Potentiality: From Dispositions to Modality (Oxford Philosophical Monographs) [1 ed.] 9780198714316, 0198714319

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Potentiality

OXFORD PHILOSOPHICAL MONOGRAPHS Editorial Committee WILLIAM CHILD,

R. S. CRISP,

A. W. MOORE, STEPHEN MULHALL,

CHRISTOPHER G. TIMPSON other titles in this series include Moral Reason Julia Markovits Category Mistakes Ofra Magidor Quantum Information Theory and the Foundations of Quantum Mechanics Christopher G. Timpson The Critical Imagination James Grant Aquinas on Friendship Daniel Schwartz The Brute Within Appetitive Desire in Plato and Aristotle Hendrik Lorenz Plato and Aristotle in Agreement? Platonists on Aristotle from Antiochus to Porphyry George E. Karamanolis Of Liberty and Necessity The Free Will Debate in Eighteenth-Century British Philosophy James A. Harris The Grounds of Ethical Judgement New Transcendental Arguments in Moral Philosophy Christian Illies Against Equality of Opportunity Matt Cavanagh Kant’s Empirical Realism Paul Abela

Potentiality From Dispositions to Modality

Barbara Vetter

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3

Great Clarendon Street, Oxford, ox2 6dp, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Barbara Vetter 2015 The moral rights of the author have been asserted First Edition published in 2015 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2014946305 ISBN 978–0–19–871431–6 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

For Benjamin

Acknowledgements What does it take for something to be metaphysically possible? This book argues that it takes no more and no less than the possession of certain properties, to wit: potentials, by individual objects. It is metaphysically possible that I finish this book because I have the potential to do so; it is metaphysically possible that this book is read because someone or other has a potential to read it. This way of thinking about modality is not only, I take it, very natural; it may well have been the consensus in the history of metaphysics from Aristotle up until the second half of the twentieth century. It has been eclipsed, however, by the spectacular success, in logic and linguistics, of possible-worlds-based accounts. My aim in this book is to rehabilitate the natural and, perhaps, traditional view by the standards of contemporary metaphysics. In so doing, I do not start from nothing. Recent metaphysics has become more hospitable to views of modality that are not based on possible worlds, and to potentials (or powers, or dispositions) in general. What is needed is a detailed statement of the view that shows how it deals with a number of examples, but also how it is set to work in the realms of logic and linguistics. This is what I aim to provide, or rather to begin to provide. The book is meant to be the beginning of a discussion, not its end. This book has been the work of nearly seven years. It started out as my DPhil thesis at Oxford, supervised by Timothy Williamson and Antony Eagle. Both contributed more than I can acknowledge here in terms of arguments and criticism, but also in terms of motivation and encouragement. Tim was my supervisor for most of my time as a graduate student, and I have learned more about philosophy and how to do it from him than from anyone else. His influence is obvious to me on almost every page of this book. Antony not only supervised my thesis, but also provided extremely helpful, in-depth comments on the penultimate version of the book. I am deeply indebted to both of them. A third substantial intellectual and personal debt I owe to my undergraduate teacher of metaphysics at the University of Erlangen, Friedemann Buddensiek. He set me on my intellectual path twice over, by introducing me to metaphysics and encouraging me to apply to Oxford; my intellectual life would have gone very differently had it not been for his intervention. Over the years, I have benefited greatly from discussions with many fellow philosophers, including Simona Aimar, Lorenzo Azzano, Katherine Dormandy, Eline Busck Gundersen, Troy Cross, Natalja Deng, Daan Evers, Ellen Fridland,

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Arno Göbel, Frank Hofmann, Rebekka Hufendiek, Gail Leckie, John Maier, Angela Matthies, Andreas Müller, Thomas Müller, Tobias Rosefeldt, Stephan Schmid, Markus Schrenk, Emanuel Viebahn, Alastair Wilson, and the participants in Thomas Krödel’s and my joint research seminar. Special thanks to all those who read all or substantial parts of the penultimate draft: a reading group in Berlin, including Romy Jaster, Catharine Diehl, Mathias Böhm, Beau Madison Mount, Tatjana von Solodkoff, Richard Woodward, and Robert Black; a reading group at Oxford, organized by Simona Aimar and Antony Eagle; Thomas Krödel and the participants in our joint research seminar; the participants in Dominik Perler’s research colloquium; and Ralf Busse, Christian Nimtz, Thomas Sattig, Peter Schulte, and Wolfgang Schwarz. Markus Schrenk and Ralf Busse provided extremely helpful comments on the almost final draft, not all of which I have been able to accommodate. Mathias Böhm and Beau Madison Mount helped greatly with the appendix and with the editing of the book; Steffan Koch helped with proof reading and produced the index. I was fortunate to be able to present and discuss central aspects of this book at a number of conferences, including the Eidos metaphysics conference in Geneva in 2008; a workshop on modality organized by the Phlox research group in Berlin in 2009; the Metaphysics of Science conference organized by the AHRC research group of the same name in Nottingham in 2009; a workshop on dispositions at the Centre for the Study of Mind and Nature (CSMN) organized by Eline Busck Gundersen in Oslo in 2010; the workshop ‘What Is Really Possible?’ organized by Thomas Müller and his research group in Utrecht in 2011; a conference on dispositions, laws, and explanations, organized by the DFG research group ‘Explanation, Causation, Dispositions and Laws’ in Cologne in 2012; and the ‘Putting Powers to Work’ conference, organized by Jon Jacobs at St Louis University in 2011. Many thanks to the organizers and participants at each of these events. During my time as a DPhil student, I was generously supported by the German Studienstiftung des deutschen Volkes and the Deutscher Akademischer Austauschdienst (DAAD); the British Arts and Humanities Research Council (AHRC); and by a Hanfling scholarship from the Faculty of Philosophy, for which I am particularly grateful to the late Oswald Hanfling and his lovely family. Since 2010, I have had the good luck to work with wonderful colleagues and students, first at the University of Duisburg-Essen and for the last three years at Humboldt-University, Berlin. I am deeply indebted to them for providing me with a stimulating, genial atmosphere in which it is a pleasure to do philosophy. I would especially like to thank Thomas Spitzley, Dominik Perler, Stephan Schmid, Romy Jaster, Mathias Böhm, Thomas Krödel, and, again, the participants in Thomas’s and my joint research seminar.

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A small portion of this book has been previously published in articles. Part of chapter 2 has appeared as ‘Multi-track dispositions’ in The Philosophical Quarterly 63 (2013), pp. 330–352; part of chapter 3 is a revised version of ‘Dispositions without conditionals’, Mind 123, (2014), pp. 129–56; and part of chapter 6 has previously appeared as ‘“Can” without possible worlds. Semantics for antiHumeans’ in Philosophers’ Imprint 13, vol. 16 (2013), pp. 1–27. I am grateful for the permissions to use these materials (or, in the case of Philosophers’ Imprint, for leaving the copyright with me in the first place). Above all, I thank my family: my parents, my son, and my husband and partner of many years, Benjamin. Without his love, support, and sense of humour, my life in the past years (and in all likelihood this book) would have been very different, and not nearly as good. This book is dedicated to him.

Contents 1. The Project 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Introduction Modality Dispositionalism about modality Three constraints Potentiality Background assumptions Outline of the book

2. Dispositions: Against the Standard Conception 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Introduction Dispositions and conditionals: the state of the debate The problem of qualitative diversity The problem of quantitative diversity Nomological dispositions Multi-track dispositions and realism Preview of chapter 3

3. Dispositions: an Alternative Conception 3.1 3.2 3.3 3.4 3.5 3.6

A fresh start Modal semantics Modal semantics, continued A background for context-sensitivity Maximal dispositions Taking stock: from dispositions to potentiality

4. Varieties of Potentiality 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Introduction Joint potentiality introduced Joint potentiality: five questions Extrinsic potentiality introduced Extrinsic potentiality systematized Iterated potentiality Taking stock: expanding potentiality

5. Formalizing Potentiality 5.1 5.2 5.3 5.4 5.5

Introduction Framing the language Difficult cases Defining iterated potentiality Introducing the logic of potentiality

1 1 4 10 15 19 23 30

33 33 35 39 43 50 53 60

63 63 67 75 79 85 94

101 101 105 108 122 130 135 139

141 141 143 148 158 161

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contents 5.6 5.7 5.8 5.9

Comparisons: possibility, essence, ability Defending the principles Interlude: potentiality in time The logic of potentiality and the logic of possibility

6. Possibility: Metaphysics and Semantics 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Possibility defined Applying the definition Three constraints Formal adequacy Semantic utility: introduction ‘Can’ and context-sensitivity Dynamic modality beyond ‘can’ Dynamic modals as predicate operators Modality: root versus epistemic Conclusion

7. Objections 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

Introduction Potentiality without possibility? A catch-all solution: the powerful world Interlude: the powerful world and possible worlds Possibilities of existence Possibilities of non-existence Abstract objects Nomic possibility and metaphysical possibility Metaphysical possibility in time Conclusion

Appendix 1 2 3 4 5

Formal adequacy

Syntax Semantics The system P The operator 3 Possibility: the operator 3∗

Bibliography Index

164 170 186 194

197 197 200 206 207 214 217 223 228 232 246

247 247 250 257 263 267 273 277 281 290 299

301 302 303 305 309 311

323 331

1 The Project 1.1 Introduction Individual objects have potentials: paper has the potential to burn, an acorn has the potential to turn into a tree, some people have the potential to run a mile in less than four minutes. Some potentials have names: a vase’s fragility is a potential to break or to be broken, a person’s irascibility is her potential to get angry. Some potentials are classified as abilities, such as your potential to read English; others are not, such as the paper’s potential to burn. Some potentials are classified, by philosophers, as ‘dispositions’; these include the vase’s fragility, a person’s irascibility, and an atom’s disposition to decay. Potentials are often ascribed simply with the auxiliary ‘can’: paper can burn, the acorn can become a tree, some people can run a mile in less than four minutes. This, in fact, is one simple way of stating what all the properties I have so far mentioned have in common: they concern what a given individual can do. I call any such property a potentiality. It goes without saying that the notion of potentiality is of Aristotelian pedigree, and I believe Aristotle’s Metaphysics, book , to be one of the most illuminating treatments of it. Starting with early modern philosophers such as Descartes, talk of potentiality was long regarded as suspect. In those strands of philosophy that led to contemporary analytic philosophy, David Hume’s empiricist criticism of ‘necessary connections’ has been particularly influential, and the notion of potentiality has been very much out of favour, a fate it shared with many other modal notions, such as essence and a metaphysically substantial notion of necessity and possibility.1 The latter pair had a revival in analytic philosophy with the development of modal logic and the discovery that a semantics of ‘possibly’ and ‘necessarily’ can be treated as a special case of the logic of the existential and 1 I use the term ‘modal’ in the wide sense that includes not just metaphysical possibility and necessity, but also essence, dispositions, and laws of nature. Thus Fine’s (1994) ‘non-modal’ account of essence is, in my terms, not non-modal: it is merely an account that rejects the reduction of one modality (essence) to another (necessity). I will say more about modality in this broad sense in section 1.2.

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universal quantifier, as long as we allow the quantifiers to range over an infinity of ‘possible worlds’. The philosophical debate about the nature of modality has subsequently focussed largely on the nature of those ‘possible worlds’. In the process, the modal notions of an object’s potentiality or its essential properties have been either neglected or explained in terms of those of possibility or necessity. To many, they have remained suspect. In recent years, the philosophical climate has changed somewhat. Kit Fine and others have provided accounts of essence as irreducible to, and in fact prior to, necessity (see Fine 1994, Lowe 2008, Oderberg 2007). Anti-Humean accounts of the laws of nature, admitting ‘necessary connections’ in nature, and some even locating these connections in dispositional properties of fundamental entities, are thriving (see, e.g., Ellis 2001, Bird 2007; David Armstrong, though not an anti-Humean himself, has been instrumental in the rise of anti-Humeanism). Dispositions in general have received a fair amount of attention in the recent metaphysical debate, though they tend to be treated in isolation from other kinds of potentiality, such as abilities. The ideas put forward in the pages to follow are part of that changed climate. This book is a plea for potentiality. It is a plea for recognizing a unified notion of potentiality instead of selectively focussing attention on only some kinds of potentiality; and most importantly, it is a plea for recognizing potentiality as an explanans in the metaphysics of modality, rather than as something in need of explanation and reduction. Potentiality, as I understand it, is closely related to possibility in ways to be explained in this book. To get a first grasp on the relation between them, we may somewhat metaphorically call it a relation between localized and non-localized modality.2 A potentiality is localized in the sense that it is a property of a particular object. That I have the potential to write this book is first and foremost a fact about me; it is a property that I possess. Possibility, on the contrary, is not localized in this way. Its being possible that such-and-such is not primarily a fact about any one particular object; it is a fact about how things in general could have turned out to be. Hence our intuitions about what is possible and what is not can be captured by postulating, for everything that is possible, an entire world that did turn out to be that way. The proper operator for ascribing a potentiality is thus a predicate operator: . . . has a potentiality to . . . (fill in a singular term for 2 In Vetter (2010) and Vetter (forthcoming), I have used the labels ‘local’ and ‘global’ instead of ‘localized’ and ‘non-localized’, but I have found those labels to evoke some misleading associations: the potentialities of the world as a whole, discussed in chapters 7.3–7.4, certainly deserve the label ‘global’, but I emphatically do not want to simply identify possibilities with those potentialities. I hope the revised labels, while somewhat less vivid, are less prone to lead to confusion.

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the first blank, and a predicate for the second). Possibility, on the other hand, is aptly expressed by a sentential operator: it is possible that . . . (fill in a sentence). The distinction is paralleled by the relation of essence to necessity. Essence, like potentiality, is localized: a property is essential to a particular object. Necessity, like possibility, is not localized: its being necessary that such-and-such is not primarily a fact about one particular object, but a fact about how the world must be. The difference between essence and necessity has been pointed out and studied by Kit Fine (1994). To cite Fine’s famous example, it is necessary that Socrates is a member of his singleton set; but this necessity does not have its source in Socrates himself: it is not essential to him.3 Potentialities, in short, are possibilities rooted in objects; they are like possibilities, but they are properties of individual objects. They stand to possibility as essence (on the Finean view) stands to necessity. Explaining the notion of potentiality in this way is not meant to provide a definition or reduction, not only because it is a mere analogy, but also because potentiality, as I will understand it, is the primitive notion in terms of which possibility will be explained. The notion of potentiality itself will be introduced as a generalization of the more familiar notion of a disposition in chapter 3 (a preview of which is given in section 1.5). My plea for potentiality is to show precisely that taking potentiality as a primitive or basic notion is philosophically fruitful; that we can say a great deal about potentiality without defining or reducing it; and that we can say a great deal about other things in terms of potentiality. In particular, we can say a great deal about possibility in terms of potentiality. This is the one main use to which I want to put the notion of potentiality: to develop an account of possibility (and, thereby, of necessity) that is based entirely on potentiality. Potentiality is, metaphorically speaking, possibility anchored in individual objects; I claim that all possibility is thus anchored in some individual object(s) or other. My notion of potentiality differs, in ways that will become more conspicuous in chapters 3–5, from contemporary assumptions about dispositions. Nevertheless, I will argue in those chapters that potentiality as I construe it is nothing but the natural generalization of the more familiar dispositions, a generalization which is required in any case by a non-reductive metaphysics of dispositions. Thus the above characterization of the account, that all possibility is anchored in 3 The distinction between localized and non-localized is not equivalent to the de re/de dicto distinction. The latter distinction applies primarily to sentences, while the former is straightforwardly metaphysical. Moreover, the distinctions are not co-extensional, as Fine’s famous examples show. Thus it is necessary for Socrates to be a member of his singleton set (de re necessity), but it is not part of his essence to be a member of his singleton set.

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some object(s) or other, will be given a more intuitive grounding in our understanding of dispositions, and the account can justly be called dispositionalist. A dispositionalist account of metaphysical modality is clearly a desideratum in the growing anti-Humean literature.4 My aim in this book will not be to argue for the theory or to compare it to its rivals. The idea of a dispositionalist theory, I take it, is interesting and plausible enough to deserve further scrutiny. (I will provide some motivation for such a theory in section 1.3, but I do not claim that any of the considerations adduced there force us into accepting the theory.) What is lacking, at least in the contemporary literature, are the details. Before the theory can be evaluated or compared, it needs to be spelled out. This is what I aim to do in this book. It goes without saying that I would not have bothered to spell it out if I did not believe that it was true. But the development of the view should be of interest even to readers who do not share that initial assumption. In the remainder of this chapter, I will situate the kind of theory to be developed in the context of contemporary modal metaphysics (1.2), provide some motivation for it and outline the basic options in developing it (1.3), and then sketch how the theory will be developed throughout the book, guided by three basic constraints on any theory of modality (1.4–1.7). The aim of this chapter is to give a feeling for how things will go and why; detailed arguments will be given in the chapters that follow. Let us begin, then, with a brief look at contemporary modal metaphysics.

1.2 Modality Modality comes in a package. There are, of course, the two familiar modalities of necessity and possibility. There are also such modal phenomena as (if we take them seriously as phenomena; otherwise, there are such modal notions as those of): laws of nature, essences, the counterfactual conditional, causation, and dispositions. A reductive approach to modality will try to describe all of these phenomena in a language that is taken from outside the modal package: the language of worlds as maximal spatiotemporally connected entities, for instance. In so doing, a reductive account may nonetheless impose some hierarchy on the modal package. David Lewis, for instance, analyses laws of nature in terms of a best system for the actual world, counterfactuals in terms of possible worlds and laws of nature, causation in terms of counterfactuals, and dispositions in terms of 4 There are, of course, historical precedents, in particular in the Aristotelian tradition: see Schmid (forthcoming) for a useful discussion of potentiality and possibility in medieval philosophy. What is needed, and what I aim to provide, is a theory that takes account of our contemporary understanding of modal metaphysics, modal logic, and modal semantics.

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counterfactuals and causation. A non-reductive account of modality need not be quietist. It will not try to capture the elements of the modal package in terms of something non-modal, but it can impose a hierarchy on the package itself, understanding parts of the package in terms of other parts. Thus Williamson (2007b) suggests that we can account for (at least our knowledge of) possibility and necessity in terms of (our knowledge of) counterfactual conditionals; Lange (2009) provides an account of various forms of necessity, from logical to nomological, in terms of primitive counterfactual conditionals or ‘subjunctive facts’; and Fine (1994) suggests that we understand necessity in terms of essence. A more traditional non-reductionist view (found, for instance, in Stalnaker 2003) is the idea that possibility and necessity are (metaphysical or conceptual) primitives, with the help of which we can give an account of possible worlds, and that possible worlds in turn provide, in one way or another, the truth conditions for statements about the rest of the modal package. The modal package can be partitioned in various ways. The partition that is relevant to my purposes is one which I have introduced above as the distinction between localized and non-localized modalities. As already indicated, we can make the distinction a little more precise by looking at the canonical expressions required for either kind of modality. The operators for the nonlocalized modalities may be one-place operators (as in the case of possibility) or two-place operators (as with the counterfactual conditional); but their argument places must always be filled by an entire sentence. Possibilities are possibilities that . . . . The operators for localized modalities, on the other hand, must have at least one argument for the object (or objects) to which the modality belongs, and another argument place for that which is intuitively the content of the modality, and which is most naturally expressed by a predicate. Thus we have: . . . is essentially . . . , and . . . has a disposition to . . . the first blank, in each case, requiring a singular (or plural) term to be filled, the second a predicate.5 In contemporary metaphysics, the focus has been on the non-localized modalities, and it has generally been assumed that the localized ones can be defined in terms of them. The conditional analysis of dispositions and the modal account of essence6 are symptoms of that general tendency. Non-localized modality, in turn, has been thought about in terms of possible worlds: thus what is possible 5 The second argument place may be construed differently for some purposes: thus Kit Fine has used a sentence operator for essence, x , read: ‘it is true in virtue of the essence of x that . . .’. Fine (1995c) provides an illuminating discussion of different constructions of the essence operator. For potentiality, the construction as a predicate operator is clearly more intuitive: an object’s dispositions are dispositions to . . . , not dispositions that . . . . 6 See Fine (1994), who rejects that account.

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is simply what is true in some possible world or other. The crucial question then becomes: what are possible worlds? Are they concrete universes, spatiotemporal totalities just like ours, as David Lewis (1986a) has it? Or are they maximal (sets of) propositions (Plantinga 1974, Adams 1974), uninstantiated properties of the world (Stalnaker 1976), recombinations of actual properties (Armstrong 1989a), or mere elements of fictions (Rosen 1990)? Why have possible worlds been so pervasive in the metaphysics of modality? The simplest and most powerful consideration in favour of possible-worlds talk is its theoretical usefulness. Possible worlds provide a powerful semantics for modal logic, reducing the operators ‘possibly’ and ‘necessarily’ to simple and well-understood existential and universal quantification. They also provide an excellent formal model for the context-sensitivity of modal expressions in natural language, by invoking mechanisms—in particular, restricted quantification— that are known to be ubiquitous in natural language already. But to have a formal model, even a powerful one, is not necessarily to have a good metaphysics. Various philosophers have expressed doubts that possible worlds really provide an insight into the nature of metaphysical modality. Roughly, the reasoning goes as follows: if we give an account of modality in terms of possible worlds, those worlds are either concrete, Lewisian worlds or some kind of abstract entities, such as sets of propositions or uninstantiated properties. As to the former, the ‘incredulous stare’7 is a strong objection; furthermore, it is hard to see what evidence could be adduced for that initially rather implausible claim; and finally, even if it were true that there are infinitely many concrete universes, that does not seem to be a fact about possibility and necessity, but rather a curious contingent fact about the one actual world, which includes all those ‘universes’. If, on the other hand, possible worlds are sets of propositions, we need some way to distinguish those sets of propositions that do from those that do not correspond to genuine possibilities; mere logical consistency is not enough. If abstract possible worlds are supposed to deliver a robust account of metaphysical modality, it is hard to see how they can avoid circularity; if not, then they are simply irrelevant to the metaphysical question of what possibility and necessity are. (See Williamson 1998 and Jubien 2007 for contemporary versions of this kind of criticism.) These considerations, brief as they are, are certainly not decisive. They provide some reason to doubt that the formal apparatus of possible worlds can be simply implanted into a metaphysics of modality, and thereby some reason for theories of metaphysical modality that are, as Contessa (2009) has put it, ‘hardcore 7

See Lewis (1986a).

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actualist’: theories, that is, which do without any appeal to possible worlds, abstract or concrete.8 (Note that none of these considerations cast any doubt on the use of possible worlds as a formal tool in logic and linguistics. It is only when we try to answer the metaphysical question what possibility and necessity consist in, that these objections apply at all.) But there is a further motivation for the appeal to possible worlds in the metaphysics of modality, one which concerns our basic metaphysical commitments. It is the widespread assumption of what David Lewis has called ‘Humean supervenience’: Humean supervenience is named in honor of the greater [sic] denier of necessary connections. It is the doctrine that all there is to the world is a vast mosaic of local matters of particular fact, just one little thing and then another. (But it is not part of the thesis that these local matters are mental.) We have geometry: a system of external relations of spatiotemporal distance between points. Maybe points of spacetime itself, maybe point-sized bits of matter or aether or fields, maybe both. And at those points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated. For short: we have an arrangement of qualities. And that is all. There is no difference without difference in the arrangement of qualities. All else supervenes on that. Lewis 1986b, ix f.

The aspect of Humean supervenience that is of interest here is its exclusion of modality—the whole modal package—from the supervenience base. The Humean world is, at root, thoroughly non-modal. If modality, in its various facets, is nonetheless to be accounted for—and it is—then we must construct it from the non-modal materials at the supervenience base, or we must find it elsewhere. Possible worlds provide a viable way for the Humean to ‘outsource’ modality: it is still a matter of deeply non-modal facts; we simply need enough such facts. One Humean world does not provide modality, but many of them do. Thus the metaphysics of modality, for the Humean, becomes a metaphysics of possible worlds.9 8

I discuss a number of such theories under the label ‘New Actualism’ in Vetter (2011b). While the rejection of Humean supervenience provides a welcome background and motivation for a dispositionalist account of modality, the account that I am going to develop is not opposed to all aspects of Humean supervenience. (Thanks to Markus Schrenk for pressing me on this point.) Maudlin (2007) distinguishes three aspects in Humean supervenience. The first is Separability, the claim that the total physical state of the world supervenes on the local, intrinsic states of each spacetime point and the spatiotemporal relation between them; ‘the world as a whole is supposed to be decomposable into small bits laid out in space and time’ (Maudlin 2007, 51). The second is Physical Statism, the view that ‘[a]ll facts about a world, including modal and nomological facts, are determined by its total physical state’ (Maudlin 2007, 51). The view that I am going to develop can accept both claims; in fact, the ‘localizing’ impetus of Separability is quite close to that of the potentiality view. What I disagree with is Lewis’s view of what the ‘small bits’ that constitute the physical state of 9

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Humean supervenience, however, is by no means a mandatory assumption. As Tim Maudlin (2007) notes, it is unclear why we should believe in Humean supervenience in the first place. Lewis’s own stated motivation is ‘to resist philosophical arguments that there are more things in heaven and earth than physics has dreamt of’ (Lewis 1994, 474). Ironically, it is precisely from the philosophy of science that Humean supervenience has recently come under considerable pressure. Many philosophers of science now argue that the fundamental physical properties, those which make up the supervenience base, are not the Humean’s ‘qualities’, that is, quiddistic properties with no modal profile. Science, as Simon Blackburn and others have argued, ‘finds only dispositional properties, all the way down’ (Blackburn 1990, 63; see also Molnar 1999, and Bird 2007). What physics tells us about a fundamental property, such as—for the sake of a, probably inaccurate, example—negative charge is how that property enables and disposes its bearers to react and interact with things that have the same or other fundamental properties. Physicists have nothing to say about any ‘underlying’ qualities or quiddities that are independent of such dispositional patterns, but such qualities are precisely what is required for the Humean’s supervenience base.10 In maintaining that there is more to the properties discovered by science, namely, a quiddistic nature, it is Humean supervenience that is guilty of supposing that ‘there are more things in heaven and earth than physics has dreamt of’. Apart from seeming unwarranted by the standards of physics, the supposition of quiddistic fundamental qualities leads to various problems. If Humean

the world are like—the third element that Maudlin discerns in Humean Supervenience, a condition on acceptable analyses that he labels the Non-circularity condition: ‘[t]he intrinsic physical state of the world can be specified without mentioning the laws (or chances, or possibilities) that obtain in the world’ (Maudlin 2007, 51). Potentiality can, presumably, be added into the bracket with chances and possibilities; and it is here that I am anti-Humean. Note that while the view developed in this book is compatible with Separability and Physical Statism and indeed sympathetic at least to the former, it is not committed to either. In chapter 4.2, I briefly discuss the possibility of there being primitive ‘joint potentialities’ arising from quantum-entangled states. It is precisely such entangled states that Maudlin takes to refute Separability. The criticisms of Humean supervenience that I discuss in the main text are all directed at the combination of Physical Statism and the Non-circularity condition. 10 In the same paper, Blackburn suggests that this finding should cause concern: ‘To conceive of all the truths about a world as dispositional, is to suppose that a world is entirely described by what is true at neighbouring worlds. And since our argument was a priori, these truths in turn vanish into truths about yet other neighbouring worlds, and the result is that there is no truth anywhere’ (Blackburn 1990, 64). Holton (1999) has shown that the worry is ungrounded if it is one of incoherence. The worry that remains is one of regress or circularity, if the fundamental properties are dispositions whose manifestations are in turn dispositions, whose manifestations . . . and so forth. As Holton points out, it is not entirely clear what is so bad about circularity in this case. For a more detailed argument, see chapter 6 of Bird (2007).

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supervenience is true, then the fundamental properties are not the dispositions that physics prima facie takes them to be. Accordingly, the name of such a fundamental property (say, ‘charge’) does not refer to a dispositional property, but to the categorical property that plays the role specified in the disposition’s description. (See Lewis 1970.) This, however, leads to a rather unattractive proliferation of possibilities: on the assumption that charge and mass are categorical properties that play their dispositional roles only contingently, it should be possible that they completely swap their roles. (See Black 2000, adapting an argument from Chisholm 1967 concerning haecceitism.) So there should be possible worlds just like the actual one in every detail except that mass plays the charge role and charge the mass role (and the same goes for any number of properties). Perhaps more worryingly, there is no reason why the same role should not be played by two, indeed by any number of, distinct categorical properties, in the actual world: Perhaps there is a possible world just like ours, not only in surface appearance, but in all that physics could ever discover, in which the dispositions have a different categorical ground, G . Perhaps in our own world G supports dispositions on Mondays and Wednesdays, while G supports them on the other days. Blackburn 1990, 64

But that would make it impossible for our theoretical terms such as ‘charge’ to refer to anything at all: if there is no one categorical property that plays the charge role, then there is no property for ‘charge’ to refer to (compare the definite description ‘the word on this page’), and the best science that we could possibly achieve would merely deal in empty words. (See Bird 2007, 76–79, where this argument is spelled out in much more detail.) Again, such considerations may not refute the claim of Humean supervenience. But they cast serious doubts on the main motivation for Humean supervenience: its being, apparently, closer to science than its competitors. Without such a motivation, it is hard to see why we should accept Humean supervenience in the first place. And without Humean supervenience, a powerful motivation for a ‘Humean’ reductive account of modality in terms of possible worlds disappears. A number of recent, ‘anti-Humean’, metaphysicians have rejected Humean supervenience for these and related reasons. The thesis of Humean supervenience had, of course, never been fully victorious; there were always different strands within metaphysics. The explicit rejection of Humean supervenience on the basis of a shared commitment to scientific realism, however, is a relatively recent phenomenon. It is championed by dispositionalists, such as C.B. Martin, George Molnar, Stephen Mumford, Brian Ellis, and Alexander Bird, who take very seriously the idea, expressed in the above argument against Humean

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supervenience, that the world at bottom is dispositional. Dispositional essentialists, most prominently Ellis and Bird, hold that the fundamental properties in nature are (all or most of them) essentially dispositional: what these properties are is simply a matter of what they enable or dispose their bearers to do. If the world is irreducibly dispositional, it is irreducibly modal. There is, then, no need to outsource all modality to other possible worlds, and reason to hope that no such outsourcing is needed. Possible-worlds talk has its place, of course, as a formal model in some areas, and perhaps as a descriptive and heuristic tool. But we should not make the mistake either of thinking that ‘possible worlds’ are genuinely worlds, or that they have any special connection with possibility. Possible worlds can be used to model a variety of phenomena, from metaphysical modality through obligation, knowledge, and belief to vagueness. For some of these phenomena, it is useful to include ‘possible worlds’ that are metaphysically impossible: as is well-known, the modelling of epistemic states in terms only of metaphysically possible worlds has the consequence that a subject is taken to know and believe all that is metaphysically necessary and never to believe anything that is metaphysically impossible. A more realistic model might differentiate between different instances of knowledge or belief concerning metaphysically necessary or impossible matters by including metaphysically impossible worlds. Thinking that there is a tight connection between metaphysical modality on the one hand, and possible worlds as a formal tool on the other, may prove unhelpful both in accounting for metaphysical modality and in using the tool of possible worlds in other areas. The world as the dispositionalist envisages it, I said, is irreducibly modal. But the modality that it fundamentally contains is localized: it is the dispositions of objects to behave thus-and-so. It becomes natural, then, to use this local modality in accounting for other phenomena that are otherwise explained in terms of possible worlds. One suitable explanandum for dispositionalists has been the laws of nature, which are thought to be fully grounded in the dispositional essences of the properties that they concern (Ellis 2001, Bird 2007). Another is causation, for which there are currently different dispositionalist proposals on the table (Mumford and Anjum 2011, Bird 2010, Hüttemann 2013). A third obvious candidate is the focus of this book: metaphysical modality.

1.3 Dispositionalism about modality Dispositionalism about modality is the view that metaphysical modality is, in some way or another, to be accounted for in terms of dispositional properties. By ‘metaphysical modality’, I mean the non-localized modalities of metaphysical

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possibility and necessity. The view has been suggested in the writings of dispositionalists such as Bird (2007, 218, fn.146) and Pruss (2002). Borghini and Williams (2008) and Jacobs (2010) have begun to spell it out. Dispositionalism about modality has a number of advantages. One is the simple fact that it is parsimonious given the dispositionalist metaphysics just outlined: if we already have dispositional properties, we do not need anything else to account for modality. Parsimony depends on background commitments: if you do not believe in irreducible dispositions, then of course a dispositionalist theory of modality is not going to be cost-efficient. There are, however, two further and less background-dependent benefits of dispositionalism about modality. One is the naturalness of the ontological picture that comes with it. As metaphysical realists, we tend to think of the world as consisting in objects that have properties; the paradigmatic cases of real, mind-independent facts are, at bottom, facts about things having properties.11 Metaphysical modality is puzzling because it does not fit into the schema of objects-with-properties. It seems to consist of facts that float free of any particular object: its being possible that there are talking donkeys, for instance. One of the attractions of Lewisian modal realism is that it anchors those free-floating facts in objects. One of its drawbacks is that the objects are otherworldly donkeys for or against whose existence we can in principle have no evidence. Dispositionalism promises to share the attraction without succumbing to the drawback: it, too, anchors possibilities in objects. But its objects are just the ordinary objects of this, the actual, world, with which we are in regular epistemic contact. It remains to be seen how exactly the view will account for the particularly free-floating possibilities such as that of there being talking donkeys. But if it succeeds, then it does so by anchoring possibilities in realistically respectable bits of the world, ordinary concrete objects. The second notable benefit of dispositionalism about modality is epistemological. Dispositionalism, I said, avoids the drawback of a possible-worlds metaphysics by anchoring possibilities in the right kind of objects: actual objects, with which we have epistemic contact. By anchoring them in the dispositions of such objects, dispositionalism promises a plausible story about the epistemology of modality. We clearly have a great deal of knowledge about the dispositions of the individual objects around us (as well as of our own). Such knowledge arises from, and is used in, both everyday and scientific contexts. We learn early on that glasses are fragile, that sugar is water-soluble, and that some people are irascible.

11 The intuitive picture may, of course, be questioned. One well-known challenge to it comes from Ladyman et al. (2007). But Ladyman et al. would have more fundamental disagreements with the kind of metaphysics in which this book is engaging.

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We use that knowledge in dealing carefully with fragile glasses and irascible people, and in putting sugar into our tea. Dispositional properties such as solubility in various substances, fitness for survival in a given environment or the ability to fly, and the various behavioural dispositions of human agents, play a crucial role in chemistry, biology, and psychology. It is an interesting question, and one which it is beyond the scope of this book to answer, how we acquire the relevant knowledge. Inference to the best explanation will certainly play a role,12 as will various mechanisms for generalizing from one observed glass, sugar cube, or bird to others that are relevantly similar. Given a sufficiently rich view of perception, it is not implausible even to think that in some cases we can perceive an object to have a certain disposition: I can see that the glass is fragile, just as I can see that it is a champagne glass. For present purposes the crucial points are, first, that we clearly have such knowledge, whatever exactly our account of it is; and second, that such knowledge is not a matter of philosophical speculation, but of both practical and scientific knowledge about the world. Systematic theorizing about dispositions in philosophy will go beyond both our ordinary grasp and our scientific understanding of dispositions in some ways (as is witnessed by chapters 3–5 of this book). But it is informed by, and continuous with, the empirical knowledge that we already have of dispositions. If metaphysical modality is based on dispositions, then our ways of knowing about dispositions are, in principle, ways of knowing about metaphysical modality. The epistemology of metaphysical modality may then be just a generalization of those empirical ways of knowing about dispositions. For those with roughly empiricist inclinations in the epistemology of modality, dispositionalism promises a good answer to what Christopher Peacocke (1999) has called the ‘integration challenge’: the challenge of providing for a given phenomenon a metaphysics and an epistemology which fit together’ that is, which describe the phenomenon metaphysically in such a way that we can know about it, and our ways of knowing about it in such a way that they can be ways of knowing about that kind of thing. Dispositionalism about modality does better in this respect than other standard views on the metaphysics of modality. Lewisian modal realism is notorious for its divorce between the metaphysics of modality—which is a matter of concrete worlds, all except one of which are inaccessible to us—and its epistemology—which is largely a matter of applying Recombination, the idea that everything can co-exist with anything. It seems like black magic that the epistemology of Recombination should just happen to get the metaphysics of possible 12

See chapter 3.5 for more on the relation between dispositions and explanations.

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worlds right, given that there is no connection by which the latter might have informed the former. Other possible-worlds-based views of modality fare hardly better: their possible worlds are abstract and rather remote from the everyday concerns in which our initial modal knowledge is embedded, such as the knowledge that I can ride a bike, that it can rain later, that my precious vase can break easily if handled without care, and so on. It is sometimes argued that, in order to meet the integration challenge, we must give up on the idea that modality is mind-independent. This is Peacocke’s own moral (Peacocke 1999, ch. 4): if modality is a matter of concepts, as he suggests, then it is hardly puzzling that we can acquire knowledge about it. Our knowledge of modality, in such a case, would be best explained by a rationalist epistemology. In this book, I will be interested in an account of modality that does take it to be entirely mind-independent. (My account, that is, is a realist one, in a well-established sense. I will use the term ‘realism’ in a stronger sense below.) Dispositionalism promises to provide such an account and a plausible empiricist epistemology for it. There may be independent reason for being suspicious about an object– property ontology, or about empiricist approaches to modal epistemology, and I am not going to argue for either of the two views. Dispositionalism about modality is part of an attractive package which features an anti-Humean view of the natural properties, an intuitive ontology, and a modal epistemology that is well equipped to solve the integration challenge. Dispositionalism about modality is worth spelling out, even if only to see more clearly what the package as a whole would amount to. There are two basic versions of dispositionalism about modality, differing in the kind of modality that is said to be grounded in dispositional properties: counterfactual conditionals or possibility. In the current philosophical literature, a disposition such as fragility is generally characterized by a counterfactual conditional such as ‘If x were struck, x would break’. Where the project was to reduce dispositional talk to something else, these counterfactual conditionals have generally been appealed to as terms of the reduction. Dispositionalists, of course, will be opposed to such a reduction. But that need not prevent them from making use of the very same link that their opponents detect between a disposition ascription and a counterfactual conditional, and merely reversing the order of explanation. A first stab (and no more than that) at a dispositionalist theory of counterfactual conditionals would be the following: (C) A counterfactual of the form ‘If x were S, then x would be M’ is true just in case x has a disposition to M if S.

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For instance, the counterfactual conditional ‘If the glass were struck, it would break’ is true just in case the glass is fragile’ and its truth or falsity is explained by the glass’s having or lacking the disposition. A theory along these lines is suggested, for instance, by Bird (2007). Jacobs (2010) has spelled it out in some more formal detail. The approach has the virtue that, given an account of counterfactual conditionals, it can then define possibility and necessity in terms of them (see Jacobs 2010). Ironically, (C) fails for the same reasons that have initially motivated dispositionalists such as Martin (1994) to reject a reductive analysis of dispositions. The case for this failure has been made by Eagle (2009) (for a closely related argument with a slightly different target see Schrenk 2010). A disposition is a matter of how things stand with a particular object—dispositions are, more often than not, intrinsic properties of their bearers.13 The truth of a counterfactual, however, depends on more than the intrinsic nature of a particular object. That is why the truth-values of a disposition ascription and the corresponding counterfactual conditional can diverge. A vase that is safely packed remains fragile, yet the corresponding counterfactual ‘If the vase were struck, it would break’ is false of it. (This is a classic case of ‘masking’ or ‘antidotes’; see Johnston 1992 and Bird 1998.) A live wire disposed to conduct electricity if touched may be equipped with a fail-safe mechanism that turns off the electricity if it were touched, thus rendering false the conditional ‘If the wire were touched, it would conduct electricity’. (This is Martin’s (1994) case of an ‘electro-fink’.) Such cases make the prospects of a reductive analysis based on (C) rather poor, no matter which way the analysis is meant to go.14 But there is another way for the dispositionalist about modality. Objects possess many dispositions without manifesting them (any fragile but unbroken glass will serve as an example); the manifestation of such a disposition is merely possible. Dispositions are thus linked not only to counterfactual conditionals, but also to possibility. This opens another route for the dispositionalist who wants to ground modality in dispositions. Schematically, the second route is as follows: (P) A possibility statement of the form ‘It is possible that x is M’ is true just in case x has a disposition to be M. For instance, it is true that the glass possibly breaks just in case the glass is fragile; and the truth or falsity of the possibility claim is explained by the 13 As McKitrick (2003) has shown, not all dispositions are intrinsic. I will discuss extrinsic dispositions in detail in chapter 4. The account that I give there may help with a semantics of counterfactual conditionals, and I return to the problem in chapter 6. 14 Maier (ms) tries to solve these problems by ascribing dispositions to the world as a whole. Although intriguing, I think that his proposal takes away much of the intuitive appeal of the original theory, which arose from its basis in our thought about ordinary middle-sized objects.

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glass’s having or lacking the disposition. A view along these lines is proposed by Borghini and Williams (2008) and briefly sketched in Pruss (2002). (P) is, to a first approximation (and only to a first approximation), the view that I will develop in this book. The official version of the view, a descendant of (P), will be stated only in chapter 6.1. (So hold the counterexamples! I will say a bit about them in the next section, and much more in the remainder of the book.) But before the view can be properly stated and defended, much work has to be done to develop the right notion of dispositions—or, as I will call it, of potentiality. In chapters 2–3, I am going to argue that we should not have expected anything other than (P) in the first place. The dispositions that we ascribe in ordinary language are not nearly as closely related to counterfactual conditionals as most philosophers have thought. The important link, rather, is between dispositions and possibility. If we wanted to reduce dispositions away, we would have to reduce them to (a special kind of) possibility; and if, on the other hand, we want to base modality in general on dispositions, we should start with possibility as (P) does. Necessity, of course, can be defined in the usual way. Counterfactual conditionals will not be at the focus of my account, though I will make some suggestions on how to integrate them in chapter 6. (P) is only a first step towards a dispositionalist theory of modality. The agenda for the theory can be brought into focus by looking at the different constraints that a theory of metaphysical modality will have to meet.

1.4 Three constraints A first constraint on any theory of metaphysical modality is extensional correctness. We have certain firm convictions about what is or is not metaphysically possible or necessary, and these had better come out mostly true on any metaphysical account of modality. There is some room for negotiation—perhaps some of our firm convictions are just an artefact of philosophical theorizing, which has happened to be opposed to dispositionalism. But negotiation must end somewhere. Otherwise we have just changed the topic (or we must embrace an error theory of our beliefs about modality, an undesirable last resort in my opinion). My objection to (C) above was based on a failure of extensional correctness. A second constraint is formal adequacy.15 Modal logic has studied various systems for formalizing modality, and there is wide agreement on some minimal 15 This constraint, and the difficulties in addressing it, have not received much attention in the literature so far; a notable exception is Yates (forthcoming). Yates’s proposal for addressing the constraint leads to a more modest dispositionalism than mine.

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conditions that a system has to meet in order to count as a formalization of (metaphysical) necessity and possibility. Thus we know that possibility, for instance, is closed under logical implication (if it is possible that p, and q follows from p, then it must be possible that q), is closed under and distributes over disjunction (it is possible that p-or-q iff it is possible that p or it is possible that q), and is implied by actuality (if p, then possibly p). The dispositionalist about modality should provide an account that is not merely compatible with, but which explains or entails those features. Otherwise it may reasonably be doubted that the dispositionalist account really is an account of modality. Again, there is some room for negotiation—philosophers have argued over which system of modal logic is best suited to characterize metaphysical modality, and while most accept that victory goes to S5, some have defended a weaker system, T, as the best formalization of metaphysical modality (see Salmon 1989). An otherwise plausible metaphysics of modality may take a stand with the minority here, if that is what its principles require. But the minimal conditions just given are not up for negotiation. A third constraint may be called semantic utility. Utterances in natural language are rife with modality. Speaking about what can, might, or must happen is crucial to human communication. In metaphysics, we use our grasp of naturallanguage modals to think about the modal claims made by philosophers. It had better turn out that these two usages of modal terms are not referring to different kinds of modal reality. Our modal metaphysics should provide the materials for a semantics of at least a significant part of natural-language modality. This is not trivial: as has often been noted, modality in natural language is highly contextsensitive (see, for instance, Kratzer 1977 and 1991). If modal metaphysics provides the semantic materials for natural-language modality, its materials must be such as to allow for contextual variation. Again there is some room for negotiation, and I will argue in chapter 6 that dispositionalism does not, and need not, provide a semantics for epistemic and deontic modals. But some parts of modal language must be accounted for. There may be other constraints (epistemic accessibility, as outlined above, comes to mind). But these three are crucial; and they are, prima facie, difficult for the dispositionalist to meet. Take extensional correctness first: it is possible of many things that they break without those things’ being fragile (a sturdy steel bridge will serve as an example). So we seem to have possibilities without a corresponding disposition—and hence a problem for extensional correctness. The semantic utility of dispositionalism, further, requires that there are contexts in which we would have to ascribe to a sturdy steel bridge the disposition to break—after all, there are contexts in which it is perfectly true to say that the bridge can break. And it requires, further, that there are contexts in which a highly

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fragile but safely packed glass does not count as disposed to break—after all, there are contexts in which it is perfectly true to say that the glass is so safely packed that it cannot break. However, the most difficult constraint for the dispositionalist to meet is formal adequacy. Dispositions do not appear to be governed by the same formal principles as possibility. If the glass is disposed to break, is it thereby disposed to be such that it is raining or it is not raining? If the manifestations of dispositions were closed under logical implication, it would have to be. But such reasoning is far removed from an intuitive understanding of dispositionality. And if it is not correct, how can the dispositionalist explain the formal structure of possibility itself? To make matters worse, it is hard to see how (P) should even be formulated at the right level of generality. As it stands, it can account only for a limited range of possibilities—those whose content can be stated with a simple predication of the form ‘a is F’. But what of the more complex or free-floating possibilities, such as the possibility that Jill and Jack are both 5 feet tall, the possibility that it is raining or sunny, or the possibility that there are talking donkeys? (P) does not even tell us what to do with them; it has the wrong logical form. And its logical form is no accident either. It arises from the fact that dispositions are dispositions of a particular object to behave in some particular way, a fact to which I have appealed in distinguishing dispositions, as a localized modality, from other parts of the modal package. One option in responding to the problem of logical form is to make no more than the more restricted claim embodied in (P). But such a move would hardly deserve the name of a theory of metaphysical possibility in general. Alternatively, we might try tinkering with the relation that is to hold, according to (P) or its improved successors, between a disposition and a possibility. Perhaps all we need to claim is that every true possibility statement has a truthmaker in some dispositional property or properties. If there is no strict ‘if and only if’ relation, then the form of disposition ascriptions and the form of possibility statements need not be made to ‘fit’ together as they do in (P). I think such a move, while more audacious than the first, is still too timid. For it gives us no obvious way of finding out whether or not the constraint of formal adequacy has been met in general. Truthmaking and similar notions do not (yet?) seem to afford the generality and rigour needed to check whether or not those constraints hold. I prefer a third way of responding to the challenge of logical form: tinkering with the form of disposition ascriptions. What we need is a general understanding of an object’s having a disposition for it to be the case that p. Given such an understanding, we can then examine the logical principles that govern those

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dispositions: are they, for instance, closed under logical implication?16 Further, given those principles, we can very easily check the logic of possibility that they imply in conjunction with a bridge principle similar to, but more general than, (P). Formally speaking, this is the path of least resistance. Metaphysically speaking, however, it is not. For it requires some hard work on the required notion of dispositionality. Most of this book will be devoted to developing such a notion which is wide enough to make sense of dispositions for it to be the case that p, for any value of p. I will drop the term ‘disposition’ and adopt, instead, that of ‘potentiality’. Potentiality, again, is intended to be simply the best generalization of our (perhaps) more intuitive notion of dispositions. It will be argued, first, that dispositions (and potentialities in general) are individuated solely by their manifestation, so any theory along the lines of (C) will be precluded from the start. Second, it will be argued that we can make sense of potentialities for it to be the case that p without giving up the paradigm of a localized modality. Third, potentiality will be argued to exhibit the minimal formal features of possibility itself: closure under logical implication, distribution over disjunction, and implication by actuality. Fourth, potentialities will be shown to be so broad as to accommodate the constraints of extensional correctness and semantic utility too. The strategy in all this will be to start with simple, intuitive examples and to generalize from there, arguing that any limit on potentiality short of those that I want to adopt would be metaphysically arbitrary. Potentiality so understood differs from possibility mainly in being relative to a particular object. We may think of potentiality as possibility that is relativized to a particular object, but on the view here proposed that would get the direction of explanation wrong. Rather, we should think of possibility as potentiality in abstraction from its bearer. A possibility is a potentiality of something or other, no matter what. An improved version of (P) therefore reads as follows: (P ) It is possible that p just in case something has a potentiality for it to be the case that p. (P ) is closer to, but still not identical to, the final version of the view that will be stated in chapter 6.1. The final version will read as follows: POSSIBILITY It is possible that p =df Something has an iterated potentiality for it to be the case that p. 16 Strictly speaking, what might or might not be closed under logical implication are not dispositions, but their manifestations. For convenience, I will continue to speak in this loose way throughout the book. Thus chapter 5 argues that potentiality—strictly speaking: the manifestations of an object’s potentialities—is (are) closed under logical implication.

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To understand, let alone evaluate, POSSIBILITY, we will need to do more work yet on the notion of potentiality. For the time being, therefore, I will stick with (P ). (P ) already makes it clear that the burden of the theory rests on its conception of potentiality. A large part of the book will be spent discharging that burden, and arguing that the required theory of potentiality is independently plausible.

1.5 Potentiality As I am going to use the terminology, every disposition is a potentiality but not every potentiality is a disposition. Some potentialities are abilities, such as my ability to walk, which may or may not have a dispositional equivalent. Some potentialities are what we might call powers or potentials, which appear to be weaker than dispositions—such as a sturdy steel bridge’s potential to break, or my potential to jump out of the window (which, fortunately, is no disposition at all). It is not always clear from the literature on dispositions how extensively we are to conceive of dispositions; perhaps there is a sense of ‘disposition’ in which the potentials just noted, and many others like them, do qualify as dispositions. Never mind; I do not want to quibble about terminology, so I will use the term ‘potentiality’ instead. It has a suitably general ring to it, and it has the virtue of being an obviously theoretical term with few pre-philosophical intuitions that would constrain its use. I will not, however, provide an explicit definition of this theoretical term; it is going to be the primitive in terms of which I define other things. You may justly wonder what the point is of a term that has neither an intuitive, pre-philosophical content nor an explicit definition. I reply that while the term itself has little usage outside philosophy, we have a rather firm prephilosophical grasp on part of its extension. That part will turn out not to be very precisely circumscribed, and I use the term ‘potentiality’ for whatever is the best and most general precise notion that includes the pre-philosophical extension. Let me be more specific. Our understanding of potentiality begins with dispositions. Fragility, solubility, irascibility, elasticity, and so on are properties that we ascribe all the time to objects in both everyday life and science. We appear to have a good pretheoretical understanding of what it takes for an object to possess such a property, and we use that understanding to guide our actions: being careful not to drop fragile objects, putting soluble sugar in our coffee, avoiding provoking behaviour around irascible people, and putting elastic bands around objects to hold them in place. As so often in metaphysics, we can start with an intuitive understanding but we cannot stop with it. We need a more general understanding of what those

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properties are that we ascribe and rely on in everyday talk, thought and behaviour. But in order to gain that general understanding, we also need to generalize the subject matter and extend our conception of dispositionality beyond the obvious, everyday examples. Such an extension will be necessary for the purpose of giving a theory of modality; the intuitive examples of dispositions are clearly not enough to account for all the possibilities there are. But it will also be necessary in order to give a non-reductive account of those properties. Here is why. Disposition terms are often context-sensitive, and they are often vague. What counts as fragile in the context of an antiquities shop may differ from what counts as fragile in the context of space travel. Whatever the context, there are borderline cases between what does and what does not count as fragile.17 An ordinary context will probably determine that champagne glasses, tea cups, and tumblers count as fragile while blocks of steel, wooden desks, and brick stones do not. But what about an ordinary plant-pot that is easier to break than the brick stone, though less easily broken than the tumbler: is it fragile or not? There may be no one right answer to that question (or if epistemicism is correct, it will be impossible to determine the right answer).18 The plant-pot is a borderline case of fragility in the given context. However, both vagueness and context-sensitivity are features of language, not the world. In the case of context-sensitivity this is obvious. What a context-sensitive term such as ‘I’ refers to varies with the context in which it is uttered. But that to which it refers in a given context—a particular individual—does not, of course, turn into something else when the word is uttered in a different context. Similarly, which property is ascribed by ‘fragile’ may vary with contexts. But the properties that are ascribed in the various contexts remain the same properties. Contextual variation concerns only which of them are ascribed. Vagueness, too, is a feature of language, not the world.19 If it is vague whether a particular stone is part of Mount Everest, that is not because there is an object, Mount Everest, which is indeterminate with regard to whether or not it includes that stone. Rather, it is because the name ‘Mount Everest’ is vague—there is no right answer to the question whether it refers to the mountain so delineated as to include the stone, or the mountain so delineated as to not include it; or, if 17 I adopt a common first characterization of vagueness: any predicate that gives rise to borderline cases and a Sorites series is vague. 18 Epistemicism is the view that vague predicates have perfectly precise boundaries, but that we cannot know where they lie. See Williamson (1994). 19 This view, while not uncontroversial, is widely accepted. See Williamson (2003) for a detailed discussion.

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epistemicism about vagueness is correct, there is a right answer (perhaps a different one in different contexts), but one which we are in principle unable to know. Similarly, if it is vague whether the plant-pot counts as fragile, that is not because there is a property, fragility, which is indeterminate with regard to whether or not the plant-pot is included in its extension. Rather, it is because the term ‘fragile’ is vague—there is no right answer to the question whether it ascribes a property that (determinately) includes, or a similar property that (determinately) does not include, the plant-pot in its extension; or if epistemicism is correct, there is a right answer (probably a different one in different contexts) which we are in principle unable to know. If disposition terms are vague and context-dependent, then we cannot simply rely on our grasp of them in spelling out the underlying metaphysics, even if that metaphysics is to be non-reductive. We must try to understand the precise, context-independent background on which the various precisifications of a vague term, and the various shifts in context, operate. Traditionally, this background has been provided by possible-worlds semantics. It is, on that framework, an entirely precise and context-independent matter which possible worlds there are. Vagueness and context-sensitivity of disposition terms (and other modal expressions) can then be modelled as indecision and contextual shifts regarding which of these worlds count as relevant, and exactly which kind of quantification is applied to them. A non-reductive metaphysics of dispositions will provide a different background for vagueness and contextual shifts, one that is dispositional, but general enough to accommodate all the borderline cases and all the contextual resolutions. Specifically, we will see in chapter 3 that the context-sensitivity of disposition terms such as ‘fragile’ is a matter of degrees of fragility: whether something counts as fragile in a given context is a matter of how fragile it is, context setting a minimum degree which needs to be satisfied for the predicate to be true of a thing.20 The context-independent metaphysical background must consist, for the dispositionalist, of a property that comes in degrees, some of which will, while others won’t, be sufficient for the true application of ‘is fragile’. We might call that property a disposition, and distinguish between the contextinsensitive and the context-sensitive use of ‘disposition’. Or we might reserve the term ‘disposition’ for the context-sensitive use and introduce a new term for the context-insensitive metaphysical background. The decision is a purely terminological one. I have chosen the second option and the term ‘potentiality’. Thus in my terminology, having a disposition such as fragility is a matter of having 20

This has been forcefully argued by Manley and Wasserman (2007).

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the right potentiality (in this case the potentiality to break or be broken) to a contextually sufficient degree. Once we have introduced potentiality in this way, it can be argued that the potentialities which objects have outrun the dispositions that we are willing to ascribe to them. A chunk of gold will not, in any context, count as fragile. (Let us suppose, in any case, that this is so.) Yet there is no deep distinction between a chunk of gold and a champagne glass with regard to fragility; the difference is one of degree. We can see this by imagining a series of objects, each just a little less fragile than the last, from a champagne glass through a tumbler, a plant pot, a rock, and a diamond, down to the chunk of gold. Apart from contextual (and hence linguistic, not metaphysical) constraints, the difference between the objects in this series, so far as fragility is concerned, is one of degree, not all or nothing. If the potentiality to break is that property whose degrees provide the metaphysical foundation for the comparative ordering, then that potentiality must be possessed all the way through the series; any stopping-point would be arbitrary.21 I have anticipated some of the argument of chapter 3 to show how it is nonreductive realism about dispositions, together with quite general metaphysical assumptions (in this case, the rejection of vague or arbitrary cut-offs in metaphysics), which forces us to expand the intuitive notions of dispositions that form our starting point. This expansion is just what we need to account for metaphysical possibility. The considerations just sketched, for instance, motivate a conception of potentiality that can account for the possibility, say, of a large block of concrete or even a chunk of gold breaking. Neither would count in an ordinary context as ‘fragile’, but each has the potentiality to break, albeit to a relatively minimal degree. Saying otherwise would require the arbitrary cut-offs that I have shunned. And so it is non-reductive realism itself, not the project of accounting for metaphysical possibility, that motivates the expansion. My theoretical term ‘potentiality’ is defined as that, whatever it is, which results from such expansion in the end. In a nutshell, the argument of this book is that a conception of potentiality which is general enough to satisfy the constraints of non-reductive realism is also general enough to satisfy the constraints on a dispositionalist theory of modality: extensional correctness, formal adequacy, and semantic utility. We have just seen some steps towards extensional correctness. Formal adequacy

21 The rejection of arbitrary cut-offs will play a role, in different ways, in chapters 3.4, 4.3, and 5.3.3. I have little to say in its defence, but I take it to be a widespread and natural line of thinking about metaphysics.

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and semantic utility both require in addition a notion of extrinsic potentiality. With that notion in place, potentiality becomes flexible enough to account for the context-sensitivity of modal expressions in ordinary language, and to exhibit the same logical structure that possibility has: closure under logical implication, distribution over disjunction, and implication by actuality. The conception of potentiality that is reached by the end of chapter 5 is an expansion and generalization of the intuitive examples of dispositions, but the expansion is considerable. We will arrive at a conception of potentiality that is very much the equivalent of Fine’s notion of essence: it stands to possibility as essence stands to necessity, the localized counterpart of a non-localized modality. You may ask at that point what is left of dispositionalism. Was the idea not to root metaphysical modality in something so mundane and well-understood as fragility, solubility, and so forth? It is indeed, but we should not forget that metaphysical possibility is already a great expansion and generalization from our ordinary understanding of possibility. Ordinary thought has little use for the possibility of there being talking donkeys, or for the inference from actuality to possibility. (Outside philosophy, to say that it is possible for the glass to be broken when we know that it is actually broken is misleading at least.) It is only to be expected that in accounting for metaphysical possibility, we need to formulate a thoroughly metaphysical conception of dispositions, or, as I have labelled that conception, of potentiality. Once we apply the same standards of generality and rigour to dispositions that we have become accustomed to applying when we talk of modality, it will be seen that dispositionality can be stretched just as far as possibility has been stretched already. (I will return to the issue of how informative my account of possibility is later in the book, in chapter 5.9.) Before I begin, I want to state explicitly some important assumptions that will be in the background throughout much of the book. They concern the status of objects, the metaphysics of properties, and the relation of grounding. I will not argue for those assumptions. We all have to start somewhere; this is where I start.

1.6 Background assumptions I will assume that the following picture of reality is roughly correct: the world consists of objects, and those objects have potentialities. The world is, moreover, ordered (or structured) by a relation of objective grounding, where the more fundamental grounds the less fundamental. (I do not assume that there is a final, fundamental level which grounds everything else—grounding might go on infinitely.) To say that the world ‘consists of’ objects is to say that no matter how far we progress from the less to the more and more fundamental (and to the ultimate

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fundamental level, if there is one), we will always find objects. To say that these objects have potentialities is to say that no matter how far we progress from the less to the more and more fundamental (and to the ultimate fundamental level, if there is one), we will never entirely get rid of potentialities. Those are the basic assumptions underlying the theory to be developed in this book. My aim is not to defend those assumptions directly, but rather to show what we can do with them—in particular, how we can develop a theory of modality based on them. Since they are crucial to the book, these assumptions deserve to be explored in just a little more depth. Objects. I will assume that the world is fundamentally constituted of things. It is not composed of free-floating tropes or states of affairs (as Wittgenstein may have held); nor is it a whole that is prior to its parts, the individual objects in it (as Schaffer 2010 holds); nor is it a structure of relations that are prior to their relata (as Ladyman et al. 2007 argue). It is individual, concrete things that come first. I will have nothing to say either about what makes a particular thing a particular thing, or about what makes a particular thing the thing that it is. We have seen above that the attraction of the dispositionalist theory of modality derives, to a considerable degree, from its smooth integration into an object-based ontology. In spelling out the view, we should allow ourselves such an ontology. Potentialities. I assume non-reductive realism about potentiality; for brevity’s sake, I will often speak of ‘realism’ with the understanding that it is to be of the non-reductive sort. (‘Realism’ has a number of usages that are not co-extensive. Thus my account of modality is realist, as opposed to anti-realist, in that it takes modality to be mind-independent; but it is not realist, as opposed to reductionist, because it takes metaphysical modality to be reducible to potentiality. My view of potentiality is realist in both senses, but I will use the term ‘realism’ in the sense opposed to ‘reductionism’.) Thus realism about potentiality is the view, first, that things have potentialities. But that is not quite enough. Even the Humean metaphysician can agree that things have potentialities. The point of disagreement is not the platitude that things are fragile, soluble, and so on; it is the underlying metaphysics. For the Humean, the underlying metaphysics might involve some such things as other possible worlds, laws of nature, and/or the ‘categorical’, Humeanly acceptable properties that form the ‘basis’ of the dispositions in question. For the purposes of this book—and for reasons that have been sketched above—I am going to assume that such a reductionist stance is mistaken. I will reserve the term realism (about dispositions, or about potentiality in general) for the anti-reductionist view that dispositions, or potentialities, are metaphysically basic, primitive, irreducible.

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Of course, the potentialities of things on the macro-level can often be explained in terms of their smaller parts and the structural arrangement of those parts. A glass’s fragility may be explained, for instance, by appeal to the molecules that make it up and the robustness (or lack of it) of the bonds between them; metaphysically speaking, it is grounded in the molecular make-up of the glass. However, the robustness of bonds (or the lack of it) is itself dispositional, a matter of potentiality. In explaining potentialities on the macro-level, we have to invoke potentialities again, though this time on the micro-level. The realist’s contention is not that every potentiality is irreducible or inexplicable. It is rather that, in reducing or explaining one potentiality, we always find ourselves saddled with new potentialities. Hence we can phrase realism about potentiality as the claim that as we progress from the less to the more fundamental levels, we will always find potentialities. It’s potentiality ‘all the way down’. Realism about potentialities is not, however, committed to the view that all, or even all of the fundamental (if such there are) properties are potentialities. There is some disagreement among dispositional realists on the question whether or not that claim should be adopted. Some, such as Ellis (2001), hold that there are both categorical and dispositional properties (potentialities). Others, the pandispositionalists such as Bird (2007), prefer to think that all properties are dispositional (potentialities). Spatiotemporal relations are a prime example of properties that are difficult to understand in dispositional terms, but Bird (2007, ch. 8) has offered an attempt to fit them into a pandispositionalist conception of properties. The claim that some or all properties are dispositional is often associated with the label ‘structuralism’, and it will be helpful to explore this label a little further. According to causal structuralism, the identity of a property is given by its place in a causal structure (see Shoemaker 1980, 1998, and Hawthorne 2001). I adopt a version of structuralism, at least for potentialities, but I want to replace causal structure by a potentiality-based structure. In causal structuralism, the structure of properties is given by a relation ‘ . . . causes . . . ’. In potentialitybased structuralism, the structure of properties is given by the relation between a potentiality and its manifestation, the relation ‘ . . . is a potentiality to . . . ’, which I shall call the manifestation relation. The manifestation relation is directed: it goes from potentiality to manifestation. It is irreflexive (no potentiality is its own manifestation), asymmetric (no potentiality is the manifestation of its own manifestation), and intransitive (a potentiality to have a potentiality to F is not thereby a potentiality to F). On the present version of structuralism, the identity of a potentiality is given by its manifestation, and nothing else. The manifestation of a potentiality may itself be a potentiality, or a complex (a conjunction, disjunction, and so forth)

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of potentialities, in which case its identity is given by its manifestation (or the manifestations of its disjuncts, conjuncts, and so forth). If there are properties that are not potentialities—in other words, if there are categorical properties— then those may still be manifestations of potentialities. Categorical properties would then be part of the structure that is given by the manifestation relation, but they would be dead ends in such a structure. The question of pandispositionalism can be phrased as the question whether there are any dead ends in the potentiality structure. Pandispositionalists and causal structuralists generally restrict their view to sufficiently natural properties—Bird (2007) is concerned only with the fundamental physical properties, and Shoemaker (1980), while aiming at a wider class of properties, explicitly excludes so-called Cambridge properties, for instance, the property of being such that grass is green. In this book, I will adopt a rather liberal view on which properties there are (more on which in a moment), and I would not wish to claim that all of the not-so-natural properties that I accept are indeed potentialities. The restricted forms of pandispositionalism are, of course, still on the table, and I have some sympathy for them. Nevertheless, what I need to assume is just that there are potentialities, and a great many of them, not that there are no other properties. Grounding. I assume that certain facts hold in virtue of or because of other facts’ holding; that some facts are grounded in, explained by, fixed by, or depend on others. Metaphorically, this is sometimes put by saying that if God had made the world so that the one set of facts held, then she would thereby have made the world so that the other set of facts held. For instance, it is plausible that truth depends on being, as it is sometimes put: if God made the world so that grass is green, there was nothing else for her to do in order to make the world so that the proposition that grass is green is true. My assumption is that such relations of ‘because’ or grounding are metaphysically real. I assume, further, that grounding is related, in some way or another, to fundamentality. If there is a fundamental level, then its fundamentality could be characterized, wholly or partly, in terms of grounding.22 More importantly, we can think of comparative fundamentality in terms of grounding: that which 22 Schaffer (2009a) defines the fundamental entities as those that are not grounded by any other entities. Fine (2001) rejects such a simple definition of what is ‘real’ (his preferred term where I would use ‘fundamental’): p might be ungrounded without being real, if p is not factual (this, Fine claims, is the status of moral propositions according to expressivists), and p might be real while being grounded if some real facts (or, for Fine, propositions) ground each other. Sider (2011) argues that ‘Structure’, which characterizes the fundamental level, is not to be characterized in terms of grounding at all. So the assumption is not entirely uncontroversial, but since I will not be appealing to absolute fundamentality much in this book it should be an innocuous idealization at worst.

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grounds is more fundamental than that which is grounded in it. If you would like to reserve the notion of fundamentality for a non-comparative or altogether different notion, feel free to rephrase everything that I say about comparative fundamentality in terms of grounding. Grounding—under this label or any of the others that I have used in introducing the notion—has received a fair amount of attention recently,23 and there is considerable controversy on how best to construe it. I will not rely on any particular conception of grounding, but I will rely on some commonly shared distinctions and precisifications, which it is worth noting here. (a) Grounding is not supervenience, on any of the standard modal definitions of supervenience. Appealing to a modal notion such as supervenience in the course of this book would beg the question, since modality is to be the output, not an input, of the present theory. But there are independent reasons, now well rehearsed, for rejecting the modal notion as a way of capturing our intuitions of grounding or dependence. Supervenience, being defined modally, is at most intensional: it can distinguish between propositions whose truth-values actually coincide, but not between propositions whose truth-values necessarily coincide. Grounding, on the other hand, is hyper-intensional: it can discriminate even between modally constant co-variants. Thus Socrates’ existence and the existence of his singleton set {Socrates} symmetrically supervene on each other—no two possible worlds could differ from each other with respect to one without differing also with respect to the other—but it is the set that depends on Socrates, and not vice versa. Further, and unlike supervenience, grounding is generally taken to be a primitive. The reasoning is that it is a concept that we can hardly avoid, of which we have a good intuitive grasp, and which has resisted all our attempts at reductive definition. It is also a primitive that we have come to understand better in recent years through an investigation of its formal features.24 (b) Grounding can be viewed as operational or relational. Relational grounding, as the label suggests, is a relation between entities and is best expressed by a two-place relational predicate ‘x grounds y’ enabling us to say such things as: the singleton set {Socrates} is grounded in Socrates, or: the truth of the proposition that Socrates is wise is grounded in the fact that Socrates is wise. (Schaffer (2009a) appeals to relational grounding.) Operational grounding, on the other hand, is 23 See especially Rosen (2010), Schaffer (2009a), Correia (2010), Schnieder (2011), Fine (2012a), Fine (2012b), and the papers in Correia and Schnieder (2012). A useful overview of the debate is given in Clark and Liggins (2012). For a defence of grounding against some criticisms, see Raven (2012). 24 See, for instance, Rosen (2010), Schnieder (2011), and Fine (2012b).

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best expressed by a two-place sentence operator completed not by terms for entities but by entire sentences. The sentence operator may be read as ‘because’ or ‘in virtue of’, enabling us to say such things as: Socrates’ singleton exists because Socrates exists, and the proposition that Socrates is wise is true because Socrates is wise; or: Socrates’ singleton exists in virtue of Socrates’ existing, and the proposition that Socrates is wise is true in virtue of Socrates’ being wise. The operational approach to grounding is explicitly adopted, for instance, by Rosen (2010), Correia (2010), Schnieder (2011), and Fine (2012b). For my purposes, the operational view will be more convenient. I will not primarily be concerned with the dependence of one class of entities on another, but with the dependence of an object’s having some potentiality on its or other objects’ having other potentialities. For convenience, however, I will sometimes speak of (the fact that) p grounding (the fact that) q. (c) Grounding can be full or merely partial. The distinction is most readily illustrated by an example: p or q is fully grounded by p; p and q is only partially grounded by p, and is fully grounded only by p and q together. Fa fully grounds ∃xFx, but only partially grounds ∀xFx.25 As Kit Fine puts it: A number of truths will fully ground another when they are sufficient on their own to ground its truth. . . . One truth partly grounds another when it is of help in fully grounding the other. Fine 2012b, 3

So far, the assumptions that I have made (with the possible exception of the last paragraph) have been indispensable for the project of this book. Another bundle of assumptions should be in principle dispensable, but I will make them for expository purposes and leave the reformulation of my account without these assumptions for another time. The assumptions all concern ontological liberalism. Liberalism takes a number of forms: about objects, about properties, and about facts. I am moderately liberal about which objects there are. I hold that there are particles, vases, dogs, trains, clouds, and nations, but I remain neutral on whether there are numbers, directions, and smiles. In short: I am liberal about concrete objects, but neutral concerning abstract objects (with the exception of properties, as we shall see in what follows). The existence of some objects may be grounded in that of others, but I do not take that as a reason to reject the grounded objects— on the contrary. 25 See Schnieder (2011, 461) for a discussion of the somewhat ‘hard to swallow’ case of universal quantification.

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Concerning properties, I am maximally liberal. Liberalism here has two dimensions. First, I assume that there are properties, and that a property is the kind of entity that different particulars can share. When two apples are both red, then there is a property that they both possess, the property of being red. Accordingly, different things can share a potentiality for the same property: the sheets of paper in this book all share the potential to burn. Nominalists about properties have different strategies in dealing with our ubiquitous talk of properties shared by things. One is to claim that when we say that two apples share a property, what we say is not in fact true, but something similar is (the two red apples have more or less exactly resembling tropes, or they are both members of the same class of particulars that resemble each other in a certain respect). Another is to claim that what we say is true, but analyse away the apparent appeal to shared properties (what we really say is that things have resembling tropes, or that they belong to the same class). I suspect that both strategies could be applied throughout the book, making only slight alterations to my central claims and arguments. (Chapter 7 would probably require the most substantial changes.) But at the moment, I have no more than that suspicion to offer to the nominalist. Second, I will be maximally liberal about which properties there are. The properties that there are include the properties of: being electrically charged and being fragile, being green and being grue, being self-identical and being identical to me, and being such that grass is green. Some of these properties, to be sure, are more natural than others. I think of the naturalness ordering among properties in comparative terms. The basic facts about naturalness are of the form: property P is more natural than property Q. Others, such as Lewis (1983), think of the ordering in absolute terms: they begin with a ground floor of perfectly natural properties, and understand a property’s degree of naturalness by the length of its definition in terms of perfectly natural ones. (For the contrast, see Sider 2011, 128.) In general, possession of the more natural properties grounds possession of the less natural ones; thus we might equally speak of more or less fundamental properties. Again, I do not take the fact that a property’s instantiation is grounded in the instantiation of other properties as a reason to reject the grounded property—on the contrary. I adopt this liberal approach mostly because modal facts, prima facie, involve properties of all degrees of naturalness, and so it will be useful to think of potentialities as involving (manifestation) properties of all degrees of naturalness. My concern is not with the properties that are ascribed in the scope of the ‘possibly . . . ’ operator, but with the operator itself. If you wish to adopt a more austere metaphysics of properties, restricting yourself to the perfectly natural

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properties only, my suspicion is that you should be able to rephrase and reformulate most of what I say in this book and still obtain a plausible account of potentiality and possibility. For my purposes, it is easier to start with the liberal view. With both objects and properties, then, I accept all levels of fundamentality, the grounding and the grounded alike. Unsurprisingly, the same goes for facts or, to be less committal, for how things are. When p is grounded in q, that is to me no reason to reject p in favour of q. Rather, it is all the more reason to accept p if we accept q; after all, we have acceptable grounds for p. In fact, I believe that where we can find respectable facts to fully ground the fact that p, we thereby have irresistible reason to accept that p. Again, if your preferences are more austere in any or all of these fields—if you want to accept only fundamental objects, perfectly natural properties, and how things are at the fundamental level—you will have found some ways of reformulating our claims about non-fundamental objects, not-so-natural properties, and how things stand with them. Feel free to apply those ways throughout this book; everything should still work out fine. While I personally accept the liberalism that I adopt quite literally, it can be seen as an expository device for the purposes of this book.

1.7 Outline of the book The main task of this book is to develop a theory of potentiality that is general enough to meet the requirements of formal adequacy, semantic utility, and extensional correctness. I do so in chapters 2–5. Those chapters provide the materials to understand and defend the account of possibility in chapters 6 and 7. An appendix will show in more detail that the requirement of formal adequacy is met. The chapters that follow come in pairs. In a first pair of chapters (2 and 3), I look more closely at dispositions. Chapter 2 argues that the standard linking of dispositions to conditionals, and the two-part structure of stimulus and manifestation that comes with it, runs into trouble if we want to be realists about dispositions, whether they are the ordinary, everyday ones such as fragility or the more controversial, scientific ones such as electric charge. Chapter 2 ends by suggesting that dispositions are to be understood not in terms of a stimulus and a manifestation which stand in a relation much like that of antecedent and consequent of a counterfactual conditional. Rather, we should think of dispositions in terms of the manifestation alone—not as a disposition to . . . if . . . , but as a disposition to . . . , full stop. Chapter 3 develops this suggestion in much more detail, drawing on the semantics of ordinary

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disposition terms and the general considerations that I have already anticipated above. The notion of potentiality is introduced at the end of chapter 3, as the suitably general realist background to the contextual variation of disposition terms. The second pair of chapters (4 and 5) deals directly with potentiality. Chapter 4 introduces important classifications of potentialities: the joint potentialities that objects possess together; the extrinsic potentialities that objects possess in virtue of possessing joint potentialities together with other objects; and iterated potentialities, potentialities to acquire potentialities (to acquire potentialities to acquire potentialities to . . . and so forth). Extrinsic potentiality will be crucial in meeting the constraints of semantic utility and formal adequacy. Iterated potentiality is indispensable in meeting the requirement of extensional correctness. Those are my motivations for introducing these types of potentiality. But chapter 4 argues that all of them are motivated independently of the project of this book: in each case, we start with intuitive cases and merely generalize from there. Chapter 5 goes on to provide a more formal treatment of potentiality. A formal language for potentiality is introduced, which allows us to express such potentiality ascriptions as ‘I have a potentiality to be such that I am sitting or you are standing’, etc. It is argued that any type of sentence that can be formed with the resources of the language is comprehensible and a candidate for truth, including the outlandish and somewhat surprising ones. This is an important step towards both extensional correctness and formal adequacy, for it allows us to formulate a corresponding potentiality ascription for any possibility, no matter what the form of the embedded sentence. The second half of chapter 5 then goes on to argue that potentiality is governed by axioms and rules that are in parallel with those for metaphysical modality: in particular, closure under logical implication, distribution over disjunction, and implication by actuality. (While addressing formal issues, the chapter itself should be readable without formal background.) In a final pair of chapters, the wealth of materials that has been developed is finally applied. Chapter 6 returns to the account of metaphysical possibility that has been given above as (P ) and makes some final improvements on it by including the case of iterated potentiality. I explore how this account deals with some standard cases of metaphysical possibility and necessity and how it meets the two constraints of formal adequacy (easily, given the work of chapter 5) and semantic utility (by construing a significant class of modal expressions as ascribing or denying potentialities). Extensional adequacy requires a longer story, and the final chapter 7 is devoted entirely to it. I go through counterexamples of two kinds: potentialities without the corresponding possibilities, and possibilities without the corresponding potentialities. The latter are the more numerous and

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take up most of the chapter. I outline different options in responding to them and argue that at least one such option is available for each putative counterexample. Incidentally, I outline a (Stalnakerian) account of possible worlds as unmanifested potentialities of the world as a whole (but I do not use it to respond to the objections). The most worrying questions for the potentiality-based account of modality stem from an issue that I cannot fully resolve in this book, and which seems to me the most pressing issue for further work: the relation between potentiality and time. I do, however, provide reasons for optimism even concerning that difficult subject. The appendix aims to provide a somewhat more rigorous and systematic framework for the argument concerning formal adequacy. It does not, however, provide a full-blown logic of potentiality; that is a task which I must leave for another time (or for another philosopher). Note, again, that I will not be concerned with showing the superiority of the potentiality-based account over alternative accounts of modality. I have provided some support for the idea that it may indeed be superior, but the case cannot be made before the account is fully developed. It is the development of the view, not its comparison with competitors, that is the goal of this book.

2 Dispositions: Against the Standard Conception 2.1 Introduction The aim of this book is to develop a theory of possibility based on the potentialities of individual objects. To do so, we need first to develop a suitably general and fully realist view of potentiality. My starting point in understanding potentiality is dispositions: properties such as fragility, solubility, irascibility, and many others that have been the subject of extensive debate in recent metaphysics. I am going to argue that the extension and generalization that I need for a theory of possibility are required for full-blown realism about dispositions, quite independently of the target theory of possibility. In this chapter and the next, I will be concerned with philosophers’ prime examples of dispositions: ordinary properties such as fragility, solubility, irascibility, and so forth, as well as (from section 2.5 on) the dispositions that are involved in the laws of nature. But we begin with ordinary dispositions such as fragility. Dispositions are modal properties. In saying of a vase that it is fragile, of a person that she is irascible, or of a disease that it is transmissible, we are not, or not primarily, saying something about what the vase, the person or the disease is actually doing, but rather about what it would or could do. Being modal sets dispositions apart from categorical properties (if there are any), while being properties sets them apart from what I have called the ‘non-localized’ modalities such as possibility, necessity, and the counterfactual conditional. This minimal characterization of dispositions gives rise to two separate questions. One is the question how we are to understand the modal aspect of dispositions. In answering that first question, it is heuristically useful—though, as we shall see, not necessarily metaphysically explanatory—to appeal to the familiar non-localized or sentential modalities. To which of those are dispositions most

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similar, or most closely linked? The standard answer that pervades the contemporary literature is that it is the counterfactual conditional. This standard answer comes with a claim about the individuation of dispositions: each disposition is specified by a stimulus condition, akin to a conditional’s antecedent, and a manifestation, akin to the consequent. Thus the following two claims deserve the label ‘the standard conception of dispositions’: 1. A disposition is individuated by the pair of its stimulus condition and its manifestation (or, if it is a multi-track disposition, by several such pairs): it is a disposition to M when S (or a disposition to M1 when S1 , to M2 when S2 , etc., if it is a multi-track disposition). 2. Its modal nature is, in some way or another, linked to or best characterized (to a first approximation) by a counterfactual conditional ‘If x were S, x would M’ (or, if it is a multi-track disposition, by several such conditionals). Note that the standard conception does not tell us how a disposition is linked to a conditional (or conditionals). The link might be reductive: for an object to have a disposition just is for the corresponding conditional(s) to be true of that object. Or it might be non-reductive: the conditional provides an illuminating, perhaps only an approximate, characterization of the disposition, but without reducing it away. Finally, dispositional realists may turn the tables and reverse the order of explanation: for a counterfactual conditional to be true just is for the relevant objects to possess a certain disposition. Each of these options has been defended in the literature.1 But it is important to see that each of them is an answer to a second question: not which (non-localized) modality dispositions are linked to, but how they are linked to that modality. The second question arises from the fact that dispositions, besides being modal, are properties. It is the question how we are to understand the propertyhood of dispositions: as reducible to a non-localized modality that is not property-like, or as a phenomenon in its own right, perhaps even one that is prior to the non-property-like sentential non-localized modalities. It is in answering this second question that the debate between realists and reductionists about dispositions takes place. Unlike the first question I have raised, there is no ‘standard’ answer to this second question in the literature. My favoured answer, of course, is a realist one. 1 Lewis (1997) gives the best-known reductive analysis. Martin (1994) holds that conditionals are just ‘clumsy and inexact linguistic gestures to dispositions and they should be kept in that place’ (Martin 1994, 8), while Manley and Wasserman (2008) seek to provide a true equivalence between disposition ascriptions and conditionals that need not be reductive. Jacobs (2010) gives truth-conditions for counterfactual conditionals in terms of dispositions.

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To properly discuss and even to formulate an answer to the second question, and to spell out either a realist or a reductionist conception of dispositions, we need already to have an answer to the first question: how are we to think of the modal aspect of dispositions, and to which non-localized modality is it connected in either a realist or a reductionist way? Due to the almost unquestioned prevalence of the standard conception, that question has been rarely asked. My goal in this chapter and the next will be to pose that question and to answer it in a nonstandard way. In chapter 3, I will propose that we replace the standard conception of dispositions with an alternative conception: 1. A disposition is individuated by its manifestation alone: it is a disposition to M, full stop. 2. Its modal nature is that of possibility, linked to or best characterized (to a first approximation) by ‘x can M’. The task of this chapter is to prepare the ground for the alternative conception, by arguing that the standard conception is less than desirable for a realist metaphysics of dispositions. The first question—which modality is involved in dispositions—to which the standard and the alternative conception provide competing answers, is prior to the second question: how is that modality involved in dispositional properties. So in drawing out the consequences of the standard conception, we can largely ignore the second question, and with it the debate between realism and reductionism. Accordingly, sections 2.2–2.5 will remain neutral on the issue of realism or reductionism. Only in section 2.6 will we return to the dispositional realism that is part of the project of this book and apply the results of the foregoing discussion to it. (Chapter 3, in spelling out the alternative conception, will proceed in a similar way.)

2.2 Dispositions and conditionals: the state of the debate Dispositional properties have traditionally been characterized in terms of conditionals. Thus a fragile vase is one that would break if it were struck, an irascible person is one who would get angry if she were provoked, and a water-soluble substance is one that would dissolve if it were immersed in water. On the simple conditional analysis, the conditional is all we need to provide a complete analysis of such a disposition. Thus the simple conditional analysis of fragility goes as follows: (SCA)

x is fragile iff, if x were struck, x would break.

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It is now agreed by most that the simple conditional analysis is false. A vase’s fragility may be finked by a sorcerer who would immediately remove its fragility should it be struck, or masked by anti-deformation packaging which would prevent it from breaking even if it were struck. In both cases the left-hand side of the biconditional in (SCA) is true, but its right-hand side is false. Moreover, a non-fragile concrete block may have a reverse fink if a sorcerer were to make it fragile as soon as it were struck, or it may mimic fragility if it is attached to an explosive that would go off and shatter the block if it were struck. In these latter two cases, the right-hand side of the biconditional is true although the left-hand side is false.2 But there is another kind of problem with (SCA) altogether, which David Manley and Ryan Wasserman have done much to establish (see Manley and Wasserman 2007 and 2008). These are structural problems with (SCA) and any other attempt at analysing a disposition in terms of a single conditional. A first structural problem is that dispositions are gradable: a champagne glass is more fragile than an ordinary tumbler, and some of us are more irascible than others. This is a problem because (SCA) does not afford the materials for an account of the gradability of ‘fragile’: unlike fragility, the truth of a conditional is an allor-nothing matter. Second, and relatedly, it appears that disposition terms are context-sensitive: what counts as fragile in the context of aeronautics may not count as fragile in the context of ordinary life. It is appealing to account for that context-sensitivity in terms of the degrees of fragility: whether something counts as fragile in a given context is a matter of how fragile it is. (SCA), again, cannot deliver such an account of context-sensitivity, because it cannot deliver an account of gradability. Third, Manley and Wasserman point out that not every disposition comes with a specified stimulus condition. Loquaciousness and irascibility, for instance, may be triggered by any condition whatsoever (Manley and Wasserman 2008, 72). But if there is no one condition to fill the place of a disposition’s stimulus, then there is no way to fill the antecedent of the related conditional. To solve these structural problems, Manley and Wasserman (2008) propose that (SCA)’s mistake lies in its focus on a single conditional. Instead, they suggest, we need to think of a disposition as correlated with a large (in fact, a non-denumerably infinite) number of conditionals, each of which specifies in its antecedent a ‘fully specific scenario that settles everything causally relevant to the manifestation of the disposition’ (Manley and Wasserman 2007, 72). We can then say that x is more fragile than y just in case more of these specific conditionals are 2 See Martin (1994), Lewis (1997), Johnston (1992), and Bird (1998). Bird speaks of ‘antidotes’ where I, following Johnston, speak of ‘masks’.

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true of x than of y, and, with this comparative ordering established, contextsensitivity can work by setting a threshold on it. Finally, for dispositions with no specific stimulus condition, we can simply take any scenario as a relevant case for the antecedent of some conditional.3 The problems are structural because they depend on a very general structural feature of (SCA): the correspondence between a disposition and single conditional. The problems arise whatever we think is the right response to the problems of finks and masks. Various responses to the problems of finking and masking have been suggested, of which Lewis’s ‘reformed conditional analysis’ (Lewis 1997) may be the best known. Typically, however, these have shared the structural feature of (SCA) which gives rise to the structural problems pointed out by Manley and Wasserman: they correlate one disposition with one conditional. Take, for instance, Lewis’s ‘reformed conditional analysis’: (RCA) Something x is disposed at time t to give response r to stimulus s iff, for some intrinsic property B that x has at t, for some time t  after t, if x were to undergo stimulus s at time t and retain property B until t  , s and x’s having of B would jointly be an x-complete cause of x’s giving response r. Lewis 1997, 157

The conditional is considerably more complicated than that on (SCA)’s righthand side, but it is a single conditional no less. Note, however, that Lewis switches (as have most subsequent authors) from ordinary disposition ascriptions to overt disposition ascriptions of the form ‘x is disposed to . . . if . . . ’. The aim is to avoid masks by incorporating their absence into the stimulus condition. But it may be observed that he thereby also evades the structural problems, insofar as they concern fragility. (RCA) carries no commitment to a correspondence between fragility and a single conditional. It does, however, carry a commitment to a correspondence between an overtly ascribed disposition and a single conditional. The structural problems are thereby not avoided; they are merely transferred to the overtly ascribed dispositions. The same goes for more recent attempts to salvage a conditional analysis. Choi (2006) defends the simple conditional analysis for overt disposition ascriptions, (SCA∗ ) x is disposed to M if S iff, if x were S, x would M. (Adapted and simplified from Choi 2006, 374; the label is mine.) 3 For a comparison between Manley and Wasserman’s view and the one that I will develop in chapter 3, see Vetter (2011a) and Manley and Wasserman (2011) as well as footnote 11 in chapter 3.

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(SCA) fails, according to Choi, not because (SCA∗ ) fails but because it specifies the wrong stimulus and manifestation for fragility. The correct specification would rely on something like ideal or normal conditions,4 which exclude masks and finks. Steinberg (2010) has offered a different emendation of the simple conditional analysis. He holds that, while finks and masks provide counterexamples to (SCA) and (SCA∗ ) alike, they can be accommodated by a simple ceteris paribus clause, yielding [A]n object is disposed to  when conditions C obtain if and only if, if conditions C were to obtain, then the object would  ceteris paribus. Steinberg 2010, 340

Steinberg’s analysis, like Lewis’s and Choi’s, tacitly requires that for any values of M and C there is a single (suitably qualified) conditional to characterize the disposition to M if C. These responses to the problems of finking and masking, thus, do not address the structural problems with (SCA); they merely transfer them to overt disposition ascriptions. A different reaction to problems of finks and masks has been that of dispositional realists such as Martin (1994) and Molnar (2003), who take these problems to show that dispositions are unanalysable, and indeed metaphysically irreducible. Insofar as they take (SCA) as their starting point, however, even these realists are subject to the structural problems. The structural problems are prior to the question of whether dispositions can be analysed or reduced. For that question arises only once we have a firm grasp of what the analysans, or the terms of the reduction, are supposed to be. If (SCA) fails for structural reasons, then the question whether it or anything that is structurally like it is an analysis or a reduction of dispositions is moot. In this chapter, I am going to add another structural problem to those which Manley and Wasserman have pointed out, and I will argue that the structural problems have wide-ranging consequences. In the first part of the chapter (sections 2.3–2.5), I provide a further argument for the conclusion that any but the most contrived dispositions are infinitely multi-track, in a sense to be precisified in section 2.3.5 I first consider the relatively uncontroversial case of 4

Choi (2006, 378, fn.3) refers to Mumford (1998) and Malzkorn (2000). Mellor (2000b) and Cross (2005) also acknowledge the infinite multi-track nature of dispositions, but for different reasons. 5

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fragility (section 2.3), and then argue that the same considerations which make it multi-track also apply to overtly ascribed dispositions, such as the disposition to break if struck (section 2.4), and in a similar though not exactly parallel way to what I will call ‘nomological dispositions’, such as (the disposition bestowed by) charge or mass (section 2.5). The conclusion I draw here is not entirely new—it has been drawn by Manley and Wasserman—but my argument is novel. More importantly, the consequences that this conclusion has for a realist metaphysics of dispositions have been severely underestimated in the literature to date. I draw out those consequences in section 2.6. My considerations, like Manley and Wasserman’s, will be purely structural, and entirely independent of issues of finks and masks. I will occasionally adopt Steinberg’s ceteris paribus clause to mark this fact. Finks, masks, and related counterexamples are a symptom of the fact that dispositions are properties: dispositions are a matter of how things stand with a particular object, and thus largely independent of how things stand with the world outside that object. In Martin’s classic case of the ‘electro-fink’, a dead wire is connected to a machine that makes it live whenever touched by a conductor. Hence the conditional ‘If the wire is touched by a conductor then electrical current flows from the wire to the conductor’ is true, although the wire is not live. This is so precisely because the wire’s being live or not is a property of the wire itself. It is a matter of how things stand with the wire, not of how things stand in the wire’s vicinity where the electro-fink is to be found. Similarly, a vase that has its fragility masked by anti-deformation packaging remains fragile, despite the falsity of the corresponding conditional, precisely because its fragility is a property of the vase. It is a matter of how things stand with the vase, not with the anti-deformation packaging. I have distinguished above between two questions concerning dispositions: the question which non-localized modality is linked to dispositions and the question how such a modality is linked to dispositions. The question of how to deal with finks and masks is part of the second question, not the first. Since my concern here is the first, not the second, question, it will be best to ignore the so-called problems of finks and masks. (I will return to them briefly in chapter 3.)

2.3 The problem of qualitative diversity A multi-track disposition, as I shall use the term, is a disposition which cannot be characterized adequately in terms of a single conditional. What is it for

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a disposition to be ‘characterized adequately’ in terms of a single conditional? I take it that it is for (at least) the following two conditions to hold: (ST) Where D is a disposition and C a conditional of the form: if x were S, then x would be M, D is adequately characterized by C (and C alone) only if (1) For all objects x that have D, if x were S, then ceteris paribus x would be M. (2) For all objects x that have D, if x manifests D at t, then x is M at t and x is S at or before t.6 Single-track dispositions are those that are characterized adequately by some one conditional. Hence if D is a single-track disposition, then there is a conditional C that fulfils conditions (1) and (2). If there is no conditional that fulfils the two conditions, and hence no conditional that alone adequately characterizes D, I shall call D a multi-track disposition. (1) is simply the condition that a conditional which alone characterizes a disposition must be true, ceteris paribus (i.e. finking and masking aside), of all those objects which possess the disposition. This seems uncontroversial, for both realists and reductionists: both can agree on the truth of (1) but then disagree on the status of the ceteris paribus clause (and the direction of explanation). Condition (2) is less familiar but should nonetheless be uncontroversial. It is based on the idea, first, that if a disposition is characterized by a single conditional, it is characterized by a single pair of stimulus condition and manifestation; and second, that if the disposition is so characterized, it is only manifested in yielding the manifestation upon being subjected to the stimulus condition. Again, this seems utterly uncontroversial, for how else would D be manifested? If D were manifested, say, in an object’s being M upon being S∗ instead of S, then C alone would not adequately characterize D; a further conditional with S∗ in its antecedent would have to be added. It is easy to see that most of our everyday dispositions are multi-track. A fragile glass may manifest its fragility in breaking upon being hit with a spoon, being dropped onto the floor, being sung to by a soprano, or being subjected to pressure over a period of time. Fragile parchments break upon being merely touched, and a fragile old wooden chair may split when transferred into a different temperature. Irascible people may manifest their irascibility by becoming angry upon being yelled at, being disagreed with, or being politely told to wait. At first look, 6 Clause (2) could be rephrased in terms of explanation: if x manifests D at t, then x is M at t because x is or was S (at or before t). Since ‘because’ is factive, the alternative clause would imply (2) as it stands and would make no difference to the argument that follows.

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what we have here are qualitatively different stimulus conditions, and accordingly several different conditionals: x would break if it were dropped onto the floor, x would break if it were touched, x would break if it were sung to by a soprano, and so forth. It is worth spelling out in some detail why this leads to fragility being a multi-track disposition according to (ST). Suppose (for reductio) that fragility were to be treated as a single-track disposition. Given the different stimulus conditions that I have cited, how might we come by a single conditional to characterize that putative single-track disposition in a way concordant with (ST)? There are two things we might do. (A) We might pick one stimulus from our list and focus on the resulting conditional. There may be worries about clause (1): given the variety among fragile objects, is there really one stimulus condition (and hence one conditional) that works for them all? Let us suppose, however, that there is. The conditional (C1)

If x were hit with a hammer, x would break

has a good chance of being true, ceteris paribus, of all fragile objects. The deeper problem lies with clause (2). By (2), any fragile object that manifested its fragility would have to do so upon being hit with a hammer. Breaking upon being hit with a spoon, being dropped onto the floor, or being transferred into a different temperature will no longer count as manifesting an object’s fragility. (C1) will classify all the right objects as fragile. But it will not classify all the right breakings as manifesting fragility: only those breakings which happen upon being struck with a hammer will count as manifesting fragility. (C1), then, falsifies clause (2). (B) The obvious solution is to generalize the stimulus by providing a condition which captures all the conditions under which fragility is ever manifested. Define being stressed as the disjunction of all the properties that may trigger a fragile object to break: to be stressed is to be hit with a spoon or dropped onto the floor or sung to by a soprano or subjected to pressure over a period of time or transferred into a different temperature or . . . . The conditional to characterize fragility is now (C2)

If x were stressed, x would break.

A disjunctive property such as being stressed is multiply realizable: one object may be stressed by being hit with a spoon, another by being transferred into a different temperature, and so on. Thus a glass’s breaking upon being hit with a spoon will count as the object’s breaking upon being stressed, as will a chair’s breaking upon being transferred into a different temperature. Clause (2) is no longer violated.

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The problem is that this strategy now falsifies clause (1). Suppose I uttered (C2), replacing ‘x’ by ‘this vase’. You may take the vase, transfer it to a different temperature, and point out that the vase has now been stressed but did not break. It may well be true that the vase would break if it were hit. But (C2) does nothing to privilege any particular one of the disjuncts that were used to define stressing. If it did, it would simply collapse back into strategy (A) and the conditional (C1). The very feature that made it superior to (C1), its generality, makes (C2) more susceptible to refutation than it should be. Of course, clause (1) required only that a conditional which characterizes fragility be true of all fragile objects ceteris paribus. But that is of little help in the present context. The ceteris paribus clause was inserted to take care of finks, masks, etc. But what we have here is not a case of finking or masking. Finks and masks interfere once the right stimulus condition has been applied. But our case is not one of interference. It is, rather, a case in which the right stimulus has not been applied. It will help to spell this out in terms of the standard semantics for the counterfactual conditional. According to the orthodox Lewisian semantics, a counterfactual A  C is true (at our world) just in case all the closest worlds in which A holds are also worlds in which C holds. (This is a simplification, but good enough for our purposes.) Note that, where A is equivalent to a disjunction A1 ∨A2 ∨. . .∨An , the truth of A  C does not follow from the truth of Ai  C for any one disjunct Ai . Even if all the closest Ai -worlds are C-worlds, there may always be a non-Ai -world among the closest A-worlds, say an Aj -world, in which C is not true. Accordingly, the truth of (C2) does not follow from the truth of ‘If x were hit, x would break’, although the antecedent of (C2) is equivalent to a disjunction one of whose disjuncts is that x is hit. This is so because there is no guarantee that the closest stressing-worlds include only hitting-worlds. It is easy to imagine that there is a world in which the vase is not hit, but transferred to a different temperature, and that this world is at least as close as the closest worlds in which the vase is being hit. (In fact, this may be so precisely because we know that the vase would break if it were hit and consequently take greater care not to hit it than we do not to transfer it to a different temperature.) But if this is so, then (C2) is false, despite the fact that the vase is fragile, and clause (1) is violated. This concludes part (B) of my argument. We now have a dilemma: in formulating a single conditional to characterize fragility, we must either pick one stimulus condition to the exclusion of others, and hence violate (2) because that one stimulus condition may not be involved in

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all the manifestations of fragility, or we must generalize the stimulus, and violate (1) because there are too many ways for the generalized conditional to be false even of a fragile object. Either way, fragility is not a single-track disposition. The argument did not rely on any features that are specific to fragility, parallel arguments can be run for irascibility and any other disposition that has multiple qualitatively different stimulus conditions. I conclude that fragility (and irascibility, and so on) is a multi-track disposition. I do not expect much controversy concerning this result. It is one of the reasons why philosophers have preferred to discuss the better-behaved, ‘overt’ disposition ascriptions such as ‘x is disposed to break if hit’ or ‘x is disposed to break if heated’. Those, it is often thought, ascribe single-track dispositions. After all, they specify a single stimulus condition. They must therefore escape the problems that I have pointed out. I am now going to argue that this thought is mistaken: the problems with (1) and (2) repeat themselves even with such qualitatively uniform dispositions.

2.4 The problem of quantitative diversity Not only dispositions, but also their typical stimulus conditions, come in degrees. A glass can be struck with a greater or lesser force, a vase can be dropped from a greater or lesser height, a person can be yelled at more or less loudly. These properties are quantities: determinable properties with a range of determinates ordered by a relation such as that of being greater than.7 Striking, for instance, if understood as the exertion of mechanical force, has determinates such as striking with a force of 8.35 N. Typically, the disposition’s degree is (inversely) correlated with that of its stimulus. Thus, in general, a glass that is more fragile will break if struck with a lesser force, and one which is less fragile will break only if struck with a greater force. In this section, I will focus exclusively on the disposition to break if struck. For convenience, I will abbreviate ‘the disposition to break if struck’ to ‘fragility∗ ’, and say that objects with this particular disposition are fragile∗ . When, in what follows, I appeal to intuitions about fragility∗ and fragile∗ objects, readers are invited to examine either their intuitions regarding the disposition to break if struck, or their intuitions concerning fragility but with the proviso that all stimulus conditions except striking are irrelevant. (I prefer the second option: as we 7 I use the terminology of ‘determinable’ and ‘determinate’ as the more general distinction and understand quantities to be a special kind of determinables, as indicated in the main text. Thus both being red and having mass are determinables, with determinates such as being scarlet or having mass of 5 kg, respectively. In addition, having mass is a quantity, while being red is not.

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will see in chapter 3.1, the ‘disposed to’ locution is a piece of technical terminology and therefore not suited to elicit pretheoretical intuitions.) I will assume for the sake of argument that being struck has the qualitative homogeneity that forestalls the argument of section 2.3, but will argue that its quantitative nature leads to problems very similar to those that faced the qualitatively different stimulus conditions in section 2.3. We may begin by noting that the determinable property of being struck relates to its determinates in much the same way as the disjunctive property of being stressed, discussed in the previous section, relates to the properties that form its disjuncts. An object may be stressed by being struck, heated, dropped, sung to, or subjected to pressure. Similarly, an object may be struck by being struck with 1.01 N, being struck with 8.35 N, or being struck with 142.56 N, and so forth. For any instantiation of being stressed, there is a particular disjunct of the property of being stressed that is instantiated; and similarly, for any instantiation of being struck, there is a particular determinate of the determinable property of being struck that is instantiated. Where we had, in the case of fragility, a multiplicity of qualitatively diverse stimulus conditions, we now have, in the case of fragility∗ , a multiplicity of qualitatively homogeneous but quantitatively diverse stimulus conditions. If we want to formulate a single conditional to characterize fragility∗ , we again have the following two options in dealing with this multiplicity. (A∗ ) We may pick one stimulus and consider only one of the many determinates, say being struck with 8.35 N, as the stimulus condition of fragility∗ . However, the corresponding conditional (C3)

If x were struck with 8.35 N, x would break

would falsify condition (2) of the characterization of a single-track disposition in (ST) above. Clearly, not all manifestations of fragility∗ are preceded by the fragile∗ object’s being struck with exactly 8.35 N. (B∗ ) Again, the obvious alternative is to generalize the stimulus. To avoid privileging any particular determinate stimulus condition, we should incorporate the determinable and characterize fragility∗ by the conditional (C4) If x were struck, x would break. However, this strategy now violates clause (1) of (ST). Take a fragile∗ vase which would break, ceteris paribus, if struck with a force of 5 N or more, but not if struck with any lesser force. Call this vase Ming. Now strike Ming very lightly, say with 2 N. It will not break, and (C4) will therefore not be true of it. Once again, this is not a case of finking or masking: finks and masks interfere once the right stimulus

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has been applied, but the right stimulus has precisely not been applied in our case. In terms of possible-worlds semantics, (C4) is true just in case x breaks in all the closest worlds where it is struck. But a fragile∗ object may be such that (i) it breaks in all the closest worlds where it is struck with 5 N or more, yet (ii) among the closest worlds in which it is struck there are worlds in which it is struck with less than 5 N and it does not break. In fact, Ming is, by stipulation, just such an object. To guarantee the truth of (C4), we would need to make sure that some of the at-least-5 N worlds count as closer than any less-than-5 N worlds; we would have to privilege the former worlds over the latter. But (C4), like (C2), provides no foothold for such a principled privilege. So much for the parallel between fragility and fragility∗ . But perhaps the privilege is easier to come by in the present case. The determinable quantity of being struck is not entirely analogous to the disjunctive property of being stressed. Unlike the disjuncts of being stressed, the determinates of being struck are ordered on a continuous scale of magnitudes, from the very slight to ever greater forces. This ordering allows for more than just the picking out of one determinate (as in (A∗ )) or the inclusion of all of them (as in (B∗ )). It allows, in particular, for the setting of a threshold and consideration of everything above or below that threshold. The problem with Ming in (B∗ ) was precisely that worlds below the 5 N threshold were allowed to count as relevant. What we need to do is exclude such worlds. The idea, then, is to incorporate into (C4) a threshold value. This connects happily with the observation that ‘fragile’, as well as ‘fragile∗ ’, is context-sensitive. Each context will determine a threshold value such that objects which would break, ceteris paribus, if struck with at least that force count as fragile∗ while others do not. As I said above, degrees of fragility∗ appear inversely correlated to the degree of the stimulus that is required for its manifestation: the more fragile∗ an object is, the smaller the force it takes to break it. By setting a threshold on the scale of forces, we determine how fragile∗ an object has to be to count as fragile∗ within a given context. For each context C, then, the property ascribed in that context by ‘x is fragile∗ ’ (or by the unabbreviated ‘x is disposed to break if struck’) is adequately characterized by a conditional of the form: (C5) If x were struck with a force of at least nC , x would break, where nC is the threshold value determined by C. It may be objected that the proposal is oversimplified. In particular, it may be claimed that a context does not simply set one threshold value for all kinds of objects. Perhaps the threshold value varies, within a given context, with the kind

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of object to which fragility or fragility∗ is ascribed. In order to avoid commitment on this question, I will continue with the simplification but restrict my examples to only one kind of object, vases. I am not convinced that thresholds are kindrelative, but this will not affect my argument. However this question is resolved, (C5) will not do either. The reason, in general, is that being struck with at least nC still has the structure of a determinable quantity. Take a context C, and suppose that the minimal force required to break those objects which count as fragile∗ in C ranges from 0.1 N to 10 N. Anything that would break, ceteris paribus, only if struck with a force greater than 10 N does not count as fragile∗ in C (though it may in a different context). Clearly, C is a context in which Ming counts as fragile∗ . After all, it would break if it were struck with at least 5 N. Now, where in the interval from 0.1 N to 10 N is nC to be set? We have three options here. (I will occasionally drop explicit relativization to C in what follows, but all ascriptions of fragility∗ are to be understood as true in the context C unless stated otherwise.) First option: set the threshold at 0.1 N (or below), with the most fragile∗ objects. The conditional that characterizes all objects that count as fragile∗ in C is: if x were struck with at least 0.1 N, then x would break. But that conditional is not true of any but the most fragile∗ objects. Being struck with at least 0.1 N exhibits the by now familiar structure of a determinable property that we have seen in the property of being struck. An object may be struck with at least 0.1 N by being struck with exactly 0.1 N, with 2.0 N, or with 8.3 N, and so forth. Accordingly, our new conditional can be refuted in just the same way. Take Ming and strike it with a force of exactly 2 N; it will not break. Since being struck with a force of exactly 2 N is a way of being struck with at least 0.1 N, the conditional at issue is not true of Ming although Ming is fragile∗ . Thus condition (1) in (ST) is violated once again. Second option: set the threshold at 10 N (or above), with the least fragile∗ objects. The conditional that holds of any object x which qualifies as fragile∗ is now: if x were struck with at least 10 N, then x would break. This conditional, to be sure, is true of all the fragile∗ objects. It does not, however, adequately capture their shared fragility∗ . Take Ming again and strike it with a force of 8 N; nothing devious is going on: Ming breaks. Has it thereby manifested its fragility∗ ? Surely we should say that it has. But on the present proposal, we cannot say that: for fragility∗ , on the present proposal, is the disposition to break if struck with at least 10 N, and Ming has not been struck with at least 10 N. Given clause (2) in (ST), if Ming is to manifest its fragility∗ , it must be struck with at least 10 N. Indeed, on the present proposal, it is precisely the

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wrong breakings that qualify as manifestations of fragility∗ : breakings upon being struck with 10.1 N, 11 N, or 100 N. Breakings upon being struck with any force less than 10 N are excluded from being a manifestation of fragility∗ ; breakings upon being struck with any force of 10 N or more may be included among the manifestations of fragility∗ . Given clause (2) of (ST), the conditional cannot adequately characterize fragility∗ . The first option, then, violated condition (1) by not classifying the right objects as fragile∗ ; the second option violated condition (2) by not classifying the right breakings as manifestations of fragility∗ . A third option is to set the threshold somewhere in between. Since we are concerned with Ming, 5 N would be a natural candidate for a threshold value. The conditional that would characterize fragility∗ in C would be: if x were struck with at least 5 N, then x would break. But rather than solving the problem of the two earlier options, this solution will incur the problem of both. As far as Ming is concerned, the third option gets everything right. But that is not enough. If Ming counts as fragile∗ in C, then so must a more delicate vase, Meissen, that would break if struck with at least 1 N. (In fact, we can ascribe fragility∗ to both of them within a single sentence: Ming and Meissen are both fragile∗ .) But now strike Meissen with 2 N; it will break; but its breaking will not count as manifesting its fragility∗ . Thus condition (2) is violated once again. Moreover, Ming’s qualifying as fragile∗ in C should not exclude a slightly less fragile∗ vase, call it Amphora, from also counting as fragile∗ in C. (In fact, we can ascribe fragility∗ to both of them within a single sentence: Ming and Amphora are both fragile∗ .) But since Amphora is, by stipulation, less fragile∗ than Ming, it will not be true of Amphora that it would break, ceteris paribus, if struck with at least 5 N; for it would not break if struck with exactly 5 N. Hence condition (1) is violated. Whatever value we fix for nC , then, (C5) will not yield an adequate characterization of fragility∗ . Perhaps (C5) made a mistake in its treatment of the threshold value. After all, what sets the fragile∗ things apart from the non-fragile∗ ones is not that they break if struck with a greater force, but that they—and only they—would break if struck with a slighter force, a force below a certain threshold. On this new proposal, fragility∗ is adequately characterized by the conditional (C6)

If x were struck with a force of at most nC , x would break,

nC being, again, the threshold value determined by the context C. But (C6) founders on condition (1) wherever the threshold value is set in context C.

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Suppose that nc is set at 0.1 N (or below). It is not true of any, not even the most fragile∗ , objects, that they would break, ceteris paribus, if struck with 0.1 N or less; so clause (1) is violated. Setting nC to be at 10 N (or above) will not help; the conditional will once again be too easily refuted, by simply striking our fragile∗ vase Ming with a force of, say, 2 N. The vase will have been struck with at most 10 N, but it does not break. So the conditional is false of it, against clause (1). And, again, setting the threshold anywhere between 0.1 N and 10 N will incur the problems of both previous options. Are there any other options? We have tried using the threshold value as the lower bound of the interval in (C5) (the upper bound being infinity) and as the upper bound of the interval in (C6) (the lower bound being 0). A further option is to let our two main candidates for threshold values—0.1 N and 10 N—specify both the lower and upper bounds, yielding (C7)

If x were struck with a force between nC and mC , x would break,

nC and mC being the lower and upper threshold values determined by the context C. By now it should be easy to see why (C7) does not work. Let nC = 0.1 N and mC = 10 N. Then (C7) violates (1) because it will not be true of any but the most fragile∗ objects that they would break if struck with any force between these two values. Setting nC and mC at values outside or inside the interval from 0.1 N to 10 N will not change the situation. A final resort may be to a somewhat more sophisticated version of (C5)–(C7). Perhaps there is no one threshold value that can accommodate all the different fragile∗ objects. But what those objects have in common is that their individual thresholds for breaking, different as they may be, are all of the right kind to make them count as fragile∗ in a given context. So we may try: (C8) For some nC ∈ S, if x were struck with a force of at least nC , x would break, S being a contextually determined set of suitable threshold values. Analogues of (C8) can be formulated for the ‘at most’ threshold (as in (C6)) and the interval option (as in (C7)), and the following remarks reply to them too mutatis mutandis.8 (C8) is not a conditional that adequately characterizes a disposition. The reason is not that it falsifies one of the conditions, (1) and (2), for adequately characterizing a disposition. Rather, it is that (C8) is not of the right form for those 8

Thanks to Wolfgang Freitag for suggesting this solution.

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conditions to apply. Its main connective is not the counterfactual conditional, but the existential quantifier. Characterizations of (C8)’s type are equivalent to infinite disjunctions of conditionals of the form of (C5) and its ilk. While (C8) may give us a single characterization of a disposition, the disposition that is so characterized is not a single-track disposition: it is not adequately characterized by a single conditional. (C8) may be a useful way of talking about a number of conditionals in one sentence, by quantifying over them. But it is not a way of collapsing those many conditionals into one. I conclude that the considered strategy fails. Setting a threshold value for the stimulus condition may help with context-sensitivity, but it does not solve the problem at hand: it does not provide us with one conditional that is both true, ceteris paribus, of all objects that count as fragile∗ in a given context, and that captures the right breaking events as manifestations of fragility∗ . If there is no one such conditional, then fragility∗ is not a single-track disposition in the sense outlined in section 2.3; it is a multi-track disposition. Further, the argument relied on no specific features of being struck other than its being a quantity and, more generally, a determinable with different determinates.9 The same is true of other stimulus conditions for fragility, or so I have suggested earlier. The same argument, then, applies to other qualitatively homogeneous ‘tracks’ of fragility, and indeed to any disposition that has a determinable stimulus condition. Let us take stock. The argument of section 2.3 showed the inadequacy of a simple conditional analysis of ‘fragile’, which takes the form (SCA )

x is fragile iff, if x were S, then x would M.

We have seen that there were too many candidates to fill the place of S, and that (SCA ) failed whether we picked only one of them (strategy (A)) or tried to generalize enough to cover them all (strategy (B)). Section 2.4 showed, further, the inadequacy of a simple conditional analysis of the ‘disposed to’ locution along the lines of (SCA∗ )

x is disposed to M if S iff, if x were S, then x would M.

(SCA∗ ) will fail, for the reasons outlined in this section, whenever S is a determinable property that has a range of different determinates. The argument applies not only to fragility and the disposition to break if struck, but to any disposition with a determinable stimulus condition. I have yet to see an ordinary disposition 9 The argument as I have presented it relies on being struck being a quantity, for it is its quantitative nature that allows for the natural setting of a threshold. Where a stimulus condition is a determinable that is not a quantity, there may be no natural equivalents to (C5)–(C8). But that would only make my argument easier.

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that does not have such a stimulus condition. Take water-solubility: an object can be immersed in more or less water for a longer or shorter period of time. Or take inflammability, with (one of) its stimulus condition(s): being heated, clearly another determinable quantity. An irascible person may be provoked more or less fiercely. And so on. Unlike (SCA ), (SCA∗ ) has some instances that escape the arguments I have adduced: those instances, that is, where S is replaced by a fully determinate property. For all that I have said, an analysis along the lines of (SCA∗ ) may succeed for such very specific dispositions as the disposition to break if struck with exactly 8.35 N, call it fragility8.35 . If we are looking for single-track dispositions, we need to look here. Note, again, that finks, masks, and so forth have played no role in my argument; we have assumed that no such factors were present in any of the cases considered. Whatever is done to accommodate such interfering factors—be it Lewis’s reference to a causal basis that is retained throughout the process, Choi’s incorporation of ideal conditions into the stimulus condition, or Steinberg’s ceteris paribus clause—will have no force against the arguments of this section and section 2.5.

2.5 Nomological dispositions So far, my argument has been about the everyday dispositions of dry middlesized objects such as vases, glasses, and ourselves. Things look a little different when we consider what I will call ‘nomological dispositions’, the more fundamental dispositions which, in one way or another, encode laws of nature. Dispositional realists, as I have characterized the view in chapter 1, hold that dispositional properties are not reducible to non-dispositional properties. If there are any fundamental or perfectly natural properties, those will include dispositions; if there are not, then the claim is just that at each successive level of fundamentality there will be dispositional properties. Some dispositional realists, the dispositional essentialists, have argued that the fundamental or near-fundamental dispositions are what ground the laws of nature: to simplify a little, it is a law (Coulomb’s Law, which we will consider again below) that like-charged objects repel each other because charge just is the disposition to repel like-charged objects (see Bird 2007, Ellis 2001). But we need not be realists to see a close connection between laws of nature and certain dispositions. To the reductionist, charge may not be identical with the disposition to repel like-charged objects, but in our world, charged objects still invariably have that disposition. Charge

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is the categorical basis for that disposition.10 But it is the categorical basis for this disposition because of its involvement in Coulomb’s Law. For the dispositional essentialist, the dispositions ground the laws; for the reductionist, the laws (partly) ground the dispositions. Either way, there is a close connection between laws of nature and certain dispositions. This section will have a closer look at the relevant dispositions and argue that they, too, are infinitely multi-track. The reason, again, has to do with the quantitative nature of the stimulus and (in this case also) manifestation properties. Like many others, I will use charge, and electric charge in particular, as my examples. My argument will concern very general features of charge, which are due entirely to the fact that the law of nature that captures the disposition which is or is conferred by charge is a functional law. If the real laws of nature are anything like the kind of laws that our best physics is seeking, the argument will apply to the properties that figure in them no matter what exactly they are. Like the ordinary dispositions that we have looked at so far, charge is gradable, as are its stimulus and manifestation conditions. Objects can have charges of different values. Electrons, for instance, have a particular determinate of charge, electric charge; protons have another. Charge is manifested in the exertion of attractive or repulsive forces, which can be more or less strong; they are triggered by other charges, which can have any determinate value, at some distance, which may be greater or lesser, from the object with the original charge. Unlike the ordinary dispositions that we have looked at, the various determinates of charge itself, as well as its stimulus and manifestation conditions, are partitioned into orderly clusters. It is not that there is one set of conditionals and differently charged objects differ in which and how many are true of them, but with great overlap; rather, for each value of charge there is one set of conditionals all of which are satisfied by objects with that particular charge. Nonetheless, I shall argue, those are many conditionals, and there is no way to unite them into one conditional that would then characterize charge, or even a determinate charge property such as electric charge. To a first approximation, charge may be characterized as the disposition to attract differently charged objects and repel like-charged ones. But that is no more than an approximation. More precisely, charge will have to be characterized in accordance with Coulomb’s Law. Coulomb’s Law says not simply that, if a particle has (say) negative charge and there is another negatively charged object in 10 See Prior et al. (1982) for an argument that dispositions and their categorical bases cannot be identical.

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its vicinity, then the particle will exert a repulsive force. It tells us, rather, what the mathematical correlations are between the two charges, their distance from each other, and the force exerted. For the sake of a definite example, let us focus on a determinate charge property: electric charge, or charge e. Where q is the charge of another particle at distance r from a particle with charge e, the force F that is exerted by the latter particle is given by Coulomb’s Law as eq

(CL) F =  r2

( is a constant.) The question then is, how is (CL) to be embedded in a conditional so as to characterize electric charge? The conditional (Charge1) If x were at distance r from a charge of q, then x would exert a eq force of F =  r2 sounds right, but is ambiguous as it stands. The best reading takes (Charge1) to universally quantify implicitly over charges and distances. The disposition that electric charge bestows on its bearers is, after all, the disposition to exert a force in accordance with (CL) given any other charges at any distance. The universal quantifier must have the entire conditional in its scope, to make sure that any values substituted for q and r in the antecedent are substituted identically in the consequent. Thus a more explicit rendering of (Charge1) reads: (Charge2) ∀r∀q: (If x were at distance r from a charge of q, then x would eq exert a force of F =  r2 ). However, like (C8) above, (Charge2) is not a conditional that adequately characterizes a disposition by the conditions introduced in section 2.3 as (ST). The reason is not that (Charge2) falsifies one of the two conditions for adequately characterizing a disposition. Rather, it is that (Charge2) is not of the right form for those conditions to apply. Its main connective is not the counterfactual conditional, but the universal quantifier. As Bird (2007, 22) notes, characterizations of (Charge2)’s type are equivalent to infinite conjunctions of conditionals, with specific values filled in for r and q in each of them. One such conditional is (Charge3) If x were at a distance of 5.3 × 10–11 m from a charge of 1.6 × 10–19 C, then x would exert a repulsive force of 8 × 10–8 N. But (Charge3), too, does not adequately characterize electric charge. It fails clearly by condition (2): it is not the case that every manifestation of the disposition which is (conferred by) electric charge consists in the exertion of a repulsive

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force of 8 × 10–8 N after being at a distance of 5.3 × 10–11 m from a charge of 1.6 × 10–19 C. There are infinitely many other ways of manifesting electric charge. (Charge3) does characterize a more specific disposition: the disposition to exert a repulsive force of 8 × 10–8 N when at a distance of 5.3 × 10–11 m from a charge of 1.6 × 10–19 C. This more specific disposition, by the lights of (ST), is a single-track disposition. In fact, it is one of the many ‘tracks’ of the multi-track disposition that is (conferred by) electric charge. Note, incidentally, that the relation between the multi-track disposition and its many ‘tracks’ is not the same as that between a determinable (such as charge) and its determinates (such as electric charge). Having a determinable property entails having one of its determinates, to the exclusion of all others. Having the multi-track disposition electric charge, on the contrary, entails having all the corresponding single-track dispositions. Mutatis mutandis, the same reasoning will apply to other nomological dispositions. The argument I have given relied on nothing more than the fact that electric charge, or the disposition it bestows, has quantities as its stimulus conditions and manifestations and relates these quantities in a way that is not merely on/off, but a mathematical correlation. It will therefore apply to any other nomological dispositions with this kind of structure—and that includes at least the dispositions which correspond to the laws of physics and chemistry. It is not a superficial phenomenon that we can hope to get rid of at the fundamental level, if there is one. For our best bet at knowledge of a fundamental level comes from physics, and the properties that physicists look for, and find, at more and more fundamental levels of reality are quantities. The dispositions that belong to, or are bestowed by, the properties at that fundamental level will have quantities as their stimulus and manifestation conditions. We have multi-track dispositions all the way down.

2.6 Multi-track dispositions and realism Sections 2.3–2.4 argued that a simple conditional analysis, even if adjusted for finks and masks, fails in all but the most contrived cases because it correlates a disposition with a single conditional. It fails for such ordinary dispositions as fragility, which have qualitatively diverse stimulus conditions. But it also fails for such apparently more tractable dispositions as the disposition to break if struck, if they have a quantity or, more generally, a determinable as their stimulus condition. Section 2.5 has added to this even the more natural dispositions that are correlated with laws of nature, the nomological dispositions. There is no one conditional to capture a nomological disposition such as electric charge, though

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there are many conditionals that must all be true of objects with electric charge. Neither such ordinary dispositions as fragility nor such nomological dispositions as electric charge can be captured by a simple conditional analysis along the lines of (SCA∗ ): (SCA∗ )

x is disposed to M if S iff, if x were S, then x would M.

We have already seen that (SCA∗ ) has some instances which escape the arguments I have adduced: those instances where S is replaced by a fully determinate property. (SCA∗ ) looks good (with the right provisions made for finks and masks) for such specific dispositions as the disposition to break if struck with exactly 8.35 N, which I have called fragility8.35 , or the disposition to exert a repulsive force of 8 × 10–8 N when at a distance of 5.3 × 10–11 m from a charge of 1.6 × 10–19 C, call it charge e∗ . If we are looking for single-track dispositions, we need to look here. In this section, I will apply these results to the project of a realist metaphysics of dispositions. I will argue that the following three claims, if not outright inconsistent, sit very uncomfortably together: realism about dispositions; the claim that all except the maximally specific dispositions are multi-track; and the standard conception of dispositions, as outlined in section 2.1. Since realism about dispositions is one of the guiding assumptions of this book, and I have argued at length that all except the maximally specific dispositions are indeed multi-track, my conclusion will be to reject the standard conception of dispositions. The standard conception, recall, was the combination of the following two claims: 1. A disposition is individuated by the pair of its stimulus condition and its manifestation (or, if it is a multi-track disposition, by several such pairs): it is a disposition to M when S (or a disposition to M1 when S1 , to M2 when S2 , etc., if it is a multi-track disposition). 2. Its modal nature is, in some way or another, linked to or best characterized (to a first approximation) by a counterfactual conditional ‘If x were S, x would M’ (or, if it is a multi-track disposition, by several such conditionals). So let us see what happens when we combine this conception with both realism and an acknowledgement of the multi-track nature of dispositions. According to dispositional realism, when we move from the less to the more natural properties, we will always find dispositions no matter how far we move along; and, perhaps equivalently, when we move from facts to other facts grounding them, we never reach a level at which dispositions entirely disappear. (If we start with a non-dispositional fact, we may find non-dispositional facts further

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down the line, of course; and if we start with a dispositional fact, we are likely to find very different dispositional facts further down the line, but we will still find dispositional facts.) I will treat these two claims as equivalent. If that is a simplification, I hope it is not a distorting one. Now, which dispositions are we more likely to find as we move to the more natural properties, or to the facts that ground other facts? Which dispositions are more fundamental—the more familiar, multi-track ones such as charge and fragility, or the maximally specific, single-track ones such as charge e∗ and fragility8.35 ? Given the standard conception of dispositions, there is considerable pressure to favour the single-track, specific dispositions. If the nature of fragility is best or adequately characterized by conditionals, then it will be infinitely complex, for it requires an infinity of conditionals. The nature of fragility8.35 , on the other hand, is as simple as a disposition can possibly be. In fact, it looks as though fragility is in some way built up from such single-track dispositions, incorporating their characterizing conditionals within its complex nature.11 The same goes for electric charge, which is infinitely more complex, on the standard conception, than such simple dispositions as charge e∗ , and looks rather to be built up from such specific dispositions as charge e∗ . The simpler building-blocks are, in whatever area, more fundamental than their complex compounds; hence fragility8.35 is more fundamental than fragility, and charge e∗ is more fundamental than electric charge. Indeed, (possession of) the complex dispositions, on this view, should be grounded in (possession of) their simpler counterparts; an object’s fragility is a matter of its possessing some, or sufficiently many, such dispositions as fragility8.35 , while a particle’s having electric charge is a matter of its possessing all the corresponding specific dispositions such as charge e∗ . (Note, again, that the relation between the complex and the simple disposition cannot be one of determinable and determinate, since any two determinates of a common determinable are mutually exclusive.) Bird (2007) explicitly embraces the picture that I have just sketched: It is my view that all impure dispositions [in our terminology: all multi-track dispositions] are non-fundamental. Fundamental properties cannot be impure [i.e. multi-track] dispositions, since such dispositions are really conjunctions of pure [i.e. single-track] dispositions, in which case it would be the conjuncts that are closer to being fundamental. Bird 2007, 22

Note, again, that this conclusion seems inevitable only given the standard conception of dispositions: the key premise in producing the complexity was the 11 This picture is suggested, I believe, by the work of David Manley and Ryan Wasserman, see Manley and Wasserman (2008).

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idea that there is a deep connection between dispositions and conditionals, as captured in the standard conception of dispositions. From the mere fact that a property can be given a complex, conjunctive or disjunctive, characterization we cannot infer that the property itself is accordingly complex. The property of being green can be characterized as a disjunction of being grue and first observed before a time t or being bleen and observed only thereafter. This does not make being green a more complex property than being grue, because we have no reason to believe that the disjunctive characterization is what gives the nature of being green. Similarly, the fact that fragility or electric charge can be characterized by a multitude of conditionals is evidence for its complexity only if that characterization is what gives the nature of fragility or electric charge. That it does, follows from the assumption that the nature of a disposition is quite generally to be characterized in terms of or in analogy to conditionals, that is, from the standard conception of dispositions. I am now going to argue that this assumption leads into trouble. In particular, I will argue that independent considerations favour the more general multi-track dispositions such as electric charge as being more fundamental.12 It is generally taken for granted that the more fundamental properties figure in the more fundamental laws. Thus electric charge is more fundamental than water-solubility because the laws of particle physics, in which the former figures, are more fundamental than—and provide the grounds of—those of chemistry, in which water-solubility has a place. Now, our best bet at near-to-fundamental laws are certain laws of physics. Those are functional laws, stating mathematical correlations between quantities. As Wilson (2012, 5) has noted, ‘the laws of physics appear to relate determinables—mass, energy, and the like—not determinates (much less maximally specific determinates)’. It is electric charge, or rather, its determinable charge, that is related in a lawlike manner to the determinables force and distance in Coulomb’s Law. If laws were only as fundamental as the properties that they relate, then the most fundamental laws would have to look quite unlike Coulomb’s Law. They would have to be very specific laws of the form: (CL∗ ) For all x, if x has charge e and is 5.3 × 10–11 m from a charge of 1.6 × 10–19 C, then x exerts a repulsive force of 8 × 10–8 N. Now, such putative laws as (CL∗ ) not only deviate from our current scientific practice. They are inferior in explanatory power to the laws that accord with scientific practice, such as (CL). Laws explain, in some way or another, 12 Note that these considerations also go against dispositional essentialism as formulated by Bird. I have argued for this point in more detail elsewhere (Vetter 2012). Chapter 7.8 will look at the consequences of this section’s arguments for dispositional essentialism.

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regularities in the behaviour of things. Laws like (CL∗ ) would explain some regularities—the regularity, that is, which consists in objects with electric charge exerting a repulsive force of 8 × 10–8 N when being 5.3 × 10–11 m from a charge of 1.6 × 10–19 C. Laws of (CL∗ )’s ilk would explain an infinity of such regularities, one by one. But they would leave entirely unexplained and inexplicable the much more striking regularity that holds between those more specific regularities of behaviour: the regularity, that is, which consists in the exerted force’s always standing in the same mathematical relation to the charges present and their distance from one another. It is no accident, then, that laws such as Coulomb’s Law and not quasi-laws such as (CL∗ ) are considered more fundamental. But if they are, then so are the properties that they relate. Charge itself, and its determinate electric charge, is therefore more fundamental than the specific, single-track dispositions.13 Similar considerations can be applied to the properties themselves, without appeal to the corresponding laws. Instantiations of the more fundamental properties ground, or ‘fix’, the instantiation of the less fundamental properties. Electric charge and the specific dispositions such as charge e∗ are so closely related that we can expect there to be a grounding relation between them one way or another. It is not intuitively clear which of them grounds or fixes which, so we must appeal to more theoretical considerations. Note, first, that facts about charge fix all the facts about the specific dispositions that are its ‘tracks’: a particle’s having electric charge guarantees its having all the specific dispositions that come with it, such as the charge e∗ . The specific dispositions, all taken together, fix facts about electric charge too: by having all of them, a particle is guaranteed to have electric charge. If the situation were entirely symmetric, parsimony should prompt us to prefer electric charge over the specific dispositions: having one dispositional property at the more fundamental level is, ceteris paribus, better than having an infinity of dispositional properties doing the same work. However weak or strong that consideration might be, the situation is not quite symmetric anyway; electric charge fixes facts that the specific dispositions, even all taken together, do not fix. Electric charge fixes facts about the more specific dispositions which are not fixed by those dispositions themselves, even taken in conjunction. These are 13 Wilson (2012) uses similar considerations to argue that determinable properties may be just as fundamental as their determinates. I sympathize with Wilson’s argument, and my argument is partly inspired by hers. However, I think I can make an even stronger case concerning the specific single-track and the general multi-track dispositions than Wilson can concerning determinates and determinables: the general dispositions are not only as fundamental as the specific ones, they are more fundamental.

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facts about the co-instantiation of the more specific dispositions: the fact, for instance, that if anything has one of the specific dispositions corresponding to electric charge, it has all the others too. (And so on for other determinate charges.) If electric charge and its ilk are the fundamental dispositions, then these co-instantiation facts are easily explained: the co-instantiated specific dispositions always come together because they are all consequences of one and the same fundamental disposition, electric charge. If, on the other hand, the specific dispositions are fundamental, their co-instantiation is hard to explain. The specific dispositions fix facts about the specific values of charge, distance, and force exerted that they are concerned with, but not about each other. I conclude that electric charge is a more fundamental property than the specific dispositions such as charge e∗ .14 My argument has not yet touched on the ordinary dispositions such as fragility and irascibility, and not everything I have said about electric charge can be straightforwardly applied to them. So far, then, it is still open to the realist to say that fragility8.35 is more natural than, and grounds, fragility, even though neither of these two properties is perfectly natural or fundamental. In fact, it would appear that fragility8.35 makes for more perfect similarity than the more general property of fragility, an often-cited criterion of comparative naturalness. So should we take a different route in the case of non-fundamental dispositions? Such a claim would fit badly with what we have seen in the case of electric charge and the specific dispositions associated with it. To begin with, it simply looks odd: at the more fundamental level, it is the more general dispositions that are more natural, but as we progress from the more natural to less and less natural dispositions, at some point (and where?) the specific dispositions become more natural; certainly a strong case would have to be made for that claim to sound plausible. More importantly, there are reasons to believe that the claim is false. For as we progress from the perfectly natural to the less and less natural dispositions, we are progressing from ground to grounded—the less natural dispositions are grounded in the more natural ones. The ‘closer’ a non-fundamental property is in the order of grounding to a fundamental property, or the more directly it is ‘fixed’ by the fundamental property, the more natural that non-fundamental property 14 Another criterion for naturalness that is often used is this: the more natural a property is, the more perfect the resemblance for which it makes. Perfectly natural, fundamental properties make for perfect resemblance. (Lewis (1983) appeals to this criterion in delineating the perfectly natural properties.) This criterion simply cuts no ice between electric charge and charge e∗ . Both, I take it, make for perfect resemblance. The former makes for more resemblance, but it is unclear whether that is relevant to its comparative degree of naturalness. See Wilson (2012) for the three criteria for fundamentality that I have appealed to: involvement in fundamental laws, fixing, and similarity.

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should be. And it is hard to see how the more fundamental general dispositions (say, electric charge) should ground the less fundamental specific dispositions ‘first’, and only then the corresponding general dispositions. Take, for the sake of an easier example, the water-solubility of salt. Salt dissolves in water because its positively charged ions attract the partially negative oxygens in the water, while its negatively charged ions attract the partially positive hydrogens in the water. Its solubility, then, is grounded in the charges of its constituent ions. Those charges, we have seen, are general dispositions, not specific ones: they are dispositions to attract or repel with whatever force is functionally determined by the circumstances. Those dispositions can directly ground the composite salt’s disposition to dissolve to whatever degree and at whatever speed is determined by the quantity of water around it, its temperature, and so forth. The salt’s much more specific disposition to dissolve at a particular rate in exactly 1 litre of water with a temperature of 24◦ C is not directly grounded in the charges of its constituent ions. Similar considerations, though with more intermediate steps, will apply to fragility and other ordinary dispositions. The result goes well with ordinary thought about dispositions. We do have some intuitive grasp on grounding relations, on which philosophers have relied in spelling out the notion. Thus it is intuition, not (just) philosophical theory, which prompts us to say that Socrates’ singleton exists because Socrates does, or in virtue of Socrates’ existing; and that a piece of cloth is red because it is scarlet, or is red in virtue of being scarlet, not the other way around. But prior to any commitment to a conditional conception of dispositions, it is not intuitive to say that the glass is fragile (even partly) because it is disposed to break if struck with 8.35 N. It sounds more intuitive to say that the glass is disposed to break if struck with 8.35 N because it is, or in virtue of being, fragile. I do not want too much weight to rest on these intuitions, but I believe they provide some confirmation for the considerations already adduced. The preference for specific, single-track dispositions over the more general multi-track ones, then, is mistaken in general. But we have seen that this preference is a consequence of the combination of three claims: realism about dispositions, the claims about multi-track dispositions that result from sections 2.3–2.5, and the standard, conditional-based conception of dispositions. Since the first of these claims is a guiding assumption of this book, and the second has been argued for extensively, the argument of this section provides strong evidence (within the project of this book) that the standard conception is mistaken. The next chapter will develop an alternative conception. But to conclude this chapter, I will give a quick preview of that conception and of how it solves the problems of the standard conception.

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2.7 Preview of chapter 3 In the next chapter, I will develop an alternative conception of dispositions that does without conditionals. For the case of ordinary dispositions, it will replace the two claims that come with the standard conception by the following pair of claims: 1. A disposition is individuated by its manifestation alone: it is a disposition to M, full stop. 2. Its modal nature is that of possibility, linked to or best characterized (to a first approximation) by ‘x can M’. The alternative conception avoids the problems which, as I have argued, beset the conditional conception. On the alternative conception, a disposition is singletrack if it is adequately characterized by a single possibility statement. Ordinary dispositions such as fragility will be single-track dispositions on that conception; hence there will be no temptation to reduce them to, or replace them with, putatively simpler, more specific dispositions. We do not need to spell out the alternative conception in any detail to see why it yields this result. First, there is in general more qualitative unity to a disposition’s manifestation than there is to its putative stimulus conditions. The fragility of a glass, a vase, a wooden chair or an old parchment is manifested in breaking even if these manifestations are triggered by such diverse factors as the glass’s being hit by a spoon, the vase’s being sung to by a soprano, the chair’s being transferred into a different temperature, and the parchment’s being merely touched. Of course, the manifestations, just like the putative stimulus conditions, will typically exhibit the quantitative diversity of determinable properties in general: things may break into two, three, or any number of pieces, their breaking may be slower or quicker, and so forth. But this is no problem, for second, possibility claims interact with disjunctions rather differently from counterfactual conditionals. The problems sketched above arose largely from a structural feature of counterfactual conditionals: the universal quantifier that is, on the standard semantics, implicit in them.15 For it to be true that x would M if it were S, x has to M in all the closest worlds where it is S. If being S is a disjunctive property (such as being stressed in section 2.3) or a determinable property (such as being struck in section 2.4), those closest worlds may include worlds where any or all of the disjuncts or determinables of S are 15 The problems cannot be solved by rejecting the standard semantics. The implicit universal quantifier is just a way of capturing the element of—variably strict—necessity that characterizes a ‘would’ counterfactual, as opposed to, say, a ‘might’ counterfactual.

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true of the object x. But, I argued above, that is too much to ask for a property such as fragility. The same problem does not arise with a possibility-based conception of dispositionality simply because there is no universal quantifier implicit in a possibility claim. According to that conception, when we say that x is fragile, this is relevantly linked to the claim that it breaks in some (contextually relevant) world. This is equally true of an object which breaks into two pieces in some such world, of an object that breaks into seven pieces in some such world, and of an object that breaks into a thousand pieces in some such world. So much, then, for ordinary dispositions. Nomological dispositions such as electric charge, however, seem to be a different matter. Surely a particle’s being electrically charged is not just a matter of its possibly exerting a certain force. Rather, the particle has to exert a certain force in such-and-such circumstances. It is important here to look at the two claims of the alternative conception separately. The first claim is that a disposition is individuated by its manifestation alone. This, I maintain, is the right view to take with regard to nomological dispositions. Recall the problem with what seemed to be an adequate formulation of electric charge: (Charge2) ∀r∀q: (If x were at distance r from a charge of q, then x would eq exert a force of F =  r2 ). The problem was that in (Charge2), the universal quantifiers take scope over the counterfactual conditional. This makes it natural to think of (Charge2) as just an infinite conjunction of its instances (of which (Charge3) is one). What we want is for the universal quantifier to occur inside the scope of the modal operator that corresponds to the modal aspect of dispositionality. The idea that is captured in (Charge2), after all, is that electric charge is the disposition to always exert a force that stands in a certain mathematical relation to other objects and their charges, whatever they are. Being represented by a two-place operator, the counterfactual conditional does not allow for a better way of capturing this than (Charge2). What we need is a one-place operator in which to embed the entire, complex manifestation of electric charge. For then we can represent the modal aspect of the disposition that is electric charge as follows ( standing proxy for whatever is the right modal operator): (Charge4) (∀r∀q: (x is at distance r from a charge of q → x exerts a force eq of F =  r2 )). Because the one-place operator  can take scope over the universal quantifiers for r and q, it is not the case that (Charge4) is best understood simply as a conjunction of its instances. It captures the idea that we wanted to capture: that

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electric charge is a disposition to exert a force which stands in a certain mathematical correlation to other objects and their charges, whatever they are. But how are we to interpret the operator ? The second claim of the alternative conception is that the modality to which dispositions are linked is possibility. If we adopted that second claim for the case of electric charge, the operator  in (Charge4) would have to be replaced by a possibility operator. But the resulting statement is obviously too weak to characterize electric charge. A better approximation would be the necessity operator: a particle with electric charge must exert a force which stands in the right correlation to other objects and their charges. But simply replacing  in (Charge4) with a necessity operator is too strong. It need not be strictly necessary that a given particle, x, exerts the forces specified in (Charge4), for x might cease to be electrically charged. (Perhaps it cannot, but that is a matter of electric charge being essential to a certain particle, and not a matter of what electric charge is.) Rather, we must think of the necessity as coming with an implicit restriction: it is necessary that, so long as x has the disposition of electric charge, it will . . . . (A similar qualification must be understood to be implicit in (Charge1)–(Charge3).) Chapter 3 will return to the question of how to adequately understand nomological dispositions such as electric charge. A pressing task will be to explain the relation between these necessity-like nomological dispositions and the possibility-like ordinary dispositions. I will argue that they are simply two ends of one spectrum: from the slightest degrees of dispositions or, as we will then call them, potentialities, to their maximal degree. But that task must be left for chapter 3. Neither the ordinary dispositions that we have started out discussing, nor the closer-to-fundamental, nomological dispositions such as electric charge, should be understood primarily in terms of conditionals. If they are, we are left with an implausible and unattractive picture of grounding relations among dispositions—the highly specific dispositions that have less explanatory power will be more fundamental than those dispositions that we started out wanting to understand, and which are explanatorily more basic. If we are to be realists about dispositions, we should set the conditional conception aside and look instead for a different way of capturing the modality that is inherent in dispositions. The one-place operators of possibility and necessity promise to be more illuminating. So far, however, they give us a rather disjunctive picture of the modality inherent in dispositions: possibility in the case of fragility and so on, and a restricted necessity for the case of electric charge. Integrating the possibility-like dispositions, such as fragility and irascibility, with the necessity-like dispositions such as electric charge will be one of the tasks of the next chapter. Let us move on, then, to developing a better conception of dispositionality.

3 Dispositions: an Alternative Conception 3.1 A fresh start In chapter 2, we have seen that the standard conception of dispositions, which links them closely to counterfactual conditionals, spells trouble for realists about dispositions. It is time for a fresh start. The fresh start that we will try in this chapter is based in the semantics of disposition ascriptions. An unprejudiced look at such ascriptions, as I hope to show, yields a rather different conception of dispositions, linking them more closely to possibility. The first part of this chapter, up until section 3.3, will be concerned with the semantics of disposition ascriptions. The point that these sections make is independent of the target metaphysics with its realism about dispositions. To show that this is so, I will offer a reductionist version of the semantics in section 3.3. My preferred realist semantics and metaphysics of dispositions will be developed in sections 3.4–3.6. Despite dominating the current literature, the preoccupation with conditionals in accounting for dispositions is oddly in tension with the linguistic means that we use to ascribe dispositions in ordinary life, adjectives such as ‘fragile’, ‘transmissible’, or ‘irascible’. Typically, those adjectives are formed from a verb (not always extant in English: ‘frag-’ is from the Latin frangere, ‘to break’; ‘irasc-’ from the Latin irasci, to get angry) and a suffix ‘-able’, ‘-ible’, or contracted forms such as ‘-ile’. These adjectives display two features that are worthy of note in the present context. First, they provide us with only one half of the putative conditional. In the cases I have cited, it is the second half, the disposition’s manifestation: breaking, being transmitted, and getting angry. Second, the most natural paraphrase for the suffixes that go into their formation is not a conditional, but ‘can’ or similar expressions. This is confirmed in lexicography as well as formal linguistics. Witness the entries provided by the Oxford English Dictionary for some of our favourite dispositional adjectives:

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Fragile: ‘liable to break or be broken; . . . easily destroyed’. Transmissible: ‘capable of being transmitted’. Irascible: ‘easily provoked to anger or resentment’. Soluble: ‘capable of being melted or dissolved’. Elastic: ‘[t]hat can be stretched without permanent alteration of size or shape’. I have selected only parts of the dictionary definition, but I have not omitted any mention of a conditional. That the entries are representative is confirmed by an empirical study (Kjellmer 1986) which provides a complete typology of entries for terms in -ble in all standard English dictionaries. While there is some variation in these entries, the conditional plays no role whatsoever. Instead, we find different expressions of what would generally be classified as possibility, ‘capable’ and ‘can,’ and in their scope a description of the disposition’s manifestation. The phenomenon has a certain stability across languages.1 Thus linguist Angelika Kratzer writes about the corresponding suffixes in German: ‘In general, the suffixes -lich and -bar express possibility’ (Kratzer 1981, 40; section 3.2 will look at Kratzer’s treatment in more detail). I am certainly not advocating that philosophy be replaced by linguistics or lexicography, but I do take these data to provide some motivation for an alternative approach. If dispositional terms linguistically express something akin to possibility, the default assumption should be that the properties ascribed with them are possibility-like. On this approach, at a first pass, a fragile vase is one that can break easily, a transmissible disease is one that is capable of being transmitted, and an irascible person is one that can easily be made angry. As in chapter 2, we can characterize the standard conception of dispositionality by the following pair of claims: 1. A disposition is individuated by the pair of its stimulus condition and its manifestation (or, if it is a multi-track disposition, by several such pairs): it is a disposition to M when S (or a disposition to M1 when S1 , to M2 when S2 , etc., if it is a multi-track disposition). 2. Its modal nature is, in some way or another, linked to or best characterized (to a first approximation) by a counterfactual conditional ‘If x were S, x would M’ (or, if it is a multi-track disposition, by several such conditionals).

1 The English suffix ‘-ble’, being derived from Latin ‘-bilis’, has close equivalents in Romance languages which, to my knowledge, are all used to express what a thing can do or have done to it. Ancient Greek has a suffix -τ oς, which functions in much the same way.

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I will continue to refer to the conjunction of these two claims as the standard conception, or sometimes as the conditional conception of dispositional properties. On the alternative conception that I am proposing, 1. A disposition is individuated by its manifestation alone: it is a disposition to M, full stop. 2. Its modal nature is that of possibility, linked to or best characterized (to a first approximation) by ‘x can M’. We may refer to this conception as the possibility conception of dispositional properties.2 Of course, the two claims are no more than a template for the view. To begin with, we may ask what kind of possibility is relevantly linked to dispositions. Surely mere logical or metaphysical possibility is too weak. So what species of possibility should we take instead to characterize dispositional properties? From the dictionary definitions cited in the introduction, we can take two suggestions. One is just plain possibility: this is the modality inherent in ‘liable to break’ or ‘capable of being melted’. Not quite metaphysical possibility, perhaps, but possibility restricted by laws of nature and some other constraints to the effect that things take their normal course—the kind of possibility that we express in ordinary language when we say such things as ‘This can break’, or ‘You can grow hydrangeas on this soil’. The second is easy possibility: this is the modality inherent in ‘easily destroyed’ or ‘easily provoked to anger’. Section 3.2 will spell out in more detail what these options are and argue for a particular answer to the question as to which of them adequately characterizes dispositions. As set out in chapter 2, a second and distinct question is that of how the modal nature of dispositions is linked to the species of possibility under consideration. Here we have the same options that we had with the conditional conception. Some may wish to provide a reductive analysis of dispositions in terms of that particular species of possibility, akin to the Lewisian project of reducing dispositions to conditionals (Lewis 1997); section3.3 will sketch how they might proceed. Others may wish to remain neutral on questions of reduction, but still provide a true biconditional with disposition ascriptions on the one hand and a, perhaps 2 To my knowledge, the only contemporary author who has explicitly argued for a similar view is Lowe (2011), whose view is very similar but not entirely identical to mine. Lowe is less concerned to spell out the positive account in as much detail as I do here, but he offers an extended and convincing diagnosis of why the conditional conception has held such attraction and where exactly its mistake lies. Lowe’s views on the metaphysics of dispositions, of course, are quite different from mine: for him, an object x is disposed to M iff x instantiates a kind that is characterized by the property M (Lowe 2006, chapter 8.5). Mumford and Anjum (2011) also seem to think of dispositions or ‘powers’ in terms of a manifestation without a stimulus, but as far as I can see they do not explicitly argue for that conception.

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complicated, possibility statement on the other. This would be akin to the project pursued, with regard to conditionals, in Manley and Wasserman (2008). Yet others will wish to remain anti-reductionist about dispositions. As with Martin (1994), issues akin to finks and masks may convince them that the reductive project is bound to fail, or they may simply think that there is no reason to seek a reductive analysis in the first place. An anti-reductive metaphysics of dispositions will still have to accommodate various features of disposition ascriptions, such as their gradability and context-dependence. Section 3.4 will spell out how to do that. Finally, there is the option of reversing the more common direction of explanation and seeing possibility as grounded in dispositional properties (an option that was taken for dispositions and conditionals by Jacobs 2010). That, of course, is the project of this book. Despite my anti-reductionist project, I will begin with a modal semantics framed in the usual terms, i.e. possible worlds. I do so because possible-worlds semantics provides a rigorous formal framework that is better developed than any other framework I know of, and because it is used in the linguistic literature that I wish to discuss. I regard the framework as a formal model with little metaphysical import, but a useful heuristic tool. The metaphysics behind disposition ascriptions, as I see it, will be developed in chapters 3.4–3.5. Before I begin, a comment on methodology is in order. In discussing properties such as fragility, solubility, and irascibility, philosophers have often found it convenient to switch from these terms to the apparently more transparent ‘disposed to . . . (if . . . )’ locution and to examine their linguistic intuitions regarding those constructions rather than regarding the terms that the debate was initially concerned with, such as ‘fragile’, ‘soluble’, or ‘irascible’. I will not follow this method. In fact, I will entirely disregard the ‘disposed to . . . ’ locution in exploring the semantics of disposition ascriptions. This may seem to beg the question, for the disregarded locution allows us to formulate disposition ascriptions which explicitly require a conditional treatment, such as ‘x is disposed to break if struck’. But there are good methodological reasons to disregard the ‘disposed to . . . ’ locution. Although philosophers tend to forget this, the locution as it is used in the metaphysical debate is almost entirely an artefact of philosophy. Outside philosophy,3 the locution ‘disposed to . . . ’ is not naturally applied to concrete inanimate objects such as vases and sugar cubes, but only to agents, animate objects and (often personified) abstracta. In the vast majority of uses, it is followed by verbs of action, or (less often) by verbs of perception or attitude, all of which are not felicitously ascribed to non-animate objects. In fact, ‘disposed 3 The following recapitulates the results of corpus-linguistic survey, more details of which can be found in Vetter (2014).

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to . . . ’ very often just means ‘willing to . . . ’. The ascription to a vase of a disposition to break (if struck), outside philosophy, has a somewhat animistic sound to it. At any rate it is so unusual that we have reason to doubt the applicability of our prephilosophical linguistic intuitions to it. Further, ‘disposed to . . . ’ is not naturally restricted to ascribing relatively permanent intrinsic properties to objects. This is obvious where ‘disposed’ simply means ‘willing’. What we are willing to do, unlike our dispositions in the philosopher’s sense, can vary greatly from one specific situation to another. Even where ‘disposed to . . . ’ is not best understood to mean ‘willing’, it is easily construed to express a rather fleeting and situation-bound tendency with little or no grounding in the individual’s more permanent intrinsic features. In sentences with a plural subject, ‘disposed to . . . ’ may also express a statistical correlation.4 ‘Disposed’ and ‘disposition’, as used in the debate about dispositions, is best understood as a theoretical term which serves as a placeholder to capture whatever it is that fragility, solubility, and so on have in common qua dispositional properties. The question of whether these theoretical terms are to be construed as a one-place operator akin to possibility, or as a two-place operator akin to a conditional, is better posed as the question of how best to characterize fragility, solubility, and other such properties. In discussing that latter question, it is perfectly legitimate to appeal to linguistic intuitions concerning ‘fragile’, ‘soluble’, and so forth. But to appeal to one’s linguistic intuitions concerning the ‘disposed to’ locution, and to use those intuitions in judging the adequacy of either the standard or the alternative conception, gets things backwards. The use of a technical term ought to be informed, ideally, by true theory, not vice versa: the use of technical terminology is not a guide to truth, though it often encodes what is held to be true. With these provisos, let us now take a closer look at the semantics of disposition ascriptions as construed in linguistics.

3.2 Modal semantics Modal semantics at the intersection between philosophy and linguistics has been shaped by the work of David Lewis and Angelika Kratzer. I will here focus on Kratzer’s work, which is spelled out in more detail. In what follows, I will provide 4

Here are two examples taken from the Corpus of Contemporary American English (Davies 2008): Liberal church leaders were disposed to look kindly on environmental issues, and global warming joined the list. Ordinary people in foreign lands are disposed to like Clinton, or at least to like the relaxed, human image they describe with reference to his smile or his saxophone.

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an outline of her semantic theory in Kratzer (1981, 1991), and her brief remarks on disposition terms. My exposition closely follows Kratzer (1991) but will skip the formal details.5 Modal expressions, for Kratzer as for Lewis, quantify over possible worlds. There are three dimensions along which different modal expressions (or their particular uses) can differ: their modal force, their modal base, and their ordering source. First, each modal expression comes with a (relatively stable) modal force. The modal force is simply the kind of quantification that is applied to the possible worlds: universal quantification for necessity as expressed, for instance, with ‘must’ or existential quantification for possibility as expressed, for instance, with ‘can’. Of course we rarely quantify over all the possible worlds. Which worlds are relevant in a particular context is determined by what Kratzer calls the ‘conversational background’. We can think of conversational backgrounds as sets of propositions. For Kratzer, there are two distinct ways in which the conversational background determines the relevant worlds. A first function of conversational background is to determine the modal base: the set of worlds that is relevant or accessible in the context. Thus when a botanist says, in a foreign country with unfamiliar vegetation, ‘Hydrangeas can grow here’, she says not just that in some metaphysically possible worlds there are hydrangeas growing on this soil. She speaks only about worlds where the biology of hydrangeas, the geology, and the climate of the country are as they are in actuality. When a detective says, ‘Mary might be the murderer’, she speaks about a different set of worlds: those that are compatible with everything she knows. In each case the conversational background selects a set of propositions: propositions about the actual facts of biology and geology, or the propositions that are known to the detective. The modal base is the set of worlds in which those propositions are true. (If worlds are sets of propositions, the modal base will be the set of those worlds which have all the relevant propositions as a member. If propositions are sets of worlds, the modal base will be the intersection of all the propositions that are relevant.) As the examples illustrate, modal bases come in two kinds, circumstantial and epistemic. Circumstantial modal bases are determined by propositions about how things stand, regardless of our epistemic access; epistemic modal bases are determined by propositions that we know to be true. Both kinds of modal bases are realistic: they always include the actual world. This is familiar territory to most philosophers. A less familiar feature of Kratzer’s semantics is the second dimension of the conversational background. 5

Kratzer’s work on the subject is now collected in Kratzer (2012).

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In addition to the modal base, Kratzer argues, conversational background must determine an ordering source. We can think of the ordering source as another set of propositions. But while being part of the modal base is an all-or-nothing matter, the ordering source imposes—as its name suggests—an ordering on worlds. It ranks the worlds of the modal base according to how close they come to a certain ideal: roughly, how many propositions out of a particular set of propositions are true in them. Ordering sources may be deontic, ranking worlds by how close they come to satisfying the demands of morality or the law, as when we say ‘Mary must pay a fine’ (because she obstructed the bike path with her car). They may be bouletic, ranking worlds by how close they come to satisfying all of a certain speaker’s desires, as when we say ‘Billy must take the A train’ (since he wants to go to Harlem).6 Or they may be stereotypical, ranking worlds by how close they come to being ‘normal’, as in the detective’s ‘Mary might be the murderer’. Kratzer has various arguments for the distinction between modal base and ordering source as two distinct features of conversational background. In our context, the most relevant motivation is based on graded notions of modality. Graded modality is expressed by such comparative locutions as ‘there is a better possibility that . . . than that . . . ’, or ‘it is more likely that . . . than that . . . ’. Graded modality is also needed to distinguish between what is said by ‘there is a good possibility that . . . ’ and ‘there is a slight possibility that . . . ’, between what might just be and what may well be the case, and so on. (See Kratzer 1981, 46ff.) In order to account for the gradable nature of many modal expressions, we need to have a dimension in the conversational background that allows for a ranking; hence the ordering source. However, we also need to account for the fact that some of the relevant facts are non-negotiable. When the detective considers who might be the murderer, she is not interested in worlds where the available evidence is not exactly as it actually is. In saying that Mary might well be the murderer, but that there is still a slight possibility that Jane did it, she is ranking worlds where Mary committed the murder as closer to some ideal of normality. Worlds in which Jane did it are further away from that ideal, but still close enough to be considered relevant. Worlds where a spaceship landed and aliens committed the murder are too far away from what is considered normal to be relevant for her modal statements, even if they too are included in the modal base. The distinction between the non-negotiable and the gradually approached, in selecting relevant worlds as the domain of quantification, motivates the distinction between modal base and ordering source. It is not, however, always the case that both of these dimensions are required. In particular, there are cases 6

The two examples are from von Fintel and Iatridou (2007).

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which do not require ranking but merely compatibility with a set of propositions. In those cases the ordering source is said to be empty. Note that Kratzer does not ever account for graded modality by altering the modal force: the idea is not that if p is more likely than q, then there are more possible worlds where p than there are worlds where q. It is rather that the next p-worlds are closer to some ideal than the q-worlds. I will argue in a moment that altering the modal force is precisely what we need to do when considering dispositions. So much for the general framework; now on to dispositional predicates. We have already seen that Kratzer identifies their modal force as that of possibility: to be fragile is to break in some relevant possible world. As to the modal base, Kratzer (1991, 647) notes, ‘any modality expressed by the suffixes -ible or -able will . . . have a circumstantial modal base’. So much seems obvious: whether a person is irascible or a disease transmissible has nothing to do with our knowledge, but with features of the person or the disease. Earlier, Kratzer (1981) had said a little more on one of the philosophical literature’s stock examples, the sentence Diese Tasse ist zerbrechlich. This cup is fragile. (Note that Kratzer’s discussion is directly concerned only with the German sentence, not the English translation.) Here is what Kratzer says: I think that this is a case of ‘pure’ circumstantial modality. It is in view of certain properties inherent in the cup, that it is possible that it breaks. The ordering source seems to be empty. Kratzer 1981, 64

Let us apply this to the English ‘fragile’ (I will return to the German original shortly). The modal force is simple existential quantification; the modal base consists of those possible worlds where the cup has certain intrinsic properties (its molecular structure, say); the ordering source is empty. Hence sentences of the form ‘x is fragile’ will be true iff, among the possible worlds in which x possesses certain intrinsic features, there is one in which x breaks. This cannot be quite right. Bricks and bridges made of steel are not fragile (not in a typical context, anyway). Yet in view of certain inherent properties of the brick or the bridge, it is possible that they break; among the worlds in which the brick or the bridge possess all of their actual intrinsic properties, there are some where they break. Similarly, not everyone who possibly, in view of their intrinsic properties, gets angry is thereby irascible. We need a stronger characterization of the truth conditions for ‘x is fragile’ and ‘x is irascible’.

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What distinguishes the fragile things from those for which it is possible to break but which are not fragile? The dictionary definitions cited in section 3.1 suggest that the former, but not the latter, can easily break or be broken. Similarly, almost everyone can be provoked to anger, but the mark of an irascible person is that they get angry easily. The naturalness of ‘easily’ in connection with dispositions is confirmed by its slipping into characterizations of a disposition even when those are officially given in the stimulus–manifestation schema: witness Alexander Bird’s characterization of fragility as ‘the disposition to break easily when stressed’ (Bird 2007, 19), or Stephen Mumford’s introductory remark that ‘although virtually all objects are breakable, it is only those which break easily . . . that are called fragile’ (Mumford 1998, 5). Notably, the term does not play any role in either author’s official characterization of dispositions. How, then, are we to understand the qualification ‘easily’? A first cue is that with ‘easily’ we have introduced an element of gradability.7 An easy possibility is, in some sense, more of a possibility than, say, a slight possibility. Moreover, easy possibility itself appears to come in degrees: a champagne glass, for instance, can be broken more easily than a coffee cup. In fact, we should not be surprised that an element of gradability is needed; after all, it is well-known that dispositions themselves are gradable, and their gradability plays an important role in the contextual variation of disposition ascriptions (as Manley and Wasserman have stressed). All of this strongly suggests that the ordering source is not, after all, empty, at least not in the case of ‘fragile’ and ‘irascible’. If there is an ordering in play here, of what kind is it? A second cue (though, as we shall see, a less promising one than the first) can be taken from philosophy rather than linguistics. Easy possibility is not a stranger in philosophy: Tim Williamson has used it in his account of knowledge (Williamson 2000, see also Williamson 1994 and Sainsbury 1997), and Christopher Peacocke has appealed to it in his compatibilist account of freedom (Peacocke 1999). For our purposes, it is enough to say that there is an easy possibility that p just in case there is at least one close world in which p, where closeness is a matter of similar initial conditions (not of the world, but of a temporal portion of it). For instance, suppose that a glass is standing close to the edge of a table, and with a careless movement I almost hit it. There is an easy possibility that the glass breaks because in some world with very similar initial conditions—that is, a world in which the glass is in more or less the same position, and I make my movement from more or less

7 For reasons given in the main text, I take the adverb ‘easily’ in the dictionary definitions to modify ‘can’. For an alternative approach, on which it modifies the verb that specifies the manifestation, see Lowe (2011).

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the same position with more or less the same lack of care—my movement is just so slightly different that I do hit the glass, it falls, and it breaks. Incorporating the conception of easy possibility into the Kratzerian framework, the ordering source will effect a ranking of worlds in the modal base according to how closely they resemble the actual world with respect to the contextually selected initial conditions. It does not matter exactly how the initial conditions are selected. One basic formal feature of easy possibility so understood is centring: the actual world is always at least as close to the ideal as any other world. After all, it is features of the actual world (and nothing else) that determine the ordering. But if that is so, then an object’s breaking in actuality will prove that there is (or was) an easy possibility of its breaking. A brick is hit with a large hammer; it breaks. On the present model of easy possibility, there was therefore an easy possibility that the brick should break. Had there been no hammer around, there would not have been an easy possibility for the brick to break. But the presence or absence of hammers is not what makes things fragile or non-fragile. Centring is a feature that we should try to avoid in characterizing dispositional predicates. Easy possibility on the Peacocke/Williamson model cannot, then, be the right model for fragility. But Kratzer offers a variety of ordering sources that are not centred on the actual world, as easy possibility on the Peacocke/Williamson model is. Stereotypical ordering sources are one example, and they promise a more intuitive starting point in understanding disposition terms. In actuality, things may not always take the most normal course of events; hitting bricks with hammers may be an extraordinary thing to do. Again, it does not matter exactly how we spell out the notion of a normal course of events. For this proposal, too, faces a basic problem. In fact, the problem with centring was only a symptom of the more general problem that will, I predict, be faced by any proposal along these lines. The problem is that one world where an object breaks is enough, on the current proposal, for that object to be fragile, and that no restriction on closeness can exclude unwanted breaking worlds from among the close worlds. To see this general problem, take the case that Manley and Wasserman (2008) call ‘Achilles’ heel’: consider a sturdy concrete block that, like Achilles, is almost entirely immune to harm. . . . But, like Achilles, the block has a weak spot. If it is dropped onto a particular corner at just the right angle with exactly the right amount of force, an amazing chain reaction will cause it to break. Manley and Wasserman 2008, 67

We can easily imagine that the block being dropped in this particular way is an entirely normal type of event—just as normal as dropping it onto any other

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corner, at any other angle, and with any other force. By our stipulation, then, there will be a world within the modal base (i.e. where the block possesses its actual intrinsic properties) which is among the highest-ranking by the ordering source (i.e. where things take their most normal course), and in which the block breaks. On the present proposal, the block would then have to count as fragile. But it seems clear that it does not. The source of the problem is not the specific characterization of the ordering source. In fact, it is hard to see how any plausible characterization of an ordering source could exclude the occurrence of Achilles’ heels. Nor is the problem usefully understood as one of finking or masking. Finking and masking, I have said in chapter 2.2, are symptoms of the fact that a disposition is a matter of how things stand with the object (the block) itself, while (easy) possibility concerns how things stand with the object and the world outside it. The block’s weak spot, however, is how things stand with the block itself; it is not an interference of its surroundings. In fact, it is no interference at all. The source of the problem is, rather, that on the present proposal it is enough for an object to count as fragile that it break in one world. However closeness is construed, one close breaking-world is not in fact enough; for an object to be fragile there need to be more ways of breaking it. This, then, is a third and final cue for understanding ‘easily’. We were right to see ‘easily’ as indicating an element of gradability. But that element of gradability, in the case of fragility, is not best understood on the level of ordering sources. Rather, it is best understood at the level of the modal force: the quantification that is applied to the relevant possible worlds. Simple existential quantification does not do justice to the modal force of ‘fragile’. The quantification needs to be of a more complex, proportional nature. On a proportional conception of degrees, x is more fragile than y just in case x breaks in more of the relevant worlds than y. x is fragile simpliciter just in case it breaks in sufficiently large proportion of the relevant possible worlds, where context may determine what counts as sufficiently large. Two points need to be made immediately. First, the sufficient proportion of worlds will still be a rather small proportion. Even a very fragile champagne glass remains unbroken in most worlds, simply because it is safely standing on a table or packed away at the back of a shelf. On the quantificational spectrum ranging from ‘at least one’ to ‘all’, the proportion of worlds where an object has to break in order to count as fragile will be close to, but not quite at, the ‘at least one’ end, bearing witness to the fact that fragility is akin to possibility, not necessity. The right proportion is best captured as ‘a few’. Second, however, we must not mistake this capturing of the right proportion for a full statement of the account. Even a sturdy brick will break in a few possible

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worlds; this is not what sets the fragile things apart from the non-fragile ones. Rather, it is that (all of) the former break in more possible worlds than (any of) the latter. This is based on two ideas: first, that the comparative ‘is more fragile than’ establishes an ordering among objects, such that to count as fragile simpliciter (in a given context) is to be above some (contextually determined) point within that ordering; and second, that to be more fragile than something else is to break in more possible worlds than it. Does the case of fragility generalize? It is instructive here to contrast ‘fragile’ with ‘breakable’. Consider our block with an Achilles’ heel again: it is not fragile, but is it breakable? Here it is much less obvious that the answer is no; I am inclined to think that it is yes. Moreover, consider the inference from an actual manifestation to the possession of the disposition that I rejected earlier: from the fact that a thing does break in actuality, we cannot infer that it is fragile; but we can, it seems, infer that it is breakable. The German ‘zerbrechlich’ is closer to ‘breakable’ than it is to ‘fragile’ in this respect, which explains Angelika Kratzer’s classification of her above-mentioned example. Nor is ‘breakable’ an isolated case. A disease’s being transmitted once in actuality is sufficient for its being transmissible. Walking a path proves that it is walkable, smashing a pot that it is smashable, winning a game that it is winnable, and (my) reading a text in fine print proves that it is readable (for me). But this inference is typical of the modal force of possibility proper: an object’s being F in actuality does not prove that it is F in a sufficient proportion of worlds, but it does prove that it is F in at least one world.8 Hence the graded possibility that we have found to characterize fragility is ruled out for those cases. Nor does ‘breakable’ require appeal to an ordering source. For an object x to count as ‘breakable’ it is enough that it is possible, in view of x’s intrinsic properties, that x breaks; or, in possible-worlds terms, it is enough that there is one breaking-world among the worlds in the modal base, the worlds where x’s relevant intrinsic properties are as they are in actuality. Having admitted gradation by proportional modal force, we no longer need to appeal to an ordering source to explain the gradability of terms such as ‘breakable’: a champagne glass and a brick are both breakable, but the glass is more breakable than the latter because it breaks in a greater proportion of relevant worlds than the brick. There is, then, some variation to be expected in the modal strength of different dispositions, between simple possibility and the graded possibility that requires manifestation in a sufficient proportion of worlds.9 The variation is not ad hoc, 8 Since I am no longer concerned with the distinction between manifestations and stimulus, I switch from ‘M’ to ‘F’ as my placeholder for manifestations. 9 Kjellmer (1986) observed this variation in his above-mentioned study on adjectives in -ble, though his take on the stronger meaning, which I have labelled ‘easy possibility’, is quite different.

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and it is not very wide-ranging. It is motivated by the different inferential behaviour of the terms we use to ascribe the dispositions in question. It is, further, a variation over only a limited part of the quantificational spectrum: it takes place between ‘at least one’ and ‘a few’. It does not, or so I have suggested, ever go near the opposite end of the quantificational spectrum, on which we find the universal quantifier that characterizes necessity. It is situated firmly within the area of possibility.

3.3 Modal semantics, continued Whatever our conception of easy possibility, it is certainly not the case that an object is fragile (at time t) just in case there is an easy possibility (at t) that it breaks. Pack a fragile glass in styrofoam, and there is no easy possibility that it breaks; place a rock in front of a bulldozer, and there is an easy possibility that it breaks. This, however, should not be a problem for the proposal: like cases of finking and masking for the conditional conception, it is merely a symptom of the fact that dispositions are properties of individual objects and, as such, relatively independent of circumstances outside that object, while the truth of a counterfactual conditional or an easy possibility claim is sensitive to those circumstances. Those who pursue a reductive analysis of dispositions have generally accepted that, because dispositions are subject to finking and masking, a disposition cannot be reduced to just any counterfactual conditional. They have then tried to manipulate and refine the conditionals accordingly; we have seen some of these attempts in chapter 2.2. On the present proposal, the same kind of strategy should be extended to easy possibility: of course a disposition ascription cannot straightforwardly be reduced to, or even be merely equivalent to, a statement of easy possibility. What needs to be done is to manipulate and refine the possibility statement accordingly. I have already made some suggestions on how such a refinement might go: the key is Kratzer’s idea that the modal base, the set of relevant worlds, is determined by the object’s intrinsic properties. If we consider all and only worlds where the glass’s or the rock’s relevant intrinsic properties, and not its extrinsic features, are held constant, then the actual presence of styrofoam or bulldozers ceases to matter.10 He also notes that the more frequent an adjective is, the more likely it is to have the stronger meaning. I would like to add the conjecture that a dispositional adjective is more likely to be understood in the simple-possibility sense when we can still hear how it is built from a verb and a suffix: ‘break-able’ is immediately understood as ‘able to break’, ‘fragile’ is not. 10 This is based on the assumption that dispositions are intrinsic, or at any rate have intrinsic bases. Strictly speaking, the assumption is false—I will have a lot more to say about extrinsic

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Interestingly, the present proposal for dispositions closely parallels Lewis’s treatment of another kind of modal property: abilities. (See Lewis 1976a, 1979.) Agentive abilities are standardly ascribed using the auxiliary ‘can’, as in ‘She can play the piano’. Traditionally, philosophers have distinguished between the ‘can’ of ability and the ‘can’ of possibility (e.g., Kenny 1976). For Lewis, however, these are not distinct senses of ‘can’. They differ merely in the kind of restriction imposed on the relevant possibility. ‘David Lewis can speak Finnish’, used to ascribe to David Lewis an ability to speak Finnish, expresses the possibility that Lewis speaks Finnish given, roughly, his intrinsic make-up. It is true just in case David Lewis does speak Finnish in at least one world in which his intrinsic make-up, in the relevant respects, is as it actually is (and in which certain further standard conditions obtain). Lewis never extended this treatment to disposition ascriptions but instead proposed the reformed version of the conditional analysis that I have cited in chapter 2.2. Others (notably, Vihvelin 2004 and Fara 2008) have tried instead to understand agentive abilities as a type of dispositions. Vihvelin, in particular, has extended Lewis’s own reformed conditional analysis of dispositions to agentive abilities, understanding them as dispositions to do X if one chooses to do X and explaining failures to do what one chooses and is able to do as cases of finking. I agree with Vihvelin and Fara that dispositions and abilities should be given uniform treatment. But instead of extending the conditional treatment of dispositions to abilities, I propose that we extend the restricted possibility treatment of abilities to dispositions: like ability ascriptions, disposition ascriptions are really expressions of possibility. The ‘property-hood’ of dispositions is built into the accessibility relation, or, in Kratzer’s terminology, the modal base: it comprises only worlds where the object’s relevant intrinsic properties are as they are in fact, while other factors are allowed to vary. We can adopt these aspects without claiming that the two cases are entirely alike: for all that I have said so far, the right modal force for abilities may be possibility proper, or easy possibility on a more standard Kratzerian conception.11 dispositions in the next chapter. But the intrinsicality assumption still seems a good idealization to begin with. Once it is dropped, the account would have to include some extrinsic features of x among the properties that determine the relevant worlds. Note, however, that while the suggested account assumes that dispositions are intrinsic, it does not assume that finks and masks are always extrinsic. What is held constant across possible worlds are the relevant intrinsic features of the object in question, the disposition’s categorical basis. Other intrinsic features may vary together with the extrinsic properties of the object. For the possibility of intrinsic finks and masks, see Clarke (2008, 2010) and Ashwell (2010). 11

See chapter 6.6 for some speculations on the nature of abilities.

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For realists, this approach is no more than a formal model of a rough approximation. For reductionists, it is the beginning of a full-blown account of dispositionality. Either way, the model will require modification in a number of respects; I will briefly mention two before we move on.12 (1) From worlds to cases. To get the proportions right, we should quantify not over worlds but over cases or centred worlds: triples of a world, a time, and an object. An irascible person will typically get angry any number of times within one world, and a transmissible disease will typically be transmitted more than once. What should count towards the proportion that makes a person irascible, or a disease transmissible, is not the number of worlds in which the person becomes angry at least once or the disease transmitted at least once, but rather the number of individual instances of anger or transmission across the relevant worlds: in other words, of cases. As with worlds, so with cases, it is crucial to provide maximal variation in the external circumstances. The proportion of cases in which a vase breaks, a person gets angry, or a disease is transmitted, should not depend on factors that are external to the vase, the person, or the disease. Otherwise we will be faced with the familiar problems of finking and masking: a dead wire that has an electrofink attached to it will conduct electricity in the same proportion of cases as a live wire, if external circumstances such as the presence or absence of electro-finks are allowed to be held fixed over the cases that count towards the wire’s being live or dead. A fragile vase that is packed in anti-deformation packaging will break in fewer cases than its intrinsic duplicate which is precariously standing at the edge of a table, if packaging and position are allowed to be held fixed over the cases that count towards the vases’ fragility. (2) Measuring proportions. The set of possible worlds, and a fortiori of cases, is non-denumerably infinite, and so, in all likelihood, are its subsets whose proportions to one another determine fragility. If the proportion of breaking-cases among the relevant cases is to be determined by comparing the cardinality of the respective sets of cases, we are faced with grave and notorious mathematical worries. Proper subsets of non-denumerably infinite sets may have the same cardinality as their supersets; and so no non-trivial comparison of cardinalities may be possible. 12 Both modifications are inspired by Manley and Wasserman (2008), who have rightly emphasized the proportional nature of dispositions. Unlike the present proposal, their view seems intended not as a rejection but as a refinement of the conditional conception. As a result, their understanding of the relevant proportions differs from mine: Manley and Wasserman’s sufficient proportions are, at least in some cases, proportions only of cases where the disposition’s stimulus has been applied, and hence presumably much greater than the proportions among all the cases that count as relevant. For a contrast between the two views, see also Vetter (2011a) and Manley and Wasserman (2011).

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I will not answer this problem here. Some suggestions for answering it can be found in Manley and Wasserman (2008 79–82), who suggest that physics might deliver a suitable measure, and Vetter (2014), where I suggest that perhaps the infinite sets of possible worlds can be replaced by maximally varied finite subsets.13 For the time being, I would like to note, first, that the problem of proportions is not one that besets only the possibility conception of dispositions. It applies equally to the standard, conditional conception, or at least to a version of the standard conception which takes seriously the problems of gradability, contextsensitivity, and examples such as the above-mentioned Achilles’ heel. It is no accident that Manley and Wasserman, whose version of the standard conception I have in mind here, have discussed those very aspects of dispositions and faced the problem of proportion too. And secondly, I will leave the problem of proportions to those who want to defend a reductionist version of the alternative conception. Those of us who are suspicious of such a conception may take problems such as the problem of proportion to be just another shortcoming of trying to account for such perfectly intelligible modal notions as that of a disposition in terms of possible worlds. We now have the outlines of a reductionist account of dispositions, or for the realist, of a formal model and rough approximation. The picture is this. We start with the graded notion: x is more fragile than y just in case the proportion of worlds where x has its relevant intrinsic features and breaks is greater than the proportion of worlds in which y has its relevant intrinsic features and breaks. For some disposition terms, such as ‘fragile’, a given context imposes a threshold: how fragile an object has to be in order to count as fragile simpliciter. An object x counts as fragile in a context C iff x is above that threshold. For other disposition terms, such as ‘breakable’, any positive proportion is suitable, and no contextual threshold is required. I am not going to address objections to this conception of dispositions here, as I have done so elsewhere (Vetter 2014).14 The task of the current chapter is, rather, to spell out the positive picture of dispositions. I have repeatedly stressed 13 Alternatively, if (and only if) the degrees of dispositions conform to the probability calculus, we might avail ourselves of the possible-worlds-based account of probability that Bigelow (1976) has proposed. Bigelow’s proposal does entirely without proportions and appeals instead to the ‘diameter’ of a set of worlds as ordered by a similarity metric. 14 The short version is this. Counterexamples may, first, take the form of ‘overt’ disposition ascriptions, e.g., the disposition to sneeze when near flowers. I reject such counterexamples for the reasons indicated above: the ‘disposed to’ locution is a piece of technical terminology, and should not be used to elicit pretheoretical intuitions. A second kind of counterexample appeals to ordinary disposition ascriptions which appear to require a stimulus condition. For instance, it may be said that acrophobia (fear of heights) and arachnophobia (fear of spiders) share a manifestation (fear) and hence must be taken to differ in their stimulus (encountering heights, and encountering

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that the semantics given so far is not, for a realist about dispositions, an account of the metaphysics of dispositions. It is time, then, to move on to a thoroughly realist version of the alternative conception of dispositions.

3.4 A background for context-sensitivity If we believe, with orthodoxy, that dispositions are closely related to conditionals, we have various options in spelling out the relation between them. The traditional route is to attempt a reductive analysis of dispositions in terms of conditionals. Realists about dispositions will resist this traditional line. They may claim either that the conditional is no more than an approximation, though the best approximation we have, for understanding dispositions (hinted at in Martin 1994, 8). Or they may reverse the traditional order of explanation and claim that dispositions are the truthmakers for, or otherwise provide for the truth of, counterfactual conditionals (e.g., Jacobs 2010). The same strategies are available when we switch from the orthodox correlation with conditionals to the alternative that I have proposed in chapter 3, the claim that dispositions are most closely associated with possibility. The traditional attempt to provide a reductive analysis will take the semantics of sections 3.2–3.3, perhaps with some refinements, as a starting point. As a realist about dispositions, however, I have further work to do: I need to provide a metaphysics of dispositions that does justice to the semantics I have sketched. With that metaphysics in hand, we may then claim that possibility provides no more than an approximation, though the best approximation we have, to dispositions; or we may claim that dispositions are the truthmaker, or otherwise provide for the truth of, the relevant possibility claims. As I have indicated already, I intend to take the second option, and I will develop it throughout the book. For the remainder of this chapter, my aim is merely to develop a realist metaphysics of dispositions that will suit either one of the realist strategies. The discussion so far sets certain desiderata for such a realist conception. First, it must provide a background for the context-sensitivity of ordinary disposition ascriptions, ideally in connection with the gradability of dispositions. Second, it should individuate dispositions by their manifestation alone, not by spiders, respectively). I respond that the typical manifestations of everyday dispositions are complex causal processes in any case, and in some cases they will have to be understood as including what has traditionally been seen as a stimulus condition. Thus acrophobia and arachnophobia differ in manifestation after all: one is the disposition to be afraid of heights, the other the disposition to be afraid of spiders. Again, I refer the reader who is interested in a more detailed story to the above-mentioned paper.

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a pair of manifestation and stimulus. And third, it should answer the question raised at the end of chapter 2.7: the question, that is, how a unified conception of dispositionality might encompass both the ordinary dispositions, which we have found to be possibility-like, and the nomological dispositions that figure in laws of nature and seem to be akin to necessity rather than to possibility. I will take these up one by one. Disposition terms such as ‘fragile’, ‘irascible’, and so forth, as we have seen earlier, are context-dependent. For that reason, realists about dispositions should not simply be realists about fragility, irascibility, and so forth. For if they were, they would have to countenance context-sensitive properties. But reality is not context-sensitive. Context-sensitivity is a matter of language, not the world. Context-sensitive expressions receive different semantic values in different contexts of utterance; reality provides the semantic values, but semantics provides the variation between them. In dealing with context-sensitivity, we need to provide a metaphysical background, a context-insensitive metaphysics from which, by the right semantic mechanisms, the semantic values of the context-sensitive expressions get selected. The possible-worlds semantics provided in sections 3.2–3.3, if taken metaphysically seriously, can be seen as a paradigm case of this. Which possible worlds there are, and what they are like, is entirely independent of any context in which we speak. Which worlds count as relevant, and which (quantificational) operation is applied to them to evaluate a given utterance, is fixed by context. ‘Fragile’ is context-sensitive because it is used to make statements about possible worlds but each context determines exactly which statement about them is made with it. In the case of possible-worlds semantics, the metaphysical background is of a rather different kind from that which we ascribe with the term. What we ascribe are properties, but the metaphysical background consists of things of a particular kind: possible worlds. There is no need, on that semantics, to include dispositional properties in our ontology (or ideology) in order to account for the truth of disposition ascriptions. I have called this kind of semantics a reductionist semantics. Clearly, it is not what the realist about dispositions has in mind. But a reductionist semantics is not mandatory at all in accounting for context-sensitivity. Consider the context-sensitive term ‘tall’. On a plausible view, reality provides a determinable—having some height—with an infinity of determinates—such as being five feet tall—and an ordering between them, which places, for instance, the property of being six feet tall above the property of being five feet tall; each context for ‘tall’ sets a threshold within that ordering, such that all and only individuals that have a height property above that threshold count

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as tall in that context. This toy semantics is clearly not reductionist in the way in which the possible-worlds semantics I have just discussed is. It provides a metaphysical background for the contextual variation of ‘tall’, but that metaphysical background consists of just the kind of properties that ‘tall’, intuitively, ascribes: height properties. Metaphorically speaking, the metaphysical background is just the property of being tall made context-insensitive; it is generalized just enough to provide the background for the context-sensitivity of ‘tall’ without itself being context-sensitive. This toy semantics-cum-background metaphysics of ‘tall’, I propose, is the right model for a realist about dispositions. What we should be realists about is not simply fragility, for it is context-sensitive which property is referred to by ‘fragility’. Rather, we should be realists about fragility made context-insensitive: about a property that is generalized just enough from the properties that are ascribed by ‘fragile’ in various contexts to provide the context-insensitive background for the context-sensitivity of ‘fragile’. The same, of course, goes for other disposition terms. So here is the realist picture. Begin with the comparative disposition ascriptions: a long-stemmed champagne glass is more fragile than an ordinary tumbler, which in turn is more fragile than, say, a plant-pot. Note that the comparative applies even where we are no longer prepared to call objects ‘fragile’: thus the plant-pot is more fragile than my desk, my desk is more fragile than a rock, but the rock is more fragile than a diamond, which in turn is more fragile than a chunk of gold. We can order objects, at least roughly, all the way from the champagne glass to the diamond. Somewhere along the line the term ‘fragile’ ceases to apply, but where exactly it does is highly sensitive to context. The ordering itself, on the other hand, is not: we may or may not count the plant-pot as fragile, but it remains less fragile than the tumbler, and more fragile than the rock, whatever the context. The distinction between fragile and non-fragile things is not given by nature. But the spectrum on which the distinction is made is. The picture is not so different from that suggested in sections 3.2–3.3: we begin with the comparative, use it to establish an ordering among objects, and then let each context set a threshold within that ordering such that objects above the threshold count as fragile in the context, while others do not. (With such dispositional terms as ‘breakable’, we do not even need the threshold: the entire spectrum counts.) The difference is merely that, with possible-worlds semantics, we had truth conditions for the comparative ‘x is more fragile than y’ in terms of possible worlds, while now we are taking the comparative and the ordering established by it as primitive.

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‘Fragile’, on this view, behaves just like ‘tall’: the comparative ‘taller than’ yields a complete ordering of objects by their height. Somewhere down the line the term ‘tall’ ceases to apply, but where exactly it does is highly sensitive to context. The ordering itself, on the other hand, is not: we may or we may not count Suzy as tall, but she remains taller than Tom and less tall than Mary, whatever the context. There is one property, call it height, that is possessed by all the objects on the spectrum, in ever decreasing degrees (or in determinates of ever smaller values). That property is independent of context, and context determines merely how much of it an object must possess to count as tall. The main disanalogy, so far as I can see, between ‘tall’ and ‘fragile’ is that in the case of ‘tall’, we obviously have an independent grasp—linguistic and otherwise—on the metaphysical base properties: we describe individuals as tall, but we also describe them as having a height of six feet. We also have a very good grasp of the ordering among those determinate properties and its extent, from the infinitesimal extension of point-sized objects (if such there are) on to infinity with ever greater heights. Some heights may be such that individuals possessing them do not in any context count as ‘tall’—the size of a proton may be one such example. Clearly this fact marks no objective difference between heights; rather, it is a result of our practical interests and limited perceptual capacities. That fact is easy to acknowledge precisely because we have an independent grasp on the spectrum of heights. With ‘fragile’, that independent grasp of the metaphysical basis is harder to come by. Our only grasp of it so far is qua metaphysical basis for the contextsensitivity of ‘fragile’. The comparative ‘more fragile than’ goes some of the way towards establishing a spectrum, but we may still wonder what its extent is. Two considerations speak in favour of extending the metaphysical base beyond—or, rather, below—what we would be prepared to call ‘fragile’. The first consideration derives from the core contention of the alternative conception: that dispositions are individuated by their manifestation alone. Since ‘fragile’ and ‘breakable’ both ascribe dispositions, in any context, whose manifestation is breaking, their respective contextual variation must have the same metaphysical background—a disposition to break, and its various degrees. We have seen in chapter 3.2 that ‘breakable’ is less demanding than ‘fragile’—in possible worlds semantics, to be breakable is to break in at least one world, to be fragile is to break in a few. For the realist, this fact is best captured by thinking of both terms as operating on the same metaphysical background, a disposition to break, but requiring different degrees. ‘Breakable’ differs from ‘fragile’ by requiring only that the dispositions be possessed to some non-zero degree; ‘fragile’

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generally requires a degree well above zero. If that is the only difference between them, then the metaphysical background for ‘fragile’—the dispositions of ever decreasing degrees—should go all the way down to what still barely counts as breakable. A second kind of consideration is more general: it appeals to the gradability of ‘fragile’ and will apply to other gradable disposition terms as well. It has been briefly prefigured in chapter 1. Fragility, we have seen, comes in degrees. Different contexts impose different (typically vague) thresholds on the degree of fragility that an object has to possess in order for the positive ‘fragile’ to be truly applied to it. Different contexts do not differ in the application of the comparative. When thinking about the metaphysics of dispositions, we try to break free of the limitations of ordinary everyday contexts; we try, some say, to create a context in which we speak unrestrictedly. Does such a context still impose a threshold on the spectrum of objects as ordered by the comparative ‘more fragile than’? I say it does not. Here are two lines of argument in favour of my claim. First, there is no natural cut-off point on the spectrum produced by the comparative. As we go down on the spectrum from more fragile to less fragile objects, there is, it seems, no natural division insofar as their fragility is concerned. To be sure, the plant-pot does not count as fragile in context C (let us suppose), but it might have—it is not so different from the tumbler which just about still counts as fragile in C. The same goes for the plant-pot and my desk, for my desk and a rock, for the rock, a diamond, and a chunk of gold. Or at any rate, we can imagine a series of objects such that each is only slightly less fragile than the foregoing, beginning with the plant-pot and continuing all the way through my desk to a rock, a diamond, and a chunk of gold. No two objects in the series are so different from each other, in any aspect that is relevant to their fragility, as to provide a non-arbitrary stopping-point between them. The (vague) cut-offs produced by everyday linguistic contexts are, of course, arbitrary in that sense. But that does not matter: they are a matter of linguistics, not metaphysics. The cut-off for which we are now looking is to be a metaphysical division: between what does, and what does not, have a disposition to break in the metaphysically least restricted sense. And such a distinction, I suggest, ought not to be arbitrary.15 Note that there is no 15 I am not saying that every distinction in dispositions must have a non-trivial explanation. If there are fundamental dispositions, the difference between having and lacking them will presumably be primitive. But in the case at hand, we have a grasp of things differing insofar as their fragility is concerned, via the comparative ‘more fragile than’. I am claiming that a gradual difference in that respect cannot make for a categorical difference between the having and lacking of a disposition.

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such pressure on reductionist views of dispositions: if dispositions are metaphysically sufficiently shallow, then there is little point in looking for a metaphysical, as opposed to a linguistic, cut-off. Another line of thought concerning gradability relies on more general observations about comparatives. Where we have an extended spectrum of gradations (such as the full, infinite range of all heights) on the one hand and a phenomenon that imposes an arbitrary cut-off on that spectrum (such as the properties ascribed, in any actual context, by ‘tall’), on the other, the former runs metaphysically deeper than the latter, and the phenomenon with the cut-off has to be understood in terms of the more extended phenomenon, that which provides the whole spectrum. Thus the comparative ‘is taller than’ orders objects by their height, beyond the heights that would qualify their bearer, in any ordinary context, as satisfying the positive ‘tall’. And it is height, not tallness, that runs deeper in metaphysics: we explain the truth-conditions for ‘x is tall’ in terms of heights, not the truth-conditions for ‘x has height’ in terms of tallness. Likewise, we must explain the truth-conditions for ‘x is fragile’ in terms of that which orders the entire spectrum, from the champagne glasses to the chunk of gold (and, perhaps, beyond both). Again, if we were in the business of providing a reductive, possible-worlds-based semantics for disposition ascriptions, there would have to be no pressure for thinking that there is a disposition that is possessed all the way down the spectrum. Just as ‘x is taller than y’ does not entail that either x or y is tall, so ‘x is more fragile/more disposed to break than y’ does not entail that either x or y is fragile, or disposed to break. But ‘x is taller than y’ does entail that both x and y have some height; and likewise, ‘x is more fragile/more disposed to break than y entails that both x and y have whatever property it is whose gradations produce the spectrum from the more to the less fragile. For a reductionist, that latter property need not be dispositional: it may be the property of having counterparts in possible worlds that satisfy some condition or other. But for the dispositionalist, dispositions are explained in terms of dispositions. The background property which produces the ordering from the more to the less fragile had better, then, be itself a dispositional property. You might not wish to call the property which is possessed all the way down the spectrum a disposition; we certainly should not call it ‘fragility’. (‘Breakability’ is a better choice, but we may need to coin words to cover other cases.) To avoid terminological dispute, I propose that we give this property a name which has philosophical tradition behind it but few linguistic intuitions from either ordinary language or contemporary philosophy. I propose that we call those properties which form the metaphysical background for disposition ascriptions potentialities.

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3.5 Maximal dispositions Potentialities in general, and dispositions in particular, come in degrees. In the previous section, we have made some progress towards understanding the spectrum of those degrees: their minimal degree, as we have seen, must be very minimal indeed. But we still lack an adequate general understanding of the degrees of potentiality. In particular, we lack a conception of their maximal degree. Understanding maximality goes hand in hand with understanding comparative potentiality ascriptions: the greater a potentiality’s degree, the closer it must be to maximality. To get our intuitions in focus, let us use the stock examples of dispositions again. Those, after all, provide our best grasp so far of potentiality. What is it to possess a potentiality to the greatest degree possible—to be maximally fragile, irascible, and so on? We can start by taking the possible-worlds semantics of section 3.2 as a heuristic tool. On that model, an object’s degree of fragility increases with the proportion of possible worlds in which it breaks. A maximally fragile object will break in a maximal proportion of possible worlds. What is it to break in a maximal proportion of worlds? It is simply to break in all worlds. The maximal degree of a disposition to be F, on this view, would be being F in all (relevant) possible worlds—being F necessarily. The same holds if we replace, as argued in section 3.3, talk of possible worlds with talk of possible cases. The degree of an object’s fragility increases, then, with the proportion of possible cases in which it breaks; to be maximally disposed to F would be to F in all possible cases, that is, to F necessarily at all times. (The relevant worlds or cases are still those where the object in question has certain of its actual intrinsic properties. The necessity, just like the possibility that characterized dispositions of lesser degrees, is restricted.)16 The realist semantics of disposition ascriptions need not take the demands of possible worlds (or cases) semantics at face value. But it should share certain structural features with the semantics that I have developed so far. So let us try and spell out the realist analogue to the idea that (restrictedly) necessary manifestation is the maximal degree of a disposition.

16 Could anything be maximally fragile according to this conception? You might think not: nothing can be broken in all cases; it must always start out unbroken. However, the answer depends on how exactly we think of the manifestation of fragility. If it is being broken in the state sense, then it looks as though nothing could be maximally fragile. But if the manifestation is breaking or being broken in the process sense, then maximal fragility would require that an object is, at any instance in any possible world, in the process of breaking; but it can be in this process without having completed it. So cases where the object is still unbroken but already beginning to break will count as cases where the disposition is being manifested.

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The realist analogue is this. A disposition is possessed to the maximal degree by an object just in case the object can do nothing other than manifest it; that is, just in case it has no potentiality not to manifest it. Necessity is the dual of possibility: it is necessary that p just in case it is not possible that not p. A maximal potentiality to F, if it is to be analogous to a necessity, should be equivalent to the lack of a potentiality not to F. Let us call this conception the necessity conception. The necessity conception of maximal potentialities has the advantage of accommodating a desideratum with which we were left at the end of chapter 2: the provision of a unified conception of dispositions that comprises the possibility-like ordinary dispositions such as breakability and fragility as well as the necessity-like nomological dispositions such as electric charge. What characterizes such nomological dispositions is not that things can behave in certain, law-conforming ways, but that they must so behave. I suggested at the end of chapter 2 that the modal nature of those dispositions is best understood as analogous to a restricted necessity. On the suggested conception of maximal degrees, nomological dispositions are simply dispositions that have maximal (or perhaps, in some cases, near-maximal) degree.17 I have presented a conception, but I have not argued that it is the correct one. I will do so in the remainder of this section. It is worth noting that even without the heuristic tool of possible-worlds semantics, there are few plausible alternatives. What else might the maximal degree of a potentiality amount to? It may be suggested that to possess a disposition to the maximal degree is simply to exercise it; the closer a disposition is to exercising, the greater its degree. Thus when a glass is struck, it becomes more and more fragile until, finally, it reaches the maximum of fragility and actually breaks. When a sugar cube is put into water, the sugar gradually becomes more soluble until, finally, it is fully dissolved. And when an irascible person gets angry, their irascibility reaches its maximum, but then returns to a more moderate degree once they calm down. Let us call this the actuality conception, because it sees maximality of dispositions in analogy with actuality, rather than necessity. Perhaps this picture holds some attraction. But it would wreak havoc with the semantics of disposition terms such as ‘fragile’ and ‘irascible’. That semantics, as we have seen above, works by imposing a threshold such that everything above that threshold on the scale of comparative fragility or irascibility counts as fragile 17 Some dispositional essentialists such as Bird (2007) will go for the maximal degree, at least concerning fundamental dispositions. Others, such as Cartwright (1989), argue that, to paraphrase the claim in current terminology, the degree of nomological dispositions is less than maximal. For more on these views, and the implications of the present view for dispositionalist accounts of the laws of nature, see chapter 7.8.

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or irascible, full stop. Anything that is maximally fragile or irascible will always be above the threshold, and hence will count as fragile or irascible in every context. On the actuality conception, anything which is actually breaking must therefore count as fragile in any context, and anyone who is actually angry will count as irascible in any context. Thus in no context is it true to say of someone that ‘She is angry, but she is not irascible’. But that seems blatantly incorrect: people do not have to be irascible to get angry every now and then, and they do not become irascible when they get angry. Some situations will prompt anger even in the most mild-tempered person, and it will still be true that, in this situation, a mild-tempered person gets angry. Moreover, a generally mild-tempered person who happens to be in an extremely vexing situation does not thereby become more irascible than a choleric who happens to have no cause for anger at the moment.18 Dispositions and their degrees are more robust over time and their manifestations than the actuality conception would suggest. The actuality conception, then, is to be rejected. This takes care of an initially plausible alternative to my own preferred conception of maximal potentiality, the necessity conception. But is there any positive reason to adopt the necessity conception? I believe that there is, but it requires a detour through some considerations about the relation between dispositions and explanation. Disposition ascriptions provide explanations, and comparative disposition ascriptions provide comparative explanations. Consider Ann and Betty. Betty gets angry much more often than Ann; Ann rarely if ever gets angry. One first step in explaining the differences of behaviour between Betty and Ann is appealing to their respective dispositions: Betty is more irascible than Ann. Given the semantics of ‘irascible’ as sketched earlier, it may be that Betty counts as irascible (in an ordinary context) while Ann does not; Betty, but not Ann, may be above the threshold required to count as irascible (in that context). The dispositional explanation, comparative or absolute, does not take us very far, but it provides a first step from which further explanations can proceed. We may wish to know how Betty came by her greater irascibility, and Ann to her lesser degree of it: is the difference genetic, or a matter of their respective upbringing? Is it related to this or that structure in their brains, or to the level of this or that hormone in their blood? Despite its modest contribution, however, the dispositional explanation is not trivial. It might just be that Betty and Ann are equally irascible, but that Betty is exposed to irritating situations all the time while Ann lives a life of calm and quiet. In that case, if we wish to explain how Betty’s behaviour differs from Ann’s, it is no use pointing to their respective dispositions. It is also no use 18

Thanks to Max Bohnet for suggesting this line of argument.

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asking about Ann’s or Betty’s genetic make-up, her upbringing, her brain, or her hormone levels. We must point to their external circumstances instead. Similarly, some salts, such as the well-known sodium chloride, are often found dissolved in water, while others, such as the lesser-known calcium sulphate, rarely are. In explaining this difference, we start (and so do chemists) by classifying sodium chloride as more water-soluble than calcium sulphate. This may seem to be a trivial step, but again it already encodes the recognition that it is the intrinsic constitution of the two substances which is responsible for their different patterns of behaviour, not the circumstances in which they are placed. From that recognition we can proceed to more informative explanations of the respective behaviour of the two kinds of salt. But the naturalness of the first step should not make us forget its value. In general, disposition ascriptions provide explanations for patterns of behaviour by attributing those patterns to the intrinsic make-up of the individuals or substances concerned; and comparative disposition ascriptions provide explanations for differing patterns of behaviour by attributing the different patterns to differences in intrinsic make-up.19 Dispositional explanations are not the only explanations there are; patterns of behaviour may have their correct explanation in the circumstances external to the individuals or substances concerned. They are, however, a particularly useful kind of explanation. The reason is that they give rise to relatively reliable predictions. The intrinsic make-up of objects is typically more stable than the relations in which the same objects stand to their external circumstances: it is easier to move a glass than to change its molecular structure, and easier to avoid provocations to Betty than to change her temperament. (I say typically, not always: a caterpillar about to turn into a butterfly is likely to experience intrinsic changes that far outrun any probable changes in its relations to its environment.) Knowing no further details about the case, we should generally be more confident that Ann’s and Betty’s degrees of irascibility will remain the same over some time to come than that their external circumstances will remain as they are. We are rightly more confident in projecting a glass’s fragility into future situations than we are in projecting its current position at the edge of a table. With salt’s solubility, this is even more obvious, since the solubility arises directly from the molecular connection that makes salt salt. Thus we can predict with relative confidence that Betty will be angry more often in the future, and that salt will continue to dissolve in water, and glass continue to break; we may adjust our actions accordingly, by treating glass carefully, putting salt in our soup, and treading carefully around Betty even if we would not do so 19

I continue to make the simplifying assumption that all dispositions are intrinsic.

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around Ann. This, I take it, is crucial to our concern, both everyday and scientific, with dispositions: being mostly intrinsic features of the objects and substances we deal with, they are relatively stable and easily projected into the future. Like explanations in general, dispositional explanations provide ceteris paribus justification for belief in the truth of the explanans given our belief in the explanandum. Thus noting that, despite being in non-extraordinary circumstances, Betty gets angry very often, we have good reason to believe that Betty is irascible. And noting that, despite both being in roughly the same external circumstances, Betty gets angry so much more often than Ann, we have good reason to believe that Betty is more irascible than Ann. Like any inference to the best explanation, these justifications provide only defeasible evidence; but if things go well, they point us towards truth rather than falsity. More generally, then, the following seems right: (a) Comparative regularities, such as y’s being F more often than x despite both being in roughly the same circumstances, provide good though defeasible evidence for comparative disposition ascriptions, such as: y is more disposed to F than x is. A second generalization that I will appeal to formulates a simple and, I hope, obvious principle about comparatives: (b) If x is disposed to F and y is more disposed to F than x is, then y is disposed to F. (b) is an instance of the valid principle that if x is G and y is more G than x is, then y is G. Here, then, is the argument. Let us start by thinking about the limiting case of a pattern of behaviour. The limiting case of a colour pattern is monochromaticity. In particular, an area that is blue throughout constitutes a limiting case of a blue-containing pattern. Similarly, the limiting case of a pattern of behaviour—and in particular, a pattern of being F—is uniformity of behaviour: being F all the time. Dispositions, we have seen, explain patterns of behaviour, and their ascription can be justified, albeit defeasibly, by that explanation. Principles (a) and (b) entail that dispositions also explain, and that their ascription can be justified by appeal to their explaining, uniformity of behaviour. Consider a third person besides Ann and Betty, Charlotte. Charlotte is angry throughout her entire existence: she is born angry, she lives angry, and she never ceases to be angry until she dies. (Never mind the unrealistic example.) Just as we have reason to say, by principle (a), that Betty is more irascible than Ann, so we have reason to say that Charlotte

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is more irascible than Betty or Ann. By the uncontentious principle (b) we can, further, conclude that Charlotte is irascible. After all, Charlotte should hardly count as less irascible than Betty! (a) and (b) together push us towards accepting permanently exercised dispositions and according them a degree that is rather high, and higher than that of most non-permanently exercised dispositions. (I say ‘most’ because (a), after all, contains a ceteris paribus clause. Charlotte’s permanent anger might be caused by external circumstances, in which case it will not count, or will count less, towards her being more irascible than Betty.) More realistic examples of permanently exercised dispositions include an animal’s disposition to metabolize, the disposition of fire to spread heat, and the above-mentioned disposition of objects with electric charge to emit forces that stand in a particular correlation to the charges and distances of objects around it. Might we just stop here and take permanent exercise to be the maximal degree of a disposition? Not quite. For permanence does not yet get to the modal character of dispositions: a disposition might be permanently exercised for contingent reasons. Rather, we need to extend the argument for permanently exercised dispositions to a stronger case: the case of a disposition that has to be permanently exercised. That, of course, is the case we are interested in, and for which electric charge is serving as an example: dispositions which are not merely permanently exercised, but which must be permanently exercised (so long as they are possessed to the same degree); in other words, dispositions such that their bearer has no potential whatsoever not to exercise them. These, according to my preferred model of potentialities’ degrees, are possessed to the maximal degree; and the fundamental dispositions, if such there are, will very likely be among them or very close to them. The argument just given for permanently exercised dispositions obviously extends to this stronger case: an object will hardly be less disposed to F in virtue of its having no potential not to F.20 The necessity conception of maximal potentiality—the view that an object is maximally disposed to F, or has the potentiality to F to the maximal degree, iff it lacks the potentiality not to F—is thus well motivated by general considerations about the degrees of dispositions and their role in explanations, and not merely 20 In possible-worlds (or cases) semantics, the necessity was restricted: the object has to permanently exercise its maximal disposition so long as it does not change in the relevant properties, the basis for its disposition. In our non-reductive metaphysics of potentiality, this thought can be mimicked: an object may have a maximal potentiality to F, hence no potentiality not to F, but also a potentiality to acquire a potentiality not to F. In chapter 4.6, we will have more occasion to look at such ‘iterated potentialities’: potentialities for the acquiring of other potentialities. Accordingly, the potentiality view of modality to be formulated in chapter 6 will not identify necessity with maximal potentiality simpliciter, but rather with maximal iterated potentiality.

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by the heuristic analogy with possible-worlds semantics. It clearly outperforms its main rival, the actuality conception sketched above. And it accommodates the nomological dispositions, which would otherwise be an entirely different kind of phenomenon than ordinary dispositions. I conclude that we should accept this conception of maximal potentiality. I have said something about minimal potentialities such as breakability, and I have offered a conception of what it is to possess a potentiality to the maximal degree. What about the degrees between these two extremes? How are we to think of them? This is a question which goes beyond the scope of the present book. In order to spell out a potentiality-based account of possibility, I will need no more than the minimal and the maximal degrees, and occasionally an appeal to our intuitive judgements of comparative potentiality ascriptions (‘x has a greater potentiality to F than y’). However, the present conception of maximal potentiality does have the further benefit that it is open to, though it does not require, formalizing degrees of potentiality with the mathematical means of the probability calculus. On the basic axioms of probability, if the probability of p is 1 (i.e. maximal), then the probability of ¬p must be 0, and vice versa. On the necessity conception of maximal potentiality, if an object x is maximally disposed to F, then it can have no potentiality not to F. Thus if we take 1 to indicate the maximal degree of a potentiality and 0 to indicate its lack, we get a perfect analogy (in this respect, at least) between potentiality and probability: to have a potentiality to F to degree 1 is to have the potentiality not to F to degree 0 (i.e. not at all).21 It is not entirely clear, however, that we should be using the probability calculus to model degrees of potentialities. Problems for this approach run in parallel to worries about the probability calculus as regimenting objective chance. Here is one classical example: Suppose that, as it is spun, a spinning pointer has an equal chance of stopping in any equal-angled sector of the circle it marks out. This makes its chance of stopping within any angle α proportional to α, and its chance of stopping at any one point zero. Yet the pointer must stop somewhere. How then can its zero chance of stopping at a point entail that it does not do so? Mellor 2000a, 22 21

The actuality conception is not strictly incompatible with a probability-like treatment of degrees of potentiality, but the combination is highly undesirable. For it would have the consequence that when an object exercises a potentiality to F, it loses the opposite potentiality not to F. That would make dispositions an even more fleeting and temporary matter than they already are on the actuality conception. But worse, it would collapse potentiality with actuality. For it would then turn out that in not exercising a disposition to F an object loses the disposition altogether, since it is thereby exercising the opposite disposition not to F, hence possessing that disposition to the maximal degree, hence not possessing the potentiality to F. Clearly, that is an unacceptable result.

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In general, when a thing has an equal chance of having any determinate Di of a determinable quantity Q, and Q is continuous, then the chance of the thing’s having any one of the determinates must be 0 even if its chance of having the determinable Q is one. By parallel reasoning, a thing’s potentiality to have any of the determinates of Q must have degree 0 even if its potentiality to have Q has maximal degree. The two problems turn out to be even more closely related if we think of objective chance as dispositional, along the lines of a Popperian propensity theory.22 Whatever the response is for chance, I suspect, will also be the right thing to say about potentiality. The potentiality theorist might, with David Lewis (1973a), deny that in the cases described the chance, and the degree of the potentiality, really is 0, and proclaim it rather to be infinitesimal. Accordingly, she might use a different measure for degrees of chances, as Thomas Hofweber (2014) has argued we should do for chance.23 Or she might, with Hugh Mellor (2000a), deny that anything ever instantiates a fully determinate property or has point-sized width, thus accepting that potentialities for fully determinate properties and for stopping at one point in the circle are 0, but denying that this is in tension with the potentialities that things actually have. I do not wish to adjudicate on these options here. I take it to be an advantage of the necessity conception that it is open to the parallel with chance. But it does not require that parallel, let alone any particular solution to the problems of spinning pointers and their ilk. Nor will the project of this book, a potentiality-based account of possibility, require a particular account of potentialities’ degrees. So we can leave the issue wide open. This completes the positive argument (as well as the tentative outlook) of this section. Before moving on, I will briefly consider three general worries that might be raised against the proposed view of maximal degrees of potentialities. First, it might be objected that dispositionality is incompatible with necessity, so we can hardly think of dispositionality—maximal degree or not—as analogous to necessity. The idea that dispositionality is incompatible with necessity has been formulated explicitly by Mumford and Anjum (2011, chapter 8.2). They argue that (B) holds (where ‘DFa’ abbreviates ‘a is disposed to F’):

22 It is customary to distinguish between long-run propensity theories and single-case propensity theories (see, e.g., Gillies 2000). The former think of chances as propensities, or dispositions, to produce a certain (limiting) frequency when repeatedly subjected to circumstances of a particular kind. Those propensities need not themselves come in degrees that correspond to the value of the chance. The latter think of chances as propensities, or dispositions, to produce a certain outcome whose strength, in any single case, equals the value of the objective probability. When associating potentiality with propensity, it should be clear that I am thinking of single-case propensities. 23 Hofweber also responds to an argument put forward by Tim Williamson (2007a) to show that infinitesimals alone do not solve the problem, and that we still need to assign 0 chance to some actual events. As far as I can see, Hofweber’s positive story about chance can be transferred to potentiality.

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(B) If DFa, then ¬Fa: the manifestation of a disposition could always be prevented. Mumford and Anjum 2011, 177

To support (B), the authors refer back to an earlier passage where, following Schrenk (2010), it was argued that dispositionality fails the antecedent strengthening test. This is because it is distinctive of causal processes in general, and dispositions in particular, that they can always be prevented, masked or finked. If that is the case, to know that something disposes towards F is to know that it does not necessitate F. Mumford and Anjum 2011, 177

As an argument for claim B above, this seems to me a case of an ignoratio elenchi. The earlier argument, which rested on the possibility of prevention, masking and finking, established that being disposed to F does not entail being necessitated to F (or F-ing necessarily). There is no argument, so far as I can see, for the much stronger conclusion that being disposed to F does entail not being necessitated to F (or F-ing only contingently). No such argument is forthcoming in the context where B is postulated, except perhaps the conjecture, implicit in the above quote, that it is something about the very nature of dispositionality which allows for finking and masking. What is in the nature of dispositionality must, of course, apply to all cases of dispositionality, but I fail to see what that nature of dispositionality might be. In fact, Mumford and Anjum themselves had said earlier, [C]ausation [and hence, by their lights, dispositionality] is consistent [my italics, BV.] with there being necessitation in the world. But the claim is that causation [and hence dispositionality] does not itself provide that necessitation. Mumford and Anjum 2011, 64

But, the objector might continue, whatever we say about Mumford and Anjum’s argument, is necessity not incompatible with dispositionality by my own lights—by the lights, that is, of the semantics sketched in sections 3.2–3.4? After all, I did say that disposition ascriptions are akin to possibility statements and not to statements of necessity. I stand by that claim. The dispositional adjectives that have been discussed in sections 3.1–3.4 ascribe potentialities of the slight degrees which, for heuristic purposes, we can think of as possibility-like. In looking at the metaphysical background for such ascriptions, however, it turns out that this background must include more, up to potentialities which we would classify, again for heuristic purposes, as necessity-like. I do not know of any dispositional expressions in ordinary language which are best construed as ascribing

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only potentialities of such high degrees. ‘Has electric charge’, for instance, is not a dispositional expression even if the property that is ascribed with it is dispositional. Still, such dispositions can be ascribed with our ordinary dispositional expressions. While ‘is soluble’, if I am right, does not require any more than a minimal or slight degree of the potentiality to dissolve, many of the substances to which the predicate applies have the disposition to a rather high degree. But to surpass a standard is not to miss it. Dispositions of all degrees have one thing in common that sets them apart from dispositions as understood on the conditional conception: they are individuated solely by their manifestation, and not by a pair of stimulus and manifestation. A final worry, and perhaps the most powerful, is that a disposition ascription simply sounds out of place when we know or believe the putative disposition to be permanently manifested, and even more so when we know or believe the object in question not to have any potentiality to do otherwise. If my account has the aim of capturing our intuitive judgements of dispositions, should it not do justice to that feeling of out-of-place-ness? This worry can be explained away by pragmatics. If an object is permanently F, and a fortiori if it must be permanently F, then there is something much more informative to say than that it has a disposition to be F. Disposition ascriptions are at their most informative when the disposition ascribed is not exercised. When confronted with two pieces of fruit, both an unripe green, the claim that one of them (an apple) is disposed to be red while the other (a lemon) is disposed to be yellow has some informative content. When confronted with a ripe red apple and a ripe yellow lemon, the same assertion is of little use. There is a stronger claim to make: that the apple is red, and the lemon is yellow. By ascribing a disposition, we implicate, though we do not assert, that the disposition may not be exercised. If it is always exercised, the implicature is not false but may be misleading; if it cannot ever fail to be exercised, the implicature is false. In neither case should we conclude that the disposition ascription itself is false. As we consider limiting cases such as the ones here discussed, disposition ascriptions become less interesting. Our interests, however, should hardly count as a guide to realist metaphysics, nor should our conversational maxims.

3.6 Taking stock: from dispositions to potentiality Beginning with ordinary and familiar dispositions such as fragility, irascibility, and solubility, we have in this chapter developed the first steps towards a general account of potentiality. Potentialities must be so conceived as to provide a sufficiently rich and systematic background for the contextual variation of

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dispositional adjectives such as ‘fragile’, ‘irascible’, and ‘soluble’. The potentiality to break stands to fragility much as the property of (having some) height stands to tallness. Like height, potentiality comes in degrees. The maximal degree of a potentiality, as I have argued in the previous section, corresponds to a lack of the opposite potentiality. So if an object possesses the potentiality to F to the maximal degree, that object must lack the potentiality not to F, and vice versa. We can think of the minimal degree (the bare possession) of a potentiality as the object’s being just barely suited to show the manifestation at all. The difference between, say, a champagne glass and a diamond in terms of their potentiality to break is a considerable difference, but it is still a matter of degree. There is no sharp boundary to be drawn between them. The difference between either the glass or the diamond, on the one hand, and a quantity of water on the other, with respect to the same potentiality, is not a matter of degree; it is a categorical difference, the difference between having and lacking a potentiality. I have not yet said anything about the relation between a given potentiality and its various degrees. We can think of that relation, again, on the model of the relation between height and the particular heights: that is, as a relation between a determinable and its determinates. (This is true even if there is no numerical measure by which we can identify the different degrees.) In order to account for metaphysical possibility, the target of this book, we will not often need to appeal to the determinates. Hence, in what follows, when I speak of the potentiality to F, I intend to refer to the (degree-wise) determinable potentiality. I will sometimes speak of that potentiality as being possessed to a certain degree, or more often, of one object as possessing a potentiality to a different degree than another. This is merely a more idiomatic way of expressing that an object has the determinable potentiality by having a given (degree-) determinate, or that one object possesses a different determinate of the same determinable potentiality than another object. (Further, since different determinates of one determinable are mutually exclusive, the fact that potentiality P is possessed to a different degree by a given object than potentiality P shows that P and P are different potentialities.) With the conception of dispositions that I have defended in this chapter, dispositions, and hence potentialities, have been divorced from conditionals, counterfactual or otherwise. A disposition, and hence a potentiality, is individuated by its manifestation, not by a pair of stimulus and manifestation. Given what I have just said about degrees, a potentiality cannot always be individuated by its manifestation alone: fully determinate potentialities may have the same manifestation but differ in degrees. In the next chapter, it will be argued that potentialities with the same manifestation can also be different if one is intrinsic

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and the other extrinsic. The statement that a disposition is individuated only by its manifestation was a simplification.24 But none of the qualifications of that claim brings back anything that looks remotely like a stimulus condition. The notion of a potentiality has been introduced as the metaphysical background to the context-dependent notion of a disposition. It will be further explored and extended in the next two chapters; only then will it be ready to serve in an account of metaphysical possibility. Before we go on, however, it is worth noting that we have divorced dispositions, and potentiality, not only from the counterfactual conditional, but also from two other elements that are often thought to be essential to them: change and causation. Given the conception that I have defended of the maximal degree of a potentiality, objects can have the potentiality to be F while already being F; in fact, they can have that potentiality when they are always F and have no potentiality to do otherwise. Potentialities, therefore, need not be potentialities to change. They include potentialities to remain as one already is. Such potentialities are rarely ascribed, and we have seen good pragmatic reasons why they are not. Pragmatic reasons do not, however, count in metaphysics: potentialities without change may be uninteresting, but they are there. Causation, on the standard conception, is built into dispositionality via the stimulus/manifestation pair: a stimulus is typically that which, given the disposition or together with it, causes the disposition’s manifestation. Now, I have argued that we should abandon the standard conception, and with it the idea that a disposition comes with a pair of stimulus and manifestation. Causation is not built into the disposition qua disposition, though it may be built into its manifestation (more on which in a moment). In rejecting the standard conception, I have also rejected a simple dispositional theory of causation: the idea that ‘A causes B when A is the stimulus of some disposition and B is the corresponding manifestation’ (Bird 2010, 161; the idea is developed further in Bird’s paper). It is obvious that such an account is not available to a conception of dispositions that dismisses stimuli altogether.25 There is a looser link between dispositions and causation even on the conception of dispositions that I have proposed: many, probably most, of the

24

Thanks to Lisa Vogt for pointing this out to me. Divorcing dispositions from causation is not, however, a prerogative of the conception of dispositions that I have proposed. Nolan (forthcoming) argues for ‘noncausal dispositions’ within the standard conception: dispositions, that is, whose stimulus condition is not a cause of their manifestation. Nolan’s examples include chancy dispositions, such as the disposition to decay with a chance of 0.5 upon the passing of a certain amount of time, and backwards dispositions, such as a volcano’s disposition to smoke prior to eruption. 25

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dispositions that we ascribe with ordinary disposition adjectives have causation built into their manifestation. As the OED entry for the dispositional suffix -able states, it is a suffix [f]orming adjectives denoting the capacity for or capability of being subjected to or (in some compounds) performing the action denoted or implied by the first element of the compound.

An adjective of the form ‘F-able’ typically, though not universally, expresses the disposition to be F-ed, or in a plausible paraphrase, to be caused to F. Thus solubility is the disposition to be (dis)solved or to be caused to dissolve, flexibility is the disposition to be bent or to be caused to bend (from the Latin flectere, to bend), to be ignitable is to be capable of being ignited or to be caused to ignite, and so on.26 Fragility, too, may well be the disposition to be broken, rather than the potentiality to break, though I am less than certain in this particular case. (Would objects that are prone to break spontaneously count as fragile? If so, then fragility is the disposition to break; if not, it is the disposition to be broken.) This observation, in fact, goes some way towards explaining the wide spread appeal that the stimulus/manifestation model has enjoyed. For with many of these manifestation processes, we have knowledge of the typical causes—ignition, for instance, is generally caused by proximity to a source of extreme heat. Other processes may reasonably be thought to be so complex as to already include the nature of the cause: to be bent is not merely to assume a ‘bent’ shape but to do so in reaction to someone’s (or something’s) bending. In some cases, the nature of a disposition’s manifestation trivially provides one specification of its manifestation’s causes: thus water-solubility, or the disposition to dissolve in water, will have as its manifestation a process that cannot be caused without the substance’s being immersed in water. But these observations should not lead us to overgeneralize. An overgeneralization would be the claim that all dispositions, and by extension perhaps all potentialities, must have as their manifestations, processes that suitably involve causation. I think that claim would be mistaken. We can begin to address it by noting that there are two ways in which the manifestation process might involve causation: it might be a process which is picked out as one that is caused, even though the cause may not be part of the process itself, such as the process of 26 Lowe (2011) adduces similar considerations, especially regarding the grammatically passive form of a disposition’s manifestation. He denies, however, that the putative stimulus can be understood as a causal factor at all: being immersed in water is a logical consequence of being dissolved in water (p. 25). Whatever we say about the particular case of water-solubility, I doubt that his observation generalizes to the other cases that I discuss in the main text.

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being ignited. Or it might be a process which, in itself, involves causal goings-on, whether or not it is caused by anything else, such as the process of being dissolved in water.27 (In fact, both of these processes are causal in both senses, but I have highlighted different aspects with respect to each of them.) Neither of these features, however, is necessary for a process to be the manifestation of a disposition. A uranium atom’s disposition (and hence, its potentiality) to decay has a manifestation which is neither caused by anything else nor complex enough to involve any causal goings-on when it occurs. An electron’s disposition (and hence, its potentiality) to exert a force which stands in a certain mathematical correlation to surrounding charges and their distance is manifested in what seems to be a simple process, with no inner causal structure—the exertion of an attractive or repulsive force—and which presumably is not, in any comprehensible sense, caused by any other process. Even some ordinary dispositions ascribed with adjectives on ‘-ible’ are not essentially causal in either or both senses. Irascibility is a disposition to get angry. While most instances of anger have some cause and involve causal goings-on in the person’s brain and organism, we can imagine a dualist philosopher who held that people sometimes get angry spontaneously, with no causes whatsoever (the dualist philosopher is also an indeterminist) and that the psychological process of getting angry was first and foremost a process in the soul, as simple as the process of a particle’s exerting a repulsive force (the physical process might then be an epiphenomenon, caused in some way by the soul). I do not find that view particularly appealing, but the point is simply this: such a dualist philosopher would still be entitled to call people ‘irascible’ if they were prone to get angry spontaneously. Irascibility, then, need not involve a causal process as its manifestation in either of the two senses I have distinguished, even if the manifestation typically occurs as a causal process in both of these senses. It is an interesting fact, but not one of metaphysical import, that most of our disposition ascriptions do involve causation, in one way or another, in their manifestation. The reason, again, seems to be pragmatic: we ascribe dispositions mostly because we are interested in how we may interact with and manipulate objects (or in how we should avoid interacting with and manipulating them). Interaction and manipulation involve causation; we are interested in what things are easily caused to do. But pragmatic interests should not determine metaphysics. Dispositionality, and potentiality in general, does not require a causal aspect from its manifestation.

27 A third way for a process to involve causation is that it causes other processes to occur, but that is clearly not a candidate interpretation for the claim that dispositions have causal processes as their manifestation.

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Note that the fact that causation is not built into potentiality does not make potentiality unsuitable for serving in an account of causation. (Indeed, it may make such accounts more attractive because they are not in danger of being circular.) Mumford and Anjum (2011), for instance, hold that [c]ausation usually occurs when powers have accumulated, so that there is enough to trigger a certain effect. There are frequently many powers involved, each disposing in the same direction, in varying degrees. It is also possible, indeed likely, that there are many countervailing powers at work, that dispose away from a certain effect, but that are also relevant to that effect. Whether and when the effect occurs will be just as much a function of those ‘subtractive’ powers, that dispose away from the effect, as it is a function of all the powers that dispose towards it. Mumford and Anjum 2011, 22

In the terminology of this chapter, the basic picture might be rephrased roughly as follows: effects occur when potentialities of sufficient degree are present and are sufficiently higher in degree than any counteracting potentialities.28 Whether an effect occurs is typically not a matter of just one potentiality but of several coming together. While this chapter does not provide the resources to understand the ‘coming together’ of potentialities (some related considerations will be put forward in chapter 4.2), the picture certainly makes sense within the present framework. Importantly, Mumford and Anjum’s account—like the present one—does not appeal to stimulus conditions: their powers only ‘dispose towards’ (or away from) a certain outcome, and they include powers which have (in my terminology) rather low degrees. Another account of causation that is compatible with the present framework has been formulated by Hüttemann (2013, see also Hüttemann 2007 and 2010). According to Hüttemann, causation is a matter of interference with a certain class of dispositions, which in turn underwrite ceteris paribus laws. Causes are simply factors that prevent the cetera from being paria.29 While Hüttemann does not explicitly reject the standard conception, he too is not concerned with the relation between a disposition’s stimuli and manifestation, but rather with that among a disposition, its manifestation, and factors that interfere with its manifestation. 28 Unfortunately, Mumford and Anjum do not say much about how exactly those degrees are to work. It is unclear whether there is a maximal degree of powers in their account, and how the sufficient degree, the threshold for triggering an effect, is determined. They do argue for the incompatibility of dispositions with necessity, as we have seen in section 3.5. However, as I have argued there, they would be better off not making that claim, and I do not think that the claim is in any way central to their overall account. 29 Obviously, the account is more sophisticated than I can indicate in these brief remarks.

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So far as I can see, his account, too, is compatible with the framework that I have developed for the metaphysics of dispositions and potentiality. Dispositional accounts of causation, and presumably also of other phenomena, will not always require the whole range of potentialities for which I have argued in this chapter and will continue to argue in the next. They may want to appeal only to potentialities with a reasonably high degree, or to those with specific kinds of manifestation, or (to allude to a theme of the next chapter) to intrinsic potentialities. My goal in this book requires that we take the whole range into consideration. To yield an account of metaphysical possibility, we need an account of what I have in chapter 1 dubbed ‘metaphysical dispositionality’: the most inclusive kind of property that shares with dispositions their characteristic feature, the fact that they have a manifestation. We have made important progress towards an account of such metaphysical dispositionality, that is: potentiality, in this chapter. The next chapter will progress further towards the same goal.

4 Varieties of Potentiality

4.1 Introduction Let us briefly take stock. My aim is to develop a roughly dispositionalist theory of possibility. In order to do so, I need a suitably general theory of dispositions or—as we can now call it—of potentiality. In the previous two chapters, we have taken some steps towards such a general conception of potentiality. We started with everyday dispositions such as fragility, solubility, and irascibility. Ascriptions of such dispositions, while familiar from our ordinary thought and talk about the world, are vague and contextdependent. In chapter 3 we developed a more general, realist understanding of the properties that underlie the ascriptions of dispositions. These properties, which I call potentialities, come in degrees which can be ordered by comparatives: the champagne glass is more fragile than the tumbler, which is more fragile than the plant-pot, which in turn is more fragile than my desk, which is more fragile than a rock, a diamond, and a chunk of gold. There may be no context in which we would call a chunk of gold ‘fragile’, but it too lies on the continuous spectrum of decreasing fragility. On that spectrum, there are no natural cut-off points; and a realist metaphysics of dispositions should not allow arbitrary cutoffs. Better, then, to say that one property, the potentiality to break, is possessed by all the objects on our spectrum, though to different degrees; and that different contexts set different (possibly vague) thresholds on that spectrum, specifying the minimal degree to which the potentiality must be possessed in order for an object to qualify as ‘fragile’ in the context. We have seen that the degrees of potentiality go all the way from what we might think of, for heuristic purposes, as corresponding to a mere possibility—such as a chunk of gold’s potentiality to break—to the equivalent of necessity: the lack of a potentiality to do otherwise, as perhaps in the case of electric charge. We have also seen, in chapters 2–3, that dispositions are individuated, not by a pair of stimulus and manifestation and a corresponding conditional, but by their manifestation alone. The manifestation

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of a potentiality is that property which the potentiality’s possessor would possess if the potentiality were to be manifested. Potentiality can now be recognized as the common genus of dispositions and such related properties as abilities. It would be odd and probably false to ascribe to a practised piano player a disposition to play the piano (in the philosophers’ sense of ‘disposition’). The piano player has an ability to play the piano. Some people with obsessive-compulsive disorder have a disposition to wash their hands every five minutes. I do not have such a disposition. But like the OCDers, I do possess the potentiality to wash my hands, and the potentiality to wash them every five minutes. It is just that my potentiality has a rather low degree. Like the OCDers, I do have the ability to wash my hands every five minutes; it is just that unlike them, I do not feel the urge to exercise this ability. I share with OCDafflicted people the potentiality to wash every five minutes, and the ability to wash every five minutes. It would be odd to think that these are two entirely distinct properties which we share. Rather, it seems that abilities, like dispositions, are a species of potentiality. They are potentialities that meet some further constraint. Just what that constraint is in the case of agentive abilities, is a very interesting question that goes beyond the scope of this book (though see chapter 6.6 for some speculations). The important point for our purposes is that abilities, too, are a species of potentiality, thus extending the scope of intuitive examples to which we can appeal in understanding potentiality. Further steps need to be taken, however, if we are to arrive at a conception of potentiality that is general enough to do its work in a theory of possibility. In chapter 1, I have listed three central constraints on a theory of possibility. Such a theory must be extensionally correct, that is, it must pronounce the right (pretheoretically known) truth-values for statements of possibility (and necessity). It must be formally adequate, that is, it must entail that possibility has the formal structure which is known from modal logic. One central aspect of this formal structure is closure under logical entailment: if q logically entails p, then the possibility of q must likewise entail the possibility of p. A third constraint is semantic utility: a theory of metaphysical modality should also provide the materials to formulate truth-conditions for at least a significant part of our ordinary modal statements. Our modal language is notoriously context-dependent. Thus I might say of a well-packaged vase that I have packaged it so well because it can break. Or I might say, of the very same vase, that it is so well packaged now that it cannot break. Two contradicting statements about the same state of affairs, and both seem to be true. The standard solution in such cases is appeal to contexts: the two contexts in which the statements were made determined different truth conditions, and so the two statements do not contradict each other after all.

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The conception of potentiality developed in the previous chapter goes some way towards meeting the constraint of extensional correctness. After all, it ought to come out metaphysically possible that my desk, a rock, a diamond, or even a chunk of gold should break. If we were to base our theory of possibility solely on dispositions such as those ascribed, in ordinary contexts, by ‘fragile’, it would turn out difficult to provide for such more remote metaphysical possibilities. Recognizing the neglected slighter degrees of potentiality avoids this problem. Extensional correctness is not fully accounted for in this way, and we have yet to see how to meet the other two constraints. This chapter will introduce some crucial apparatus in doing so. To see where we need to go, consider first the constraint of semantic utility. In the above-cited case of context-dependence, the difference between the two contexts is plausibly construed as a matter of how we think of the vase’s packaging: in saying that the vase can break, we think of the vase on its own, which is often un-packaged; in saying that it cannot break, we take for granted that it is safely packaged. In general, such contextual shifts can be seen as a matter of regarding or disregarding circumstances that are external to the sentence’s subject. I will argue later that in both cases, ‘can’ is used to ascribe potentialities. The difference is that in some contexts, the potentialities ascribed are intrinsic: they concern nothing other than the object which possesses them. In other cases, the potentialities are extrinsic: they depend not only on their bearer’s intrinsic nature, but also on external circumstances. Different contexts determine different degrees of extrinsicality for the potentialities that are ascribed in them, just as different contexts determine different minimal degrees to which the potentiality to break has to be possessed if an object is to count as ‘fragile’. In order to develop this proposal and assess its viability, we need an account of extrinsic potentialities. Our examples of dispositions so far have been intrinsic, and intrinsic potentialities remain the paradigm of potentiality. After all, potentiality has been explicitly characterized as a local modality: unlike the nonlocalized modality of metaphysical possibility, it is a matter of how things stand with a particular object. Part of the aim of this chapter is to show how extrinsic potentialities fit into this conception of potentiality. To see another application, consider again the programmatic definition of possibility as given already in chapter 1: (P ) It is possible that p just in case something has a potentiality for it to be the case that p. The definition was programmatic because it is, prima facie, saddled with a number of problems, which it is the task of this chapter and the next to solve. The

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problem here is that potentiality is not potentiality ‘for it to be the case that . . . ’. Potentialities are potentialities to do or to be something. So the definition of possibility does not even seem to be well-formed. (If, on the other hand, it is wellformed, then it will end up being too narrow: it would account only for possibilities of a simple predicative form. This was the case of (P) as stated in chapter 1.) We could stipulate, of course, that a potentiality for it to be the case that . . . is simply a potentiality to be such that . . . , thus making the sentence grammatical. But in the absence of further explication, such a manoeuvre is no more than grammatical trickery. What we need is an account of how and in what sense an object can have a potentiality to be such that p, or for it to be the case that p, whatever the logical form of p; how, for instance, I might have a potentiality to be such that you are sitting, or to be such that someone is sitting, or to be such that someone is sitting and 2 + 2 = 4. In understanding the above definition of possibility, we just need a way to make sense of such odd potentiality ascriptions; we do not need to understand them in such a way that they are true. But things get more demanding when we look more closely at the constraint of formal adequacy. Possibility, I have said, is closed under logical entailment. How does a potentiality-based account do justice to this fact? Formally, the easiest way would be for potentiality itself to be closed under logical entailment. Given (P ) as a bridge principle, the same is then easily shown to hold of possibility. Taking potentiality itself to be closed under logical entailment is, thus, formally the route of least resistance. Metaphysically, it is quite probably the opposite. For consider the simple fact that a tautology is logically entailed by anything. My writing this book entails your sitting or not sitting. Thus by having a potentiality to write this book, I would, as a matter of mere logic, have a potentiality to be such that you are either sitting or not sitting. On the face of it, such an entailment may seem absurd. Yet I will argue that it holds. The metaphysics behind it will, again, invoke extrinsic potentiality. For the trouble with my potentiality to be such that you are sitting or not sitting is primarily that its manifestation concerns you, and not me; for me, it is a mere Cambridge property. I will argue that such potentialities are indeed possessed by objects, but that they are extrinsic potentialities which are fully grounded in the more intuitive intrinsic properties of the objects involved. Extrinsic potentialities, then, are not among the sparse or natural properties; but that is no reason to deny the truth of sentences ascribing them. Part of my argument will have to wait for the next chapter. This chapter has the important task of laying the metaphysical foundations, in giving a precise and comprehensive account of extrinsic potentiality. The constraint of extensional correctness does not require extrinsic potentialities, but it requires a further extension of potentiality, also to be accomplished

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in this chapter. It is possible that my great-granddaughter be a painter. Whose potentiality provides the grounds for this possibility now? I do not have a greatgranddaughter, and I may never have one. The relevant potentiality seems clearly to be located in me. But what exactly is that potentiality? We might say that it is the potentiality to have a great-granddaughter who is a painter, and be done with it. But to say just that would be unsatisfactory. For how do I manifest the potentiality? I may be long dead by the time it is manifested (if it ever is). Better to say that I have an iterated potentiality: a potentiality for another potentiality, which might itself be for another potentiality, and so forth. Thus I have a potentiality to have a child who has the potentiality to have a child who has the potentiality to have a daughter who has the potentiality to be a painter. In thus iterating potentialities, we have again reached a potentiality for it to be the case that p: my iterated potentiality has as its ultimate manifestation not my possession of some property, but my great-granddaughter’s being a painter. We have yet to understand how exactly this can be. I will begin the chapter with an examination of what I call joint potentialities: potentialities possessed by several objects together. Such potentialities have not gone entirely unnoticed, as the quotations in chapter 4.2 will show. The explanatory value of joint potentialities, however, has been severely underrated. A more systematic understanding of joint potentialities will provide the required account of extrinsic potentiality, as well as a deepened understanding of iterated potentialities. It is thus the heart of the general conception of potentiality that we are seeking.

4.2 Joint potentiality introduced Individual objects have potentialities; so do several objects taken together. I have the ability to see; together, you and I have the ability to see each other. If Hannah has the ability to play the piano and Jane has the ability to play the flute, then Hannah and Jane together have the ability to play a duet for flute and piano. A key has the disposition to open doors with locks of a particular shape S; the key and a particular door, d, which happens to have a lock of shape S, together have a potentiality to stand in the relation of opening.1 Each person in a crowd has some potentiality to panic and run; together, the people in the crowd have a disposition to stampede. 1 The example goes back to early modern authors; in the contemporary debate, see Shoemaker (1980) and McKitrick (2003) for discussion.

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Such joint potentialities have not gone entirely unnoticed in the literature on dispositions. Philosophers of science, and scientists themselves, speak of ‘systems’ that consist of any number of objects. Such systems are often joint possessors of potentialities. Thus the solar system, consisting of our sun and its planets, is disposed to have the earth move in an elliptical orbit. The earth on its own is not disposed to such an orbit.2 David Lewis adopts the terminology of ‘systems’ when illustrating the idea of dispositional partners: For example, I and a certain disc are so disposed that if I and it came together, it would cause in me a sensation of yellow. We could say that it is disposed to influence me; or that I am disposed to respond to it. Or both. Or we could say that the two-part system consisting of me and the disc is disposed to respond to the coming together of its parts. Lewis 1997, 144f.

Lewis, of course, is speaking within the conditional conception of dispositions, which I have rejected in the last two chapters. Without conditionals, we can make the same point though we need to rephrase it. As a perceptually ordinary human being, David Lewis is disposed to receive, from objects with a yellow surface, a sensation of yellow. The disc, having a yellow surface, is disposed to produce a sensation of yellow in ordinary human observers. These matching dispositions make Lewis and the disc ‘dispositional partners’, as C.B. Martin has put it;3 or, as Lewis puts it here, they endow the two-part system with a disposition for one of its parts, the disc, to produce a sensation in its other part, David Lewis. Other examples are of greater practical interest. ‘A nuclear pile’, Alexander Bird writes, which is above critical mass has a disposition to chain-react catastrophically. However, the pile has attached to it a fail-safe mechanism. Heat and radiation sensors detect large increases in radioactivity and allow boron moderating rods to penetrate the pile and by absorbing the radiation to prevent the catastrophic chain-reaction. Bird 1998, 229

And he continues: To see these cases correctly it is important to be precise about what object the disposition in question belongs to. In the nuclear reactor case we can distinguish three combinations: (a) the uranium pile alone (b) the uranium pile plus the boron rods (c) the uranium pile plus the boron rods plus the fail-safe mechanism (i.e. the complete reactor). 2 3

The example is from Hüttemann (2010, 8f.) Martin uses the terminology in various places, including Martin (1994, 1996, and 1998).

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These possess quite different dispositions. The uranium pile, (a), retains the disposition to chain-react all the time. The combination of pile and boron rods, (b), does have a disposition to chain-react when the rods are outside the pile, but loses this disposition when the rods are in the pile. Indeed, in the presence of the fail-safe mechanism (regarded as external to the pile-plus-rods), combination (b) with the rods out has its disposition to chain-react finkishly. Whenever this combination is about to chain-react, the fail-safe mechanism causes it to lose that disposition. The reactor as a whole, (c), i.e. including the fail-safe mechanism, as long as the mechanism is effective has no disposition to explode at all. Bird 1998, 229f.

The ‘combination’ or system of uranium pile and boron rods, that is, the uranium pile and the boron rods together, possess dispositions that differ from those possessed by the uranium pile alone. When the rods are in the pile, the pile alone does, but the pile and rods together do not, have a disposition to chain-react catastrophically. As Bird’s example shows, the dispositions that are possessed jointly by the objects that constitute a system may differ from the dispositions possessed by the constituting objects. However, in the present context we have to be careful to differentiate disposition ascriptions from potentiality ascriptions. In general, ascribing to an object a disposition to F requires that the object possess the potentiality to F to some non-negligible degree, or so I have argued in chapter 3. Thus when we contrast objects with or without a particular disposition, the contrast might not actually be a categorical one, of having or lacking a particular underlying potentiality. In many cases, it will really be one of degrees: a contrast, that is, between things that have a given potentiality to a degree that is sufficient to warrant a disposition ascription, and those which have the same potentiality to a lower degree. Thus the uranium pile and the boron rods together have some potentiality to chain-react. But that potentiality’s degree is too low to count as a disposition. Because the joint potentiality of the pile-plus-rods differs in degree from the individual potentiality of the uranium pile, the former cannot be straightforwardly reducible to the latter. More examples will be offered shortly, but these should suffice for an informal introduction of the idea. In the next section, I am going to call potentialities that are possessed by a number of objects together or in combination joint potentialities, and I will explore some general features of, and distinctions among, such joint potentialities. In particular, I will ask and answer five questions about joint potentialities: first, the question of how to think of their possessors; second, the question of how to think of their manifestations; third, the question of how to think of the joint potentialities themselves; fourth, the question of how they

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relate to the individual potentialities of the objects involved in them; and, fifth and finally, the question of what the conditions are quite generally for objects to possess joint potentialities.

4.3 Joint potentiality: five questions First question: how are we to think of the bearers of joint potentialities? Talk of ‘systems’ or ‘combinations’ is metaphysically non-committal. But what are those systems that possess joint potentialities: are they composite objects, sets of objects, or simply pluralities of objects? These are not obviously equivalent. A plurality of objects does not automatically compose a further, composite object unless a principle of unrestricted mereological composition holds. A plurality of objects does automatically compose a set if it is of the right kind and cardinality; nonetheless, as Richard Cartwright has pointed out, ‘[i]t is one thing for there to be certain objects; it is another for there to be a set, or set-like object, of which those objects are members’ (Cartwright 1994, 8). For our purposes, pluralities will serve better than sets. Sets are abstract entities, intuitively unfit to possess the potentialities here at issue: potentialities to chain-react, to play a duet, to stampede. (Of course, there is a reformulation for each of these potentiality ascriptions: the set may have potentialities to have chain-reacting, duet-playing, or stampeding members. But those potentialities are at best derivative from those possessed by the set’s members.) Sets are implausible bearers of the potentialities here at issue; composite objects are not. If you are a believer in unrestricted mereological composition, then by all means think of the possessors of joint potentialities as composite objects. But it is a controversial question, and one best kept distinct from questions about potentiality, whether any number of objects such as you and me, the people in the crowd, the uranium pile and the boron rods, or David Lewis and the yellow disk each compose a further composite object. The metaphysics of modality should not hinge on that question. So it will be best to steer clear of commitments here and to speak, very simply, of objects, in the plural, as possessing joint potentialities. Second question: how are we to think of the manifestations of joint potentialities? In the examples mentioned above, the manifestation of the joint potentiality was a relation between the possessors (one of them seeing the other, or producing a sensation in the other), a plural property (stampeding, chain-reacting), or a property possessed by only one of them (moving in an elliptical orbit or, again, chain-reacting).

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The first two kinds of manifestation are unproblematic and closely related. When the manifestation of a potentiality consists in a number of objects’ jointly possessing a plural property, or standing in a certain relation, it is only natural that the objects which would together manifest the potentiality also possess the potentiality together. Plural properties differ from relations in not being ordered: where P is a plural property, there is no difference between a and b having P on the one hand, and b and a having P, on the other. When Hannah and Jane play a duet for flute and piano, they possess the plural property of playing a duet and the plural property of playing a duet in which one of them plays the piano and the other plays the flute. But that plural property does not distinguish between which of them plays which instrument. Relations are more discriminating in this respect: if Hannah is on the piano, she stands to Jane in the relation of accompanying-on-the piano; if the instruments are exchanged, then it will be Jane who stands in that relation to Hannah. A second difference between plural properties and relations concerns adicity: the same plural property can be borne by any number of suitable subjects, but the same relation can hold only between a fixed number of things. Thus Hannah and Jane may share with the musicians of the London Symphony Orchestra the plural property of playing a piece of music; it does not matter that there are just two of Hannah and Jane while there are many more of the Symphony Orchestra musicians. Relations, on the other hand, are standardly taken to come with fixed adicity, that is, a fixed number of places: Hannah and Jane cannot stand in the same relation to each other as do, say, Peter, Tom, and Harry: for the former relation would have to be two-place and the latter three-place. When adicity is not at issue, relations can often stand in for plural properties but not vice versa: when Hannah and Jane have the plural property of playing a duet, they will thereby stand in the symmetrical relation plays a duet with. In what follows, for brevity’s sake, I will often speak of relations only. Unless otherwise specified, these are implicitly to include plural properties. The more interesting, and more troublesome, case of joint potentiality is the third: joint potentialities whose manifestation consists in some, but not all, of the potentiality-possessing objects’ having a particular property. Bird’s uranium pile and boron rods can be construed to be of this third variety. The uranium pile and the boron rods together (when appropriately arranged) have a lesser potentiality to have the plural property of chain-reacting catastrophically. But they also have a lesser potentiality for the uranium pile to chain-react catastrophically. Presumably, the degree of the former potentiality derives from that of the latter. Bird’s example is used in a discussion on dispositional antidotes (which I have earlier called masks). One thing’s disposition to F—the uranium pile’s disposition

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to chain-react—is masked because the thing is part of a larger system—the uranium pile plus boron rods and fail-safe mechanism—which does not have a disposition for that thing to F. The schema can be applied to masked (and, in reverse, to mimicked) dispositions quite generally. Take a glass wrapped in styrofoam. The glass has a potentiality to break, to a sufficiently high degree (let us suppose) to count as fragile in an ordinary context. Things are different when we consider a somewhat more complex system than the glass: the glass and the wrapping styrofoam together. The glass and styrofoam together have some potentiality to break, but its degree is clearly lower than that of the glass’s potentiality; hence the glass counts as fragile, but not the glass-cumstyrofoam. However, the glass and styrofoam taken together differ from the glass alone not only in their potentiality to break, which might be manifested by a breaking of the glass, or the styrofoam, or both of them. They differ further in the degree of their potentiality for the glass to break, which is manifested only in a breaking of the glass. Another example, though taken from a different context, is Jennifer McKitrick’s: A military target, a city, is protected by a Star-Wars like defence system. The system has sensors that bring out defences when there is a threat, rendering the city invulnerable. However, the sensors and anti-aircraft weapons are located outside the borders of the city and are built, maintained, and staffed by a foreign country. McKitrick 2003, 161

When considered on its own, the city remains as vulnerable as it was. But its intrinsic vulnerability—the disposition to be harmed by an attack—is masked by the defence system. The city and the defence system together are not disposed for the city to be harmed. Again, they may possess some potentiality to that effect, but it will be too slight to qualify as a disposition. An interesting range of cases where the degree of the joint potentiality is higher than that of the relevant individual’s potentiality concerns catalysts.4 Hydrogen peroxide (H2 O2 ) is disposed to turn into water and oxygen (H2 O and O2 ). Adding manganese to the hydrogen peroxide significantly speeds up that process; we may say that hydrogen peroxide and manganese together have a disposition of a greater degree for the hydrogen peroxide to turn into water and oxygen rapidly than hydrogen peroxide does on its own. As a catalyst, the manganese remains unaffected by the process; the manifestation is one that concerns the hydrogen peroxide alone (or, perhaps, the various atoms that make up the hydrogen peroxide molecules). 4

Thanks to Antony Eagle for suggesting these examples to me.

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Here is one final example. A bull has a certain potentiality (which may well classify as a disposition) to get angry; a fully equipped torero has a certain potentiality (which we might not wish to call a disposition) to get hurt. A bull and a fully equipped torero together have a variety of potentialities: the potentiality to perform a bull fight (a plural property), the potentiality to fight each other (a relation). They also jointly have a potentiality for the bull to get angry, and it is to be assumed that this potentiality has a greater degree than the bull’s own potentiality to get angry; and a potentiality for the torero to get hurt, which may also safely be assumed to have a greater degree than the torero’s individual potentiality to get hurt. In these cases, as in the case of the protected city and the uranium pile, a jointly possessed potentiality is manifested in a property that belongs to only some of the potentiality’s joint possessors. (In fact, in the cases described, the manifestation property belonged to only one of the potentiality’s joint possessors. But we can easily generalize this to any number of objects that does not exhaust the potentiality’s joint possessors.) These kinds of potentiality are clearly of interest, both practical, as in the case of the uranium pile plus rods and fail-safe mechanism, and theoretical, as in the case of masked dispositions. How exactly are we to understand the manifestations of these potentialities? Potentialities are manifested by their bearers; their manifestation is the property that their bearers possess, or the relation in which their bearers stand, when exercising the potentiality. Which property do the glass and styrofoam, the bull and the torero, or the city and the defence system possess when manifesting the joint potentialities just described? Given what we have said earlier, it must be a plural property or a relation. The plural property which the glass and styrofoam possess when the glass breaks has no very natural expression in ordinary language. We may describe it as the property of being such that the glass breaks. Similarly, the relation in which glass and styrofoam stand when the glass breaks is the relation in which two things stand just in case one of them is identical to the glass and breaks; again, the relation may be described as being such that the glass breaks.5 Again, it does not matter which way we go. For the sake of uniformity, I will treat the manifestations again as relations. As far as I see, nothing hinges on that decision. The glass and styrofoam’s joint potentiality, then, is a potentiality to be such that the glass breaks. The bull and torero have a potentiality to be such that the torero gets hurt, the hydrogen peroxide and the manganese together have 5 A more precise formulation will have to wait until the next chapter. In the meantime, those familiar with lambda notation will recognize that the plural property is the property ascribed with λxx.(g breaks), g standing for the glass, while the relation is ascribed with λx.λy.(g breaks).

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a potentiality to be such that the hydrogen peroxide turns into water and oxygen, the city and defence system have a slight potentiality to be such that the city is harmed, and the uranium pile plus borons and fail-safe mechanism have a very slight joint potentiality to be such that the uranium pile chain-reacts. ‘Such that’ constructions are often used to formulate so-called mere Cambridge properties, properties that have no basis in the intrinsic nature of the objects that have them.6 Thus being 50 miles from a burning barn is a mere Cambridge property, as is being such that Angela Merkel is chancellor of Germany. But the ‘such that’ properties that I am invoking here are not mere Cambridge properties. (I will get to the mere Cambridge properties later, but for now we can put them aside.) If instantiated, they do have a basis in the intrinsic nature of the objects concerned (the glass and styrofoam, the bull and the torero, and so on), even if not in the intrinsic nature of all the objects concerned. It may be helpful to consider an analogy, though it is—given my answer to the first question—no more than an analogy. Complex objects have properties that belong, strictly speaking, only to some of their parts. I have the property of having a beating heart, and the closely related property of being such that this particular heart, which is mine, is beating. The two properties differ: I would retain the first, but not the second, if I underwent heart surgery and had a new heart implanted. It is the second property that is useful for our analogy. It, too, is difficult to ascribe without a ‘such that’ locution. Yet given that this heart is mine, being such that this heart is beating is hardly a Cambridge property of me. That I have the property in question is just a matter of my heart having a particular property, namely, beating. My having the property that my heart is beating is reducible, it would seem, to my heart’s having the property of beating. But my potentiality to be such that this heart is beating is not straightforwardly reducible to my heart’s potentiality to beat. In fact, my heart on its own hardly has that potentiality to a significant degree. I, including the whole of my body with its blood vessels, neural pathways, etc., have a much greater potentiality to be such that my heart is beating: I have what it takes to keep it going. Given the answer to my first question, the members of a plurality are not simply to be identified with the parts of a whole, but my claims here are entirely analogous. As with my being such that this heart is beating, the plurality of glass and styrofoam’s being such that the glass breaks is no Cambridge property, but 6 See Shoemaker (1980), who introduces the term and contrasts the properties so characterized with ‘genuine’ properties.

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is reducible to the glass’s breaking. As with my potentiality to be such that my heart is beating, the plurality of glass and styrofoam’s potentiality to be such that the glass breaks is not simply reducible to the glass’s potentiality to break; it has a different degree. (In this case, the degree is lower; in that of the bull and torero, it is higher. In either case, it is different.) Both potentialities, it would seem, must be accepted as a genuine addition to the individual potentialities of my heart or the glass; and likewise in the other cases. This classification of manifestations provides a derived classification of the joint potentialities themselves, which it will be worth labelling for reference in what follows: Type 1 joint potentialities are joint potentialities whose manifestation is a relation between, or a plural property of, all its possessors. Type 2 joint potentialities are joint potentialities whose manifestation consists in a property or relation of only some, but not all, of its possessors. Third question: what is the ontological status of joint potentialities? Since they are possessed by pluralities of objects, joint potentialities cannot be individual one-place properties. They must be either relations between the objects that possess them, or plural properties of those objects. As we have seen above, relations are more fine-grained than plural properties. The key and the door’s joint potentiality to stand in the (directed) relation of opening differs from the door and the key’s joint potentiality to stand in that same relation. The former potentiality is manifested by the key opening the door, the latter would be manifested by the door opening the key. Where the manifestation is a directed relation, then, the joint potentiality must be such a relation too. Some joint potentialities, as we have seen, have plural properties as their manifestation: the plural property of stampeding, or of playing a duet, for instance. In those cases it will be best to think of the joint potentialities themselves as plural properties, to make them, like their manifestations, insensitive to the number of their possessors. For brevity’s sake, again, I will often speak of joint potentialities only as relations, but unless indicated otherwise this is meant implicitly to include plural properties. Fourth question: how do joint potentialities relate to the potentialities that their possessors have individually? In many cases, it is intuitively plausible that the individual potentialities contribute, in some way, to the joint potentialities. Thus the key and the door jointly possess a potentiality to stand in the opening relation because the key has shape

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S1 and a power to open doors with a lock that has shape S2 , and the door has a lock of shape S2 and a power to be opened by keys of shape S1 . The city and its defence system have a lesser potentiality for the city to be harmed by an attack because the defence system has potentialities which, intuitively, ‘detract’ from the city’s own potentiality to be harmed by an attack. The uranium pile and boron rods have a lesser potentiality to chain-react because the boron rods contribute their disposition to absorb radiation; and so on. In all of these cases, it appears that the joint potentialities are grounded in the individual potentialities of their possessors. This is not to say, however, that there must be a straightforward reduction of joint potentialities to the individual potentialities of the participating objects. The key’s having shape S1 and a potentiality to open doors with a lock of shape S2 , together with the door’s having a lock of shape S2 and a potentiality to be opened by keys of shape S1 , are neither sufficient nor necessary for the door and the key’s possession of the joint potentiality to stand in the opening relation. They are not necessary because that joint potentiality is multiply realizable: other shapes, of key and lock, would do just as well, as long as they fit together. They are not sufficient either because other potentialities of door or key might interfere. For instance, while having the potential to open doors of shape S2 , the key might have no potentiality at all to open things of material M (perhaps it would explode as soon as it touches anything made of M); and the door’s lock, in addition to being of shape S2 , might be made of material M.7 Generally speaking, when things have a joint potentiality, they will have some individual potentialities (and perhaps other properties) that ‘fit’ together, such as the door’s and the key’s. But that ‘fitting’ relation, whatever exactly it is, does not preclude the objects having other potentialities which would interfere with the joint potentiality’s manifestation. To guarantee that things have a particular joint potentiality, all of their individual intrinsic potentialities (as well as their other properties, if there are any) have to ‘fit’; but it is difficult to spell out in sufficiently general and non-metaphorical terms what that ‘fitting’ might amount to. Still, it may seem plausible that individual and joint potentialities relate to each other much as the physical and the mental do on some non-reductive versions of physicalism: while there is perhaps no algorithm to compute the joint potentialities of a number of objects from their individual potentialities, the joint potentialities of any number of objects are grounded in their individual potentialities taken together. Even that modest priority of individual potentialities may be challenged, however, if there are primitive joint potentialities: emergent plural properties 7

Lewis (1997) gives a similar variation on the disc example.

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or quantum-entangled states, which are not, and not even defeasibly, determined by the participating objects’ individual potentialities. I take it that these would be the exception rather than the rule, but we should not exclude them by fiat. What, then, is the metaphysical primitive of my theory: individual potentiality, or joint potentiality? Luckily we need not decide on that. (I take it to be an open empirical question, not one to be decided by philosophical argument in this book, whether there are primitive joint potentialities.) For we can treat individual potentialities as a limiting case of joint potentialities: since a single object is a limiting case of objects, the potentialities of a single object are a limiting case of potentialities possessed by objects. We can therefore say that the (joint) potentialities of objects are the metaphysical primitive, and leave it open whether those objects are ever more than one at a time. Note that whether we have one object or any number of objects, joint potentialities will be intrinsic to their bearers taken collectively; they are a matter of how things stand with the key and the door, or the uranium pile and the boron rods, of the people in the crowd, and so forth.8 They are not a matter of how things stand outside the participating objects. The task of this chapter will be to derive other features of potentiality from this primitive: the intrinsic joint potentialities of things, of which the intrinsic individual potentialities of a particular thing are a special case. For the sake of clarity and natural expression, however, I will continue to reserve the term ‘joint potentiality’ for those joint potentialities that are possessed by more than one object. Fifth question: what are the conditions for objects to possess a joint potentiality? We have just seen that it is difficult at least to give sufficient conditions for the possession, by some objects, of a particular joint potentiality in terms of the objects’ individual potentialities. In chapter 5.7.1, I am going to suggest a very simple sufficient condition for the possession of a certain class of joint potentialities. In particular, I will suggest that any objects whatsoever (on the trivial sufficient condition of their existence) have joint potentialities to have tautological properties and relations. But this will require a great deal of further set-up. For now, I will address rather the question whether there are any non-trivial necessary 8 Intrinsic relations need not be internal relations. Internal relations depend only on the intrinsic properties of their relata; intrinsic relations depend only on how things stand with and between their relata. As Langton and Lewis put it, a ‘relation of match in intrinsic respects, for example congruence of shape, is an internal relation. A spatio-temporal distance relation is an intrinsic relation (unless nature holds surprises), but not an internal relation. The relation of aunt to niece is not an intrinsic relation at all’ (Langton and Lewis 1998, 343).

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conditions for some objects to possess any joint potentiality whatsoever. This requires a more extensive discussion than the first four questions, and will take up the remainder of this section. We have seen some, I hope, intuitive examples of joint potentialities so far. But how are we to generalize from them? Might there be such things as a joint potentiality, possessed by the tip of my nose and the Eiffel tower, to be at a distance of precisely 50 miles from each other? Or worse still, a joint potentiality possessed by these two things to be such that the tip of my nose is tanned? Or might the glass on my desk have a joint potentiality together with some trees in the Himalayas to be such that the glass breaks? These seem rather odd. Yet if no restriction is imposed on the objects with which the glass or the tip of my nose may jointly possess potentialities, then we might have to answer each of these questions in the affirmative. And in fact, I will argue that we do. But this answer is not obvious. I will argue for it by rejecting five natural candidates for non-trivial necessary conditions on things having joint potentialities with each other and suggesting how my argument can be generalized if there are any other candidates. Our question, and the candidate answers to it, have some (limited but useful) analogy with the well-known debate about composition. While we are asking about the necessary conditions for any objects to jointly have some potentiality, the so-called Special Composition Question (van Inwagen 1990) asks what the necessary and sufficient conditions are for any objects to jointly compose a further object. The analogy between the two questions is that both ask for the conditions under which a number of objects are sufficiently intimately related to enter into a particular relation (composition, or the possession of a joint potentiality). The disanalogy is that the Special Composition Question explicitly asks for necessary and sufficient conditions, while we are at present restricting ourselves to necessary conditions. Nevertheless, the analogy with the better-known and well-mapped question about composition will be helpful in exploring our current territory. In response to the Special Composition Question, some, such as Lewis (1986a) and Sider (2003), have adopted unrestricted composition; the answer, that is, that any objects compose a further object, the necessary and sufficient condition being just that they exist. Others, such as van Inwagen (1990), have proposed nontrivial necessary and sufficient conditions for composition to occur; famously, van Inwagen’s condition is that the objects’ activities constitute a life (or else, that there is only one of them). Note that proponents of unrestricted composition can agree about any of these conditions being sufficient for composition; the debate between these two groups is rather whether a given set of proposed conditions is

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also necessary for composition to occur. Finally, there is the option of nihilism: the view that no objects ever compose a further object. For the nihilist, a necessary and sufficient condition for objects to compose a further object is the trivially unfilled contradiction, say, that the former objects are not identical to themselves.9 Analogously, in response to our question about necessary conditions we might hold the analogue of unrestricted composition—that any objects might in principle jointly possess a potentiality, on no more than the trivial condition that they exist. Alternatively, van Inwagen’s general strategy might be followed and specific necessary conditions proposed to restrict the kinds of things that may together have a joint potentiality. Let us call proponents of the first view universalists, and those of the second restrictionists. The debate between universalists and restrictionists is, by stipulation, about necessary conditions, not about sufficient conditions. I bracket the analogue to nihilism, because we are already working on the hypothesis that some things have joint potentialities. Now, I wish to defend universalism: the claim that any objects can together have joint potentialities. I will go through various candidate restrictionist views in what follows and argue that they are implausible. My arguments carry no commitment to a corresponding answer about composition. I am using the case of composition merely as a useful foil, since it comes with a range of well-defined answers and arguments. A first restriction is immediately suggested by my comparison with the Special Composition Question: perhaps the answers to the two questions are simply identical. This would give us: (R1) Objects jointly possess a potentiality only if they compose a (further) object. But (R1) is easily dispelled. For the supposed restriction is either no restriction at all, or it is too narrow. It is no restriction at all if we adopt unrestricted 9 Markosian (1998) has suggested instead that while composition is restricted (i.e. unrestricted composition is false), it is simply a brute fact that it occurs in some cases and not others, so there is no true, non-trivial, and finite answer to the special composition question. The analogous claim about joint potentialities would be desperate and hardly plausible. Markosian argues (Markosian 2007) that composition may be a candidate for brutality because it meets three criteria: it resists reduction, there are clear instances of its obtaining as well as of its not obtaining, and it is pretheoretically well understood. While joint potentialities seem to meet the first criterion, it may be too early to be too confident. The second and third criterion are related and are, I believe, not met by joint potentialities. ‘Joint potentiality’ is a term of art that I have introduced. While there are intuitive examples that can be used to introduce the concept, there is no pretheoretically understood term or concept of joint potentiality that is mundane enough to be appealed to for intuitive judgements in difficult cases.

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composition; it is too narrow if we adopt any other answer to the Special Composition Question. We do not need to go through particular answers to see why this is so. Any non-trivial restriction on composition will try to respect our intuitive judgements regarding composition; if anything, it might be more restrictive than intuitive judgements (as witnessed by van Inwagen’s claim that there are only simples and living beings, but no tables or chairs). Whatever exactly the restriction is, therefore, it will not classify a glass and the styrofoam wrapped around it, or McKitrick’s city and its defence system, or a bull and a torero, as composing one object. But since these were among our paradigm examples for jointly possessed potentialities, this means that any intuitively compelling restriction on composition will be too narrow for our purposes. A second way of restricting joint potentialities arises from the observation that, in our examples, the objects that jointly possessed potentialities were in relative spatial proximity to each other. Wrap the glass in a piece of styrofoam, and it is natural to say that they have a joint potentiality for the glass to break; but place the glass in my kitchen, and the styrofoam on the moon, and it becomes much less natural to ascribe to them any joint potentialities. So perhaps the answer to my question is: (R2) Objects jointly possess a potentiality only if they are in each other’s vicinity. My objection to (R2) relies on considerations of non-arbitrariness akin to those sketched in chapter 1 and applied in chapter 3. Intuitively, on the proposed criterion, things that are 1 mm apart should qualify as possessors of joint potentialities, while things that are 500,000,000 km apart should probably not. But where between these two values is the division to be made between things that may, and those that may not, jointly possess a potentiality? Note that we are not looking for a division that applies in this or that context, but may be shifted with a change in context; for contextual shifts are a matter of semantics, but we are concerned with metaphysics. We are looking for a division between what might in principle, and what may never, jointly possess potentialities. Nor should that division be vague, for vagueness, like contextual shifts, is a matter of semantics, not reality. But given the continuous nature of spatial distances, any sharp cutoff would be arbitrary. (In fact, a vague cut-off would presumably be arbitrary too, though perhaps less obviously so.) But such arbitrariness, I have argued earlier, is to be shunned. This is not to say that every difference in potentiality must have an explanation in terms of something other than the potentiality concerned; the having or lacking of a fundamental potentiality may well be just a brute fact. But ex hypothesi, given (R2), we are now considering differences that are partially

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explicable, namely, by degrees of spatial proximity. Potentiality, so our guiding assumption has it, cuts nature at its joints. But nature has no joints between 1 mm and 500,000,000 km; nature’s joints are not a matter of degree. Hence potentiality cannot be restricted by the difference between 1 mm and 500,000,000 km, or by spatial proximity in general. Nor would it be of any help to adjust the crude measure I have given of spatial proximity and say, for instance, that the relevant proximity is to be measured not in absolute numbers but relative to the objects concerned: 1 mm is an enormous distance between quarks; 500,000,000 km is a small distance between galaxies. For any such adjusted measure would still have to locate a cut-off point: does the candidacy for possessing a joint potentiality stop at being at a distance equal to, or twice as large, or 100 times as large, as the diameter of the objects concerned? This and similar questions can be asked whatever the measure is and they will always lead to the same problems. Considerations such as these can be generalized beyond (R2); they apply to every restriction that admits of degrees without a natural cut-off point.10 A third restriction is perhaps more intuitive. It says that (R3) Objects jointly possess a potentiality only if they can interact with one another. In the present framework, (R3) amounts to the requirement that if objects have any joint potentialities, they must have some joint potentiality to interact with one another. A strong reading of the requirement would relativize it to the potentiality’s manifestation: it is that all joint potentialities must be such that their manifestation is an interaction between its possessors. That strong reading is clearly false: the city and its defence system, in manifesting their joint potentiality for the city to be protected, need not interact. The city may remain completely unaffected while the defence system is doing its protective work. A weaker reading should thus be preferable: in order for certain objects to have a joint potentiality to be F, it is not necessary that being F would constitute an interaction between the objects; what is necessary is that the same objects have at least 10 The argument, it will be noted, is very similar in spirit to Sider’s (2003) ‘Argument from Vagueness’. However, Sider’s argument concerns putative necessary and sufficient conditions for composition to occur. If those are vague, then the boundary between things that do, and things that do not, compose a further object, must be vague; and since composing a further object results in the existence of that object, it would be vague what exists and what does not. My argument cannot work in quite the same way: vague necessary conditions do not have to result in vague boundaries between what does, and what does not, have joint potentialities. My argument, therefore, is not an argument from vagueness. It is, rather, an argument from non-arbitrariness. Thanks to Thomas Sattig for helping me see this more clearly.

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a potentiality to be G, and being G would constitute an interaction between them. But again, McKitrick’s city serves as a counterexample: while it is plausible that the city and the defence system will have some joint potentiality to interact with one another, this need not in any way be part of the story’s set-up. It is not because the city and the defence system have a joint potentiality to interact in some way that they have joint potentialities for the city to be harmed, and for the city to remain unharmed. There is no prima facie ground for saying that the latter potentialities are based on some potentiality for interaction between city and defence system, or that they would be lost if there were no such potentiality for interaction. A fourth, and last, candidate restriction arises from the way in which I have introduced joint potentialities. Like the strong version of the interaction-based view, it would restrict joint potentialities relative to their manifestations. When I introduced joint potentialities above, I repeatedly appealed to their having a different degree from the individual potentialities that had closely related manifestations. This was particularly important in the introduction of Type 2 joint potentialities, such as the glass-cum-styrofoam’s joint potentiality to be such that the glass breaks, whose chief interest lies in the fact that its degree is much lower than the glass’s individual potentiality to break. If no such difference of degree is found, then why accept a joint potentiality in addition to the individual ones that are already given? The intuitive idea behind this line of reasoning may be that in order for certain objects to have a joint potentiality over and above their individual potentialities, that joint potentiality must ‘make a difference’ to the overall distribution of potentiality.11 So we get: (R4) Objects jointly possess a potentiality to F only if that potentiality differs in degree from at least one of their individual potentialities to F. In the case of Type 1 joint potentialities, the proposed restriction would be trivially met. After all, the manifestations of these joint potentialities are always relations or plural properties which involve all of their possessors; and a single object has no potentialities to have plural properties or relations. Otherwise we might truly ascribe to me a potentiality to play a duet together, or a potentiality to be married to (not to be confused with the perfectly intelligible ascriptions of the potentiality to play a duet with someone, or to be married to someone). Such ascriptions, so far from being true, seem to be unintelligible. Hence the joint potentialities for the possession of such plural properties or relations are trivially possessed to a different degree than any individual potentialities for the same 11

Thanks to Catharine Diehl and Romy Jaster for suggesting this restriction.

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manifestation, since the latter are not possessed at all.12 The interesting application, thus, is to Type 2 joint potentialities. Applied to Type 2 joint potentialities, (R4) becomes: objects have a joint potentiality to be such that some of them, say a, is so-and-so only if that joint potentiality’s degree differs from that of a’s own individual potentiality to be so-and-so. The requirement was met by the cases with which I introduced Type 2 joint potentialities. The burden is thus on me to argue that there may be such joint potentialities even when the requirement is not met. To see why the requirement will not work, consider a variation on McKitrick’s protected city. Let the defence system be a badly designed one which, while protecting the city from attacks in a number of ways, also makes it more vulnerable to attack in others. (Perhaps it is prone to misfire and attack the city itself; since the city relies on it for defence, it would be utterly defenceless against an attack from its own defence system.) The two aspects of the defence system may cancel each other out so perfectly that the city’s own potentiality to be harmed by an attack and the city-plus-defence-system’s joint potentiality to be such that the city is harmed by an attack are exactly alike in degree. Nevertheless, a different story must be told about the joint potentiality than about the city’s individual potentiality; it seems unreasonable to deny the former just because it happens to be alike to the latter in its degree. I see no better way non-trivially to restrict joint potentialities than (R1)–(R4), and I predict that any other answers that might be put forward will be subject to similar criticisms: that is, they will either come in degrees and thus violate the principle of non-arbitrariness, or they will be too narrow to account for the paradigmatic examples (or both). Indeed, given that we already accept that some objects have joint potentialities of both Type 1 and Type 2, and given what has been said about the degrees of potentialities in chapter 3, the lack of a joint potentiality should be reserved for cases where the objects in question are principally prevented from exercising said potentiality. Thus my nose and the Eiffel tower may have the joint potentiality to be such that my nose is tanned—an uninteresting potentiality no doubt, but the 12 An exception might be formulated if we take plural properties to apply to individual objects, as a limiting case. Thus I might have a potentiality to dance a sirtaki, a dance that can be performed alone or in groups of arbitrary size: and you and I together might also have a potentiality to dance a sirtaki. If the same property of dancing a sirtaki is involved in both cases, the proposed restriction might apply. But in this case it seems just wrong. It might be that we are both equally inclined and equally able to dance a sirtaki, and that our joint potentiality to dance a sirtaki thus has the same degree as do our individual potentialities to dance a sirtaki. But the joint potentiality to dance a sirtaki would be manifested by our both dancing a sirtaki, while my individual potentiality would be manifested also if I danced a sirtaki alone. Clearly, we have a joint potentiality in addition to our individual potentialities to dance a sirtaki.

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Eiffel tower is not, after all, in the way of my nose’s getting tanned. They do not, on the other hand, have a joint potentiality to be such that my nose does not exist, for a joint potentiality must be exercised by its possessors together, and there is no way for my nose to exercise such a potentiality together with anything at all. The answer to my fifth question, then, is this. Any objects might in principle jointly possess some potentialities, on the sole and trivial condition that they exist.13 Perhaps that answer is not so counter-intuitive after all. Take the styrofoam-wrapped glass, and unpack it. Can the glass and the styrofoam nonetheless still have potentialities together? To begin with, they have a potentiality to be in close spatial proximity (the glass can be wrapped again in the future). They also have a potentiality to be in less close proximity; to be, say, exactly 500 m apart. Their joint potentiality to be such that the glass breaks has a higher degree than it had when they were in close contiguity; the styrofoam’s potentiality to absorb the relevant forces is now no longer detracting from the glass’s potentiality to break, or at least not as much as it used to. Is the glass and styrofoam’s joint potentiality for the glass to break now equal to the glass’s own potentiality to break? Or does the glass and styrofoam’s joint potentiality for the glass to be packed in the styrofoam still detract somewhat from that potentiality? These are tricky questions, which I will not be able to answer here. All I want to show for the moment is that these questions make sense; and for them to make sense, it must be possible to ascribe the potentialities in question to the glass and styrofoam jointly even when they are at opposite ends of the universe. Once we have recognized this, examples of potentialities that are possessed jointly by objects with no current spatial contiguity or causal interaction abound: the objects on my desk jointly have a potentiality to be in disarray; the Eiffel tower and I have a potentiality to stand in the seeing relation; all the trees on earth jointly have a potentiality to blossom simultaneously; humankind, that is, all human beings, jointly have the potentiality to make Earth uninhabitable. There is no reason to stop short of the totality of all objects whatsoever: all the objects that exist jointly have, at least, a potentiality to co-exist with one another.

4.4 Extrinsic potentiality introduced Extrinsic properties are properties that depend on features of the world outside the object that has them. A property P is extrinsic if it would be lost or gained (or, 13 That condition may seem less trivial if we are concerned with debates about what exists. Do past and future objects exist? If so, do they have joint potentialities with present objects and with each other? If they do exist, then my answer implies that they might in principle have such joint potentialities. Discussing this in more detail would lead too far in this chapter; but see chapter 7.9 for a more detailed discussions of this question and related difficulties for my theory of possibility.

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when the property comes in degrees, increased or decreased) through a change in circumstances that is entirely external to the object that has the property. In Langton and Lewis’s familiar definition, intrinsic properties are shared between duplicates, where duplicates are those objects that share all their basic intrinsic properties (Langton and Lewis 1998); a property is extrinsic iff it is not intrinsic. The definition is not fully reductive, since it begins with basic intrinsic properties and merely defines the non-basic ones in terms of them. Still, the three different characterizations—in terms of dependence, of counterfactual changes, and of duplicates—should give us enough of a grip on the notion to be used as heuristic tools. A thing’s shape and size are presumably intrinsic properties;14 being 2 m from a palm tree, or being an uncle, is an extrinsic property. Relations give rise to extrinsic properties. If you and I are 2 m apart, then I have the extrinsic property of being 2 m from you, and I have that property because you and I are standing in the being-2 m-apart relation. My extrinsic property of being 2 m from you depends on, or is explained by, our standing in the relation of being 2 m apart. That is why the property is extrinsic: it depends on factors external to its possessor. If you were to move away further, or to cease to exist altogether, I would lose the property of being 2 m from you. Intrinsic duplicates of me can differ with respect to the property of being 2 m from you, precisely because they may not be standing in the right relation to you. Not all extrinsic properties arise from relations in this way. The extrinsic property of being lonely, that is, co-existing with no other object, is a case in point. But any property which does arise from a relation to a disjoint object or objects is an extrinsic property. (The same goes for plural properties: if I am one of the people in a crowd which is instantiating the plural property of stampeding, then I acquire the extrinsic property of stampeding-in-a-crowd.) So far, so uncontroversial. Now on to the application. Having a joint potentiality is just another relation (or plural property) in which objects can stand. My claim in this section and the next is that they give rise to extrinsic properties which are themselves potentialities. This section aims to substantiate, by looking closely at examples, the idea that there are extrinsic potentialities, and that they are extrinsic precisely because they depend on their possessors’ having certain joint potentialities together with other objects. In the following section (4.5) I will attempt to systematize the example-driven considerations of this section. Here is a simple case. If a particular key and door together have a joint potentiality for the key to open the door, then the key thereby has an extrinsic 14

But see Skow (2007) for doubts about the intrinsicality of shape.

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potentiality to open the door. The potentiality is extrinsic because it depends on the key’s standing in a particular relation to the door, and the holding of that relation in turn depends not just on intrinsic features of the key itself, but also on the door’s existence and some of its intrinsic features (such as the shape of its lock). Accordingly, the key’s potentiality to open that door would be lost if the door ceased to exist; it would be lost or considerably decreased in degree if the door had its lock changed; and it is not shared, or not shared to the same degree, by the key’s intrinsic duplicates in possible worlds where the door is lacking or has a different lock. Just as the extrinsic property of being 2 m from you depends on its possessor’s standing in a certain relation to you, so the extrinsic potentiality to open a particular door depends on its possessor’s standing in a certain relation to that door—the relation, that is, of having a joint potentiality for one to open the other. Note, first, that we must distinguish between the key’s potentiality to open doors of a particular type, namely with a lock of shape S1 , and the key’s potentiality to open this particular door, d. The former potentiality is unaffected by d’s coming into and going out of existence, having its lock changed, and so forth. It is independent even of there being any door with a lock of shape S1 . The former potentiality is intrinsic, while the latter is extrinsic. Note, second, that the situation is symmetric between the key and the door: the door has an intrinsic potentiality to be opened by keys of a particular shape S2 , and it has an extrinsic potentiality to be opened by this particular key. The latter potentiality, too, is extrinsic because it depends on the holding of the same relation on which the key’s extrinsic potentiality depended: the joint potentiality of door and key, which in turn depends (in part) on the existence and intrinsic features of the key. In the contemporary debate about dispositions, it was long assumed unanimously that dispositions must be intrinsic.15 However, that assumption has been challenged by Jennifer McKitrick, who forcefully argued for the existence of extrinsic dispositions.16

15

See, for instance, Lewis (1997) and Bird (1998). See McKitrick (2003). I do not know of any arguments against McKitrick’s conclusion, and I will accept it as a given. Bird (2007) suggests that the fundamental dispositions are still intrinsic, and McKitrick’s paper makes no claims to the contrary. Prior to the publication of McKitrick (2003), Molnar (2003) had argued against extrinsic dispositions (or ‘powers’) on the basis that they are reducible to the intrinsic powers of objects. I agree with the premise if ‘reducible to’ is read as ‘grounded in’, and I will spell it out in much more detail below; but I draw a different conclusion from it. As I have noted in chapter 1, my concern is with all potentialities, grounded or fundamental; and if extrinsic potentialities are grounded in intrinsic ones, to me that is just another reason to accept them. 16

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We have so far gone along with the orthodox assumption: our stock examples of potentialities have all been intrinsic. There is something very appealing about the thought that dispositions, and potentialities in general, are intrinsic. The point of a disposition is, after all, that it is a localized modality: it is a matter of how things stand with a particular object, and not of how things stand outside that object. This makes dispositions practically interesting: intrinsic properties are generally more stable over time, hence knowing a thing’s intrinsic potentialities allows for better long-term predictions of its behaviour. Their intrinsicality is also what makes dispositions theoretically appealing: dispositional facts are modal facts that are suitably anchored, as properties of objects, in an objectproperty ontology. If a potentiality is, after all, partly a matter of how things stand outside its bearer, it becomes harder to draw the distinction between localized and non-localized modality. The schema that I have sketched for the case of the key and the door provides a way to accommodate extrinsic potentialities within the framework of this book. An extrinsic potentiality, on that schema, is based on an intrinsic but jointly possessed potentiality. The key’s extrinsic potentiality to open door d is based on the key and the door’s joint and intrinsic potentiality for the former to open the latter. At the basis, we have only intrinsic potentialities. Extrinsic potentialities are innocent because they are entirely derivative from that basis. In the remainder of this section, I will apply this view to some less obvious examples, taken from McKitrick’s paper. One of McKitrick’s examples is the above-discussed potentiality (or ‘power’, as she says) of a key to open a particular door. However, it will be instructive to look at some of her other examples. Here are three. Vulnerability. This is where McKitrick uses the example of the city quoted above in chapter 4.3. Here is the quote with a little more context: Something is vulnerable if it is disposed to suffer harm as a result of an attack. . . . A military target, a city, is protected by a Star Wars-like defence system. The system has sensors that bring out defences when there is a threat, rendering the city invulnerable. However, the sensors and anti-aircraft weapons are all located outside the borders of the city and are built, maintained, and staffed by a foreign country. Should the defence system be disabled, or should the foreign power withdraw its protection, the city would change from being invulnerable to being vulnerable. However, the city might remain intrinsically the same, or internally the same in all ways that are relevant to its vulnerability. McKitrick 2003, 161

The city’s vulnerability is extrinsic because it is dependent on factors outside the city itself; it is affected by the defence system that is not part of the city.

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Intrinsic duplicates of the city can differ from it with respect to whether or not they are vulnerable. Weight. McKitrick assumes an analysis of weight along the following lines: x has weight n iff x has a disposition to depress a properly constructed scale so as to elicit a reading of n pounds in x’s gravitational field. McKitrick 2003, 16017

Having weight n is not to be confused with having mass n—the disposition to depress a properly constructed scale so as to elicit a reading of n pounds in a gravitational field of (a fixed) strength f . An object’s weight can change without any change in the object’s intrinsic constitution, if the object is moved from one gravitational field to another. Intrinsic duplicates can differ with respect to their weight if they are situated in different gravitational fields. Hence weight, so understood, is an extrinsic disposition. (The example works just as well if weight and mass are not identical to, but merely bestow on all their bearers, the relevant dispositions. For simplicity and ease of reference, I will go along with the assumption of identity.) Recognizability. Bill Clinton is recognizable, but he could cease to be so without undergoing any intrinsic change, by being placed in an environment where people have never heard of him or seen his picture. His recognizability depends on the social setting in which he is situated. An intrinsic duplicate of Bill Clinton in different surroundings may easily not be recognizable. McKitrick’s concern is with dispositions, mine is with the more general notion of potentiality. Thus her examples, in the context of the present theory, require an important proviso. We have seen in chapter 3 that the possession of a disposition such as fragility is a matter of degree, in particular, a matter of the degree to which the relevant potentiality is possessed. Many non-fragile objects have a potentiality to break; fragile objects differ from non-fragile ones only by having that potentiality to at least a given minimal, contextually specified, degree. The same may be said of vulnerability and recognizability: the city still has a potentiality to be harmed by an attack even once the defence system is in place, but that potentiality’s degree is so slight as not to merit ascription by the predicate ‘is vulnerable’. Bill Clinton has a potentiality to be recognized even in very different circumstances, but that potentiality’s degree, again, may be too slight for him to count as recognizable. At the level of potentiality, then, it is not the possession (or lack) of the potentiality at issue that depends on objects other than its possessors, as it was in the key/door case. Rather, it is the degree of the potentiality that is so dependent: the degree 17

As McKitrick (2003) notes, the example is from Yablo (1999).

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of the city’s potentiality to be harmed by attack is affected by the defence system, and the degree of Bill Clinton’s potentiality to be recognized is dependent on the state of people in his environment. That is enough for the potentiality to count as extrinsic. Our more general concern with potentiality also provides us with a second type of example: extrinsic abilities. Theorists of abilities standardly distinguish between general and specific abilities. John Maier illustrates the distinction as follows: Consider a well-trained tennis player, equipped with ball and racquet, standing at the service line. There is, as it were, nothing standing between him and a serve: every prerequisite for his serving has been met. Such an agent is in a position to serve, or has serving as an option. Let us say that such an agent has the specific ability to serve. In contrast, consider an otherwise similar tennis player who lacks a racquet and ball, and is miles away from a tennis court. There is clearly a good sense in which such an agent has the ability to hit a serve: he has been trained to do so, and has done so many times in the past. Yet such an agent lacks the specific ability to serve, as that term was just defined. Let us say that such an agent has the general ability to serve. Maier 2010, §1.3

A general ability is independent of how things stand outside its possessor. A specific ability, on the other hand, depends on its possessor’s circumstances: the presence of a ball and racquet, for instance, in the case envisaged by Maier. Intrinsic duplicates will share their general abilities but may differ with respect to their specific abilities—for instance, if one is on a tennis court and fully equipped, while the other is not. Whatever else may be said about the case of general and specific abilities, then, it appears to be another case of intrinsic versus extrinsic potentiality. These four examples—the city’s vulnerability, weight n, Bill Clinton’s recognizability, and the tennis player’s specific ability to serve—are not as easily linked to an underlying joint potentiality as our simple case of the key and the door. To consider them in more detail, let me first introduce some vocabulary. Extrinsic properties in general, and extrinsic potentialities in particular, depend on aspects of the world outside the object that has them. Where those aspects are other objects (and how things stand with them), let us call those objects the dependees of the extrinsic property. Further, where an object x possesses, together with other objects yy, a joint potentiality, let us call yy18 the co-possessors of x’s joint potentiality. Talk of co-possessors is simply a side-effect of singling out some among the joint possessors of such a potentiality. 18 I follow standard convention in using double letters, xx, yy, etc., (for variables) and aa, bb, etc., (for constants) to refer to pluralities.

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My claim is that an object’s possession of an extrinsic potentiality is a matter of the object’s having a joint potentiality whose co-possessors are the dependees of the extrinsic potentiality. In the case of the key’s extrinsic potentiality to open the door, the dependee and co-possessor was easy to determine: it was part of the potentiality’s manifestation. In general, where a potentiality’s manifestation consists in a relation to a particular other object—such as opening this particular door, as opposed to opening some door with a lock of shape S1 —the potentiality will be extrinsic, and the object involved in the manifestation will function as its dependee and as the co-possessor of the relevant joint potentiality. But that is only the simple case. The four examples that I have just given are not so simple. Their manifestations may consist in relations—depressing a scale, serving a ball, being hurt by an attacker, or recognized by a bystander—but they do not specify the other relatum. In that respect, they are like a key’s intrinsic potentiality to open doors with a particular type of lock. We must look elsewhere for their dependees, the co-possessors of the underlying joint potentialities. Here it is crucial to remember the two types of manifestations that a joint potentiality can have. In type 1 joint potentialities, the manifestation is a relation (or plural property) holding non-trivially between all of its possessors; such was the case of the key and door’s joint potentiality for the one to open the other. In type 2 joint potentialities, however, the manifestation of a joint potentiality concerned only some of its possessors; such were the cases of the uranium pile plus the rods and fail-safe mechanism, the glass and the styrofoam, and the city and its defence system. The manifestation, we said, was the relation of being such that the uranium pile, the glass, or the city is so-and-so. Where an extrinsic potentiality is derived from a type 2 joint potentiality, it is no wonder that the dependees and co-possessors cannot be determined simply from the potentiality’s manifestation. They were not part of it in the first place—the ‘first place’ being the joint potentiality whose co-possessors they are. This still leaves us with the question of how the dependees of these extrinsic potentialities are determined. The answer is: it depends. It depends, in particular, on situations of the objects to which the potentiality is ascribed: the city’s, the weighty object’s, Bill Clinton’s, the tennis player’s. Different objects, or the same object at different times, may have different dependees for the extrinsic potentialities, and hence different co-possessors for the relevant underlying joint potentiality. The city’s vulnerability depends for its degree on the defence system that McKitrick has imagined; another city’s vulnerability may depend on altogether different objects, and my vulnerability when walking down a dark alleyway depends for its degree on various features of the street, the surrounding area, and

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the people who are on the street with me. Bill Clinton’s recognizability depends for its degree on the people in his immediate environment. If he were to wear a mask, thus becoming less recognizable, his recognizability would depend for its degree on the mask in addition to the people in his environment. Different objects, or the same object at different times, may be in different gravitational fields; so weight n has different dependees in those different cases. And similarly for the specific ability to serve, which may depend in different cases on different balls, racquets, or other objects in the surrounding of its possessor. If the dependees of those extrinsic potentialities are different, then the extrinsic potentialities derive from different joint potentialities, or rather from joint potentialities’ holding of different objects. An object’s vulnerability may be based on its joint potentiality, with the defence system, to be harmed by an attack; or on its joint potentiality, with the dark street and the people in it, to be harmed by an attack; and so forth. So what is the extrinsic potentiality that we ascribe with a predicate such as ‘is vulnerable’ or ‘has weight n’? The answer, again, is: it depends. It depends on the situation of the object to which the potentiality is ascribed, and perhaps also on the context of the ascription. The predicates ‘has weight n’, ‘is vulnerable’, and so on are used to ascribe to an object that extrinsic potentiality which is based on the joint potentiality whose co-possessors are most salient in the object’s current circumstances. In some cases, such as that of weight n, the determination of that joint potentiality’s co-possessors is straightforward: it is always the current gravitational field of the object with the extrinsic potentiality. In other cases, such as those of ‘vulnerable’ and ‘recognizable’, the co-possessors of the relevant joint potentiality vary more widely; they are objects of a certain kind in close vicinity to the extrinsic potentiality’s possessor (such as the people in Bill Clinton’s immediate environment), or objects that are otherwise relevant to its manifestation (such as the city’s defence system). It is important to note that the joint potentiality which is singled out as relevant by a given context does not thereby acquire special ontological status. As we have seen in chapter 4.2, any objects can together possess joint potentialities. Thus I and the city’s defence system may together possess a joint potentiality to be such that I am harmed, and the city together with the people in the dark alleyway which I am walking possess a joint potentiality to be such that the city is harmed. It is just that those are not the interesting, contextually relevant joint potentialities.19

19 Note that I am in a situation here which is no better or worse than that of the reductionist sketched in chapter 3.3. The reductionist too will have to acknowledge contextual variations in the features on which the possession of an extrinsic disposition depends. She will take those to be the

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4.5 Extrinsic potentiality systematized We started (as most philosophers do) with the basic idea that dispositions, and by extension potentialities, are intrinsic to the objects that have them. It is distinctive of dispositions, and by extension of potentialities, that they are a matter of how things stand with a particular object itself, not of how things stand outside that object. From this it is natural to conclude that dispositions, and potentialities, must be intrinsic to their bearers. McKitrick’s examples have shown that this conclusion is unwarranted. Yet a feeling of discomfort remains: how do extrinsic potentialities fit into an overall metaphysics of potentialities that is still motivated by the intuitive idea that potentialities are a matter of how things stand with a particular object? The account I have given should dispel this feeling of discomfort. What is more, it provides a systematic understanding of the relation between intrinsic and extrinsic potentiality. It is intrinsic potentialities, possessed individually and jointly, that are basic; extrinsic potentialities are merely an innocent by-product of the intrinsic potentialities that things have. The aim of this section is to further generalize our understanding of extrinsic potentiality by arguing for the following two claims: Claim 1 Whenever an object, x, possesses an extrinsic potentiality, that potentiality is fully grounded in a joint potentiality which the object possesses together with other objects (and whose co-possessors are the dependees of the extrinsic potentiality). Claim 2 Whenever a number of objects together possess a joint potentiality, each of the objects thereby possesses an extrinsic potentiality which is fully grounded in that joint potentiality (and whose dependees are the co-possessors of the joint potentiality). Taken together, Claims 1 and 2 entail that the possession of a joint potentiality by an object x together with other objects is both necessary and sufficient for the possession of an extrinsic potentiality by x. It is the second claim that will be crucial to my project in this book; but let us have a look at the first claim first. Might there be extrinsic potentialities which, unlike the ones discussed in section 4.4, are not grounded in, and do not require the possession of, a joint potentiality? If so, what would they be?

features that are held constant across possible worlds when the disposition ascription is evaluated; I take them to be features of objects that are the underlying joint potentiality’s co-possessors. We differ in the underlying metaphysics, but not in the classification of the phenomena.

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Bauer (2011) has distinguished between two types of grounding for extrinsic properties (and hence for extrinsic dispositions). An extrinsic property of an object a is ‘extrinsically grounded [object]’ iff it is ‘grounded at least partially in some property or property-complex of another object b (or multiple objects b, c . . .)’; it is ‘extrinsically grounded [environment]’ iff it is ‘grounded at least partially in some property or property-complex of the environment e inhabited by a’ (Bauer 2011, 86). Now, extrinsic dispositions that are extrinsically grounded in an object’s environment might appear to constitute a counterexample to my first claim: if they are grounded in the features of the object’s environment, then they need not be grounded in other objects. They need not have other objects as their dependees, and hence as co-possessors of any relevant joint potentiality. Or so the reasoning might go. It is one of the central assumptions of this book—one of the assumptions that motivate the potentiality approach in the first place—that the world is composed ultimately of objects: if there is a fundamental level, it consists of objects (with potentialities); if there is no such level, you will find objects (with potentialities) however far you go ‘down’. Given that assumption, for which I have not argued but which is deeply ingrained, dependence on an object’s environment will always boil down to dependence on the objects in its environment (together with the properties that those objects have and the relations in which they stand). There is, then, no sharp boundary between the two types of extrinsic grounding that Bauer distinguishes, and no objection to my first claim.20 Let us, then, accept claim 1 and move on the more important claim 2. First, let me be more specific about claim 2. Joint potentialities, as we have seen, have two basic types of manifestation. Type 1 joint potentialities are those whose manifestation is a relation or plural property that belongs non-trivially to all of the potentiality’s possessors; such was the key and door’s potentiality to stand in the opening relation. Type 2 joint potentialities are those whose manifestation consists in only some of their possessors’ having a property or standing in a relation—whose manifestation, to apply to all of its possessors, must be phrased with the somewhat awkward ‘such that’ locution. The city and its defence system had a joint potentiality of this type: to be such that the city is harmed by an attack.

20 Interestingly, Bauer himself does not explain the distinction between objects and environments, and when it comes to a case of environmental grounding, he treats that environment as being an object itself. He offers weight as an environmentally grounded extrinsic disposition because an object’s weight is grounded in its gravitational field which ‘is plausibly considered part of [the object’s] total environment’ (Bauer 2011, 86). Later it is claimed that a field is best categorized ‘as a particular that possesses a set of properties’ (Bauer 2011, 89), such as having a certain extension and being disposed to give objects a particular weight depending on their mass.

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The key’s extrinsic potentiality was based, as we have seen, on a joint potentiality of type 1. While the context-sensitivity of many dispositional expressions complicates classification somewhat, the city’s vulnerability and its low degree in McKitrick’s example are clearly based on a joint potentiality of type 2: the defence system is no participant in the property that constitutes the potentiality’s manifestation. In the former case, the manifestation of the extrinsic potentiality involves its possessors and all its dependees (or in this example, its one and only dependee). In the latter case, the manifestation of the extrinsic potentiality involves its possessor but not all its dependees: if the defence system is the only dependee, then the manifestation involves only its possessor and no dependee. Nonetheless, both extrinsic potentialities are based on joint potentialities of their possessor together with all the dependees. Claim 2 applies to both types of joint potentialities. As it stands, claim 2 does not tell us which extrinsic potentiality is possessed in virtue of a given joint potentiality, but we can make this explicit too, as follows. Type 1—Relations If x and y have a joint potentiality to stand in relation R, then x has an extrinsic potentiality to stand in relation R to y and y has an extrinsic potentiality to stand in relation R to x (with adjustments made for the direction of relation R). Type 1—Plural properties If the objects xx have a joint potentiality to be F, and x is one of xx, then x has an extrinsic potentiality to be F with the rest of xx. Type 2—Relations If x and y have a joint potentiality to be such that x is F, then x has an extrinsic potentiality to be such that it, x, is F (and hence an extrinsic potentiality to be F) and y has an extrinsic potentiality to be such that x is F. Type 2—Plural properties If objects xx have a joint potentiality to be such that one of them, x, is F, then any objects which form a proper subplurality of the xx have an extrinsic potentiality to be such that x is F. What is important about these generalizations is that they are symmetrical: they apply to each of the possessors of a joint potentiality. I said at the outset that the case of key and door was symmetrical: in virtue of their jointly possessed potentiality, the key has an extrinsic potentiality to open the door, and the door has an extrinsic potentiality to be opened by the key. That much seems intuitively obvious. To be fully general in the account, we should expect symmetry for the type 2 cases too: the city has an extrinsic potentiality to be harmed, with the defence system as its dependee, and the defence system has an extrinsic potentiality to be such that the city is harmed, with the city as its obvious dependee. Both

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extrinsic potentialities seem to arise naturally from the same joint potentiality. The latter and its ilk are rarely ascribed, and they are generally of little interest. But the current account of extrinsic potentialities, and in particular claim 2 above, predicts that there are such extrinsic potentialities too. For it says that the implication from joint potentiality to extrinsic potentiality holds for each of the joint potentiality’s possessors. In the case of type 1 joint potentialities, this seems independently plausible: for how would we discriminate between the joint potentiality’s bearers in this case? In the case of type 2 joint potentialities, it is less obvious but nonetheless no more than a systematization of the notion of extrinsic potentiality. What is the status of the above four principles? I would be happy to treat them as constitutive of the very notion of extrinsic potentiality, in much the same way in which the inference from p and q to p and to q, and the inference from p, q to p and q are constitutive for the notion of conjunction.21 The important point for my purposes, however, is the metaphysics that underlies them: each of the extrinsic potentialities in the consequents of the above principles is fully grounded in the joint potentialities mentioned in the antecendent. There is no more to any object having an extrinsic potentiality than the possession, by this object together with the relevant other objects, of a joint potentiality. In chapter 1, I have explicitly made the assumption that once we have the full grounds for p, we get p automatically or, if you will, ‘for free’. Since full grounding is transitive,22 anything that grounds the possession of a joint potentiality thereby grounds the possession of the relevant extrinsic potentialities; and anything that is grounded in an extrinsic potentiality’s possession is grounded in the having of a joint potentiality. Given, further, that grounding is a kind of explanation—and specifically, the kind of metaphysical explanation that metaphysicians are after—the possession of extrinsic potentialities brings nothing new into the world as either explanandum or explanans. Indeed, while I am very generous with both ontology and ideology and will speak of extrinsic potentialities as respectable, albeit derivative, properties, those with more austere tastes may as well understand the ascription of extrinsic potentiality as a mere façon de parler, made true (where it is true) by the possession of joint potentialities. Nothing depends on generosity or austerity here; all that matters is that we can truly ascribe extrinsic potentialities and that such ascription is underwritten entirely by joint potentialities.23 21

Thanks to Ralf Busse for the suggestion. See Rosen (2010), Schaffer (2009a). Schaffer (2012) argues that partial grounding may not be transitive, but his argument does not affect full grounding. 23 Note that it is important that we use the notion of grounding here, rather than (some standard modal definition of) supervenience. Extrinsic potentialities supervene on intrinsically possessed 22

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Given this picture, I see no objection to the symmetry exhibited by the above principles or to the claim that a joint potentiality endows each of its joint possessors with extrinsic potentialities of the relevant kind. The tricky case is, of course, the ascription of extrinsic potentialities to all possessors of a type 2 joint potentiality: the defence system’s potentiality to be such that the city is harmed; the styrofoam’s, to be such that the glass breaks; the bull’s, to be such that the torero is hurt; and so forth. My justification for covering these cases is twofold: first, in so doing we are giving the most systematic generalization of the cases that we have seen, since extrinsic potentiality is generated symmetrically in the case of type 1 potentiality too.24 Second, extrinsic potentiality in general is metaphysically innocent (in ways that will be cashed out differently depending on how generous or austere our tastes), since it is fully grounded in joint potentiality. If this is so, then why all the fuss about extrinsic potentiality? Because it will prove extremely useful in formulating general principles for the logic of potentiality. Just as predicate logic is not concerned with natural properties only, but with the truth or falsity of any predicative statement, so the logic of potentiality will be concerned, not with the most natural potentialities only, but with the truth or falsity of any potentiality ascriptions, including ascriptions of extrinsic potentiality. And once we recognize this, it will be easier to recognize also that potentiality is indeed subject to the very logical principles that make a potentiality-based account of possibility formally adequate. But that will be the topic of the next chapter. One final generalization is needed. It should be obvious and has been bracketed so far only for the sake of expository simplicity. Not only individual objects possess extrinsic potentialities, but any number of objects, xx, can together possess an extrinsic joint potentiality if xx have a relevant joint potentiality together joint potentialities; but the same holds vice versa. Given the extrinsic potentialities that objects have (with their dependees), it is fixed which joint potentialities they possess together with other objects; there can be no difference in joint potentialities without a difference in extrinsic potentialities. (Let worlds w1 and w2 differ merely in that, in w1 , a, b, and c jointly possess a potentiality to stand in relation R, while in w2 , only a and b possess that joint potentiality, and c has no part in it. Then in w1 , but not in w2 , c has an extrinsic potentiality to stand in R to a and b.) The intrinsic potentialities that individual objects possess singly, rather than jointly, do not supervene either on their joint or their extrinsic potentialities. But the supervenience base can consist of those individual intrinsic potentialities plus the extrinsic potentialities, or of the individual intrinsic plus the joint intrinsic potentialities. Supervenience does not discriminate between extrinsic potentialities and jointly possessed potentialities; their relation is symmetric, so far as supervenience is concerned. 24 Indeed, in chapter 5, we will be able to formulate the principles in a more general way, so that they cover type 1 and type 2 joint potentialities simultaneously. See footnote 9 in chapter 5.3.3.

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with other objects distinct from xx. So in the above four principles, we could replace each individual variable x and y with plural variables and obtain a generalized and valid set of principles regarding extrinsic potentiality.

4.6 Iterated potentiality In recognizing extrinsic potentialities, we have considerably extended the reach of our initial notion of potentiality. An object, x, can have potentialities to be such that p, where p is entirely about objects other than x. Another way of extending the reach of potentiality is the recognition of iterated potentiality. Things have potentialities to possess properties. Potentialities themselves are properties. So, prima facie, things should have potentialities to have potentialities. And the latter potentialities might themselves be potentialities to have potentialities. So there is nothing to prevent things from having potentialities to have potentialities to have potentialities, or potentialities to have potentialities to have potentialities to have potentialities . . . and so forth. I will call any such potentiality an iterated potentiality. Iterated potentialities have been recognized in the literature on dispositions. Borghini and Williams (2008) appeal to them, as I will too, in giving a dispositional theory of possibility. They call them ‘higher-order’ dispositions.25 I prefer to reserve that expression for dispositions of dispositions, if such there are; hence the term ‘iterated potentiality’. Here are some intuitive examples of such iterated potentialities, taken from the paradigm examples of potentiality: dispositions and abilities. I do not have an ability to play the violin. Nor does my desk. Unlike my desk, however, I possess the ability to learn to play the violin—the ability, that is, to acquire the ability to play the violin. That latter ability distinguishes me from my desk, insofar as playing the violin is concerned. I have, while the desk lacks, an iterated ability to play the violin. 25 As they put it, ‘the “higher-order” dispositions are dispositions for further dispositions’ (30, fn. 23). That characterization, however, appears only in a footnote. In the main text, Borghini and Williams appear to introduce the term not for what I would call iterated dispositions themselves, but for the dispositions which, in my terminology, are the manifestations of the iterated dispositions: ‘Let us call those dispositional properties we find instantiated at any point in the (complete history of the) world “first-order dispositional properties”. . . . The manifestations of the first-order dispositions— regardless of whether they ever come about—are states of affairs that if manifested would include other dispositional properties: call these “second-order dispositional properties”.’ (30). I prefer the definition given in the footnote, since it is a contingent matter which dispositions are instantiated, and hence it would be a contingent matter which dispositions are first- or second-order on the main text characterization.

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Whether or not colours are secondary qualities, i.e. dispositions to appear in a certain way to a normal observer, coloured objects certainly have these dispositions. Now take an apple about to ripen and turn a bright shade of red. The apple is disposed to become red; it is thereby disposed to acquire the disposition to look red to a normal observer. It has an iterated disposition to look red to a normal observer. Water, I have said earlier, has no potentiality to break. But water has a potentiality to be frozen and turn into ice, which does have a potentiality to break. So water has a potentiality to acquire (by freezing) a potentiality to break. Let us fix terminology a little. The immediate manifestation of an iterated potentiality, such as the apple’s disposition above, is another potentiality, in this case the disposition to look red to normal observers; that is how we have defined the term. Its ultimate manifestation is the manifestation of the potentiality that it is a potentiality to have, or the potentiality that it is a potentiality to have a potentiality to have . . . and so forth. Thus the apple’s iterated disposition above has as its immediate manifestation the disposition to look red to normal observers; its ultimate manifestation, however, is looking red to normal observers. Since the interest of iterated potentialities lies in their relation to the latter, I will drop the term ‘ultimate’ and simply speak of the iterated potentiality’s manifestation. Moreover, it is useful to distinguish between different numbers of iterations. The same potentiality may be an iterated potentiality for different (ultimate) manifestations, given that there is a difference in the number of iterations that are needed to ‘reach’ the manifestation. It will be useful to include what we would otherwise think of as non-iterated potentiality as the limiting case of one iteration. Thus I have a once-iterated ability to learn to play the violin and a twice-iterated ability to play the violin; the apple has a once-iterated disposition to turn red and a twice-iterated potentiality to look red to a normal observer; and water has a once-iterated potentiality to turn into ice and a twice-iterated potentiality to be broken.26 The manifestation of a (once-iterated) potentiality can be a relation in which the potentiality’s bearer stands to another object. If the manifestation is a relation to a particular other object, the potentiality is extrinsic: recall the key’s potentiality to open the particular door d. If the potentiality’s manifestation is simply standing in a given relation to some object or other, then the potentiality may 26 Although abilities and dispositions provide intuitive examples to introduce the notion of iterated potentiality, the fact that I have only a (twice-)iterated ability to play the violin does not entail that I have only a (twice-)iterated potentiality to play the violin, and likewise for dispositions. In fact, it seems that I have both a once-iterated and a twice-iterated potentiality to play the violin, but only the latter qualifies as an ability.

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well be intrinsic: recall the key’s intrinsic potentiality to open doors with a lock of shape S1 . Jointly having a potentiality is just another relation in which things can stand. If there are potentialities to have other potentialities, as well as potentialities to stand in relations to other things, then there should be potentialities to stand in the relation of jointly having a potentiality: iterated joint potentialities, as we might call them. A door may have the potentiality to be opened by a particular key k, but it may also have the potentiality to acquire (through an exchange of its lock) the potentiality to be opened by another key, k . Both the ordinary and the iterated potentiality are extrinsic, because both depend on the existence and the shapes of objects distinct from the door, keys k and k respectively. But the door also has iterated joint potentialities that are intrinsic to it. In addition to its intrinsic potentiality to be opened by keys of shape S1 , it may have a potentiality to acquire the joint potentiality, with keys of shape S2 , to stand in the opening relation. This latter potentiality, too, does not depend on anything other than the door. It is an intrinsic iterated potentiality, whose immediate manifestation would occur if the door had its lock changed, thereby acquiring the potentiality to be opened by keys of shape S2 . There is, again, no limit on the number of iterations. Moreover, by repeated iteration an object may have potentialities that concern objects entirely distinct from it. Let us look at the case schematically first, before we introduce examples. One object, a, may have a potentiality to stand in a certain relation to another object, b; and that relation may be the relation of having a joint potentiality together. As we have seen, the manifestation of a joint potentiality need not concern all of the objects that jointly possess it. Hence the joint potentiality that a has a potentiality to have together with b may be one whose manifestation concerns only b and not a; in particular, it may be a potentiality to be such that b is F. So here we have the chain linking a, by iterating potentialities, to b’s being F: a has a potentiality to have, with b, a joint potentiality to be such that b is F. We can go further, of course: a may have a potentiality to have, with b, a joint potentiality to be such that b has a potentiality to be F; or, a may have a potentiality to have, with b, a joint potentiality to be such that b has a joint potentiality, with c, to be such that c is F; and so on. These may concern particular objects b and c, or else any objects x and y that meet some condition. In the former case, the iterated potentiality is extrinsic because it depends on the existence and various features of b and c; in the latter case, it may well be intrinsic. In either case, the manifestation of the iterated potentiality takes place outside its possessor; if the chain of iterations is long enough, a may even have ceased to exist by the time the iterated potentiality is manifested. Let us now look at some examples.

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A violin teacher possesses a particular skill: the ability to teach students how to play the violin. The manifestation of this ability consists in another individual, a student, acquiring an ability: the ability to play the violin. Or take a colouring agent, used to turn white textiles red. When that potentiality is manifested, a piece of textile becomes red and thereby acquires the disposition to look red to normal observers. My freezer has the potential to turn water into ice. When that potentiality is manifested, a quantity of water becomes frozen and thereby acquires the disposition to break. The textile dye has a potentiality to stand in a certain relation to a piece of textile: the relation of possessing a joint potentiality, which typically comes with the dyeing process. This is a joint potentiality whose manifestation consists in only one of its possessors, the piece of textile, acquiring a property: the property of being red, and thereby the potentiality to look red to normal observers. The textile dye has a potentiality to have, together with some piece of textile, a joint potentiality to be such that one of them, the piece of textile, has a potentiality to appear red to normal observers. This is an intrinsic, three-times iterated potentiality, the manifestation of which is that a piece of textile looks red to a normal observer. The case of the violin teacher and that of the freezer is perfectly analogous. The teacher has an ability to enter, with any student, into a joint potentiality to be such that the student acquires the ability to play the violin. Hence the teacher has an intrinsic, three-times iterated ability, the manifestation of which is that a student plays the violin. My freezer has a potentiality to enter, with any appropriatelysized quantity of water, into a joint potentiality to be such that the water (freezes and thereby) acquires the potentiality to be broken. Hence the freezer has an intrinsic, three-times iterated potentiality, the manifestation of which is that a quantity of water is broken. It is important to note that iterated potentialities are potentialities ‘for p’; their manifestation is not a property. The reason is straightforward: the ultimate manifestation of the iterated potentiality may concern something other than the iterated potentiality’s possessor itself. The above-mentioned dyeing agent does not have an iterated potentiality to appear red to normal observers; it has an iterated potentiality for a textile to appear red to normal observers. The iterated potentiality’s manifestation is not, then, appearing red to normal observers; it is that a textile appear red to normal observers. It is not even the property of being such that the textile appear red to normal observers; for that property would still have to be possessed by the dyeing agent when the iterated potentiality is manifested. But the dyeing agent itself may have ceased to exist by the time a dyed textile appears red to anyone, in which case it cannot have the property of being such that the textile appears red to normal observers, yet its iterated potentiality

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for the textile to appear red is still being manifested. Likewise, mutatis mutandis, for the violin teacher and the freezer. Iterated potentiality thereby provides us with a veritable extension of the ‘reach’ of an object’s potentialities. An object x may have an iterated potentiality for it to be the case that p, where p is entirely about objects distinct from x. We have seen a similar extension taking place with the introduction of extrinsic potentiality. Iterated potentiality constitutes a further extension in two respects. First, the manifestation of an iterated potentiality need not even be of the predicative form ‘being such that p’. As we have seen, iterated potentialities are potentialities for p. Second, iterated potentialities can be intrinsic and still have manifestations that do not concern their possessors at all. Iterated potentiality will be invaluable in formulating the theory of possibility in chapter 6. With extrinsic potentialities and iterated potentialities of the variety that we have just considered, potentiality has become a much more flexible tool. We will be able to use it in accounting for remote possibilities such as the possibility of my great-granddaughter’s being a painter, for the context-sensitivity of our modal language, and for the logic of modality. Before we can set this tool to work, we need to spell out some of its features in a formally more rigorous way than has been required for the purposes of this chapter. That will be the task of chapter 5.

4.7 Taking stock: expanding potentiality With this chapter and the last, we have reached a conception of potentiality that goes far beyond its beginnings in our ordinary understanding of dispositions. In chapter 3, we have seen that dispositions come in degrees and that those degrees outrun the cases in which we are willing to apply dispositional adjectives such as ‘fragile’. They outrun them in two directions. They may be below the threshold for a dispositional adjective to apply, as with my desk’s or a rock’s potentiality to break. And they may be so high that we would refrain from the ascription of a disposition for pragmatic purposes, when an object lacks the disposition to do otherwise. To avoid terminological dispute, I have reserved the term ‘disposition’ for that which we ascribe with ordinary dispositional adjectives. Dispositions so understood are not metaphysically basic: they have context-sensitive and, in any ordinary context, metaphysically arbitrary cut-off points. Therefore I have introduced the term ‘potentiality’ for that class of properties which includes dispositions but is conceived widely enough to count as metaphysically basic, and hence serve in a dispositionalist metaphysics. In this chapter, we have looked at three varieties of potentiality so conceived: potentialities that are possessed jointly by a number of objects; extrinsic

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potentialities, which I have argued arise from joint potentialities; and iterated potentialities. I have argued, again, that the range of examples with which we started calls for extension if we are serious realists about potentiality. There is no sharp, non-arbitrary cut-off to be found between the intuitive cases that I have used to introduce the phenomena in question and those cases that may seem, at first, a little outlandish. A systematic, realist metaphysics of potentiality had therefore better accept the whole range, including the seemingly outlandish cases. Outlandish cases include joint potentialities between objects that have little to do with one another, such as my nose and the Eiffel tower. They include, further, extrinsic potentialities, possessed by one thing, to be such that another, entirely disjoint object, has a certain property. In almost every context, such potentialities are of little or no interest. In the present context, however, they are significant: they will be crucial in formulating and defending the potentiality-based account of possibility that is the project of this book. Let me stress, again, that the outlandish cases of extrinsic potentiality are not introduced as primitives: they are grounded in, they fully depend on, the intrinsic joint potentialities of their bearers and certain further objects. The metaphysics that has been developed so far will be put to use in formulating the logic of potentiality (chapter 5) and thereby in meeting the challenge of formal adequacy; in formulating the first steps of a modal semantics based on potentiality (chapter 6), thereby meeting the challenge of semantic utility; and in meeting various objections to the extensional correctness of the potentiality-based account of possibility (chapter 7).

5 Formalizing Potentiality 5.1 Introduction The aim of this chapter is to provide the grounds on which the potentiality account of possibility can meet the challenge of formal adequacy. That challenge has two parts. One part concerns logical form: potentiality, as I have repeatedly pointed out, has the structure of a predicate modifier: it is always a potentiality to. . . . Possibility, on the other hand, has the structure of a sentence modifier: it is always a possibility that. . . . This confronts us, first, with the challenge of finding a suitable formulation for the potentiality-based account: how do we get from the predicates in the scope of ‘potentiality to . . . ’ to the sentences that we need in the scope of ‘possibility that . . . ’? We have made some progress towards answering that question in the previous chapter, in particular with the introduction of iterated potentialities. Even so, the challenge of logical form is not fully addressed, for sentences, and a fortiori the sentences in the scope of a possibility operator, can have a great variety of forms and a great deal of complexity. They can be negated, disjunctive or conjunctive, existentially or universally quantified, or any combination of these, combinations of combinations of them, and so forth. To give a full account of possibility in terms of potentiality, we must find a way to link potentialities to sentences of arbitrary form and complexity. My strategy will be to take the path of least formal resistance and solve the problem on the level of potentiality, by allowing sentences of arbitrary form to be embedded in the scope of a potentiality operator. Sections 5.2–5.4 will meet this first part of the challenge by introducing a formal language that has the requisite expressive power and arguing that this language is adequate to the metaphysics of potentiality. The second part of the challenge concerns logical structure: that is, the rules and axioms that govern valid inferences concerning possibility. The basic logic of metaphysical possibility is one of the few subjects on which philosophers are almost universally agreed: metaphysical possibility, if there is such a thing, is governed by a normal modal logic—one which allows for the substitution of logical

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equivalents and in which possibility is closed under, and distributes over, disjunctions; and among the various systems of normal modal logics, it validates at least system T, on which actuality implies possibility.1 It is at least not obvious that potentiality is governed by the same logical rules. But in one way or another, a potentiality-based account must explain why possibility is governed by these logical principles. I will, again, take the path of least formal resistance, and solve the problem at the level of potentiality, by arguing that potentiality itself is governed by logical principles that are analogous to those which govern possibility (modulo its status as a predicate operator). Sections 5.5–5.9 will provide the required arguments. In the next chapter, their results will be used to show how the second part of the challenge is met. Before we turn to a formalization of potentiality, let me rehearse some of the features of a theory of potentiality that the previous chapters have established. First: Potentialities include dispositions and abilities, but they extend beyond both (see chapter 3.3–3.4). Thus a steel bridge has the potentiality to break, and I have the potentiality to play the violin, though neither of these potentialities qualifies as either a disposition or an ability. The formal language and the logic to be developed in this chapter concern potentiality in general, not dispositions or abilities in particular. Second: Potentialities come in degrees; the maximal degree of a potentiality to F consists in the lack of a potentiality not to F (see chapter 3.5). The formal language and logic to be developed will concern potentialities irrespective of their degree. Third: Potentialities can be intrinsic or extrinsic; the extrinsic potentialities, I have argued, are grounded in (intrinsic) joint potentialities that their possessors have together with other objects (see chapter 4.4). The formal language and logic of potentiality developed in this chapter is to apply to all of these: intrinsic and extrinsic potentialities possessed by individual objects or jointly by any number of objects. This means that we are not formalizing the potentialities that are metaphysically basic. We are formalizing any potentiality, no matter how derivative. Note that this is not at all unusual: predicate logic, too, does not care about the metaphysical status of the properties expressed by its predicates; it, too, applies to any property ascriptions whatsoever. Fourth: Potentialities can be iterated (see chapter 4.6). The formal language and logic developed in this chapter will be a formalization only of what

1 It is customary to start with the necessity operator rather than the possibility operator. For my purposes, it is obviously possibility that we should start with. A more precise formulation of what a normal modal logic involves will be given in chapter 5.6.

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is intuitively non-iterated potentiality (though it is better to think of it as once-iterated potentiality), i.e. potentiality simpliciter. Section 5.4 will show how iterated potentiality is defined in terms of it. My strategy in this book has been so far, and will continue to be in this chapter, to generalize potentiality as far as we can. It is worth stressing that it is this generalized notion of potentiality which will be formalized and logically systematized in this chapter. I make no claim that this is the only logical formalization and system which can be given of potentiality. Others might focus on the metaphysically basic potentialities, or on intrinsic potentiality, or might look at the degree of the potentialities at issue. For other purposes, such systems may well be more useful and more illuminating. My purpose is to show that, given the most general notion of potentiality, an account of possibility can be formulated which meets the constraints of extensional correctness, formal adequacy, and semantic utility. To show this, I must concern myself with the most general notion of potentiality, which includes any potentiality regardless of its degree and of its intrinsicality or extrinsicality. Chapters 3–4 have provided the metaphysics which builds up from the more intuitive, and from the more basic, potentialities to this general account. The task of the formalization is not to encode the building work, but to systematize its result. The formal language to be developed in sections 5.2–5.4, then, as well as the principles to be defended in sections 5.5–5.9, will concern ascriptions of potentiality regardless of their degree and their extrinsicality or intrinsicality. So much for a preface; now let us begin.

5.2 Framing the language To ask the right questions about potentiality ascriptions, we need a language in which we can phrase them. Our basis will be the language of standard predicate logic with identity, consisting of individual constants a, b, . . . , and variables x, y, . . . , as well as plural constants aa, bb, . . . , and variables xx, yy, . . . ; individual n-place predicates R, S, and so on, as well as the more familiar individual one-place predicates F, G, and so on;2 and the usual logical connectives with disjunction and negation (∨ and ¬) as primitives, the identity sign =, and the quantifiers ∃ and ∀, with ∃ as a primitive and ∀ defined in terms of it in the usual way. We deviate from standard predicate logic (with plurals) only by not allowing zero-place predicates, or sentence letters, since for our purposes it is important 2 I omit plural predicates because we will not use them in the examples in this chapter, but the language will, of course, contain plural n-place predicates as well.

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that we can ‘see into’ the predicative structure of a sentence. As metavariables, I will use φ, ψ, . . . , for sentences of any form, , , . . . , for (n-place, singular or plural) predicates of any form, t or t1 , t2 , . . . , for terms (that is, constants or variables), singular or plural; and x, y, . . . , for variables only, singular and plural. (I will not always keep object language and metalanguage properly apart. For present purposes, this should do no harm and help to enhance readability.) For expository purposes, we will adopt three conventions concerning the semantics of the terms and predicates. First, we assume that a predicate letters expresses what Fine (1995a, 243) calls ‘a pure property, i.e. one which does not involve any objects’. Being squared is a pure property, being loved by John or being 10 m from a palm tree is not. This assumption will help us keep track of what Fine calls ‘objectual content’ in later sections. Since I will ultimately reject the idea that objectual content plays a role for the logic of potentiality, the assumption can be dropped in our final formulation of the logic of potentiality. Second, we assume that each singular term refers to one object in the domain and that no two singular terms refer to the same object. The first part serves to avoid the question—which should be of little consequence to the logic of potentiality—of empty names. The second part ensures, again, that we can keep track of objectual content. When we use two terms, say a and b, it is implicitly understood that they denote different objects, i.e. that a = b. Like the first assumption, this one can be dropped in the final formulation of the logic of potentiality. The third convention is the most substantial. We do not use tense operators or time indices in the language, and we will assume that our predicates are not implicitly tensed or time-indexed. Thus no predicate can be interpreted as meaning, say, ‘was red’ or ‘is walking on 30 October 2012’. This restricts the expressive power of our language somewhat, but it is a useful idealization in our first construction of a logic of potentiality. Eventually, we will want a logic that incorporates the interaction between potentiality ascriptions, tenses, and times, but I leave that as a task for future developments of the view. I will, however, address the most salient issues (as I see them) in section 5.8, albeit in a somewhat tentative manner. Despite this limitation in the expressive resources of our language, of course, all formulas are evaluated for their truth at a time (not an interval, but a point). After all, potentialities, like other properties, are possessed (and lacked) at times: they can be lost and gained, so there is no sense in asking whether a potentiality ascription is true or false without specifying a time at which it is to be true or false. (This will be relevant below in section 5.7.4.) Now that we have our base language, we introduce into it an operator POT to express potentiality. I have said repeatedly that potentiality is properly expressed

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by a predicate operator. Potentialities are potentialities to . . . , individuated by their manifestation property. Potentialities are, moreover, properties of individual objects. POT must therefore be a predicate operator which takes a predicate—specifying the potentiality’s manifestation—to form another predicate, which can then be used to ascribe the specified potentiality to an object. The syntax of POT is as follows. Where  is an n-place singular predicate and t1 , . . . , tn are singular terms, or  is an n-place plural predicate and t1 , . . . , tn are plural terms, POT[](t1 , . . . , tn ) is a well-formed sentence. Semantically, it ascribes to the object or objects denoted by t1 , . . . , tn a potentiality to have the property or stand in the relation denoted by . In what follows, with the sole exception of section 5.4, I will focus exclusively on the case of singular monadic ascriptions of potentialities: the ascription, to a single object, of a potentiality to have a particular property, or syntactically the case where  is a singular predicate and there is only one of t1 , . . . , tn . The generalization to the plural case, as well as the case of relations (with  a manyplace predicate), is naturally made, and for the logic of possibility we will require no more than this special case of singular, monadic potentiality ascriptions (as we shall see in chapter 6.1). So far, we can only form predicates with POT by putting an atomic predicate into the square brackets. In order to extend the range of our predicates, and therefore the range of expressions that can be in the scope of POT, we need to introduce a standard predicate-forming operator, λ. Where φ is a sentence, open or closed, and φ[t/x] is the result of substituting a term t for any free occurrence of x in φ, the sentence λx.φ (t) is true just in case φ[t/x] is true. Intuitively, λx.φ turns the sentence φ into a predicate meaning ‘is such that φ’ (with any free occurrences of x in φ becoming ‘gaps’ in the predicate). The λ operator has two main functions. One is to express logically complex predicates; the other is to turn closed sentences into ‘such that’ predicates. We will look at both of these in turn. First, it is used to express complex predicates with the help of the sentence connectives ¬, ∨ and so on. Thus λx.¬Fx expresses the property of not being F; λx.(¬Fx)(a) is true just in case a is not F. λx.(Fx ∨ Gx) expresses the property of being F-or-G; λx.(Fx ∨ Gx)(a) is true just in case a is F or a is G. Thus the λ operator allows us to formulate potentiality ascriptions such as the following: (1)

a. POT[λx.¬Fx](a), b. POT[λx.(Fx ∨ Gx)](a).

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The former might be used to express that I have a potentiality not to walk, the latter to express that I have a potentiality to walk or run. Such potentiality ascriptions will be indispensable in formulating the potentiality-based account of possibility (after all, possibilities come in arbitrary degrees of logical complexity). But they are quite independently motivated. We often find ourselves wanting to talk about opposed potentialities, that is, the potentiality to do something and the potentiality not to do it. I did so in chapter 3 when speaking of the maximal degree of a potentiality, which I argued to be the lack of the opposing potentiality. Those in the debate on free will often talk about the ability not to do what one actually did, and some (such as Steward 2012) claim that abilities are ‘two-way powers’: to have an ability to do one thing (to F), one must also have the ability not to do it (not to F, i.e. to λx.¬Fx). Potentiality ascriptions of the form of (1-a), therefore, should be expressible in an adequate language of potentiality. In thinking about potentiality ascriptions with complex manifestations, such as (1-b), it is useful to begin with the case of conjunction. Having the ability to do a headstand requires having the ability to support one’s body weight with one’s arms, as well as the ability to balance one’s legs in the air. But it won’t do to have each of these abilities separately; one must be able to exercise them together. If I can either rest my weight on my arms (as long as someone stabilizes my legs) or balance with my legs in the air (as long as someone helps me support my weight), I do not have the ability to do a headstand. What is required for the ability to do a headstand, then, is an ability with a conjunctive manifestation: the ability to support one’s body weight with one’s arms and (at the same time) balance one’s legs in the air. So we must allow for logically complex manifestations. Examples of disjunctive manifestations may be less natural, but they can hardly be excluded. A fragile glass has a disposition to break into two or more pieces, though it may not have a disposition to break into two pieces or a disposition to break into more than two pieces. (Its potentiality to break into two pieces, and its potentiality to break into more than two pieces, may lack the degree required to count as a disposition). How does the glass exercise its potentiality to break into two pieces or break into more than two pieces? By breaking into two pieces, or by breaking into more than two pieces, as the case may be.3 3 The potentiality to F-or-G is easily understood in natural language not as a potentiality with a disjunctive manifestation, but as a conjunction of two potentialities. As Kenny (1976) points out, sentences such as ‘I can take it or leave it’ are naturally read as ascribing not a disjunctive ability to take-it-or-leave-it, but rather an ability to take it and an ability to leave it. When I speak of potentialities to F-or-G, I intend the literal, disjunctive reading.

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Since we follow standard usage in defining conjunction by way of negation and disjunction ((φ ∧ ψ) is equivalent to ¬(¬φ ∨ ¬ψ)), this already provides a nice example of how these two kinds of complex manifestations, negative and disjunctive, may occur in combination. A λ operator, therefore, is an independently motivated useful tool when we combine the languages of standard predicate logic and of potentiality. The second use to which we can put λ operators is less obviously motivated, though I will argue that it is well motivated nonetheless. This second use is to turn a closed sentence into a predicate. Thus, where φ is a closed sentence expressing, for instance, that Tina is tall, λx.φ will express the property of being such that Tina is tall. This is a property which everything possesses so long as Tina is tall, and which nothing possesses otherwise. Typically, though not universally, such predicates express ‘mere Cambridge properties’, properties whose possession has no grounding in the intrinsic nature of their bearers. The property of being such that Tina is tall is a Cambridge property for most objects, but not for Tina herself. Introducing a λ operator into our language gives us the resources to express such potentiality ascriptions as the following: (2)

a. POT[λx.∃xFx](a),4 b. POT[λx.Fb](a). (Recall the convention that a = b.)

The former might ascribe to me a potentiality to be such that something is tall, or the potentiality to be such that something is a horse. The latter might ascribe to me a potentiality to be such that Tina is tall. Such potentiality ascriptions, no doubt, are odd. For now, my point is merely that we can express them with the resources of the language that I have introduced. The next section will discuss whether we should be able to express them and whether they might ever be true. To have a convenient means of referring to them, I shall call potentiality ascriptions with quantifiers in the square brackets (such as (2-a)) ‘quantified potentiality ascriptions’, and those which contain in the square brackets names other than that of the potentiality’s bearer (i.e. other than those that occur in the round brackets, as in (2-b)) ‘Cambridge potentiality ascriptions’, and likewise for the ascribed potentialities. We will begin, however, with a slightly different but also prima facie puzzling case: potentialities with tautological manifestations, or ‘tautological potentialities’. 4

Note that x in Fx is bound by ∃ and therefore not by λ.

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5.3 Difficult cases The aim of this section is to establish that all, even the most surprising, potentiality ascriptions that can be formulated in the language I have sketched can be made sense of, and (with some obvious exceptions, such as contradictions) are true under some interpretation. There are two questions that need to be addressed with respect to each of these potentiality ascriptions. A first one is the syntactic question whether we should regard the sentence as well-formed, and hence as even a candidate for truth or falsity. The second is the semantic question whether there are true instances of the sentence. The syntactic question is, of course, informed by semantic considerations in the broader sense: namely, the question whether we can understand or make sense of the relevant potentiality ascription in the first place. (Your first reaction to the claim ‘I have a potentiality to be such that something is red’ may be: I do not even understand what that means.) I will argue that we should accept each of the apparently problematic forms of potentiality ascriptions as well-formed because we can make sense of them, and I will support the claim that we can make sense of them by offering for each of them some interpretation and some situation such that the sentence under that interpretation would be true in the situation.

5.3.1 Tautological potentialities Before I turn to the problematic potentiality ascriptions with which the last section ended, I will discuss another case that might worry some readers and will be crucial in the discussion to come: potentialities with tautological manifestations. There are various ways of formulating such potentialities. I will concentrate on two simple cases, but my arguments are easily transferred to any other potentialities with tautological manifestations. The two cases are (3)

a. POT[λx.(Fx ∨ ¬Fx)](a), b. POT[λx.x = x](a).

The former might ascribe to me a potentiality to dance or not to dance, the latter a potentiality to be self-identical. Does anything have such potentialities? We may want to resist potentiality ascriptions such as (3-a) and (3-b) on the grounds that such potentialities, if they were possessed, would have to be constantly exercised, or on the related grounds that the bearer of such a potentiality could not ‘make a difference’ to the exercise of those potentialities. Nothing has a potentiality not to dance-or-not-dance, or not to be self-identical. So what’s the point of potentiality ascriptions such as (3-a) and (3-b)?

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In chapter 3, we have already dealt with potentialities that have to be constantly exercised because their possessors have no potentiality for the opposed manifestation.5 There I argued that we should not only accept such potentialities, but that we should think of them as potentialities possessed to the maximal degree. For the arguments to that effect, the reader is referred back to chapter 3.5. In fact, the view developed in that chapter concerning degrees of potentialities yields a principle which should lead us directly to accept the potentiality ascriptions above. On that view, an object x has the potentiality to F to the maximal degree just in case x lacks the potentiality not to F (and vice versa). It follows directly that for any pair of opposing potentialities—the potentiality to F and the potentiality not to F—every object must possess at least one member of that pair. For the lack of one member of the pair guarantees that the other member is possessed to the maximal degree. We have, thus, a law of excluded middle for potentiality: (EM) POT[λx.φ](x)∨ POT[λx.¬φ](x). Suppose that someone wished to maintain that potentiality statements such as (3-a) and (3-b) were never true: that nothing could have a potentiality to dance or not to dance, or a potentiality to be self-identical. By (EM), it would follow that everything had the potentiality not to dance-or-not-dance, or not to be selfidentical. Ascribing those potentialities is no better, and indeed seems much worse, than ascribing the potentialities to be self-identical or to dance-or-notdance. Hence we must accept that things have the original potentialities. Such an argument would indeed make short work of the opposition to any contested potentiality ascriptions. In many cases, however, it would be unfair. The opponent of the relevant potentiality ascriptions may not want to assert either the ascriptions themselves or their negations. She may hold that those ascriptions are not so much false as meaningless, and so they should not be allowed to be well-formed sentences.6 In that case, the syntactic clause for the POT operator would have to contain certain restrictions, allowing, for instance, no identity sign in the square brackets that follow POT. Then (EM) would not apply, since clearly it holds only for well-formed potentiality ascriptions. 5 Note that the tautological potentialities discussed here differ from the examples in chapter 3.5 in two ways. First, they are possessed, to the maximal degree, by everything that there is. Second, the tautological potentialities can be iterated any number of times and are still possessed to the maximal degree: just as I have no potentiality not to be self-identical, so I do not have a potentiality to have a potentiality not to be self-identical, or a potentiality to have a potentiality to have a potentiality not to be self-identical . . . and so forth. 6 I am inclined to adopt this line on potentiality ascriptions that involve empty names, so I am not in principle opposed to it.

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The argument from (EM) should be judiciously applied. But in the case of (3-a), it can be applied. For there is nothing syntactically wrong with (3-a): we have seen that negation and disjunction should be allowed in the scope of POT. The tautological potentiality ascription is of the same form as (1-b), which is clearly acceptable. If syntactic acceptability is built up compositionally (as it should be), then we cannot reject (3-a) on syntactic grounds. The parallel with the more ordinary disjunctive potentiality ascriptions extends from the syntactic features of (3-a) to its truth-conditions. Recall the potentiality to break in two pieces or in more than two pieces. The potentiality is exercised whenever its bearer breaks. It may be exercised by breaking in two pieces or breaking in more than two pieces, as the case may be. The disjunctive potentiality is exercised whenever a potentiality for one of its disjuncts is exercised: the potentiality to break in two pieces or the potentiality to break in more than two pieces. Likewise for (3-a). The potentiality to dance or not to dance may be exercised by dancing or by not dancing, as the case may be. It is exercised whenever a potentiality for one of its disjuncts is exercised: the potentiality to dance or the potentiality not to dance. Of course, one of these two potentialities is always exercised. But that makes the potentiality to dance or not to dance a limiting case of disjunctive potentialities, not a case that is unlike the disjunctive potentialities we have already accepted. We feel that in order to exercise a potentiality, an object has to do something, though at this stage we can require this only in a very minimal sense of ‘do’. But as the case of disjunctive potentialities in general shows, an object need not always do the same thing in manifesting one potentiality. It may break in two pieces, or in 1027 pieces; it may dance, or it may not dance. Given that (3-a) must be accepted, (3-b) cannot easily be rejected simply because the ascribed potentiality is always exercised. We might still wonder what an object does in exercising this potentiality. The answer is: anything whatsoever. To be self-identical, an object must do something, in the minimal sense at issue, for it must at least exist. In chapter 3.5, I offered a pragmatic explanation of our reluctance to ascribe potentialities of maximal degree: after all, there is a more informative thing to say, namely, that the object has the manifestation property. In the present case, even the ascription of the manifestation property is a rather uninformative thing to say, because everything has those tautological properties. But again, this does not make the ascription of such properties false; it merely makes it uninformative. A similar line of thought is widely accepted for tautological properties such as dancing-or-not-dancing or being self-identical. For the same reasons it should be accepted for the corresponding potentialities: potentialities

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to dance-or-not-dance, to be self-identical, and any other potentiality with a tautological property as its manifestation. So much, then, for tautological potentiality ascriptions. I have argued that we have good reason to accept that such potentiality ascriptions can be, and indeed are, true. We now turn to the examples that were pointed out at the end of the previous section: quantified and Cambridge potentiality ascriptions.

5.3.2 Quantified potentialities We begin with the case of existential quantification. Here are some examples of true ascriptions of such potentialities. I have an ability to walk. Suppose I exercise that ability. I am walking, and in virtue of my walking it is true that something is walking. We can truly ascribe to me the (no doubt abundant) property of being such that something walks. Suppose, implausibly of course, that nothing else is walking. Then I acquired the property of being such that something walks by manifesting my ability to walk. Suppose, on the contrary, that other individuals are also walking. Then I have not acquired the property to be such that something is walking, but I have it on different grounds now, and I do so, again, by manifesting my ability to walk. I have the property of being such that something walks, and I have it in virtue of manifesting my own potentialities. Should we not say that, by having a potentiality to walk, I have a potentiality to be such that something walks? Another kind of case is less easily phrased in formal terms. It concerns, roughly, the potentiality to produce something which is F. I have a potentiality to give birth to a daughter; hence, it would seem, I have a potentiality to be such that something is my daughter. A craftsman has the potentiality to make a beautiful chair; hence, it would seem, he has a potentiality to be such that there is a beautiful chair. We can bring this out by contrast. I have a potentiality to walk; I have no potentiality to lay eggs. Thus I am differently related to the property of walking than I am to the property of laying eggs. I am also thereby differently related to the property of being such that something walks than I am to the property of being such that something lays eggs. In the former case, the contrast is between having and lacking a potentiality. What else should it be in the latter case?7 7 This will be modified in a moment: I have an extrinsic potentiality to be such that something lays eggs, in virtue of having the extrinsic (Cambridge) potentiality to be such that a given hen lays eggs. The point still stands: I am differently related to the property of being such that something walks than I am to the property of being such that something lays eggs: namely, by having an intrinsic potentiality for the former but only an extrinsic potentiality for the latter. The same qualification applies to the next example.

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A craftsman has the potentiality to make a beautiful chair, but a squirrel does not. The craftsman and the squirrel are differently related to the property of making a beautiful chair; they are also differently related to the property of being such that there is a beautiful chair. Given that the former is a contrast between having and lacking a potentiality, what else should the latter be? Generalizing from this case, we should say that (at least) anything which has a potentiality to be or to produce an F thereby has a potentiality to be such that something is F. There are, then, at least two kinds of cases in which an object, a, can be said to have a potentiality to be such that something is F: when a itself has a potentiality to be F, and when a has a potentiality to produce an F. Anticipating the topic of section 5.3.3, a third and fourth kind of case can be added: a may have a potentiality to be such that something is F in virtue of having an (extrinsic) potentiality to be such that a different object, b, is F; or in virtue of having an (extrinsic) potentiality to be such that b produces an F.8 Given potentiality ascriptions that involve existentially quantified sentences, we can make sense of those that involve universally quantified sentences. For applying the standard interdefinability of ∃ and ∀, an object a will have a potentiality to be such that everything is F just in case a has a potentiality to be such that it is not the case that something is not F; that is, we can understand (4) in terms of (5): (4) POT[λx.∀xFx](a), (5) POT[λx.¬∃x¬Fx](a). Further, by (EM), it is sufficient—though not necessary—for the truth of (5) (and thereby the truth of (4)) that x does not possess the potentiality to be such that something is F, i.e. that (6) ¬ POT[λx.∃x¬Fx](a). The same, of course, holds where the sentences in the scope of the quantifiers are more complex than Fx. For instance, assuming the necessity of origin,9 I have no potentiality to be such that something that is derived from a different sperm and egg is identical with me; hence, by (EM), I do have a potentiality to be such that nothing derived from a different sperm and egg is identical with me; and hence, 8 I believe that those are all the cases that there are, with the addition (if it is an addition and not a case of production) of potentialities to constitute an object that is F. Chapter 7.5 will return to this limitation. In this chapter, my claim is not a restrictive one but rather the positive claim that there are at least these cases. 9 For more on this thesis, and its potentiality-based version, see chapter 6.2.

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by the interdefinability of ∃ and ∀, I have a potentiality to be such that everything derived from a different sperm and egg is distinct from me.

5.3.3 Cambridge potentialities: extrinsic potentiality revisited We now turn to our final case: ascriptions of Cambridge potentialities of the form exemplified in (2-b): POT[λx.Fb](a), or ‘I have a potentiality to be such that you are sitting.’ Such potentiality ascriptions we have already encountered in chapter 4, as a special case of extrinsic potentialities. Let us briefly recapitulate the central points from that chapter. Chapter 4 introduced a notion that will be crucial for the logical and semantic applications of potentiality: extrinsic potentiality. Extrinsic potentialities, it was argued, are grounded in the (intrinsic) joint potentialities which their bearers possess together with other objects. Those other objects we have called the joint potentiality’s co-possessors and the extrinsic potentiality’s dependees. It was also argued that there are two different kinds of joint potentiality: those of type 1, whose manifestation consists in a relation between, or a plural property of, all the objects which jointly possess it; and those of type 2, whose manifestation consists in a property or relation which, if the potentiality were manifested, would belong to only some of the potentiality’s joint possessors. A key and a door’s joint potentiality to stand in the relation of opening is of the first kind; a glass and styrofoam’s joint potentiality to be such that the glass breaks is of the second. Further, these two types of joint potentiality give rise to three types of extrinsic potentiality. Type 1 joint potentialities yield (a) extrinsic potentialities whose manifestation consists in a relation to (or a plural property with) the potentiality’s dependees. Type 2 joint potentialities yield (b) extrinsic potentialities whose manifestation consists in a property of the potentiality’s bearer itself, as well as (c) those whose manifestation consists in a property or relation, not of the extrinsic potentiality’s possessor, but of the dependee(s). Schematically (and using the more familiar case of relations rather than that of plural properties), the three types may look as follows: (7)

a. POT[λx.Rxb](a), b. POT[λx.Fx](a), c. POT[λx.Fb](a).

Recall, again, the convention that a = b. Statements of type (7-c) are precisely what is at issue now: ascriptions of Cambridge potentialities. Note that, due to the occurrence of another object’s name, b, in the specification of the manifestation, (7-a) and (7-c) can only express extrinsic potentialities, while (7-b) may be used to ascribe either an intrinsic or an extrinsic potentiality.

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Cases of the form (7-a) or (7-b) (read as an extrinsic potentiality) are relatively uncontroversial, or so I have argued in chapter 4. (7-a) might express a key’s potentiality to open a particular door, and (7-b) a city’s extrinsic potentiality to be harmed by attacks, a potentiality whose dependees include an external defence system if there is one. Cases of the form (7-c) fall out of the metaphysics for free, but are much less intuitive: they include the defence system’s potentiality to be such that the city is harmed, the styrofoam’s potentiality to be such that the glass breaks, or my potentiality to be such that you are sitting. I have stated my metaphysical motivation for accepting such cases in chapter 4, and the metaphysics of that chapter can stand on its own.10 Nevertheless, the resources of this chapter allow us to see from a different angle why such a metaphysics is needed. Suppose that we wanted to reject, as a matter of principle, any ascriptions of Cambridge potentialities, such as (7-c). We might do so on the level of syntax, or else on the level of semantics. That is, we might exclude the relevant sentences from the well-formed formulas of our language (on the intuitive grounds that they are meaningless), or else we might accept that they are well-formed but claim that they are all false. The syntactic restriction would not be easy to formulate. It cannot require that there is a free occurrence of x in the scope of λx in the square brackets, for we have just seen some admissible cases in which there is no such occurrence, quantified potentiality ascriptions such as (2-a). Nor can the requirement be that there are no occurrences of names other than a, the name of the potentiality’s bearer, for there are clearly admissible (extrinsic) potentiality ascriptions in which there are such occurrences, such as (7-a) above. Nor does it help to combine the two requirements disjunctively, for both are violated by true potentiality ascriptions of the form (8) POT[λx.∃xRxb](a), 10 With the resources of this chapter, we can now formulate the transition principles from joint potentialities to extrinsic potentialities, given in chapter 4.5, in a more general way, covering type 1 and type 2 joint potentialities together. The principles go as follows:

Plural Properties If POT[λxx.φ](aa) is true, a is one of aa, and φ ∗ is the result of replacing any free occurrence of xx in φ by aa, then POT[λx.φ ∗ ](a) is true. Relations If POT[λx1 . . . λxn .φ](a1 , . . . , an ) is true, 1 ≤ i ≤ n, and φ ∗ is the result of replacing each free occurrence of any of x1 , . . . , xn except xi by the corresponding a1 , . . . , an , POT[λxi .φ ∗ ](ai ) is true. In the case of type 2 joint potentialities, φ will concern only some of the aa (for plural properties) or relate only some of a1 , . . . , an (for relations). The two transition principles still apply in those cases. A Cambridge potentiality ascription results when a (for plural properties) or ai (for relations) is among those of the joint potentiality’s possessors that are not concerned or related by φ. In the relational case, this will be visible by there being no free occurrence of xi in φ (nor, as a consequence, in φ ∗ ).

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of which there are true instances: for instance, given a key’s potentiality to open a particular door b, the key has a potentiality to be such that something opens that particular door, a potentiality that may be ascribed by (8). Note, again, that absent a syntactic restriction to exclude the ascription of Cambridge potentialities, we can argue by (EM) that if potentiality ascriptions such as (7-c) are false, their negative counterparts must be true. But the negative counterparts, i.e. sentences such as (9) POT[λx.¬Fb](a), appear no more acceptable than the original Cambridge potentiality ascriptions. Rather than try to devise the right restriction to delimit the ‘good’ potentiality ascriptions, we should reconsider where that restriction is to be placed and why. I hold that there is no good answer to this question. I would like to substantiate this claim by presenting, in the remainder of this section, a ‘slippery slope’. I shall start with potentiality ascriptions that are clearly (or have been argued to be) acceptable and proceed by introducing potentiality ascriptions that are relevantly similar so that, on pain of drawing arbitrary boundaries, we should find them acceptable too. By repeating the process, we finally reach Cambridge potentiality ascriptions. My challenge to the opponent of such ascriptions is to locate, in that spectrum of cases, a non-arbitrary boundary between the acceptable and the non-acceptable cases. I do not see how the challenge is to be met, and I reject arbitrary boundaries in the metaphysics of potentiality. Thus I accept Cambridge potentiality ascriptions. My range of cases starts with tautological potentialities. We have seen that objects may have such potentialities, such as the potentiality to dance or not to dance. My earlier examples involved potentialities that were most naturally read as intrinsic, but there is no reason not to apply the argument to (the unproblematic examples of) extrinsic potentiality. Thus we might say that a key has a potentiality to open-or-not-open a particular door, a potentiality which it exercises by opening the door or by not opening it, as the case may be. Thus we have a true interpretation of (10) POT[λx.(Rxb ∨ ¬Rxb)](a). In addition, we need to consider another kind of logically complex extrinsic potentiality. I have a potentiality to walk on High Street, Oxford. This potentiality, expressed in sentences such as (7-a) above, can be exercised in various conditions: I might exercise it by walking on High Street while it is clean and dry, or by walking on High Street while it is wet. It seems right to say, then, that I have a

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potentiality to walk on High Street while it (High Street) is wet. By contrast, I have no potentiality to walk on High Street while it is flooded. Flooded streets simply do not allow for walking on them; a flood on High Street works as a mask to our joint potentiality to stand in the walking-on relation, and hence to my resulting extrinsic potentiality. (I am assuming that by ‘flooded’ we mean ‘very seriously flooded’). Thus the sentence (11) POT[λx.(Rxb ∧ Fb)](a) has both a true interpretation (with a referring to me, b to High street, R ascribing the walking-on relation, and F the property of being wet) and one that is presumably false (as before, but with F ascribing the property of being flooded). We can now start the slippery slope. The interesting feature of (10) is the tautological nature of the manifestation; the interesting feature of (11) is the second conjunct in its manifestation. Of course, the object referred to in that conjunct is one to which the potentiality’s bearer is intuitively ‘linked’ by the relation of the first conjunct. We can now combine the interesting features of both. Neither feature is objectionable on its own; there is no prima facie reason why they should be objectionable when combined. Hence a sentence of the form (12) POT[λx.((Rxb ∨ ¬Rxb) ∧ Fb)](a) should be unobjectionable too. It might ascribe to me a potentiality to walk or not to walk on High Street while it is wet. (12) already shares one feature with Cambridge potentiality ascriptions: whether or not the potentiality is exercised depends, intuitively, not on what I (the potentiality’s possessor) do, but on whether or not High Street is wet. But we should not reject (12) either because this exercise of the potentiality that it ascribes depends on something not done by its bearer; the same is true of many unobjectionable extrinsic potentialities, such as my potentiality to dance with you, or a key’s potentiality to open a particular door. In both cases, some doing is required from another object—some dance movements on your part, and certain changes in the position of its lock on the part of the door—for the potentiality to be exercised. Nor should we reject (12) on the grounds that its manifestation does not require any doing on my part. It does; but, as with straightforward cases of tautological potentialities such as (10), the doing can be done in different ways. My contribution to the exercise of the potentiality in (12) may consist in walking on High Street or in not walking on High Street, as the case may be. Given the acceptability of (12), it would be arbitrary to reject (13) POT[λx.(x = x ∧ Fb)](a). The only difference between (13) and (12) is that the tautological first conjunct in (13), unlike that in (12), no longer provides a ‘link’ to the object in the

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second conjunct. But given the thin nature of the link in (13), this seems an odd place to draw the boundary. (13), like (12), is exercised by its bearer’s doing something, albeit in a minimal sense of ‘do’, namely, by the bearer’s existing; and by another object b’s doing something in a slightly less minimal sense, say, being wet. The potentiality is never exercised without a’s doing what it has to do: a could not be self-identical without existing. The final step is from (13) to (14): to exercise a potentiality such as the one ascribed in (13), an object has to do no more than to exist. But that minimal doing, made explicit in the first conjunct in (13), is required for any potentiality to be exercised anyway. Hence we might as well drop the first conjunct, yielding our target Cambridge potentiality ascription (14) POT[λx.Fb](a). Expression (14), like (13), ascribes a potentiality whose exercise requires some minimal doing on the part of its bearer—even to bear a ‘such that’ property, an object has at least to exist—and some, perhaps less minimal, doing on the part of something else, the referent of b. If (13) is acceptable, then so is (14). Let me stress again that the slippery slope which I have just presented is not intended as a demonstrative argument. Rather, it is a challenge, to anyone who holds that such sentences as (14) are never true, to specify where the boundary is drawn between acceptable and unacceptable potentiality ascriptions and to show that the boundary is not arbitrary. (Analogous slippery slopes can be constructed, and so the same challenge holds, for Cambridge potentiality ascriptions of greater complexity than (14).) We have now seen three reasons to think that sentences of the form POT[λx.Fb](a) are sometimes true. First, the potentialities which they ascribe are part of a systematic metaphysics of extrinsic potentiality: that metaphysics, as given in chapter 4, delivers such potentialities as a side-product in accounting for the more obviously plausible examples of extrinsic potentiality. Second, a principled restriction on predicates in the scope of POT that appeals to objectual content is bound to rule out too much. Third, any alternative restriction is faced with the prospect of drawing what seems, on the face of it, an arbitrary line within the spectrum of cases from (10) to (14), or at least with the challenge of showing that such a line would not be arbitrary. Both formal and metaphysical considerations, then, support the truth-eligibility of sentences ascribing potentialities for Cambridge manifestations. The truth of some such sentences in turn is indispensable for a logical principle that I will defend below: the closure of potentiality under logical implication.

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The language that we have developed has sufficient expressive power to account for possibilities that φ, for any logical form of φ, in terms of a corresponding potentiality. Thus the first part of the challenge of formal adequacy is met. Before we move on to the second part—capturing the logical structure of possibility in terms of potentiality—we should note an important application of the language that we have developed: defining iterated potentiality.

5.4 Defining iterated potentiality The formal language that we now have at our disposal provides us with the resources to express iterated potentiality formally. Note that the operator POT expresses only potentiality simpliciter or, strictly speaking, once-iterated potentiality. But we can now use it to define an operator POT∗ to express iterated potentiality quite generally. Iterated potentialities are potentialities for the possession of further potentialities, such as my ability to learn to play the violin, a green apple’s disposition to acquire the disposition to appear red to a normal observer, or liquid water’s potentiality to turn to ice and acquire the potentiality to break. Iterated potentialities include those potentialities whose manifestation consists in the acquisition by another object of a potentiality, such as a violin teacher’s ability to teach me to play the violin, a textile dye’s disposition to endow a piece of cloth with the disposition to look red to a normal observer, or my freezer’s potentiality to turn water fragile by freezing it. How are we to formulate such iterated potentialities? In the simplest case, they might have the form (15) POT[λx. POT[F](x)](a), ascribing to a an iterated potentiality to be F. But as we have seen, the ultimate manifestation of an iterated potentiality may concern objects other than the bearer of the initial potentiality. (The initial potentiality is the one that is ascribed by the entire formula; the ultimate manifestation is given by the predicate that occurs in the scope of a POT operator without itself containing any further POT operators; in the case of (15), the ultimate manifestation is being F). In chapter 4, I have argued that such iterated potentialities, for instance that of the violin teacher, involve three steps: the teacher has an ability to enter, with any student, into a joint potentiality to be such that the student acquire the ability to play the violin. Hence the teacher has a three-times iterated ability, the ultimate manifestation of which is that a student plays the violin. Where a stands for the teacher, b for the student, cc for the plurality of them both, and F ascribes the property of playing the violin, we can formulate this case as

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(16) POT[λx. POT[λyy. POT[F](b)](cc)](a).11 In (16), we have focussed on a particular student (b is an individual term). Of course, the teacher’s skill is more general: she has a potentiality not just for this particular student, but for some student or another, whoever they are, to acquire the ability to play the violin. To cover this case, we need existential quantification to replace both the individual term b (for the student) and the plural term cc (for the teacher-and-student). Instead of (16), then, we get: (17) POT[λx.∃yy POT[λyy.∃z. POT[F](z)](yy)](a). (17) ascribes to the piano teacher (a) a potentiality to be such that some things have a potentiality to be such that something has a potentiality to play the violin. (17), of course, is ‘an unlovely mouthful’.12 But it seems to cover the case of iterated potentiality in the most general way: iterated potentialities are potentialities to be such that something has a potentiality (to be such that something has a potentiality, and so on). The ‘something’ might be the iterated potentiality’s possessor, as in (15); or a particular other object, as in (16); or it may be unspecified, as in (17). I propose that we adopt the existentially quantified version as the most general expression of iterated potentiality. It is not, to be sure, always the most nuanced or informative way to ascribe a given iterated potentiality. But it has the virtue of covering the different cases, such as (15)–(17), precisely by prescinding from their differences. And this, in turn, will allow us shortly to define a general operator for iterated potentiality, POT∗ , that is easier to wield than ascriptions such as (17). One last modification is required, however, before we can do so. I have argued in chapter 4 that iterated potentialities are potentialities ‘for (it to be the case that) p’: their ultimate manifestation is not always, and hence should not in general be understood as, a property of the initial potentiality’s bearer, since the bearer need no longer exist when the ultimate manifestation occurs. If we tried to read the iterated potentiality’s manifestation off from (17), we would find only a property: playing the violin, the property ascribed by F. But the ultimate manifestation of this iterated potentiality should be that someone plays the violin. In the earlier case exemplified by (16), on the other hand, the ultimate manifestation should be that b plays the violin. Strictly speaking, even in the simplest case expressed with (15) the manifestation of the iterated potentiality is not playing the violin, but that a plays the violin. To make potentiality ascriptions in our formal language conform to the canonical expressions developed in chapter 4.6, I propose that we rewrite (15)–(17) as follows: 11 12

I use plural predicates and variables, but the idea is easily generalized to relations. As David Lewis said about his ‘Reformed Conditional Analysis’ of dispositions.

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(18) POT[λx.∃y POT[λy.Fa](y)](a), (19) POT[λx.∃yy POT[λyy.∃z POT[λz.Fb](z)](yy)](a), (20) POT[λx.∃yy POT[λyy.∃z. POT[λz.∃zFz](z)](yy)](a). The addition of existential quantifiers throughout the various stages of these iterated potentialities is a way of disregarding the identity of any particular object that might be involved in those stages; it ensures that we focus not on the intermediate stages, but on the iterated potentiality’s ultimate manifestation. That ultimate manifestation is going to be all we need when we put iterated potentiality to use in formulating an account of possibility. Once more, my investigation into potentiality is driven by my target, an account of possibility. But it is so driven only in its focus, not in its content. We can now define an operator to express iterated potentiality quite generally, with a focus only on its ultimate manifestation. We do so in two steps. First, we define an operator for n-step potentiality, as follows. An object x has a one-step potentiality for φ just in case x has a potentiality to be such that φ; x has an (n + 1)-step potentiality for φ just in case x has a (one-step) potentiality to be such that something has (or some things have) an n-step potentiality to be such that φ; x has an iterated potentiality for φ just in case for some n, x has an n-step potentiality for φ. An iterated potentiality is simply an n-step potentiality for any value of n. Formally, we introduce the operators POTn for n-step potentiality, and POT∗ for iterated potentiality in general, by stipulating that, for any natural number n greater than zero, individual or plural term t and individual or plural variables x, and open or closed sentences φ, the following are sentences: POTn [φ](t); POT∗ [φ](t). POTn [φ](x) is defined inductively in terms of POT: 1. POT1 [φ](t) =df POT[λx.φ](t), 2. POTn+1 [φ](t) =df POT[λx.∃x POTn [φ](x)](t). The truth conditions for POT∗ [φ](t) are then defined in terms of POTn : (POT∗ )

POT∗ [φ](t) is true just in case for some n, POTn [φ](t).

Some comments are in order. First, the definition of n-step potentiality is inductive and can be applied to any (natural) number n. There is no upper bound on the number of times that a potentiality can be iterated.

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Second, I have quantified over numbers in defining iterated potentiality. I am not, however, thereby committed to the existence of numbers. The definition could be given as an infinite disjunction of n-step potentialities; quantifying over n is merely a matter of convenience and brevity. (For a suggestion of how to spell this out more formally, see the appendix.) Third, as before, I have included potentiality simpliciter as once-iterated potentiality; but my definition explicitly includes only potentialities to be such that φ. For the sake of having a catch-all phrase, I will adopt the following convention: when an object, x, has a potentiality to be , I shall say that x has an iterated potentiality for x (itself) to be . This is innocuous, for if x has a potentiality to be , x thereby has a potentiality to be such that x is , and vice versa. (Mutual entailment may or may not amount to identity either of the potentialities or of the state of affairs of x possessing them, but it is good enough for my purposes.) That potentiality can then be expressed, in accordance with my definition, as a one-step potentiality for x to be . (Note that we can apply this little trick even when ‘’ is already of the form ‘such that φ’: thus x’s having a one-step potentiality for φ entails x’s having a one-step potentiality for x to be such that φ.) We now return to the main thread of this chapter and move on to the second part of the challenge of formal adequacy: capturing the logical structure of possibility. To do so, we need to examine the logical structure of potentiality. This is the aim of the remaining sections of this chapter.

5.5 Introducing the logic of potentiality We now have at our disposal a formal language with sufficient flexibility and expressive power to serve in a logic of potentiality. In the remainder of the chapter, I will be concerned to develop and defend this logic. The formal statement of the logic can be found in the appendix. Here, I will provide a metaphysical defence of its principles. The ultimate goal, of course, is to demonstrate formal adequacy: the logic of potentiality, together with the definition of the possibility operator in terms of it, will yield the right logic for possibility. Formally, my strategy in showing this is to take the path of least resistance: I argue that potentiality already has the formal features that make for the right logic (modulo its status as a predicate operator). Metaphysically, this is by no means the path of least resistance, and the aim of my remarks in what follows is to make my claims metaphysically palatable. Much of the metaphysical work, however, has been done in chapters 3–4 and the first half of this chapter. I will heavily draw on what has already been said. I will continue to focus, as in sections 5.2–5.3, on the potentialities of individual objects, joint potentialities being covered by simple generalization.

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To begin, let me put my cards on the table. I will argue that the logic of potentiality is governed by a closure rule and three axioms: CLOSURE Potentiality is closed under logical equivalence: If being  is logically equivalent to being , then having a potentiality to be  is logically equivalent to having a potentiality to be .13 DISJUNCTION Potentiality distributes over, and is closed under, disjunction: An object has a potentiality to be -or- if and only if it has a potentiality to be  or a potentiality to be . ACTUALITY Potentiality is implied by actuality: Anything which is  must also have a potentiality to be . NON-CONTRADICTION Nothing has a potentiality to be such that a contradiction holds. To argue for both CLOSURE and the right-to-left direction of DISJUNCTION, I will appeal to a closure principle that is stronger than CLOSURE: the principle that potentiality is closed even under one-directional logical implication. That principle will be defended in section 5.7.1, but it is worth stating here: CLOSURE1 Potentiality is closed under logical implication: If being  logically implies being , then having a potentiality to be  logically implies having a potentiality to be . In the formal language that we have developed, the four principles for potentiality can be formulated as follows. With a suitable semantic definition of , and holding fixed the reading of POT as an operator ascribing potentiality, the following are validated. The CLOSURE rule becomes: ClosurePOT

If  t ≡ t, then  POT[](t) ≡ POT[ ](t).

The following three are axioms, which correspond to DISJUNCTION, ACTUALITY, and NON-CONTRADICTION: POT∨ POT[λx.(φ ∨ ψ)](t) ≡ POT[λx.φ](t) ∨ POT[λx.ψ](t), TPOT t → POT[](t), NCPOT ¬ POT[λx.⊥](t). 13 Logical equivalence between properties (or, more formally, between predicates) is less familiar than logical equivalence between sentences, but easily defined in terms of it: being  is logically equivalent to being just in case t is logically equivalent to t. An analogous definition can be used for logical implication between properties, as in CLOSURE1. Note also that, again, I am using the locution ‘potentiality is closed under logical implication (or equivalence)’ as a shorthand to say that the manifestations of any given object’s potentialities are closed under logical implication (or equivalence).

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(⊥ stands for an arbitrary contradiction.) The formal statements will be important in the more formal development of the logic, to be provided in the appendix and sketched briefly in chapter 6.4. In the remainder of this chapter, the informal statements will serve us well enough, and my (informal) reasoning will be directed at them. I will speak indiscriminately of the four principles that govern the logic of potentiality, ignoring (when I do so) the difference between rule and axioms and between the informal and the formal statements. The four principles above will guarantee that possibility, on the potentialitybased view, has the right logical structure. They do so by endowing potentiality itself with much the same logical structure as that possessed by possibility. As we shall see in the next section, the principles for POT correspond exactly (modulo its status as a predicate operator) to the logical principles which the possibility operator 3 must minimally obey, if we are to read it as metaphysical possibility. In defending the four principles, I will proceed as follows. For each principle in turn, I will begin by stating its intuitive motivation and then proceed to consider, and reject, challenges to it. In defending the principles, it will be useful to draw on and contrast with existing logical systems. There is as yet no literature on the logic of potentiality as I conceive it (unsurprisingly, since I have introduced the term as a technical notion in chapter 4). The closest phenomenon to potentiality that is discussed in the literature is dispositionality. But the logic of dispositions and dispositionality, too, is not a subject that has attracted much attention, mostly because dispositions are standardly assumed to be counterfactual-like, and hence to follow the logic of counterfactual conditionals if any. I have argued for a different conception of dispositions in chapters 2 and 3: dispositions are individuated by their manifestation alone, not by a pair of stimulus and manifestation, and whether a potentiality counts as a disposition is a matter of its having a contextually specified minimum degree. In the following section, I will occasionally contrast the general logic of potentiality with the special case of dispositions. But there is no well-established logic of dispositions that we could draw on. In some ways, therefore, my discussion in what follows is a discussion between me and myself. To keep it from becoming esoteric, I have chosen to relate my discussion to three related phenomena whose logic has been studied in the literature. The objections to my favoured principles for potentiality will stem largely from the parallels with those related phenomena, transferring objections and alternatives that have been suggested in their case to the case of potentiality. In no case does my discussion constitute decisive proof that the principle at issue holds. Decisive proofs are rare in philosophy in any case. However, I would like to put particular emphasis on this point for the present chapter. The discussion that follows aims to be the start of a debate, not its endpoint.

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Before going through the logical principles that I have proposed, the next section will introduce the three phenomena whose logic has been studied in sufficient detail to serve as a point of contrast and reference here: possibility, essence, and ability. They are a rather mixed bunch, held together for present purposes by their dialectical relation (or rather, by their having some dialectical relation) to the logic of potentiality.

5.6 Comparisons: possibility, essence, ability The following table provides a synopsis of how the logic of potentiality compares to logical systems aimed at treating related phenomena: possibility, essence, and ability. In addition, I provide a column on what will shortly be introduced under the name ‘E-potentiality’, the dual of Finean essence; as well as one on dispositions to make explicit any contrasts where they are relatively uncontroversial. Since they lack a worked-out logic so far, dispositions will not be among the points of comparison. Brackets mark deviations from the logic of potentiality that I do not take to be interesting for comparison. In the remainder of this section, I will then sketch the other systems for possibility, essence, and ability, and their motivation, insofar as it is relevant for our purposes. (1) Possibility. The most familiar and, given the project of this book, the most obvious point of comparison is modal logic: the logic of possibility and necessity. There are various systems of modal logic, but it is generally acknowledged

Potentiality

Possibility

Essence

(System T)

(Fine)

E-Potentiality Ability Dispositions (Dual of

(Kenny)

essence) Closure under logical

yes:

yes

restricted:

restricted:

CLOSURE

no objectual

no objectual

content lost

yes:

yes

equivalence Closure under logical

over

?

no

?

no

no

content added CLOSURE1

implication Distribution

yes

yes:

yes

restricted:

restricted:

no objectual

no objectual

content added

content lost

(no)

DISJUNCTION

restricted: no objectual

disjunction

content lost

Implication

yes:

by actuality

ACTUALITY

Exclusion of

yes:

contradiction

NONCONTRADICTION

yes

(no)

yes

no

no

yes

yes

no

yes

yes

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that the weakest system which has any claim to capture metaphysical (as opposed to, say, deontic) modality is system T. There are a number of alternative ways to minimally characterize system T. One of them, which is best suited to our purposes, consists in a rule of closure under logical equivalence plus three axioms that correspond exactly (again, modulo the distinction between predicate operator and sentence operator) to DISJUNCTION, NON-CONTRADICTION, and ACTUALITY: possibility is closed under and distributes over disjunction; no logical falsehood is ever possible; and possibility implies actuality.14 This parallelism is of great importance for the project of a potentialitybased theory of possibility. Given the bridge principle that it is possible that p just in case something has an iterated potentiality for p, it can be shown that the principles for potentiality entail the corresponding principles for possibility. For the requirement of formal adequacy to be met, it is therefore of paramount importance that these principles can be argued to govern potentiality. The main challenge arising from the comparison between potentiality and possibility is not their difference but their very similarity, which may seem to threaten not the truth, but the interest and informativeness of a potentialitybased account of possibility. This challenge will be taken up in section 5.9, once the principles for potentiality have been developed and defended. Until then, the comparison with possibility is postponed; it will play no role in the defence of the four principles throughout 5.7. (2) Essence and its dual. Finean essence, I have said in chapter 1, stands to necessity much as potentiality, on my view, stands to possibility. Essence and potentiality are both localized: they are properties of particular objects. Necessity 14 Formally, with the standard operator 3 for possibility, the relevant rule and axioms are as follows:

Closure3

If  φ ≡ ψ, then  3φ ≡ 3ψ,

(3∨)

3(φ ∨ ψ) ≡ (3φ ∨ 3ψ),

(T3)

φ → 3φ,

(3⊥)

¬3⊥.

Given the interdefinability of 3 and the necessity operator , this is equivalent to the more usual way of characterizing system T in terms of the necessitation rule and two axioms for : Nec

If  φ, then  φ,

(K)

 (φ → ψ) → (φ → ψ),

(T)  φ → φ. See Chellas (1980, 114–118) for the different axiomatizations that are available for a normal modal logic.

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and possibility, on the other hand, are not localized: they are just the necessity or the possibility that so-and-so, with no relativization to any object or objects in particular. It might be expected, then, that there are some close analogies between the logic of essence and that of potentiality, in particular as they relate to the logic of necessity and possibility. So let us have a brief look at the logic of essence. (In the interest of brevity and easy comprehensibility, I will skip over a number of formal details that are irrelevant for my purposes.) Essence, in Fine’s logic, is expressed by a sentence operator x , to be read ‘it is true in virtue of the essence of x that . . . ’.15 The logic of essence, according to Fine, is structurally very similar to the logic of necessity. It is not, however, identical with it. (We cannot, that is, take the axioms and rules that are given for systems of normal modal logic and simply replace the necessity operator  throughout with an essence operator x ). Fine notes one important difference at the outset of Fine (1995a): like propositions that are necessarily true, the propositions true in virtue of the nature of given objects are taken to be closed under logical implication; any logical consequence of such propositions is also to be such a proposition. . . . However, this closure condition is subject to a certain constraint. For we do not allow the logical consequences in question to involve objects which do not pertain to the nature of the given objects. Let us suppose, for example, that the empty set does not pertain to the nature of Socrates, then we do not allow the proposition that the empty set is self-identical to be true in virtue of the nature of Socrates even though this proposition is a consequence of any set of propositions whatsoever. Fine 1995a, 242

Despite the first sentence of the quote, then, what stands in the scope of the essence operator x is not closed under logical implication. Or, to be more precise, it is closed under logical implication subject to one constraint: the implication must not ‘involve objects which do not pertain to the nature of the given objects’. (The same will hold for closure under logical equivalence.) To make this a little more precise, we need to introduce Fine’s notion of objectual content. The objectual content of a formula is the set of objects which, intuitively, the formula is ‘about’. Within the language that we have introduced in chapter 5.2, the objectual content of a formula, given an interpretation of the language and an 15 I deviate slightly from Fine (1995a), who uses predicates rather than individual variables in the subscript, i.e. F , read ‘it is true in virtue of the essence of the Fs that . . . ’. This allows for a uniform treatment of essence statements about individual objects and pluralities of objects. Since I am restricting myself here to the case of individual objects, this provision is not needed and will be ignored. Note also that x , unlike POT, is a sentence operator. Fine (1995c) provides a useful discussion of the respective benefits of using a sentence or a predicate operator to express essence.

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assignment to the variables, is the set of objects that are referred to by the names and free variables occurring in the formula.16 The objects that ‘pertain to the nature of an object’ x, in the passage quoted above, are those objects that are included in the objectual content of sentences true in virtue of x’s essence. Fine has argued elsewhere (Fine 1995b) that those are the objects on which x ontologically depends. In the logic of essence, ontological dependence is introduced as a primitive in its own right but systematically connected to essence through the axiom that x depends on y just in case y is among the objectual content of φ, and φ is true in virtue of the essence of x. We can refer to the set that contains all and only those objects on which x depends as the closure of x (not to be confused with the closure of a class of truths under logical implication!). The closure of x will include x itself. The basic idea behind Fine’s restriction on closure under logical implication is that logical implication by itself cannot engender ontological dependence; it cannot make an object pertain to the essence of another object. That Socrates is human entails that the empty set is self-identical. But that in itself is not enough to make Socrates ontologically depend on the empty set, or in other words, to make the empty set pertain to Socrates’s essence. To prevent such spurious dependence relations, we must restrict closure under logical implication to cases in which no ‘new’ objects are introduced. If the logical implication introduces only objects (if any) on which Socrates already depends, then it can further the true essence claims. Given the notion of an object’s closure, we can thus formulate the restricted closure principle for essence: CL-Essence If φ logically implies ψ, and if ψ’s objectual content does not include objects that exceed both the objectual content of φ and the closure of x, then x φ logically implies x ψ.17 Informally, x’s essence is closed under logical implication unless the implied sentence introduces new objectual content that is not already in x’s closure. Now, the logic of essence might serve as a model for the logic of potentiality in two ways. One is the indirect way: we might think that, since essence stands 16 Fine has, in addition to individual terms, a category of ‘rigid predicates’ which also contribute to objectual content. For our purposes, no such category is needed. 17 Here is Fine’s own, properly formal, statement of CL-Essence (cF stands for the ‘closure of the Fs’, the set of all objects that the Fs depend on, including the Fs themselves, |A| is the object-language expression for the objectual content of A, and all other symbols should be self-explanatory): [Theorem 4] (ii) If  A → B, then  |B| – |A| ⊆ cF → (F A → F B). Fine (1995a) 253

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to necessity much as potentiality stands to possibility, the logic of potentiality should differ from the logic of possibility in much the same way as the logic of essence differs from that of necessity. So, in particular, we might try to build into CLOSURE a qualification that runs in parallel to the restriction that is built into CL-Essence. I will consider, and reject, this approach in section 5.7.1. Alternatively, the logic of essence might serve as a model for us in a more direct way: just as the two non-localized modalities, possibility and necessity, are duals, so we might think that potentiality and essence should be duals as well. What an object has a potentiality to do would then just be that which is not excluded by its essence. The logic of potentiality could be derived from that of essence (while yet leaving open the question which of the duals, if any, is metaphysically prior). Indeed, in Fine (1995a), Kit Fine defines an operator that is the dual of his essence operator, as follows:

3x φ =df ¬x ¬φ.18 Let me call the dual of Finean essence, i.e. whatever is expressed by 3x , ‘Epotentiality’ (E, of course, standing for essence). If potentiality (as I understand it) were the dual of Finean essence, then it would be identical with E-potentiality. Of course, unlike POT, the operator for E-potentiality is a sentence operator, but we could read 3x φ as ‘x has a potentiality to be such that φ’, thus allowing a large subclass of potentiality ascriptions to be formulated with the operator 3x . E-potentiality, like essence, has a qualified closure principle, which can be derived from CL-Essence via the duality of x and 3x : CL-Epot If φ logically implies ψ, and if φ’s objectual content does not include objects that exceed both ψ’s objectual content and the closure of x, then 3x φ logically implies 3x ψ.19 Informally, x’s E-potentiality is closed under logical implication, unless the implied sentence ‘loses’ objectual content that belongs to x’s closure. Section 5.7.1 will argue that this strategy, too, fails as an attempt to characterize the logic of potentiality. Section 5.7.3 will provide further argument against 18 Fine (1995a), 245, (I)(ii). As before, Fine’s language differs slightly from mine, and he uses predicates rather than terms as the subscripts of  and 3. I have adapted his definition to my language. Nothing of consequence for the present argument is lost. 19 Here is Fine’s own, properly formal, statement of CL-Epot (for explication of the symbols, see footnote 18):

[Theorem 4] (iii) If  A → B, then  |A| – |B| ⊆ cF → (3F A → 3F B). Fine (1995a) 253

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the duality of essence and potentiality, based on the failure of E-potentiality to validate NON-CONTRADICTION. As the table and the above discussion suggest, essence and its dual, Epotentiality, will concern us primarily in discussing CLOSURE and NONCONTRADICTION. (3) Ability. The connection between ability and potentiality is naturally made: some of my paradigmatic examples of potentialities were abilities, such as the ability to play the piano, or the ability to walk. Kenny (1976), in an influential paper, has challenged the idea that abilities are a species of possibilities, precisely by pointing out that the logic of abilities cannot be identical to, or even precisely parallel with, the logic of possibility. (See also Kenny 1975). Kenny’s arguments have not gone unchallenged (see Carr 1979 for some forceful objections). But if they are successful, then they show at least that the operator POT, according to the first column of the table in this section cannot be read as the predicate operator ‘has an ability to’, nor the operator 3, if it follows the second column, as the sentence operator ‘has an ability to make it the case that . . . ’. We will need to carefully distinguish potentiality from ability to make sure that Kenny’s objections do not threaten the four principles as principles for the logic of potentiality. In what follows, I will grant for the sake of the argument that Kenny’s arguments go through for abilities but argue that, even so, they do not apply to potentiality in general. According to Kenny, ability cannot satisfy CLOSURE, DISJUNCTION, or ACTUALITY. CLOSURE fails because, while everything entails a tautology, we have no abilities to bring about tautologies. As Kenny puts it, The President of the United States has the power to destroy Moscow, i.e. to bring it about that Moscow is destroyed; but he does not have the power to bring it about that either Moscow is destroyed or Moscow is not destroyed. The power to bring it about that either p or not p is one which philosophers, with the exception of Descartes, have denied even to God. Kenny 1976, 214

This is easily adapted to serve as a counterexample to DISJUNCTION when read from right to left. But DISJUNCTION also fails for abilities when read from left to right, according to Kenny. Counterexamples are provided by the limitations of what Kenny calls ‘discriminatory skill’: Given a pack of cards, I have the ability to pick out on request a card which is either black or red; but I don’t have the ability to pick out a red card on request nor the ability to pick out a black card on request. Kenny 1976, 215

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ACTUALITY, finally, fails because success may be due to luck rather than ability: A hopeless darts player may, once in a lifetime, hit the bull, but be unable to repeat the performance because he does not have the ability to hit the bull. . . . Counterexamples similar to these will always be imaginable whenever it is possible to do something by luck rather than by skill. Kenny 1976, 214

For the sake of my argument, I will grant that Kenny is right about abilities. I will argue that, even if that is the case, the result for the special case of abilities cannot be carried over to the general case of potentiality. Note that, while Kenny concluded from his examples that no modal logic of abilities was to be had, others have provided (non-normal) modal logics of abilities that conform to all or most of Kenny’s requirements.20 Abilities and their logic (or rather, the negative points that Kenny has made about their logic) will play a role primarily in discussing DISJUNCTION and ACTUALITY. I will argue that, even if Kenny is right about abilities, his arguments cannot be transferred onto potentialities in general. The characteristic mark of abilities on which Kenny’s examples trade is an element of control which is characteristic of a subset of potentialities—the abilities—but not of potentiality in general. The logic of essence and the logic of ability will serve as points of comparison and contrast in the sections to come. The table at the beginning of this section shows where the contrasts lie. I will now discuss each principle in the logic of potentiality, after some initial motivation, by reference to these contrasts and the challenges that they pose.

5.7 Defending the principles 5.7.1 Defending CLOSURE The principle that I wish to defend is CLOSURE Potentiality is closed under logical equivalence: If being  is logically equivalent to being , then having a potentiality to be  is logically equivalent to having a potentiality to be . 20 Brown (1988) proposes a non-normal logic of abilities based on the idea that we can quantify not just over worlds, but over clusters of worlds. Brown’s logic does validate the ascription of tautological abilities and accordingly a rule that corresponds to CLOSURE, but other than that it conforms to Kenny’s arguments. A similar logic of abilities is developed, in the context of a logic of agency, by John Horty and Nuel Belnap (Horty and Belnap 1995; see also Belnap and Perloff 1988 and Belnap 1991), who add a special clause to preclude tautological abilities.

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To argue for CLOSURE (which is all that we will need for the logic of possibility), I will appeal to a stronger principle: CLOSURE1 Potentiality is closed under logical implication: If being  logically implies being , then having a potentiality to be  logically implies having a potentiality to be . Here is quick argument to show that CLOSURE follows from CLOSURE1. Suppose that being  and being are logically equivalent. This is just to say that each logically implies the other. Given that being  logically implies being , having a potentiality to be  logically implies having a potentiality to be . The same holds in the other direction. So given the assumption that being  is logically equivalent to being , it follows that having a potentiality to be  and having a potentiality to be imply each other, hence are logically equivalent. For the remainder of this section, I will be concerned to defend CLOSURE1, or closure under logical implication. I will refer to this principle, as well as any restricted variation of it, as a ‘closure principle’. Why should we adopt CLOSURE1? Here is some intuitive motivation. Suppose that an object, x, has a potentiality to be F, and that being F, as a matter of logic, implies being G. Suppose, further, that x manifests its potentiality to be F. Being F, x cannot then fail to be G. So x cannot have lacked a potentiality to be G. We have seen one instance of this kind of reasoning above, when considering sentences of the form ‘a has a potentiality to be such that something is F’. Being such that something is F is a logical consequence of being F; and an object which has a potentiality to be F cannot therefore fail to have a potentiality to be such that something is F. But there are many more examples. If I have a potentiality to read Middlemarch, then I have a potentiality to read something; if you have a potentiality to walk while singing, then you must have a potentiality to walk; if the glass has a potentiality to break, then it has a potentiality to break or not to break. So much for the motivation; now on to the defence. The worrisome applications of CLOSURE1 will be those that concern tautologies. Note, first, that anything logically implies a tautology. We have seen that Kenny (1976) rejects the equivalent of CLOSURE1 for abilities for this very reason: being F logically implies being F-or-not-F, but nothing, in Kenny’s view, has an ability to bring about a tautology. We have considered tautological potentialities at length in section 5.3, so Kenny’s argument will not concern us further in this instance. But by CLOSURE1, any object not only has potentialities to possess any tautological property, such as being red or not red; it is also the case that every object

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possesses potentialities to be such that any other object possesses any tautological properties. A tomato’s being red implies, as a matter of logic, the tomato’s being such that Doris is dancing or is not dancing. But Doris’s dancing or not dancing has nothing to do with a tomato’s potentiality to be red. Why, then, should the tomato’s potentiality to be red endow it with a potentiality to be such that Doris is or is not dancing? This is the main worry about CLOSURE1. (Note that the worry arises for our target principle CLOSURE too: given that the tomato has the tautological potentiality to be red or not red, and that being red or not red is logically equivalent to being such that Doris is dancing or not dancing, it follows that the tomato has a potentiality to be such that Doris is dancing or not dancing). We need to be careful, however, about what exactly the worry is. It should not be a worry about the entailment itself. For, of course, a tomato’s having a potentiality to be red does not ‘endow’ the tomato with any potentialities regarding Doris in any metaphysically interesting sense of ‘endow’. Likewise, being red does not ‘endow’ the tomato with the property of being such that it’s raining or not raining, yet the tomato’s being red entails its being such that it is raining or not raining. Entailment is not grounding. The worry, then, should not be that by CLOSURE1 potentialities imply other potentialities with which they have nothing to do; this is what logical entailment is like. Rather than being about the relation between what entails and what is entailed, the worry should be about the nature of that which is entailed. Leaving aside the potentiality to be red, why should a tomato have a potentiality to be such that Doris is dancing or not dancing? My strategy in defending CLOSURE1 against this worry will be as follows. I consider two options for restricting it, both inspired by Kit Fine’s logic of essence, and I argue that both fail in their application to potentiality. The argument against the restrictions will point us towards a more direct answer to the worry: an argument to the effect that everything does have the apparently problematic potentialities, such as a potentiality to be such that Doris is dancing or is not dancing. Fine’s restriction on the closure principle for essence is that the entailment is not to introduce the wrong kind of objectual content. We might think that the same kind of restriction applies to potentiality: the entailment is not to introduce new objects, unless those are already included in the ‘reach’ of the object’s potentialities. Alternatively, we might think that the restriction that applies to potentiality is of the opposite kind: it is not the introduction of new objects, but the loss of objectual content, that is to be excluded. This latter thought goes with the idea that potentiality is identical to E-potentiality, the dual of essence, i.e. that

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an object has a potentiality to be F just in case it is not the case that the object is essentially not F. Such an equivalence would reverse the closure restriction for potentiality as compared to that of essence. The second line of thought is quickly disposed of. Take the entailment, which I have argued to be unproblematic, from POT[λx.Fb](a) to POT[λx.∃xFx](a). Clearly, objectual content is lost here, but the entailment is no worse for it. This holds whether or not b is among the objects on which a depends. Not so for Epotentiality: the corresponding entailment, from 3a Fb to 3a ∃xFx, goes through only if no objectual content is lost that is part of what a depends on. To have a concrete example, assume the essentiality of origin, let a be a human being and b the mother of a. Then a might well have a potentiality and an E-potentiality to be such that b is happy; and a should thereby count as having a potentiality to be such that something is happy. But not so for E-potentiality: the entailment does not go through because b drops out of the objectual content. Even if we thought that CLOSURE1 had to be restricted, surely this is not the restriction that we are looking for. Let us return, then, to the first line of thought. Here the strategy is not to restrict CLOSURE1 by direct appeal to the logic of essence, as we would if we thought of potentiality as the dual of essence. Rather, the strategy is to restrict CLOSURE1 in a way that is relevantly parallel to the restriction in the logic of essence. The motivation for Fine’s restriction on closure under logical implication for essence was that an object, y, should not come to matter for the essence of another object or objects, x, just as a matter of logic. y must already be involved in the essence of x, in order for logical implication to yield new essential truths for x that concern y. Fine made sure that this was so by appealing to the notion of dependence and the closure of x: logical implication affects the essence of x only if it concerns no more than the objects that x already depends on or, equivalently, the objects that are already involved in its essence: the closure of x. With potentiality, added objectual content brings not dependence, but extrinsicality. For an object x to have a potentiality whose manifestation involves another object, y, is for x to stand in a relation to or have a plural property with y: having a joint potentiality with it. Now the thought may go as follows. The property or relation of having a joint potentiality, like the relation of ontological dependence, should not be generated by logical implication alone. Rather, x and y must already possess some joint potentialities if logic is to add further joint potentialities between them. In parallel to the Finean restriction, then, we might try the following. Define a relation of co-possession, to hold between any objects

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which jointly possess a potentiality. (As a trivial limiting case, the relation holds between each object and itself.) The co-possession closure of an object x is the set which contains all and only those objects to which x stands in the relation of co-possession. Then we might say that logical implication yields further potentialities only if it does not introduce any objects that are not already in the co-possession closure of x. The resulting restricted Closure Principle is parallel to CL-Essence: CL-Potentiality If being  logically implies being , and if x has a potentiality to be , and if ’s objectual content does not include objects that exceed both the objectual content of  and the co-possession closure of x, then x has a potentiality to be .21 Whatever its merits or disadvantages, the proposed restriction will not function as a substantial restriction on closure under logical implication for POT. The reason is that, whatever x is, anything whatsoever is already a member of x’s co-possession closure. Hence any new objects that might introduce over and above those contained in ’s objectual content will always satisfy the criterion that they form a subset of the co-possession closure of x. And so the restricting clause will not, in practice, ever make a difference. In which case it is just as well to keep the unrestricted closure principle. The crucial premise in this short argument is, of course, that every object is already a member of x’s co-possession closure. This claim requires a more extended defence, which will take up the remainder of this section. My claim is reminiscent of, but goes beyond, my answer in chapter 4.3 to the question under what conditions a range of objects jointly possess a potentiality. There I argued that there are no non-trivial necessary conditions for the possession of a joint potentiality: any objects might, in principle, have joint potentialities with any other objects. Hence anything might, in principle, have extrinsic potentialities concerning any other object. My claim here is stronger: it concerns sufficient conditions for the possession of a joint potentiality. I say that any objects do in fact have some joint potentiality. The argument for that claim is simple: once we have allowed joint potentialities with tautological manifestations, there is no limit to be imposed on them. 21

More formally, rendering the co-possession closure of x as CP(x), we might put the principle

thus: If  x → x, then  | | – || ⊆ CP(x) → (POT[](x) → POT[ ](x)). (This is exactly parallel to the formulation of CL-Essence in Fine 1995a, 253; see the earlier footnote on CL-Essence for comparison).

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Mary and Jane have a joint potentiality to sing a duet. They also have a joint potentiality to sing or not to sing a duet, which they exercise by singing or by not singing, as the case may be. Mary, accordingly, has an extrinsic potentiality to sing a duet with Jane, and a potentiality to sing or not to sing a duet with Jane. To some degree—lesser perhaps due to our different musical abilities—Mary has a potentiality to sing a duet with you, with me, with Barack Obama, and so forth. Mary has no potentiality to sing a duet with, say, her copy of Middlemarch, since books do not have the intrinsic potentialities required to contribute to a joint potentiality for duet-singing. Still, Mary has a joint potentiality with her copy of Middlemarch to sing-or-not-to-sing a duet. That joint potentiality is exercised by the two of them not singing a duet, and never by their singing a duet. It is a highly uninteresting potentiality, but a potentiality it is no less. Generally speaking, it does not take much for two or more objects to stand in a tautological relation to each other; in fact, it takes no more than their existence. Likewise, no more than existence is required for a potentiality to stand in a tautological relation to each other. This shows that any object whatsoever belongs to the co-possession closure of any other object; the restriction on CLOSURE1 suggested above does not function as a restriction at all. Indeed, it shows more than this. It shows that the lack of a restriction on CLOSURE1 is innocuous. For by the same reasoning, we can see that the apparently problematic potentialities that CLOSURE1 yields are no more problematic than the joint potentialities that I have just appealed to. Recall Claim 2 in chapter 4.5: every joint potentiality gives rise to, and fully grounds, corresponding extrinsic potentialities of all its possessors. Thus it can be argued that if everything has potentialities to stand in tautological relations to any other thing, then everything has potentialities to be such that any other thing has any tautological property whatsoever. The reasoning goes just as it did before. Ann has a potentiality to walk, and a potentiality to walk-or-not-to-walk. Ann and Betty jointly have a potentiality to be such that Ann walks and a potentiality to be such that Ann walks or does not walk. Hence Betty has an extrinsic potentiality to be such that Ann walks and an extrinsic potentiality to be such that Ann walks or does not walk. And there is nothing special about Ann and Betty here. If Betty has such potentialities concerning Ann, then so do you, I, Barack Obama, and her own copy of Middlemarch. If Betty has such potentialities concerning Ann, then she also has the potentiality to be such that I am walking or not walking, to be such that you are walking or not walking, and to be such that her copy of Middlemarch is walking or not walking. Unlike the former potentialities, the last one is never exercised by Betty’s being such that her copy of Middlemarch

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is walking; books cannot walk. The potentiality is exercised, rather, by Betty’s being such that the book is not walking. When I introduced joint potentialities, of a number of objects, to be such that some (but not all) of them were such-and-such, I appealed to examples of masked or mimicked dispositions. A glass together with the styrofoam that is wrapped around it has the potentiality to be such that the glass breaks to a lesser degree than the glass on its own; a city together with its external defence system has a lesser joint potentiality to be such that the city is harmed than the city has on its own. On the other hand, the glass and styrofoam together have the joint potentiality to be such that the glass remains intact to a greater degree than the glass on its own, and the city and its defence system have the joint potentiality to be such that the city is successfully defended to a greater degree than the city does on its own. Starting with the glass or the city, we ‘add’ further objects to be the co-possessors of a joint potentiality, and as we do so we get potentialities with different degrees, depending on how the potentialities of the ‘added’ objects contribute to or detract from the degree of the potentiality in question. Styrofoam and defence systems detract from the degree of the glass’s potentiality to break and the city’s potentiality to be harmed, respectively, but add to the degree of their respective potentialities to remain intact or be successfully defended. For any objects to have no joint potentiality whatsoever, by this way of thinking, is for them to detract so much from each other’s potentialities that the degree which is reached by ‘adding’ them in is zero. On this way of thinking, the lack of a joint potentiality (of the kind at issue) becomes a limiting case of masked potentiality. But how should a potentiality with a tautological manifestation be masked in the first place? How should any other object ‘detract’ from my potentiality to be walking-or-not-walking, from your potentiality to be self-identical, or from the glass’s potentiality not to be broken and unbroken (at the same time)? It appears that it is the lack, not the possession, of joint potentialities that would require an explanation. The burden of proof now rests on my opponent. If the opponent wishes to deny the claim that all objects have joint potentialities with, and hence (at least tautological) extrinsic potentialities regarding, all other objects, she needs to produce a counterexample. I have not given a decisive proof for CLOSURE1, but I have provided reasons to think that counterexamples are hard, indeed impossible, to come by. Given the defence of CLOSURE1, we have reason to adopt the target principle CLOSURE, the closure of potentiality under logical equivalence.22 22 The most promising line in the search for counterexamples, it seems to me, would be to appeal to incompatible objects. If x and y are incompatible, how should they have any joint potentialities?

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5.7.2 Defending DISJUNCTION We now turn to a second crucial principle in the logic of potentiality: DISJUNCTION Potentiality distributes over, and is closed under, disjunction: An object has a potentiality to be -or- if and only if it has a potentiality to be , or a potentiality to be . Closure under disjunction (the right-to-left direction of DISJUNCTION) follows straightforwardly from CLOSURE 1: being  entails being -or- ; hence by CLOSURE 1, having a potentiality to  entails having a potentiality be -or- . Incompatibility may be temporal or modal. A cup and a statue may be made from the same material at different times. Assuming that their material is essential to both, they could not exist at the same time; they are temporally incompatible. A sperm s and two eggs e and e∗ intuitively make for two possible human beings: h, originating in s and e, and h∗ , originating in s and e∗ . Since s can only combine with one egg, only one of h and h∗ can come into existence; h and h∗ are modally incompatible. (For the second example, and others of the same kind, see Williamson 2010). How we construe these cases depends on whether we are eternalists or presentists, in the temporal case, and on whether we are necessitists or contingentists, in the modal case. According to presentism, everything is always present; according to (weak) eternalism, some things are not present, that is, there are merely past or merely future objects. According to actualism, necessarily everything is actual; according to possibilism, possibly some things are not actual, that is, there are merely possible objects. On this way of introducing the divisions, eternalism and possibilism are merely the negations of presentism and actualism, respectively. In general, eternalists and possibilists want to make a stronger claim. According to strong eternalism, it is always true that everything is always something. In other words, everything that exists at any time exists at all times. According to necessitism—or ‘strong possibilism’, as we might call it—it is necessarily true that everything is necessarily something. In other words, everything that possibly exists exists necessarily. In what follows, I will be concerned with the contrast between presentism and actualism, on the one hand, and strong eternalism and necessitism, on the other hand. (For the first pair of contrasts, see, for instance, Sider 1999, 326. For the definition of necessitism, see Williamson 2010.) Given their temporal incompatibility, at any time only one of the cup and the statue can be present. For the presentist, this means that only one of them exists. Given their modal incompatibility, only one of h and h∗ can be actual. For the contingentist, this means that only one of them exists. My claim does not concern joint potentialities with non-existents. So on the assumption of presentism or contingentism the respective cases do not provide counterexamples to my claim. For eternalists and necessitists respectively, the cup and statue or h and h∗ do exist. Given their respective incompatibilities, at most one of the cup and statue can be present at any one time, and at most one of h and h∗ can be concrete. But for the eternalist, both cup and statue exist at any time t, and both have at least such tautological properties as the property of breaking-or-not-breaking; if one of them, say the cup, is present, this does not in any way prevent the other from having such tautological properties; jointly, the cup and statue have a potentiality to be such that one of them, say the statue, breaks or does not break. (If breaking at t requires being present at t, then the cup’s being present prevents the statue’s breaking, and so cup and statue do not have a joint potentiality to be such that the statue breaks; but that is no counterexample to my claim.) Likewise for the case of h and h∗ , on the assumption of necessitism. If both exist, then both have tautological properties such as the property of dancing-or-not-dancing, and nothing about either one of them prevents the other from having that property. Jointly, they have a potentiality to be such that one of them, say h∗ , is dancing or not dancing. (If dancing requires being concrete, then they have no joint potentiality to be such that h∗ is dancing. But again, this is no counterexample to my claim.)

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The same holds for a potentiality to . Hence if x has either a potentiality to  or a potentiality to , then by CLOSURE 1, x has a potentiality to -or- . Since CLOSURE 1 has been defended at length in the previous section, this section will discuss the other direction of DISJUNCTION, distribution over disjunction. We begin with an intuitive motivation for the principle. To get a feel for disjunctive manifestations, we may think of a determinable property and its determinates, e.g., being red (the determinable) and being any particular shade of red: red1 , red2 , etc. (the determinates). Take an apple’s potentiality to be red. That potentiality is intimately linked to, and at least entails, a potentiality with the following (disjunctive) manifestation property: to be red1 or to be red2 or . . . Does it follow that the apple has a potentiality to be red1 or a potentiality to be red2 or . . . ? I say, it does: if the apple manifests its potentiality to be red, it will inevitably do so by being some particular shade of red or other. Whichever shade it is, call it redn , the apple then has the property of being redn , and it has that property through a manifestation of a potentiality of its own. There is, then, no reason to deny to the apple the potentiality to be redn . But redn must, by hypothesis, be one of the determinates red1 , red2 , . . . Hence there must be at least one disjunct in the long disjunctive property which is entailed by being red that the apple has a potentiality to have: it must have a potentiality to be red1 or a potentiality to be red2 or . . . . Now for the generalization. The crucial feature of the determinates / determinables relation is that to possess the determinable property, an object must possess exactly one of the determinate properties; in fact, what was crucial to my argument was merely that to possess the determinable, an object must possess at least one of the determinates. But that is precisely what is true of a disjunctive property. If an object, x, has a potentiality to be F-or-G, that potentiality could be manifested in no other way than by x’s being F, or x’s being G. Suppose, then, that the potentiality is manifested by x’s being G. Then x is F, and it is so by exercising its own potentialities. Surely, then, x must have had a potentiality to F; and the same goes, mutatis mutandis, for F. Hence x must have at least one of the corresponding potentialities: a potentiality to be F, or a potentiality to be G. So much for the motivation. The challenge to DISJUNCTION comes from the logic of ability. For the corresponding axiom for abilities has been challenged, as we have seen earlier, by Kenny (1976). Here is Kenny’s counterexample again: Given a pack of cards, I have the ability to pick out on request a card which is either black or red; but I don’t have the ability to pick out a red card on request nor the ability to pick out a black card on request. Kenny 1976, 215

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The example requires two quick comments. First, the scope of the disjunction is not entirely clear in the quote, but Kenny clearly intends it as follows. Let R denote the property of picking a red card, and B the property of picking a black card. Being presented with the pack of cards, I have the ability to R-or-B (formally, to λx.(Rx ∨ Bx)). But I have neither the ability to R, nor the ability to B. Second, it may be observed that Kenny’s example concerns a rather situationspecific ability. I have argued earlier that such abilities are extrinsic potentialities. But that is not an essential feature of the example. We may return to the stock examples of disjunctive manifestations, the case of determinables and determinates. I have the ability to lift my arm. To lift my arm is to lift it at one or another particular angle. Hence I have the ability to lift my arm at an angle of 180 degrees or to lift it at an angle of 200 degrees or . . . . I do not, however, have the ability to lift my arm at an angle of exactly 180 degrees, for I do not have control over the exact angle at which my arm goes up. Kenny’s example, therefore, need not appeal to the very situation-specific, extrinsic ability that he does invoke. My claim, as I have said earlier, is that even if Kenny’s argument applies to agentive abilities, it does not apply to potentialities in general. Having a potentiality does not require control over its manifestation in the way that an ability appears to do. Glasses, earthen pots, and steel bridges have the potentiality to break. Being inanimate, not one of them has control (in any ordinary sense) over its breaking. People have dispositions, and hence potentialities, to catch diseases, to break out in tears, or to get a migraine from certain weather conditions. Here, again, there is no sense in which we exercise control over the manifestation of such potentialities; on the contrary, they catch us against our will and often by surprise. The label ‘disposition’, as I have argued in chapter 3, is applied to a potentiality only when that potentiality has a certain minimal degree. Potentiality is that which provides for those degrees; it is present even when the degree is too low to qualify for dispositionality. Even those of us not prone to crying have some potentiality to break out in tears, and even with a healthy immune system we have some potentiality to catch a cold. Likewise, we all have some potentiality to pick a red card from a down-facing deck, or to lift our arms at an angle of exactly 180 degrees. Having control over the exercise of a potentiality is a further condition that a potentiality may or may not fulfil, and it may (if Kenny is right) be a condition which a potentiality must meet if it is to qualify as an ability. It is not a condition on the potentiality being a potentiality in the first place. Kenny’s counterexamples fail to affect potentiality because they require too much. The logic of ability, if it respects Kenny’s examples, is not the logic of potentiality. It may be, indeed on the present view it must be, the logic of an

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important species of potentiality. But its distinctive features, such as the denial of DISJUNCTION, stem from the additional conditions that a potentiality has to fulfil in order to count as an ability. They do not stem from the nature of potentiality itself. It is worth noting that not only abilities (probably) fail to distribute over disjunction. Dispositions do too. A fragile glass is disposed to break into two pieces or break into more than two pieces, but it need not be disposed to break into two pieces, nor disposed to break into more than two pieces. The degree of the glass’s potentiality to break into two or more pieces may be just high enough to qualify as a disposition, even if neither of the other two potentialities has the required degree. Abilities and dispositions, then, fail to distribute over disjunction. But the explanation lies in the additional conditions that a potentiality must fulfil in order to qualify as an ability or a disposition: a control condition, or a condition of sufficient degree. In neither case does the explanation lie in the nature of potentiality itself. Neither case, therefore, does anything to discredit DISJUNCTION.

5.7.3 Defending NON-CONTRADICTION We turn next to the axiom NON-CONTRADICTION Nothing has a potentiality to be such that a contradiction holds. This should be the most obvious of the four principles: nothing ever has a potentiality for a contradictory manifestation, for instance, a potentiality to be red-and-not-red. Such a contradictory potentiality must be sharply distinguished from different potentialities with mutually contradictory manifestations: I have an ability to walk and an ability not to walk, but I have no ability to walkand-not-walk. It must also be sharply distinguished from potentialities whose manifestations involve changing from one pair of a contradiction to another. Thus I have a potentiality to start walking, whose manifestation involves, first, my not walking and, at a later stage, my walking. But the manifestation is not contradictory. In specifying it, we use contradictory predicates, but they are temporally specified. To manifest a potentiality to be walking-and-not-walking, one would have to have the predicate ‘is walking and not walking’ true of oneself at some time or other. But that conjunctive predicate, unlike the predicate ‘is first walking and then not walking’, is never true of anything. Intuitively, then, it seems clear that nothing has a potentiality to do anything contradictory, or to be such that a contradiction holds. Unsurprisingly, the principle is not challenged by most of the logical systems that I have chosen as points

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of comparison. The only challenge comes from the logic of E-potentiality, the dual of Finean essence. An object’s E-potentiality might include contradictions. This result (while not, to my knowledge, made explicit in Fine’s logic) follows from the definition of E-potentiality as the dual of Finean essence, together with the fact that not every tautology holds in virtue of the essence of a given object. Recall that a statement, φ, is true in virtue of the nature of an object, x, only if the objectual content of φ does not outrun the closure of x, that is, the objects on which x depends. This holds even when φ is a tautology. Thus while it is true, and tautologically true, that Socrates is not simultaneously a philosopher and not a philosopher, this truth does not hold in virtue of the essence of Plato (assuming, plausibly, that Plato does not ontologically depend on Socrates). Using a for Plato, b for Socrates and F to ascribe the property of being a philosopher, we have (21) ¬a ¬(Fb ∧ ¬Fb). By the definition of 3x , we therefore also have (22) 3a (Fb ∧ ¬Fb), in English: Plato has an E-potentiality to be such that Socrates is a philosopher and is not a philosopher. Since on our interpretation (21) is true, (22) must also be true; hence E-potentiality does not validate NON-CONTRADICTION. Intuitively, E-potentiality captures compatibility with an object’s essential nature. An important motivation for introducing the Finean notion of essence was the desire to be able to discriminate where necessity does not allow for discriminations: Socrates’s being simultaneously a philosopher and not a philosopher is excluded, not by Plato’s essence, but by Socrates’. It is Socrates, not Plato, who is the source of the impossibility of his being a philosopher and not a philosopher. Not being the source of the impossibility that Socrates be a philosopher and not a philosopher, Plato’s nature is compatible with Socrates’ having that contradictory property, or indeed any other. It should be clear that this is not how we ought to think of potentiality, in the sense at issue in this book. Potentialities that concern other objects are not, on the present picture, mere compatibilities in the weak sense just described. They are extrinsic potentialities, which depend on a joint potentiality with the other objects concerned, and thereby on the potentialities of those objects themselves. If Socrates has no potentiality to be a philosopher and not a philosopher, then nothing else can have a potentiality to be such that Socrates is a philosopher and not a philosopher. I have said earlier that a potentiality is, metaphorically speaking, the suitability of an object for a particular property. But with extrinsic potentialities, this suitability is a matter not only of the object that has the potentiality, but also

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of those objects on which its potentiality depends. Since Socrates is not suited to have a contradictory property, nothing else is suited to be such that he does. Given the metaphysics of extrinsic potentiality, therefore, we can see how potentiality in the sense developed in this book differs from E-potentiality. As a consequence, the failure of E-potentiality to validate NON-CONTRADICTION does not affect NON-CONTRADICTION as an axiom for potentiality itself. As an added bonus, this section has shown conclusively that potentiality, as it has been developed in this book, is not the converse of Finean essence. The converse of Finean essence is E-potentiality, and for the reasons just given E-potentiality is not potentiality as I understand it.

5.7.4 Defending ACTUALITY ACTUALITY is the axiom that, whenever an object, x, is  (no matter the logical form and complexity of ), x has a potentiality to : ACTUALITY Potentiality is implied by actuality: Anything which is  must also have a potentiality to be . I want to offer a very simple argument for adopting this axiom (the work, as so often, lies in defending the premises). It goes as follows: Premise 1: Whenever an object x s, x thereby exercises its potentiality to . Premise 2: Whenever an object x exercises a potentiality to , then x must (simultaneously) possess the potentiality to . Conclusion: Whenever an object x s, x must (simultaneously) possess the potentiality to . Why do we need premise 2? Premise 1 amounts to the claim that every property (in the widest possible sense of the term) that an object has is the exercise of a potentiality of that object. This might be true and ACTUALITY still have false instances. For it might then be that every property which an object has is the exercise of a potentiality that the object possesses at some time or other. But the implication in ACTUALITY must be evaluated at a time, and to be valid it must come out true at all times. And if there is a time t at which x is , hence exercising its potentiality to be , yet the potentiality is (or was) possessed only at some time t  = t, then at t ACTUALITY will be false. I need to provide some defence of both premise 1 and premise 2, therefore. Let me start with premise 1. Premise 1, I submit, is highly plausible given the notion of potentiality that has been developed so far. To have a counterexample to premise 1, we would need

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an example of an object’s possessing a property that is not plausibly construed as the manifestation of a corresponding potentiality. Why would this be the case? It could not be because the object’s possessing that property was highly unlikely given its intrinsic constitution. As we have seen in chapter 3.5, potentialities can come in very low degrees, so even a steel bridge’s breaking is the manifestation of a potentiality (of slight degree) to break. It could not be because the manifestation is not sufficiently causal; for we have seen in chapter 3.6 that the manifestations of potentialities need not be causal processes. It could not be, either, because the property is too complex, i.e. because the predicate replacing  is of high logical complexity. We have seen, in section 5.3, that we can make sense of potentialities with complex manifestations, and of logically complex predicates in the scope of the potentiality operator POT. Nor could it be because the property is of the wrong kind in another way, a ‘Cambridge’ property, concerning objects other than x. For we have seen, in chapter 4 and again in section 5.3.3, that such potentialities are simply extrinsic potentialities that may be uninteresting but should not for that reason be rejected. Kenny (1976) has argued against the equivalent of ACTUALITY in the case of abilities: a hopeless darts player, Kenny said, may hit the bull by mere luck, and yet lack the ability to hit the bull. In general, the ability version of ACTUALITY is refuted, according to Kenny, by the possibility of achievements ‘by luck rather than by skill’ (Kenny 1976, 214). Given my discussion of DISJUNCTION, it should now be clear what my response is to Kenny’s example. Granting that Kenny is right about abilities, I reject any application of his remarks to potentiality in general. Having a potentiality does not require control over its manifestation in the way that an ability appears to do, just as it does not require a certain minimum degree in the way that some dispositions do. A steel bridge does not have the disposition of fragility, but it does have a potentiality to break. Likewise, even the hopeless darts player has some potentiality to hit the bull’s eye. To be sure, it is not in her power to ensure that she hits the bull’s eye; she does not have control over her hitting or missing the bull’s eye in the way that an accomplished darts player does. But control is an additional feature of a potentiality, not one that is required for its very status as a potentiality. Like DISJUNCTION, ACTUALITY and, in particular, premise 1 fails for dispositions too. A steel bridge that breaks does not exercise its fragility because it is not fragile in the first place, and a generally healthy person who catches a cold does not exercise her disposition to catch colds because she need not be so disposed. Potentialities may get exercised despite having a degree that is too low for them to qualify as dispositions. But a high enough degree, like

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the element of control, is an additional feature of some potentialities, not a requirement that they have to meet in order to qualify as potentialities in the first place. So much, then, for premise 1. What about premise 2? Let me start with a simple and straightforward argument for premise 2. In order to exercise a potentiality, an object must have the potentiality in the first place. I cannot exercise potentialities which I do not possess, even if I have possessed them or will possess them at another time. I once had some potential to be a child prodigy, but sadly I have lost that potential in growing up. Given that loss, there is no way for me now to exercise the potential to be a child prodigy. How would I exercise it if I do not have it? By contraposition, if I exercise a potentiality I must at least possess that potentiality. But the simple argument is too simple. Exercises of a potentiality may take time. A flammable match takes a while to burn, a soluble piece of sugar takes a while to dissolve, an able piano player takes a while to play the Goldberg Variations. The potentiality has to be there to get the process going: the match must be flammable when it is struck, the sugar soluble when it is immersed in water. But perhaps, it may be said, that is all we need: once the process is started, the potentiality has done its work and might as well disappear. In fact, is there not reason to think that it will disappear? After all, the dissolved sugar (assuming we can still refer to it) is no longer soluble, and a match, once burnt, is no longer flammable. So here we have the manifestation of an object’s potentiality—being burnt, or being dissolved—that is very likely not accompanied by the relevant potentiality. And there is no riddle about a potentiality’s being exercised in absentia. For the potentiality was present to set the process going, although it disappeared in the course of the manifestation.23 Distinguo. Potentialities are individuated by their manifestation. The following potentialities are distinct, albeit systematically related: the potentiality to catch fire (to begin to burn); the potentiality to burn (to be in the process of burning); the potentiality to be burnt (to be in the end state of said process). The first is exercised while the match catches fire; once the fire is caught, and the match is burning, it is no longer exercised. The second is exercised in the process of the match’s burning, not before that process begins nor after it is over. The third is exercised in the state of the match’s being burnt, but not before that state sets in or after it has ceased to obtain.24 23

Thanks to Markus Schrenk for pressing me on this kind of counterexample. This fine-grained distinction between potentialities depends on the idea that no potentiality ever has two or more distinct properties as its manifestation(s). If we thought that the potentiality to burn and the potentiality to be burnt are the same potentiality, premise 2 would indeed seem to 24

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The response to my simple argument had two aspects. First, it had an explanation of how a potentiality is exercised after it is lost: the potentiality sets the process going, but it need not accompany it. I reject the explanation. A potentiality is not a cause that sets its effect going. It is the object’s suitability for a process or state. If the object loses its potentiality, it is no longer suited to sustain the process or state. If the match lost its potentiality to burn during the process of burning, then how would it continue to burn? And if it lost the potentiality to be burnt after that process had reached its end-point, how would it continue to be burnt? Second, the response had examples of the kind of phenomenon that I am rejecting: a match’s being burnt counts as the exercise of its flammability, but the match is no longer flammable when it is burnt. To this, I say: we need to distinguish the potentialities with which we are concerned. It is a linguistic question, and not one that I will endeavour to answer, whether ‘flammable’ ascribes the potentiality to burn or to be burnt, or indeed the potentiality to catch or be set on fire. But the potentiality to burn is manifested in the process of the match’s burning, during which the match retains the potentiality to burn; the potentiality to be burnt is manifested in the match’s being burnt, which sets in at the end of said process. The fact that the match has no potentiality to burn when it is burnt is not a counterexample to Premise 2, or to ACTUALITY in general. The objection was correct, however, in locating the difficulty for ACTUALITY in temporally extended processes. Here is what I take to be the most pressing objection to ACTUALITY. Many processes are heterogeneous: they involve subprocesses of rather different kinds. Walking is a heterogeneous process, as is breaking. Now, an object may go through the first sub-processes of such a heterogeneous process and lose, half-way through, its potentiality to go through them again from the start. Take the heterogeneous process of a glass’s breaking, as described by Mark Johnston: Packing companies know that the breaking of fragile glass cups involves three stages: first a few bonds break, then the cup deforms and then many bonds break, thereby shattering the cup. Johnston 1992, 233

By the time when many bonds are breaking, the glass may no longer have a potentiality (it may no longer be suitable) to undergo the deformation that was the fail. For suppose that these were one and the same potentiality, and call that potentiality P. Then by premise 2, the burnt match, by exercising P, would also have to possess P; and since P is also its potentiality to burn, the burnt match would have to have the potentiality to burn, which it does not. Note, however, that premise 2 seems fine for P qua the potentiality to be burnt: surely, by the considerations so far adduced, the match still has that potentiality in its burnt state. But the very fact that premise 2 seems fine for P qua the potentiality to be burnt but not for P qua the potentiality to burn should incline us to resist the identification of these potentialities.

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earlier stage of its breaking. But to break is to go through the entire process. If the glass has lost the potentiality to go through some parts of the process, then it has lost the potentiality to break. And yet, at the time when many bonds are breaking, the glass is breaking, and its breaking is a manifestation of its potentiality to break. Do we, then, have a counterexample to Premise 2 after all? We do not. Recall that we are evaluating (as is standard) sentences at a time. At the time when the many bonds are breaking, it is not true of the glass that it is undergoing the deformation; that has already taken place. The potentiality at issue cannot be a potentiality to have a few bonds break and to deform and to have many bonds break, for the italicized conjunction is not true of the glass at the time. What is true of the glass is that it is in the course of undergoing a process which involves, first, the breaking of a few bonds, then deformation, and then the breaking of many bonds. To be in the course of undergoing such a process is not to begin it from scratch, or indeed to perform any particular one of the stages involved. Hence the potentiality to be in the course of undergoing such a process does not entail a potentiality to start the process from scratch, nor a potentiality to undergo a deformation of the relevant kind.25 I conclude that ACTUALITY is an attractive principle, and one to which no successful objection has been made.

5.8 Interlude: potentiality in time In introducing the language for the logic of potentiality, I stipulated that no expressions of that language be interpreted so as to involve tenses or reference to time. To the astute reader, this may have seemed like a cheap way of avoiding counterexamples: counterexamples, that is, which involve tensed or dated predicates, and in particular those which involve past-tensed predicates or predicates dated to the past. Suppose, then, that we enrich the object language of this chapter, or at least its admissible interpretations, so that we can form tensed predicates such as ‘was red’, ‘will be dancing’, and dated predicates such as ‘is red before 1 January 2014’, or ‘is dancing on 5 August 2013’. We are then able to formulate, in the enriched language, two types of apparent counterexamples to the core principles for the logic of potentiality. Take ACTUALITY: it is true of me now that I was once a child, but it would be odd to say that I now have a potentiality to have been a child. Generalizing from 25 There are difficult issues involved here, concerning the ascription of process predicates at a time. I will not go into those issues. Note also that our language does not include tense operators or time indices. The next section will address some of the issues that may arise from introducing such operators.

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the example, tensed (or dated) predicates, and in particular those that concern the past, can be truly applied to individuals, but it is far less clear that we can truly (or even meaningfully) ascribe potentialities to satisfy them. So any pasttensed or past-dated predicate true of an individual will suffice to produce a counterexample to ACTUALITY. Or take CLOSURE1: as we saw in section 5.7.1, this principle entails that anything has a potentiality to satisfy any tautological predicate. If we allowed for tensed or dated predicates, it would follow that I have a potentiality to have been a child or not to have been a child; or a potentiality to have been in London on 1 January 2000 or not to have been in London on 1 January 2000. But if we cannot truly or meaningfully ascribe any potentialities whose manifestation concerns the past, then even these tautological potentialities should be ruled out. Moreover, by DISJUNCTION, my having a potentiality to have been a child or not to have been a child entails that I have a potentiality to have been a child or else a potentiality not to have been a child—or generally, that I must have at least one of two potentialities with contradictory, past-concerning manifestations. But again, it might seem that the two candidate potentiality ascriptions should both be false (or meaningless). In general, it is natural to think that potentiality is essentially bound up with time in a fundamentally asymmetric manner: the manifestations of potentialities can concern the future or (as I have argued in section 5.7.4) the present, but not the past. Potentiality, to put it metaphorically, is ‘forward-looking’ in time; or so the thought goes. It is worth noting the import of these considerations. Suppose we accepted that there are no true, or even no meaningful, potentiality ascriptions with manifestations concerning the past. What is past now was once present and, before that, future. So we need not exclude that I ever had a potentiality, say, to be in London on 1 January 2000; I did, presumably, have such a potentiality up until 1 January 2000. Metaphysical possibility, as we shall see in chapter 6, is a matter of something’s possessing a relevant potentiality at some time or other. Metaphysical possibility is, in a certain sense, timeless. Even if nothing now has an (iterated) potentiality for me to be in London on 1 January 2000, then, it is still metaphysically possible even now that I should have been in London on 1 January 2000, since it is still true that I once possessed the potentiality to be in London on 1 January 2000. As this suggests, the import of a general ban on potentialities with past-concerning manifestations is limited when it comes to a potentialitybased theory of metaphysical modality. Nevertheless, I am now going to argue that such a ban is not justified, at least not for potentialities with past-dated manifestations. Until further notice, I will consider only this case of past-concerning

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potentialities: those whose manifestations involve a date in the past. (The case of undated past-tensed manifestations is a little more complicated, and I will return to it at the end of this section). Now, what exactly is objectionable about my having the potentiality, say, to have been in London on 1 January 2000? An initial response might be: if the potentiality is manifested at all, it has been manifested prior to my possession of it; but a potentiality’s manifestation cannot predate its possession. This cannot be quite right, however, at least if we think that a potentiality to be F is manifested in its possessor’s being F. The manifestation of my potentiality to have been a child, if there is any such potentiality, would consist in my having the manifestation property, that is, in my having been a child; likewise, the manifestation of my potentiality to be in London on 1 January 2000 consists in my possessing, right now, the property of being-in-London-on-1 January 2000. These are properties which I possess now, simultaneous with my potentiality (if any) to possess them.26 A better characterization of the relevant potentialities is that the possession of their manifestation property would have to be (and, if they are manifested, is) grounded in the possession of properties in the past: I now have the property of having been a child once because I once had the property of being a child; and if I now had the property of having been in London on 1 January 2000 (which I do not), I would have it because on 1 January 2000 I had the property of being in London. Let us reserve the term ‘past-concerning potentiality’ for those potentialities whose manifestation property, if possessed, must be grounded in the possession of some property in the past. (Obviously, whether a property is pastconcerning may change over time: the property of being in London on 1 January 2000 became a past-concerning property on 2 January 2000.) So what exactly is wrong with past-concerning potentialities? I would like to suggest that the objectionable feature of such potentialities is this: we cannot change the past; hence, it seems, we cannot exercise potentialities concerning the past; hence, it is concluded, we do not have such potentialities. If I was not in London on 1 January 2000, there is nothing I can do now to acquire the property of having been in London on 1 January 2000; if I was in London on 1 January 2000, there is nothing I can do now either to change or to contribute to my having the property of having been in London on 1 January 2000; that property is, as it were, fixed. 26 Moreover, while it is true that a potentiality cannot be possessed only after it is manifested, this is no useful way of capturing a temporal asymmetry in potentiality. For it is equally true that a potentiality cannot be possessed only before it is manifested; as I have argued in section 5.7.4, a potentiality must be possessed simultaneously with its exercise. Of course, potentialities may be, and very often are, possessed for a while prior to their manifestation; but it is equally true that potentialities may be, and often are, retained for a while after their manifestation.

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The intuitive idea that the past is fixed can be accounted for, however, in a different way. We might allow for past-concerning potentialities but insist that they are always trivial: where being F is a property whose possession would be grounded in the past, an object x will have the potentiality to be F if and only if the object is F. We have no potentialities for the past to have been different, though we have potentialities for the past to have been just as it was. There may not be much that we can now contribute to the manifestation of such potentialities, save our existence; but that, we have seen, is a feature shared by other potentialities, such as the potentiality to be self-identical. (This is not to say that past-concerning potentialities are just like tautological potentialities: my potentiality to be in London on 1 January 2000 had a non-trivial degree until it became past-concerning.) Accordingly, the degrees of past-concerning potentialities would have to be trivial, that is, maximal or zero. For as I argued in chapter 3, to have a potentiality to the maximal degree just is to lack the potentiality for the contradictory manifestation altogether. Since only one of two contradictory properties is ever possessed, it follows that of two potentialities with contradictory past-concerning manifestations, one is always lacked, and the other therefore possessed to the maximal degree. This too captures the idea, which motivated a general ban on manifestation properties dated to the past, that we cannot change the past: unlike most other potentialities, our past-concerning potentialities fix whether or not we have the relevant manifestation properties. It also offers a ready explanation of why we are so reluctant to ascribe such potentialities: as with other maximal-degree potentialities, their ascription is pragmatically odd because we have something more informative to say (namely, that the object has the past-dated property, say the property of having been in London on 1 January 2000). Here, then, is an alternative to a ban on past-concerning potentialities: the triviality thesis says that past-concerning potentialities are possessed if and only if their manifestation properties are, and hence are possessed to maximal degree if they are possessed at all. The triviality thesis, I maintain, does no worse than a general ban on past-concerning potentialities in capturing the intuitive reluctance to ascribing such potentialities. I will now suggest that it is indeed preferable to such a ban. Let us look, first, at potentialities whose manifestations involve dates that are not in the past. I have an ability to do about 60 miles a day by bike. It is now 10 a.m. on 1 January 2014; given that I am in Berlin, about 55 miles from the Polish border, I now also have a potentiality to be in Poland tonight, that is, by 8 p.m. on 1 January 2014. A sapling has the disposition to grow, say, one inch per month over the next six months; given that it is 20 inches tall in February 2014, it is

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disposed to be about 26 inches tall by August 2014. Note that these potentialities are extrinsic: the degree of their possession depends, in part, on their possessor’s position in time. If I do not start cycling towards the Polish border, my ability to be in Poland by 8 p.m. rapidly decreases in degree. If the sapling were to remain intrinsically unchanged from February to July 2014, then in July it would no longer be disposed to be 26 inches tall come August. (I have argued earlier that extrinsic potentialities are always grounded in joint potentialities with something else. What the ‘something else’ might be in this case depends on the metaphysics of time: it might be instances and stretches of time itself for an absolutist about time, or for a relationist it might be other objects whose configuration constitutes instances and stretches of time.) As we approach the evening, I don’t move towards the Polish border but remain where I am. My potentiality to be in Poland at 8 p.m. gradually decreases in degree; at 8 p.m., plausibly, it reaches degree zero: I no longer have the potentiality. But this is hardly an argument for the conclusion that nothing has potentialities whose manifestations are dated to the past. Take my potentiality not to be in Poland by 8 p.m. this evening. As we approach the evening and I remain where I am, that potentiality increases in degree; at 8 p.m., the increase should reach its culminating point. Or take my potentiality to be in Berlin at 8 p.m.: as we approach 8 p.m. and I make no attempt to leave the city, this potentiality, too, increases in degree while my potentiality not to be in Berlin at 8 p.m. decreases in degree; at 8 p.m., I have lost the latter potentiality, but have I also lost the former? It seems rather that I now possess the potentiality to be in Berlin at 8 p.m., and the potentiality not to be in Poland at 8 p.m., to the maximal degree: there is nothing I can do to lose the manifestation properties of these potentialities. I do not claim that these considerations constitute conclusive proof of anything. But they suggest that within the present framework, it is more natural to treat potentialities with manifestations dated to the past as being trivial, rather than to rule them out in principle. The triviality thesis, I have argued, provides a good way of capturing the intuitive difference between those potentialities that concern the past and those that do not. It is not entirely clear, however, that this intuitive difference must be captured: it might be mistaken. Here are two kinds of considerations which point in that direction. First: time travel. Suppose that time travel is possible. In that case, it would seem that I have some (perhaps slight) potentiality to travel back to the year 1914, and to have all sorts of properties dated to 1914: say, the property of witnessing

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how Franz Ferdinand was shot in 1914.27 Such potentialities are of exactly the kind that the triviality thesis would have excluded: potentialities, possessed at a time t, to be Ff-at-t  , where t  is earlier than t and the individual was not actually FD at t  (or indeed potentialities to have been a certain way, if the tense is understood to express what Lewis (1976a) calls ‘external time’, rather than ‘personal time’). Second: the nature of time. It is a much-discussed question whether time itself is asymmetric with respect to its past and future direction. It certainly seems that way to us in ordinary life: the future is largely unknown and open to be influenced, while the past appears fixed, inaccessible to causal influences, and better known than the future. It is highly debatable, however, whether this apparent asymmetry really is part of the nature of time itself. Horwich (1987), for instance, argues that while an asymmetry in the laws of nature would constitute good evidence for an intrinsic asymmetry of time itself, ‘there have emerged no compelling reasons to adopt time-asymmetric laws (or to postulate de facto intrinsic differences between the past and future directions)’ (Horwich 1987, 54f.). Now suppose that time itself exhibits no asymmetry; the seeming asymmetries are the product of our own psychological make-up, or of certain contingent boundary conditions of the universe, or something along those lines. Then we would have reason at least to doubt that potentiality exhibits temporal asymmetry. As I have pointed out above, potentialities with manifestations dated to the past appear to be based on joint potentialities with time itself, or with whatever its metaphysical basis might be. If that basis itself is asymmetric, it is plausible to assume that such an asymmetry will carry over to the relevant joint potentialities, and hence to the extrinsic potentialities grounded in them; if the basis is not asymmetric, or only contingently so, it is plausible that there is no asymmetry, or only contingent asymmetry, in the relevant potentialities too. To pursue this line of thought in any detail, however, goes beyond the scope of the present chapter. I will note only that any defender of the symmetry of time will have to provide some explanation of the obstinate appearance that the future is very different from the past. She will have to show how and why the seeming asymmetry is only contingent, or holds only locally, or is an illusion based on some features of human cognition. (Again, Horwich 1987 provides a useful model: he appeals to certain features of human cognition and phenomenology together with the de facto irreversability of certain processes.) Whatever exactly such an explanation looks like, it will most likely do double duty in 27 Thanks to Antony Eagle for pressing me on the case of time travel and on the relation between potentiality and time in general.

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explaining how and why the seeming temporal asymmetry of potentiality is only contingent, holds only locally, or is an illusion. What these considerations suggest is that the asymmetry stipulated in the triviality thesis may not hold, or may hold only approximately. It might be that past-concerning potentialities seem to us, mistakenly, to be trivial, for the same reasons for which time seems to us, mistakenly, to be asymmetric, and because time travel has not (yet) been invented. Or it might be that, for some reason to do not with the nature of time and potentiality, but with certain facts about (our portion of) the universe that are not essential to either time or potentiality, pastconcerning potentialities are typically almost trivial, i.e. tend only ever to have (nearly) maximal or very slight degrees. Fortunately, we need not settle the question here whether past-concerning potentialities are trivial, or nearly trivial, or only apparently trivial. So long as anything has at least potentialities to have those past-concerning properties that it actually has, the counterexamples with which I opened this section can be avoided. For in that case, clearly, we can infer from the actual possession of a past-concerning property to the possession of a potentiality to have it, hence saving ACTUALITY. We can do so, a fortiori, for tautological past-concerning properties, vindicating CLOSURE1. And since for any pair of contradictory pastconcerning properties, an individual will possess exactly one of them, everything will possess a potentiality for one of a pair of contradictory past-concerning properties, as required by CLOSURE1 in combination with DISJUNCTION. In my discussion so far, I have focussed on past-dated predicates and properties, such as the property of being in London on 1 January 2014. But what to do with tensed predicates? What, for instance, are we to say about the potentiality to have been a child? If we adopt a B-theory of time, then there is no separate question about tensed predicates or expressions. On such a theory, tensed predicates and statements are reducible, very roughly, to dated predicates and statements;28 thus a potentiality to have been red, as possessed at t, amounts to nothing other than a potentiality to be red at some time t  before t. On this interpretation, the considerations regarding dated predicates carry over directly to tensed predicates. For A-theorists, tensed predicates and statements are not so reducible, and it is a difficult question how A-theoretic tenses should be embedded within a potentiality operator. What exactly should a potentiality to have been F amount to? If we take it literally, it should be a potentiality to have the property of having been 28 That is not quite accurate. For a B-theorist, a tensed predicate or statement is typically reduced to relations between times plus an indexical element: hence ‘It was raining’ is true at any time t iff it is raining (timelessly) at a time t  prior to t.

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an F, that is, being a past F. Unlike the past-dated properties discussed above, the property of being a past F is always a past-concerning property. And for that very reason, the property of being a past F, unlike the past-dated properties we have discussed, will typically be one with respect to which we can change. For even if the past is fixed, it is not fixed what the past is; tomorrow is now in the future, but will soon be in the past. If I have a potentiality to climb Mt Everest tomorrow (and to survive the feat for some time), I thereby have a potentiality to acquire the property of having climbed Mt Everest; I would acquire that property as soon as the manifestation of the former potentiality is completed. There is no pressure, on this understanding of tensed predicates in the scope of the potentiality operator, towards an equivalent of the triviality thesis. The past-tensed manifestations, on this reading, concern the past relative to their time of manifestation; but the triviality thesis seemed plausible for potentialities whose manifestations concern the past relative to the time of their possession.29 If, however, there is a reading of irreducibly tensed predicates in the scope of a potentiality operator that makes them past-concerning in the latter way—along the lines of: a potentiality to have, now while the potentiality is possessed, the property of having been 30 —then such a reading, I suggest, should be open to much the same lines of argument as the case of dated predicates, or reducible tenses, discussed above. When in the remainder of the book I discuss ‘past-concerning potentialities’, I will be referring to those potentialities for which the triviality thesis has been argued to be natural (though not obligatory): potentialities to have a property whose possession would have to be grounded in the past relative to the time of their possession. In what follows, I will take it that we do not impose a general ban on pastconcerning potentialities; but I will remain neutral between the triviality thesis and the alternative view that there are non-trivial past-concerning potentialities. Mostly I will argue on the basis of the triviality thesis, for two reasons. First, the triviality thesis makes my life harder in accounting for metaphysical modality, by giving rise to apparent counterexamples (see chapter 7.6). And second, in the absence of a properly developed alternative view, the triviality thesis is the more clearly defined thesis; for if there are non-trivial past-concerning potentialities, the above considerations have not given us a good sense of which ones there are and how they behave. 29 The former way of understanding tensed predicates in the scope of a potentiality operator can be mimicked in B-theoretic terms as well: it is the potentiality to have, at some time t, the property of being F at a time t  prior t. 30 It seems to me difficult to formulate this reading without the use of at least some B-theoretic vocabulary.

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The relation between potentiality and time is a difficult one, and I will return to it and the questions that it raises at the end of chapter 7. For the time being, we have reason to think that this relation does not, at least, pose any insuperable obstacles to the logic of potentiality as proposed in this chapter.

5.9 The logic of potentiality and the logic of possibility In section 5.7, I have defended the four principles which, I hold, govern the logic of potentiality. The comparisons with essence and ability served to highlight where the logic of potentiality differs from the logic of these related phenomena and to formulate the main challenges to my four principles by analogy with these related phenomena. In comparing the logic of potentiality to that of possibility, we are faced with a challenge not of dissimilarity, since the two are hardly dissimilar at all, but rather with a challenge of similarity. Given that the logic of potentiality is governed by the rule of CLOSURE and the axioms DISJUNCTION, NON-CONTRADICTION, and ACTUALITY, does it not simply collapse into the logic of possibility? And if it does, then how is potentiality itself different from possibility: is it not just de re possibility by another name? And if that is the case, then how is a potentiality-based account of possibility to provide any illumination of or insight into the nature of possibility? These are fair questions, but they are also questions to which the materials of this book so far provide answers. Put briefly, the answer is this: the crucial point about the present account is not just where we are now, with the logic of potentiality. It is how we got there; it is the direction of explanation. The account of potentiality that we have now reached is indeed very close to what is commonly known as de re possibility. (Potentiality differs from de re possibility, formally, in being expressed by a predicate operator; and extensionally with respect to a few cases: thus, pace Williamson (2010), it is possible that I do not exist, but I have no potentiality not to exist. Never mind those two differentiations; my point here is more general). But that large-scale agreement should hardly come as a surprise: after all, the goal was to generalize potentiality so as to make it suitable for an account of possibility. The important point is that this general notion of potentiality has not been formulated directly, or simply stipulated out of nowhere; it has taken me four chapters to reach the generalized notion. We started with ordinary dispositional properties as we know them from everyday and scientific discourse. We have seen how a metaphysically realist account of such properties must be more generous

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in the disposition-like properties—which we have called potentialities—that it accepts (chapter 3). We have further extended the conception of potentiality in chapter 4, embracing extrinsic potentialities. The generosity of chapter 3 has been central in arguing for DISJUNCTION and ACTUALITY; the extrinsic potentialities of chapter 4 have been central in arguing for CLOSURE. But the former is needed simply to have a systematic, realist metaphysics of dispositional properties such as fragility; and the latter we have seen to be grounded in the intrinsic potentialities that objects possess, individually or jointly. The metaphysical story that we can tell about the four principles starts with the intrinsic dispositions of objects, generalizing just enough to yield a suitably realist account of them. It continues to build up from them extrinsic potentialities, which are fully grounded in the intrinsic potentialities with which we started. With this building work done, we have reached a conception of potentiality that is not metaphysically basic but formally well regimented: potentiality that may be intrinsic or extrinsic, fundamental or derivative. It is this conception that comes close—though no more than close—to de re possibility. The next chapter will take the last steps to reach possibility, de re as well as de dicto, by only a little more building work. What is distinctive of the potentiality-based account of possibility is not where it ends: if successful, it ends at possibility. (Where else?) It is, rather, where it starts. Heuristically, it starts with dispositions such as fragility. Metaphysically, it starts with (i.e. takes as its primitive) intrinsic potentialities, possessed individually and jointly, where ‘potentiality’ is understood in the generous sense that was reached at the end of chapter 3. All else is built on that. If we were to talk of potentiality for its own sake, or for the sake of a different explanatory project, some of the features that I have highlighted in this chapter and the last—such as Cambridge potentialities, or closure under logical implication—might be utterly uninteresting. The discussion of potentiality has been tailored to my needs in developing a theory of possibility. However, it has been so tailored, I hope, not by being prejudiced in favour of certain answers, but only in identifying the relevant questions. The target account of possibility has set the agenda, but it has not influenced the outcome. I said in chapter 1 that, in order to capture metaphysical possibility, we need to develop a more metaphysical conception of dispositionality or, as we have come to call it, potentiality. That conception is now fully developed. It may not serve all purposes that a dispositionalist might wish to pursue. For some purposes—such as a theory of causation, of laws, or perhaps of action—it will be more useful to focus on a narrower class of potentiality. For my purpose, we should impose no restrictions; every potentiality counts.

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The comparison with possibility raises one final point that does, after all, concern dissimilarity. The axioms and rule that we have given for potentiality correspond to those of system T of modal logic, which I have said is the minimal requirement on a logic of metaphysical modality. Of course, many believe that the logic of metaphysical modality ought to be stronger than system T. In particular, it is generally assumed that metaphysical modality validates the characteristic axioms of the systems S4 and S5: (S4) 33φ → 3φ, (S5) 3φ → 3φ. I have not given any corresponding axioms for potentiality and the operator POT. Given the usual interdefinability of 3 and , the POT analogues of (S4) and (S5) would be as follows: (POT4) POT[λx.POT[](t)](t) → POT[](t), (POT5) POT[](t) → ¬ POT[λx.¬POT[](t)](t). In English, (POT4) says that if an object has a potentiality to have a potentiality to , then it also has a potentiality to : iterated potentialities to  collapse to non-iterated ones. (POT5) says that if an object has a potentiality to , then it has no potentiality not to have the potentiality to : nothing has a potentiality to lack (or lose) the potentialities that it actually has. Neither principle seems particularly plausible to me. Iterated potentialities do not seem to collapse in the way envisaged by (POT4): liquid water, by having a potentiality to turn to ice, has an iterated potentiality to break; but it does not have a potentiality to break. This is a counterexample to (POT4). A piece of ice, by having the potentiality to melt, has a potentiality to lose its potentiality to break; a child with the potential to be a child prodigy has the potentiality to lose that potential, as is witnessed by the fact that the potential is in fact lost when the child grows up. These are counterexamples to (POT5). From the rejection of (POT4) and (POT5) it does not follow that (S4) and (S5) must be rejected on a potentiality-based account of metaphysical possibility. For the account has some other resources: it appeals to iterated potentiality, not potentiality simpliciter, and it prefixes the potentiality ascription with an existential quantifier. It is time to look at the account of possibility in more detail.

6 Possibility: Metaphysics and Semantics 6.1 Possibility defined Until now, we have examined potentiality. I have argued that a suitably realist understanding of potentiality is already committed to all that is required for a theory of possibility. Most of the work, therefore, has already been done. What remains for this chapter and the next is to draw the different threads together and spell out in some detail the emerging picture of metaphysical modality. Given the understanding of potentiality that has been achieved, we are now in a position to provide a definition of the possibility operator. Intuitively, the idea is this. When we speak of possibility, we speak of potentiality in abstraction from its possessor; a possibility is a potentiality somewhere or other in the world, no matter where. To make this more precise, we express the idea of abstracting from the potentiality’s possessor by an existential quantifier. Of course, we want to include potentialities with any number of iterations: potentialities to have, or for something to have, further potentialities (to have, or for something to have, yet further potentialities, and so on). Iterated potentiality also takes care of the logical form that we need in order to correlate potentiality statements with possibility claims. So we get: POSSIBILITY It is possible that p =df Something has an iterated potentiality for it to be the case that p. Note that POSSIBILITY provides a reductive account of possibility in the strictest sense: the possibility operator could, in principle, be eliminated everywhere and be replaced by its definiens. I have no ambition to eliminate the operator or possibility talk in general. The purpose of such talk is precisely to abstract from the bearer of a potentiality. I will argue shortly that this is not something that we do very much in ordinary modal talk: when we speak modally we do generally

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bother to specify the bearer of the potentiality in question. But this is not to say that we have no use for such abstraction; in philosophy, we certainly do. The account of possibility that I am offering is reductive in that it provides a straightforward definition of the possibility operator. It is not reductive in the sense of ‘reducing the modal to the non-modal’. The notion of potentiality is a modal one if anything is. The account does not expel modality from the fundamental level, if there is one. Rather, it tells us where to look for modality: in the properties of objects. We can think of the account as providing an explanatory re-ordering within the package of modal concepts, putting the notion of potentiality at the basis, and building other notions, in particular, that of possibility, on top of it. The theoretical weight of POSSIBILITY rests firmly on the account of potentiality. Iterated potentiality, as we have seen earlier, provides not only an extension of the ‘reach’ of an individual’s potentialities, but also the right kind of syntactic structure to define possibility: while potentiality simpliciter is always a potentiality to . . . (hence expressed by an operator that takes a predicate to form another predicate), iterated potentiality is a potentiality for it to be the case that . . . (hence expressed by an operator that takes a sentence to form a predicate; for the sake of brevity, I often abbreviate this to ‘an iterated potentiality for p’). Iterated potentiality includes the case of potentiality simpliciter as once-iterated potentiality. Iterated potentiality, and potentiality in general, includes both intrinsic and extrinsic potentiality. If we wanted to include joint potentiality (that is, potentiality possessed by a number of objects together), we might amend the right-hand side of POSSIBILITY to read ‘Something has, or some things have, an iterated potentiality for it to be the case that p’. For the sake of defining possibility, this is not strictly needed: whenever a number of objects xx jointly have an iterated potentiality for p, any one of xx thereby has an extrinsic iterated potentiality for p (or so I have argued in chapter 4.6). However, in applying POSSIBILITY, it will be useful to keep this qualification in mind. For in so doing, we will be looking for witnesses to the existential claim on its right-hand side—that is, objects which verify the existential claim—in order to establish the truth (or suggest the falsity) of its left-hand side. While the bearer of an extrinsic potentiality will do the job, the real metaphysical story will often have to appeal to joint potentialities. For instance, the possibility that Mary marries John might have Mary as its witness, in virtue of her possessing the extrinsic potentiality to marry John. That extrinsic potentiality in turn is grounded in Mary and John’s joint potentiality to marry. So the full metaphysical story will appeal to joint potentialities; but the definition itself, just for the sake of extensional correctness, need not do so.

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Indeed, given the flexibility of extrinsic potentiality that I have proposed in chapter 4, there will be an abundance of totally irrelevant witnesses to any given possibility. I might serve as a witness for the possibility that you are standing, if I have an extrinsic potentiality to be such that you are standing. Again, there is a grounding story to be told about how I come by that extrinsic potentiality, and it will involve our joint potentiality to be such that you are standing. But the relevant potentiality is, of course, yours and yours alone: it is simply your potentiality to stand. And it is this potentiality that grounds, partly but largely, our joint potentiality. When applying POSSIBILITY to different cases, I will be concerned to point to relevant witnesses. The abundance of irrelevant witnesses, due to the flexibility of extrinsic potentiality, is no threat to the definition. As far as metaphysical explanation goes, it is the relevant witnesses—in the case just described: you, not me—that are basic. Grammatically speaking, the two sides of POSSIBILITY are phrased in the present tense, but they are meant to be read as timeless. Metaphysical possibility is not subject to change over time. Potentiality, of course, is. This is why we must read the locution ‘has a potentiality’ in the definition as untensed. We might make this explicit by slightly rephrasing POSSIBILITY as POSSIBILITY∗ It is possible that p =df Something has, had, or will have a potentiality for it to be the case that p.1 The fact that potentiality itself is not timeless enables us to define tensed notions of possibility as well. As a brief excursion, I would like to define two such notions: weak and strong tensed possibility. The tensed notions of possibility will be indicated by a subscript T (for ‘tensed’) to ‘possible’ and cognate expressions. Unlike metaphysical possibility, they are thought to hold at a time; hence the two sides of the biconditionals have to be read as properly tensed. (WTP) It is (was, will be) weakly possibleT that p just in case something has (had, will have) an iterated potentiality for p. (STP) It is (was, will be) strongly possibleT that p just in case everything jointly has (had, will have) an iterated potentiality for p.

1 I use tenses, rather than quantifiers over times, to remain neutral on issues surrounding the presentism/eternalism debate. Participants in the debate agree on the legitimacy of tenses but disagree about the legitimacy of quantifying over times other than the present. See, however, chapter 7.9 for considerations that favour a combination of the potentiality view with eternalism rather than presentism.

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Weak possibilityT takes into account all the iterated potentialities of objects at a given time. Strong possibilityT takes into account only the joint iterated potentialities of everything there is at a given time, but not the iterated potentialities of anything short of the totality of all the objects there are. Weak possibilityT is simply metaphysical possibility restricted to a time. It discounts potentialities that were possessed in the past, such as my potentiality (now lost) to be a child prodigy, or Socrates’s never exercised potentiality to be a carpenter, or my once-possessed potentiality to be Germany’s first female chancellor (now lost, though for reasons extrinsic to me). Possibility in this temporally restricted sense is what concerns us when we wonder how things could ‘go from here’. Strong possibilityT is a matter of the potentialities that all things together have. It is the kind of possibility that is at stake in debates about determinism. We might think of determinism as the thesis that for each pair of mutually exclusive future courses of events, only one of them is strongly possibleT . By the definition, this means that the joint potentialities of all things together have either the maximal degree or none at all; in other words, all the objects that there are have an iterated potentiality for p, they have no potentiality for its negation. Strong possibilityT is possibility ‘all things considered’. Incompatibilists about freedom and responsibility often hold that for an action to be free and to vindicate attributions of responsibility, it must have been strongly possibleT for the agent to do otherwise; some compatibilists respond that weak possibilityT is enough to vindicate the ascription of freedom and responsibility. Or so, at any rate, a reconstruction of the debate in terms of potentiality should say. Neither strong nor weak possibilityT is metaphysical possibility proper, and neither of them will be my concern in what follows. Both notions are useful for other purposes, however, such as the free will debate. A potentiality-based account of them can be used in those areas. However, this is a different project from the one that I am pursuing here.

6.2 Applying the definition To give a feel for the account of possibility, I will go through a few examples of metaphysical possibilities and show how they are accounted for. (These are the relatively easy cases; potential counterexamples will be discussed in chapter 7.) The simplest kind of case is that of predicative possibilities of the form: it is possible that a is F. For instance, it is possible that I sit, and it is possible that I stand. Both possibilities are simply a matter of my having the relevant potentialities: a potentiality to sit and a potentiality to stand. In virtue of having the

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potentiality to sit, I have a potentiality to be such that I am sitting, and hence (by our definition) a once-iterated potentiality for it to be the case that I sit. The appeal to iterated potentiality is unnecessary for a case like this, but it does no harm either. The potentiality that ensures the truth of a possibility claim might be a joint potentiality. Thus it is possible (though not actually true) that you and I sing a duet together. The relevant potentiality, of course, is our joint potentiality to sing a duet. Note, again, that POSSIBILITY is formulated with a singular quantifier; so the witness for the possibility of our singing a duet together cannot be you and me, a plurality. It might be our mereological fusion, or, to be mereologically uncommitted, it might be either one of us: you, in virtue of your extrinsic potentiality to sing a duet with me, or I, in virtue of my joint potentiality to sing a duet with you. If this is thought problematic, we can easily rephrase POSSIBILITY with a plural quantifier instead, as indicated above. Since every joint potentiality gives rise to an extrinsic potentiality and every extrinsic potentiality must be based on some joint potentiality, such rephrasing is not required for extensional correctness. The two formulations are extensionally equivalent. Iterated potentiality is needed when we deal with such cases as the following. It is possible that my great-granddaughter will be a painter. I do not have a great-granddaughter. Perhaps my great-granddaughter is a future existent; but perhaps I will never have a great-granddaughter. (I might have no greatgrandchildren, or they might all be male.) It is still possible that I should have a great-granddaughter who becomes a painter. The relevant potentiality in this case is mine, and it is iterated: I have a potentiality to have a child who has the potentiality to have a child who has the potentiality to have a daughter who has the potentiality to be a painter. Hence something (to wit, I) has an iterated potentiality for it to be the case that my great-granddaughter is a painter. If it is indeed true that there will be a great-granddaughter of mine, then it might be my future great-granddaughter herself who serves as witness to the existential claim that something has an iterated potentiality for my greatgranddaughter to be a painter. By POSSIBILITY∗ , it is possible that p if there is, was, or will be an object with an iterated potentiality for p. So if there will be a great-granddaughter of mine with a (once iterated) potentiality (for herself) to be a painter, then she counts as a witness to the future-tensed existential claim that there will be an object with an iterated potentiality for my great-granddaughter to be a painter. Her claim, however, would not cancel out mine. There may be any number of witnesses to an existential claim. While future objects are often dispensable as witnesses to the truth of the existential claim that something has an iterated potentiality for p, past objects

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may well be indispensable (depending on how matters turn out concerning the triviality thesis in chapter 5.8). The possibility that Socrates should have been a carpenter is not explained by the iterated potentialities of any object now existing. It is because it was the case that there was an object with an iterated potentiality for Socrates to be a carpenter, that this metaphysical possibility holds now. The relevant witness for the past-tensed existence claim is (or rather, was) Socrates himself in virtue of his having possessed the (once-iterated) potentiality to be a carpenter. So far, we have focussed on possibilities de re, arguably the easier case for the potentiality-based conception of possibility. Given the notion of potentiality that I have developed, however, possibilities de dicto can be accounted for as well. Let us start, again, with the simplest kind of case. It is possible that there is a woman president of the US. A relevant witness (certainly not the only one) is Hillary Clinton, who has a potentiality to be a woman president of the US, and thereby a potentiality for there to be a woman president of the US. Next: it is possible that there be a human space station on Mars. A relevant witness might be any of the engineers who have a potentiality to build a space station on Mars (and the equipment for getting there). The potentiality must be extrinsic, based on a joint potentiality possessed at least with Mars. It is sometimes said that it is metaphysically possible that humans should have three legs instead of two. Here it is less easy to identify a witness. What we would have to do is roll back evolutionary history and look at our pre-human ancestors. They had an iterated potentiality for there to be human beings: a potentiality to have offspring with a potentiality to have offspring with a potentiality to have . . . and so on, to the potentiality to have offspring that is human. They also, presumably, had a potentiality to have offspring with a potentiality to have offspring with a potentiality to have . . . and so on, all the way to the potentiality to have offspring that is human and three-legged. I said ‘presumably’; and in fact, it is a feature of the potentiality-based account that what is possible and what is not is hostage to the way things actually are. It is thereby also hostage to matters that we can find out (at best) by empirical enquiry. The same considerations hold, mutatis mutandis, for other alleged de dicto possibilities, such as the possibility of there being talking donkeys, the possibility of there being unicorns, and so forth. That empirical enquiry is relevant for modal knowledge should not be news to metaphysicians who have learned Kripke’s and Putnam’s lessons. But the potentiality-based account certainly intensifies the dependence of modality on what is actually the case and of modal knowledge on empirical knowledge. However, on the metaphysics that I have developed throughout this book, what is

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actually the case and what we can find out about by empirical enquiry, is itself abundantly modal. The actual world is rich with potentialities. It is, moreover, modal on a large scale: the manifestations of the potentialities that there are go far beyond our first intuitive examples. I take its sensitivity to empirical discovery to be a virtue of the account, though to spell out the full epistemology of modality on the present view is a subject for another book. Given an account of possibility, we can define necessity in the usual way: NECESSITY

It is necessary that p =df It is not possible that not p.

Strong and weak possibilityT likewise have their corresponding kinds of necessity. NECESSITY links necessities to potentialities indirectly, via possibility. Intuitively, necessities mark the limits of the potentialities that objects have. More precisely, it is necessary that p just in case nothing has, or had, or will have a potentiality to be such that not-p. It will be useful to look at two standard examples from the literature on modality. One example is the necessity of identity. Let us begin with the simple case: say, the necessity that Hesperus is Hesperus. By the logic of potentiality as laid out in chapter 5 (in particular, NON-CONTRADICTION), nothing has a potentiality to be such that a contradiction holds; and hence nothing has a potentiality to be such that Hesperus is not Hesperus. So the simple case is easily accounted for. The more interesting case, of course, is the identity claim ‘Hesperus = Phosphorus’. (Note that the formal language of chapter 5 by stipulation excluded the possibility of coreferential names, so identity claims such as ‘Hesperus = Phosphorus’ cannot even be formulated in it. But that was a mere convention, designed only to highlight objectual content, and as a consequence easily dropped.) We can see informally that nothing will have a potentiality to be such that Hesperus = Phosphorus. Any appearance to the contrary is based, after all, on the difference between the names ‘Hesperus’ and ‘Phosphorus’, or on their meaning (in some appropriate sense of ‘meaning’ that does not identify the meaning of a name with its referent). But potentiality is not a matter of words, nor of their meaning. It is the object itself, however we name it, that possesses or fails to possess a potentiality; and it is the object itself, however we name it, that is involved in a manifestation property such as being identical to Phosphorus. There is no difference, at the level of metaphysics, between Hesperus having a given potentiality, and Phosphorus having it; or between something’s having a potentiality to be such that Hesperus = Phosphorus and something’s having a potentiality to be

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such that Hesperus = Hesperus. Since nothing has the latter potentiality, nothing has the former potentiality.2 The same goes for iterated potentiality. Our second example, and a more complex one, is the necessity of origin. Many have been convinced by Kripke (1980) that a given individual could not have a different origin from the one that it actually has.3 For instance, I could not have originated from different parents, and indeed not even from a different sperm and egg than I actually did. This kind of necessity encodes a modal asymmetry in the temporal career of an individual: while it is contingent where and when an individual’s life ends (and indeed how it develops once the individual has originated), its beginning is settled, or else it would not be that same individual.4 Note, however, that temporal asymmetry cannot be the whole story, as McGinn (1976, 130) has pointed out: for not every feature of an individual’s origin is plausibly considered necessary. Even if I could not have originated from different parents, I might have been born in a different place, or in a different way (assuming, for the sake of the example, that I originated in my birth, rather than my conception or somewhere between those two). Penelope Mackie has accordingly distinguished two questions: the ‘why origin (rather than development)?’ question and the ‘why these features of origin (rather than others)?’ question (Mackie 1998, 64). Ideally, the potentiality account should answer both questions. The description of the necessity of origin thesis as, in part, a matter of temporal asymmetry already suggests where a potentiality-based answer to the ‘why origin?’ question should start: with temporal asymmetries in potentiality itself. I have discussed such asymmetries in chapter 5.8. The (tentative) conclusion was that there may be some temporal asymmetry between potentialities whose 2 Formally, we can argue for the necessity of identity in much the same way as others have, namely by substitution of identicals:

(1)

Assume a = b.

(2)

¬POT[λx.x = a](a).

(3)

Therefore, by substitution of identicals (b for a): ¬POT[λx.x = a](b).

And so on for any number of iterations. 3 The necessity of origin is not uncontroversial: Dummett (1981) and Wiggins (1980) have expressed doubts about it; Robertson (1998) argues against it. The potentiality view, as we shall see in this paragraph and the next, is not strictly committed to the necessity of origin. Whether that necessity holds depends on whether potentiality has certain features to be pointed out in a moment. It is natural, but not absolutely necessary, to think that potentiality has those features. 4 Kripke himself notes, albeit somewhat implicitly, that the necessity of origin is related to an asymmetry between divergent future possibilities and convergent past possibilities (Kripke 1980, 115, fn. 7). Mackie (1974) has argued more explicitly for an explanation of the necessity of origin intuition that appeals to this symmetry; Mackie (1998) has linked it to a branching picture of possibility. The potentiality account is rather similar in spirit, though not quite in letter, to a branching-worlds model of possibility.

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manifestations concern the future (or the present) and those which concern the past. In particular, it seemed a natural assumption that past-concerning potentialities, such as the potentiality to be F-at-t as possessed at times after t, or the potentiality to have been F (on one reading: see chapter 5.8), are possessed only trivially: where G is a past-concerning property, any individual x has the potentiality to G to the maximal degree if x is G, and lacks it otherwise. That natural assumption, however, is not obligatory: cases of time travel and considerations concerning the nature of time suggest that there may be past-concerning potentialities with non-trivial degrees. Perhaps the degrees of such past-concerning potentialities are only nearly trivial, or perhaps they only appear to be trivial (or nearly trivial) because we do not usually consider time travel or related scenarios. For present purposes, it is good enough if the appearance of a temporal asymmetry in potentiality explains the appearance of the necessity of origin; the latter appearance will be veridical if, and plausibly only if, the former is. Suppose, then, that there really are no non-trivial past-concerning potentialities. How would that explain the necessity of origin? Take the impossibility of my having different parents. If it were possible for me to have different parents, then something, at some time, would have to have an (iterated) potentiality for me to have different parents. We can see the problem for such an alleged possibility if we ask at what time the potentiality would be possessed. Suppose, first, that it was possessed at or after the time of my origin (be it birth, conception, or anything in between). But in that case it would have to be a pastconcerning potentiality: to have different parents, in the sense at issue, just is to have been originated, in the right ways, by different people. If such potentialities are always trivial, then anything’s potentiality for me to have had my actual parents has maximal degree, while nothing has a potentiality for me to have had different parents. (Note that, if time travel convinces us of the possibility of non-trivial past-directed potentialities, these need not include the potentiality to have different parents; such a potentiality is hardly a matter of time travel.) Suppose, then, that the relevant potentiality is possessed prior to the time of my origin. But then, who or what would possess it? Who or what, that is, would be the relevant witness to the possibility of my having different parents? Here it helps to keep the distinction between Mackie’s two questions in mind and think about the difference between those features of origin that are allowed to be contingent and those that appear to be necessary. In accounting for the possibility that, say, I had originated (i.e. been born or conceived or . . . ) in a different place, or in a different way, the relevant witnesses will always be or at least include my

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actual parents: my mother had the potentiality to give birth in a different hospital, or to make different decisions about my birth, and so forth. (Similar considerations hold for the sperm and egg from which I originated.) Certain individuals involved in our origin, most saliently the parents, have a special role to play in explaining the possibility that some features of that origin had been different. It is unsurprising, then, that the possibility of individuals other than these very ones to have been thus involved in our origin is a special case. Indeed, it is hard to find a convincing story about such a possibility: how would those very individuals, my parents, be witness to the possibility that not they but someone else play that role in my origin? And if not they, then who or what else could be? These are open questions, not conclusive arguments. They do not prove the necessity of origin thesis, even given the assumptions of a potentiality-based theory of modality and the temporal asymmetry of potentiality. But they show how to debate that thesis on a potentiality-based theory, and they provide some prima facie reasons for accepting it. There is, of course, room for debate. But that is true on any account of metaphysical modality. This should suffice as a start. The potentiality view can account for a variety of cases in a natural way. Like any philosophical theory, it is faced with a number of challenges. I have mentioned those challenges at various points in the book. It is time to begin to answer them directly and systematically.

6.3 Three constraints In chapter 1, I listed three constraints for a potentiality-based account of possibility. I propose that the success (or failure) of the account is a matter of how well (or badly) it does at meeting those constraints. The rest of the book will be concerned with arguing that it does succeed, and showing how it does, in meeting them. So it is worth restating the three constraints. (1) Extensional correctness: The potentiality-based account must respect, at least to a large extent, our prior judgements about what is or isn’t metaphysically possible. Where it does not respect them, it should offer an explanation for why those judgements went wrong. (2) Formal adequacy: The potentiality-based account must provide the right logical form (as an operator that takes sentences to form sentences) and the right logical structure (as captured at least in system T of modal logic) for possibility statements. (3) Semantic utility: Potentiality must provide the resources to formulate a semantics for ordinary modal language.

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The constraint of extensional correctness will be addressed in the next chapter, where we will go through difficult cases and putative counterexamples and address them one by one. The constraints of formal adequacy and of semantic utility will be addressed in this chapter. Both are part of the positive picture that comes with the potentiality account, and the goal of this chapter is to develop the positive picture. Section 6.4 will begin by showing how the view meets the constraint of formal adequacy. The bulk of the work has been done, of course, in chapter 5, where it was argued that potentiality can not only afford the introduction of a sentence-modifying operator for iterated potentiality, but is also governed by logical principles that correspond exactly (modulo the difference between sentence and predicate modifier) to those of system T for possibility. What remains to be done is simply this: to show how these features are implemented, given the bridge principle POSSIBILITY, in deriving the relevant features of possibility. Sections 6.5–6.9 then go on to sketch the application of potentiality to modal semantics. I will focus on what is sometimes called ‘dynamic’, and sometimes ‘circumstantial’, modality, and in particular on the paradigmatic case of the modal auxiliary ‘can’. The basic idea is that such modal expressions straightforwardly ascribe potentialities to objects. Modal language is notoriously contextdependent, and so I take the chief challenge to be the formulation of a mechanism for this context-dependence. It is here that extrinsic potentialities will, once again, be indispensable: much of the contextual variation in modal language will be attributed to the degree of intrinsicality or extrinsicality that a given utterance requires of the potentiality ascribed.

6.4 Formal adequacy The challenge of formal adequacy came in two parts. A first part concerned the logical form of potentiality ascriptions and possibility statements: potentialities are potentialities to . . . , while possibilities are possibilities that . . . . The second concerned the logical structure of metaphysical possibility, which is characterized by such principles as the rule of closure under logical implication and the T axiom. The challenge is simply to show that the potentiality-based account can accommodate, and ideally explain, this logical structure. Let me take up the two parts of the challenge in turn. In the first half of chapter 5, I showed that we can make sense of potentialities to be such that φ, for any logical form of φ (in the language of standard predicate logic). I then defined iterated potentiality in terms of potentiality simpliciter, as follows. We use POT as the operator that expresses potentiality simpliciter, such

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that where  is a singular or plural predicate of arbitrary complexity and t a singular or plural term, POT[](t) ascribes to t the potentiality to . Next, we define an operator POTn [φ](t), for any natural number n ≥ 1, sentence φ, and term t, to express ‘n-step potentiality’: 1. POT1 [φ](t) =df POT[λx.φ](t), 2. POTn+1 [φ](t) =df POT[λx.∃x(POTn [φ](x)](t)). When n = 1 n-step potentialities are potentialities simpliciter; otherwise they are potentialities to be such that something has an (n – 1)-step potentiality to be such that φ. Finally, iterated potentiality itself is simply a generalization over n-step potentiality: it is n-step potentiality, for any n (greater than 0). Thus we can understand POT∗ in terms of POTn , as follows: (POT∗ ) is true.

POT∗ [φ](t) is true iff, for some natural number n ≥ 1, POTn [φ](t)

We thus have a notion of potentiality available that is properly expressed as an operator on sentences. The definition of a possibility operator 3, in accordance with POSSIBLITY above, then becomes: (3) 3φ =df ∃x POT∗ [φ](x) (for x the first variable not free in φ). The first part of the challenge is met: the definition of possibility is well-formed. Now, if we plug into (3) the definitions of POT∗ and POTn , we get an interesting result. For the definition of possibility merely repeats outside the scope of the potentiality operator what the definition of iterated potentiality has already done inside its scope: the addition of existential quantifiers prefixing POT. Thus if 3φ is true, at least one statement out of the following list must be true (the list is really infinitely long, so I can only gesture at it, but the continuation should be clear): (3LIST) 1. ∃x POT[λx.φ](x), 2. ∃x POT[λx.∃x POT[λx.φ](x)](x), 3. ∃x POT[λx.∃x POT[λx.∃x POT[λx.φ](x)](x)](x), ... Informally, it is possible that φ just in case something has a potentiality to be such that φ, or something has a potentiality to be such that something has a potentiality to be such that φ, or something has a potentiality to be such that something

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has a potentiality to be such that something has a potentiality to be such that φ, and so forth. This observation will help us understand why the logical structure of potentiality, as studied in chapter 5, can be transferred directly to possibility. Recall the principles that govern the logic of potentiality: CLOSURE Potentiality is closed under logical equivalence: If being  is logically equivalent to being , then having a potentiality to be  is logically equivalent to having a potentiality to be . DISJUNCTION Potentiality distributes over, and is closed under, disjunction: An object has a potentiality to be -or- if and only if it has a potentiality to be  or a potentiality to be . ACTUALITY Potentiality is implied by actuality: Anything which is  must also have a potentiality to be . NON-CONTRADICTION Nothing has a potentiality to be such that a contradiction holds. In the formal language of POT, these become: ClosurePOT POT∨ TPOT NCPOT

If  t ≡ t, then  POT[](t) ≡ POT[ ](t),

POT[λx.(φ ∨ ψ)](t) ≡ POT[λx.φ](t) ∨ POT[λx.ψ](t), t → POT[](t), ¬ POT[λx.⊥](t).

Now, it is generally agreed that a logic for metaphysical modality must be at least a so-called normal modal logic satisfying axiom (T). There are a number of ways in which a normal modal logic can be characterized; typically, this is done in terms of the necessity operator. Then a normal modal logic is any extension of classical logic that includes the rule of necessitation (any theorem remains a theorem when it is prefixed by the necessity operator) and axiom (K) (if the implication from p to q is necessary, then the necessity of p also implies the necessity of q). The (T) axiom, in terms of necessity, is simply the implication from necessity to actuality: if necessarily p, then p. For our purposes, it will be better to choose an alternative but equivalent formulation that uses the possibility operator. Chellas (1980, 114–118) provides a list of combinations of rules and axioms that yield a normal modal logic. For our purposes, the best combination is one that corresponds to CLOSURE, DISJUNCTION,

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and NON-CONTRADICTION: possibility is closed under logical equivalence, distributes over and is closed under disjunction, and is never truly applied to a contradiction. The relevant (T) axiom, of course, is that actuality implies possibility: if p, then possibly p. Put formally, we have Closure3 If  φ ≡ ψ, then,  3φ ≡ 3ψ, (3∨) 3(φ ∨ ψ) ≡ (3φ ∨ 3ψ), (T3) φ → 3φ, (3⊥) ¬3⊥. This parallelism in structure should already be a cause of optimism in the face of the challenge of formal adequacy. Let me briefly outline how the challenge is met. Given the definition of possibility in POSSIBILITY and (3), together with the principles for potentiality, we can see why the logic of possibility must be (at least) as we have just outlined it. For the logical structure of potentiality and possibility are not only alike (modulo the difference between predicate operator and sentence operator), they are also relevantly similar to that of the existential quantifier. The quantifier ∃ is closed under logical implication and equivalence; it distributes over, and is closed under, disjunction; it is never truly applied to a contradiction; and it is truth-preserving when prefixed to a true sentence (assuming a non-empty domain). That should be no surprise; after all, standard possible-worlds semantics concurs with the potentiality view that the truth conditions for possibility claims have the form of existence claims. Now, if we look at (3LIST) above, it turns out that any item in the sequence involves just an increasingly long and complex combination of the potentiality operator, POT, and the existential quantifier, ∃. Both, as we have just seen, exhibit basically the same logical structure; and that structure, as is known from classical predicate logic, is one that is preserved through iterations of the operators. Given these two observations, we should expect every item on the list to be governed by principles corresponding to CLOSURE, DISJUNCTION, ACTUALITY, and NON-CONTRADICTION. And so they are. Hence possibility, which by our definition consists in the holding of at least one item on (3LIST), is itself governed by those principles. We can look at each of the principles in a little more detail to see why this is so (readers less interested in the details can skip the remainder of this section; readers more interested in the details are referred to the appendix). First, Closure3. Take any two logically equivalent sentences, φ and ψ. Being such that φ will also be logically equivalent to being such that ψ. Hence by CLOSURE, having a potentiality to be such that φ is logically equivalent to having a

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potentiality to be such that ψ. So if something has a potentiality to be such that φ, something (i.e. that same thing) has a potentiality to be such that ψ. (CLOSURE, like any equivalence, remains valid when we insert an existential quantifier on each side of the biconditional in its consequent.) Thus we have Closure3 for the case of once-iterated potentiality, item 1 on (3LIST). By repeating the same argument, we can show it to hold for subsequent items too. Second, (3∨). Given the logic of the existential quantifier, we can insert an existential quantifier on each side of the biconditional in DISJUNCTION: something has a potentiality to be -or- if and only if something (namely, the same object) has a potentiality to be  or a potentiality to be . The second existential quantifier may, in turn, be distributed over the disjunction in its scope: something has a potentiality to be -or- if and only if something has a potentiality to be  or something has a potentiality to be . Now let  and be of the form ‘is such that . . . ’, each completed with some sentence, say φ and ψ respectively. Then we have (3∨) for the case of once-iterated potentiality, item 1 on (3LIST). Again, subsequent items on the list are covered by repeating the same argument. Third, (3⊥). According to NON-CONTRADICTION, nothing has a potentiality to be such that a contradiction is true. Hence it is not the case that something has a potentiality to be such that a contradiction is true; this is item 1 on (3LIST). Since that is itself an axiom, its negation is a contradiction, which allows for another application of NON-CONTRADICTION to yield the next item on (3LIST), and so on. Fourth, (T3). Take any truth: for instance, the truth that I am sitting. In a trivial application of the ‘such-that’ locution, I am such that I am sitting. By ACTUALITY, I have a potentiality to be such that I am sitting. Hence something (namely, I) has a once-iterated potentiality to be such that I am sitting; hence, by our definition, it is possible that I am sitting. The same argument can be run for any other truth. (In this case, we need not even go down the list of iterations.) It will be seen that the sketched proofs are rather simple, relying on the principles for POT and some basic principles that govern the existential quantifier (the introduction rule and the above-mentioned principles that are structurally analogous to features of POT and 3: closure under logical implication and equivalence, distribution over disjunction, the introduction rule which corresponds closely to the (T) axiom, and a non-contradiction axiom resembling the ⊥ axioms). Formally, we have chosen the path of least resistance in showing formal adequacy, leaving the hard work for the metaphysics and logic of potentiality. That hard work was done in chapters 3–5, and it now pays off in the simple proofs that we can give for the principles governing possibility.

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Another feature of my account that is paying off now is the fact that possibility has been straightforwardly defined. If the claim were merely that every truth of the form ‘possibly φ’ was made true by some potentiality, we would have much less stringent standards for showing formal adequacy. With the definition in place, we can see exactly what is required of potentiality in order to meet the constraint of formal adequacy; and we have seen that potentiality meets the requirements. A final question remains regarding the logic of possibility on the potentiality account: does metaphysical possibility validate any of the standard systems beyond system T? At the end of chapter 5, I raised the question whether the logic of potentiality should include axioms akin to the S4 and S5 axioms of modal logic: (S4) 33φ → 3φ, (S5) 3φ → 3φ. I rejected the corresponding axioms for potentiality: having a potentiality to have a potentiality to  does not imply having a potentiality to  (the analogue of S4); and having a potentiality to  does not imply lacking the potentiality to lack the potentiality to  (the analogue of S5). Thus the following are not axioms in the logic of potentiality: (POT4) POT[λx.POT[](t)](t) → POT[](t), (POT5) POT[](t) → ¬ POT[λx.¬POT[](t)](t). However, what of (S4) and (S5) themselves? Does the potentiality account validate these axioms? In their potentiality translation, (S4) says that if something has an iterated potentiality for something to have an iterated potentiality for φ, then something has an iterated potentiality for φ, i.e. that iterating iterated potentialities yields simply iterated potentialities. (S5) says that if something has an iterated potentiality for φ then nothing has an iterated potentiality for it to be the case that nothing has an iterated potentiality for φ, i.e. that, given an iterated potentiality, there are no potentialities for that iterated potentiality never to be possessed. Formally, these are expressed not as (POT4) and (POT5), but rather as (S4∗ )

∃xPOT∗ [∃yPOT∗ [φ](y)](x) → ∃xPOT∗ [φ](x),

(S5∗ )

∃xPOT∗ [φ](x) → ¬∃yPOT∗ [¬∃xPOT∗ [φ](x)](y).

(S4∗ ) looks plausible, though it is somewhat complicated in the case of infinitely iterated potentiality (for which case, see the appendix): given the insensitivity of

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iterated potentiality (expressed by POT∗ ) to the number of iterations, embedding iterations within iterations of potentiality will yield nothing other than iterated potentialities. With (S5∗ ), matters are more complicated. Whether that principle should be adopted depends on matters that go beyond the pure metaphysics of potentiality. Suppose there are some things whose potentialities are ‘fixed’, in the sense that nothing has a potentiality for those things to have different potentialities. ((POT5) would hold of those things, but the assumption is stronger: it is that nothing else has a potentiality for them to have different potentialities.) Such things might be the objects (presumably particles) that existed at the very first moment of the universe. Appealing to the triviality thesis from chapter 5.8, much as we did for the necessity of origin, it may be argued (and indeed I will argue, again tentatively, in chapter 7.9) that nothing ever has a potentiality for those things to be different at the very first moment. Suppose, in addition, that those objects, at the very first moment, already had iterated potentialities for every potential development of the universe. Then for every iterated potentiality that is ever possessed by anything, an iterated potentiality for the same ultimate manifestation is possessed by those objects at the first moment, and fixedly so. Since, by assumption, nothing has an (even iterated) potentiality for those objects to have different potentialities at that time, it would follow that for every iterated potentiality, nothing has an even iterated potentiality for it to be the case that nothing has an iterated potentiality for the same ultimate manifestation; and thus that (S5∗ ) and hence (S5) holds. This is only one metaphysical assumption that would validate the S5 axiom; there may well be others. The above assumption is not one to which the potentiality account is obviously committed. Nor is it one that would obviously go against the spirit or the letter of the potentiality account. So the matter, I take it, is genuinely open. I leave open the question of whether or not the potentiality view validates the S4 and S5 axioms. The systems generated by these principles, to be sure, are attractive and widely accepted in the logic of metaphysical modality. But unlike system T, characterized by axiom (T3), S4 or S5 is not a nonnegotiable requirement for any metaphysics of modality.5 I also leave open the question of how exactly quantified modal logic will turn out on the potentiality account. These 5 Salmon (1989) has disputed S4 as a logic for metaphysical possibility, and suggested that we should content ourselves with system T. More recently, Wedgwood (2007) has argued that S5 is not at all mandatory for a metaphysics of modality ‘unless we believe that the modal concepts possible and necessary can be defined in terms of a conceptually prior notion of something’s being the case at a world’ (Wedgwood 2007, 220). Wedgwood argues that S5 is undesirable because it excludes that necessary truths may have contingent grounds.

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are matters for further debate, given that the potentiality account is a serious candidate for a theory of modality. That it is a serious candidate, I hope to have shown, at least so far as formal adequacy is concerned. I conclude that the constraint of formal adequacy has been met. Possibility, as defined in POSSIBILITY, has the requisite logical structure. We can now move on to another constraint.

6.5 Semantic utility: introduction We now turn to the constraint of semantic utility and the task of constructing a modal semantics based on potentiality. The basic idea of the semantics will be that (the relevant part of) modal language is used to ascribe or deny potentialities to objects. In chapter 3, I developed a view of dispositions which linked them closely to ‘can’ statements. A fragile object, on that view, is one which can easily break or be broken; a soluble substance is one which can be dissolved. On the possible-worlds semantics with which I began, both disposition ascriptions and ‘can’ statements were construed as existential quantification (or something near enough) over possible worlds or cases. On the dispositionalist view, disposition ascriptions are more naturally construed as doing no more than what they appear to do: ascribing dispositional properties. Those properties, as we saw in chapter 3, must be construed more broadly than dispositions properly speaking, if they are to provide a metaphysically realist background to account for the context-sensitivity of disposition ascriptions: they must be potentialities, in the sense that I have developed in this book. The next move for the dispositionalist is to apply these insights to ‘can’ statements themselves: they, too, are to be construed not as existential quantifications over possible worlds, but as ascriptions of potentialities. In short, chapter 3 linked disposition ascriptions to ‘can’ statements. That link can be used in either direction. On the standard possible-worlds semantics we start with an understanding of ‘can’ as quantifying over possible worlds, and apply the same understanding to disposition terms. On the dispositionalist or potentiality-based view, we now start with an understanding of disposition terms as ascribing potentialities, and apply the same understanding to the modal auxiliary ‘can’. The basic idea of such a semantics is that ‘can’, like disposition terms, is used to ascribe potentialities. Hence the truth-conditions for ‘can’ statements will look roughly as follows: (CAN) ‘x can F’ is true just in case x has a potentiality to F. (CAN) will form the starting point for the account of linguistic modality that will be sketched in the remainder of this chapter.

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We can begin both to develop and to defend the account by considering three successive attempts at generalizing (CAN): first, to account for the contextsensitivity of ‘can’ itself; second, to account for other expressions of dynamic or circumstantial modality; and third, to account for all expressions of linguistic modality. I will argue that the first two generalizations are feasible and indeed natural given (CAN), but that the third fails, and indeed fails for the right reasons. Here I will make only a few remarks on how the argument will proceed. (CAN) applies very naturally where ‘can’ is used to ascribe abilities, which are among our paradigmatic examples of potentialities. But ‘can’ is not only used to ascribe an ability (‘I can play the piano’). It is used also to express that an ability is possessed and conditions are suitable for its exercise (‘You can buy a kettle in that store’), or just that there is a possibility of something coming about (‘You can fall off the cliff if you’re not careful’). What is more, one and the same sentence can express all these different things in different contexts. Is it true to say of me, right now, that I can swim? Yes and no. Yes: I have learned to swim, my muscles are in working order, of course I can swim. Then again, no: there is no body of swimmable water anywhere near me. How should I swim if there’s no water? Clearly, I cannot swim. Or suppose that I am celebrating my birthday on the beach and have had one glass of wine too many. Can I swim? Yes and no. Yes: not only have I learned to swim, but there is plenty of water around for me to swim in. Then again, no: given the amount of wine I have had, I would fail miserably if I attempted to swim here and now, so I cannot swim. The same sentence (‘I can swim’), applied to the same situation, may with equal right be either affirmed or denied, held true or false. This is witness to the fact that ‘can’ is context-sensitive: it is used to express different things in different contexts. We have shifted the context of assertion by focussing, first, on my muscles, etc., and second on the availability of water; or in the second case, by first focussing on my training and the opportunity to exercise it, and second, on my temporary state of inebriation. Some of these aspects appear to be a matter of my intrinsic abilities, but others concern matters that are extrinsic to me (such as the presence or absence of water in the vicinity) or too temporary to count towards or against my abilities (such as the fact that I have had too much wine). Our first task will be to account for all of these uses, and in particular to do so in a way that makes the context-sensitivity of ‘can’ statements comprehensible— in other words, to provide a mechanism for context-dependence. This will be the task of section 6.6. To provide a serious modal semantics, we must, of course, generalize beyond the case of ‘can’. Linguistic modality comes in a great variety of syntactic categories: there are modal verbs (‘have to’, ‘be able to’) and auxiliaries (‘can’, ‘must’,

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‘may’); modal adjectives (‘possible’, ‘necessary’) and adverbs (‘probably’, ‘possibly’), as well as the suffixes that we have already encountered in chapter 3 (‘-able’ and its derivatives). Semantically, these are used to express different ‘flavours’ of modality, which are standardly partitioned into epistemic, deontic, and dynamic.6 Roughly, epistemic modality is about what is compatible (or not) with our knowledge, deontic modality is about permission and obligation, and dynamic or circumstantial modality is about developments that are open (or not) given how things really are. Let me briefly illustrate these three types of modality with three typical examples (two of which should be familiar from chapter 3). First example: a detective reviews the inconclusive evidence in a murder case and says ‘Mary might be the murderer’. The detective expresses an epistemic possibility of Mary’s being the murderer; it is compatible with the detective’s evidence that Mary is the murderer. Second example: a father tells his son ‘you may go out and play now’. ‘May’ here, as often, expresses deontic possibility: the son is permitted to go out and play. Third example: a botanist analyses the soil in a foreign country, thinking about which plants to import. She informs her colleague ‘Hydrangeas can grow on this soil’, even though she is fully aware that no hydrangeas are growing on it now. This is a dynamic or circumstantial possibility. ‘Can’, our paradigmatic case, is used primarily to express dynamic modality, though it has deontic uses expressing permission (‘Can I go now?’) and even occasionally an epistemic reading (especially when negated: ‘This cannot be true!’). The majority of uses, however, are dynamic, and it is those dynamic uses with which (CAN) was concerned. Indeed, ‘can’ appears to be our primary expression of dynamic modality: it is one of the most common modal auxiliaries, and the only one that has a clear majority of dynamic uses.7 We will see below, in section 6.9.1, that modal adverbs and adjectives are used primarily for epistemic modality. Dynamic modality, quite generally, seems to be the natural province of a potentiality-based semantics. Unlike deontic or epistemic modality, and like potentiality, dynamic modality is a matter simply of how things really are, not how they ought to be or how we know them to be. It is also the closest relative of metaphysical modality, if the latter is understood in a realist way (indeed, some linguists include metaphysical or ‘alethic’ modality, as they sometimes call it, within dynamic modality). 6 See Kratzer (1991), Palmer (1990), Collins (2009). As we have seen in chapter 3, Kratzer treats deontic modality differently from dynamic/circumstantial and epistemic. That difference will not concern me in this chapter. 7 Namely, 81% according to a recent corpus survey (Collins 2009, 98). In contrast, ‘may’ was found to be used dynamically in only 8.1%, ‘must’ in 6.3%, and ‘would’ in 22.9% of occurrences (Collins 2009, pp. 34, 92, 140).

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The semantics for ‘can’ that is indicated in (CAN) is extended easily to other expressions of dynamic modality, as section 6.7 will suggest. It is less easily extended to expressions of epistemic or deontic modality. This may seem a limitation of the account. It is joined by other seeming limitations: in particular, since it construes ‘can’ as a predicate modifier, (CAN) does not account for de dicto statements even of dynamic possibility (see section 6.8). Section 6.9 will argue that these two seeming limitations are not only closely linked, but indeed well motivated.

6.6 ‘Can’ and context-sensitivity As we have seen, the basic idea of a potentiality-based semantics for ‘can’ is that ‘can’ is used to ascribe potentialities: (CAN) ‘x can F’ is true iff x has a potentiality to F.8 However, not any kind of potentiality can be ascribed with any ‘can’ statement, let alone in any context. This is the key to understanding the context-sensitivity of ‘can’ statements, as outlined in section 6.5. The quantification (‘a potentiality’) on the right-hand side of (CAN) must be read as contextually restricted; or we might rephrase (CAN) to make this more explicit: (CAN∗ ) ‘x can F’ is true in a context C iff x has a potentiality to F which is relevant in C. In what follows, I will assume that (CAN) is read as (CAN∗ ), and I will highlight three conditions on relevant potentialities: degrees, intrinsicality/extrinsicality (shortly to be generalized to ‘granularity’), and agency. The first two are by now largely familiar; the third is a matter of what distinguishes agentive abilities from other types of potentialities. I will introduce all three via the contrast with dispositional adjectives such as ‘fragile’. Degrees On a potentiality-based semantics, ‘x can break’ works in much the same way as does ‘x is fragile’. Both ascribe to x a potentiality to break, but both are selective about the kind of potentiality that they ascribe. Just as ‘My desk is fragile’ is false in many contexts, so ‘My desk can break’ will not be true in all contexts: both 8 I am taking a cavalier attitude to the object language/metalanguage distinction for ‘x’ and ‘F’ here. Properly, (CAN) should read: ‘x can F’ is true iff the object referred to by ‘x’ has a potentiality to have the property ascribed with ‘F’. Note also that ‘x’ and ‘F’ are used here to stand in for ordinary language expressions and no longer as part of the formal language developed in chapter5.

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statements require the ascribed potentiality to have a certain minimal degree. Thus an engineer might say of a steel bridge that ‘it cannot break’, truly (it would seem) in her context, although the bridge does have some potentiality to break. It seems, however, that the threshold set by ‘x can break’ is lower than that for ‘x is fragile’. In many ordinary contexts, I would be prepared to say that my desk can break though it is not fragile. Perhaps a closer analogy in this respect is with more regularly formed disposition terms such as ‘breakable’. Granularity A notable and, for our purposes, more fruitful difference between ‘can’ and disposition ascriptions relates to intrinsicality. Typically, dispositional terms (terms of the form ‘F-able’ and related terms) ascribe intrinsic potentialities: ‘x is breakable’ does not become false when the object x is packed in anti-deformation packaging, nor does it become true when the object is put in front of a bulldozer, but in some contexts the truth value of ‘x can break’ may change with such circumstances. Dispositional terms, it seems, typically come with a strong and relatively stable implication of intrinsicality which is held fixed across contexts. Even terms for extrinsic dispositions, such as ‘vulnerable’, will be quite selective and, more importantly, quite fixed in the kinds of external circumstances that are relevant for the property they are used to ascribe. ‘Can’ is much more flexible in this respect, and accordingly more sensitive to our interests in a given situation. Suppose I am moving to a new flat and considering where to store my valuable vase on the moving van. Since I have taken care to pack the vase in antideformation packaging, it is perfectly natural to say ‘The vase cannot break; it is so safely packed’, thus denying the vase an extrinsic potentiality (of a contextually relevant degree) to break. In another context, say, considering where to put the vase in my new flat, we may then switch to the vase’s intrinsic potentialities and say ‘The vase should be in a safe place; it can break so easily’. In the first case, my practical interest is in the extrinsic potentialities of the vase. After all, with an extrinsic potentiality, more possible interferences with its manifestation are taken account of. My vase may have the potentiality to break to a great degree but since it is so safely packed, I need not worry about its breaking for the moment. As things stand, we interact not with the vase on its own but with the vase plus packaging; and it is our interactions with the things I am moving that are currently of significance. Hence our interest is not so much in the individual, intrinsic potentialities of the vase, but in the joint potentialities of the vase-plus-packaging, and the extrinsic potentialities with which they endow the vase.

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In the second case, on the other hand, the long-term interest brings with it an interest in the vase’s intrinsic potentialities because an object’s intrinsic properties are generally more stable over time than its extrinsic properties. So it is not surprising that the nature of our interests, long or short-term, is one determining factor in selecting whether it is intrinsicality or extrinsicality that is required of a relevant potentiality in a given context. To account for more cases, we need to look at a plausible generalization of the intrinsic/extrinsic distinction. In chapter 4.3, I compared the joint potentialities of several objects together, and their relation to the individual potentialities of the participating objects, to the potentialities of composite objects and their relation to the potentialities of their parts. In both cases, the picture is one of objects coming together, each equipped with its own potentialities; given those potentialities and the objects’ other properties and relations, the resulting whole or plurality will have certain potentialities of its own (as well as, perhaps, some emergent potentialities, an option that I left open in chapter 4.3). So far, we have made use only of the second party in that comparison, the joint potentialities. But now we need to look a little more closely into the other party, composite objects. Remember the drunk swimmer from the previous section: she is equipped with an ability to swim, but her drinking has also induced certain more temporary dispositions, such as the disposition to get disoriented and erratic in her movements.9 Perhaps, indeed very plausibly, these different potentialities belong to different parts of the individual, or they might belong to the same parts in virtue of different properties of those parts. In either case, the different potentialities within an individual can contribute to or distract from one another, yielding the individual’s overall intrinsic potentialities. This suggests that the ‘intrinsic’ part of our intrinsic/extrinsic distinction was simplified in this respect: there are finer structures of potentiality within the intrinsic makeup of most objects. The same goes for the ‘extrinsic’ part of the distinction. Suppose I raised my arm. Did I have an ability not to raise my arm? Plausibly, I possess the general ability to not raise my arm; very likely, I also have an overall (intrinsic) ability not to raise my arm, since no disposition that is intrinsic to me prevents me from not raising my arm. Did I have an extrinsic ability not to raise my arm? That depends: which jointly possessed potentiality is to ground the extrinsic potentiality at issue? If it is just potentialities that I possess jointly with everyone and everything in this room, and if no one in this room was intrinsically necessitated to make me raise my arm, then I did have the extrinsic ability not to raise my arm. 9 This is very plausibly an example of an intrinsic mask. See Clarke (2010) and Ashwell (2010) for a defence of intrinsic masks.

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But we may think bigger. In particular, we may wonder whether I had an extrinsic ability not to raise my arm grounded in my joint potentialities with everything else: could I have not raised my arm, given the way the world is as a whole?10 That is (one version of) the question of determinism, and nothing in my metaphysical story answers it. The important point is that extrinsicality comes, as it were, in degrees: the possession of an extrinsic potentiality may depend on more or less of the world surrounding the object in question. It appears, then, that we have a spectrum: from the most finely-grained potentialities of individual objects through their overall (normally so-called) intrinsic potentialities, to the extrinsic potentialities arising from their joint potentialities with bigger and bigger parts of the world. Let us call this the spectrum from fine-grained (intrinsic) potentialities to the coarse-grained (extrinsic) potentialities, or more generally of granularity. Granularity is a matter of degree, with a minimum and maximum value. This more general way of thinking about granularity, rather than just the dyadic distinction of intrinsic and extrinsic, allows us a more nuanced picture of context-sensitivity. Thus of the drunk swimmer, we may say ‘She can swim’, ascribing to the swimmer the maximally fine-grained (and otherwise relevant) potentiality to swim; or we may say, ‘She cannot swim’, denying her the overall intrinsic (and otherwise relevant) potentiality to swim. Of a well-trained and sober swimmer who is tied to a chair, we may say ‘She can swim’, ascribing to her the intrinsic (and otherwise relevant) potentiality to swim, or we may say ‘She cannot swim’, denying her the extrinsic (and otherwise relevant) potentiality to swim that is based in her joint potentialities with relevant objects of her surrounding. Finally, as philosophers interested in determinism, we may say of a competent, sober, and unimpeded swimmer who did not, at a particular occasion, swim: ‘She could have swum’, ascribing to her an intrinsic or a mildly coarse-grained potentiality, taking into account only objects of her closer surroundings at the time. Or we may say, assuming the truth of determinism and the fact that she did not in fact swim, ‘She could not have swum’, denying her the maximally coarse-grained potentiality to swim. Our interests may play a role, again, in which level of granularity is selected. But in some cases, including perhaps the ones I have just outlined, our practical interests underdetermine the requirements on relevant potentialities. In such contexts, the guiding principle may simply be what Lewis has called a ‘Rule of Accommodation’: ceteris paribus, context fixes the relevant values so that the utterances made in it come out true (Lewis 1979, 347; see also Kratzer 1981). 10 This kind of maximally extrinsic potentiality is closely related to, but not identical with, the strong tensed possibilities of section 6.1.

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Not all sentences vary as freely in the required granularity as does our example ‘She can swim’. Consider our earlier examples of intuitively opportunityor possibility-expressing sentences: (1)

a. You can buy a kettle in that store. b. You can fall off the cliff if you’re not careful.

Neither of these has a plausible reading that ascribes to the sentence’s addressee an intrinsic potentiality (to buy a kettle or to fall off the cliff). But that should not come as a surprise. Both sentences ascribe potentialities whose manifestations consist in a particular relation (buying a kettle in . . . or falling off . . . ) to a particular object (‘that’ store, the cliff that is contextually relevant). Potentialities of this kind are generally extrinsic. Consider a key’s potentiality to open a particular door, or my potentiality to see you. The key would lose its potentiality if the door ceased to exist or merely had its lock changed, and I would lose my potentiality if you no longer existed or became invisible. A potentiality to stand in relation R to a particular object b, as possessed by an object a distinct from b, is always extrinsic, for it depends on the existence of b, on the intrinsic potentialities that b has, and on the relations that hold between a and b. Both our examples are of this kind: they ascribe extrinsic potentialities that depend on the existence and the potentialities of another object (the store and the cliff), as well as the relation in which the addressee stands to them. In general, the present approach treats ‘can’ statements that appear to express opportunities or possibilities as ascriptions of extrinsic potentiality. Degrees and, more importantly, granularity account for a great deal of contextsensitivity, but not for all. We need to consider a third factor. Agency In a great many contexts, when confronted with a sentence ‘x can F’ where ‘x’ denotes an agent and ‘F’ is a verb of action, we understand ‘can’ to ascribe to an agent not just any potentiality, but an ability, capability, or rational capacity in a stronger sense. Thus when I say ‘I can play the piano’, what I say is at least misleading, and more likely false, if I possess merely the general potentiality to play the piano but not the skill or know-how that is acquired by lessons and practice. And when I say ‘I can hit the bull’s eye’, what I say is taken to be false if I’m a hopeless darts player who hits the bull’s eye every now and then by sheer accident (Kenny 1976). This way of construing the relevant ‘can’ sentences (with agentive subject terms and predicates) is natural but not mandatory. I am indeed a hopeless darts player; I cannot hit the bull’s eye even if I stand close to it. But look at it this way:

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I have arms and the muscles and bodily make-up required to throw a dart, and there is nothing to prevent me from throwing a dart that lands in the bull’s eye; I have been known to hit it by luck now and then. Surely I can hit the bull’s eye, even if I am no good at it. (Compare Lewis’s (1976a) remarks on the ability to speak Finnish.) I have argued in chapter 5.7 that potentialities, even where the subject and the manifestation are both agentive, are less demanding than abilities; not every potentiality is an ability. Thus I have a potentiality to hit the bull’s eye even though that potentiality fails to qualify as an ability. ‘Can’ statements can ascribe such ‘mere’ potentialities, with a little contextualization, even when the subject and predicate are agentive. But the default reading in such cases is the ability reading. This third factor, again, contrasts ‘can’ with typical disposition ascriptions ending in ‘-able’. Where ‘can’, through the dimension of agency, has a certain affinity to what are traditionally called active powers, adjectives ending in ‘-able’ or ‘-ible’ are closely linked to passive potentialities or dispositions. To be F-able, typically, is to have a potentiality to be F-ed. Thus to be washable, soluble, or readable, is to have a potentiality to be washed, dissolved, or read; and so on for most (not for all11 ) adjectives of the kind. Abilities in this strong sense, on the metaphysics developed in this book, are a kind of potentiality. But what kind? What sets them apart? This is a difficult question, which cuts across the distinction between a potentiality-based metaphysics and its competitors. On one tradition that goes back at least to G.E. Moore, abilities are related to conditionals of the form ‘If x wanted (decided, chose, intended, or tried) to F, then x would F’. The conditional approach has faced a number of objections, most famously from Austin (1961) and Lehrer (1968). It has recently made a comeback in the form of a ‘New Dispositionalism’, defended by Vihvelin (2004) and Fara (2008). For the New Dispositionalism, an ability is tantamount to a disposition to F if one wants (decides, chooses, intends, or tries) to F.12 If my arguments in chapters 2–3 have been correct, the problem with the New Dispositionalism is that it is based on a mistaken view of dispositions as akin to conditionals. In the present context, therefore, the New Dispositionalism is not a viable option. Another tradition, which goes back at least to Ryle (1949), holds that abilities are characterized by being particularly multi-track. An ability is not simply an 11 Exceptions include: ‘honorable’ and ‘payable’, which appears to have a deontic meaning, the latter atypically expressing deontic necessity; and ‘capable’, ‘feasible’, and others that have lost, as it were, their compositional nature. 12 For a thorough criticism, see Clarke (2009), who also introduces the label ‘New Dispositionalism’.

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ability to do one thing; it is systematically related to abilities to do similar things and to abilities to do the same thing in a number of different circumstances. As Smith (2003) points out, if I have the ability to answer one logic question correctly, I will have the ability to answer other logic questions. I will also have the ability to answer the question in a number of different kinds of circumstances. While Smith spells this out in terms of possible worlds, the present metaphysics can easily accommodate the basic idea: to have an ability (to F) is not just to have one potentiality (the potentiality to F), but to have a cluster of related potentialities (the potentiality to F and the potentiality to F , the potentiality to F in conditions C and the potentiality to F in conditions C , and so forth). A third tradition, going back at least to Thomas Reid, thinks of abilities as ‘twoway powers’ (for more recent proponents see, for instance, Horty and Belnap 1995, Mayr 2011, and Steward 2012). To have an ability to F requires having an ability not to F, or an ability to refrain from F-ing. Again, this can be spelled out in terms of possible worlds or in terms of potentialities. On the potentiality version of this view, to have an ability to F is not just to have a potentiality to F but also to have a potentiality not to F, or to refrain from F-ing. It does not matter for present purposes which, if any, of these conceptions is correct. What matters is that some potentialities—such as a dart player’s ability to hit the bull’s eye—count as abilities in some stronger sense, while others—such as my own potentiality to hit the bull’s eye—do not. Exactly how that distinction is captured is a question that is largely independent of the metaphysics of potentiality in general. And whatever it is that sets abilities apart from other potentialities will, in some contexts, be among the conditions for a potentiality to count as relevant. This I call the factor of agency. So much, then, for the context-sensitivity of ‘can’. We will briefly consider how the semantics sketched so far is transferred to other expressions of dynamic modality, before considering in more detail some of its structural features.

6.7 Dynamic modality beyond ‘can’ ‘Can’ is the most common, but not the only, expression of dynamic modality. But the semantics of (CAN) is easily transferred. Expressions that are usually classified as expressing dynamic possibility include the modal verb ‘be able to’, the suffixes ‘-able/-ible’, the adjectival construction ‘it is possible for . . . to . . . ’, and some uses of ‘may’ and ‘might’, especially in the counterfactual ‘might have’. (Note, however, that ‘may’ and ‘might’ are primarily expressions of epistemic

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modality, and as such not the subject of this section13 .) Where they express dynamic possibility, these expressions can be given truth-conditions of the same form as (CAN): (∗ )

‘ . . . ’ is true in a context C iff x has a potentiality to F that is relevant in C,

with the right kind of construction replacing the blank ‘ . . . ’. Where these other expressions differ from ‘can’ and from each other, they do so in the general conditions that they impose on relevant potentialities. Thus it appears that ‘be able to’ has a more robust implication of agency than ‘can’ (we say ‘the vase can break’, but not ‘the vase is able to break’), while ‘possible for . . . to . . .’ and ‘might (have). . .’ appear to be relatively neutral with regard to agency (‘it is possible for Naomi to play the piano’ is more easily accepted than ‘Naomi can/is able to play the piano’ when Naomi’s potentiality does not amount to an ability, but it is equally true in the presence of a full-blown ability). The suffixes ‘-able’ ‘-ible’, on the contrary, have an implication of passivity: as we have seen earlier, adjectives of the form ‘F-able’ are typically paraphrased as ‘can be F-ed’. In the previous section, we have seen some further difference between the suffixes and ‘can’ regarding granularity and degrees. Modals that are normally classified as expressing dynamic necessity will be naturally accommodated, within the present framework, via their duality with the modals of dynamic possibility. A sentence of the form ‘x must F’ or ‘x has to F’ is true just in case x lacks a contextually relevant potentiality not to F. The requirements on relevant potentialities appear to be much the same as they are for ‘can’. Consider, by way of example,14 (2)

a. I must sneeze. b. She is obsessive-compulsive; she just has to wash her hands once every hour. c. Like-charged particles must repel each other; it’s a law of nature.

Especially where the subject is agentive, a natural paraphrase uses ‘can’: what an individual must do is what she cannot help doing, or what she cannot refrain from doing. The aspect of agency, which we have seen to arise with ‘can’, is relevant here too. Here, however, it is not the agent’s control over her actions but her lack of control that is expressed in sentences (2-a) and (2-b). What the subject of these sentences is said to lack is the (intrinsic or extrinsic) ability to do other than 13

Collins (2009, 92, Table 4.2; 118, Table 4.10). We must be careful to prise apart the dynamic reading from epistemic and deontic ones. ‘Must’ and ‘have to’ are most often used deontically (‘He has to go to school’) or epistemically (‘This must be where she lives’). See Collins (2009, 34, Table 3.2; 60, Table 3.12). 14

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sneeze or wash her hands. (2-c) does not carry such an implication of agency, but it too can be understood as the negation of a ‘can’ statement: like-charged particles cannot fail to repel each other; they lack the potentiality to do so. This is no more than an indication of how the semantics should be developed; I have provided a little more detail in Vetter (2013). But I want to end this section by discussing the case which presents the main difficulty for a potentiality-based semantics as I have sketched it: counterfactual conditionals. For a more standard account of dispositions, counterfactuals would have been the natural starting point in the dispositionalist project. If dispositional properties were indeed individuated by a stimulus as well as a manifestation, relating to each other in much the same way as a counterfactual’s antecedent to its consequent, then the semantics of ‘would’ counterfactuals would naturally proceed in terms of such dispositions. I have argued, in chapters 2 and 3, that such a conception of dispositions is mistaken. Dispositions, and by extension potentialities, are individuated by a manifestation and akin to ‘can’ statements rather than to counterfactual conditionals; hence ‘can’ statements were the natural starting point for my semantics. A dispositionalist semantics of counterfactuals along the lines of the standard conception also faces a further problem, which I have sketched in chapter 1. The problem was that, because of the possibility of masks and mimics, the disposition was neither sufficient nor necessary for the truth of the corresponding counterfactual conditional: a fragile vase may be wrapped in styrofoam, so that it is not true that it would break if it were struck; a non-fragile concrete block might be attached to an explosive so that it is true that it would break if it were struck. We can now see the beginnings of a natural answer to that challenge, based on the observations we have made about ‘can’: while ‘would’ counterfactuals do ascribe dispositions, they are tied not so much to intrinsic but to suitably extrinsic ones. Thanks to its joint potentialities with the packaging material, the vase lacks an extrinsic disposition (of a sufficiently high degree) to break if struck; thanks to its joint potentialities with the attached explosive, the concrete block has such an extrinsic disposition. The moral to draw is not that counterfactuals do not ascribe dispositions, but that they do not (or only rarely) ascribe intrinsic dispositions. While this strategy may meet the challenge of masks and mimics, the account of potentiality that I have presented here cannot make use of the solution. If there is no stimulus involved in a disposition, then we get no straightforward semantics for the counterfactual conditional in terms of dispositions (or, by extension, potentialities). But the semantics has other drawbacks too: at least on the standard way of thinking about the relation between a disposition’s stimulus

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and manifestation, it is less than plausible for any non-causal counterfactual, for instance: ‘If I were brighter, I would be better at maths.’15 How else might we, then, capture counterfactual conditionals? To answer this question satisfactorily would go beyond the scope of this book. But I would like to indicate one promising avenue for a potentiality-based semantics of counterfactuals. It begins, not with ‘would’ counterfactuals, but with ‘could’ or ‘might’ counterfactuals, and construes them as ascribing iterated potentialities to individuals. It comes with commitments and questions that I can only indicate, not answer or defend here. Let us start with an example of a ‘could’ counterfactual: (3) If I were brighter, I could prove that theorem. We can take (3) to ascribe to the speaker an iterated potentiality: a potentiality to acquire a property, being brighter, which in turn brings with it a (contextually relevant) potentiality to prove the theorem in question. A similar construal would apply to (3) with ‘could’ replaced by ‘might’ (on a dynamic reading)— except, perhaps, that the conditions for contextual relevance at the second stage of the iterated potentiality are different: in the ‘could’ version, it is easy to read the sentence as requiring that the potentiality to prove that theorem would be an ability, while with the ‘might’ version that reading is less natural. Generalizing from the idea, we get the following first sketch. We take a counterfactual of the form ‘If it were that A, it could/might be that C’ to ascribe an iterated potentiality for C to be the case, where C is the counterfactual’s consequent. The antecedent specifies an earlier stage within that iterated potentiality. So at least for the case where both antecedent and consequent are of a simple predicative form and share their subject, we get (COULD) ‘If x were F, then x could/might be G’ is true iff x has an iterated potentiality to be G, and being F is an earlier stage in that iterated potentiality.16 Past-tense counterfactuals (‘if x had been F, x could/might have been G’) can be read as ascribing such iterated potentialities in the past. Cases where antecedent and consequent do not share their subject will presumably require only a simple generalization, where the iterated potentiality in question is a joint potentiality of their different subjects. Context-dependence can function, once 15 But see Nolan (forthcoming), who argues that a disposition’s stimulus and manifestation need not be causally related. 16 This implies that the ‘might/could’ counterfactual is false where it is impossible for x to be F. Note that this is a standard assumption, which falls out, for instance, from Lewis’s treatment of ‘might’ counterfactuals as the dual of ‘would’ counterfactuals in Lewis (1973b).

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again, by requiring x’s potentiality to be of a certain minimal degree, or to be of a certain granularity, or perhaps even (in the case of the ‘could’ counterfactual) of the right ‘agentive’ kind. ‘Would’ counterfactuals may be defined, as has been suggested by several authors (e.g. Lewis 1973b), as the dual of ‘could’ or ‘might’ counterfactuals: ‘if it were that A, it would be that C’ is simply shorthand for ‘not: if it were that A, it might be that not C’. Note, however, that (COULD) and any more general account based on it is promising only if we subscribe to a strongly structuralist picture of properties. I have assumed merely that some properties are potentialities, and hence individuated by their relation to other properties, their manifestations. According to a stronger form of structuralism, every property and every relation is at least a potentiality or perhaps (following the spirit, though not the letter, of Shoemaker 1980) a cluster of potentialities. (I say ‘at least’ because the view is meant to be compatible with the idea that each property has, in addition, a ‘qualitative aspect’.17 ) On this strong structuralist view, every property or property complex endows the objects that instantiate it with potentialities to instantiate a number of further properties and property complexes. The relation between a property (complex) P1 and another property (complex) P2 for which P1 is or includes a potentiality is naturally expressed in terms of a ‘might’ or ‘could’ counterfactual: if an object, x, had P1 , then x might or could have P2 . Now suppose strong structuralism to be false. Then there are properties or property complexes that are not (not even in part) potentialities or clusters thereof. Then the ascription of such a property (complex) in a counterfactual’s antecedent would immediately falsify the counterfactual itself. Among the best candidates for being non-structuralist properties are spatiotemporal properties and relations, such as being 10 cm long or being 2 m apart. But there seem to be clearly true ‘might’ and ‘could’ counterfactuals with such antecedents, for instance (4)

a. If this string were 10 cm long, it could be used as shoelace. b. If I were 2 m from you, I might still hear you whispering.

The semantics expressed in (COULD) had better, therefore, come with the full picture of strong structuralism, and maintain that the antecedents of (4) ascribe to the objects mentioned in each (clusters of) potentialities. Whether that is a problem for the iteration strategy is a question that goes beyond the scope of this book. 17 This claim has been made repeatedly by C.B. Martin and John Heil; see, for instance, Martin’s contribution in Armstrong et al. (1996), as well as Martin and Heil (1999), and Heil (2003).

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6.8 Dynamic modals as predicate operators I have argued so far that (CAN) is a promising semantics for ‘can’, both in accounting for context-sensitivity and in serving as a starting point for the semantics of other dynamic modals. You may wonder how the semantics accommodates modality beyond the dynamic. Before we can answer that question, however, we need to examine a crucial feature of (CAN) and, by extension, of the semantic clauses inspired by it. The relevant feature is this. (CAN), and the semantics that is based on it, takes ‘can’ to be a predicate modifier, not a sentence modifier. It thereby deviates from standard semantics, such as Kratzerian possible-worlds semantics as outlined in chapter 3. For Kratzer, ‘on a level of logical form . . . all modal operators are sentential operators’ (Kratzer 1981, 44, fn. 3). Take a simple ‘can’ statement, such as ‘Ann can swim’. According to the standard semantics, ‘can’, like ‘possibly’, is a sentence modifier, which is applied to a whole sentence (‘Ann swims’) to yield a new sentence (‘Can(Ann swims)’). This goes well with the idea that ‘can’ expresses, roughly, truth in some possible world: for it is the entire sentence in its scope (‘Ann swims’) that is evaluated for truth at another possible world. According to (CAN), ‘can’ is a predicate modifier, which is applied to a predicate (‘swim’) to yield a complex predicate (‘can swim’), which is then applied to the sentence’s subject (‘Ann (can swim)’). At the level of surface syntax, the predicate-modifier view seems correct. Sentence modifiers, such as ‘possibly’, can be prefixed to a declarative sentence, yielding another declarative sentence, but this is not true for ‘can’: ‘possibly, Ann swims’ is a declarative sentence, ‘Can Ann swims’ is not. (CAN) implies, while standard Kratzerian semantics denies, that the surface syntactic structure is also the deeper syntactic structure, i.e. that ‘can’ is a predicate modifier at the level of logical form, and that the semantics has to be construed accordingly. Of course, the same will go for other expressions of dynamic modality if they are to be construed in analogy to (CAN), but in this section I will again focus on ‘can’. The predicate-modifier construal has a number of implications, which at first sight may seem to make the semantics overly restrictive. This section will highlight the implications; their vindication has to wait until the next section. A first implication is that, if tenses are sentence modifiers, then tenses will always take scope over dynamic modals such as ‘can’. (It is not entirely uncontroversial that tenses are sentence modifiers; hence the hypothetical form of this implication.) Thus a sentence such as ‘Ann could have come to the party’ cannot be construed as expressing the dynamic possibility that, in the past, Ann did come to the party. Rather, it must be construed to express the past possession, by Ann, of a potentiality to come to the party.

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In general, the logical form of ‘x could F’ or ‘x could have F-ed’, on this view, has the tense take scope over ‘can’: roughly, it is ‘it was the case that x can F’, not ‘x can have-Fed’. It is hardly surprising, on the semantics sketched in (CAN), that we can tense ‘can’ statements in this way. Potentialities, after all, are possessed at times, and they may be lost (recall my long-lost potentiality to be a child prodigy). So, given that ‘can’ is used to form potentiality ascriptions, we should expect there to be tensed forms of ‘can’. The simplest such form is ‘could’, as in ‘Mozart could play the piano blindfolded’: here we have a straightforward ascription of a potentiality (indeed, an ability) in the past tense. ‘Could have’ is a little more complicated, since it carries an implication or implicature that the potentiality was not exercised: we would not say ‘Ann could have come to the party’ if Ann did indeed come to the party. It is a difficult question, and not one to be answered here, whether that difference between ‘could’ and ‘could have’ is a difference in truth-conditions or in pragmatic implicature.18 Taking tense to scope over ‘can’, then, seems a viable option. I have not shown that it is the right construal. But we shall see in a moment that it is indeed the right construal for dynamic modals in general, although not for epistemic modals. A second implication is that we cannot form dynamic de dicto statements. Given (CAN), every ‘can’ statement needs a subject.There may be quantifiers in the scope of ‘can’, but only as part of a complex predicate (for instance, ‘I can dance with anyone who has a sense of rhythm’). This seems to fly in the face of natural de dicto readings of such dynamic statements as the already mentioned (5) Hydrangeas can grow on this soil. (read: it is possible that there be hydrangeas growing on this soil), or sentences such as (6) Someone can see us. (read: it is possible that someone sees us). What is the potentiality-based semantics to say about such cases? I say that the indicated readings are not available, at least not for dynamic ‘can’. If they seem available, that is only because we switch to an epistemic reading. This response will be vindicated in the next section. A third seeming implication is related to the second. Every ‘can’ statement, according to the semantics, needs a subject to which it ascribes a potentiality. But some sentences seem to have the wrong kind of grammatical subject: one which 18 There is no separate future tense for ‘can’, but it sometimes functions as its own future tense and sometimes borrows forms of ‘be able to’, as in ‘This time next year, I can (I will be able to) travel to Maroc’.

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does not appear to refer to the kind of object which plausibly has any potentialities, either because it does not refer to an object at all or because it refers to an abstract object. The semantics would seem to imply that such sentences cannot be true. Consider, for instance, (7)

a. The debt rate can rise next year. b. The average tax payer can still afford a yearly vacation.

Neither the debt rate nor the average tax payer is plausibly construed as an object, or at any rate not as the kinds of object that might be a bearer of potentialities. And yet both sentences may well be true. How is the potentiality semantics to capture this? The answer is that (CAN) applies to such sentences as the starting point of an analysis. The sentences in (7) are no more and no less problematic than other ascriptions to such ontologically suspect subjects of ordinary properties, such as (8)

a. The debt rate is rising. b. The average tax payer is suffering from the recession.

Superficially, the sentences in (8) ascribe properties—rising and suffering, respectively—to objects that seem hardly suited to be their bearers (if they are objects at all). Clearly they are ripe for analysis. Such analysis will typically proceed by substituting in the analysans a different subject or subjects—debtors and creditors, or tax payers—and ascribing to it or them (perhaps in a quantificational guise) a possibly different but suitably related property—lending and owing larger sums of money, or indeed suffering from the recession. The same strategy can be applied to what are, by the lights of the present account, ascriptions of potentialities. Only here, the analysans, too, will ascribe potentialities. Thus in the analysantia of (7), we are ascribing (perhaps in a quantificational guise) to different subjects—debtors and creditors, or tax payers—possibly different but suitably related potentialities—a contextually relevant potentiality to lend and owe larger sums of money, or a contextually relevant potentiality to afford a vacation. (CAN), therefore, remains the official statement of the potentiality-based semantics, and the appearance that it rules out ‘can’ statements with the wrong kind of grammatical subject is misleading. ‘Can’ sentences with the wrong kind of subject should not, then, be a cause of specific worries for the potentiality semantics. The first two implications remain, however, and they remain in need of vindication. I have indicated (and no more than indicated) that their vindication will appeal to a distinction between dynamic and epistemic modality. But on the face of it, that distinction merely

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provides another undesirable limitation for the potentiality semantics. Modality, as we have seen above, comes in at least three different flavours: epistemic, deontic, and dynamic. The standard, possible-worlds based, semantics accounts for all three (and any others) in one go, by taking them simply as differently restricted quantifications over possible worlds. But we have so far only seen how the potentiality-based semantics accounts for dynamic modality. How is it to cover epistemic and deontic modality? The answer is: not easily. The connection between potentiality and dynamic modality, beginning with the ability-ascribing uses of ‘can’ and the suffixes ‘-able’/‘-ible’, is intuitively close. But permission and obligation, and what is compatible with our state of knowledge—the respective subject matters of deontic and epistemic modality, to a first approximation—have no equally natural connection with potentiality. Or if they do have a connection, that connection is an altogether different one. Consider, by way of example, DeRose’s (1991) truth-conditions for a typical statement of epistemic modality: S’s assertion ‘It is possible that P’ is true if and only if (1) no member of the relevant community knows that P is false, and (2) there is no relevant way by which members of the relevant community can come to know that P is false. DeRose 1991, 593f., my italics

We need not endorse DeRose’s view, but it illustrates how dynamic modality and thereby, on the present approach, potentiality comes into the truth-conditions for epistemic modals. Clause (2) contains the dynamic modal ‘can’. The potentiality semanticist will construe the sentence as denying members of the relevant community the relatively coarse-grained (extrinsic) ability to come to know that P is false. In this way, epistemic possibility becomes a matter of our extrinsic abilities to rule out a hypothesis. But this is very different from the role that potentialities play in the truth conditions for dynamic modality. Contrast the following two sentences: (9)

a. Frank can run four-minute miles. b. It is possible that Frank runs four-minute miles.

(The example is DeRose’s.) (9-a), which is clearly dynamic, ascribes to Frank, the subject of the sentence, a potentiality: the ability to run four-minute miles, on the most natural construal. (9-b), which expresses an epistemic possibility, may be used (on DeRose’s account) to deny to a very different subject, the relevant community (presumably including the speaker, but not Frank), a very different potentiality: the potentiality to rule out that Frank runs four-minute miles. Similar observations will apply to other accounts of epistemic modals. We have here

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yet another implication of the construal, central to the potentiality semantics, of ‘can’ as a predicate modifier. For as a predicate modifier ‘can’ in (9-a) must be read to ascribe a property to the sentence’s subject, Frank. But no such construal is plausible for (9-b), which concerns the epistemic status of the whole proposition that Frank runs four-minute miles, and ascribes or denies potentialities, at most, to subjects that stand in some epistemic relation to that proposition. What, then, is the potentiality semantics to do with epistemic (and deontic) modality? There are two options. One is to argue that the potentiality-based semantics can, after all, be extended to epistemic and deontic modality. The second is to argue that a unified account of dynamic, epistemic, and deontic modality is not so desirable in the first place. In the next section, I will take and defend this latter option.

6.9 Modality: root versus epistemic 6.9.1 Syntax The potentiality-based semantics appears to divorce dynamic modality from epistemic and deontic modality. While initially this may appear to be a drawback, I will now argue that a unified semantics for the different flavours of modality is not so desirable after all. Many philosophers accustomed to thinking about modality in possible-worlds terms will find this strategy rather surprising, to say the least. Possible-worlds based semantics provides for a natural unification of the three flavours of modality: the difference between, say, epistemic and dynamic modality is just a matter of which worlds are being quantified over, those compatible with actual fact (whether known or unknown) or those compatible with what we know. But linguists have long known that such unification, while theoretically attractive, is a gross oversimplification. (Kratzer herself thought of her model as a simplification, albeit of course a ‘rewarding’ one: see Kratzer 1981, 33.) In particular, there is a deep and well-known divide between so-called root modality (including dynamic modality) and epistemic modality, which can be seen at the level of surface syntax, of logical form, and arguably also of semantics. In this section and the next, I will present the relevant linguistic literature on this distinction and discuss the consequences. It will turn out that this literature supports not only a split between dynamic and epistemic modality, but also certain features of the potentiality-based semantics that I have highlighted in section 6.8: the exclusion of dynamic statements that are de dicto and of tenses embedded under ‘can’. Indeed, these are some of the very features that make a fully unified semantics of dynamic and epistemic modality look less than plausible.

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To begin, let us look at the root/epistemic distinction in a little more detail. The term ‘root modality’ was originally introduced negatively, as comprising any modality that is not epistemic. It is standardly taken to include circumstantial/dynamic and deontic modality, and not much else. Here is a preliminary statement of the distinction from Cook (1978): [E]pistemic modal[ity] modif[ies] a sentence and deal[s] with the truth value of that sentence; root modal[ity] relate[s] . . . [a subject] to an activity and deal[s] with permission, obligation and ability. Cook 1978, 6, cited with parentheses from Butler (2003)

Root modality is located more deeply inside the structure of a sentence: it does not take a sentence as a whole to yield a new sentence. Rather, it looks into the structure of the sentence and relates its subject and predicate to each other. Epistemic modality, on the other hand, applies to a sentence as a whole, ascribing to it a certain epistemic status or positioning the speaker in relation to it. We can bring out the difference in a way that is at once more precise and more useful for our purposes if we look at the place of root and epistemic modals in the syntax of sentences. In what follows, I neglect deontic modality and focus on the contrast between epistemic modality on the one hand and the dynamic part of root modality on the other. I will start with the surface syntax of the respective expressions and then move on to their logical form. At the level of surface syntax, as we have seen earlier, some modal expressions, in particular, the modal auxiliaries such as ‘can’, function as predicate modifiers: they take a predicate (‘swim’) to form a complex predicate (‘can swim’), which is then applied to a subject (‘Ann can swim’). Others, in particular, modal adverbs such as ‘possibly’, function as sentence modifiers: they take a sentence (‘Ann is swimming’) to form a complex sentence (‘Possibly, Ann is swimming’). Surface syntax need not correspond exactly to logical form (the ‘deep-down’ syntax, which is used for compositional semantics). But we can note already that there is an asymmetry between dynamic and epistemic modality here. At least in English, modal expressions that are syntactically clearly sentencemodifiers seem to be reserved for epistemic modality. Sentence-adverbs such as ‘possibly’ and sentence-modifying constructions such as ‘It is possible that’ are used, outside the philosopher’s vernacular, to express epistemic possibility. Dynamic modality is expressed exclusively by expressions which, at the level of surface syntax, are predicate modifiers. Prominent among them are the modal verbs and auxiliaries, such as ‘can’ and ‘be able to’. Where an expression such as ‘possible’ is used to express dynamic rather than epistemic possibility,

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it tends to have predicate-modifier structure as well, as in the construction ‘it is possible for . . . to . . . ’. Compare the following four sentences (the example is Keith DeRose’s): (10)

a. b. c. d.

It is possible that Frank runs four-minute miles. Possibly, Frank runs four-minute miles. It is possible for Frank to run four-minute miles. Frank can run four-minute miles.

(10-a) and (10-b) express an epistemic possibility of Frank’s running a fourminute mile, while (10-c) and (10-d) express a dynamic possibility. As DeRose (1991) points out, when Frank’s new friends begin to suspect that he is a track star, they may truly and adequately utter (10-a) or (10-b) even though Frank in fact, and unbeknownst to them, is quite incapable of running a four-minute mile (‘his only event is throwing the javelin’, DeRose 1991, 602). But given Frank’s incapability to run four-minute miles, (10-c) as well as (10-d) are false. Conversely, (10-c) and (10-d) may be adequately asserted by someone who knows that Frank has never run a four-minute mile, judging merely from the constitution of his legs, lungs, etc. On the other hand, (10-a) and (10-b) are not adequately asserted by a subject who knows that Frank does not actually run four-minute miles. These preliminary observations do not establish anything about the deep syntax, let alone the semantics of modal expressions. But they suggest already that there is a link between sentence-modifying expressions and epistemic modality, on the one hand, and predicate-modifying expressions and dynamic modality, on the other. Kratzer (1981) observes the same pattern in the case of German: ‘Sentence adverbs like wahrscheinlich or möglicherweise [‘perhaps’ and ‘possibly’] . . . always express epistemic modality—if they express modality at all’ (Kratzer 1981, 56). The step from surface syntax to logical form is taken when we look at the respective scope of expressions for epistemic modality (or epistemics, for short) and those for root modality (or roots, for short). It is well-known that epistemics invariably take scope over roots. Not only that: epistemics take scope over tense and aspect, while tense and aspect take scope over roots. In generative syntax, epistemics will be found in a higher position, above tense, aspect, roots, and the verb phrase; roots will be found below epistemics, tense, and aspect, and just above the verb phrase, as illustrated in the following diagram (the left-hand side shows the position of an epistemic modal, the right-hand side that of a root modal):

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Mod

T

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T Asp

VP

Asp

Mod

VP

The scoping hierarchy between epistemics, tense, aspect, and roots, is not an idiosyncratic feature of English. Cinque (1999) argues that this hierarchy holds, primarily for adverbs but also for other types of modal expression, across languages, even those of different families (Cinque (1999, 33–43) includes English, Bosnian/Serbo-Croation, Hebrew, and Chinese.) The scoping hierarchy that Cinque found can be put briefly as follows: (11) Modepis > Tense > Aspect > Modroot (Cinque’s own hierarchy is much more complex, and comprises adverbials that go far beyond the realm of the modal. The short version reproduced here, as well as the diagram above, follows Hacquard 2010.) The different scope of roots and modals with respect to tense is visible, for instance, in the following two sentences on their natural readings: (12)

a. Mary had to sneeze. (Dynamic: she couldn’t help it.) b. Mary had to be home at the time of the crime. (Epistemic: all the evidence points in this direction.)

In (12-a), which is naturally read as dynamic, tense takes scope over the modality: it was true at some point in the past that there was nothing for Mary to do but to sneeze. In (12-b), on its natural epistemic reading, the scope is reversed: it is necessary now, given the evidence, that in the past (at the time of the crime) Mary was home. That epistemics take scope over roots, even disregarding tense, can be seen directly in examples such as the following. Note that the scope is insensitive to the order of the words in the sentence and to the modal force of the modals (possibility or necessity):19 19 The range of possible examples is somewhat restricted due to the fact that English does not allow for two auxiliaries to follow one another. Cinque (1999) has examples from Scots English, which does allow for a sequence of modal auxiliaries, which confirm the same point; and I can confirm that German (where modal verbs take more grammatical forms than the English modal auxiliaries) displays the same structure.

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(13)

a. I might be able to help you with that. (epistemic might > dynamic be able to) b. Mary can probably come tonight. (epistemic probably > dynamic/deontic can) c. I might have to cancel the meeting. (epistemic might > deontic have to) d. John must have to work a lot, we rarely see him. (epistemic must > deontic have to)

This shows that, as suggested in Cook’s characterization with which we started, roots are more deeply ingrained in the structure of the sentence that they modify than epistemics. The epistemic modal applies to a whole sentence, complete with tense, aspect, and root modals (if any). We do not yet have decisive evidence about the level at which the root applies. It might apply to a proto-sentence, lacking tense and aspect but otherwise complete; or it might itself already be part of that proto-sentence. In the former case, we should think of roots as sentence operators: they take a (proto-)sentence and yield another (proto-)sentence. (This approach is taken and defended in Hacquard 2010.) In the latter case, we should think of roots as predicate operators: they take a predicate and form another predicate, relating the original predicate in a new way to the sentence’s subject. Arguments for the latter kind of construction are provided by Brennan (1993). Brennan argues that roots take scope over a sentence’s verb phrase, relating it to the sentence’s subject, while epistemics take scope over the entire sentence.20 Intuitively, roots are used to ascribe modal properties to objects; epistemics are used to ascribe modal status to a sentence. Brennan has a number of arguments to this effect, of which I will recount two: a further argument from scope, and the argument from symmetric predicates. (i) The argument from scope. Briefly put, Brennan’s claim is ‘that in epistemic modal sentences, the relative scope of the subject and modal is systematically ambiguous whereas in dynamic modal sentences the subject 20 To use a common bit of terminology, Brennan argues that the modal auxiliaries that express root modality are control verbs, which have two complements: a noun phrase and a verb phrase; while those that express epistemic modality are raising verbs, which have an entire sentence as their complement. Compare: ‘try’ is a control verb, ‘seem’ is a raising verb. At the level of surface syntax, both look alike: ‘George tries to escape’ appears to have the same form as ‘George seems to escape’. At the level of logical form, however, ‘tries’ takes scope over ‘to escape’, while ‘seems’ is moved up (‘raised’) above the other components, qualifying the entire sentence. No such movement happens with ‘tries’. Evidence for the different behaviour is found, for instance, in the fact that we can paraphrase ‘George seems to escape’ as ‘It seems that George escapes (is escaping)’, but we cannot equally paraphrase ‘George tries to escape’ as ‘It tries that George escapes (is escaping)’. If we apply the test to ‘can’, we get a mixed result: ‘Frank can run four-minute miles’ may be turned without loss of grammaticality into ‘It can be that Frank runs four-minute miles’. However, the operation effects a change in modal flavour: the latter sentence, unlike the former, expresses an epistemic possibility.

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always takes wide scope’ (Brennan 1993, 92). Take the following pair of examples (Brennan 1993, 93): (14)

a. Every radio may get Chicago stations and no radio may get Chicago stations. b. # Every radio can get Chicago stations and no radio can get Chicago stations.

(14-a) is ambiguous between two readings, one of which is contradictory (if there are radios), while the other is not. The contradictory reading says that it is true of every radio that it may get Chicago stations, and then adds that it is true of no radio that it may get Chicago stations. Assuming that there are radios and the quantified statements are therefore not vacuously true, the conjunction of these two propositions is a contradiction. The non-contradictory reading says that it is possible that every radio gets Chicago stations, and it is also possible that none does. This reading is non-contradictory since there may be opposing possibilities, as long as not both of them are actualized. The contradictory reading, of course, is a de re reading of the modal operator ‘may’; the consistent reading is de dicto. Contrast (14-b): this is flat out contradictory. The de dicto reading, which made (14-b) consistent, is not available for (14-b). The fact that we cannot help but read (14-b) as a contradiction (again, assuming that there are radios) is evidence that ‘can’, unlike ‘may’, cannot take scope over the (quantified) noun phrase; only the de re reading is available for ‘can’. Now, ‘can’ is root modal, while ‘may’ generally, and certainly in (14-a), is epistemic. The relevant generalization from the example is that while epistemics can take scope over or under a quantified noun phrase, i.e. allow for both de re and de dicto readings, roots only take scope under a quantified noun phrase, i.e. allow only for de re readings. This, in turn, can be explained if we take epistemics to be sentence-modifiers and roots to be predicate-modifiers. Then (14-a) allows for an ambiguity between the two readings formalized in (15), while (14-b) allows only for one reading, as exhibited in (16): a. ∀xMAY(R(x) → C(x)) ∧ ∀x¬MAY(R(x) ∧ C(x)) [Contradictory, if there are radios] b. MAY ∀x(Rx → Cx)∧ MAY ∀x¬(Rx ∧ Cx) [Consistent] (16) ∀x(R(x) → CAN[C](x)) ∧ ∀x¬(R(x)∧ CAN[C](x)) [Contradictory, if there are radios] (15)

The differences in scope over quantified noun phrases are widely accepted and not restricted to the universal quantifier that was used in our example.21 21 See ch. 2.1.2 of Brennan (1993) for definite descriptions, and Butler (2003, 980–3) for bare plurals.

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Brennan’s observation vindicates the exclusion, discussed in section 6.6, of de dicto ‘can’ statements. The argument from scope does not prove that roots are predicate modifiers while epistemics are sentence modifiers; it might be that both are sentence modifiers, but that there is some other explanation for the fact that roots take only the narrow-scope position in logical form. The hypothesis that roots are predicate modifiers is a good explanation that avoids ad hoc restrictions, but it is not mandatory. Brennan’s second argument is more forceful in that respect. (ii) The argument from symmetric predicates. The basic observation here is that a root can turn a symmetric relation into an asymmetric one. The relation R is symmetric, of course, just in case R(xy) entails R(yx), for any x and y. Getting the same score as is clearly a symmetric relation. Now take the two claims in (17) (from Brennan 1993, 129): (17)

a. Peter can get the same score as Joan. b. Joan can get the same score as Peter.

(See Cross 1986 for structurally similar examples.) As Brennan observes, (17-a) does not entail (17-b): ‘While Peter may have the ability to get the same score as Joan (no matter what score she gets), Joan may not be able to get the same score as Peter’ (Brennan 1993, 129). This finding is not only explained by, but is more direct evidence for, the predicate-operator view of the root modal ‘can’. For if ‘can’ applied to the sentence as a whole, the inference from (17-a) to (17-b) should be valid: after all, the sentences that would be in the scope of ‘can’ are logically equivalent given the symmetry of the relation getting the same score as.22 Brennan concludes: Given a symmetric predicate, R, we know that Rab expresses the same proposition as Rba, for any a and b. Thererfore [sic], we cannot assume that, for . . . dynamic [(17)], the modal operator is concatenated with the same argument in [(17-a)] and [(17-b)]; that is, we cannot assume that in both cases, the semantic structure is: Modal[Rab]. Assuming instead that the modal in these examples combines with the VP to form a modal VP, we expect that the resulting modal property expression will have a different meaning depending on the meaning of the VP . . . . Brennan 1993, 129f.

Note that Brennan’s expectation is indeed met by a potentiality-based semantics: for that semantics, (17-a) ascribes an extrinsic potentiality to Peter while (17-b) ascribes an extrinsic potentiality to Joan. Both extrinsic potentialities are 22 An alternative would be to claim that ‘can’ is not congruent, i.e. not closed under logical implication. That, however, would wreak havoc with any standard semantics for the modals.

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grounded in the same joint potentiality, possessed by Joan and Peter, so either both or none of Joan and Peter will possess the relevant extrinsic potentiality. The difference lies in where the conditions of relevance applies: as Brennan’s remarks indicate, the natural reading for both sentences is the ability reading, and it may well be that Peter’s extrinsic potentiality qualifies as an ability while Joan’s does not, or vice versa. Contrast (17) with (18), which replaces the dynamic ‘can’ with an epistemic ‘might’: (18)

a. Peter might get the same score as Joan. b. Joan might get the same score as Peter.

In this case, the inference from (18-a) to (18-b) (and vice versa) is clearly valid, confirming again the difference between predicate-modifying dynamic and sentence-modifying epistemic modals. These and related considerations have led Brennan and other linguists to construe roots, unlike epistemics, as operating on a sentence’s verb phrase (VP), that is, as predicate operators. We will now go on to consider the implications of this construal for a semantics of root modals, such as ‘can’.

6.9.2 Semantics So far, we have collected evidence that there is a real difference between roots, and in particular dynamic modals, on the one hand, and epistemics on the other. (The case of deontic modality is actually more complicated: it is generally classified as part of root modality, but Brennan argues that deontic modality divides into sentence-modifying ‘ought-to-be’ modals and predicate-modifying ‘ought-to-do’ modals. I will continue to focus on the more clear-cut contrast between dynamic and epistemic modality.) The difference, as we have considered it so far, concerns syntax, at the level of logical form: epistemics and roots have different positions in the logical form of a sentence, and apply to expressions of different forms: whole sentences for epistemics, and verb phrases (including complex predicates) for roots. How are we to deal with this difference in the semantics of modal language? There is a tension here between uniformity and diversity that linguists have struggled to resolve. Hacquard (2010) expresses the tension thus: [O]n the one hand, given that the same modal words can express both epistemic and root modality, we want to give them the same lexical entry; on the other, the fact that epistemics and roots differ systematically in height of interpretation suggests that they should be treated as separate elements. Hacquard 2010, 81

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It will be useful to label the two sides of the tension: Uniformity The same modal words can express both epistemic and root modality. Diversity Epistemics and roots differ systematically in their scope (the ‘height of interpretation’). Orthodox Kratzerian semantics falls squarely to the Uniformity side of this tension, assuming that ‘on a level of logical form . . . all modal operators are sentential operators’ (Kratzer 1981, 44, fn. 3). Others have since tried to take a more nuanced approach that does justice to both Uniformity and to the depth of the root/epistemic distinction. Hacquard (2010), for instance, suggests a revision that makes modals relative not to a possible world, but to an event of evaluation. In the case of epistemics, the event is the event of the sentence’s utterance; in the case of roots, it is the event described in the sentence’s verb phrase. Hacquard further uses this idea to explain the difference in syntactic position between roots and epistemics. We can adopt her explanation without going into the details of her proposal. The basic idea is that, since epistemic modals express the compatibility (or otherwise) of a certain propositional content with a given information state (p.105), they can only be applied to something that is suited to represent a propositional content. A modal in the ‘low’, narrow-scope position does not range over anything that is rich enough to represent propositional content, lacking as it does aspect, tense, and (on the reading suggested by Brennan) even the noun phrase that is the sentence’s subject.23 This explains why epistemic modality is only ever expressed by wide-scope modals. Hacquard gives no further explanation of the fact that root modality is only ever expressed by narrow-scope modals. However, an explanation along similar lines is natural, given the way that the root/epistemic distinction has been introduced: roots ascribe modal properties (typically) to the sentence’s subject. In order to do so, they must be located further down in the syntactic hierarchy, between the noun and verb phrase. A modal expression that is higher up, i.e. has wider scope, has no noun phrase to which it can be applied, ascribing to the phrase’s referent a modal property; so a modal in the wide-scope position cannot be given a root reading.

23 Hacquard’s own solution is more complex, and much more precise. It relies on the idea that there are different event variables in the syntactical representation of a sentence, including one for the event of the sentence’s utterance and another as part of the verb phrase. The utterance event is a ‘contentful’ event, i.e. one that has propositional content; the verb phrase-event typically is not. To be interpreted epistemically in a coherent way, a modal must be relativized to a contentful event. So a modal that is relativized to the verb-phrase event cannot be coherently interpreted as epistemic.

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Brennan explicitly draws from her above-cited considerations the conclusion that roots (or rather, ‘VP-modals’: dynamic and deontic ought-to-do modals) are intuitively construed by language users as ascribing modal properties, while epistemics (or rather, ‘S-modals’: epistemic and deontic ought-to-be modals) are not The intuition behind the . . . arguments for VP-scope modal rules is that there are such things as modal properties—these are (at least) our abilities, dispositions, rights and obligations. I argue that what we have in English root modal + VP concatenations is a kind of complex verb formation, of the sort attested in many languages: the modal in dynamic modal sentences (and ought/allowed-to-do deontic modal sentences) combines with a property-denoting expression to form a modal-property expression. Brennan 1993, 122f.

Brennan goes on to give a broadly Kratzerian, possible-worlds semantics for both S-modals (epistemics) and VP-modals (roots), according to which they differ only in the accessibility relation imposed on relevant possible worlds. The distinction between the two kinds of modals, to her, is not a matter of metaphysics. It is merely ‘a consequence of the fact that the community of language users has made certain ontological commitments and not others’ (Brennan 1993, 196). The metaphysics that I have been developing in this book, however, enables us to take the distinction more seriously. Indeed, insofar as dynamic modality is concerned, the properties ascribed—dispositions and abilities—are among the building blocks of our metaphysics. Unlike Brennan, we are therefore in a position to take seriously the commitments of the language community, and not only to take them seriously as commitments of the community, but to fully endorse them. This opens up the possibility of a semantics which links dynamic modals quite generally to the modal properties which they are used to ascribe; more specifically, which treats dynamic modals as ascribing potentialities. That, of course, is precisely the semantics that I have proposed and developed in this chapter. The semantics, to be sure, falls squarely on the Diversity side of the tension, described above, between Uniformity and Diversity. For it seems unlikely that a semantics can be given for epistemic modals that runs in exact parallel to the potentialitybased semantics for dynamic modals. So, in settling on the latter, we will divorce dynamic from epistemic modality at the level of semantics. Is this a problem? Recall that an orthodox Kratzerian semantics, too, is one-sided in its dealing with the tension. Where a potentiality-based semantics one-sidedly favours Diversity, Kratzerian semantics one-sidedly favours Uniformity. Those working in the Kratzerian tradition have made proposals that aim to accommodate both sides of the tension, while remaining within a broadly Kratzerian framework.

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An analogous strategy may be pursued by the potentiality theorist, though I do not at present see how it would go. Here, I will defend instead a more radical strategy. Instead of striving for full Uniformity, the potentiality-theorist, I suggest, should point out that the deep differences which the semantics sees really are deep differences in the kinds of phenomena that are being referred to. Metaphysically speaking, dynamic and epistemic modality appear to be two very different kinds of animals: one concerns reality, the other concerns our knowledge of it. The distinction is not an artefact of the potentiality-based semantics. Rather, it was what motivated the semantics’ initial focus on dynamic modality. In sharply distinguishing between the two (as well as between the dynamic and the deontic) types of modality, the potentiality semanticist is guided by plausible metaphysical considerations. Is this enough to vindicate the potentiality-based semantics vis-à-vis a more unified semantics, such as Kratzerian possible-worlds semantics? I will now argue that it is. To see why, we need first to take a step back. When confronted with competing semantic theories, there are (at least) two things that we can do. One is to take their competition seriously: only one of the theories can be the right theory. Of course, it is quite likely that of two competing theories neither one is the correct theory, and it may be worth developing both for the sake of one day finding a third, better theory that combines the virtues of both. Nevertheless, on this approach, at most one of two (or more) competing theories can be right. A different approach is to treat both theories as highlighting, if each is successful in its own right, different features of language. Perhaps no grand unified theory of a given language can ever be produced; perhaps language is just too complex a phenomenon for such a theory. What we can do, on this approach, is provide semantic theories as useful idealizations of certain aspects of language, no more than an approximation to the real thing, but good enough as an approximation. If we insist that only one of them can be right, we are like the proverbial people touching different parts of an elephant in the dark. If we take the latter, more liberal approach, then a potentiality-based semantics of dynamic modality is not actually in direct competition with, say, Kratzerian possible-worlds semantics. It makes systematic sense of some aspects of dynamic modality that are not so much in the focus of standard Kratzerian semantics: the semantic aspect that dynamic modals are thought to ascribe modal properties, and the syntactic aspect that dynamic modals function as predicate operators. It further links our modal language to what is, on the metaphysics that I have developed in this book, the underlying modal reality: modal properties and, in particular, potentiality. That is enough for it to be worth our while.

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Still, we might want an explanation for Uniformity. I will return to that desideratum in a moment. If we take the former, less liberal approach, then a potentiality-based semantics is in competition with standard semantics, possible-worlds based or otherwise. In that case, we will have to carefully weigh the costs and benefits of the different competitors. The benefits of a potentiality-based semantics should be clear now: it links our modal language to what is, on the metaphysics that we have assumed throughout the book, the structure of modal reality; and it provides a straightforward explanation for Diversity. Its main cost, too, should be clear: the lack of a natural explanation of Uniformity. This cost should not lead us to reject the semantics outright. Instead, I suggest that we try to provide a deflationary explanation for Uniformity: an explanation which operates on a different level from that of the semantics and which requires no basis in the truth-conditions. Such explanations are typical for a type of ambiguity that is usually called polysemy, and is to be contrasted with homonymy. Ambiguous terms whose different meanings have no relation to each other, such as ‘bank’, are homonymous; ambiguous terms whose different meanings are related to each other, such as ‘healthy’ (exhibiting health/contributing to health) or ‘since’ (temporal succession/causal relation), are polysemous. My suggestion, then, is that the potentiality-based semantics should take modal expressions to be polysemous. While one modal expression such as ‘can’ or ‘might (have)’ is ambiguous between the dynamic, epistemic, deontic (and any other) ‘flavours’, hence providing different truth-conditions for sentences in which they occur with different flavours, these different meanings are not accidentally united in one word. (For a more detailed exposition and defence of the polysemy view, see Viebahn and Vetter 2014.) In what follows, I will offer two such explanations for Uniformity that operate on a different level than the truth-conditions; one is practical, one historical. The two explanations are compatible with each other, and indeed may complement each other. They should satisfy the desideratum of an explanation for Uniformity for both the liberal and the less liberal approach. A first explanation for Uniformity is practical. (I would say ‘pragmatic’, if that did not suggest the usual semantics/pragmatics distinction.) Unity of linguistic expression may have different sources. One source is unity in the underlying reality: all the phenomena described with the relevant type of expression have something in common. Another source may be unity in the practical significance of the underlying reality: while the phenomena described with the relevant type of expression differ from each other substantially, they play the same role for our deliberation, planning, and action. If we take

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possible-worlds semantics seriously, its diagnosis of the unity in modal language is of the first type. The potentiality semanticist should instead opt for the second type of diagnosis. Knowing that the vase can break (dynamic modality) or not being able to rule out that it will break (epistemic modality) both lead to the same result: I will pack the vase safely. A child’s knowing that she is unable to do a cartwheel in the classroom (dynamic modality) and her knowing that she is not allowed to do a cartwheel in the classroom (deontic modality) both have the same result, at least in a rational and obedient child: she will not attempt to do a cartwheel in the classroom. Dynamic, deontic, and epistemic modality alike play the role of delimiting the space of options in our practical deliberation. Their metaphysics may be very diverse, and they may easily come apart in more sophisticated deliberation. But the basic function of the different types of modal knowledge is the same. It is not surprising, then, that a unified idiom has developed to express and share these different types of modal knowledge. But this need not imply that the idiom must be given the same semantics in all its uses; not, at any rate, if semantics is a matter of specifying truth conditions and thereby linking language to those aspects of the world that it is about. A second explanation for Unity is historical. The historical development of modal expressions in a large number of languages has been studied in much detail by Bybee et al. (1994), whose results I can only report in the roughest outline.24 Their findings suggest that modal language starts with certain root meanings and is gradually extended to cover more and more, and in particular, to cover epistemic modality too. One starting point is formed by verbs that express ability, and which are typically connected to verbs that signify knowledge (‘can’, for instance, is etymologically connected with ‘know’) or physical power/strength (as, for instance, the Latin verb posse, ‘to know’, from which the English adjective ‘possible’ is derived). The claim is that the meaning of those words is then extended to include other meanings, such as the epistemic one. Thus with a modal such as ‘can’, the use of an originally ability-ascribing expression is extended to include not only the subject’s mental or physical prowess, but also the external enabling conditions, yielding ability-plus-opportunity uses of ‘can’; then to include among the external enabling conditions social conditions, yielding the deontic meaning of ‘can’. The epistemic senses of modal expressions develop later than, and out of, the agent-oriented [i.e. root] senses. In fact, for the English modals, where the case is best documented, the epistemic uses do not become common until quite late. Bybee et al. 1994, 195 24 For a more recent summary and discussion of the results, see also van der Auwera and Plungian (1998).

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While these developments in the meaning of modal expressions are well documented, the mechanism behind them remains a matter of debate. The practical considerations that I have offered above may well play a role (hence my claim that the two explanations complement each other). Sweetser (1990) has offered an influential model based on metaphor. Epistemic modality is metaphorically related, on this view, to dynamic modality via a phenomenology of ‘forces and barriers’, imposed, in the case of dynamic and deontic modality, on the sentence’s subject by the physical or social world and, in the case of epistemic modality, on the sentence’s speaker by the realm of the rational. Similar metaphorical extensions from the external to the internal world happen, according to Sweetser, when verbs of perception become internalized, as for instance when the verb ‘to see’ acquires the meaning of ‘to understand’. An alternative explanation is defended by Traugott (1989) and largely endorsed by Bybee et al. It is based on an idea called ‘pragmatic strengthening’: [A]t the first stage the early meaning [pragmatically] implies the new meaning, then cases arise in which both meanings are expressed, and finally, cases arise where the new meaning is used alone. Bybee et al. 1994, 197

A similar mechanism is found in the equally polysemous word ‘since’, whose original meaning is temporal succession but which, by an inference from succession to causation (the classical fallacy of post hoc ergo propter hoc), acquires a causal meaning akin to ‘because’. The suggestion is that a similar inference happens in the development from root to epistemic modality. For instance, ‘It can take me up to four hours to get there’ (root), in estimating an arrival time, implies ‘it may take me up to four hours to get there’ (epistemic) (Bybee et al. 1994, 198).25 These mechanisms, and others like them, can be combined with the practical explanation that I have offered. Both of them provide a semantically deflationary account of Uniformity—deflationary in the sense that it does not require a deeper, semantic explanation in terms of uniform truth-conditions. Nor are such 25 The development involves a shift from the narrow-scope root modal to the wide-scope epistemic modal, as Bybee et al. explicitly recognize:

A shift from agent-oriented [i.e. root] to epistemic meaning involves a change in scope. The agent-oriented modal is part of the propositional content of the clause and serves to relate the agent to the main predicate. The epistemic modal, on the other hand, is external to the propositional content of the clause and has the whole proposition in its scope. Bybee et al. 1994, 198f. The shift, Bybee et al. (1994, 198f.) suggest, is effected via cases of narrow-scope, root modal sentences such as ‘someone can sneak in here’ or ‘one can hide one’s misfortunes’, where the sentence’s subject makes only a minimal semantic contribution, so that the narrow-scope modal already ranges over most of the propositional content that must be in the scope of the wide-scope, epistemic modal.

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explanations ad hoc: both metaphorical extension and pragmatic strengthening are used to explain a large range of polysemies. Given either of these explanations, it is not surprising that a more unified semantic theory can also be formulated, as witnessed by standard Kratzerian semantics. All that the present semantics is committed to is the claim that such a unified semantics, while providing a useful model of the shared structure of different aspects of modal language, is not a theory of how this language relates to the world it describes, and not a theory that gives the truth-conditions in any interesting sense. Rather, it—and any other semantic theory that likewise begins from Uniformity—is at most an interesting structural model.

6.10 Conclusion On the account of metaphysical possibility that I have offered in this chapter, it is possible that p just in case something has (or had, or will have) an iterated potentiality for it to be the case that p. Given the logic of potentiality as developed in chapter 5, possibility so defined meets the minimal criteria of formal adequacy. Given, further, the metaphysics of potentiality developed in chapters 3–4, potentiality provides the basis of a well-supported and promising research programme in modal semantics, which does justice to the context-sensitivity of modal expressions such as ‘can’ as well as to the logical form of dynamic modality statements. What seemed at first a drawback, the exclusion of epistemic modality and of certain constructions such as dynamic de dicto statements, turns out to be well supported by linguistic research into the root/epistemic distinction. Several questions remain open for the potentiality-based semantics, including, for instance, the semantics of the counterfactuals, the precise relation between dynamic and epistemic modality, and the status of deontic modality (which I have largely bracketed in my discussion). But a well-supported and promising research programme should be enough to meet the requirement of semantic utility. In the next and final chapter, we will look in more detail at the third of the constraints on a theory of possibility: extensional correctness.

7 Objections 7.1 Introduction In chapter 6, I spelled out the positive picture that comes with the potentiality view of possibility. I have argued that the view meets the constraints of formal adequacy, yielding at least the minimal standard logic T for metaphysical modality, and of semantic utility, providing a framework for a semantics of modal language. A third constraint will be my concern in this final chapter: the constraint of extensional correctness. As I noted in chapter 1, we have certain firm convictions about what is or is not possible. The potentiality account should vindicate sufficiently many of those convictions, or provide a plausible explanation where it does not. The potentiality-based account of possibility is captured in the simple definition POSSIBILITY It is possible that p =df Something has an iterated potentiality for it to be the case that p. (Note that the right-hand side of POSSIBILITY is to be read timelessly, as: something has, had, or will have an iterated potentiality . . . .) Challenges to its extensional correctness will take the form of counterexamples to POSSIBILITY. Such counterexamples, in turn, can take two forms: potentialities without the corresponding possibilities or possibilities without any corresponding potentialities. The former are challenges to the sufficiency of the account of possibility, the latter to its necessity. Section 7.2 will address objections of the first kind, potentialities without the corresponding possibility. I am aware of only one extant challenge of this kind, formulated in a recent paper by Jenkins and Nolan (2012); so section 7.2 will be concerned to address their arguments. The more pressing, and in my experience the more widespread, challenges are of the second kind: metaphysical possibilities for which it appears that no corresponding potentiality, or no witness to the right-hand side of POSSIBILITY, can be found. Those come in many different forms, and will be addressed from section 7.3 onwards.

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Here I will provide a quick overview of the types of counterexamples that will be discussed, and of the candidate strategies in addressing them.1 Possibilities of existence. It seems metaphysically possible that there should have existed individuals other than the ones that there actually are. Some such possibilities are easily accounted for in terms of potentialities: my parents, for instance, once had the potentiality to have another child, thus accounting for the possibility that I have another sibling. But the reach of potentialities is limited in this respect. It might seem possible that there should have been objects which no actual object has or had a potentiality to produce or otherwise bring into existence. How would the potentiality view account for those? This question I will take up in section 7.5. Possibilities of non-existence. Conversely, for any contingent existent, it is metaphysically possible that it should not have existed. But nothing has a potentiality not to exist; for each contingent object, then, we must find something else with an iterated potentiality for that object not to exist. Worse, it seems metaphysically possible that none of the actual contingent existents ever existed; but which actual object should have a potentiality for that to be the case? These questions will be my concern in section 7.6. Abstract objects and related problems. Many metaphysical possibilities concern concrete objects, which are, at least in principle, suitable as bearers of potentiality. But what of possibilities that concern abstract objects? Such possibilities are legion given that, on the standard modal logic which I have argued to hold of metaphysical possibility, necessity implies possibility. So it is possible, since it is necessary, that 2+2=4. But what might have an iterated potentiality for it to be the case that 2+2=4: the numbers 2 and 4? I will discuss this issue in section 7.7. Nomic and metaphysical possibility. Two standard assumptions seem to be in tension with one another, given POSSIBILITY. The first is that metaphysical possibility outstrips nomic possibility: it is metaphysically possible for the laws of nature to be different. The second is that dispositions as well as abilities are constrained by the laws of nature: nothing has a disposition, or an ability, to act against the laws of nature. If both assumptions are true, then metaphysical possibility must outstrip the dispositions and abilities that objects have; and if we can generalize from dispositions and abilities to potentiality in general, the present account will be extensionally inadequate. This objection is discussed in section 7.8. 1 A shorter and slightly different version of these objections and my response to them is given in Vetter (forthcoming), which despite its publication date was written earlier than this chapter.

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I believe that these objections and questions can all be satisfactorily answered. There are, in principle, two ways of answering them: one by a catch-all solution, which addresses all objections at once; the other in a piecemeal fashion. Catch-all solutions include appeal to an omnipotent god, who has all the potentialities not found in the more mundane objects that I have so far focussed on (see, for instance, Pruss 2002); appeal to so-called mere possibilia (as defended by Williamson 1998 and Zalta 2006)2 with potentialities to exist, to exist without any of the actual objects, or to be governed by different laws; and appeal to the world as a whole as a bearer of potentialities. I will not consider the deistic version of the catch-all strategy; it smacks too much of a deus ex machina solution. Mere possibilia, despite the label, are not objects whose existence is merely possible; according to Williamson, their main contemporary defender, they exist actually and are distinguished from other objects only by the range of properties that they have: namely, only logical properties such as existence and self-identity, as well as the property of being possibly so-and-so (being possibly my sister, for instance). Within a potentiality-based framework, this would need slight reformulation: the mere possibilia are those objects which have only logical properties and potentialities.3 I will not, however, pursue this strategy any further, for the simple reason that it detracts from one of the core motivations for the potentiality view: the idea that modality is a matter of how things stand with concrete individuals, those that we are happy to accept into our everyday ontology. Instead, then, I will explore the third option: the world as a bearer of potentialities. This option fits more plausibly into the metaphysical picture of this book, and it has the added interest of allowing for an excursus (in section 7.4) into a theory of possible worlds on the basis of the potentiality view. The powerful world as a catch-all solution will be discussed in section 7.3. The upshot of that discussion, however, will be that it is worth pursuing instead a piecemeal approach: that is, looking at each counterexample in its own right and either pointing out that it is, after all, accommodated by the potentiality view, or biting the bullet and arguing that there is, indeed, no such metaphysical possibility as the objector claimed. This is the approach which I will apply throughout sections 7.5–7.8. Section 7.9 will, finally, tentatively address issues to do with the connection between potentiality and time. It will end in an optimistic, if not entirely conclusive, spirit. The potentiality-based view of metaphysical modality, which I have developed and defended throughout this book, is a serious contender. 2 Both Williamson and Zalta defend the existence of mere possibilia, but of course neither of them does so within a potentiality-based framework. 3 This strategy would require some further work: if radical structuralism about properties is correct, all objects have only (properties that are) potentialities.

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So much for a preview. Now let us begin with the first kind of counterexample: potentialities that are apparently without a corresponding possibility.

7.2 Potentiality without possibility? In a recent paper, Jenkins and Nolan (2012) have argued that there are ‘unmanifestable dispositions’: dispositions whose stimulus or manifestation condition is logically, metaphysically, and/or nomically impossible. Their examples provide a formidable challenge to the view that dispositions or, more generally, potentialities, are sufficient for the obtaining of a possibility (Jenkins and Nolan note that this is a consequence of their view: see Jenkins and Nolan 2012, 751). It is not, however, a challenge that cannot be met; or so, at any rate, I will now argue. To be more precise, Jenkins and Nolan’s claim is that there are non-trivially true disposition ascriptions of the form ‘x is disposed to F in (circumstances) C’ where either F or C or both are logically, metaphysically, or nomically impossible. One motivation for this view, which need not concern us in the present context, is the thought that dispositions are closely related to counterfactual conditionals, coupled with the view that there are non-trivially true counterfactual conditionals with impossible antecedents or consequents (or both). Having rejected the link between dispositions and conditionals, I am obviously not moved by this consideration. What is more troubling for my view are their specific examples. I will list three central examples. My response to them generalizes to other, analogous examples that are mentioned in the paper or can be constructed along similar lines. 1. Unmanifestable dispositions of agents. This kind of case is illustrated by the example of Heidi: Suppose Heidi is the best mathematician of her age, and X is some complex mathematical conjecture, of the kind with which Heidi is most competent, whose truth-value is as yet unknown to Heidi and her community but which is in fact false [and hence, metaphysically impossible]. We can then say with some plausibility that Heidi is disposed to produce a proof of conjecture X on the condition that there is one. Jenkins and Nolan 2012, 739

Nor is this disposition ascription trivial, because there is a parallel disposition ascription that is false. Heidi, we may suppose, is not disposed to produce a proof of some other unproved conjecture Y from an area of mathematics with which Heidi is unfamiliar and in which she has a marked aversion to working. (Suppose Y is in fact also false.) Jenkins and Nolan 2012, 739

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Such disposition ascriptions are not without interest: when assigning tasks in the mathematical community, Heidi’s disposition to produce a proof of X if there is one makes it reasonable to assign the task of attempting to prove X to Heidi rather than to Hilda, who lacks such a disposition. (See Jenkins and Nolan 2012, 741.) Similar cases can be constructed for logical, nomic, and non-mathematical metaphysical impossibilities. 2. Unmanifestable dispositions without agents. Jenkins and Nolan note that it is easier to construct cases (as in 1) that involve agents and their responses in impossible situations. To provide a non-agentive case, they focus on the case of nomic impossibility. Suppose that it is a law of nature that p. In establishing that p is the true law, scientists may use an apparatus that can detect whether p or some closely related hypotheses p , p , etc., are true. In order to successfully distinguish between the different hypotheses, the scientists’ apparatus ought to have a disposition to reliably indicate that p is true if it is true, that p is true if p is true, and so forth. Of course, if p is the true law, then the rival hypotheses p , p , etc., are false, and hence nomically impossible. (At any rate, the case can be constructed so that they are.) Nor are the corresponding disposition ascriptions trivial, for there are various parallel ascriptions involving other nomically impossible hypotheses q which the apparatus has no disposition to detect. Again, such dispositions are certainly not without interest; in fact, as Jenkins and Nolan argue, apparatuses of the kind described have been employed, and their usefulness derives precisely from the dispositions just described. (Jenkins and Nolan give the example of Millikan’s oil drop experiment designed to test the atomic theory of charge: see Jenkins and Nolan 2012, 743 ff.) 3. Scientific idealizations. Scientific idealizations ascribe dispositions to objects or collections of objects whose manifestation is, strictly speaking, nomically or even metaphysically impossible. In a given idealization, a rabbit population may be disposed to ‘increase by 0.1 of a rabbit in a given period of time’ (Jenkins and Nolan 2012, 746), but it may well be nomically or even metaphysically impossible for this to happen. Nor will the disposition ascription be trivial, since parallel idealized disposition ascriptions—involving, perhaps, the disposition to increase by 0.8 of a rabbit over the same period of time—will be false. And again, the disposition ascriptions are clearly not without interest: scientific idealizations are an extremely useful tool. Jenkins and Nolan do not claim these examples to be indisputable, but they take the burden of proof to be on their opponent. In responding to them, the opponent has two strategies at her disposal.

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A first strategy is to provide a general reason for thinking that there are no unmanifestable dispositions (or, in the present context, no unmanifestable potentialities). Within the present context, this strategy is inadvisable. For it amounts to defending precisely what is at issue: that whenever something has a potentiality to F, it is metaphysically possible for that thing to F. How is this claim to be defended? By assuming an understanding of both potentiality and of possibility that is perfectly general and already given and then providing for a link between the two. I cannot do that, for a general understanding of possibility is precisely what I am defending. To assume it in countering the examples would be circular. A second strategy is to proceed piecemeal and reject the counterexamples one by one. Since each counterexample claims that there is a disposition without a corresponding possibility, rejecting the counterexamples amounts to either denying that things have the dispositions at issue or arguing that their manifestation is indeed possible. This is the option that I will take: I will deny the disposition ascriptions. In choosing this second strategy, I run the risk of being less than comprehensive: unlike the first strategy, the present strategy defeats counterexamples one by one and cannot, therefore, preclude that new counterexamples will be adduced. To minimize that risk, it is important to defeat the counterexamples in as systematic a way as is possible. Our reasons for rejecting the counterexamples should stem from very general considerations about dispositions (or potentialities), not specific features of the examples, thus giving us reason to hope that they can be used against any new counterexamples as well. One way in which we may try to argue against the relevant disposition ascriptions is by appeal to parsimony about dispositional properties (or potentialities). This is a strategy which Jenkins and Nolan suggest at the end of their paper: even if it is true that, say, Heidi is disposed to prove X if X is true, this is not the core characterization of any of Heidi’s dispositional properties. The dispositional property which makes true the sentence ‘Heidi is disposed to prove X if X is true’ may be Heidi’s disposition to prove true theorems in the area to which conjecture X belongs. Or perhaps it is, instead, a variety of dispositional properties: her disposition to come up with creative solutions to problems, her disposition to persevere with difficult proofs, the various dispositions that constitute her mastery of the area to which X belongs, all of which have clearly metaphysically possible manifestation. It should go without saying that this parsimonious approach is not the one I have adopted in this book. So this particular solution is of little use in the present context.

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Fortunately, the conception of potentiality that I have been developing in this book allows us to respond to examples of type 1–3 in ways that are both general and independently motivated. Let me go through them one by one. First: Heidi’s disposition to prove conjecture X if it is true. Similar examples are: Jane’s disposition to be surprised if presented with a round square object (Jenkins and Nolan 2012, 738), or Graham’s disposition to be pleased if there is a true contradiction. Examples of this type follow a simple recipe: take a metaphysically impossible stimulus condition (conjecture X being true, the existence of a round square object, the truth of a contradiction) and a plausible reaction that an agent would show to such a stimulus (proving the truth of X, being surprised or pleased). The examples do not depend on any particular conditional analysis of dispositions; but they do depend on what I have called the conditional conception of dispositions in chapter 3: the idea that a disposition is to be individuated by a stimulus and a manifestation, that dispositions are always dispositions to . . . if . . . . I have rejected that conception and adopted, instead, a conception of dispositions and, more generally, of potentialities, as individuated solely by their manifestation. This conception makes Jenkins and Nolan’s examples harder to formulate and much less compelling. On the present account of dispositions, Heidi’s disposition must be specified in terms of its manifestation alone. So what is the manifestation? If it is simply proving conjecture X, it becomes much easier to deny Heidi that disposition: how should she be disposed to prove a conjecture when that conjecture is not true? The manifestation of that disposition includes more than just Heidi’s performing certain activities. It requires cooperation from the world outside Heidi: the world must be such that conjecture X is true. Heidi’s disposition to prove conjecture X would have to be an extrinsic disposition. But there is nothing in the world to have, together with Heidi, the joint potentiality to be such that Heidi proves conjecture X—that is, to be such that conjecture X is true and Heidi performs activities that constitute proving it. So we have good independent reason to deny that Heidi has the disposition, or any potentiality, to prove conjecture X. Instead of dropping the putative stimulus condition, we might try to incorporate it into the manifestation. Heidi’s disposition, then, would be the disposition to prove-X-if-X-is-true. That disposition she possesses, but only trivially. Its manifestation consists in satisfying the conditional ‘. . . proves X if X is true’, which is trivially satisfied by everyone at all times; not by proving anything, but simply by being such that the antecedent is false. (The disposition is, again,

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extrinsic.) This disposition does not set Heidi apart from Hilda, or from anyone, indeed anything, else. The best reading of what is intended by the disposition ascription takes the ‘if’ clause to have a different scope. It is not that Heidi is disposed to prove-theoremX-if-X-is-true. Rather, the following conditional holds: if X were true, then Heidi would be disposed to prove it. If Jane were presented with a round square object, she would be disposed to be surprised.4 Jenkins and Nolan generously offer this strategy to their opponent but point out that they themselves are not at all confused about the difference between X’s being disposed to  in C and X’s being such that were C the case X would be disposed to  (under certain circumstances). When we say that the claims about things actually having unmanifestable dispositions sounded plausible to us, we say that in full awareness of the non-conditional nature of those attributions. Jenkins and Nolan 2012, 747

In the absence of independent motivation, the re-scoping that I have suggested begs the question against Jenkins and Nolan. However, I have offered independent motivation for that re-scoping by abandoning the conditional conception (for reasons that are entirely unrelated to the present issue). I have also given reasons for mistrusting linguistic intuitions concerning the ‘disposed to’ locution quite generally (see chapter 3.1 and, for more details, Vetter 2014), which explain away some of the implausibility of taking their intuitions to be simply mistaken. The first kind of case, then, need not worry us. Second: experimental apparatuses. This kind of case differs from the first in two ways: the alleged disposition’s bearers are non-agentive; and the impossibility of the alleged disposition’s manifestation is nomological, not metaphysical (or logical). Because of this last feature, the cases of this kind are already less worrisome to me than the first, since my concern is metaphysical, not nomological possibility. If there are dispositions, and hence potentialities, with nomologically impossible manifestations, then the metaphysically possible surpasses the nomologically possible, a plausible idea in any case (and one to which I will return later). In verifying this plausible idea, Jenkins and Nolan’s cases would work in my favour. 4 Such counterpossibles, of course, will sit uneasily with any potentiality-based semantics for the counterfactual conditional. Still, the present strategy would have reduced one problem—apparently unmanifestable dispositions—to another one that we already have: the semantics of counterpossibles. I conjecture that counterpossibles, where true, are used epistemically, hence outside the purview of a potentiality-based semantics.

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Within the present framework however, the second kind of case fails for the same reasons as the first: it relies on the conditional conception of dispositions, which I have rejected in chapters 2–3. For Jenkins and Nolan, the experimental apparatus is disposed to indicate that p is true if p is true, to indicate that p is true if p is true, and so forth. Suppose that p is the true hypothesis, and p is nomically impossible. If we were to individuate these dispositions by their manifestation alone, as I have argued we should, what would be the manifestation of Jenkins and Nolan’s disposition to indicate that p if p is true? First, it might be simply indicating that p . But that disposition will clearly not do. An experimental apparatus with a disposition to indicate that p , with no sensitivity to whether p is true, is clearly useless. A better option would be accurately indicating that p . But as with Heidi’s disposition to prove conjecture X, it is doubtful that anything has such a disposition. After all, in order for the apparatus to accurately indicate that p , p would have to be true. Accurate indication, like proving a conjecture, requires cooperation from the world outside: it requires that the world be such that p is true. The disposition to accurately indicate that p would, therefore, have to be an extrinsic disposition. Whether an apparatus possesses it depends on whether some appropriate object or objects possess potentialities for p to be true. This is a separate question, which we will discuss again in chapter 7.8. The present example does not force us to go either way. Alternatively, we might take the conditional of Jenkins and Nolan’s formulation of the case seriously. One way of doing so is to assign to the experimental apparatus the disposition to indicate-that-p -if-p -is-true. The manifestation of this disposition would consist in the apparatus’s satisfying the conditional ‘ . . . indicates that p if p is true’. But that conditional is satisfied trivially if p is false. So the apparatus has the disposition in question, but everything else does too. Clearly, this is not the disposition ascription we are looking for. Finally, another way of taking seriously the conditional is to take the ‘if’ clause as having a scope outside the disposition ascription. What we need to know about an experimental apparatus is that if p were true, the apparatus would be disposed to indicate that p . Again, Jenkins and Nolan will insist that they are aware of the scope ambiguity in ‘The apparatus is disposed to indicate that p if p is true’, and that it is the narrow-scope reading (with ‘if’ in the scope of ‘disposed to’) which they find intuitively plausible. Again, I respond by pointing out that there is independent evidence for the kind of re-scoping that I am suggesting and that our linguistic intuitions concerning the ‘disposed to’ locution are not to be trusted. I conclude that Jenkins and Nolan’s second kind of example succeeds if there is independent reason to believe that the nomically impossible p is metaphysically

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possible, but does not otherwise. In either case, the examples pose no threat to the potentiality-based account of metaphysical possibility. Third: scientific idealizations. Here the conditional model of dispositionality is not playing much of a role, but my earlier observations on the ‘disposed to’ locution (again, see chapter 3.1 and Vetter 2014) still apply. ‘Disposed to’ is often used, in everyday and scientific parlance, to express statistical generalizations which need not translate into disposition ascriptions (in the philosopher’s sense) to the individuals that are being generalized over.5 If scientific idealizations expressed with ‘disposed to’ are of this kind, then whatever problems are raised by them are not problems about dispositions, nor about potentialities in general. But there is a more direct way of dealing with the examples. Consider the manifestations of the putative dispositions: increasing by 0.1 of a rabbit, for instance. Assume, further, that it is metaphysically impossible for the putative disposition’s bearer, in our case the rabbit population, to have that property. Then we are confronted with a case that is even more puzzling than the seeming ascription of a disposition for a metaphysically impossible manifestation. For the scientific idealization will ascribe to a given rabbit population not only the disposition to increase by 0.1 rabbit in a given period of time. In many cases, it will contain outright ascriptions of the metaphysically impossible property itself. Thus a given rabbit population may be said to have increased by 0.1 rabbit per month. If it is impossible for the population to do so, then a fortiori it will be false that it did; and hence we must find some way of dealing with the idealization that does not require the literal truth of what it says. We might take the statements made within scientific idealizations to be read non-literally, and perhaps give a translation or analysis. Or we might count them as straightforwardly false but useful. Whichever way we go on the non-dispositional claims of metaphysical impossibilities within scientific idealizations, we can in a second step apply the same strategy to the disposition ascriptions (if such they are) cited by Jenkins and Nolan. We might say that they are not to be read literally, but need to be translated or analysed. Or we might count them as straightforwardly false but useful. Either way, we need not envisage dispositions with metaphysically impossible manifestations. Given the account of dispositions in particular, and potentiality in general, that has been developed in this book thus far, each of Jenkins and Nolan’s

5 To repeat an example from chapter 3.1: ‘Ordinary people in foreign lands are disposed to like Clinton, or at least to like the relaxed, human image they describe with reference to his smile or his saxophone.’

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counterexamples can be explained away, and in each case appeal was made to very general considerations—the rejection of the conditional model for dispositions, the characterization of extrinsic potentialities, the unreliability of linguistic intuitions concerning ‘disposed to’, and so forth. We can expect, therefore, that the strategies I have offered can be extended to any counterexamples that are minimally similar to Jenkins and Nolan’s. We now move on to objections to the account that focus on the opposite direction: possibilities without potentialities. An overview of the counterexamples, as well as the main overall strategies in responding to them, has been given in section 7.1. We will now look at them in more detail.

7.3 A catch-all solution: the powerful world In section 7.1, we have seen four kinds of counterexamples to the necessity of a potentiality account of possibility: possibilities of existence, possibilities of nonexistence, possibilities concerning abstract objects, and possibilities that outstrip the nomic possibilities. This section and, as a brief excursion, the next will deal with what might be a catch-all solution to such and perhaps other examples: appeal to a powerful world. The world itself, it was suggested, might be the bearer of potentialities and as such serve as a catch-all solution to the counterexamples that I have outlined: by having potentialities to contain different objects or properties, potentialities not to contain the objects or properties that it does contain, potentialities to exhibit certain structural features (e.g. being such that 2+2=4), and potentialities to be governed by different laws of nature. In this section and the next, I will investigate this catch-all solution and come to a tentatively sceptical conclusion: while the catch-all solution might work, it is in dire need of a suitable understanding of what kind of object ‘the world’ is supposed to be. This section will go through various candidates and show that they are either not as universally applicable as one might have hoped or else simply underdeveloped. The following section 7.4 will follow a sideline and examine the relation between potentialities of the world and possible worlds on a Stalnakerian conception as uninstantiated properties of the world. It will be argued, first, that if we can make sense of potentialities of the world, then we can supply a candidate for the role of possible worlds; and second, that in so doing we do not run the risk of collapsing the potentiality account into a possible-worlds account. But first, to the task at hand: formulating a catch-all solution. If the world is to be a bearer of potentialities, we need a better understanding of the kind of object that is ‘the world’. There are a number of different options.

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To begin with, we must ask what the relation is between the world and the things that are ‘in’ it. Does the world have smaller objects that are ‘in it’ as its parts, or does it not? If it does, then we can think of the world as a whole, the whole that has everything as its parts. In this case, objects are ‘in’ the world like raisins are in a cake: by being part of it. If the world does not have objects as its parts, then we can think of the world as a container, which contains but is disjoint from everything else that there is. In that case, objects are ‘in’ the world like water is in a bottle: by being enclosed in it. Take, first, the conception of the world as a whole, or as a composite object. What kind of composite object is the world supposed to be? In general, a composite object consists of parts that are arranged in a certain structure. We might think of a composite object primarily in terms of its parts and not its structure and individuate a composite object by its parts and not the structure in which they are arranged. If we do so, we take the object to be a mere sum of its parts: an object that ceases to exist if any of its parts is lost, but that survives as long as the parts do, no matter how they are arranged. Alternatively, we might think of a composite object primarily in terms of its structure and not its parts, and individuate it by its structure and not the specific parts that go into the structure. If we do so, we take the object to be a structured object: an object that ceases to exist if its structure is too severely interrupted, but that survives changes in some or all of its parts. (We might think of an object as individuated by both its parts and its structure, but that rigidity is scarcely motivated. If we thought of a composite object neither in terms of its parts nor its structure, it would become difficult to see how we should think of it at all.) The first way of thinking about a composite object is appropriate, or so most would say,6 for the purposes of classical mereology. To think in this way of a composite object is to think of it as a mereological sum. The second way of thinking is more common, and arguably more appropriate, in everyday contexts; it is how we think of ordinary objects such as animals and artefacts. I have left ‘structure’ deliberately vague to accommodate a great variety of views about such objects. We will see below some ways in which this notion of structure may be spelled out when the object in question is the whole world. If we take the world to be a whole in the mereological sense, a mere sum, then it will have no potentialities of the kind that help with possibilities of existence and non-existence. I said that a mereological sum is individuated by its parts. We may put this in terms of essence: it is essential to the sum that it has all and only the parts that it does have. Or, more relevantly for present purposes, we may put 6

An exception is van Inwagen (2006).

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it in terms of potentialities: a mereological sum has no potentiality, no matter how many times iterated, to lack any of its actual parts or to have any parts other than the ones that it actually has. What is more, if the world is nothing but the sum of its parts, its potentialities are nothing over and above those possessed, individually and jointly, by its parts. But we introduced the world as a catch-all solution precisely to accommodate possibilities for which the potentialities of the objects in the world, individually and jointly, seemed not to be enough. If the world as a composite object has nothing to offer over and above those potentialities, then it will not do as a solution to our problems. This is not to say that we should reject the notion of the world as a mereological sum quite generally; perhaps there is such a thing, and perhaps it can be put to explanatory use elsewhere. But for our purposes, it will not do. If, on the other hand, we think of the world as a structured object, we may obtain better results. A structured object in general can exchange its parts while remaining the same object; it has potentialities, that is, to possess different parts from the ones it does possess. Thus the world as a structured object might, in principle, have potentialities not to contain some of the objects that it does contain, as well as potentialities to contain other objects. There may be principled limitations on the kinds of things that a structured object can contain as its parts: I have some potentiality for my heart to be replaced by another heart, perhaps even an artificial one, but no potentiality for it to be replaced by an entirely un-heart-like object such as the Eiffel tower. In general, the potentialities of a structured object are limited not by its actual parts, but (at least) by certain structural features. Those are, we might say, essential to the object; in current terminology, it is enough to say that they limit the object’s potentialities. In the case of ordinary objects, which many take to be structured objects in this (vague) sense, it is often difficult to say just what the structural features are that the object must possess so long as it exists; on the other hand, we seem to have a good pre-theoretical grasp, since we can trace dogs and watches across space and time through most of their changes in parts. In the case of the world as a structured object, it is the other way around: we have no such intuitive grasp—what would it even be to trace the world ‘across space and time’?—and are therefore in dire need of explicating the structural features that give this object, the world, its nature and the scope of its potentialities. So what might the ‘structure of the world’ be? Talk of ‘structure’ and, in particular, the structure of the world, has recently been made popular by Ted Sider (2011). Sider is not, however, interested in our present question: the status and nature of the world as an object that can itself be

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the bearer of properties, and in particular of potentialities. Structure, for Sider, is closely linked to fundamentality: what is structural, in Sider’s sense, just is what is fundamental. If we think of structure as consisting, roughly, of fundamental facts,7 then this approach will be of no use in the present context. For fundamental facts can be entirely contingent: ‘Facts about the positions of subatomic particles would be, on most views, fundamental facts, whereas the fact that some people smile when they eat candy would presumably not be’ (Sider 2011, 105). Both kinds of fact, however, should come out contingent: the subatomic particles could have been located elsewhere, just as the correlation between candy and smile might have universally failed to hold. Such contingent, albeit fundamental, facts should not restrict the potentialities of the world. Sider’s own approach is less restrictive. ‘Structure’, for him, is subpropositional (Sider 2011, 128, 147f.): it comprises properties (such as, perhaps, electric charge) as well as what we might think of as operators (such as negation). We might think of structure as being a matter of the world’s building blocks— with the precaution that Sider is explicitly rejecting commitment to those building blocks being entities. They are building blocks of ideology, not ontology. Sider’s view of the structure of the world (the fundamental ideology) is that it comprises elements corresponding to first-order quantification theory, set theory, and fundamental physics, as well as the property of being structural itself (Sider 2011, 292). He takes these elements to be non-modal and offers a ‘Humean’ reduction of modality to a diverse set of non-fundamental phenomena (Sider 2011, ch.12), a reduction to which the present account obviously will not subscribe. But we might take Sider’s overall view of structure and yet disagree with him about the ideology required for fundamental physics: if the fundamental physical properties are dispositional or, in our terminology, potentialities, then modality is part of the structure of the world, and in just the way in which it ought to be. So will this revised Siderian picture of ‘the world’ serve as a catch-all solution to all our problems? Not quite. If the fundamental physics is part of the structure of the world, then the world has no potentialities to have a different physical outlook. In particular, if the properties of fundamental physics are part of the world’s structure, then the world will have no potentiality either to lack any of these properties or to contain any others. If physics is thus encoded into the world’s structure, then the world will presumably have no potentiality to have different physical laws or contain objects with altogether alien fundamental properties. Similar problems beset earlier approaches that have explicitly taken the world to be an object and investigated its structure without linking structure 7

This appears to be Kit Fine’s view: see Fine (2001).

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to fundamentality. Again, the problems are not immanent to the views as such, but are merely problems with their use in the present context as a catch-all solution to the objections which face the potentiality account. Bigelow et al. (1992) have proposed that we treat the world as ‘one of a kind’, that is, an object which belongs to a natural kind and thereby has certain essential properties. Their concern is to provide a grounding in natural kinds for laws of nature that seem to concern the very structure of the world: for instance, the conservation laws, the principles of relativity, and the symmetry principles (Bigelow et al. 1992, 371). Bigelow, Ellis, and Lierse themselves are open to its being possible that instead of the world that we inhabit, there might have been a world of a different natural kind; indeed, that is the only way for them to make sense of the possibility that the laws had been different. However, I am not concerned to ground laws of nature in the essential nature of the world, but to explain possibilities in terms of potentialities, and if certain laws of nature were essential to the world, then the world could have no potentiality to be without those laws, and there is nothing that might have a potentiality for there to be another kind of world altogether. Hence, if we understood the structural features of the world to be those laws that are essential to it (on this view), then potentialities of the world might take care of the possibilities of existence and of non-existence, as well as the possibility of certain necessary truths (e.g. that 2+2=4), but not of the possibility that there be radically different laws of nature. Joseph Almog has suggested a rather different view of the ‘structure of the world’: it is described by logic (Almog 1989). Structural facts or ‘pre-facts’, on Almog’s view, are those that are ‘permutation-resistant’: they are facts that would still remain facts if the objects or properties that constitute them were ‘permuted’ with any others. On this view, ‘Quine exists’ is not only a truth of logic, but a structural fact about the world (substitute any other object for Quine, and the fact remains a fact—the example is Almog’s, and the intended sense of ‘exists’ is timeless). Of course, Quine’s existence is contingent, and in fact Almog explicitly divorces logic from modality (Almog 1989, 203: ‘the falsehood of a proposition in a counterfactual situation is no hold against its truth in virtue of the structural traits of this, very actual, world’). Again, this view does not appear very promising for my present purposes: after all, I moved to the second sense of ‘the world’ precisely to find a sense in which the world might have existed without containing, or consisting of, the very objects that it does. The upshot is not that the world as a structured object cannot serve as a catchall solution. It is, rather, that a new conception of the world as a structured object would have to be formulated if it is to serve as such a solution; the extant conceptions will not do. I leave this task to the sympathizer of the catch-all solution.

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My own preferred reaction is to take the piecemeal approach, as I will do from section 7.5 onwards. Given the partial solutions that were offered by appeal to the world in the various senses discussed, we might even use the powerful world as part of a piecemeal solution in one place or another; but my focus will be on different responses. I started with the question whether the world has the objects that are in it as its parts. The positive answer, which takes the world to be a composite object (be it a mereological sum or a structured object), has been seen to be less than promising. We now turn to the negative answer and the idea that ‘the world’ is no more than a container for the things that are in it. What kind of thing is the world, conceived as a container? Substantivalists about spacetime might have an answer: the container is spacetime itself. Here is Carl Hoefer’s characterization of substantivalism: all that it [substantivalism] signifies is a belief that space (or space-time) is something real. To be slightly more precise, a modern-day substantivalist thinks that space-time is a kind of thing which can, in consistency with the laws of nature, exist independently of material things (ordinary matter, light, and so on) and which is properly described as having its own properties, over and above the properties of any material things that occupy parts of it. Hoefer 1996, 5

Spacetime, on this account, sounds like an excellent candidate for ‘the world’ as we need it for a catch-all solution: an object which has its own properties, and which is disjoint from and independent of the things that are in it. Spacetime, on this account, may very well have potentialities not to contain any of the things that it does contain, and to contain very different ones, perhaps with very different properties. However, the motivation for modern-day substantivalism again imposes restrictions that stand in the way of its being used in a catch-all solution. To cite Hoefer again, Why be a substantivalist? The answer has to do . . . with the dramatic successes of Einstein’s General Theory of Relativity (GTR) in accounting for gravitational phenomena. . . . GTR describes space-time in a way that allows it to exist and have determinate properties not reducible to the properties and relations of the material contents of space-time. For this reason, just about everyone who writes on GTR (philosopher or physicist) is, broadly speaking, a substantivalist. Hoefer 1996, 5f.

Spacetime conceived (and motivated) in this way is characterized by the attributes ascribed to it in GTR, and it is hard to see why and how it should have

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a potentiality to be otherwise characterized.8 If this is so, we are again threatened by a collapse of metaphysical into nomic possibility (at least insofar as GTR is concerned). Perhaps that is all right; but it makes the world in this sense less of a catch-all than we might have hoped for. Do we perhaps have a sense of the world-as-container that is more general still than the spacetime conception, and not committed to any essential structures stemming from physics? I do not know of such a sense, and I do not claim that it cannot be developed. But the friend of a catch-all solution has her work cut out for her: to develop an account of the world, as a composite object or a container, that is neither ad hoc nor too limited to serve as a solution. I do not know how to go about solving that task. Hence my own approach will be piecemeal (which, again, does not preclude a piecemeal appeal to the world in one sense or another). But the result of this section is not that the catch-all solution is bound to fail. The upshot is merely that it is not an easy solution.

7.4 Interlude: the powerful world and possible worlds In the previous section, I have discussed the ascription of potentialities to the world as a way of answering objections to the potentiality account. I have expressed some doubts on the feasibility of this answer but have not rejected it outright. In this section, I am going to pursue a sideline which would open up if the proposal of the previous section were accepted after all: an account of possible worlds as (a certain kind of) potentialities of the actual world. The proposal will be close to Stalnaker’s view of possible worlds. Stalnaker (1976), following a well-known argument in Lewis (1973b, 84), identifies possible worlds as ‘ways a world might be’ but interprets that identification differently from Lewis. Stalnaker argues that [t]he argument Lewis gives for . . . identifying possible worlds with ways things might have been, seems even to be incompatible with his explanation of possible worlds as more things of the same kind as I and all my surroundings. If possible worlds are ways things might have been, then the actual world ought to be the way things are rather than I and my surroundings. The way things are is a property or a state of the world, not the world 8 There is some debate over exactly how to characterize substantivalist spacetime: is it merely the manifold, as suggested in Earman and Norton (1987), or the manifold plus a metric, as Maudlin (1988) holds; or is it the metric field, as argued by Hoefer (1996)? However that debate is resolved, the general issue in the main text remains as it is. There is, further, the Spinozist view that spacetime itself is a substance and therefore not a container of things but itself the bearer of all properties (Schaffer 2009b). That view, with its denial of ordinary things as primary vis-à-vis the world, is too remote from the assumptions of this book to figure in a solution to our problems.

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itself. The statement that the world is the way it is is true in a sense, but not when read as an identity statement (compare: ‘the way the world is is the world’). This is important, since if properties can exist uninstantiated, then the way the world is could exist even if a world that is that way did not. One could accept [the] thesis . . . that there really are many ways things could have been . . . while denying that there exists anything else that is like the actual world. Stalnaker 2003, 28, first published as Stalnaker 1976

Stalnaker’s suggestion that possible worlds, rather than being concrete things like this world, should be identified with uninstantiated properties of this world, is clearly and explicitly non-reductionist. After all, those uninstantiated properties of the world that should be counted as possible worlds are those ways that the world might or could have been. Stalnaker is not interested in a reduction of modality: modal notions are basic notions, like truth and existence, which can be eliminated only at the cost of distorting them. One clarifies such notions, not by reducing them to something else, but by developing one’s theories in terms of them. Stalnaker 2003, 7

I agree with Stalnaker that modal notions are basic; but I stress that there are more modal notions than possibility, necessity and the counterfactual, and that developing our theories in terms of modal notions may involve imposing an explanatory hierarchy on them; and I claim further that the fundamental element in that hierarchy is potentiality. For Stalnaker, possible worlds are a certain class of possibly but not actually instantiated properties of the world; presumably we must think of the relevant properties as maximal in some suitable sense. For the potentiality theorist, possible worlds can be construed analogously as a certain class of unmanifested potentialities of the world; presumably we must think of the relevant potentialities’ manifestations as maximal in the same sense that would be suitable for Stalnaker. (Note, incidentally, that Stalnaker, too, would need a suitable conception of ‘the world’ to ensure that it is the right kind of object to possibly have those properties. So the view is no better off, in this respect at least, without appeal to potentialities.) If the Stalnakerian strategy succeeds, the potentiality account will be able to provide a conception of possible worlds to do the work that possible worlds do in formal semantics and logic. This would certainly be a benefit. But an objection to the Stalnakerian strategy is not far off either: would the potentiality account, if it allowed for and appealed to possible worlds in this sense, not simply collapse into Stalnaker’s own account? Why bother with the potentialities of individual objects if we could just ascribe the suitably maximal

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potentialities to the world itself and be done with it? In the remainder of this section, I will argue that even given a commitment to potentialities of the world and the possibility of identifying those potentialities with possible worlds, the potentiality account is still best construed as appealing to the potentialities of individual objects, rather than the potentialities of the world. This constitutes a substantial difference from Stalnaker’s account and prevents a simple collapse of my view into Stalnaker’s. The key difference between the two approaches lies in their respective explanatory ambitions regarding possibility. While Stalnaker takes possibility as a given and shows how to construct possible worlds with that given, the potentiality account aims to explain possibilities in terms of potentialities. Let me spell out this difference with respect to each of the two overall conceptions of ‘the world’. First, the world as a structured composite object (I omit the mereological conception, which we have seen to be of very little use for our purposes). If we follow the analogy with smaller composite objects, the potentialities of the whole, the world, are constituted at least in part by the potentialities of its parts—the things in the world. True, the whole might have had (has a potentiality to have) different parts; nonetheless, the parts that it does have contribute to its potentialities. Thus it is true that the world in this sense has a potentiality to be such that I am sitting. However, the world has that potentiality in virtue of my having the potentiality to be sitting, not vice versa. The world also has a potentiality to be in various total states, some of which include my sitting while others do not. Let S be such a total state which includes my sitting. (S is a ‘possible world’, if you like.) How is it that the world has a potentiality to be in S? That potentiality is constituted in part by the potentialities of the things in the world, such as my potentiality to be sitting, just as my potentiality to be sitting is in turn constituted in part by the potentialities of my legs and back, or my bones and muscles, or my molecules, etc. Of course, not all the potentialities of the world will be like this. If the world has a potentiality to contain nothing but unicorns, this will not be grounded in any particular thing’s potentiality (other than the world itself). Note here one disanalogy with the case of a complex object. I (a complex object) can have an (extrinsic) potentiality to have some particular other object(s) as my part(s). I have a potentiality to eat and digest this apple, making its molecules become part of me. That potentiality, like any potentiality concerning another object, is extrinsic and grounded in a joint potentiality that I possess together with the molecules of the apple. The world-as-a-whole does not have extrinsic potentialities to contain any particular objects other than the ones it does contain, for there are no other objects for it to have a joint potentiality with. It can only have potentialities to

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contain other objects of a certain kind, just as I can have a potentiality to contain other objects of a certain kind even if nothing of that kind exists at present, e.g. an artificial replica of my heart. Second: the world as a container. You, I, and the apple in front of me are not parts of the world in this sense; we are merely in it. The world has the property of containing me, but that property is not, as it was for the previous sense, an intrinsic property (as my property of having this particular heart is intrinsic). It is an extrinsic property grounded in a relation between two distinct objects: the world, on the one hand, and me, on the other. The ‘total states of the world’ or ways things as a whole could have been are, in this sense, to be construed slightly differently. There is the way things could have been—a total state of things in the world, call it S; and there is the state of the world which is, strictly speaking, not S but the property of containing all and only things that are in S. The world in this sense has a potentiality to be such that I am sitting, and a potentiality for S to be the total state of the things in it, where S is a particular state that includes my sitting. Now, if the world has a potentiality (as it should) to contain me sitting, then that potentiality is an extrinsic one: it involves an object disjoint from the world itself, me. It is a potentiality that is grounded in a joint potentiality which I have with the world: to stand in the contains-sitting relation. On its own, the world as a container could have no such potentiality. Its intrinsic potentialities can not only involve no particular non-existent object (as with the world as a composite object), but also no particular object existing in it. It can have an intrinsic potentiality to contain unicorns, or to contain unicorns and nothing else, or to contain someone who is sitting and is F (where F is a complete list of my non-haecceitistic properties); but it can have no intrinsic potentiality to contain me. Whichever of the two suggested senses of ‘the world’ we adopt, then, talk of individual objects’ potentialities will not be idle. In both cases, the potentialities of the individual objects in the world will provide the metaphysical grounds for an important class of the potentialities of the world: those potentialities that concern any particular object at all. Unlike Stalnaker, I am looking for a substantial metaphysical account of where in the world modality is to be found. On the present version of the view I am proposing, modality is to be found in the potentialities of the world itself, and to the extent that we accept Stalnaker’s identification of possible worlds, it is to be found in possible worlds. But that is not, metaphysically, the most enlightening thing to be said about it. For the source of much of the modality to be found in potentialities of the world is the potentialities of the particular objects in the world. That is why a potentiality-based account of modality, even if it accepts potentialities of the world (conceived as a composite object or as

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a container), should nonetheless appeal to the potentialities of particular objects in the world wherever that is an option. So much for the excursion. We will now return to the objections outlined in section 7.1 and go through them one by one.

7.5 Possibilities of existence I am an only child, but that is clearly a contingent fact: my parents might have had children other than me. Our solar system has eight planets, but that too is contingent: there might have been a ninth planet in the system. Many, though not all (see Kripke 1980), think that there might have been unicorns, or the monster of Loch Ness, or even talking donkeys. All of these are unactualized possibilities of existence. We have seen an account for some such possibilities in chapter 6. Sometimes it is possible that something is F because there is something which is possibly F. Thus it is possible that there be a woman President of the US because, for instance, Hillary Clinton is possibly a woman President of the US. For the present account, those are the easiest cases: it is possible that there be a woman President of the US because Hillary Clinton (among others) has the potentiality to be a woman President of the US, and thereby a potentiality to be such that something is a woman President of the US. The possibilities that I have mentioned in the previous paragraph do not look the same way: nothing is possibly (or has a potentiality to be) a sibling of mine, or a unicorn, or the monster of Loch Ness. At any rate, let us assume for the sake of the argument that this is so. If the examples are not quite right, there will be others. For adherents of the Barcan formula, there are no such examples. The Barcan formula says that whenever it is possible that something is F, then something is possibly F; and that something will then be, for the potentiality-based account, the bearer of the relevant potentiality. However, subscribers to the Barcan formula as it stands must either be committed to the existence of mere possibilia, a view briefly discussed in section 7.19 —my possible younger sister, possible unicorns, and so on—or deny many unactualized possibilities of existence. Both options are in principle available to the potentiality-based account, and would cut short the discussion of this chapter. However, I prefer to take the more 9 Tim Williamson has made this a live option (see Williamson 1999, 1998, 2002, and 2010). Note, incidentally, that the potentiality account does validate the much less controversial Converse Barcan Formula: If something is possibly F, then it is possible that something is F. For something to be possibly F, on the present account, is just for something to have an iterated potentiality (for itself) to be F; the same thing will thereby automatically have an iterated potentiality for something to be F.

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conservative path: to accept that there are unactualized de dicto possibilities of existence in the absence of a corresponding de re possibility. In what follows, I restrict myself to precisely those possibilities when I say ‘possibilities of existence’. What, then, are the bearers of the relevant potentialities for such possibilities of existence? In many cases, it is clearly the objects that have a potentiality to produce an object of the relevant kind: my parents, who had the potentiality to have another child, and perhaps evolutionary ancestors of present-day animals, if they had a potentiality to develop so as to yield unicorns, Loch Ness monsters, or talking donkeys. There is no guarantee that they did have such potentialities; once again, we are subject to empirical fortune in the claims that we can make about modality. This is a feature of the present approach, and I take it to be an advantage rather than a drawback. But potentialities for production do not exhaust the possibilities of existence, even on the potentiality account. Objects also have potentialities to constitute other objects. Take two knives, k1 and k2 , each consisting of a handle and a blade.10 It is possible that there should have been a knife made of k1 ’s handle and k2 ’s blade. This is so not only because there are craftsmen able to put such a knife together, but also, and perhaps more basically, because k1 ’s handle and k2 ’s blade have a joint potentiality to constitute a knife together. Similarly, it may be possible for there to be unicorns even if nothing in evolutionary history had the right potentialities for evolutionary development, as long as there are particles that have the joint potentiality to constitute a unicorn. (Whether this is so depends on the criteria for unicorn-hood. If we require that an object be a member of a species that has developed by evolution, then the particles would perhaps not have the potentiality to constitute a unicorn, but only to constitute a unicorn-like object.) The strategy is obviously applicable, in an equally fallible way, to many other cases, from talking donkeys to additional stars and planets. Yet even this strategy is not universally applicable, or so the objection will go. (We have now, finally, reached the core of the objection.) It does not apply to what have come to be called, in the wake of Lewis’s treatment (Lewis 1986a), ‘aliens’. Lewis discusses the possibility of ‘extra natural properties, alien to actuality’ (Lewis 1986a, 159). Lewis’s own discussion is aimed at linguistic ersatzers, whose theory can countenance the possibility of such properties, but is unable to distinguish between them and thus ‘conflate[s] possibilities that differ only in 10

The example is Williamson’s (1999), though he is putting it to a different use.

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respect of their alien natural properties’ (Lewis 1986a, 159). The charge against the potentiality account is different: it is that the account cannot countenance the possibility of any genuinely alien properties. Note that, on Lewis’s conception, the possibility of alien properties is eo ipso the possibility of alien objects: a property, for Lewis, exists at a world only if it is instantiated at that world. The possibility of alien properties, therefore, is the possibility of there being objects that have properties and belong to kinds which are entirely alien to actuality. Can I accommodate such possibilities? Yes, but within limits. If an alien natural property is simply a natural property that is never actually instantiated, then there is at least the epistemic possibility that some things—the matter at the beginning of the universe, if nothing else—have or had the potentiality to produce or constitute objects with some such properties. Whether that is so is, again, a matter to be settled, if at all, by physics and not philosophy; but nothing in the present account excludes the possibility that such alien properties be instantiated. The opponent will not be pacified, of course. Is it not possible, she will ask, that there be objects with (natural) properties that no actual thing ever had a potentiality to have, to produce, or to constitute? Call such properties super-alien properties. Is it possible for there to be objects with super-alien properties? And, if so, how is the potentiality account to explain this possibility? It is here that I believe I have to bite the bullet. I do not know of any way for an object to have a potentiality to be such that something is F, other than the ones I have just mentioned. (And if there is another way, it too can be simply excluded by a revised definition of super-alien properties.) There is, then, no possibility of super-alien properties being instantiated. But it will be useful, first, to see how we might try to accommodate such a possibility; and, second, to spell out the general picture that comes with bullet-biting at this juncture. Let being F be a super-alien property. By definition, we will not be able to point to any objects with a potentiality to be F, to produce an F, or to constitute an F. If the potentiality theorist wanted to accommodate the possibility of there being Fs, she would have to come up with another way of construing an object’s potentialities to be such that there are Fs. Here are two options. Suppose we believed in universals, and thought of them along roughly Platonist lines. Such universals might then count as objects in a strong enough sense to qualify as bearers of potentialities themselves; and the universal F-hood might exist (pace Lewis) despite being never instantiated, and even if nothing ever has a potentiality to instantiate, produce something which instantiates, or constitute something which instantiates it. We should then be able to say that, although nothing else has a potentiality to instantiate the universal F-hood, the universal

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itself has a potentiality to be instantiated. Indeed, that potentiality is arguably what makes it a universal (rather than some other kind of abstract object). Hence it would be the universal, F-hood, itself that grounds the possibility of there being Fs. Suppose, alternatively, that we believed in propositions, and thought of them along roughly Platonist lines. Such propositions might then count as objects in a strong enough sense to qualify as bearers of potentialities themselves (they might, for instance, have the potentiality to be thought about by me on 1 January 2020). The proposition that there are Fs will exist even if it is false. We should then be able to say that the proposition that there are Fs, despite being false, has a potentiality to be true and thereby grounds the possibility that there are Fs.11 At least for present purposes, I do not want to adopt either solution to the problem (if it is a problem) of super-alien properties. The picture of metaphysical modality that I am offering derives its attraction, I believe, not merely from the fact that it locates modality in properties, viz. potentialities, but also from the fact that it thereby locates modality in objects of the ordinary kind: concrete objects, as distinguished from their properties or the propositions in which they occur. I will argue shortly (in section 7.7) that, unless we are nominalists, we can and should indeed ascribe potentialities to abstract objects. But we should not use those abstract objects as another catch-all solution. Where possibilities concern concrete objects, even if only de dicto, appeal should be made first and foremost to the potentialities of concrete objects. If the properties and propositions just described have the requisite potentialities, those should derive from potentialities of concrete objects. Hence my preference for biting the bullet when it comes to super-alien properties. And it is not too hard to bite the bullet here, since the bullet is part of an attractive picture of properties that is much more in line with the approach taken in this book than the appeal to Platonist universals or propositions as potentiality-bearers. The Platonist approach, I have said, goes against the motivation of this book: the idea that modality is ultimately a matter of how objects, in the ordinary, concrete sense, are. I have been assuming, however, that there are properties, in the sense of universals that can be multiply instantiated. What is the status of such universals, and how do they relate to the objects which, on the picture I am proposing, are the bedrock of reality? 11 Or suppose, third, that we believed in ‘the world’ as an object and a bearer of potentialities, along the lines of Bigelow et al. (1992), or perhaps in the ‘container’ sense. Then we may appeal to the world, in this sense, as having potentialities to contain objects of super-alien kinds. I am not sure exactly how to evaluate the claim that the world, in either of these two senses, does have such potentialities, so I am not going to discuss this option in any more detail.

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It will not be surprising that my preferred view of properties is what is often labelled the ‘Aristotelian’ view. I hold that there are properties, but that they are in rebus: their existence derives from how things are. The opposed, Platonist, picture has it that properties exist independently and that objects are as they are in virtue of partaking in, or instantiating, the relevant universals. This is not the place to go into the distinction in much detail, but I will spell out at least briefly what the Aristotelian view amounts to for a potentiality theorist. Perhaps the best known contemporary defender of an Aristotelian approach to universals is David Armstrong. Armstrong’s Aristotelianism is condensed in his ‘Principle of Instantiation’, which demand[s] that every universal be instantiated . . . for each property universal [it must] be the case that it is a property of some particular[, and f]or each relation universal [it must] be the case that there are particulars between which the relation holds. Armstrong 1989b, 75

According to the Principle of Instantiation, there are no uninstantiated properties. The Principle of Instantiation, so understood, might seem to spell trouble for the strategy that I have outlined earlier in this section. If properties are universals, and there are no uninstantiated universals, then there are no uninstantiated properties.12 If there are no uninstantiated properties, then there can be no potentialities to have (or to produce or constitute something which has) an uninstantiated property. The potentiality to be F, quite generally, exists and can be instantiated only if there is a property of being F. For the potentiality is individuated by (or, metaphorically, it is built up from) its manifestation property. If there is no property to function as its manifestation, then the potentiality never gets to be the potentiality that it is supposed to be. But if there are no potentialities for uninstantiated properties, then we cannot account for the possibility of other properties being instantiated by ascribing to actual objects the potentiality to have (or to produce or constitute something which has) those properties. And indeed, the Principle of Instantation as it stands is ill-motivated for the Aristotelian about properties who also believes in potentialities. For the believer in potentialities, instantiation is not the only interesting relation in which an object can stand to a property. Potential instantiation is another, as is potential 12 Armstrong, strictly speaking, is not committed to all properties’ being universals: he only stipulates universals for the perfectly natural properties, and constructs the rest from them. I have been taking a more liberal approach in this book, but the main text could easily be formulated to accommodate Armstrong’s more discriminating stance.

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production or constitution of something which instantiates the property. The basic idea that the existence of universals derives from how things are can take a more liberal form on the present approach. Instead of a Principle of Instantiation, we should subscribe to a Principle of Potential Instantiation, as follows: PPI Every universal must be at least potentially instantiated: there is a property universal of being F only if there is some particular thing which is F, is potentially F, or is potentially such that something is F; there is a relation universal of R-ing only if there are some particular things which R, or which potentially R, or which are potentially such that some things R. PPI, like Armstrong’s original Principle of Instantiation, is formulated as a necessary rather than a sufficient condition on property existence, but once it is accepted as such, I see no problems with accepting it as a sufficient condition as well, given a general commitment to the existence of universals. Accepting PPI, and according it the status of a necessary and sufficient condition for the existence of universals, solves the problem with the original Principle of Instantiation: the claim that something has a potentiality to have (or produce or constitute something which has) the actually uninstantiated property of being F is not in jeopardy because there might be no property of being F. Rather, that claim, if true, guarantees that there is such a property, because this is precisely what it takes for there to be a property of being F. While being more liberal than Armstrong’s approach, the Principle of Potential Instantiation is more restrictive than Lewis’s. It provides a coherent and well-motivated framework for denying the possibility of super-alien properties being instantiated, because it denies that there are such properties in the first place. Lewis subscribes to the Principle of Instantiation as a necessary condition on the existence of universals at a world (see Lewis 1986a, 161), but since for him there are (simpliciter) more things than there are actual things, there can also be (simpliciter) more properties than are actually instantiated. Moreover, there is no reason for the Lewisian to believe that all the properties instantiated in other possible worlds are within the reach of the actual objects’ potentialities; so there may be, for the Lewisian, super-alien properties (though they are instantiated in other worlds than ours). With the Principle of Potential Instantiation, we have tied property existence more tightly to the things that exist in actuality; which is to say (since we are actualists), we have tied them more tightly to the things that there are simpliciter. In so doing, we have motivated the restriction on metaphysical possibilities of existence to which, on the version that I am developing, the potentiality view is committed. For the Principle of Potential Instantiation tells

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us that, if nothing has a potentiality to be such that something is F (i.e. a potentiality to be F, to produce an F, or to constitute an F), there is no property of being F. And if there is (simpliciter) no such property, then it becomes difficult to make sense of the possibility that such a property be instantiated: that there be Fs. The framework that I have offered is one that goes nicely with, but is not required by, the potentiality-based approach to modality. If the potentiality approach were spelled out in either a more nominalistic spirit, thus doing away with PPI as a sufficient condition, or else in a more Platonistic spirit, thus doing away with PPI as a necessary condition for property existence, then the response to the challenge of super-alien properties would have to be reconsidered. But the picture that I have proposed is, I have argued, an independently attractive one and thus a well-motivated framework for denying the possibility that there be things with super-alien properties.

7.6 Possibilities of non-existence In the previous section, we discussed possibilities that such-and-such objects exist, where it is not the case that such-and-such objects do exist. In this section, we will look at the opposite kind of case: possibilities that so-and-so do not exist, when so-and-so do in fact exist. We will find that, at least given certain (tentative) assumptions about potentiality’s relation to time, there is a principled restriction, on the potentiality account, to possibilities of non-existence. This will not be palatable to all critics of the account, but I will suggest that the possibilities thereby excluded are too far-fetched for us to have reliable intuitions to the contrary. Cameron (2008) has discussed two close relatives of the present account, Borghini and Williams’s (2008) and Pruss’s (2002) dispositionalist accounts of modality, and objected that they cannot make sense of certain possibilities of non-existence, or at any rate not without costs. The criticism applies, with only a slight change in terminology, to the present view. Here it is: Intuitively, I am a contingent being—I might not have existed. What, for the Aristotelian, grounds this possibility? Presumably, it is my parents; for just as it was within their power to beget me, it was also within their power not to, and had they exercised the latter power I would not have existed. And the truthmaker for the truth that my parents might not have existed is, in turn, their parents. But what about the highly intuitive possibility that none of the actual contingently existing substances existed—what is the truthmaker for the truth that this situation is possible? It can’t be any of the actual contingently existing beings, for none of these beings has the capacity to bring it about that it itself never existed. Cameron 2008, 273

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As Cameron rightly grants, most of the more local possibilities of non-existence can be accounted for by the potentiality account. The strategy is parallel to that which we employed in the previous section to account for possibilities of existence: a contingent object will generally have developed in some way out of other things or will be constituted by some other things. Those are the things that would have had potentialities for the object to exist if it did not in fact exist; those are also the things which, at least prior to the development of the object in question, had a potentiality for things to develop differently. Note that we are not here concerned with the possibility of something’s ceasing to exist. Contingent objects generally have the potentiality to cease to exist: people have a potentiality to die, stars have a potentiality to implode, and so forth. A potentiality to cease to exist is not a potentiality to fail to exist: no object has a potentiality not to exist, because the object can never be there to exercise such a potentiality. Nor are we concerned with the possibility of something’s no longer existing. While no object, for the reason just given, has a potentiality to no longer exist, other objects may have a potentiality for it to no longer exist: any potentiality to destroy or to survive a given object will serve as an example. But we are concerned here with the possibility of an object’s not existing, where the phrase in italics is read timelessly. Once an object x has come into existence, it seems, it is too late for anything to have a potentiality for x’s non-existence. Hence we must appeal to the potentialities that other objects, such as x’s producers or constituents, had before x came to be. The possibilities with which we are concerned, then, are possibilities that soand-so never existed, possibilities for non-existence at all times. In what follows, I will discuss those possibilities exclusively and reserve the term ‘possibilities of non-existence’ for them. Some of those possibilities, as we have seen, can be accounted for. But those do not amount to the global possibility which Cameron raises: the possibility that none of the actual contingently existing substances (i.e. objects) had existed. That this global possibility indeed holds is, as Cameron says, ‘highly intuitive’. After all, for each contingently existing object, it is possible that it, this object, did not exist—that is simply what it is for an object to be a contingent existent. Now, for any contingently existing objects x and y, if it is possible that x does not exist and it is possible that y does not exist, it should be possible that neither x nor y exist. After all, how should the non-existence of one contingent object force another into existence? But if we can draw this conclusion for x and y, we should be able to draw it for x, y, and z; and for x, y, z, and w; and so forth. In general, if x1 , . . . , xn are contingently existing objects, then for each of them it is possible that it does not exist; and if that is so, then it should be possible that none of x1 , . . . , xn exist.

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Hence, since for each individual contingent object it is indeed possible that it does not exist, it must be possible that none of these contingent objects should exist.13 So the possibility, for each contingent object, that it does not exist, together with what we might call a principle of independence—that the non-existence of contingent objects can never force other contingent object into existence—yields the global possibility that none of the actual contingent objects exist. (As Cameron points out, this is not to be confused with the more controversial possibility that no contingent objects exist. If none of the actual contingent objects existed, other contingent objects might take their place.) I will not challenge the argument or its premises in what follows. Rather, I will show that I can accept the conclusion. It is easily seen that a crucial assumption in setting up the problem for the potentiality theory is the temporal asymmetry of potentiality: once x exists, I said above, it is too late for anything to have a potentiality for x never to have existed. It is indeed ‘too late’ if the triviality thesis from chapter 5.8 is true. This was the thesis that all past-concerning potentialities are trivial, or, more precisely: where being F is a property whose possession would have to be grounded in the possession of properties in the past, anything has a potentiality to be F if and only if it actually is F. Once an object has come into existence, everything is such that the object has come into existence, and nothing is such that it has not; hence nothing has a potentiality for the object not to have come into existence. I have adduced some considerations, in chapter 5.8, to question the triviality thesis, appealing to the possibility of time travel or the nature of time itself. If these considerations are accepted, then the problem would not arise quite so easily: at least the opponent would have to provide further argument to exclude these particular past-concerning potentialities. However, I will choose a different route in this section. Let us accept, for the time being, that the triviality thesis holds, and hence that potentiality is temporally asymmetric in the way required by the setup of the problem. Nevertheless, I shall argue, we can accept the conclusion that it is possible for none of the actual contingent objects to have existed. Let x1 , . . . , xn be all the actual contingently existing objects. Then it is possible that none of x1 , . . . , xn existed. Something, therefore, must have a potentiality to be such that none of x1 , . . . , xn existed; but that something cannot itself be any of x1 , . . . , xn . Since x1 , . . . , xn are all the contingently existing objects that there are, the bearer of the relevant potentiality cannot itself be a contingently existing object. Therefore it must be a necessary existent. 13 Contessa (2009) sketches a very similar argument and provides an alternative strategy in responding to it. I want to thank Lorenzo Azzano for pressing me on this objection.

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Which necessary existents might have the potentiality that we are looking for, to be such that none of x1 , . . . , xn exist? As in the previous section, a Platonist might appeal to abstract entities (paradigmatically necessary existents) as bearers of potentialities. Perhaps the universal property of being identical to one of x1 , . . . , xn has a potentiality to be uninstantiated. Or perhaps the proposition that none of x1 , . . . , xn exist has the potentiality to be true.14 But as in the previous section, I prefer not to take that route. My project is to anchor metaphysical possibility in objects of the ordinary, concrete kind. But then where are we to find the required necessary existents? The solution (to this problem, at least) lies in the very assumption of potentiality’s asymmetry with respect to time. I have used this assumption to explain, partially and hypothetically, the necessity of an object’s origin in chapter 6: an object’s potentiality to have a different origin would have to be a non-trivial past-concerning potentiality and there are no such potentialities. A similar observation then applies to the temporal beginning of the universe. Nothing has or ever had a potentiality for the beginning of time to be different than it was: for there was never a time at which such a potentiality might have been possessed. Hence, whatever entities existed at the beginning of the universe are, on this view, necessary existents: nothing has or ever had a potentiality, iterated or otherwise, for them not to have existed. You may find this outcome worrisome in and of itself. It does, however, solve the problem at hand: providing a bearer for the potentiality to be such that none of x1 , . . . , xn ever existed. Whatever the entities at the beginning of time were, they had potentialities to develop in the way they did, yielding the universe as we know it; and presumably they had potentialities to develop differently, yielding a different kind of universe. Moreover, those entities are ex hypothesi not among the contingent existents x1 , . . . , xn . Thus, presumably, they had iterated potentialities for any or all of x1 , . . . , xn to exist and potentialities for any or all of x1 , . . . , xn not to exist. The solution to our problem comes at a price. Not only must we accept that the very first entities in the history of the universe are necessary existents. We also get a restricted range of possibilities of non-existence. It is possible that none of the actual contingent objects had ever existed. But it is not possible that there had always been different (concrete!) entities, from the beginning of time. 14 And again, we may try appeal to ‘the world’: perhaps the world in the spacetime-substantivalist sense has or had a potentiality not to contain any of the objects that it does contain, even at the first moment of its existence; and perhaps the world in that sense is not itself a contingent existent. Again, I am not entirely sure how to evaluate that claim, so I am not going to appeal to it in what follows.

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Is the price worth paying? Never mind, for present purposes: the price has already been paid, if we accept the triviality thesis. The necessary existence of the very first entities in time is not an assumption that we made in order to meet Cameron’s challenge, and which has to prove its worth by its explanatory power. It is a consequence of the present theory of possibility, together with the claim that past-concerning potentialities are always trivial (a claim which we have found to be useful in explaining the necessity of origin). The real problem, if there is one, is not the possible non-existence of all the actually existing contingent objects. It is the assumption of a temporal asymmetry of potentiality, and thereby, on the present account, of possibility, with respect to time. We have seen in chapter 5.8 that this assumption is not obligatory; if it is dropped, then a different solution to Cameron’s problem becomes available. But for now, let me only add a quick comment on the solution that I have given here. Does the consequence that the entities at the beginning of time are necessary existents constitute a reductio ad absurdum of the potentiality account as conjoined with the triviality thesis? I do not think so. The possible non-existence of the very first entities in the universe, plausible as it may seem at first look, is a somewhat marginal intuition and not one to which we have much opportunity to appeal in real life. It is not strong enough or central enough to carry the weight of a reductio ad absurdum. Moreover, at the basis of my denying that intuition lies a principle which many philosophers have implicitly found appealing: the idea that metaphysical possibility is not insensitive to the direction of time, as is seen in the widely accepted necessity of an object’s origin. Nevertheless, the task for any aspiring potentiality theorist unwilling to pay the price of temporal asymmetry is clear: it is to provide a reasoned alternative to the triviality thesis. I conclude, for the reasons given, that possibilities of non-existence do not in themselves present a serious problem for the potentiality account. However, one reason why they do not is related to what might be considered a worse problem— the relation between potentiality and time, to which I will return in section 7.9.

7.7 Abstract objects Some metaphysical possibilities concern abstract objects. It is necessary that 2+2=4; necessity entails possibility; hence it is possible that 2+2=4.15 On the 15 That necessity entails possibility is a feature of system T and I have argued in chapter 6 that this system is validated by the potentiality account of possibility. It is necessary that p iff it is impossible that not p. By axiom (T), actuality entails possibility; by contraposition, impossibility implies nonactuality. Hence if it is impossible that not p, it follows that not not p, i.e. that p. Applying (T) again without the contraposition, we get that it is possible that p.

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potentiality account, then, something must have an iterated potentiality for it to be the case that 2+2=4. But what might a relevant witness be? The numbers 2 and 4? Those are certainly not the most plausible candidates to be bearers of potentialities, if they are to count as objects at all. How the potentiality account deals with such cases depends largely on matters external to the account itself—namely, on the view that is taken of the ontological status of abstract objects. According to Platonists (in the widest sense of the term), there are abstract objects. I will argue below that Platonists should indeed ascribe potentialities to abstract objects, thus answering the questions of the previous paragraph in the affirmative. According to nominalists, there are no abstract objects. I will argue that nominalists, insofar as they take statements that are apparently about abstract objects to be literally true (since, on their construction, such statements are only apparently, and not in fact, about abstract objects), automatically provide alternative bearers for the relevant potentialities. I will not, however, consider nominalist views that do not take at least some statements apparently about abstract objects to be literally and nonvacuously true. It is not entirely clear whether the statements at issue would even qualify as metaphysically necessary on such a view, and discussing this would take us too far here. Also, for the sake of simplicity I will speak rather sweepingly of abstract objects, covering numbers, sets, universals, propositions, and whatever other abstract objects might be held to exist. It is, of course, perfectly possible to adopt different stances towards these different types of putative objects. Quine, for instance, was known to accept sets but reject universals. However, since my concern is to outline general strategies, I will speak of abstract objects quite generally. Different strategies may be applied to different kinds of objects. Let us start with Platonism. In its simplest form, Platonism claims that abstract objects exist as sui generis entities without any dependence on the realm of concrete objects. Platonism is compatible, however, with certain reductionist or otherwise explanatory projects within the realm of the abstract. It is compatible, for instance, with the identification of numbers or universals with sets (problematic as both may be for well-known, independent reasons; see the classics Benacerraf 1983 and Quine 1969). It is also compatible with a certain brand of structuralism, such as that developed in Shapiro 1997, where numbers as objects are abandoned in favour of pure structure, but that structure is itself an abstract object which exists independently of anything instantiating it. Platonism in the broad sense in which I am here using the term is even compatible with theories that construct abstract entities out of concrete ones, for instance via abstraction principles (as in neo-Fregeanism; see Wright 1983 and Hale and

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Wright 2001). For such views, there are abstract objects such as numbers (and, on related views of them, universals, sets, and so forth). Those objects may be in some sense derivative from concrete objects—to use the terminology of this book instead of that of the neo-Fregeans, abstract objects and facts about them may be (at least partially) grounded in concrete objects and facts about them— but that is no obstacle to their existence. Similarly, Platonism in this broad sense includes the Aristotelian view of properties sketched in section 7.5 (where I contrasted it with Platonism more narrowly construed), according to which the existence of universals is a matter of, or is grounded in, how concrete objects are. Again, their status as grounded is no obstacle to the existence of such abstract objects. Now, if any version of Platonism is combined with a potentiality-based metaphysics, the best thing to say is that abstract objects, just like concrete ones, have potentialities. In sections 7.5–7.6, I have rejected appeal to universals as bearers of potentialities. But there we were still concerned with possibilities that ultimately concerned concrete objects, even if only de dicto. In the cases now at hand, the possibilities directly concern the abstract objects themselves. Ascribing potentialities to them, and appealing to those in explaining the possibilities at issue, is a better option here than it was there. But do abstract objects have potentialities? We have already seen the notion of potentiality stretched far beyond the initial examples, dispositions, and abilities. I have argued (in chapter 5.7.4) that for an object to be some way or other is sufficient for that object to (simultaneously) possess the potentiality to be that way. When, outside the philosophy room, we ascribe potentialities to objects, we usually ascribe those whose manifestation would constitute a change in the object. There is a good reason for that: if exercising the potentiality would not constitute a change in the object, we can say something stronger and more relevant by ascribing to it not the potentiality, but the manifestation property itself. That is no reason to deny to objects the potentialities that are being manifested, even if they are being manifested throughout an object’s existence. We have also seen (in chapter 3.5) that potentialities come in a spectrum of degrees, the maximum of which consists in the lack of a potentiality for the opposite manifestation. Hence if I have no potentiality, say, not to be human, I thereby have a potentiality (of the maximal degree) to be human. Again, such potentialities, like the corresponding possibilities, are rarely ascribed outside the philosophy room, for we have stronger and more informative things to say in such cases. In denying potentialities to abstract objects, our motivation would most likely be either that they do not change, or that they do not have any (intrinsic)

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contingent features, hence no potentialities to be other than they are. Given what I just said, these two observations point to a rather different conclusion: not that we should not ascribe potentialities to abstract objects, but rather that we have an explanation for why we do not usually ascribe potentialities to them: we have stronger and more interesting things to say about them. Where an abstract object is grounded (partially) in concrete objects, as for instance the singleton set of Socrates is grounded in Socrates, then some of its potentialities will derive from the potentialities of the concrete object(s) in question. The singleton’s potentiality to contain a carpenter, for instance, is grounded in Socrates’s potentiality to be a carpenter; and, on the Aristotelian view of properties, the potentiality of a given universal to be instantiated is grounded in the iterated potentialities of concrete objects for it to be instantiated. So the potentialities of abstract objects need not be mysteriously sui generis, and in some cases the appeal to their potentialities may or should be translated into an appeal to the potentialities of the concrete objects in which they are grounded. But even so, abstract objects will plausibly have some potentialities that are sui generis: the potentiality to be a set, for instance. In those cases, there is nothing for us to do but appeal to the potentialities of abstracta. I conclude that if there are abstract objects, the potentiality-based account of possibility will fare best by ascribing to them potentialities, and that our initial resistance to such ascriptions should not mislead us into thinking that they express falsehoods. There is a good alternative explanation for that resistance: propositions ascribing potentialities to abstract objects are not false, but whenever we are in a position to assert them, we are typically also in a position to make a more informative statement, and we have a well-known resistance to making the less informative statement. For nominalists, there are no abstract objects. A fortiori, there are no potentialities possessed by abstract objects. The potentialities that verify a possibility claim such as ‘possibly, 2+2=4’ must be located elsewhere. I have said earlier that I focus on nominalist theories according to which at least some statements that are apparently about abstract objects, such as ‘2+2=4’, are non-vacuously true. Thus I exclude certain kinds of fictionalism, on which (say) mathematical statements are true only ‘according to fiction’. (Again, this kind of stance may be adopted locally: a nominalist might accept statements of mathematics as true but reject statements about universals as false.) If so, then such statements cannot really be about abstract objects. Rather, they must be about unobjectionably concrete portions of reality. They might be about particular concrete objects that play the role which we thought was played by abstracta (thus concrete inscriptions or perfectly resembling concrete objects

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may play the role of universals; and according to Field (1980), regions of space can occupy the role of numbers at least for the purpose of Newtonian physics). Or the statements in question might be about concrete objects in a more indirect way: by quantifying over them. For the resemblance nominalist, for instance, true statements apparently about universals, such as ‘Orange is brighter than Red’, are really about resemblance relations between concrete objects, in this case, orange and red objects. And on a nominalist version of structuralism about mathematics, statements of mathematics are really about concrete objects, but they are not about any particular concrete objects such as regions of space. Rather, they are about any objects that instantiate a certain structure. Unlike Shapiro’s Platonist structuralism, nominalist structuralism allows into its ontology only the concrete items in a structure, rather than the abstract structure itself. Each of these views can, again, be combined with a potentiality-based metaphysics. The question where to locate the potentialities that verify possibility statements such as ‘possibly, 2+2=4’ then has a simple answer: the bearers of the potentialities are precisely those unobjectionably concrete objects which are to provide for the truth of the apparently abstract statements quite generally. If certain regions of space, for instance, are constituted so as to verify ‘2+2=4’, then (by ACTUALITY) those very regions will have a potentiality to be just so, and hence a potentiality for it to be the case that 2+2=4. In general, if what appeared to be facts about abstract objects are really facts about concrete objects, then the possibly true propositions that are apparently about abstract objects should be understood as possibly true propositions that really concern concrete objects. The potentialities that account for the possibility of those propositions will then be located in some or all of those concrete objects; there is no special problem of the possibility of propositions about abstracta, since there are (on that view) no abstracta. Whether we are Platonists or nominalists about (a given kind of) abstract objects, then, possibilities that concern abstracta are not threatening to the potentiality account of possibility.

7.8 Nomic possibility and metaphysical possibility The problem we are about to discuss can be put as the tension between two widely accepted views. The first is that the metaphysically possible outruns the nomically possible: that it is metaphysically possible for (some or all of) the actual laws of nature not to hold. The second is that dispositions and abilities are constrained by what is nomically possible: that nothing has a disposition or an ability to violate

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the laws of nature.16 It is natural, at least prima facie, to generalize the second view as a view about potentiality: nothing has a potentiality to violate any actual law of nature. If the generalized view applies to iterated potentialities too, then together with POSSIBILITY it is clearly incompatible with the first claim: that at least some laws of nature are metaphysically contingent. If we adhere to POSSIBILITY, there are only two ways in which the tension can be resolved: by rejecting the first claim, that the laws are metaphysically contingent, or by rejecting the second, that nothing has an (even iterated) potentiality for an actual law of nature not to hold. The natural move for the potentiality view, it would seem, is the first. After all, the potentiality view is a close ally of dispositional essentialism, and dispositional essentialism is known to reject the metaphysical contingency of the laws of nature (see Bird 2007). To join forces with dispositional essentialists would be to bite the bullet of a putative counterexample: to say that, when philosophers claim that the laws of nature are contingent, they are simply mistaken. But surely there are worse bullets to bite. Ellis (2001) and Bird (2007), and before them Shoemaker (1980) and Swoyer (1982), have argued that the laws of nature are metaphysically necessary, or in other words, that metaphysical possibility entails nomic possibility. The appearance, at least to philosophers, to the contrary may be put down to a confusion between conceivability and metaphysical possibility, and the potentiality account has good reason to drive a wedge between these two. (Note, however, that even on the dispositional essentialist view the world might have been governed by different laws, as long as those laws would have involved different properties. What is excluded is the metaphysical possibility of different laws involving the same properties as our actual laws; the possibility of different laws becomes a matter of the possibility of different properties, and the problem of this section turns into the problem of section 7.5.) However, at closer look it is not at all clear that the potentiality account must, or indeed should, bite that bullet and reject the view that the laws are metaphysically contingent. We can see why this is so if we have a closer look at the main views about the metaphysics of laws, including the above-mentioned thesis of dispositional essentialism. 16 With regard to dispositions, the second view is commonly found as an aside. Langton and Lewis (1998), for instance, adapt their definition of intrinsicality so that dispositions can be intrinsic despite their apparent dependence on the laws. Armstrong (1996) takes the laws of nature to be among the truthmakers of a disposition ascription, while dispositional essentialists such as Bird (2007) take dispositions to be the metaphysical basis of the laws. With abilities, the idea that no one has an ability to act against the laws of nature is part of the standard set-up of the problem of free will; see the classic statement in van Inwagen (1975). See, however, Lewis (1981) for a challenge to the view about abilities.

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According to Humeans, who abhor ‘necessary connections’ in nature, laws of nature are metaphysically shallow. They do not govern the way things behave; rather, they are an ex post facto generalization of the way things behave. Of course, not every true generalization qualifies as a law of nature; but for the Humean, what sets the law-like generalizations apart from the rest is, again, not a deep metaphysical matter. On the best-known Humean view of the laws, Lewis’s Best Systems Account, the laws of nature are those generalizations that enter into the best system of true generalizations about the world; the best system is that which achieves the best balance between informativeness and simplicity (see Lewis 1973b; Lewis 1983 adds the requirement that the generalizations must be phrased with predicates that refer to sufficiently natural properties; Lewis 1994 discusses the difficult problem of how to incorporate chances). The laws of nature, on this view, are metaphysically contingent: had the particular matters of fact over which the laws generalize been different, the laws themselves would have been different. A second view, which is sometimes labelled ‘semi-Humean’ (see Bird 2007), rejects the Humean view that the laws are metaphysically shallow, but still holds that they are contingent. On this view, which is defended by Armstrong (1983), Tooley (1977), and Dretske (1977), laws are a matter of a certain second-order relation holding between universals. Armstrong calls this relation ‘nomic necessitation’. It is characterized by the fact that its holding between two universals F and G entails that Fs are always G (in Armstrong’s terminology: N(F,G) entails ∀x(Fx → Gx)). The laws are metaphysically deep or substantial on this view because they are a matter of a real relation holding between real universals, and because this relation, via the stated entailment, provides a ‘governing’ role for the laws: they determine how things can actually behave. However, it is metaphysically contingent, on this view, whether N holds between a given pair of universals. Hence the laws themselves are metaphysically contingent. Dispositional essentialism, the view championed by Ellis (2001) and Bird (2007), takes laws to be both metaphysically deep and metaphysically necessary. The basic idea is that, first, the fundamental (or perhaps all natural) properties have an essence; that, second, this essence consists in their being a certain disposition; and that, third, it is this dispositional essence of the relevant properties which metaphysically grounds any given law of nature. To oversimplify a favourite example of dispositional essentialists, charge is a property which, by its very nature, is the disposition to repel like charges and attract opposite ones; and that is why it is a law (Coulomb’s Law, in fact) that like charges repel each other, while opposite charges attract. Because they are grounded in essences, the laws of nature are metaphysically necessary. Coulomb’s Law could not be violated because

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to violate it, an object would have to be charged but not behave in accordance with Coulomb’s Law; but to be charged just is to behave in accordance with Coulomb’s Law. Unlike nomic necessitation, a property’s dispositional essence is not externally imposed on it and in principle separable from it; it is built into the property itself. The dispositional essentialists provide a dispositionalist account of the laws; I provide a dispositionalist account of modality. Clearly, the projects are natural allies. However, given the conception of dispositions and of potentiality that I have defended in this book, I cannot endorse dispositional essentialism quite as it stands. For as it stands, paradigmatically in Bird (2007), dispositional essentialism crucially relies on the conditional conception of dispositions, which I have rejected in chapters 2–3. Bird uses the following statement of a property’s (P’s) dispositional essence: (I) Px → (Sx  Mx), that is, P’s dispositional essence is a matter of its entailing a counterfactual conditional whose antecedent and consequent state, respectively, that the disposition’s stimulus holds and that its manifestation occurs. From (I), Bird is able to derive (by classical logic plus modus ponens for the counterfactual) what he takes to be the general statement of a law of nature: (V) ∀x((Px ∧ Sx) → Mx). (V) says that if the property (say, electric charge) and its stimulus (say, being in the vicinity of another negative charge) both occur, then the manifestation (the exertion of a repulsive force) also occurs. Because (I) is a statement of essence, it holds of necessity; and so does (V), in virtue of being derived from (I). Now, I have argued in chapter 2 that conditionals such as the schematic one in (I) are not adequate characterizations of what I have called the nomological dispositions: that is, precisely such dispositions as charge, which occur in laws of nature. Because of the quantitative nature of the involved properties, I have argued there, we must think of the nomological dispositions as being individuated not by a separate stimulus and manifestation property, but rather by a single, yet complex, manifestation. Electric charge, for instance, is (or brings with it) the disposition to exert a force F whose value stands to other charges q and their distance eq r in the precise mathematical relation F =  r2 . Moreover, what is characteristic for those nomological dispositions, I further argued in chapter 3, is their degree: they are possessed to maximal, or perhaps near-maximal, degree. In terms of the familiar modal operators, they are akin to a (qualified) necessity: the necessity

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of the object’s exerting a force F whose value stands to the surrounding charges eq q and their distance r in the precise mathematical relation F =  r2 , conditional on its being electrically charged. In terms of potentiality alone, the potentiality’s being possessed to a maximal degree entails that no opposite potentiality is possessed. Dispositions in general, I argued, are simply potentialities of a sufficiently high degree; the nomological dispositions are their limiting case, potentialities possessed to the maximal degree (or nearly so—a qualification to which I will return in a moment). The conception of dispositions in general, and of nomological dispositions in particular, that I have developed in chapters 2–3 is independent of dispositional essentialism, though it assumed dispositional realism (the view that dispositions are not reducible to non-dispositional properties). I argued that it was simply the conception most adequate to what we know and what we, as dispositional realists, believe about dispositions, ordinary and nomological. The question now is: how does this view square with dispositional essentialism? According to dispositional essentialism as characterized by Bird, at least some fundamental properties have dispositional essences. I have not made much of the idea that properties have essences, but it is part of the metaphysics of this book that we find potentialities ‘all the way down’. So if there are any fundamental properties, at least some of them have to be—and not merely to endow their bearers with—potentialities. If properties can be said to have essences at all, those properties will be essentially potentialities. So far, then, I agree with the dispositional essentialists, despite a bit of reformulation. The crucial difference lies in our different conceptions of dispositions, or potentialities. Saying, as we must on my conception, only that the fundamental nomological potentialities are possessed to maximal degree leaves open the key question of why things behave in accordance with the laws. Here is why. On my conception, objects that have electric charge have the potentiality to exert a force F whose value stands to other charges q and their distance r in the eq relation F =  r2 , and they have this potentiality to the maximal degree (again, I put aside the alternative option that their degree is only near-maximal, to which I will return in a moment). This does not yet tell us what charge is. In answering that question, we have two basic options. Option (i): Electric charge = the potentiality to exert a force F whose value eq stands to other charges q and their distance r in the relation F =  r2 . Option (ii): Electric charge = the maximal potentiality to exert a force F whose value stands to other charges q and their distance r in the relation eq F =  r2 .

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Take option (i) first. The potentiality that it identifies with electric charge, like other potentialities, allows for a variety of degrees. For the dispositional essentialist thesis, as it is currently construed, to work, it is crucial that this potentiality is always possessed to the maximal degree. For to say that objects have the nomological dispositions to the maximal degree is to say that they lack the potentiality for the opposite manifestation. And it is because they lack this opposite potentiality that they cannot do otherwise than behave in accordance with, in this case, Coulomb’s Law. However, on the present construal, the fact that the potentiality which is electric charge is always possessed to the maximal degree is an additional, external fact about electric charge; it is not part of its nature. So why are there no objects (particles, or otherwise) that have, or at least have a potentiality to have, electric charge to a lesser degree? If there were, then those objects would have a potentiality (or a potentiality to have a potentiality, or . . .) to violate Coulomb’s Law. For only the maximal degree of a potentiality to conform to the law precludes the object from also having a potentiality not to conform to it. If anything had an (iterated) potentiality to have electric charge to any other degree than the maximum, then, Coulomb’s Law would be possibly violated and hence would not hold of metaphysical necessity. Option (i), thus, is left with a central question: given that electric charge is eq the potentiality to exert force F =  r2 , why is it that everything which has this potentiality at all has it to the maximal degree? As long as this question remains unanswered, the modal force of Coulomb’s Law—and by similar arguments, of other laws that are to be grounded in nomological dispositions—has not been explained, let alone shown to be that of metaphysical necessity. Let us, then, turn to option (ii). Does it do any better? In chapter 3.6, I pointed out that that we can think of potentialities to F with a specific degree as determinates of the determinable potentiality to F. The determinable potentiality has been my concern for most of this book. Option (ii) is to identify a nomological disposition such as electric charge with a determinate of the potentiality in question, namely, its maximal determinate. From the identification in option (ii), we can indeed derive the required law (Coulomb’s Law as it applies to electric charge): that everything which has electric charge always exerts a force F whose value stands to other charges q and their eq distance r in the relation F =  r2 . Assume that an arbitrary object x is electrically charged; it follows that x has the maximal-degree potentiality to exert a force eq of F =  r2 . By the characterization of maximal degree, it follows that x has no eq potentiality not to exert a force of F =  r2 . By ACTUALITY and modus tollens, eq we get that x does exert a force of F =  r2 . So, if x is electrically charged, it exerts eq a force of F =  r2 , with the above specifications of q and r.

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The law is metaphysically necessary if the identification from which it is derived is metaphysically necessary: if nothing has an iterated potentiality for electric charge not to be the above-mentioned potentiality, then nothing has an iterated potentiality for electrically charged objects to behave other than Coulomb’s Law states. And the identification, like any identity, must be necessary (see chapter 6.2). But why should the identification in option (ii) be adopted? If it is merely a matter of how we use the word ‘electric charge’, then the necessity we get is of the wrong kind to underwrite a law of nature. There must be something about the maximal-degree potentiality that we refer to with the term ‘electric charge’ that makes it, rather than (say) the determinable potentiality to exert a force of eq F =  r2 , the right referent for the term ‘electric charge’. A natural move in such cases is to say that the term ‘electric charge’ is used so as to express a natural property. But why should the maximal determinate potentiality be more natural eq than its determinable, the potentiality to exert a force of F =  r2 (or any of its non-maximal determinates)? All that we can tell, on the dispositional essentialist story sketched thus far, is that the latter determinable potentiality is only ever instantiated as the former, i.e. to the maximal degree. But that is precisely what cries out for explanation. Simply stipulating the one to be more natural than the other is ad hoc. Whichever way we go on the identification of electric charge, there is a significant gap in the explanation provided by the present version of dispositional essentialism: it cannot explain why nothing has the potentiality to exert a force eq of F =  r2 to any other than the maximal degree. And in failing to explain this, it fails to provide the status of metaphysical necessity for Coulomb’s Law. For that status depends on the modal status of whatever it is that explains the fact that the potentiality is only ever possessed to the maximal degree: if the explanans is metaphysically necessary, then so is the law; if it is contingent, then the law is too. So long as we do not know what the explanans is, we do not know whether the law, on the revised dispositional essentialist picture, is contingent or necessary. Much the same argument, I suspect, can be applied in other cases of (putative or real) fundamental dispositions and laws. The upshot is that dispositional essentialism cannot simply adopt the present framework for thinking about dispositions and potentialities and then continue on the familiar path. New requirements for explanation turn up, and it is not at all clear how they are to be met and whether, if they are met, dispositional essentialists will still be warranted in according laws the status of metaphysical necessity. And note, again, that the present framework was motivated precisely by the observation that characterizations such as (I) do not adequately capture the

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dispositional nature of nomological dispositions such as charge, or even electric charge. The revised version of dispositional essentialism is, for that reason, a better version. But it leaves wide open questions about the modal status of the laws of nature. I have assumed so far that the nomological dispositions are indeed possessed to the maximal degree, and never to any lesser degree; we have seen that this assumption must be made to underwrite the claim of metaphysical necessity for the laws. One alternative is to abandon the assumption. Perhaps some or all nomological dispositions are possessed only to near-maximal degree; or perhaps their degrees vary widely, and their scientific interest lies in something other than their degree. (Either way, the nomological dispositions will be identified along the lines of (i) above.) Some such view may be attributed to Nancy Cartwright, who sees laws as generalizations over the dispositions (or ‘capacities’, or causal powers) of objects, but argues that such dispositions can be interfered with, and hence that the laws are no more than idealizing generalizations. (See, for instance, Cartwright 1989, as well as Schrenk 2007 and Schrenk 2010. Lowe 2006 also offers a roughly dispositionalist and non-necessitarian view of laws.) A disposition or potentiality that can be interfered with, on the present framework, must be a potentiality that is possessed to less than the maximal degree: for if the potentiality to F is interfered with, the object will not F, and by ACTUALITY (see chapter 5.7.4), it follows that the object has or had some potentiality not to F. Since maximality of degree has been characterized as a lack of the opposite potentiality, such an object can possess the potentiality to F only to a non-maximal degree. On this broadly Cartwrightian construal, we can think of laws as being grounded in potentialities without thinking of them as metaphysically necessary. For clearly there are objects with potentialities not to behave in accordance with the laws, and hence with a (once-)iterated potentiality for the laws to be violated: they are the very same objects which have potentialities to behave in accordance with the laws. Even a dispositionalist view of the laws, then, need not be committed to the laws of nature being metaphysically necessary, or in other words, to the view that metaphysical possibility collapses into nomic necessity. Moreover, a dispositionalist metaphysics does not automatically commit one to a dispositionalist account of the laws. If, as dispositionalists, we want to think of the laws as metaphysically deep or substantial, then the natural way to go surely is a dispositionalist account: dispositionality, or potentiality on the present view, is metaphysically substantial, and the more we can explain in terms of it the better. (Thus the Armstrong/Tooley/Dretske view, which takes laws to be substantial but grounded in a relation of nomic necessitation external to the properties

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themselves, is not an attractive option for the dispositionalist.) But it is open to the dispositionalist to take a shallow view of the laws of nature, especially on the present version of dispositionalism, where dispositionality (or rather, potentiality) is conceived as generally closer to possibility than to any kind of necessity, nomic or otherwise. (Mumford 2004 can be classified as a dispositionalist with a shallow conception of laws.) Indeed, a broadly Lewisian Best Systems Account, despite its original motivation which may seem antagonistic to a dispositionalist metaphysics, can in principle be combined with such a metaphysics. All we need is the assumption (which is neither made nor excluded by the above statement of the Best Systems Account, though it was certainly rejected by Lewis himself) that among the natural properties which are fit to enter into a true, simple, and informative system there are potentialities. There is, so far as I can see, no general problem with adding this assumption. And prima facie a dispositionalist who holds a Best Systems Account (or indeed any shallow conception of the laws) has no reason to assume that the laws constrain the potentialities that things have—simply because the laws, on such a view, do not constrain anything at all. But we can be more specific about the kind of scenario which, given the potentiality account of possibility and the Best Systems Account of laws, would ensure the metaphysical possibility of the laws being different. For it to be possible that an actual law is violated, it must be the case that, for some p, it is a law that p and it is possible that not p. On the present combination of views, this is to say that p is a theorem of the best system and that something has an iterated potentiality for not-p. Since the best system must at least be true, that iterated potentiality will not ever be manifested. (I take the manifestation of an iterated potentiality to include the manifestation of each iteration, that is, of all embedded potentialities.) In fact, its failure to manifest must be sufficiently systemic that it merits inclusion in the best system. How might that be? Recall some of my examples for joint potentialities (chapter 4.2): a fragile glass may possess the potentiality to break to a very high degree yet be packed in styrofoam with potentialities that ‘cut across’ the glass’s; their joint potentiality for the glass to break is of a rather low degree. If all glasses in the world were always packed in styrofoam, they would rarely, and perhaps never, get to exercise their potentiality to break. The best system, which describes the distribution of properties actually possessed by objects, might then contain an axiom to the effect that all glasses are unbroken. Of course, glasses and their potentialities to break are unlikely to figure in any best system describing our world. But I see no reason why the same should not happen at the level of perfectly natural properties too: some things may have a (perfectly natural) potentiality to be F without being

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G but those potentialities may be constantly masked by stronger counteracting potentialities, the equivalent of the styrofoam’s potentiality to absorb forces acting on it, so as never to be manifested. Which potentialities are manifested is a function not of any one thing’s potentialities alone, but of the joint potentialities of all things together. Many potentialities of particular things may well be ‘lost’ in the overall scheme of things: their contribution to the overall joint potentialities is entirely swallowed up by counteracting potentialities, and hence they systematically fail to be manifested. The best system may then codify that failure in the form of a law that all Fs are Gs. Whether or not we should think of metaphysical possibility as collapsing into nomic possibility, then, is a matter that is left wide open by the potentiality account.

7.9 Metaphysical possibility in time The relation between potentiality and time, and consequently on the present account between metaphysical modality and time, has been touched on several times. In this section, I will draw together worries that are connected to it. A first point concerns potentiality’s seeming asymmetry with regard to time, as captured in the ‘triviality thesis’ discussed, though not fully endorsed, in chapter 5.8. One surprising, and perhaps troubling, consequence of that thesis has been seen in section 7.6. If potentiality is indeed asymmetric with respect to time in the way that the triviality thesis claims—that potentialities concerning the past are always trivial, i.e. possessed if and only if they are manifested—then we cannot accept the possibility that there were never any of the actually existing objects. Thus, potentiality’s temporal asymmetry would turn some objects, which are distinguished by their temporal position in the universe—the very first objects that existed—into necessary existents. One option, of course, is to reject the triviality thesis. I have indicated some reasons for doing so in chapter 5.8, but I have not settled on any alternative view. Instead, I have pointed to some alleviating considerations concerning the triviality thesis’s implications in section 7.6, and I will have nothing to add to them here. The intuition that there could always have been different objects is, I believe, not at the centre of our modal intuitions, and like many philosophical intuitions it may well be theory-driven. The very same temporal asymmetry was shown to explain a modal principle that is, I believe, more central and accepted by many philosophers: the necessity of origin. Indeed, this further consequence of temporal asymmetry can be seen as another application of the necessity of origin. It is the origin of the universe itself, in precisely the objects which originally constituted it, that is necessary on the present view. Nor

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does the view have to be entirely counter-intuitive. As Mackie (1998) has noted, the plausibility of the necessity of origin is a symptom of our general tendency to think of possibility in terms of what she calls a branching model: unactualized possibilities have to branch, or to have branched, off the actual development of things at some point in time. Given the asymmetry assumption, the potentiality account, while not using the language of branching, makes sense of this picture, which in turn makes sense of the idea that the beginning of time is modally ‘special’. Intuitions to the contrary—intuitions that the universe could have always contained different objects—can then be argued to be based rather on the conceivability of such a state of affairs. But the potentiality account has good reason to divorce possibility from conceivability, just as the potentiality-based semantics had good reason to divorce dynamic from epistemic modality. In this case, therefore, I am prepared to conditionally bite the bullet: if potentiality really is asymmetric with respect to time, then the universe’s origin is necessary. In the absence of a properly developed alternative to the triviality thesis, I suspect that this will be considered the greatest theoretical cost of the theory; and quite generally I believe that the relation between potentiality and time is one of the most pressing questions for further research. The bullet may be thought to be bigger than I have made it look so far, if we combine it with the view (not established, but not refuted either, in the previous section) that the laws of nature are metaphysically necessary. For suppose that the laws, in addition to being necessary, are deterministic: given any total state of the universe, they necessitate the following total states. If the beginning, hence the first total state of the universe, is metaphysically necessary, and it is metaphysically necessary that this state necessitates such-and-such following states, will not everything be necessary and nothing contingent? That would be a dire consequence, not least for a view that is so keen to locate (non-trivial) possibilities in the world. But it is not a consequence of the potentiality view at all. Metaphysical possibility, on this view, resides in the potentialities of any objects, big or small. Even in a deterministic universe individual objects will have potentialities to act otherwise than they do and are determined to act. They will just have no opportunities to exercise those potentialities; or, in the terminology of this book, they will have the individual, perhaps even joint potentialities to act so-and-so even when they do not have the joint potentiality together with everything there is to act so-andso. (Recall the distinction between weak and strong possibilityT in chapter 6.1: the latter, but not the former, is affected by determinism.) But individual potentiality, or joint potentiality of objects however much they fall short of being the totality of all objects, is enough for metaphysical possibility on the potentiality

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account. The status of the beginning of the universe as necessary or contingent does not affect these. Whatever our stance on the seeming asymmetry of potentiality with respect to time, a second set of questions arises from a much less debatable connection between potentiality and time. Potentialities, like any other properties, are possessed by objects at times (whatever the right metaphysical construal is of property possession at a time); they can, in principle, be lost and gained, and they require the existence of a possessor in order to be possessed. These commonplace observations pose a problem for a certain class of possibilities: possibilities of relations between objects across time. I will adduce a few examples and then discuss at some length how they can or cannot be accommodated given further assumptions about the nature of time and the existence of objects in it. Here are the examples. 1. It is possible that Jack admires Socrates. 2. It is possible that I should be taller than Socrates. 3. It is possible that I meet my great-grandmother. Suppose that all of these possibilities are counterfactual: Jack has in fact never heard of Socrates, I am (and always will be) shorter than Socrates was, and my great-grandmother died long before I was born. Still, something must have iterated potentialities for Jack to admire Socrates, for me to be taller than Socrates, or for me to meet my great-grandmother, for evidently these are (metaphysical) possibilities. The question is, who or what possesses those potentialities, and at which time? Given that each of 1–3 expresses de re possibilities concerning two distinct objects, the relevant potentialities should be, ultimately, joint potentialities of the objects concerned. And here is the problem. I argued in chapter 4.2 that there is no necessary condition that two or more objects have to meet in order to be joint bearers of such a potentiality—except, of course, their existence. In order to bear any relation to any other thing, and hence in order to be the co-possessor of a joint potentiality, an object must exist. This seems trivial. Now the objects in 1–3 do or did exist, but they never exist(ed) at the same time. At which time, then, are they to be the joint possessors of a joint potentiality? This, in short, is the challenge. How we can respond to it depends on our further commitments in the philosophy of time, and in particular on whether we are eternalists or presentists. Let me briefly outline the options. According to presentism, everything is always present; according to (weak) eternalism, some things are not present, that is, there are merely past or merely

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future objects.17 On this way of introducing the division, eternalism is merely the negation of presentism. In general, however, eternalists want to make a stronger claim. According to strong eternalism, it is always true that everything is always something. In other words, everything that exists at any time exists at all times. I will use the label ‘eternalism’ to refer to strong eternalism. Other forms of weak eternalism will share the fate of presentism in some parts and that of eternalism in others. Presentism has the consequence18 that only present objects can be bearers of properties (singular or plural) or relata of relations. There are no past or future objects, so clearly there are no past or future objects available as bearers of properties or relations. This goes for all properties and relations, including those that are potentialities. However, the presentist can and should accept (what look like) property ascriptions to past or future objects, as long as they are properly prefixed by a tense operator. Thus, it was once the case that Socrates was snub-nosed; and equally, it was once the case that Socrates had a potentiality to be a carpenter. Presentists are, however, known to have trouble with so-called cross-temporal relations. It would appear that I stand in certain relations to Socrates: I admire him, I am less tall than him, I am influenced by him (being part of a tradition that he helped initiate). But for the presentist it cannot be true now that I admire, am less tall than, or am influenced by Socrates, if we think of these as genuine relations, for Socrates is not available as a relatum. Nor can we say that it was once the case that I admired Socrates (and so forth). For although that might, metaphorically, take us back to a time when Socrates existed, ipso facto it takes us back to a time when I did not exist, and so I was not available as a relatum. Those troubles apply directly to the (apparently) cross-temporal relations and plural properties that are joint potentialities. Jack or I cannot have joint potentialities to admire, or to be taller than, Socrates, nor can I have a potentiality to meet my great-grandmother. For Socrates and my great-grandmother are not available as a co-possessors of such a joint potentiality. Nor can we say that it was once the case that Socrates or my great-grandmother had such joint potentialities with Jack or with me. For although that might, metaphorically, take us back to a time when Socrates or my great-grandmother existed, ipso facto it takes us back to a time when Jack or I did not exist, and so we were not available as co-possessors of joint potentialities. 17 To be precise, the characterization of presentism should be prefixed with ‘necessarily’, since the claim is not intended to capture a contingent fact. The characterization of weak eternalism is so weak that it includes the ‘Growing Block’ theory, according to which there are past objects but no future ones. See footnote 21 in chapter 5.7.1 for more details. 18 Under the plausible assumption that having a property or standing in a relation implies existence; however, see Hinchliff (1996) for an alternative view.

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Presentists have found ways and means to deal with the lack of cross-temporal relations.19 In general, they involve the construction of a surrogate for the apparent cross-temporal relation, by way of paraphrasing the objectionable relation ascriptions. Admiring, for the presentist, cannot be a genuine relation between the admirer and the admired. Rather, ‘I admire Socrates’ is to be paraphrased in much the same way as ‘I admire Sherlock Holmes’ would be, that is, in some such way as the following: There are various properties, p1 – pn , such that (i) I associate p1 – pn with the name ‘Socrates’, and (ii) thoughts of either p1 –pn or the name ‘Socrates’ evoke in me the characteristic feeling of admiration. Markosian 2004, 63

Intentional relations to past (or future) objects, for the presentists, thus aren’t relations at all. But not all cross-temporal relations are intentional relations. Thus, for instance, I stand in the relation of being the great-granddaughter of to my great-grandmother who died before I was born. Here, too, we have a relation between objects that have no temporal overlap, but clearly it cannot be analysed along the lines of ‘I admire Socrates’. Markosian (2004) suggests that such relations are indirect. What makes true the statement ‘I am the great-granddaughter of X’ is a series of direct relations between objects that do or did co-exist: it was the case that X was the mother of Y, and it was the case that Y was the mother of Z, and it was (and is) the case that Z is my mother. The same story, though with a longer chain of direct relations, may then be told about the influence that Socrates has on me. Yet other cases may be analysed by factoring the apparent cross-temporal relation between objects such as Socrates and myself into two relations with a common relatum that co-exists or co-existed with both. Thus Socrates’ being taller than me is a matter of there being a height, n, such that it was the case that Socrates was taller than n and it is the case that I am less tall than n. And so it goes on. My present concern is not to evaluate the success or failure of these presentist strategies, but to see whether they might be applied to the presentist-cum-potentiality-theorist problem of cross-temporal joint potentialities, to secure the truth of possibility claims such as 1–3. In judging the success of these applications, keep in mind that this is only the presentist version of the current theory, and that any worries that it might engender should be held against current theory of modality only in combination with presentism, and only if they go beyond the worries with which presentists are already faced. As it will turn out, there are indeed such additional worries. 19

See, for instance, Markosian (2004) and Sider (1999). Of course, Sider is not a presentist.

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Example 1 may be paraphrased in much the same way as its potentiality-less counterparts. The possibility of Jack’s admiring Socrates is the possibility that an apparently cross-temporal relation holds between Jack and Socrates. For presentists, this cannot be a genuine possibility of a cross-temporal relation holding. Rather, it must be a possibility for some such paraphrase as the one that I have quoted above to be true. Consequently, on the present view, this possibility must derive from an iterated potentiality for that paraphrase to be true. And in particular, it might be a matter of Jack’s having a potentiality to be such that there are various properties, p1 – pn , such that (i) Jack associates p1 – pn with the name ‘Socrates’, and (ii) thoughts of either p1 – pn or the name ‘Socrates’ evoke in him the characteristic feeling of admiration. This particular presentist strategy, then, is applicable in the context of a potentiality-based theory of modality. Other presentist strategies are not so easily transferred, as a closer inspection of cases 2 and 3 will show. With example 2, as with 1, we will try to analyse the apparently cross-temporal potentiality by analysing its apparently cross-temporal manifestation. We have seen an analysis for the manifestation, my being taller than Socrates, above. The natural analysis of the apparent joint potentiality might then be: there is a height n such that it was the case that Socrates had a potentiality to be no taller than n, and it is the case that I have a potentiality to be taller than n. (Whether these are individual potentialities of Socrates and me, respectively, or joint potentialities that each of us possesses together with the height n, depends on the ontology of heights. For present purposes, it does not matter.) The problem with the strategy is that it does not yield the possibility of my being taller than Socrates. Instead, it gives us a conjunction of two possibilities: for some height n, it is possible that Socrates was no taller than n and it is possible that I am taller than n. What we need, however, is the possibility of the conjunction: we want there to be a height n such that it is possible that Socrates was no taller than n and I am taller than n. There is no way to infer the possibility of the conjunction from the conjunction of possibilities: 3p ∧ 3q does not entail 3(p ∧ q).20 The problem generalizes. In every case where it is seemingly possible that p and q but at no time are there relevant witnesses for the possibilities of both p and q, we will be able to account only for the conjunction that it is possible that p and possible that q, but not for the possibility of the conjunction that p and q. 20 It does not help to appeal to only one potentiality and keep the other height fixed as it is in actuality. For instance, we might try saying that there is a height n such that it was the case that Socrates was no taller than n and it is now the case that I have a potentiality to be taller than n. But this only provides for the possibility of my being taller than Socrates actually was, not for the possibility of my being taller than Socrates.

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Intuitively, however, it seems that there are such possibilities; the possibility of my being taller than a given height n and Socrates’ being no taller than n is just one example. These, then, are problems for the combination of presentism and the potentiality account of modality. The close relation between modality and time entailed by the latter, together with the restrictions imposed on the metaphysics of time by the former, yield unwelcome restrictions on the metaphysics of modality. The extant strategies of the presentist, or at least those that I have outlined above, do not yield a suitable substitute or analysis for the potentialities that we need to account for common-sense examples of many cross-temporal possibilities. A presentist-cum-potentiality theorist will have to either give up on a great deal of extremely plausible possibility claims, or come up with new strategies for analysis or substitution. What about our third example, the possibility of my meeting my greatgrandmother? Which potentiality, and whose, might be the basis of this possibility? The natural way to go would be to mimic the presentist’s indirect relations strategy by appealing to iterated potentialities. My great-grandmother, let us say, had a potentiality to live longer and an iterated potentiality to enter (by living longer) into a joint potentiality with me, namely, the joint potentiality to meet. (I am assuming that the necessity of origin holds and imposes some restrictions on an individual’s time of origin, so my great-grandmother’s living longer than she in fact did is a necessary condition for our meeting.) Can the presentist tell this story? I doubt it. Even a more than once iterated potentiality cannot concern objects that do not exist. I do not have a however many times iterated potentiality to meet Sherlock Holmes, since (or if) Sherlock Holmes does not exist. Likewise, my great-grandmother cannot have had a however many times iterated potentiality to meet me, since (for the presentist) I did not exist in the relevant past. What she may have had, and presumably did have, was a potentiality to live long enough to see her great-grandchildren, and an iterated potentiality for herself to have and to meet a great-granddaughter who is F, where ‘F’ is a complete (non-haecceitistic) description of me. But by POSSIBILITY, such an iterated potentiality does not yield the de re possibility for my great-grandmother and me to meet. Again, the combination of the potentiality view with presentism appears less than desirable: it is committed to rejecting a number of highly plausible possibility claims. The best bet for the potentiality view, it seems, is to join forces with eternalists rather than presentists. For the eternalist, there is no general problem of cross-temporal relations. Socrates exists simpliciter, as do I. Socrates is a bearer of properties even today: for instance, the property of being admired. Socrates and

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I can stand in cross-temporal relations: for instance, the relation of admiring. Similarly, it would seem, nothing should stand in the way of Socrates’ and my standing in the cross-temporal relation of having a joint potentiality to be such that I am taller than him. Recall, however, that potentialities, like all properties, are possessed at times. At which time, then, are cross-temporal potentialities possessed?21 Note that, even for the eternalist, not just any property can be possessed by an object, and not just any relation can hold between objects, at any time. Although all objects exist at all times, not all objects are present at all times. (What I call ‘being present’, Sider (2003) calls ‘existing-at’, in contrast with existing simpliciter.) Socrates exists, but he is no longer present, while I am both existent and present now. Some properties, such as being tall, require that the object be present at a 21 For a perdurantist, this question makes little sense. Perdurantists hold that objects persist through time by having different parts at different times; endurantists hold that objects persist by being wholly present at different times. (The terminology is due to Lewis 1986a, 202.) Perdurantists will typically hold that persisting objects have properties ‘at a time t’ by having temporal parts or stages at t which have the property. The ultimate possessors of properties and relata of relations are the stages themselves. Stages possess properties simpliciter, not at a time; and they stand in relations simpliciter, not at a time. So Socrates’s temporal part at a given moment in 405 bc might stand to my present temporal part in the taller-than relation. The question at which time they stand in this relation is simply the wrong question. The same applies to the relations or plural properties that are joint potentialities: potentialities are possessed ‘at a given time’, for the perdurantist, only in the sense that they are possessed, like any other properties and relations, by temporal parts or stages of objects. My question, then, would be: which stages are the joint possessors of the potentiality? According to endurantism, on the other hand, objects are wholly present at different times, and their possession of properties is indeed a matter of the object (the whole object) possessing a given property, or standing in a given relation, at a particular time. On this view, we are faced with the question: at what time are objects to possess cross-temporal joint potentialities? In what follows, I will use the endurantist language. Not only is it more natural; it also appears to tie in better with a metaphysics of potentiality. Potentialities, in general, are manifested by their bearers: one glass’s fragility, for instance, cannot be manifested in another glass’s breaking. For the endurantist, this can be a straightforward matter: it is in every respect one and the same thing which possesses, perhaps at an earlier time, the potentiality and which, at a later time, manifests it. For the perdurantist, we are ultimately dealing with two distinct things, the earlier and the later stage. The idea that objects manifest their own potentialities would then have to be rephrased in terms of stages: stage s can manifest a potentiality of stage s only if s and s are parts of the same persisting object. This formulation not only loses much of the intuitive appeal of the original. What is more, most perdurantists think that whether s and s are parts of the same persisting object is not a deep metaphysical matter; it supervenes on a number of other features of s and s , and one stage can in principle be part of more than one persisting object (see Lewis 1976b, Sider 2003). The tie between an object, its potentialities and their manifestations becomes a somewhat superficial matter on this view. Note that, in principle, a perdurantist might hold that while objects exist at a time by having a temporal part located at that time, some properties of the composite, temporally extended object are emergent, i.e. are not reducible to the properties possessed by individual stages. Such a perdurantist would also have a more natural way of accounting for the idea that potentialities are manifested by their possessors, and nothing else; but I know of no perdurantist who holds such a view. (Thanks to Thomas Sattig and Antony Eagle for helpful discussion of these issues.)

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time, while others, such as being admired, do not. Some relations, such as seeing, can only hold between objects that are present. Others, such as admiring or being more famous than can hold when only one, or even when none, of the relata is present.22 The same plausibly goes for potentialities: Socrates does not now have a potentiality to be taller than he was, though he does have a potentiality to be admired. How, then, are we to deal with joint potentialities whose co-possessors are never simultaneously present? The answer is that it depends on the joint potentiality in question. Let us, again, look briefly at our initial examples 1–3. First, the possibility of Jack’s admiring Socrates. The relevant contribution here is to be made by Jack, and not by Socrates. Jack has the relevant potentialities during his lifetime, and not during Socrates’. The relevant time for the joint potentiality is 2014 ad, not 405 bc. Second, the possibility that I be taller than Socrates. During Socrates’ lifetime, he had a potentiality to be shorter, but I did not yet have a potentiality to be taller; such potentialities are plausibly reserved to objects that are present. But Socrates may well have had, and indeed I believe did have, an iterated potentiality for me to be taller than him: a potentiality, that is, to be shorter than he was and thereby enter with me into a joint potentiality to be such that I am taller than him; or, indeed, a potentiality to be as tall as he actually was and still enter with me into a joint potentiality to be such that I am taller than him. When might that potentiality be exercised, and the joint potentiality possessed? Answer: today, in my lifetime rather than his. In addition, Socrates and I now have a joint potentiality to be such that I am taller than him. However, assuming again the triviality thesis about past-concerning potentialities is true (see chapter 5.8), we have no joint potentiality for me to be taller than him through his being shorter than he actually was; his height is fixed now. If the triviality thesis is correct, then the relevant contribution to our joint potentiality is now made by me: Socrates only contributes his property of having been as tall as he actually was, while I contribute my potentialities to have various different heights from the one that I do have. (If the triviality thesis is false, Socrates may be contributing in other ways now, but that makes no difference for present purposes.) While his contribution is fixed now, Socrates had a potentiality during his lifetime to make a different contribution to our present-day joint potentiality by being taller or shorter than he in fact is. (Without the triviality thesis, both Socrates and I may at any time have joint potentialities to be such that I am taller than he in any of 22 The distinction is not germane to endurantism: the perdurantist, too, can recognize that certain relations hold only between stages that share their temporal location.

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the ways specified.) This explains the possibility of my being taller than Socrates through his being shorter than he actually was. Third, the possibility of my meeting my great-grandmother. Here, too, the eternalist can appeal to precisely the iterated potentiality outlined above, which was precluded for the presentist. My great-grandmother (and Socrates too) had iterated potentialities that concerned me. Note, however, that the iterated potentialities strategy may not yield, say, the possibility that I should meet Socrates. For it requires an iterated potentiality on the part of the earlier individual, my great-grandmother or Socrates, to live long enough; or at least an iterated potentiality on the part of something else for those individuals to live long enough. If neither Socrates nor anything else ever had an iterated potentiality for him to be born significantly later or to live over 2400 years, then nothing has an iterated potentiality for him to meet me, and it is impossible that we should meet. This consequence, palatable or not, would be a consequence not primarily of the potentiality view but of a particular aspect of the necessity of origin thesis: if an object’s origin is necessary, if that imposes at least some rough restrictions on an object’s time of origin, and if it is impossible that Socrates (or humans in general) live over 2400 years, then it is impossible for Socrates and me to meet. I have gone through the examples piecemeal to see how, with careful consideration of the details, they can be accommodated at least in an eternalist framework. I have given no guarantee that the same or similar strategies will work for all cross-temporal cases, though I hope that the relative diversity between the three cases has inspired some optimism. To conclude this section, the relation between possibility and time, on the potentiality view, is a tricky one. Further work is required to explore in some detail alternatives to the triviality thesis. The combination of a potentiality view with presentism has turned out to be surprisingly unattractive, but an eternalist version of the potentiality view seems able to deal with intuitive cases without any undue costs.

7.10 Conclusion The potentiality conception of possibility has, I hope, been shown to be a plausible view and a serious contender in the metaphysics of modality. It is formally adequate, providing the right (minimal) modal logic for metaphysical modality; it is semantically fruitful, offering the materials for a semantics of dynamic modals that respects the context-sensitivity of those modals as well as their logical form and their difference from epistemic modals; and it has a number of responses available to objections against its extensional correctness. It is, to some extent,

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revisionary, but not so much as to distort the notion of metaphysical modality. It may well be that some of the so-called intuitions that philosophers have about what is or is not metaphysically possible are shaped by the prevalent ways of theorizing about metaphysical modality: by appeal to possible worlds, and by appeal to conceivability. On the potentiality account, neither of these should be expected to be a particularly good way of thinking about metaphysical modality. Some deviation, then, is to be expected; but the deviations are few and not unacceptable, or so I have argued in this chapter. In the course of seven chapters, we have seen a development of the notion of potentiality from its familiar and everyday beginnings in dispositions such as fragility and irascibility, through the introduction of a great spectrum of degrees, the distinction between intrinsic and extrinsic potentiality, and the introduction of iterated potentiality, to a notion that largely shares the logic of possibility, as set out in chapter 5. This notion, I have argued in this chapter and the previous one, provides a simple and successful account of metaphysical possibility as captured in the definition POSSIBILITY. Some questions remain open. But that is as it should be. This book is meant to be the beginning of a debate, not its end.

Appendix Formal adequacy In chapters 5 and 6, I have argued that a potentiality-based account of possibility meets the criterion of formal adequacy: it validates the minimal logic of metaphysical modality. My arguments to this effect have been mostly informal, sometimes semi-formal. The aim of this appendix is to reformulate those very arguments with a greater degree of rigour and detail than the main text could provide. The aim is not, however, to go beyond what those chapters have provided in substance. Thus I am not going to give independent logical motivation for the (fragment of a) system to be developed; I am not going to fully develop the syntax beyond the formation rules, and I am not going to provide an extensional model-theoretic semantics, let alone proofs of soundness or completeness. The semantics for the formal language uses the resources that are given by the metaphysics of this book, rather than those of standard model theory. I am going to treat the potentiality operator POT as a logical operator whose interpretation is fixed, rather than looking at the logic more generally. Rather than as a logical system proper, the following can be read as a fragment of a logical system together with an exploration of the structure of its intended model, the realm of potentiality. It would be very interesting to go beyond these limitations and develop a full-blown logic. But this book has been about metaphysics and concerned logic only insofar as it constrains metaphysics. For those purposes, the more modest aim of this appendix is enough. As we have seen in chapter 6.4, a modal logic’s normality can be guaranteed by various minimal combinations of axioms and rules of inference (Chellas 1980, 114–118, provides a list of such combinations.) One, though not the standard, option is this. The possibility operator 3 must 1. be closed under logical equivalence: if  φ ≡ ψ, then 3φ ≡ 3ψ; 2. be closed under, and distribute over, disjunction:  3(φ ∨ ψ) ≡ (3φ ∨ 3ψ); 3. and in addition validate the theorem  ¬3⊥. (Cf. Chellas 1980, 118, Theorem 4.5(5).) In addition to yielding a normal modal logic, any logic of 3 that purports to formalize metaphysical possibility must validate the T theorem:  φ → 3φ We will begin by restating the logic of potentiality that has been developed in chapter 5 and then using it to prove (within the system, and given its intended model) that a properly defined possibility operator meets these constraints. In addition, I will provide a conditional proof of the axiom typical of S4, but say nothing about S5. For the time being, I am content to show that the minimal requirement is met.

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1 Syntax 1.1 Vocabulary The following are primitive descriptive expressions of the language L: a denumerable set of sentence letters p0 , p1 , p2 , . . . ; a denumerable set of singular n-place predicates for all n ≥ 1, F0 , F1 , F2 , . . . ; a denumerable set of plural n-place predicates for all n ≥ 1, G0 , G1 , G2 , . . . ; a denumerable set of mixed (singular and plural) n-place predicates for all n ≥ 1, H0 , H1 , H2 , . . . ; a denumerable set of individual constants a0 , a1 , a2 , . . . ; and a denumerable set of plural constants aa0 , aa1 , aa2 , . . . . In addition, there is a denumerable set of individual variables y0 , y1 , y2 , . . . , as well as a denumerable set of plural variables yy0 , yy1 , yy2 , . . . ; as well as the following syncategorematic expressions: ∃, ∀, ¬, ∨, ∧, →, ≡, λ, POT, and the parentheses ‘(’ and ‘)’. A singular term is either an individual constant or an individual variable; a plural term is either a plural constant or a plural variable; a term is either a singular term or a plural term. I use t, t1 , t2 as a metavariables for terms (singular and plural, names and variables), x as a metavariable for variables (individual or plural), , as metavariables for predicates (singular, plural, or mixed), and φ, ψ, etc., as metavariables for sentences (open or closed).

1.2 Formation rules There are two kinds of complex expressions of L, sentences and one-place predicates, defined by the following rules:

• Every sentence letter is a sentence. • If  is an n-place predicate with m places for singular terms and (n – m) places for

• • • • •

plural terms (where n ≥ m), t1 , . . . , tm are singular terms and tm+1 , . . . , tn are plural terms, t1 . . . tn is a sentence. (The clause applies in an obvious way to mixed predicates; ‘pure’ singular and plural predicates are included as special cases: where  is a singular predicate, we have n = m; where  is a plural predicate, we have m = 0. I adopt the convention that the singular and plural terms are ordered so that the predicate is followed first by all the singular terms (if any), which are then followed by all the plural terms (if any).) If φ is a sentence, then so is ¬φ. If φ and ψ are sentences, then so are (φ ∨ ψ), (φ ∧ ψ), (φ → ψ), and (φ ≡ ψ). If φ is a sentence and x a variable, then ∃xφ and ∀xφ is a sentence. If φ is a sentence and x a variable, then λx.φ is a one-place predicate. (If x is an individual variable, λx.φ is a singular predicate; if x is a plural variable, λx.φ is a plural predicate.) If  is a one-place predicate, then so is POT[]. (If  is a singular predicate, then so is POT[]; if  is a plural predicate, then so is POT[].)

Outer brackets may be dropped. Free and bound variables, open and closed sentences are defined in the usual way.

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2 Semantics 2.1 Models and assignments A model is a pair D, v, where

• D is a (non-empty) set of individuals, • v is a valuation that assigns values to primitive descriptive expressions as follows: – – – –

to each individual constant an element of D, to each plural constant a subset of D1 , to each predicate a property or relation of the appropriate type, to each sentence letter a proposition.

All well-formed expressions receive values in a model relative to an assignment of values to the variables. An assignment s is a function that assigns to each individual variable an element of D and to each plural variable a subset of D. Closed sentences and predicates are assigned the same values relative to all assignment functions. If s is an assignment and x a variable, s[d/x] is the assignment that differs from s only (if at all) in assigning d to x.

2.2 Semantic rules The semantic rules extend the valuation function to give values, relative to an assignment, to complex expressions. Let v s be the function that assigns values to expressions in accordance with v, relative to the assignment function s. Then if φ is a primitive descriptive expression, v s (φ) = v(φ); if x is a variable, v s (x) = s(x). Next, we need to define a function that delivers for every object, sentence, and variable the proposition that is obtained by, metaphorically, sticking the object into the place of the variable of the sentence: Let (φ, x, v, s) be the propositional function that takes every object d to the proposition v s[d/x] (φ), where φ is a formula, x a variable, v a valuation function, and s an assignment. (If x is a singular variable, the function will take elements of D to propositions; if x is a plural variable, the function will take subsets of D to propositions.) Here are the semantic rules: Sentences • If φ is of the form t1 . . . tn ,  is a mixed predicate, t1 , . . . , tm (0 ≤ m ≤ n) are the singular terms of φ and tm+1 , . . . , tn are its plural terms, P = v s () and d1 = v s (t1 ), . . . , dn = v s (tn ), then v s (φ) = the proposition that d1 , . . . , dm and the members of dm+1 , . . . , the members of dn instantiate P. (Again, the clause applies in an obvious way to mixed predicates, but singular and plural predicates are included as special cases.) 1 I assign sets to plural constants (and variables) purely for reasons of convenience. Different treatments of the logic of plurals should be compatible with everything I say in this chapter, but I have not checked this.

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formal adequacy • If φ is of the form ¬ψ, then v s (φ) = the negation of v s (ψ). • If φ is of the form (ψ1 ∨ ψ2 ), (ψ1 ∧ ψ2 ), (ψ1 → ψ2 ), or (ψ1 ≡ ψ2 ), v s (φ) =

the disjunction of v s (ψ1 ) and v s (ψ2 ); the conjunction of v s (ψ1 ) and v s (ψ2 ); the material implication from v s (ψ1 ) to v s (ψ2 ); or the biconditional of v s (ψ1 ) and v s (ψ2 ), respectively. • If φ is of the form ∃xψ and f = (ψ, x, v, s), then (i) if x is an individual variable, v s (φ) = the (infinite) disjunction of f (d) over all d ∈ D; or (ii) if x is a plural variable, v s (φ) = the (infinite) disjunction of f (d) over all d ⊆ D. • If φ is of the form ∀xψ and f = (ψ, x, v, s), then (i) if x is an individual variable, v s (φ) = the (infinite) conjunction of f (d) over all d ∈ D; or (ii) if x is a plural variable, v s (φ) = the (infinite) conjunction of f (d) over all d ⊆ D. Predicates • If  is of the form λx.φ and f = (φ, x, v, s), then (i) if x is an individual variable, v s () = the property of being a d such that f (d) is a true proposition; (ii) if x is a plural variable, v s () = the property of being objects xx, where each of the xx is a member of D, such that f ({d : d is one of the xx}) is a true proposition. • If  is of the form POT[ ] and P = v s ( ), then v s () = the property of having a potentiality to instantiate P. I do not assume any particular theory of properties or propositions, but I do assume the following: A1 An n-place property and an ordered n-tuple of objects (or ordered n-tuple of sets of objects) together uniquely determine a proposition (that is, propositions are Russellian, not Fregean). A2 All propositions are either true or not true. The negation of a proposition p is true iff p is not true, the disjunction of propositions p and q is true iff at least one of p and q is true, etc. A3 For any objects xx and proposition p, xx instantiate the property of being such that p is true just in case p is true. Further, I assume that the metaphysics developed in this book is correct. So there are objects, and there are objects that possess potentialities, so the semantic clauses will not be vacuously true. Lemma 1 For every model M = D, v, where φ = ∃xψ and (i) x is an individual variable, v s (φ) is true just in case for some d ∈ D, v s[d/x] (φ) is true; or, if (ii) x is a plural variable, v s (ψ) is true just in case for some d ⊆ D, v s[d/x] (ψ) is true. Proof. (i) Let x be an individual variable. If φ = ∃xψ and f = (ψ, x, v, s), then v s (φ) = the infinite disjunction of f (d) over all d ∈ D. A disjunction, finite or infinite, is true just in case at least one of its disjuncts is. So for at least one d ∈ D, f (d) must be a true proposition. f (d) = vs[d/x] (ψ); so, for at least one d ∈ D, v s[d/x] (ψ) is a true proposition. (ii) Let x be a plural variable. If φ = ∃xψ and f = (ψ, x, v, s), then v s (φ) = the infinite disjunction of f (d) over all d ⊆ D. A disjunction, finite or infinite, is true just in case at least one of its disjuncts is. So for at least one d ⊆ D, f (d) must be a true proposition. f (d) = vs[d/x] (ψ); so, for at least one d ⊆ D, v s[d/x] (ψ) is a true proposition.

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Note: As illustrated by the proof of Lemma 1, the proofs for singular and plural terms run exactly in parallel. In the following, I will ignore plural terms in my proofs. All of them are easily adapted to cover plural terms and predicates as well, by simply replacing the relevant expressions in accordance with the following schema: for d ∈ D, write d ⊆ D; for ‘d instantiates property P’, write ‘the members of d instantiate property P’, etc.

3 The system P 3.1 Rules and theorems The system P is the set of its theorems. A sentence φ of the language L is a theorem of P just in case for all models M = D, v and assignments s, v s (φ) is a true proposition. I write P φ, abbreviated to  φ, for ‘φ is a theorem of P’. I accept classical logic: Theorems If φ is a truth-functional tautology, then  φ. (This is guaranteed by the assumptions I made about propositions.) Modus Ponens

If  φ and  φ → ψ, then  ψ.

Proof. Suppose  φ and  φ → ψ. This means that for all models M = D, v and assignments s, v s (φ) is a true proposition and v s (φ → ψ) is a true proposition, i.e. the material implication from v s (φ) to v s (ψ) is a true proposition. By A2, that material implication is a true proposition just in case, if v s (φ) is a true proposition, then so is v s (ψ). Since v s (φ) is, by assumption, a true proposition, v s (ψ) must be true too. Since M and s were arbitrary, the same goes for every model and assignment: ψ must be assigned a true proposition in every model, relative to every assignment. Hence  ψ. This proof of Modus Ponens should give a feel for how my non-standard semantics validates classical truth-functional logic. Given any theorem of the form  φ → ψ, Modus Ponens may be used to formulate a derived rule of the form: If  φ, then  ψ. For convenience, here is a list of such rules, as well as one theorem, of classical logic that will be appealed to in proofs below: (I) If  φ ≡ ψ,  φ ≡ φ1 and  ψ ≡ ψ1 , then  φ1 ≡ ψ1 . (Closure of ≡ under logical equivalence) (II) If  φ ≡ ψ, then  ∃xφ ≡ ∃xψ. (Closure of ∃ under logical equivalence) (III)  ∃x(φ ∨ ψ) ≡ (∃xφ ∨ ∃xψ). (Closure under, and distribution over, disjunction of ∃). (IV) If  φ ≡ ψ, then  φ → ψ. (V) If  φ → ψ and  ψ → χ, then  φ → χ. (Transitivity of →) (VI) If  ¬φ and  ¬ψ, then  φ ≡ ψ. (Logical equivalence of all contradictions) In addition, there are theorems and rules validated by the semantics of POT and λ. The following are schemas for theorems of P. First, a rule and a theorem for λ:

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formal adequacy If  φ ≡ ψ, then  λx.φ(t) ≡ λx.ψ(t).

Closureλ

(Tλ)  φ ≡ λx.φ(t), where x is not free in φ. And secondly, a rule and three theorems containing POT: ClosurePOT

If  t ≡ t, then POT[](t) ≡ POT[ ](t).

(POT∨)  POT[λx.(φ ∨ ψ)](t) ≡ (POT[λx.φ](t) ∨ POT[λx.ψ](t)). (TPOT ) (NCPOT )

 t → POT[](t).  ¬ POT[λx.⊥](t).

(⊥ is an abbreviation for an arbitrary contradiction.)

3.2 Proofs I will begin with the proofs for (Closureλ) and (Tλ). They will be very explicit in their appeal to the semantics set out above. After that, my proofs will take a slightly more condensed form. In particular, I will omit reference to the semantic clauses and to assumptions A1 to A3. Axioms containing POT will be proved by relating the semantics for expressions of L to the metaphysics of potentiality as stated in chapter 5. As a reminder, here are the informal principles: CLOSURE Potentiality is closed under logical equivalence: If being  is logically equivalent to being , then having a potentiality to be  is logically equivalent to having a potentiality to be . DISJUNCTION Potentiality distributes over, and is closed under, disjunction: An object has a potentiality to be -or- if, and only if, it has a potentiality to be  or a potentiality to be . ACTUALITY Potentiality is implied by actuality: Anything which is  must also have a potentiality to be . NON-CONTRADICTION Nothing has a potentiality to be such that a contradiction holds. For our purposes, it will be useful to reformulate DISJUNCTION so as to have the disjunctive ‘or’ function as a sentential connective: DISJUNCTION∗ Potentiality distributes over, and is closed under, disjunction: An object has a potentiality to be such that φ-or-ψ if, and only if, it has a potentiality to be such that φ or a potentiality to be such that ψ. Proof of Closureλ. Suppose  φ ≡ ψ, i.e. for all models M = D, v and all assignments s, v s (φ ≡ ψ), the biconditional of v s (φ) and v s (ψ), is a true proposition. By A2, then, for all models M = S , v and assignments s, v s (φ) is true iff v s (ψ) is true, i.e. v s (φ)

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and v s (ψ) are either both true or both false. In particular, for all d ∈ D, v s[d/x] (φ) is true iff v s[d/x] (ψ) is true, i.e. v s[d/] (φ) and v s[d/x] (ψ) are either both true or both false. Take an arbitrary model M = D, v and an arbitrary assignment s. Let f =

(φ, x, v, s), the function that takes every d ∈ D to the proposition v s[d/x] (φ), and let f  = (ψ, x, v, s), the function that takes every d ∈ D to the proposition v s[d/x] (ψ). Since for every d ∈ D, v s[d/x] (φ) is true iff v s[d/x] (ψ) is true, it follows that for all d, f (d) is a true proposition just in case f  (d) is a true proposition. Now, v s (λx.φ) = the property of being a d ∈ D such that f (d) is true. By A3, every d ∈ D instantiates that property just in case f (d) is true. By exactly parallel reasoning, every d ∈ D instantiates the property v s (λx.ψ) just in case f  (d) is true. So, for every d ∈ D: d has v s (λx.φ) if and only if f (d) is true; f (d) is true if and only if f  (d) is true; f  (d) is true if and only if d has v s (λx.ψ); hence d has v s (λx.φ) if and only if d has v s (λx.ψ). Furthermore, for every term t, v s (λx.φ(t)) is true just in case v s (t) instantiates s v (λx.φ), and v s (λx.ψ(t)) is true just in case v s (t) instantiates v s (λx.ψ). Take a term t and object d ∈ D such that d = v s (t). Then v s (λx.φ(t)) is true just in case d instantiates v s (λx.φ); d instantiates λx.φ if and only if d instantiates v s (λx.ψ); and d instantiates v s (λx.ψ) just in case v s (λx.ψ(t)) is true; hence v s (λx.φ(t)) is true if and only if v s (λx.ψ(t)) is true. Hence the biconditional of v s (λx.φ(t)) and v s (λx.ψ(t)) is true; hence v s (λx.φ(t) ≡ λx.ψ(t)) is true. Since M and s were arbitrary, the same holds for all models and assignments:  λx.φ(t) ≡ λx.ψ(t). Proof of (Tλ). Take a model M = D, v, an assignment s, and a variable x such that x is not free in φ, and let f = (φ, x, v, s). That is, for any d ∈ D, f (d) = vs[d/x] (φ). Since x is not free in φ, v s[d/x] (φ) = v s (φ), and hence f (d) = v s (φ), for every d ∈ D. Now, v s (λx.φ(t)) is true just in case the object v s (t) instantiates the property s v (λx.φ). v s (λx.φ) = the property of being a d ∈ D such that f (d) is true. By A3, any d ∈ D instantiates that property if and only if f (d) is true; since f (d) = v s (φ), it follows that every d ∈ D instantiates v s (λx.φ) if and only if v s (φ) is true; in particular, v s (t) instantiates v s (λx.φ) if and only if v s (φ) is true. Hence v s (λx.φ(t)) is true just in case v s (φ) is true. So the biconditional of v s (φ) and v s (λx.φ(t)) is true; hence v s (φ ≡ λx.φ(t)) is true. Since M, s were arbitrary, the same holds for all models and assignments:  φ ≡ λx.φ(t). Proof of ClosurePOT . Suppose  t ≡ t. Take an arbitrary model M = D, v and assignment s and let P1 = v s (), P2 = v s ( ) and d = v s (t). Then as a matter of logic (the logic of system P), d instantiates P1 if and only if d instantiates P2 ; d’s instantiating P1 is logically equivalent to d’s instantiating P2 . Now suppose that v s (POT[](t)) is true. Then d has the potentiality to instantiate P1 . Since d’s instantiating P1 is logically equivalent to d’s instantiating P2 , it follows by CLOSURE that d has the potentiality to instantiate P2 , and hence that v s (POT[ ](t)) is true. So if v s (POT[](t)) is true, then so is v s (POT[ ](t)). By exactly parallel reasoning, if v s (POT[ ](t)) is true, then so is v s (POT[](t)). Hence v s (POT[](t))

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is true if and only if v s (POT[ ](t)) is true; so v s (POT[](t) ≡ POT[ ](t)) is true. Since M, s were arbitrary, the same holds for all models and assignments:  POT[](t) ≡ POT[ ](t). Proof of (POT∨). Take a model M = D, v and an assignment s. Let f (d) =

(φ, x, v, s), f  (d) = (ψ, x, v, s), and f  (d) = ((φ ∨ ψ), x, v, s). For any d ∈ D, f  (d) is a disjunction with exactly two disjuncts: f (d) and f  (d). Given DISJUNCTION∗ , an object d ∈ D has a potentiality to be such that f  (d), the disjunction of f (d) and f  (d), is true just in case d has a potentiality to be such that f (d) is true or d has a potentiality to be such that f  (d) is true. Take d ∈ D such that d = v s (t). Then we can again reason by a chain of equivalences: v s (POT[λx.(φ ∨ ψ)](t) is true iff d has a potentiality to be such that f  (d) is true iff d has a potentiality to be such that f (d) is true or d has a potentiality to be such that f  (d) is true iff v s (POT[λx.φ](t)) is true or v s (POT[λx.ψ](t)) is true iff v s (POT[λx.φ](t))∨ POT[λx.ψ](t)) is true. So: v s (POT[λx.(φ ∨ψ)](t)) is true iff v s (POT[λx.φ](t))∨ POT[λx.ψ](t)) is true; hence v s (POT[λx.(φ ∨ ψ)](t) ≡ (POT[λx.φ](t) ∨ POT[λx.ψ](t))) is true. Since M, s were arbitrary, (POT∨) is a theorem. Proof of (TPOT ). Take a model M = D, v and an assignment s. Let d = v s (t) and P = v s (), and suppose that v s (t) is true. Then d instantiates P. By ACTUALITY, if d instantiates P, it follows that d has a potentiality to instantiate P, and hence that v s (POT[](t)) is true. So if v s (t) is true, then so is v s (POT[](t)); v s (t →POT[](t)) is true. Since M, s were arbitrary, (TPOT ) is a theorem. Proof of (NCPOT ). Take a model M = D, v and an assignment s. Let f = (⊥, x, v, s). Since a sentence that is a contradiction remains a contradiction under any assignment, f (d) will be a contradiction for all d ∈ D. Let d = v s (t). By NON-CONTRADICTION, nothing has a potentiality to be such that a contradiction is true. Hence it is not true that d has a potentiality to be such that the contradiction f (d) is true. So v s (¬POT[λx.⊥](t)) is true. Since M and s were arbitrary, the same holds for all models and assignments; (NCPOT ) is a theorem.

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4 The operator 3 I now introduce an operator 3, which can serve as a first approximation to possibility. (The qualification is important: metaphysical possibility will be defined as an iteration of 3.) It is defined as an existential generalization on POT: (Def3)

3φ =df ∃x POT[λx.φ](x), x the first variable not free in φ.

Intuitively, 3φ says that something has a potentiality to be such that φ.

4.1 Theorems and a rule Closure3

If  φ ≡ ψ, then  3φ ≡ 3ψ,

(3∨)  3(φ ∨ ψ) ≡ (3φ ∨ 3ψ), (T3)  φ → 3φ, (3C)  ¬3⊥. The following corollary captures an intuitive connection between potentiality and possibility: if an object has (or some objects have) a potentiality to , then it is possible that that object ’s (or that those objects ): Corollary 1

 POT[](t) → 3(t).

4.2 Proofs Proof of Closure3. Suppose  φ ≡ ψ, and let x be the first variable not free in φ or ψ.2 By (Tλ), we have in addition  φ ≡ λx.φ(x) and  ψ ≡ λx.ψ(x). By classical logic (I),  λx.φ(x) ≡ λx.ψ(x). And then by ClosurePOT , it follows that  POT[λx.φ](x) ≡ POT[λx.ψ](x). By classical logic (II), it follows that  ∃x POT[λx.φ](x) ≡ ∃x POT[λx.ψ](x), so by (Def3)  3φ ≡ 3ψ. 2 There may be no one variable that is both the first variable not free in φ and the first variable not free in ψ. To apply (Def3) below, we will then need an extra step of substituting logical equivalents. The same goes for the proof of (3∨) below.

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Proof of (3∨). Given (POT∨), we already have  POT[λx.(φ ∨ ψ)](x) ≡ (POT[λx.φ](x)∨ POT[λx.ψ](x)). for any x, including x the first variable not free in φ or ψ. By classical logic (II), it follows that  ∃x POT[λx.(φ ∨ ψ)](x) ≡ ∃x (POT[λx.φ](x)∨ POT[λx.ψ](x)). The right-hand side of this equivalence being logically equivalent, by classical logic (III), to ∃x POT[λx.φ](x)∨∃x POT[λx.ψ](x), it follows again by classical logic (I), that  ∃x POT[λx.(φ ∨ ψ)](x) ≡ (∃x POT[λx.φ](x) ∨ ∃x POT[λx.ψ](x)). Then by (Def3),  3(φ ∨ ψ) ≡ (3φ ∨ 3ψ). Proof of (T3). Take a model M = D, v and an assignment s, and suppose that v s (φ) is a true proposition. Let x be the first variable not free in φ. Then by (Tλ), we know that v s (φ → λx.φ(x)) is also true; that is, if v s (φ) is true, then so is v s (λx.φ(x)). Given the initial assumption, then, v s (λx.φ(x)) must be true. By (TPOT ), we know further that v s (λx.φ(x) → POT[λx.φ](x)) is true; that is, if v s (λx.φ(x)) is true, then so is v s (POT[λx.φ] (x)). Hence from the initial assumption that v s (φ) is true, it follows that so is v s (POT [λx.φ](x)). Now let d = v s (x). Then v s[d/x] (POT[λx.φ](x)) = v s (POT[λx.φ](x)), hence v s[d/x] (POT[λx.φ](x)) is true. By lemma 1, then, v s (∃xPOT[λx.φ](x)) is true; hence (by definition) v s (3φ) is true. So: if v s (φ) is true, then so is v s (3φ); hence v s (φ → 3φ) is true. Since M and s were arbitrary, the same holds for all models relative to all assignments:  φ → 3φ. Proof of (3C). Suppose (for reductio) that for some model M = D, v and some assignment s, v s (3⊥) was true. By definition, that is to say that v s (∃x POT[λx.⊥](x)) is true, for x the first variable not free in ⊥. By lemma 1, there must be a d ∈ D such that v s[d/x] (POT[λx.⊥](x)) is a true proposition. But by (NCPOT ), we also know that v s[d/x] (¬POT[λx.⊥](x) is true, and consequently that v s[d/x] (POT[λx.⊥](x) is not true. Hence the initial assumption must be rejected: v s (3⊥) cannot be true, and its negation, v s (¬3⊥), must be true. Since M and s were arbitrary, the same holds for all models and assignments:  ¬3⊥. Proof of Corollary 1. By (Tλ), we have  t ≡ λx.(t)(t), And from that by ClosurePOT it follows that  POT[](t) ≡ POT[λx.t](t).

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By classical logic (IV), we also have the weaker theorem  POT[](t) → POT[λx.t](t). Now, take a model M = D, v and an assignment s, and suppose that v s (POT[](t)) is a true proposition. Since POT[](t) → POT[λx.t](t) is a theorem, v s (POT[](t) → POT[λx.t](t)) must be true. So if v s (POT[](t)) is true, then so is v s (POT[λx.t](t)). We have assumed that the antecedent holds; so it follows that v s (POT[λx.t](t)) is true. Now take d ∈ D such that v s (t) = d. Then v s (POT[λx.t](t)) = v s[d/x] (POT[λx.t](x)). Since, given our initial assumption, v s (POT[λx.t](t)) is true, it follows that v s[d/x] (POT[λx.t](x)) is true. And then by lemma 1, v s (∃x POT[λx.t](x)) is true; hence, by definition, v s (3t) is true. So: if v s (POT[](t)) is true, then so is v s (3t); v s (POT[](t) → 3t) is true. And since M and s were arbitrary, the same holds for all models and assignments:  POT[](t) → 3t.

5 Possibility: the operator 3∗ 5.1 Introduction: possibility and iterated potentiality The operator 3 is not an adequate expression of metaphysical possibility: it is an existential generalization on non-iterated potentiality, while metaphysical possibility is to be understood as an existential generalization on iterated potentiality (or so I have argued). One way of formally rendering that understanding would be by defining an operator for iterated potentiality and then defining the new possibility operator in terms of it. While this will not ultimately be my strategy, it will be instructive to see how it would be done. An iterated potentiality, I have said, is a potentiality whose manifestation consists in something (perhaps the iterated potentiality’s possessor, but perhaps not) having a potentiality (whose manifestation may in turn consist in something having a potentiality, etc.) to be such that p (chapters 4.6, 5.4). We can count the number of iterations on a potentiality: a potentiality to be F is once iterated, a potentiality for something to have a potentiality to be such that p is twice iterated, etc. Here is a suggestion for formally capturing the notion of an n-times iterated potentiality: Syntax of POTn : if φ is a sentence and x a variable not free in φ, then for any natural number n ≥ 1, POTn [λx.φ] is a predicate. Definition of POTn : 1. POT1 [λx.φ](t) =df POT[λx.φ](t), 2. POTn+1 [λx.φ](t) =df POT[λx.∃x POTn [λx.φ](x)](t). We can then define a generalized operator for potentiality that is iterated any number of times, say POT∗ : Syntax of POT∗ : if φ is a closed sentence and x a variable not free in φ, then POT∗ [λx.φ] is a predicate.

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Semantics of POT∗ : If v is a valuation, s an assignment, φ a sentence and x a variable not free in φ: Let f be the function that takes every natural number n ≥ 1 to the property v s (POTn [λx.φ]). Then v s (POT∗ [λx.φ]) = the property of possessing at least one of the f (n), for all natural numbers n ≥ 1. (Note that the introduction of POT∗ , which is not directly defined in terms of POTn , would constitute an expansion of our language.) The idea that possibility is an existential generalization on iterated potentiality would then be captured by a definition such as the following: (∗ )

3φ =df ∃x POT∗ [λx.φ](x), for x the first variable not free in φ.

Why, then, am I not going to go for this strategy? Because we can make things easier than this. Consider that, if 3φ is to be true according to (∗ ), then at least one of the following will be true (by the semantic clause for POT∗ and the definition of POTn ):

• ∃x POT[λx.φ](x), • ∃x POT[λx.∃x POT[λx.φ](x)](x), • ∃x POT[λx.∃x POT[λx.∃x POT[λx.φ](x)](x)](x). ....

In representing an iterated potentiality, we apply an existential generalization to a POT sentence in very much the same way as we do in defining the possibility operator; only in the former case we do so only inside the scope of a λ operator which is itself embedded in the scope of a POT operator. Formally, that difference in treatment seems spurious. Let us try an alternative treatment then. We can retain the original definition of the operator 3 in terms of non-iterated potentiality: (Def3)

3φ ≡ ∃x POT[λx.φ](x), for x the first variable not free in φ.

Substituting 3 in accordance with this definition, we could rewrite the above sequence of sentences as follows:

• ∃x POT[λx.φ](x), • ∃x POT[λx.3φ](x), • ∃x POT[λx.33φ](x), ...

That is to say, we can understand an iterated potentiality as a potentiality whose manifestation is a possibility. Or indeed, we could leave talk of iterated potentialities completely out of the picture and apply (Def3) more thoroughly, thus transforming our sequence of sentences into

• 3φ, • 33φ, • 333φ, ...

It seems, then, that we need not define an operator for iterated potentiality to capture that sequence; all we need to iterate is the possibility operator that has already been defined.

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In omitting the formalization of iterated potentialities, we are not omitting the iterated potentialities themselves: given the metaphysics developed in chapters 4 and 5, iterated potentialities are what make a sentence of the form 3φ, 33φ, etc., true: intuitively, these sentences say that something has a once-, twice-, or however-many-times-iterated potentiality for it to be the case that φ. In giving up the formalization of iterated potentiality, we merely relinquish the opportunity to ascribe an iterated potentiality to a particular object (or to particular objects): we are left only with a device for expressing that something has an iterated potentiality. In the long run, we may wish to ascribe iterated potentialities to particular objects, and I have indicated how we should go about doing so. My present concern, however, is the definition of a possibility operator based on the logic of potentiality outlined in the previous sections of this appendix. For that concern it will be perfectly sufficient to iterate 3 as defined by (Def3). I will now go on to spell out this idea in detail.

5.2 Syntax and semantics I begin by introducing an operator 3n , which can be defined inductively as follows: (Def3n )

1. 30 φ = φ, 2. 3n+1 φ = 33n φ.

Intuitively, 3n is merely an abbreviation for a sequence of exactly n occurrences of 3. Semantically, 3∗ is a generalization over 3n . Syntactically, it is an addition to our language. Let L∗ be L with 3∗ added to the vocabulary. The syntax of the new operator is obvious:

• If φ is a sentence, then so is 3∗ φ. Now we need to introduce a further function that will facilitate formulating the semantics: Let (φ, v, s) be the function that takes every natural natural number n to the proposition v s (3n φ). Then we can add the following clause to the semantic rules given in section 2.2:

• If φ is of the form 3∗ ψ and f = (ψ, v, s), then v s (φ) = the disjunction of f (n) over all n ∈ N.

5.3 Theorems and a rule for 3∗ 3∗ , like 3, is closed under logical equivalence and generates exactly similar theorems.

Here they are: Closure3∗ If  φ ≡ ψ, then  3∗ φ ≡ 3∗ ψ, (3∗ ∨)  3∗ (φ ∨ ψ) ≡ (3∗ φ ∨ 3∗ ψ), (T3∗ ) φ → 3∗ φ, (3∗ C) ¬3∗ ⊥. A corollary parallel to Corollary 1 can also be proved: Corollary 2

 POT[](t) → 3∗ (t).

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In addition, there is reason to believe that the semantics of 3∗ validates the theorem typical of S4: (43∗ ) 3∗ 3∗ φ → 3∗ φ. (43∗ ) can be proved conditional on certain formal and metaphysical assumptions, to be highlighted below.

5.4 Proofs My general strategy in proving those theorems containing 3∗ that correspond to theorems containing 3 will be the following. In a first part, I show that the theorem in question holds when 3∗ is replaced throughout by 3n , for every natural number n. This is done by showing (A) that the theorem containing 3n is valid for n = 0; and (B) that for every natural number n, if the theorem is valid for n, it is also valid for n + 1. By mathematical induction, (A) and (B) together deliver the desired result, that the theorem in question holds for every value of n. In a second part, then, I use that result to prove the theorem containing 3∗ . As in section 3.2, I will begin with a relatively explicit proof, which should demonstrate the kind of reasoning involved in proofs of this kind. The rest of the proofs will be more abbreviated. Proof of Closure3∗ . Part 1: (Closure3n ) If  φ ≡ ψ, then  3n φ ≡ 3n ψ. (A) φ ≡ ψ is definitionally identical to 30 φ ≡ 30 ψ. Since every formula follows from itself: If  φ ≡ ψ, then  30 φ ≡ 30 ψ. (B) Suppose that Closure3n already held for a given fixed value of n ∈ N. Suppose further that  φ ≡ ψ; it follows that  3n φ ≡ 3n ψ. By Closure3, it follows that  33n φ ≡ 33n ψ. Then by definition,  3n+1 φ ≡ 3n+1 ψ. So: If  φ ≡ ψ, then  3n+1 φ ≡ 3n+1 ψ. By mathematical induction, (A) and (B) together entail that Closure3n holds for every value of n ∈ N. Part 2: Closure3∗ . Suppose  φ ≡ ψ. By Closure3n , it follows that  3n φ ≡ 3n ψ, for every n ∈ N. Now take a model M = D, v and an assignment s. Let f = (φ, v, s) and f  = (ψ, v, s). Suppose v s (3∗ φ) is true. Since v s (3∗ φ) = the disjunction of f (n) over all n ∈ N, at least one disjunct f (k) = v s (3k φ) of that disjunction must be true. By Closure3n , v s (3k φ ≡ 3k ψ) is true. It follows that, since v s (3k φ) = f (k) is true, v s (3k ψ) = f  (k) must be true too. Hence v s (3∗ ψ), the disjunction of f  (n) for all n ∈ N has at least one true disjunct, f  (k), and must therefore itself be true. So: if v s (3∗ φ) is true, then so is v s (3∗ ψ). By exactly parallel reasoning, if v s (3∗ ψ) is true, then so is v s (3∗ φ).

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Taking both directions together, v s (3∗ ψ) is true iff v s (3∗ φ) is true. Hence ≡ 3∗ ψ) is true. Since M and s were arbitrary, the same holds for all models relative to all assignments:  3∗ φ ≡ 3∗ ψ. v s (3∗ φ

Proof of (3∗ ∨). Part 1: (3n ∨) 3n (φ ∨ ψ) ≡ (3n φ ∨ 3n ψ) (A) 30 (φ ∨ ψ) =df (φ ∨ ψ) =df (30 φ ∨ 30 ψ). Hence  30 (φ ∨ ψ) ≡ (30 φ ∨ 30 ψ). (B) Suppose we have (3n ∨), for a given fixed value of n ∈ N. By Closure3,  33n (φ ∨ ψ) ≡ 3(3n φ ∨ 3n ψ). Further, by (3∨):  3(3n φ ∨ 3n ψ) ≡ (33n φ ∨ 33n ψ). Applying classical logic (I) to these two formulas, we get  33n (φ ∨ ψ) ≡ (33n φ ∨ 33n ψ). So by (Def3n ),  3n+1 (φ ∨ ψ) ≡ (3n+1 φ ∨ 3n+1 ψ). By mathematical induction, (3n ∨) holds for all n ∈ N. Part 2: Proof of (3∗ ∨). Take a model M = D, v and an assignment s, and let f = (φ, v, s), f  = (ψ, v, s), f  = ((φ ∨ ψ), v, s). Given (3n ∨), for all n ∈ N, f  (n) is true iff f (n) is true or f  (n) is true. The truth of v s (3∗ (φ ∨ ψ) ≡ (3∗ φ ∨ 3∗ ψ) is proved by proving both directions of the biconditional. (i) Suppose v s (3∗ (φ ∨ ψ)) is true. Then for at least one n ∈ N, f  (n) must be true. If f  (n) is true, then f (n) or f  (n) must be true. If f (n) is true, it follows that v s (3∗ φ) must be true; if f  (n) is true, it follows that v s (3∗ ψ) must be true. Since at least one of f (n) and f  (n) must be true, it follows that v s (3∗ φ ∨ 3∗ ψ) is true. (ii) Suppose v s (3∗ (φ ∨ψ)) is not true. Then for no n ∈ N, f  (n) is true; accordingly, for no n ∈ N either f (n) or f  (n) is true. So neither v s (3∗ φ) nor v s (3∗ ψ) can be true. Therefore their disjunction, v s (3∗ φ ∨ 3∗ ψ), cannot be true either. Taking (i) and (ii) together: v s (3∗ (φ ∨ ψ)) is true iff v s (3∗ ψ ∨ 3∗ ψ) is true, and so ≡ (3∗ ψ ∨ 3∗ ψ)) is true. Since M, s were arbitrary, (3∗ ∨) is a theorem.

v s (3∗ (φ ∨ψ)

Proof of (T3∗ ). The proof of (T3∗ ) is trivial: take a model M= D, v and an assignment s, and let f = (φ, v, s). If v s (φ) is true, then by definition v s (30 φ) is true, hence f (0) is true, hence the disjunction of f (n) over all n ∈ N is true; hence v s (3∗ φ) is true. So v s (φ → 3∗ φ) is true. M and s were arbitrary, so  φ → 3∗ φ. This triviality should not cause any concern, however; given (T3), (T3∗ ) could be proved in a non-trivial way. Proving it in this way simply speeds things up.

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Proof of (3∗ C). Part 1: (3n C) ¬3n ⊥ (A) 30 ⊥ is identical, by definition, to ⊥, and ¬⊥ is a theorem of classical logic. Hence  ¬30 ⊥. (B) Suppose that we have, for a given fixed value of n ∈ N,  ¬3n ⊥. Given  ¬⊥, it follows by classical logic (VI) that  ⊥ ≡ 3n ⊥. Hence by Closure3,  3⊥ ≡ 33n ⊥, which is by definition identical to  3⊥ ≡ 3n+1 ⊥. So for all models M = D, v and assignments s, v s (3⊥) is true iff v s (3n+1 ⊥) is true. Given (3C), v s (3⊥) cannot be true; so v s (3n+1 ⊥) is not true and v s (¬3n+1 ⊥) is true. Since M and s were arbitrary, it follows that  ¬3n+1 ⊥. By mathematical induction, (3n ⊥) holds for every value of n ∈ N. Part 2: Proof of (3∗ ⊥). Take a model M = D, v and an assignment s, and let f = (⊥, v, s). Then v s (3∗ ⊥) is the disjunction of f (n) over all n ∈ N. By part 1, for all n ∈ N, v s (¬3n ⊥) is true, and so f (n) = v s (3n ⊥) is not true. Accordingly, v s (3∗ ⊥), the disjunction of f (n) over all n ∈ N, is not true; hence, v s (¬3∗ ⊥) is true. Since M, s were arbitrary,  ¬3∗ ⊥. Proof of Corollary 2. Take an arbitrary model M = D, v and an assignment s, and let f = (t, v, s). Then v s (3∗ t) is the disjunction of f (n) over all n ∈ N. Suppose that v s (POT[](t)) is true. By Corollary 1, it follows that v s (3t) is true. Since v s (3t) = v s (31 t) = f (1), it follows that the disjunction of f (n) over all n ∈ N has one true disjunct, hence is itself true. So if v s (POT[](t)) is true, then v s (3∗ t) is true; v s (POT[](t) → 3∗ t) is true. Since M, s were arbitrary,  POT[](t) → 3∗ t. Note that the converse of Corollary 2 does not hold. 3∗ t may be true because of a (more than once) iterated potentiality that the referent of t possesses, without there being a corresponding (once-iterated) potentiality to provide the relevant true interpretation of POT[](t). Conditional proof of (43∗ ). To prove (43∗ ), we first need to prove two lemmas. Lemma 2 3n+1 φ = 3n 3φ. Proof. By definition, 3n+1 φ = 33n φ = 333n–1 φ = . . . etc. until a formula is reached where the final superscript is 1. That formula will be a sequence of exactly n + 1 occurrences of 3, followed by φ. Also by definition, 3n 3φ = 33n–1 3φ = 333n–1 3φ = . . . , etc. until the superscript 1 is reached. Again, that formula will be a sequence of exactly n + 1 occurrences of 3, followed by φ. Hence the two formulas are the same. Lemma 3 3n 3m φ = 3m+n φ. Proof. By lemma 2, 3n 3m φ = 3n–1 33m φ, which by (Def3∗ ) is identical to 3n–1 3m+1 φ. By alternating applications of lemma 2 and (Def3∗ ), the formula can be repeatedly transformed in this way until 3n–n 3m+n φ = 30 3m+n φ = 3m+n φ is reached.

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Second, I need to make an assumption. To begin, assume that we can introduce infinitary disjunction into the language L∗ without causing any problems. (I will not discuss this background assumption any further.) Then the main assumption is this: Assumption: For any number n, 3n distributes over infinite disjunctions, i.e.  3n (φ1 ∨ φ2 ∨ . . . ) → (3n φ1 ∨ 3n φ2 ∨ . . . ) holds even when there are infinitely many disjuncts φi . Why should we believe this assumption to be true? It is true if 3 distributes over infinite disjunctions, which in turn will be true if POT does. So the truth of the assumption depends on the behaviour of potentialities with infinitely disjunctive manifestations. Intuitively, the assumption that potentiality distributes over infinite disjunction seems correct: given that we can think of determinables as (entailing, if not identical to) disjunctions of their determinates, and many determinables have infinitely many determinates, the considerations adduced in favour of DISJUNCTION in chapter 5.7.2 suggest that potentiality distributes over infinite disjunctions. However, I do not wish to fully endorse this principle until the relation between degrees of potentialities and the probability calculus has been more thoroughly investigated. For we can think of having a potentiality as having it to some degree greater than zero. If degrees of potentialities behave like probabilities, then they may be faced with familiar problems as interpretations of the probability calculus (see chapter 3.5; having a non-zero probability does not, or at least not without problems, distribute over infinite disjunctions. I have suggested that we should take that observation to tell against a probabilistic model for degrees of potentialities, rather than against the distribution of potentiality over infinite disjunction. If that suggestion can be maintained, then the stated assumption is defensible. But for the time being, my argument for (43∗ ) will be hypothetical: if the stated assumption is true, then (43∗ ) is a theorem. Here is the proof. Take a model M = D, v and an assignment s. For all n ∈ N, let f (n) = (φ, v, s) and f  (n) = (3∗ φ, v, s). Suppose v s (3∗ 3∗ φ) is true. v s (3∗ 3∗ φ) is the infinite disjunction of f  (n) over all n ∈ N. Each disjunct of that disjunction, however, is itself an infinite disjunction: the disjunction of f (n) over all n ∈ N. Schematically, then, 3∗ 3∗ φ is a formula of the form 30 (30 φ ∨ 31 φ ∨ . . .) ∨ 31 (30 φ ∨ 31 φ ∨ . . .) ∨ 32 (30 φ ∨ 31 φ ∨ . . .) ∨ . . .

where ‘. . .’ indicates continuation ad infinitum. If my hypothetical assumption is true, then the n-step possibility operator at the beginning of each disjunct distributes over the infinite disjunction in its scope. Schematically, that is to say that our infinite disjunction is equivalent to this infinite disjunction: (30 30 φ ∨ 30 31 φ ∨ . . .) ∨ (31 30 φ ∨ 31 31 φ ∨ . . .) ∨ (32 30 φ ∨ 32 31 φ ∨ . . .) ∨ . . . And using lemma 3, we can then do away with the duplication of diamonds and contract each pair of consecutive diamonds into one with a new superscript: (30 φ ∨ 31 φ ∨ . . .) ∨ (31 φ ∨ 32 φ ∨ . . .) ∨ (32 φ ∨ 33 φ ∨ . . .) ∨ . . .

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This infinite disjunction, though infinitely repetitive, is logically equivalent to the disjunction of f (n) over all n ∈ N, and hence to v s (3∗ φ). So if the stated assumption holds, then v s (3∗ 3∗ φ → 3∗ φ) is true. Since M, s were arbitrary, (43∗ ) is a theorem, conditional on the assumption of distribution over infinite disjunctions.

5.5 Possibility and Necessity I have introduced two distinct possibility operators, 3 and 3∗ , and shown that both are rightly associated with possibility, since both satisfy the formal requirements for an operator to express possibility: closure under logical equivalence, closure under and distribution over disjunction, implication by actuality (T), and non-applicability to contradictions (3C); 3∗ in addition probably yields the characteristic S4 theorem. I have suggested that 3∗ expresses metaphysical possibility. My task, it would seem, is finished. But there are two further interesting lines that are worth pursuing for a moment. One, of course, is necessity. Not that I needed to prove anything about it: the normality of my modal logic has been established through the theorems and rules for 3 and 3∗ , so we can be assured that introducing their corresponding necessity operators in the usual way will preserve that normality. Nonetheless, it may be interesting to have a closer look. The second line is the relation between the starred and the non-starred operators. I will discuss that relation in the next section and also prove an interesting theorem, putting to use the observations about necessity. Let me, then, begin with necessity. We can define two necessity operators in the standard way: φ =df ¬3¬φ, ∗ φ =df ¬3∗ ¬φ. Since we have seen that both 3 and 3∗ yield a normal modal logic, we can be assured that the two necessity operators defined in this standard way will behave in an equally normal way. In particular, we know that the following will be theorems and rules of P: Nec

If  φ, then  φ,

(K)  (φ → ψ) → (φ → ψ), (T)  φ → φ, Nec∗

If  φ, then  ∗ φ,

(K∗ )

 ∗ (φ → ψ) → (∗ φ → ∗ ψ),

(T)  ∗ φ → φ, (4∗ )

 ∗ φ → ∗ ∗ φ.

All of these can be derived from the theorems given for 3 and 3∗ , respectively, in the usual way. It is the operator ∗ that expresses metaphysical necessity. Note that, once we have defined , we could from there go on to introduce ∗ without appeal to 3∗ , but in a way that is entirely parallel to the way in which 3∗ has been introduced. Here it is.

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First, I define n : 1. 0 φ = φ, 2. n+1 φ = n φ. Intuitively, n is merely an abbreviation for a sequence of exactly n occurrences of . ∗ is a generalization over n . Its syntax is obvious: If φ is a sentence, then so is ∗ φ. Let ρ(φ, v, s) be the function that takes every natural natural number n to the proposition v s (n φ). Then we could add the following clause to the semantic rules: (∗ ) If φ is of the form ∗ ψ and f = ρ(ψ, v, s), then v s (φ) = the conjunction of f (n) over all n ∈ N. I will, however, not add that clause to the semantic rules; rather, I will understand ∗ as defined in terms of 3∗ . But importantly, both strategies are equivalent when it comes to the truth conditions for ∗ φ, though they may (depending on one’s view of propositions) be seen as assigning different semantic values to the formula. The following lemma states their truth-conditional equivalence, which I will appeal to in the next section. Lemma 4 For all models M = D, v and assignments s, if f = ρ(φ, v, s), v s (∗ φ) is true just in case the conjunction of f (n) over all n ∈ N is true. Proof. Take an arbitrary model M = D, v and assignment s, and let f = ρ(φ, v, s) and f  = (¬φ, v, s); i.e. for all n ∈ N, f (n) = v s (n φ) and f  (n) = v s (3n ¬φ). Then by the definition of ∗ , v s (∗ φ) = v s (¬3∗ ¬φ) = the negation of v s (3∗ ¬φ) = the negation of the disjunction of f  (n) over all n ∈ N. The negation of a disjunction is true iff the conjunction of the negations of all its disjuncts is true; hence v s (∗ φ) is true iff the conjunction of the negation of f  (n), over all n ∈ N, is true. Since for every n ∈ N, f  (n) = v s (3n ¬φ), the negation of f  (n), for any n ∈ N, is v s (¬3n ¬φ) = v s (n φ). Hence v s (∗ φ) is true iff the conjunction of v s (n φ) = f (n) over all n ∈ N is true.

5.6 3 and 3∗ ,  and ∗ Now to the next line: how does 3∗ relate to 3, and ∗ to ? A model for their relationship can be found in dynamic logic, which has the modal operators a and [a] with roughly the following semantics: let a be a programme, that is, an operation which takes states of the world to (potentially different) states of the world. Let s, t stand for such states. Then aφ is true at a state s iff there is some state t such that a can take s to t and at t, φ is true; [a]φ is true at s iff at all states t such that a can take s to t, φ is true. The interesting parallel, however, is not with this semantics (which is a species of possible-worlds semantics), but with the further operators a∗  and [a∗ ]. Where aφ means, informally, that running the programme a could take the world to a state where φ (or its semantic value) is true, a∗  means, informally, that running the programme a any

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(nonnegative) number of times could take the world to a state where φ (or its semantic value) is true; equally, mutandis mutatis, for [a] and [a∗ ]. As with my 3∗ and ∗ operators, the asterisk is used in dynamic logic to indicate iteration. Rather than pursuing this intuitive parallel, however, I would like to point out that my semantics validates the same theorems containing the operators 3, 3∗ , and , ∗ as does dynamic logic, mutatis mutandis. In particular, the system and semantics as given so far validate the ‘induction’ axiom of dynamic logic, which provides an interesting connection between the starred and non-starred necessity operators. Here is the original (from Harel 1983, 512): (Ind) [a∗ ](φ → [a]φ) → (φ → [a∗ ]φ). (Ind) can be reformulated, via the interdefinability of [a] and a and classical logic, as (Ind )

a∗ φ → (φ ∨a∗ (φ ∧aφ)).

I will now prove the theorem corresponding to (Ind) for the necessity operators of my system P. However, to do so, I first need to derive some preliminary results. Preliminary Results This proof requires two preliminary results. The first is simple: Lemma 5 n+1 φ = n φ. Proof. The proof is exactly parallel to that of Lemma 2 in section 5.4. The second result that I will need is the K axiom for n , for every value of n ∈ N. The proof will proceed in the familiar way, showing the theorem first to hold for n = 0, and then to hold for n + 1 whenever it holds for a number n. (Kn )  n (φ → ψ) → (n φ → n ψ). Proof. (A) For n = 0, (Kn ) amounts to  (φ → ψ) → (φ → ψ), which is a truthfunctional tautology and therefore a theorem of P. (B) Suppose we had, for a given value of n,  n (φ → ψ) → (n φ → n ψ). By Nec,  (n (φ → ψ) → (n φ → n ψ)). Furthermore, as an instance of theorem (K),  (n (φ → ψ) → (n φ → n ψ)) → (n (φ → ψ) → (n φ → n ψ)). By modus ponens, we can detach the consequent and get  n (φ → ψ) → (n φ → n ψ). Another instance of theorem (K) is  (n φ → n ψ) → (n φ → n φ).

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Applying classical logic (V), we get:  n (φ → ψ) → (n φ → n φ). Which is, by Lemma 5, identical to  n+1 (φ → ψ) → (n+1 φ → n+1 φ).

Proof of (Ind). I can now prove the theorem that corresponds to (Ind): (Ind) ∗ (φ → φ) → (φ → ∗ φ). Take an arbitrary model M = D, v and assignment s and let f = ρ(φ, v, s) and f  = ρ((φ → φ), v, s). Then by lemma 4, v s (∗ (φ → φ) is true iff the conjunction of f (n) over all n ∈ N is true; and v s (∗ φ) is true iff the conjunction of f  (n) over all n ∈ N is true. Now make two assumptions: (a) v s (φ) = f (0) is true; (b) v s (∗ (φ → φ) is true. The following is an instance of theorem (Kn ) and holds for every natural number n: n (φ → φ) → (n φ → n φ). By Lemma 5, it can be rewritten as n (φ → φ) → (n φ → n+1 φ). From assumption (b) it follows, as we have seen, that the conjunction of f  (n) over all n ∈ N is true; then f  (n) = n (φ → φ) must be true for every natural number n. Hence we can detach the consequent of our instance of (Kn ) and conclude that for every n ∈ N, v s (n φ → n+1 φ) must be true. So for every n ∈ N, if f (n) = v s (n φ) is true, then so is f (n + 1) = v s (n+1 φ). Together with assumption (a), this entails by mathematical induction that f (n) is true for all n ∈ N. So the conjunction of f (n) over all n ∈ N is true, and then v s (∗ φ) is true. By double conditionalization, we get: if v s (∗ (φ → φ) is true, then if v s (φ) is true, v s (∗ φ) is true; i.e. v s (∗ (φ → φ) → (φ → ∗ φ)) is true. Since M, s were arbitrary, the same holds for all models and assignments:  ∗ (φ → φ) → (φ → ∗ φ).

Bibliography Adams, Robert Merihew. 1974. ‘Theories of Actuality.’ Noûs 8:211–31. Almog, Joseph. 1989. ‘Logic and the World.’ Journal of Philosophical Logic 18:197–220. Armstrong, David. 1983. What is a Law of Nature? Cambridge: Cambridge University Press. Armstrong, David. 1989a. A Combinatorial Theory of Possibility. Cambridge: Cambridge University Press. Armstrong, David. 1989b. Universals: An Opinionated Introduction. Boulder, CO: Westview. Armstrong, David. 1996. ‘Dispositions as Categorical States.’ In David Armstrong, C.B. Martin, and U.T. Place (eds.), Dispositions. A Debate, 15–18. London/New York: Routledge. Armstrong, David, Martin, C.B., and Place, U.T. 1996. Dispositions. A Debate. London/ New York: Routledge. Ashwell, Lauren. 2010. ‘Superficial Dispositionalism.’ Australasian Journal of Philosophy 88:1–19. Austin, J. L. 1961. ‘Ifs and Cans.’ In J.O. Urmson and G. Warnock (eds.), Philosophical Papers, 205–32. Oxford: Oxford University Press. Bauer, William A. 2011. ‘An Argument for the Extrinsic Grounding of Mass.’ Erkenntnis 74:81–99. Belnap, Nuel. 1991. ‘Backwards and Forwards in the Modal Logic of Agency.’ Philosophy and Phenomenological Research 51:777–807. Belnap, Nuel and Perloff, Michael. 1988. ‘Seeing to it that: a Canonical Form for Agentives.’ Theoria 54:175–99. Benacerraf, Paul. 1983. ‘What Numbers Could Not Be.’ In Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics. Selected Readings, 272–94. Cambridge: Cambridge University Press. Bigelow, John. 1976. ‘Possible Worlds Foundations for Probability.’ Journal of Philosophical Logic 5:299–320. Bigelow, John, Ellis, Brian, and Lierse, Caroline. 1992. ‘The World as One of a Kind: Natural Necessity and Laws of Nature.’ British Journal for the Philosophy of Science 43:371–88. Bird, Alexander. 1998. ‘Dispositions and Antidotes.’ The Philosophical Quarterly 48: 227–34. Bird, Alexander. 2007. Nature’s Metaphysics. Oxford: Oxford University Press. Bird, Alexander. 2010. ‘Causation and the Manifestation of Powers.’ In Anna Marmodoro (ed.), Powers. Their Grounding and their Manifestations, 160–68. Abingdon: Routledge. Black, Robert. 2000. ‘Against Quidditism.’ Australasian Journal of Philosophy 78:87–104. Blackburn, Simon. 1990. ‘Filling in Space.’ Analysis 50:62–5. Borghini, Andrea and Williams, Neil E. 2008. ‘A Dispositional Theory of Possibility.’ Dialectica 62:21–41.

324

bibliography

Brennan, Virginia. 1993. Root and epistemic modal auxiliary verbs. Ph.D. thesis, University of Massachusetts. Brown, Mark A. 1988. ‘On the Logic of Ability.’ Journal of Philosophical Logic 17:1–26. Butler, Jonny. 2003. ‘A Minimalist Treatment of Modality.’ Lingua 113:967–96. Bybee, Joan, Perkins, Revere, and Pagliuca, William. 1994. The Evolution of Grammar: Tense, Aspect and Modality in the Languages of the World. Chicago: The University of Chicago Press. Cameron, Ross. 2008. ‘Truthmakers and Modality.’ Synthese 164:261–80. Carr, David. 1979. ‘The Logic of Knowing How and Ability.’ Mind 88:394–409. Cartwright, Nancy. 1989. Nature’s Capacities and their Measurement. Oxford: Oxford University Press. Cartwright, Richard. 1994. ‘Speaking of Everything.’ Noûs 28:1–20. Chellas, Brian F. 1980. Modal Logic: An Introduction. Cambridge: Cambridge University Press. Chisholm, Roderick. 1967. ‘Identity through Possible Worlds: Some Questions.’ Noûs 1:1–8. Choi, Sungho. 2006. ‘The Simple vs. Reformed Conditional Analysis of Dispositions.’ Synthese 148:369–79. Cinque, G. 1999. Adverbs and Functional Heads: A Crosslinguistic Perspective. Oxford Studies in Comparative Syntax. Oxford: Oxford University Press. Clark, Michael J. and Liggins, David. 2012. ‘Recent Work on Grounding.’ Analysis 72:812–23. Clarke, Randolph. 2008. ‘Intrinsic Finks.’ The Philosophical Quarterly 58:512–18. Clarke, Randolph. 2009. ‘Dispositions, Abilities to Act, and Free Will: The New Dispositionalism.’ Mind 118:323–51. Clarke, Randolph. 2010. ‘Opposing Powers.’ Philosophical Studies 149:153–60. Collins, Peter. 2009. Modals and Quasi-Modals in English. Amsterdam/New York: Rodopi. Contessa, Gabriele. 2009. ‘Modal Truthmakers and Two Varieties of Actualism.’ Synthese 174:341–53. Cook, W.A. 1978. ‘Semantic Structure of English Modals.’ TESOL Quarterly 12:5–17. Correia, Fabrice. 2010. ‘Grounding and Truth-functions.’ Logique et analyse 53:251–79. Correia, Fabrice and Schnieder, Benjamin. 2012. Metaphysical Grounding: Understanding the Structure of Reality. Cambridge: Cambridge University Press. Cross, Charles B. 1986. ‘ “Can” and the Logic of Ability.’ Philosophical Studies 50:53–64. Cross, Troy. 2005. ‘What is a Disposition?’ Synthese 144:321–41. Davies, Mark. 2008. ‘The Corpus of Contemporary American English: 450 Million Words, 1990–Present.’ Available online at . DeRose, Keith. 1991. ‘Epistemic Possibilities.’ Philosophical Review 100:581–605. Dretske, Fred. 1977. ‘Laws of Nature.’ Philosophy of Science 44:248–68. Dummett, Michael. 1981. Frege. Philosophy of Language. London: Duckworth, 2nd edition. Eagle, Antony. 2009. ‘Causal Structuralism, Dispositional Actualism, and Counterfactual Conditionals.’ In Toby Handfield (ed.), Dispositions and Causes, 65–99. Oxford: Oxford University Press.

bibliography

325

Earman, John and Norton, John. 1987. ‘What Price Substantivalism? The Hole Story.’ British Journal for the Philosophy of Science 38:515–25. Ellis, Brian. 2001. Scientific Essentialism. Cambridge: Cambridge University Press. Fara, Michael. 2008. ‘Masked Abilities and Compatibilism.’ Mind 117:843–65. Field, Hartry. 1980. Science without Numbers. Oxford: Blackwell. Fine, Kit. 1994. ‘Essence and Modality.’ Philosophical Perspectives 8:1–16. Fine, Kit. 1995a. ‘The Logic of Essence.’ Journal of Philosophical Logic 24:241–73. Fine, Kit. 1995b. ‘Ontological Dependence.’ Proceedings of the Aristotelian Society 95: 269–89. Fine, Kit. 1995c. ‘Senses of Essence.’ In Walter Sinnott-Armstrong (ed.), Modality, Morality, and Belief. Essays in Honour of Ruth Barcan Marcus, 53–73. Cambridge: Cambridge University Press. Fine, Kit. 2001. ‘The Question of Realism.’ Philosophers’ Imprint 1:1–30. Fine, Kit. 2012a. ‘Guide to Ground.’ In Fabrice Correia and Benjamin Schnieder (eds.), Metaphysical Grounding. Understanding the Structure of Reality, 37–80. Cambridge: Cambridge University Press. Fine, Kit. 2012b. ‘The Pure Logic of Ground.’ Review of Symbolic Logic 5:1–25. Gillies, Donald. 2000. ‘Varieties of Propensity.’ British Journal for the Philosophy of Science 51:807–35. Hacquard, Valentine. 2010. ‘On the Event Relativity of Modal Auxiliaries.’ Natural Language Semantics 18:79–114. Hale, Bob and Wright, Crispin. 2001. The Reason’s Proper Study: Essays Towards a NeoFregean Philosophy of Mathematics. Oxford: Oxford University Press. Harel, David. 1983. ‘Dynamic Logic.’ In Dov Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, volume II, chapter 10, 497–604. Dordrecht: Reidel. Hawthorne, John. 2001. ‘Causal Structuralism.’ Philosophical Perspectives 15:361–78. Heil, John. 2003. From an Ontological Point of View. Oxford: Oxford University Press. Hinchliff, Mark. 1996. ‘The Puzzle of Change.’ In James Tomberlin (ed.), Philosophical Perspectives, 10, Metaphysics, 119–36. Cambridge, MA: Blackwell. Hoefer, Carl. 1996. ‘The Metaphysics of Space-Time Substantivalism.’ The Journal of Philosophy 93:5–27. Hofweber, Thomas. 2014. ‘Infinitesimal Chances.’ Philosophers’ Imprint 14:1–34. Holton, Richard. 1999. ‘Dispositions All the Way Round.’ Analysis 59:9–14. Horty, John F. and Belnap, Nuel. 1995. ‘The Deliberative Stit: A Study of Action, Omission, Ability, and Obligation.’ Journal of Philosophical Logic 24:583–644. Horwich, Paul. 1987. Asymmetries in Time: Problems in the Philosophy of Science. MIT Press. Hüttemann, Andreas. 2007. ‘Causation, Laws and Dispositions.’ In Max Kistler and Bruno Gnassounou (eds.), Dispositions and Causal Powers, 207–19. Aldershot: Ashgate. Hüttemann, Andreas. 2010. ‘Eine dispositionale Theorie der Kausalität.’ Lebenswelt und Wissenschaft. Deutsches Jahrbuch Philosophie 2:451–67. Hüttemann, Andreas. 2013. ‘A Disposition-based Process Theory of Causation.’ In Stephen Mumford and Matt Tugby (eds.), Metaphysics and Science, 101–22. Oxford: Oxford University Press.

326

bibliography

Jacobs, Jonathan D. 2010. ‘A Powers Theory of Modality: or, How I Learned to Stop Worrying and Reject Possible Worlds.’ Philosophical Studies 151:227–48. Jenkins, C.S. and Nolan, Daniel. 2012. ‘Disposition Impossible.’ Noûs 46:732–53. Johnston, Mark. 1992. ‘How to Speak of the Colors.’ Philosophical Studies 68:221–63. Jubien, Michael. 2007. ‘Analyzing Modality.’ Oxford Studies in Metaphysics 3:99–139. Kenny, Anthony. 1975. Will, Freedom, and Power. Oxford: Blackwell. Kenny, Anthony. 1976. ‘Human Abilities and Dynamic Modalities.’ In Juha Manninen and Raimo Tuomela (eds.), Essays on Explanation and Understanding, 209–32. Dordrecht: Reidel. Kjellmer, Göran. 1986. ‘Legible but not Readable: On the Semantics of English Adjectives in -ble.’ Studia Neophilologica 58:11–38. Kratzer, Angelika. 1977. ‘What “Must” and “Can” Must and Can Mean.’ Linguistics and Philosophy 1:337–55. Kratzer, Angelika. 1981. ‘The Notional Category of Modality.’ In H. J. Eikmeyer and H. Rieser (eds.), Words, Worlds, and Contexts. New Approaches in Word Semantics. Berlin/New York: de Gruyter. Kratzer, Angelika. 1991. ‘Modality.’ In Arnim von Stechow and Dieter Wunderlich (eds.), Semantik: Ein internationales Handbuch der zeitgenössischen Forschung, 639–50. Berlin/New York: de Gruyter. Kratzer, Angelika. 2012. Modals and Conditionals: New and Revised Perspectives. Oxford: Oxford University Press. Kripke, Saul. 1980. Naming and Necessity. Cambridge, Mass.: Harvard University Press. Ladyman, James, Ross, Don, Spurrett, David, and Collier, John. 2007. Every Thing Must Go. Oxford: Oxford University Press. Lange, Marc. 2009. Laws and Lawmakers. Science, Metaphysics, and the Laws of Nature. Oxford: Oxford University Press. Langton, Rae and Lewis, David. 1998. ‘Defining “Intrinsic”.’ Philosophy and Phenomenological Research 58:333–45. Lehrer, Keith. 1968. ‘Cans without Ifs.’ Analysis 29:29–32. Lewis, David. 1970. ‘How to Define Theoretical Terms.’ Journal of Philosophy 67:427–44. Lewis, David. 1973a. ‘Causation.’ Journal of Philosophy 70:556–7. Lewis, David. 1973b. Counterfactuals. Oxford: Blackwell. Lewis, David. 1976a. ‘The Paradoxes of Time Travel.’ American Philosophical Quarterly 13:145–52. Lewis, David. 1976b. ‘Survival and Identity.’ In Amelie Oksenberg Rorty (ed.), The Identities of Persons, 17–40. University of California Press. Lewis, David. 1979. ‘Scorekeeping in a Language Game.’ Journal of Philosophical Logic 8:339–59. Lewis, David. 1981. ‘Are we Free to Break the Laws?’ Theoria 47:113–21. Lewis, David. 1983. ‘New Work for a Theory of Universals.’ Australasian Journal of Philosophy 61:343–77. Lewis, David. 1986a. On the Plurality of Worlds. Oxford: Blackwell. Lewis, David. 1986b. Philosophical Papers, Volume II. Oxford: Oxford University Press. Lewis, David. 1994. ‘Humean Supervenience Debugged.’ Mind 103:473–90. Lewis, David. 1997. ‘Finkish Dispositions.’ Philosophical Quarterly 47:143–58.

bibliography

327

Lowe, E.J. 2006. The Four-Category Ontology. A Metaphysical Foundation for Natural Science. Oxford: Oxford University Press. Lowe, E.J. 2008. ‘Two Notions of Being: Entity and Essence.’ Philosophy 83:23–48. Lowe, E.J. 2011. ‘How Not to Think of Powers: A Deconstruction of the “Dispositions and Conditionals” Debate.’ The Monist 94:19–33. McGinn, Colin. 1976. ‘On the Necessity of Origin.’ Journal of Philosophy 73:127–35. Mackie, John L. 1974. ‘De What Re Is De Re Modality?’ Journal of Philosophy 71:551–61. Mackie, Penelope. 1998. ‘Identity, Time, and Necessity.’ Proceedings of the Aristotelian Society 98:59–78. McKitrick, Jennifer. 2003. ‘A Case for Extrinsic Dispositions.’ Australasian Journal of Philosophy 81:155–74. Maier, John. 2010. ‘Abilities.’ In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Spring 2010 edition. Maier, John. ms. ‘A Dispositional Theory of Counterfactuals.’ Manuscript. Malzkorn, Wolfgang. 2000. ‘Realism, Functionalism and the Conditional Analysis of Dispositions.’ The Philosophical Quarterly 50:452–69. Manley, David and Wasserman, Ryan. 2007. ‘A Gradable Approach to Dispositions.’ The Philosophical Quarterly 57:68–75. Manley, David and Wasserman, Ryan. 2008. ‘On Linking Dispositions and Conditionals.’ Mind 117:59–84. Manley, David and Wasserman, Ryan. 2011. ‘Dispositions, Conditionals, and Counterexamples.’ Mind 120:1191–227. Markosian, Ned. 1998. ‘Brutal Composition.’ Philosophical Studies 92:211–49. Markosian, Ned. 2004. ‘In Defense of Presentism.’ In Dean W. Zimmerman (ed.), Oxford Studies in Metaphysics, volume 1, 47–82. Oxford: Oxford University Press. Markosian, Ned. 2007. ‘Restricted Composition.’ In John Hawthorne, Theodore Sider, and Dean Zimmerman (eds.), Contemporary Debates in Metaphysics, 341–64. Oxford: Oxford University Press. Martin, C.B. 1994. ‘Dispositions and Conditionals.’ The Philosophical Quarterly 44:1–8. Martin, C.B. 1996. ‘How It Is: Entities, Absences and Voids.’ Australasian Journal of Philosophy 74:57–65. Martin, C.B. 1998. ‘Rules and Powers.’ Philosophical Perspectives 12:284–312. Martin, C.B. and Heil, John. 1999. ‘The Ontological Turn.’ Midwest Studies in Philosophy 23:34–60. Maudlin, Tim. 1988. ‘The Essence of Space-Time.’ In A. Fine and M. Forbes (eds.), Philosophy of Science Association 1988, volume II, 82–91. Philosophy of Science Association. Maudlin, Tim. 2007. The Metaphysics within Physics. Oxford: Oxford University Press. Mayr, Erasmus. 2011. Understanding Human Agency. Oxford: Oxford University Press. Mellor, D.H. 2000a. ‘Possibility, Chance and Necessity.’ Australasian Journal of Philosophy 78:16–27. Mellor, D.H. 2000b. ‘The Semantics and Ontology of Dispositions.’ Mind 109:757–80. Molnar, George. 1999. ‘Are Dispositions Reducible?’ The Philosophical Quarterly 49:1–17. Molnar, George. 2003. Powers. A Study in Metaphysics. Oxford: Oxford University Press. Mumford, Stephen. 1998. Dispositions. Oxford: Oxford University Press. Mumford, Stephen. 2004. Laws in Nature. London/New York: Routledge.

328

bibliography

Mumford, Stephen and Anjum, Rani Lill. 2011. Getting Causes from Powers. Oxford: Oxford University Press. Nolan, Daniel. forthcoming. ‘Noncausal Dispositions.’ Noûs. Oderberg, David. 2007. Real Essentialism. London/New York: Routledge. Palmer, Frank. 1990. Modality and the English Modals. London: Longman, 2nd edition. Peacocke, Christopher. 1999. Being Known. Oxford: Oxford University Press. Plantinga, Alvin. 1974. The Nature of Necessity. Oxford: Oxford University Press. Prior, Elizabeth W., Pargetter, Robert, and Jackson, Frank. 1982. ‘Three Theses about Dispositions.’ American Philosophical Quarterly 19:251–7. Pruss, Alexander R. 2002. ‘The Actual and the Possible.’ In R.M. Gale (ed.), Blackwell Guide to Metaphysics, 317–33. Oxford: Blackwell. Quine, W.V.O. 1969. ‘Natural Kinds.’ In Nicholas Rescher (ed.), Essays in Honor of Carl G. Hempel. Dordrecht: Reidel. Raven, Michael J. 2012. ‘In Defence of Ground.’ Australasian Journal of Philosophy 90:687–701. Robertson, Teresa. 1998. ‘Possibilities and the Arguments for Origin Essentialism.’ Mind 107:729–50. Rosen, Gideon. 1990. ‘Modal Fictionalism.’ Mind 99:327–54. Rosen, Gideon. 2010. ‘Metaphysical Dependence: Grounding and Reduction.’ In Bob Hale and Aviv Hoffman (eds.), Modality: Metaphysics, Logic and Epistemology. Oxford: Oxford University Press. Ryle, Gilbert. 1949. The Concept of Mind. London: University of Chicago Press. Sainsbury, Mark. 1997. ‘Easy Possibilities.’ Philosophy and Phenomenological Research 57:907–19. Salmon, Nathan. 1989. ‘The Logic of What Might Have Been.’ Philosophical Review 98:3– 34. Schaffer, Jonathan. 2009a. ‘On What Grounds What.’ In David J. Chalmers, David Manley, and Ryan Wasserman (eds.), Metametaphysics. New Essays on the Foundation of Ontology, 347–83. Oxford: Oxford University Press. Schaffer, Jonathan. 2009b. ‘Spacetime the One Substance.’ Philosophical Studies 145:131– 48. Schaffer, Jonathan. 2010. ‘Monism: The Priority of the Whole.’ Philosophical Review 119:31–76. Schaffer, Jonathan. 2012. ‘Grounding, Transitivity, and Contrastivity.’ In Fabrice Correia and Benjamin Schnieder (eds.), Metaphysical Grounding. Understanding the Structure of Reality, 122–38. Cambridge: Cambridge University Press. Schmid, Stephan. forthcoming. ‘Potentialities in the Late Middle Ages: The Latin Tradition.’ In Kristina Engelhard and Michael Quante (eds.), Handbook Potentiality. Springer. Schnieder, Benjamin. 2011. ‘A Logic for “Because”.’ Review of Symbolic Logic 4:445–65. Schrenk, Markus. 2007. ‘Can Capacities Rescue Us from Ceteris Paribus Laws?’ In Max Kistler and Bruno Gnassounou (eds.), Dispositions and Causal Powers, 221–47. Aldershot: Ashgate. Schrenk, Markus. 2010. ‘The Powerlessness of Necessity.’ Noûs 44:725–39. Shapiro, Stewart. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press.

bibliography

329

Shoemaker, Sydney. 1980. ‘Causality and Properties.’ In Peter van Inwagen (ed.), Time and Cause, 109–35. Dordrecht: Reidel. Shoemaker, Sydney. 1998. ‘Causal and Metaphysical Necessity.’ Pacific Philosophical Quarterly 79:59–77. Sider, Theodore. 1999. ‘Presentism and Ontological Commitment.’ Journal of Philosophy 96:325–47. Sider, Theodore. 2003. Fourdimensionalism. An Ontology of Persistence and Time. Oxford: Oxford University Press. Sider, Theodore. 2011. Writing the Book of the World. Oxford: Oxford University Press. Skow, Bradford. 2007. ‘Are Shapes Intrinsic?’ Philosophical Studies 133:111–30. Smith, Michael. 2003. ‘Rational Capacities, or: How to Distinguish Recklessness, Weakness, and Compulsion.’ In Barry Stroud and Christine Tappolet (eds.), Weakness of Will and Practical Irrationality, 17–38. Oxford/New York: Oxford University Press. Stalnaker, Robert. 1976. ‘Possible Worlds.’ Noûs 10:65–75. Stalnaker, Robert. 2003. Ways a World Might Be. Oxford: Oxford University Press. Steinberg, Jesse R. 2010. ‘Dispositions and Subjunctives.’ Philosophical Studies 148:323–41. Steward, Helen. 2012. A Metaphysics for Freedom. Oxford: Oxford University Press. Sweetser, Eve. 1990. From Etymology to Pragmatics. Cambridge: Cambridge University Press. Swoyer, Chris. 1982. ‘The Nature of Causal Laws.’ Australasian Journal of Philosophy 60:203–23. Tooley, Michael. 1977. ‘The Nature of Laws.’ Canadian Journal of Philosophy 7:667–98. Traugott, Elizabeth. 1989. ‘On the Rise of Epistemic Meanings in English: An Example of Subjectification in Semantic Change.’ Language 65:31–55. van der Auwera, Johan and Plungian, Vladimir A. 1998. ‘Modality’s Semantic Map.’ Linguistic Typology 2:79–124. van Inwagen, Peter. 1975. ‘The Incompatibility of Free Will and Determinism.’ Philosophical Studies 27:185–99. van Inwagen, Peter. 1990. Material Beings. Ithaca: Cornell University Press. van Inwagen, Peter. 2006. ‘Can Mereological Sums Change their Parts?’ Journal of Philosophy 103:614–30. Vetter, Barbara. 2010. Potentiality and possibility. D Phil thesis, Oxford. Vetter, Barbara. 2011a. ‘On Linking Dispositions and which Conditionals?’ Mind 120:1173–89. Vetter, Barbara. 2011b. ‘Recent Work: Modality without Possible Worlds.’ Analysis 71:742–54. Vetter, Barbara. 2012. ‘Dispositional Essentialism and the Laws of Nature.’ In Alexander Bird, Brian Ellis, and Howard Sankey (eds.), Properties, Powers and Structures. Issues in the Metaphysics of Realism. Routledge. Vetter, Barbara. 2013. “‘Can” without Possible Worlds. Semantics for anti-Humeans.’ Philosophers’ Imprint 13:1–27. Vetter, Barbara. 2014. ‘Dispositions without Conditionals.’ Mind 123:129–56. Vetter, Barbara. forthcoming. ‘From Potentiality to Possibility’. In Kristina Engelhard and Michael Quante (eds.), Handbook Potentiality. Springer. Viebahn, Emanuel and Vetter, Barbara. 2014. ‘How Many Meanings for “May”? The Case for Modal Polysemy.’ Manuscript.

330

bibliography

Vihvelin, Kadri. 2004. ‘Free Will Demystified: A Dispositionalist Account.’ Philosophical Topics 32:427–50. von Fintel, Kai and Iatridou, Sabine. 2007. ‘What to Do If You Want to Go to Harlem: Anankastic Conditionals and Related Matters.’ Manuscript. Wedgwood, Ralph. 2007. The Nature of Normativity. Oxford: Oxford University Press. Wiggins, David. 1980. Sameness and Substance. Oxford: Blackwell. Williamson, Timothy. 1994. Vagueness. London/New York: Routledge. Williamson, Timothy. 1998. ‘Bare Possibilia.’ Erkenntnis 48:257–73. Williamson, Timothy. 1999. ‘Existence and Contingency.’ Proceedings of the Aristotelian Society 73:181–203. Williamson, Timothy. 2000. Knowledge and its Limits. Oxford: Oxford University Press. Williamson, Timothy. 2002. ‘Necessary Existents.’ In A. O’Hear (ed.), Logic, Thought and Language, 233–51. Cambridge: Cambridge University Press. Williamson, Timothy. 2003. ‘Vagueness in Reality.’ In Michael J. Loux and Dean W. Zimmerman (eds.), The Oxford Handbook of Metaphysics, 690–715. Oxford: Oxford University Press. Williamson, Timothy. 2007a. ‘How Probable is an Infinite Sequence of Heads?’ Analysis 67:173–80. Williamson, Timothy. 2007b. The Philosophy of Philosophy. Oxford: Blackwell. Williamson, Timothy. 2010. ‘Necessitism, Contingentism, and Plural Quantification.’ Mind 119: 657–748. Wilson, Jessica. 2012. ‘Fundamental Determinables.’ Philosophers’ Imprint 12:1–17. Wright, Crispin. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press. Yablo, Stephen. 1999. ‘Intrinsicness.’ Philosophical Topics 26: 590–627. Yates, David. forthcoming. ‘Dispositionalism and the Modal Operators.’ Philosophy and Phenomenological Research . Zalta, Edward N. 2006. ‘Essence and Modality.’ Mind 115: 659–93.

Index abilities 1, 2, 19, 76, 89, 116, 136, 142, 146, 169, 215, 217, 221–4, 226, 239, 241, 244, 248, 281 extrinsic 127, 179, 219–20, 224, 231 general and specific 127, 179 intrinsic 215, 219, 224 logic of 164, 169–71, 178–80, 183 ‘-able’ (suffix) 63, 216, 222, 231 abstract objects 6, 28, 66, 108, 230, 248, 257, 270, 276–81 Achilles’ heel 72–4, 78 Adams, Robert Merihew 6 Aimar, Simona vii–viii ‘aliens’ 260, 268–73 Almog, Joseph 261 Anjum, Rani Lill 10, 65, 92–3, 99 antidotes. see masks Aristotle vii, 1, 4, 271, 273, 279–80 Armstrong, David 2, 6, 227, 271, 282–3, 288 Ashwell, Lauren 76, 219 A-theory of time. see time Austin, J. L. 222 Azzano, Lorenzo vii, 275 Barcan formulas 267 Bauer, William A. 131 Belnap, Nuel 170, 223 Benacerraf, Paul 278 Bigelow, John 78, 261, 270 Bird, Alexander 2, 8–12, 14, 25–6, 36, 50, 52, 56, 71, 86, 96, 106–7, 109, 124, 282–4 Blackburn, Simon 8–9 Black, Robert viii, 9 Böhm, Matthias viii Borghini, Andrea 11, 15, 135, 273 Brennan, Virginia 236–41 Brown, Mark A. 170 B-theory of time. see time Buddensiek, Friedemann vii Busck Gundersen, Eline vii–viii Busse, Ralf viii, 133 Butler, Jonny 233, 237 Bybee, Joan 244–5 Cameron, Ross 273–5, 277 ‘can’ 63, 103, 215–16, 221–2, 224–5, 229, 237–9 potentiality-based semantics for 207, 214, 217, 230–1 as a predicate modifier 217, 228, 232–3 Carr, David 169

Cartwright, Nancy 86, 288 Cartwright, Richard 108 categorical properties. see properties causation 79, 93, 96–100, 183, 185, 191, 195, 245, 288 chance. see probability Chellas, Brian F. 165, 209, 301 Chisholm, Roderick 9 Choi, Sungho 37–8, 50 Cinque, G. 235 Clarke, Randolph 76, 219, 222 Clark, Michael J. 27 Collins, Peter 216, 224 composition 7, 24, 59, 108, 116–19, 156, 219, 258–9, 262–3, 297 conceivability 282, 291, 300 Contessa, Gabriele 6, 275 context-sensitivity, context-dependence 6, 20–1, 23, 36, 45, 49, 79–81, 96, 102–3, 132, 139, 207, 214–15, 220, 226, 228, 246 contingency, contingent 6, 9, 90, 93, 135, 177, 191–2, 204–5, 213, 248, 260–1, 267, 273–7, 280, 282–3, 287, 291–3 Cook, W. A. 233, 236 Correia, Fabrice 27–8 counterfactual conditional 4–5, 13–15, 30, 33–4, 42, 49, 52, 54, 60–1, 63–4, 75, 79, 95–6, 163, 225–7, 246, 250, 254, 284 Cross, Charles B. 238 Cross, Troy vii, 38 Davies, Mark 67 Deng, Natalja vii DeRose, Keith 231, 234 Descartes 1, 169 determinates/determinables 43–6, 49–58, 66, 80, 82, 92, 95, 129, 178–9, 262, 286–7, 317 Diehl, Catharine viii, 120 ‘disposed to’ 44, 49, 67, 78, 254–7 dispositional essentialism. see laws of nature dispositionalism 4, 7, 9–17, 21–6, 84, 86, 101, 139, 195, 214, 222, 225, 273, 284, 288–9 dispositional partners 106 dispositions ascriptions, semantics of 63, 66–7, 85 conditional/standard conception of 34–5, 54, 59, 63–5, 106, 253, 255, 284 degrees of 62, 78, 83, 85, 90

332

index

dispositions (continued) extrinsic 14, 75, 124, 126, 129, 131, 218, 225, 253, 255 finkish. see finks higher-order 135 intrinsic 195, 225 masked. see masks maximal 85, 86, 90–2, 288 multi-track 34, 38–41, 43, 49, 51, 53–7, 59, 64, 222 nomological 39, 50, 53–4, 61–2, 80, 86, 91, 254, 284–6, 288 ordinary 33, 37, 49, 51, 53–4, 58–62, 78–80 possibility/alternative conception of 65, 78–9, 82 realism about 22, 33, 54, 59, 63 reductive analysis of 14, 65, 75, 79 Dormandy, Katherine vii Dretske, Fred 283, 288 Dummett, Michael 204 Eagle, Antony vii–viii, 14, 110, 191, 297 Earman, John 263 easy possibility/‘easily’. see possibility Ellis, Brian 2, 9–10, 25, 50, 261, 282–3 endurantism/perdurantism 297–8 essence 1–5, 10, 23, 62, 96, 98, 164, 166–70, 172–4, 177, 181–2, 187, 192, 194, 258–9, 261, 283–8 dual of (E-potentiality) 164, 168–9, 172–3, 181–2 logic of 166–8, 170, 172–3 eternalism 177, 199, 292–3, 296–7, 299 Evers, Daan vii extensional correctness 15, 16, 18, 22, 30–1, 102–4, 140, 143, 198, 201, 206–7, 246–7, 299 Fara, Michael 76, 222 Field, Hartry 281 Fine, Kit 1–3, 5, 23, 26–28, 144, 164, 166–168, 172–4, 181–2, 260 finks 14, 36–40, 42, 44, 50, 53–4, 66, 73, 75–7, 93, 107 formal adequacy 102, 104, 134, 140–1, 143, 158, 161, 165, 206–7, 210–12, 214, 246–7, 299, 309 Fridland, Ellen vii Gillies, Donald 92 Göbel, Arno vii grounding 9–10, 13–14, 23, 25–30, 50–1, 54–9, 62, 66–7, 104–5, 114, 124, 130–1, 133–4, 134, 140, 142, 147, 153, 172, 175, 188–91, 193, 195, 198–9, 213, 219–20, 239, 261, 265–6, 270, 273, 275, 279–80, 283, 286, 288 Hacquard, Valentine 235–6, 239–40 Hale, Bob 278

Hanfling, Oswald viii Harel, David 320 Hawthorne, John 25 Heil, John 227 Hinchliff, Mark 293 Hoefer, Carl 262–3 Hofmann, Frank viii Hofweber, Thomas 92 Holton, Richard 8 Horty, John F. 170, 223 Horwich, Paul 191 Hufendiek, Rebekka viii Humean supervenience, Humeanism 7–9, 13, 24, 260, 283 Hüttemann, Andreas 10, 99, 106 Iatridou, Sabine 69 Jacobs, Jonathan viii, 11, 14, 34, 66, 79 Jaster, Romy viii, 120 Jenkins, C. S. 247, 250–7 Johnston, Mark 14, 36, 185 Jubien, Michael 6 Kenny, Anthony 76, 146, 164, 169–71, 178–9, 183, 221 Kjellmer, Göran 64, 74 Koch, Steffen viii Kratzer, Angelika 16, 64, 67–70, 72, 74–6, 216, 220, 228, 232, 240–2, 246 Kripke, Saul 202, 204, 267 Krödel, Thomas viii Ladyman, James 11, 24 lambda operator (λ) 111, 145, 147, 312 Lange, Marc 5 Langton, Rae 115, 123, 282 laws of nature 1–2, 4, 10, 24, 33, 50–1, 53, 65, 80, 86, 191, 224, 248, 251, 257, 261–2, 281–4, 287–9, 291 best systems account of 4, 283, 289–90 dispositional essentialist account of 10, 50–1, 56, 86, 282–8 as necessary or contingent 248, 261, 281–3, 288, 291 nomic necessitation account of 283–4, 288 Leckie, Gail viii Lehrer, Keith 222 Lewis, David 4, 6–9, 11–12, 29, 34, 36–8, 42, 50, 58, 65, 67–8, 76, 92, 106, 108, 114–16, 123–4, 159, 191, 220, 226–7, 263, 268–9, 272, 282–3, 289, 297 Lierse, Caroline 261 Liggins, David 27 Lowe, E. J. 2, 65, 71, 97, 288

index Mackie, John L. 204 Mackie, Penelope 204–5, 291 Madison Mount, Beau viii Maier, John viii, 14, 12 Malzkorn, Wolfgang 38 manifestation/exercise (of a disposition or potentiality) complex 61, 146–7, 183, 284 quantified. see potentiality, quantified tautological 147–8, 174, 176, see also potentiality, tautological Manley, David 21, 34, 36–9, 55, 66, 71–2, 77–8 Markosian, Ned 117, 294 Martin, C. B. 9, 14, 34, 36, 38–9, 66, 79, 106, 227 masks 14, 36–40, 42, 44, 50, 53–4, 66, 73, 75–7, 93, 109–11, 129, 156, 176, 219, 225, 290 Matthies, Angela viii Maudlin, Tim 7–8, 263 Mayr, Erasmus 223 McGinn, Colin 204 McKitrick, Jennifer 14, 105, 110, 118, 120–1, 124–6, 128, 130, 132 Mellor, D. H. 38, 91–2 mimics 36, 90, 110, 176, 193, 225, 296 modality the epistemology of 11–13, 203 graded 69–70 linguistic: circumstantial/dynamic 68, 70, 207, 215–17, 223–4, 226, 228–36, 238–9, 241, 246, 291, 299; deontic 16, 69, 216–17, 222, 224, 231–3, 236, 239, 241–6; epistemic 16, 68, 217, 223–4, 229–46, 269, 291, 299; root 232–41, 244–6 localized vs. non-localized 2–3, 5, 7, 10, 17–18, 23, 33–5, 39, 103, 125, 166, 168 metaphysics of 2, 6–7, 12, 16, 108, 213, 296, 299 modal logic 1, 4, 6, 15–16, 102, 141–2, 165–6, 170, 196, 206, 209, 212–13, 248, 299, 301, 318 normal 141–2, 165–6, 170, 209, 301, 318 system S4 196, 212–13, 301, 314, 318 system S5 16, 196, 212–13, 301 system T 16, 142, 164–5, 206–7, 212–13, 277 Mumford, Stephen 9–10, 38, 65, 71, 92–3, 99, 289 Müller, Andreas viii Müller, Thomas viii necessitism 177 necessity defined 203 necessary existents 275–7, 290, see also necessitism of identity 203–4 of origin 152, 204–6, 213, 277, 290–1, 296, 299 Nimtz, Christian viii Nolan, Daniel 96, 226, 247, 250–7 Norton, John 263

333

objectual content 144, 157, 164, 166–7, 168, 172–4, 181, 203 Oderberg, David 2 ontological dependence 167, 173, 181 Palmer, Frank 216 Peacocke, Christopher 12–13, 71–2 perdurantism. see endurantism/ perdurantism Perler, Dominik viii Perloff, Michael 170 Plantinga, Alvin 6 Plungian, Vladimir A. 244 possibility, possible, possibly approximate definitions of: (P), (P ) 13–14, 17–19, 31, 103 de dicto 3, 195, 202, 217, 229, 232, 237–8, 246, 268, 270, 279 defined 18–19, 197–9, 201, 207, 210, 214, 247–8, 282, 296, 300 de re 194–5, 268, 292, 296 easy 65, 71–6 logic of. see modal logic nomic 248, 257, 263, 281–2, 290 tensed (possibilityT ) 199–200, 203, 220, 291 possible cases 85–6, 90 possible worlds 2, 4–7, 9–11, 13, 21, 24, 27, 32, 45, 66, 68, 70, 73–4, 76–8, 80–2, 84–6, 90–1, 124, 130, 210, 214, 223, 228, 231–2, 240–4, 249, 257, 263–6, 272, 300, 319 reconstructed as potentialities of the world 2, 32, 257, 260–1, 264–6 -semantics 21, 45, 66, 80–1, 85–6, 91, 210, 214, 228, 241, 244, 319 potentiality ascriptions, the logical form of 134, 145, 147, 150, 154, 168, 207 Cambridge 147, 151, 153–7, 195 closed under logical equivalence 162, 165–6, 170, 176, 209, 306 closed under logical implication 23, 31, 157, 162, 164, 166–7, 171, 173, 195, 207, 211 degrees of 21–2, 24, 29, 36, 43, 45, 63, 72, 85–6, 90–6, 99–103, 107, 109–13, 119–24, 126–9, 142–3, 146, 149–50, 163, 175–6, 179–80, 183, 189–90, 192, 200, 205, 207, 217–18, 220–1, 224, 227, 279, 285–9, 300–1, 317 extrinsic 23, 31, 103–5, 122–5, 127–30, 132–5, 139–40, 142, 151–7, 174–6, 179, 181–3, 190–1, 195, 198–9, 201, 207, 218–21, 238–9, 257, 265, 300 introduction of 19–20, 84 iterated 18, 31, 90, 105, 135–43, 158–61, 165, 187, 196–202, 204–5, 207–8, 211–213, 226, 246–8, 267, 276, 278, 280, 286–9, 292, 295–6, 298–300, 311–13, 316

334

index

potentiality (continued) joint 8, 31, 105–24, 127–34, 137–8, 140, 142, 153–4, 156, 158, 161, 173–7, 181, 190–4, 198–202, 218–20, 225–6, 239, 253, 265–6, 268, 289–298; type 1/type 2, 113, 120–1, 128, 131–4, 153–4 logic of; actuality/TPOT 162, 164–5, 169–70, 182–3, 185–7, 194–5, 209–11, 281, 286, 288, 306, 308; closure, closurePOT 162, 164, 168–72, 176, 194–5, 209–11, 306–7, 309–10; closure1 162, 164, 171–3, 175–8, 187, 192; disjunction/POT∨ 162, 164–5, 169–70, 177–8, 180, 183, 187, 192, 194–5, 209–11, 306, 308, 310, 317; (EM) 149–50, 152, 155; non-contradiction/NCPOT 162, 165, 169, 180–2, 194, 209–10, 211, 306, 308, 310 maximal 62, 85–7, 90, 92, 95–6, 99, 142, 146, 149–50, 189–90, 192, 200, 205, 220, 264–5, 279, 285–8 past-concerning 188–9, 192–3, 205, 275, 277, 298 quantified 147, 151, 154 tautological 147–51, 155–6, 171–2, 187, 189 POT (operator) 144–5, 147–50, 152–60, 162–3, 166, 168–9, 173–4, 183, 196, 204, 207–13, 301–2, 304–13, 316–17 power 19, 65, 99, 114, 124–5, 146, 222–3, 244, 288 presentism 177, 199, 292–6, 299 Principle of Instantiation (PI) 271–2 Principle of Potential Instantiation (PPI) 272–3 Prior, Elizabeth W. 51 probability, chance 78, 91–2, 236, 317–18 properties Cambridge 26, 104, 112, 147, 183 categorical 7–9, 26, 33, 136, 227 existence of 26–30, 271–2 extrinsic 76, 122, 123–4, 127, 131, 219, 266 intrinsic 7, 14, 37, 67, 70, 73–6, 85, 104, 115, 123, 125, 219, 266 natural/non-natural 13, 26, 29–30, 50, 54–5, 58, 104, 134, 168–9, 271, 283, 287, 289 plural 108–9, 111, 113–14, 120–1, 123, 128, 131–2, 153, 173, 293, 297 structuralism about 25, 227, 249, 278 propositions 6, 26–8, 68–70, 166, 232, 237–8, 240, 245, 260–1, 270, 276, 278, 280–1, 303–7, 310–11, 313, 319 Pruss, Alexander R. 11, 15, 249, 273 qualities. see properties, categorical Quine, W. V. O. 278 Raven, Michael J. 7 relations 7, 24–6, 52, 56, 59, 62, 88, 109, 111, 113, 115, 120, 123, 128, 131–2, 137, 145, 153–4, 159, 175, 219, 221, 227, 262, 292–4, 296–8

Robertson, Teresa 204 Rosefeldt, Tobias viii Rosen, Gideon 6, 27–8, 133 Ryle, Gilbert 222 Sainsbury, Mark 71 Salmon, Nathan 16, 213 Sattig, Thomas viii, 119, 297 Schaffer, Jonathan 24, 26–7, 133, 263 Schmid, Stephan viii, 4 Schnieder, Benjamin 27–8 Schrenk, Markus viii, 7, 14, 93, 184, 288 Schulte, Peter viii Schwarz, Wolfgang viii Semantic utility 16, 18, 22–3, 30–1, 102–3, 140, 143, 206–7, 214, 246–7 Shapiro, Stewart 278, 281 Shoemaker, Sydney 25–6, 105, 112, 227, 282 Sider, Theodore 26, 29, 116, 119, 177, 259–60, 294, 297 Skow, Bradford 123 Smith, Michael 223 Spitzley, Thomas viii Stalnaker, Robert 5–6, 32, 257, 263–6 Steinberg, Jesse R. 38–9, 50 Steward, Helen 146, 223 stimulus (of a disposition) 30, 34, 36–8, 40–5, 49–51, 53–4, 60, 64–5, 71, 77–80, 94–7, 99, 101, 163, 225–6, 250, 253, 284 structuralism 25–6, 227, 249, 278, 281 supervenience 7–9, 27, 32, 133–4, 297 Humean. see Humean supervenience Sweetser, Eve 245 Swoyer, Chris 282 tense 144, 186–8, 191–3, 199, 201–2, 220, 226, 228–9, 232, 234–6, 240, 293 time A-theory/B-theory of 192 direction of 277 -travel 190–2, 205, 275 Tooley, Michael 283, 288 Traugott, Elizabeth 245 triviality thesis 189–93, 202, 275, 277, 290–1, 298–9 universals 269–72, 276, 278–81, 283 vagueness, vague 10, 20–2, 83, 101, 118–19, 258–9 van der Auwera, Johan 244 van Inwagen, Peter 116–18, 258, 282 Vetter, Barbara 2, 7, 37, 56, 66, 78, 225, 243, 248, 254, 256 Viebahn, Emanuel viii, 243 Vihvelin, Kadri 76, 222

index Vogt, Lisa 96 von Fintel, Kai 69 von Solodkoff, Tatjana viii Wasserman, Ryan 21, 34, 36–9, 55, 66, 71–2, 77–8 Wedgwood, Ralph 213 Wiggins, David 204 Williams, Neil E. 15, 135, 273 Williamson, Timothy vii, 5–6, 20, 71–2, 92, 177, 194, 249, 267–8

335

Wilson, Alastair viii Wilson, Jessica 56–8 Woodward, Richard viii world, the (as an object) 32, 249, 257–67, 270, 276, 282–3 Wright, Crispin 278–9 Yablo, Stephen 126 Yates, David 15 Zalta, Edward N. 249