relations are Gaussian. Before turning to the treatment of distance in general relativity, we need to further develop the Gaussian conception. Thus far, length of path has been left unanalyzed. If length of path is taken as primitive, then the Gaussian and the intrinsic conception are both global conceptions of distance: both apply primitive metrical notions to pluralities of points, in one case to paths, in the other to pairs. I now want to develop

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

 .    ()

a local version of the Gaussian conception according to which the only primitive metrical notions are local properties of points. (I postpone an exact definition of ‘local property’ until the next section.) Let us again assume that the points of space have a manifold structure in terms of which paths can be characterized as continuous and smooth. Now, assign to each point of space a metric tensor g, or ds², which supplies information about distances within an “infinitesimal neighborhood” of the point; call the tensor g at a point p the local metric at p.²³ Given two points p and q (no matter how “close together”), the distance between them is not determined by the local metric at p and the local metric at q. But given a path from p to q, the length of that path is determined by the local metric at each point along the path: it is the result of integrating ds along the path from p to q.²⁴ (In effect, the local metric at a point provides a set of “infinitesimal measuring rods,” one for each direction, to be used for determining lengths of infinitesimal portions of paths passing through the point; integration then corresponds to measuring the length of a path by laying (continuum-many!) appropriately directed measuring rods end to end.) On the local Gaussian conception, the properties of having such-and-such local metric are taken as primitive. These properties then suffice to determine the length of any path through space, and so the Gaussian distance between any two points. There are no primitive global metrical properties or relations. Now, I claim that general relativity is a local Gaussian theory. (Of course, here the local metric has Lorentz signature, since it provides information about infinitesimal intervals in spacetime, rather than infinitesimal distances in space; the spacetime interval between two points is given by a longest, rather than a shortest, path.) A key insight behind general relativity is that all physics takes place by local action.²⁵ When applied to gravitation, this led to Einstein’s field equation, the fundamental law of gravitation in general relativity. Einstein’s equation—G = πT—tells how the local mass-energy density, given by the stress-energy tensor T, relates to the local curvature of spacetime, given by the Einstein curvature tensor G. (G is analyzable in terms of the metric tensor g.) More generally: when the fundamental laws of physics are formulated within general relativity, the metrical notions that occur in the laws are all local metrical properties, including the metric tensor g. (This contrasts sharply with Newtonian physics: according to the fundamental law of gravitation, the force of gravity varies inversely as the square of the distance.) Now, I suppose that all and only the perfectly natural properties and relations instantiated at a law-governed world occur in the fundamental laws of that world. It follows that the properties of having such-andsuch local metric are perfectly natural properties instantiated at general relativistic worlds, and that general relativity, as formulated by Einstein, is a local Gaussian theory. ²³ More exactly, the metric tensor at a point is an inner product on the tangent space of the point; and the metric tensor field is differentiable, it varies “smoothly” from point to point. ²⁴ The information carried by ds² is coordinate-independent, though of course calculations of length of path will be done by representing ds² and the path in question relative to some chosen coordinates. (Coordinate-free geometric objects are represented in boldface.) In the case of a three-dimensional Euclidean space, there will be x,y,z-coordinates under which ds² = dx²+dy²+dz². In general, however, ds² will be a more complicated quadratic function of dx, dy, and dz, for any x,y,z-coordinates. ²⁵ For an elaboration on this theme, see the introductory chapter of Misner, Thorne, and Wheeler (: ).

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    



I do not deny that one could give an empirically equivalent reformulation of general relativity in terms of global metrical relations; under specifiable conditions, global and local metrical relations are interdefinable with aid of the calculus. But that would not show global metrical relations to be perfectly natural, any more than, say, a reformulation of the laws of color (if there were such laws) in terms of grue and bleen would show grue and bleen to be perfectly natural. In general relativity, the local metric at a point is a dynamic object, a prime mover: it tells objects at that point how to move. It has the same claim to perfect naturalness as the other prime movers, such as the electromagnetic field at a point. It would be arbitrary and absurd to hold that some prime movers are perfectly natural, but not others. It is well known that Einstein’s general relativity eliminates primitive action at a distance. I have been arguing what is perhaps less well known, that general relativity eliminates primitive “distance at a distance.” The reduction of global relations to local properties in general relativity applies to metrical relations as well. This has implications for the formulation of philosophically interesting supervenience theses. David Lewis has defended the viability of Humean supervenience, according to which all facts, other than facts about spatiotemporal distance, supervene on local matters of particular fact. At worlds of Humean supervenience: “We have geometry: a system of external relations of spatiotemporal distance between points. . . . And at the points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated. . . . All else supervenes on that.”²⁶ But have we not just seen that, at least at local Gaussian worlds, even the relations of distance between points supervene on local matters of fact? Do we have, then, a sweeping elimination of all primitive global notions at local Gaussian worlds, a grand supervenience of the global on the local? That would be too much to ask. On the local Gaussian conception, global distance relations supervene not on local metric alone, but on local metric plus manifold structure. Without manifold structure, no integration. Without integration, no analysis of global distance relations in terms of local metric. Manifold structure is in part topological structure, and topological structure, it is easy to see, is irreducibly global. Consider a two-dimensional Euclidean plane and (the surface of) an infinite cylinder. They are locally indistinguishable: each consists of continuum-many points that are locally Euclidean. But the plane and the cylinder differ topologically. For example, the plane, but not the cylinder, is simply connected: all closed paths can be continuously contracted to a point. Just as there are irreducibly global topological features of space, so also of spacetime at relativistic worlds. Thus, general relativity suggests no grand supervenience of everything on local matters of particular fact; it suggests something more modest. Call it Einsteinian supervenience. At worlds of Einsteinian supervenience: we have a manifold of spacetime points, and a distribution of perfectly natural local properties over those points; all else supervenes on that. Of course, Einsteinian supervenience, like its Humean cousin, is philosophically controversial. I here claim only that it is the right supervenience thesis to consider at general relativistic worlds. Under pluralism,

²⁶ From the Preface to Lewis (e: –).

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

 .    ()

Einsteinian and Humean supervenience are not in conflict. Each holds contingently, and governs its own region of logical space. These regions differ with respect to the instantiation of perfectly natural spatiotemporal relations of distance. Given the success of general relativity, I suspect we are nearer to, if not within, the region of Einsteinian supervenience.

. Modal Recombination and the Non-Standard Continuum I have argued that modern physics of spacetime is based on a local Gaussian conception of (spatiotemporal) distance. In this final section, I ask whether the local Gaussian conception is metaphysically suspect. I claim that the local metric at a point, as standardly conceived, is not an intrinsic property of the point.²⁷ Thus, the local Gaussian appears to be committed to perfectly natural, extrinsic properties. That would introduce necessary connections between distinct co-inhabitants of local Gaussian worlds, namely, between points and their surrounding space; it would violate a modal “principle of recombination” that I, for one, would be loath to give up. I take this as a challenge not to the physicist—I am not so bold—but to the metaphysician: provide a coherent metaphysical foundation for modern spacetime theories. First, we need a precise characterization of local properties. The characterization requires topological structure. Say that a part of space, N, is a neighborhood of a point p iff some part of N includes p and is open in the topology of the space. (For example, in a three-dimensional Euclidean space, N is a neighborhood of p iff N includes some open ball around p, that is, all the points less than some positive distance r from p.) A property of points P is local iff, for any points p and q, for any neighborhood N of p and any neighborhood M of q, if N is a duplicate of M and p is a (N, M)-counterpart of q, then P holds of p iff P holds of q. Note that if a property of points is intrinsic, then it is local; for counterparts, being duplicates, share all their intrinsic properties. But in general local properties need not be intrinsic. Call a property of points that is local but not intrinsic neighborhood-dependent.²⁸ The most familiar examples of neighborhood-dependent properties come from elementary calculus: derivatives of functions. Consider the position of some point-sized object as a function of time; suppose at time t it is located at point p. The instantaneous velocity of the object at t is the derivative of the position function evaluated at t. This derivative at t depends on the object’s position not only at t, but also at “neighboring” times. Or, turning this around, the derivative at t depends on when the object is located not only at p, but also at “neighboring” points. The object’s instantaneous velocity at t is thus a neighborhood-dependent property of both the time t and the point p. ²⁷ David Lewis, in the passage just quoted, requires that local properties be intrinsic properties of points (or their point-sized occupants); but I do not think that is how ‘local’ is standardly used in mathematics or physics. ²⁸ Neighborhood-dependent properties may be exclusive or inclusive: those are exclusive that exclude information about the intrinsic nature of points that instantiate them. Thus, a property P of points is exclusively neighborhood-dependent iff, for any points p and q, for any neighborhood N of p and any neighborhood M of q, if N – p is a duplicate of M – q, then P holds of p iff P holds of q.

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    - 



(In spacetime, of the “event” .) In two or more dimensions, position is given by a vector, and so instantaneous velocity is a vector as well, having both a magnitude (speed) and a direction (if non-zero). Both an object’s speed and its direction of motion are neighborhood-dependent properties of points and of times. Now, I claim that the local metric at a point p, as characterized in differential geometry, is a neighborhood-dependent property of p. That is because the local metric at p is an inner product on the tangent space at p: it takes a pair of tangent vectors as input, and gives a real number as output. (If the inputs are one and the same, the output is the squared length of the tangent vector.) The tangent vectors at p are defined as the derivatives of “smoothly” parametrized paths through p. (A parametrized path is a function from an interval of real numbers to the points of the path. If one thinks of the parameter as “time,” then a “smoothly” parametrized path through p is a trip through p with no jolts or stops, and the tangent vectors at p are all the possible “velocities,” or “states of motion,” when passing through p.) These tangent vectors, being derivatives, give information not just about p, but about the space immediately surrounding p. For example, the dimensionality of the tangent space is the dimensionality, not of p which is zero, but of the immediately surrounding space. In short: the tangent vectors provide neighborhood-dependent information about p. Since the local metric at p is an operator on tangent vectors, it inherits neighborhood-dependence from its operands.²⁹ Thus, the local metric at a point, as standardly conceived in differential geometry, is neighborhood-dependent; and that is trouble for the local Gaussian conception. For, on the local Gaussian conception, the local metric is also perfectly natural. Apply the definitions from Section .. If perfectly natural, then shared by duplicates; if shared by duplicates, then intrinsic. So both neighborhood-dependent and intrinsic. Contradiction. Perhaps we should revise our definitions, not our conception of distance. It was simply built in to the definitions that all perfectly natural properties are intrinsic. What was built in can be built out. For a cost. We need to introduce a primitive distinction between the perfectly natural properties that are intrinsic and those that are not. (So we no longer can analyze ‘intrinsic’ just in terms of perfectly natural properties and relations.) Now the definition of ‘duplicate’ bifurcates: X and Y are intrinsic duplicates iff there is a one-one correspondence between the parts of X and the parts of Y that preserves all intrinsic perfectly natural properties and relations; X and Y are local duplicates iff there is a one-one correspondence that preserves all perfectly natural properties and relations.³⁰ A local property is now defined simply as ²⁹ This is a bit fast and loose. Unless the paths through p are embedded in some higher-dimensional Euclidean space, the derivatives in question are not defined, and tangent vectors are instead identified with (directional) derivative operators. See O’Neill (: –). But the argument is essentially unchanged, since derivative operators, which require manifold structure, are no less neighborhood-dependent than derivatives. ³⁰ One might ask, independently of the question whether perfectly natural properties can be extrinsic, whether ‘duplicate’ in ordinary usage means ‘local duplicate’ or ‘intrinsic duplicate’, or is indeterminate. Test case. Consider a cube with sides of two feet and a sphere with a diameter of one foot, each composed of (the same kind of) homogeneous continuous matter. The sphere has continuum-many intrinsic duplicates among the parts of the cube; but the sphere has no local duplicates, since no interior point of the cube is a

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

 .    ()

a property that can never differ between local duplicates. That perfectly natural properties are local is now built in to the definitions. Now we face a dilemma. I suppose we will want the revised theory to incorporate the Humean denial of necessary connections between distinct existences, in particular, between a point p and its surrounding space: any other point q could have taken p’s place. Of course, at no world is q itself in the place of p. We need to formulate the principle in terms of duplicates: for any points p and q, perhaps from spaces of different worlds, there is a world whose space is a duplicate of the space of p except that it contains a duplicate of q where the duplicate of p would be.³¹ But, on the revised theory, we must decide: do we mean local duplicate or intrinsic duplicate? Does the principle require that there be a local duplicate of q where the duplicate of p would be? It had better not. For example, suppose that p is surrounded by positively curved space, q by negatively curved space. Then, a world whose space is a duplicate of the space of p but with a local duplicate of q in p’s place must be both positively curved and negatively curved in the immediate neighborhood of q. No world is like that. So the principle requires only that there be an intrinsic duplicate of q where the duplicate of p would be. More generally, the Humean denial of necessary connections is formulated in terms of intrinsic duplicates, not local duplicates. On the revised theory, however, that will be too weak to capture the spirit of the Humean denial. It rules out necessary connections between the intrinsic natures of distinct things. But, on the revised theory, there may be more to a thing than is given by its intrinsic nature. Thus, formulating the Humean denial in terms of intrinsic duplicates fails to rule out necessary connections between the distinct things themselves, in particular, between a point and its surrounding space.³² I suggest we drop the revised theory, and pursue a different tack. Although the local metric, as standardly conceived, is an extrinsic property of points, and therefore not perfectly natural, perhaps the extrinsic local metric is “grounded” on an intrinsic, perfectly natural property of points. To illustrate the sort of grounding I have in mind, consider mass density. If one assumes that each neighborhood of a point has some determinate (finite) mass and volume, then the mass density at a point may be characterized as the limit of the ratio of mass to volume, as volume shrinks to zero. So characterized, mass density is an extrinsic property of points. But it is customary in physics, when considering a continuous matter field, to instead take mass density to

local duplicate of any boundary point of the sphere. Using our ordinary notion of duplicate, how many duplicates of the sphere are there in the cube? It seems to me one can answer either way. ³¹ This is an instance of the principle of recombination put forth by David Lewis. See Lewis (a: –). (Of course, the points in question must be of the same kind, be it Newtonian or spatiotemporal.) [See Chapter  for precise versions of Lewis’s principle of recombination from which the claim in the text follows.] ³² The argument is especially compelling if one holds that perfectly natural properties correspond to immanent universals or classes of tropes. For immanent universals or tropes are present in their instances. Now consider a neighborhood-dependent, perfectly natural property of p. The corresponding universal, or a corresponding trope, is present at p. And, unlike a dyadic universal or trope, it is wholly present at p. (Remember: its holding at p tells one nothing about any point other than p, not even something relational.) Intrinsic or not, how can one deny that it is part of the nature of p, and so must be “recombined” along with p?

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    - 



be a primitive scalar field: a function that assigns to each point a real number representing (given appropriate units) the intrinsic mass density at the point. Given intrinsic mass density, and an assumption about its smooth distribution, mass can be defined by integration. Extrinsic mass density then supervenes on intrinsic mass density. And, thanks to a fundamental theorem of integral calculus, the values of extrinsic and intrinsic mass density coincide. Note that the smoothness of the intrinsic mass density field is a contingent feature of worlds with continuous matter. Given a principle of recombination for points, there will be worlds whose intrinsic mass densities (perhaps no longer properly so-called) are jumbled up in such a way that no (finite) masses (and no extrinsic mass densities) exist at the world.³³ The suggestion, then, is to say something analogous about the local metric: the extrinsic local metric supervenes on an intrinsic local metric (plus manifold structure). It is the intrinsic local metric properties that are perfectly natural. That is on the right track, I think; but there is a problem. Whereas the mass density at a point is a simple scalar quantity, the local metric at a point is a tensor quantity. How can a tensor be intrinsic to a point? Points are spatially simple. Tensors, being operators on vectors spaces, are spatially complex. It is repugnant to the nature of a point to suppose that a local metric, which is a tensor, could be intrinsic to a point. If we hope to ground the extrinsic local metric on an intrinsic local metric, the latter had better be intrinsic not to a point, but to something spatially complex.³⁴ No sooner said than done. If we are willing to posit perfectly natural properties on theoretical grounds, we should be willing to posit appropriate entities to instantiate those properties: in this case, entities that are spatially complex. I propose that we reify talk of the “infinitesimal neighborhood” of a point. The tangent space at a point is now conceived as the infinitesimal neighborhood of the point “blown large,” as viewed through a “microscope” with infinite powers of magnification; it no longer depends for its existence on the manifold structure. Tensor quantities are intrinsic not to points, but to the infinitesimal neighborhoods of points. At local Gaussian worlds, space (or spacetime) has a “non-standard” structure. There are “standard” points, and there are “non-standard” points that lie an infinitesimal distance from standard points. The points along a path in space are ordered like the non-standard continuum of Abraham Robinson’s non-standard analysis.³⁵ Let us take stock. The local Gaussian conception of distance, if founded on standard differential geometry, is committed to local metric properties that are both extrinsic and perfectly natural. I propose founding the local Gaussian conception instead on non-standard differential geometry. That allows the perfectly natural local metric properties to be intrinsic, though not to points, but to their infinitesimal neighborhoods. The intrinsic local metric at a point now comprises a family of ³³ Michael Tooley has argued that extrinsic (or “Russellian”) velocity should be grounded in this way on primitive velocities that are intrinsic to points. But the theoretical reasons for positing primitive velocities, at least at worlds approximating ours, seem to me much weaker than the theoretical reasons for positing primitive local metrics. See Tooley (). ³⁴ Denis Robinson () asks whether vectors could be intrinsic to points, and answers “no.” I concur. Although vectors are spatially less complex than tensors, they have a “tail” and a “tip”: too much to fit within a single point. ³⁵ Non-standard analysis is applied to differential geometry, for example, by Abraham Robinson ().

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

 .    ()

infinitesimal distance relations. So there turn out to be perfectly natural distance relations after all; but they are local, not global, because they hold only among points within an infinitesimal neighborhood of a standard point. (‘Local’ is defined with respect to the topology of the standard points.) The local Gaussian is no longer committed to perfectly natural, extrinsic properties. Metaphysical worries about necessary connections have been resolved. When non-standard analysis gained mathematical and logical respectability some thirty odd years ago, the question naturally arose whether the non-standard continuum is instantiated at any possible world, or even at the actual world. Perhaps the mere consistency of non-standard analysis already gives reason to suppose that the non-standard continuum is possibly instantiated. The role that non-standard differential geometry can play in firming up the metaphysical foundations of physical theory gives reason all the more—including reason to suppose the non-standard continuum is actual.

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 Is There a Humean Account of Quantities? () . Introduction Humeans have a problem with quantities. A core principle of any Humean account of modality is that fundamental entities can freely recombine. But determinate quantities, if fundamental, seem to violate this core principle: determinate quantities belonging to the same determinable necessarily exclude one another. Call this the problem of exclusion. Prominent Humeans have responded in various ways. Wittgenstein (), when he resurfaced to philosophy, gave the problem of exclusion as a reason to abandon the logical atomism of the Tractatus with its free recombination of elementary propositions. Armstrong () and (a) promoted a mereological solution to the problem of exclusion; but his account fails in manifold ways to provide a general solution to the problem. Lewis studiously avoided committing to any one solution, trusting simply that, since Humeanism was true, there had to be some solution. Abandonment; failure; avoidance: we Humeans need to do better. It is high time we Humeans confronted and dispatched this elephant in the room. It won’t be easy. Whether it is possible at all I leave to the reader to judge. In this chapter, I present what I take to be the best account of quantities, tailoring it where needed to meet Humean demands as well as my own prior commitment to quidditism, and my own comparativist inclinations. In short: determinables, not determinates, are the fundamental properties, and freely recombine; determinates arise from the instantiation of determinables in an enhanced world structure; determinate quantities may be local (in a sense to be explained), but they are not intrinsic. Is the account I end up with Humean? Not, unfortunately, as it stands: the problem of exclusion still rears its ugly head. After dismissing a failed attempt at a solution, I consider in the final section the two viable Humean options. One attributes the source of the necessary exclusions to conventional definition, the other attributes it to logic. The first is safe and familiar, but not a response I can accept given my other commitments. The second is more radical and less familiar; but I am convinced it is

First published in Philosophical Issues  (): –. Reprinted with the permission of John Wiley and Sons, Inc. Versions of this chapter were presented at the Seven Hills Workshop at the College of Holy Cross in November  and at the conference “Modal Metaphysics: Issues in the (Im)Possible” before the Slovak Metaphysical Society in Bratislava in August, . For comments and discussion, I thank Cameron Gibbs, Chris Meacham, and Alejandro Perez Carballo. Modal Matters: Essays in Metaphysics. Phillip Bricker, Oxford University Press (2020). © Phillip Bricker. DOI: 10.1093/oso/9780199676569.001.0001

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

      ? ()

on the right track. I don’t have space to develop it much here, but I put it out for future research.

. Quantities By quantities, I will mean quantitative properties. Perhaps there are also quantitative relations; but they are beyond the scope of the present inquiry. I understand ‘quantitative’ broadly: what matters for my purposes is that quantities come in families with a determinate-determinable structure; much of what I say could be applied also to the problem of exclusion for qualitative determinates, such as determinates of color. I call both the determinates and the determinables “quantities.”¹ On my usage, ‘determinate’ is an absolute, not a relative, term: determinates are maximally specific. I will use mass as a paradigm example. There is the determinable property, having mass. And there are the determinate properties: having  kg mass, having  kg mass, etc. In the case of mass, the determinates have definite magnitudes that are represented by the non-negative real numbers (relative to the chosen unit). But there are other possibilities: for example, the determinates of charge may have just two magnitudes, one positive and one negative. Both mass and charge are scalar quantities. There are also vector quantities, such as force, the determinates of which have both a magnitude and a direction. And there are other, more complex, sorts of quantities. Although I have space here only to discuss scalar quantities, I take it to be a desideratum of any account of scalar quantities that the account generalizes naturally to vectors, and more complex quantities. I will confine my attention to worlds that are quantitative at the fundamental level, although whether it is the determinate quantities or the determinable quantities that are fundamental, or something that undergirds them both, is yet to be decided. As a realist, I suppose that fundamental quantities, no less than other fundamental properties, correspond to immanent universals or tropes. (In fact, I am a trope theorist, but I will remain neutral in what follows.) As I note below, however, the Humean does not escape the problem of exclusion by endorsing nominalism. I will also confine my attention to worlds at which the fundamental quantities are instantiated by pointlike entities, entities that, though they may not be simple, have no finite spatiotemporal extent. Thus, for the case of mass, I have in mind something like the ideal point particles of classical physics. But, perhaps unlike classical physics, I will also confine my attention to worlds at which there is at most one object located at any point of spacetime. That allows me to speak indifferently of a fundamental property being instantiated by an object or by the spacetime point occupied by the object.

¹ On an alternative usage, ‘quantity’ refers, not to the individual determinates and determinables, but to the family or type to which they belong, as when we say that mass is a quantity; I trust when I use ‘quantity’ in this way it will cause no confusion.

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



. Humeanism Humeans, in the sense relevant to the metaphysics of modality, accept what is often called Hume’s Dictum: (HD) There are no necessary connections between distinct existents. The necessity in question is some sort of metaphysical or broadly logical necessity, not nomological or epistemic necessity. Entities are distinct if they have no parts or constituents in common. Hume famously applied his dictum to events in his arguments about causation. Contemporary Humeans may apply it to concrete objects, or to properties, or both. Hume’s Dictum, on its face, is an assertion of modality de re. But when applied to objects it must be given a special interpretation to be plausible. It is not meant to rule out, for example, the necessity of origins, that I could not have existed without my parents. (Although for the Humean such de re necessity involves nominal, not real, essences.) When applied to objects, Hume’s Dictum must be understood in terms of intrinsic duplicates: a duplicate of me could have existed without any duplicate of my parents existing. When applied to tropes or universals, no special interpretation is needed: the posited possibilities are naturally understood to involve duplicates of the tropes, or the universals themselves. It has become standard to interpret Hume’s Dictum as a principle of recombination stated in terms of possible worlds. I do not need to give the principle in its most general form.² The part of the principle that will be relevant to my discussion is just the part that rules out necessary exclusions, as follows: (HPR)

Consider any class of distinct elements (perhaps taken from different worlds). Consider any possible world structure, and (category-preserving) arrangement of those elements within that world structure. There is a possible world in which (duplicates of) those elements are arranged in that way.³

We can say that the elements in question can be freely recombined. Of course, the import of this principle depends on what the “elements” are; all that is presupposed by (HPR) is that the elements are fundamental entities. If the elements are all simple, then ‘distinct’ can be omitted; if the elements are all universals, then ‘duplicate’ can be omitted. The qualification ‘category-preserving’ will be needed if the Humean countenances more than one fundamental category that the elements belong to. For example, if there are particulars and universals among the elements, then the Humean will only allow that the particulars all freely recombine with one another and the universals all freely recombine with one another. Or, for another example, if there are relations of different adicity among the elements, the Humean will only allow that relations of the ² On how to formulate general principles that together rule out all necessary connections and exclusions, see Chapter . ³ An arrangement is a many-many mapping from the elements to be recombined into the places of the world structure. The left ‘many’ is to allow that different elements may occupy the same place; the right ‘many’ to allow that a single element may occupy, or have duplicates that occupy, multiple places. See below for more on world structures.

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

      ? ()

same adicity freely recombine. This qualification requires that each place in a world structure be assigned the category of an entity that occupies that place in a world having that world structure. (HPR) then says: for any arrangement that assigns elements to places that match their category, there is a possible world in which the elements are arranged in that way. Note that Humean principles of recombination are only part of an account of modal plenitude, of what possibilities there are. We also need an account of what structures are possible world structures, and what elements are possible inhabitants of worlds, in particular, whether there are elements that are alien to our world. Then the recombination principle tells us that all ways of assigning the elements to (appropriate) places in a possible world structure represent a possible world. An account of modal plenitude that permits violations of Humean recombination principles will, on its face, be committed to primitive modality. But there are other ways to be so committed, for instance, by providing modal characterizations of the possible worlds, or possible elements, or possible structures. A Humean account of modal plenitude, I will assume, eschews all forms of primitive modality. Or, being a bit more careful, the Humean eschews all modality that cannot be reduced to logic and definitions. A weaker version of Humeanism would accept (HD) and (HPR), but allow that primitive modality may be needed to express the Humean account. On this weaker version, although there is no primitive modality “in the world,” primitive modality is needed for our theorizing about the world. It is the stronger version that I accept and will attempt to defend below.⁴ Humeanism is sometimes taken to include Lewis’s doctrine of Humean supervenience, that worlds that agree with respect to the distribution of local qualities over spacetime agree with respect to the truth or falsity of all (qualitative) propositions. But Humeanism, as I understand it, is known a priori if known at all, whereas the doctrine of Humean supervenience, for all we know, is false at the actual world. Indeed, even for the region of logical space that is my target, where classical physics reigns, I do not accept that part of the doctrine that asserts: the only fundamental relational structure is spatiotemporal structure. I do however hold that, at such classical worlds, all truth supervenes on the distribution of fundamental local qualities over the fundamental relational structure, whatever that structure may be. In particular, all facts about quantities so supervene. In this sense, one can say that facts about quantities are local facts.

. The Problem: Quantities Appear to Stand in Necessary Connections and Exclusions First off, there appear to be necessary connections between the determinates of a family and their determinable. For example, necessarily, if something has the determinable of mass, then it has some determinate mass property; and necessarily, if ⁴ Lewis (a: ) seems to recognize this distinction, holding that some ways of accepting primitive modality—for example, accepting immanent modal relations—are “especially repugnant.” On different sorts of primitive modality, see Section ..

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  



something has a determinate mass property, then it has the determinable of mass. This problem is easily solved, however, by noting that Humean recombination applies only to fundamental properties (if it applies to properties at all), and so does not prohibit necessary connections between non-fundamental properties, or between fundamental properties and non-fundamental properties. So if either the determinate properties or the determinable properties are not fundamental, the above necessities are no violation of (HPR). The natural response—the response I will explore at the start—is to say that the determinate properties are fundamental, and the determinable property is analyzed as a(n infinite) disjunction of the determinate properties: to have mass is either to have  kg mass or  kg mass or . . . (where the ellipsis covers disjuncts corresponding to every non-negative real number).⁵ But, second, the determinates themselves appear to stand in necessary connections (or, more precisely, necessary exclusions). For example, necessarily, nothing instantiates both being  kg mass and being  kg mass. This seems to be a clear violation of the recombination principle if fundamental properties are among the elements to which it applies,⁶ and the determinate properties are taken to be fundamental. My focus for the remainder of this chapter will be on whether the Humean can develop an account of quantity that successfully responds to the problem raised by these necessary exclusions.

. First Humean Response According to the first response I consider, the Humean recombination principle does not apply to properties; it applies only to objects (or things, or so-called thick particulars). This was Lewis’s response in On the Plurality of Worlds. His recombination principle was explicitly restricted to “spatiotemporal parts” of worlds. Universals or tropes, which (if they exist) are non-spatiotemporal parts of worlds for Lewis, are no counterexample to free recombination because they do not recombine at all.⁷ This also seems to be Russell’s response in his logical atomist period. At any rate, he never endorses the logical independence of properties or elementary propositions, only of particulars. He writes: “each particular has its being ⁵ Rosen (: ) suggests for epistemic reasons an alternative analysis that quantifies over properties and does not require an infinite analysandum: to have mass is to have some determinate of mass. Both analyses leave open the hard question (if determinates are fundamental): in virtue of what are two determinate properties determinates of the same determinable? ⁶ This is loose talk. Following Lewis (a: –), I take properties to be abundant and not immanent. Fundamental properties are just some of the abundant properties, and so also not immanent. Strictly speaking, then, it is the universals or tropes that are among the elements to which (HPR) applies, not the properties to which the universals or tropes correspond. For convenience I persist in this loose talk throughout, trusting that it will not mislead. ⁷ Lewis does not explicitly characterize this restriction on recombination as a response to the problem of exclusion, but I suspect the problem was an unstated motivation. The only reason he gives for not endorsing a recombination principle that applies to all parts of worlds is not cogent. He writes: “such a principle, unlike mine, would sacrifice neutrality about whether there exist universals or tropes” (Lewis a: ). But I fail to see why accepting a principle that applies to all parts of worlds (or all fundamental parts) should commit one to non-spatiotemporal parts. By the time we get to Lewis (), recombination is taken to apply to all the elements, whatever the elements are.

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

      ? ()

independently of any other and does not depend on anything else for the logical possibility of its existence” (Russell : ).⁸ This response, however, is inadequate, for two reasons. First, if one is a realist about (sparse) properties, taking them to be immanent universals or tropes, then it is arbitrary to apply the principle of recombination to some fundamental entities and not others. As a trope theorist, I cannot avail myself of this response. But, second, even for the nominalist this response is inadequate. Recombinatorial reasoning applies to determinates of different determinables. For example, if determinates of mass and determinates of charge are fundamental properties, then for any determinate of mass and determinate of charge, it is possible that something have that combination. This places the following demand on any acceptable Humean account of quantities: it must entail (D)

There are no necessary exclusions between determinates of different determinables.

The first Humean response does nothing to meet this demand and, therefore, should be rejected.

. Second Humean Response A second Humean response is to baldly deny that there are metaphysically necessary exclusions between determinates of a single determinable. This response comes in two versions. The first takes the exclusions to be only nomically (or causally) necessary. Thus, it is metaphysically possible for something to be both  kg mass and  kg mass, just not nomically possible; it is ruled out by the laws of nature. This response is floated, though not endorsed, by Lewis. What he endorses is the unknowability of whether the exclusions are metaphysically, or only nomically, necessary. He writes: On some other questions . . . we just have to confess our irremediable ignorance. I think one question of this kind concerns incompatibility of natural properties. Is it absolutely impossible for one particle to be both positively and negatively charged? Or are the two properties exclusive only under the contingent laws of nature that actually obtain? I do not see how we can make up our minds . . . . (Lewis a: )

But I do not find this response at all plausible. The problem isn’t that it accepts metaphysical possibilities that go beyond anything we ordinarily take to be possible. The principle of recombination already commits the Humean to a slew of bizarre possibilities. For example, it allows one to patch together parts of worlds with different laws, resulting in a schizophrenic world in which induction would be utterly unreliable. But distinguish: for all these bizarre worlds, it seems right to say that it is an a posteriori matter whether or not our world is one of them. Worlds at which two determinates of a determinable are co-instantiated, on the other hand, seem to be ruled out a priori. Do I really need to make observations to discover whether or not ⁸ For discussion of how Russell’s logical atomism compares to Wittgenstein’s post-Tractarian view, see Bell and Demopoulos ().

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  



something is both  kg and  kg? That sounds absurd. I know that nothing is both  kg and  kg simply in virtue of understanding the concepts involved. I conclude, then, that another demand on an acceptable Humean account is that: (D)

The exclusions between determinates of a single determinable are knowable a priori.

This suggests a better way to deny that the exclusions are metaphysically necessary: take them to be conceptually necessary, and thus a priori, and claim that a conflation between conceptual and metaphysical necessity is responsible for our mistaken judgment that the exclusions are metaphysically necessary. One must distinguish between the concepts of determinate quantities, and the determinate quantities themselves, that is, the properties picked out by those concepts. Thus, consider the concepts of being  kg mass and being  kg mass, and the determinate properties that these concepts pick out (at a classical world). On the response now being considered, there will be a possible world in which these two determinate properties are coinstantiated in accord with the principle of recombination; but in this world, the concepts do not apply to the properties, the properties are not properly called “determinates of mass.” This could be because it is analytic to ‘determinate’ and ‘determinable’ that distinct determinates of a determinable are never co-instantiated, or because it is analytic to ‘mass’ that distinct determinates of mass are never co-instantiated. Either way, on the response being considered the necessary exclusion is de dicto, not de re. Although I will revisit a version of this Humean response below, it should be rejected in its present form. Any attempt to explain away the necessary exclusion of mass determinates by packing the exclusions into the concept of determinate or the concept of mass is bound to fail because the problem of exclusion persists when expressed using different concepts that lack any exclusionary nature. Suppose that your favorite property is being  kg mass and my favorite property is being  kg mass. There is a clear sense in which it is true to say: my favorite property and your favorite property are necessarily not co-instantiated. But no plausible implementation of the current strategy can capture this sense. And that is just to say that the necessity in question is de re, not de dicto: the properties themselves, independently of how we pick them out, necessarily exclude one another. If the Humean is to attribute the source of the necessary exclusions to some sort of conceptual necessity, it must derive not from the concepts used to pick out the determinates of mass, but from the concept of necessity itself. I return to consider this sort of response in the final section.

. Third Humean Response The first two responses, in one way or another, denied that there are necessary exclusions. The third response instead accepts the necessary exclusions and tries to explain them. It takes the fundamental determinates to be structural properties, properties that are instantiated by composite entities in virtue of the properties of and relations among their parts. The structural properties are themselves complex entities that have those properties and relations as constituents. The idea, then, is that in some way this complexity can be used to explain the exclusions.

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

      ? ()

One version of this response is due to Armstrong. He focuses on the case of extensive quantities, such as mass. He holds that whenever an object has a determinate mass, it divides into distinct parts whose masses sum to the mass of the whole. Thus, the determinate, being  kg, will be associated with the structural property, being composed of five distinct  kg proper parts. But, of course, it is no less associated with infinitely many other structural properties, such as being composed of two distinct proper parts of  kg and  kg. This suggests that, to avoid arbitrariness and capture all relations between the mass determinates, the determinate, being  kg mass, should be identified with the conjunction of all of these associated structural properties. It then follows that, for any two determinates of mass, the smaller mass determinate and all of its constituents are also constituents of the larger mass determinate. This sharing of constituents, according to Armstrong, can then be invoked to explain the resemblance between the determinates of mass. My concern here, however, is not with Armstrong’s attempted explanation of the resemblance between determinates of a determinable, but his attempted explanation of their incompatibility.⁹ In this respect, the account is clearly inadequate. As best I can tell, the explanation is simply this. Consider, for reductio, an object that instantiates two determinates of mass, say,  kg and  kg. In virtue of being  kg, it has a proper part that is  kg. But then the object will share a structural property, the property of being  kg, with one of its proper parts, and that, Armstrong thinks, is impossible.¹⁰ But why? Granted, if an object shares a structural property with one of its proper parts, the object will have to contain an infinite sequence of smaller and smaller proper parts. But that is to be expected, on Armstrong’s mereological account, if there is no quantum of mass. (And note that this infinite nesting of proper parts doesn’t require that the objects be gunky, and not composed of extensionless points; it only requires that all (non-zero) mass determinates be instantiated by extended objects.) I conclude that Armstrong’s account fails to provide an explanation for the incompatibility of determinates.¹¹ Could Armstrong instead claim that it is the sharing of constituents that explains why determinates of mass are incompatible? Indeed, according to Humean recombination principles, only elements that are distinct, that share no parts or constituents, freely recombine. But failure of distinctness only explains necessary connections, not necessary exclusions. When elements are not distinct, the existence of (a duplicate of ) one necessitates the existence of (a duplicate of) a part or constituent of the other. But I see no reason why the existence of (a duplicate of) one should necessitate the non-existence of (a duplicate of) the other, or any part or constituent of the other. If this third Humean response is to solve the problem of exclusion, it needs an

⁹ The account of resemblance has many problems, not least of which is its failure to generalize to fundamental quantities that differ in their structure from mass: intensive quantities, vector quantities, quantities with both positive and negative magnitudes. For a decisive critique of Armstrong’s account of resemblance between quantities, see Eddon (). ¹⁰ See, for example, Armstrong (: ). In Armstrong (a: ) he says only this: “it becomes clear why the very same thing cannot be both five and one kilogram in mass. To attempt to combine the two properties in the one thing would involve the thing’s being identical with its proper part.” ¹¹ This problem for Armstrong’s explanation of incompatibility is briefly raised in Lewis (: ).

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  



explanation of incompatibility that does not rest on whether or not the determinates are distinct. Or does it? Could Armstrong claim that he does not need to provide an explanation of the exclusions, that it is enough if his account does not entail any violations of the Humean principle that distinct entities freely recombine? But an account of modal plenitude is incomplete if it does not say, for entities that are not distinct, how they recombine. A demand on a successful account of Humean recombination is that it entails all the possibilities for recombining entities, whether distinct or not. We need, for example, an account of why being  kg and being  kg do not exclude one another if they are instantiated at different spacetime locations. More generally, any acceptable Humean account of quantities must entail: (D) There are no necessary exclusions between a determinate being instantiated at one spacetime location and a different determinate being instantiated at a different location. If Armstrong’s account of Humean recombination fails to entail anything about the recombination of entities that are not distinct, his Humean account of quantities will not be able to meet this demand. Perhaps, however, there is a better version of the response that determinates are structural properties. Surely sometimes entities necessarily exclude one another in virtue of their differing structure. Consider shapes. A cube and a sphere could not be co-located: in virtue of their differing spatial structure, they make incompatible demands on the regions of space that they occupy. So a cube occupying a region necessarily excludes a sphere occupying that very same region. Similarly, if being  kg and being kg are differently structured entities, then the Humean is not committed to holding that they freely recombine. It may be that no arrangement could assign them to the same place in a world structure. Complex elements, as well as the places they occupy, must first be divided into categories: elements belong to the same category iff they have matching internal structure. We then invoke the qualified version of (HPR) that requires only that elements of the same category freely recombine: any arrangement that assigns elements to places that respect their category represents a possible world. Determinates of the same category, with matching structure, freely recombine, but not determinates of different categories. This is the natural way to apply Humean recombination to elements that are complex. Although it allows that there may be necessary connections between complex elements, the core of the Humean view is that all necessary connections be explained. The focus has often been put too much on explanations that invoke distinctness.¹² But explanations that invoke differing structure may be no less in accord with the Humean view.

¹² Indeed, if Humean recombination is not understood in terms of duplicates, and mereological essentialism is rejected, then failure of distinctness is irrelevant to explaining necessary connections. Ross Cameron (b) has argued that the core of Humeanism is not that there are no necessary connections between distinct existences, but that all necessary connections can be explained. I concur, but would add: can be explained without invoking primitive modality.

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

      ? ()

There is a problem, however: how, on this view, can we satisfy the desideratum, (D), that there are no necessary exclusions among determinates of different determinables, whether or not those determinates have a different internal structure? There is no way of dividing determinate properties into categories such that a determinate of one determinable freely recombines with every determinate of any other determinable without it immediately following that all determinates belong to a single category, and so freely recombine with one another. In any case, taking determinates to be structural properties faces a decisive objection. Structural properties can only be instantiated by composite entities. But simples can have determinate quantities, at least if classical physics with its attribution of mass to point particles is not impossible. The only way to make the account of determinates as structural properties compatible with classical physics is to suppose that its point particles are really mereologically complex, indeed, have infinitely many parts. That is to demand too much. It is one thing to hold that an account of quantities demands that there be additional structure to the world; I will endorse that below. But to demand that the additional structure be internal to what instantiates the quantities, with the accompanying multiplication of entities, does not seem to me to be defensible. I conclude: the internal structure of determinates is not what explains the exclusions.

. Fourth Humean Response A fourth Humean response claims that neither the determinates nor the determinables are fundamental entities. In that case, (HPR) simply won’t apply to quantities. Since determinates are not fundamental, there is no reason, in principle, why they cannot necessarily exclude one another. Determinates will still be properties in the abundant sense. But abundant properties, of course, can necessarily exclude one another without violating Humeanism; just consider a property and its negation. What combinations of quantities are possibly co-instantiated will be a consequence of some theory of quantities that we posit. Let us simply stipulate that it will follow from the theory that determinates of the same determinable do, and determinates of different determinables do not, exclude one another. As long as we take this theory of quantities to be necessary, we can say that it explains the recombinatorial facts. In order to be strongly Humean, the necessity of the theory will somehow need to be reduced. But set that aside for now. Indeed, there are many such relational accounts of quantities on the market, socalled comparativist views, that implement this strategy. A familiar version comes from measurement theory.¹³ What is fundamental are relations between objects such as x is at least as massive as y and x together with y is equally massive as z. The facts about determinate properties, including how they can be represented by numbers, are then derived by way of proving representation and uniqueness theorems. On this view, an object has a determinate property, such as being  kg mass, in virtue of the mass relations it stands in to other objects in the world. We can still say that mass is ¹³ For measurement-theoretic approaches to quantities, see Krantz et al. ().

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  



fundamental in a derivative sense if the mass relations that ground the facts about mass are fundamental. But here I sense a confusion between how we get evidence for statements about determinate mass, and what the content of those statements should be taken to be. The mass relations are indeed closer to the empirical operations by which we measure mass, for example, using balances; we measure the mass of one body by comparing its behavior to the behavior of other bodies. But that is not a good reason to take the mass relations to be fundamental. In any case, the measure-theoretic approach has unacceptable consequences. It allows that an object’s mass would be different if the masses of objects elsewhere had been different. Indeed, if there are not enough objects in the world to prove the uniqueness theorem, then no object has a determinate mass. We need an account of determinate mass that makes it local in this sense: the determinate mass of one object does not depend on the determinate mass of its worldmates, or even whether it has worldmates at all. More generally, I claim that an account of quantities should affirm this: (D)

Determinates of fundamental quantities are local properties.

(D) does not entail that the determinate quantities are intrinsic: to say that the determinate properties had by one object do not depend on the existence or nature of other objects is not to say that it does not depend on anything external to the object. Should we take the further step of affirming that determinates are intrinsic? On the one hand, if determinates are intrinsic, that would explain why material objects within the same world can be compared with respect to their mass independently of the masses of other material objects in that world. The mass relations would be internal relations, supervening on the intrinsic natures of their relata taken separately. But it would also require that material objects in different worlds can be compared with respect to their mass; and that seems to me a step too far. For example, I do not allow that two worlds could differ only in that the objects of one have double the mass of the corresponding objects of the other. How can these claims all be reconciled? Say that a relation is world-internal iff it supervenes on the intrinsic natures of its relata taken separately, together with the structure of the world the relata inhabit and their location in that structure. Worldinternal relations need not be internal because the structure they depend on is not intrinsic to the relata. But whether a world-internal relation holds does not depend on the intrinsic nature of any material objects beyond the relata. Plausible examples of world-internal relations come from physical geometry. Consider the relation of having the same orientation, or handedness. This relation is not plausibly taken to be internal: whether it holds depends on the structure of the surrounding space. A rightand left-handed glove are differently oriented in three-dimensional Euclidean space, but not if embedded in a four-dimensional Euclidean space in which one can be “flipped” onto the other. Similarly, having the same length is arguably not internal: it depends on whether the surrounding space would allow one object to be superimposed on another. For objects in different worlds, there is no fact as to whether or not they have the same length. Similarly, we can say that there may be a determinate fact as to whether objects in the same world have the same mass, or as to what the ratio is between their masses. But for objects in different worlds, there are no such

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

      ? ()

determinate facts: they can only be compared by introducing some sort of counterpart relation that is based on global similarities between the worlds, or based on stipulation.¹⁴ This leads to the following demand on an account of quantities: (D)

The relation having the same determinate quantity is world-internal, but not internal. Determinate quantities are not intrinsic.

Taking determinate quantities in general, and determinate masses in particular, not to be intrinsic is controversial to be sure. I have space here just to say a few words in its defense. Note, first, that it won’t do to say that determinate quantities are always intrinsic because determinate vector quantities are not. Vector quantities have a magnitude and a direction. It might seem, intuitively, that both the magnitude and direction could be intrinsic features of a vector. But that can’t be right: it makes no sense to say that vectors in different worlds are pointing in the same direction. It only makes sense to compare the directions of vectors if they are embedded in a spatial (or spatiotemporal) structure with an affine connection. Then, in terms of that connection, one can define a notion of “parallel transport” along paths connecting the locations of the vectors, and one can compare the direction of the vectors by parallel transporting one to the other. Note, second, that in a non-Euclidean space whether two vectors have the same direction is path-dependent. Two vectors may agree in direction when brought together by parallel transport along one path, but disagree in direction when brought together by another path. Now, it seems to me clearly possible for magnitudes to be path-dependent as well. For example, it seems to me that there are worlds where two remote objects have the same mass relative to one path for transporting one object to the other, but different masses relative to another path. If a physical theory were to propose this, I would not respond: “But that’s impossible!” A general account of quantity, then, needs to allow for this possibility. And that in brief is why I endorse (D). On this approach, the intuition that mass is intrinsic can be explained by the fact that, in classical worlds, having the same mass is not path-dependent, just as an intuition that vector quantities are intrinsic can be explained by the fact that, in Euclidean worlds, having the same direction is not pathdependent. These intuitions can be accommodated within a general account of quantity that endorses (D).¹⁵ Let’s take stock. When objects are embedded within a world structure, they may have intrinsic properties that, together with the world structure and their location in that structure, determine their determinate quantities. For example, supposing mass is not path-dependent (as I will henceforth), the determinate mass of an object supervenes on the object’s intrinsic nature and how the object is embedded in its world structure. Duplicate objects within the same world may have the same or

¹⁴ On the use of counterpart relations for accommodating the intuition that objects in different worlds can be compared with respect to their mass, see Dasgupta (). ¹⁵ See Maudlin (: –) for some discussion of path-dependence within gauge theory. Maudlin claims that fundamental quantities at the actual world, such as quark colors, are path-dependent. Note that once we allow that determinate quantities may be path-dependent, we must distinguish between local determinates and global determinates. There are no global determinates for path-dependent quantities.

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  



different quantities of mass depending on the surrounding structure in which they are embedded. On this account, whatever the determinate mass properties are, they are not fundamental. What is fundamental is the intrinsic property that together with the world structure determines the determinate mass properties. But this intrinsic property, it seems, is just the determinable having mass. And this leads to a rejection of the fourth Humean response, which held that neither the determinates nor the determinables are fundamental.

. Fifth Humean Response We have arrived at a fifth Humean response, some version of which I accept: (D)

Determinables are intrinsic and fundamental.

I have argued that we can’t take both determinates and determinables to be fundamental, and that we can’t take determinates to be fundamental; but now there is no reason not to reconsider our original assumption that determinables should be analyzed in terms of determinates, and to take the determinables to be what is fundamental.¹⁶ This accomplishes two things. First, it allows us to explain why any two massive objects are qualitatively similar, and any two charged objects are qualitatively similar, but an object that only had mass and an object that only had charge would not be qualitatively similar. The sharing of fundamental properties grounds qualitative similarity. Second, the determinables freely recombine, just what the Humean expects of the fundamental properties. It is possible for an object to have one or both or neither of the determinables having mass and having charge. Moreover, some of the features traditionally taken to characterize the fundamental properties apply to determinables, not determinates. For example, when we say that physics posits only a few fundamental properties, such as mass, charge, and spin, we seem to be taking the determinables, not the determinates, to be fundamental.¹⁷ In taking the determinables to be intrinsic and fundamental, I mean to endorse a form of quidditism: two worlds can be structurally isomorphic but qualitatively discernible in virtue of having different fundamental properties occupying corresponding places within the world structure.¹⁸ The Humean could, in theory, accept combinatorialism with respect to fundamental properties without endorsing quidditism. In that case, permuting or replacing fundamental properties within a possible world would always result in a possible world—just not always a different possible ¹⁶ Wilson () defends the view that determinable properties are fundamental, although she allows that determinates may be fundamental as well. Denby () takes determinables to be more fundamental than determinates as a way of capturing the logical relations between them. ¹⁷ Hawthorne () discusses ways in which Lewis’s characterization of fundamental, or perfectly natural, properties sometimes better applies to determinables rather than determinates. But I don’t think there can be any doubt that Lewis’s considered opinion was that the determinates are fundamental. Lewis (: ) gives this characterization of the fundamental properties: “They are not at all disjunctive, or determinable, or negative.” ¹⁸ I distinguish quidditism from the weaker haecceitism about properties which holds only that structurally isomorphic worlds may differ by a permutation of properties, not that they may differ qualitatively. See Hildebrand () for discussion. (He calls the two views qualitative quidditism and bare quidditism.) [For arguments supporting quidditism, see: Section .; Section .; and Section ...]

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

      ? ()

world. But such a Humeanism would be toothless. The Humean recombination principles applied to fundamental properties would not generate new possibilities, just new ways of representing the same old possibilities. I will assume, henceforth, that the Humean endorses quidditism. It should be noted that the move to take determinables to be fundamental has been driven thus far not specifically by Humean concerns, but by my own commitments to quidditism and (a version of) comparativism. In the last section of this chapter, however, it will be apparent that taking determinables to be fundamental is integral to the solution to the problem of exclusion that I prefer. Claiming that it is the determinables that are fundamental, and not the determinates, is just the beginning of this fifth response. The Humean still needs an account of determinate quantities, and needs to show how the account explains the necessary exclusions. And it is natural at this point to try this: determinates arise from the way in which the fundamental determinables are instantiated, where the different ways of being instantiated correspond to the determinable occupying different locations in the world structure. But now it is high time I say more about world structures.

. World Structure: Horizontal and Vertical For any possible world, we can ask what the (total) structure of that world is. A complete description of the world is given by saying how the fundamental elements, whatever they may be, are arranged in the world structure. This shouldn’t be controversial. Indeed, world structures are tacitly presupposed by all recombination principles. It isn’t enough to say just that, for any elements, there is a world in which those elements all coexist. To capture the full plenitude of possibilities, one must also say that, for any arrangement of the elements, there is a world that arranges those elements in that way. And talk of arrangements only makes sense against a backdrop of structures within which to do the arranging. What is controversial is how world structures are to be understood. In particular, one can ask: Does the world structure supervene on facts about the relations among the elements? Or is structure an independent feature of worlds, in which case places in the structure may be unoccupied by any element? To grasp what is at stake, consider the case of spatial structure and responses to the problem of empty space, that is, space unoccupied by matter or energy. I suppose that it is possible for a Euclidean world to have a region of empty space. The problem is: in virtue of what can we say that that empty region is itself Euclidean in shape? There are three main views. One view grounds the shape of empty space in brute modality of some sort: were an object located in that empty region, it would be Euclidean in shape. But that view, of course, is anathema to Humeans. A second view grounds the shape of empty space in the spatial relations, not between material objects, but between unoccupied points of physical space that compose the empty region. To be empty of matter is not to be empty of any substance. This is the familiar response of the substantivalist about space. It upholds the supervenience of structure by positing additional elements. A third response grounds the shape of empty space in irreducible facts about the structure of the world. On this view, the world structure

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 :   



does not supervene on relations among the elements; structures are sui generis and abstract.¹⁹ Here I will remain neutral between these two views of world structure. Either view is compatible with an epistemology of geometry according to which we may have good reason to posit facts about the geometric structure of empty space. Similarly, either view is compatible with an epistemology of quantities according to which we have good reason, as I suppose, to posit a quantity structure whose places may or may not be occupied by determinable elements. The spatial or spatiotemporal structure of a world are instances of what I call horizontal structure, the structure that unites the objects of a world to make a single world. In our world, according to general relativity, it is four-dimensional spatiotemporal structure that is locally Minkowskian, but with variable curvature. According to Lewis, the horizontal structure had by all worlds is spatiotemporal structure, structure determined by fundamental relations that are either spatiotemporal, or analogous to the spatiotemporal relations. I have argued elsewhere (in Chapter ) that this conception of horizontal structure is too narrow. There are possible worlds whose horizontal structure is not spatiotemporal but is instead provided by other sorts of fundamental external relations. But I will continue to focus on worlds with spatiotemporal structure in what follows. Vertical structure is, perhaps, less familiar; but it must be posited by any account of the world that is realist about properties, and so allows that there are multiple elements located at a single point of spacetime. Vertical structure unites the properties, or the properties and a particular, into a single object. In the simplest case, the vertical structure could just be a cardinal number assigned to each point of spacetime that represents how many properties are, or might be, instantiated at that point. (Compare Armstrong’s metaphor of a “layer cake,” where the properties are “stacked up” one on top of the other.) If the properties in question are scalar quantities, however, more structure is needed: at each spacetime point, we need the structure of the non-negative reals to give the magnitude of the determinable that is instantiated at that point. For vector quantities, it seems, we need even more vertical structure: at each spacetime point, we need a four-dimensional space in which vectors can live, and have determinate magnitude and direction, the tangent space associated with the point. And for more complex quantities, perhaps more structure is needed. But let us focus, for now, on the case of (extensive) scalar quantities. The simplest way to think about how vertical and horizontal structure make up the world structure is to take the world structure to be the product of vertical and horizontal structure. For example, if horizontal structure is a four-dimensional spacetime and vertical structure (to account for scalar quantities) is a onedimensional half real line, then the location of a scalar quantity is given by giving five coordinates: four to give where the quantity is located in spacetime, and one to give its magnitude. Thus, taking account of vertical structure (for scalar quantities)

¹⁹ The distinction between these latter two views tracks the debate between in re and ante rem structuralism in the philosophy of mathematics; see Shapiro (). There is also a mixed view according to which points of physical space exist only where no matter or energy exists. [I endorse this mixed view in Section .. A place in a structure that is unoccupied by any qualitative element is occupied by a “bare particular.” That allows for a nominalist account of structure.]

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

      ? ()

makes the world structure five-dimensional. More generally, every location in the world structure can be divided into two components, a location in the horizontal structure and a location in the vertical structure. This simple product account of horizontal and vertical structure, however, is insufficiently general; it posits too much structure. On this account, whether or not two instantiations of a determinable at different spacetime points have the same magnitude supervenes on the locations of those two instances. I want an account of world structure according to which locational properties carry less information.²⁰ Such an account must distinguish between the local vertical structure, the structure at each spacetime point, and a global structure that connects all the local vertical structures. The local structure, we are supposing, must at least have the structure of the half real line so as to capture the magnitudes of extensive quantities. The connection then tells us when one point of a half real line in one local structure is to be identified with a point of a half real line in another local structure, thereby calibrating units across different points of spacetime. But without the calibration, there are no facts as to whether, when the mass determinable is instantiated at two different spacetime points, the determinate masses are the same or different. In other words, the relation, having the same magnitude, does not supervene on the locational properties of its relata. Now, in terms of this account of world structure, we can give an analysis of what it is for a point object o located at a point of spacetime p to have the determinate property, having magnitude m, of some determinable d. Identify magnitudes with classes of vertical locations that, according to the connection, have the same value. Then, o has m of d iff, for some vertical location v, d is located at and v is in m.²¹ How does this account relate to comparativist accounts that take relations such as having the same mass, having the same charge, etc. to be primitive relations holding between objects or properties? For one thing, the relevant relations on my account hold between locations in the world structure, and so do not depend on the existence of objects or properties to provide the relata. Moreover, speculatively, I would argue that, at each spacetime point, there is a single local vertical structure in which all quantities live, and there is a single global connection; there are not separate structures and relations for each quantity. Thus, having the same mass reduces to having mass and having the same magnitude. Even more speculatively, I would identify the single local vertical structure with the tangent space associated with a point of spacetime interpreted as the infinitesimal neighborhood of that point, and the global connection with the connection as characterized by the spatiotemporal structure. On this account, the vertical structure is reinterpreted as an enhanced horizontal structure; at worlds with spacetime, all world

²⁰ Often it won’t matter to the quantitative truths at the world because that information will be available by other means. But when comparisons of quantities are path-dependent, it matters. ²¹ Note two things. First, the magnitudes are not numbers; rather, we represent a magnitude by a number, its coordinate, by conventionally choosing one of the magnitudes to serve as the unit. Second, if a quantity is path-dependent, then there are no determinate magnitudes for that quantity. In that case, we can say there are local determinates in virtue of the determinable for that quantity occupying a location in the vertical structure at spacetime points; but there are no global determinates for that quantity.

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   :  



structure is spatiotemporal.²² Much more would need to be said about the conceptual and technical underpinnings of such as account. But for reasons of space, I need now to return to the problem for Humeanism with which we began: the problem of exclusion.

. Refining the Fifth Response: Identifying Arrangements Whatever the merits of the account I have given of determinable and determinate quantities, with respect to the problem of exclusion it appears to be a total failure. The Humean must claim that the fundamental properties, the determinables, can be freely recombined within the world structure. And that gives the wrong results. For there are multiple distinct vertical locations corresponding to each location in spacetime, and the Humean will be committed to saying that a single determinable can occupy more than one such vertical location. But that would correspond to a location of spacetime instantiating different determinates of a determinable. We have made no progress with the problem of exclusion. We have no explanation, for example, for why the determinable of mass can occupy different horizontal locations, but not different vertical locations at the same horizontal location. The Humean seems to be in a bind: either there are multiple locations at a spacetime point for the determinable to occupy, or there is only one. If there are multiple locations, then it seems that the Humean is committed to allowing different determinates of a single determinable to be instantiated at a spacetime point. If there is only one, then it seems the Humean will have to posit fundamental relations holding between instances of a determinable at different spacetime points to capture the ratios of their magnitudes, relations such as having the same mass as, having twice the mass as, etc. But then these posited fundamental relations will violate (HPR): if two instances of the mass determinable stand in one mass ratio, they cannot stand in any other. Or perhaps, to avoid a violation of (HPR), the Humean could go adverbial: the uncountably many ways for the mass determinable to be instantiated at a spacetime point do not correspond to different recombinations of the elements. But one still needs modality to say why, if a mass determinable is instantiated -kg-ly at a spacetime point, it cannot be instantiated -kg-ly at that point. Strong Humeanism is still threatened. Let’s take stock once again. The problem with the present account is that there are too many arrangements, resulting in too many possibilities. Now, there are two general strategies for cutting down on the arrangements. The most obvious strategy is to rule out some arrangements as impossible. That strategy immediately raises the specter of primitive modality. I will return to it below. But there is a different general strategy that I want to consider first. On this strategy, we say that distinct arrangements need not correspond to distinct possibilities. This is no violation of (HPR) which says only that each arrangement corresponds to some possibility. The arrangements, after all, were introduced to represent possibilities; they are not the ²² I argued for this view on somewhat different Humean grounds in Chapter .

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

      ? ()

possibilities themselves. On this strategy, we can say that we are “identifying” arrangements that correspond to the same possibility. It will be useful first to see how this strategy of identifying arrangements is needed to solve a different problem with the present account. The problem is this. On some arrangements, determinables for different quantities will occupy the same place in the vertical structure at a spacetime point (assuming, as I do, that there are not separate vertical structures for the different quantities). But that is too much information: there isn’t any genuine fact as to whether, say, the mass at one point does or does not have the same magnitude as the charge at that point; and the global connection gives no information as to how the magnitude of mass at one point compares to the magnitude of charge at another point. To eliminate this fake news, we need to identify arrangements that differ only in that all the instantiations of a single determinable have had their magnitude uniformly scaled up or down. Arrangements that differ only in this way represent the same possibility.²³, ²⁴ Now, the thing to notice about this solution for our purposes is that it cuts down on the arrangements without invoking modality. In the course of identifying arrangements, no arrangement is deemed impossible. Can something similar work to solve the problem of exclusion? Call an arrangement bad if it ever assigns a determinable to multiple places at a single spacetime point; call it good if it doesn’t. What we need is a way to divide the arrangements into equivalence classes so that every bad arrangement gets identified with exactly one good arrangement. To illustrate: consider the case where the vertical structure at each spacetime point is a half real line. Then we could say: arrangements are equivalent iff, for any spacetime point and any determinable, the magnitudes assigned to the determinable at the point by the arrangements have the same greatest lower bound. We can then say that equivalent arrangements represent the same possibility, the possibility represented by the good arrangement that assigns each determinable at each point to that greatest lower bound. So no possibilities are represented in which a determinable takes on multiple values at a single spacetime point. The problem of exclusion is thus solved without invoking modality. Or is it solved? I hope it is obvious that, in this case, the method is a cheat. What we have is a gimmick for ignoring bad arrangements. There is no natural sense in which the bad arrangements represent the same possibility as the good arrangement. The proffered representation relation between arrangements and possibilities depends on an arbitrary choice. Moreover, when we switch to consider vector quantities, there

²³ Note bene. If different arrangements represent the same possible world, then the underlying structure of the world is not the same as the structure within which we do the arranging. In this case, the underlying world structure is sensitive to the number and types of quantities instantiated at the world. I see no way to avoid this consequence without an unfortunate multiplying of possibilities. ²⁴ The strategy of identifying arrangements can also be used to solve the following problem: how can a single vertical structure be used to represent both scalar and vector quantities? For the vector quantities, we need the vertical structure at a spacetime point to be the tangent space at the point. But that is too much structure for the scalar quantities. If we assign a scalar determinable to a location in the tangent space, we wrongly attribute a direction to the scalar quantity. But the strategy in question gives us a natural way, in effect, to ignore the direction. We can identify arrangements that assign a scalar determinable to locations in the tangent space with the same magnitude but with different directions.

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   :  



will be nothing analogous to the greatest lower bound that would allow this method to be adapted to the problem of exclusion for directions. The strategy of identifying arrangements will not solve the problem of exclusion.

. Refining the Fifth Response: Omitting Arrangements Time for the Humean to face the music. The Humean must remove the bad arrangements from the scope of (HPR). And that implicitly invokes modality. We can say which arrangements are bad without invoking modality, but not why they are bad. What makes an arrangement bad is that it is not metaphysically possible; it doesn’t represent the way any possible world could be. But all is not lost: to invoke modality is not to be committed to primitive modality. The Humean does not reject modality when it can be reduced to logic and definitions. And the Humean can allow that there are substantive, objective facts as to what the true logic is, and what the linguistic conventions are. Indeed, I am a Humean with a robust conception of logic, and a robust notion of absolute modality that comes with it. I am a realist about mathematical systems and possible worlds and mixtures and generalizations of the two. Whatever is not ruled out by logic (in a broad, non-formal sense of ‘logic’) exists somewhere in reality.²⁵ Quantifying over this expansive reality leads to a Humeanly acceptable notion of absolute modality according to which only logic and definitions are absolutely necessary. The question before us is: does this realist account make the exclusions between determinates absolutely necessary? That depends. There are two ways for the Humean to approach this question. One approach takes the exclusions to be metaphysically necessary in virtue of conventions that govern the meaning of ‘metaphysical necessity’; but they are not taken to be absolutely necessary. The other approach understands logic in a way that makes the exclusions absolutely necessary. I consider these approaches in turn. On the first approach, it is not a deep fact that determinates necessarily exclude one another; it tracks no joint in reality. It is a superficial fact that reflects a particular conventional definition of ‘necessity’, what philosophers refer to as ‘metaphysical necessity’. Some philosophers, ancient and modern, have thought that there are deep facts about ‘metaphysical necessity’ having to do with real essences, and what not; but they are deluded in this, and the delusion has become widespread. The Humean succumbs to no such delusion. Starting from absolute necessity, she can define a restricted notion of necessity to match these deluded philosophers’ usage. That determinates necessarily exclude one another can simply be a part of this definition, and in that way be true by convention. I need this strategy to be available in any case, whether or not I use it to solve the problem of exclusion. My notion of absolute modality does not agree with any notion of modality typically used by philosophers or the folk. On my notion, much more is ²⁵ See Section ., and Chapter  together with its postscript. One thing that makes my logic “broad” is that it includes mereology and higher-order plural quantifiers. Another is introduced below.

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

      ? ()

possible, much less is necessary. Mathematical truths, for example, are true at some mathematical systems and not others, and therefore not absolutely necessary. Moreover, I believe that there are world-like entities that have any sort of mathematical structure as their horizontal structure. The existence of such entities is absolutely possible—they are part of reality—but their existence would not be deemed metaphysically possible by most modal metaphysicians. To speak their language, I need to define a notion of modality that matches their usage.²⁶ Why not just include as part of the meaning of ‘metaphysical necessity’ that determinates of a determinable necessarily exclude one another? In that way, I get the necessary exclusions for cheap. Note that this is a version of the second Humean response considered above: the exclusions are conceptually necessary, following from conventional definitions, but not absolutely necessary. There are world-like entities in reality at which, say, the determinable of mass is badly arranged. But since these world-like entities are beyond the range of our defined necessity operator, we can truly say that, necessarily, determinates of a determinable exclude one another. Moreover, because the necessity of good arrangements is analytic to ‘necessary’, not to ‘determinable’ or to terms for quantities such as ‘mass’, we can truly say de re of a determinable that it necessarily never has two determinate values at a single point.²⁷ What more could a Humean want? I am not content. The badly arranged world-like entities are now deemed impossible by definition, but they are still a part of reality. And so it still makes sense to wonder whether I am located in one of them. I cannot know a priori that that is not the case, and so this response fails to meet (D), one of the desiderata on an acceptable account. Nor would it help to say: I know I am located in a possible world a priori in virtue of knowing a priori that I am actual and that whatever is actual is possible. For on the view in question, I cannot know a priori that whatever is actual is possible. Would it help if the Humean rejects my realism about the absolutely possible? Realism makes the problem more vivid, perhaps. But any Humean who accepts a robust notion of absolute possibility according to which the bad arrangements are absolutely possible will be in my boat; the bad arrangements cannot be ruled out a priori. At best, only a Humean who takes all logic and modality to be conventional should consider availing herself of this response. That leaves the second approach: take the logic that governs the instantiation of determinables to be the source of necessity. On this approach, we don’t divide arrangements into the good and the bad and then throw out the bad. We use a non-traditional logic to reconfigure what the possible world structures are so that it follows that there are no bad arrangements. The problem of exclusion turns out to be an artifact of our reliance on traditional logic. It results from the overly restrictive assumption that properties (and relations) are all or nothing, that there is only one way for a property to be instantiated. Instead, I propose, we should take determinables to belong to different logical categories depending on how they are instantiated. In the simplest case, determinables of extensive scalar quantities, we can make use of a many-valued logic whose values come from the non-negative reals. Thus, the ²⁶ That is what I took myself to be doing, for example, in Chapter . ²⁷ Sider (: –) makes a similar point in connection with Armstrong’s reductive account of modality.

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   :  



determinable of mass is assigned a value at each spacetime point that, in ways already discussed, gives rise to a determinate magnitude. The exclusions come, not from any linguistic conventions, but from logic itself.²⁸ I expect the following objection. “The problem of exclusion has not been solved, but merely shifted to another domain. We can now ask: why is it necessary that a determinable have exactly one value assigned by the logic relative to a point in the domain? What explains that necessity?” But this objection misunderstands the Humean’s commitments. It was no part of Humeanism that combinatorialism be applied even to the elements of logic, thereby generating alternative logics. No Humean principle of recombination entails, for example, that a proposition can be both true and false. The Humean is entitled to hold, as I do, that there is one true logic, and that the values assigned by that logic hold exclusively. Since a determinable having multiple values relative to a point in the domain is ruled out by logic, it is absolutely impossible. There is a terminological issue that needs to be resolved. Up to now, ‘determinable’ has been taken to refer to all-or-nothing properties. These all-or-nothing properties are still around (on an abundant conception), but they are not what our present account takes to be fundamental. So perhaps it would be better to call the fundamental, many-valued elements ur-determinables: they ground the instantiation of the all-or-nothing determinables that, necessarily, are co-instantiated with them. Terminology aside, what matters is that the fundamental quantities, on the present account, are of a different logical type than the fundamental quantities posited by traditional accounts. Will this account generalize to vector quantities, and more complex sorts of quantity? I see no reason why not. The values that the logic assigns to a determinable, say, of a vector quantity at a point of spacetime will have both a magnitude and a direction; there will be not just a continuum of values, but a space of values with a more complex structure. Whatever vertical structure we posited previously to account for some quantity can now be taken to derive from the logic of the determinable associated with that quantity. There are delicate and substantial questions, however, that I leave here unanswered, questions as to how logic relates to world structures, and, in particular, as to how vertical and horizontal structure are integrated with one another. For this approach to work, the Humean must hold that logic alone determines what world structures, and what arrangements of elements within those structures, are absolutely possible. Answers to these questions about logic and structure will be needed to configure the boundary of the absolutely possible.

²⁸ Interestingly, this appears to be Wittgenstein’s solution to the problem of exclusion, in which case he was not abandoning Humeanism (as I understand it) when he abandoned logical atomism. He claims that the truth table assignment to two elementary propositions that exclude one another will be seen not to be a possible assignment once we have developed the proper logic, the proper “syntax” of the elementary propositions. See Wittgenstein (: –).

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OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

Index a priori knowledge , –, –, , –, –, –, –, –, , –, , , –, –,  absolutism (or absolute realism) –, ,  abstract entities –, , , , , , , , ; see also concrete, vs. abstract abstract particulars, see tropes abstract worlds, see ersatzism abstraction , ,  abstractionism, see ersatzism abundant conception of properties or propositions , –, , , , , –, , , –,  accessibility relation ,  acquaintance , , , –,  actual by courtesy ,  actual world, more than one , –; see also island universes actualism , , , , , , , , , – actualist quantifier, see quantifier, inner and outer actuality , –, –, –, –, – absolute , , –, , –, –, , ,  categorial , , –, ,  concept vs. property of –, – indexical , –, , –,  relative  sparse property of  see also actualization actuality operator – actualization: primitive ,  unconditional , ,  universal –, , –, ,  see also actuality Adams, Robert , , , , , – agnosticism about metaphysical theories ; see also skepticism about metaphysical theories alephs, as an absolute measure of size , –, – algebra, abstract ,  alien individuals and properties , , , , –, , –, –, –, –; see also plenitude of world contents analytic , , , , , , , , , ,  anti-haecceitism, see haecceitism

Armstrong D. M. , , , , , , , , , , –, –, , –, , , , , , –, –, , –, , , , , –, ,  arrangements: categorical – category-preserving , ,  consistent  definition of , ,  full ,  identifying – non-overlapping ,  atom, mereological , –, –, –; see also gunk atomic propositions, see propositions, atomic bad company objection  Balaguer, Mark , , ,  bare particular –, , , ,  barely many –, , ; see also maximally many Baxter, Donald , ,  Bell, John L.  Bergson, Henri – best system analysis and laws of logic – Bigelow, John , , , , – Bohn, Einar , , , ,  Boolean algebra , , , , ,  Boolean operators , , , , , – Boolos, George , , ,  boundary, in logical space – boundary points, in space or spacetime – Boyd, Richard  Bradley, F. H. –, ,  brute counterfactuals and dispositions , –, ,  Burgess, John , ,  Cameron, Ross , , , ,  Campbell, Keith  Cantor, Georg , ,  cardinality argument: against linguistic ersatz worlds , – against modal realism –,  see also Forrest-Armstrong argument Carnap, Rudolf , – Carroll, Lewis  categorical properties, vs. modal –, –, –,  categorical theories , , –

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi





categories, see ontological category Chalmers, David , ,  charge, relational – Chisholm, Roderick  circularity objection to linguistic ersatzism , –, – circularity objection to modal realism – classical logic, see logic, classical classical mereology , , , , –, , ,  classical physics –, , , , ; see also scientific theories coherence: as sufficient for truth –, –, –, – vs. consistency , ,  coincide, applied to portions of reality –; see also coincident entities; identity, generalized coincident entities , –, –, ; see also counterpart relations, multiple Collapse , , ,  co-location –, –,  color , – combinatorial conception of modality , , , , –, , , , , , ; see also Humeanism; plenitude, Lewis’s principle of recombination comparativism (with respect to quantities) , ,  completeness (of truth at a world) –; see also determinate; witnessing (of truth at a world) component, general notion of , – composition, as analogous to identity, see composition as identity, weak composition, deflationary notion of , , ,  composition, unmereological , , , , –, , –,  composition as identity: formal theory – informal characterization –, – moderate –, – strong –, , –, –, – weak –, , ,  see also identity, generalized comprehension principles – conceptualist reduction of worlds  concrete, vs. abstract –, , –, , –, ,  congruence (of parts of space) , , ; see also distance, intrinsic conception consciousness , –, ; see also experience consistent –, , –, –, –, –

contingency: of actuality –, –,  of mathematics – of mereology – continuum hypothesis , ,  contradictions , –, , –; see also dialethism conventionality: of logic  of metaphysical modality , , , – of spatial metric  copies, see recombination correspondence theory –, ,  Cotnoir, Aaron ,  counterfactuals , , , –, , – counterpart relations: admissible –,  multiple , , , , – truthmaking – see also essence; modality, de re counterpart theory , –, –, , –, , –, ,  Cowling, Sam , ,  creation myth , –, , – creation version of Leibnizian realism, see Leibnizian realism, creation version crossworld identity, see transworld identity cube worlds –, –, – curvature, of space or spacetime , , –, –, , ,  Darby, George , , – Dasgupta, Shamik  Davidson, Donald ,  Davis, Cruz  de dicto, see modality, de dicto de re, see modality, de re; plural de re de se, see knowledge de se demarcation of worlds , , , , –, –,  dialethism ,  Democritean worlds , –, , ,  Demopoulos, William  determinables, see determinates and determinables determinates and determinables –, , , , , – determination, see ontological determination difference-making ,  differential geometry , , , , , , – discernibility, absolute vs. relative ,  disconnected: externally , , ,  spatially or spatiotemporally , , , , , , ,  see also island universes

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

 discrete space, time, or spacetime , –, , , , – distance: Gaussian conception of – intrinsic conception of –, – naïve conception of – distinct, vs. non-identical , , – Divers, John  Domain (in Lewis’s theory of classes)  Dorr, Cian ,  dualism (about the mental) , ,  Dummett, Michael  duplicates: defined in terms of perfectly natural properties –, , , , ,  internal vs. external , ,  intrinsic vs. local – E Pluribus Unum (EPU) –; see also Unrestricted Composition economy , , ; see also Occam’s razor; ontological parsimony Eddon, Maya  Efird, David , – Einstein, Albert –, , – Einsteinian supervenience – Eklund, Matti , , – elasticity principle ,  emergent, see properties, emergent empty space , – empty world –, ; see also nothing, possibility of Enderton, Herbert  English, see ordinary language entailment: and supervenience –, , –, ,  and truthmaking , –, , ,  entailment principle (EP) – epistemic chauvinism , – epistemic counterpart –, – ersatzism: linguistic , –,  magical , , , ,  pictorial , , ,  see also Leibnizian realism especially rich world –,  essence and essential properties , –, –, , , , –, , –, –, , , , , , –, , ; see also counterpart theory essentialism, mereological , , , , –, ,  Etchemendy, John  exclusion, problem of –, –, –, –; see also determinates and determinables



experience –, , –, –,  explanation (in metaphysics) , , , –, – extended simple –, ,  external relation, see relation, external extrinsic properties and relations: and spacetime , , –, –, – and truthmaking –, , , – see also intrinsic properties facts, see states of affairs factualism –; see also states-of-affairs theory fictional entities  Fine, Kit , , , , , , , , ,  Fischer, John  Forbes, Graeme ,  formal (applied to logical relations) , ,  formalism , , , – formalization (of modal discourse) – Forrest, Peter , , , , ,  Forrest-Armstrong argument , , , –, , –, , , – foundation (epistemological) , , , , –, , ,  framework, principles of , , , –, , –, –, , , , – free lunch , –, , ,  Frege, Gottlob , , , , –, , –, , ,  French, Steven  Friedman, Michael  Fritz, Peter ,  fully determinate , , –, , –, ; see also maximal consistent fundamental properties and relations , , , –, , , , –, –, , , , –, , , –, –; see also natural properties and relations Fusion vs. Weak Fusion ; see also composition as identity Gaussian conception of distance, see distance, Gaussian conception of Geach, Peter  general facts and propositions , –, – general relativity , , , , – generalization, natural (in mathematics) –, , – generalized identity, see identity, generalized geometrodynamics ,  geometry , –, –, –, ; see also differential geometry Gibbard, Alan 

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi





Gibbs, Cameron ,  Giganto  Global Choice  global supervenience , , – global vs. local unification (of worlds) –, –, –,  Gödel, Kurt ,  Goodman, Nelson ,  grasping –; see also acquaintance grounding –, –, –, ,  Grünbaum, Adolf ,  gunk , , –, –, , –, –, ,  haecceity –, ,  haecceitism , –, , –, –, –, , , ,  Hale, Bob  Hallett, Michael  handedness , , ,  Hare, R. M.  Hawthorne, John , , , , , –, ,  Hazen, Alan , , , , , , , , ,  Hempel, Carl  heterogeneous world  Hilbert, David –, ,  Hildebrand, Tyler  Hintikka, Jaakko  Hodes, Harold  homogeneous matter , , ; see also cube worlds homogeneous world , , – Horgan, Terence  horizontal structure, see structure, horizontal Hume’s Dictum , –, –, , , –; see also Humean denial of necessary connections Humean denial of necessary connections , , –, , –, , , , –, , –, ; see also Hume’s Dictum Humean supervenience –,  Humeanism , –, , –; see also combinatorial conception of modality; plenitude, Lewis’s principle of recombination hyper-co-location – hypergunk – idealism , ,  ideally rational thinker – identity , , , –, –, , , , , , , , –, – broadened sense –,  generalized , , , –, –

kind of –, –; see also composition as identity, moderate many-many and many-one –, –,  numerical , –,  partial , , ,  plural , –, ,  relative  total , –,  universality of –,  see also composition as identity identity of indiscernibles , , ; see also worlds, indiscernible ideology , , , –, , ,  immortality – impossible worlds , , –, , ,  inconstancy, see counterpart theory indefinite extensibility , , , ,  indeterminacy , , , –, –, , , , ; see also vagueness indexicality –, , –, –,  indiscernibility: qualitative , , –, , , , –,  structural – indiscernibility of identicals; –, , , –; see also Leibniz’s Law indiscernible worlds, see worlds, indiscernible infinitary languages –, ,  infinitesimals, see non-standard analysis infinity machine – innocence, see ontological innocence instantial structure, see structure, instantial instantiation (of structures): elementary  model-theoretic – natural – see also structure, instantial instantiation relation –, , , , ; see also relation regress intensional entities , , –, , ; see also propositions; representation, realm of intentionality , , , –, –, – internal relation, see relation, internal intimacy –; see also composition as identity, moderate intrinsic properties , , , , –, , –, , – intrinsic nature –, , –, –, , , , , , , –, , , –, , – intrinsically distinct – intuitions (as support for metaphysical theorizing) , –, –, –, , , , , –, , , , , , , , , –,  islands of reality , –, –, , –, ; see also island universes

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

 island universes –, , , –, –, –, , –, , , , , , , –, , ,  Ismael, Jenann  isolation , –, , , –, , –, , –, –, , –, , ,  isometry –, – iterated modality, see modality, iterated iterative conception , –, –, –; see also set theory Jackson, Frank ,  Jubien, Michael ,  Kamp, Hans ,  Kant, Immanuel , , – Kaplan, David , , , ,  Kim, Jaegwon , , ,  Kitcher, Philip  Kripke, Saul , , –, , , , ,  knowledge de se , , , –; see also perspectivalism Kruskal, Martin  Ladyman, James  Lange, Marc  Langton, Rae , , ,  large world ; see also Size of Worlds laws of logic, see logic, laws of Lechthaler, Manuel  Leibniz, Gottfried –, , , , ,  Leibniz’s Law , , , , , , –; see also substitutivity of identicals Leibnizian realism –, –, –, – creation version , –, – one-property version –, ,  traditional –, –, – transformation version , , –, ,  see also actuality, absolute Leslie, John  Lewis, David: on actuality , –, –, –, – on classes , , –, , –, , , – on mereology , –, –, –, –, – on modality , , –, –, –, –, –, –, , , , , , , –, –, –, – on plenitude –, –, –, –, –, – on properties , , , , , , , , , 



on realism , , , –, –, , –,  Lewisian realism , –, –, –, –, –, , –, –, –,  Limitation of Size – linguistic ersatzism, see ersatzism, linguistic Linnebo, Øystein ,  local properties –,  local unification, see global vs. local unification (of worlds) logic – alternative , ,  classical , , , ,  external and internal conception of – laws of , –, , , ,  many-valued – mereology and , , , –, ,  logical constants – logical modality, see modality, logical vs. metaphysical logical positivism ,  logical space , –, –, , –, , –, –, , , –, ; see also modal space logicism , ,  Lovejoy, Arthur  Lycan, William  MacBride, Fraser ,  McDaniel, Kris , , , , , – MacFarlane, John ,  McGinn, Colin  McLaughlin, Brian  magical ersatzism, see ersatzism, magical magnitudes – manifold structure , , ,  Markosian, Ned  Martin, C. B.  mass , , –, , –, , , –, – mass density – match (between domains and structures) , ,  materialism , , – mathematics: contingency of – modal knowledge and –,  physics reduces to – transfinite longevity and – mathematical realm (or reality) , , –, , –, –, ; see also mathematical systems; modal realm (or reality) mathematical systems , , , –, –, , , , , , , –, , , , , – mathematical theories –, 

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi





Maudlin, Tim  maximal consistent –, –, , , – maximalism , , –; see also plenitudinous realism maximally many , ; see also barely many measurement (in geometry) , – megethology , , , ,  Meinongianism – memory (and personal identity) –,  Menzel, Chris  mereology, see composition as identity mereological atoms, see simples mereological essentialism , , , , , ,  mereological harmony, see simples, extended metaphysical reality (contrasted with physical reality) –, –, – metric (in space, time, or spacetime) –, –, , –; see also distance metric space , – Miller, Richard  modal discourse –, – modal intuition –, , , , , –, , , , , , –,  modal operators, analysis of , , –, –, , , – modal realism, see Lewisian realism modal realm (or reality) , , , –, –, , , , –, –, ; see also logical space; mathematical realm (or reality) modality: absolute –, , –; see also modality, logical de re –, –, –, , –, , , , , , ; see also counterpart theory; plural de re analysis of –, , –, , ; see also modal operators, analysis of conceptual , , , , , , ,  epistemic –, , – epistemology of – iterated – logical –, –, –, –, , –, , – metaphysical (contrasted with logical) –, , –,  natural (nomological, physical) –, –, , ,  primitive, see primitive modality moderate composition as identity, see composition as identity, moderate monism, priority , 

natural break (in set-theoretic hierarchy) , – natural language, see ordinary language natural numbers , , , –, , , , , ; see also Peano arithmetic natural properties and relations , , , –, , , , –, , , –, , –, –, –, ; see also fundamental properties and relations naturalness condition (on reduction) , ,  necessary connection , –, , , , –, –, , , , , –, , –, –, ; see also Humean denial of necessary connections necessitism (Williamson)  negative existentials, truthmakers for , –, , , – Nerlich, Graham  Newtonian spacetime , , , , , , ,  Nolan, Daniel , , , , , –, , , , , ,  nominalism , –, –, , , , , , , , , , , , ,  non-classical logic, see logic, alternative; logic, classical Non-Entailment Thesis –, ,  non-Euclidean geometry –, , –,  non-standard analysis – nothing, possibility of –, , ,  numbered class , –, – numerical identity, see identity, numerical objective chance – Occam’s razor ,  Oliver, Alex  ontological base – ontological category , , –, , –, –, –, , –,  ontological commitment , , , , , –, , ,  ontological determination –, –,  ontological innocence , , , , , , , ,  ontological kind, see ontological category ontological parsimony , , , , , ; see also Occam’s razor ontological pluralism –, , ; see also ontological category ordinal numbers –, –, , –, , – ordinary language , , , , , , , , , , , , ,  orientation, see handedness

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

 Parfit, Derek  Pargetter, Robert  parochialism , –, , –, ,  parsimony, see ontological parsimony Parsons, Josh ,  Parsons, Terrence , ,  Parthood Principle (in truthmaking)  partial identity, see identity, partial particulars, thick and thin –, , –, , , – particulate structure – path (in space or spacetime) , , –, ; see also distance path-dependent quantities ,  pattern of instantiation (of fundamental properties and relations) –, , , , , , , , , , , –; see also structure, instantial Peacocke, Christopher  Peano arithmetic , , , –; see also natural numbers perdurantism  perfectly natural properties, see natural properties permutation invariance  Perry, John , ,  perspectival concept –, –,  perspectivalism , –, , –; see also epistemic chauvinism phenomenal content , , ,  phenomenalism – physical reality, see metaphysical reality (contrasted with physical reality) physics and physical theories , –, , , , –, –, , , –, , –, , – pictorial ersatzism, see ersatzism, pictorial Plantinga, Alvin ,  platonism , , , , , ,  plenitude, law of –, , , –, ,  plenitude, principles of , –, , –, –, –, –, –, – generalized principle of solitude (GPS) –, , – generalized principle of solitude for particulars (GPSP)  Lewis’s principle of recombination (LPR) , –, –, –, –, –, – principle of alien individuals (PAI) , ,  principle of atomic diversity (PAD) – principle of contingent existence (PCE) , – principle of duplication (PD) –, , –,  principle of heterogeneity (PH) –



principle of interchangeable parts (PIP) , –, , , –,  principle of plenitude for structures (PPS) –, –,  principle of solitude (PS) –, , , , , –, –,  principle of structural diversity (PSD) – substructure principle (SP) – unqualified principle of recombination, see unrestricted principle of recombination unrestricted principle of recombination (UPR) , –, –, – plenitude of world contents , –, , –, , ; see also plenitude, principle of interchangeable parts plenitude of world structures –, , –, , , –, , , –, , ; see also plenitude, principle of plenitude for structures plenitudinous realism –, –, –, ; see also maximalism; plenitude, law of plural de re – plural identity, see identity, plural plural logic and quantifiers , , , , , , –, –, –, , , , , –, , , –, , –, – higher-order , , , , ,  pluralism (vs. singularism) ,  Poincaré, Henri  point particles –, ,  points of matter –, , – point-sized –, , , ,  portion of reality (as plural term) , ,  possibilism, vs. Meinongianism  possibility, see modality possible individual, see spatiotemporal part possible worlds, see worlds pragmatic criteria (for theory choice) , –, –, –, ,  pragmatism (about truth) , – predications, truthmakers for: essential  intrinsic –, , –, – presumption (in favor of possibility) – Priest, Graham , ,  primitive modality –, –, , , , –, –, –, , , , , ,  principles of the framework , , –, ,  principles of plenitude, see plenitude, principles of problem of exclusion (for quantities) – projective geometry –

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi





proper classes , , , , , , , , ,  properties: all-or-nothing , , – directional – distributional  emergent , , , , – extrinsic, see extrinsic properties and relations intrinsic, see intrinsic properties non-qualitative –, , ,  qualitative , , , , , , , , –, , , –, , –,  relative –,  slice-sensitive , –, –, , – structural , –, – see also natural properties and relations propositions: and logic – atomic –, , , , , –, –, –, – existential , , , , , , –,  general, see general facts and propositions mathematical , –, – plenitude of –, ; see also abundant conception of properties or propositions singular –,  structured vs. unstructured , , , –, , , –, –,  Putnam, Hilary  qua-names (in truthmaking) –, –, , – qualitative character , –, –, , –, , , , ; see also properties, qualitative quantified modal logic – quantifier, inner and outer –, , –,  quantities – quantum mechanics , , , –, ,  quidditism –, , – quiddity –, –, , , –, , ,  Quine, W. V. , –, , , , , , , , , , , , , , ,  Ramsey, Frank  rationality – –, ,  Rayo, Agustín , ,  real numbers , , –, , , –, , –, ,  realism, see absolutism; Leibnizian realism; Lewisian realism; plenitudinous realism

reasonable language (for reduction) , , , ,  recombination , , , , , , , , –, , –, , , , –, –, , –, –, –, –, –, , , ; see also plenitude, Lewis’s principle of recombination reduction, necessary conditions for – reduct (of a world structure) , – redundancy (theory of truth) – reflexive individual ,  Reichenbach, Hans  relations: external , , , , , , , –, , , , , , , , ,  internal , , , , ,  world-internal – relation regress – relative identity, see identity, relative relativity (in physics) , , , –, , –,  relevant entailment ,  representation, realm of –, , , –, , ; see also intensional entities representation de re –, , , ; see also counterpart theory, transworld identity Resnik, Michael , , ,  Restall, Greg , , , ,  Riemann, Bernhard – rigidity (semantic) , ,  Robinson, Abraham  Robinson, Denis  Rosen, Gideon , , , ,  Routley, Richard  Russell, Bertrand , , , , , , , , – Russell, Jeffrey Sanford , , ,  scalar quantity , , , ; see also vector quantity Schaffer, Jonathan , , ,  scientific theories , , –, ; see also physics and physical theories second-order theories , , , , , ,  Segal, Aaron  semantics, for modal language , –, , , , , , , , ; see also quantified modal logic separation, see isolation set theory –, , , , –, –, –, , , , , , , , – sets and classes, existence of –, , , –, , –, , –, –, , –; see also proper classes shape (intrinsic or extrinsic) –; see also distance

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi

 Shapiro, Stewart ,  Sher, Gila  Shoemaker, Sydney ,  Sider, Ted , , , , , , , , , , , , , , , , , , , , , ,  similarity map , ; see also duplicates Simons, Peter  simples , –, , , , , , , , ,  extended simples –, , ,  simplex structure  singletons , , –, , , –, –, , ,  singularism, see pluralism (vs. singularism) Size of Reality –, , ,  Size of Worlds – skepticism: about being actual –, , –, –, , – about being a set –,  about external world –,  about metaphysical theories –,  Skyrms, Brian ,  slice-sensitive properties, see properties, slice-sensitive slicing –, – Smart, J. J. C.  Smiley, Timothy  sortal – space and spacetime: discrete , –, – mathematical vs. physical , ,  structure of , , , , –, –, –, – spatiotemporal part , –, , –, , , ,  spatiotemporal relations, analogical , , , ; see also worlds, analysis of special composition question ,  Stalnaker, Robert , , , , , , , ,  state descriptions – states of affairs: and correspondence theory –,  and mereology , , –, , ,  and necessary connections –, – and truthmaking –, – atomic , , – conjunctive , ,  disjunctive ,  negative , ,  totality –, , , , ,  states-of-affairs theory –; see also factualism stipulation (and transworld identity) – Stoneham, Tom , –



Strawson, P. F.  strong composition as identity, see composition as identity, strong structural pluralism , ; see also isolation structure: horizontal and vertical , , –, –, – instantial , , , , , – underlying –, –, – see also plenitude of world structures structuralism (mathematical) , , , , , , , , , ,  subject matter –, – Subject Matter Principle , , –, – substantivalism , ,  substitutivity of identicals –, , , , ; see also indiscernibility of identicals; Leibniz’s Law substructure (or a world structure) , , , , –, –, –; see also plenitude, substructure principle suchness, see quiddity sui generis entities –, – super-ordinal numbers ,  supervaluations , , , , , ,  Supervenience of the General on the Particular –,  supervenience of truth on being –, , ,  tangent space , , , –,  Tarski, Alfred , –,  Tegmark, Max  Teller, Paul  temporal parts , –,  tensor quantity , ; see also vector quantity thing theory , –, –, –, –, – thisness, see haecceity Thomason, Richmond  Thomasson, Amie  Tooley, Michael  topology , , , , , , , ,  total identity, see identity, total transparent structure , – transworld identity –, , , , , , ; see also counterpart theory; haecceitism transworld individual , , –, , ,  triangle inequality – tropes –, , –, –, , , –, , , , , – Truthmaker argument (for states of affairs) –, 

OUP CORRECTED PROOF – FINAL, 13/3/2020, SPi





Truthmaker Principle –, –, –, –, –; see also Subject Matter Principle; supervenience of truth on being Truthmaker Maximalism ,  Truthmaker Necessitarianism – truthmaking counterpart relation – Turner, Jason ,  underlying structure, see structure, underlying unification (of worlds) , , , , , –, , –, –, , , –, ; see also isolation; worlds, analysis of Uniqueness of Composition , –, , , , ,  universal applicability (of logical notions) – universality (of identity), see identity, universality of universality (of mathematics)  universals , –, , , , , –, –, –, , , , – and Humeanism – and mathematics ,  structural , , ,  uninstantiated ,  see also natural properties and relations; particulars, thick and thin; properties Unrestricted Composition , –, , –, –, , , , – urelement , , , –, –, ,  Urelement Set Axiom –,  Uzquiano, Gabriel , , –,  vagueness , , , , , ; see also indeterminacy Van Cleve, James  van Inwagen, Peter , ,  van Fraassen, Bas ,  variants (of Giganto)  vector quantity , , , , –, ; see also scalar quantity; tensor quantity vector space , 

velocity –,  vertical structure, see structure, vertical Vlach, Frank , ,  von Neumann, John ,  Vulcan (planet) ,  Wallace, Megan ,  Watson, Duncan , , – weak composition as identity, see composition as identity, weak Weingard, Robert  well-ordering –, , –, , ; see also ordinal numbers Wheeler, John , ,  Williams, Bernard  Williams, Donald C.  Williamson, Timothy ,  Wilson, Jessica , ,  wishful thinking (in theory choice) ,  witnessing (of truth at a world) – Wittgenstein, Ludwig , , , ,  world contents, see plenitude of world contents world structures, see plenitude of world structures worldbound individual , , ; see also counterpart theory; transfinite individual worldmate relation –, , –, , , –, – worlds: ageless – analysis of –, – indiscernible , , , , , , , , , , , , ,  small vs. large , , see also Size of Worlds with ordinal time –, , ,  wormhole , , – Wright, Crispin  Yablo, Stephen ,  Yi, Byong-uk ,  Zermelo, Ernst , ,  Zermelo-Fraenkel set theory, see set theory zombies –