A Chance for Possibility: An Investigation into the Grounds of Modality 9783110338232, 9783110334890

Citation preview

Alexander Steinberg A Chance for Possibility

Philosophische Analyse / Philosophical Analysis

Herausgegeben von / Edited by Herbert Hochberg, Rafael Hüntelmann, Christian Kanzian, Richard Schantz, Erwin Tegtmeier

Volume / Band 51

Alexander Steinberg

A Chance for Possibility An Investigation into the Grounds of Modality

ONTOS

ISBN 978-3-11-033489-0 e-ISBN 978-3-11-033823-2 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.dnb.de abrufbar. © 2013 Walter de Gruyter GmbH, Berlin/Boston Druck und Bindung: Hubert & Co. GmbH & Co. KG, Göttingen ∞ Gedruckt auf säurefreiem Papier Printed in Germany www.degruyter.com

Für Brigitte und Stefanie

Acknowledgements This book is based on my PhD thesis that was submitted to University College London in 2010. The thesis was supervised by Mark Kalderon and examined by Dorothy Edgington and Fraser MacBride. To all three I am greatly indebted for many valuable comments on the material that has made its way into the book (and some that hasn’t). I started working on the material at UCL, joining the Berlin based Phlox research group around half way. Both places provided as stimulating environments for studying and discussing philosophy as they were enjoyable for just being in general. For creating this happy combination of features I would particularly like to thank Steinvör Árnadóttir, Tim Crane, Liz Cripps, Rod Farningham, Katharina Felka, Marie Lundstedt, Brent Madison, Howard Peacock, Polly Pantelides, Tobias Rosefeldt, Arthur Schipper, Robert Schwartzkopff, Laura Valentini, and Lee Walters as well as the other members of Phlox Nick Haverkamp, Miguel Hoeltje, Moritz Schulz, and Benjamin Schnieder. Further, I am very grateful to my teacher Wolfgang Künne whose clarity of style, attention to detail and exegetical charity is something to aspire to, and without whom I would not have ended up doing philosophy at all. Work on the thesis was supported by a UCL Graduate School Research Studentship, an Arts and Humanities Research Council Doctoral Award and a Departmental Research Studentship from the University of St. Andrews, all of which I gratefully acknowledge. The thesis was awarded the 2012 ontos-Award by the Gesellschaft für Analytische Philosophie (GAP). I am very grateful to everyone involved. Last but not least, I wish to thank my family for their continuing support. Hamburg, 27th June 2013

A.S.

Contents 1

Introduction

7

2 Supervenience 2.1 Supervenience . . . . . . . . . . . . . . . . . 2.2 Covariance . . . . . . . . . . . . . . . . . . . 2.3 Covariance and Ontological Priority . . . . . 2.4 Covariance and Modal Supervenience . . . . 2.5 Modal Supervenience and Explanation . . . 2.6 Explaining Explanations . . . . . . . . . . . . A Strong and Global Supervenience . . . . . . A.1 Kim’s Attempted Equivalence Proof A.2 Non-Equivalence Examples . . . . . A.3 Restricted Entailment? . . . . . . . . A.4 Against Restricted Entailment . . . . 3

Concrete Possible Worlds 3.1 Possibility and Possible Worlds . 3.2 The Analysis Claim . . . . . . . 3.3 Lewis Worlds . . . . . . . . . . . 3.4 The Irrelevance Objection . . . . 3.4.1 Against IO1 . . . . . . . 3.4.2 Against IO2 . . . . . . . 3.4.3 Against IO3 . . . . . . . 3.5 Counterparts . . . . . . . . . . . 3.6 Actual Problems for Lewis . . . 3.6.1 Possible Non-Existence 3.6.2 Surprises . . . . . . . . . 3.6.3 Actuality . . . . . . . . . 3.7 Lewisian Explanations . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

13 15 19 26 31 36 44 55 56 58 61 63

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

69 . 70 . 72 . 73 . 76 . 77 . 81 . 86 . 89 . 96 . 96 . 99 . 101 . 107

4 Abstract Possible Worlds 4.1 Something from Nothing . . . . . . . . . . . . . . . . 4.1.1 Pleonastic Properties . . . . . . . . . . . . . . 4.1.2 Something-from-Nothing Entailment Claims 4.1.3 Property Concepts . . . . . . . . . . . . . . . 4.1.4 True SNECs . . . . . . . . . . . . . . . . . . . 4.2 Pleonastic Possible Worlds . . . . . . . . . . . . . . . 4.3 Objections and Clarifications . . . . . . . . . . . . . . 4.3.1 Existence . . . . . . . . . . . . . . . . . . . . . 4.3.2 Explanation . . . . . . . . . . . . . . . . . . . 4.3.3 Competitors . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

113 . 115 . 117 . 120 . 127 . 135 . 152 . 167 . 168 . 171 . 177

5 Possibility and Probability 5.1 Initial Motivation . . . . . . . . . . . . . 5.2 Different Kinds of Probability . . . . . 5.2.1 Epistemic Probabilities . . . . . 5.2.2 Objective Probabilities . . . . . 5.3 The Temporal Structure of ‘Might’s . . 5.3.1 Might . . . . . . . . . . . . . . . 5.3.2 Might Have . . . . . . . . . . . 5.3.3 Different Readings . . . . . . . 5.4 DeRose on ‘Might’ Sentences . . . . . 5.5 Supervenience . . . . . . . . . . . . . . 5.6 Objective and Metaphysical Possibility

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

181 . 183 . 188 . 189 . 194 . 201 . 201 . 206 . 210 . 214 . 224 . 229

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

6 Conclusion

237

Appendix A Non-Nominal Quantification

241

Name Index

257

Bibliography

261

Chapter 1

Introduction In philosophy and in ordinary life we are not only interested in how things are, but also in how they might have been, could not have been and must have been, in how they might be, cannot be and must be. That is, we are not only interested in what is the case, but also in what is possible, impossible, and necessary. Socrates was a philosopher. But he might also have become a carpenter instead. If Origin Essentialism is correct,1 he could not have been born by different parents, he must have been born by his actual parents, if he is to exist at all. This tritium atom did not decay yesterday, but it might have. Neither did it turn into a prime number. But contrary to decaying, it could not have turned into a prime number. So, yesterday I should not have predicted that the tritium atom would not decay. But I was well within my rights to predict that it wouldn’t turn into a prime number. If Bernhard Langer had won the 2007 BMW Open, he would have been the first German golfer to have won all German PGA tournaments. He didn’t, but he might have. In fact, had he not hooked his last tee shot he probably would have. So, it was fairly reasonable to bet on him at the time, even though in retrospect I’m glad that I didn’t. This morning I took my umbrella, not because I knew that it would rain, but because I knew that it might. Modality comprises possibility and necessity. Both may come in different flavours. A fundamental distinction is that between epistemic and nonepistemic modality. When wondering whether Goldbach’s Conjecture—the thesis that every even number greater than two is the sum of two primes—is true or false, we might well say that Goldbach’s Conjecture might be true and it might be false. When we do, we are most naturally understood to have ascribed epistemic possibility both to the truth and falsity of Goldbach’s Conjecture. The truth of what we say turns on our epistemic position vis-à-vis Goldbach’s Conjecture. Roughly, what we say is true just in case we are not certain that Goldbach’s Conjecture is true and we are not certain that Gold1

See e.g. Kripke (1980: 112) and Forbes (1985: ch. 6).

8

Chapter 1. Introduction

bach’s Conjecture is false. For those of us who are thus uncertain, it is epistemically possible that Goldbach’s Conjecture is true and it is epistemically possible that Goldbach’s Conjecture is false. On the other hand, according to the orthodoxy in the philosophy of mathematics, if a proposition of pure mathematics is true, it is non-epistemically necessary that it is true, and if such a proposition is false, it is non-epistemically necessary that it is false. Since Goldbach’s Conjecture is a proposition of pure mathematics, if it is true, it is non-epistemically necessary that it is true, and, consequently, nonepistemically impossible that it is false. Likewise, if Goldbach’s Conjecture is false, it is non-epistemically necessary that it is false, and, consequently, non-epistemically impossible that it is true. Either way it is either epistemically possible but non-epistemically impossible that Goldbach’s Conjecture is true, or it is epistemically possible but non-epistemically impossible that Goldbach’s Conjecture is false. Epistemic and non-epistemic modality do not coincide. Epistemic modality is an interesting topic in its own right. This book is concerned almost exclusively with the non-epistemic kind, with the kind of modality expressed by modal sentences when they are not to be understood epistemically.2 It is plausible that a great many of the sentences containing the modal auxiliaries ‘might’ and ‘could’ we use in everyday discourse about what is possible express non-epistemic modality. When I said in the first paragraph that Langer might have won the 2007 BMW Open, I was most naturally understood to have said something whose truth does not turn on my—or, indeed, anyone’s—epistemic situation. I am certain now that Langer did not win the tournament. And what I said would have been true even if I had been certain all along—unreasonably, but certain nonetheless—that Langer wouldn’t win. The tritium atom might have decayed and could not have turned into a prime number regardless of what anyone believed or knew about it. Michael Dummett once wrote The philosophical problem of necessity is twofold: what is its source, and how do we recognize it? (Dummett 1959: 327)

In this passage, Dummett speaks only of necessity. But surely things are no different for possibility, and, thus, for modality in general. We may thus paraphrase Dummett as asking for (i) a supervenience base for modal truths, and (ii) an account of their epistemology. 2

The distinction between epistemic and non-epistemic modality will be discussed in chapter 5. For now, I rely on an intuitive understanding of this distinction.

9

Epistemic modality does not seem to pose dramatic problems here.3 Whether what is said with epistemic modal sentences is true or false depends on the epistemic states of epistemic subjects, and we can know what is epistemically possible and necessary in so far as we have epistemic access to our (and relevant others’) epistemic states. Things are not as clear cut for non-epistemic modality. On the face of it, there is something mysterious about non-epistemic possibility and necessity, though it is not easy to put one’s finger on it.4 This sense of mysteriousness is only exacerbated by epistemological considerations. Clearly, we do know that Langer might have won the BMW Open but can’t any more or that this tritium atom might decay within the hour. But how? At least relatively common and unproblematic ways of coming to know things, like sense perception, do not seem to help here. We can hear that Langer didn’t strike the ball cleanly, but we are not able to hear that Langer might still win. We can see (let’s suppose) that the atom isn’t currently decaying, but we can’t see that it might do so in a minute. Perhaps we can introspect that it is epistemically possible (for us now) that Langer will win his next tournament, but introspection is useless as a guide to non-epistemic possibility. (Smell and touch are equally unsuitable for detecting non-epistemic possibilities). But maybe we can take a leaf from epistemic modality. Whether there is some epistemic possibility that things are a certain way supervenes on facts about what we know or are certain of. Since these facts are as unproblematic a thing as you are likely to find in philosophy, epistemic possibility itself isn’t a mysterious notion. Since, moreover, we have epistemic access to some such facts, there is no principled obstacle to a successful epistemology of epistemic possibility either. An answer to Dummett’s first question may facilitate an answer to Dummett’s second question. The hope is that if we can find unproblematic (or less problematic) truths for non-epistemic modal truths to supervene on, we can dispel the sense of mysteriousness here as well, and, furthermore, explain how we can know them. This, then, is the 3

4

I don’t mean to imply that there are no problems. They invented a whole branch of philosophy surrounding them: epistemology. The claim is just that there are no dramatic problems specific to epistemic modality (and even that claim would have to be qualified). I take the attempt of some philosophers, for instance Blackburn (1993), to construct expressivist (or ‘quasi-realist’) theories of modality to be evidence for this feeling of mysteriousness. Not even those tempted by expressivism in some areas, I take it, would put forward expressivist accounts of, say, observation reports.

10

Chapter 1. Introduction

guiding question of this book: granted that non-epistemic modal truths are not basic,5 on what do non-epistemic modal truths supervene? What is the place of modal truths in the structure of reality? In order to answer the supervenience question, I will proceed as follows. Chapter 2 will sharpen the question by discussing in some detail the notion of supervenience at issue. What exactly is supervenience? How should we understand the claim that modal truths supervene on other truths of some kind? In particular, is there a workable notion of supervenience according to which the supervenience of modal truths on other truths is a substantive rather than a trivial claim? In the supervenience debate it is common, though not uncontroversial, to equate supervenience with covariance, a purely modal notion. Roughly, A-truths covary with B-truths just in case there cannot be a difference in A-truths without a difference in B-truths. Equivalently, A-truths covary with B-truths just in case for every A-proposition there must be some B-propositions such that if the latter are true, so is the former. The chapter argues that we should not equate supervenience with covariance if supervenience claims are to capture the dependency structure of reality. In particular, we should not equate supervenience with covariance if the supervenience of the modal is not to be a trivial thesis. Instead, we should enrich covariance with explanatory notions in order to get a notion of supervenience with which to formulate interesting supervenience claims. Roughly, in addition to asking whether for every modal proposition there are some B-propositions that must be true if the modal proposition is, we should ask whether the truth of the B-propositions also explains the truth of the modal proposition. If so, modal truths supervene on B-truths, if not, not. The subsequent chapters rely on this explanatorily enriched understanding of supervenience in evaluating the prospects of particular supervenience theses for modal truths. Possible worlds play an important role in philosophical discussions of modality. It is commonly acknowledged that sentences of the form ‘it is possible that p’ and the corresponding sentences of the form ‘there is a possible world at which p’ are equivalent, and so are, correspondingly, ‘it is necessary that p’ and ‘at all possible worlds, p’. Because of this equivalence, we can give extensional semantics for the modal fragments of natural languages that employ the machinery of quantification over possible worlds. Also, in conveying modal ideas in philosophy, we may help ourselves to possible worlds talk. It is agreed on all sides that such a switch has eminent heuristic value. 5

Dummett, for instance, clearly presupposes that they are not. In this book I will likewise assume that they are not basic as a working hypothesis.

11

But, perhaps, something stronger is true. Perhaps, possible worlds truths provide a suitable supervenience base for modal truths. Perhaps, it is possible that p because there is a possible world at which p, and it is necessary that q because at all possible worlds, q. There are two fundamentally different conceptions of possible worlds. One camp, whose major proponent is David Lewis, conceives of possible worlds as certain mereological sums of spatio-temporally related concrete individuals much like distant planets. The other makes possible worlds out to be some sort of abstract entities, ontologically on a par with properties and propositions. Since the prospects of the corresponding supervenience theses differ significantly, I discuss them in two separate chapters. In chapter 3 I will investigate the claim that modal truths supervene on possible worlds truths in the light of Lewis’s conception of possible worlds. It will be argued that Lewis is indeed committed to the supervenience claim, since, according to him, the possible worlds biconditionals—‘it is possible that p just in case there is a possible world at which p’ and ‘it is necessary that p just in case at all possible worlds, p’—are such that their right-hand sides analyse their left-hand sides. One strand of criticism against Lewis’s account is directed against consequences of the supervenience claim, the so-called Irrelevance Objection. The chapter looks at various forms of the Irrelevance Objection and concludes that, although it does not go through as it stands, it can be strengthened to prove fatal for the supervenience claim. Modal truths do not supervene on truths about possible worlds as Lewis conceives of them. Or so I will argue in chapter 3. Chapter 4 looks to the other class of conceptions of possible worlds. It will be argued that the claim that modal truths supervene on truths about abstract possible worlds is hopeless. Instead, it is a consequence of a plausible conception of abstract objects in general, that the direction of supervenience is reversed. There is a possible world at which p because it is possible that p, and at all possible worlds, q because it is necessary that q, not the other way around. An account of possible worlds that has this consequence is developed and defended against objections. Possible worlds, whether concrete or abstract, are of no help for answering the guiding question of this book. Abstract possible worlds are part of the superstructure instead of being the grounds of modal truths. Nevertheless, recognising their elevated position in the structure of reality is part of providing an attractive overall picture of the modal realm. So, should we give up our initial assumption that modal truths are not ba-

12

Chapter 1. Introduction

sic and learn to live with the sense of mysteriousness that surrounds them as well as our unclear epistemic access to them? Chapter 5 argues that this pessimism would be premature. Instead, taking a closer look at how possibility and necessity are expressed in natural languages suggests a more promising supervenience claim. Modality and probability are intimately related. In particular, impossibility and necessity correspond to non-minimal and maximal probability respectively. Although there are different kinds of probability, there is linguistic evidence that modal truths expressed by natural language non-epistemic modal sentences go with objective chances. The chapter argues that objective chances also provide an attractive supervenience base for non-epistemic modal truths. Langer might have won the BMW Open because there was such-and-such an objective chance that he would win, and Langer cannot win any more because there is a zero chance that he will win. My overall conclusion is, thus, that the (non-epistemic) modal realm is tripartite. Truths about objective chances form the ground level. Non-epistemic modal truths supervene on them. And truths about possible worlds supervene on them in turn. In the course of this book, I will sometimes use non-standard quantification. In particular, I will not only use quantifiers that bind variables that occupy the syntactic positions of singular terms but also quantifiers binding variables in general term and sentence position. Appendix A discusses these non-standard forms of quantification. It is argued that we should accept quantification into general term position as primitive, while quantification into sentence position may be elucidated with the help of quantification into general term position. Although the book presupposes the results of the appendix, most of its arguments are independent of a particular understanding of non-standard quantification.

Chapter 2

Supervenience Reality is structured. Some aspects of reality depend on other aspects, and some of the former are completely determined by the latter. In this fashion, things have their place in the ontological hierarchy. Investigating the nature of the relation that makes up the links in the ontological hierarchy as well as charting the structure that emerges are important tasks of metaphysics. This chapter is concerned with the former, in preparation for a small but interesting fragment of the latter. When do aspects of reality depend on, and are completely determined by, other aspects? Although there are no uncontroversial examples, the following two can at least illustrate the phenomenon: wholes and events. Arguably, wholes depend on their parts in such a way that everything true of a composite object is determined by what goes on with its parts. For instance, the colour of a bicycle is determined by the colour of its frame, and how fast this car can go, depends on the relevant features of its parts, in particular its engine. If this generalises, composite objects are farther up in the ontological hierarchy than the parts they are composed of: what features the former have depends on, and is completely determined by, what features the latter have. In philosophers’ terminology: wholes supervene on their parts. One attractive view of events—endorsed, e.g., in Bennett (1988)—is that what is true of them depends on, and is completely determined by, what is true of non-events (usually their subjects, if there are any). Whether the launch of the space shuttle took place depends on whether the space shuttle launched, whether its journey to the space station took four days, depends on whether the space shuttle journeyed for four days, and so forth. If this generalises, events are farther up in the ontological hierarchy than their subjects: whether they take place at all and what features they have depends on, and is completely determined by, what features the latter have. In philosophers’ terminology: events supervene on their subjects. Thus Bennett: Events are not basic items in the universe: they should not be included in any

14

Chapter 2. Supervenience fundamental ontology. […] I shall say that events are supervenient entities, meaning that all truths about them are logically entailed by and explained or made true by truths that do not involve the event concept. (Bennett 1988: 12)

Supervenience relations are the links in the ontological hierarchy. There is the foundation of things that do not depend on anything else—these are traditionally called substances—and, if Bennett is right, events are not among them. There is also the ontological superstructure consisting of everything else. There may be structure within the superstructure. Perhaps, Fs are basic, Gs supervene on them and Hs supervene on Gs in turn. Or there is additional structure within the Gs: perhaps some Gs depend on others just as the Gs as a whole depend on the Fs. One job of metaphysics is to uncover the structure of the ontological hierarchy.1 This book is concerned with the question of how modality fits into the structure of the world, whether the world’s modal aspects supervene on other aspects, and, if so, on which. To start with, then, we need to get clearer on what exactly supervenience is. In this chapter I will follow some strands of the current discussion in order to extract supervenience notions that can be used to formulate interesting supervenience theses concerning modality, theses that place modality within the ontological hierarchy. In order to do so, I will first sketch what philosophers take to be the common denominator of notions of supervenience in section 2.1. One thing that is commonly taken to be implied by supervenience is covariance, i.e. that there cannot be a difference in supervenient items without a difference in the subvenient ones. Different kinds of covariance and their relations are discussed in section 2.2. Sometimes it is thought that covariance exhausts supervenience. In section 2.3, I sketch why covariance of any kind does not guarantee another essential aspect of supervenience, namely that supervenient items depend on their subvenient base. If supervenience were merely covariance, it would not structure reality in the way we envisaged. In section 2.4, I will argue that this is especially grave in the case of modal supervenience, since covariance is to be had far too easily here. In section 2.5, I will draw the consequences of what went on before and propose a provisional definition of supervenience in explanatory terms, which will be further discussed and refined in section 2.6. Finally, in an appendix, I will trace the debate about the relation between two different kinds of covariance, strong covariance and global covariance, 1

For a spirited defence of the view that this is what metaphysicians should be primarily concerned with instead of being preoccupied with virtually trivial existence questions, see Schaffer (2009).

2.1. Supervenience

15

and argue that there are lessons to be learned from the modal case. 2.1

Supervenience

Supervenience is a technical concept of philosophy. It does not seem to correspond closely to uses of ‘supervene’ in ordinary English.2 Since this is so, natural language places hardly any constraints on the definition of a concept of supervenience. Rather, it is philosophical utility and broad agreement with prior philosophical usage which set the standards. Since different notions may be useful for different purposes, different notions of supervenience have surfaced.3 This chapter is specifically concerned with finding a supervenience concept that can be of use in a discussion of modal supervenience.4 Though there are different concepts of supervenience, there is an overlap between them. First of all, it is commonly agreed that supervenience is a relation between two pluralities—mental facts are supposed to supervene on physical facts, determinable properties on determinate ones, wholes on their parts, for example.5 What kinds of items—facts, truths, states of affairs, properties, events or even individuals—are supposed to be related by supervenience varies. Although in the remainder of the book I will discuss whether modal truths supervene on other truths, in this chapter I will follow the bulk of the work on supervenience in concentrating on property supervenience.6 2

3 4 5

6

According to the OED, ‘supervene’ means to come on or occur as something additional or extraneous; to come directly or shortly after something else, either as a consequence of it or in contrast with it; to follow closely upon some other occurrence or condition. Thus, ordinary use seems to focus on causal consequences (in close temporal vicinity), while the philosophical use focuses, if anything, on conceptual or metaphysical consequences. In both cases, what supervenes (or how it is) is, arguably, explained by that on which it supervenes. Vernacular and philosophical use may thus have more in common than would appear at first glance. Cp. Kim (1990: 1f.). Lewis laments an ‘unlovely proliferation of non-equivalent definitions’ (Lewis 1986a: 14). For a general introduction to supervenience see Steinberg (2013b). Cf. e.g. Correia (2005: 132) and Kim (1990: 7). Kim represents this in terms of set talk. The difference is insignificant in this case. Incidentally, just as singleton sets are very small sets, I take it that the limiting case of a plurality is a ‘plurality’ consisting of only one thing. So, we can talk of a single item supervening on some items or even a single item. The supervenience debate tends to employ an abundant conception of properties, so that existence questions are not at issue. See e.g. McLaughlin (1995: 21). This will also be assumed here. For my favourite spelling out of such a conception see chapter

16

Chapter 2. Supervenience

Thus, what is at issue is what it means for properties of a certain kind to supervene on properties of another kind. This is not as much of a detour as it may appear because one kind of property supervenience (global property supervenience) turns out to be equivalent to truth supervenience, and, in any case, the lessons to be learned from the property case are easily transposed to the truth case. The other overlap concerns what is entailed by supervenience claims. First of all, supervenience entails covariance.7 Roughly, if A-properties supervene on B-properties, there cannot be a difference in A-properties without a difference in B-properties. Different notions of covariance spell out this rough characterisation differently.8 Some philosophers have even made out the connection between supervenience and covariance to be extremely intimate: they just define supervenience as covariance.9 As I explain later, this would have disastrous consequences for modal supervenience theses. Secondly, supervenience is commonly taken to entail some sort of dependence. If A-properties supervene on B-properties, A-properties depend on Bproperties.10 In fact, something stronger seems to be true. Not only does supervenience entail dependence, it even entails one way dependence, or ontological posteriority.11 This is part of why supervenience theses are so attractive: they promise an ontology of all kinds of things we may want to include in the inventory of the world while keeping the ontological ground level tidy. Mental facts, for instance, are easier to swallow if this acceptance is accompanied by the claim that they arise in virtue of ontologically more basic physical facts—this is one reason to go in for mental/physical supervenience. Let me add a note of caution. As I said two paragraphs back, some philosophers just take supervenience to be defined as covariance, and, thus, purely modally. Since supervenience is a philosophical concept, without too many 7 8 9 10

11

4 and cp. Steinberg (2012). Cf. Kim (1990: 9) and Correia (2005: 132). This will be the topic of the next section. For instance Kim (1984). He changed his mind in Kim (1990: 10), acknowledging that the dependence component of supervenience is not ensured by covariance. Kim (1990: 8f.) claims ‘virtual analyticity’ for the idea that supervenience is a kind of dependence after Davidson (1970: 214) mentioned the two notions in one breath. See also Paull and Sider (1992: 833). Cf. Correia (2005: 132). Ontological posteriority is the converse of ontological priority.

2.1. Supervenience

17

ties to everyday thought, such modal definitions are simply taken to introduce notions of supervenience by stipulation. Lewis, for instance, clearly thinks that there is no more to supervenience than covariance: To say that so-and-so supervenes on such-and-such is to say that there can be no difference in respect of so-and-so without difference in respect of such-andsuch. […] A supervenience thesis is, in a broad sense, reductionist. But it is a stripped-down form or [sic] reductionism, unencumbered by dubious denials of existence, claims of ontological priority, or claims of translatability. (Lewis 1983a: 358, my italics)12

In this chapter, I want to argue that covariance is not enough for supervenience, since covariance cannot make up the links in the ontological hierarchy. If supervenience just is covariance by fiat, this argument is an obvious non-starter. I think that philosophers who accept the antecedent of this conditional rather too quickly may be misled by the fact that there are no natural language constraints on a definition of supervenience into thinking that there are no constraints at all. But supervenience has an—admittedly rather short—philosophical history, and it is anything but clear that a purely modal definition does justice to its use in informal philosophical discussions. For instance, the passage from R.M. Hare’s The Language of Morals which is typically credited to have introduced the term ‘supervenience’ into philosophical discussions13 reads as follows: Let me illustrate one of the most characteristic features of value-words in terms of a particular example. It is a feature sometimes described by saying that ‘good’ and other such words are the names of ‘supervenient’ or ‘consequential’ properties. Suppose that a picture is hanging upon the wall and we are 12

13

It is somewhat surprising that Lewis should complain about ‘dubious claims of ontological priority’ here. After all, the passage is from an article in which he capitalises on the natural-unnatural divide between properties, a distinction that is, arguably, best viewed as a distinction between fundamental and derivative properties, and, thus, as one that relies on an ontological priority ordering on properties. Cp. Schaffer (2009: 353). In a similar vein, McLaughlin (1995: 18) complains about strengthening the ‘core idea of supervenience’ in such a way that it would yield an ontological priority ordering because such a strengthening would rely on ill-understood notions. However, just a few pages later, he has no problems to discuss what the primary relata of the supervenience relation are, such that all other things are only relata of supervenience in a derivative way. See McLaughlin (1995: 20). One can’t help but think that these philosophers scorn definition proposals for interesting sorts of supervenience for employing notions they themselves are happy to rely on elsewhere. This chapter testifies to the fruitfulness of parting company with them here. Though Hare insisted later (in his 1984: 1) that he does not deserve the credit.

18

Chapter 2. Supervenience discussing whether it is a good picture; that is to say, we are debating whether to assent to or dissent from, the judgement ‘P is a good picture’. […] Suppose that there is another picture next to P (I will call it Q). Suppose that either P is a replica of Q, or Q is a replica of P, and we do not know which, but do know that both were painted by the same artist at about the same time. Now there is one thing that we cannot say; we cannot say ‘P is exactly like Q in all respects save this one, that P is a good picture and Q not’. If we were to say this, we should invite the comment, ‘But how can one be good and the other not, if they are exactly alike? There must be some further difference between them to make one good and the other not.’ (Hare 1952: 80f., the last italics are mine)

If we take the last sentence of the quotation seriously, Hare takes it to be a salient consequence of the supervenience of A-properties on B-properties that for any A-property there are some B-properties that make things have the former. Now, making something some way (true, good, a criminal mastermind) is neither obviously itself a purely modal notion nor entailed by one.14 Hare’s introduction of the term, then, does not support a dismissive attitude towards the project of this chapter, and neither do many other uses of the concept.15 Consequently, I think that we may reasonably ask whether supervenience is a purely modal notion, and that, if the answer is ‘no’, the definition proposals (e.g. in Kim 1984) that define supervenience as covariance share the fate of purely modal definitions of other important philosophical concepts with slightly more pre-theoretical content—like making something true/false and being ontologically dependent on:16 they are simply inadequate. However, this is in large part a terminological issue. If you stick to your guns and insist that ‘[t]he core idea of supervenience is captured by the slogan, “there cannot be an A-difference without a B-difference”’ (McLaughlin and Bennett 2008: §1),17 read ‘covariance’ as ‘supervenience’ and ‘supervenience’ as whatever you think may figure as the links in the ontological hierarchy—‘being groun14

15

16 17

It is, thus, somewhat surprising that philosophers cite the above passage and go on to assert that ‘[s]upervenience, then, is a modal notion’ (Horgan 1993: 555). (It is clear from the surrounding text that Horgan means that it is a purely modal notion). For a discussion of the philosophically important notion of making something true, arguing for the inadequacy of a purely modal characterisation, see Schnieder (2006c). Recall for instance the quotation from Jonathan Bennett above (on pages 13–14): ‘[E]vents are supervenient entities, meaning that all truths about them are logically entailed by and explained or made true by truths that do not involve the event concept’. For ontological dependence see Correia (2005) and Schnieder (2006b). For truthmaking see the references in fn. 14. McLaughlin and Bennett go on to argue (in §3.5) that supervenience does not entail

2.2. Covariance

19

ded in’, perhaps18 —in the remainder of this book. 2.2

Covariance

Covariance is a modal notion. It holds between A-properties and B-properties iff there cannot be a difference in A-properties without a difference in B-properties.19 For an uncontroversial example of covariance stripped of its modal aspects consider a digital thermometer that is fully operational throughout its existence. Its readings covary with the surrounding temperature: there is never a difference in its readings without a difference in surrounding temperature. On the other hand, the surrounding temperature may not covary with the thermometer’s readings. Some temperature changes are too subtle to be picked up by the thermometer: sometimes, there are differences in temperature without differences in thermometer readings. Once we bring back in the modal component, different non-equivalent notions of covariance become available. Kim (1990) discusses three: weak, strong and global covariance. Weak and strong covariance concern differences of individuals, global covariance differences of whole possible worlds. Kim defines weak and strong covariance as follows:20 Weak Covariance (WC) A-properties weakly covary with B-properties ↔df. There are no world w and possible individuals x and y such that x and y have the same B-properties in w but differ in their A-properties in w. Strong Covariance (SC) A-properties strongly covary with B-properties ↔df. There are no worlds w and w′ and no possible individuals x and y such that x has the same B-properties in w as y has in w′ , yet x has different A-properties in w than y has in w′ .

18 19

20

dependence with arguments similar to the ones I will use to show that supervenience is not exhausted by covariance. As, e.g., Jonathan Schaffer believes (see Schaffer 2009). For simplicity’s sake I will mainly be concerned with monadic properties whose exemplification is not relative to time. Also, I will allow myself to the sentential operators ‘◻’ and ‘◊’ to express necessity and possibility respectively, and sometimes to possible worlds talk. All of these simplifications are standard in the literature on supervenience. Cf. e.g. Correia (2005: 134). Cf. Kim (1990: 10). Kim speaks of (in)discernibility. However, this has unwanted epistemic connotations which talk of qualitative sameness and difference avoids.

20

Chapter 2. Supervenience

It is easy to see that Strong Covariance logically entails Weak Covariance— just let w = w′ in (SC). On the other hand, Weak Covariance does not entail Strong Covariance. For Weak Covariance imposes only intra-world constraints on differences in A-properties, while all inter-world differences are allowed. Strong Covariance imposes inter-world constraints as well. If intraworld constraints are satisfied while inter-world constraints are violated, we have Weak Covariance without Strong Covariance. A relatively uncontroversial example of Strong Covariance is the property of being a prime number and the set of number identity properties (i.e. {x ∶ ∃n (x = the property of being identical with the number n)}). The former strongly covaries with the latter, since no number is prime in one world but fails to be prime in another.21 Since the property of being a prime number strongly covaries with number identity properties it also weakly covaries with them. An uncontroversial example of Weak Covariance without Strong Covariance is harder to find, but easy to make up. Consider the thermometer again, but suppose that in some possible worlds it is fully operational while in all others it has an offset by a few degrees, i.e. in the offset worlds it shows a reading of 2 degrees higher, say, than the surrounding temperature. The thermometer readings weakly covary with the surrounding temperature, since there are no worlds in which there is a difference in thermometer readings without a difference in surrounding temperature. But thermometer readings do not strongly covary with surrounding temperature, since there is an offset world w and a non-offset world w′ such that the temperature surrounding the thermometer is the same in w and w′ while the thermometer reading is two degrees higher in w than it is in w′ . Kim (1990: 10) proposes alternative definitions of weak and strong covariance that are equivalent to the original definitions on the assumption that the B-properties are closed under Boolean operations.22 I present them here 21 22

However, number identity properties neither strongly nor weakly covary with the property of being prime: there is more than one prime number. To be more explicit, the claim is that, necessarily, if the B-properties are closed under Boolean operations, then A-properties weakly/strongly covary with B-properties just in case the former weakly/strongly covary* with the latter. I will explain what the assumption of Boolean closure amounts to when it is needed. Incidentally, McLaughlin (1995: 29f.)—rightly, though perhaps somewhat overestimating the fact’s significance—points outs that this still does not mean that the relations of covariance and covariance* are equivalent, since they can hold between properties that are not closed under Boolean operations, and may then come apart. We will soon see

2.2. Covariance

21

as well, since they (or, rather, slight modifications) will be used later on as the basis of our definitions of supervenience. Here are Kim’s alternative definitions of covariance:23 WC* A-properties weakly covary* with B-properties ↔df. ◻∀x, P [(P is an A-property & x has P) → ∃Q (Q is a B-property & x has Q & ∀y (y has Q → y has P))]. In words: A-properties weakly covary* with B-properties just in case it is necessary that anything with an A-property also has a B-property that is materially sufficient for having the A-property. SC* A-properties strongly covary* with B-properties ↔df. ◻∀x, P [(P is an A-property & x has P) → ∃Q (Q is a B-property & x has Q & ◻ ∀y (y has Q → y has P))]. In words: A-properties strongly covary* with B-properties just in case it is necessary that anything with an A-property also has some B-property that is strictly sufficient for having the A-property. The starred variants of (WC) and (SC) will be the bases for the definition proposal for supervenience I will eventually propose. Note that they only differ in the strength of the property connections they require (i.e. material vs. strict sufficiency). Here is the proof that (WC*) and (SC*) are really equivalent with (WC) and (SC) respectively on the assumption of Boolean closure:24 For the proof that (SC*) entails (SC) we don’t need the assumption of Boolean closure: Suppose that S1 A-properties strongly covary* with B-properties; i.e. in every possible world in which anything has an A-property it also has a B-property that is strictly sufficient for having the A-property.25 Suppose further that S2 There are worlds w and w′ such that something, x, has different Aproperties in w than something, y, has in w′ . 23 24 25

how a simple modification deals with this. I use upper case letters starting with ‘P’ as (singular) variables ranging over properties. Cp. Kim (1984: 163ff.). To make the transition from boxes to universal quantification over possible worlds and back again, I assume that it is necessary that p just in case at all worlds, p. I take it that this is not up for debate. Also, I assume that, necessarily, A-properties are essentially A-properties, i.e. that if something is an A-property in some world, it is an A-property in all worlds, or at least in all worlds in which it exists at all.

22

Chapter 2. Supervenience

Then either (i) x has an A-property P in w that y lacks in w′ or (ii) vice versa. Suppose (i). By (S1), x also has a B-property Q in w which is strictly sufficient for having P. Since y does not have P in w′ , it does not have Q either there. So, x also has different B-properties in w than y has in w′ . The same holds, mutatis mutandis, for (ii). Consequently, discharging (S1), if x has different A-properties in w than y has in w′ , x also has different B-properties in w than y has in w′ . Hence, Strong Covariance* entails Strong Covariance. If we substitute ‘strictly’ in ‘strictly sufficient’ with ‘materially’ and let w = w′ , we get a proof that Weak Covariance* also entails Weak Covariance. For the other direction we need the assumption of Boolean closure of Bproperties. If we think of properties as somehow exhibiting structure comparable to conjunction and negation, then the assumption of Boolean closure is that if some properties are B-properties, their conjunction is also a B-property, as is the negation of each. On the other hand, we may think of properties as sets of pairs of possible individuals and worlds—intuitively, the pairs ⟨x, w⟩ such that x has the property at w.26 Boolean closure of the B-properties then means that for any subset C of the B-properties, the intersection of C, ⋂ C, is also a B-property, and for any B-property P, P’s complement (with respect to the universal property, I × W; I is the set of possible individuals and W is the set of possible worlds, so that I × W is the set of all individual/world pairs), ¯ is also a B-property. P, Now assume that A

B-properties are closed under Boolean operations;

and suppose that S1

A-properties strongly covary with B-properties;

i.e. that there are no worlds w and w′ and individuals x and y such that x has the same B-properties in w as y has in w′ , but x has different A-properties in w than y has in w′ . Suppose further, for reductio, that S2

There is some world w in which something x has some A-property P but does not have any B-property that is strictly sufficient for having P.

By (A), there is such a B-property as x’s B-w-profile property—a property which is such that if anything has it in any world, it has exactly those B26

As, for instance, in Lewis (1986a: §1.5). Since, according to Lewis, possible individuals inhabit exactly one world, he can take properties to be sets of possible individuals, instead of pairs of possible individuals and worlds.

2.2. Covariance

23

properties in that world which x has in w.27 Since none of x’s B-properties is strictly sufficient for having P, x’s B-w-profile property is not sufficient for having P. This means that there is some world w′ and individual y such that y has x’s B-w-profile property in w′ but does not have P in w′ , i.e. y has the very same B-properties in w′ as x has in w, but different A-properties, contra (S1). Thus, by reductio, on the assumption of Boolean closure, if A-properties strongly covary with B-properties, then if anything has an Aproperty in any world, it also has a B-property which is strictly sufficient for having the A-property, i.e. A-properties strongly covary* with B-properties. Again, if we substitute ‘strictly’ in ‘strictly sufficient’ by ‘materially’ and let w = w′ throughout, we get a proof that Weak Covariance also entails Weak Covariance* —on the assumption of Boolean closure. Some philosophers argue that the assumption of Boolean closure is very problematic for many substitution instances of ‘B’.28 Otherwise numbers would have weight properties, for example, simply because none of them weighs 65 kg. However, as Correia (2005: §6.3) points out, the proof makes clear that the assumption of Boolean closure is only needed in order to justify (WC*)’s and (SC*)’s singular existential quantification ‘∃Q (Q is a Bproperty & …)’. If we use plural quantification instead,29 Boolean closure need not be assumed. There is no need for Boolean closure in order to prove the equivalence of (WC) and WC** A-properties weakly covary** with B-properties ↔df. ◻∀x, P [(P is an A-property & x has P) → ∃QQ (QQ are the B-properties x has & ∀y (QQ are the B-properties y has → y has P))].30 27

28 29 30

We may think of it as a conjunctive property which, for any atomic B-property, either has it (if x exemplifies it in w) or its negation (if x does not exemplify it in w) as a conjunct. Alternatively, we may think of it as the intersection of all B-properties x has in w. See e.g. McLaughlin (1995: 28). I use variables consisting of two letters to indicate plural quantification. The locus classicus for plural quantification is Boolos (1984). See also Rayo (2007). To prove the equivalence we do not need full blown Boolean closure but the more modest assumption that the property of having no (other) B-properties is itself a Bproperty. Alternatively, we can just add the disjunct ‘∨ (¬∃Q (Q is a B-property & x has Q) & ∀y (¬∃Q (Q is a B-property & y has Q)) → y has P)’—saying that the relevant individual does not have any B-properties while not having any B-properties

24

Chapter 2. Supervenience

In words: A-properties weakly covary** with B-properties just in case it is necessary that anything with an A-property also has B-properties that are such that having them and no other B-properties is materially sufficient for having the A-property. Likewise, Boolean closure need not be assumed in order to prove the equivalence of (SC) and SC** A-properties strongly covary** with B-properties ↔df. ◻∀x, P [(P is an A-property & x has P) → ∃QQ (QQ are the B-properties x has & ◻ ∀y (QQ are the B-properties y has → y has P))]. In words: A-properties strongly covary** with B-properties just in case it is necessary that anything with an A-property also has B-properties that are such that having them and no other B-properties is strictly sufficient for having the A-property. Since covariance and covariance** thus defined can be shown to be equivalent notions,31 we may drop the double stars when thinking of covariance in terms of (WC**) and (SC**) from here on. Both Weak Covariance and Strong Covariance concern differences of individuals and require property connections (i.e. material and strict sufficiency). However, there is another kind of covariance, Global Covariance, which is silent on individuals and property connections. Global Covariance is instead defined via sameness and differences in A- and B-respects of whole worlds:32 Global Covariance (GC) A-properties globally covary with B-properties ↔df. ∀w, w′ (w and w′ agree in B-respects → w and w′ agree in A-respects). Different authors have different views on how the phrase ‘w and w′ agree in A-respects’ should be spelled out. Clearly, it should not mean that the worlds have the same A-properties. For, in many interesting candidate cases, the relevant properties are not exemplified by worlds at all.33 Mental properties may non-trivially globally supervene on physical properties. But worlds do not have mental properties, their inhabitants do. Thus, trivially, no two worlds have different mental properties.

31 32 33

is materially sufficient for having the A-property in question—to the consequens of the definiens. (Likewise, mutatis mutandis, for the formulation of Strong Covariance to follow). In order to avoid clutter I left it at the formulation in the main text. The proof is an obvious variant of the one presented above. Cf. Correia (2005: 137). See Kim (1984: 168). Cf. Paull and Sider (1992: 834f.).

2.2. Covariance

25

For the purposes of this chapter, I will follow Kim (1984: 167f.)’s proposal to understand the phrase in terms of property distributions at the relevant worlds: w and w′ agree in A-respects just in case whichever A-properties are exemplified by an individual in one of the worlds are also exemplified by that individual in the other world. Thus, on this understanding of its definiens, (GC) says that A-properties globally supervene on B-properties just in case whenever worlds agree on which individuals have which B-properties, they also agree on which individuals have which A-properties. Other authors have proposed more lenient understandings of (GC)’s definiens. We will come across one of those in the appendix to this chapter. Kim’s understanding of Global Covariance has the advantage that Global Covariance turns out to be equivalent to a certain sort of truth-covariance that will concern us in the chapters to follow.34 Two worlds agree in their distribution of A-properties in the sense indicated just in case they agree in which A-propositions are true at them. Thus, Global Covariance of A-properties on B-properties goes along with covariance of A-truths on B-truths:35 Strong Truth Covariance (SCT ) A-truths strongly covary with B-truths ↔df. There are no worlds w and w′ such that the same B-propositions but different A-propositions are true at w and w′ . Again, there is an equivalent formulation in terms of boxes: Strong Truth Covariance* (SCT *) A-truths strongly covary* with B-truths ↔df. ◻∀x [x is an A-truth → ∃yy (yy are B-truths & ◻ (yy are true → x is true))].36 In words: Necessarily, for any A-truth there are some B-truths that entail the A-truth. The proof that Strong Truth Covariance and Strong Truth Covariance* mirrors that for the equivalence of Strong Covariance and Strong 34 35 36

Again, we may distinguish between weak and strong covariance. This will become obvious in the alternative formulation to be given shortly. This was already argued in Kim (1984: 168f.), although he did not mention that this is a kind of strong supervenience thesis. The same caveat as in fn. 30 applies, as well as the assumption that, necessarily, Apropositions are essentially A-propositions. Incidentally, we get Weak Truth Covariance by deleting the second box. The result is then equivalent to the result of deleting the whole last conjunct, i.e. to ‘◻∀x (x is an A-truth → ∃yy (yy are B-truths))’—which defines a rather uninteresting notion, especially once we incorporate the caveat of fn. 30.

26

Chapter 2. Supervenience

Covariance**.37 Since Strong Truth Covariance and Strong Truth Covariance* turn out to be equivalent notions, we may drop the star when thinking of Strong Truth Covariance in terms of (SCT *) from here on. Moreover, we may think of Global Covariance in terms of (SCT *)’s definiens. Some philosophers have found Global Covariance more attractive than both Weak Covariance and Strong Covariance because it appears to avoid commitment to property connections.38 However, the relation between Global Covariance and the other sorts of covariance—and correlatively the question of whether Global Covariance really can avoid commitment to property connections—is much debated. I will trace the debate in an appendix. 2.3 Covariance and Ontological Priority As I mentioned before, some philosophers think there is nothing more to supervenience than covariance. In this section, I will argue that this conflicts with Kim’s ‘virtually analytic’ claim that supervenience entails dependence or ontological posteriority—because covariance does not. In the next section I will argue that it is essential that we respect this desideratum if we want to end up with a notion of supervenience that allows us to make interesting supervenience claims concerning modal properties and truths. Structural properties of covariance might already make us doubt that covariance of any kind entails dependence. For convenience, the characterisations of the structural properties of binary relations that will be relevant in this chapter are given below: Reflexivity

R is reflexive ↔df. ∀x (x stands in R to x);

Irreflexivity

R is irreflexive ↔df. ∀x ¬(x stands in R to x);

37

38

Sketch. (SCT *) ⇒ (SCT ): Suppose there are worlds w and w′ at which different Apropositions are true. Let x be an A-proposition true at one of them, at w say, and not true at the other. If A-truths strongly covary* with B-truths, there are some Bpropositions yy true at w which entail x. But yy cannot also be true at w′ , otherwise x would be true there, too. Thus, w and w′ also differ in their B-truths. (SCT ) ⇒ (SCT *): Suppose that A-truths do not strongly covary* with B-truths. This means that there is a possible world w at which some A-proposition x is true while no B-propositions are true at w that entail x. Thus, there is some world w′ at which all of w’s B-propositions are true while x is not. But then w and w′ differ in their A-truths without also differing in their B-truths. Cf. Kim (1984: 167).

2.3. Covariance and Ontological Priority

Symmetry

R is symmetric ↔df. ∀x, y (x stands in R to y → y stands in R to x);

Asymmetry

R is asymmetric ↔df. ∀x, y (x stands in R to y → ¬(y stands in R to x));

Antisymmetry

R is antisymmetric ↔df. ∀x, y ((x stands in R to y & y stands in R to x) → x = y);

Transitivity

R is transitive ↔df. ∀x, y, z ((x stands in R to y & y stands in R to z) → x stands in R to z).

27

According to Kim (1990: 13), dependence is asymmetric, while covariance is not. The easiest way to see this is to note that covariance of any kind is clearly reflexive: A-properties weakly and strongly covary with themselves, A-truths strongly covary with themselves, and, thus, A-properties globally covary with themselves. Reflexivity excludes asymmetry. Thus, if dependence is asymmetric, it is not entailed by covariance. However, it is not clear that dependence is asymmetric. Kim notes that we can speak of ‘mutual functional dependence’ (Kim 1990: 13)—if x = y + 2, y = x − 2, and, hence, the value of x depends on the value of y and the value of y depends on the value of x—but alleges that this is an extended sense of ‘dependence’. What is entailed by supervenience, according to him, is ontological or metaphysical dependence and this kind of dependence does not allow for mutual dependence. I am not so sure. After all, even conceptual dependence appears to be able to be mutual. It is at least not absurd to claim that the concepts of elementhood and set are mutually dependent. Why would ontological dependence always have a clear-cut preference for the chicken or the egg?39 This is a moot point. Though ontological dependence may not be asymmetric, ontological posteriority, that is, non-mutual ontological dependence, clearly is—if ontological dependence does not have a preference between two items then neither is ontologically prior to the other. If any of the supervenience relations can provide the links in the ontological hierarchy, it must 39

Fine (1995a: 282ff.), for instance, seriously considers the possibility that (some kind of) ontological dependence may be mutual. He goes on to distinguish, as we will do presently, between dependence and priority, where the former allows, but the latter precludes, reciprocity.

28

Chapter 2. Supervenience

not only entail ontological dependence but ontological posteriority. Thus, the asymmetry argument stands. We might think that the asymmetry worry is easy to deal with if we just fiddle around with covariance a bit. For one, although covariance is reflexive, we can easily define an irreflexive relation on the basis of any relation whatsoever. Maybe supervenience is not just covariance but covariance between different sets of properties. This hope is spoiled by the observation that covariance of any kind is not even anti-symmetric—sometimes A-properties covary with B-properties and vice versa even though A ≠ B. Consequently, covariance between different sets of properties will still not be asymmetric. Kim himself gives an argument he borrows from Lombard (1986: 225ff.) to the effect that strong (and, thus, weak) covariance is not antisymmetric: perfect sphere surface area properties strongly covary with perfect sphere volume properties—no perfect sphere has the same volume (in any world) as another (in any world) but a different surface area. But perfect sphere volume properties also strongly covary with perfect sphere surface area properties. ‘And we don’t want to say that either determines, or depends on, the other, in any sense of these terms that implies an asymmetry’ (Kim 1990: 13). Now, there is room for debate whether Kim’s example shows what it is supposed to show. Some philosophers think that properties are individuated intensionally, i.e. that the following is true:40 Property Individuation

P = Q ↔ ◻∀x (x has P ↔ x has Q).

In words: No two properties are necessarily co-exemplified. These philosophers will hold that in Kim’s example the allegedly two sets of properties, i.e. the set of properties of being a perfect sphere with such-and-such a surface area and the set of properties of being a perfect sphere with such-and-such a volume, is one set, since for any surface area property in the first set there is a necessarily co-exemplified volume property in the second set and vice versa. Consequently, those philosophers would deny that Kim’s example shows that covariance is not antisymmetric (we knew all along that it is reflexive). Kim’s point can be made nevertheless. Consider any two properties P and Q that are modally independent of each other, i.e. P and Q are such that any combination of them is possibly exemplified by some individual—being red and being a car, for instance: it’s possible for there to be a red car, a non-red 40

This is implied, for instance, by Lewis’s conception of properties as sets of (worldbound) individuals, and, of course, by the adequacy of thinking of properties as sets of possible-individual/world pairs. See Lewis (1983a). Also, the conception discussed in chapter 4 has this consequence.

2.3. Covariance and Ontological Priority

29

car, a red thing that is not a car, and a non-red non-car. Consider the sets ¯,P ¯ The A- and B-properties ¯ ∩ Q, P ¯ ∩ Q}. A = {P, Q} and B = {P ∩ Q, P ∩ Q mutually strongly covary with each other: there cannot be a difference in the ‘conjunctive’ properties between any individuals (whether in the same world or not) without a difference in the ‘conjuncts’, but neither can there be a difference in the conjuncts without a difference in the conjunctive properties. Still, A and B are different sets, even on an intensional individuation of properties. P and Q are modally independent, so there are four properties ¯,P ¯ name the same ¯ ∩ Q, P ¯ ∩ Q’ in B (no two entries on the list ‘P ∩ Q, P ∩ Q property). Since A has only two members, A ≠ B. Consequently, strong covariance is not antisymmetric, and, thus, strong covariance between different sets of properties is not asymmetric.41 The non-antisymmetry of covariance gives unacceptable results for the proposal to view covariance as providing the links in the ontological hierarchy. For instance, it is easy to show that if A-properties strongly covary with B-properties and the B-properties are closed under Boolean operations, then there is a subset of the B-properties which strongly covaries with the A-properties. Here is how it goes: Assume that A1

A-properties strongly covary with B-properties;

and that A2

B-properties are closed under Boolean operations.

Then, for any A-property P there are some B-properties, Q1 , . . . , Qn , such that having each of them is strictly sufficient for having P and no other B-property 41

However, we can show that, given an intensional individuation of properties, if Aproperties mutually covary with B-properties, then the Boolean closure of A = the Boolean closure of B. Think of properties as drawing boundaries into logical space (thought of as consisting of possible-individual/world pairs). The intensional individuation of properties ensures that no two properties draw exactly the same boundaries. The mutual covariance of A- and B-properties means that the properties in A—collectively—and the properties in B—collectively—draw the same boundaries. This does not yet mean that A and B contain the same properties (as witness the example in the main text). But now consider the Boolean closures of A and B, B(A) and B(B). What closure ensures is that every (possibly scattered) area bounded by the boundaries drawn by the properties in A corresponds to exactly one property in B(A), and likewise for B. Since the properties in A and B draw the same boundaries, every property in B(A) is also in B(B) and vice versa. Thus, if A- and B-properties mutually covary, the Boolean closure of A = the Boolean closure of B. However, the argument at the end of this section spoils the hope of taking advantage of this result.

30

Chapter 2. Supervenience

is strictly sufficient for having P. By (A2) there is a disjunctive B-property, having any of Q1 , . . . , Qn , such that having P is strictly sufficient for having it. For, suppose it were not, then something could have P without having the disjunctive property. But this contradicts (A1) and the construction fact that the Qi ’s are all the B-properties that are each strictly sufficient for having P. Since the choice of P was arbitrary, there is such a disjunctive B-property for any A-property. The set B∗ which has all of those disjunctive B-properties as members will be a subset of the set of the B-properties and strongly covary with the A-properties. If covariance were to provide the links in the ontological hierarchy, and, thus, were to entail ontological posteriority, it would be possible for B-properties to be ontologically prior to A-properties, while some B-properties are also ontologically posterior to the A-properties. But this is surely absurd. Kim (1990: 13f.) considers another proposal to get around the difficulty. The proposal is to define supervenience in terms of non-mutual covariance, i.e. A- properties supervene on B-properties iff A-properties covary with Bproperties and B-properties do not covary with A-properties. However, as he notes himself, non-mutual covariance seems to be neither necessary nor sufficient for supervenience, since it is neither necessary nor sufficient for ontological posteriority. Non-mutual covariance is not necessary because neither ontological posteriority nor supervenience seem to exclude mutual covariance.42 Recall the ¯,P ¯ The current pro¯ ∩ Q, P ¯ ∩ Q}. two sets A = {P, Q} and B = {P ∩ Q, P ∩ Q posal would exclude that the B-properties supervene on the A-properties, or that B(A) ∼ A—the Boolean closure of A without P and Q—supervenes on A, just because the sets mutually covary. However, arguably, being truthfunctionally constructed out of is one relation which can be responsible for some properties to be less natural than others, and, thus, for the former to supervene on the latter.43 Furthermore, it is plausible that if A-properties supervene on B-properties and B-properties are closed under Boolean operations, A-properties also supervene on the subset of B-properties mentioned in the penultimate paragraph. But A-properties do not non-mutually covary with the subset. Non-mutual covariance is not sufficient because properties necessarily ex42 43

Ontological priority excludes mutual dependence. But covariance does not entail dependence. See e.g. Schaffer (2009: 353) for the view that for some properties to be more natural than others just is for the former to be ontologically prior to the latter.

2.4. Covariance and Modal Supervenience

31

emplified by everything or nothing (necessarily universal and necessarily empty properties) covary with any properties. But no properties that are neither necessarily universal nor necessarily empty covary with them. If non-mutual covariance were sufficient for supervenience we would have to conclude that any set of necessarily universal or necessarily empty properties supervenes on any set of contingent properties. But, arguably, not any set of contingent properties is ontologically prior to all necessarily universal or necessarily empty properties.44 Suppose, for instance, that physical properties are ontologically prior to mental properties. If non-mutual covariance were sufficient for supervenience it would be true that, still, the property of having or not having electrical charge, say, is ontologically posterior to all contingent mental properties. 2.4

Covariance and Modal Supervenience

In the last section we saw that supervenience cannot be exhausted by covariance if it is to make up the links in the ontological hierarchy. Roughly, it is generally too easy for properties to covary, and it is sometimes too hard for properties to covary non-mutually. In this section I will argue that covariance of modal properties (and truths) in particular is a trivial thesis. Roughly, the modal aspects of the supervenience candidates interact with the modal aspects of the covariance theses in undesired ways. Hence, we should want more from modal supervenience than modal covariance. The most unspecific thesis of modal supervenience is that modal properties (or truths) supervene on non-modal properties (or truths). It appears not to be a trivial thesis. It has attracted many proponents who certainly do not take it to be a trivial thesis, but rather think that it can be fruitfully used to cast doubt on philosophical views that conflict with the thesis.45 The corresponding covariance theses are: 44 45

See Correia (2005: §4) for these and further problems for defining supervenience just in terms of covariance. This is what happens, for instance, in a strand of the debate over constitution and identity. Roughly, proponents of the identity view argue that if the statue and the lump of clay that wholly constitutes it were different objects, their respective modal properties would not supervene on their non-modal properties, since lump and statue differ in the former but agree in the latter. See e.g. Zimmerman (1995: §9) and Bennett (2004b).

32

Chapter 2. Supervenience

Weak Modal Covariance (WMC) There are no world w and no possible individuals x and y such that x and y have the same non-modal properties in w but differ in their modal properties. Strong Modal Covariance (SMC) There are no worlds w and w′ and individuals x and y such that x has the same non-modal properties in w as y has in w′ but x has different modal properties in w than y has in w′ . Global Modal Covariance (GMC) There are no worlds which are the same with respect to non-modal truths but differ with respect to modal truths. All of these covariance theses are trivially true, on the assumption that S5 is the correct modal logic. Let’s start with Global Modal Covariance. In S5 we have both 1

◊p → ◻◊p;46

and 2

◻p → ◻ ◻ p.47

Partly transposed into possible worlds talk: whatever is possible or necessary is possible or necessary respectively in every world. Thus, if S5 is the correct modal logic, there are no worlds that differ with respect to modal truths full stop. Hence, any set of truths covaries with modal truths. But, surely, even if S5 is the right modal logic we can have non-trivial modal supervenience claims. A supervenience claim that is exhausted by Global Modal Covariance is not one of them. Now consider Weak Modal Covariance. Though there are no differences between worlds in modal truths, there can be, and are, differences between individuals in modal properties. My chair may be necessarily non-human, but I am certainly not. However, here the worry is that the proposed subvenient base, non-modal properties, is too broad to make for an interesting covariance thesis.48 It might be, for instance, that intra-world sameness in physical properties is enough to ensure intra-world identity. Since physical properties are presumably non-modal properties, intra-world sameness in non-modal properties is enough to ensure intra-world identity. But then we have 46 47 48

The characteristic S5 axiom. The characteristic S4 axiom. Cf. Divers (1992).

2.4. Covariance and Modal Supervenience

33

3

For every world w and individuals x and y: if x has the same non-modal properties in w as y, x = y. Since, necessarily, if x is the same as y, x has all the same properties as y, a fortiori x has all the same modal properties as y. Hence, intra-world sameness in non-modal properties ensures intra-world sameness in modal properties, but trivially. Anything weakly covaries with non-modal properties. Blackburn (1985: 50f.) recognises the problem of supervenience bases that are too broad to make for interesting weak covariance theses, and proposes to place a Limitation Constraint on supervenience: This will say that whenever a property F supervenes upon some basis, there is necessarily a boundary to the kind of G properties which it can depend upon. For example, the mental may supervene upon the physical, in which case the [constraint requires] that necessarily there are physical properties of a thing which are not relevant to its mental ones. (Blackburn 1985: 51)

This suggests a solution to our problem. Though modal properties trivially weakly covary with non-modal properties, they may non-trivially covary with a subset of the non-modal properties—intra-world sameness in a subset of non-modal properties may not ensure intra-world identity. The passage quoted is actually ambiguous. The first part suggests that modal properties supervene on non-modal ones just in case for every modal property there is a subset of the non-modal properties with which it weakly covaries. The second part suggests that modal properties supervene on nonmodal properties just in case there is a subset of the non-modal properties such that modal properties weakly covary with it. The difference between the two claims is that in the first case every non-modal property may be used, so to say, in a singular modal supervenience claim (though they may not all be used at once). In the second case, on the other hand, some non-modal properties need to remain unused over all—there need to be non-modal properties which are irrelevant to every singular modal supervenience claim. However, none of this helps.49 There is a very small subset of non-modal properties, the set of identity properties, which by itself makes weak cov49

Both interpretations also have quite generally implausible consequences. If we have true general supervenience claims—mental properties supervene on physical ones, for instance—we presumably also have true singular supervenience claims—the property of being in such-and-such a pain supervenes on physical properties P1 , . . . , Pn . But both interpretations of the Limitation Constraint would require that at least one of the Pi ’s has to be irrelevant to being in the pertinent kind of pain if the singular supervenience claim is to be true. The general recipe for singular supervenience claims would be to throw in some irrelevant properties in order to meet the Limitation Constraint.

34

Chapter 2. Supervenience

ariance trivial. We can leave out any other non-modal property in all singular modal supervenience claims and still get trivial results. Thus, for every modal property there is a subset of the non-modal properties, the identity properties, with which it weakly covaries. And also, there is a subset of the non-modal properties, the identity properties, such that modal properties weakly covary with it. This is so, since the following is true: 4

For any world w and individuals x and y: if x and y agree on their identity properties in w, x = y.

Since, necessarily, if x is the same as y, x has all the same properties as y, a fortiori x has all the same modal properties as y. Hence, intra-world sameness in identity properties ensures intra-world sameness in modal properties, but trivially. Anything weakly covaries with identity properties. Further, we cannot just exclude identity properties from the non-modal supervenience base. Arguably, the property of being necessarily identical with Socrates, say, does supervene on the property of being identical with Socrates. In any event, this should not be ruled out by fiat. Consequently, no version of Blackburn’s Limitation Thesis helps with the triviality issue in the case of Weak Modal Covariance. However, as we will see later, I think Blackburn was right on target with his observation that, often, not all supervenience base properties are relevant to specific singular supervenience claims. Though modal properties may supervene on non-modal properties, very many non-modal properties (identity properties in particular) are irrelevant for specific modal supervenience claims: weighing 65 kg is presumably irrelevant for possibly dying an unnatural death. But this insight is not well captured by the Limitation Constraint—nor indeed by any beefed up version of Weak Covariance. Finally, consider Strong Modal Covariance. Things have different properties in different worlds. So, inter-world sameness in identity properties does not ensure inter-world sameness in all properties. Socrates was a philosopher and he might have decided not to be. Suppose in w he did so decide. Then, though Socrates has the same identity property in the actual world as he has in w, he has different properties in the actual world than he has in w (for one, he exemplifies the property of being a philosopher in the actual world but does not do so in w). Thus, though everything weakly covaries with identity properties (and, thus, with non-modal properties in general), some things do not strongly covary with identity properties.

2.4. Covariance and Modal Supervenience

35

However, we can combine the problems for Global Modal Covariance and Weak Modal Covariance to show that Strong Modal Covariance is still not what we want. The predicate calculus versions of (1) and (2) are 5 ◻∀x ◻ (◊Fx → ◻◊Fx); and 6 ◻∀x ◻ (◻Fx → ◻ ◻ Fx). Thus, if any possible individual has a modal property, it has it in every world, according to S5. But this, together with 7 For any worlds w and w′ and individuals x and y: if x has the same identity properties in w as y has in w′ , x = y; ensures that Strong Modal Covariance holds. For, suppose that x has the same non-modal properties—and, thus, the same identity properties—in w as y has in w′ . Then, by (7), x = y. By (5) and (6), x has the same modal properties in w as y has in w′ . Although not everything strongly covaries with identity properties (and modal properties do not strongly covary with everything)50 , the combination of the two makes for a strong covariance thesis that is true for purely formal reasons. Thus, Strong Modal Covariance is trivially true, if S5 is the right modal logic. But, surely, even if S5 is the right modal logic, we can have substantive modal supervenience claims. A supervenience claim that is exhausted by Strong Modal Covariance is not one of them. The arguments in this section rely on the assumption that S5 is the correct modal logic. Although this is often assumed in philosophical discussions, it is not uncontested. Let me end this section by indicating why I think it is legitimate to rely on S5 nonetheless. First, the triviality of a general modal supervenience thesis should arguably not depend on results in the philosophy of modal logic. It would be surprising, to say the least, if modal supervenience theses could only be saved from triviality by arguing that S5 is too strong. Second, the most widely considered argument against the adequacy of S5 is the Chandler/Salmon argument.51 The argument turns on the plausibility of certain fairly localised failures of transitvity of the accessibility relation. Chandler, for instance, asks his readers to accept that being initially composed of most of its parts is essential to a certain bicycle, while being initially composed of all of its parts is not. By gradually ‘replacing’ initial bicycle 50 51

I may have the same mass as my (very solid) desk but while I am possibly human my desk is not. Hence, modal properties do not strongly covary with mass properties. See Chandler (1976) and Salmon (1982: §28).

36

Chapter 2. Supervenience

parts through a chain of possible worlds, we get a case in which: (i) neighbouring worlds have access to (i.e., they are possible with respect to) each other—(necessarily,) the bicycle might have been originally made of slightly different parts—, and (ii) the endpoint worlds (and presumably a great many more) do not have access to each other—(necessarily,) the bicycle could not have been originally made of entirely (or even largely) different parts. One of the more conservative reactions to this case—and the one both Chandler and Salmon advertise—52 is to say that accessibility is not transitive, and that, thus, neither the characteristic S5 nor the characteristic S4 axioms ((1) and (2) above) are valid (in the pertinent understanding of their boxes and diamonds).53 Even if such examples show that accessibility is not in general transitive, the reason that it is not transitive seems to be fairly localised. For most instances of (1) and (2) there is no reason to doubt that they will still be true, even if transitivity fails. If so, similar arguments to the ones of this section go through for a slightly restricted class of modal properties. I conclude that my reliance on S5 in this section is harmless, and that, whether or not there are independent reasons for mistrust in S5, the above arguments show that interesting modal supervenience claims cannot be exhausted by modal covariance. 2.5 Modal Supervenience and Explanation In the last section we saw that if supervenience were exhausted by any kind of covariance, the thesis that the modal supervenes on the non-modal would be trivially true. If modal supervenience is to do any serious philosophical work, there must be more to modal supervenience than covariance. A passage of one of the early proponents of moral/non-moral supervenience, Henry Sidgwick, may help here:54 There seems, however, to be this difference between our conceptions of eth52 53

54

But see Forbes (1985: ch. 7) and Lewis (1986a: 244f.) for a different conclusion to draw. S4 is sound and complete with respect to all possible world models in which accessibility is reflexive and transitive, S5 is sound and complete with respect to all possible world models in which accessibility is an equivalence relation (i.e. reflexive, symmetric and transitive). Quoted in Kim (1990: 5f.). For some reason the passage leads Kim to ascribe a covariance thesis to Sidgwick. Recall also the earlier quotations from Jonathan Bennett (on pages 13–14) and R. M. Hare (on pages 17–18) in which similar views on supervenience surface.

2.5. Modal Supervenience and Explanation

37

ical and physical reality: that we commonly refuse to admit in the case of the former—what experience compels us to admit as regards the latter—variations for which we can discover no rational explanation. In the variety of coexistent physical facts we find an arbitrary element in which we have to acquiesce, […]. But within the range of our cognitions of right and wrong, it will generally be agreed that we cannot admit a similar unexplained variation. We cannot judge an action to be right for A and wrong for B, unless we can find in the nature or circumstances of the two some difference which we can regard as reasonable ground for difference in their duties. (Sidgwick 1884: 206)

Sidgwick writes about what we can or cannot judge55 and about finding the grounds of certain differences. However, if we strip Sidgwick’s description of its epistemic elements, we get a quite plausible (partial) characterisation of supervenience: if A-properties supervene on B-properties, and two things differ in their A-properties, there must be an explanation of this difference (whether we can find it or not) in terms of their B-properties.56 Don’t get hung up on the ‘difference’: if A-properties supervene on B-properties and something has an A-property, there must be an explanation of why it has that A-property in terms of its B-properties.57 This is what is so unsatisfactory about Global Modal Covariance, Weak Modal Covariance and Strong Modal Covariance. Though (GMC) is true because there cannot be a difference in modal truths between worlds full stop, this leaves it open whether every—or, indeed, any—modal proposition which is true (at some world) is explained in terms of the non-modal truths (of that world). Similarly, though identity properties ensure that (WMC) and (SMC) are true, there most certainly are some modal properties such that that something has them is not explained in terms of its identity properties. Consequently, I propose to combine both covariance and explanatory aspects in a definition of supervenience that is suitable for modal supervenience. We can have different kinds of supervenience, depending on the kind of covariance on which it is based. Let’s start with weak and strong supervenience. We 55 56

57

Quite implausibly, at least if the ‘cannot’ is not supposed to mean the same as ‘should not’. Otherwise any epistemic subject can prove Sidgwick wrong. This fact about the existence of explanations could be the descriptive basis of the normative epistemological claim I ascribed to Sidgwick in footnote 55, namely that we should only judge that there is a difference if we can find the pertinent explanation for it. Then, of course, there is also an explanation of why two things differ in their Aproperties, if they do, in terms of their B-properties.

38

Chapter 2. Supervenience

may provisionally define along the lines of (WC**) and (SC**):58 Weak Supervenience (WS) A-properties weakly supervene on B-properties ↔df. ◻∀x, P [(P is an A-property & x has P) → ∃QQ (QQ are B-properties & x has QQ & ∀y (y has QQ → (y has P because y has QQ)))]. Strong Supervenience (SS) A-properties strongly supervene on B-properties ↔df. ◻∀x, P [(P is an A-property & x has P) → ∃QQ (QQ are B-properties & x has QQ & ◻ ∀y (y has QQ → (y has P because y has QQ)))]. Let’s say that Q1 , . . . Qn ground P just in case whatever has the Qi ’s has P because it has the Qi ’s. And let’s also say that Q1 , . . . , Qn determine P just in case, necessarily, whatever has the Qi ’s has P because it has the Qi ’s. Then we can put (WS) and (SS) into words easily. (WS) says that A-properties weakly supervene on B-properties just in case, necessarily, anything with an A-property also has some B-properties which ground the A-property. (SS) says that A-properties strongly supervene on B-properties just in case, necessarily, everything with an A-property also has some B-properties which determine the A-property. In a similar way we may also define a notion of Global Supervenience which is up to the task of yielding interesting global modal supervenience theses. Our basis will be (SCT *) for which I earlier argued that it is a definition of Strong Truth Covariance which is equivalent with Global Covariance: Global Supervenience (GS) A-properties globally supervene on B-properties ↔df. ◻∀x [x is an A-truth → ∃yy (yy are B-truths & ◻ (yy are true → (x is true because yy are true))]. In words: A-properties globally supervene on B-properties just in case, necessarily, any A-truth is explained by some B-truths which explain the A-truth in any world in which they are true. Every kind of supervenience entails the corresponding kind of covariance. That (WS) and (SS) entail (WC) and (SC) respectively is a straightforward 58

Similar definitions are given in Correia (2005: 143). A problem for all provisional definitions of this section will be the topic of the next section.

2.5. Modal Supervenience and Explanation

39

consequence of the (necessary) factivity of ‘because’:59 Factivitybecause

◻[(p because q) → (p & q)].

(GS) also entails (GC), since, again by the factivity of ‘because’, (GS) entails (SCT *), and Strong Truth Covariance of A-truths on B-truths is equivalent with Global Covariance of A-properties on B-properties. I will briefly sketch how the newly defined supervenience notions deal with the triviality problem cases for modal covariance. The problem for Global Modal Covariance was that no worlds differ in their modal truths full stop, and that, hence, modal truths covary with any truths whatsoever. This is not a problem for global modal supervenience, defined according to (GS), because it is certainly not true that just any truths explain a modal truth. For instance, modal truths do not supervene on truths about the University of London, say, because nothing about the University of London explains why Socrates might have decided to become a carpenter. So, the fact that no worlds differ with respect to their modal truths does not trivialise global modal supervenience. The problem for Weak Modal Covariance and Strong Modal Covariance were identity properties. But although modal properties weakly covary with identity properties, this does not guarantee that they weakly supervene on identity properties. In fact, they most certainly do not. Identity properties are irrelevant to many modal properties. For instance, though Socrates might have decided to become a carpenter, that he is Socrates does not seem to be able to explain why he might have so decided. Since Strong Supervenience is stronger than Weak Supervenience, identity properties do not trivialise strong modal supervenience either. It is generally assumed that ‘because’ is (necessarily) asymmetric:60 Asymmetrybecause 59 60

◻[(p because q) → ¬(q because p)].

I use lowercase letters starting with ‘p’ as sentential variables. See appendix A for more on sentential quantification. This is an extended use of ‘asymmetric’. We typically speak of asymmetric relations (e.g. the relation of being greater than), or, derivatively, of asymmetric expressions that signify relations (e.g., the general term ‘greater than’). ‘Because’ is a sentential connective. I do not mean to imply that sentential connectives are in the business of signifying relations. However, I will continue to use ‘asymmetric’ and other adjectives that typically signify structural properties of relations as applied to sentential connectives throughout this thesis. These uses should be understood in analogy with that made explicit by (Asymmetrybecause ).

40

Chapter 2. Supervenience

From the asymmetry of ‘because’ the following is an immediate consequence: 8

◻∀xx◻∀yy◻[(xx are true because yy are true) → ¬(yy are true because xx are true)].

If some propostions are true because some other propositions are true, it is not also the case that the latter are true because the former are. A slightly stronger principle is plausible as well: 9

◻∀xx ◻ ∀yy ◻ ∀yy′ ◻ [(yy′ ⊆ yy & (xx are true because yy are true)) → ¬(yy′ are true because xx are true)].

In words: If some propositions are true because some others are, it is not also the case that some of the latter explain the former.61 The feature of explanation stated in (9) gets rid of the asymmetry worries raised in section 2.3. Thus, any kind of supervenience defined above may be a dependence relation and a relation that ensures ontological posteriority.62 Three words about explanation and ‘because’ as it occurs in (WS), (SS) and (GS). First, an explanation may be complete or partial. That I bought a new laptop in early 2008 is part of the reason why I went into overdraft then. High living costs in London is another. These and a few other factors conspired to make it the case that my bank account went into overdraft in early 2008. That I bought a new laptop, that my day to day life was costly and a few other things together completely explain my financial mini crisis. Each by itself provides merely a partial explanation. In general, whenever some things xx completely explain y and z is one of xx, z partially explains y.63 ‘Because’ sentences can correctly be used to give both complete and partial explanations. However, if B-truths merely partially explain A-truths, this is not enough for an A-B supervenience claim to be true. This is why the ‘because’ sentences in the definientia of supervenience are stipulatively required to express complete explanations. Second, ‘because’ in the here pertinent sense is used to express objective explanations, explanations whose correctness does not turn on what the 61

62 63

A principle very similar to (9) is defended in Correia (2005: 63). (The principle is (P11), stating that ‘partial grounding’ is asymmetric). (8) is the special case of (9) in which yy′ are all of the yy’s. For more on the logic of ‘because’ see also Schnieder (2011) and Fine (2012b). In fact, Correia (2005) and Schnieder (2004a) argue that dependence and ontological priority should be defined in explanatory terms. See e.g. Fine (2012a: §5).

2.5. Modal Supervenience and Explanation

41

speaker, the hearer or anyone knows or believes (unless, of course, this is the topic of explanans or explanandum).64 Whether the explanation given by uttering a certain ‘because’ sentence is particularly helpful or even whether the act of explaining is successful may depend on a great many epistemic features of speakers and audience (e.g. on whether the relation between explanandum and explanans is already known to the audience), but whether the content of the ‘because’ sentence thus uttered is true does typically not thus depend on the epistemic states of thinkers. The explanations at issue here are such contents of objectively used because sentences. Third, in natural language we can use ‘because’ objectively to give causal explanations. The crowd cheered because the singer took the stage; the dog barked because it scented the burglar; the bomb exploded because the president pushed the button. Causal explanations make up a significant part of our explanatory repertoire. However, they do not make up all of it. Some philosophers have argued for this view in detail.65 I will be content with giving a few examples drawn from everyday life, science and philosophy: When Socrates drank the hemlock, Xanthippe became a widow because Socrates died;66 HCl is an acid because HCl is a proton donor;67 and if utilitarianism is right, some act may be good because it maximises overall happiness. All of these seem to be correct explanations, but they are not causal explanations. Causal explanations are backed by cases of causation.68 But Socrates’ death does not cause Xanthippe’s becoming a widow: for one, the death and the change in marital status occur at the very same time. Rather, the fact that Xanthippe became a widow consists in the fact that her husband died. Similarly, HCl’s being an acid is not a causal effect of HCl’s being a proton donor: for one, each HCl molecule is an acid for as long as it exists. Rather, HCl’s being an acid consists in its being a proton donor. And, of course, utilitarianism does not have the consequence that comparative facts about happiness 64

65 66 67 68

This might contrast with epistemic uses of ‘because’ as in ‘They are home, because the lights are on’, in which, roughly, the sentence following the ‘because’ cites epistemic reasons for believing what the sentence preceeding the ‘because’ says. Such epistemic uses are not relevant here. I trust that they are easy enough to spot so as to not engender confusion. See e.g. Achinstein (1983: ch. 7) and Ruben (1990: ch. 7). Kim (1974) already uses this example to draw attention to non-causal explanations. The example is Rosen’s. See Rosen (2010). In the simplest case, roughly, a sentence of the form ‘p because q’ provides a causal explanation only if the event described by ‘q’ causes the event described by ‘p’.

42

Chapter 2. Supervenience

distributions are causally responsible for the moral status of certain acts. In the case of supervenience we are dealing exclusively in such objective but non-causal explanations. The ‘because’ in the definientia of our supervenience definitions should be understood accordingly. Many philosophers recognise cases of objective but non-causal explanation in the light of examples like the ones just mentioned. However, there does not seem to be a generally accepted account of non-causal explanations —of what the talk of something’s consisting in something else amounts to— nor is it clear that there is a unified account to be given. Perhaps, there are various relations that back non-causal explanations in the way causation backs causal explanations. Our first example suggests that cases of conceptual analysis can back explanations.69 It is a conceptual truth revealed by analysis of the concept of a widow that a wife becomes a widow when her husband dies. This conceptual truth seems to back the non-causal explanation that Xanthippe became a widow because Socrates died in the same way that the fact that the president’s push of the button caused the explosion backs the causal explanation that the bomb exploded because the red button was pushed. In the next section we will have reason to take a closer look at explanations backed by conceptual analyses. However, examples like the two others suggest that things may not always be so. According to one prominent conception of conceptual analysis, conceptual analyses are epistemically transparent. Someone who doubts the statement of a correct analysis reveals himself as incompetent with at least one expression that occurs in the statement.70 Now, while it may be a broadly conceptual matter, discovered by armchair reflection, that those and only those acts that maximise happiness are good (let’s suppose), this will not be revealed merely by an analysis of the concept of a good act on such a view. After all, non-utilitarians will not ipso facto reveal themselves as incompetent in the use of ‘good’ and cognate expressions, even if utilitarianism turns out to be true. Rather, if it is a purely conceptual explanation that a certain 69 70

See e.g. Sharvy (1972: §3), Achinstein (1983: ch.7.8), Ruben (1990: 220), Künne (2003: 155) and Schnieder (2006b: §5.2). The discussion in Burge (1979) may give proponents of this view pause. Recently, Williamson (2007b: ch. 4) has forcefully argued against ‘understanding-assent links’. However, the main point to be made presently is that some non-causal explanations are not backed by epistemically transparent determination relations. If Burge and Williamson are right, this does not exclude conceptual analysis.

2.5. Modal Supervenience and Explanation

43

act is good because it maximises happiness, this will be due to conceptual connections of a whole cluster of concepts, not easily surveyable and not obvious to the average thinker. Further, in the case of HCl it is hard to see how it could be a conceptual matter at all. It seems that one could master all the relevant concepts and draw all conclusions that follow without realising that acids are proton donors.71 It seems that empirical investigation is necessary to discover this, while conceptual analysis is beside the point. But then the explanation that HCl is an acid because it is a proton donor is not backed by conceptual analysis. According to Achinstein (1983: ch.7.9), such cases of non-causal and nonconceptual explanation are backed by property identities. The property of being an acid just is the property of being a proton donor, even though conceptual analysis cannot reveal that all and only acids are proton donors (compare: Hesperus just is Phosphorus, even though conceptual analysis cannot reveal that all and only the things identical with Hesperus are identical with Phosphorus).72 However, this view cannot do justice to the non-reciprocity of the relevant explanations. If the property of being an acid just is the property of being a proton donor, the property of being a proton donor just is the property of being an acid. But it is not the case that HCl is a proton donor because it is an acid. Thus, at the very least, more would have to be said under what conditions property identities are able to back non-causal explanations. More promising in this regard is a view recently put forward by Gideon Rosen (in Rosen 2010). According to Rosen, some non-causal explanations are backed by real definitions, in this case of the property of being an acid. Philosophical analyses typically do not aim to provide analyses of concepts but rather of properties or states of affairs. Whether or not the analysis of knowledge is in the end a particularly fruitful research project, it would seem that right after the Platonic analysis attempt proved unsatisfactory, philosophers should have abandoned it, had they aimed at an analysis of the concept of knowledge. The proposed analysanda had a tendency of becoming very involved very quickly, and certainly conceptually more demanding than a concept that is used frequently by the man in the street can plausibly be taken to be.73 However, this does not mean that they did not reveal what knowledge consists in, what it takes for something to be a piece 71 72 73

There is a homonymous technical term of chemistry that is defined in this way. However, this is clearly not the definition of the English term ‘acid’. See also Ruben (1990: 218ff.). For an overview see e.g. Shope (1983).

44

Chapter 2. Supervenience

of knowledge.74 That is, they were still in the running for providing real definitions of knowledge. Such real definitions may be plausible candidates for backing some of those non-causal explanations whose epistemic intransparency counts against their being backed by conceptual analyses. Of course, the topic of real definitions is a difficult one, and quite a few philosophers may doubt that real definitions are well enough understood to play any worthwhile philosophical role.75 These philosophers will not find Rosen’s suggestion of non-causal explanations backed by real definitions particularly helpful. But this is not the place to decide between accounts of non-causal explanation. For present purposes I merely hope that the examples have rendered it plausible (i) that there are non-causal explanations, (ii) that some are backed by conceptual analyses, analyses that uncover the structure of concepts that occur in the explanandum, and (iii) that some noncausal explanations are backed by determination relations that need not be epistemically transparent. Summing up, the ‘because’ in our new definitions of supervenience has to be understood in a special way: it is restricted to complete objective but non-causal explanations.76 Before we can go on to make use of the defined concepts in more specific modal supervenience claims, though, we have to deal with a problem generated by the sensitivity of ‘because’ to subtle differences in its embedded sentences. The aforementioned sensitivity threatens the legitimacy of our provisional definitions of supervenience. This is the topic of the next section. 2.6 Explaining Explanations In the last section I proposed to define supervenience notions that do not only require covariance but a necessary explanatory link between subvenient and supervenient items. This proposal is reminiscent of what Terence Horgan (in Horgan (1993)) calls superdupervenience, though our supervenience notions and Horgan’s superdupervenience are not the same. Nevertheless, the question that needs an acceptable answer for a covariance claim to target a case of superdupervenience can be asked of supervenience claims in our sense as 74 75 76

The counterexamples took care of that. Pace Fine (1994) who appeals to real definitions in an attempt to elucidate his unorthodox conception of essence. In the literature, this brand of explanation sometimes figures under the title full grounding. See Fine (2012a).

2.6. Explaining Explanations

45

well. One answer we may want to give points to a problem with our provisional definitions, and shows that they require fine-tuning. But let’s start with superdupervenience. In Horgan (1993), Terence Horgan capitalises on the fact that covariance theses by themselves are a bit unsatisfactory. If A-properties covary with B-properties, it would be nice to know why they covary. Horgan discusses this desideratum in the context of physicalism and gives covariance theses accompanied by a ‘materialistically acceptable’ explanation of why they hold the honorific title of superdupervenience theses. Note that Horgan’s superdupervenience is not the same as any of the notions of supervenience defined in the last section, even if we forget about materialistic acceptability. For, the target phenomenon of superdupervenience theses is covariance, with the added benefit of there being some explanation for why the entities in question covary. The supervenience notions defined in the last section, on the other hand, require not merely covariance, but a (necessary) explanatory link, while leaving it open whether the fact that there is such an explanatory link can be explained in turn. The difference between our supervenience notions and superdupervenience shows up with respect to the phenomena at hand. For instance, there is an explanation (and, presumably, a materialistically acceptable one, although I am unsure about what the standards for materialistic acceptability are) of why modal truths covary with mass truths: the explanation is simply that no worlds differ in their modal truths full stop. On the other hand, as we have just seen, modal truths most certainly do not supervene in our sense on mass truths. Although there are no worlds that differ in modal truths without differing in mass truths, mass truths do not explain modal truths—not in the actual world, and, thus, a fortiori not in all possible worlds. Consequently, modal truths superdupervene (in Horgan’s sense) on mass truths, while they do not supervene (in our sense) on mass truths. Supervenience and superdupervenience thus differ in extension. Moreover, the shortcomings of covariance seem to carry over to superdupervenience. Superdupervenience does not ensure that the superdupervenient properties (or truths) depend on, and are completely determined by, their bases. Superdupervenience does not ensure ontological posteriority, while supervenience as defined in the last section arguably does. Nevertheless, just as we can ask whether there is something that explains the truth of a covariance thesis, we can also ask whether there is something that explains the necessary explanatory link required by supervenience.77 77

We may then give those supervenience theses for which there is an explanation of why

46

Chapter 2. Supervenience

Consider, for instance, the thesis that properties like weighing at least 60 kg strongly supervene on specific weight properties like weighing 67.1 kg. Plausibly, it is necessary that whatever has the latter property has the former because it has the latter: necessarily, whatever weighs 67.1 kg, weighs at least 60 kg because it weighs 67.1 kg. Since this generalises, the supervenience thesis is true. Now, there is a simple explanation for why it should be true: weighing at least 60 kg is a determinable property whose determinate properties are the specific weight properties of weighing exactly 60+n kg, for 0 ≤ n ≤ ∞. Again, this generalises: weighing at least m kg, is a determinable property whose determinates are the specific weight properties of weighing exactly m + n kg, for 0 ≤ n ≤ ∞. Consequently, the fact that the properties in question stand in the determinable-determinate relation explains why the supervenience thesis holds: it explains why there is a necessary explanatory link between at-least weight properties and specific weight properties. Different supervenience theses may be true for different reasons. As we have just seen, a supervenience thesis may be underwritten by determinabledeterminate relations. In the next chapter, we will consider the view that modal truths supervene on possible worlds truths, since there is a conceptual analysis running from the latter to the former. That is, the view under consideration will be Possible Worlds Supervenience (?) ◻∀x [x is a modal truth → ∃yy (yy are possible worlds truths & ◻ (yy are true → (x is true because yy are true)))].78 Moreover, the truth of Possible Worlds Supervenience is supposed to be underwritten by an analysis of modal truths in terms of possible worlds truths. In particular, it is claimed, (all instances of) the possible worlds biconditionals ◊-Bi It is possible that p ↔ there is a possible world at which p; and ◻-Bi It is necessary that p ↔ at all possible worlds, p; remain true, when we replace the biconditional operator ‘↔’ with the analysis operator ‘↔df. ’ (where the left-hand side is the analysandum and the righthand side the analysans).

78

they hold the honorific title of superdupervenience theses, perhaps renaming what Horgan calls superdupervenience ‘coolvariance’. I only have aesthetic objections. Or, rather, something close to Possible Worlds Supervenience (?), hence the question mark in the title.

2.6. Explaining Explanations

47

Reasons for believing or disbelieving Possible Worlds Supervenience as well as the analysis claim are the topic of the next chapter. What matters for now is that it should be possible to combine a supervenience claim of A-truths on B-truths with the corresponding analysis claim. In fact, it should be possible for the latter to underwrite the former. Since this will be important for the chapters to come, the remainder of this section will first explain why analysis claims plausibly underwrite (some) supervenience theses and then deal with a problem which presents itself. Since the remainder of this book will be concerned with (strong) supervenience theses of truths on other truths, this is what I will focus on from now on. Similar considerations apply, mutatis mutandis, to property supervenience theses. Take the stock case of conceptual analysis: the concept of a bachelor is analysable in terms of the concepts of being male and being unmarried. This systematically gives rise to analyses stated with the help of sentences in which the term ‘bachelor’ occurs, for instance 10 Ludwig is a bachelor ↔df. Ludwig is an unmarried man. As we have noted before, many philosophers accept that to a conceptual analysis as the one stated by (10) correspond conceptual explanations: if the analysandum is true, it is true because the analysans is, and if the analysandum is false, it is false because the analysans is.79 Assuming that Ludwig is indeed a bachelor, (10) thus gives rise to the following explanation: 11 Ludwig is a bachelor because Ludwig is an unmarried man. In general, whenever, we have a true analysis claim, there is a corresponding true explanatory claim, according to the following schema: A⇒E If p ↔df. q, then (i) if p, p because q; and (ii) if ¬p, ¬p because ¬q. If Ludwig is a bachelor, he is a bachelor because he is an unmarried man. If Ludwig isn’t a bachelor, he is not a bachelor because he is not an unmarried man. If Donald is a bachelor, he is a bachelor because he is an unmarried man. And so forth. 79

Most pertinently, this is why Lewis can claim, as he does, explanatory benefits from an analysis of modal truths in terms of possible worlds truths. His most thorough exegete, John Divers, explicitly refers to conceptual analysis as a species of explanation (cf. e.g. Divers 2002: 112). For approval of the analysis-explanation link independent of Possible Worlds Supervenience see e.g. Künne (2003: 155).

48

Chapter 2. Supervenience

Moreover, if (10) is true at all, it is not merely contingently true, but necessarily so. It is not as if the concept of a bachelor just happens to be analysable in terms of the concepts of being male and being unmarried, but could have been either unanalysable or analysable in a different way. The structure of the conceptual hierarchy is non-contingent. Furthermore, if there is an analysisexplanation link as envisaged by A⇒E, this is not a contingent fact either. This suggests that we may necessitate (i) and (ii) of A⇒E:80 A⇒◻E If p ↔df. q, then (i) ◻ (if p, p because q); and (ii) ◻ (if ¬p, ¬p because ¬q). Clearly, a fairly demanding notion of analysis has to be in play in order for A⇒E (and A⇒◻E) to be true. If you think, for instance, that analyses merely provide synonymous expressions, there is no reason to subscribe to A⇒E. Indeed, there is every reason not to. For one, if e is synonymous with e′ , e′ is synonymous with e (synonymy is symmetric). Since ‘because’ is asymmetric, there will be many ‘analyses’ which do not entail corresponding ‘because’ sentences, at least one for each analysis pair ‘p ↔df. q’ and ‘q ↔df. p’.81 Demanding, but not unreasonable. Sharvy (1972), for instance, ascribes just such a notion to Socrates, in reconstruing Socrates’s famous argument against Euthyphro. Euthyphro had proposed an analysis of piety as being loved by the Gods. Socrates objects to that analysis for the reason that it would entail that the pious are pious because they are loved by the Gods, whereas Euthyphro himself accepts that the explanation runs in the reverse direction: the pious are loved by the Gods because they are pious.82 In his reconstruction of the argument, Sharvy defends a variant of A⇒E,83 and also 80

81 82 83

More explicitly: I find it plausible that we may necessitate the whole conditional of A⇒E and, furthermore, that if A⇒E’s antecedent is true, it is necessarily true. Therefore, if A⇒E’s antecedent is true, its consequent is necessarily true, by the Distribution Axiom which is valid in the weakest normal modal logic K and allows us to move from ‘◻p’ and ‘◻ (p → q)’ to ‘◻q’. Also, of course, every ‘analysis’ of the form ‘p ↔df. p’ has a false corresponding ‘because’ sentence. See Plato’s Euthyphro Dialogue: 10a–11e (Stephanus pagination). Sharvy is concerned with analyses of predicates (or, more accurately, of properties those predicates signify), not sentences. Therefore, the principle he defends is A⇒E* ∀x (Fx ↔df. Gx) → ∀y (Fy → (Fy because Gy)).

2.6. Explaining Explanations

49

notes that certain principles concerning analyses can be seen as consequences of the structural properties of ‘because’, i.e. irreflexivity, asymmetry and transitivity. To the principles that concern Sharvy we may add, for instance, that the fact that circular analysis proposals are unacceptable would be a fairly straightforward consequence of the asymmetry and transitivity of explanation.84 Instances of A⇒E, of course, do not give causal explanations. Ludwig’s being an unmarried man does not cause him to be a bachelor. Rather, if ‘bachelor’ is indeed analysable in terms of ‘unmarried’ and ‘man’, 11

Ludwig is a bachelor because Ludwig is an unmarried man;

provides what may be called a conceptual explanation: for Ludwig to be an unmarried man would be what Ludwig’s being a bachelor consists in. Given A⇒◻E, it is easy to see how analyses can underwrite supervenience claims. The last things left to note are that analyses systematically pair up analysanda and analysantia and that analysanda and analysantia of (correct) analyses are intensionally equivalent, that is if p ↔df. q, ◻ (p ↔ q). Take the example of the concept of a bachelor again. Its analysis in terms of the concepts of being male and being unmarried systematically gives rise to analyses stated with the help of whole sentences in which the term ‘bachelor’ occurs. Let pi be placeholders for such sentences and let qi be replacable with the corresponding analysantia. Then, by A⇒◻E and the intensional equivalence of analysantia with analysanda, for each pi we have ◻ (qi → (pi because qi )). This seems to straightforwardly entail the supervenience of bachelor truths on truths about unmarried men, i.e.

84

Cf. Sharvy (1972: 129). Since analyses of predicates give rise to analyses of sentences in which they occur, every instance of ‘Fy → (Fy because Gy)’ that is licensed by A⇒E* will also be licensed by A⇒E. Together with principles linking complex sentences via ‘because’ with their compounds—for a systematic treatment see Schnieder (2011). An illustration: Suppose ‘p ↔df. (q & r)’ is proposed as an analysis while it is also held that ‘q ↔df. (s & p)’. Presumably, this is a case in which a charge of circularity would be levelled against the analysis proposal. Suppose ‘p’ is true. According to A⇒E and the factivity of ‘because’, we would have (i) ‘p because (q & r)’ and (ii) ‘q because (s & p)’. The principle that a conjunction is (partially) explained by either of its compounds, further yields (iii) ‘(q & r) because q’ and (iv) ‘(s & p) because p’. By transitivity applied to (i) and (iii), we have (v) ‘p because q’. By transitivity applied to (ii) and (iv) we have (vi) ‘q because p’. But, by the asymmetry of ‘because’, (v) and (vi) cannot both be true. So, on the assumption that some instance of the analysandum is true, an absurdity follows. This is ample reason to dismiss a proposed analysis.

50

Chapter 2. Supervenience

12

◻∀x [x is a bachelor truth → ∃y (y is an unmarried man truth & ◻ (y is true → (x is true because y is true)))].85 Similarly we seem to be able to argue from the analyses corresponding to (◊-Bi) and (◻-Bi) to the truth of Possible Worlds Supervenience, i.e. Possible Worlds Supervenience (?) ◻∀x [x is a modal truth → ∃y (y is a possible worlds truth & ◻ (y is true → (x is true because y is true)))]. Now, here’s the rub.86 Many philosophers believe that analysans and analysandum of a correct analysis express the same concept. For instance, they believe that the concept of a bachelor is nothing but the concept of an unmarried man. From this it follows that 13 the proposition that Ludwig is a bachelor is nothing but the proposition that Ludwig is an unmarried man. Still, we claimed that 11 Ludwig is a bachelor because Ludwig is an unmarried man; is true. Doesn’t that square badly with my insistence on the asymmetry, and, therefore, irreflexivity, of explanation? Not really, but it may make a problem for our provisional definition of supervenience. Let me first explain why it is not really a problem for the irreflexivity of explanation. What I meant when I spoke about the irreflexivity of explanation was that all instances of Irreflexivitybecause

¬ (p because p);

are true. But (Irreflexivitybecause ) is not in conflict with (13) and (11). It correctly entails that 14 Ludwig is a bachelor because Ludwig is a bachelor; is false. But to move from (13) and (14) to the falsity of (11), we would have to assume that it is legitimate to replace expressions with the same content in the context of ‘because’. Example pairs like (11) and (14) may just be taken to show that it is not. If so, ‘because’ is what may be called hyper-sensitive:87 85

86 87

The added flexibility of plural quantification seems to be irrelevant in the case of supervenience theses underwritten by analyses. Consequently, I revert to singular quantification here. Cp Schnieder (2010). The term is chosen as a reminder of the sensitivity of ‘because’ to more than the propositions, or Fregean senses, expressed by the sentences it governs. It is borrowed from Schnieder (2010).

2.6. Explaining Explanations

51

substitution of expressions with the same content in the scope of ‘because’ does not always preserve truth-value. Let’s turn to why the hyper-sensitivity of ‘because’ is problematic for our provisional definitions of supervenience. Consider the unboxed version of (12):88 15

∀x [x is a bachelor truth → ∃y (y is an unmarried man truth & (y is true → (x is true because y is true)))].

How are we to understand (15)? If propositions satisfy ‘x is a bachelor truth’, then the proposition that Ludwig is a bachelor is surely among them. If propositions satisfy ‘y is an unmarried man truth’, then the proposition that Ludwig is an unmarried man is surely among them. Moreover, the latter is a natural candidate for explaining the former. It is the only propositional candidate for an explanation underwritten by analysis. But how are we to understand ‘x is true because y is true’ when the proposition that Ludwig is a bachelor is assigned to the variable ‘x’ and the proposition that Ludwig is an unmarried man is assigned to the variable ‘y’, given that (13) is true? After all, if (13) is true, the very same proposition is assigned to both variables in this case. But 16

the proposition that Ludwig is a bachelor is true because the proposition that Ludwig is an unmarried man is true;

is true, while 17

the proposition that Ludwig is a bachelor is true because the proposition that Ludwig is a bachelor is true;

is false. The hyper-sensitivity of ‘because’ would render quantification into ‘because’ contexts problematic.89 Consequently, our provisional definitions of supervenience which use such quantification would be problematic as well. I see two ways of dealing with the problem.90 The first is to just deny that the entities quantified over in our definition of supervenience of A-truths on B-truths are as coarsely individuated as presently envisaged. If they are individuated finely enough, the hyper-sensitivity of ‘because’ is unproblematic. One pertinent view is that propositions themselves are extremely finely individuated. Precisely because of the possibility of conceptual explanations 88 89 90

Recall that (12) would say that bachelor truths supervene on unmarried man truths. The problem is compounded, of course, for intensionally individuated properties. Short of denying that supervenience theses may be underwritten by analyses, that is.

52

Chapter 2. Supervenience

underwritten by analyses (and because he takes propositions to be the relata of explanation), Sharvy (1972: 134f.), for instance, denies the commonly held view that analysans and analysandum of a correct analysis express the same concept. Consequently, he would deny (13), and the problem for truth supervenience theses underwritten by analyses disappears.91 Another pertinent view would be that, although we may want to have propositions that are individuated coarsely enough to make (13) true, there also are more finely individuated entities, perhaps Schnieder’s (2004a: 355f.) propositions taken in expression, that are the proper relata of the supervenience relation in these cases. This variant of the first way of dealing with the problem would accept (13), but would claim that ‘is a so-and-so truth’ in the properly understood supervenience theses refers not to propositions but to their more finely individuated cousins. Again, since they are finely individuated enough so that substituting different names for variables assigned to one and the same proposition taken in expression does not lead to ‘because’ sentences with different truth-values, the problem for truth supervenience theses underwritten by analyses disappears. For the purposes of this book I propose a different, less committal, way of dealing with the sensitivity of ‘because’ to (apparently) more than the propositions expressed by the embedded sentences. It is compatible with the view that propositions are extremely fine-grained and with the view that there are other proposition-like entities that are the relata of the explanation relation. But it is not committed to the truth of either, or, indeed, to the truth of the disjunction. The proposal is to move from nominal quantification— quantification into singular term position—over propositions (or propositionlike entities) to sentential quantification—quantification into sentence position.92 Thus, instead of considering, e.g., Possible Worlds Supervenience (?) ◻∀x [x is a modal truth → ∃y (y is a possible worlds truth & ◻ (y is true → (x is true because y is true)))]; we should consider 91 92

See also Künne (2003: 369f.). For property supervenience theses, the analogous move would be, of course, to use quantification into general term position instead of nominal quantification over properties. For more on quantification into sentence and general term position see appendix A.

2.6. Explaining Explanations

53

Possible Worlds Supervenience ◻∀p [it is a modal truth that p → ∃q (it is a possible worlds truth that q & ◻ (q → (p because q)))]. Reconsider the unboxed version of the claim that bachelor truths supervene on unmarried man truths, 15

∀x [x is a bachelor truth → ∃y (y is an unmarried man truth & (y is true → (x is true because y is true)))].

(15)’s variant in which nominal quantification makes room for sentential quantification is 15*

∀p [it is a bachelor truth that p → ∃q (it is an unmarried man truth that q & (q → (p because q)))].

The problem with (15) was that 16

the proposition that Ludwig is a bachelor is true because the proposition that Ludwig is an unmarried man is true;

is true, and 17

the proposition that Ludwig is a bachelor is true because the proposition that Ludwig is a bachelor is true;

is false, while (13) assures us that the same proposition is expressed in all four sub-sentences. Consequently, it was hard to see how the truth of (16) should make (15) true, if the falsity of (17) does not make (15) false at the same time. The correlative worry was that the quantification in (15)—and, consequently, in our provisional definitions of supervenience—is of dubious cogency. The ‘because’ sentences relevant to (15*) that correspond to (16) and (17) are 16*

Ludwig is a bachelor because Ludwig is an unmarried man; and

17*

Ludwig is a bachelor because Ludwig is a bachelor;

respectively.93 Again, (16*) is true, while (17*) is false. How then does the move from (15) to (15*) help? Obviously, we have to say that even if the proposition that Ludwig is a bachelor is nothing but the proposition that Ludwig is an unmarried man, ‘it is a bachelor truth that Ludwig is a bachelor’ is true, while ‘it is a bachelor truth that Ludwig is an unmarried man’ is not. The 93

(16*) and (17*) highlight another advantage of supervenience theses using sentential quantification instead of nominal quantification over proposition: we are not forced into ‘propositional ascent’ anymore.

54

Chapter 2. Supervenience

idea here is that the operator ‘it is a φ-truth that’ is—just like ‘because’— hyper-sensitive: it is not only sensitive to the proposition expressed by the appended sentence, but also to the way the sentence expresses it. Roughly, if a true sentence contains φ-vocabulary, the result of appending it to the operator ‘it is a φ-truth that’ is true, if not, not.94 Thus, (16), but not (17), is relevant for the truth or falsity of (15). In general, even if analysans and analysandum of a correct analysis express the same proposition, our modified definition of truth supervenience allows us to deal with supervenience underwritten by analysis. In particular, the hyper-sensitivity of ‘it is a φ-truth that’ allows us to single out the intended instances of a supervenience thesis even if intended and unintended instances are indistinguishable with respect to the propositions expressed by the sentences that instantiate the sentential variable. Consequently, the problem for truth supervenience theses underwritten by analyses disappears. In the next two chapters I will evaluate the chances of a supervenience claim that is quite popular in the philosophy of modality: the claim that modal truths supervene on truths about the existence of possible worlds, in the sense of Possible Worlds Supervenience. A discussion of the relation between global and strong covariance follows in an appendix.

94

We may express this in an ontologically loaded way by saying that, at least in one legitimate use of the phrase, ‘is a φ-truth’ applies to Schnieder’s propositions taken in expression. We may but we need not.

A. Strong and Global Supervenience

A

55

Strong and Global Covariance

Global covariance is the topic of an ongoing debate. The notion was introduced into the supervenience debate, since it promised an interesting relation between respects in which worlds may differ which allows more flexibility than the local property connections required by weak and strong covariance. These hopes were raised and crushed depending on the state of the discussion about the relation between Global Covariance and Strong Covariance. In his 1984 article, Jaegwon Kim argued that Global Covariance is in fact equivalent to Strong Covariance, and that, hence, both require the very same property connections. Responding to Kim, Petrie (1987) gave what he took to be a counterexample to the claim that Global Covariance entails Strong Covariance (the Entailment Claim). Kim (1987: §2) concedes, with reference to Petrie (1987), that he was wrong to claim earlier that Global Covariance is merely Strong Covariance in a different guise. He goes on to argue, in Kim (1987: §3), that Global Covariance does not even entail Weak Covariance and that, hence, Global Covariance is not strong enough to be a kind of dependence relation. Consequently, no notion of supervenience that deserves its name should be exhausted by Global Covariance. A few years later, Paull and Sider (1992) argue that Petrie (1987)’s counterexample was defective and does not show what it aims to show. However, they go on to provide their own counterexample to the Entailment Claim but argue that the existence of counterexamples of this type does not preclude Global Covariance from being a dependence relation. Nowadays, the agreed view seems to be that Global Covariance does not entail Strong Covariance, because there are Paull/Sider-like cases, but that Paull/Sider-like counterexamples to the Entailment Claim rely on particular features of the properties in the covariant set.95 In particular, they only go through if the covariant set includes extrinsic properties. Generalizing from this observation it is claimed that if we require that covariant and base set only include intrinsic properties, Global Covariance is equivalent to Strong Covariance.96 95

96

See e.g. McLaughlin (1995: §4) and Bennett (2004a). Bennett distinguishes three global covariance relations and argues that one of them is equivalent to Strong Covariance, while the other two are too weak to be (complete) dependence relations. More precisely: necessarily, if all A- and B-properties are intrinsic, A-properties globally covary with B-properties iff A-properties strongly covary with B-properties. An argument is provided in Bennett (2004a: Appendix I). As always, Kim had already been there. In Kim (1993: 170) he conjectured that Global Covariance only fails to entail Strong Covariance when ‘extrinsic properties are present in the [covariant] set

56

Chapter 2. Supervenience

My view on the issues are a direct consequence of what I said in this chapter.97 My argument is very simple: modal truths globally covary with any kinds of truths, for instance with mass truths (cf. p. 32 above). But modal properties do not strongly covary with every kind of properties (cf. p. 34 above). In particular, they do not strongly covary with mass properties. Thus, Global Covariance does not entail Strong Covariance. Furthermore, modal properties are not extrinsic qua modal properties.98 So, Global Covariance does not entail Strong Covariance even if we require covariant and base set to only include intrinsic properties—the view nowadays agreed upon is false. Further, Global Covariance is not a kind of dependence relation, but this is not dependent upon whether or not it entails Strong Covariance: I argued above (pp. 26ff.) that no covariance relation is a dependence relation, not even Strong Covariance. However, the non-equivalence of Global Covariance and Strong Covariance opens the door for Global Covariance’s being the basis99 of a covariance relation that is not just Strong Supervenience in disguise—Global Covariance has as strong a status as the two other notions of covariance considered in this chapter. The modal case shows that Global Covariance does not entail Strong Covariance, whether we restrict covariant and base set to intrinsic properties or not. However, strong arguments to the contrary have been given. In the remainder of this appendix I will present these arguments and indicate where they go wrong. A.1 Kim’s Attempted Equivalence Proof In Kim (1984: 168), Jaegwon Kim attempts to give a proof to the effect that Global Covariance and Strong Covariance are equivalent. Since I don’t disagree with the claim that Strong Covariance entails Global Covariance, I only present the other half of the proof. Recall that the generic definition of Global Covariance says that A-properties globally covary with B-properties just in case no worlds differ in A-respects without also differing in B-respects. Above I said that Kim attempts 97 98

99

but disallowed from the [covariance] base’. Cp. also Steinberg (2013a). Some modal properties are certainly extrinsic. They are extrinsic because their nonmodal cousins are. But the argument does not rely on this feature of theirs: we could just exclude them from the covariant set. Like in the other two cases, Global Covariance is enriched by explanatory notions to yield Global Supervenience; recall p. 38 above.

A. Strong and Global Supervenience

57

to spell out what it means for worlds to (dis-)agree in A-respects in terms of which individuals have which A-properties at these worlds. That is, according to Kim, the generic definition of Global Covariance should be spelled out as follows: Global Covariance—Kim (GCKim ) A-properties globally covary with B-properties ↔df. ∀w, w′ , x [∀P (P is a B-property → (x has P at w ↔ x has P at w′ )) → ∀Q (Q is an A-property → (x has Q at w ↔ x has Q at w′ ))] In words: If worlds agree on which individuals have which B-properties, they also agree on which individuals have which A-properties. Other authors opt for a more lenient characerisation of what it is for worlds to agree in certain respects, and, thus, for a different spelling out of Global Covariance. We will deal with such an alternative characterisation in section A.3. Here is Kim’s argument: Suppose S1 A-properties do not strongly covary with B-properties. Then there is an A-property P which something x has in some world w such that all the B-properties x has in w are not jointly strictly sufficient for having P. Now, (!) since the B-properties x has in w are not jointly strictly sufficient for having P, there is a world just like w in its distribution of B-properties— everything has the very same B-properties in it that it has in w; in particular, x has all the B-properties it has in w, and no more—but in which x does not have P. Call it w′ . w′ shows that A-properties do not globally covary with B-properties, since, by construction, w and w′ agree in their distribution of B-properties, but they disagree in their distribution of A-properties—x has P at w and x fails to have P at w′ . Thus, discharging (S1), if A-properties do not strongly covary with B-properties, A-properties do not globally covary with B-properties. By contraposition, if A-properties globally covary with B-properties, A-properties strongly covary with B-properties. Global Covariance entails Strong Covariance. The problem with the attempted proof is that the ‘since’ in the sentence marked with an exclamation mark is not warranted. That some B-properties QQ are not jointly strictly sufficient for having P entails that there is a world in which something has QQ without having P—formally: ◊∃y (y has QQ & ¬y has P). But this does not guarantee that some particular thing, in this case x, can have QQ without P.100 Thus, from the fact that QQ are not jointly 100 Nor does it seem to guarantee that there is a world in which something has QQ without

P while the distribution of B-properties over all other individuals is the same as in

58

Chapter 2. Supervenience

w1 x has P ¬(y has P) x has Q y has Q

w2 ¬(x has P) ¬(y has P) x has Q ¬(y has Q) Figure 1

strictly sufficient for having P, it does not follow that there is a world in which x has QQ without P, but only that there is a world in which something has QQ without P. On the other hand, that there is a world in which x has QQ without P would follow from the fact that everything is such that there is a world in which it has QQ without P—formally: ∀y ◊(y has QQ & ¬y has P). But this does not follow from the fact that QQ are not jointly strictly sufficient for having P. Either way, the second part of (!) does not follow from the first part of (!). Therefore, the ‘since’ is not warranted and Kim’s attempted proof fails. Our counterexample to entailment is, of course, a case in point. Modal properties do not strongly covary with mass properties, since, for instance, my (very solid) desk has the same mass as I do, but I am possibly a philosopher while my desk is not. Thus, having a mass of 70 kg is not strictly sufficient for being possibly a philosopher. But from this it clearly does not follow that there is a world in which I lack the property of possibly being a philosopher. Since Kim’s attempted proof assumes that it does, the attempt fails. A.2 Non-Equivalence Examples In Petrie (1987: §1), Bradford Petrie argued that Global Covariance does not entail Strong Covariance by giving a purported counterexample. He asks us to consider two possible worlds, w1 and w2 , two properties, P and Q, and two individuals, x and y, such that x has both P and Q at w1 and Q but not P at w2 , while y has Q but not P at w1 and neither of them at w2 . This is represented in figure 1: P does not strongly covary with Q, since having Q is not strictly sufficient for having P—for instance, at w1 , y has Q but not P. On the other hand, if things are as the example says they are, P globally covaries with Q, since although the worlds w1 and w2 differ with respect to which individuals have P (x has P at w1 , while it lacks P at w2 ), they also differ with respect to which individuals have Q (y has Q at w1 , while it lacks Q at w2 ). some other world.

A. Strong and Global Supervenience

59

Kim (1987: §2) retracts the Entailment Claim, referring to the counterexample just presented. However, Paull and Sider (1992) challenge the pertinence of Petrie’s example to the most pressing question at issue: whether global covariance metaphysically entails strong covariance, i.e. whether it is metaphysically possible for there to be sets A and B such that A-properties globally covary with B-properties without strongly covarying. What Petrie’s example shows is merely that there is a (strictly) logical possibility, but not all logical possibilities are metaphysically possible: if Kripke (1980: 112f.) is right, for instance, it is logically possible but metaphysically impossible that Elizabeth II. should be born from any other than her actual parents. More pertinently, it is logically possible but metaphysically impossible that there are just two possible worlds.101 Further, Paull and Sider argue that there are good reasons to think that, typically, if there are worlds like w1 and w2 —which by themselves do not violate Global Covariance—there are also some other worlds which do violate Global Covariance. They rely on a variant of the following principle: Isolation Principle (I) For any world w and individual x there is a world w′ such that only x (, its parts and the things whose existence is entailed by x’s existence) exists at w′ and for any intrinsic property P, x has P at w′ just in case x has P at w.102 101 This

is a somewhat charitable reconstruction of the debate. In their paper, Paull and Sider claim that Petrie must have intended to answer the question of metaphysical possibility, albeit using inadequate means (p. 836f.). However, given that Petrie’s target, Kim (1984), aimed to show that the Entailment Claim holds drawing only on logical resources and closure conditions of the base set, Paull and Sider’s claim about Petrie’s intentions appears doubtful. 102 This is not quite the principle they rely on, since they employ (and (mis)represent Petrie as employing) a more lenient characterisation of what it is for worlds to be the same in A-respects than both Kim and I. This allows them to use a weaker principle than (I) in order to challenge Petrie. However, the relevant points can be made while sticking with the stricter understanding. For simplicity’s sake, I will pretend in the main text that Paull and Sider actually did. Once we introduce the more complex understanding, it should be clear how the arguments can be transposed to do justice to it. Incidentally, Bennett (2004a: 524, fn.) implies that our (I), as opposed to Paull and Sider (1992)’s, ‘entail[s] that nothing has any essential extrinsic properties’. But this is false. Presumably, if something has an essential extrinsic property, perhaps an origin property like being NN’s child, then its existence will entail the existence of whatever it is it is essentially related to. (I) does not say that everything can be the sole inhabitant of a world, but that it can inhabit a world together with things whose

60

Chapter 2. Supervenience

w4 ¬(x has P) x has Q

w3 x has P x has Q Figure 2

(I) tells us that whenever we have some possible world we can slice it up into mini-worlds inhabited by one individual each that has all the same intrinsic properties in both worlds. Intrinsic properties are those properties that ‘can never differ between duplicates’, and duplicates are ‘exactly qualitatively similar considered “as they are in themselves” and not in relation to other things’ (Paull and Sider 1992: 838). Without restriction to intrinsic properties the quite plausible (I) would be false. The actual world is one in which Prince Harry has Prince William as a brother. But there is no Harry mini-world in which he has William as a brother. However, being a brother of William is not an intrinsic property—duplicates of Harry differ in it. So, (I) is not threatened by the fact that there are no such mini-worlds. Now, assume that both P and Q are intrinsic properties and that the existence of x is independent of the existence of y. Then the Isolation Principle tells us that we can divide w1 and w2 in half to get new worlds whose existence is ensured by the existence of w1 and w2 . In particular, the left halfs of w1 and w2 yield the worlds depicted in figure 2: w3 and w4 differ in Prespects while not differing in Q-respects. Thus, P does not globally covary with Q. Petrie considered only a portion of logical space which (I) ensures us is richer in relevant ways. In particular, if it contains w1 and w2 , it also contains w3 and w4 . w1 and w2 , though they conflict with a strong covariance thesis, do not conflict with a global covariance thesis. But w3 and w4 do. Consequently, Petrie’s example as it stands does not show that Global Covariance does not entail Strong Covariance. Although Petrie’s example fails to establish what it aims to establish, the discussion why it fails makes it clear how we should modify it in order to get a better example.103 The idea is to just have an extrinsic property in the covariant set. existence is entailed by its existence. Thereby, it clearly allows for essential extrinsic properties. Incidentally, I will suppress the bracketed provisos in what follows in order to avoid irrelevant complication. 103 Paull and Sider (1992: §3) propose the modified example.

A. Strong and Global Supervenience

61

w1 x has P ¬(y has P) y has Q ¬(x has Q) x has R ¬(y has R)

w2 x has P ¬(x has Q) ¬(x has R)

Figure 3

Let P and Q be any mutually independent intrinsic properties, for instance being tall and being a philosopher, and let R be a property that is such that, necessarily, something has R iff it has P and there is something which has Q, i.e. R ◻∀x (x has R ↔ (x has P & ∃y (y has Q))). Now, R globally covaries with {P, Q}, i.e. there are no worlds which differ in R-respects without also differing in {P, Q}-respects. For, suppose w and w′ differ in which individuals have R, i.e. either (i) there is something, x, which has R in w and does not have it in w′ or (ii) vice versa. Suppose (i). Given (R), x has P in w, something has Q in w and either x does not have P in w′ or nothing has Q in w′ . Either way, w′ also differs from w in which individuals have which members of {P, Q}. The same holds, mutatis mutandis, if (ii) is the case. Hence, R globally covaries with {P, Q}.104 On the other hand, R does not strongly covary with P and Q. Consider the two worlds pictured in figure 3: x has R at w1 because it has P and y has Q. x does not have R at w2 because nothing has Q at w2 . Obviously, w1 and w2 show that R does not strongly covary with {P,Q}, since x has the same pattern of P/Q exemplification at w1 —x has P and lacks Q at both worlds— but a different pattern of R exemplification—x has R at w1 but lacks R at w2 . Consequently, the example shows that Global Covariance does not entail Strong Covariance. A.3

Restricted Entailment?

The counterexample of Paull and Sider (1992) is now generally considered to be successful. It relies essentially on a peculiar feature covariant properties may have which then allows global and strong covariance to come apart: is easy to see that the Isolation Principle cannot come to the rescue here, since it is restricted to intrinsic properties, but R is extrinsic. Suppose there is a world at which something x has P but not Q, while something else has Q. In that world, x has R. But there is no x-mini-world in which x has P and R but not Q: since nothing has Q at any x-mini-world in which x does not have Q, x does not have R in such a world.

104 It

62

Chapter 2. Supervenience

extrinsicness. In response, Kim (1993: 170) conjectures that, though the Entailment Claim fails, it only fails in cases in which covariant properties are extrinsic (and the properties with which they covary aren’t). Thus, we should restrict the Entailment Claim to intrinsic properties: necessarily, if all A- and B-properties are intrinsic and A-properties globally covary with B-properties, A-properties strongly covary with B-properties (the Restricted Entailment Claim). Recently, Karen Bennett has given an argument for the Restricted Entailment Claim (Bennett 2004a).105 The remainder of the appendix argues that the modal counterexample to the Entailment Claim also applies to its restriction. In order to make room for it, I will show that the only argument for the Restricted Entailment Claim, provided by Bennett, is inconclusive. In preparation, I will present the analysis of (GC)’s definiens operative in Bennett’s argument. Bennett’s favoured understanding of (GC)’s definiens is (at least in one way) more permissive than Kim’s.106 Kim takes worlds to be the same in Arespects just in case whenever something has an A-property in one, it also has it in the other. Bennett, on the other hand, only requires there to be a bijection between the inhabitants of the worlds that preserves A-properties.107 If we have worlds in which the same individuals with the same A-properties exist, the identity mapping is such a bijection. But there may also be others. With this understanding of ‘agreement in A-respects’, we can give various non-equivalent characterisations of global covariance.108 Since Bennett argues that even the weakest kind of global covariance she considers entails strong covariance, I will give only its definition here: Global Covariance—Bennett (GCBennett ) A-properties globally covaryB with B-properties ↔df. For any worlds w and w′ , if there is a bijection from the inhabitants of w to the inhabitants of w′ which is B-preserving, then there is at least one bijection from the inhabitants of w to the inhabitants of w′ which is A-preserving. 105 In

fact, Bennett gives an argument for a slightly stronger claim. However, as Moyer (2008: fn. 26) shows, it fails as an argument for the stronger claim for reasons that do not touch its cogency as an argument for the Restricted Entailment Claim. 106 In this she follows Paull and Sider (1992) and McLaughlin (1995). The idea is due to Horgan (1982). 107 A bijection is a one-one mapping from one set onto another. 108 See Bennett (2004a: 503).

A. Strong and Global Supervenience

63

Bennett’s argument is a generalisation of Paull and Sider’s critique of Petrie’s counterexample:109 suppose that S1

A- and B-properties are intrinsic;

and S2

A-properties do not strongly covary with B-properties.

Then, by (S2), there is an A-property P, world w1 , and individual x such that x has P at w1 but the B-properties x has are not strictly sufficient for having P. Thus, there is a world w2 and individual y such that the B-properties y has at w2 are the B-properties x has at w1 , but y lacks P at w2 . Now, according to the Isolation Principle, there is an x-mini-world w3 such that only x exists at w3 and x has all the same intrinsic properties at w3 as at w1 . In particular, by (S1), x has all the same A- and B-properties at both worlds. Likewise, according to the Isolation Principle, there is a world w4 at which only y exists and has all the same A- and B-properties as at w2 . But then, x at w3 has the same B-properties as y at w4 , but different A-properties. Since x and y are the sole inhabitants of w3 and w4 respectively, there is exactly one bijection from the inhabitants of w3 to the inhabitants of w4 —the function f that maps x to y. f is B-preserving, but not A-preserving. Hence, A-properties do not globally covaryB with B-properties. So, discharging (S1) and (S2) and contraposing, if A- and B-properties are intrinsic and A-properties globally covaryB with B-properties, they strongly covary with B-properties—the Restricted Entailment Claim is true. A.4

Against Restricted Entailment

The argument is unassailable, if we grant its presuppositions. But I think we shouldn’t, since the kind of global covariance it employs is deficient for being unable to deal with existence indifferent properties.110 Existence indifferent properties are properties that do not guarantee that things that exemplify them exist, i.e. Existence Indifference (EI) A property P is existence indifferent ↔df. ¬ ◻ ∀x ◻ (x has P → x exists). Taking existence to be temporally modified, being famous may be an existence indifferent property—Socrates is still famous, though he no longer 109 cf.

Bennett (2004a: Appendix I). My restatement does not engage with the partdiscussion prompted by one of the bracketed provisos in (I). Nothing hinges on this. 110 I borrow the label from Künne (1983: 23).

64

Chapter 2. Supervenience

exists. In an atemporal understanding, modal properties are good candidates for being existence indifferent. Suppose for a moment that there are existence indifferent properties in the atemporal sense. Clearly, Global Covariance—Bennett cannot deal with them because it only considers mappings from the inhabitants of a world to the inhabitants of another. Inhabitants of a world are things that exist at that world. Global Covariance—Bennett just ignores potential differences in existence-indifferent properties between the non-existents of worlds. Consider as a toy example the property signified by the open sentence ‘x exists or Socrates exists’. Call it P. Clearly, there are no worlds that disagree on whether the objects that exist at them have P or not—all of them do. But some worlds don’t agree on whether the things that don’t exist at them have P. Some thing that does not exist at w will have P at w just in case Socrates exists at w. Since Socrates exists at some but not all worlds, some worlds disagree on whether their non-existents have P at them. Global Covariance—Bennett simply ignores this disagreement. Thus, Global Covariance—Bennett is not an adequate covariance relation for existence indifferent properties.111 It is fairly obvious how we should deal with the problem. Instead of only considering bijections between world domains, we have to consider bijections from the universal domain (i.e. the union of all world domains) to itself. That is, we should replace (GCBennett ) with Global Covariance* (GC*) A-properties globally covary* with B-properties ↔df. For any worlds w and w′ , if there is a bijection f from the universal domain to itself such that for every object x and B-property P, x has P at w iff f(x) has P at w′ , then there is at least one bijection g from the universal domain to itself such that for every object y and A-property Q, y has Q at w iff g(y) has Q at w′ . Global Covariance* can deal with existence indifferent properties, since it does not ignore differences between a world’s non-existents. But Bennett’s attempted proof does not go through for Global Covariance*. For, nothing assures us that only the facts about w3 and w4 that Bennett considers are relevant for Global Covariance*. Only x exists at w3 and only y exists at w4 , which is why there is no bijection from the domain of w3 to the domain of w4 which is A-preserving. But this is consistent with there being an A-preserving bijection from the universal domain to the universal domain. Thus, Bennett’s 111 This

point is spelled out in more detail in Steinberg (2013a).

A. Strong and Global Supervenience

65

argument does not show what it aims to show on an understanding of global covariance which is suitable for dealing with existence indifferent properties. One may be wary of the existence of existence indifferent properties. After all, modal properties and slightly unnatural property constructs appear to be the only good candidates. To be sure, there are sentences which are true at a world even though one of their singular terms does not refer—Venus is depicted in the Pantheon, for instance. But it is not clear that such true sentences give rise to property ascriptions—does Venus have the property of being depicted in the Pantheon, although she does not exist?—much less to property ascriptions to contingently non-existent individuals. But if ‘Venus is depicted in the Pantheon’ does not give rise to an ascription of an existence indifferent property, perhaps modal truths do not give rise to such an ascription either. That is, though instances of e.g. ‘a is possibly F’ are true at every world if they are true at any, they at best give rise to an ascription of the property of possibly being F at worlds at which a exists. Thus, the claim that any two worlds agree on which individuals have which modal properties is far from being the triviality I made it out to be—it is most certainly false. However, disputing the existence of existence indifferent properties would be to miss the point. First of all, the covariance debate tends to employ a very thin conception of properties.112 But even if there were no attractive conception of properties that allowed existence indifferent properties, this would only cast doubt on the spelling out of the global covariance slogan in terms of properties. Worlds can clearly differ in depiction-respects even if their inhabitants have the very same depiction-properties (perhaps Venus is depicted in the Pantheon in one of the worlds, while the Pantheon houses only a statue of Raphael in the other). This case is a paradigm example of a difference between worlds which we should expect to covary with more mundane differences (perhaps with differences in the intentions of painters and sculptors). On the other hand, if S5 is the correct modal logic, worlds just cannot differ in modal respects. If they can differ in which individuals have which modal properties at them, so much the worse for spelling out the global covariance slogan in terms of properties. Talk of existence indifferent properties only serves to keep the terms of engagement as close as possible to those my opponents have laid out.113 112 See

e.g. McLaughlin (1995: 21).

113 In the modal case, there is another option.

Williamson (2002) argues that only constant domain models represent modal reality accurately. In this case, of course, there is no reason to prefer (GC*) over (GCBennett ) because there are no existence indifferent

66

Chapter 2. Supervenience

That Bennett’s argument does not go through does not come unexpectedly. After all, modal properties provide a counterexample to the claim that global entails strong covariance even if covariant and base properties are restricted to intrinsic properties. For, modal properties are not extrinsic qua modal properties. To be sure, there are extrinsic modal properties, e.g. the property of possibly being married. But such modal properties plausibly inherit their extrinsicness from the properties they are modalizations of, and, more importantly, intrinsic properties plausibly have intrinsic modalizations.114 For instance, there seems to be no reason to think that the property of possibly being a philosopher is extrinsic (assuming that the property of being a philosopher isn’t). Firstly, it is in no obvious way a relational property, as opposed to such paradigm examples of extrinsic properties as the property of being married. The only reason I can see for thinking otherwise is the popular doctrine that the best semantics for modal sentences explains their truth in terms of quantification over possible worlds. But this does not make modal properties relational, otherwise virtually all properties would be. For, it is also commonly held that the best semantics for simple subject-predicate sentences explains their truth in terms of property ascriptions. That is, a sentence of the form ‘a is F’ is often supposed to be true (if it is) because a has the property of being F. ‘a has the property of being F’ ascribes a relational property, to wit: having the property of being F. Thus, if we are forced to take sentences of the form ‘a is possibly/necessarily F’ to be ascriptions of relational properties, we are also forced to take simple subject-predicate sentences to be ascriptions of relational properties. But then the distinction between relational and non-relational properties breaks down. Secondly, duplicates cannot differ in the property of possibly being a philosopher: if a thing has it, all of its duplicates do, and if a thing lacks it, all of its duplicates lack it. Thus, the property of possibly being a philosopher has as good a claim to being an intrinsic property as the property of being a philosopher. properties. However, on this suggestion, Bennett’s argument does not go through because of her use of the Isolation Principle. According to Williamson, you cannot subtract anything from the domain of any world and still get a world. A fortiori you cannot subtract everything but one object. 114 This claim is stronger than it needs to be. For the argument to go through, we only need one intrinsic modal property that is not shared by everything.

A. Strong and Global Supervenience

67

Now consider the intrinsic modal properties. They trivially globally covary* with any set of intrinsic properties, e.g. with mass properties. For, the S5 axioms 1 ◊p → ◻◊p; and 2 ◻p → ◻ ◻ p; ensure that the identity mapping, i, is such that for any worlds w and w′ and object x, x has the same intrinsic modal properties at w as i(x) has at w′ . Again, intrinsic modal properties do not strongly covary with mass properties—the property of possibly being a philosopher is an intrinsic modal property, my desk and I have the same mass, but only one of us has it. My verdict still stands: global does not entail strong covariance whether or not the covariant set is restricted to intrinsic properties. The arguments to the contrary are inconclusive, while modal covariance provides a counterexample even to the Restricted Entailment Claim.

Chapter 3

Concrete Possible Worlds In the last chapter I argued that supervenience should not be understood to be a purely modal notion. Following Kim and Correia, we distinguished it from covariance, the latter a purely modal notion, and remarked that if S5 is correct, modal truths trivially co-vary with everything. Enriching the definition of covariance with an explanatory notion—expressed by ‘because’— coheres well with the point supervenience claims are supposed to have and saves modal supervenience theses from triviality. Since, typically, modal truths, not modal properties, are at issue, we will concentrate on truth supervenience claims, claims that have the following form: Modal Truth Supervenience Modal truths supervene on φ-truths ↔df. ◻∀p [it is a modal truth that p → ∃q (it is a φ-truth that q & ◻ (q → (p because q)))]. On the face of it, there is something mysterious about possibility, though it is not easy to put one’s finger on it. This sense of mysteriousness is only exacerbated by epistemological considerations. Clearly, we do know that Bernhard Langer might have won the 2007 BMW Open but can’t any more or that this tritium atom might decay within the hour. But how? At least relatively common and unproblematic ways of coming to know things, like sense perception, do not help here. We can hear that Langer didn’t strike the ball cleanly, but we cannot hear that Langer might still win. We can see (let’s suppose) that the atom isn’t currently decaying, but we cannot see that it might do so in a minute. (Smell and touch are equally unsuitable for detecting possibilities). The hope is that if we can find unproblematic (or less problematic) truths for modal truths to supervene on, we can dispel the sense of mysteriousness as well as explain how we can know them. This, I take it, is one of the major driving forces to look for a suitable supervenience base for modal truths.

70

Chapter 3. Concrete Possible Worlds

Since possible worlds semantics was fruitfully used by Kripke and others to illuminate the properties of and relations between various systems of modal logic,1 the idea suggests itself that truths about possible worlds constitute our desired supervenience base, i.e. that the following thesis is true: Possible Worlds Supervenience ◻∀p [it is a modal truth that p → ∃q (it is a possible worlds truth that q & ◻ (q → (p because q)))]. This chapter and the next will explore the merits of this idea. Although the results are mainly negative, a discussion of Possible Worlds Supervenience will pave the way for a more satisfactory modal supervenience thesis simply by discarding possible worlds as irrelevant. It will show that (i) possible worlds as concreta have no place in the supervenience hierarchy of modality, and (ii) an acceptable conception of abstract possible worlds will locate truths about them on a higher level. Thus, Possible Worlds Supervenience is false, and we are free to look for a satisfactory supervenience thesis elsewhere. 3.1 Possibility and Possible Worlds Everyone who believes in the existence of more than one possible world—a realist about possible worlds, for short—thinks that their existence is systematically related to the truth and falsity of possibility and necessity claims:2 Biconditional for Possibility Sentences (◊-Bi) ∀p (it is possible that p ↔ there is a possible world at which p); and Biconditional for Necessity Sentences (◻-Bi) ∀p (it is necessary that p ↔ at every possible world, p). (◊-Bi) and (◻-Bi) express the familiar idea that possible truth is (equivalent to) truth at some possible world, while necessary truth is truth at all of them. A proponent of Possible Worlds Supervenience takes his cue from (◊-Bi) and 1

2

See e.g. Kripke (1963). The important contribution of this paper is a semantic proposal for predicate modal logic that neither validates the Barcan Formula (instances of ‘◊∃x φx → ∃x◊ φx’) nor the Converse Barcan Formula (instances of ‘∃x◊ φx → ◊∃x φx’). At least for one reasonable non-epistemic interpretation of ‘possible’ and ‘necessary’. It is usually thought that other interpretations are covered by suitably restricting the range of the right-hand side quantifier.

3.1. Possibility and Possible Worlds

71

(◻-Bi) and claims that there is a necessary explanatory relation between their right-hand sides and their true left-hand sides. For instance, he claims that, necessarily, if it is possible that McCain wins the elections, it is possible that he does so because there is a possible world at which he wins. Thus, (◊-Bi) and (◻-Bi) would provide instances for Possible Worlds Supervenience in a systematic way. Whether this is a particularly plausible claim crucially depends on what kind of things possible worlds are supposed to be. In the philosophical literature, two fundamentally different conceptions of possible worlds can be found. One camp, represented most notably by David Lewis,3 conceives of possible worlds, roughly, as, perhaps composite, concrete objects, some made up of people and talking donkeys (their ‘inhabitants’). The other camp conceives of possible worlds as, perhaps complex, abstract objects, ontologically on a par with properties and propositions. Concrete possible worlds are the topic of this chapter, abstract possible worlds that of the next. In order to evaluate the two variants of Possible Worlds Supervenience I will assume that (◊-Bi) and (◻-Bi) are true—at least for one reasonable sense of the modal idioms and some understanding of what possible worlds are. Also, I assume that their right-hand sides have the semantic structure they seem to have—that they are indeed statements whose main operator is a quantifier ranging over possible worlds. Since some things are possible but not compossible, it follows that realism about possible worlds is correct.4 Further, I will make the simplifying assumption that the discussion about natural language modal discourse can be conducted in terms of the philosophers’ favourites ‘it is necessary that’ and ‘it is possible that’. Although I find this assumption dubious,5 it significantly simplifies discussion of the relevant literature. Finally, I will concentrate on possibility claims in order to avoid tedious repetitions. 3 4

5

See particularly Lewis (1973, 1986a). For an extensive discussion of both camps— with sympathies for the Lewisian side—see Divers (2002). An anti-realist about possible worlds can still hold on to (◊-Bi) and (◻-Bi) as long as he either does not take their right-hand sides at face value or allows revisionary verdicts with respect to their left-hand sides. Modal fictionalists (e.g. Rosen 1990), and some modalists (e.g. Forbes 1985) try to secure the advantages of possible worlds talk while doing away with possible worlds. Divers (2004, 2006) makes a case for the second strategy. Chapter 5 is an attempt to exploit the closer to home ‘might’s and ‘could’s in arriving at a more satisfactory supervenience thesis.

72

Chapter 3. Concrete Possible Worlds

The present chapter deals with the claim that modal truths supervene on truths about possible worlds as Lewis conceives them (Lewis worlds, for short). I will first sketch Lewis’s theory of possible worlds, focusing on their nature and their relation to modal statements. Then, I will discuss three variants of an objection that has been levelled against Lewis and which also pertains to Possible Worlds Supervenience: the Irrelevance Objection. It will turn out that the original Irrelevance Objection is unsuccessful as it stands, since Lewis can counter it by appeal to the theoretical benefits of his view. However, in the last section I will argue, drawing on recent work by Fara and Williamson, that there are good reasons to resist the Lewisian counter. As I have already indicated, it will turn out that the chances of Possible Worlds Supervenience—understood to concern possible worlds as Lewis conceives of them—are dim. 3.2 The Analysis Claim I’ve already pointed out that realists (and some anti-realists) about possible worlds take (◊-Bi) and (◻-Bi) to be true. In fact, the truth of (◊-Bi) and (◻Bi) is what makes talk about possible worlds attractive, since it allows us to replace talk involving modal idioms with something that is often both more suggestive and easier to parse.6 The heuristic value of possible worlds talk is widely recognised by realists and anti-realists alike. If (◊-Bi) and (◻-Bi) were false, it would be hard to account for. Belief in the truth, or even necessary truth, of (◊-Bi)7 is one thing, believing that its left-hand side is analysed by its right-hand side is another. For one, ‘↔’ is symmetric, i.e. ∀p, q [(p ↔ q) → (q ↔ p)].8 Unless you think that analysis is symmetric as well, you should not take the (necessary) truth of (◊-Bi) to be sufficient for the corresponding analysis claim. Also, 1 ∀p [p ↔ (p and everything is self-identical)]; is true, and necessarily so, but claiming that the right-hand side of (1) is an 6

7 8

How widely it can be replaced depends, of course, on the status of the biconditionals. If they are true, they license substitution in extensional contexts. If they are necessarily true, we may also substitute in intensional contexts. If both sides express the same proposition, we may also substitute in standard hyper-intensional contexts. In the last chapter we saw that there may even be ‘hyper-sensitive’ contexts in which substitution of sentences that express the same proposition is not safe. and (◻-Bi), of course. From now on, I will concentrate, as indicated before, on possibility, since everything I say can be easily transposed to fit necessity. Recall that this is an extended use of ‘symmetric’ (see fn. 60 of chapter 2).

3.3. Lewis Worlds

73

analysis of its left-hand side would be to put too much faith in the conceptual ubiquity of self-identity (and conjunction). Thus, the (necessary) truth of (◊-Bi) by itself is compatible with the hypothesis that the order of analysis is the other way around as well as with the hypothesis that neither sentence is analysed by the other.9 Friends of (◊-Bi) are not forced to accept the corresponding analysis claim. Forced or not, Lewis is willing to make it. According to him, possibility sentences can be analysed in terms of quantification over possible worlds— we may just replace ‘↔’ with ‘↔df. ’: Analysis for Possibility Sentences (◊-A) ∀p (it is possible that p ↔df. there is a possible world at which p). For instance, 2

It is possible that McCain wins the 2008 presidential elections.

is analysed as follows, according to (◊-A): 3

There is a possible world at which McCain wins the 2008 presidential elections.

By (possibly repeated) application of (◊-A) and its analogue for necessity, every modal sentence is paired up with a possible worlds analysans. ‘So’, says Lewis, ‘modality turns into quantification’ (Lewis 1986a: 5). 3.3

Lewis Worlds

According to Lewis, the realm of possible worlds is the philosopher’s paradise. Once we recognise that there are such things, they can do all kinds of formerly undreamt of explanatory work. In particular, they afford conceptual analyses of many notions in the repertoire of the metaphysician, thereby significantly reducing the metaphysician’s basic vocabulary. (◊-A) may be the central case,10 but there is a battery of notions that can be analysed, according to Lewis, in terms of possible worlds: various restricted kinds of possibility and necessity, counterfactual conditionals, properties and propositions. 9

10

There may be other reasons that exclude one option or another. For instance, we may think that the expressive richness of possible worlds talk excludes the hypothesis that the direction of analysis is from right to left. I am not at all sure that this is true. In any case, the claim made in the main text is only that the necessary truth of a biconditional leaves open the three options with respect to analysis outlined, and, thus, just believing that (◊-Bi) and (◻-Bi) are necessarily true does not commit the believer to one of them. Cp. Divers (2002: 47).

74

Chapter 3. Concrete Possible Worlds

Lewis attempts to show that once we accept the notion of possible worlds as conceptually basic (or, at least, locate it on a sufficiently low level of the conceptual hierarchy), it suffices to ground a significant part of the conceptual superstructure needed in order to do metaphysics. With this alleged conceptual priority there has to go along a theory of possible worlds that can sustain the Lewisian analysis claims. In particular, talk of possible worlds cannot itself be analysed in terms of modal notions, counterfactuals, properties or propositions on pain of circularity. Either the notion of a possible world can be analysed without recourse to the standard resources of the metaphysician, or it is unanalysable and can at best be elucidated. This is what Lewis suggests in the following passage from Counterfactuals (Lewis 1973: 85): I can only ask [someone who asks what sort of thing possible worlds are] to admit that he knows what sort of thing our actual world is, and then explain that other worlds are more things of that sort, differing not in kind but only in what goes on at them.

In Plurality of Worlds (Lewis 1986a: 2), he is more explicit: Our world consists of us and all our surroundings, however remote in time and space; just as it is one big thing having lesser things as parts, so likewise do other worlds have lesser other-wordly things as parts. The worlds are something like remote planets; except that most of them are much bigger than mere planets, and they are not remote. Neither are they nearby. They are not at any spatial distance whatever from here. They are not far in the past or future, nor for that matter near; they are not at any temporal distance whatever from now. They are isolated: there are no spatiotemporal relations at all between things that belong to different worlds.

Thus, Lewis relies in his elucidation of what possible worlds are on our understanding of what our world is—something that, plausibly, does not require a prior grasp of concepts analysable in terms of possible worlds. Just as our world is made up of Lewis and his surroundings, so are other Lewis worlds mereological sums of concrete things, some of them composed of talking donkeys and, perhaps, people very similar to me except for having already finished a chapter on Lewis worlds. Lewis worlds are spatio-temporally isolated from each other. That is, if some thing is part of some world, it is not spatio-temporally related to anything that is part of some other world. Also, Lewis worlds are maximal with respect to spatio-temporal relatedness. That is, if something is part of some world, everything spatio-temporally related to it is also part of that world.

3.3. Lewis Worlds

75

Lewis worlds are, according to Lewis, what the quantifiers in the analysantia of (◊-A) quantify over. We may put this by saying that, according to Lewis, truths about Lewis worlds make true the (true) analysantia of (◊-A), just as, perhaps, truths about glasses of H2 O make true 4

There are three glasses of water in the room. For instance,

3

There is a possible world at which McCain wins the 2008 presidential elections;

is made true by there being a Lewis world—a maximal sum of spatio-temporally related individuals—at which McCain wins the 2008 presidential elections. Thus, we should understand the Lewisian theory of modality as postulating two distinct steps. First, there is the analysis of modal sentences in terms of quantification over possible worlds. Then, there are claims about what makes these analysantia true.11 Although the second kind of claims may make the analysis claim plausible (or implausible, as the case may be), they are independent of each other. One could, for instance, consistently hold that modal sentences are analysable in terms of quantifications over possible worlds, but reject the view that these in turn are made true by truths about a certain sort of mereological sum. Likewise, one could think that claims about possible worlds are made true by truths about Lewis worlds, while resisting the view that modal sentences are analysable at all.12 In the course of the further discussion we will have reason to take a closer look at the supposed correlation between modal sentences and their Lewis world truth-makers.13 11

12

13

In what follows, I will sometimes skip the intermediate step and talk about the truth makers of the modal sentences themselves, assuming that those are whatever is supposed to make true the corresponding possible world quantifications. Sometimes the Lewisian truth-maker claims are represented as a further step in analysing modal sentences. That’s fine as well, as long as we keep in mind that these ‘analyses’ may lack some of the features paradigm cases of conceptual analysis have. For instance, someone who affirms the sentence ‘there are possible worlds at which donkeys talk’ while not willing to affirm the sentence ‘there are mereological sums some of whose parts are talking donkeys’ does not betray a misunderstanding of either of the sentences (or of ‘there could have been talking donkeys’, for that matter). Just as someone who affirms ‘there are glasses of water in the room’ while not willing to affirm ‘there are glasses of H2 O in the room’ does not betray a conceptual deficiency (though presumably a scientific one). Some objections to the Lewisian theory may be defused by highlighting the difference between the two steps in the analysis, or, as I’d rather put it, the difference between the analysis claims and the truth-maker claims. In section 3.5 we will see a systematic way of correlating a broad range of modal

76

Chapter 3. Concrete Possible Worlds

For now, it suffices to say that, according to the Lewisian theory, there is some systematic correlation between Lewis worlds and their inhabitants and the modal sentences for which they provide the truth-makers. In the last chapter (on pages 46–50) we saw that analyses may underwrite supervenience theses. If Lewis is right, (◊-A) and its analogue for necessity are true. So, Possible Worlds Supervenience is true as well. As we have seen in this section, Lewis thinks that the analysantia of (◊-A) are made true by truths about Lewis worlds, maximal sums of spatio-temporally related possible individuals. This combination of Possible Worlds Supervenience with a view of what makes talk about possible worlds true, has been the focus of a class of objections to Lewis’s theory that go under the name Irrelevance Objection. In particular, the objections are directed against the specific explanatory consequences of (◊-A) together with Lewis’s understanding of what possible worlds are. The original Irrelevance Objection targets the following consequence of Lewis’s theory: IO ◻∀p (It is possible that p → (it is possible that p because there is a Lewis world at which p)). Since any objection to (IO) is ipso facto an objection to Possible Worlds Supervenience as motivated by Lewis’s theory, the Irrelevance Objection will be discussed in the next section. In section 3.6, we will see how the original Irrelevance Objection can be improved upon by focusing on the companion principle of (IO): IO′ ◻∀p (It is impossible that p → (it is impossible that p because there is no Lewis world at which p)). 3.4 The Irrelevance Objection The Irrelevance Objection claims that what goes on at Lewis worlds is irrelevant to whether something is possible or not. There are at least three ways of fleshing out the objection, which can be seen as targeting the following three consequences of (IO): IO1 ∀p (It is possible that p → there is a Lewis world at which p); IO2 ◻∀p (It is possible that p → there is a Lewis world at which p); IO3 ∀p (It is possible that p → (it is possible that p because there is a Lewis world at which p)). sentences with their Lewis world truth-makers. In section 3.6 we will see why this may not be possible for all modal sentences.

3.4. The Irrelevance Objection

77

(IO2 ) is a consequence of (IO) because of the factivity of ‘because’.14 (IO3 ) is a consequence of (IO) because every necessary truth is also a truth, and (IO1 ) is a consequence of (IO) because of the two principles combined. The three variants of the Irrelevance Objection will be presented and discussed in the following subsections.15 3.4.1 Against IO1 In van Inwagen (1986: 221ff.), Peter van Inwagen argues that the two sides of (◊-A) do not even have the same truth-value, provided that the right-hand side is to be understood in terms of Lewis worlds. In particular, whenever it is merely possible that p—whenever it is possible but not the case that p—the corresponding instance of IO1 ∀p (It is possible that p → there is a Lewis world at which p); is false. The example van Inwagen discusses is 5

There is no million-carat diamond but there could have been one.

(5), we may assume, is true.16 But, according to van Inwagen, 6

There is no million-carat diamond (at this world), but there is a Lewis world at which there is one;

is not. Since this is so, (IO1 ) is false. 14

That is, if IO ◻∀p (It is possible that p → (it is possible that p because there is a Lewis world at which p)); then IO2 ◻∀p (It is possible that p → (it is possible that p & there is a Lewis world at which p)).

15

Perhaps, arguments against (IO3 ) have the best claim to the title Irrelevance Objection. I extend the honour to (IO1 ) and (IO2 ) here as well, partly to have a common name for them, partly because Irrelevance Objectors took themselves to argue against relevance at least by arguing against (IO2 ). According to astronomers at the Harvard-Smithonian Center for Astrophysics, there is a dwarf star, Lucy, in the constellation of Centaurus whose core is a 1034 carat diamond (see http://news.bbc.co.uk/1/hi/sci/tech/3492919.stm). So, van Inwagen may have been off by some quintillion carats. Such are the perils of armchair empiricism.

16

78

Chapter 3. Concrete Possible Worlds

Now is the time to have a closer look at how Lewis wants his claims about Lewis worlds to be interpreted, in order to better understand (6). According to Lewis, the phrase ‘at world w’ works like ordinary restrictors—‘in Australia’, ‘in the 19th century’, ‘at home’ and the like: roughly, it restricts the range of the quantifiers in its scope to things suitably related to what its argument denotes. For instance, according to Lewis, the restrictor ‘in Australia’ in 7

In Australia, there are black swans;

manages to restrict the range of the quantifier ‘there are’, roughly, to things that are part of Australia. Likewise, ‘at world w’ in 8

At world w, there is a million-carat diamond;

manages to restrict the range of the quantifier ‘there is’ to things that are part of w. Of course, if there are black swans in Australia, there are black swans full stop. Likewise, if there is a million-carat diamond at some Lewis world, there is a million-carat diamond full stop. Does not this consequence of (6) already conflict with (5)—one of whose conjuncts is, after all, ‘there is no million-carat diamond’? Not really. According to Lewis, when we utter quantified sentences, the context of utterance manages to impose restrictions on the quantifiers, so that what is said has a chance of being true. When I utter ‘everyone came to the party’, what I said is not shown to be false by pointing out that the Russian Prime Minister didn’t attend. At least according to one reasonable theory of contextual restriction, this is so because the context in which I uttered the sentence managed to restrict the range of the quantifier, perhaps, to people actually invited.17 Likewise for ‘there is no million-carat diamond’ and other quantified sentences devoid of modal vocabulary. As uttered in a normal context, the context manages to restrict the range of the quantifiers occurring in them, according to Lewis, to things that are part of the actual world.18 But 17

18

It is an interesting question whether this is a semantic or a pragmatic phenomenon. On an alternative theory, what I said is strictly speaking false (since millions of people didn’t come to the party) but nevertheless saliently implies something true (that everyone who was invited came). Since judgements concerning truth and falsity will then typically go with what is saliently implied, the discussion in the main text could be easily amended to fit such a theory. For such a view see, for instance, Bach (2005), especially the discussion of thesis 4. The parenthetical remark was added to (6) in order to make the context do as little work as possible. Also, arguably, stating (6)’s second conjunct that mentions Lewis

3.4. The Irrelevance Objection

79

that there is no million-carat diamond at the actual world does not conflict with the claim that there is a million-carat diamond at some Lewis world, just as the claim that there is no black swan in Britain conflicts with the claim that there is a black swan in some country or other. Given that Lewis worlds are maximal sums of spatio-temporally related individuals, and that you and I are parts of the actual world, (6) implies 9

There is no million-carat diamond spatio-temporally related to us, but there is a million-carat diamond;

where both quantifiers are supposed to be unrestricted. But (9) is simply false, according to van Inwagen: 10

Though indeed there could have been a million-carat diamond, there simply is [absolutely unrestricted quantifier] none.

[…] At least, this may well be true. I believe it, and I see no reason to feel uneasy about believing it, though I can’t prove it. 11

Nothing [absolutely unrestricted quantifier] is spatiotemporally unrelated to me (unless, like a number or proposition, it is not spatiotemporally related to anything).

At least, this may well be true. I believe it, and I see no reason to feel uneasy about believing it, though I can’t prove it. (van Inwagen 1986: 222, my numbering)

Since (9) is false and (6) implies (9), (6) is false as well. Since (5) is true, we assumed, (IO1 ) has a false instance. Further, if van Inwagen is right and there is unrestrictedly nothing spatio-temporally unrelated to us, there is no Lewis world other than the actual world. So, if it is not the case that p at the actual world, it is not the case that p at any Lewis world. Consequently, any instance of (IO1 ) is false, if the corresponding instance of ‘it is possible but not the case that p’ is true. Whether van Inwagen’s attack on (IO1 ) is successful depends on whether he has good reasons to believe (10) and (11) and, thus, good reasons to deny (9). Surely, the autobiographical remark that van Inwagen believes something inconsistent with (IO1 ) is not enough to discredit it, especially since Lewis himself offers an argument to the contrary. The argument is Lewis’s argument from theoretical utility that has been alluded to before. According to Lewis, we have good reasons to believe in the existence of a plurality of Lewis worlds because of the eminent theoretical utility they afford. If there are Lewis worlds, they provide the explanatory basis for worlds may result in the context’s not being ‘normal’.

80

Chapter 3. Concrete Possible Worlds

all kinds of phenomena metaphysician is concerned with. Moreover, there is, according to Lewis, no better or equal explanatory basis for these phenomena. Given their superior explanatory potential, we should believe that Lewis worlds exist, at least in the absence of serious counterevidence. But if they do, there are things spatio-temporally unrelated to us (the inhabitants of other worlds), so (11) is false. Further, given that Lewis is right about one central explanatory success of his theory, (◊-A), among those things spatiotemporally unrelated to us, there is a million-carat diamond. So, (10) is false. One way to resist Lewis’s reasoning is to challenge his claim that Lewis worlds really do provide a superior explanatory basis. This is a strategy that will be discussed in the subsection on (IO3 ). Another way is to point to serious counterevidence against the existence of Lewis worlds. Perhaps the existence of a further planet, Vulcan, was the best potential explanation for deviations in the orbit of Mercury available at the time of Le Verrier. Both the realisation that Le Verrier’s was not the best theory to explain Mercury’s orbit after all,19 and the observations of stronger telescopes trumped this reason for believing in the existence of Vulcan. But is there any counterevidence against the existence of Lewis worlds, or the existence of million-carat diamonds, comparable to observations with telescopes strong enough to detect an extra planet if there were one? Here, a feature of Lewis worlds that is the focus of another widely discussed objection to the Lewisian theory, the Epistemic Objection,20 actually helps Lewis. Given that other Lewis worlds are spatio-temporally unrelated to us, neither their existence nor their non-existence is epistemically accessible via straightforward empirical means. Of course, there is no chance of discovering another Lewis world by hunting around in the universe. Since Lewis worlds are spatio-temporally separated from each other, any other Lewis world is spatio-temporally separated from the actual world, and, thus, from us. But, by the same token, our failure to have discovered other Lewis worlds is no evidence against their existence. Never having seen invisible things lends no credibility to the view that everything is visible. Likewise never having been in touch with spatio-temporally unrelated things should not lend any credibility to the view that everything is thus related. On the other hand, it is difficult to see what non-empirical counterevidence against the existence of million-carat diamonds or whole Lewis worlds is 19 20

In fact, it turned out that the theory of relativity could account for this phenomenon. For more on the Epistemic Objection see, e.g., Richards (1975); Skyrms (1976) and Lewis’s own discussion in Lewis (1986a: §2.4).

3.4. The Irrelevance Objection

81

supposed to look like. Perhaps, some variant of Ockham’s Razor could be appealed to, i.e. some principle barring the postulation of ‘unnecessary’ entities. But, clearly, appeal to such a principle would presuppose that one could show that Lewis worlds are indeed unnecessary. This would come down to denying their superior explanatory potential. Again, this strategy is the topic of subsection 3.4.3. Let me sum up. Van Inwagen attempts to cast doubt on (IO1 ) by considering a case in which it is possible but not the case that p. He claims that there is no Lewis world at which p, and that, therefore, (IO1 ) is false. However, Lewis’s argument from theoretical utility gives Lewis, at least initially, the upper hand in the dispute. It cannot just be claimed that there are no non-actual Lewis worlds, there needs to be evidence against their existence. Since there can be no empirical evidence, the best bet of his opponent is to find fault with Lewis’s argument from theoretical utility. Since this would be to argue against (IO3 ), the discussion is postponed to subsection 3.4.3. 3.4.2 Against IO2 The second way in which (IO) may be attacked is by challenging (IO2 ), i.e. IO2 ◻∀p (It is possible that p → there is a Lewis world at which p). The following passage by Michael Jubien can be interpreted to attempt just this:21 Consider this thought-experiment. Suppose God created everything that exists, including all the worlds. Now imagine two ways in which the creating might have been done. One of the ways is just the way things happen to have been done. Now suppose that this happens to include worlds at which there is someone enough like me to be my ‘counterpart’ but who is a cab driver. Now let the second way be just like the first except that no cab driver worlds get created. […] Even if things were like this it would still seem true that I could have been a cab driver. […] As long as other worlds are entirely distinct, independent things (at least physically), it is hard to see how they can have anything at all to do with the truth or falsity of simple statements of possibility. (Jubien 1988: 304) 21

Another way to interpret the passage is that Jubien just raises an epistemic possibility. However, Jubien takes the point made in the passage ‘to be a metaphysical point’ (Jubien 1988: 304)—which discourages this alternative interpretation. Whatever Jubien may have had in mind, Lewis has a straightforward response to the epistemic worry on the basis of fn. 12 above.

82

Chapter 3. Concrete Possible Worlds

In this passage, the Lewisian notion of counterparthood occurs. Since nothing hangs on it, I will, for simplicity, just replace it with the less explicit but more familiar talk of worlds at which Jubien is a cab driver.22 So, Jubien’s main claims come down to the following: 12

It is possible that there is no Lewis world at which Jubien is a cab driver;

and 13

If there were no Lewis worlds at which Jubien is a cab driver, it would still be possible that Jubien is a cab driver.

Since from ‘◊p’ and ‘p € q’ it follows that ◊(q & p), the following is a consequence of (12) and (13): 14

◊ (It is possible that Jubien is a cab driver & there is no Lewis world at which Jubien is a cab driver);

Since (14) is the negation of an instance of (IO2 ), (IO2 ) is false. That Jubien uses the hedging phrase ‘it would still seem true that’ suggests that he would like us to take his word for (13).23 However, there is also a straightforward argument for it, when we assume that the characteristic axiom of S5, S5

◊p → ◻◊p.

is correct.24 For, suppose that it is possible that Jubien is a cab driver. By (S5), it is necessary that it is possible that Jubien is a cab driver. Since ‘∀p (p € q)’ is true, as long as ‘◻q’ is, it would have been possible that Jubien is a cab driver, had anything been the case. A fortiori, it would have been possible that Jubien is a cab driver, had there been no Lewis world at 22

23

24

Lewis is happy to talk that way, as long as talk of non-actual possible worlds at which Jubien is so-and-so, is understood to be concerned not with worlds that have Jubien as a part but with worlds that have counterparts of Jubien as parts. Cf. Lewis (2001: 604f.). At least I interpret it to be a hedging phrase with the same import as ‘it would still be true, it seems, that’. In the alternative—admittedly more natural—interpretation, Jubien talks about what would appear to us to be true in the counterfactual circumstances. But whether or not it would appear to us to be possible that Jubien is a cab driver if there were no cab driver worlds is irrelevant for the truth of (IO2 ). As should become clear in section 3.5, Lewis cannot subscribe to (S5) in its full generality. However, even according to Lewis, (S5) holds, when ‘p’ does not contain a name or a free variable (when ‘◊p’ is de dicto, that is). The discussion in the main text could have easily been modified to involve only such de dicto possibilities.

3.4. The Irrelevance Objection

83

which he is a cab driver.25 Consequently, the contentious premiss of the argument is (12). Unfortunately for Jubien, it may express two things, depending on whether ‘there is’ is read as a restricted or an unrestricted quantifier. In the remainder of this subsection I will explain why the first reading of (12) is true but irrelevant (no pun intended), and the other false—at least according to Lewis. Consider the true reading first. Lewis worlds are things of the same kind as the actual world. According to Lewis, they are mereological sums, mostly, if not entirely, made up of contingently existing individuals. Presumably, if sufficiently many parts of a mereological sum don’t exist, the sum itself doesn’t.26 Thus, just as the actual world may not have existed (because you and I and some other things failed to exist), so can other worlds fail to exist. In particular, presumably, every Lewis world at which Jubien is a cab driver may have failed to exist.27 This claim of possible non-existence of certain worlds is justified by the contingent existence of their inhabitants. Accordingly, the former should receive a similar treatment as the latter. We have already seen that, according to Lewis, unembedded existential quantifications are such that their quantifiers are restricted to parts of the actual world. This may be brought out in semi-formal renderings of them. For instance, 15 There are talking donkeys; can be formalised as 15* [∃x ∶ x is a part of w] x is a talking donkey. 25

26 27

Even Lewis could accept (13). For, he accepts the principle that ‘p € q’ is true if it has an impossible antecedent. As we will see presently, this is what Lewis may want to say about (13). ‘Strict’ mereological essentialism holds that ‘sufficiently many’ here means at least one. Jubien himself argues slightly differently: 0.0ptthey [other possible worlds] came about in ways similar to the way ‘the [actual] world’ did, however that may have been. […] They owe their existence to divine conjuration or perhaps to certain relatively cosmic and cataclysmic singularities. (Jubien 1988: 305) His thought seems to be that the relevant conjuration or cosmic singularity need not have taken place, which is why the existence of many worlds is contingent. However, it is not clear whether the actual world, as Lewis understands the term, owes its existence to anything, let alone to some cosmic singularity which would, presumably, itself be part of the actual world.

84

Chapter 3. Concrete Possible Worlds

Here, the quantifier phrase ‘[∃x ∶ x is a part of w]’ makes it clear that the quantifier ‘there are’ in (15) is restricted, and the variable ‘w’ is free to be assigned by the context to the actual world. When we put a modal operator in front, a quantifier is introduced which binds the free variable in the restricting expression of the original quantifier. For instance, 16

It is possible that there are talking donkeys;

may be formalised as 16*

∃w[∃x ∶ x is a part of w] x is a talking donkey.

Likewise, 17

It is possible that there are no Lewis worlds at which Jubien is a cab driver;

should receive the formalisation 17*

∃w ¬[∃x ∶ x is a part of w] x is a Lewis world at which Jubien is a cab driver.

In other words, on this reading (12) comes down to the truth that there are a great many Lewis worlds which do not have parts that are Lewis worlds at which Jubien is a cab driver. Since every Lewis world has only one Lewis world, namely itself, as an (improper) part,28 this is true of any Lewis world at which Jubien is not a cab driver. However, (12) on a reading on which it may be formalised as (17*) does not count against (IO2 ). This can be seen by looking at (17*) itself. The possibility operator in (17) introduces an unrestricted quantifier into (17*), while the second quantifier is restricted. This is not an idiosyncrasy of my formalisation of (17) but a general feature of Lewis’s account. Otherwise, the account would not stand a chance of being true. For, consider what would happen if possibility operators introduced restricted quantifiers. In that case, (16)’s formalisation would not be (16*) but 16?

[∃w ∶ w is a part of w′ ][∃x ∶ x is a part of w] x is a talking donkey.

(16?) contains a free variable in the restricting expression of its first quantifier, namely ‘w′ ’. In accordance with what happens, according to Lewis, to other unbound world variables, we should expect the context to assign the actual world to ‘w′ ’ in an utterance of (16?). But then, what is said by an utterance of (16?) is true just in case there is a world that is part of the actual 28

Recall that no part of a world is spatio-temporally related to any part of another world.

3.4. The Irrelevance Objection

85

world and which has talking donkeys as parts. Since only the actual world is an (improper) part of the actual world, (16?) is true just in case the actual world has talking donkeys as parts, i.e. if there (restrictedly) are talking donkeys. Obviously, this is not what anyone would want as a formal rendering of a consequence of an analysis of the sentence ‘It is possible that there are talking donkeys’. Likewise, the analysans (◊-A) assigns to 18 It is possible that Jubien is a cab driver; is to be formalised as 18* ∃w (at w, Jubien is a cab driver); rather than as 18? [∃w ∶ w is a part of w′ ] at w, Jubien is a cab driver. Since (17*) is an existential quantification of the negation of (18?) instead of (18*), its truth is irrelevant to whether (IO2 ) is true—at least on a reading on which (IO2 ) follows from Lewis’s proposed analysis (◊-A). What is relevant, on the other hand, is whether it is possible that (18*) is false, i.e. whether 19 It is possible that ¬∃w (at w, Jubien is a cab driver). According to the Lewisian theory, (19) has a formal rendering in which ‘it is possible that’ is replaced by a world quantifier which binds any free world variable in (18*), i.e. 19* ∃w′ ¬∃w (at w, Jubien is a cab driver). But since (18*) does not contain any free variables, the first quantifier in (19*) has no variable to bind and is, thus, redundant. Consequently, (19*) is true just in case (18*) is false, i.e. if there (unrestrictedly) is no Lewis world at which Jubien is a cab driver.29 Of course, Jubien could claim that (18*) is false. But this is a strategy already discussed in subsection 3.4.1 and found to be not particularly promising. In any case, it would presuppose a successful challenge to (IO1 ). But if there were a successful challenge to (IO1 ) this would already be enough to discredit (IO). Consequently, Jubien’s argument against (IO2 ) is either invalid or redundant. It relies on a premiss, (12), which has a true reading that cannot sustain an argument against (IO2 ). The premiss also has a reading 29

This is one spelling out of Divers’s claim that ‘extraordinary’ modal claims are cases of redundant quantification and, thus, give rise to modal collapse (◊p ↔ ◻p ↔ p). See Divers (2002: 47ff.).

86

Chapter 3. Concrete Possible Worlds

which can sustain the argument. But on that reading its truth is not easier to show than anything that would already count against a weaker consequence of (IO). To end our discussion of the Irrelevance Objection, let us take a look at (IO3 ). 3.4.3 Against IO3 The last way in which the Irrelevance Objection may be levelled is by objecting to (IO3 ), i.e. to IO3 ∀p (It is possible that p → (it is possible that p because there is a Lewis world at which p)). This is what van Inwagen may have had in mind when he wrote that Lewis face[s] the problem of […] explaining what these things [Lewis worlds] would have to do with modality if there were any of them. (Van Inwagen 1985: 203)

To this, Lewis replied—rather brusquely—as follows: You might think that I have often explained what these things have to do with modality, for instance by saying that the modal operators are quantifiers over them. (Lewis 1986a: 98)

He then went on to suggest that van Inwagen might have been mistaken in thinking that Lewis worlds are ‘subdivisions of actuality’, and, as a result, thought—justifiably but incorrectly—that modal sentences cannot be explained in terms of them. Perhaps. But I think it more likely that van Inwagen wanted to deny the explanatory claim, whether or not Lewis worlds are subdivisions of actuality. If so, the ‘exchange’ between van Inwagen and Lewis is rather curious. One of them (van Inwagen) simply denies an obvious consequence of the other’s theory, while the other (Lewis) responds by simply reaffirming that part of his theory of which it is a consequence. Again, Lewis seems to initially have the upper hand, since (IO3 ) is backed by a systematic theory which includes—but is not restricted to—modality. Are there any good reasons van Inwagen could have offered for denying (IO3 ) nevertheless? If a case could be made against (IO1 ) and (IO2 ), this would be an excellent reason to deny (IO3 ). However, as was discussed in subsections 3.4.1 and 3.4.2, it seems that an objection against (IO2 ) stands and falls together with one against (IO1 ), and that, furthermore, a successful objection against (IO1 ) would have to rely on a prior argument against (IO3 ). Consequently, an objection to (IO3 ) should be directed straight at (IO3 )’s explanatory component.

3.4. The Irrelevance Objection

87

This is what Charles Chihara attempts by comparing Lewis’s argumentative strategy with that of the hypothetical Barney Bailley: Barney Bailley advances an analysis of grammaticality, according to which a sequence of English words σ is a grammatical sentence of English iff σ has ever appeared in the imagination of the angel Gabriel. The Grammatical Irrelevance Objection is raised: ‘You need to explain, Bailley, what a sequence of English words appearing in the imagination of the angel Gabriel has to do with grammaticality’. Suppose Bailley responds in the way Lewis did: ‘You would think that I have explained what a sequence of English words appearing in the angel Gabriel’s imagination has to do with grammaticality, for instance by saying that a grammatical sequence is one that appears in that angel’s imagination’. The inappropriateness of this reply is obvious […]. (Chihara 1998: 94f)

Like Lewis, Chihara’s’s Bailley proposes an analysis which commits him to a corresponding explanatory claim. When a doubt is raised regarding the truth of the explanatory claim, he does nothing but to reaffirm his theory from which the explanatory claim follows. ’s suggestion is that, just as we should not let Bailley get away with unfounded explanatory claims linking grammaticality with angelic imagination, neither should we let Lewis get away with (IO3 ). ’s bad company objection is only as strong as the analogy between Lewis and Bailley. But there are good reasons to think that the cases are disanalogous in crucial respects. For one, Chihara’s exposition does nothing to suggest that Bailley’s explanatory claim is backed up by a systematic theory of linguistic phenomena. Surely, grammaticality of whole sentences is not a phenomenon cut off from others. For instance, one would expect there to be a similar explanation for the grammaticality of subsentential expressions. Also, the syntactic structure of sentences is presumably determined by whatever determines their grammaticality. It is not clear, however, how these related explanatory aims can be achieved with the resources available to the hypothetical Bailley. It seems likely that this unembeddedness of Bailley’s proposed analysis is one major reason why it appears preposterous. On the other hand, Lewis appears to have provided a systematic and wide ranging theory of modality and related phenomena. Since Chihara does nothing to show that appearances are deceptive, this appears to be quite a relevant difference between Bailley and Lewis. Secondly, there are alternative explanations of grammaticality which prima—and, presumably, ultima—facie look far more promising than a proposed explanation involving archangels. Again, these alternatives, unlike

88

Chapter 3. Concrete Possible Worlds

Bailley’s proposal, embed grammaticality of sentences within a network of related phenomena. The existence of alternatives does not refute an explanatory claim, since one phenomenon may have various non-competing explanations. But it does put pressure on anyone who would like to add to them. Very likely, a proponent of Bailley’s hypothesis has not much to say in response to the charge that his preferred explanation is superfluous. Not so in the case of Lewis. After all, a major part of Lewis (1986a) is dedicated to a critique of other accounts of modality, weighing their advantages and disadvantages against those of his own theory. Consequently, I think that Chihara’s bad company argument against (IO3 ) is unsuccessful as it stands. It highlights the fact that Lewis was overly brief in his response to van Inwagen. In addition to reaffirming his analysis claim he should have pointed out, as he does elsewhere, that it is part of a systematic theory with great explanatory potential which is, furthermore, more successful than its rivals. However, since Chihara’s Bailley cannot defend his explanatory claim in a similar way, the ‘obvious inappropriateness’ of Bailley’s sticking to his analysis on being challenged, does not seem to carry over to Lewis. Let me briefly sum up the discussion of this section. We have seen that Possible Worlds Supervenience cannot easily be dismissed by rejecting (IO). All versions of the Irrelevance Objection ultimately run up against the claim that the Lewisian theory provides a systematic and wide ranging account of matters modal. This gives Lewis the initial upper hand in the debate. None of the arguments was strong enough to take this initial advantage away from him. This suggests that a successful form of the Irrelevance Objection would have to target the claim that the Lewisian theory provides such a systematic and wide ranging account of modality. This may not look like a particularly promising task. However, recent work by Fara and Williamson suggests that there may still be good reasons to doubt that the Lewisian theory is both systematic and appropriately wide ranging. This is so, since there is reason to doubt that the Lewisian theory can adequately deal with a class of modal truths that does not figure prominently in the original Irrelevance Objection, namely de re modal truths. Fara and Williamson (2005) argue that the Lewisian theory cannot adequately deal with de re modal truths, a fact that is brought out by considering what might or might not have been actually the case. In a nutshell, they argue that any way of systematically correlating de re modal sentences (involving actuality) with sentences about counterparts (Lewis’s way of dealing with de re modality)

3.5. Counterparts

89

fails, since it predicts that certain inconsistent sentences are in fact consistent. They can be taken to argue against the companion principle of (IO): IO′

◻∀p (It is impossible that p → (it is impossible that p because there is no Lewis world at which p)).

If successful, we would have a variant of the Irrelevance Objection which at the same time undermines Lewis’s response to the original Irrelevance Objection. I think it is fair to say that if Fara and Williamson are right, Possible Worlds Supervenience as motivated by the Lewisian theory is no longer a live option. In the remainder of the chapter I first introduce counterpart theory, Lewis’s way of dealing with de re modal truths. Then I present (some of) Fara and Williamson’s objections. Finally, I explain how they bear on Possible Worlds Supervenience. 3.5

Counterparts

Recall that Lewis worlds are maximal spatio-temporally related mereological sums. Consequently, if something is part of one Lewis world, it is not also part of any other. This immediately raises the question of how an analysis of modal sentences in terms of possible worlds is supposed to deal with de re modal sentences, sentences that, intuitively, say of an object that it is modally so-and-so. It is common to give a (quasi-)30 syntactic characterisation of de re modal sentences. Following Forbes (1985: 48) we may say that an English sentence S under an interpretation I is de re just in case some rendering of S under I in the language of quantified modal logic (QML) is such that it has a modal subformula—i.e. a subformula whose main operator is ‘◊’ or ‘◻’—which contains either an individual constant or a variable not bound by a quantifier within that subformula.31 Modal sentences which are not de re are de dicto. For instance, 30 31

It is a purely syntactic question only if it is a purely syntactic question whether a given expression is to be treated as an individual constant—which seems doubtful. Mention of sentences under an interpretation is meant to deal with syntactically ambiguous sentences—which may have non-equivalent renderings corresponding to their syntactic disambiguations. Also, a syntactically unambiguous but complex sentence has many non-equivalent QML renderings (e.g. ‘◊Fa’ ‘◊p’ and ‘q’). In what follows I will ignore these complications. When I give examples of the QML rendering of S in the main text, I will pick a sufficiently specific rendering of a natural interpretation of S.

90

Chapter 3. Concrete Possible Worlds

20

Socrates might not have been a philosopher; and

21

Necessarily, whenever some thing is different from another it is necessary that they are different;

are de re, since their QML renderings 20*

◊ (¬Fa);32 and

21*

◻∀x, y (x ≠ y → ◻(x ≠ y));

have modal subformulae 20** ◊ (¬Fa);33 and 21** ◻ (x ≠ y); which contain an individual constant (‘a’) and unbound variables (‘x’ and ‘y’) respectively.34 22

Necessarily, everything is identical with itself;

on the other hand, is de dicto, since the only modal subformula of its rendering 22*

◻∀x (x = x).

is (22*) itself which only contains a bound variable. Back to Lewis. According to Lewis, (20) is analysed (recall (◊-A)) as 20A

There is a possible world at which Socrates is not a philosopher;

which in turn is made true by there being a Lewis world at which Socrates is not a philosopher. But what does this come down to? A Lewis world at which Socrates is not a philosopher cannot be a Lewis world one of whose parts is Socrates and is such that, at it, that part is not a philosopher. The actual world has Socrates as a part. Since Lewis worlds are spatio-temporally isolated, no Lewis world except the actual world has Socrates as a part. But at the actual world, it is not the case that Socrates is not a philosopher. So, there is no 32

33 34

This assumes the thesis that natural language proper names like ‘Socrates’ are formalisable as individual constants and are not, e.g., covert quantifier phrases. Nothing in the discussion hinges on this. Note that (20**) just is (20*). Subformulae do not need to be proper subformulae. Note that a formula may be de re with respect to a position without being de re with respect to some other position. (21*), for instance, is de re with respect to the positions of the third occurrences of ‘x’ and ‘y’ while not being de re with respect to the positions of their second occurrences. Given our characterisation, an English sentence is de re just in case its rendering is de re with respect to some position.

3.5. Counterparts

91

Lewis world which has Socrates as a part and is such that at it, Socrates is not a philosopher. According to Lewis, though there are no other Lewis worlds which have Socrates as a part, there are many Lewis worlds which have counterparts of Socrates as parts. Counterparts of Socrates resemble him closely in content and context in important respects. They resemble him more closely than do the other things in their worlds. But they are not really him. For each of them is in his own world, and only Socrates is here in the actual world. Indeed, we might say, speaking casually, that his counterparts are him in other worlds, that they and him are the same; but this sameness is not a literal identity […]. It would be better to say that Socrates’ counterparts are men he would have been, had the world been otherwise. (Lewis 1968: 28, with slight adjustments to fit our case)

So, Socrates’ counterparts are parts of non-actual Lewis worlds which are similar to him in important35 intrinsic and extrinsic36 respects. According to Lewis, it is these counterparts which can be used to make sense of the claim that there is a non-actual world at which Socrates is so-and-so. For instance, there is a non-actual world at which Socrates is not a philosopher because there is a non-actual world that contains a counterpart of Socrates who is not a philosopher. Generally speaking, possible worlds analyses of de re modal sentences are made true, according to Lewis, by truths about counterparts. In Lewis (1968: 30f.), Lewis provides a translation scheme for translating sentences of QML into the language of Counterpart Theory (CT), where CT is a first-order predicate logical theory that contains additional axioms constraining the interpretation of its theoretical terms, most importantly ‘xCy’ (to be read as ‘x is a counterpart of y’) and ‘x < y’ (‘x is in y’ or ‘x is a part of 35

36

In Lewis (1971), Lewis argues that there is a variety of different counterpart relations differing in how much importance they assign to different respects of similarity. For instance, a counterpart relation which assigns much importance to position in space (and almost none to anything else) may have a (non-actual) coffee mug as one of my counterparts. Since nothing in the discussion turns on the question of whether there are one or many counterpart relations, I will pretend that there is only one. This is meant to paraphrase Lewis’s slightly obscure ‘content and context’. If you do not find the intrinsic/extrinsic distinction particularly transparent either, you may be helped by Lewis’s explanation that an object’s intrinsic properties are shared by all (unrestrictedly) its duplicates. My coffee mug’s shape, for instance, will presumably be shared by all of its duplicates, while its position in space is not. Similarity in important intrinsic and extrinsic respects ensures that not all of my coffee mug’s non-actual duplicates are also its counterparts. Similarity in important intrinsic and extrinsic respects ensures that not only its (near) duplicates are its counterparts.

92

Chapter 3. Concrete Possible Worlds

y’37 ).38 w Let [ψ ] be a function which assigns to a formula ψ of QML and a worldterm w a formula of the language of CT (≈ ‘ψ is true at the world denoted by w’). The translation of a sentence of QML ψ is the sentence of CT that this @ function assigns to ψ and a name for the actual world @, i.e. [ψ ] (≈ ‘ψ is w true at the actual world’). [ψ ] is defined recursively as follows:39 w CTat [φ] is φ, if φ is atomic; w

w

CT∃

[∃x φ] is ⌜∃x (x < w & [φ] )⌝;40

CT∀

[∀x φ] is ⌜∀x (x < w → [φ] )⌝;

CT◊

[◊φ] is ⌜∃v, y1 , . . . yn (y1 < v & y1 Cα1 . . . & yn < v & yn Cαn & v [φ] (y1 , . . . yn ))⌝, given that φ is φ(α1 , . . . αn ); where for any formula φ (with n free terms)41 and terms α1 , . . . αn , φ(α1 , . . . αn ) is the result of uniformly replacing φ’s free terms with α1 , . . . αn in the order of their first occurrence;42

CT◻

[◻φ] is ⌜∀v, y1 , . . . yn ((y1 < v & y1 Cα1 . . . & yn < v & yn Cαn ) → v [φ] (y1 , . . . yn ))⌝, given that φ is φ(α1 , . . . αn ).

w

w

w

w

(CTat ) tells us that an atomic sentence is true at a Lewis world just in case it is true simpliciter.43 (CT∃ ) and (CT∀ ) tell us, in conformity with our discussion in subsection 3.4.2, to restrict the quantifiers when evaluating existentially 37

38

39 40 41 42 43

In 1968 Lewis hadn’t worked out his full blown conception of Lewis worlds as mereological sums of their inhabitants yet. However, given later developments, the suggestive notation ‘ Pp1 and ∀q (Pp1 ⊩ q ∨ Pp1 ⊩ ¬q), ∃q(Pp2 ⊩ q & Pp2 ⊩ ¬q). But then Pp2 is not a possibility, since ∃q(◻(p2 → q) & ◻(p2 → ¬q)) only if ¬◊p2 . Thus, if a possibility leaves no question undecided, it is a possible world, according to our definition. Given the expressive limitations of natural languages, it is very likely that no canonical designator of the form ‘Pp’ constructable in a natural language

164

Chapter 4. Abstract Possible Worlds

designates a possible world. Perhaps, there could not have been any language which contains some, let alone all, such canonical designators. This means that there is, and, perhaps, could have been, no instance of the definition schema (Df-IPP) which states the definition of an individual concept a possible world falls under. However, this does not mean that, on the present account, there are no possible worlds, or only those for which a definition can be stated in some possible language. It is not as if, as Schiffer (2003: 71) claims, ‘we can make sense of [a proto-possibility] just by virtue of our ability to make sense of there being a language’ which has the resources to express a proposition which entails everything that is forced by it.82 Rather, we can make sense of such a proto-possibility by grasping the easy instances of (Df-IPP) and realising that many propositions may not be expressible in natural languages, nor, perhaps, in any possible language. In any case, I do not see that the present account forces me to say otherwise.83 Let me sum up. In this section I applied Schiffer’s account of pleonastic entities to proto-possibilities. Just as the property of being wise is whatever must be had by all and only the wise things, so the proto-possibility that Socrates is wise is whatever must force all and only those things entailed by 82

83

Schiffer speaks not about proto-possibilities but about properties. The contention is an inessential and rather unfortunate strand in his account, which is also exemplified by the oxymoronic title of an earlier article ‘Language-Created Language-Independent Entities’ (Schiffer 1996). It is unfortunate, because it suggests that a pleonastic account of properties is a form of what Armstrong (1978: ch. 2) calls Predicate Nominalism, which it isn’t. (Instead, it is a form of ‘Ostrich Realism’. See subsection 4.3.2 below). Roughly, the essential part of Schiffer’s view is that we acquire concepts of pleonastic entities by engaging in linguistic practices which incorporate their something-from-nothing transformations. But this does not mean that these concepts, or the entities falling under them, are in any good sense ‘language created’, nor is there any reason to think that only those of them exist whose transformations can be carried out in some language. Rather, we pick up on these concepts by participating in certain linguistic practices. This is compatible with the view that there are some individual concepts for pleonastic entities we could not have picked up on because there is some limit to the expressive richness of any language. In Humberstone (1981), Humberstone shows how a semantics for modal languages can be given that uses possibilities in the present sense—the miniature worlds of school probability, as Kripke may say—without making any assumptions about the existence of maximal refinements thereof. Even if my contention in the main text is wrong and there should be good reasons to think that no possibility counts as a possible world, the existence of maximally refined possibilities, aka possible worlds, may be inessential for other philosophical purposes as well.

4.2. Pleonastic Possible Worlds

165

the proposition that Socrates is wise. Like properties, proto-possibilities are abstract entities whose existence may be inferred from premisses that do not mention them via something-from-nothing transformations whose conclusion involves their characteristic operator. What having is for properties, forcing is for proto-possibilities. Since the existence and characteristic properties of proto-possibilities can be inferred via mastery of the proto-possibility concepts from propositions that can be known independently, it is no mystery how we can come to know about them. Just as we may validly infer that Socrates has the property of being wise from the proposition that Socrates is wise, so we may infer that the proto-possibility that die A shows a 5 and die B a 6 forces that the total number thrown is eleven from the proposition that it is necessary that if die A shows a 5 and die B a 6, the total number thrown is eleven. Of course, we are only interested in proto-possibilities in so far as they are also possibilities. But again, whether a proto-possibility is a possibility can be inferred from something that can be known independently. That the proto-possibility that die A shows a 5 and die B a 6 is a possibility may be inferred from the fact that it is possible that die A shows a 5 and die B a 6. Thus, pleonastic possibilities seem to be just the things we want as an epistemically accessible tool in semantics and metaphysics. Let me end this section by briefly indicating why the simplifying assumption that we may neglect iterations of modal operators is relevant to the current proposal and how the proposal may be varied so that the simplifying assumption can be dropped. Suppose that the characteristic S4 axiom is invalid, so that, e.g., S4☇ ◻¬p1 & ◊◊p1 . Intuitively, what we want in this case is that, although no possibility accessible84 from the actual possibility forces that p1 , there is some possibility that is accessible from a possibility accessible from the actual possibility which forces that p1 . However, given (Df-IPP) and the first conjunct of (S4☇ ), the only proto-possibility that forces that p1 is the proto-possibility that forces everything, the impossibility. Thus, when accessibility enters the picture, the current proposal yields too few possibilities. In fact, it only yields those possibilities accessible from the actual possibility. The idea of how to deal with this complication is simple: start with the actual possibility and then build outwards. In order to do so, we will have 84

Accessibility is defined as usual, i.e. Accessibility x has access to y ↔df. ∀p (y ⊩ p → x ⊩ ◊p) See e.g. Kripke (1963: 64).

166

Chapter 4. Abstract Possible Worlds

to exploit the modal information possibilities themselves contain. Where the original proto-possibility that p forced whatever is entailed by the proposition that p, the new proto-possibilities that p will force whatever is entailed by the proposition that p, according to some other possibility. Let me explain. Let’s start by giving the definition for the individual proto-possibility concept expressed by ‘the actual proto-possibility’ or ‘@’: Df. The Actual Proto-Possibility (Df-@) x = @ ↔df. ◻∀p (x ⊩ p ↔ Ap) The actual proto-possibility is whatever must force everything that is actually the case and nothing else. Given that the actual proto-possibility is a proto-possibility, @ is also a possibility, since it does not force everything. Moreover, given that ∀p (Ap ∨ A¬p), @ is even a possible world. Now, @ forces many things non-modal: that Socrates is a philosopher, that Socrates is not a carpenter, perhaps that Obama is president in 2014. But it also forces modal truths. For instance, since it is actually possible that Obama is president in 2014, @ forces that it is possible that Obama is president in 2014. Since it is actually necessary that Obama is or is not president in 2014, @ ⊩ ◻(Obama is or is not president in 2014), and so forth. Starting from the actual possibility, then, we can exploit @’s modal information to get our first round of proto-possibilities by emulating our initial strategy. Df. Schema: Individual @-Proto-Possibilities (Df-@IPP) x = the @-proto-possibility that p ↔df. ◻∀q (x ⊩ q ↔ @ ⊩ ◻(p → q)) The @-proto-possibility that p—P@ p, for short—must force whatever is entailed by the proposition that p, according to @. It is easy to see that for any of our original proto-possibilities we have exactly one @-proto-possibility that forces the very same things and vice versa. For, consider Pp, the protopossibility that p as originally defined. Pp ⊩ q just in case ◻(p → q). If ◻(p → q), A◻(p → q). So, by (Df-@), @ ⊩ ◻(p → q). Thus, by (Df-@IPP), P@ p forces that q. On the other hand, if Pp ⊩ / q, then ¬◻(p → q). So, ¬A◻(p → q), and, by (Df-@), @ ⊩ / ◻(p → q). Thus, by (Df-@IPP), P@ p ⊩ / q. The other direction is similar. Now that we have the @-proto-possibilities, nothing stops us from exploiting their modal information to get further proto-possibilities and so on. Let’s use ‘w’ as a variable for proto-possibilities. Then the general definition schema can be put as follows:

4.3. Objections and Clarifications

167

Df. Schema: Individual w-Proto-possibilities (Df-wIPP) x = the w-proto-possibility that p ↔df. ◻∀q (x ⊩ q ↔ w ⊩ ◻(p → q)) Clearly, (Df-@IPP) is just a special case of (Df-wIPP). I merely introduced it to make the idea of (Df-wIPP) more transparent. The outlines, then, of the modified proposal are to take (Df-@) as the basis and get all other protopossibilities via (Df-wIPP). Reconsider (S4☇ ). Because of its first conjunct we get only possible @proto-possibilities which force that ¬p1 . However, because of its second conjunct, some possible @-proto-possibilities will force that ◊p1 . Let’s call one of them p. Since p ⊩ ◊p1 , there will also be a possible p-proto-possibility which forces that p1 , and, thus, a proto-possibility which forces that p1 . The latter will just not be accessible from @. A lot more would have to be said about the modified proposal. However, my aim was only to indicate how the initial account could be modified in order to deal with iterations of modal operators. A detailed exposition will have to wait for another occasion. I will now turn to some possible objections to the initial account which are meant to clarify its details and its relation to other accounts. 4.3

Objections and Clarifications

In this chapter I presented an account of abstract possible worlds as pleonastic entities. It starts from the observation that possible worlds—and possibilities in general—seem to be subject to something-from-nothing transformations. In this respect they seem are on a par with other abstracta many philosophers are happy to believe in, like properties, sets, concepts, propositions, states of affairs and others. For all of these, the hypothesis that they are pleonastic entities explains (i) how we can come to know about them even though they are abstract, and, thus, causally inaccessible, and (ii) how it can be that facts about them must line up with facts ostensibly not about them. The explanation for both (i) and (ii) relies on the cornerstone of the pleonastic account of those entities—that the validity of their something-from-nothing transformations is a conceptual matter. For instance, it is a conceptual truth that the property of being wise is whatever must be had by all and only the wise things. Since this is so, we can come to know that Socrates has the property of being wise simply by drawing on our knowledge that Socrates is wise and on our mastery of the concept of the property of being wise. Further, since it is a conceptual truth that Socrates has the property of being wise just in case he is wise, it couldn’t have been that those two facts come apart.

168

Chapter 4. Abstract Possible Worlds

The last section presented an account of possible worlds according to which the same thing can be said about possibilities. How can it be that we know about possibilities even though they are abstract and, thus, causally inaccessible? How can it be that facts about possibilities must line up with facts about what is possible? Well, the validity of their something-fromnothing transformations is a conceptual matter. For instance, it is a conceptual truth that the proto-possibility that Socrates is a famous carpenter forces that Socrates is a carpenter just in case Socrates’ being a famous carpenter entails that Socrates is a carpenter. Furthermore, it is a conceptual truth that the proto-possibility that Socrates is a famous carpenter is a possibility just in case it is possible that Socrates is a famous carpenter. This is how we can come to know about proto-possibilities. In particular, this is how we can come to know about what is forced by this or that proto-possibility and which proto-possibilities are possibilities. For instance, we can come to know that there is a possibility that forces that Socrates is a carpenter by drawing on our knowledge that it is possible that Socrates is a famous carpenter and that it is necessary that if Socrates is a famous carpenter, he is a carpenter. Since it is a conceptual truth that there is a possibility that forces that Socrates is a carpenter just in case it is possible that Socrates is a carpenter, it couldn’t have been that those two facts come apart. In this section I will try to clarify aspects of the proposal and relate it to the guiding question of this and the previous chapter, namely whether modal truths supervene on truths about possible worlds. I will do so by discussing three potential objections. The first objection doubts the existence of protopossibilities. The second finds fault with their (lack of) explanatory potential. And the last reminds us that there are plenty of other competitor accounts of possible worlds as abstracta, accounts that are more informative and, perhaps, otherwise preferable to the current proposal. 4.3.1 Existence Objection: This is surely too easy. You can’t just define proto-possibilities into existence. I agree that it is necessary that if Socrates is a famous carpenter, he is a carpenter. But I disagree that there is any such thing as P(Socrates is a famous carpenter) of which we can know that it stands in some mysterious relation of forcing to the proposition that Socrates is a carpenter, simply by knowing your cooked up ‘definition’ of ‘its’ individual concept. Response: Nothing gets ‘defined into existence’. Last section’s proposal

4.3. Objections and Clarifications

169

is a proposal of how the proto-possibility concept cluster could look like in order for proto-possibilities to be pleonastic entities, subject to somethingfrom-nothing transformations, and available for the quantifier in (◊-bi)—as well as (◊-bi2 ) and (◻-bi2 )—to range over. There is no claim to the effect that proto-possibilities exist, somehow, thanks to my giving some definition or other. Proto-possibilities, if they exist, exist independently of that, although knowledge of the definitions will give us epistemic access to them. Still, the worry may be rather that none of the definitions for the individual proto-possibility concepts have been shown to be adequate in the usual way.85 In order to show that the definition proposal for the individual concept expressed by ‘P(Socrates is a famous carpenter)’ is adequate in the usual way, we would have to show that there is exactly one thing that forces all and only the things entailed by the proposition that Socrates is a famous carpenter. In general, since a definition cannot ensure that there is a thing that falls under it, while the definitions of individual concepts presuppose that there is exactly one such thing each, everyone proposing such definitions has to show that their presuppositions are satisfied.86 But this has not been done. Indeed, this cannot be done in the case of pleonastic entities: it is a central tenet of pleonasticism that our epistemic access to these entities has to proceed via mastery of their concepts, i.e. those very concepts whose legitimacy is at issue. Instead, a different condition of adequacy is proposed: the satisfaction of conservativeness. But why should we accept this replacement? Let me start with an irenic response. Although somewhat of a disappointment after 58 pages of discussion, it would not be a total disaster if there were no such things as (proto-)possibilities. For, the guiding question of this and the previous chapter is whether Possible Worlds Supervenience is correct, i.e. whether modal truths supervene on truths about possible worlds. If there are no proto-possibilities, there are no possibilities either. If there are no possibilities there are no possible worlds. Thus, if there are no protopossibilities, the answer to the question is a straightforward ‘no’. Since, as discussed in the next subsection, Possible Worlds Supervenience is false even if there are proto-possibilities, denying the existence of proto-possibilities is a quick route to the same conclusion. What would be a disaster is if there 85 86

Recall for the following the discussion on pages 135–136 above. This is one of Hartry Field’s main worries concerning Neo-Fregeanism adapted to the case at hand. See Field (1984: 661) and cp. Wright (1990). For an overview of the debate see e.g. MacBride (2003: §5).

170

Chapter 4. Abstract Possible Worlds

were no proto-possibilities but other abstracta some of which count as possible worlds and make true Possible Worlds Supervenience. This possibility will be discussed in subsection 4.3.3. Having said this, let me still try to take up the cudgels for the existence of proto-possibilities by sketching why conservativeness may be a plausible condition of adequacy in the case at hand. The basic idea is that Occam’s razor is plausible only for fundamental, non-derivative entities, but that pleonastic entities are ipso facto derivative. In the case of derivative entities only illegitimate intrusion could prevent their existence, a possibility excluded by the satisfaction of conservativeness. This kind of permissivism is not unheard of in other areas of ontology. In his Events and their Names, Jonathan Bennett maintains that events are ‘supervenient entities, meaning that all the truths about them are logically entailed by and explained or made true by truths that do not involve the event concept’ (Bennett 1988: 12).87 He goes on: “Stop shilly-shallying! Are there such things as events, or aren’t there?” Well, there was an earthquake in California last week and a birth in China this morning, so there are events. One might express the supervenience thesis in the form “Basically there are no events”, but that is a needlessly provocative way of saying something rather humdrum. And I have no grounds for saying that [adverb] there are no events, with any of the other usual adverbs. (Bennett 1988: 19)

I recommend the same attitude towards possibilities. It is possible that Socrates is a philosopher and it is possible that he is not. So, there is a possibility that forces that Socrates is a philosopher and there is a possibility that forces that Socrates is not a philosopher. So, there are possibilities. We may say “Basically there are no possibilities”, but that is a needlessly provocative way of saying that possibilities do not belong to the fundamental furniture of the world, the ‘basic layer of reality’, as it were. And, I think, there are no grounds for saying that [adverb] there are no possibilities, with any of the other usual adverbs. In particular, just like properties, they seem to pass the Conservative Extension test, and, given that the last section is on the right track, there are no epistemological or metaphysical mysteries about them. They are just like the other abstracta that are subject to something87

Cp. also Balaguer’s Plenitudinous Platonism concerning numbers (in Balaguer 1998: ch. 3) and Schaffer (2009) who recommends permissivism about existence questions, while focusing on what grounds what and how. In the case of possibilities, the answer proposed in this chapter is that modal truths ground truths about possibilities and that it is a kind of conceptual grounding.

4.3. Objections and Clarifications

171

from-nothing transformations. We can come to know about possibilities by drawing on our modal knowledge and our mastery of their concepts. And facts about them must covary with modal facts because it is a conceptual truth (whose source is the proto-possibility concept cluster) that they do. Thus, I see no reason to deny that there are possibilities.88 4.3.2 Explanation Objection: Your proto-possibilities may be alright if all we want is to have things for the quantifier to range over, so that, e.g., (◊-bi2 ), (◻-bi2 ), and ‘Being a philosopher is an essential property of Socrates just in case Socrates has it at every world at which he exists’ are literally true. But we want more. We want truths about possible worlds to explain modal truths. For instance, we want it to be the case that Socrates could have been a carpenter because there is a possible world (or possibility) at which he is a carpenter. Your account of proto-possibilities does not allow us to say so. If that’s not already obvious, consider a generalised version of A⇒E (of chapter 2 on page 47): A⇒E* If ⌜∎t1 ,…tn φ(t1 , . . . tn ) ↔ p⌝ is true, then (i) if φ(t1 , . . . tn ), φ(t1 , . . . tn ) because p; and (ii) if ¬φ(t1 , . . . tn ), ¬φ(t1 , . . . tn ) because ¬p. 88

There is a more specific existence worry concerning possibilities that force things about actual non-existents. Targeting a view of possible worlds as propositions or states of affairs, Fine argues that the proposition that Socrates is a philosopher, say, wouldn’t have existed if Socrates hadn’t existed, since the latter is a constituent of the former. Since there could have been things that don’t actually exist, and those things then would have been modally so-and-so, there simply aren’t enough propositions to go around. See Prior and Fine (1977: ch. 8, §4). For a reply see Plantinga (1979: 146ff.). It seems that a similar worry may be raised concerning proto-possibilities. These are difficult issues that I have brushed over in the last section, and that would have to be addressed in a fuller account. At this point let me just note that (i) they are not specific issues about proto-possibilities but concern a whole range of conceptions of possible worlds as abstracta, so that it is reasonable to assume that I can just take on board whatever reaction is adequate on behalf of these other accounts—perhaps, we may just deny Fine’s premiss that there could have been things that don’t actually exist, as argued in Williamson (2002); and that (ii) it is not clear that questions about the modal profile of propositions are pertinent at all, as I have already suggested in the course of the discussion of (BF-∀q) on pages 158–159 above.

172

Chapter 4. Abstract Possible Worlds

Whether conservativeness is all that’s needed or not, surely it is a conceptual truth flowing from the proto-possibility concept cluster that, say, P(Socrates is a famous philosopher) exists, if it is a truth at all. But then, corresponding to (◊-bi2 ), we have all instances of 58

∎Pp, proto-possibility, possibility ∃x (x is a possibility & x ⊩ p) ↔ ◊p.

The same goes, mutatis mutandis, for (◻-bi2 ). But given (58) and A⇒E*, we also have all instances of 59

If ∃x (x is a possibility & x ⊩ p), ∃x (x is a possibility & x ⊩ p) because ◊p; and

60

If ¬∃x (x is a possibility & x ⊩ p), ¬∃x (x is a possibility & x ⊩ p) because ¬◊p.

Since explanation is asymmetric, it cannot also be the case that truths about possibilities explain truths about what is possible. In particular, on your account, it cannot be the case that Socrates could have been a carpenter because there is a possibility according to which he is a carpenter. Thus, your protopossibilities cannot do the explanatory work we want them to do.89 Response: To speak with the Rolling Stones, you can’t always get what you want. The hypothesis that possibilities are pleonastic entities explains how we can come to know about them and why truths about them must line up with truths about what is possible. It requires that the latter truths are conceptually more basic than the former, and, thus, that the former can’t explain the latter. That is, it avoids epistemological and metaphysical mysteries concerning possibilities at the cost of denying them any fundamental explanatory role. In this respect, pleonastic possibilities are no different from pleonastic properties, sets, propositions, and so forth. Compare with the case of properties. The hypothesis that properties are pleonastic entities explains how we can come to know about them and why truths about them must line up with truths about how things are. It requires that truths about how things are are conceptually more basic than truths about what corresponding properties they have. For instance, it requires that the 89

We can reach the same conclusion with a principle slightly weaker than A⇒E*, the No Upstream Explanations principle, which says, roughly, that there are no explanations running against the conceptual flow: NUE If ⌜∎t1 ,…tn φ(t1 , . . . tn ) ↔ p⌝ is true, then (i) (ii)

¬(p because φ(t1 , . . . tn )); and ¬(¬p because ¬φ(t1 , . . . tn )).

4.3. Objections and Clarifications

173

truth expressed by ‘Socrates is a philosopher’ is conceptually more basic than the truth expressed by ‘Socrates has the property of being a philosopher’. Thus, it cannot also be true that the latter explains the former, i.e. that Socrates is a philosopher because Socrates has the property of being a philosopher. If your account of properties is meant to discharge fundamental explanatory ambitions, it had better not be pleonastic. Note, however, that this does not mean that pleonastic properties can’t figure in any kind of explanation. They can, but there will always be a more fundamental explanation to be given. Even if properties are pleonastic entities, it may well be true that 61

Ann and Ben are similar because they have many properties in common.90

It’s just that we don’t have to stop there: (61) does not state an ultimate explanation. Rather, (61)’s explanans may be explained in turn: 62

Ann and Ben have many properties in common, because they both have the property of being a philosopher, the property of living in Berlin, the property of being clever …; and

63

Ann and Ben both have the property of being a philosopher, the property of living in Berlin, the property of being clever … because they are both philosophers, they both live in Berlin, they are both clever ….

Given the transitivity of ‘because’ we will also have 64

Ann and Ben are similar because they are both philosophers, both live in Berlin, both are clever …;

and, thus, a ‘property-free’ explanation of (61)’s explanandum. But that does not mean that there is anything wrong with (61). Not all proper explanations are ultimate explanations. Note that properties’ being pleonastic does not prevent them from doing informative systematising work in philosophy, for instance in a semantics for natural languages. In fact, it may even be true that 65

90

‘Socrates is a philosopher’ is true because (‘Socrates is a philosopher’ means that Socrates is a philosopher and) Socrates has the property of being a philosopher. What can’t be true is that Ann and Ben are similar because they stand in the relation of similarity.

174

Chapter 4. Abstract Possible Worlds

It’s just that (the second half of) the explanans of (65) itself allows of an explanation, i.e. 66 Socrates has the property of being a philosopher because Socrates is a philosopher. Chaining them together, we get a ‘property-free’ explanation of (65)’s explanandum, namely 67 ‘Socrates is a philosopher’ is true because (‘Socrates is a philosopher’ means that Socrates is a philosopher and) Socrates is a philosopher. But that does not mean that there is anything wrong with (65). How could a proponent of pleonastic properties respond to the demand that grander explanatory ambitions be satisfied by properties? Well, he could highlight the epistemological and metaphysical benefits of a conception of properties as pleonastic. He could also ask back for a good reason to expect any such kind of systematic explanation. Clearly, there are good answers to the question 68 Why is Ann clever? We can perhaps cite her upbringing, her genetic make-up and other factors. But why think that there are correct answers to all91 questions of the form 69 Why is a F? systematically correlated to these questions? In particular, why think that the corresponding answer of the form 70 Because a has the property of being F; is a particularly good candidate for being a correct answer? The question, of course, is rhetoric. I see no good reason for expecting there to be such systematic explanations of simple predications. I thus recommend being what Armstrong (1978: ch. 2) may call an Ostrich Realist, although being an Ostrich Realist does not demand burying your head in the sand as far as properties are concerned. Armstrong characterises a close relative of the Ostrich Realist, the Ostrich Nominalist, in his well-known discussion of the so-called One Over Many Problem thus: I have in mind those philosophers who refuse to countenance universals but who at the same time see no need for any reductive analyses of [sentences of the form ‘a has the property of being F’]. There are no universals but the proposition that a is F is perfectly all right as it is. (Armstrong 1978: 16) 91

All questions of that form such that the corresponding sentence ‘a is F’ is true, that is.

4.3. Objections and Clarifications

175

The Ostrich Realist on the other hand (i) countenances properties, (ii) gives an account of property ascriptions, but (iii) also thinks that ‘the proposition that a is F is perfectly all right as it is’, at least if being perfectly all right means that there is no systematic explanation of why those that are true are true. As Lewis has emphasised, (ii) is not the opposite of (iii). That Armstrong seems to think so may be excused by the ease of something-fromnothing transformations (or, rather, in this case, the corresponding ‘nothingfrom-something transformations’). It is a mistake nonetheless: there being a systematic account of property ascriptions—most pertinently, sentences of the form ‘a has the property of being F’—does not entail that there is a systematic account of simple predications—sentences of the form ‘a is F’. Lewis makes the point succinctly: [When Armstrong’s] analysandum switched, from Moorean facts of apparent sameness of type to predication generally, then I say that the question ceased to be answerable at all. The transformed problem of One over Many deserves our neglect. The ostrich that will not look at it is a wise bird indeed. (Lewis 1983b: 352)

Return to possibilities. Note that the claim that possibilities are pleonastic entities, and, thus, don’t figure in any fundamental explanations, does not mean that they don’t figure in any explanations at all. It may well be true that 71

Being human is an essential property of Socrates because Socrates has the property of being human at every possibility at which he exists.

It’s just that we don’t have to stop here. If (71) is true, (71)’s explanans allows of a further explanation: 72

Socrates has the property of being human at every possibility at which he exists because it is necessary that Socrates has the property of being human, if he exists.

Chaining them together, we get a ‘possibility-free’ explanation of (71)’s explanandum, namely 73

Being human is an essential property of Socrates because it is necessary that Socrates has the property of being human, if he exists.92

But that does not mean that anything is wrong with (71). 92

In a further step, we can, of course, give a possibility- and property-free explanation.

176

Chapter 4. Abstract Possible Worlds

Also, possibilities’ being pleonastic does not prevent them from doing informative systematising work in philosophy, for instance in an intended model of an extensional semantics for the modal fragment of a natural language. In fact, it may even be true that 74 ‘It is possible that Socrates is a carpenter’ is true because (‘It is possible that Socrates is a carpenter’ means that it is possible that Socrates is a carpenter and) there is a possibility at which Socrates is a carpenter. It’s just that (the second half of) the explanans of (74) itself allows of an explanation, i.e. 75 There is a possibility at which Socrates is a carpenter because it is possible that Socrates is a carpenter. Chaining them together we get a ‘possibility-free’ explanation of (74)’s explanandum. But that does not mean that there is anything wrong with (74). Why not go for grander explanatory ambitions for possibilities to satisfy? First, because a pleonastic conception of them avoids epistemological and metaphysical mysteries about them. Second, because I can’t see a good reason to expect there to be any such kind of systematic explanation. Of course, there are good answers to the questions 76 Why is it impossible for die B to show a 6 after the throw? 77

Why is it still possible for Langer to win?

One may cite the fact that die B is a trick die, none of whose sides displays a 6, and that Langer is only one shot behind with one hole to go, having excelled at the last two holes while his opponent faltered. But why think that there are correct answers to all questions of the form 78 Why is it possible that p? and 79

Why is it impossible that q?

systematically correlated to these questions? In particular, why think that the corresponding answers of the form 80 Because there is a possibility at which p; and 81

Because there is no possibility at which q;

are particularly good candidates for being correct answers? Of course, in the case of modal truths there may be a greater pull than in the case of truths expressed by simple predicative sentences to look for some explanation of them. Also, the epistemological benefit of a pleonastic account of possibilities may not seem all that great. How does it help to be

4.3. Objections and Clarifications

177

told that we can come to know about possibilities by drawing on our modal knowledge and some conceptual resources, when it is anything but clear how we get our modal knowledge in the first place? But these are reasons to look for some supervenience basis for modal truths, not reasons to put a load on possibilities they are not able to bear. The next chapter will present a more promising alternative which complements the present conception of possibilities. 4.3.3 Competitors Objection: There are many other accounts of possible worlds as abstracta out there. Many of these tell us something informative about what possible worlds—and, perhaps, possibilities in general—are. According to them, possible worlds are (i) certain states of affairs (e.g. Plantinga 1974, 1976, 1987) or (ii) certain (sets of) propositions (e.g. Adams 1974, 1981), (iii) certain properties (e.g. Stalnaker 1984: ch. 3), or perhaps (constructions out of) other abstract objects we are already familiar with. Arguably, we need states of affairs, sets, propositions and properties anyway. So, an account which says that possible worlds are these kinds of things will be part of a more parsimonious overall theory and, thus, is surely to be preferred over one that employs purpose-built proto-possibilities. Response: Let me start by noting that there is a certain embarras de richesses. As the objector says, possible worlds are states of affairs or (sets of) propositions or properties or …. But which? If there is nothing that makes one of these more suitable to be possible worlds than any of the others, being purpose-built may actually be an advantage. For instance, proto-possibilities are as finely individuated as they have to be and as coarsely individuated as they can be in order to register modal differences (recall (IC1 )). On the other hand, Plantinga, for instance, is happy to say that there are many different logically equivalent states of affairs, and, thus, for any one possibility there is a plethora of states of affairs lining up to do its job. For instance, according to Plantinga, there are many ‘actual worlds’.93 On the face of it, proto-possibilities are a better fit for the job. Since they are purpose-built to fill it, they are at least as good a fit as any of the other candidates. Second, pleonastic proto-possibilities come with little or no epistemological or metaphysical costs. Consequently, it is not obvious why parsimony considerations are pertinent at all. Thus Schaffer: 93

See e.g. Plantinga (1985b: 89).

178

Chapter 4. Abstract Possible Worlds [O]ne might object that permissivism [about existence questions] violates some crucial methodological, epistemological, or metaphysical dictum. […] I answer that there need be no conflict with any reasonable dictum. Occam’s Razor should only be understood to concern substances: do not multiply basic entities without necessity. There is no problem with the multiplication of derivative entities—they are an “ontological free lunch”. (Schaffer 2009: 361)

If proto-possibilities are pleonastic, they are ‘derivative entities’—how they are depends on and is completely determined by how things are modally. Moreover, the hypothesis that they are pleonastic entities explains how this is possible. Thus, I see no pull towards parsimony. Let me end on a concessive note, though. Perhaps there is some reason or other to prefer a conception of possible worlds as states of affairs or as sets of propositions or as properties over a conception of possibilities as pleonastic proto-possibilities. We might still ask in turn what states of affairs, propositions and properties are. An attractive answer is that they are themselves pleonastic entities. In the case of properties I have already indicated the attractions of a pleonastic account of them. But, similarly, states of affairs and propositions seem to be subject to something-from-nothing transformations and their being pleonastic would explain how we can come to know about them so easily and why facts about them must line up with more basic, epistemically accessible, facts.94 I conclude by giving a rough sketch for an argument to the effect that Possible Worlds Supervenience will still turn out to be either false or uninteresting if possible worlds are these kinds of things. Suppose possible worlds are pleonastic abstract entities of some kind K with a certain property P. Either P is to be specified, ultimately, in modal terms or not. Assume the former.95 Then the right-hand sides of instances of 94

Plantinga, for instance, only introduces states of affairs by example and remarks that it is ‘obvious, I think, that there are such things as states of affairs’ (Plantinga 1976: 257). When he speaks about states of affairs, he seems to rely on the validity of paradigm cases of something-from-nothing transformations, namely, in the simplest case, a is F. a is not F. (O) (¬O) The state of affairs of a’s The state of affairs of a’s being being F obtains. F does not obtain.

95

Lewis (and Divers) argue extensively that it is very likely that in order to assure adequacy (i.e. the truth of (◊-bi)) an account of possible worlds as abstracta will have to specify P modally. See Lewis (1986a: ch. 3) and Divers (2002: ch. 11). A paradigm example is Plantinga’s account, according to which possible worlds are maximally inclusive states of affairs that possibly obtain. See e.g. Plantinga (1976: 258). In

4.3. Objections and Clarifications

179

(◊-bi), i.e. ◊-bi ∀p (it is possible that p ↔ there is a possible world at which p); will have explanations in terms of how Ks are modally. Since Ks are pleonastic, there will be an explanation of these facts as well. How exactly they look will depend on our account of Ks. But whatever they are, they will be in terms of how non-Ks are modally. At least in some cases, the explanans of such an explanation will simply be the left-hand-side of (◊-bi). By transitivity of explanation, the right-hand side of some instances of (◊-bi) will be explained by the corresponding left-hand side. By asymmetry, the former cannot also explain the latter. Thus, Possible Worlds Supervenience will be false.96 Suppose now that P is not to be specified modally. Suppose also that the left-hand-sides of instances of (◊-bi) are explained by the corresponding right-hand-sides. Since possible worlds are, we assumed, Ks and Ks are pleonastic, these explanations will not be fundamental. Instead, their explanans will in turn be explained by how non-Ks are, though not by how non-Ks are modally. By transitivity, the left-hand-sides of (◊-bi) will be explained by non-modal truths about non-Ks. But if this is so, the really interesting thing is not the connection between modal truths and truths about Ks, but rather the connection between modal truths and non-modal truths about non-Ks, and, thus, non-modal truths that aren’t about possible worlds. Consequently, Possible Worlds Supervenience, though perhaps true, will merely reflect an interesting connection between modal and non-modal truths that has nothing to do with possible worlds. Either way, with Possible Worlds Supervenience we haven’t reached the ground floor of the ontological hierarchy. Either, as

96

later writings, Plantinga adds the further condition of ‘non-transience’ (see Plantinga (1985a: 327), prompted by Pollock (1985: 122)). All of maximal inclusiveness, possible obtaining and non-transience are modal notions. Plantinga is happy to accept this: Such modal notions as possibility and necessity, then, are not to be defined or explained in terms of possible worlds; the definition or explanation must go the other way around. (Of course it does not follow that the idea of possible worlds cannot help us deepen our grasp of these modal notions; it has obviously been a splendidly fertile source of modal insights; it has enabled us to explore these notions in a much more penetrating way.) Model [sic] discourse, therefore, cannot be reduced to non-modal discourse; it is none the worse for that. (Plantinga 1985b: 89) See also Plantinga (1976: 258).

180

Chapter 4. Abstract Possible Worlds

I believe, because truths about possibilities supervene on modal truths—we have gone up in the hierarchy instead of going down. Or because Possible Worlds Supervenience merely reflects a more fundamental supervenience relation between modal and non-modal truths between which truths about possible worlds are sandwiched—we haven’t reached the base level. Let me sum up. In this and the previous chapter we explored the thesis that modal truths supervene on truths about possible worlds, Possible Worlds Supervenience. The result is that if possible worlds are Lewis worlds, Possible Worlds Supervenience fails, since some modal truths involving what is actually the case do not line up with truths about Lewis worlds on any plausible account of how they should line up with them. On the other hand, if possible worlds are abstracta then a plausible view of abstracta in general makes possible worlds out to be explanatorily non-fundamental. In section 4.2 I gave a non-reductive account of abstract possible worlds for the case in which we may disregard iterations of modal operators and outlined a modification of the account which can drop this simplifying assumption. On that account, it is fairly obvious that Possible Worlds Supervenience fails, since the order of explanation is the other way around: modal truths do not supervene on truths about possible worlds, but truths about possible worlds supervene on modal truths. I also argued, in section 4.3, that any plausible reduction of possible worlds to better known abstracta will either have the same consequence or at least imply that Possible Worlds Supervenience is not the last word on the issue, since there is a more fundamental explanation to be given. I thus conclude that truths about possible worlds are not the supervenience base for modal truths we were looking for. This takes us back to where we started, but at least excludes one initially attractive destination: modal truths do not supervene on possible worlds truths. Instead, arguably, possible worlds truths are located at a higher level of the ontological hierarchy than modal truths. The next chapter will take a fresh look at modal supervenience.

Chapter 5

Possibility and Probability In the last two chapters we explored the prospects of the thesis that modal truths supervene on truths about possible worlds. Since possible worlds play a prominent role in philosophical discussions of modality, this was a natural place to start. Perhaps, the idea was, the possibility and necessity biconditionals ◊-bi ∀p (it is possible that p ↔ there is a possible world at which p); and ◻-bi ∀p (it is necessary that p ↔ at all possible worlds, p); are not only true but exhibit an explanatory relationship between (the true instances of) their left-hand sides (the potential explananda) and the corresponding right-hand sides (the potential explanantia). However, this hope turned out to be premature. In the penultimate chapter I argued that if possible worlds are Lewis worlds—roughly, maximal sums of spatio-temporally related concreta—the supervenience thesis fails, since truths about them do not even line up with the allegedly corresponding modal truths (on any reasonable account of the correspondence), something brought out by considering certain modal truths involving what is actually the case. In the last chapter I argued that if possible worlds are abstracta the most plausible account of them will have the consequence that truths about possible worlds supervene on modal truths and not vice versa. Perhaps there is a third way. Perhaps possible worlds are not abstract without being Lewisian. Or they are some sort of magical abstracta—abstracta truths about which do not supervene on more pedestrian truths. Perhaps. This third way has a competitor, though, the account of possible worlds I presented in the last chapter. According to it, possible worlds are less powerful than we may want them to be—though they are still there for the nominal quantifiers to range over and available for natural language semantics, they can do no fundamental explanatory work. But they are also less problematic, epistemically and metaphysically, than we may have expected them to be. In my view, this strikes the right bal-

182

Chapter 5. Possibility and Probability

ance. All the more so, if we can find a more plausible supervenience base for modal truths than truths about possible worlds. The resulting overall picture of modality is then that possible worlds truths supervene on modal truths which in turn supervene on still other truths. Finding these other truths is the task of this chapter. But where to start? I suggest taking a closer look at those idioms that are actually used when we voice our judgments about what is possible in everyday talk. When philosophers introduce their readers to the topic of possibility, they often help themselves to natural language formulations involving ‘might’ and ‘could’. Nixon could have become a Democrat instead of a Republican; it might not have rained today, even though it did; Bernhard Langer could have won the BMW Open in 2007, thereby becoming the first German golfer to have won all German PGA tournaments—unfortunately, he only came second. However, in philosophical jargon ‘might’s and ‘could’s are usually supplanted pretty quickly by the philosophers’ favourite, the sentence operator ‘it is possible that’: from ‘Nixon could have become a Democrat’ we switch to ‘It is possible that Nixon was a Democrat’ and get on with it. Unfortunately, the latter way of talking is relatively rare in non-philosophical contexts.1 Thus, it stands to reason that there are insights to be gained from focusing instead on the closer to home ‘might’s and ‘could’s. In this chapter I will focus on the modal auxilliary ‘might’ for definiteness. On the face of it, ‘might’ has both epistemic and non-epistemic readings, and in both readings ‘might’ and expressions of probability like ‘probably’ or ‘it is (un-)likely that’ have an intimate relationship. In section 5.1 I will motivate this claim. This leads up to the conjecture that, roughly, a sentence of the form ‘a might φ’, in a given reading, is true just in case there is a nonzero probability (in the same reading) that a φ’s. Section 5.2 gives a brief overview of the different kinds of probability that may be relevant. Section 5.3 deals with the temporal structure of ‘might’ and ‘might have’ sentences. It seems that utterances of the very same ‘might’ sentence can vary in truth value when they take place at different times and that what is said with a ‘might have’ sentence may be true while the corresponding ‘might’ 1

Worse, ‘it is possible that’ may only allow an epistemic reading. In linguistics, it is common to see ‘it is possible that’ classified as an epistemic modal, while ‘could’ has a predominant ‘root’ (≈ non-epistemic) modality reading. Cf. Coates (1983: chs. 5, 6). Note, however, that it is not clear that the linguists’ classification as epistemic coincides with the philosophers’ classification. For instance, Condoravdi (2002: 59, fn.) suggests that ‘the class of so-called epistemic modals includes modals expressing metaphysical as well as epistemic modality’. More on this later.

5.1. Initial Motivation

183

sentence is false. But what exactly is the relation of such sentences to time? The section will suggest an answer and relate it to the availability of epistemic and non-epistemic readings of such sentences. Section 5.4 then deals with a contention of Keith DeRose’s which is potentially fatal to my project. DeRose argues that, despite initial appearances, ‘might’ invariably expresses epistemic possibility. Since my discussion of ‘might’ sentences is meant to shed light on non-epistemic possibility, it is doomed to failure if DeRose is right. Fortunately, his argument is far from conclusive. Or so I will argue in that section. We may thus take the lessons we learned from our investigations of non-epistemic ‘might’s seriously. They point to an answer to the question of what these other truths are on which (non-epistemic) modal truths supervene: modal truths supervene on truths about probability. Section 5.5 will defend this view. In the last section of the chapter, section 5.6, I will round up the discussion by considering how what has emerged thus far relates to what philosophers call metaphysical possibility. Incidentally, it is common to attribute the epistemic and non-epistemic readings of ‘might’ sentences to an ambiguity in ‘might’ (and, correspondingly, in ‘possible’, ‘possibility’ and so forth). Kratzer (1977) argues that this is a mistake, and that, rather, the difference is due, roughly, to an extra argument in ‘might’ that is typically filled in by the context but may be explicitly specified with the help of ‘in view of’ phrases.2 My discussion does not commit itself to either of these options, as long as the same option is taken for explicit probability ascriptions. Consequently, I will speak of different readings of the relevant sentences, without attributing these different readings to an ambiguity in the modal auxiliary. 5.1

Initial Motivation

Natural language ‘might’ sentences express possibility. But there are at least two substantially different kinds of possibility for them to express: epistemic and objective possibility. Roughly, epistemic possibility is possibility relative to the epistemic state of epistemic subjects. Objective possibility, on the other hand, has nothing essentially to do with what is known or believed (or would be known or believed under certain conditions) by anyone, but is 2

An epistemic reading may come about when possibility ‘in view of what NN knows at t’ is at issue, a non-epistemic reading when possibility ‘in view of how the world is at t’ is, for instance.

184

Chapter 5. Possibility and Probability

rather possibility relative to how the world is.3 This is a rather rough characterisation, but for now it will have to do. On the face of it, natural language ‘might’ sentences can express both kinds of possibility, epistemic and objective. When someone utters the sentence 1 Goldbach’s Conjecture might be true, but it might also be false; she isn’t thereby denying the orthodox view in the philosophy of mathematics that if something is a truth of pure mathematics its negation is objectively impossible. Rather, she seems to correctly give voice to her ignorance or uncertainty concerning both Goldbach’s Conjecture and its negation. On the other hand, when someone utters the sentence 2 Langer might have won the tournament, although he didn’t; she doesn’t thereby express her current ignorance or uncertainty about the outcome of the tournament and immediately retracts. Rather, she seems to correctly point to the fact that there was an objective possibility that Langer would win, in the face of her knowledge that that possibility wasn’t realised. Indeed, the standard view amongst philosophers seems to be that many modal auxiliaries including ‘might’ and other modal expressions are ambiguous. As a witness I cite George Bealer who states succinctly: The modal expressions ‘could’, ‘can’, ‘might’, ‘possible’ are used in diverse ways that fall into two broad classes: (i) epistemic and (ii) nonepistemic. (An analogous distinction holds for ‘must’ and ‘necessary’.) (Bealer 2002: 77)4

Ideally after noting the different readings, metaphysicians then standardly move on to presuppose an objective reading of ‘might’, while the discussion of epistemic modals in the philosophy of language tends to ignore the possibility of an objective reading and presupposes the epistemic reading instead.5 3 4

5

Since epistemic facts are facts about how the world is, some objective possibilities turn on what is known or believed, but only because of their subject matter. That Bealer has ambiguity in mind when he talks of different uses of the modal idioms is apparent from the fact that he contrasts this with the univocity of ‘contingent’ (Bealer 2002: 78). Another example is Forbes (1985: 1) who writes ‘There is an ambiguity [in the phrase “it is possible that”] of which one should beware’. For the metaphysics side see, e.g., Kripke (1980) and Mackie (2006). For the epistemic side see, e.g., DeRose (1991) and Egan et al. (2005). If there are these two readings of ‘might’ sentences, both sides would seem to be well advised to pay more attention to the other reading. For instance, DeRose (1991) would have profited from at least considering the option that some of the cases that motivate his rather complex assignment of truth conditions to epistemic ‘might’s may instead be expressions of non-epistemic possibility. For a treatment which considers both kinds of possibility

5.1. Initial Motivation

185

Both the view that there are genuinely different readings and the view that, if there are these two readings, it is a genuine case of ambiguity may be disputed. However, the former is a topic for a later section. The latter I don’t have to decide here. What is less often noted in the philosophical discussion of possibility, is that there seems to be a close connection between possibility and probability.6 Possibility and probability talk just go very smoothly together. Consider 3

They might still be at home. In fact, I saw them there only ten minutes ago, so they are probably still at home;

4

They might already be at the station. But since I saw them at home only ten minutes ago, they probably aren’t at the station yet;

5

Langer might have won the tournament. In fact, until he hooked his last tee shot, it was quite likely that he would win;

6

Even after his horrible tee shot Langer might still have won the tournament. But it was very unlikely then that he would go on to win.

The ‘might’ in the sentence pair (3)/(4) is naturally taken to express epistemic possibility, that in the sentence pair (5)/(6) objective possibility. But in both cases, the explicit probability assertions (‘probably (not) …’/‘it is (un-)likely that …’) seem to enforce what has been said before (in (3) and (5)) or contrast with it (in (4) and (6)). Compare them in this respect with 7

There are some tickets left. In fact, quite many; and

8

There are some tickets left, but only a few.

In the case of (7) and (8), there is an underlying numerical scale, somewhere on which the number of remaining tickets is located. The first part of the sentences indicates that it is not at the ‘zero’ position (and, perhaps, not at the ‘one’ position either; I will ignore this in what follows). The second part of (7) reinforces this by indicating that it is quite far away from this endpoint. That is why it is appropriate to use ‘in fact’. The second part of (8), on the other hand, indicates that, though the number of tickets isn’t zero, it is quite close to zero. This is why it is appropriate to use ‘but’ in this situation. Assuming an initial supply of one hundred tickets, the underlying scale and the regions in which ‘some’, ‘many’ and ‘a few’ may be used can 6

and their interrelations see Edgington (2004). But see e.g. Edgington (2004: §III), Strawson (1997) and Schulz (2010). Coates (1983: 28) includes ‘probably’ in her list of modal expressions.

186

Chapter 5. Possibility and Probability

be pictured as in figure 4. a few some

. 0

many

50

100

Figure 4: An underlying numerical scale

The propriety of ‘in fact’ in (7) and of ‘but’ in (8) corresponds to the propriety of these locutions in the corresponding sentences that explicitly refer to the underlying scale, i.e. 7* The number of remaining tickets is greater than zero. In fact, it is much greater than that; and 8*

The number of remaining tickets is greater than zero. But it is not much greater than that;

respectively. I suggest that something analogous is going on in the case of the sentence pairs (3)/(4) and (5)/(6). There is an underlying probability scale in both cases. The ‘might’ sentences indicate that the relevant probability is not zero. The second parts of sentences (3) and (5) reinforce this by indicating that it is quite far away from zero. This is why it is appropriate to use ‘in fact’. The second parts of (4) and (6), on the other hand, indicate that, though the probability is not zero, it is quite close to zero. This is why it is appropriate to contrast with ‘but’. Assuming that probability comes on a scale from 0 to 1, the underlying scale and the regions in which ‘might’, ‘probably’ and ‘it is unlikely that’ may be used can be pictured as in figure 5. it is unlikely that . 0

might

probably

1/2

1

Figure 5: An underlying probability scale

The propriety of ‘in fact’ in (3) and (5) and of ‘but’ in (4) and (6) corresponds to the propriety of these locutions in the corresponding sentences that explicitly refer to the underlying scale, i.e. 3* The probability that they are at home is greater than zero. In fact, it is much greater than that;

5.1. Initial Motivation

187

5*

The probability that Langer would win was greater than zero. In fact, it was much greater than that;

4*

The probability that they are at the station is greater than zero. But it is not much greater than that; and

6*

The probability that he would win was greater than zero even after his horrible tee shot. But it was not much greater than that;

respectively. In view of the previous considerations, I conjecture that ‘might’ sentences are true just in case the embedded sentence has non-zero probability, i.e. MIGHT Might(p) ↔ Prob(p) > 0.7 (MIGHT) is meant as a statement of conceptual links between possibility and probability without a claim to conceptual priority of either side. The significance of tense within ‘might’ sentences is the topic of section 5.3. It should be noted that, given that there are at least two readings of ‘might’ sentences, an epistemic and an objective reading, (MIGHT) is open to misunderstandings. Which kind of possibility is supposed to correspond to nonzero probability? Or do they both? But in which sense are they then different kinds of possibility? The answer is, of course, that corresponding to epistemic and objective possibility, there is also epistemic and objective probability.8 Epistemic possibility corresponds to non-zero epistemic probability while objective possibility corresponds to non-zero objective probability. Since ‘might’ sentences may express both kinds of possibility, (MIGHT) should be understood to cover both of these claims.9 The conjunction of these claims is motivated by the observation that both sentence pairs (3)/(4) 7

8

9

Or, perhaps, ignoring reference to times, ⌜Might(p)⌝ is true in context C just in case Prob(p) > δ , where δ is a positive number close to zero somehow determined by C, as Strawson (1997: 180f.) seems to suggest. I will ignore this possibility. But see Schnieder et al. (2010) for reasons against this less elegant variant of the proposal in the main text. See e.g. Carnap (1945); Lewis (1980); Edgington (2004) for the view that there are two kinds of probability, one of which may be labelled epistemic, the other objective (though nothing hinges on the labels). More on this in the next section. Schnieder (2009) argues, in response to a puzzle raised by Yalcin (2007) concerning the impossibility of certain embeddings, that sentences containing epistemically used modals don’t contribute to the truth-conditions of sentences in which they occur, but are rather speech-act modifiers. According to Schnieder, the sentences so used don’t have truth-conditions, but only sincerity- or acceptability-conditions (see also Schnieder et al. 2010). I’m not at all certain that Schnieder’s considerations are per-

188

Chapter 5. Possibility and Probability

and (5)/(6) suggest a relation between possibility and probability regardless of whether ‘might’, ‘probably’ and ‘it is likely that’ uniformly receive an epistemic or an objective reading. If there is a genuine ambiguity in ‘might’, this ambiguity carries over to (or is carried over from) ‘probably’ and other explicit expressions of probability. If the different readings result from an implicit argument place, this argument place is filled in the same way for ‘might’s and expressions of probability in the corresponding readings. 5.2 Different Kinds of Probability There are various accounts of probability on the market, which are thought by some to be competitors. However, Carnap (1945) argues that this is a mistake. According to him, there are two concepts of probability philosophical explications have aimed at, roughly an epistemic and an objective one. Lewis agrees: Carnap (1945) did well to distinguish two concepts of probability, insisting that both were legitimate and useful and that neither was at fault because it was not the other. (Lewis 1980: 83)

But even within the epistemic and the objective camp, there are considerable differences, which makes one wonder whether there are not rather three or four legitimate and useful notions, some of which epistemic and some of which objective.10 It is not my aim here to decide this question. Rather, I’d like to give a map of the prima facie distinct kinds of probability that are candidates for providing the underlying scale for distinct kinds of possibil-

10

tinent to all uses of epistemic modals, in particular, to all epistemic uses of ‘might’. Since I’m not primarily concerned with epistemic possibility here, however, I just mention this view. Schnieder’s account is compatible with the claim that both sides of (MIGHT) have the same acceptability-conditions if ‘Might’ and ‘Prob’ receive an epistemic reading. If pressed, I would be happy to settle for that. Nevertheless, in the main text I will go on to pretend that epistemic ‘might’ sentences, just like objective ‘might’ sentences, have truth-conditions. The Lewis quotation continues: I do not think Carnap chose quite the right two concepts, however. In place of his ‘degrees of confirmation’ I would put credence or degree of belief ; in place of his ‘relative frequency in the long run’ I would put chance or propensity, understood as making sense in the single case. (Lewis 1980: 83f.). However, he does not explain why a replacement is called for. Perhaps, we should just add Lewis’s concepts to Carnap’s list of legitimate probability concepts.

5.2. Different Kinds of Probability

189

ity. I’ll start with the epistemic end of the spectrum, degrees of belief (or credences for short) and evidential probability. 5.2.1 Epistemic Probabilities Beliefs come in degrees. I believe that Robert paid for all of his drinks, but I’m not certain that he did. My degree of belief in the proposition that Robert paid for all of his drinks is reasonably high but not maximal. I wouldn’t be flummoxed to learn that he forgot to pay for one of them. Assuming that degrees of belief come on a scale from 0 to 1, that high credences are credences greater than .6, and that certainty marks the upper endpoint of the scale, we may express this state of affairs as follows (letting CrS, t abbreviate ‘the credence of S at t that …’ and p ‘Robert paid for his drinks’): 9 .6 < CrAS, now (p) < 1. If I later learn that the whole bill has been covered, my credence may get even closer to one. If the patron starts chasing Robert when we leave the bar, my credence may approach zero. Your credence may have been close to zero—or even zero—all along, if you were clear-headed, kept count of Robert’s drinks and saw him pay for one too few. Thus, CrS, t (p) may not be CrS′ , t′ (p) if S ≠ S′ or t ≠ t′ —credences in the same proposition may be different for different epistemic subjects or different times. This is anything but surprising, since, of course, outright belief in the same proposition may vary with the time and the epistemic subject as well. In general, different subjects, or the same subject at different times, may be in different total epistemic states, which determine whether the subject believes a certain proposition at that time or has this or that credence in it. Epistemic subjects do not assign credences to propositions independently of their logical relationships. If I am fairly confident that Robert paid for all of his drinks I shouldn’t also be fairly confident that he forgot to pay for one of them. Even after a long night my degrees of belief conform to some such rule. In the ideal case, they would obey the axioms of the Probability Calculus (‘P’ is a place-holder for a term signifying a probability function; a probability function obeys the axioms if the result of replacing ‘P’ with a term that sifnifies it has only true instances):11 11

The axioms in the main text are not quite the axioms of the (standard) Probability Calculus. The former make non-trivial assumptions about the interrelations of the truth-functional connectives with the elements of Ω in the real axioms: for a non-empty set Ω, the so-called set of possible outcomes or sample space, and A1 , A2 ∈ ℘(Ω) (A1 and A2 are so-called events):

190

Chapter 5. Possibility and Probability

Nonnegativity

P(p) ≥ 0

Finite Additivity

If p1 and p2 are inconsistent, P(p1 ∨ p2 ) = P(p1 ) + P(p2 )

Normalisation

If p is a logical truth, P(p) = 1

Mine don’t. For instance, I am not certain of many complex logical truths (nor, more annoyingly, of some not so complex ones). Nor do I believe all other logically equivalent propositions to the same degree. If my credences conformed to the three axioms I would. But sometimes wishful thinking or just plain ignorance of logical facts get in the way. The credences of perfectly rational (and, one may add, logically omniscient) epistemic subjects would conform to the axioms of the Probability Calculus (and, perhaps, satisfy additional constraints). Let’s call those rational credences. Perhaps there is even a way of constructing rational credences from the (imperfectly rational) actual degrees of belief of epistemic subjects like myself, thus idealising less than ideal epistemic states. These may then have important roles to play in philosophy.12 On the face of it, credences are a good candidate for providing the probability scale for at least some uses of epistemic modals. Let ‘p’ express a fairly complex logical truth. When I say 10

Perhaps, p, and perhaps, not p;

I seem to express my present uncertainty about whether p,13 i.e. 10*

I’m not certain that it’s not the case that p and I’m not certain that (its not the case that it’s not the case that) p.

If I say something true with (10), moreover, I express my actual credences, not some idealisation thereof. Some epistemic uses of ‘might’ seem to be mere stylistic variants of such ‘perhaps’ sentences, as witness our earlier example Nonnegativity P(A1 ) ≥ 0, Finite Additivity If A1 ∩ A2 = ∅, P(A1 ∪ A2 ) = P(A1 ) + P(A2 ) Normalisation P(Ω) = 1 12 13

Edgington (2004), for instance, suggests that the a priori truths are those truths that receive rational credence 1 against any epistemic background. But see Yalcin (2007).

5.2. Different Kinds of Probability

1

191

Goldbach’s Conjecture might be true, but it might also be false.14

Nevertheless, in philosophical and linguistic discussions of epistemic modals, possibility (for S at t) corresponding to non-zero degree of belief (of S at t) does not take centre stage.15 Instead, epistemic possibility is typically identified with, roughly,16 what is not known not to be the case or—sometimes indiscriminately—with what is consistent with what is known.17 That is, the two proposals are EP1 It is epistemically possible (for S at t) that p ↔ S does not know at t that ¬p; and EP2 It is epistemically possible (for S at t) that p ↔ what S knows at t does not logically entail that ¬p. The right-hand side of (EP1 ) is clearly weaker than the right-hand side of (EP2 ), since knowledge is not closed under logical entailment. Suppose that it does not follow from what S knows at t that ¬p. Then, S does not know that ¬p, since, trivially, ¬p ⊧ ¬p. Consequently, if the right-hand side of (EP2 ) is satisfied, so is the right-hand side of (EP1 ). On the other hand, even if it follows from what S knows that ¬p, S may still not know that ¬p. Some logical consequences of what we know are just too hard to detect. So, if the right-hand side of (EP1 ) is satisfied, the right-hand side of (EP2 ) need not be satisfied. According to (EP2 ), calling something an epistemic possibility even for ourselves is a bit more risky than according to (EP1 ): usually we’re not too bad at detecting what we know and what we don’t know. Detecting the logical consequences of our knowledge is a different matter. According to these proposals, (at least some) epistemic uses of ‘Might p’ express that it is an epistemic possibility for S at t that p, where S and t are contextually determined. The context may simply determine the speaker and the time of utterance. But observations, e.g., about grounds for criticism of utterances of epistemic ‘might’s suggest that S may often be a group of epistemic subjects, the speaker, the audience and/or relevant experts, for 14 15 16 17

Cp. also Coates (1983: 133). But see Strawson (1997), Edgington (2004: §3), Schnieder et al. (2010: §1), Schnieder (2009) and Schulz (2010). DeRose (1991) motivates a modification. I will ignore this detail until it becomes relevant in section 5.4. For the former see e.g. DeRose (1991). For the latter see e.g. Kratzer (1977); von Fintel and Gillies (2007). Egan et al. (2005) oscillates between both characterisations.

192

Chapter 5. Possibility and Probability

instance.18 Since I’m not primarily concerned with epistemic possibility, I just mention this purported flexibility of (some uses of) epistemic ‘might’s. Nothing in what follows turns on this. The kind of probability that corresponds to this kind of epistemic possibility is evidential probability as spelled out in Williamson (2000: ch. 10). According to Williamson, the evidential probability of the proposition x (for an epistemic subject S at time t) is, roughly, the probability of x given S’s total evidence at t. Since, according to Williamson, one’s total evidence at a time is simply what one knows at that time (Williamson 2000: ch. 9) this comes down to the following: EV EvS, t (p) = IPS, t (p ∣ kS, t ); where EvS, t (p) is the evidential probability for S at t that p, kS, t is the conjunction of what S knows at t, and IPS, t is an initial probability distribution which conforms to the Probability Calculus and ‘measures something like the intrinsic plausibility of hypotheses prior to investigation’ (Williamson 2000: 211).19 IP(p ∣ k)—the conditional probability of the proposition that p on the & k) proposition that k—is defined as usual as IP(p ∣ k) = IP(p IP(k) , for IP(k) ≠ 0. Note that, just like credences, the evidential probability of a given proposition may be different for different epistemic subjects and different times, since epistemic subjects don’t all know the same and lose and gain knowledge over time. Different epistemic subjects, or the same epistemic subject at different times, may be in different total epistemic states, which determine what the subject knows at that time and, thus, the evidential probability of a proposition for the subject at the time. I will not say much more about evidential probability, since our understanding of epistemic possibility does not seem to gain very much from an 18

19

Their knowledge would then be pooled in order to determine whether it is an epistemic possibility for them. See e.g. DeRose (1991). More recalcitrant cases may even be taken to suggest that ‘might’ sentences have assessment relative truth conditions— roughly, that one and the same utterance of ‘Might p’ is true relative to context of assessment A, but false relative to A′ , because relative to A epistemic possibilities for S at t are relevant, whereas relative to A′ epistemic possibilities for S′ at t′ are. For a defence of this view see e.g. Egan et al. (2005). For a critique see von Fintel and Gillies (2008). Since, according to Williamson, ‘the notion of intrinsic plausibility can vary in extension between contexts’ (Williamson 2000: 211), I subscripted IP in (EV) as well. However, I will ignore the possibility that different initial probability distributions are relevant for different epistemic subjects or different times and, thus, drop the subscript in what follows.

5.2. Different Kinds of Probability

193

understanding of evidential probability. I only mention evidential probability here to point out that there is a notion of probability such that something has non-zero probability just in case it is epistemically possible in the sense of ‘epistemically possible’ favoured in the philosophical debate of epistemic ‘might’s. To show this, we have to assume a ‘natural constraint’ (Williamson 2000: 225) on IP, regularity. Roughly, IP is regular just in case IP(p) = 0 only if ‘p’ is inconsistent—any consistent hypothesis has a modicum of initial plausibility prior to investigation.20 Suppose, then, that it is epistemically possible (for S at t) that p. According to (EP2 ), it does not follow from what S knows at t that ¬p. Let k be the conjunction of what S knows at t. Then ‘p & k’ is consistent. Thus, by regularity of IP, IP(p & k) > 0. Since k is true, it is consistent. Thus, IP(k) > 0, again by regularity. Putting both facts together, IP(p ∣ k) is defined and > 0. Consequently, if it is epistemically possible that p, the evidential probability of the proposition that p is nonzero. On the other hand, suppose that it is not epistemically possible that p. Then it follows from what S knows that ¬p. Consequently, ‘¬(p & k)’ is a logical truth. Thus, IP(¬(p & k)) = 1 by Normalisation. By Finite Additivity, IP(p & k) = 0. Consequently, if the evidential probability of the proposition that p is non-zero, it is epistemically possible that p. Thus, evidential probability (for S at t) can provide the underlying probability scale for the epistemic uses of ‘might’ which express epistemic possibility in the sense of (EP2 ). 20

In the main text I ignore the quite general problem with uncountably infinite Ω. Roughly, if Ω is uncountably infinite, not all singletons of members of Ω can get a positive real-numbered probability, given that probability obeys the laws of the Probability Calculus (with Sigma Additivity, a generalisation of Finite Additivity to the infinite case; since all singletons are pairwise disjoint, we can just add up their probabilities to get the probability of their union, according to Sigma Additivity; since there are more than uncountably many such singletons, the probability of some such union, and thus the probability of Ω, would have to be greater than 1, contra Normalisation). Still, regularity is an intuitive desideratum, so some authors help themselves to infinitesimal probabilities (e.g. Lewis 1980). But see Williamson (2007a) for an argument that this move is unsatisfactory. I don’t have anything to add to this discussion, apart from noting that there may be technical reasons for not requiring regularity for IP (or objective chances, say), but these reasons are not reflections of how we think about the relation of probability and possibility. Thus, I am inclined to think that whenever positive probability and possibility seem to come apart because we are forced to give up regularity, this is a harshness of the Probability Calculus, rather than a reason to give up on the conceptual link between possibility and probability as articulated by (MIGHT).

194

Chapter 5. Possibility and Probability

I can’t offer a kind of epistemic probability that provides the underlying scale for epistemic possibility in the sense of (EP1 ). If there is none, this counts against spelling out epistemic possibility as in (EP1 ). Some authors (e.g. DeRose 1991) simply don’t consider (EP2 ) as an alternative. Perhaps the thought is that sentences like 10

Perhaps, p, and perhaps, not p;

where ‘p’ expresses a logical truth, should sometimes come out true, even in view of the fact that logical truths follow from the empty set of premisses, and, thus, a fortiori from whatever is known by any epistemic subject.21 (EP1 ) could account for such cases, since epistemic subjects don’t always know the logical consequences of their knowledge. But this reasoning overlooks the option that there may be two epistemic readings for expressions of possibility: one in which it expresses non-zero evidential probability and one in which it expresses non-zero credence. Though (10) will come out false on an evidential probability reading, it may still come out true when it is read in terms of the actual credences of the epistemic subject. In any case, I will not dwell on this issue but proceed to the main topic of this chapter, objective probabilities. 5.2.2 Objective Probabilities Apart from epistemic probabilities—credences and evidential probability— relative to the epistemic states of epistemic subjects, there is also a kind of probability that has nothing essentially to do with what people believe or know. As Lewis emphasises (1980: 83), we recognize objective chances in science and in everyday life. There is some objective chance that a certain tritium atom will decay within a year or that Theseus, who chooses directions with the help of a randomisation device (a coin, say), will reach the centre of the maze by noon. How high these chances are is independent of anyone’s degree of belief or of the support the proposition that Theseus reaches the centre by noon (that atom a will decay) receives relative to anyone’s body of knowledge. It may well be that rational credences track objective chances.22 When I’m certain that the objective chance that Theseus reaches the centre of the maze by noon is .8, say, my credence in the proposition that Theseus will reach the centre by then should be .8 (in the absence of any evidence 21 22

See e.g. Huemer (2007: 125) and Hale (1997: 488). This is a rough and ready statement of Lewis’s Principal Principle. See Lewis (1980: 86ff.).

5.2. Different Kinds of Probability

11am

11.20am

195

11.40am

12pm

t

Figure 6: Objective chances for Theseus

bearing more directly on whether Theseus will make it). But that does not make objective chance any more of an epistemic notion than truth. Rational degrees of belief vary with our best information about how the world is. This sometimes includes not only information about what determinately is or is not the case but also about objective chances. As epistemic probabilities vary with the epistemic states of epistemic subjects, objective chances vary with the state of the world. Since the state of the world only varies with time, we may as well say that objective chances vary with time. To take Lewis’s (1980: 91) story of Theseus in the maze: when he started his quest, say at 11am, there was some chance that he would reach the centre by noon (depending on how intricate the maze is, how fast he can walk etc.). At 11:20am, after he had taken some unlucky turns, this chance has decreased. At noon and from then on, the objective chance is either 1 (if he reached the centre in time) or 0. Interesting (i.e. non-limit) objective chances are always future-directed, since the past and the present are already settled. Let’s say that something is objectively possible at a time just in case it has non-zero objective chance at that time. When Theseus started his quest it was objectively possible for him to reach the centre in time. At 11.20am it was still possible, though rather unlikely. At noon that ship had sailed: it was impossible for Theseus then to have reached the centre by noon and remained impossible thereafter. We may picture the situation as in figure 6. Each fork in the diagram represents a fork in the maze. The bold line on the time line represents the

196

Chapter 5. Possibility and Probability

path Theseus actually took. Paths that end in a check mark are ones that get Theseus to the centre in time. Those with a cross do not take him to the centre in time. If we assume that Theseus used a fair coin to choose directions, so that taking a right was no more likely than taking a left, his chance of reaching the centre by noon at 11am was 1/2. Since he took a left at the very beginning, his chance decreased dramatically to 1/4 (if he had taken a right, his chance would have gone up to 3/4). Taking a right at around 11.20am temporarily raised his chance of getting to the centre by noon (taking a left would have been fatal: it would have been impossible then for Theseus to reach the centre by noon). But already at a quarter to 12 it was settled that he wouldn’t get to the centre in time. By then, it was objectively impossible for Theseus to reach the centre in time and remained impossible thereafter. What goes for Theseus goes for the world at large. As the world develops possibilities close off. Many things that were once possible (that this book would be published in 2012, for instance) no longer are. Their negations have turned into necessities. Objective possibilities don’t really ‘open up’.23 But things that were once very unlikely may be quite likely now. Suppose, for instance, it was very unlikely that Theseus would even make it to the entrance of the maze before noon. Then it was even less likely that he would reach its centre by noon. By some extremely lucky turn of events he made it to the entrance at 11am. There had always been some minute chance that he would make it to the centre in time—since there was a chance that the extremely lucky turn of events would come about and he would take the right turns— but that chance had dramatically increased by 11am. The above diagram will still do for a representation of the chances for the world at large, just extend the time line, add a few branches and mark the chances at each fork. If you add the right branches and label them appropriately, you have a picture of the objective possibilities sub specie aeternitatis. Something is objectively possible at a given time just in case it is true on a branch branching off from the actual course of events at a later time.24 23

24

Suppose there will be a non-zero chance that p in the future. There is now a nonzero chance that the world develops as it actually will. There is also now a non-zero (indeed, 1) chance that if it so develops, there will be a non-zero chance that p. Thus, there is now a non-zero chance that p. In general: whenever there is a non-zero chance that p at a time later than t, there is a non-zero chance that p at t as well. There is no harm done in thinking of the branches as partly overlapping possible worlds, as long as we think of those worlds along the lines outlined in the previous chapter.

5.2. Different Kinds of Probability

197

On the face of it, objective chance is a good candidate for providing the underlying scale for at least some uses of non-epistemic modals. Up until 11.45am it was still correct to say 11 Theseus might still reach the centre by noon. From then on (at least until shortly before noon) it was correct to say 12 Theseus can’t reach the centre by noon. It seems that the correctness of (a natural reading of) an utterance of (11) at 11.30am, say, is due to there being a positive objective chance then that Theseus will reach the centre. Likewise, the correctness of an utterance of (12) at 11.50am is due to there being no such chance any more. Suppose (universal) determinism is true. Then the state of the world at any given time determines the state of the world at any future time. The picture of the possibilities for the whole world would look like the stretch from 11.45am till 12pm in our initial picture. There simply would not be any non-trivial chances, at least after the initial conditions have been settled. If there are no non-trivial chances, there is no non-trivial objective modality either. Possibility and necessity at any time (apart from, perhaps, the beginning) will—in extension—collapse into truth, and impossibility at that time collapses into falsehood. David Lewis says, in response to the question of how he can square his conception of chance with determinism, that he can’t and shouldn’t: ‘there is no chance without chance’ (Lewis 1980: 120). Well, there is no non-trivial objective modality without chance either. If determinism is true at all times, we better remain ignorant of that fact, lest our use of ‘might’ in which it expresses objective possibility becomes superfluous along with our talk about chances.25 Perhaps there is also a kind of non-epistemic probability which does not go trivial when determinism is true. An obvious candidate is statistical probability which conforms to some variant of a ‘frequentist interpretation’ of probability.26 The relative frequency of a certain outcome O within a finite set of trials T is just the ratio of the number of O-outcomes over the number 25

26

Non-trivial chances are compatible, however, with long phases of determinism, say from the moment after the Big Bang onwards. Even if the state of the world at any given time after the Big Bang determines the state of the world at any later time, there were non-trivial chances before then, and, thus, non-trivial objective possibilities. Thus, a quite strong thesis, universal determinism, needs to be true in order for possibility at some time to collapse into truth. Recall that Carnap’s second legitimate notion of probability was ‘relative frequency in the long run’. Cf. Carnap (1945: 517). See also von Mises (1972: 11ff.).

198

Chapter 5. Possibility and Probability

of trials. If you throw a fair die 10 times, and a 6 comes up at the fifth and sixth throw, the frequency of sixes in your throws of the die is 2 in 10, or 1/5. Frequentists claim that the only workable notion of (non-epistemic) probability is some generalisation of such local relative frequencies: perhaps actual (past, present and future) relative frequencies in the long run or hypothetical such frequencies, the frequencies of sixes in throws of the die if there had been sufficiently many.27 Never mind about the exclusivity claim, and suppose that some frequentist account of statistical probability gives acceptable results.28 Let’s say, for simplicity, that it’s the account in terms of actual frequencies. Statistical probabilities vary with the choice of the set of trials, the so-called reference class. For, suppose the fair die has been and will be29 thrown a great many times, so that the frequency of sixes in throws of the die is close to 1/6. However, as the die is located in England for its whole carreer, only very few of these times are times at which it is sunny and bright outside. As it happens, a six falls very often then, let’s say 1/3 of the times. Then, the statistical probability that the die shows a six is 1/3 with respect to throws of the die while it’s sunny, but 1/6 with respect to throws of the die.30 27

28

29 30

Hypothetical frequencies are characterised with the help of a counterfactual. They may, thus, not be an initially appealing candidate for some sort of analysis of modality. Since the aim of this section is only to present different kinds of probability that may serve as the underlying scale for ‘might’ sentences, this fact is irrelevant for my current purposes. This is anything but obvious. For instance, depending on what ‘sufficiently many’ means in the rough statement of hypothetical frequentism, there may be no such thing as the relative frequency of sixes in throws of the die, had it been thrown sufficiently many times, since worlds with different relative frequencies may be equally close. I will ignore times in what follows, since they are not relevant here, and speak in the atemporal present. You may suspect a slight of hand here. The reason that the two relative frequencies could come apart drastically was that one of the reference classes was sufficiently small to allow for a significant deviation from a 1/6 relative frequency of sixes, while the other was large enough to counterbalance the deviation. Perhaps examples like these just count against a naïve spelling out of statistical probability in terms of actual frequencies. However, this is an accidental feature of my example, largely due to the fact that I wanted throws of this die to constitute the reference classes for expository purposes. Once we drop this constraint—as in other cases we must, simply because some ‘trials’ are not repeatable with the same subject (think of the trial that consists of living your life until you’re 40, and then checking whether you die within one year)—it is clear that there will be divergent actual and hypothetical frequencies: the

5.2. Different Kinds of Probability

199

Statistical probability (with respect to some reference class) may have nontrivial values in cases where objective chance does not—when things are already settled. Take any deterministic process with n different outcomes that occur equally often. For instance, a very boring computer program may show the numbers 1, 2 and 3 repeatedly and in that order on the screen. Ignoring the possibility of program failure, power outages and the like, the objective chance at any given time that the next number shown will be a 3 (or that the previous number shown was a 1) is either 0 or 1, but the statistical probability that the next number shown is a 3 (with respect to the reference class of all displays, supposing there to be sufficiently many of them) is 1/3.31 Sometimes statistical probabilities are very important to us. Suppose John is suspected to have cancer.32 A test is run. Clinical trials confirm that whenever the test results are negative, the person tested does not have cancer. On the other hand, 60% of people with positive test results have cancer. The test comes back and it’s positive. Jane, John’s wife, might well say 13

It is a bit more likely than not that John has cancer.

At least initially, it appears that the statistical probability of John’s having cancer is relevant to (a natural interpretation of) Jane’s utterance: at the time of utterance it’s already settled whether John has cancer, and, thus, the objective chance that John has cancer is either 1 or 0 (and the objective chance of John’s not having cancer is either 0 or 1 respectively). But then, the objective chance that John has cancer is either maximally greater or maximally smaller than the objective chance that John does not have cancer. If (13) can be true in the described circumstances, the expressions of probability can’t be interpreted in terms of objective chance. On the other hand, the truth or falsity of Jane’s utterance does not seem to turn on anyone’s epistemic states.

31

32

frequency of sixes in throws of a die, the frequency of sixes in throws of a six-sided die, the frequency of sixes in throws of a fair six-sided die, etc. The phenomenon that statistical probabilities are relative to the choice of reference class is known as the Reference Class Problem. For a more thorough discussion see e.g. Hájek (2007). However, with respect to the class of displays following the display of a 2 the statistical probability is 1, and with respect to the class of displays following the display of anything other than a 2 the statistical probability is 0. The claim is not that, if determinism is true, any relevant statistical probability is non-trivial, but that some are. In fact, it is plausible that the statistical probabilities we are most interested in are those that estimate objective chances. The example, but not the diagnosis (no pun intended), is DeRose’s. See DeRose (1991).

200

Chapter 5. Possibility and Probability

Suppose Jane was misinformed about the ratio of cancer cases among people with positive test results. Instead of 60% as she had heard, it’s only 10%. It seems that Jane should retract her utterance of (13) upon learning the truth, regardless of how much support John’s having cancer received from her or anyone else’s evidence at the time of utterance and of how strongly she then believed that John had cancer.33 Consequently, it’s a natural thought that (13) is best interpreted as saying that John’s having cancer has a greater statistical probability (with respect to the reference class of people with positive test results) than his not having cancer. But then it would seem that a good candidate for providing the underlying scale for (a natural interpretation of) an utterance of 14

John might have cancer;

in the same circumstances, is statistical probability.34 Nevertheless, in what follows I will focus on an objective chance reading of non-epistemic ‘might’s. This is motivated by the fact that the structure of objective chances conforms better to the behaviour of non-epistemic ‘might’ sentences. For one, statistical probability does not seem to vary with time. That there are n O-outcomes in past, present and future T-trials, or that the ratio of O’s to T’s would have been n/m, had there been enough T’s, does not seem to be the kind of thing that can vary with time.35 But the truth-value of non-epistemic ‘might’ sentences can.36 The discussion of the temporal structure of ‘might’ sentences 33

34 35

36

These phenomena can also be dealt with in a framework that allows assessment relativity. Recall fn. 18. All else being equal, however, I take the recognition of a further, plausible, reading of expressions of probability and possibility to be preferable to a major deviation from standard semantics. DeRose (1991) overlooks this option. The first ratio doesn’t change at all. The second may, if the closeness ordering of possible worlds changes over time. But, arguably, it does not change drastically enough to explain the falsity of ‘might’s concerning formerly very likely states of affairs. That is, the same ‘might’ sentence—read non-epistemically—may express a truth when uttered at one time, a falsehood when uttered at another time. One may reply that this may be so even if non-zero statistical probabilities are expressed, since the relevant reference class may vary between utterances. However, unless reference classes are systematically determined (by context, one assumes) to mirror objective chances, the flexibility afforded by statistical probability’s relativity to reference classes rather points against its relevance for most ‘might’ sentences and explicit expressions of probability. Neither ‘might’ sentences nor other expressions of probability are as flexible as statistical probability with respect to various reference classes would allow.

5.3. The Temporal Structure of ‘Might’s

201

in the next section gives indirect evidence for the view that objective chance readings are predominant among non-epistemic ‘might’s. 5.3

The Temporal Structure of ‘Might’s

‘Might’ and ‘might have’ sentences exhibit an interesting and, at first sight, baffling, temporal structure.37 In this section I will first discuss the temporal structure of ‘might’ (in subsection 5.3.1) and ‘might have’ (in subsection 5.3.2) sentences. Then, I will discuss how the different temporal structures relate to the availability of different readings for these sentences (in subsection 5.3.3). 5.3.1 Might Consider 15

He might be sick;

and contrast with 16

He might get sick.

(16), but not (15), seems to have a ‘future orientation’: for (16) to be true, it seems that it has to be possible (now) that he will be sick. For (15) to be true, on the other hand, it must be possible (now) that he is now sick—(15) seems to have a ‘present orientation’. This present orientation can be overruled with so-called frame adverbials like ‘tomorrow’ and ‘next month’, but the future orientation of (16) can’t (with ‘now’): 15a

He might be sick tomorrow/next month/now;

16a

He might get sick tomorrow/next month/*?now.38

37 38

Throughout this section I draw heavily on Condoravdi (2002). It’s important to get the scoping right. Might (Now (he get sick)) is relevant here, not Now (Might (he get sick)). ‘He might get sick now’ is only acceptable under the second reading. That it is acceptable might explain why (16a) with ‘now’ does not sound as bad as either of (15b) and (16b). Further, ‘he might be getting sick now’ may not be aesthetically pleasing, but it is acceptable. However, because of the presence of the progressive, ‘he might be getting sick’ belongs together with (15) instead of (16). Incidentally, I follow Condoravdi (2002) in assuming that Might always takes whole (tensed or untensed) sentences, instead of verb phrases. This is debatable, but nothing hinges on it in the present discussion.

202

Chapter 5. Possibility and Probability

On the other hand, neither (15) nor (16) allow a past orientation, as the deviance of the following sentences shows: 15b *He might be sick yesterday; 16b *He might get sick yesterday. Condoravdi (2002: §3) argues that these facts can be best accounted for by the hypothesis that one job of modals like ‘might’ is to modify the time of evaluation for the embedded sentence. In this respect, then, ‘might’ is like the auxiliary ‘have’, expressing the perfect in English (Perf ).39 The deictic tenses Pres, Past and Fut pick up a time of evaluation for the sentences they govern from the (linguistic or non-linguistic) context. This time of evaluation, t, is then available to be modified by Perf : Perf shifts the time of evaluation to some time before the time it gets as input, i.e. to some t′ ≺ t.40 In the case where Perf occurs in the scope of Pres, Perf takes the time of evaluation set by Pres—the time of utterance, now, for short—and shifts the time of evaluation to some t ≺ now. So, e.g., ‘He has been sick’—which we may represent as ‘Pres (Perf (he be sick))’—41 is true just in case he be sick is true at some t ≺ now. The hypothesis that Perf shifts the time of evaluation rather than picking it up from the context like Past explains the temporal structure of sentences in which Perf occurs within the scope of tenses other than Pres. For instance, ‘He will have been sick by the end of the year’ is true just in case he be sick is true at some t ≺ the end of the year, not necessarily at some time in the past. According to Condoravdi, Might takes the time of evaluation, t, and extends it indefinitely into the future.42 Let us use interval notation for perspicuity. [t, t′ [ is the interval from, and including, t to, but excluding, t′ . Sometimes, t and/or t′ may be intervals themselves. Suppose t′ is. Then, e.g., ]t, t′ [ is the interval from, but excluding, t to, but excluding, all of t′ . Incidentally, I assume that time intervals are sets of continuous instants, so 39 40 41 42

For more on this see e.g. Partee (1984: §4). The range of the quantifier ‘some’ may well be contextually restricted. Pres is introduced by the verb ending of ‘have’ in ‘He has been sick’. What is meant is the future from the point of view of t. Less misleadingly we could have said that Might extends the time of evaluation forward. Might extends the time of evaluation, rather than shift it like Perf, because t is a subinterval of the open interval from t onwards.

5.3. The Temporal Structure of ‘Might’s

203

that [t, t′ [= {x ∶ t ⪯ x ≺ t′ }.43 We may thus use set operations in specifying intervals. Condoravdi’s proposal, then, is that Might takes the time of evaluation t and extends it to [t, ∞[. This is why (15a) (with ‘tomorrow’, let’s assume) is acceptable, but (15b) is deviant. We may represent them as follows: 15a* Pres (Might (Tomorrow (he be sick))); and 15b*

Pres (Might (Yesterday (he be sick))).

Pres is the morphologically unmarked but obligatory present tense in both sentences. It sets the time of evaluation for its scope to the time of utterance, now. Might extends the time of evaluation (for its scope) to [now, ∞[. The frame adverbials ‘tomorrow’ and ‘yesterday’ restrict the time of evaluation (for their scope) to the intersection with the day after and before now respectively, i.e. to [now, ∞[ ∩ tomorrow and [now, ∞[ ∩ yesterday.44 Since the latter is empty, (15b) is deviant. The same goes, of course, for (16b). The hypothesis that ‘might’ extends the time of evaluation into the future explains the difference in acceptability between (15a) and (15b), but it doesn’t yet explain why ‘He might get sick now’ should be deviant as well. For this, we need to add a standard hypothesis of the difference between stative and eventive verb phrases with respect to the truth at a time of evaluation of sentences of which they are the predicate. I assume a rough understanding of the distinction between stative and eventive verb phrases—the former ‘describe’ states, the latter events. It may be sharpened a bit with the help of the following observations.45 Eventive but not stative verb phrases can occur in the progressive: 17

He is getting sick;

17*

*He is being sick.

Eventive but not stative verb phrases can occur in the imperative: 18

Get to know Kate!

18*

*Know Kate!

43 44 45

For unification, if the relevant time is an instant, t should be thought of as the unit set of that instant, although, for simplicity, I will brush over the difference. Tomorrow and yesterday are intervals comprising the instants of the day after and before the time of utterance respectively. See Larson and Segal (1995: 490). Cp. also Vendler (1967: ch. 4).

204

Chapter 5. Possibility and Probability

Eventive but not stative verb phrases can occur in pseudocleft constructions: 19 What he did was get to know Kate; 19*

*What he did was know Kate.

The standard hypothesis is that sentences with a stative predicate are true at a time of evaluation t just in case, roughly, the state they describe holds at an interval which overlaps t, while sentences with an eventive predicate are true at t just in case the event they describe occurs at a time included in t.46 Given our understanding of intervals, inclusion may just be represented with the help of the subset relation, i.e. Inc

t is included in t′ ↔df. t ⊆ t′

and overlap is non-empty intersection, i.e. Ov

t overlaps t′ ↔df. t ∩ t′ ≠ ∅.47

If we let τ be a function that takes sentences to the times of holding or occurring of the states or events they describe,48 we can apply the hypothesis to the sentences in question as follows: 20 he be sick is true at time of evaluation t ↔ τ (he be sick) ∩ t ≠ ∅; and 21

he get sick is true at time of evaluation t ↔ τ (he get sick) ⊆ t.

Letting the time of evaluation be [now, ∞[, the states of affairs pictured in figure 7 are allowed in order for he be sick and he get sick to be true at the time of evaluation. What is excluded is that all instants of τ (he be sick) and some instant of τ (he get sick) is earlier than now.49 46 47

48

49

See again Partee (1984: §4). Incidentally, Partee speaks of overlap but then goes on to treat it as if it were reverse inclusion—i.e. as if ‘t overlaps t′ ’ meant the same as ‘t′ is included in t (t′ ⊆ t)’. See Partee (1984: 255). Although she does not say, this may be an acceptable simplification because she thinks of the relevant times of evaluation as instants, so that t′ is always a unit set (see fn. 43). If t′ has only one element, then (t ∩ t′ ≠ ∅) ↔ t′ ⊆ t. Cp. Condoravdi (2002: §3.2). This is very rough, since my presentation tries to do without quantification over events and states in the semantics. Clearly, if he never gets sick, then there just is no event described by ‘he get sick’. We might then let τ (he get sick) be the empty set and require additionally that it is not the empty set in (21). It is important to note that aspect like Perf and the progressive (Prog) may act on sentences with an eventive predicate. In the evaluation of the whole sentence, it will then be relevant whether Prog (p) is true at a time of evaluation. But Prog turns an eventive sentence into a stative one, so to say. Thus, while ‘he get sick’ may not be

5.3. The Temporal Structure of ‘Might’s

205 time of evaluation

he be sick

he be sick .

he get sick now

tomorrow

t

Figure 7: Time of evaluation and the eventive/stative distinction

Now consider what happens when we introduce the frame adverbials ‘tomorrow’ and ‘now’. Recall that they modify the time of evaluation for the sentence in their scope to the intersection of the previous evaluation time— [now, ∞[ for our current purposes—and tomorrow and now respectively. This is not a problem for he be sick: the intersections are non-empty (they are just tomorrow and now respectively) and τ (he be sick) may overlap them. In fact, in the above diagram the first case is one in which Now (he be sick) is true at [now, ∞[, while Tomorrow (he be sick) is false, and the second case is one in which matters are reversed. However, Now (he get sick) is not true at [now, ∞[ according to the diagram, since τ (he get sick) is not a subset of now. Neither is Tomorrow (he get sick) true at [now, ∞[, since τ (he get sick) is not a subset of tomorrow either. But while the latter is only an artefact of our diagram (we could have just drawn the ellipsis a bit farther to the right), the former is not. Assuming that getting sick takes some time, we cannot fill in the diagram in such a way that τ (he get sick) is a subset of now, since now is either an instant or a very short interval. This explains, according to Condoravdi, why ‘He might get sick now’ is deviant. In general, the hypothesis that Might extends the time of evaluation to the future, together with the difference between sentences with stative and eventive predicates explains the observed future orientation of sentences like 16

He might get sick;

and the present or future orientation of sentences like 15

50

He might be sick.50 true at t because τ (he get sick) is not a subset of t, Prog (he get sick) may still be true at t, since τ (he get sick) overlaps t. This is why I said, in fn. 38, that ‘he might be getting sick’ belongs with ‘he might be sick’ rather than with ‘he might get sick’. See Condoravdi (2002: §3.2). I find Condoravdi’s general diagnosis extremely appealing. Nevertheless, I should add that it also seems to have some lacunae. First, nothing in her account explains why (15) cannot be read as having a future orientation

206

Chapter 5. Possibility and Probability

Pres Might

.

now

he get sick t

Figure 8: Temporal structure of ‘might’ sentences

5.3.2 Might Have To recapitulate, ‘might’ sentences like (15) and (16) may be represented as follows: 15*

Pres (Might (he be sick)); and

16*

Pres (Might (he get sick)).

That Might is governed by Pres ensures that present possibilities are at issue. It also determines the starting point of the interval relevant for the evaluation of the embedded sentence, since Might takes a time of evaluation as input and extends it forward. Differences in orientation between (15*) and (16*) are due to the former containing a stative, the latter an eventive predicate. Focusing on the latter case, we can picture its temporal structure as in figure 8. What about ‘might have’ sentences? Although some would like to take ‘might have’s en bloc,51 they appear to have a semantic structure—they seem to be built up of the modal auxiliary ‘might’ and the perfect auxiliary ‘have’.

51

unless we insert a frame adverbial like ‘tomorrow’ (or there is strong pressure from the context). Perhaps, she could say that the default reading of (15) is one with an unpronounced Now. Second, her account does not explain why ‘might’s with sentences that (potentially) describe instantaneous (or nearly instantaneous) events in their scope are still future directed. If τ (p) is a unit set or a very short interval, τ (p) may be a subset of now (Condoravdi says that now, the time of utterance, is an interval, albeit a very short one). Still, ‘he might pull the trigger’, say, seems to be just as future directed as ‘he might get sick’. Perhaps, she could claim that eventive predicates (but not stative ones) in the scope of ‘might’ move the starting point of the evaluation time interval forward, so that it becomes ]now, ∞[. This move would be similar to an attractive view concerning the difference between eventive and stative predicates in past narratives (see Partee 1984: §4). See the references in Condoravdi (2002: §2) for linguists who succumb to that temptation. With philosophers, it is harder to tell. But it seems reasonable to ascribe some such view to those who want to drive a huge wedge between ‘might’s on the one hand and ‘might have’s on the other—usually assuming the former to be univocal expressions of epistemic possibility and the latter to be univocal expressions of metaphysical possibility.

5.3. The Temporal Structure of ‘Might’s

207

There is also a neat argument for the compositionality of ‘might have’:52 sometimes adverbs like ‘still’ and ‘already’ intervene between ‘might’ and ‘have’, as in 22

They might have already returned.

In (22) ‘already’ cannot be read to have scope over they might have returned. If it could, we should expect it to be able to have scope over they might return in 23

*They might already return;

as well, which it can’t. Also, (22) has no natural reading in which it is roughly equivalent to ‘It’s already possible that they have returned’. The two remaining scoping options are that ‘already’ intervenes between ‘might’ and ‘have’ and that ‘might have’ has scope over ‘already’. The latter option is excluded, since ‘already’ would then have to have they return in its immediate scope. But ‘already’ selects against eventive predicates—it cannot have sentences with eventive predicates in its immediate scope. Cp. 24

*They already return/returned; vs.

25

They are/were already at home.

However, ‘already’ can have scope over Perf (p), even if p’s predicate is eventive, as in 26

They have already returned.

So, in (22) ‘already’ has to intervene between ‘might’ and ‘have’—which it could not if ‘might have’ were a semantic unit. Now, given that ‘might have’ decomposes into Might and Perf, there are still two scoping options for many ‘might have’ sentences: Might (Perf (p)) and Perf (Might (p)). Both are realised. 27

They might have returned;

may be (structurally) disambiguated as 27*

Pres (Might (Perf (they return))); and

27** Pres (Perf (Might (they return))). There is both inter- and intra-language evidence that both scopings are realised. English syntax requires the surface order modal auxiliary > perfect auxiliary. But the syntax of other languages is more flexible. In German, (27) may be translated as 52

See Condoravdi (2002: §3.1).

208

Chapter 5. Possibility and Probability

Pres

27*

Might they return .

now Pres

27**

Might

Perf

they return

t

Perf

Figure 9: Temporal structure of ‘might have’ sentences

28*

Sie könnten zurückgekehrt sein; or as

28** Sie hätten zurückkehren können. (28*) corresponds in its surface structure to (27*) and (28**) corresponds to (27**).53 In English, inserting the adverbs ‘already’ and ‘still’ (in its temporal as opposed to its concessive reading) forces the different scopings: 29*

They might have already returned;

29** They might still have returned. (29*) can only be read with the scoping of (27*), while (29**) requires the scoping of (27**). Applying the hypotheses of the last subsection, we may picture the temporal structure of both scopings of (27) as in figure 9. In the first case, the morphologically unmarked but obligatory Pres sets the initial time of evaluation to now. So, the time of evaluation for Might is now. Then Might— superfluously, in this case—extends the time of evaluation to [now, ∞[. Perf shifts the time of evaluation to some time t before the time of evaluation it gets as input. Thus, some t ≺ [now, ∞[ is the time of evaluation for they return.54 Recall that τ (they return) has to be a subset of t in order for they return to be true at t. Consequently, our hypotheses predict correctly that (27), with the scoping of (27*), is true just in case it is a present possibility that their return took place entirely in the past. 53 54

Egan et al. (2005: fn. 3) mention that the same goes for Spanish. Again, the quantifier may be contextually restricted.

5.3. The Temporal Structure of ‘Might’s

209

In the second case, Pres again sets the initial time of evaluation to now. Perf shifts the time of evaluation to some t ≺ now.55 Thus, the time of evaluation for Might is some t, t ≺ now. Might then extends the time of evaluation to [t, ∞[. Again, τ (they return) has to be a subset of [t, ∞[ in order for they return to be true at [t, ∞[. Thus, our hypotheses predict correctly that (27), with the scoping of (27**), is true just in case there was a past possibility that they would return, i.e. that there is a time in the past at which it was possible that they return at a (then) future time. The same goes, mutatis mutandis, for ‘might have’ sentences with a stative predicate. In the scoping Pres (Might (Perf )), we have a present possibility concerning states overlapping the past.56 In the scoping Pres (Perf (Might)), we have a past possibility concerning states that overlap the (then) future. Just as in the case of ‘might’ sentences, ‘might have’s with a stative predicate may, thus, lack the forward orientation that is obligatory for ‘might have’s with an eventive predicate in that scoping. The temporal structure of ‘might’ and ‘might have’ sentences discussed in this and the previous subsection, together with the predominance of objective chance readings of non-epistemic ‘might’s, can explain the (un)availability of non-epistemic readings for different ‘might’ sentences. This is the topic of the next subsection. 55

56

Contextual restriction of the temporal quantifier introduced by Perf is, arguably, the source of (at least some) cases of a phenomenon known as modal inconstancy (cp. Lewis 1986a: ch. 4.5). Consider the following scenario (due to Louis deRosset): wanting to express some piece of American conventional wisdom you say ‘Hoover couldn’t have won the 1932 election’. I tease you by pointing out that, surely, Hoover did not have to be president during the Great Depression. That, in fact, a Democrat might have been president then. So, Hoover might well have won the 1932 election. If you belong to the inconstancy camp, you will want to maintain that ‘Hoover could not have won the 1932 election’ expressed something true when you uttered it, and something false in the context in which I uttered its negation. A very natural thought is that this is due to different contextual restrictions on the quantifier over times: when you uttered it, it was restricted to times reasonably close to the 1932 election. My remarks broadened its range a little. And it may well both be true that it was impossible in 1931 that Hoover would win the 1932 election, but possible in 1928, say. Sometimes possibilities expire prematurely. And so it may well have been for Hoover and the 1932 presidency. Which may, thus, still hold. The truth of ‘they might have been at home’ read with the relevant scoping does not exclude the truth of ‘they might (still) be at home’.

210 30

Chapter 5. Possibility and Probability Pres Might Now they be at home now

31

t

Pres Might

Theseus reach the centre

Figure 10: Objective chances and ‘might’ sentences

5.3.3 Temporal Structure and Different Readings Some ‘might’ sentences only have an epistemic reading. Consider 30

They might be at home (now).

(30) only has a reading in which the present epistemic possibility that they are at home now is relevant for its truth. On the other hand, 31

Theseus might reach the centre;

seems to allow for a non-epistemic reading: one in which it is irrelevant for its truth or falsehood what the present epistemic state of this or that epistemic subject is. The temporal structure of both sentences, together with the claim that the non-epistemic reading would have to be one concerning non-zero objective chances, can explain why this should be so. Recall that non-trivial objective chances are always future-directed, since the past and the present are already settled. We pictured this in section 5.2 with the help of a forward branching tree diagram. Now consider what happens when we insert the temporal information of the two sentences as in figure 10. In an objective possibility reading, (30) would be true just in case at the time of utterance there is a non-zero objective chance that they are at home

5.3. The Temporal Structure of ‘Might’s

211

at the time of utterance. On the other hand, (31) would be true just in case at the time of utterance there is a non-zero objective chance that Theseus reaches the centre sometime after the time of utterance.57 In the second case, there may be non-trivial objective chances, for instance when the time of utterance is right before Theseus enters the maze. But in the first case there can’t be. Whatever the time of utterance, whether they are at home then is already settled at the time of utterance. Thus, the objective chance (then) that they are at home (then) is either 1—if they are at home—or 0—if they aren’t. Faced with an utterance of (30), the best explanation of why the speaker still chose to utter a ‘might’ sentence instead of the simpler 32

They are at home (now);

will be that he didn’t have a non-epistemic but an epistemic reading in mind. Even if it is already settled whether they are at home, the speaker may not know which way it is settled, or be uncertain about it. Thus, the present orientation of some stative ‘might’ sentences forces an epistemic reading of them, while the future orientation of all eventive ‘might’ sentences—and some stative ones—58 allows a non-epistemic reading. Condoravdi (2002: §4) even claims that an objective reading is the default reading for those ‘might’ sentences that allow it. Although, as far as I can tell, she does not fully justify this claim, an interesting observation in this connection is that in (31) we cannot felicitously insert ‘already’: 33

*Theseus might already reach the centre.

Because ‘already’ selects against eventive predicates, the scoping has to be 33*

Pres (Already (Might (Theseus reach the centre)));

so that (33) is roughly equivalent to ‘it’s already possible that Theseus will reach the centre’. Now, Already(p) introduces the presupposition that at a time before the time of evaluation, ¬p.59 This is why sentences like 34 57 58 59

*She is already young; Recall that this is due to the future orientation of ‘might’s with an eventive predicate and the possibility of a present orientation of ‘might’s with a stative predicate. E.g. ‘They might be at home tomorrow’. Cf. Löbner (1989: 181f.). Incidentally, I follow Löbner and Condoravdi in assuming that sentences that suffer presupposition failures are neither true or false. Nothing hinges on this.

212

Chapter 5. Possibility and Probability

are never acceptable: whatever the time of evaluation, one cannot be young at that time while not having been young at an earlier time. Consequently, either the sentence is false (since she is not young), or it is neither true nor false, since its presupposition is not fulfilled. Since objective possibilities don’t open up, the structure of objective possibilities would afford a similar explanation for the unacceptability of sentences like (33): no matter how things stand, either there is no objective chance that p at the time of evaluation—Already (Might (p)) is false—or there is, but the presupposition that there was an earlier time at which there was no such chance is false— Already (Might (p)) is neither true nor false. On the other hand, nothing in the structure of epistemic possibilities seems to allow for such an explanation of the unacceptability of (33), since epistemic possibilities may well open up.60 Let’s turn to ‘might have’s. Consider 35 Theseus might have reached the centre by noon; and its two possible scopings 35* Pres (Might (Perf (Theseus reach the centre))); and 35** Pres (Perf (Might (Theseus reach the centre))). Suppose the time of utterance is sometime in the afternoon. Putting the temporal information for the two scopings and the objective chances together, we can picture the situation as in figure 11. In an objective possibility reading, (35) with the scoping of (35*) would be true just in case at the time of utterance there is a non-zero objective chance that Theseus reached the centre before the time of utterance. On the other hand, (35) with the scoping of (35**) would be true just in case sometime before the time of utterance there was a non-zero objective chance that Theseus would reach the centre. If things are as pictured in the diagram, there was indeed a non-trivial objective chance that Theseus would reach the centre by noon, although it’s settled now that he didn’t.61 But in the first 60

61

I take this to be rather weak evidence for the claim that an objective reading of (31), say, is the default reading. Although epistemic possibilities may open up—people forget things, and lose certainty—, it would be peculiar, to say the least, to presuppose that the negation of the embedded sentence was once an epistemic necessity when uttering an epistemic ‘might’. However, I take the above considerations to be quite strong evidence for the predominance of objective possibility readings among the non-epistemic readings of ‘might’ sentences. Mondadori and Morton (1976: 11, fn.) claim that ‘“might have” is not exactly a past (perfect) tense of “might.” The past truth of “might” does not entail the truth of

5.3. The Temporal Structure of ‘Might’s

213

Theseus reach the centre

35*

Perf Might Pres

11am

11.20am

11.40am

12pm

now

t

Pres

35** Might

Theseus reach the centre

Perf

Figure 11: Objective chances and ‘might have’ sentences

case there are no non-trivial objective chances. Since the time of utterance is past noon—which it has to be in order for the utterance to be acceptable—, whether Theseus has reached the centre by noon is already settled at the time of utterance. Thus, the objective chance (then) that Theseus has reached the centre (before then) is either 1—if he has—or 0—if he hasn’t. Faced with an utterance of (35) with the scoping of (35*), the best explanation of why the speaker still chose to utter a ‘might’ sentence instead of the simpler 36

Theseus reached the centre by noon; “might have” until the time at which the event in question happens or does not’. See also Mondadori (1978: 223f.) Let’s not worry too much about the mysterious time at which some event ‘does or does not happen’. In our present example, it is clear that they mean 12pm. Their claim is that (35) (with the scoping of (35**)) is not true when uttered before noon on the day in question. But this is clearly wrong. (35) is both true and assertable at 11.45am (when, recall, it’s already settled that Theseus won’t make it). (35) may not be assertable before things are settled. But this isn’t because it is false then, but merely because uttering it saliently implies a falsehood: that there no longer is a non-zero objective chance that Theseus will reach the centre in time. Cp. ‘He once lived in London’ said of someone who still lives in London.

214

Chapter 5. Possibility and Probability

will be that he didn’t have a non-epistemic but an epistemic reading in mind. Even if it is already settled whether Theseus reached the centre by noon, the speaker may not know which way it is settled, or be uncertain about it. Thus, ‘might have’s in which Might has scope over Perf force an epistemic reading, while those in which Perf has scope over Might allow a non-epistemic reading. This concludes my discussion of the temporal structure of ‘might’ sentences. We saw that differences in orientation of ‘might’ sentences can be explained by a difference between stative and eventive predicates. Further, ‘might have’s are compositional. Differences in the temporal interpretation of different ‘might have’ sentences can be explained by different scopings of Might and Perf. Finally, the (un)availability of non-epistemic readings for certain ‘might’ sentences lines up neatly with the structure of objective chances. Consequently, the hypothesis that non-epistemic ‘might’ sentences predominantly express non-zero objective chances can be fruitfully brought to bear to explain the behaviour of non-epistemic ‘might’s. The hypothesis thus receives indirect evidence from the considerations of subsection 5.3.3. However, this straightforward picture of the interrelations between ‘might’ and ‘might have’ sentences and their different readings is challenged by an argument given by Keith DeRose (in DeRose 1998) who defends the view that ‘might’ always expresses epistemic possibility. In the next section I present his argument and show that there are good reasons to resist it. 5.4 DeRose on ‘Might’ Sentences Keith DeRose defends a thesis he calls the General Epistemic Thesis: General Epistemic Thesis (GET) Present tense ‘might’ sentences unambiguously express epistemic possibilities. (Cf. DeRose 1998: 67f.) He argues that even utterances of ‘might’ sentences that at first sight look as if they expressed non-epistemic possibility turn out to be epistemic at closer inspection.62 In particular, he shows how he can handle cases that appear to 62

The 1998 paper may thus be viewed as filling in a lacuna of his earlier discussion of epistemic possibility (in DeRose 1991) mentioned in section 5.2. There I pointed out that DeRose proposes a rather complicated account of epistemic possibility without discussing the suspicion that some of the examples that are supposed to motivate it do not express epistemic possibility at all. DeRose’s contention that ‘might’ sentences can only express epistemic possibility is, predictably, well received by philosophers concerned with epistemic modality. See e.g. Egan et al. (2005: fn. 3).

5.4. DeRose on ‘Might’ Sentences

215

be problematic for his thesis and gives a positive argument for (GET). Let me start by noting that (GET) as stated only concerns ‘might’ and not ‘might have’ sentences. This may be a relief to those philosophers interested in non-epistemic modality who mainly rely on ‘might have’ sentences in their informal discussions. This, however, would be short sighted. For, as the discussion in the last section suggests, (prima facie) non-epistemic ‘might have’ sentences are merely ones in which Perf has scope over Might. Thus, when such a ‘might have’ sentence is true at a time, the corresponding ‘might’ sentence was true at an earlier time. Consequently, if ‘might’ sentences only have an epistemic reading,63 so do ‘might have’ sentences.64 In the relevant scoping they would then exclusively express past epistemic possibility. DeRose’s contention, thus, threatens to undermine the intuitive basis for a significant amount of work on non-epistemic modality. DeRose’s strategy is to start from what looks like a paradigm case for a non-epistemic reading of a ‘might’ sentence. He shows how an epistemic interpretation of the example still yields the right results and explains why it looks as if non-epistemic possibility is expressed. He then modifies the example slightly so that, according to him, only epistemic possibility is an interpretative option. From the modified example he draws the resources for a general case against non-epistemic readings of ‘might’ sentences and, thus, for (GET). In this section, I will first present his argument, then slightly modify the example myself, so that only non-epistemic possibility is an interpretative option. I will end by giving an alternative diagnosis of the data DeRose cites in support of the General Epistemic Thesis. My conclusion is that DeRose’s case against non-epistemic readings of ‘might’ sentences is far from conclusive and that, therefore, we have no reason to doubt that things are as they appear to be: many ‘might’ and ‘might have’ sentences have a non-epistemic reading, which is, moreover, the natural reading in many cases. We saw in our discussion in section 5.3.3 that there is no good reason to express present- or past-directed current objective chances. So, DeRose chooses his target well when he makes out that the main challenge for (GET) 63

64

Recall that, in order to not commit myself to the view that ‘might’ is a genuine case of ambiguity, I prefer to talk about different readings of ‘might’ sentences, without attributing the availability of such readings to an ambiguity in the modal auxiliary. DeRose (1998: 67) realises this, but promises discussion of ‘might have’s (in which Perf has scope over Might) for another occasion. As far as I am aware, this occasion has not presented itself yet.

216

Chapter 5. Possibility and Probability

are future-directed ‘might’ sentences. His example are two scientists who send a ball on its way through a ‘zone of indeterminism’65 such that it is causally undetermined whether it will veer left or right (but it is determined that it will either veer left or right, but not both).66 They might well say 37 The ball might veer left. When they say so, they don’t seem to talk about epistemic possibilities. Rather, DeRose is willing to grant, they appear to mean ‘something like that it is not causally determined that the ball will not veer left’, or that there is a (present) non-zero objective chance that it will veer left.67 DeRose strengthens the case against (GET) by supposing further that the set up is such that, after a certain point, it is causally determined whether the ball will veer left or right. At that point the scientists may have the following sensible conversation A: Is it still true that the ball might veer left and it might veer right? B:

No. Now it can only veer left or it can only veer right (but I don’t know which).

Here, it seems a very good guess that scientist A asks whether it’s still the case that both objective possibilities are open, which scientist B denies while admitting that he doesn’t know which one is closed off. After all, the ‘I don’t know which’ admission is incompatible with a straight-forward epistemic interpretation. The admission of ignorance is incompatible with a straight-forward epistemic interpretation, but not with DeRose’s favoured one. DeRose conceives of epistemic possibility (for a subject S at a time t) as (partly) determined by what S knows at t. But rather than accepting, e.g., 65

66 67

Again, DeRose is right. There is only a non-limit objective chance that p if it’s not already settled whether p. If determinism is true (on some level of description, about some things), then there are no interesting objective chances (on that level of description, about those things). However, we may doubt that we need a scientific apparatus to set up ‘zones of indeterminacy’. Macro-events seem to be indeterminate enough (or at least we typically take them to be so, which is enough to make ‘might’ talk worthwhile). Inicidentally, in the discussion I will use the noun ‘indeterminacy’ instead of DeRose’s ‘indeterminism’ because the latter, but not the former, has a use in which it designates a particular philosophical theory. Cf. DeRose (1998: 68f). DeRose does not speak about non-zero objective chances but about what is causally determined. For stage setting purposes, I will assume that there is no relevant difference.

5.4. DeRose on ‘Might’ Sentences

217

EP2 It is epistemically possible (for S at t) that p ↔ what S knows at t does not logically entail that ¬p; he argues in DeRose (1991)—for reasons independent of our present case— that we should slightly strengthen its right-hand side.68 Roughly, what’s epistemically possible does not only depend on what we know but also on what we could come to know by contextually salient means. That is, DeRose proposes to replace (EP2 ) with EPD It is epistemically possible (for S at t) that p ↔ what S knows at t together with what S could come to know by φ-ing does not logically entail that ¬p; where ‘φ-ing’ is a place-holder for a description of a contextually salient way of knowledge acquisition. Thus, while we may still say that epistemic possibility that p is non-zero evidential probability that p, the relevant proposition conditional on which the proposition that p is evaluated for initial plausibility may be more inclusive than we may have previously thought.69 Instead of its being a conjunction of what S knows at t, it is a conjunction of what S knows at t together with the propositions that S could come to know by φ-ing. Now, according to DeRose, what the ‘might’ sentences express in our apparent paradigm example of a non-epistemic reading is (present) epistemic possibilities for the two scientists, where the contextually relevant way of coming to know things is finding out about everything that is causally determined at the time.70 It may well be causally determined at a time that the ball will not veer left—and, thus, follow from what the scientists would find out were they to find out about everything that is causally determined at that time—without scientist B actually knowing which way the ball will veer. Consequently, the admission of ignorance is not inconsistent with an epistemic interpretation of the scientists’ discourse along the lines of (EPD ). Furthermore, it is easy to see that DeRose’s epistemic possibility reading (with the very far-reaching way of coming to know allegedly fixed by context) will agree with an objective possibility reading of ‘might’ sentences in 68

69 70

In fact, DeRose starts with EP1 It is epistemically possible (for S at t) that p ↔ S does not know at t that ¬p; and modifies it in a way analogous to what is described in the main text. Since, in section 5.2.1, I gave reasons to prefer (EP2 ), I will conduct the discussion in terms of a modification of (EP2 ) instead. Nothing hinges on this. Recall the discussion of evidential probability on pages 192–193 above. See DeRose (1998: 76f.).

218

Chapter 5. Possibility and Probability

the example described. This, according to DeRose, is why it looks as if a non-epistemic reading is appropriate. But it merely looks that way, since there is a general reason to think that the possibility expressed by ‘might’ sentences is always epistemic. Take the initial case, but assume further that scientist A believes he has some direct connection to God, who, being both benevolent and omniscient, informed him that the ball will veer right (or so A believes). In that situation, DeRose claims, it is appropriate for A to say either of the following: 38

I know the ball won’t veer left, though it isn’t now causally determined that it won’t;

39

The ball won’t veer left, though it isn’t now causally determined that it won’t.

However, the following sentences cannot be appropriately uttered by A: 38*

I know the ball won’t veer left, but it might;

39*

The ball won’t veer left, but it might.

That (38) and (39) can be appropriately uttered by A is supposed to show that we don’t just balk at A’s peculiar epistemic situation (‘God told you?’). On the other hand, that (38*) and (39*) are inappropriate shows that ‘might’ cannot here express causal indeterminacy (to be precise: that it is not causally determined that not …)—otherwise (38*) and (39*) should be just as acceptable as (38) and (39) are. But if ‘might’ could ever express non-epistemic possibility, we should expect it to express non-epistemic possibility here. After all, (38*), in an epistemic possibility reading, is inconsistent. (39*) epistemically read, on the other hand, is Moore-paradoxical: by asserting its first conjunct one represents oneself as having a certain piece of knowledge (that the ball won’t veer left) while what one asserts with the second conjunct is inconsistent with one’s having that very piece of knowledge.71 71

This—an inconsistency between the content of the representation involved in asserting one conjunct and the content of the assertion of the other—is DeRose’s preferred diagnosis of what goes on in Moore-paradoxical cases. Cf. DeRose (1991: 597f.). It relies on the principle that by asserting that p the speaker represents himself as knowing that p. Given that principle, a slightly different diagnosis of the cases is also possible: that by asserting the conjunction one represents oneself in an inconsistent way—as knowing that the ball won’t veer left and as knowing that it is compatible with what one knows (and could come to know by φ-ing) that the ball won’t veer left. In what follows I will sometimes speak as if the latter diagnosis were intended. This is simply for the sake of brevity, however.

5.4. DeRose on ‘Might’ Sentences

219

If charitable interpretation counts for anything, any available alternative interpretation should be preferred. That (38*) and (39*) sound as bad as they do is evidence that there is no such alternative interpretation. Or so DeRose argues. This brings us to DeRose’s ‘main [positive] reason for accepting’ the General Epistemic Thesis (cf. DeRose 1998: 72f). According to him, it is always ‘wrong’ to conjoin a sentence of the form ‘a might φ’ with the corresponding ‘a won’t φ’ or ‘I know that a won’t φ’ sentences.72 That is, any instance of 40

(I know that) a won’t φ, but a might;

‘sounds wrong’. This suggests, according to DeRose, that someone who utters an instance of (40), like our scientist A, does not merely make a factual error. His error is linguistic. What is it? If there are non-epistemic readings of ‘might’ sentences, there is no obvious answer to this question. Nothing can be linguistically wrong with uttering both conjuncts of (40), at least if neither linguistic nor non-linguistic context force us to interpret the second epistemically. As we have already seen, DeRose, on the other hand, has an answer readily available: in the knowledge case, the mistake is to utter a conjunction whose conjuncts are (more or less obviously) inconsistent, in the other, it is to represent oneself in certain (more or less obviously) inconsistent ways. So, endorsing (GET) affords an explanation of the general impropriety of instances of (40), while DeRose’s account of epistemic possibility is able to explain why some ‘might’ sentences appear to have a natural nonepistemic reading. Consequently, in the absence of any successful challenge to the General Epistemic Thesis, we should endorse it and deny that there are non-epistemic readings of ‘might’ sentences. This concludes DeRose’s argument. My strategy in replying to DeRose is as follows: first, I will set up yet another modification of the scientists example that cannot be interpreted to be about epistemic possibility in DeRose’s sense. This should motivate us to find a flaw in DeRose’s argument. I will then propose an alternative diagnosis of conjunctions of the form of (40): contrary to what DeRose suggests it is typically not acceptable to combine a claim to the effect that the occurrence of some event is genuinely chancy with a claim to nevertheless know that it 72

This is only meant to be true for ‘all in one breath, flatfooted’ assertions, as DeRose calls them. Any ‘unusual or strong stress, intonation or emphasis’ may be a sign of shifting conversational parameters in mid-sentence. Such utterances are irrelevant because we can’t be sure that their acceptability isn’t just a result of those shifts. Cf. DeRose (1998: 70f).

220

Chapter 5. Possibility and Probability

will or won’t occur. Thus, in the objective possibility reading of instances of (40) one openly violates a plausible epistemic principle. No wonder, then, that the availability of a non-epistemic reading does not interfere with the fact that such instances are, typically, odd.73 Consider the following situation involving our two scientists. Scientist A has changed the set up. Now, two balls are sent through a zone of indeterminacy. But, as before, not everything is left to chance. In particular, it is causally determined that one of the balls will veer right and the other left. A may well explain the new set up to B by saying Look. It’s causally determined that one of the balls will veer left and the other right. But there’s still indeterminacy involved. At the moment, it’s still left open which ball will veer which way: the ball that will veer left (whichever it is) might still veer right (and the ball that will veer right might still veer left).

Take the sentence 41 Whichever ball will veer left might still veer right. Clearly, A said something true when uttering (41) in the above explanation of the new set up. Moreover, as in the initial case, it at least looks as if A’s utterance should be read to express non-epistemic possibility. I will now try to show that the only reading in which it expresses a truth is an objective possibility reading. We should first note that, on the face of it, (41) allows for two structural disambiguations, one that is usually called a de dicto and the other a de re reading:74 41* Might (whichever ball will veer left (it will veer right)); and 41** Whichever ball will veer left (Might (it will veer right)). In the de dicto reading, (41*), (41) is hopeless whether it expresses epistemic or objective possibility. In the envisaged scenario, it is not an epistemic possibility (for the scientists then) that whichever ball will veer left will veer right. After all, both know that none of the balls will veer in both directions. But neither is it an objective possibility: scientist A has set up the experiment in such a way that the objective chance that whichever ball will 73

74

That they are not always odd is old news for friends of Modest Mouse: ‘I might / And you might / But neither of us do, though / And neither of us will’ (Modest Mouse: Might (1996)). Thanks to Benjamin Schnieder for pointing this song out to me. Recall the discussion of the distinction between de dicto and de re in chapter 3 (on pages 89–90).

5.4. DeRose on ‘Might’ Sentences

221

veer left will also veer right is zero. So, (41) in the scoping of (41*) does not express a truth regardless of whether it expresses epistemic or objective possibility. This leaves the de re option, i.e. the scoping of (41**). On the face of it, both readings are sensible. Even though the scientists know that whichever ball will veer left will not also veer right, they do not know of whichever ball will veer left that it will not veer right. Likewise, even though there is no non-zero chance that whichever ball will veer left will also veer right, there is a non-zero objective chance for whichever ball will veer left to veer right. At first, thus, it seems that the fact that (41) can express a truth—and is naturally taken to do so in the situation—is due to the distinction between de dicto and de re readings of the sentence, and not to the difference between epistemic and objective possibility. This irenic diagnosis assumes that both scoping options are available for epistemic and objective possibility readings alike. However, this is not clearly true. Indeed, von Fintel and Iatridou argue for what they call the Epistemic Containment Principle: ‘A quantifier cannot have scope over an epistemic modal’ (von Fintel and Iatridou 2003: 174). If the Epistemic Containment Principle holds,75 (41**) is not a genuine scoping option for a reading of ‘might’ in which it expresses epistemic possibility. Instead of repeating von Fintel and Iatridou’s arguments for the general thesis, let me just present a local consideration in favour of the view that (41**) is not a scoping option for an epistemic reading of (41). If (41**) were a scoping option, we should expect that the following sentence, read epistemically, has a structural disambiguation on which it is true in the envisaged situation: 75

The truth of the Epistemic Containment Principle would admittedly be somewhat surprising—perhaps, its truth can be taken as evidence for a thesis like Schnieder’s, discussed in fn. 9, according to which epistemic ‘might’s have a radically different function from non-epistemic ‘might’s. There is a debate (cp. e.g. Huitink 2009) whether the Epistemic Containment Principle holds cross-linguistically and in full generality. However, the evidence Huitink (2009) cites for de re readings of ‘might’ sentences are ones where the modal auxiliary may naturally be read objectively or statistically (Huitink herself conjectures that something like this is responsible for the availability of de re readings). Instead of casting doubt on my use of the Epistemic Containment Principle—and the principle itself, when the ‘epistemic’ in ‘epistemic modal’ is taken seriously—, it would seem that the apparent counterexamples to the Epistemic Containment Principle strengthen my case to the effect that (41) needs to be read non-epistemically for it to express a truth.

222

Chapter 5. Possibility and Probability

42 Both balls might veer right. After all, both balls are such that the scientists do not know of them that they do not veer right. However, (42) clearly does not have a true epistemic reading. Since there is no true epistemic reading of (42), there is no epistemic reading of (42) in which ‘both balls’ has scope over ‘might’. The only available scoping is the one corresponding to (41*), namely 42* Might (both balls (they will veer right)). Since everyone concerned knows that only one of the balls will veer right, (42)—in an epistemic reading—is false. But if no de re epistemic reading of (42) is available, this is good reason to think that there is no de re epistemic reading of (41) either: (41**) is not a genuine scoping option for (41). Since (41) can only be true on a de re reading and epistemic ‘might’s do not have a de re reading, (41) needs to be able to express non-epistemic possibility for A to have said something true by uttering it. Consequently, some ‘might’ sentences do not merely appear to express non-epistemic possibility, they do. If this is correct, the typical unacceptability of sentences of the form 40 (I know that) a won’t φ, but a might; cannot be due to the fact that they only have an epistemic reading and, in that reading, are either inconsistent or Moore-paradoxical. They also have an objective possibility reading, and, in that reading, they are neither inconsistent nor Moore-paradoxical. Why, then, are they still typically unacceptable? I suggest that this is because, on the objective reading, the speaker openly flouts an epistemic principle relating knowledge of chanciness with knowledge of outcomes which is generally presumed to hold. The principle I have in mind is what Hawthorne (2003: 93) calls the Chance-Knowledge Principle: Chance-Knowledge Principle If at t, S knows that there is a non-zero objective chance that p at t, then, at t, S does not know that ¬p. Whether or not the Chance-Knowledge Principle is true as it stands, it seems plausible enough to suppose that we generally take it to be true.76 In fact, we 76

It follows from two principles that Hawthorne (2003: 112) deems to be ‘prima facie attractive principles, which it would be natural [for an account of knowledge] to try to respect’. As Hawthorne and Lasonen-Aarnio (2009: §3) point out, the principle as it stands (as well as the strengthening I also claim to be plausible in the main text) falls prey to cases of the contingent a priori. I take these to be peripheral enough, however, not to impinge on the plausibility of the Chance-Knowledge Principle.

5.4. DeRose on ‘Might’ Sentences

223

generally take an even stronger principle to be true, namely one that merely talks about non-zero objective chances in the antecedent instead of knowledge thereof. When things are chancy—and, perhaps, the chances are not too close to 0 or 1—, we generally take people not to know them. Now, what happens when we assert an instance of (40) and intend it to be read objectively is that we present ourselves as knowing that there is a nonzero objective chance that a will φ by asserting the second conjunct. But in asserting the first conjunct we say that we know, or represent ourselves as knowing, that a will not φ. Consequently, when we assert an instance of (40) we either say something inconsistent with the Chance-Knowledge Principle or represent ourselves in a way inconsistent with it. Since we generally presume something like the Chance-Knowledge Principle to hold, this is reason enough for our assertion to be odd. To be sure, the oddness is not due to an outright inconsistency in what we say or in how we represent ourselves by asserting something, but rather in an inconsistency of either with a plausible epistemic principle. But DeRose’s method of checking for what ordinary speakers find unacceptable without regard to what the facts are does not seem to be sensitive enough to decide between an outright inconsistency and an inconsistency with a generally endorsed further assumption.77 Thus, the availability of a non-epistemic reading is no obstacle to an explanation of the oddness of instances of (40): both readings are unacceptable. I, thus, disagree with DeRose’s claim that the acceptability of 38

I know the ball won’t veer left, though it isn’t now causally determined that it won’t; and

39

The ball won’t veer left, though it isn’t now causally determined that it won’t;

shows that the strange epistemic situation scientist A believes himself to be in does not affect our judgements of acceptability. My suspicion is that an assertion of causal indeterminacy leaves some loopholes an assertion of nonzero objective chance does not, so that we are willing to give scientist A the benefit of the doubt when he asserts either of (38) or (39), while no such latitude is available for assertions of (38*) and (39*). This suspicion is borne 77

Arguably, for instance, cases of so-called category mistakes—‘green ideas sleep furiously’—sound odd because they are inconsistent with generally accepted further truths—that ideas aren’t coloured, that sleeping isn’t a kind of thing that can be done furiously, and so forth—, not because something is intrinsically wrong with the sentences.

224

Chapter 5. Possibility and Probability

out when we compare (38) and (39) to their relatives that explicitly mention objective chances: 38** I know that the ball won’t veer left, although there is a 50:50 chance that it will veer left; and 39** The ball won’t veer left, although there is a 50:50 chance that it will veer left. To my ears, contrary to (38) and (39), these sound just as odd as the corresponding ‘might’ sentences. It’s not the oddity of (38*)/(39*) but the acceptability of (38)/(39) that is surprising! The best explanation is, I think, that the second conjunct of (38) and (39) leaves open the possibility that the direction the ball will veer is determined, though not causally, in some other way, so that the objective chances for the ball to veer left would be zero after all. I conclude that we should reject the General Epistemic Thesis. It is not forced on us by DeRose’s argument, since there is an alternative explanation for DeRose’s data. Further, the General Epistemic Thesis conflicts with our judgments concerning cases like the modified scientists case. Consequently, there is no reason to think—indeed, some reason to doubt—that appearances are deceptive with respect to non-epistemic ‘might’ sentences. Some ‘might’ sentences do not only appear to express non-epistemic possibility, they do. Consequently, the view of ‘might’ and ‘might have’ sentences presented in the last section is not threatened by DeRose’s arguments. Focusing on ‘might’ sentences can shed light on non-epistemic possibility. In particular, they suggest an answer to the question of what the supervenience base for modal truths may be. This is the topic of the next section. 5.5 Supervenience In section 5.1 I argued that there is a conceptual link between possibility and probability. Roughly, there is an underlying probability scale for modal sentences, and the latter are true just in case the probability is not located at the lower endpoint of the scale. In section 5.2 we distinguished different kinds of probability that are candidates for providing the underlying scale for some ‘might’ sentences. It turned out, in section 5.3, that it is objective chances that provide the underlying scale for non-epistemic ‘might’s, since the temporal structure of objective chances would best explain the temporal behaviour of ‘might’s and ‘might have’s as well as the (un)availability of nonepistemic readings for them. This picture was defended, in the last section, against a contention of Keith DeRose. Pace DeRose, there are non-epistemic

5.5. Supervenience

225

uses of ‘might’ sentences, and these uses can shed light on non-epistemic possibility. In particular, the non-epistemic uses of ‘might’ can shed light on the supervenience question, the question of what the supervenience base for non-epistemic modal truths is.78 In section 5.3 I argued that ‘might’ sentences express current non-zero objective chance, while ‘might have’ sentences express past non-zero objective chance. This suggests an answer to the supervenience question. Truths expressed by ‘might’ sentences supervene on truths about the objective chance distribution at the time of utterance, while the truths expressed by ‘might have’ sentences supervene on truths about the objective chance distributions at times earlier than the time of utterance. In general, then, modal truths supervene on truths about objective chances: Objective Chance Supervenience ∀p [it is a modal truth that p → ∃q (it is an objective chance truth that q & ◻ (q → (p because q)))]. As an illustration of Objective Chance Supervenience recall Theseus and the maze. It’s 11.20AM. Theseus might still reach the centre. He might still reach the centre because there is a 25% objective chance that he will reach the centre. Further, necessarily, whenever there is a 25% objective chance that Theseus will reach the centre, he might reach the centre because there is a 25% objective chance that he will. It’s 12.20pm. Theseus didn’t reach the centre. But he might have reached the centre. He might have reached the centre by noon because, for instance, at 11.20am there was a 25% objective chance that he would reach the centre by noon. Further, necessarily, whenever there was a 25% objective chance at 11.20am that Theseus would reach the centre by noon, an hour later he might have reached the centre by noon because at 11.20am there was a 25% objective chance that he would reach the centre in time. Objective Chance Supervenience says that this generalises. Whenever it is a modal truth that p, there is an objective chance truth, the proposition that q, such that, necessarily, whenever the proposition that q is true, it explains the proposition that p. 78

From now on, when I speak about ‘might’ and ‘might have’ sentences, I have nonepistemic uses in mind. Consequently, when I talk about ‘might have’ sentences the scoping in which Perf has scope over Might is relevant. Likewise, when I speak of modal truths, I mean non-epistemic modal truths.

226

Chapter 5. Possibility and Probability

Objective Chance Supervenience qualifies as a superdupervenience thesis, a supervenience thesis whose truth is further explicable.79 There is an answer to the question of why there should be a necessary explanatory link between modal truths and objective chance truths. In the case of Possible Worlds Supervenience, the Lewisian contention was that modal truths supervene on possible worlds truths because the latter analyse the former. In the case of Objective Chance Supervenience, I suggest, the reason is slightly different. Recall the case of at-least weight properties, like weighing at least 60 kg, and specific weight properties, like weighing 67.1 kg.80 The former supervene on the latter, since, necessarily, whenever someone weighs 67.1 kg, he weighs at least 60kg because he weighs 67.1 kg. Moreover, there is a straightforward explanation for why there should be a necessary explanatory link between at-least weight properties and specific weight properties: the latter are determinates of the former—weighing 67.1 kg is one way of weighing at least 60 kg, and so forth. Something similar is going on in the case of modal truths and objective chance truths. The particular distribution of objective chances over propositions (at a time) determines which of them have a positive objective chance of being true (at that time). Thereby, it also determines which propositions’ negations have zero objective chance of being true (at that time). That is to say, the particular distribution of objective chances (at a time) determines which propositions are possible and necessary (at that time). Particular objective chance truths stand to modal truths as particular weight properties stand to at-least weight properties. Stretching the meaning of the term a little, we may say that particular objective chance truths are determinates of modal truths. Consequently, modal truths supervene on objective chance truths because the latter are determinates of the former. In chapter 4 I argued that possible worlds truths supervene on modal truths. Since modal truths in turn supervene on objective chance truths, the picture of the modal realm that emerges is tripartite as pictured in figure 12. Objective chance truths form the base level—they are those truths modal truths supervene on, while they themselves neither supervene on modal truths nor on possible worlds truths. Modal truths comprise the intermediate level. They supervene on objective chance truths, but possible worlds truths supervene on them in turn. Possible worlds truths, finally, comprise the top level. They supervene on modal truths, and, since supervenience is transitive and modal 79 80

Cp. section 2.6 of chapter 2 above. See again section 2.6 of chapter 2 above.

5.5. Supervenience

227

Possible worlds truths Modal truths Objective chance truths Figure 12: The modal realm

truths supervene on objective chance truths, possible worlds truths supervene on objective chance truths as well. I take this to be an attractive view of modal truths. First, it avoids the metaphysical mysteries of Possible Worlds Supervenience. Modal truths do not supervene on truths about epistemically inaccessible and metaphysically dubious objects like possible worlds. Instead the latter supervene on the former which both makes possible worlds metaphysically unproblematic and explains our epistemic access to them. A possible world at which Socrates is a carpenter, say, is just whatever must force that Socrates is a carpenter and decides any other question without forcing everything.81 We can know about possible worlds by drawing on our modal knowledge—our knowledge that Socrates might have become a carpenter, for instance—and our mastery of the possible worlds concepts. Possible worlds are, thus, legitimate tools in the philosopher’s toolbox. As supervenient entities they are there to be used in extensional semantics for our modal discourse and for the nominal quantifiers to range over, e.g., in our philosophical explanations of modal notions like the notion of an essential property. Second, the proposed thesis about the structure of the modal realm elucidates our epistemic access to modal truths themselves. We can know the modal truths we do know by knowing what the present and past objective chances of the relevant propositions are, at least approximately. No doubt we sometimes know these things. Waving doubts about universal determinism, I know that, just before his last tee shot, there was a non-zero chance that Langer would win the tournament. Indeed, I know that there was a significant chance of Langer’s winning the tournament then. I know now and I knew then that this chance was not maximal, that there was also a chance that 81

See section 4.2 of chapter 4 above.

228

Chapter 5. Possibility and Probability

Langer would not go on to win the tournament. Since I know these things, and modal truths supervene on objective chance truths, I am in a position to know the correlated modal truths. In order for a ‘might’ sentence to be true it is enough for the chances not to be at the lower endpoint of the probability scale. Consequently, rather rough and ready knowledge of objective chances suffices to know a whole lot about how things might have been. For instance, although I have enough empirical information about Langer’s situation to know that the objective chance that he would go on to win was non-zero, my knowledge is the result of rather rough extrapolations from similar cases. However, knowledge about chances is not always such a rough and ready affair. Our knowledge may be the result of detailed empirical investigation. For instance, science tells us that the chance of this tritium atom’s decaying within the next 4500 days is very close to 1/2 or that the half-live of this tellurium atom is 7×1024 (7 quadrillion) years. Since science tells me, I know as well. So, as a result of scientific investigations, I know that, although, perhaps, no tellurium atom has ever decayed, any of them might do so tomorrow. Let me end this section by forestalling a misunderstanding. In this section and elsewhere when I aim at generality I speak as if truths that ascribe certain properties—to have such-and-such an objective probability—to propositions constitute the proposed supervenience base for modal truths. This is only for ease of exposition.82 The truths I really have in mind (and cite when I give examples) are expressed by sentences of the form ‘It is so-and-so likely that p’ (understood objectively). Now, in line with a pleonastic conception of abstracta in general83 I am inclined to hold that the corresponding truths expressed by sentences of the form ‘the proposition that p has a likelihood of so-and-so’ are explained by the former: the proposition that Theseus reaches the centre by noon has an objective chance of 25% because it is 25% likely that Theseus reaches the centre, for example. This leaves it open whether the derivative truths about propositions (‘the propositon that p has such-and-such an objective chance’) are sandwiched between the intended objective chance truths on the one hand (‘it is such-and-such likely that p’) and the modal truths on the other (‘Might(p)’), or whether the truths about propositions fail 82

83

Talk of assigning numbers to propositions or, as will become relevant presently, possibilities may also have other perks when it comes to systematisation. But, as was pointed out in chapter 4 section 4.3.2 above, being able to do systematising work does not secure explanatory priority. Recall ch. 4.

5.6. Objective and Metaphysical Possibility

229

to stand in an explanatory relationship with modal truths at all. This is a question I do not have to decide here. A very similar question does get decided, however: the question of whether modal truths supervene on certain truths about possibilities, namely truths about how likely it is that a certain possibility is realised. Since I argued that possibilities are pleonastic entities, I am committed to the view that truths about them are explained by truths which do not mention them. In particular, every truth expressed by a sentence of the form ‘The possibility that p is so-and-so likely to be realised’ (a realisation truth, for short) is explained by something else: the truth expressed by the corresponding sentence of the form ‘It is so-and-so likely that p’. Now, given that I also hold that truths about possibilities supervene on modal truths and supervenience is asymmetric and transitive, these specific truths about possibilities—unlike those about propositions—cannot be sandwiched between the intended objective chance truths on the one hand and the modal truths on the other. But I see no pressure to think otherwise. One and the same thing—in this case: the intended objective chance truths—may explain different things—realisation truths on the one hand and modal truths on the other—which do not stand in any direct explanatory relationship themselves. To sum up: in this section I argued that modal truths supervene on objective chance truths, and that the resulting picture of the modal realm is an attractive one. It allows us to keep possible worlds in our philosophical tool box, while not putting too much of an epistemological and metaphysical burden on them. The burden, rather, lies on objective chances. But they are much better equipped to shoulder the burden, since they are metaphysically innocuous features of the physical world and epistemically accessible to us via empirical investigation. To round up my discussion, I will remark briefly on the relation between the kind of possibility that is expressed by many non-epistemic natural language ‘might’ sentences—objective possibility, possibility that goes with non-zero objective chance and, thus, changes over time—and what philosophers call metaphysical possibility. 5.6

Objective and Metaphysical Possibility

I will end this chapter with some remarks on how possibility conceived of as non-zero objective chance, or objective possibility for short, relates to metaphysical possibility. Dorothy Edgington (2004: 5f.) suggests that what philosophers have in mind when they talk about metaphysical possibility is objective possibility

230

Chapter 5. Possibility and Probability

at some time,84 i.e. MP ◊M p ↔ ∃t (there is a non-zero objective chance at t that p). Given (MP), something is metaphysically possible just in case the corresponding ‘might’ sentence is true at some time.85 Thus, a claim of metaphysical possibility could be made with the help of a ‘might have’ sentence when it is understood that the quantifier over times introduced by Perf is absolutely unrestricted86 and implicatures are ignored.87 For, suppose it is metaphysically possible that p. Then, according to (MP), there is a time at which there is a non-zero objective chance that p. Since objective possibilities don’t open up, there is some time in the past at which there was a non-zero objective chance that p. So, the ‘might have’ sentence is true. On the other hand, if the ‘might have’ sentence is true then there is a time in the past at which there was a non-zero objective chance that p. A fortiori, there is a time at which there is a non-zero objective chance that p, and, thus, according to (MP), it is metaphysically possible that p. Consequently, (MP) would justify the philosophers’ use of ‘might have’s in discussions of metaphysical possibility. (MP) also conforms well to certain essentialist theses held by many philosophers. The orthodox view about essentialist theses is that they are analysable in terms of metaphysical necessity and existence:88 Aristotle is essentially human just in case it is metaphysically necessary that he is human provided he exists at all. Aristotle essentially has the parents he actually has just in case he must have them if he is to exist. This last thesis is an instance of a kind of Origin Essentialism, defended by, e.g., Saul Kripke and Graeme Forbes:89 Origin Essentialism Things that come into existence by a process of natural generation are (metaphysically) necessarily such that if they exist, they have the progenitors they actually have. 84 85

86 87 88 89

See also Mackie (1974). Recall that I assume that it’s understood that we deal with objective possibility readings of ‘might’ and ‘might have’ sentences, and, thus, with readings of ‘might have’ sentences in which Perf has scope over Might. Recall the discussion in fn. 55 above. In particular, the implicature that there no longer is a non-zero objective chance needs to be ignored. Recall the discussion in fn. 61. Cf. e.g. Mackie (2006: 3). For the heterodoxy see Fine (1994). See Kripke (1980: 112) and Forbes (1985: ch. 6).

5.6. Objective and Metaphysical Possibility

231

Arguably, Origin Essentialism is very plausible when it is understood to concern non-zero objective chances. Though there once was a chance that Socrates would become a carpenter, say, there never was a chance that he would be born of different parents.90 However, there are also reasons to doubt (MP). One has to do with the status of the laws of nature, another with our (alleged) epistemic access to metaphysical possibility. First, laws of nature. It is very plausible that objective chances are constrained by the laws of nature. Consequently, no law of nature ever has a non-zero objective chance of not holding. If generalisations are true but chancy, they simply are not laws of nature.91 So, there is no time at which it is an objective possibility that any given law of nature does not hold (though it certainly is an epistemic possibility with respect to many epistemic states). So, laws of nature are objectively necessary at all times. By (MP), laws of nature would be metaphysically necessary. However, as Edgington (2004: 2) admits, when metaphysicians assist their audiences in latching onto metaphysical possibility, they typically contrast it both with epistemic92 as well as with causal or natural possibility. In this connection, they often cite as a paradigm case of metaphysical possibility something whose possibility implies that the laws of nature are not metaphysically necessary. Thus Plantinga on the very first pages of The Nature of Necessity: But what exactly do these words—‘necessary’ and ‘contingent’—mean? […W]e must give examples and hope for the best. […T]he sense of necessity is wider than that captured in first order logic. On the other hand, it is narrower than that of causal or natural necessity. […I]t may be necessary—causally necessary—that any two material objects attract each other with a force proportional to their mass and inversely proportional to the square of the distance between them; it is not necessary in the sense in question. (Plantinga 1974: 1f.)

It is hard to see how this guidance could ever get anyone to latch onto objective possibility at some time as the target of discussion. Moreover, it is equally hard to see how so many philosophers could be so wrong in what they take to be metaphysically possible. Surely, if (MP) were correct, we 90

91 92

Mackie (1974) defends the Kripkean essentialist theses by appeal to the structure of objective possibilities. See also Brody (1980) and the reconstruction of Strawson (1997) in Schnieder et al. (2010: §3). Edgington (2004: §7) argues that Kripke’s discussion of metaphysical modality is best understood in terms of (MP). See, e.g. Edgington (2004: 19) and Lewis (1980: 124f.). I assume, with Edgington, that so-called logical or conceptual possibility is a brand of (idealised) epistemic possibility.

232

Chapter 5. Possibility and Probability

should expect the view that the laws of nature are metaphysically necessary to be the majority view. But it is far from that.93 Second, epistemic access. It is not entirely clear what philosophers mean when they talk about conceivability, and, in fact, doubtful that they all and always mean the same. Let’s assume nevertheless that there is some sense to be made of conceivability that fits philosophical discussions of it. Perhaps, following Yablo (1993: §10), we can say that it is conceivable (for S at t) that p just in case S is, at t, able to imagine a situation which S takes to verify that p.94 Many philosophers think that conceivability considerations provide (perhaps, defeasible) evidence for what is and what is not metaphysically possible.95 However, if metaphysical possibility is non-zero objective chance at some time, it is far from obvious why the fact that someone can conceive that p should be even prima facie evidence for its metaphysical possibility. It is clear that (present) conceivability, say, concerning the proposition that the die will show a 6 at its next toss, does not tell us a whole lot about what the objective chance is that the die will show a 6. Checking how well one can imagine a situation in which the die shows a 6 is not a particularly good substitute for the pivot test, for instance—a test used by gamblers to determine whether a die is loaded. But just as strength (or vividness or what have you) of imaginings does not seem to be particularly good evidence for the determination of particular values of objective chances, neither seems someone’s ability to imagine that the die shows a 6 at its next toss to be particularly 93

94

95

Notable exceptions are, of course, Edgington (2004: §2) herself, Swoyer (1982), Shoemaker (1998) and Bird (2001). However, though the other three authors agree with Edgington’s contention that metaphysical possibility is constrained by the laws of nature, they do not do so because they take metaphysical possibility to be positive objective chance at some time. Rather, they argue that (at least some) laws of nature are metaphysically necessary on the grounds of (non-standard) views on the nature of properties in the case of Swoyer and Shoemaker—roughly, that the causal profile of a property is essential to it—and by citing considerations (that do not obviously generalise to all laws) concerning the essential properties of chemical substances in the case of Bird. Incidentally, some authors (e.g. Leeds 2001: 178) think that laws of nature are best described as ‘metaphysically contingent physical necessities’—a contradictio in adiecto if (MP) is true. I give this proposal only for definiteness. Nothing hinges on its details. I trust that the point I make in the main text can be made with any reasonable account of what ‘philosophical’ conceivability is. For more on conceivability see the papers collected in Gendler and Hawthorne (2002). See again Yablo (1993) and the references therein.

5.6. Objective and Metaphysical Possibility

233

good evidence that there is a chance greater than zero that it will show a 6. If it is a trick die without any side displaying a 6, the chance that it will show a 6 at its next toss is 0. Present ability to imagine that it shows a six (or a seven!) is just not correlated very tightly with present objective chances.96 But what goes for present objective chance seems to go for objective chance at some time as well. We have no difficulty in imagining situations we take to falsify one or another of the laws of nature.97 And, even if determinism were true, we still wouldn’t have a problem imagining situations patently at odds with how things really are. Conceivability is in no obvious way correlated with non-zero chances at some time, nor do we have reason to suspect that it is. I take this to be fairly obvious. But because it is so obvious, it is hard to see how philosophers could have seriously considered the possibility—let alone endorsed the view—that conceivability is our prime epistemic route to metaphysical possibility. Unless, of course, they did not take metaphysical possibility to be non-zero objective chance at some time. (MP) is at odds both with what many philosophers believe to be paradigm cases of metaphysical possibility and with what they take to be good evidence for metaphysical possibility. Since both divergences are rather obvious, I find (MP) at best doubtful as an explication of metaphysical possibility. Although it conforms to some aspects of philosophical discussions of metaphysical possibility, it disagrees with other, central, ones. There is only so much disagreement with common lore—which, in the case of metaphysical possibility, seems to be almost exclusively constituted by the views of philosophers—an explication can take.98 96 97

98

We can also imagine all kinds of falsehoods concerning the past, in spite of the fact that their present objective chance is 0. Sidelle (2002) argues on that basis that the laws of nature are not metaphysically necessary, by showing that post-Kripkean caveats with respect to conceivability are irrelevant in the case of laws of nature. Perhaps, we should read Edgington (2004) rather as advising to give up on the concept of metaphysical possibility and focus on objective and epistemic possibility, and notions definable therefrom, instead: [I]t is not uncommon, in explaining the allegedly important, philosophers’ sense of the modal notions, to do so by contrast with ‘mere’ epistemic possibility, or with ‘mere’ physical possibility (or impossibility). I try to show, on the contrary, that we can get by with just these two families of modal notions […]. (Edgington 2004: 2) The discussion of this section is not meant to disagree with such advice.

234

Chapter 5. Possibility and Probability

So, where does that leave objective possibility? Well, it is the kind of possibility that is firmly rooted in our ordinary ways of speaking and thinking about what, non-epistemically, might or cannot happen and about what might have or could not have happened. It is also the non-epistemic kind of possibility an understanding of which we bring into philosophical discussions of modality, and by which we judge the plausibility of philosophical examples. As such it should be the topic of investigation in philosophical discussions of modality. Edgington’s (MP) is the only proposal that would straightforwardly link this familiar kind of non-epistemic possibility with metaphysical possibility. Its apparent failure leaves open two possible strategies for the friends of metaphysical possibility as a kind of non-epistemic possibility. They could claim (a) that philosophers latch onto metaphysical possibility independently of their pre-philosophical competence with the non-epistemic readings of natural language modal idioms or (b) that metaphysical possibility is still linked to objective possibility, although in a way more complicated than (MP) envisages. Both strategies seem less than promising. For one, if we don’t rely on our competence with natural language modal idioms, it is hard to see how else we manage to get in touch with metaphysical possibility. Further, what is so special about the philosophy class room that makes metaphysical possibility salient enough to override our everyday use of the modal idioms, while whatever it is is inoperative elsewhere? But if the modal idioms don’t take on a special philosophers’ sense when they are used to introduce metaphysical possibility, this counts against option (b) as well. Recall that (MP) has its intuitive basis in how philosophers actually speak. As noted before, (MP) renders legitimate the philosophers’ use of ‘might have’ and ‘couldn’t have’, as long as it is understood that the modal auxilliaries are used non-epistemically, that, consequently, Perf takes scope over the modal, and that all contextual restrictions and implicatures are suspended. No such intuitive basis is available for a more complicated proposal, even if there were one that is extensionally adequate. A more promising diagnosis would start from the observation that the modal idioms have uses in which they express epistemic possibilities. Although it has never been objectively possible that a certain law of nature would fail to hold, it is possible against many epistemic backgrounds: most of us are ignorant of many laws of nature, and many of our predecessors were ignorant even of laws all of us are now acquainted with. So, in an epistemic use of ‘Two material objects might not attract each other with a force

5.6. Objective and Metaphysical Possibility

235

proportional to their mass’ in which such backgrounds are relevant it will express a truth. Further, although conceivability is a poor guide to objective possibility, imagining a situation and checking what we would believe about it appears to be an excellent guide to epistemic possibility against epistemic backgrounds other than our own.99 So, typical philosophical views about what is metaphysically possible and about how we come to know it suggest that an—appropriately restricted—epistemic reading of natural language modal idioms may be pertinent. This is a thesis that will not appeal to many friends of metaphysical possibility, and much more would have to be done to substantiate it.100 So, I leave it as a challenge to my opponent to render either of options (a) and (b) above sufficiently plausible—in particular, more plausible than the alternative suggestion that metaphysical possibility is epistemic in kind.

99

Cp., e.g., Shoemaker (1998: 72ff.) who suggests that conceivability considerations are mainly pertinent to epistemic, rather than metaphysical, possibility. 100 Perhaps most pressingly, something would need to be said about the functioning of the actuality operator on such a proposal.

Chapter 6

Conclusion In this book I argued that the modal realm is tripartite. The non-epistemic modal truths are sandwiched between objective chance truths on the one hand and non-epistemic possible worlds truths on the other. Rather than providing a supervenience base for modal truths, possible worlds truths supervene on modal truths. It is not the case that this atom might decay because there is a possible world at which it does. Rather, there is a possible world at which the atom decays because it might decay. Possible worlds, like other abstract objects, are pleonastic entities. What is true about pleonastic objects is completely determined by how more pedestrian things are. In particular, whether there is a certain possible world and what it forces is completely determined by how things stand modally. This explains why truths about possible worlds must line up with truths about what is possible and necessary, and how we can know about possible worlds even though they are abstract, and, thus, causally inaccessible. It also makes possible worlds particularly bad candidates for providing a supervenience base for modal truths. Modal truths do not supervene on possible worlds truths, since the latter supervene on the former. Objective chance truths are a much better candidate for providing a supervenience base. What Langer might or could not have done, and what this tritium atom might or cannot become, depends on the distribution of objective chances. Langer might have won the tournament because there was a nonzero objective chance that he would. Langer cannot win anymore because there is no such chance now. The atom might decay in the next hour because there is a non-zero objective chance that it will. It cannot turn now—indeed, it could never have turned—into a prime number because there has never been, nor will there ever be, a non-zero chance for it to turn into an abstract object. Modal truths supervene on objective chance truths. Since possible worlds truths supervene on modal truths and supervenience is transitive, the former also supervene on objective chance truths. Objective chance truths

238

Chapter 6. Conclusion

form the ground level on which everything else in the modal realm supervenes. This, I argued, is the answer to the first question Michael Dummett posed in the following passage already quoted at the very beginning of this book: The philosophical problem of necessity is twofold: what is its source, and how do we recognize it? (Dummett 1959: 327)

The view that modal truths supervene on objective chance truths leads to further questions. In particular, one may ask whether we have reached bedrock with objective chance truths. Do objective chance truths supervene on still other truths? If objective chance truths supervene on still other truths, modal truths will supervene on these other truths as well, and there may be a more informative supervenience thesis concerning modal truths available than that argued for in this book. For instance, it seems plausible that facts about an atom’s chance of decay—that the chance of this tritium atom’s decaying within the next 4500 days is about .5, for instance—depend on facts about that atom’s micro-structure. If this generalises, a full-fledged physicalism may be defensible with respect to objective chances, and, in turn, with respect to modal truths: How things might or cannot be supervenes, according to such a view, on how they are non-modally.1 Although I find the emerging picture very attractive, I have to leave discussion of this issue to future research. Also, in this book I have only dealt cursorily with Dummett’s second question, the question of how we can come to know what is possible and necessary. Since modal truths supervene on objective chance truths, a more detailed discussion would have to focus on the question of how we can come to know about the distribution of objective chances. How do we know that this tritium atom has a chance of about .5 to decay within the next 4500 days or that this tellurium atom has the same chance to decay within the next 7 × 1024 (7 quadrillion) years? Determining the frequency of atoms that decayed within the pertinent time may be a viable option in the first case, but it certainly isn’t in the second. Still, even if no tellurium atom has ever decayed, scientists know that it is possible that one will decay tomorrow. But how do they know? Langer may never have been in quite the same situation as he was before he hooked his last tee shot of the 2007 BMW Open. Still, everyone who watched knew that it was possible for him to win the tournament then, that there was a significant chance that he would win. But how did they 1

For some steps in this direction, though leaving out the connection with objective chance, see Mondadori and Morton (1976).

239

know? Again, it seems plausible that the epistemological question may turn on an answer to the supervenience question. Plausibly, chances of decay of tellurium atoms supervene on truths about their micro-structure. Those latter facts may well be epistemically accessible even when no frequency information is available. If this generalises, there may be a story to tell about how we can come to know about how things might be by knowing about how things are. Again, I have to leave further discussion to future research. In this book I defended the thesis that possible worlds truths supervene on modal truths, which in turn supervene on objective chance truths. These supervenience facts fall well short of answering all interesting questions that may be asked about modality. However, I hope to have made it plausible that they provide a fruitful framework in which answers to further questions can be given.

Appendix A

Non-Nominal Quantification Throughout this book I sometimes use non-standard quantification. In particular, I use quantifiers that bind variables that occupy not only the syntactic positions of singular terms but also the syntactic positions of general terms and sentences. How are such formalisms to be understood? This appendix attempts to give a tentative answer. Consider the following sentences: 1

Socrates is wise;

2

Plato is wise.

(1) and (2) invite generalisation into singular term position. The imprudent among us may want to universally generalise on the position of ‘Socrates’ and ‘Plato’. We are all well acquainted with the following semi-formal rendering of this generalisation: 3

∀x (x is wise).

It is less risky to negate the universal generalisation of the negation of the embedded sentence: 4

¬∀x ¬(x is wise); or simply

5

∃x (x is wise).

Both (3) and (5) are semi-formal quantifications into singular term position that have, inter alia, (1) and (2) as instances. They differ in that (3) entails all of its instances, while (5) is entailed by any of its instances. (4), on the other hand, is a negation of a quantification into singular term position that has, among others, the negations of (1) and (2) as instances. The quantification of which (4) is the negation entails all of its instances, so that the negation of any of its instances, e.g. (1) or (2), entails its negation. (3), (5) as well as the embedded sentence of (4) are quantifications into singular term position, since the variables bound by their initial quantifiers occupy syntactic positions that may be filled by a singular term, and their instances

242

Appendix A. Non-Nominal Quantification

are constructed by replacing these variables with singular terms, as in (1) and (2). For brevity I will sometimes call quantification into singular term position nominal quantification (since it is quantification into the position of a name). Sometimes we may want to generalise into positions other than those of a singular term. Consider for instance 6

If Socrates is wise, Plato is wise;

7

If Socrates is a carpenter, Plato is a carpenter.

(6) and (7) invite generalisation into general term position. The imprudent among us may want to universally generalise on the position of ‘wise’ and ‘a philosopher’. We may want to render this generalisation semi-formally as 8

∀F (Socrates is F → Plato is F).

It is less risky to generalise existentially: 9

∃F (Socrates is F → Plato is F).

Both (8) and (9) are quantifications into general term position that have, inter alia, (6) and (7) as instances.1 They differ in that (8) entails (6) and (7), while (9) is entailed by (6) and by (7). (8) and (9) are quantifications into general term position, since the variables bound by their quantifiers occupy syntactic positions that may be filled by a general term, and their instances are constructed by replacing these variables with general terms, as in (6) and (7). Or consider the sentences 10

If Socrates says that snow is white, snow is white;

11

If Socrates says that grass is red, grass is red.

1

General terms are to be distinguished from predicates in the modern, Fregean, sense of the term. Predicates are expressions which combine with singular terms to form sentences. For instance, ‘is wise’ is the predicate in ‘Socrates is wise’. General terms on the other hand are terms which combine with the copula to form predicates, e.g., ‘wise’ in ‘is wise’. Not all predicates have the surface structure ‘be + general term’. For instance, ‘jumps’ is the predicate in ‘Socrates jumps’. Still, even in full verbs we may distinguish between a part which fulfils the function of a general term, the verb stem, and another part which fulfils the function of the copula, the verb ending. To see this, note that ‘Socrates jumps’ is a stylistic variant of ‘Socrates does jump’ in which terms fulfilling the two functions are syntactically separated just as in sentences containing the auxiliary ‘be’. In what follows ‘a is F’ is a stand-in for sentences of the surface grammatical form of ‘Socrates is wise’, but also for sentences like ‘Socrates does jump’ or simply ‘Socrates jumps’. Cp. e.g. Künne (2006: §1).

243

(10) and (11) invite generalisation into sentence position. The imprudent among us may want to universally generalise on the position of ‘snow is white’ and ‘grass is red’. We may want to render this generalisation semiformally as 12

∀p ((Socrates says that p) → p).

It is less risky to generalise existentially: 13

∃p ((Socrates says that p) → p).

Both (12) and (13) are quantifications into sentence position that have, inter alia, (10) and (11) as instances.They differ in that (12) entails (10) and (11), while (13) is entailed by (10) and by (11). (12) and (13) are quantifications into sentence position, since the variables bound by their quantifiers occupy syntactic positions that may be filled by a sentence, and their instances are constructed by replacing these variables with sentences, as in (10) and (11). In this book I assume that quantification into general term position, such as (8) and (9), and quantification into sentence position, such as (12) and (13), are in order. It is, of course, an interesting question how best to understand such quantifications. There are at least three options: (i) non-nominal quantification is to be understood objectually, (ii) non-nominal quantification is to be understood substitutionally, (iii) non-nominal quantification is sui generis—non-nominal quantification is neither objectual nor substitutional nor reducible to anything else.2 To illustrate the three options, let’s focus on quantification into general term position. How should we understand semi-formal sentences like 8

∀F (Socrates is F → Plato is F)?

One obvious way is to explain such quantifications as covert nominal quantifications over objects suitably related to general terms, perhaps sets or properties—the extensions and intensions of general terms respectively. (8) would then say the same3 as the more explicit 2

3

Another option is to follow Boolos in regarding quantification into general term position as some sort of plural quantification. See e.g. Boolos (1984). However, if quantification into general term position is acceptable in its own right, as I will suggest in what follows, understanding the formalism via plural quantification is just as uncalled for as understanding it objectually or substitutionally. One may wonder whether the thesis of sameness of content is stronger than necessary. For instance, couldn’t the objectualist merely claim that nominal quantification over objects suitably related to general terms will figure in an adequate (illuminating, the best) semantics for sentences involving quantification into general term position?

244

Appendix A. Non-Nominal Quantification

8O1

∀A (Socrates ∈ A → Plato ∈ A); or

8O2

∀P (Socrates has P → Plato has P);

where ‘A’ is a variable ranging over sets and ‘P’ is a variable ranging over properties. What looks like non-nominal quantification into general term position would become nominal quantification whose variables range over sets or properties (while the remainder of the embedded sentences is suitably adjusted to allow for a nominal variable; obviously, we couldn’t just put ‘Socrates is P’ as the antecedent of (8O2 ), for instance, since the result of replacing ‘P’ with a singular term is either ungrammatical—in the predicative reading of ‘is’—or does not say what it should say—taking the ‘is’ to be the ‘is’ of identity). Another obvious way to explain sentences like (8) is to understand them meta-linguistically. Instead of being about Socrates and Plato, (8) should be understood to be about a range of sentences, the substitution instances of its embedded sentence. In this understanding, the variable ‘F’ is associated with a set of expressions, its substituends. (8) would then say the same4 , as the more explicit 8S

∀x ∈ S(⌜Socrates is x → Plato is x⌝ is true);

where ‘S’ designates the set of admissible substituends for ‘F’, perhaps the set of English general terms, and the corner quotes, ‘⌜’ and ‘⌝’, are a device for selective quotation.5 Both options so far introduced try to make quantification into general term

4

5

Discussing this variant of the objectualist position would require to get clearer on how two sentences need to be related for one to be associated with the other by an adequate semantic theory. If the relation is sufficiently loose the three options discussed—so modified—may not even be in opposition. Cp. Wright (2007: 162ff.). Friends of substitutional quantification may not agree that the characterisation presented here captures the meaning of sentences in which substitutional quantification occurs, but only their truth-conditions. Substitutional quantification, they may say, is truth-conditionally equivalent to but not propositionally equivalent to nominal quantification over expressions. See van Inwagen (1981). Unfortunately, I am in the position of van Inwagen here: if substitutional quantification is not to be explained in meta-linguistic terms, I don’t understand it. In any case, for the purposes of this discussion the truth-conditionally equivalent formulation given in the main text will suffice. That is, ‘∀x ∈ S(⌜Socrates is x → Plato is x⌝ is true)’ is short for ‘For all elements x of the set of substituends S: the concatenation of “Socrates is ” with x with “→ Plato is ” with x is true’.

245

position into some kind of nominal quantification either over objects suitably related to general terms or over general terms of English themselves. This has the advantage that the apparently non-standard kind of quantification turns into a kind of quantification—quantification into singular term position whose variables range over a restricted class of objects: sets, properties, or certain expressions respectively—we have been familiar with all along. As an explication of a non-standard formalism there is nothing obviously wrong with either of these options.6 In fact, when I use non-nominal quantification in this book, there is no harm done in interpreting it substitutionally throughout or objectually for the most part.7 However, one may wonder whether an understanding of the formalisms ‘∀F(. . . F . . .)’ and ‘∃F(. . . F . . .)’ that turns apparent quantification into general term position into some kind of covert nominal quantification is forced on us. I believe that it is not. An important reason we are happy with semi-formalisms such as 3

∀x (x is wise).

is that they are straightforward formalisations of sentences to be found in natural languages we already understand. (3) is just the semi-formal counterpart of the English sentence 3NL

Everything is wise;

where ‘everything’ fulfils the function both of the initial quantifier ‘∀x’ as well as that of the bound variable. Slightly more stilted, we may separate these functions even in natural language by saying something on the order of ‘Everything is such that it is wise’ instead, having the initial ‘everything’ do the work of the quantifier and the pronoun ‘it’ do the work of the bound nominal variable in (3). The intelligibility of (3NL ) is, thus, part of what licenses our uncritical use of such semi-formalisms as (3). The same goes, of course, for 5

∃x (x is wise).

which is the straightforward semi-formal counterpart of the natural language sentence 5NL

Something is wise.

Likewise, (8O1 ), (8O2 ) and (8S ) are legitimate, which is partly why they may 6 7

There may be less than obvious reasons for dissatisfaction, however, as we will see later on. Some things I said at the end of chapter 2 will not be genuine alternatives on a straightforward objectual reading of quantification into sentence position.

246

Appendix A. Non-Nominal Quantification

provide an acceptable understanding of (8).8 For an analogous reason, we may be happy with semi-formalisms such as 8

∀F (Socrates is F → Plato is F);

as well. For, it seems that (8), too, has a straightforward counterpart in natural languages like English: 8NL

Plato is whatever Socrates is; or

8NL*

Everything Socrates is, Plato is as well.

Likewise, 9

∃F (Socrates is F → Plato is F).

seems to have a straightforward, if somewhat stilted, natural language counterpart: 9NL

There is something such that if Socrates is it, Plato is it, too; or

9NL*

There is some way such that if Socrates is that way, Plato is that way as well.

If so, semi-formalisms such as (8) and (9) can draw their legitimacy from the availability and intelligibility of quantification into general term position in natural languages we already speak and understand well enough. But are the NL-sentences really natural language quantifications into general term position? They employ quantificational idioms like ‘whatever’, ‘everything’, ‘something’, the anaphoric ‘it’, ‘some way’ and ‘that way’. We are accustomed to view such idioms as signalling nominal quantification. However, this is not so in the sentences in question, which can be seen by considering the acceptable namely-riders for existential generalisations such as (9NL ). Since (9NL ) is slightly stilted,9 let’s look at the generalisation which uses conjunction instead: 14

∃F (Socrates is F & Plato is F);

14NL

Socrates is something Plato is as well.

8

9

Another important reason we are happy with (3) and its ilk is, of course, that there are acceptable semantics for the machinery of nominal quantification. However, these semantics themselves employ the resources of nominal quantification, and can, thus, not give legitimacy to nominal quantification. As many existential generalisations—whether nominal or non-nominal—with an embedded conditional seem to be. Perhaps, because they are too weak to be of much use.

247

When we know of instances of an existential generalisation that make the generalisation true, we sometimes specify such an instance with the help of a ‘namely’-clause. The following is an example of the phenomenon involving nominal quantification: 15

There is a property such that both Socrates and Plato exemplify it, namely wisdom.

The phrase that follows the ‘namely’ here can be used to construct a verifying instance of our generalisation in a straightforward way by plugging it into the open place that results from deleting the quantificational idiom: 16

Both Socrates and Plato exemplify wisdom.

We can be assured that (15) is a case of nominal quantification by noting that the expression following the ‘namely’ in an acceptable namely-rider is indeed a singular term, or, in any case, is used as a singular term in the constructed instance.10 When we turn to (14NL ) we see that an acceptable namely-rider has to consist (in addition to ‘namely’) not of a singular but of a general term: 17

Socrates is something Plato is as well, namely wise, a philosopher, …; not

18

Socrates is something Plato is as well, namely wisdom, the property of being a philosopher, ….

Plugging the general terms in (17) into the open places that result from deleting the quantificational idiom in (14NL ) yields verifying instances of (14NL ): 19

Socrates is wise and Plato is wise;

20

Socrates is a philosopher and Plato is a philosopher.

Plugging in the singular terms in (18), on the other hand, produces grammatical garbage:11 21

Socrates is wisdom and Plato is wisdom;

22

Socrates is the property of being a philosopher and Plato is the property of being a philosopher.

10 11

Some expressions may have singular term uses and non-singular term uses, as, for instance, ‘green’ in ‘her favourite colour is green’ and ‘her shirt is green’ respectively. If we read the ‘is’ in the instances in the same way as the ‘is’ in the generalisation: as the ‘is’ of predication. If we read it as the ‘is’ of identity, the purported verifying instances are neither instances (since meanings are not held constant) nor true.

248

Appendix A. Non-Nominal Quantification

We can thus be assured that (14NL ) is, despite its employment of quantificational idioms also known from nominal quantification, a natural language quantification into general term position. Since we already understand these quantifications, we can mobilise this understanding to understand semiformal sentences like (14) as well. Consequently, I see no need to resort to alternative characterisations of them in terms of nominal quantification. Quantification into general term position is acceptable in its own right and has no need to draw its legitimacy from quantification into singular term position. Both kinds of quantification are already present in natural language.12 Given the availability in natural language of quantification into general term position, we can see that the alternative characterisations of the semiformal sentences sometimes disagree with the intuitive understanding of such sentences. Take first the objectual characterisation which explains apparent quantification into general term position as covert nominal quantification over sets and consider the following sentence:13 23

∃F ∀x (x is F ↔ x ∉ x).

(23) is the semi-formal rendering of a fairly straightforward natural language quantification into general term position, namely 23NL

There is something all and only the self-membered things are.

If we understand (23) via (23NL ), we can see that it should be true. After all, the namely-rider ‘namely not self-membered’ supplies the (trivial) verifying instance. On the other hand, understanding (23) in terms of nominal quantification over sets turns it into a falsehood. According to it, (23) means the following: 23O

∃A ∀x (x ∈ A ↔ x ∉ x);

where ‘A’ ranges over sets. But we know that there is no such set: if there were, it would have to be a member of itself just in case it was not a member of itself—a logical impossibility. Consequently, (23O ) is false, and, thus, (23) would have to be false as well if it had to be understood in terms of nominal quantification over sets.14 An analogous argument can be run with 12 13 14

For more on this, in particular on how to understand ‘polyadic second-order quantifiers’ as, e.g., in ‘∃R ∀x, y (xRy → yRx)’ see Rayo and Yablo (2001). Cp. Rayo and Yablo (2001: §7). This is why I said in fn. 6 that there may be reasons for dissatisfaction with the alternative characterisations. This case, in particular, does not merely turn on our intuitive judgements of natural language quantifications into general term position. It could equally start with the observation that an existential quantification (whether into gen-

249

respect to an understanding of apparent quantification into general term position as covert nominal quantification over properties.15 Therefore, (23)’s characterisation in terms of nominal quantification over objects suitably related to general terms disagrees with its intuitive understanding in certain cases. Likewise for the substitutional characterisation. Sometimes existential generalisations are true even though they have no true instances, due to the expressive limitations of the language in which they are formulated. This may well be so in the case of ‘There is a pebble at the beach of Brighton’, if, as a matter of empirical fact, no one has ever bothered to name a pebble at the beach of Brighton. If so, there is no singular term of English such that the concatenation of it with ‘is a pebble at the beach of Brighton’ is a true sentence. Consequently, ‘There is a pebble at the beach of Brighton’ would be a true existential generalisation that, as a matter of empirical fact, has no true instances.16 Some existential generalisations cannot even have true instances, even though the existential generalisation itself may well be true, as, for instance ‘There is a pebble at the beach of Brighton which has never been, and will never be, referred to’. The same appears to be true of existential generalisations into general term position. The following may be an example: 24NL

There is some way Socrates is, and there is no general term of English that applies to all and only those things that are that way.

(24NL ) is meant to be the natural language counterpart of the following semiformal sentence: 24

∃F (Socrates is F & ¬∃x (x is a general term of English & ∀y (x applies to y ↔ y is F))).

In all likelihood, (24NL ) is true. Just as English does not have a name for everything there is, so, for some way things are, English does not have a

15

16

eral or singular term position) should follow from any of its instances. ‘All and only the self-membered things are self-membered’ seems to be a true instance of (23). Thus, (23) itself should be true. However, (23O ), its alleged objectual analysis, is not. Consider ‘∃F ∀x (x is F ↔ ¬x exemplifies x)’ instead of (23) and note that there is no property exemplified by all and only those things that do not exemplify themselves. If there were, it would have to exemplify itself just in case it did not—a logical impossibility. Things get more complicated once we take account of indexical singular terms. Nothing essential turns on this.

250

Appendix A. Non-Nominal Quantification

general term that applies to those and only those things that are that way.17 But clearly, (24NL ) does not have true instances. For, its instances are formed by deleting the initial quantifier and uniformly replacing the newly unbound variables with a general term of English. For instance, 25 Socrates is wise & ¬∃x (x is a general term of English & ∀y (x applies to y ↔ y is wise)); is an instance of (24NL ). (25) is false, since its nominal quantifier ranges, inter alia, over the general terms of English which are available for substitution for the general term variable, ‘wise’ in particular. Consequently, the negated nominal quantification ‘∃x (x is a general term of English & ∀y (x applies to y ↔ y is wise))’ is true, and, thus, (25)’s whole second conjunct false. This generalises: whenever we can construct an instance of (24NL ), the general term with which the instance is constructed will make the negated nominal quantification true, and, thus, the instance’s whole second conjunct false. Thus, (24NL ) seems to be a true existential generalisation into general term position that, nevertheless, cannot have true instances. If we understand (24) via (24NL ), we can see that it is very likely that it should be true. On the other hand, understanding (24) substitutionally turns it into a falsehood. Understood substitutionally, (24) means the following: 24S ∃x ∈ S (⌜Socrates is x & ¬∃y (y is a general term of English & ∀z (y applies to z ↔ z is x))⌝ is true). For the same reason that (24NL ) cannot have true instances, (24S ) is false. S has all and only the general terms of English as members. Trivially, the concatenation of ‘z is ’ with any of them is true under an assignment of an object to the variable ‘z’ just in case ‘y applies to z’ is true under an assignment of the same object to the variable ‘z’ and the general term in question to the variable ‘y’. Thus, the substitutional characterisation of (24) disagrees with its intuitive understanding in certain cases. In particular, the substitutional characterisation must disagree with the intuitive verdict that there may be true existential generalisations that do not have true instances.18 17

18

Cp. e.g. Hugly and Sayward (1996: 245f.). One way they argue for the claim is by extrapolation. In the 17th century, English lacked the general term ‘electron’, and any other general term that applies to all and only electrons. Thus, English as spoken in the 17th century was expressively limited in the pertinent way. It would be very surprising if the English we speak today were not expressively incomplete in the very same way, although, of course, we cannot cite a general term it lacks. There are substitutional characterisations on the market that are more sophisticated than the one I presented here. See e.g. Hugly and Sayward (1996: ch. 13). Unlike

251

So far I have argued that my use of the formalisms ‘∃F’ and ‘∀F’ which bind variables in the position of general terms is unproblematic in so far as they are merely formalisations of quantificational idioms we already find in natural languages, and that we are fairly familiar with. In explaining formal devices that bind variables in general term position we do not need to resort to nominal quantification over objects suitably related to general terms like sets or properties or to nominal quantification over expressions. Just as in the case of nominal quantification we may rely on our understanding of such devices available in the language we already speak. I also gave examples in which—due to the set-theoretic paradoxes or the expressive limitations of natural languages—the alternative characterisations yield different results from the intuitive understanding available to us. However, these divergences are fairly local and, at least in the case of the substitutional characterisation, unlikely to be relevant to my use of non-standard quantification in this book. No great harm is done if the so disposed reader continues to read the nonnominal quantifiers that occur in this book substitutionally, or, for the most part, objectually. What should we say about quantification into sentence position? Some authors argue that we should take an exactly parallel stance. According to them, sentential quantification is already present in natural languages. Thus, there is no need to offer characterisations of the formal devices ‘∃p’ and ‘∀p’ that make them out to be covert nominal quantification over objects suitably related to sentences—states of affairs or propositions, perhaps—or covert nominal quantification over sentences themselves. We may then try to bolster our case by giving examples in which such alternative characterisations yield results that are in disagreement with the intuitive understanding of sentential quantification that derives from our use of such idioms in the language we already speak. This is, for instance, the strategy of Hugly and Sayward who offer 26NL

Bill says that things are a certain way, and things are that way;

as the natural language sentential quantification that may be formalised as the naïve one, they may well not disagree with the intuitive understanding on the truth-values of particular generalisations. Even though I cannot help but think that quantification into general term position is about how things are and not about how they can be described to be in (extensions of) this or that language, for the purposes of this book there is no harm done in understanding it according to some extensionally adequate version of the substitutional characterisation if such there be.

252

Appendix A. Non-Nominal Quantification

26 ∃p (Bill says that p & p).19 In (26NL ) we are to understand ‘things are a certain way’ as the natural language quantificational idiom that corresponds to (26)’s initial quantifier and the first occurrence of the thus bound variable, while ‘things are that way’ functions as a ‘pro-sentence’ corresponding to the second occurrence of the bound variable in (26). For sentential quantification, so-called pro-sentences are meant to fulfil the anaphoric function that pronouns fulfil in the case of nominal quantification. Just as the ‘it’ in ‘something is such that it is wise’ is meant to be formalisable by a variable bound by the quantifier introduced by the quantificational phrase ‘something is such that’, so ‘things are that way’ is meant to be formalisable by a bound variable, albeit one that occupies the syntactic position of a sentence, not that of a singular term. As in the case of nominal quantification, we may expand the quantificational idiom ‘things are a certain way’ so that its two functions—introducing a sentential quantifier and binding the first occurrence of the sentential variable—become apparent. Perhaps the following will do: 26NL* For some way things are-or-aren’t, Bill says that things are that way, and things are that way.20 In a similar vein, Wolfgang Künne writes the following: Is there also a foothold for [sentential quantification] in some common idiom or other? Yes, there is […]. Everyday language provides us with prosentences such as ‘Things are thus’, ‘This is how things are’, ‘Things are that way’ (‘Es verhält sich so’). So we can explain ‘∃p (. . . p . . .)’ by ‘For some way things may be said to be. . .things are that way. . .’. (Künne 2003: 364)

I would be happy to follow the lead of these authors, if I thought that sentential quantification were already present in natural languages. However, I am not convinced that it is. In particular, I am not convinced that natural languages contain genuine pro-sentences. Consider ‘things are that way’, the phrase explicitly offered by Künne as a natural language pro-sentence 19 20

See Hugly and Sayward (1996: 252). This is an application of Rayo and Yablo (2001: §VIII)’s strategy to the sentential case. The cumbersome ‘are-or-aren’t is due to the fact that ‘For some way things are, things aren’t that way’ is quite obviously not a felicitous rendering of ‘∃p¬p’. As will become apparent in the next quotation, other candidates are available. For instance, as witnessed by the quotation to follow, Künne puts his money on ‘For some way things may be said to be’ instead as a natural language counterpart of the existential sentential quantifier. Since I will argue that ‘things are that way’ is not a genuine pro-sentence anyway, this difference is irrelevant for the discussion to follow.

253

and used by Hugly and Sayward in their natural language rendering of (26). When I argued above that, for instance, 14NL Socrates is something Plato is as well. is a genuine case of natural language quantification into general term position I offered the results of the ‘namely’ test as evidence. When we know of verifying instances of an existential generalisation, we can specify such an instance with the help of a namely-rider. The phrase that follows the ‘namely’ in such namely-riders will be terms that go in the syntactic position suitable for variables bound by the quantifier in question. The acceptability of 17 Socrates is something Plato is as well, namely wise, a philosopher, …; I argued, is evidence that (14NL ) is indeed quantification into general term position, since ‘wise’ and ‘a philosopher’ are general terms. The unacceptability of 18 Socrates is something Plato is as well, namely wisdom, the property of being a philosopher, …; on the other hand, is evidence that (14NL ) is not a case of quantification into singular term position, since appending singular terms like ‘wisdom’ and ‘the property of being a philosopher’—and, indeed, any other singular term—to the ‘namely’ produces grammatical garbage. How do things stand with the alleged pro-sentence ‘things are that way’? Reconsider 26NL Bill says that things are a certain way, and things are that way. According to Hugly and Sayward, the following is a true instance of (26NL ): 27 Bill says that snow is white, and snow is white. Since (27) is a true instance of (26NL ) we should expect that we should be able to felicitously append the sentence ‘snow is white’ to the ‘namely’ of a namely-rider for (26NL ): 28? Bill says that things are a certain way, and things are that way, namely snow is white. But like (18), and unlike (17), (28?) is grammatical garbage. The ‘namely’ test suggests that (26NL ) is not a genuine case of quantification into sentence position, and, correlatively, that ‘things are that way’ is not a genuine prosentence. I should point out that I agree with Künne and Hugly and Sayward that (26NL ) is an acceptable piece of natural language quantificational discourse.

254

Appendix A. Non-Nominal Quantification

I only disagree with them in their classification of ‘things are that way’— and, for that matter, of the other phrases Künne offers—as natural language pro-sentences. But if ‘things are that way’ is not a genuine pro-sentence, how does it work instead? Again, I suggest, the ‘namely’ test points us in the right direction. For, the following is an acceptable specification: 28!

Bill says that things are a certain way, and things are that way, namely such that snow is white.

The phrase following the ‘namely’ in the namely-rider of (28!)—‘such that snow is white’—is a general term: it combines with the copula to form predicates, expressions that combine with singular terms to form sentences.21 This suggests that ‘things are that way’ is complex in the way we may have expected it to be all along. It consists of ‘things are’ and a certain pro-form. When marking the quantificational idiom and pro-forms in (26NL ) we should have marked them as follows: 29NL

Bill says that things are a certain way, and things are that way.

Here ‘a certain way’ corresponds to a quantifier and the first occurrence of a bound variable, while ‘that way’ corresponds to the second occurrence of that variable. But it is not a variable in sentence position but in the position of a general term. The more explicit semi-formal variant of (29NL ) is thus: 29

∃F (Bill says that things are F & things are F).

(29NL ) aka (26NL ) is not a case of natural language quantification into sentence position. Instead, it is a case of natural language quantification into general term position—a device I have already argued to be available in natural language. ‘Things are’ in (29NL ) also functions as it appears to. ‘Are’ is the copula, and ‘things’ functions as the subject expression that is needed to make the complex phrase into a sentence. Arguably, however, ‘things’ is a dummy subject in (29NL ): it is present for grammatical reasons but has no semantic import, as, for instance, the ‘it’ in ‘it is raining’, and unlike the ‘it’ in ‘it is in the living room’. This explains why it may appear as if (29NL ) is a case of natural language quantification into sentence position, and that (27) is one of its instances. For, (27) and 21

‘Such that snow is white’ contains a sentence, but it is not itself a sentence, just as ‘the philosopher who is wise’ contains a predicate but is not itself a predicate.

255

30

Bill says that things are such that snow is white, and things are such that snow is white; do not seem to differ significantly in what they say.22 If what has been said so far is correct, and we cannot understand ‘∃p (. . . p . . .)’ and ‘∀p (. . . p . . .)’ via natural language sentential quantification we are already familiar with—because natural language does not provide for quantification into sentence position, or, at any rate, we have not seen clear examples that show that it does—how else should we understand the formalisms? Our discussion of the shortcomings of ‘things are that way’ as a pro-sentence suggest a way out. When I write ‘∃p (. . . p . . .)’ and ‘∀p (. . . p . . .)’ in this book, this can be read as an abbreviation for ‘things are’ quantifications into general term position in which ‘things’ is a dummy subject. This ensures that the potential verifying instances of these quantifications are formed exclusively with the help of ‘such that’ general terms. The proposal is thus: Sentential Quantification ‘∀p (. . . p . . .)’ is shorthand for ‘∀F (. . . thingsdummy are F . . .)’ ‘∃p (. . . p . . .)’ is shorthand for ‘∃F (. . . thingsdummy are F . . .)’ This stipulation concludes my discussion of the non-standard quantification that is used throughout this book. Quantification into general term position is a kind of quantification we are already familiar with through our mastery of natural language quantificational idioms. These idioms can be used to characterise the unfamiliar formalism that apparently quantifies into sentence position as well.

22

(27) is, of course, a much more agreeable piece of English prose.

Name Index Achinstein, Peter, 41–43 Adams, Robert, 177 Armstrong, David, 117, 164, 174–175 Bach, Kent, 78 Balaguer, Mark, 170 Bealer, George, 184 Benacerraf, Paul, 118 Bennett, Jonathan, 13, 14, 18, 36, 170 Bennett, Karen, 15, 18, 31, 55, 59, 62–64, 66 Bird, Alexander, 232 Blackburn, Simon, 9, 33 Boolos, George, 145, 243 Brody, Baruch, 231 Burge, Tyler, 42 Carnap, Rudolf, 187, 188, 197 Chandler, Hugh, 35, 36 Chihara, Charles, 87, 88, 109 Coates, Jennifer, 182, 185, 191 Condoravdi, Cleo, 182, 201–207, 211 Correia, Fabrice, 15, 16, 18, 19, 23, 24, 31, 38, 40, 69 Cresswell, Max, 153 Crossley, John, 101 Davies, Martin, 101 DeRose, Keith, 183, 184, 191, 192, 194, 199, 200, 214–224

Divers, John, 32, 47, 71, 73, 85, 114, 178 Dummett, Michael, 8–10, 238 Edgington, Dorothy, 120, 185, 187, 190, 191, 229, 231–234 Egan, Anthony, 184, 191, 192, 208, 214 Fara, Michael, 72, 88, 89, 92, 97, 101, 102, 105–109, 111, 112 Field, Hartry, 143, 169 Fine, Kit, 27, 40, 44, 121, 123, 124, 146, 171, 230 Forbes, Graeme, 7, 71, 89, 94, 97, 101, 102, 105, 124, 184, 230 Frege, Gottlob, 242 Gillies, Anthony, 191, 192 Gupta, Anil, 128 Hájek, Alan, 199 Hale, Bob, 115, 130, 131, 136, 143, 194 Hawthorne, John, 184, 191, 192, 208, 214, 222 Hodes, Harold, 101–103 Horgan, Terence, 18, 44–46 Horwich, Paul, 128 Huemer, Michael, 194 Hughes, George Edward, 153 Hugly, Philip, 250–253 Huitink, Janneke, 221 Humberstone, Lloyd, 101, 163, 164

258

Iatridou, Sabine, 221 Jubien, Michael, 81–83, 85 Kim, Jaegwon, 15, 16, 18–21, 24–28, 30, 36, 41, 55–59, 62, 69 Kratzer, Angelika, 183, 191 Kripke, Saul, 7, 59, 70, 94, 153, 154, 162, 164, 165, 184, 230, 231 Künne, Wolfgang, 42, 47, 52, 63, 128, 158, 242, 252–254 Larson, Richard, 203 Lasonen-Aarnio, Maria, 222 Leeds, Stephen, 232 Lewis, David, 11, 71–76, 78–112, 114, 117, 175, 178, 187–188, 193–195, 197, 209, 226, 231 Löbner, Sebastian, 211 MacBride, Fraser, 169 Mackie, John, 230, 231 Mackie, Penelope, 184, 230 McLaughlin, Brian, 15, 17, 18, 20, 23, 55, 62, 65 Modest Mouse, 220 Mondadori, Fabrizio, 212, 213, 238 Morton, Adam, 212, 238 Mulligan, Kevin, 152 Partee, Barbara, 202, 204, 205 Paull, Cranston, 55, 59–63 Petrie, Bradford, 55, 58–60, 63 Plantinga, Alvin, 113, 171, 177–179, 231 Plato, 48 Pollock, John, 179 Prior, Arthur, 124 Ramachandran, Murali, 101, 105

Name Index

Rayo, Augustín, 248, 252 Richards, Tom, 80 Rosen, Gideon, 41, 43, 44, 71 Ruben, David H., 41–43 Sadock, Jerrold, 131 Salmon, Nathan, 35, 36 Sayward, Charles, 250–253 Schaffer, Jonathan, 170, 177–178 Schechter, Josh, 146 Schiffer, Stephen, 114–116, 119–121, 123, 125–127, 129–134, 136–147, 149–152, 164 Schnieder, Benjamin, 18, 40, 42, 49, 50, 52, 54, 129–133, 135, 157, 187, 188, 191, 221, 231 Schulz, Moritz, 185, 187, 191, 231 Segal, Gabriel, 203 Sharvy, Richard, 42, 48, 49, 52 Shoemaker, Sydney, 232, 235 Sidelle, Alan, 233 Sider, Ted, 55, 59–63 Sidgwick, Henry, 36–37 Skyrms, Brian, 80 Stalnaker, Robert, 177 Strawson, Peter, 185, 187, 191, 231 Swoyer, Chris, 232 van Inwagen, Peter, 77, 79, 81, 86, 88, 244 Vendler, Zeno, 203 von Fintel, Kai, 191, 192, 221 von Mises, Richard, 197 Weatherson, Brian, 184, 191, 192, 208, 214 Williams, Donald, 116 Williamson, Timothy, 42, 72, 88, 89, 92, 96, 97, 101, 102, 105–109,

Name Index

111, 112, 171, 192, 193 Wright, Crispin, 115, 130, 131, 136, 143–145, 169, 244 Yablo, Stephen, 232, 248, 252 Yalcin, Seth, 187, 190 Zimmerman, Dean, 31 Zwicky, Arnold, 131

259

Bibliography Achinstein, P. 1983: The Nature of Explanation. Oxford: Oxford University Press. Adams, R. M. 1974: ‘Theories of Actuality’. Noûs 8, pp. 211–231. —— 1981: ‘Actualism and Thisness’. Synthese 49, pp. 3–41. Armstrong, D. M. 1978: Nominalism and Realism. Universals and Scientific Realism Volume I. Cambridge: Cambridge University Press. Bach, K. 2005: ‘Context ex Machina’. In Szabó, Z. G. (ed.): Semantics versus Pragmatics, Oxford: Oxford University Press, pp. 15–44. Balaguer, M. 1998: Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press. Bealer, G. 2002: ‘Modal Epistemology and the Rationalist Renaissance’. In Gendler and Hawthorne (2002), pp. 71–125. Benacerraf, P. 1973: ‘Mathematical Truth’. The Journal of Philosophy 70, pp. 661–679. Bennett, J. 1988: Events and their Names. Oxford: Clarendon Press. Bennett, K. 2004a: ‘Global Supervenience and Dependence’. Philosophy and Phenomenological Research 68, pp. 501–529. —— 2004b: ‘Spatio-Temporal Coincidence and the Grounding Problem’. Philosophical Studies 118, pp. 339–371. Bird, A. 2001: ‘Necessarily, Salt Dissolves in Water’. Analysis 64, pp. 267– 274. Blackburn, S. 1985: ‘Supervenience Revisited’. In Hacking, I. (ed.): Exercises in Analysis: Essays in Honour of Casimir Lewy, Cambridge: Cambridge University Press, pp. 47–69. —— 1993: ‘Morals and Modals’. In Essays in Quasi-Realism, New York and Oxford: Oxford University Press, pp. 52–74.

262

Bibliography

Boolos, G. 1984: ‘To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)’. The Journal of Philosophy 81, pp. 430–449. —— 1990: ‘The Standard of Equality of Numbers’. In Boolos, G. (ed.): Meaning and Method: Essays in Honor of Hillary Putnam, Cambridge: Cambridge University Press, pp. 261–277, reprinted in Boolos (1998: 202– 219). Page references are to the reprint. —— 1998: Logic, Logic, and Logic. Cambridge, MA and London: Harvard University Press. Brody, B. 1980: Identity and Essence. Princeton, NJ.: Princeton University Press. Burge, T. 1979: ‘Individualism and the Mental’. Midwest Studies in Philosophy 4, pp. 73–121. Carnap, R. 1945: ‘The Two Concepts of Probability’. Philosophy and Phenomenological Research 5, pp. 513–532. Chandler, H. S. 1976: ‘Plantinga and the Contingently Possible’. Analysis 36, pp. 106–109. Chihara, C. 1998: The Worlds of Possibility. Modal Realism and the Semantics of Modal Logic. Oxford: Clarendon Press. Coates, J. 1983: The Semantics of the Modal Auxiliaries. London, Canberra: Croom Helm. Condoravdi, C. 2002: ‘Temporal Interpretation of Modals. Modals for the Present and for the Past’. In Beaver, D. I., L. D. Casillas Martínez, B. Z. Clark and S. Kaufmann (eds.): The Construction of Meaning, Stanford: CSLI Publications, pp. 59–88. Correia, F. 2005: Existential Dependence and Cognate Notions. Munich: Philosophia Verlag. Crossley, J. N. and L. Humberstone 1977: ‘The Logic of “Actually”’. Reports on Mathematical Logic 8, pp. 11–29. Davidson, D. 1970: ‘Mental Events’. In Foster, L. and J. W. Swanson (eds.): Experience and Theory, London: Duckworth, reprinted with a postscript in Davidson (1980: 207–225). References are to the reprint. —— 1980: Essays on Actions and Events. Oxford: Clarendon Press.

Bibliography

263

Davies, M. and L. Humberstone 1980: ‘Two Notions of Necessity’. Philosophical Studies 38, pp. 1–30. DeRose, K. 1991: ‘Epistemic Possibilities’. The Philosophical Review 100, pp. 581–605. —— 1998: ‘Simple “Might”s, Indicative Possibilities and the Open Future’. The Philosophical Quarterly 48, pp. 67–82. Divers, J. 1992: ‘Modal Supervenience and Modal Realism’. Theoria 58, pp. 99–115. —— 2002: Possible Worlds. London, New York: Routledge. —— 2004: ‘Agnosticism About Other Worlds: A New Antirealist Programme in Modality’. Philosophy and Phenomenological Research 69, pp. 660–685. —— 2006: ‘Possible-Worlds Semantics Without Possible Worlds: The Agnostic Approach’. Mind 115, pp. 187–225. Dummett, M. 1959: ‘Wittgenstein’s Philosophy of Mathematics’. The Philosophical Review 68, pp. 324–348. Edgington, D. 2004: ‘Two Kinds of Possibility’. Proceedings of the Aristotelian Society. Supplementary Volume 78, pp. 1–22. Egan, A., J. Hawthorne and B. Weatherson 2005: ‘Epistemic Modals in Context’. In Preyer, G. and G. Peter (eds.): Contextualism in Philosophy: Knowledge, Meaning, and Truth, Oxford: Oxford University Press, pp. 131–170. Fara, M. and T. Williamson 2005: ‘Counterparts and Actuality’. Mind 114, pp. 1–30. Field, H. 1980: Science Without Numbers. A Defence of Nominalism. Princeton, NJ: Princeton University Press. —— 1984: ‘Platonism for Cheap? Crispin Wright on Frege’s Context Principle’. Canadian Journal of Philosophy 14, pp. 637–662. Fine, K. 1978: ‘Model Theory for Modal Logic Part I – The De Re/De Dicto Distinction’. Journal of Philosophical Logic 125–156, pp. 125–156. —— 1994: ‘Essence and Modality’. Philosophical Perspectives 8, pp. 1–16.

264

Bibliography

—— 1995a: ‘Ontological Dependence’. Proceedings of the Aristotelian Society 95, pp. 269–290. —— 1995b: ‘Senses of Essence’. In Sinnott-Armstrong, W. (ed.): Modality, Morality, and Belief. Essays in Honor of Ruth Barcan Marcus, Cambridge: Cambridge University Press, pp. 53–73. —— 2012a: ‘Guide to Ground’. In Correia, F. and B. Schnieder (eds.): Metaphysical Grounding. Understanding the Strcuture of Reality, Cambridge: Cambridge University Press, pp. 37–80. —— 2012b: ‘The Pure Logic of Ground’. The Review of Symbolic Logic 5, pp. 1–25. Forbes, G. 1982: ‘Canonical Counterpart Theory’. Analysis 42, pp. 33–37. —— 1985: The Metaphysics of Modality. Oxford: Clarendon Press. —— 1989: Languages of Possibility. Oxford: Blackwell. Gendler, T. S. and J. Hawthorne (eds.) 2002: Conceivability and Possibility. Oxford: Clarendon Press. Gupta, A. 1993: ‘Minimalism’. Philosophical Perspectives 7, pp. 359–369. Hájek, A. 2007: ‘The Reference Class Problem is Your Problem Too’. Synthese 156, pp. 563–585. Hale, B. 1997: ‘Modality’. In Hale, B. and C. Wright (eds.): A Companion to the Philosophy of Language, Oxford: Blackwell, pp. 487–514. Hale, B. and C. Wright 2000: ‘Implicit Definition and the A Priori’. In Boghossian, P. and C. Peacocke (eds.): New Essays on the A Priori, Oxford: Clarendon Press, pp. 286–319. —— 2001: The Reason’s Proper Study. Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press. —— 2009: ‘Focus Restored: Comments on John MacFarlane’. Synthese 170, pp. 457–482. Hare, R. M. 1952: The Language of Morals. Oxford University Press. —— 1984: ‘Inaugural Address: Supervenience’. Proceedings of the Aristotelian Society, Supplementary Volumes 58, pp. 1–16. Hawthorne, J. 2003: Knowledge and Lotteries. Oxford: Oxford University Press.

Bibliography

265

Hawthorne, J. and M. Lasonen-Aarnio 2009: ‘Knowledge and Objective Chance’. In Greenough, P. and D. Pritchard (eds.): Williamson on Knowledge, Oxford: Oxford University Press, pp. 92–108. Hodes, H. 1984a: ‘Axioms for Actuality’. Journal of Philosophical Logic 13, pp. 27–24. —— 1984b: ‘Some Theorems on the Expressive Limitations of Modal Languages’. Journal of Philosophical Logic 13, pp. 13–26. Horgan, T. 1982: ‘Supervenience and Microphysics’. Pacific Philosophical Quarterly 63, pp. 29–43. —— 1993: ‘From Supervenience to Superdupervenience: Meeting the Demands of a Material World’. Mind 102, pp. 555–586. Horwich, P. 2001: ‘A Defense of Minimalism’. In Lynch, M. P. (ed.): The Nature of Truth, Cambridge, MA: MIT Press, pp. 559–577. Huemer, M. 2007: ‘Epistemic Possibility’. Synthese 156, pp. 119–142. Hughes, G. E. and M. J. Cresswell 1996: A New Introduction to Modal Logic. London: Routledge. Hugly, P. and C. Sayward 1996: Intensionality and Truth: An Essay on the Philosophy of A.N. Prior. Dordrecht: Kluwer. Huitink, J. 2009: ‘Scoping Over Epistemics in English and Dutch’. In Current Issues in Unity and Diversity of Languages. Collection of the papers selected from the CIL18, Seoul: The Linguistic Society of Korea, pp. 2077– 2089. Humberstone, L. 1981: ‘From Worlds to Possibilities’. Journal of Philosophical Logic 10, pp. 313–339. Jubien, M. 1988: ‘Problems with Possible Worlds’. In Austin, D. F. (ed.): Philosophical Analysis. A Defense by Example, Dordrecht: Kluwer, pp. 299–322. Kim, J. 1974: ‘Noncausal Connections’. Noûs 8, pp. 41–52. —— 1984: ‘Concepts of Supervenience’. Philosophy and Phenomenological Research 45, pp. 153–176. —— 1987: ‘“Strong” and “Global” Supervenience Revisited’. Philosophy and Phenomenological Research 48, pp. 315–326.

266

Bibliography

—— 1990: ‘Supervenience as a Philosophical Concept’. Metaphilosophy 21, pp. 1–27. —— 1993: ‘Postscripts on Supervenience’. In Supervenience and Mind. Selected Philosophical Essays, Cambridge: Cambridge University Press, pp. 161–171. Kratzer, A. 1977: ‘What “Must” and “Can” Must and Can Mean’. Linguistics and Philosophy 1, pp. 337–355. Kripke, S. A. 1963: ‘Semantical Considerations on Modal Logic’. Acta Philosophica Fennica 16, pp. 83–94, reprinted in Linsky (1971: 63–72). Page references are to the reprint. —— 1980: Naming and Necessity. Cambridge, MA: Harvard University Press. Künne, W. 1983: Abstrakte Gegenstände. Semantik und Ontologie. Frankfurt am Main: Suhrkamp. —— 2003: Conceptions of Truth. Oxford: Clarendon Press. —— 2006: ‘Properties in Abundance’. In Strawson, P. F. and A. Chakrabarti (eds.): Universals, Concepts and Qualities: New Essays on the Meaning of Predicates, Aldershot: Ashgate, pp. 184–249. —— 2007: Abstrakte Gegenstände. Semantik und Ontologie. 2nd ed., Frankfurt am Main: Klostermann. Larson, R. and G. Segal 1995: Knowledge of Meaning. An Introduction to Semantic Theory. Cambridge, MA and London: MIT Press. Leeds, S. 2001: ‘Possibility: Physical and Metaphysical’. In Gillett, C. and B. Loewer (eds.): Physicalism and Its Discontents, Cambridge: Cambridge University Press, pp. 172–193. Lewis, D. 1968: ‘Counterpart Theory and Quantified Modal Logic’. The Journal of Philosophy 65, reprinted with a postscript in Lewis (1983b: 26– 46). Page references are to the reprint. —— 1971: ‘Counterparts of Persons and their Bodies’. The Journal of Philosophy 68, reprinted in Lewis (1983b: 47–54). Page references are to the reprint. —— 1973: Counterfactuals. Oxford: Blackwell.

Bibliography

267

—— 1980: ‘A Subjectivist’s Guide to Objective Chance’. In Jeffrey, R. C. (ed.): Studies in Inductive Logic, vol. 2, Berkeley and Los Angeles: University of California Press, reprinted with a postscript in Lewis (1986b: 83–133). Page references are to the reprint. —— 1983a: ‘New Work for a Theory of Universals’. Australasian Journal of Philosophy 61, pp. 343–377. —— 1983b: Philosophical Papers, vol. I. New York and Oxford: Oxford University Press. —— 1986a: On the Plurality of Worlds. Oxford: Blackwell. —— 1986b: Philosophical Papers, vol. II. New York and Oxford: Oxford University Press. —— 2001: ‘Truthmaking and Difference-Making’. Noûs 35, pp. 602–615. Linsky, L. (ed.) 1971: Reference and Modality. London: Oxford University Press. Löbner, S. 1989: ‘German “Schon – Erst – Noch” : An Integrated Analysis’. Linguistics and Philosophy 12, pp. 167–212. Lombard, L. B. 1986: Events: A Metaphysical Study. London: Routledge and Kegan Paul. Loux, M. J. (ed.) 1979: The Actual and the Possible. Readings in the Metaphysics of Modality. Ithaca and London: Cornell University Press. MacBride, F. 2003: ‘Speaking with Shadows: A Study of Neo-Logicism’. British Journal for the Philosophy of Science 54, pp. 103–163. Mackie, J. L. 1974: ‘De What Re is De Re Modality?’ The Journal of Philosophy 71, pp. 551–561. Mackie, P. 2006: How Things Might Have Been. Individuals, Kinds, and Essential Properties. Oxford: Clarendon Press. McLaughlin, B. and K. Bennett 2008: ‘Supervenience’. In Zalta, E. N. (ed.): The Stanford Encyclopedia of Philosophy, Fall 2008 ed. Available at http://plato.stanford.edu/archives/fall2008/ entries/supervenience/ McLaughlin, B. P. 1995: ‘Varieties of Supervenience’. In Savellos, E. E. and Ü. D. Yalçin (eds.): Supervenience: New Essays, Cambridge: Cambridge University Press, pp. 16–59.

268

Bibliography

Mondadori, F. 1978: ‘Remarks on Tense and Mood: the Perfect Future’. In Guenthner, F. and C. Rohrer (eds.): Studies in Formal Semantics. Intensionality, Temporality, Negation, Amsterdam: North-Holland Publishing Company, pp. 223–248. Mondadori, F. and A. Morton 1976: ‘Modal Realism: The Poisoned Pawn’. The Philosophical Review 85, pp. 3–20. Moyer, M. 2008: ‘Weak and Global Supervenience are Strong’. Philosophical Studies 138, pp. 125–150. Mulligan, K. 2006: ‘Ascent, Propositions and Other Formal Objects’. Grazer Philosophische Studien 72, pp. 29–48. Partee, B. H. 1984: ‘Nominal and Temporal Anaphora’. Linguistics and Philosophy 7, pp. 243–286. Paull, R. C. and T. Sider 1992: ‘In Defense of Global Supervenience’. Philosophy and Phenomenological Research 52, pp. 833–854. Petrie, B. 1987: ‘Global Supervenience and Reduction’. Philosophy and Phenomenological Research 48, pp. 119–130. Plantinga, A. 1974: The Nature of Necessity. Oxford: Clarendon Press. —— 1976: ‘Actualism and Possible Worlds’. Theoria 42, pp. 139–160, reprinted in Loux (1979: 253–273). Page references are to the reprint. —— 1979: ‘De Essentia’. In Sosa, E. (ed.): Essays in the Philosophy of Roderick M. Chisholm, Amsterdam: Rodopi, pp. 101–121, reprinted in Plantinga (2003: 139–157). Page references are to the reprint. —— 1985a: ‘Replies to my Colleagues’. In Tomberlin and van Inwagen (1985), pp. 313–396. —— 1985b: ‘Self-Profile’. In Tomberlin and van Inwagen (1985), pp. 3–97. —— 1987: ‘Two Concepts of Modality: Modal Realism and Modal Reductionism’. Philosophical Perspectives 1, pp. 189–231. —— 2003: Essays in the Metaphysics of Modality. Oxford: Oxford University Press. Pollock, J. L. 1985: ‘Plantinga on Possible Worlds’. In Tomberlin and van Inwagen (1985), pp. 121–144.

Bibliography

269

Prior, A. N. and K. Fine 1977: Worlds, Times and Selves. London: Duckworth. Ramachandran, M. 1989: ‘An Alternative Translation Scheme for Counterpart Theory’. Analysis 49, pp. 131–141. Rayo, A. 2007: ‘Plurals’. Philosophy Compass 2, pp. 411–427. Rayo, A. and S. Yablo 2001: ‘Nominalism Through De-Nominalization’. Noûs 35, pp. 74–92. Richards, T. 1975: ‘The Worlds of David Lewis’. Australasian Journal of Philosophy 53, pp. 105–118. Rosen, G. 1990: ‘Modal Fictionalism’. Mind 99, pp. 327–354. —— 2010: ‘Metaphysical Dependence: Grounding and Reduction’. In Hale, B. and A. Hoffmann (eds.): Modality. Metaphysics, Logic, and Epistemology, Oxford: Oxford University Press, pp. 109–135. Ruben, D.-H. 1990: Explaining Explanation. London, New York: Routledge. Salmon, N. 1982: Reference and Essence. Oxford: Basil Blackwell. Schaffer, J. 2009: ‘On What Grounds What’. In Chalmers, D. J., D. Manley and R. Wasserman (eds.): Metametaphysics. New Essays on the Foundations of Ontology, Oxford: Clarendon Press, pp. 347–383. Schiffer, S. 1996: ‘Language-Created Language-Independent Entities’. Philosophical Topics 24, pp. 149–167. —— 2003: The Things We Mean. Oxford: Clarendon Press. Schnieder, B. 2004a: Substanzen und (ihre) Eigenschaften. Eine Studie zur analytischen Ontologie. Berlin, New York: de Gruyter. —— 2004b: ‘Once More: Bradleyan Regresses’. In Hochberg, H. and K. Mulligan (eds.): Relations and Predicates, Frankfurt: Ontos Verlag, pp. 219–256. —— 2005: ‘Property Designators, Predicates, and Rigidity’. Philosophical Studies 122, pp. 227–241. —— 2006a: ‘Canonical Property Designators’. American Philosophical Quarterly 43, pp. 119–132.

270

Bibliography

—— 2006b: ‘A Certain Kind of Trinity: Dependence, Substance, Explanation’. Philosophical Studies 129, pp. 393–419. —— 2006c: ‘Truth-Making Without Truth-Makers’. Synthese 152, pp. 21– 47. —— 2007: ‘Eigenschaften und ihre Namen’. In Rami, A. and H. Wansing (eds.): Referenz und Realität, Paderborn: Mentis, pp. 179–208. —— 2009: ‘Expressivism Concerning Epistemic Modals’. The Philosophical Quarterly, published online. —— 2010: ‘A Puzzle About “Because”’. Logique et Analyse 53, pp. 317– 343. —— 2011: ‘A Logic for “Because”’. Review of Symbolic Logic 4, pp. 445– 465. Schnieder, B., M. Schulz and A. Steinberg 2010: ‘What Might Be and What Might Have Been’. In Conrad, S.-J. and S. Imhof (eds.): P. F. Strawson: Ding und Begriff, Frankfurt: Ontos Verlag, pp. 135–162. Schulz, M. 2010: ‘Wondering What Might Be’. Philosophical Studies 149, pp. 367–386, published online. Sharvy, R. 1972: ‘Euthyphro 9d-11b: Analysis and Definition in Plato and Others’. Noûs 6, pp. 119–137. Shoemaker, S. 1998: ‘Causal and Metaphysical Necessity’. Pacific Philosophical Quarterly 79, pp. 59–77. Shope, R. K. 1983: The Analysis of Knowledge: a Decade of Research. Princeton, NJ.: Princeton University Press. Sidelle, A. 2002: ‘On the Metaphysical Contingency of Laws of Nature’. In Gendler and Hawthorne (2002), pp. 309–336. Sidgwick, H. 1884: The Methods of Ethics. London: Mackmillan. Skyrms, B. 1976: ‘Possible Worlds, Physics and Metaphysics’. Philosophical Studies 30, pp. 323–332. Stalnaker, R. C. 1984: Inquiry. Cambridge, MA and London: MIT Press. Steinberg, A. 2012: ‘Pleonastic Possible Worlds’. Philosophical Studies online first. —— 2013a: ‘Defining Global Supervenience’. Erkenntnis online first.

Bibliography

271

—— 2013b: ‘Supervenience: A Survey’. In Hoeltje, M., B. Schnieder and A. Steinberg (eds.): Varieties of Dependence. Ontological Dependence, Grounding, Supervenience, Response-Dependence, Basic Philosophical Concepts, Munich: Philosophia Verlag, pp. 123–166. Strawson, P. F. 1997: ‘May Bes and Might Have Beens’. In Entity and Identity and Other Essays, Oxford: Clarendon Press, pp. 179–190. Swoyer, C. 1982: ‘The Nature of Natural Laws’. Australasian Journal of Philosophy 60, pp. 203–223. Tomberlin, J. E. and P. van Inwagen (eds.) 1985: Alvin Plantinga. Dordrecht: D. Reidel. van Inwagen, P. 1981: ‘Why I don’t understand substitutional quantification’. Philosophical Studies 39, pp. 281–285. —— 1985: ‘Plantinga on Trans-World Identity’. In Tomberlin, J. E. and P. van Inwagen (eds.): Alvin Plantinga, Dordrecht: D. Reidel, reprinted in van Inwagen (2001: 186–205). Page references are to the reprint. —— 1986: ‘Two Concepts of Possible Worlds’. Midwest Studies in Philosophy 9, reprinted in van Inwagen (2001: 206–242). Page references are to the reprint. —— 2001: Ontology, Identity, and Modality. Essays in Metaphysics. Cambridge: Cambridge University Press. Vendler, Z. 1967: Linguistics in Philosophy. Ithaca and London: Cornell University Press. von Fintel, K. and A. S. Gillies 2007: ‘An Opinionated Guide to Epistemic Modality’. In Gendler, T. S. and J. Hawthorne (eds.): Oxford Studies in Epistemology, vol. 2, Oxford: Oxford University Press, pp. 32–62. —— 2008: ‘CIA Leaks’. The Philosophical Review 117, pp. 77–98. von Fintel, K. and S. Iatridou 2003: ‘Epistemic Containment’. Linguistic Inquiry 34, pp. 173–198. von Mises, R. 1972: Wahrscheinlichkeit, Statistik und Wahrheit. 4th ed., Wien, New York: Springer. Williams, D. C. 1953: ‘On the Elements of Being: I’. Review of Metaphysics 7, pp. 3–18.

272

Bibliography

Williamson, T. 2000: Knowledge and Its Limits. Oxford: Oxford University Press. —— 2002: ‘Necessary Existents’. In O’Hear, A. (ed.): Logic, Thought and Language, Cambridge: Cambridge University Press, pp. 233–251. —— 2007a: ‘How Probable is an Infinite Sequence of Heads?’ Analysis 67, pp. 173–180. —— 2007b: The Philosophy of Philosophy. Oxford: Blackwell. Wright, C. 1990: ‘Field and Fregean Platonism’. In Irvine, A. (ed.): Physicalism in Mathematics, Dordrecht: Kluwer, pp. 73–93. —— 1997: ‘On the Philosophical Significance of Frege’s Theorem’. In Heck, R., Jr. (ed.): Language, Thought and Logic: Essays in Honour of Michael Dummett, Oxford: Clarendon Press, pp. 201–244, reprinted in Hale and Wright (2001: 272–307). Page references are to the reprint. —— 1999: ‘Is Hume’s Principle Analytic?’ Notre Dame Journal of Formal Logic 40, pp. 6–30. —— 2007: ‘On Quantifying Into Predicate Position’. In Leng, M., A. Paseau and M. Potter (eds.): Mathematical Knowledge, Oxford: Oxford University Press, pp. 150–174. Yablo, S. 1993: ‘Is Conceivability a Guide to Possibility?’ Philosophy and Phenomenological Research 80, pp. 1–42. Yalcin, S. 2007: ‘Epistemic Modals’. Mind 116, pp. 983–1026. Zimmerman, D. W. 1995: ‘Theories of Masses and Problems of Constitution’. The Philosophical Review 104, pp. 53–110. Zwicky, A. M. and J. M. Sadock 1975: ‘Ambiguity Tests and How To Fail Them’. In Kimball, J. P. (ed.): Syntax and Semantics, vol. 4, New York: Academic Press.