Oxford Studies in Metaphysics [10] 0198791976, 9780198791973

Table of contents :
Preface
Contents
The Sanders Prize in Metaphysics
DISPOSITIONS
1 Teleological Dispositions • Nick Kroll
2 Indirect Directness • Jennifer McKitrick
3 Dispositions without Teleology • David Manley and Ryan Wasserman
ANALYTICITY REVISITED
4 Devious Stipulations • John Horden
5 Stipulations and Requirements: Reply to Horden • Louis deRosset
WHAT REALITY IS LIKE
6 Colors as Primitive Dispositions • Hagit Benbaji
7 Are There Ineffable Aspects of Reality? • Thomas Hofweber
8 The Metaphysics of Quantities and Their Dimensions • Bradford Skow
MODALITY AND EXISTENCE
9 Vague Existence • Alessandro Torza
10 Ersatz Counterparts • Richard Woodward
GROUNDING AND EXPLANATION
11 The Principle of Sufficient Reason and Probability • Alexander Pruss
12 Grounding Ground • Jon Erling Litland
Author Index

Citation preview

OXFORD STUDIES IN METAPHYSICS

OXFORD STUDIES IN METAPHYSICS Editorial Advisory Board David Chalmers (New York University and Australasian National University) Andrew Cortens (Boise State University) Tamar Szabó Gendler (Yale University) Sally Haslanger (MIT) John Hawthorne (University of Southern California) Mark Heller (Syracuse University) Hud Hudson (Western Washington University) Kathrin Koslicki (University of Alberta) Kris McDaniel (Syracuse University) Brian McLaughlin (Rutgers University) Trenton Merricks (University of Virginia) Kevin Mulligan (Université de Genève) Laurie Paul (University of North Carolina–Chapel Hill) Theodore Sider (Cornell University) Timothy Williamson (Oxford University)

Managing Editor Peter van Elswyk (Rutgers University)

OXFORD STUDIES IN METAPHYSICS Volume 10

Edited by Karen Bennett and Dean W. Zimmerman

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors 2017 The moral rights of the authors have been asserted First Edition published in 2017 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2004233307 ISBN 978–0–19–879197–3 (hbk.) 978–0–19–879198–0 (pbk.) Printed in Great Britain by Clays Ltd, St Ives plc Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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

CONTENTS The Sanders Prize in Metaphysics

ix DISPOSITIONS

1 Teleological Dispositions

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Nick Kroll 2 Indirect Directness

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Jennifer McKitrick 3 Dispositions without Teleology

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David Manley and Ryan Wasserman ANALYTICITY REVISITED 4 Devious Stipulations

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John Horden 5 Stipulations and Requirements: Reply to Horden

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Louis deRosset WHAT REALITY IS LIKE 6 Colors as Primitive Dispositions

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Hagit Benbaji 7 Are There Ineffable Aspects of Reality?

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Thomas Hofweber 8 The Metaphysics of Quantities and Their Dimensions

171

Bradford Skow MODALITY AND EXISTENCE 9 Vague Existence

201

Alessandro Torza 10 Ersatz Counterparts

Richard Woodward

235

Contents

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GROUNDING AND EXPLANATION 11 The Principle of Sufficient Reason and Probability

261

Alexander Pruss 12 Grounding Ground

279

Jon Erling Litland Author Index

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

Thomas Hofweber, “Inexpressible Properties and Propositions”, Vol. 2; Matthew McGrath, “Four-Dimensionalism and the Puzzles of Coincidence”, Vol. 3; Cody Gilmore, “Time Travel, Coinciding Objects, and Persistence”, Vol. 3; Stephan Leuenberger, “Ceteris Absentibus Physicalism”, Vol. 4; Jeffrey Sanford Russell, “The Structure of Gunk: Adventures in the Ontology of Space”, Vol. 4; Bradford Skow, “Extrinsic Temporal Metrics”, Vol. 5; Jason Turner, “Ontological Nihilism”, Vol. 6; Rachael Briggs and Graeme A. Forbes, “The Real Truth About the Unreal Future”, Vol. 7; Shamik Dasgupta, “Absolutism vs Comparativism about Quantities”, Vol. 8; Louis deRosset, “Analyticity and Ontology”, Vol 9;

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

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The Sanders Prize in Metaphysics Nicholas K. Jones, “Multiple Constitution”, Vol. 9; Nick Kroll, “Teleological Dispositions”, Vol. 10; Jon Erling Litland, “Grounding Grounding”, Vol. 10.

Enquiries should be addressed to Dean Zimmerman at: [email protected]

DISPOSITIONS

1 Teleological Dispositions Nick Kroll 1. INTRODUCTION Some things are disposed to break when struck. Some things are disposed to bend when stressed. Some things are disposed to dissolve in water. Things have dispositions. But dispositions need not manifest. A vase disposed to break when struck may meet its end by melting rather than breaking. A rod disposed to bend when stressed might succumb to rust without ever bending. And a chunk of salt disposed to dissolve in water may be used up while making hydrochloric acid. These are mundane observations. Reflecting on them, however, leads to two important intuitions about dispositions. The first is that dispositions have some kind of directedness. The second is that dispositions have some kind of connection with conditionals. Consider the vase that meets its end by melting rather than breaking. The vase doesn’t break. Yet it seems that in virtue of being disposed to break when struck, the vase is in a state that is in some sense directed at a state of affairs in which the vase breaks. Likewise, even though the wire never bends, it seems that in virtue of being disposed to bend when stressed, the wire is in a state that is in some sense directed at a state of affairs in which the wire bends. Generalizing, the idea is that something with a disposition is in a state that is in some sense directed at a manifestation of the disposition. Put simply: a disposition need not manifest but it is in some sense directed at manifesting. I’ve characterized the directedness intuition as one that needs an explanation. What needs to be explained is the sense in which dispositions are directed at manifesting. Keeping this in mind, let’s turn to the intuition that dispositions have some kind of connection to conditionals. What if the vase had been suitably struck instead of being placed in the furnace? Presumably, it would have broke and so its disposition to break when struck would have manifested. Likewise, what if the wire had been

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suitably stressed before corroding? Presumably, it would have bent and so its disposition to bend when stressed would have manifested. Generalizing, it seems that while a disposition need not manifest, it nonetheless would manifest if certain conditions were to obtain. There thus appears to be some kind of connection between dispositions and conditionals. Suppose the connection between dispositions and conditions is a strong connection. In particular, suppose that we have an informative and counterexample free conditional analysis of dispositions. Then perhaps we could explain the directedness intuition: the sense in which a disposition is directed at manifesting is that the disposition would manifest if conditions C were to obtain (where conditions C are specified by the given conditional analysis). Orthodoxy would have it that the correct account of dispositions rests on a conditional analysis of dispositions and so such an explanation of the directedness intuition is the right explanation. I argue otherwise. In particular, I argue for a teleological account of dispositions. According to this account of dispositions, the connection between dispositions and conditions is explained in terms of the directedness of dispositions and the directedness of dispositions is a teleological directedness. We begin by undermining orthodoxy. Following Molnar (2003) and Fara (2005), I believe the project of analyzing dispositions in terms of conditionals is a lost cause. But the purpose of our overview of conditional analyses is not to establish such a strong conclusion. The purpose is rather to motivate a turn towards a teleological account of dispositions.

2. CONDITIONAL ANALYSES

2.1 The simple conditional analysis Our starting point is the so-called simple conditional analysis of dispositions. The basic idea behind this analysis is that every disposition has a stimulus condition, and if that stimulus condition were to obtain, the disposition would manifest. More explicitly, the proposal is this: (SCA)

Necessarily: x is disposed to M when C iff x would M if C were the case.1

For example, if (SCA) is correct, a vase is disposed to break when struck just in case the vase would break if it were struck. 1

See Ryle (1949), Goodman (1954), and Quine (1960).

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(SCA) is obviously compatible with the observation that a disposition need not manifest: if the vase is never struck, its disposition to break when struck never manifests. But (SCA) has some problems. We’ll mention two of the most notorious. First, there is the problem of finks. The problem here is that the stimulus condition for a disposition may also be a condition for losing or acquiring the disposition. The classic example is C. B. Martin’s electro-fink.2 It’s connected to a wire that is not disposed to conduct electricity when touched by a conductor. However, the fink ensures that if the wire were to be touched by a conductor, the wire would acquire the disposition. Thus, while the wire is not disposed to conduct electricity when touched by a conductor, it would conduct electricity if it were touched by a conductor. Martin also provides an example going the other way. Suppose the electro-fink has a reverse cycle. On reverse cycle, the electro-fink ensures that any wire connected to it that is disposed to conduct electricity when touched by a conductor loses this disposition when it is touched by a conductor. Consider such a wire. While the wire is disposed to conduct electricity when touched by a conductor, it would not conduct electricity if it were touched by a conductor. On the contrary, if it were touched by a conductor, it would lose this disposition. So, we have counterexamples to (SCA) in both directions. Second, there is the problem of masks. A mask is something that prevents a disposition from manifesting when the stimulus condition obtains, and it does so without taking away the disposition.3 A standard example involves an antidote for a poison.4 The poison is disposed to kill when ingested. But when ingested, the poison takes some time to do its work. During that time, if you were to take the antidote, you would be saved. The antidote, however, does not remove the poison’s disposition. It simply prevents the poison from doing any more damage. Put it this way: even though the poison is killing you, it need not kill you. You could take the antidote. So, there are situations where the poison wouldn’t kill if it were ingested even though the poison is disposed to kill when ingested. We have another counterexample to (SCA).5

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3 See Martin (1994). Masks owe their name to Johnston (1992). This example is due to Bird (1998). 5 Choi (2008) denies that (SCA) is subject to counterexample from finks and masks. Somehow, the right-hand side of the relevant instance of (SCA) is such that the possibility of finks and masks doesn’t arise. Like many others, I’m not convinced. One reason why I’m not convinced is that if Choi is right, it’s hard to see why so many have had the intuition that (SCA) is subject to counterexample by situations involving finks and masks. 4

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2.2 Appealing to the exclusion of external interference One response to situations involving finks and masks is that they involve some kind of external interference.6 In the case of finks, some kind of external interference causes an object to acquire or lose the relevant disposition when the stimulus condition for the disposition obtains. In the case of masks, some kind of external interference prevents the manifestation of a disposition from obtaining even when disposition’s stimulus condition obtains and the disposition remains. Perhaps, then, we can avoid the problem of finks and the problem of masks by appealing to a clause which excludes external interference. (ECA)

Necessarily: x is disposed to M when C iff x would M if C were the case and nothing external were to interfere.

(ECA) should only be seen as a first attempt. For it raises the following question: nothing external to what interferes with what? More needs to be said. I’m going to attempt to fill in the details. We’ll begin with a conditional analysis of dispositions inspired by Lewis (1997). (LCA)

Necessarily: x is disposed to M when C iff x has some intrinsic property I in virtue of which: if C were the case and x were to retain I , x would M .

One might ask: why the appeal to intrinsic properties? Lewis answers, in effect, by claiming that dispositions are intrinsic properties of their bearers.7 This claim is controversial. Some have argued that some dispositions are extrinsic properties.8 But we need not concern ourselves with whether dispositions are intrinsic properties. More important for present purposes is that the appeal to intrinsic properties seems to solve the problem of finks. Consider once again the wire connected to the electro-fink that is not disposed to conduct electricity when touched by a conductor. This wire doesn’t have an intrinsic property I in virtue of which the wire would conduct electricity if it were touched by a conductor and retain I . The wire does have such an extrinsic property: namely, being connected to the electro-fink. And, sure enough, if the wire were touched by a conductor, the electro-fink would make the wire acquire such a intrinsic property. But this is neither here nor there as far as (LCA) is concerned. It correctly predicts

6 8

See Johnston (1992). See McKitrick (2003).

7

See Lewis (1997, p. 155).

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that, in this scenario, the wire is not disposed to conduct electricity when touched by a conductor. The scenario involving the electro-fink running on reverse cycle is handled in a similar manner. In this case, the electro-fink makes the wire, when touched by a conductor, lose some intrinsic property, and consequently lose the disposition to conduct electricity when touched by a conductor. But what would happen if the wire were touched by a conductor and it were to retain this intrinsic property? It would conduct electricity, so it seems. So, (LCA) correctly predicts that the wire is disposed to conduct electricity when touched by a conductor. Masks, however, are still a problem. You ingest the poison but take the antidote. The antidote doesn’t remove the poison’s disposition to kill when ingested. It just prevents the manifestation of the disposition from obtaining. So, the poison retains whatever intrinsic property grounds its disposition. (LCA) thus predicts that you die. But you don’t, thanks to the antidote. There is a modification of (LCA) that avoids this counterexample. Consider (LCA+) and one of its instances. (LCA+)

(1)

Necessarily: x is disposed to M when C iff x has some intrinsic property I in virtue of which: if C were to obtain and x were to retain I , there would be a process p such that if nothing external to p were to interfere with p, x would M (as a result).

Necessarily: the poison is disposed to kill when ingested iff: the poison has some intrinsic property I in virtue of which: if it were to ingested by x and it were to retain I , there would be a process p such that if nothing external to p were to interfere with p, the poison would kill x (as a result).

Now take some time t after you have ingested the poison but before you have taken the antidote. The poison is causing damage to your organs at t. Let this process be p. The question, then, is whether p is such that if nothing external to it were to interfere with it, the poison would kill you as a result. It seems plausible, at least from the rather limited description of the case, that p is such a process. Shortly after t, you take the antidote. In doing so, another process obtains, one which is external to p. This process prevents x from developing into one in which the poison kills you as a result. But if it weren’t for this external interference, p would have developed into such a process. Thus, (1) correctly predicts that the poison is disposed to kill you when you ingest it. (LCA+) is the best I can do to spell out the “provided nothing external interferes” response. But my best is not good enough: the analysis is subject to counterexample.

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Our counterexample is a variant of one due to Manley and Wasserman (2008).9 Suppose there is a concrete block that withstands any damage done to it by a sledgehammer unless it is hit in just the right spot with just the right amount of force. Furthermore, it is nearly impossible to hit the block in this spot with the right amount of force. But if the block is hit in the right spot with the right amount of force, it shatters as a result. The block has an Achilles’ heel. Now while this block could shatter from being struck, it seems that we would quite rightly think that it is not disposed to shatter when struck. Now suppose that, due to some freak occurrence and despite no intrinsic change to the block, conditions are temporarily just right for the block to be hit in the just the right spot with just the right amount of force, provided someone is around to strike the block. For the next two seconds, if anyone were to pick up a sledgehammer and strike the block, the block would be struck in just the right spot with the right amount of force. But no one is around to strike the block. While conditions did momentarily obtain such that the block would shatter if it were struck, these conditions did not thereby render the block disposed to shatter when struck. We thus have a counterexample to (LCA+). During the above described two second interval, it is not the case that the block is disposed to shatter when struck. However, it is the case that the block has some intrinsic property (its Achilles’ heel) in virtue of which, if the block were struck and retain this property, there would be a event such that if this event were to continue without interruption, the block would shatter (as a result). So my best is not good enough.10 9 Manley and Wasserman (2008) use their example to refute conditional analyses that reply to the problem of finks and the problem of masks by appealing to hyper-specific stimulus conditions. The basic idea behind such analyses is that when we say, for example, “the vase is disposed to break when struck,” the disposition ascribed is not simply being disposed to break when struck but being disposed to break when struck in some hyperspecific way. (This seems to be Lewis’s response to the problem of masks: see Lewis 1997, p. 153.) I don’t consider “hyper-specific” conditional analyses here, mainly because they seem to be ad hoc responses to the problems of finks and masks. Manley and Wasserman offer a more substantive (and conclusive) argument against hyper-specific conditional analyses. 10 Contessa (2013) offers an “interference free” conditional analysis of disposition ascriptions. But his best is no better than mine: it is subject to counterexample as well. Bypassing the details of his analysis, it suffices to note that his analysis predicts that if x is intrinsically disposed to M when C and y is something that interferes with x’s being intrinsically disposed to M when C , then it’s not the case that x would M if C were the case. But this prediction is incorrect. Here’s why. Suppose x is intrinsically disposed to break into pieces when struck. Further suppose that y is something that interferes with x’s disposition in the following way. If x were struck, a process P of x breaking into pieces would begin as a result. However, this process P

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2.3 Appealing to normal conditions One might worry that the counterexamples we’ve considered so far invoke situations that are in some sense abnormal or atypical. Certainly, the situation involving the block with an Achilles’ heel is not a normal situation. Likewise, it seems that wires, whether or not they are disposed to conduct electricity when touched by a conductor, are not normally connected to some sort of electro-fink. And in the situation where you ingest the poison but take the antidote, it would be natural to say something like “The poison didn’t kill you, but normally it would have.” Perhaps, then, we should consider a conditional analysis of dispositions that explicitly appeals to normality. (NCA)

Necessarily: x is disposed to M when C iff in normal conditions, x would M if C were the case.

Supposing that situations involving finks, masks, and Achilles’ heels are not normal situations, the hope is that (NCA) allows us to properly ignore these situations. But in what sense are such situations abnormal? Fara (2005) points out there need not be anything bizarre about a mask: Dispositions of objects are being masked all the time. I’m disposed to go to sleep when I’m tired; but this disposition is sometimes masked by too much street noise. Cylinders of rubber are disposed to roll when placed on an inclined plane; but this disposition can be masked by applying a car’s brakes . . . [T]he masking of dispositions is such a humdrum occurrence that any adequate account of [dispositions] must accommodate it.11

Similar remarks apply to Achilles’ heels: they are so common that any adequate account of dispositions must accommodate them. In short, then, wouldn’t culminate because some state of affairs S involving y would obtain that stops the process. So, while x would have started cracking when S obtains, it wouldn’t yet be broken into pieces. But due to S obtaining another state of affairs S 0 would obtain. Due to S 0 obtaining, another process P 0 of x breaking into pieces would begin and culminate. So, if x were struck, it would (after all and despite y) break into pieces. Now, since S is a state of affairs that prevents the manifestation of x’s disposition to break into pieces when struck without taking away x’s disposition, y is some kind of mask. Furthermore, since S 0 is not a state of affairs that involves x being struck, x being broken into pieces in S 0 is not a manifestation of x’s disposition to break into pieces when struck. Putting this all together, we have a counterexample to Contessa’s analysis: (i) x is intrinsically disposed to break into pieces when struck, (ii) y is something that interferes with x being disposed to break into pieces when struck, and (iii) x would break into pieces if it were struck. 11 Fara (2005, p. 50).

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unless the defender of (NCA) further specifies the sense of normality she is appealing to, the threat of counterexample remains. A defender of (NCA) may protest that she does not need to specify the exact sense of normality in (NCA). She may simply rest her case on the intuitive plausibility of the “Yeah . . . but normally . . . ” response to situations involving finks, masks, and Achilles’ heels. But this will not do. To see why, let us begin with an observation due to Fara (2005). This is the observation that situations that are normal with respect to x being disposed to M when C are situations where x would M if C were the case. Following Fara, we further note that it follows that if the sense of normality appealed to in (NCA) is one that is relativized to x being disposed to M when C , the right-hand side of (NCA) is subject to trivialization. It amounts to the trivial claim that situations where x would M if C were the case are situations where x would M if C were the case. Now recall the incident involving the poison and the antidote. It may be natural to respond with “Yeah, the poison didn’t kill you, but normally it would have.” But if asked why the poison would normally would kill you, it seems to me that one would say “Because it is disposed to kill when ingested.” But then the sense of normality appealed to is one that is relativized to the poison being disposed to kill when ingested. If so, the “Yeah . . . but normally . . . ” response in this case actually undermines (NCA). It suggests that the notion of normality appealed to in (NCA) is one that is relativized to x being disposed to M when C , and thus the righthand side of (NCA) is subject to trivialization. So, it would be a mistake to a defender of (NCA) to rest her case on the “Yeah . . . but normally . . . ” response to situations involving finds, masks, and Achilles’ heels. Furthermore, we’ve seen that unless the defender of (NCA) gives us some reason to think otherwise, we have reason to believe that the sense of normality appealed to in (NCA) is one that is relativized to x being disposed to M when C . Consequently, we have reason to believe that (NCA) is subject to trivialization.12 Similar remarks apply to appeals to ideal conditions, typical conditions, or ceteris paribus conditions.13 Moline (1975) nicely captures the typical attitude towards such appeals: [T]hey are fundamentally dodges . . . They amount to muddled ways of disguising from ourselves more or less serious ignorance of the dispositional properties of individual things or persons and of types of things or persons.14

12 13

See Hauska (2008) for further concerns about appealing to normality. 14 See Mumford (1998) and Steinberg (2009). Moline (1975, pp. 244–5).

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I agree, ceteris paribus. That is, unless an “escape-clause” conditional analysis of dispositions can show that it is legitimate to appeal to its escape-clause(s), it is fundamentally a dodge. Thus, the worry isn’t just that such analyses are subject to trivialization. It is also that, trivialization aside, they are uninformative.15

2.4 Appealing to proportionality The conditional analyses we’ve considered so far face either the threat of counterexample, the threat of trivialization, or a worry about informativeness. Finks, masks, and Achilles’ heels provide the threat of counterexample. Adding some qualification or “escape-clause” to avoid this threat brings with it either the threat of trivialization or a worry about informativeness. Perhaps, then, we should try to avoid the threat of counterexample some other way. This is what Manley and Wasserman (2008) try to do with (PROP). (PROP)

Necessarily: x is disposed to M when C iff x would M in a suitable proportion of C -cases.

The basic idea behind (PROP) is that instead of looking at what would happen at the closest world(s) where C obtains, we look at what would happen in situations (some actual, the rest merely possible) where C obtains. If a suitable proportion of these situations are situations where x M s, then x is disposed to M when C . The converse is alleged to hold as well.16 An interesting feature of (PROP) is that situations involving finks and masks are not ignored. To illustrate, consider a vase disposed to break when struck and the following instance of the (PROP). (2)

Necessarily, the vase is disposed to break when struck iff the vase would break in a suitable proportion of cases where it is struck.

15 It should be noted that Moline’s complaint isn’t avoided by inventing a new type of conditional with a semantics that is supposed to model normal conditions, ideal conditions, or ceteris paribus conditions. So, the conditional analyses of Maurreau (1997), Gundersen (2002), and Bonevac et al. (2006) do not avoid Moline’s complaint. Similar remarks apply to an appeal to context to avoid the counterexamples (see Fara 2005). 16 One might wonder whether (PROP) is a conditional analysis. Certainly, the righthand side of (PROP) is not a subjunctive or indicative conditional and no such conditional is embedded in (PROP). So, how exactly is it a conditional analysis? Perhaps the idea is that the truth or falsity of the right-hand side depends upon a bunch of counterfactual facts, facts like x would M if C1 were the case, x wouldn’t M if C2 were the case, x would M if C3 were the case, and so on. In any case, Manley and Wasserman take it to be a conditional analysis, and we’ll follow suit.

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There is no shortage of (nomologically) possible cases where the vase is struck and its disposition is masked by something or other. There is also no shortage of (nomologically) possible cases where the vase is struck but some fink makes it lose the disposition to break when struck. The right-hand side of (2) takes these cases into consideration. If Manley and Wasserman are correct, such cases are outweighed by those cases where the vase is struck and breaks: there is a “suitable” proportion of cases where the vase is struck and breaks. So, Manley and Wasserman’s strategy for avoiding the problem of finks and the problem of masks is not to add at some qualification to SCA so that such situations are properly ignored. Rather, the strategy is to take such situations into consideration but maintain that there are enough non-finkish and non-masking situations where the relevant disposition manifests. Similar remarks apply to problems with Achilles’ heels. I’m not convinced that this is a successful strategy.17 It’s clear enough what a counterexample going from left to right would look like. We would need x to be disposed to M when C yet across the relevant region of modal space C -cases are, by and large, cases where x’s disposition to M is masked or finked and so x doesn’t M . My computer’s CPU provides such a counterexample. The CPU is disposed to overheat when running a large number of (tasking) processes. That’s why the computer has a heatsink and fans. When the CPU is running a large number of processes, the heatsink and fans mask the CPU’s disposition to overheat when running a large number of processes. Now creatures smart enough to design such a CPU are also smart enough to realize that its disposition to overheat when running a large number of processes needs to be masked/finked when the CPU is running a large number of processes. And this isn’t an accident. It holds across the relevant region of modal space that cases where the CPU is engineered are, by and large, cases where the engineers realize that its disposition to overheat when running a large number of processes needs to masked/finked in some way. So, the relevant region of modal space is such that cases where the CPU is running a large number of tasking processes are predominantly cases where the CPU’s disposition to overheat when running a large number of processes is masked or finked. So, on any reasonable understanding of “suitable proportion,” it’s not the case the CPU would overheat in a suitable proportion of cases where it is running a large number of tasking processes.18 Yet the CPU is disposed 17

I should also say that I’m not convinced by an argument Manley and Wassermann make against competing conditional analyses. Since this argument also applies to my teleological account of dispositions, I leave discussion of it for the Appendix. 18 One might worry that there are at least continuum-many cases where the CPU’s disposition is neither masked nor finked, the CPU is running a large number of processes,

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to overheat when running a large number of tasking processes. That’s why the heatsink and fans are there. We have a counterexample to (PROP).19 Here is similar but more fanciful counterexample. Suppose advanced creatures have engineered an artifact that is disposed to shatter when struck. While the blueprints for this artifact were being drawn, the creatures realized that the artifact will be disposed to shatter when struck. So, not only did they engineer a mask, they also made sure that the mask would be applied upon creation of the artifact. One more bit of fantasy. This artifact is so sophisticated that it can only be engineered by creatures smart enough to realize that the artifact will be disposed to shatter when struck and also realize that some kind of mask or fink will need to be engineered and applied upon creation. Well, maybe lesser creatures could somehow “accidentally” engineer the artifact but the probability is miniscule. The important point is that across modal space there are hardly any cases where the artifact is struck and shatters. Granted there are cases where the mask/fink is defective, cases where the mask/fink is not applied, and cases where lesser creatures create the artifact. But such cases are hardly worth noticing. What is worth noticing is that on any reasonable understanding of “suitable proportion,” it’s false that the artifact would shatter in a suitable proportion of cases where it is struck. Yet the artifact is disposed to shatter when struck. That’s why the mask was applied upon creation. Some may not be convinced by these alleged counterexamples. Regarding the alleged counterexample involving my computer’s CPU, one might object that because of the heatsink and fans, the CPU is not disposed to overheat when running a large number of processes. This objection, however, seems (to me anyway) to confuse my CPU with my computer. The latter is not disposed to overheat when the CPU is running a large number of processes. The former is so disposed. That’s why the heatsink and fans are there. One might also worry about my claim that, across the relevant region of modal space, cases where the CPU is engineered are, by and large, cases where the engineers realize that its disposition to overheat when running a large number of processes needs to be masked/finked in some way. Maybe and so its disposition manifests. We are, after all, taking modal space into consideration. So, let x be the set of such cases. Surely the set of cases where the CPU’s disposition is masked or finked but the CPU is running a large number of processes has the same cardinality as x. So, how can it be that there are more of the latter cases than the former cases? Good question. It is, however, a question for Manley and Wasserman to answer. They offer some suggestions. Whatever metric they use to justify the appeal to proportionality, it had better come out that my CPU wouldn’t overheat in a suitable proportion of cases where it is running a large number of (tasking) processes. 19 This scenario is also a counterexample to the variant of (PROP) found in Wasserman (2011).

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engineers (across modal space) don’t have to be that smart to engineer the CPU. I don’t think so, but suppose I’m wrong. What about the counterexample involving the super-sophisticated artifact? I find it hard to believe that such an artifact is impossible.20

2.5 A different direction The problems raised for the above conditional analyses do not show that there is no satisfactory conditional analysis of dispositions. But they do motivate a move in a different direction. That is, they at least provide some reason for considering an account of dispositions that does not rest on a conditional analysis of dispositions. So, I propose that we look in a different direction. In particular, I propose that we look towards a teleological account of dispositions.

3 . A T E LE O L OG I C A L A C C O U N T O F D I S P O S I T I O N S

3.1 Preliminary remarks The move towards a teleological account of dispositions is not motivated solely by the shortcomings of the above conditional analyses. It’s also motivated by the intuition that dispositions are, in some sense, directed at their manifestations. Some may claim to not have this intuition. To them, I point to Goodman’s famous characterization of dispositions in terms of “threats and promises.”21 Goodman’s metaphor captures an important intuition about dispositions. And it seems pretty clear that this intuition is the directedness intuition. So, if you are not sure whether you have the directedness intuition, check whether you get the metaphor. If you get it, you have the intuition. Some may doubt that the directedness intuition should be given much weight when evaluating an account of dispositions. Perhaps the stronger intuition is that there is an important connection between dispositions and conditionals, and if we could only get this connection straight, we would 20 Manley and Wasserman (2008) suggest that (PROP) may have to be revised so that some C -cases are weighed more heavily than others. So, perhaps this is a way to avoid the above counterexamples. But then, as Cross (2013) notes, it’s hard to see how (PROP) is an improvement over appeals to normal conditions, ideal conditions, or ceteris paribus conditions. 21 Goodman (1954, p. 40).

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have a conditional analysis of dispositions that explains, in some deflationary way, intuitions about the directedness of dispositions. As a first step in developing a teleological account of dispositions, I turn this concern around and into an argument for appealing to teleology in giving an account of dispositions. In particular, I argue that by appealing to teleology we can actually provide a counterexample free and informative connection between dispositions and conditionals.

3.2 The first step Consider the following passage from Lewis (1997). Sometimes it takes some time for a disposition to do its work. When stimulus S arrives and the disposition is present, some process begins . . . When the process reaches completion, then that is, or that causes, response r. But if the disposition went away part-way through, the process would be aborted.22

To foreshadow what is to come, I claim that the processes Lewis describes are teleological processes. But before saying anymore about this, we need to slightly amend what Lewis says. Masking cases show that even if the stimulus condition arrives and the disposition remains, there need not be some process that begins and ends with a manifestation of the disposition. So, taking masks into consideration, suppose Lewis had said something slightly different. Something like this: Sometimes it takes some time for a disposition to do its work. When the disposition is activated, some process begins. When the process reaches completion, then that is, or that causes, response r. But if the disposition went away part-way through, the process would be aborted. Nonetheless, if nothing were to interfere with this process, there would be response r.

We would then have the following activation principle in place: (AV)

If x’s disposition to M when C is activated, then either x immediately M s or there is some process such that if the process were to continue without interruption, x would M .

I claim that (AV) is counterexample free. Before offering support for this claim, I need to clarify a distinction (AV) relies upon. This is the distinction between a disposition being activated and the stimulus condition for the disposition obtaining. Certain masking cases illustrate the distinction. Distinguish between two types of masking cases: those in which a disposition is manifesting but does 22

Lewis (1997, p. 146).

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not manifest because of a mask, and those in which a mask prevents even a partial manifestation of the disposition even though the stimulus condition obtains. Call the former “Type-1” masking cases and the latter “Type-2” masking cases. The case of the poison that is killing you but doesn’t kill you because of the antidote is an example of a Type-1 masking case. For an example of a Type-2 masking case, consider a vase disposed to break when dropped but wrapped in bubble-wrap. It’s dropped and doesn’t break. It seems plausible that the bubble-wrap not only prevented a manifestation of the disposition but also a partial manifestation of the disposition. Furthermore, it seems plausible that the bubble-wrap prevented even a partial manifestation of the disposition because it prevented the disposition from being activated in the first place. The disposition remained dormant even though the stimulus condition obtained. Similar remarks apply to my CPU’s disposition to overheat when running a large number of processes. The fans and heatsink prevent the disposition from being activated when the CPU is running a large number of processes. Generalizing, Type-2 masking cases are cases where the stimulus condition obtains, the disposition remains, but the disposition is not activated. Cases involving finks also illustrate the distinction. When does the electro-fink running on reverse cycle make the wire lose its disposition to conduct electricity when touched by a conductor? Well, when the wire is touched by a conductor. But does the fink do its work instantaneously?23 If so, the instant the wire is touched by a conductor, it is not disposed to conduct electricity. If not, there is an instant or interval of time where the wire is touched by a conductor and disposed to conduct electricity when touched by a conductor. In the first case, the disposition is not activated because it’s no longer present. In the second case, it’s not clear whether the disposition is momentarily activated before the fink does its work. If it isn’t, then we have a case where the stimulus condition obtains but the disposition isn’t activated.24 Generalizing, it seems that (reverse) finks can work in one of three ways. They can make it so that the disposition goes away the instant the stimulus condition obtains, and so the disposition is not activated because it’s no longer there. They can make it so that the disposition goes away when the stimulus condition obtains but before the disposition is activated. And they can make it so that the disposition goes way after the disposition is activated but before the disposition manifests. Instances of the first two ways bring out the distinction between a disposition being activated and the stimulus condition of the disposition obtaining.

23 24

Lewis (1997) calls this “a dilemma about timing.” My intuitions, though not entirely clear, are that the disposition isn’t activated.

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It should be clear, then, that there is a distinction between a disposition being activated and the stimulus condition of the disposition obtaining. Of course, there is a connection between the two. For a disposition to be activated, its stimulus condition must obtain. Likewise, though perhaps this is obvious, the activation of the disposition requires that the disposition is present. Let’s return to (AV). I claim that (AV) is counterexample free. Type-1 masking cases pose no threat. In such cases, the disposition is activated but doesn’t manifest because of the mask. Nonetheless, when the disposition was activated, there was thereby a process such that were that process to continue without interruption, there would be a manifestation of the disposition. Type-2 masking cases pose no threat. In such cases, the disposition isn’t activated. Finks pose no threat. Either the fink makes it so that the disposition isn’t activated when the stimulus condition obtains or it makes it so that the disposition goes away after it has been activated but before it manifests. In the first case, there is obviously no threat to (AV). In the second case, there is no threat to (AV) because if it weren’t for the fink the disposition would have manifested. And this is so because when the disposition was activated, there was thereby a process such that if the process were to continue without interruption, there would be a manifestation of the disposition. So, (AV) has no problems with the problem of finks or the problem of masks. What about Achilles’ heels? My intuitions suggest that the concrete block with the Achilles’ heel has two dispositions. Because it is sturdy, it is disposed to remain intact when struck. Because of its Achilles’ heel, there is a particular spot s and a particular amount of force f such that the block is disposed to shatter when struck with force f in spot s. Suppose the block is struck with force f in spot s and its disposition to shatter when struck with force f in spot s manifests. My intuitions suggest that even though the block was struck, its disposition to remain intact when struck wasn’t activated. What was activated was its Achilles’ heel. But some may have the intuition that if the block is disposed to shatter when struck with force f in spot s, then it can’t be disposed to remain intact when struck. To them, I say consider Achilles. The greatest warrior of the Trojan War was not disposed to fall when struck. He was disposed to withstand harm when struck. However, because of his Achilles’ heel, he was also disposed to fall when struck in the just the right spot. When Achilles was struck in just the right spot, his disposition to fall when struck in just the right spot was activated and subsequently manifested. Was his disposition to withstand harm when struck activated? My intuitions suggest that it wasn’t. If I’m right, there is no threat to (AV). If I’m wrong, there is a threat to (AV) only if when Achilles was struck in just the right spot, there wasn’t thereby a

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process such that if the process were to continue without interruption, Achilles’ disposition to withstand harm when struck would manifest. My intuitions aren’t entirely clear because I don’t have the intuition that Achilles’ disposition to withstand harm when struck was activated in the first place. But it does seem to me that if this disposition was activated, there was such a process. And what interfered with it was the activation of Achilles’ Achilles’ heel. So, in either case, there is no threat to (AV). Generalizing, we can think of cases involving Achilles’ heels as special cases of masking. If the case is a Type-1 masking case, then the relevant disposition is manifesting but doesn’t manifest because of the activation of an Achilles’ heel. If the case is a Type-2 masking case, then the activation of an Achilles’ heel prevents the relevant disposition from being activated in the first place. In either case, there is no threat to (AV). So, since I can think of no other potential threat to (AV), I conjecture that (AV) is counterexample free.25 However, there is still an issue that needs to be resolved. The issue is that (AV) is an “escape-clause” account of the relationship between dispositions, their activation, and their manifestation. To give a few simple examples: • If D is activated, then, ceteris paribus, it manifests. • In normal/ideal/typical circumstances, if D activated, it manifests. • If D is activated, then, provided nothing interferes, it manifests. Moline’s remark about such proposals bears repeating. [T]hey are fundamentally dodges . . . They amount to muddled ways of disguising from ourselves more or less serious ignorance of the dispositional properties of individual things or persons and of types of things or persons. (Moline 1975, pp. 244–5)

As I said above, I agree, ceteris paribus. With respect to (AV), this means that if we don’t have some reason for thinking that it is legitimate to appeal to the notion of a process continuing without interruption, (AV) is simply a way to disguise more or less serious ignorance of the relationship between dispositions, their activation, and their manifestation. (AV), however, can be legitimized by appealing to teleology. Consider the following passage from Makin (2006):

25 Jenkins and Nolan (2011) argue that it is possible for there to be dispositions with impossible manifestations. However, their alleged examples pose no threat to (AV). They offer no example of disposition such that it’s possible for the disposition to be activated but impossible for there to be even a partial manifestation of the disposition.

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[I]f it is appropriate to view a [process] teleologically, it is therefore also appropriate to apply other notions: interference, interruption, hindrance, and a normal outcome. It makes sense to talk of a teleological process being interrupted . . . That is because a teleological process has a privileged stage to which it runs in normal conditions, unless interfered with or hindered: the [end] to which it is directed.26

Makin is offering an explanation of why Aristotle often appeals to teleological notions in discussing the nature of his distinction between potentiality and actuality and, in particular, why actuality is prior to potentiality. Putting Aristotle aside, it shouldn’t be too difficult to see how Makin’s insight is relevant to (AV). Makin’s insight is that if a process is a teleological process, then it is legitimate to appeal to the notion of the process continuing without interruption. It’s legitimate because of the following principle governing teleological processes. (TP)

If a process p is directed at end E, then: in virtue of p being directed at end E, if p were to continue without interruption, E would be the case.

(AV) follows from the conjunction of (TP) and (T1). (T1)

If x’s disposition to M when C is activated, then either (a) x immediately M s, or (b) there is some process directed at the end that x M s.

Given (T1), we thus have good reason for thinking (AV) isn’t fundamentally a dodge. On the contrary, it’s a consequence of (T1), which is a substantive thesis about the relationship between dispositions, their activation, and their manifestation. So I offer (T1) to those looking for an interesting and counterexample free connection between dispositions and conditionals. If you accept my offer, you’ll get (CDC). (CDC)

Necessarily: if x is disposed to M when C , x’s disposition to M when C is activated, and x doesn’t immediately M , then there is some process such that: (*) if the process were to continue without interruption, x would M .

(T1) may seem like a steep price to pay for an interesting and counterexample free connection between dispositions and conditionals. However, the point remains that by appealing to teleology, we can actually provide an interesting and counterexample free connection between dispositions and 26

Makin (2006), p. 194.

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conditionals. Thus, those who think that the directedness of dispositions can be explained, in some deflationary way, by some interesting connection between dispositions and conditionals have a worry to contend with if we can offer an account of dispositions that explains the directedness of dispositions and, in doing so, explains why (T1) holds. The worry is that the directedness of dispositions is what explains why there is some interesting connection between dispositions and conditionals. The stage is now set for a teleological account of dispositions.

3.3 The proposal Following Molnar (2003), I claim that directedness is what sets dispositions apart from non-dispositional properties. There seems to be no sense in which a triangular object is, in virtue of being triangular, in a state directed at the occurrence of some event.27 On the other hand, there seems to be some sense in which a vase disposed to break when struck is, in virtue of being so disposed, in a state directed at the occurrence of an event in which the vase breaks. Such intuitions provide some initial justification for the claim that directedness is what sets dispositions apart from non-dispositional properties. But, of course, more needs be said about exactly what type of directedness is alleged to distinguish dispositions from non-dispositional properties. I say that teleological directedness is what sets dispositions apart from nondispositional properties. (T2) spells out the details. (T2)

Necessarily: a property P is a disposition iff there is a condition C and event-type M such that necessarily, P is the property of being in a state directed at the end that one M s when C .

(T2.1) follows from (T2). (T2.1)

Necessarily: a property P is a disposition iff there is a condition C and event-type M such that: necessarily, x has P iff x is in a state directed at the end that x M s when C .

So, what makes a disposition a disposition is that the property just is the property of being in a state directed at a certain teleological end. Consequently, to have a disposition just is to be in a state directed at a certain teleological end. 27 Inspired by Mellor (1974), some might claim that a triangular object is, in virtue of being triangular, in a state directed at the occurrence of an event in which its sides are counted and the result is three. I don’t have this intuition.

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(T2) does not specify the relation between the property of being disposed to M when C and the property of being in a state directed at the end that one M s when C . Sure enough, the property of being disposed to M when C is a disposition. But all that follows from (T2) is that there is some C * and M * such that the property of being disposed to M when C is the property of being in a state directed at the end that one M * s when C * . I propose that the identity relation holds between the property of being disposed to M when C and the property of being in a state directed at the end that one M s when C . (T3)

Necessarily: the property of being disposed to M when C just is the property being in a state directed at the end that one M s when C .

It follows from (T3) that necessarily, x is disposed to M when C just in case x is in a state directed at the end that x M s when C . With (T3) in hand, we turn to the question: what happens when C obtains and x is disposed to M when C ? My answer is that what happens depends on whether x’s disposition is activated when C obtains. If x’s disposition isn’t activated, then nothing of interest happens with respect to x being disposed to M when C . That is, the disposition remains dormant. However, if x’s disposition is activated, something of interest does happen. In particular, in virtue of x being disposed to M when C, either x immediately M s or there is a process directed at the end that x M s. Given (T3), we thus get (T4). (T4)

If x’s disposition to M when C is activated, then in virtue of x being in a state directed at the end that x M s when C , either x immediately M s or there is a process directed at the end that x M s.

So, the teleological directedness of a disposition does some work when the disposition is activated. It also explains why (T1) holds. The conjunction of (T2), (T3), and (T4) constitute my teleological account of dispositions: (TAD), for short. A concrete example may help clarify (TAD). Consider the property of being disposed to dissolve when in water. (T3) tells us that this property just is the property of being in a state directed at the end that one dissolves when in water. (T2), then, tells us the property of being disposed to dissolve when in water is a disposition. (Nothing new.) But it also tells us that what makes this property a disposition is that it is the property of being in a state directed at the end that one dissolves when in water. Now consider a chuck of salt disposed to dissolve when in water. Suppose the salt is placed in water and it’s disposition is activated. (T4) tells that in virtue of the salt being in a state directed at the end that it dissolves when in water, either the salt immediately dissolves or there is a process directed at the end that the salt dissolves.

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Supposing the salt does not instantaneously dissolve, there is then a process directed at the end that the salt dissolves. If this telos of the process were to obtain, there would be a manifestation of the salt’s disposition. If this telos doesn’t obtain, there is no manifestation in this particular circumstance (but if the process had continued its normal course without interruption, the telos would have obtained). The argument for (TAD) is that it explains the directedness of dispositions, and in doing so, provides an interesting and counterexample free connection between dispositions and conditionals. This wouldn’t be much of an argument if a competing account of dispositions provides both a better explanation of the directedness of dispositions and an interesting and counterexample free connection between dispositions and conditionals. So, we’ll have to see what the competition has to offer. Before comparing (TAD) to its rivals, though, we should address a concern that I’m sure has been gnawing away at some.

4. A DETOUR

4.1 Seriously? The concern can be put like this: “You can’t be serious.” We can categorize those with such a concern into three groups. First, there are those who, setting aside the analogy, would dogmatically agree with Francis Bacon’s remark that “inquiry into final causes is sterile, and, like a virgin consecrated to God, produces nothing.”28 Second, there are those who think that inquiry into final causes has a place, provided that place concerns the goals of agents or the purposes of the artifacts they design. For them, a teleological account of dispositions is committed to projecting mental states to properties or treating them as artifacts, and so is absurd. Third, there are those who allow for so-called “natural teleology” but restrict it to the function or proper function of features of organisms. For them, a teleological account of dispositions is, at best, committed to treating dispositions as biological functions, and so is hardly worth considering. The dogmatists can be ignored since nothing can be said to make them change their minds. The simple response to the other two groups is that while teleology is often tied up in talk of goals, purposes, design, function, proper function, and sometimes talk of certain outcomes being better than others, there is no reason to assume that teleology must be tied up in such 28 De Augmentis Scientiarum, Bk. iii, Ch. 5, quoted in Woodfield (1976, p. 3) and Hawthorne (2006, p. 268).

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talk. So, don’t make this assumption when considering (TAD). There is a general notion of teleological directedness that outstrips talk of goals, purposes, design, and function. It is this general notion at play in (TAD). Some might not be convinced by the simple response. They might want some reason, independent of my argument for (TAD), to take seriously a general notion of teleological directedness that outstrips talk of goals, purposes, design, and function. To this end, we take a detour from dispositions and turn to sentences in the progressive aspect.

4.2 The progressive aspect and events in progress Here are some examples of sentences in the progressive aspect. (3a) Steve is driving to Boston. (3b) A chunk of salt is dissolving. (3c) The universe is expanding. (3a) says that there is an event of Steve driving to Boston in progress; (3b) says that there is an event of some salt dissolving in progress; and (3c) says that there is an event of the universe expanding in progress. So it is in general: a sentence in the progressive aspect says that there is an event of some type in progress. Indeed, this is the core semantic intuition about the progressive aspect. The orthodox approach to capturing the core intuition is to offer a modal analysis of the progressive.29 The appeal to modality usually starts with the observation that an event in progress need not culminate.30 Steve could be driving to Boston but be forced to turn around due to car troubles. A chunk of salt could be dissolving but be taken out of water before it (fully) dissolves. In such cases, there is an event in progress that doesn’t culminate. But what if Steve’s drive hadn’t been interrupted by car troubles? Presumably, he would have driven to Boston. And what if the salt hadn’t been taken out of the water? Presumably, it would have (fully) dissolved. Generalizing, the basic idea is that while an event in progress need not culminate, it nonetheless would culminate if it were to continue without interruption. As 29 See Dowty (1979), Asher (1992), Landman (1992), Bonomi (1997), Portner (1998), Hallman (2009), and Higginbotham (2009). 30 This observation is one way of putting what is called the “imperfective paradox” in the literature on the progressive. Another way of putting the imperfective paradox is that a past progressive does not, in general, entail its perfective correlate. For instance, “Steve was driving to Boston” does not entail “Steve drove to Boston.”

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far as I can tell, this “no-interruption” intuition is what spurs modal analyses of the progressive. Bypassing the details of a semantics for the progressive that captures the no-interruption intuition, let’s just focus on the account of events in progress suggested by the no-interruption intuition. It’s helpful here to appeal to resultant states. So, let me say a word about resultant states, and then offer the account of events in progress suggested by the no-interruption intuition. A resultant state of an event is a state of the event having occurred or taken place. For example, suppose Mirah drew a circle. As a result of this event taking place, a state of Mirah having drawn a circle obtains. This state of Mirah having drawn a circle is a resultant state of the event. If this example doesn’t help, here is a heuristic that may. Take a sentence in the simple past which describes an event and form its present perfect correlate. Then, think of the present perfect sentence as describing the (relevant) resultant state of the event described by the simple past sentence. For example, in the sentences that follow think of each b-sentence as describing the (relevant) resultant state of the event described by the a-sentence. (4a) Steve drove to Boston. (4b) Steve has driven to Boston. (5a) A chunk of salt (fully) dissolved. (5b) A chunk of salt has (fully) dissolved. (6a) The universe expanded. (6b) The universe has expanded. To put it yet another way, think of (4b) as saying that a state of Steve having driven to Boston now holds, think of (5b) as saying that a state of the chunk of salt having (fully) dissolved now holds, and think of (6b) as saying that a state of the universe having expanded now holds. Such states are resultant states of particular events.31 With resultant states suitably clarified, we can now offer the account of events in progress suggested by the no-interruption intuition. (NI)

Necessarily, e is a φ event in progress at t iff e would bring about a resultant of a φ event if it were to continue past t without interruption.

31 There are analyses of the perfect that involve quantification over resultant states, and so take the above paraphrases to be semantically significant. See Parsons (1990), Kratzer (2000), and Higginbotham (2009).

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For instance, an event is an event of the universe expanding in progress (at t) just in case the event would bring about a state of the universe having expanded if it were to continue (past t) without interruption. There are, however, two problems with (NI). The first is that the right-toleft direction is subject to counterexample. The second is that the left-toright direction is subject to a worry about informativeness. Our counterexample is a variant of a scenario due to Landman (1992). Suppose Mary is delusional and thinks she needs to swim to the other side of the Atlantic to save her soul. Mary is not only delusional but also a very bad swimmer. So, she enters the Atlantic around Boston, swims for an hour, and then drowns. Mary was trying to swim to the other side of the Atlantic but she wasn’t actually swimming to the other side of the Atlantic. Indeed, this case is a nice example of the difference between trying to do something and actually doing what you are are trying to do. But what if Mary’s swim had continued without interruption. Well, she wouldn’t have drowned. Likewise, she wouldn’t have been eaten by a shark or saved by a fishing boat. In short, the swim would have continued on until she miraculously reached some place on the other side. So, we have an event that is not an event in progress of Mary swimming to the other side of the Atlantic but one that would bring about a state of Mary having swum to the other side of the Atlantic if it were to continue without interruption. Thus, we have a counterexample to the right-to-left direction of (NI). The worry about informativeness is that unless more is said about what counts as an interruption of a φ event in progress, we have no reason not to think of an interruption of a φ event in progress as simply something that prevents it from bringing about the resultant state of a φ event. We should just as well treat an interruption of an event in progress of Steve driving to Boston as simply something that prevents the event from bringing about a state of Steve having driven to Boston. So, the claim that an event in progress of Steve driving to Boston would bring about a state of Steve having driven to Boston if it were to continue without interruption amounts to the uninformative claim that this event in progress would bring about such a state unless something prevented it from doing so. In short, unless something more is said about what counts as an interruption of a φ event in progress, the left-to-right direction of (NI) amounts to the uninformative claim that a φ event in progress would bring about a resultant state of a φ event unless something prevents it from doing so. Perhaps there is a way to modify (NI) so that it avoids the threat of counterexample and the worry about informativeness. Suffice it to say that as in the case of conditional analyses of dispositions, there is no modal analysis of the progressive that is widely recognized to be informative and

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counterexample free.32 So, someone who is sympathetic to a modal account of events in progress has her work cut out for her. There is, however, an account of events in progress that not only explains the no-interruption intuition but also explains another intuition about events in progress. This is the intuition that an event in progress is in some sense directed at bringing about a certain state of affairs. Surely, an event in progress of Steve driving to Boston is in some sense directed at bringing about a state of affairs in which Steve has driven to Boston. Surely, an event in progress of a chunk of salt dissolving is in some sense directed at bringing about a state of affairs in which the salt has (fully) dissolved. Surely, an event in progress of the universe expanding is in some sense directed at bringing about a state of affairs in which the universe has expanded. So, it is in general: events in progress have some kind of directedness. I claim that the directedness of events in progress is a teleological directedness: for an event in progress to be directed at a certain state of affairs is for the relevant state of affairs to be a telos of the event in progress. More specifically, I propose the following teleological account of events in progress. (EIP)

Necessarily, e is a φ event in progress at t iff e is, at t, directed at the end that it cause the resultant state of a φ event to obtain at some t 0 >t.

So, for example, an event is an event in progress of Steve driving to Boston (at t) iff it is an event that is (at t) directed at the end that it cause a state of Steve having driven to Boston to obtain (at some later t 0 ).33 (EIP) explains the directedness intuition about events in progress. It also explains the no-interruption intuition. Recall Makin’s insight.

32 See Szabó (2004), Szabó (2008), and Kroll (2015) for arguments against modal analyses of the progressive. 33 (EIP) should not be understood as a complicated way of saying that an event in progress is directed at the end that it culminate. To see why, let e be a sufficiently extended event in the progress of the universe expanding. For each expansion of the universe during this time, there is a corresponding resultant state of the universe having expanded that is brought about by e. But e doesn’t culminate each time it brings about a state of the universe having expanded. Indeed, as an event in progress of universe expanding, e is not associated with any kind of culmination. What (EIP) basically tells us is that even if e brings about a state of the universe having expanded at some moment t, it is still, at t, directed at bringing about a later state of the universe having expanded. So, while e is not directed at any kind of culmination, it is still, at each moment, directed at bringing about further expansion of the universe. In short, some events in progress are (teleologically) directed at culminating, others aren’t, but all are (teleologically) directed at bringing about a later resultant state.

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If a process p is directed at end E, then: in virtue of p being directed at end E, if p were to continue without interruption, E would be the case.

(EIP) treats events in progress as teleological processes. Thus, (E1) follows from (EIP). (E1)

If e is a φ event in progress at t, then e would bring about a resultant state of a φ event if it were to continue past t without interruption.34

So, (EIP) explains the intuition that spurs modal accounts of events in progress. Furthermore, the informativeness worry is no longer a worry. (E1) is a consequence of (EIP), which is a substantive thesis about events in progress. Here, then, is an argument, independent of my argument for (TAD), to take seriously a general notion of teleological directedness that outstrips talk of goals, purposes, design, and function. (EIP) is an account of events in progress that explains the directedness intuition and explains the intuition that motivates its rivals (i.e. modal accounts). So, we have good reason to take (EIP) seriously. The notion of teleological directedness in (EIP) is a general notion that outstrips talk of goals, purposes, design, and function. For example, bringing about a state of the universe having expanded is not a goal, purpose, or function of an event in the progress of the universe expanding, and I see no reason to think such an event in progress was designed to bring about such a state. 34 (E1) is simply the left-to-right direction of (NI). So, I am committed to this direction of (NI) being counterexample free. The only alleged counterexample I aware of is due to Szabó (2008). Szabó asks us to consider a young boy, Frank, who starts enumerating prime numbers in sequence: two, three, five, seven, eleven, thirteen, and so on. According to Szabó, uttering “Frank is enumerating the primes” is an accurate description of what Frank is doing. But it’s not possible for Frank to enumerate the primes. So, it’s not possible for there to be a resultant state of an event in which Frank enumerates the primes. Thus, if “Frank is enumerating the primes” is an accurate description of what Frank is doing, we would appear to have a counterexample to (E1). I’m not convinced. Suppose Frank’s mother utters “Frank is enumerating the primes” to describe what Frank is doing and you overhear the utterance. Suppose you ask: “All of the primes or some of the primes?” If the mother were to respond with “All of the primes,” she would be saying something false. On the other hand, if the mother were to respond with “Some of the primes,” she would by saying something true. It seems, then, that taking the mother to be saying something true when she utters “Frank is enumerating the primes” rests on taking her to be communicating the proposition that Frank was enumerating some of the primes. If this is right (and I think it is), then the scenario is not a counterexample to (E1). To be such a scenario Frank would have to be enumerating all of the primes, which he obviously is not doing. Rather, the scenario is one where Frank is enumerating some of the primes, and he obviously can enumerate some of the primes.

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So, since we have reason to take (EIP) seriously, we have reason to take seriously a general notion of teleological directedness that outstrips talk of goals, purposes, design, and function. I have not provided a detailed defense of (EIP) over its rivals.35 But the purpose of our detour isn’t to provide a detailed defense of (EIP). It’s to provide some independent reason to take seriously a general notion of teleological directedness. Let us, then, return to dispositions by comparing (TAD) to its rivals.

4.3 Against the competition The competition falls into two camps. The first camp consists of accounts of dispositions that rest upon a conditional analysis of dispositions. The second camp consists of accounts of dispositions that appeal to a different type of directedness.36 We have yet to see a satisfactory (i.e. counterexample free and informative) conditional analysis of dispositions. This is a problem for the first camp. It also makes comparing (TAD) to the competition from the first camp rather difficult. Furthermore, there doesn’t seem to be much motivation for continuing the search for a satisfactory conditional analysis if we can offer an account of dispositions that explains both the directedness of dispositions and provides a connection between dispositions and conditionals. (TAD) is such an account of dispositions. Those sympathetic to conditional analyses might respond by claiming that we want a deflationary account of dispositions, and we get what we want only if there is a counterexample free and informative conditional analysis of dispositions. They might remind me of something else Goodman said—namely, that it would be ideal to explain what dispositions are without “any reference to occult powers.”37 I have two things to say in response. 35

See Kroll (2015) for such a defense of (EIP). There is a possible third camp. Following Fara (2005), one might reject conditional analyses but offer a “habitual” account of dispositions in which being disposed to M when C just is having an intrinsic property in virtue of which one M s when C . I do not consider this possible third camp to provide actual competition to (TAD) because the counterexamples I offered to (PROP) serve as counterexamples to a habitual account of dispositions: my CPU does not have an intrinsic property in virtue of which it overheats when running a large number of processes, yet it is disposed to overheat when running a large number of processes (that’s why the heatsink and fans are there). If you are skeptical of my counterexamples to (PROP), see Wasserman (2011) for reasons to be skeptical of the prospects of a habitual account of dispositions. 37 Goodman (1954, p. 40). 36

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First, a conditional analysis of dispositions does not tell us what dispositions are.38 All it provides is a schematic biconditonal whose instances are alleged to be truths. Recall (SCA). (SCA)

Necessarily: x is disposed to M when C iff x would M if C were the case.

As Manley (2012) notes, you could accept (SCA) and hold a reductive functionalist account of dispositions in which the property of being disposed to M when C just is the second-order property of having a non-dispositional property p in virtue of which one would M if C were the case. Or you could hold a similar but non-reductive functionalist account of the dispositions. Or you could accept (SCA) yet hold that the disposition to M when C just is the property of being such that one would M if C were the case. Of course, it would be a mistake to accept (SCA). But the point is that if you are worried about occult powers, a conditional analysis of dispositions by itself isn’t enough to relieve your worries. You need an account of dispositions that does not require occult powers. Second, we don’t want a deflationary account of dispositions if a deflationary account of dispositions is one that rejects the possibility of a disposition being fundamental property (i.e. a property that “carves the world at its joints”). Dispositional essentialists argue that any account of dispositions that rejects the possibility of fundamental dispositions is badly mistaken.39 According to such theorists, our best science tells that at least some fundamental properties of the actual world are dispositions. Just consider spin, charge, and mass. Our best science tells us that these properties are fundamental properties. And surely spin, charge, and mass are dispositions. Witness Ellis and Lierse (1994): With few exceptions, the most fundamental properties that we know about are all dispositional . . . Therefore, we must either suppose that these basic properties are not truly fundamental . . . or else we must concede that categorical realism is false.40

We’ll understand categorical realism is the view that no disposition is a fundamental property (i.e. no property that carves the actual world by its joints is a disposition). Ellis and Lierse’s challenge to the categorical realist is straightforward. Either show that charge, for instance, is not a disposition or show that it’s not a fundamental property. Good luck with either disjunct. I will not take a stand on whether or not luck is on the categorical realist’s side. Maybe our best science will change. Or maybe there is some interpretation of 38 Manley (2012) stresses the importance of this point, and, I should say, inspired this first response. 39 See Ellis (2001), Molnar (2003), and Bird (2007). 40 Ellis and Lierse (1994, p. 32).

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our current best science in which charge is not a fundamental property or at least not a disposition. I will, however, take a stand on the possibility of there being something like charge that is a disposition and a fundamental property. Let’s stipulate d-charge is the property the fundamental particles of the actual world would have if what the dispositional essentialist says about charge is correct. In other words, if the dispositional essentialist is right about charge, then d-charge is charge. Could there be a world w in which the fundamental entities of w have d-charge as a fundamental property? The dispositional essentialist says the actual world is such a world. I say that such a world is possible and so it’s possible for there to be a disposition that is a fundamental property. Let’s now put aside whether (TAD) undermines the motivation for analyzing dispositions in terms of conditionals. We’ll just suppose that we have a counterexample free and informative conditional analysis. I argue that even with such an analysis, there is reason to favor (TAD), and the reason comes from the possibility of a basic or fundamental disposition. For the sake of concreteness, suppose (PROP) is a counterexample free and informative analysis of dispositions. Now suppose that x is disposed to M when C . Then, (7) follows from (PROP). (7)

x would M in a suitable proportion of C -cases.

Further suppose that the property of being disposed to M when C is a basic or fundamental property. Presumedly, then, there is something about the disposition itself that explains why (7) is the case. What could it be? One option is that the property of being disposed to M when C just is the property of being such that one would M in a suitable proportion of C3 -cases.41 Taking this approach, we would be committed to primitive counterfactual facts, and so would be committed to denying the plausible principle that what something would do depends on how it is. But maybe that principle needs to be rejected once we consider the possibility of fundamental dispositions. Another option is to add (PROP) to (TAD) but deny that the property of being disposed to M when C just is the property of being such that one would M in a suitable proportion of C -cases. Taking this approach, we would be committed to primitive teleological facts that explain certain counterfactual facts. In other words, the position would be that (7) is the case in virtue of (8) being the case. (8)

x is in a state directed at the end that x M s when C . 41

See Manley (2012).

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So, by appealing to (TAD), we don’t have to reject the principle that what something would do depends on how it is. Thus, there is reason to favor the appeal to (TAD). Generalizing, I make the following objection to the first camp. We want an account of dispositions that does not rule out the possibility of a disposition being a fundamental property. The first camp promises to provide a counterexample free and informative conditional analysis of dispositions. Suppose they provide such an analysis. There is still reason to favor (TAD) so long as (TAD) is compatible with the analysis. The reason is that (TAD) provides an account of dispositions that allows us to retain the principle that what something would do depends on how it is even in the case where that thing has a fundamental disposition.42 I conclude, then, that there isn’t much of a challenge from the first camp. First, they have yet to provide a satisfactory conditional analysis from which they can issue a challenge. Second, even if they can provide such an analysis, there is a challenge only if they can show that the analysis is incompatible with (TAD) or show that it is impossible for a disposition to be a fundamental property. Let’s turn to the second camp. The competition from the second camp appeals to a different type of directedness in providing an account of dispositions. Whereas I claim that the directedness of dispositions is teleological, the second camp claims that the directedness of dispositions has something to do with intentionality. For lack of a better name, we’ll call members of the second camp “intentionality-based” accounts of dispositions.43 Intentionality-based accounts of dispositions are inspired by some parallels between intentional mental states and dispositions. Suppose Johnny believes that Santa brings him presents, lives at the North Pole, has red cheeks, etc. There is a certain sense in which these beliefs of Johnny’s are directed at Santa. Since Santa doesn’t exist, there is also a certain sense in which these beliefs are directed at something that doesn’t exist. And so it is for intentional mental states in general: there is a certain sense in which they are directed at

42 It should be noted that accepting (TAD) does not imply accepting the possibility of fundamental dispositions. Suppose we were to add (T5) to (TAD): (T5) Necessarily: the property of being in a state directed at the end that one M s when C just is the second-order property of having a non-dispositional property I in virtue of which: if C were to obtain, either x would immediately M or there would be a process directed at the end that xM s.

(TAD) would now entail that it is impossible for a disposition to be a fundamental property. Of course, (T3) would be subject to counterexample, and (T4) would be subject to overdetermination worries. But that seems to go with the territory. 43 See Place (2005), Martin and Heil (1998), and Molnar (2003).

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something, and they can be directed at something that doesn’t exist.44 But dispositions have these two features as well: there is a certain sense in which a disposition is directed at its manifestation, and, since a disposition need not manifest, there is a certain sense in which a disposition can be directed at something that does not exist. So, since some dispositions are not mental properties, this suggests that intentionality might not be the mark of the mental. It also suggests that maybe we should try to offer an account of dispositions in which the directedness of dispositions is explained in terms of the directedness of intentionality. In other words, perhaps we should offer an intentionality-based account of dispositions. There is an obvious objection to this line of thought. The sense in which an intentional mental state is directed at something is that the mental state is about or represents something. Johnny’s belief that Santa brings him presents is directed at Santa in the sense that his belief is about or represents Santa. But dispositions are neither about nor represent their manifestations. Thus, the directedness of dispositions cannot be explained in terms of the directedness of intentionality.45 If it weren’t for (TAD), I don’t think this objection would be decisive. A defender of an intentionality-based account of dispositions could respond by saying that the numerous failed attempts to provide a satisfactory conditional analysis of dispositions provide some motivation for looking in a different direction. Perhaps the parallel between the directedness of intentional mental states and the directedness of dispositions isn’t perfect. But the parallel doesn’t have to be perfect for us to grasp a more general notion of intentionality, one in which there is both mental intentionality and physical intentionality. (TAD) undermines this response. It provides an explanation of the directedness of dispositions, and it is not motivated by appealing to any parallel between intentional mental states and dispositions. As illustrated by the above detour, part of the motivation for (TAD) is that there is some parallel between events in progress and dispositions. But this seems to be correct. Contrast the following: (i) the sense in which an event in progress of Steve driving to Boston is directed at its culmination. (ii) the sense in which a vase’s disposition to break when struck is directed at its manifestation. (iii) the sense in which Johnny’s belief that Santa has red checks is directed at Santa. 44

This observation is famously due to Brentano (1874). Bird (2007) argues that other alleged parallels between intentional mental states and the directedness of dispositions are weak at best. 45

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It strikes me that (i) and (ii) are very similar but (iii) is very different. So, I think it is safe to appeal to a parallel between events in progress and dispositions as part of the motivation for (TAD). In any case, (TAD) offers an explanation of the directedness of dispositions that does not require postulating a more general notion of intentionality. Thus, the burden is on the second camp to show that this explanation is inadequate but an explanation that rests on (ii) and (iii) being similar in some way is adequate. I’m skeptical. The second camp might respond by claiming that my teleological notion of directedness falls under the general notion of intentionality. If this can be shown, then I join the second camp.

4.4 Taking stock Let’s take stock. The first camp needs to provide a satisfactory conditional analysis that is incompatible with (TAD) or they need to provide a satisfactory conditional analysis and show that it is impossible for a disposition to be fundamental property. The second camp needs to provide a convincing argument that a teleological explanation of the directedness of dispositions is inadequate but an explanation that appeals to similarities between intentional mental states and dispositions is adequate. I’ve argued that neither camp has provided what it needs to provide. I conclude we have good reason to favor (TAD) over the competition.

5 . C O N C L U D I N G R E M AR K S I’ve argued that whenever something has a disposition, something is in a state with a telos. Along the way, I’ve also sketched an argument for the claim that whenever there is an event in progress, there is an event with a telos. Absent a strong argument that nothing has any disposition at any time and that nothing is ever happening at any time, I take it that there is thus good reason to think that Bacon was terribly mistaken about teleology. Inquiry into final causes is far from sterile. On the contrary, it reveals the nature of dispositions and events in progress. Or so I’ve argued.46 Nick Kroll Franklin & Marshall College 46 If we conjoin (TAD) with a dispositional essentialist account of the laws of nature, we also have an argument that teleology reveals something about the laws of nature. According to a dispositional essentialist account of the laws of nature, the laws, or at least some of the laws, are grounded in those dispositions that are fundamental properties. (See Ellis 2001 and Bird 2007.) Combining this account of laws with (TAD), it follows that at

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Nick Kroll ACKNOWLEDGMENTS

I would like to thank George Bealer, Keith DeRose, Tamar Gendler, and Raul Saucedo for discussion and comments on earlier versions of this paper. My deepest thanks to Zoltán Szabó.

A PP E N D I X Manley and Wasserman (2008) argue that a problem with competing conditional analyses is that they cannot account for the fact that dispositions come in degrees. Simplifying somewhat, their argument runs as follows. Take the adjectives “fragile,” “sturdy,” and “soluble.” These adjectives denote dispositions. And these adjectives are clearly gradable adjectives. Witness: “This vase is more fragile than that one,” “The concrete block is very sturdy,” “How soluble is salt?” So, since gradable adjectives denote gradable properties, it follows that some dispositions are gradable properties. Thus, some dispositions come in degrees. The problem, then, for conditional analyses like (SCA) and (LCA) is that while dispositions come in degrees, conditionals do not. (PROP), however, doesn’t have this problem because proportions come in degrees. One problem with this argument is that on what is perhaps the standard semantics for gradable adjectives—such adjectives do not denote properties of individuals. Rather, they denote measure functions: functions from individuals to degrees on a scale. (See Kennedy 2007.) For example, “tall” is taken to denote a function from individuals to degrees on a scale of height and “cold” is taken to denote a function from individuals to degrees on a scale of temperature. Comparative morphemes, then, are taken to establish an ordering relation between degrees on the relevant scale. Bypassing the compositional details, the upshot is that “x is taller than y” is true just in case x’s degree of height is greater than y’s degree of height. A covert morpheme is postulated for the positive form of a gradable adjective (occurrences in clauses without any overt degree morphology like “x is G” where “G” is a gradable adjective). Semantically, this covert morpheme takes a measure function and returns a context sensitive function from individuals to truth values. Relative to a context, this function takes an individual and returns the value True just in case the the value of the measure function applied to the individual is a degree on the relevant scale that “stands out” in the context. Bypassing the compositional details, the upshot is that “x is tall” is true relative to a context c just in case x’s degree of height stands out in c. Let’s go back to the gradable adjective “fragile.” Under the above semantics for gradable adjectives, “fragile” does not denote a disposition. It denotes a measure function from individuals to degrees on a scale. But what is this scale measuring?

least some of the laws of nature are grounded in teleological properties. That is, it follows that at least some of the laws are teleological laws.

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Assuming that fragility has something to do with breaking and striking, it is probably further safe to assume that the scale is measuring the degree to which something is disposed to break when struck. Suppose the degrees on this scale are dispositions. At the bottom of the scale are things that are disposed to withstand any damage when struck. Going up, we find things that are disposed to crack a little when struck. Going further up, we finds things that are disposed to shatter when struck. So, x is more fragile than y just in case x’s position on this scale is higher than y’s position on this scale. Likewise, “x is fragile” is true relative to a context c just in case x’s position on this scale stands out in c. The defender of (SCA) can now claim that the degrees of the scale (i.e. the dispositions) correspond to certain counterfactual properties. Anything at the bottom of the scale is such that it would withstand any damage if struck. Going further up, we find things that would crack a little if struck. Further up yet, we find things that would shatter if struck. Generalizing, the degrees on the scale can be mapped onto certain counterfactual properties, and so (SCA) has no problems with the fact that “fragile” is a gradable adjective. On the other hand, it could be that the degrees on the scale are dispositions, but towards the top of the scale are things that are disposed to break when struck ever so lightly. Going down are things disposed to break when struck not ever so lightly but with a moderate amount of force. Going further down are things disposed to break when struck not with a moderate amount of force but with a great deal of force. At the bottom, we find the unbreakable: things that are disposed to withstand any striking. So, x is more fragile than y just in case x position on this other scale is higher than y’s position on this other scale. Likewise, “x is fragile” is true relative to a context c just in case x’s position on this other scale stands out in c. And, once again, the defender of (SCA) can now claim that these degrees on this other scale correspond to certain counterfactual properties. One can devise other scales, but so long as the degrees on this scale are dispositions, the defender of (SCA) will not have a problem. Similar remarks apply to any account of dispositions and any gradable adjective that is usually taken by philosophers to denote a disposition. So, Manley and Wasseman’s objection has no force against (TAD).

R E F E REN C E S Asher, N. (1992). “A Default, Truth Conditional Semantics for the Progressive.” Linguistics and Philosophy 15(5), 463–508. Bird, A. (1998). “Dispositions and Antidotes.” Philosophical Quarterly 48, 227–34. Bird, A. (2007). Nature’s Metaphysics: Laws and Properties. New York: Oxford University Press. Bonevac, D., J. Dever, and D. Sosa (2006). “The Conditional Fallacy.” The Philosophical Review 115(3), 273–316. Bonomi, A. (1997). “The Progressive and the Structure of Events.” Journal of Semantics 14(2), 173–205. Brentano, F. (1874). Psychology from an Empirical Standpoint. London: Routledge.

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Choi, S. (2008). “Dispositional Properties and Counterfactual Conditionals.” Mind 117(468), 795–841. Contessa, G. (2013). “Dispositions and Interferences.” Philosophical Studies 165(2), 401–19. Cross, T. (2013). “Recent Work on Dispositions.” Analysis 72(1), 115–24. Dowty, D. (1979). Word Meaning and Montague Grammar: The Semantics of Verbs and Times in Generative Semantics and in Montague’s PTQ. Dordrecht, Netherlands: Springer. Ellis, B. (2001). Scientific Essentialism. Cambridge: Cambridge University Press. Ellis, B. and C. Lierse (1994). “Dispositional Essentialism.” Australasian Journal of Philosophy 72(1), 27–45. Fara, M. (2005). “Dispositions and Habituals.” Noûs 39(1), 43–82. Goodman, N. (1954). Fact, Fiction, and Forecast. Cambridge, MA: Harvard University Press. Gundersen, L. (2002). “In Defence of the Conditional Account of Dispositions.” Synthese 130(3), 389–411. Hallman, P. (2009). “Proportions in Time: Interactions of Quantification and Aspect.” Natural Language Semantics 17, 29–61. Hauska, J. (2008). “Dispositions and Normal Conditions.” Philosophical Studies 139(2), 219–32. Hawthorne, J. (2006). Metaphysical Essays. Oxford: Oxford University Press. Higginbotham, J. (2009). Tense, Aspect, and Indexicality. Oxford: Oxford University Press. Jenkins, C. and D. Nolan (2011). “Disposition Impossible.” Noûs 46(4), 732–53. Johnston, M. (1992). “How to Speak of the Colors.” Philosophical Studies 68(3), 221–63. Kennedy, C. (2007). “Vagueness and Grammar: The Semantics of Relative and Absolute Gradable Adjectives.” Linguistics and Philosophy 30(1), 1–45. Kratzer, A. (2000). “Building Statives.” In L. Conathan (ed.), Proceedings of the 26th Annual Meeting of the Berkeley Linguistic Society. Berkeley: Berkeley University Press. Kroll, N. (2015). “Progressive Teleology.” Philosophical Studies 172(11), 2931–54. Landman, F. (1992). “The Progressive.” Natural Language Semantics 1(1), 1–32. Lewis, D. (1997). “Finkish Dispositions.” Philosophical Quarterly 47(187), 143–58. McKitrick, J. (2003). “A Case for Extrinsic Dispositions.” Australasian Journal of Philosophy 81(2), 155–74. Makin, S. (2006). Aristotle: Metaphysics Book Θ. Oxford: Clarendon Press. Manley, D. (2012). “Dispositionality: Beyond the Biconditionals.” Australasian Journal of Philosophy 90(2), 321–34. Manley, D. and R. Wasserman (2008). “On Linking Dispositions and Conditionals.” Mind 117(465), 59. Martin, C. B. (1994). “Dispositions and Conditionals.” Philosophical Quarterly 44(174), 1–8. Martin, C. B. and J. Heil (1998). “Rules and Powers.” Noûs 32(12), 283–312.

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Maurreau, M. (1997). “Fainthearted Conditionals.” The Journal of Philosophy 94(4), 187–211. Mellor, D. H. (1974). “In Defense of Dispositions.” Philosophical Review 83(2), 157–81. Moline, J. (1975). “Provided Nothing External Interferes.” Mind 84(1), 244–54. Molnar, G. (2003). Powers: A Study in Dispositions. Oxford: Oxford University Press. Mumford, S. (1998). Dispositions. New York: Oxford University Press. Parsons, T. (1990). Events in the Semantics of English. Cambridge, MA: MIT Press. Place, U. T. (2005). “Intentionality as the Mark of the Dispositional.” Dialectica 50(2), 91–120. Portner, P. (1998). “The Progressive in Modal Semantics.” Language 74(4), 760–87. Quine, W. (1960). Word and Object. Cambridge, MA: MIT Press. Ryle, G. (1949). The Concept of Mind. Chicago, IL: University of Chicago Press. Steinberg, J. R. (2009). “Dispositions and Subjunctives.” Philosophical Studies 148(3), 323–41. Szabó, Z. G. (2004). “On the Progressive and the Perfective.” Noûs 38(1), 29–59. Szabó, Z. G. (2008). “Things in Progress.” Philosophical Perspectives 22(1), 499–525. Wasserman, R. (2011). “Dispositions and Generics.” Philosophical Perspectives 25(1), 425–53. Woodfield, A. (1976). Teleology. Cambridge: Cambridge University Press.

2 Indirect Directness Jennifer McKitrick In “Teleological Dispositions,” Nick Kroll appeals to teleology to account for the way that dispositions seem to be directed toward their merely possible manifestations. He argues that his teleological account of dispositions (TAD) does a better job of making sense of this directedness than rival approaches that appeal to conditional statements or physical intentionality. In this short critique, I argue that, without satisfactory clarification of a number of issues, TAD does not adequately account for the directedness of dispositions. I focus on two aspects of TAD: the Activation Principle, and the proposed necessary and sufficient conditions for being a dispositional property. It is common in the dispositions literature to say that a disposition has a trigger, also known as a stimulus, stimulus condition, or circumstance of manifestation.1 For example, a stimulus condition for fragility is said to be ‘being struck.’ Insofar as a counterfactual conditional statement is true of an object in virtue of having a certain disposition, the antecedent of that conditional characterizes the stimulus condition for that disposition. For example, if the conditional “if it were struck, it would break” is true of the fragile glass in virtue of its being fragile, then “being struck” characterizes the stimulus for its fragility. As Kroll points out, some philosophers analyze “the glass is fragile” in terms of a conditional such as “if the glass were struck, it would break.” Kroll effectively characterizes many of the reasons conditional analyses have been on the defensive in recent years. Many philosophers have recognized numerous scenarios in which the truth values of the disposition ascription and that of counterfactual diverge.2 To take a simple 1 Some dispute the claim that dispositions have stimuli. See Vetter, Barbara. “Dispositions without Conditions.” Mind 123 (2014): 129–56 and Mumford, Stephen and Rani L. Anjum, Getting Causes from Powers. Oxford: Oxford University Press (2011): 37. 2 Numerous objectors to conditional analyses of disposition ascriptions include Vetter (op. cit.) as well as Smith, A. D. “Dispositional Properties.” Mind 86 (1977): 439–45; Martin, C. B. “Dispositions and Conditionals,” Philosophical Quarterly 44 (1994): 1–8;

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example, the counterfactual “if it were struck, it would break” is not true of the fragile glass when it is wrapped in bubble wrap. At this point, of course, there are many moves available to defenders of the conditional approach,3 but I will not rehearse them here. Rather, I want to focus on Kroll’s alternative to conditional accounts of dispositions—his teleological account. I begin with a key principle of Kroll’s account, the Activation Principle: “(AV)

If x’s disposition to M when C is activated, then either x immediately Ms or there is some process such that if the process were to continue without interruption, x would M.”

On the face of it, this principle seems subject to the same kinds of counterexamples as conditional analyses. If the glass’s disposition to break when struck is activated by striking while it is wrapped in bubble-wrap, the glass does not immediately break, and quite possibly, there is no process which is such that, if it were to continue without interruption, the glass would break. However, Kroll claims that the activation principle is counterexample free. How could this be? Kroll claims that his account avoids such counterexamples by making a distinction between a disposition being “activated” and its stimulus condition obtaining. Accordingly, a disposed object can be subject to the stimulus condition for its disposition without that disposition being activated. In other words, the disposition can be “stimulated” without being “activated.” Kroll gives the following examples to illustrate stimulation without activation: a vase wrapped in bubble wrap is dropped, and is consequently subject to the stimulus condition for “the disposition to break when dropped,” but the disposition is not activated; a computer’s disposition to overheat is subject to the stimulus condition— running a large number of processes—but the computer’s disposition to overheat is not activated due to the computer’s cooling mechanisms. The distinction between “a disposed object being in stimulating circumstances” on the one hand, and “an object having its disposition activated” on the other, makes some intuitive sense. But I suspect that part of this intuitive Bird, Alexander. “Dispositions and Antidotes,” The Philosophical Quarterly 48 (1998): 227–34; Molnar, George. “Are Dispositions Reducible?” Philosophical Quarterly 49 (1999): 1–17, 145–67; Clarke, Rudolph. “Intrinsic Finks,” The Philosophical Quarterly 58 (2008): 512–18; Everett, Anthony. “Intrinsic Finks, Masks, and Mimics,” Erkenntnis 71 (2009): 191–203; Schrenk, Markus. “Hic Rhodos, Hic Salta: From Reductionist Semantics to a Realist Ontology of Forceful Dispositions,” in Damschen, Gregor, Robert Schnepf, and Karsten R. Stüber, eds., Debating Dispositions: Issues in Metaphysics, Epistemology and Philosophy of Mind. Berlin: Walter de Gruyter, 2009. 3 See Choi, Sungho. “The Simple vs. Reformed Conditional Analysis of Dispositions.” Synthese 148 (2006): 369–79; Gundersen, L. (2002). In Defence of the Conditional Account of Dispositions. Synthese, 130, 389–411.

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appeal draws on the plausible distinction between the stimulus condition partially obtaining and the stimulus condition fully obtaining. Arguably, the stimulus condition for the disposition to break when struck does not fully obtain if something is dropped while wrapped in bubble wrap. If the distinction between partial and complete stimulus conditions is not the kind of distinction Kroll has in mind, then it must be that every aspect of the stimulus condition could fully obtain, and yet the activation of the disposition does not occur. What, then, is this activation? What kind of thing is it? What are the grounds or truth-makers for “x’s disposition to M when C is activated”? Clearly, an activation is supposed to be something that happens at a time—prior to, or simultaneous with the disposition’s manifestation, as (AV) suggests. I assume that an activation happens in a place as well, somewhere in the vicinity of the disposed object. Since an activation has a duration and a location, it seems like an event. But if the activation is an event, it is clearly supposed to be an event that is distinct from the stimulus event. So, consider a case where an unprotected glass is struck and it breaks. If the striking started a process that continued without interruption, when and where did the activation happen? One possibility is that it happened at the same time and place as the striking. If that is so, we need some criteria for event-identity beyond spatio-temporal location to differentiate the stimulus event from the activation. Perhaps these two supposedly different events involve instantiation of different properties. But what properties does an activation event have? Another possibility is that the activation happens after the stimulus occurs. So, the striking happens, and then the disposition to break when struck is activated, initiating a process which continues uninterrupted until the glass breaks. And if it just so happens that the process never starts, that’s no problem for the account, because we can just say that, while stimulated, the disposition was never activated. And what’s the reason for thinking the disposition was not activated? Perhaps we should think that the activation did not occur because neither the manifestation, nor a process leading to the manifestation, occurred. It’s not clear if there could be any independent empirical evidence that an activation did or did not occur. What if the manifestation happened immediately upon the occurrence of the stimulus? Suppose the glass shatters instantaneously upon being struck. Then the activation must be simultaneous with both the striking and the breaking. If the activation is a third event happening at the same place and time, again it is hard to see any independent empirical evidence for its occurrence. Either an activation is a mysterious and ad hoc third event (in addition to the stimulus and the manifestation) or it is indistinguishable from the initial stage of the process leading to the manifestation, or it is

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indistinguishable from the manifestation itself. If the activation is construed as either the manifestation, or as the initial stage of the process leading to the manifestation, then the account becomes circular and trivial. It would essentially say: If x’s disposition to M when C is such that x immediately Ms or some process leading to x M-ing commences, then either x immediately Ms or there is some process such that, if the process were to continue without interruption, x would M.

None of these options seem very attractive, so I assume Kroll will want to say something else about the nature of activation, or the truth-makers of activation claims. In doing so, perhaps he could address another concern about his stimulus/activation distinction. Clearly, on Kroll’s view, the stimulus is not sufficient for the activation of the disposition. Nor is the stimulus sufficient for the manifestation, nor for any process that would lead to the manifestation. However, the activation is sufficient for the manifestation, or it is sufficient for the commencement of a process that would lead to the manifestation. Consequently, it seems like activation is where the action is at. So, then what is the role of the stimulus? Is it possible to activate a disposition without the stimulus occurring? If so, then the stimulating circumstance C seems irrelevant. Something that has “a disposition to M in C ” may or may not M in C, but it must M (or commence a processes leading to M-ing) when it is activated. Perhaps “the disposition to M in C ” should be called “the disposition to M when activated.” Cases in which fragile glasses stay intact when struck would be irrelevant, since striking is merely a stimulus, and the thing that matters for manifesting is activation. This makes the questions about the nature of activation more pressing, because it seems like the only rationale for positing the idea that an activation occurred (or did not occur) is the occurrence (or nonoccurrence) of a manifestation (or a process leading to a manifestation). Such a rationale for making claims about when activations occur makes the Activation Principle counterexample-free by fiat. An alternative to making the stimulus incidental to the manifestation process is to say that activation does not occur unless the stimulus occurs— that the stimulus is necessary for the activation. But what sort of necessity could this be? Perhaps the activation is grounded in the stimulus? But seeing as most accounts of grounding consider grounds to be sufficient for the grounded,4 this would have the consequence that a stimulus is sufficient for an activation, contrary to Kroll’s account. Then perhaps the activation

4 Metaphysical Grounding: Understanding the Structure of Reality, ed. F. Correia and B. Schnieder. Cambridge: Cambridge University Press, 2012.

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causally depends on the stimulus? In other words, perhaps the stimulus causes the activation. If this is right, it raises the question: (1)

Why does a stimulus sometimes cause an activation, and sometimes not?

Maybe sometimes a stimulus has all that it takes to cause an activation, and other times the stimulus lacks something. Note the similarities to the earlier discussion about partial and complete stimulus conditions. Also, note the similarity between question (1) and a question implicitly considered earlier with respect to conditional analyses: (2)

Why does a stimulus sometimes cause a manifestation, and sometimes not?

Kroll might answer question (2) by saying that, sometimes, stimulus causes an activation, and sometimes it doesn’t. But if there’s no answer to question (1), then this answer to question (2) is unsatisfying. And answering question (1) would seem to require getting specific, or appealing to ideal conditions, or normal conditions, or any of the other moves that defenders of conditional analyses have tried—moves which Kroll criticizes. Another way to put the point is as follows. According to a simple account of the manifestation process, when a disposition is stimulated, it manifests. We noted a problematic mismatch between stimulated dispositions and manifesting dispositions. Kroll’s alternative offers a perfect match between activated dispositions and manifesting dispositions.5 Yet the account entails an unexplained mismatch between stimulated dispositions and activated dispositions. So, the introduction of “activation” adds another element or step to the manifestation process, thereby relocating, but not solving, the problem with the simpler account. Kroll goes on to develop his Teleological Account of Dispositions (TAD) beyond the Activation Principle. Insofar as dispositions are directed at their manifestations, a teleological directedness seems like a plausible way to go. However, the details of the account warrant clarification. One of the key tenets of TAD states necessary and sufficient conditions for a property to be a disposition: “(T2)

Necessarily: a property P is a disposition iff there is a condition C and event-type M such that necessarily, P is the property of being in a state directed at the end that one Ms when C.”

5 I am simplifying slightly. Kroll’s perfect match is between activated dispositions and manifestations (or interruptible processes leading to manifestations).

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One implication of this analysis is that the existence of the stimulating condition is necessary for a property to be a disposition. It is implausible to think that this means that a particular occurrence of the stimulus is necessary for the disposition to exist. So, it must mean that the stimulus-type is necessary for the disposition to exist. I am not sure what the existence conditions for stimulus-types are. Perhaps, in order for a type of event to exist in a world, an instance of that type must occur in that world. That would have the implication that, say, in a world where immersion in water has never occurred, there is no such thing as water-solubility. Also, in the actual world, there would no dispositions to manifest in merely possible kinds of circumstances. This implication aside, my first question about (T2) is, what does Kroll mean by “a property of being in a state”? To answer that, one would have to say what a state is. Perhaps a state is a state of affairs, which some philosophers construe as a particular instantiating a property.6 Then “a property of being in a state” would be “a property of being a particular instantiating a property,” and the account does not say what this further property is. If a disposition is a property of being a particular with a certain property, this suggests that a disposition is a second-order property—a property that a thing has in virtue of having some other property. This further suggests that a disposition must have some sort of basis, or grounds. This claim has been disputed by a number of dispositions theorists.7 Whatever a state is, according to (T2), some states are directed at an end. So then a disposition is a property of being in a state, and this state is directed at an end. According to Kroll, it is the state that is directed at the end, not the disposition. So, despite being promised an account of a disposition’s directedness, what’s directed is not the disposition, but a state. Is the disposition indirectly directed at the end in virtue of being a property of a state that is directed at an end? Then dispositions are at best indirectly directed, so to speak. Perhaps there’s nothing problematic here, but featuring “states” in this analysis adds apparently unnecessary complications, since the motivation for doing so is unclear. My second line of questions about (T2) are about the end at which the state is directed—“that one Ms when C.” What does it mean to be directed at M-ing when C ? Does it mean that, when C happens, one is directed at

6 See, for example, Armstrong, D. M. A World of States of Affairs. Cambridge: Cambridge University Press, 1997. 7 See, for example, McKitrick, Jennifer. “The Bare Metaphysical Possibility of Bare Dispositions.” Philosophy and Phenomenological Research 96 (2003): 349–69; Mumford, Stephen. “The Ungrounded Argument.” Synthese 149 (2006): 471–89; Bird, Alexander. “The Regress of Pure Powers?” The Philosophical Quarterly 57 (2007): 513–34.

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M-ing ? Or, does it mean that one is directed at C happening so that one can M ? Or does it mean that one is directed at both C happening and M-ing? For example, if the end of “the disposition to break when struck” is that one breaks when struck, does that mean that one aims at getting struck and consequently breaking? If so, then fragile things are, in part, directed at getting struck. It is implausible to think that disposed objects are in a state such that they are directed towards triggering their dispositions, even in part. Perhaps, instead, one aims at breaking only when one is struck. This suggests that when one isn’t struck, one isn’t aiming at breaking. This would have the consequence that when dispositions aren’t in the stimulating circumstances, they are not directed at their manifestations. If so, then this analysis does not account for the directedness of dispositions when stimulating circumstances do not obtain. Furthermore, when Kroll writes that the state is directed at the end that “one Ms when C,” what is the referent of “one”? Anything? So, perhaps the state is directed at an existential fact that something Ms when C ? We get some clarification in (T2.1), which is said to follow from (T2): “(T2.1):

Necessarily: a property P is a disposition iff there is a condition C and event-type M such that: necessarily, x has P iff x is in a state directed at the end that x Ms when C.”

So, the object that has the disposition is the “one” that Ms, if the end is realized. Since (T2) does not specify what thing Ms in the end, it is not clear how (T2.1) is supposed to follow from (T2). At any rate, it is questionable whether we should accept (T2.1), for it entails that the locus of manifestation is always the disposed object. Consequently, it rules out the possibility that a thing can have a disposition for something else to M. But examples of such dispositions are common: being lethal, poisonous, soporific, attractive, or provocative, for example. Perhaps Kroll would want to say that, in such cases the manifestation is causing death, causing poisoning, causing sleep, etc. and these are things that the disposed object does when its disposition is activated. But is “causing death” an event-type? It sounds like a causal process, and various kinds of causal processes can be causings of death. Furthermore, “M-ing when C ” is supposed to be “the end.” If the end is “causing death,” the end at which the state is directed is itself a causal process, and not the end of that causal process. But if, instead, the manifestation is the end of that causal process—sleeping, dying, being angry, etc.—then the particular that is “M-ing” is not the particular that had the disposition in question, and (T2.1) should be rejected. Even if these questions about (T2.1) have satisfactory answers, there are further questions to consider about ends. Kroll approvingly quotes Makin: “a teleological process has a privileged stage to which it runs in normal

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conditions.” If the end is truly a privileged state, then not just any M-ing will do—only M-ing when C. So the characterization of the stimulating circumstances matters. And M-ing is said to be one event-type. Typing events has its own challenges: does shattering into thousands of shards, chipping, cracking, and splitting in two all count as instances of the event-type “breaking?” Regardless, a surprising consequence of this account is that it rules out multi-track dispositions. Multi-track dispositions manifest via different types of events in different types of circumstances.8 For example, courage can manifest by rushing into a burning building, or by standing up for an unpopular political position. One manifestation of electrical charge is attraction (in the stimulating circumstances of being in proximity to certain kinds of particles) and another manifestation of electrical charge is repulsion (in the stimulating circumstance of being in proximity to other kinds of particles). Kroll could say that attraction and repulsion are really the same type of event, but then the criteria for event-typing looks suspiciously ad hoc. Kroll could side with Alexander Bird and say that charge is merely a cluster of different dispositions with different manifestations.9 And Kroll could say that, while there are many different dispositions to do many different kinds of dangerous or frightening things, there is no such thing as courage. Such costs follow from defining dispositions as exclusively “single-track.” Furthermore, recall that (T2.1) says that when x has a dispositional property to M in C, “ . . . x is in a state directed at the end that x Ms when C.” So, consider a puddle of water. It has many dispositions: to freeze when cold, to evaporate when hot, to dissolve salt when salt is immersed in it, and many others. So, according to Kroll’s account, the puddle is in a state directed at the end that it freezes when cold, and it is in a state directed at the end that it evaporates when hot, and it is in a state directed at the end that it dissolves salt when salt is immersed in it. If each particular pair corresponds to a different disposition, and we differentiate event-types in a relatively finegrained way, this list is innumerably long. So, how many states is the puddle in? Is it just one state that is directed at all of these different ends? Then the puddle would be in one state simultaneously directed at innumerably many ends, most of which could not be jointly realized. If there

8 See Ryle, Gilbert. The Concept of Mind. New York: Barnes and Noble, 1949, for an introduction to the multi-track/single-track distinction. See Vetter, Barbara. Potentiality: From Dispositions to Modality. Oxford: Oxford University Press, 2014, pp. 36–46, for arguments that all dispositions are massively multi-track. 9 Bird, Alexander. Nature’s Metaphysics: Laws and Properties. Oxford: Oxford University Press, 2007, pp. 21–4. Vetter points out that this conflicts with our best scientific understanding of such properties (Potentiality: From Dispositions to Modality).

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is a different state for each stimulus-manifestation pair, the puddle would simultaneously be in innumerably many states, and simultaneously directed at innumerably many ends, most of which could not be jointly realized. And the simple puddle would be no anomaly, in terms of its massively-multi-directedness. Summing up, the main reasons why TAD does not adequately account for a disposition’s directedness are the following. First, TAD depends on the idea that dispositions lead to their manifestations when they are activated, but it is unclear what it means to say a disposition is activated as opposed to being stimulated. Second, TAD does not attribute directedness to the disposition itself. Third, TAD does not account for directedness when the stimulating circumstances do not obtain. Fourth, if TAD does give us directedness, it gives too much, for it seems to entail that everything is always directed in innumerably many different directions. Jennifer McKitrick University of Nebraska, Lincoln

3 Dispositions without Teleology David Manley and Ryan Wasserman In “Teleological Dispositions,” Nick Kroll offers a novel theory of dispositions in terms of primitive directed states. Kroll is clear that his notion of directedness “outstrips talk of goals, purposes, design, and function” (23), and that it commits him to “primitive teleological facts” (30). This notion may strike some as outdated and unscientific, but Kroll argues that it has an important theoretical role to play. In particular, he holds that a primitive notion of directedness can provide a theory of dispositions, an explanation of the link between dispositions and conditionals, and an account of the progressive aspect in English. In this paper, we raise some worries for each of these claims.

1. DIRECTEDNESS AND DISPOSITIONS Kroll’s first and most important claim is that a primitive notion of directedness can provide a plausible theory of dispositions. He summarizes this theory as follows: (T3)

Necessarily: the property of being disposed to M when C just is the property being in a state directed at the end that one Ms when C.

According to this view, something is disposed to dissolve in water, for example, just in case it is in a state that is directed at the end that it dissolves when placed in water. On Kroll’s view, directed states of this sort are not reducible to non-teleological facts.1 1 Note that one could accept (T3) without accepting this last claim. For example, one could reduce facts about dispositions to facts about directedness, and then reduce facts about directedness to facts about singular causation, or primitive laws, or counterfactuals. (See the following section on Directedness, Dispositions, and Conditionals for more on this last suggestion.) Kroll rejects these reductions on the grounds that “it’s possible for there to be a disposition that is a fundamental property” (30). But he offers no argument

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We have two main concerns for this account. First, (T3) is only a plausible account of dispositions if the proffered explanans is less mysterious than the explanandum. We are not sure this is the case. Suppose, for example, that Adam is disposed to order an extra shot of espresso in the morning whenever he’s been up late the night before (unless it’s the weekend or a holiday). Does this mean that Adam always walks around “in a state directed at his ordering an extra shot of espresso in the morning when he’s been up late, provided that it’s not a weekend or a holiday”? We have no idea how to understand this question—not, that is, unless we understand it as a roundabout way of asking whether Adam has the relevant disposition. To us, this suggests that directedness is more naturally understood in terms of dispositions, rather than the other way around. Our second worry for (T3) is that it is unclear whether it can account for three interrelated features of our talk about dispositions. First, our talk about dispositions can take the comparative form. We can say that a substance is volatile, for example, but we can also say that one substance is more volatile than another. Second, our talk about dispositions is gradable. We can say that an object is fragile, for example, but we can also say that it is highly fragile, or that it is somewhat fragile. Third, our talk about dispositions is sensitive to context. The standards for counting as “irascible” in a context, for example, depend on who else is relevant in that context, and how short their fuses are. These three features of dispositions are closely related. In fact, if we follow the standard approach to gradable adjectives, the second and third features can be analyzed in terms of the first. We begin, for example, with the relation of being more fragile than. This relation provides an ordering of objects along a scale. We can then say that an object is fragile just in case it meets a contextually-determined cutoff point on that scale, and that it is highly fragile (for example) just in case it easily meets that cutoff point. In this way, we can provide a unified account of the comparativity, gradability, and context-sensitivity of our disposition talk.2 However, this approach is in tension with many theories of dispositions. Consider, for example, the simple conditional analysis of dispositions:

for the claim that fundamental dispositions are possible, and we confess to lacking any direct intuitions about whether it’s true. As a result, we think this question should be left as spoils to the victor. 2 Elsewhere we have argued that things are a little more complicated, because there is more than one dimension to the context-dependence of gradable adjectives (see Manley and Wasserman 2008: 78–9).

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(SCA) Necessarily, N is disposed to M when C iff N would M if C.3 According to (SCA), an object is disposed to break when dropped, for example, just in case it would break if it were dropped. One problem for this idea is that that it does not allow for comparisons. Two objects might be such that they would both break if dropped, but it’s not as if one of these counterfactual facts is somehow “more of a fact” than the other. Since conditionals do not come in degrees, (SCA) does not establish a scale. And, since it does not establish a scale, it does not allow for the standard approach to gradability and context-sensitivity.4 Kroll’s account seems to face the same problem. As he elucidates his view, it appears that a state either has a certain telos or it does not—there is no such thing as having this telos to a greater degree than some other object does. However, if directedness does not come in degrees, it is unclear how (T3) could establish a scale. And without a scale, there is no way to apply the standard treatment of gradability and context-sensitivity. Kroll anticipates this objection, and offers a response in the appendix to his paper.5 There, he focuses on the case of fragility, which (he says) has something to do with striking and breaking. Since both of these things admit of degrees, Kroll says that we can establish an appropriate ordering for fragility. More specifically, we can establish a ranking of more specific dispositions—there is, for example, the disposition to break when struck very hard, the disposition to break when struck with at least moderate force, the disposition to break when struck ever so lightly, and so on (where each of these more specific dispositions can be analyzed in accordance with (T3)). He then says that one object is more fragile than another just in case the first

3 Conditional “analyses” are often formulated as biconditionals, sometimes prefixed by a necessity operator. However, biconditionals and modal claims only report patterns across modal space, whereas an analysis should arguably explain such patterns. For more on this point, see below. See also Wasserman (forthcoming a and forthcoming b) and Manley (2012). 4 Manley and Wasserman (2007, 2008). 5 We have some concerns about the way Kroll characterizes the standard account. He writes that “on what is perhaps the standard semantics for gradable adjectives, such adjectives do not denote properties of individuals. Rather, they denote measure functions. . . . A covert morpheme . . . takes a measure function and returns a context sensitive function from individuals to truth values” (34). This strikes us as misleading. It is true that some semanticists—like Christopher Kennedy (2007)—take gradable adjectives to express measure functions rather than properties (i.e. functions from individuals to truth values). But everyone including Kennedy agrees that the verb phrase as a whole—including any covert elements—expresses a property relative to a context. For example, in an utterance of “Amal is tall,” the function of the verb phrase, saturated by whatever contextual contribution, is to attribute a property to Amal. The same thing goes for dispositional predicates.

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object’s more specific disposition ranks higher than the second’s. For example, if one object is disposed to break when struck with at least moderate force and the other is not, then the first is more fragile than the second. With this kind of comparison in place, we can go on to give the standard analysis of gradability and context-sensitivity for “fragile.” Unfortunately, there are two interrelated problems with this approach.6 The first is that the dispositions on Kroll’s scales will themselves come in degrees. For example, two objects might both be disposed to break when lightly struck, and yet one of those objects might be more so disposed. After all, there are lots of different angles at which something might be lightly struck, a lot of different places where it might be struck, and lots of different environmental factors that might be in place when it is lightly struck.7 Intuitively, all of these things (and more) matter when it comes to making comparisons of fragility. Suppose, for example, that one object is disposed to break when lightly struck at any angle, and that a second is disposed to break when lightly struck at many (but not all) angles. All else being equal, this would mean that the first object is more fragile than the second. What this goes to show is that Kroll will have to get much more specific about the dispositions that appear on his scales—indeed, he will have to get maximally specific, in the sense that the “stimulus condition” for each disposition will have to specify not just the exact amount of force that is applied, but also the exact angle at which it is applied, the exact place it is applied, and so on. This leads directly to the second problem: Achilles’ heels. Let’s suppose that we specify a particular angle, place, etc. in which a force is to be applied. We then allow the specific amount of force to vary in order to establish a scale of dispositions—there is the disposition to break when struck with a force of 1.425 millinewtons at angle a, in place p, etc., the disposition to break when struck with a force of 2.386 millinewtons at angle a, in place p, etc. and so on. Now imagine an unusual block-shaped object. If you hit the block at almost any angle, in almost any place, with almost any force, it will not break. In fact, it will not even be scratched. However, if you hit it at exactly angle a, in exactly place p, with any force whatsoever, it will shatter. This is the block’s Achilles’ heel. Here is the problem: given this block’s unusual nature, its disposition to break when struck will rank at (or near) the top of the relevant scale—the scale in which we hold fixed place p, angle a, etc. and allow the amount of force to vary. So, if an object’s fragility is determined by where its disposition appears on this particular scale, our unusual block will turn out to be more fragile than a delicate crystal vase. But it clearly is not. 6 These problems are raised, in a more general form, in Manley and Wasserman (2008). 7 Including the presence or absence of various finks and masks.

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This kind of case shows that we cannot analyze fragility as meeting a specific threshold on a specific scale of specific dispositions. Rather, in order to understand fragility—in all of its forms—we must somehow take all of the different stimulus conditions from all of the more specific dispositions into account. Kroll does not offer any suggestions about how this is to be done, so his reply is, to that extent, incomplete.8

2 . D I R E C T E D N ES S , D I S P O S I T I O N S, AND CONDITIONALS One task for a theory of dispositions is to explain the evident link between dispositions and conditionals. In most cases, a device is disposed to circulate air when turned on just in case it would circulate air when turned on. So too, a person is disposed to smoke when nervous just in case he would smoke if nervous. There are, of course, exceptions to these rules,9 but an adequate theory of dispositions should explain why there is an exception-admitting rule at all—that is, it should explain the link between dispositions and conditionals. According to Kroll, his account of dispositions suggests the following “interesting and counterexample free connection between dispositions and conditionals:” (CDC)

Necessarily: if x is disposed to M when C, x’s disposition to M when C is activated, and x doesn’t immediately M, then there is some process such that: if the process were to continue without interruption, x would M.

Here, the notion of continuing without interruption is to be understood in terms of directedness (22): to say that x would M if the relevant process were to continue without interruption is to say that the process is directed at x’s M-ing. So, for example, if something is disposed to kill when ingested, and this disposition is activated (by, for example, someone’s ingesting it), then it will either immediately kill the person, or there will be a process in place that is directed at killing that person. More generally, an object with a disposition to M when C will M when C if the disposition is activated and there is no interruption. According to Kroll, this is the link between dispositions and conditionals. 8 We suggest an alternative approach to these issues in the final section on Directedness and Progressives. 9 For a discussion of these exceptions, see Martin (1981), Lewis (1997), Fara (2005), and Manley and Wasserman (2008).

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We have three general worries for this suggestion. First, we worry about Kroll’s notion of teleological processes. However, we will put off discussion of this point until the final section of this paper, Directedness and Progressives. Second, we worry about Kroll’s notion of “activation.” In the example just given, a poison’s disposition to kill when ingested is activated when the relevant stimulus condition obtains—i.e. when someone ingests the poison. However, Kroll tells us that these two things can come apart. For example, he tells us that a vase’s disposition to break when dropped is not activated when it is dropped while in protective packing. In this case, he says, the disposition is “stimulated” without being “activated.” But, given this distinction, we are unsure whether we have an adequate grasp of the latter concept. Here is one way of putting pressure on this idea. Let’s suppose that there are fundamental probabilistic dispositions. More specifically, let’s suppose that there are A-particles and B-fields, and that A-particles are probabilistically disposed to decay when they enter B-fields. Most of the time, when an A-particle enters a B-field, it decays instantaneously. However, sometimes— by chance—an A-particle enters a B-field without decaying. (This is not a case in which a decay-process begins and is interrupted—it is simply a case in which no decay-process begins.) Now, suppose that an A-particle has just entered a B-field and that—by chance—it does not decay. Was this particle’s disposition to decay activated? To the extent that we understand this question, we think that the answer is left open. On the one hand, it seems like indeterminism might enter in at the activation stage—i.e. it might be indeterminate whether the disposition will be activated when the stimulus condition obtains. But it also seems as if indeterminism could enter in at the manifestation stage—i.e. the disposition might always be activated when the stimulus condition obtains, but it might be indeterminate whether the activated disposition manifests. If this is indeed possible, we have a counterexample to (CDC)—the particle’s disposition to decay when entering a B-field is activated, the particle doesn’t immediately decay, and yet there is no process such that: if the process were to continue without interruption, the particle would decay. Here is a second, related example.10 Suppose that magical spells involve direct causation at a distance, so that there is no process connecting the casting of a spell to its effect. Suppose further that the Froschkönig spell is disposed, when cast, to turn the prince into a frog at midnight. Suppose, finally, that Merlin casts the Froschkönig spell at noon and the prince turns 10

This case is inspired by Schaffer (2000).

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into a frog at midnight. To the extent that we understand talk of “activation,” this strikes us as a clear case of a disposition being activated. After all, we are talking about the disposition to frogify the prince when cast, the spell has been cast, and the prince does turn into a frog. It may be possible for stimulus conditions and activation conditions to come apart, but we do not see how this is possible in this case. And, given this, we have yet another counterexample to (CDC)—the spell’s disposition is activated (at noon), the prince does not instantaneously turn into a frog, and yet there is no process such that: if the process were to continue without interruption, the prince would turn into a frog. Of course, Kroll might claim that these kinds of cases are all impossible, but this strikes us as a cost.11 Our third and final worry about (CDC) is that we think it understates the connection between dispositions and conditionals. This is because we think the following principle is correct: (PROP) Necessarily, N is disposed to M when C iff N would M in a suitable proportion of C-cases. We have provided a detailed explanation of this principle elsewhere.12 Here we will limit ourselves to one quick example involving two fragile objects. Holding fixed the actual laws of nature, consider all of the possible situations in which an object could be dropped. If one object is such that, for a greater proportion of those situations, it would break if it were in them, then it is more disposed to break when dropped than the other. And if it would break in a suitable proportion of dropping-cases, then it is disposed to break when dropped simpliciter (where what counts as a “suitable proportion” is determined by context). Finally, if the relevant object would break in an especially high proportion of dropping-cases, it will count as “highly” disposed to break when dropped. In this way, PROP explains the link between dispositions and conditionals, while also accounting for the comparability, gradability, and context-sensitivity of our disposition talk.13 11 One could rule out our kind of case by treating “x’s disposition to M when C is activated” as synonymous with “either x immediately Ms, or there is some process such that: if the process were to continue without interruption, x would M.” However, this would render (CDC) an uninformative truism about a term of art. 12 See Manley and Wasserman (2007, 2008, 2011). 13 Much more could be said about PROP, but we will stress just one point. PROP is nothing more than a modal claim—it says that there is a necessary connection between dispositions and conditionals, but it does not say why this connection holds. For this reason, we do not take PROP to be a theory of what dispositions are. Of course, one could strengthen the principle by replacing “iff” with an “iff and because,” or converting PROP into a claim about what it is for N to be disposed to M in C. (See Manley 2012.) An alternative approach would be to take certain dispositional facts as basic, and to use those

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Kroll, however, raises the following objection to PROP: My computer’s . . . CPU is disposed to overheat when running a large number of (tasking) processes. That’s why the computer has a heat sink and fans. When the CPU is running a large number of processes, the heat sink and fans mask the CPU’s disposition to overheat when running a large number of processes. Now creatures smart enough to design such a CPU are also smart enough to realize that its disposition to overheat when running a large number of processes needs to be masked/finked . . . And this isn’t an accident. It holds across the relevant region of modal space that cases where the CPU is engineered are, by and large, cases where the engineers realize that its disposition to overheat when running a large number of processes needs to be masked/finked in some way . . . So, on any reasonable understanding of “suitable proportions,” it’s not the case the CPU would overheat in a suitable proportion of cases where it is running a large number of tasking processes. (22)

The idea is that CPUs like this only lack masks in a small minority of nomological possibilities—in a sense of “minority” that incorporates objective probability.14 So there is not a “suitable proportion” of cases in which they overheat. There are two problems with this argument. The first involves the jump from “small minority” to “not a suitable proportion;” the second involves the gap between Kroll’s notion of a “case” and our notion of a “C-case”. Here is the first problem. As we have argued elsewhere, it is sometimes sufficient for a disposition to manifest in a pretty small proportion of stimulus cases.15 (Consider, for example, the disposition of a disease to spread upon contact.) So it is not enough to argue that cases with the relevant kind of CPU are, “by and large,” cases where the engineers mask the dispositions. What counts as a “suitable proportion” is a contextdependent matter, after all, and often involves a comparison with other salient objects. In the context of comparing various kinds of CPU, to say that one’s own is disposed to overheat may require only that it is more

facts to explain the truth of certain conditionals (rather than the other way around). Yet another option would be to take certain teleological facts as basic, and to use those facts to explain both the truth of certain conditionals and the presence of various dispositions. For this reason, we take PROP to be compatible—at least in principle—with a teleological approach to dispositions. 14 There are infinitely many cases of both kinds, but presumably the idea is that in imposing a measure on the space of cases, we should take objective probability into account. We are happy to grant all of this, even though we’re not exactly sure how to cash out the relevant notion of probability if the laws are deterministic (so that it can’t be understood directly in terms of objective chance), or how to compare the probabilities of various sets of initial conditions compatible with the laws. 15 Manley and Wasserman (2008: 75; 2011: 1203).

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disposed to overheat than some other salient CPUs. (And clearly this disposition comes in degrees: the new Intel Core M, for example, is less disposed to overheat than its predecessors.) There is no fixed minimum for what proportion of cases is required to count as “suitable.”16 But this does not get to the heart of the issue. Indeed, Kroll could avoid this objection by recasting his argument in terms of comparatives. Suppose CPU-1 is highly sophisticated—say, a quantum processor—while CPU-2 is extremely basic. And suppose that intuitively they are equally disposed to overheat—for every fully specific situation in which one would overheat, the other would as well. Still, let’s suppose that across nomological possibilities, processors like CPU-1 are more likely to have their disposition masked. (Engineers capable of creating a processor like CPU-1 will be especially motivated to protect it with masks.) So, probabilistically speaking, there are more possibilities in which CPU-1 would overheat than in which CPU-2 would overheat. But they are equally disposed to overheat—a violation of our account of the comparative. This leads us to our second response. In the description of the revised objection we wrote: (1) For every fully specific situation in which one would overheat, the other would as well. But we also wrote: (2) There are more possibilities in which CPU-1 would overheat than in which CPU-2 would overheat. These claims are consistent only because we mean different things by “situation” and “possibility.” To illustrate, suppose we ask what Donald would do in a very specific situation—say, meeting President Obama by chance in a café for example. We can also ask what Ted would do in that very same situation. Even a highly specific situation involves more than one (centered) possibility—Donald’s meeting Obama in just that way, Ted’s meeting Obama in just that way, etc. These possibilities are comprised of the same situation saturated by different individuals.

16 Kroll may have in mind a context in which one is explaining the presence of the heat sink and fan. This adds a layer of complication, because it is very natural to interpret an utterance like “There’s a fan because the CPU is disposed to overheat” as eliding the word “otherwise”: it is otherwise disposed to overheat. (In another context it’s perfectly fine to say that the CPU is not disposed to overheat—because of the heat sink and fan.) Arguably the function of “otherwise” in such contexts is to hold fixed the absence of the explicitly mentioned mask. (Compare: “Why did I buy insurance? Well, I’d go broke if there were a fire.” Here the lack of insurance is held fixed in evaluating the counterfactual.)

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The point is that our C-cases are situations, not possibilities.17 Think of each C-case as corresponding to an open sentence describing all the specific factors that are causally relevant to the disposition. For example, comparing the fragility of two objects, we evaluate each one with respect to the truth of a huge list of counterfactuals of the form “N would break if it were dropped from 1 meter onto a surface with a Shore hardness of 90A, through air with a density of 1.2 kg/m3 . . . etc.” And we ask in what proportion of these situations each object would break. It will make no difference to this comparison whether, for some of those situations, one object is more likely to be in that situation than the other. Thus a very fancy vase and a very plain vase might be such that they would break in all the same situations, even though the fancy vase is less likely to be in many of the situations that induce breakage. Even if this were true across nomologically possible worlds, it would make no difference to their relative fragility on our view—and this seems like the right result.18,19

3 . D I R E C T E D N E S S A N D PR O G R E S S I V E S At this point, we have addressed Kroll’s first two reasons for believing in primitive directedness: the claim that it provides a successful account of dispositions and the claim that it explains the link between dispositions and conditionals. We now turn to his third and final reason: the claim that primitive directedness can provide an account of the progressive aspect in English. A sentence in the progressive says that a certain type of event is in progress. For example, “Steve is driving to Boston” says that a drive to Boston by Steve is in progress. The challenge is to say what it is for a given type of event to be “in progress,” and to do this without presupposing that this 17 See Manley and Wasserman (2008: 74–5): “We can introduce the term ‘stimulus condition case’ or ‘C-case’ for every precise combination of values for heights, Shore measurements, densities of the medium, and so on . . .” 18 It might make a difference if, though they would otherwise break in all the same situations, one object would break in a more probable situation, while the other would break in a less probable situation. When imposing a measure on situations, we may take into account the objective probability of the situations, independently of the objects that saturate them. (For example, the situation of being struck by an iron hammer may be more probable than being struck by a francium hammer—even across nomologically possible worlds.) 19 The comparative version of this argument does seem to cause trouble for Barbara Vetter’s version of PROP, since she explicitly treats cases as “triples of a world, a time, and an object” (2012: }2; 2014: }2.3).

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event will culminate (since Steve could be driving to Boston, for example, even if he will never reach his destination). Kroll suggests the following: (EIP)

Necessarily, e is a φ event in progress at t iff e is, at t, directed at the end that it cause the resultant state of a φ event to obtain at some t0 > t.

So, for example, what makes it the case that there is a “drive to Boston by Steve” in progress is that that there is an event going on that is directed at bringing it about that Steve has driven to Boston. Crucially, such an event could be going on, even if this telos is never achieved—this, Kroll claims, is what makes teleological talk ideal for understanding the progressive aspect. We have three main worries for this claim. First, we worry that Kroll’s approach requires too much of events in progress. Of course we agree with Kroll that there is some sense in which Steve’s drive is directed toward his having driven to Boston. Similarly, if Sanjay is building a house, then there is some sense in which his activity is directed toward his having built a house. However, once we get beyond these kinds of examples, teleological talk seems far less natural. Suppose, for example, that Sela is laughing, or standing still, or shaking uncontrollably. All of these statements are in the progressive aspect, but none of them need have anything to do with teleology. There is no sense—or, at least, no intuitive sense—in which Sela’s laughter is “directed at the end that it cause Sela to have laughed.” Since (EIP) requires this to be the case, that principle strikes us as implausible. Our second worry is that Kroll’s approach seems to require too little of events in progress. Suppose Bill kills Bob with his gun. While he is firing the gun, Bill is killing Bob; but while he is planning the murder and buying the gun, he is not yet killing Bob. But why not? On Kroll’s view, it must be that no event has begun that has Bill’s killing Bob as its end. But that seems false, as far as we can understand it. It seems that Bill’s planning is directed at this end in exactly the same sense in which his firing the gun is. But Kroll will need to draw a sharp distinction between these events in order to get the data right. This strikes us as a difficult position to maintain. Our third and final worry concerns the relation between progressives, outcomes, and directedness. To begin, consider the following case, which Kroll discusses elsewhere: Suppose Mary needs to cross a minefield. Besides the first few feet, the minefield is densely filled. So, while there is some probability that she would cross it if she tried, the probability is minuscule. Also, the mines are spread out fairly evenly: besides the

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first few feet, for each step she could take forward, there is roughly the same, and very high, probability that she would step on a mine. Mary, however, is unaware that the field is a minefield. All she knows is that she needs to make it to the other side. So, she steps out into the minefield, showing absolutely no caution. She takes a few steps and then, not surprisingly, steps on a mine. Let t be some time after Mary first steps out into the minefield but before she steps on the mine that kills her. (6a) is true at t. But what about (6b)? (6a). Mary is walking in the minefield. (6b). Mary is crossing the minefield. It seems to me (and many others) that (6b) is false at t. Mary was trying to cross the minefield but wasn’t crossing it. (2015: 2934)

We agree with Kroll’s judgment in this case: (6b) is false in the case where Mary will soon step on a landmine. In order to generate this result, Kroll will have to deny that there is (at that time) an event that is directed at bringing it about that Mary crossed the minefield. So far, so good. But now what about the case in which Mary does—by a string of lucky steps—happen to cross the minefield? In that case, Kroll says that (6b) is—at some point—true (2015: 2935, fn 9). Indeed, even if Mary had said after the first few steps, “I’m crossing the field!”, we would have to admit in retrospect that she was speaking the truth. But, in this case, Kroll will have to say that there is an event that is directed at bringing it about that Mary crossed the minefield. In other words, whether or not there is a directed event of the relevant kind will depend upon the occurrence (or non-occurrence) of certain events in the future.20 This strikes us as a strange thing for the teleologist to say—one would think that whether or not a primitive teleological fact obtains at t would be independent of what goes on after that time. In fact, this final worry seems to bring out a tension between Kroll’s theory of dispositions and his account of the progressive. After all, the fact that directedness is outcome-independent is precisely what was supposed to make it suitable for analyzing dispositions in the first place: whether or not an object is disposed to break when dropped—or whether it is in a state directed at breaking when dropped—does not turn on whether it will actually be dropped or broken. But, in that case, directedness seems illsuited for analyzing the progressive since some progressive statements differ from disposition ascriptions in exactly this respect. David Manley University of Michigan, Ann Arbor Ryan Wasserman Western Washington University 20 Suppose Bill is firing his gun at Bob and intending to kill him. Is he killing Bob? If Bill happens to miss every time—however unluckily—the answer is “no.” If he hits Bob—however luckily—and causes Bob to die, the answer is “yes.”

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R E F E REN C E S Fara, Michael. 2005. “Dispositions and Habituals.” Noûs 39(1): 43–82. Kennedy, Christopher. 2007. “Vagueness and Grammar: The Semantics of Relative and Absolute Gradable Adjectives.” Linguistics and Philosophy 30(1): 1–45. Kroll, Nick. 2015. “Progressive Teleology.” Philosophical Studies 172(11): 2931–54. Kroll, Nick. 2016. “Teleological Dispositions” [in this volume, 3–37]. Lewis, David. 1997. “Finkish Dispositions.” The Philosophical Quarterly 47(187): 143–58. Manley, David. 2012. “Dispositionality: Beyond the Biconditionals.” Australasian Journal of Philosophy 90(2): 321–34. Manley, David and Wasserman, Ryan. 2007. “A Gradable Approach to Dispositions.” The Philosophical Quarterly 57(226): 68–75. Manley, David and Wasserman, Ryan. 2008. “On Linking Dispositions and Conditionals.” Mind 117(465): 59–84. Manley, David and Wasserman, Ryan. 2011. “Dispositions, Conditionals, and Counterexamples.” Mind 120(480): 1191–227. Martin, C. B. 1994. “Dispositions and Conditionals.” The Philosophical Quarterly 44 (174): 1–8. Schaffer, Jonathan. 2000. “Overlappings: Probability-Raising without Causation.” Australasian Journal of Philosophy 78(1): 40–6. Vetter, Barbara. 2012. “On Linking Dispositions and Which Conditionals?” Mind 120(480): 1173–89. Vetter, Barbara. 2014. “Dispositions without Conditionals.” Mind 123(489): 129–56. Wasserman, Ryan. Forthcoming a. “Theories of Persistence.” Philosophical Studies. Wasserman, Ryan. Forthcoming b. “Vagueness and the Laws of Metaphysics.” Philosophy and Phenomenological Research.

ANALYTICITY REVISITED

4 Devious Stipulations John Horden 1. INTRODUCTION Traditionally, analytic truths have been thought, in some good sense, to require nothing of the world. And it is a corollary of this thought that combining a theory with its analytic consequences never produces a theory that requires more of the world than the original. Rudolf Carnap (1950) employed this idea in an attempt to reconcile his empiricism with a liberal acceptance of abstract entities; claiming that the existence of numbers, properties, etc. can be trivially deduced by the rules governing our expressions for such things, once those expressions and rules are introduced into our language. In a highly influential response, Willard Van Orman Quine (1951) argued that the analytic/synthetic distinction Carnap relied on for that result is untenable. Thereafter many so-called analytic philosophers, reputedly a majority for a while, followed Quine in disavowing analyticity altogether. However, since then, analyticity has slowly crept back into philosophical respectability. Nowadays, it seems, analytic philosophers are mostly inclined to accept both that there are analytic truths (see Bourget and Chalmers, 2014), and that such interpreted sentences are characteristically metaphysically undemanding, at least in the sense of being neutral with respect to our position in logical space. Rather than rejecting analyticity outright, most would simply deny that there are any metaphysically interesting analytic truths. Meanwhile, going against the grain somewhat, there have in recent years been several notable attempts to answer ontological questions through conceptual analysis (see Hale and Wright, 2001; Schiffer, 2003; Thomasson, 2007, 2015; Hirsch, 2011; Hofweber and Velleman, 2011; Steinberg, 2013). And, by and large, all sides agree that if the existence of certain things analytically follows from sentences we already accept, then explicitly acknowledging the existence of those things cannot sensibly be regarded as an extra theoretical cost.

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In his ‘Analyticity and Ontology’, Louis deRosset challenges this consensus. He tries to refute two principles (2015: 131, 139): DAO

GAO

If P analytically entails the existence of certain things, then a theory that contains P but does not claim that those things exist is no more ontologically parsimonious than a theory that also claims that they exist. If P analytically entails Q, then (P ∧ Q) requires nothing more of the world than does P.

(DAO is what deRosset calls ‘the doctrine of analyticity in ontology’ and is his main target. GAO is a generalization thereof.) He glosses these principles as follows: ‘a sentence f is analytic iff it is entailed by true sentences ψ1, ψ2, . . . such that failure to accept any ψn constitutes some measure of linguistic incompetence’ (2015: 133); ‘a sentence f analytically entails a sentence ψ iff the material conditional (f ) ψ) is analytic’ (2015: 133); ‘the parsimony of a theory is given by what the truth of the theory requires of the world with respect to what there is’ (2015: 138). So the more ontologically parsimonious a theory is, the less it requires of the world with respect to what there is. DeRosset does not define what it is for a sentence or theory to require something of the world, but makes clear that ‘the world’ here rigidly designates the actual world: The argument [against GAO] here relies on the assumption that actually: f requires no less of the world—the actual world, that is—than does f. [ . . . W]hat’s required of the actual world for actually: grass is green to be true is just for the actual world to meet whatever requirements there are for grass is green to be true. (2015: 143; see also n. 20)

Perhaps we should accept something along the following lines: a sentence f requires of the world that R iff (i) ‘R ’ does not logically follow from sentences that are true entirely because of their meanings, and (ii) whichever world is actual, if f is true, this is partly because R. And then: a sentence f requires nothing more of the world than a sentence ψ iff, however ‘R ’ is replaced, if f requires of the world that R, then ψ requires of the world that R. Thus ‘Actually grass is green’ requires nothing more or less of the world than ‘Grass is green’. I won’t rely on the correctness of the above definitions exactly as stated; just as deRosset, so he tells us (2015: 135), does not rely on his ‘rough and ready’ definition of analyticity. However, I shall make two assumptions. First: (schematically) if a sentence requires of the world that R, then, whichever world is actual, that sentence is true only if R. Second: if a sentence is true

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entirely because of its meaning, then it requires nothing of the world. (Notice that I say ‘entirely because’ here, not ‘only because’. This is to allow for overdetermination. Plausibly, ‘Everything is self-identical’ is true partly because everything is self-identical (see Quine, 1954: 113; Boghossian, 1996: 364). Nonetheless, that sentence is fully determined to be true by its meaning; so in the relevant sense, it requires nothing of the world (see Russell, 2008: }1.2).) Now, to see the prima facie appeal of DAO and GAO, consider: (M1) (M2)

Someone is married. Someone is married to someone else.

(M1) explicitly affirms the existence of one person only, whereas (M2) explicitly affirms the existence of two. Nonetheless it seems clear that the conjunction of (M1) and (M2) requires nothing more of the world (and in particular, is no less ontologically parsimonious) than (M1) alone, because (M1) analytically entails (M2). And likewise, we might think, for more metaphysically interesting analytic entailments, if such there be. However, deRosset denies that DAO and GAO hold in full generality. He suggests that in the metaphysically interesting cases just alluded to, these principles are liable to break down. So even if, to take his leading example, ‘There are particles arranged tablewise in location L’ analytically entails ‘There is a table in L’, it remains plausible that the conjunction of these sentences is less ontologically parsimonious than the former sentence alone. (Here he targets Amie Thomasson in particular. For the relevant sense of ‘arranged tablewise’ and the like, see her 2007: 16–17.) Thus while unQuineanly granting the tenability of the analytic/synthetic distinction, deRosset disputes its metaphysical significance. To this end, he purports to produce counterexamples to DAO and GAO by means of linguistic stipulations. He offers two main candidate counterexamples just to GAO (though he admits that the first of these is not decisive) and one specifically to DAO. I aim to show where these arguments go wrong. 2. ‘VERDANTLY*’ Here is deRosset’s first candidate counterexample to GAO. (I bypass his initial, illustratively unsuccessful ‘verdantly’ stipulation.) Suppose we introduce a new sentential operator ‘verdantly*’ by stipulating the following rules of inference: P ├ Verdantly* P Verdantly* P ├ Actually grass is green Verdantly* P ├ Possibly P

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‘Actually’ here has its familiar interpretation from two-dimensional semantics: ⌜Actually f⌝ is true at a world iff f is true at the actual world. Hence what deRosset calls Stevenson’s constraint on linguistic stipulation (a generalized version of the constraint proposed by Stevenson, 1961) initially appears to be satisfied: [A] linguistic stipulation succeeds only if there is a consistent way to assign truth conditions to sentences containing the introduced term that makes the content of the stipulation true. (2015: 141)

To see this, suppose grass were not green. Then by the above introduction rule, ‘Verdantly* grass is not green’ would be true, and so by the first elimination rule, ‘Actually grass is green’ would be true. But even if grass were not green, grass would still be green at the actual world, so no inconsistency results. If his ‘verdantly*’ stipulation succeeds, deRosset tells us, we have a counterexample to GAO. For then ‘Snow is white’ analytically entails ‘Actually grass is green’. But ‘Snow is white and actually grass is green’ requires more of the world than ‘Snow is white’. For the latter only requires that snow be white, whereas the former also requires that grass be green. However, deRosset admits some uncertainty as to whether this stipulation succeeds (2015: 143–4). For perhaps when applying Stevenson’s constraint we are not entitled to take for granted a posteriori necessities such as the fact that grass is actually green. Perhaps we should also consider how to evaluate sentences containing ‘verdantly*’ on the assumption that grass is not actually green. But then consider what truth value should be assigned to Verdantly* grass is either green or not green. If it is true, the first elimination rule for ‘verdantly*’ is invalid, because ‘Actually grass is green’ is by hypothesis false. If, on the other hand, the evaluated sentence is false, the introduction rule is invalid, because ‘Grass is either green or not green’ is (we may assume) logically true. So the ‘verdantly*’ stipulation fails to meet Stevenson’s constraint on this more stringent construal. DeRosset ultimately leaves open how stringently Stevenson’s constraint should be construed, and so admits that his ‘verdantly*’ stipulation does not provide a decisive counterexample to GAO. Fair enough; though whichever way we construe Stevenson’s constraint, I wonder how useful it will be for adjudicating on attempted stipulations in cases where the bounds of conceptual possibility are in dispute. Should we consider how to evaluate sentences containing ‘number’ or ‘proposition’ on the assumption that there are only finitely many things, for example? If so, then the broadly Fregean and Carnapian views of abstracta targeted by

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deRosset are in trouble; but any such objection to those views—which, to be fair, deRosset does not himself raise—seems question-begging. In any case, there is another reason to think that the ‘verdantly*’ stipulation fails. For ‘verdantly*’, as defined, is by reasonable standards clearly and fatally nonconservative. Nuel Belnap (1962) recommended a conservativeness constraint on stipulation in response to Arthur Prior’s (1960) parody of implicit definition, wherein ‘tonk’ is defined by the rules: P├ P tonk Q P tonk Q ├ Q Thus any sentence analytically entails any other; hence every sentence is analytic. Obviously the stipulation fails; the only question is how. J. T. Stevenson (1961) proposed one constraint in response: every truthfunctional connective must have a consistent truth table (notice that this is significantly weaker than the constraint deRosset names after Stevenson, on either construal of the latter). Belnap proposed another constraint: the introduction of any new vocabulary must yield a conservative extension of the language. That is to say, the rules stipulated to govern new vocabulary cannot allow the derivation of any sentence of the old language (i.e. any sentence without the new vocabulary) that was not already derivable.1 In footnotes, deRosset describes this conservativeness constraint, and observes that it is met by his subsequent stipulations—presumably regarding this as a point in their favour—but conspicuously does not claim the same for his ‘verdantly*’ stipulation. And it is easy to see why: ‘Actually grass is green’ is a synthetic sentence of English before this stipulation, but would become derivable from any sentence whatsoever (and hence analytic) were the stipulation to succeed. Thus Belnap’s constraint is manifestly violated. Admittedly, many regard Belnap’s constraint as too strong (see Read, 1988: }9.3; 2000: 125–7; Peacocke, 1993: }}3–4; 2004: 18–21; Prawitz, 1994: 374; Shapiro, 1998: }3). For example, it seems legitimate to add second-order quantifiers or a truth predicate to a first-order system of arithmetic, though doing so enables the derivation of the original system’s Gödel sentence. Also, abstraction principles such as Hume’s Principle The number of Fs = the number of Gs iff there are just as many Fs as Gs if successfully stipulated, allow us to derive results about the (infinite) size of the domain of first-order quantification that were previously statable but perhaps unprovable. Hence Bob Hale and Crispin Wright endorse a weaker 1 This principle of conservativeness, or ‘noncreativity’, was previously endorsed by others; see e.g. Frege, 1914: 208. For a historical discussion, see Urbaniak and Hämäri, 2012.

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version of conservativeness (cf. Field, 1980: ch. 1; Wright, 1997: }9; 1999: }2.5; Schiffer, 2003: }2.2), also mentioned by deRosset (2015: 141, n. 19), with the following proviso attached: It is our view that a stipulation may have consequences which can be expressed in the antecedent language, and to which there need have been no previous commitment, without compromise of its legitimacy provided the truth of these consequences makes no demands on the previously recognized ontology. (Hale and Wright, 2000: 302, n. 32)

However, the truth of ‘Actually grass is green’ clearly makes demands on the previously recognized ontology. For it demands that grass be green. So the ‘verdantly*’ stipulation fails to meet even this weaker version of conservativeness. So if either Belnap’s or Hale and Wright’s version of conservativeness is correct, this stipulation cannot provide a genuine counterexample to GAO. Moreover, regardless of the specific constraints on stipulation, deRosset’s ‘verdantly*’ example faces a ruinous dilemma. First horn: ‘Actually grass is green’ retains its prior meaning. Then the nonanalyticity of this sentence, given its empirical status, is far more certain than the success of the stipulation. Given our understanding of analyticity, regrettably imprecise as that may be, it is absurd to think that we could make this sentence analytic without changing its meaning. So we should conclude, in Moorean fashion, that the stipulation somehow fails. If there were no known constraint that prohibited this stipulation, we would have to posit one. That, mutatis mutandis, is the lesson of ‘tonk’. Second horn: ‘Actually grass is green’ loses its prior meaning. Then we don’t know what, if anything, this sentence means after the stipulation: its subsequent meaning seems to be radically underdetermined. So we have no reason to accept that its conjunction with ‘Snow is white’ requires more of the world than ‘Snow is white’ alone. Either way, we do not get a genuine counterexample to GAO. 3 . ‘G R A S S G R E E N ’ Let us now examine deRosset’s second candidate counterexample to GAO: Suppose we stipulate that ‘grassgreen’ is to be a predicate that expresses the property being green if, as a matter of fact, grass is green, and not being green otherwise. (2015: 144)

This stipulation, deRosset tells us, meets Stevenson’s constraint even on its more stringent construal, as well as meeting other proposed constraints including conservativeness, generality, and harmony.2 2 The latter two constraints are proposed by Hale and Wright (2000)—in addition to their version of conservativeness—and are described by deRosset thus: ‘Generality: the

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Now, he continues, we have a more decisive counterexample to GAO. For if this stipulation succeeds, then ‘Grass is grassgreen’ is thereby guaranteed to be actually true, no matter what colour grass actually has.3 So ‘Snow is white’ analytically entails ‘Grass is grassgreen’. But ‘Snow is white and grass is grassgreen’ requires more of the world than ‘Snow is white’. For since grass is actually green, ‘grassgreen’, according to the stipulation, expresses the property of being green. So whereas ‘Snow is white’ only requires that snow be white, ‘Snow is white and grass is grassgreen’ also requires that grass be green. Clearly something fishy is going on here. If ‘Grass is grassgreen’ is guaranteed to be actually true, no matter what colour grass actually has, then it cannot require of the actual world that grass be green. Even if we grant that the stipulation succeeds, it seems sensible to examine the semantics of the novel term ‘grassgreen’ a little more closely before we embrace inconsistency. So what is its stipulated meaning? That, I think, depends on whether the stipulation is conditional or unconditional. This yields another dilemma. First horn: ‘is grassgreen’ is conditionally stipulated to mean is green if grass is actually green, and is not green otherwise. Then the stipulation does not by itself settle the meaning of this predicate. The inscription-type ‘Grass is grassgreen’ is thereby guaranteed to express some truth at the actual world in our extended language, but which truth it expresses depends on the actual colour of grass. In fact, since grass is actually green, the resulting interpreted sentence is exactly synonymous with ‘Grass is green’. So we have no reason to think that it is analytically entailed by ‘Snow is white’. Second horn: ‘is grassgreen’ is unconditionally stipulated to mean is green iff grass is actually green.4 Then ‘Grass is grassgreen’ does not require of the actual world that grass be green. Nor does it require anything else of the world. So its conjunction with ‘Snow is white’ requires nothing more of the world than ‘Snow is white’ alone. Either way, we do not get a genuine counterexample to GAO. DeRosset concedes that it is unclear what, exactly, we should take ‘grassgreen’ to mean, but insists that this does not matter for his purposes, so long as his stipulation succeeds (2015: 148). On the contrary, his argument here apparently rests on an equivocation between two intensionally equivalent stipulation should enable the interpretation of a wide enough range of relevant sentences’; ‘Harmony: the introduction and elimination rules should not allow us to infer more (or problematically less) than our warrant for the premises allows us to infer’ (2015: 141, n. 19). 3 At least, ‘Grass, if it exists, is grassgreen’ is guaranteed to be actually true; like deRosset, I set aside this nuance. Also I grant that analytic truths can be contingent. As deRosset explains in his paper, he regards that assumption as inessential to his arguments anyway. 4 Or, more longwindedly: is such that either (a) it is green and actually grass is green or (b) it is not green and it is not the case that actually grass is green.

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interpretations of the introduced predicate, either of which is consistent with the explicit content of the stipulation. (Likewise for deRosset’s third candidate counterexample, as we shall shortly see.) On one interpretation, ‘Grass is grassgreen’ requires something of the world, but is not guaranteed by its meaning to be actually true. On another interpretation, ‘Grass is grassgreen’ is guaranteed by its meaning to be actually true, but requires nothing of the world. And yet, on an intensional individuation of properties, ‘grassgreen’ expresses the same property on either interpretation: the property of being green. As it happens, deRosset anticipates the second interpretation, though he relegates this point to a brief footnote: It is plausible, perhaps, to think that the truth of [‘Grass is grassgreen’] requires that grass be green iff grass is actually green; this is the view most naturally suggested by the content of the ‘grassgreen’ stipulation. But this requirement clearly goes beyond the requirement for the truth of [‘Snow is white’], as evidenced by the fact that the requirement is not satisfied in circumstances in which grass is purple but snow is still white. (2015: 161–2, n. 47)

This response is clearly inadequate, however. For deRosset, recall, takes the expression ‘requires nothing more of the world’, as it appears in GAO, to be equivalent to ‘requires nothing more of the actual world’. And though ‘Grass is green iff grass is actually green’ plausibly requires of nonactual worlds that grass be green, it requires nothing of the actual world (see Evans, 1979: }4; Davies and Humberstone, 1980: }2). So if ‘Grass is grassgreen’ merely abbreviates that biconditional, then its conjunction with ‘Snow is white’ requires nothing more of the world than ‘Snow is white’ alone. It is irrelevant that at some other possible world the biconditional is false while ‘Snow is white’ is true. 4. ‘ P R I M A N B E I N G S ’ Now for deRosset’s third candidate counterexample to GAO, which he also intends to serve as a counterexample to DAO: Suppose we stipulate that ‘priman being’ is to be a predicate that expresses the property being a human being if, as a matter of fact, there are more than seven billion human beings, and being a prime number otherwise. (2015: 146)

The stipulation here closely parallels that of deRosset’s previous example. And as with his previous example, he assures us that this stipulation meets all the constraints mentioned so far. If this stipulation succeeds, he continues, then the material conditional ‘If there are more than 7 billion prime numbers, then there are more than

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7 billion priman beings’ is thereby guaranteed to be actually true, no matter how many humans there actually are, hence it is analytic. So ‘There are more than 7 billion prime numbers’ analytically entails ‘There are more than 7 billion priman beings’. But ‘There are more than 7 billion prime numbers and there are more than 7 billion priman beings’ requires more of the world (and in particular, is less ontologically parsimonious) than ‘There are more than 7 billion prime numbers’. For since there are actually more than 7 billion humans, ‘priman being’, according to the stipulation, expresses the property of being human. So whereas ‘There are more than 7 billion prime numbers’ does not require the existence of any humans, ‘There are more than 7 billion prime numbers and there are more than 7 billion priman beings’ requires the existence of more than 7 billion of them. The analogy with deRosset’s ‘grassgreen’ example should be clear; hence I shall respond analogously. If, given the existence of more than 7 billion prime numbers, ‘There are more than 7 billion priman beings’ is guaranteed to be actually true, no matter how many humans there actually are, then it cannot require of the actual world that there be more than 7 billion humans. Even if we grant that the stipulation succeeds, we should examine the semantics of the novel term ‘priman being’ a little more closely before we embrace inconsistency. So what is its stipulated meaning? Again, that depends on whether the stipulation is conditional or unconditional. This yields a dilemma. First horn: ‘is a priman being’ is conditionally stipulated to mean is human if there are actually more than 7 billion humans, and is a prime number otherwise. Then the stipulation does not by itself settle the meaning of this predicate. Provided that there are more than 7 billion prime numbers, the inscription-type ‘There are more than 7 billion priman beings’ is guaranteed by the stipulation to express some truth at the actual world in our extended language, but which truth it expresses depends on how many humans there actually are. In fact, since there are actually more than 7 billion humans, the resulting interpreted sentence is exactly synonymous with ‘There are more than 7 billion humans’. So we have no reason to think that it is analytically entailed by ‘There are more than 7 billion prime numbers’. Second horn: ‘is a priman being’ is unconditionally stipulated to mean is human if there are actually more than 7 billion humans, and a prime number otherwise.5 Then ‘There are more than 7 billion priman beings’ requires nothing more of the actual world than whatever is required for there to be more than 7 billion prime numbers. So its conjunction with 5 Or, more longwindedly: is such that either (a) it is human and actually there are more than 7 billion humans or (b) it is a prime number and it is not the case that actually there are more than 7 billion humans.

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‘There are more than 7 billion prime numbers’ requires nothing more of the world, and is no less ontologically parsimonious, than ‘There are more than 7 billion prime numbers’ alone. Either way, we do not get a genuine counterexample to either GAO or DAO.

5 . C O N C LU S I O N So, as we have seen, none of deRosset’s stipulations yields a genuine counterexample to either of the principles he targets. Despite his efforts to discredit them, the doctrine of analyticity in ontology and its generalization remain as plausible as ever.6 John Horden National Autonomous University of Mexico

REFERENCES Belnap, N. D., 1962, ‘Tonk, Plonk and Plink’, Analysis, 22(6): 130–4. Boghossian, P. A., 1996, ‘Analyticity Reconsidered’, Noûs, 30(3): 360–91. Bourget, D. and D. J. Chalmers, 2014, ‘What Do Philosophers Believe?’, Philosophical Studies, 170(3): 465–500. Carnap, R., 1950, ‘Empiricism, Semantics, and Ontology’, Revue Internationale de Philosophie, 4: 20–40; reprinted in his 1956, Meaning and Necessity: A Study in Semantics and Modal Logic, 2nd edn, Chicago, IL: University of Chicago Press. Davies, M. and L. Humberstone, 1980, ‘Two Notions of Necessity’, Philosophical Studies, 38(1): 1–30. deRosset, L., 2015, ‘Analyticity and Ontology’, in K. Bennett and D. W. Zimmerman, eds, Oxford Studies in Metaphysics, vol. 9, Oxford: Oxford University Press. Evans, G., 1979, ‘Reference and Contingency’, The Monist, 62(2): 161–89. Field, H. H., 1980, Science Without Numbers: A Defence of Nominalism, Princeton, NJ: Princeton University Press. Frege, G., 1914, ‘Logic in Mathematics’, trans. P. Long and R. White, in H. Hermes, F. Kambartel, and F. Kaulbach, eds, 1979, Gottlob Frege: Posthumous Writings, Oxford: Blackwell. Hale, B. and C. Wright, 2000, ‘Implicit Definition and the A Priori’, in P. Boghossian and C. Peacocke, eds, New Essays on the A Priori, Oxford: Oxford University Press; reprinted in Hale and Wright, 2001. 6

Thanks to Philipp Blum, Aurélien Darbellay, Miguel Hoeltje, Dan López de Sa, Giovanni Merlo, Sven Rosenkranz and Louis deRosset for comments and discussion. My research was supported by projects CSD2009-0056, FFI2012-35026, and FFI201566372-P, Gobierno de Espan~a, and the LOGOS group, grant 2014SGR-81, Generalitat de Catalunya.

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Hale, B. and C. Wright, 2001, The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics, Oxford: Oxford University Press. Hirsch, E., 2011, Quantifier Variance and Realism: Essays in Metaontology, Oxford: Oxford University Press. Hofweber, T. and J. D. Velleman, 2011, ‘How to Endure’, Philosophical Quarterly, 61(242): 37–57. Peacocke, C., 1993, ‘Proof and Truth’, in J. Haldane and C. Wright, eds, Reality, Representation, and Projection, Oxford: Oxford University Press. Peacocke, C., 2004, The Realm of Reason, Oxford: Oxford University Press. Prawitz, D., 1994, review of M. Dummett, The Logical Basis of Metaphysics, Mind, 103(411): 373–6. Prior, A. N., 1960, ‘The Runabout Inference-Ticket’, Analysis, 21(2): 38–9. Quine, W. V., 1951, ‘Two Dogmas of Empiricism’, Philosophical Review, 60(1): 20–43. Quine, W. V., 1954, ‘Carnap and Logical Truth’, in his 1976, The Ways of Paradox and Other Essays, 2nd edn, Cambridge, MA: Harvard University Press. Read, S., 1988, Relevant Logic: A Philosophical Examination of Inference, Oxford: Blackwell. Read, S., 2000, ‘Harmony and Autonomy in Classical Logic’, Journal of Philosophical Logic, 29(2): 123–54. Russell, G., 2008, Truth in Virtue of Meaning: A Defence of the Analytic/Synthetic Distinction, Oxford: Oxford University Press. Schiffer, S., 2003, The Things We Mean, Oxford: Oxford University Press. Shapiro, S., 1998, ‘Induction and Indefinite Extensibility: The Gödel Sentence is True, but Did Someone Change the Subject?’, Mind, 107(427): 597–624. Steinberg, A., 2013, ‘Pleonastic Possible Worlds’, Philosophical Studies, 164(3): 767–89. Stevenson, J. T., 1961, ‘Roundabout the Runabout Inference-Ticket’, Analysis, 21(6): 124–8. Thomasson, A. L., 2007, Ordinary Objects, Oxford: Oxford University Press. Thomasson, A. L., 2015, Ontology Made Easy, Oxford: Oxford University Press. Urbaniak, R. and K. S. Hämäri, 2012, ‘Busting a Myth about Leśniewski and Definitions’, History and Philosophy of Logic, 33(2): 159–89. Wright, C., 1997, ‘On the Philosophical Significance of Frege’s Theorem’, in R. Heck, ed., Language, Thought, and Logic: Essays in Honour of Michael Dummett, Oxford: Oxford University Press; reprinted in Hale and Wright, 2001. Wright, C., 1999, ‘Is Hume’s Principle Analytic?’, Notre Dame Journal of Formal Logic, 40(1): 6–30; reprinted in Hale and Wright, 2001.

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5 Stipulations and Requirements: Reply to Horden Louis deRosset

In (deRosset, 2015) (henceforth, AO), I argued that there are counterexamples to each of the following two principles: (DAO)

(GAO)

If P analytically entails the existence of certain things, then a theory that contains P but does not claim that those things exist is no more ontologically parsimonious than a theory that also claims that they exist. If P analytically entails Q, then ðP∧QÞ requires nothing more of the world than does P.

John Horden has offered interesting criticisms of that argument. In this short note, I reply to the criticisms and briefly indicate what I take their lesson to be. The putative counterexamples to GAO and DAO are of essentially the same sort. For the sake of simplicity, then, I will focus on a putative counterexample to GAO. In AO I asked readers to suppose that we had stipulated that ‘grassgreen’ is to be a predicate that expresses the property being green if, as a matter of fact, grass is green, and not being green otherwise. To simplify the discussion, let’s dispense with suppositions: I do hereby so stipulate. I trust you know enough to recognize that grass is grassgreen. Also, as you may know, the flesh of kiwi fruit, clover, and most freeway signs in the U.S. are grassgreen. Neither roses nor violets are grassgreen. My argument that this stipulation yields a counterexample to GAO relied on three premises. A B

My stipulation is successful, where “success” in this sense entails that, as a result of the stipulation, its content is true. “Disquoting” the content of a successful stipulation yields an analytic truth.

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Requirements for the truth of a sentence of the form ‘τ is grassgreen’ are given by which individual (if any) is the referent of τ and which property is expressed by “grassgreen” in a straightforward way: the requirement is that that very individual have that very property.

Given (A), we may conclude that, if grass is, as a matter of fact, green, then “grassgreen” expresses being green, and, otherwise, “grassgreen” expresses not being green. Application of (B) then yields the analyticity of (1)

if, as a matter of fact, grass is green, then something is grassgreen iff it is green; and, otherwise, something is grassgreen iff it is not green.

It is then very simple to show that (2)

snow is white

analytically entails (3)

grass is grassgreen.

Application of (C), together with the fact that grass is, as a matter of fact, green, yields the conclusion that (3) requires for its truth that grass have the property being green. Since that requirement clearly goes beyond requirements for the truth of (2), we have a counterexample to GAO. In his comment on AO, Horden offers two reasons for thinking this argument unsound. First, he contends that its conclusion is inconsistent. Second, he contends that my argument faces a dilemma, depending on which of two hypotheses about the meaning of (3) turns out to be true. I will discuss each contention in turn. Horden’s charge of inconsistency is made on p. 69: “If ‘grass is grassgreen’ is guaranteed to be actually true, no matter what colour grass actually has, then it cannot require of the actual world that grass be green. Even if we grant that the stipulation succeeds, it seems sensible to examine the semantics of the novel term ‘grassgreen’ a little more closely before we embrace inconsistency” (emphasis original). It is a little difficult to see what is supposed to be inconsistent here. The guarantee is epistemic, while requirements for the truth of a claim concern the metaphysical or semantic question of what the world has to be like for the claim to be true. Given (C), any residual air of inconsistency dissolves once we realize that the success of the stipulation entails that the truth of (3) requires one thing if, as a matter of fact, grass is not green and another thing if, as a matter of fact, grass is green. Horden does not explicitly say why an epistemic guarantee of the truth of a sentence is supposed to be inconsistent with there being substantial

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requirements for its truth. But he provides a clue when he suggests a schematic biconditional governing requirements: “a sentence f requires of the world that R iff (i) ‘R’ does not logically follow from sentences that are true entirely because of their meanings, and (ii) whichever world is actual, if f is true, this is partly because R” (p. 64).1 Horden only tentatively advances this biconditional, but says that his argument assumes the following consequence of its left-to-right direction (p. 64): L1

if a sentence requires of the world that R, then, whichever world is actual, that sentence is true only if R.

With (L1) in hand, it’s pretty clear why Horden thinks the argument against GAO entangles us in inconsistency. Consider the relevant instance of (L1): L2

If (3) requires of the world that grass is green, then, whichever world is actual, (3) is true only if grass is green.

If the argument from (A)–(C) is sound, then, because grass is in fact green, (3) in fact requires of the world that grass is green. However, if grass turns out as a matter of fact to be purple—if, that is, a world in which grass is purple turns out to be actual—then the soundness of the argument implies that (3) does not require that grass be green. This would contravene (L2), which entails that (3)’s requiring of the world that grass be green be insensitive to how things turn out to be as a matter of fact. So, the contention that the argument involves us in an inconsistency can be made good if we assume (L1). But (L1) faces counterexamples. I have just flipped a coin. I hereby stipulate that “TrueThat” is a syntactically atomic sentence that requires for its truth that grass be green if the coin landed heads, and requires for its truth that grass not be green otherwise. The stipulation passes all of the standard tests for successful stipulation. In particular, it is conservative and harmonious, and passes what in AO (pp. 141–4) I called Stevenson’s constraint. I have gone on to use “TrueThat” in conversation with my colleagues, friends, and family. As you will see, I will even be using it in this paper. Given its success, the content of my “TrueThat” stipulation is true. So, (L1) has a false instance: L3

If “TrueThat” requires of the world that grass is green, then, whichever world is actual, “TrueThat” is true only if grass is green.

It turns out that the coin landed heads. This provides enough information for you to recognize that TrueThat iff grass is green. If, however, a world in 1

I have benefited here from personal correspondence with Horden.

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which the coin landed tails is actual, then TrueThat iff grass is not green. In general, we appear to be able, under the right circumstances, to successfully stipulate the requirements for the truth of new sentences. What’s more, we have at our disposal devices for conditionalizing on how things are as a matter of fact in the content of such stipulations. So, (L1) appears to implausibly restrict our stipulative powers. It thus provides no reason to think that the argument from (A)–(C) against GAO entangles us in any inconsistency. Horden’s second objection to the argument from (A)–(C) is that it faces a dilemma, depending on what we take (3) to mean. In AO (p. 148), I argued that the force of the putative counterexamples was independent of which of several plausible hypotheses about the meaning of, e.g. (3) turned out to be correct, and otherwise avoided explicit discussion about the meanings of my stipulated terms. A word of explanation for why we might want to avoid claims about what (3) means may, however, be appropriate. The issue is that there are a dizzying array of different semantic values that might be assigned to (3). Assume for illustration that the occurence of “grass” in (3) is a singular term, so that (3) has the form “a is F ,” where a is a term and F is a predicate.2 Then it is plausible to claim that the sentence expresses a Russellian proposition represented by hgrass, being greeni. Certainly, a proponent of DAO is in no position to deny that sentences like (3) express such Russellian propositions, given how “Russellian proposition” and related vocabulary are used in the philosophical community. But such a proponent is also in no position to deny that (3) has other semantic values, including a truth value, a carnapian intension, a character (Kaplan, 1989), or perhaps even a primary intension (Chalmers, 2002). Which, if any, of these semantic values should we hold gives the meaning of (3)? It seemed to me that nothing interesting hung on which of many plausible answers we might give to this question. Horden disagrees. Here is his dilemma. If (3) means (4)

grass is green

then, Horden contends, (B) is false: though the stipulation succeeds and the truth of (3) requires that grass have the property being green, (3) is not analytic. Alternatively, if (3) means (5)

grass is green iff grass is actually green

then, Horden contends, (C) is false: the requirements for the truth of (3) are trivially satisfied and so cannot include that grass have the property being

2 If (3) lacks such a structure, then it will express some slightly more complicated Russellian singular proposition, but the point in the main text would be unaffected.

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green. More generally, Horden’s argument here might be taken to suggest that there is no way of fixing a meaning for any sentence on which both (B) and the analogue of (C) are true. That is, there is no way of fixing a meaning for a sentence so that, like (4), it requires for its truth that grass be green, but, like (5), it is analytic. Let’s consider each horn of the dilemma in turn. On the first horn, the defense of GAO would require that (3) be synthetic, even though it has what in AO (p. 158) I called the trappings of analyticity: the conventions governing the use of (3) guarantee that it is entailed by certain sentences such that failure to accept any of these sentences constitutes some measure of linguistic incompetence. For instance, one of the conventions governing the use of (3) is given by the content of my “grassgreen” stipulation. From those conventions, instances of certain disquotation principles, and a little logic, it’s easy to work out that (3) is true. This certainly makes it look as if (3) is analytic, and it seems to me that there is little reason to deny that it really is analytic, short of antecedent commitment to GAO. Thus, I would argue that the most plausible verdict on this horn of the dilemma is that my “grassgreen” stipulation has fixed the meaning of (3) in such a way as to make both (B) and (C) true. Suppose, however, that I’m completely wrong about that. Suppose, that is, that appearances in this case really are misleading in the way that Horden contends: (3) has the trappings of analyticity but is not analytic. Horden’s contention here is an instance of a more general strategy for defending DAO and GAO from the putative counterexamples. This defense denies that the possession of the trappings of analyticity is enough to guarantee analyticity. But the defense thereby gives up an important methodological advantage on which the philosophical applications of DAO rely. If a sentence can have the trappings of analyticity without being analytic, then we can’t rely on our appreciation of the trappings of analyticity in cases like (6)

If particles p1 ; . . . ;pn are arranged table-wise in L, then there is a table in L

to warrant the application of DAO. Thus, a radical ontologist who agrees that particles p1 ; . . . ;pn are arranged table-wise in L but denies that there is a table there can admit that (6) has the trappings of analyticity while reasonably denying that it is analytic. In AO (p. 159), I concluded on this basis that, on this response, “DAO is saved, but it’s rendered toothless.”3

3

Please see AO }4.2 for more extended discussion.

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A defender of DAO and GAO might reply by distinguishing mere uninterpreted strings from meaningful expressions of an interpreted language.4 The idea is that (3) fails to be analytic because “grassgreen” is a mere uninterpreted string: the content of my stipulation gives us reason to think that that string will be true when it acquires a meaning, but no reason to think that any meaningful interpreted sentence is true. It is only meaningful, interpreted sentences that are analytic or synthetic, so (3) is not even the right sort of thing to be analytic. This reply runs afoul of the fact that “grassgreen” is already a term being used meaningfully (if only by me) to characterize things. In the second paragraph of this paper, for instance, I used it three times. On each occasion, I asserted something true. Perhaps you count yourself as falling short of ideal competence with the expression, but you know enough to recognize the truth of (3). So, (3) is already a meaningful sentence of an interpreted language, and the content of my stipulation provides an epistemic guarantee of its truth. It won’t, then, fail to be analytic for want of semantic significance. Consider now the second horn of Horden’s dilemma, on which we assume that (3) means the same thing as (5)

grass is green iff grass is actually green.

The defender of GAO holds that (5) requires nothing of the world. If this claim is granted, then the semantic equivalence of (3) and (5) yields the falsity of (C): like (5), (3) requires nothing of the world. Horden is correct to think that (C) entails that (3) and (5) are semantically discernible. In particular, the biconditional (5) won’t bear an interpretation on which its truth requires that grass have the property being green. So, if (3) and (5) were semantically indiscernible, then (C) would fail because (3), like (5), would also fail to require that grass have the property being green. I would think, however, that the claim that (3) and (5) are semantically indiscernible is independently implausible. (3) has semantic values that (5) lacks, including the Russellian proposition represented by hgrass, being greeni. So, Horden is correct to contend that (C) is, plausibly, false if (3) means the same thing as (5); but it is not plausible to hold that (3) means the same thing as (5). In any case, we can tweak the argument a little to avoid this horn of the dilemma entirely, using a stipulation much like my “TrueThat” stipulation. I hereby stipulate that “GreenThat” is a syntactically atomic sentence that requires for its truth that grass be green if, as a matter of fact, grass is green; and that grass not be green otherwise. I have gone on to use “GreenThat” in conversation with my colleagues, friends, and family. You know enough about grass to recognize that GreenThat iff grass is green, so you could use it 4

Here again I benefit from correspondence with Horden.

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too if you were so inclined. My stipulation passes all of the standard tests for successful stipulation. In particular, it is conservative and harmonious, and passes Stevenson’s constraint. The stipulation takes the analogue of the second horn of Horden’s dilemma off the table. In fact, it takes the denial of the analogue of (C) off the table if the success of my stipulation is granted. If the “GreenThat” stipulation is successful, then its content is true, so it requires for its truth that grass be green. For this reason, insofar as one is willing to countenance the idea that linguistic conventions or stipulations fix meanings for sentences, my “GreenThat” stipulation illustrates a way to fix a meaning for a sentence on which both (B) and the analogue of (C) appear to be true. I have argued that neither the objection based on (L1) nor the dilemma ultimately threatens the cogency of the putative counterexamples to DAO and GAO. But we shouldn’t let these difficulties for the objections obscure the more general lessons of Horden’s discussion. Horden is admirably explicit about the assumption (L1) governing requirements for the truth of a sentence on which his first objection depends. Though, as I have argued, (L1) itself is implausibly restrictive, I believe that this is a fruitful line of inquiry for the analyticity theorist to pursue. (L1) is part of Horden’s proposal for a partial theory of requirements. I think (L1) itself is false, but there are other aspects of that theory that are quite attractive. For instance, on Horden’s tentative proposal, the requirements for the truth of a sentence are canonically given by a specification of the form (7)

Sentence S requires that f.

Further, the requirements for the truth of a given sentence are conceived by Horden as playing an explanatory role: they specify the conditions in virtue of which S is true (if it is). These seem like fruitful starting points for a theory of requirements. To my mind, the most promising line of defense for GAO and DAO will follow Horden’s lead: first, articulate, motivate, and defend a (perhaps partial) theory of requirements; then, show that, on that theory, the proposed counterexamples fail. Moreover, the discussion of Horden’s dilemma has outlined the strategy that the defense must follow. The theory of requirements on which the defense relies must be supported well enough to motivate the rejection of the otherwise plausible claim that such stipulations as the ones introducing “grassgreen” and “GreenThat” are successful.5 Louis deRosset University of Vermont 5 Thanks are due to Mark Moyer and Kate Nolfi for comments on earlier drafts of this paper. Special thanks are due to John Horden for helpful correspondence. Thanks also to Amanda Lowe and Caley Millen-Pigliucci for help in preparing the text.

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R E F E RE N C E S Chalmers, David J. “On Sense and Intension.” In James E. Tomberlin, ed., Philosophical Perspectives: Language and Mind, volume 16, pp. 135–82. Oxford: Blackwell, 2002. deRosset, Louis. “Analyticity and Ontology.” In K. Bennett and D. Zimmerman, eds., Oxford Studies in Metaphysics, volume 9, pp. 129–70. Oxford: Oxford University Press, 2015. Kaplan, David. “Demonstratives.” In Joseph Almog, John Perry, and Howard Wettstein, eds., Themes From Kaplan, pp. 481–563. New York: Oxford University Press, 1989.

WHAT REALITY IS LIKE

6 Colors as Primitive Dispositions Hagit Benbaji

This paper offers an account of colors that combines primitivism with dispositionalism. “Primitivist dispositionalism,” as I shall call it, may sound like an oxymoron. Primitivism takes redness to be a sui generis simple property of an object, wholly borne on its surface, and fully revealed to us in visual experience.1 Dispositionalism takes redness to be the disposition of an object to look red to normal perceivers under normal perceptual conditions.2 But it is puzzling how a sui generis property, wholly borne—like shape—on the surface of a physical object, is essentially connected to experiences, and, from the other direction, puzzling how a disposition to cause experiences is entirely revealed in these experiences. Primitivist dispositionalism is not only hardly intelligible, but also difficult to offer a motivation for. Perhaps primitivism is the view of colors that best captures the phenomenology of color experience3—colors do indeed appear to be simple properties wholly borne on the surface of objects—but there are no such properties; only in Eden, it seems, was the apple “gloriously, perfectly, and primitively red ” (Chalmers 2006, 49). By contrast, while no one can deny that objects are disposed to look colored, dispositionalism has been criticized for misrepresenting the phenomenology of color experience.4 However we might want to analyze or unpack the concept 1 Although primitivism emerged relatively recently, there are now many different variants: McGinn’s impressionism (1996); Campbell’s simple view (1993); Johnston’s hylomorphism (unpublished manuscript); Yablo’s “naïve objectivism” (1995); Westphal’s phenomenalism (1987, 2005); Broackes’s “ways of changing light” (1997, 2007); Gert’s “unmysterious” primitivism (2008); Kalderon’s pluralism (2008, 2013); Allen’s selectionism (2007, 2009); Watkins’s “a posteriori” primitivism (2010). 2 Dispositionalism has put forward by Locke. Contemporary advocates include Ryle 1949, Peacocke 1984, McDowell 1985, Evans 1980, McGinn 1983, Wiggins 1987, Johnston 1992. All page references to articles reprinted in Byrne and Hilbert 1997 refer to that edition. 3 See Chalmers 2006, Johnston manuscript, McGinn 1996. 4 McGinn 1996, 541. See also Johnston 1992, 140–2.

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of color, experientially speaking, the critics insist, colors simply do not look like dispositions. Indeed, colors seem to be stubbornly non-dispositional. Combine this phenomenological charge—colors don’t look like dispositions—with the metaphysical extravagance of primitive properties, and primitivist dispositionalism would seem to be a non-starter. This paper has three goals. First, to show how primitive dispositions are possible. Second, to claim that colors are primitive dispositions. Third, to argue that if the primitive property that is directly revealed to us in experience is none other than the apple’s disposition to look red, then objects do have these Edenic colors after all. The key to achieving all three goals is the notion of an “appearance property,”5 such as my aunt’s looking young, her youthful appearance. When I look at my aunt, I see that she looks young, not that she is young. That is, I see an appearance property: the property of looking youthful. On my conception, appearance properties are to be understood, not in terms of mental effects they generate in an observer—sensations, qualia, and the like—but as genuine properties of physical objects. This paper will argue that to see a color is to see an appearance property, just as to see my aunt’s youthful appearance is to see an appearance property. I begin by articulating the basic insights worth saving from dispositionalism and primitivism, as well as the tension raised by their conjunction (section 1). In the main body of the paper, I then outline a model for appearance properties that establishes the possibility of primitive dispositions (section 2); spell out what colors have to be, namely, primitive dispositions, for this model to be applicable to them; and address the metaphysical implications of primitivist dispositionalism (section 3). I close by replying to the phenomenological objection, showing that colors do not look non-dispositional (section 4), and furthermore, colors do indeed look like dispositions (section 5). Thus my argument may also shed light on our inclination “to feel the pull of Revelation,”6 Russell’s famous claim that “I know the colour perfectly and completely when I see it” (1912, 47).

5

Suggested (for different purposes) in Pettit 2003, Noë 2004, Phillips 2006. Johnston 1992, 139. Several characterizations of this thesis have been offered. Campbell takes it as saying that experience provides us with “knowledge, for each particular color, of which particular property it is—the qualitative characters of the colors” (2005, 4); according to Johnston, Revelation is the view that, “Canary yellow is counted as having just those intrinsic and essential features which are evident in an experience of canary yellow” (1992, 139); Byrne and Hilbert (2007) argue that Revelation implies that any necessary truth P about the nature of colors is given in experience, so that if it seems to us (under normal viewing conditions) that P, then P is true (Infallibility), and if P is true, then it seems to us that P is true (Self-Intimation). 6

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The resulting account of color is dispositional, in that there is nothing to being red beyond looking red. Nevertheless, by giving due credit to the phenomenology of color experience that makes primitivism appealing, it offers a way for us to remain on Earth, yet feel like we’re in Eden.

1. MOTIVATIONS AND PUZZLES Primitivism claims that colors are more than dispositions. Regardless of what that extra something consists in, this claim flatly contradicts any version of color-dispositionalism. Likewise, a reductive dispositional account is straightforwardly incompatible with primitivism, since it purports to identify redtype experiences without invoking the concept of redness. Thus, primitivist dispositionalism can accommodate only non-reductive dispositionalism, which denies that experiences of red can be identified without invoking the concept of redness, and some inspirations of primitivism. Specifically, I extract each view’s core motivation: from dispositionalism, the objective–subjective duality of colors (section 1.1), and from primitivism, the claim that colors are experienced directly (section 1.2). These motivations are, I will show, compatible. To say that these motivations are compatible is not to eliminate the initial conflict their conjunction may generate. Clarifying the tenets of dispositionalism and primitivism does, in fact, bring to the fore a certain tension between them (section 1.3).

1.1 The motivation for dispositionalism Colors, like shapes, are properties of material objects, with all that this familiar belief implies; for instance, that colors do not disappear the minute we stop looking at them, when it gets dark, or when lighting conditions change, and that they leave room for error and illusions. In this sense, colors are objective. In another sense, colors are, unlike shapes, subjective. We understand what it is for something to be a triangle if, say, we take it to be a closed three-sided figure, that is, by having a non-perceptual description of it. By contrast, a full understanding of what it is for something to be red requires an understanding of what it is for that object to look red (McDowell 1985, 203). Setting aside the rubric of “dispositions,” which may create the impression that the underlying insight is excessively complicated, all that is implied by the dispositional account is that for an object to be red, say, is for it to

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look red. The connection between colors and experiences is vindicated by commonplace facts such as the following: 1. All we can do, and all we need do, in reply to the question, “What does ‘red’ mean?”, is point to the apple and say “Look!”, whereas shapes can be taught by touch or geometry. 2. It seems to be the case that the congenitally blind cannot understand the concept of redness, though they can understand shape concepts. Why can’t the congenitally blind understand what redness is? It does not suffice to reply that redness implies looking red, for it is also true that squareness implies looking square, and those with congenital blindness can understand what squareness is. To explain this phenomenon we have to acknowledge that what it is for something to be red is simply for it to be such as to look red. This—namely, that being colored is solely a matter of appearances, of how things look to us, rather than a matter of having some underlying geometric structure, or non-perceptual, hidden (physical, chemical) structure—is the grain of truth common to all versions of the dispositional account. Hence so many believe that “nothing can be seen as a colour without being seen as essentially connected with vision” (Boghossian and Velleman 1989, 96). This is what I call the ‘objective–subjective duality’ of colors, which is the core claim of dispositionalism. What it amounts to is that colors, inasmuch as they are visible properties of objects, are objective, while inasmuch as they are essentially connected to experience, they are subjective. Although colors are, in a sense, mind-dependent, they are, as Allen aptly puts it, “still properties that exist without—in the spatial sense of being outside of—the mind” (2007, 142).

1.2 The motivation for primitivism On the face of it, primitivism is focused on what colors are, that is, on the nature of colors, only insofar as that nature is derived from how colors are perceived—what it’s like to see redness. Indeed, primitivism seems to follow from Revelation: the nature of colors is wholly revealed in experience, but since it is not revealed to us in experience that colors have microstructures, or (allegedly) that they are dispositions, colors must be primitive, viz. sui generis properties.7 Granting that Revelation implies primitivism, why should this mean that primitivism is true? Does the phenomenology of color experience really 7

This argument is discussed in Byrne and Hilbert 2007.

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make any difference to our philosophical analysis of color?8 For it might be argued, so what if dispositions or physical properties are not fully revealed to us in experience, or cannot be seen at all? If an otherwise-well-supported view, whatever it is, does not mesh well with color experience—so be it. A better motivation for primitivism than the radical requirement that the nature of colors be fully revealed in experience is the more modest requirement that at least part of the nature of colors be revealed in experience. Although weaker than Revelation, “Manifestation,” as Kalderon calls it, is a strong claim. Manifestation claims that when we perceive redness, we ipso facto perceive at least part of its nature; we perceive what red is like (e.g. that it is reddish and not at all bluish). Manifestation claims that colors are what they appear to be. To say that redness is what it appears to be is to say that redness determines the phenomenal character of our experience of it—what it’s like to see redness. Thus, Manifestation implies “Inheritance” (Kalderon 2007),9 the view that “the qualitative character of a colour-experience is inherited from the qualitative property of the colour” (Campbell 1993, 189).10

8 Some have taken phenomenological considerations seriously enough to withdraw their support for dispositionalism and instead endorse primitivism. Johnston, unpublished manuscript; McGinn 1996. Other philosophers are not bothered by the charge that their view violates our commonsense experience of color; see Johnston 1992, Boghossian and Velleman 1989, Chalmers 2006, Miscevic 2007. 9 As characterized by Johnston (1992) and Kalderon (2007), my contention that Manifestation implies Inheritance is, perhaps, controversial. Johnston argues that color dispositions satisfy Manifestation in virtue of the fact that experiences of colors are the manifestations, i.e. the triggering, of the dispositions. “About any disposition of objects to produce a given experience, it is plausible to hold that if one has an experience of the kind in question and takes that experience to be a manifestation of the disposition in question, one thereby knows the complete intrinsic nature of the disposition” (Johnston 1992, 167). On Johnston’s view, Manifestation is satisfied by any account of visual experience one accepts, whether construed in terms of mere sensations, mental representations, or primitive relations of subjects to objects. However, I am arguing that without a proper understanding of visual experiences, Manifestation cannot be true. If experiences are mere sensations without intentionality, a disposition to cause red sensations is manifested only in the way a disposition to cause pain or nausea is manifested (Johnston’s example). This raises the concern that experiences of redness cannot be distinguished from bodily sensations (Johnston 1998). 10 Inheritance alone is satisfied by reductive representationalism that accepts that the phenomenal character of the experience of something’s looking red is simply a matter of its representing an object as red, representation being a matter of the perceiver’s causal or teleological connection to red objects. A physical property might determine the phenomenology of color experience in this sense, without that property’s manifesting its nature in experience. Tye and Dretske argue that physical redness determines the phenomenal character of experience, and thereby eliminates qualitative properties not only from the world, but from the mind as well. Thus, according to Tye and Dretske, Inheritance does not imply Manifestation.

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Since the claim that Manifestation implies Inheritance is the main rationale for primitivism, let me accept the following constraint on colors: MI (Manifestation cum Inheritance): a color is a property of an object, such that by being manifested in experience determines the phenomenal character of that experience. MI seeks to uphold both the irreducibility of colors and the idea that they are directly perceived. First, like Revelation, MI implies primitivism. On the one hand, Manifestation implies that colors cannot be solely a matter of physical structure, since experience does not reveal to us any physical aspects of colors. On the other, Inheritance implies that colors cannot be reducible dispositions, since reductive dispositional accounts identify color experiences independently of any putative color property of things in the world, and thus no such property can determine the phenomenology of color experience. Hence, colors are sui generis. Second, MI explicates what it means to see a property of an object directly. Hence, I take primitive colors to be properties of objects that satisfy MI. Thus far, it might seem that the basic reasoning behind the “so what?” retort to Revelation has not been undermined, but merely reformulated: “so what if the nature of colors is not manifested to us in experience? The nature of water is not manifested in experience, yet this is no reason to claim that water is more than H2O.” The nature of color might not, it could be argued, be manifest in perception at all, because it is the upshot of either a scientific discovery, or philosophical analysis (McGinn 1983, 135; Cohen 2010a).11 Nevertheless, the “so what?” retort is not apt in the case of colors. We might not care if the nature of water is not manifest to us in experience, but if colors are not manifest to us in experience, no property is. Primitivism is driven by the insight that were colors not directly revealed to us in experience, we would be unable to gain any direct knowledge of the mindindependent world (Campbell 2002), and worse, we could not even take our experiences to supply reasons for beliefs about the world (Brewer 1999). The underlying idea here is that perception of color is not merely a form of “Morse code transmission” (Johnston 1992, 166), “a reliable effect of properties whose nature remains unknown” (Kalderon 2007, 587), as perception of water might be. I cannot even begin to fully justify these epistemic claims here. My point is to emphasize that primitivism is rooted in an epistemological or cognitive outlook, rather than solely in the ideal of phenomenological adequacy. To

11 Some argue that we can see the wrongness of an act, though it might turn out that this wrongness is entirely a matter of some complex consequentialist feature.

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recap, the epistemic insight is that if colors are not directly perceived, the empirical world itself becomes invisible, unknowable, or even unjustified.

1.3 The tension The central claim of this paper is that colors are primitive dispositions. Primitive dispositions are irreducible dispositions that satisfy MI. Let me explain why the core motivation for primitivism, namely MI, seems to conflict with the core motivation for dispositionalism, namely, the subjective aspect of color, whereby colors are essentially connected to experience. Both primitivists and dispositionalists accept the following conditional: if colors are essentially connected to experiences of them, colors cannot explain these experiences. The concern that underlies this conditional is that if we explain what it is for something to be red by saying that it looks red, “we cannot explain why something looks red by saying that it is red, for this involves an explanatory circle” (McGinn 1983, 15). Dispositionalism accepts the antecedent, that is, the claim that colors are essentially connected to experiences, hence, the consequence: it deems colors to be explanatorily redundant. The implication is that colors can neither satisfy MI, nor be directly perceived. 1. Inheritance is an explanatory thesis. It claims that the phenomenal character of a color experience “depends on and derives from the qualitative character of the presented color” (Kalderon 2007, 3). To render colors explanatory powerless is to deny Inheritance. 2. Since Manifestation implies Inheritance, it is false as well. 3. Since MI gives content to the idea of direct perception, colors are not directly perceived. Indeed, we see solubility by seeing a manifestation of solubility—the sugar’s dissolving, just as we see water by seeing its liquidity, transparency, etc. If Manifestation is false, the manifestations of color dispositions do not reveal the nature of the color any more than the visible properties of water reveal its nature. Thus, we do not see colors directly. The tension between MI and the subjectivity of colors seems to be grounded in the idea of a property that explains the very experiences which constitute it, not in any particular construal of these experiences, i.e. reductive or non-reductive. The dispositionalist’s modus ponens is the primitivist’s modus tollens. Since the thrust of primitivism—Inheritance, Manifestation, and the very idea of direct experience—is lost if we reject the explanatory force of colors, primitivism concludes that colors are not essentially connected to experiences. This explains why it is so important for primitivism to reject the consequence

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of the conditional. If the way things look depends on their colors as much as it depends on their shapes, colors are prior to, and independent of, our experiences of them.

2. A P P E A R A N C E P R O P E R T I E S A S D IS P O S I T I O N S T O A P P E A R F To explain the possibility of primitive dispositions, let me introduce the notion of an appearance property (section 2.1) and claim that to see a property as an appearance property is to see it as a disposition of an object that is inherently connected to visual experience (section 2.2).

2.1 Appearance properties I look at my colleague, who is only thirty, and it occurs to me that he looks old. He does not appear to be old; he appears to me to look old—he has the timeworn look older people have. Being aware of his actual age, the property I notice when I see my colleague is not that of “being old,” but that of “looking old.” Once we become aware of the phenomenon of attending to the appearances of things, it seems ubiquitous. As Phillips (2006) remarks, we spend much time glancing at our appearance in car and shop windows, and each time we look quite different. My face looks more animated, more natural, from this angle, or in daylight as opposed to fluorescent light. We note how much she looks like her sister, how vodka looks like water, or how a certain face looks familiar. In all these cases, we intuitively take appearances, the way things look, to be bona fide properties of the things in question. “Looking like water” is taken to be a property of vodka, just as “looking old” is a property I ascribe to my colleague. As O’Shaughnessy puts it, “It was the look of Helen’s face, rather than its chemical or electrical or pheronomic properties, that caused such a horror” (2000, 570). This talk of appearance properties raises an immediate objection, namely, that appearance properties are not genuine properties at all, but mere illusions. Upon seeing a stick in water, I do not see a new property of the stick—“looking bent”—but rather, an illusion: a straight stick that appears bent. Similarly, my seeing my colleague’s aged appearance is not seeing a property, but being deluded as to his real age. More generally, seeing an appearance property is not, on this view, genuine perception of a property. Of course, I need not be misled by the illusory experience, and I may be sophisticated enough not to believe that the stick is bent. But it is

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nonetheless an illusion. The objection, then, is that any statement asserting the occurrence of an illusory experience (IE), such as (IE)

It merely seems to me that the stick is bent

can be systematically converted into a statement asserting the occurrence of a veridical experience (VE) of an appearance property: (VE)

I see that the stick is bent-looking.

Yet clearly, this linguistic maneuver cannot endow the stick with a genuine property—so goes the objection. Hence the challenge confronting one who seeks—as I do—to uphold the existence of appearance properties is to distinguish them from illusions. This challenge can be met by showing that we can distinguish between real and illusory appearance properties. Let us deem an object’s “normal” appearance to be the way the object looks under normal perceptual conditions.12 In the case of the stick, since the normal appearance of the shape always manifests the stick’s real shape, the only way the appearance can be misleading is if the stick’s normal appearance changes due to its being viewed under unusual conditions, e.g. submerged in water. This is indeed a case of illusion, inasmuch as the water alters the stick’s normal appearance so that it appears bent. On the other hand, we might be misled about someone’s age even without any change in his normal appearance, as in the case of my colleague. Purely observational properties, such as shapes, cannot be separated from their appearances, that is, from the way they look, whereas properties that are not purely observational are separable from their appearances. Inasmuch as the connection between being old and appearing old is not as close as that between being straight and appearing straight, the property of being old is less observational than that of being straight. It is observational, for there is indeed an appearance associated with it, but this appearance can be instantiated even in cases where the instantiating individual is not old. In such a case, I see the look associated with oldness, without seeing oldness; the look has a life of its own, as it were. What makes the submerged stick a case of illusion is the fact that water alters the stick’s normal appearance. By contrast, my colleague’s normal appearance has not changed at all, and thus, though we cannot see that he’s not as old as he looks, making this a case in which we might be misled by an appearance, we do nevertheless see something else—namely, the look of agedness, which is a genuine appearance property.

12 I am assuming that there is no non-circular description of the best perceptual conditions for seeing colors and shapes.

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This reply establishes a distinction between an illusion and perception of an appearance property, thereby making itself vulnerable to the deflationist concern about hypostatization of “looks.” After all, there is a simpler way to distinguish the appearance of agedness from the bent-stick illusion: when I see that my colleague looks old, I do not see any special kind of property, any putative “look;” rather, I see wrinkles, a slight shuffle, and other visible properties on which “looking old” supervenes. There is no need to reify appearances in order to distinguish the bent-stick illusion from my oldlooking colleague. The bold reply to the deflationist concern is that it involves phenomenological falsification. When I see that my colleague looks old, I do not merely see lines or brown marks, but wrinkles and age spots. What is it to see the lines as wrinkles or the brown marks as age spots? It is to see them as expressions of oldness, signs of age. And this is precisely to see my colleague’s aged “look.” The modest reply points out that even if talk of “looks” is redundant in the case of oldness, appearance properties cannot be thoroughly eliminated. Consider first the famous example of aspect seeing. We can shift at will from seeing the duck–rabbit as a duck (Figure 6.1), to seeing it as a rabbit, and we cannot specify the content of the experience of that shift without ascribing to the picture a certain “look.” We see the picture first as rabbit-shaped and then as duck-shaped, so we see that something in the figure changes. But we do not see the figure as changing its shape; rather, part of the experience is that the figure remains unchanged when we change perspective. What changes is the way it looks, the picture’s appearance: from one point of view it looks like a duck, from another it looks like a rabbit. The language of appearance properties is, then, indispensable to a plausible account of ambiguous figures. Aspect shifting is exemplified (in a way) in the ordinary phenomenon of constancy. I look at a round coin, and then view it from another angle. The coin looks round when it is in front of me, or when I look at it from above,

Figure 6.1 A famous example of aspect seeing

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but when I see it from an oblique angle, it looks elliptical. In contrast to the case of the stick in water, in which the real shape cannot be seen, in this case I see all along that the shape of the coin has not changed, that is, I see that the coin remains round through all the changes in viewing perspective. Indeed, it is an integral part of the capacity to perceive shapes that we can keep track of this property through movements in egocentric space. Note: we do not merely believe the coin to be round through all the changes in the observer’s position, but rather, we see its shape as constant through these changes. Yet in gazing at the coin, something does seem to change; what is it? The only property that can be experienced as changing here, I submit, is the look of the shape. Let me briefly consider deflationist strategies that try to explain away the reification of looks, and why these strategies fail. (a) Denial of constancy: we do not really experience the coin as having one constant shape, but as having changing shapes. Reply: to reinforce the claim that the coin’s shape is not perceived as changing, Siewert points to cases in which things clearly appear to change shape. “Gaze at the shapes in a lava lamp. Watch as a balloon is inflated. See someone’s mouth break into a smile” (2006, 10). By contrast, the coin does not appear to change its shape. Phenomenologically speaking, there is a difference between experiencing something’s changing shape, and experiencing a constant shape the appearance of which fluctuates. (b) Denial of changing appearances: we do not see that the coin usually looks elliptical when viewed from an angle; we cannot see that the coin remains round all along, and at the same time experience it as elliptical. Rather, we can only experience it as elliptical from a “detached” point of view.13 Alternatively, it might be conceded that there is an elliptical aspect to the situation, that is, the projection of the coin against a hypothetical screen, though it takes some practice to be aware of this elliptical aspect. After all, we are “not inclined, the tiniest bit to take the tilted penny to be elliptical” (Smith 2002, 182). Reply: notice that even those who deny that the coin usually looks elliptical do not do so unreservedly—they offer excuses (conflicting appearances imply different points of views), and alternatives (an elliptical aspect to the situation). But the excuses are unnecessary, and the alternatives are not phenomenologically apt. On the view I am arguing for, there are no conflicting experiences—we do not experience the coin as round and as elliptical at the same time; rather, we experience the coin as round and as elliptical-looking at the same time, hence there is no need to invoke a shift in perspective. The suggested elliptical aspect of the situation is not a 13

Kelly 2004, Smith 2002.

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satisfactory alternative, because phenomenologically, it is not merely that the situation has an elliptical aspect, but that it has an elliptical appearance.14 As to the epistemic argument that we are not inclined, even “the tiniest bit,” to believe that the coin is elliptical, this is fully in line with my view: this disinclination is rooted in the fact that we experience the coin as looking elliptical, not as being elliptical.15 (c) The changing property is internal: it is a property of our experience (sense-data theorists), or a sensory property of regions of the visual field (Peacock 2008). Reply: even apart from the contentious appeal to sense data, this answer does not accord with the phenomenology of experience. It is the coin’s properties that we experience as changing, not sense data or any other objects that might be distinct from us, yet internal, or minddependent. (d) The changing property is the coin’s location relative to the perceiver: the round coin looks round and tilted away from me. Reply: I do not deny that egocentric locations are represented in experience; I deny that egocentric location exhausts the content of the visual experience of the coin as viewed from that location. It is true that in one case the coin appears to be facing straight ahead of me, whereas in the other, it appears to be tilted away from me. But something else differentiates the content of the two experiences: in the one case, experience presents the coin as round, in the other, experience presents it as looking elliptical. Many figures are not impacted in this way by movement in egocentric space. Thus, a tilted “A” changes place relative to the perceiver, but as opposed to a tilted square or a tilted circle, still looks like an A (Macpherson 2006, 91) (Figure 6.2). By contrast, viewed from an oblique angle, the coin’s shape does have a different appearance than when viewed frontally. So a rotated circle looks like an ellipse, and a tilted square looks like a diamond, but a tilted A still looks like an A.

14 Someone congenitally blind can know that the projection of the coin against a hypothetical screen is elliptical. However, she would presumably not know what it is for the coin to be experienced as both round and as having an elliptical appearance. 15 A “parsimonious view of looks” has recently been suggested by Martin (2010). This view reduces appearance properties to objects’ visible properties: colors, shapes, sizes, etc. On this view, since the coin does not change shape, it also doesn’t change in appearance. The parsimonious view is thus an example of strategy (b). What changes is how the coin strikes us, that is, our psychological state, and the assertibility of different claims about the coin that express those psychological states, e.g. the assertibility of the claim that the coin looks elliptical when tilted (Martin 2010, 221–2). I cannot do justice to Martin’s rich semantic proposal here, but the general arguments against strategy (b) apply to his view as well.

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Figure 6.2 Tilted ‘A’

(e) The changing property is the disposition “to generate in us an instance of the type of perceptual state we undergo when perceiving an ellipse straight on” (Cohen 2010b, 17). This property is the input of “the computation whose output is a representation with the content that there is a distal instance of round” (17). What seems to change is not the distal shape, i.e. roundness, but the net perceptual signal transduced (to which distal shape and visual angle are among the contributing factors). Reply: even apart from its controversial assumption that “causally intermediate states in the subpersonal visual system” (17) have representational content, the reply falsifies the phenomenology of the experiences in question. As I already emphasized (in (c) above), the experience presents aspects of the coin as changing, not our sensations or perceptual signals at sub-personal levels. (f ) The changing property is “situation-dependent” (Schellenberg 2008). Reply: indeed, the coin is elliptical only in a given situation, and it is round independently of the situation. Yet in the situation where the coin is tilted, the coin is both elliptical and round. But admitting that one property is situation-dependent and the other situation-independent does nothing to resolve the puzzle of contradictory properties. Adding “in the given situation” doesn’t make it okay to say that the coin is elliptical. (g) The changing property is “apparent or perspectival shape” (Noë 2004): we experience a change in apparent shape, together with a constant (plain old) shape; we experience the circularity of the coin “in its merely [apparent] elliptical shape” (167). Reply: the question is what the property of being elliptical in “apparent” shape is. If it is simply being elliptical from here (164), the puzzle remains, because the coin also looks round from here. So it has to be “the shape that would be exactly hidden by an elliptical patch placed in a plane perpendicular to my current line of sight” (Siewert 2006, 7). But that shape is none other than elliptical! Siewert suggests that talk of an “apparent” shape is redundant; all we need is the following counterfactual: “if you were to block the appearance of the plate [coin], you would need to put differently shaped patches in the way, given the change in how the plate [coin] appears to you as it turns” (2006, 8). If I understand Siewert’s construal of Noë’s suggestion correctly, it implies that the appearance of the coin

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changes—it changes in a way that is captured by differently-shaped patches. That is to say, what is changing is the coin’s elliptical look. Once we concede the existence of appearance properties, the answer to the puzzle of contradictory properties is straightforward: we experience the coin as round, and as elliptical-looking. Let me emphasize that my purpose here is not to provide a full or even a partial account of either constancy or aspect shift. Nor do I utilize the phenomena to argue for the perceptual or epistemic primacy of “looks” (as do Noë 2004 and Schellenberg 2008). My appeal to constancy is made solely to lend credence to the possibility of seeing something’s “look”—an aged look, an elliptical look, and so on. The capacity to perceive the same shape through changes in its appearance presupposes the distinction, within visual experience, between visible properties themselves— squareness, roundness, etc.—and their more fine-grained appearances, for example, the way round coins look from an oblique angle. Constancy assists in demonstrating that visible properties such as shape and size can be associated with finer-grained appearances, which are sensitive to changes in spatial location. My conclusion is that to solve the puzzle of contradictory properties we must recognize in the existence of appearance properties that explain our experience of constancy and change, and concede that we see these finergrained properties as appearance properties.

2.2 Appearance properties are dispositions to appear F So far, I’ve shown that some properties are appearance properties, and that we see them as such. I now want to argue that an appearance property is a disposition. Reflection on experiencing appearance properties brings to the fore two of their most distinctive characteristics. (1) Objectivity. The ellipse-like look seems to be a property of the coin, not a mental experience of some kind. We do not experience such “looks” as properties of our experiences (Noë 2004, 85). Similarly, my colleague’s aged look is not experienced as a mental entity, but as an objective property inhering in my colleague. Neither the coin’s elliptical look, nor my colleague’s aged look, disappears when I avert my gaze. The elliptical look is the same for any perceiver viewing the coin from that specific angle (Schellenberg 2008, 11). I may fail to see the coin’s elliptical look (because it is dark, or because I’m viewing the coin from here, as opposed to there), but it can be seen when conditions are right.16 16 In the other direction, something can appear to have a certain look, though it does not in fact have that look, so there is a gap between seeming to have a certain appearance

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As Austin puts it: “I am not disclosing a fact about myself, but about petrol, when I say that petrol looks like water” (1962, 43). (2) Subjectivity. Even so, the experience of seeing looks, that is, things’ appearances, is not just like the experience of seeing shapes, say. The second distinctive characteristic of experiencing an appearance property—the coin’s elliptical look, say—is that the property is experienced only as a look, as a property whose nature is to appear a certain way. This is the import of constancy: we experience the changing appearances as appearances (looking round, looking elliptical). We don’t experience the coin as being elliptical, but as looking elliptical.17 To recapitulate, the two essential features of appearance properties are objectivity (appearances are—and are experienced by us as—genuine properties of physical objects) and connection to experience (an appearance property is—and is experienced by us as—a way of appearing). But for a property to have these two features amounts to its being, and being experienced as, a response-disposition. As I’ve already emphasized (in section 1.1), all that is implied by the dispositional account is that redness is a property of objects that is nonetheless essentially connected to visual experience. Hence, to experience the coin as looking elliptical is to experience it as looking a certain way (viz. elliptical) under certain conditions, or in other words, to experience it as being disposed to look elliptical.18 I suggest, then, that an elliptical appearance is a disposition, of an object, to appear elliptical, and it is experienced as such. Yet the identification of an elliptical appearance with a disposition to look elliptical might seem confusing. If, as I suggest, the look is the appearance, how can it be a disposition to have an appearance? The appearance seems to be the manifestation of the disposition; if not, i.e. if it is the disposition itself, what, exactly, is the manifestation of this disposition? Furthermore, if something is a disposition, then there is a property that is exemplified when and having that appearance (she doesn’t really look old, she just hasn’t slept for three nights in a row). 17 Notice that this claim, viz. that “looks” are properties whose nature is simply to be manifested visually in some specific way whenever the conditions are right, describes how we conceive “looks,” insofar as it describes how we experience “looks.” It is the special nature of being a property whose essence is just its appearance that we cannot conceive F to be a “look” without seeing it as such. 18 Although Schellenberg 2008 argues that these relational and perspectival properties are—crucially—not to be analyzed in terms of how things look. Her reason for this denial seems to be that it is “easy to imagine” (72) the illusion of an intrinsic property (ellipticalness) when what is perceived is a situation-dependent property (looking elliptical). But the fact that we might confuse an appearance property with an intrinsic one does not show that in ordinary cases we do not perceive the former as a “look.”

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the characteristic manifestation of the disposition occurs. That property cannot be the very disposition itself, for were that the case, it would be a disposition that is always triggered. So on this reasoning, it isn’t the disposition itself that appears to us, but rather, its characteristic manifestation. This objection is the key to understanding the relevance of the appearance-property model for the idea of experiencing a disposition directly, and thus, to understanding the possibility of primitive dispositions. The objector rightly assumes that when the characteristic manifestation of a disposition occurs, say, sugar is put in water and dissolves, then the property that is exemplified is “dissolving in water,” that is, the manifestation of solubility. But a visual disposition is unlike solubility, in that it is necessarily manifested when its manifestations occur. To clarify this point, we need to disambiguate the term “manifestation.” A property, the table’s squareness, say, is manifested to me in visual experience, meaning, it is presented or revealed to me in visual experience. In the more technical sense, the term “manifestation” is used to refer to the triggering or exercise of the disposition (e.g. dissolving in water is a manifestation of solubility).19 Since the manifestations—in the “triggering” sense—of a visual disposition are the experiences of that visual disposition, the disposition manifests—in the “presenting” sense—itself when its manifestations (in the triggering sense) occur. The disposition to look elliptical can be an appearance property of say, a coin, without always being triggered, because the disposition to appear elliptical manifests itself (in both senses of “manifestation”) only under specific conditions (when tilted, for example). So what happens when I see a coin from an oblique angle twice within, say, an hour? I have two different experiences of the disposition to look elliptical. But I don’t see two different manifestations (in the triggering sense) of the disposition in question (the disposition to look elliptical) as I would vis-à-vis the disposition to be breakable, were a window to be cracked twice during that time, while I’m looking at it. Rather, in having different (token) experiences, I see the same elliptical appearance, i.e. a property of the coin. The same property is revealed to me twice. We now have a model for what it is for an object to have an appearance property—a look—as, for example, when we see the coin’s elliptical look. a coin’s disposition to look elliptical = a coin’s elliptical appearance manifestations of the disposition to look elliptical = experiences of the coin as looking elliptical 19 Both the term “appearance” and the term “look” are also ambiguous—they can denote a property of an object, as well as a property of a subject (that is, an experience).

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According to this model, the disposition is manifested (in the sense of “triggered”) by causing experiences of itself, and these experiences manifest to us (in the sense of “reveal to us”) that disposition, namely, the coin’s elliptical look. The disposition is simply the coin’s elliptical appearance, which lies wholly on its surface. Let me emphasize how far this model is from the traditional view, on which the relevant dispositions are dispositions to affect experience. On the traditional view, there is no genuine appearance property on the object’s surface—the appearance property is reduced to the shape of the coin and internal experiences of the perceiver. On the model proposed here, there is, literally, an elliptical appearance, a dispositional property, that is manifested (i.e. presented) on the surface of the coin when I look at it. Looking at the coin, I have an experience of it, which is a manifestation (i.e. a trigger) of the disposition; but the experience does not somehow come between me and the coin. Rather, the experience reveals to me the coin’s elliptical appearance. On my model, an ellipse’s disposition to look elliptical is not identical to a (tilted) coin’s disposition to look elliptical. The manifestations of the former disposition are experiences of the ellipse; they present an ellipse to us. The manifestations of the latter disposition are experiences of an elliptical look; they present the disposition itself. This might seem to be an unwelcome conclusion. Under some circumstances one might be unable to tell the difference between the tilted coin and a frontal ellipse, and the explanation for this inability is that the experience of the coin from an angle and the experience of a frontal ellipse are introspectively indistinguishable. However, as I’ve already argued, the intuition regarding the phenomenon of constancy is that the tilted coin does not look like an ellipse viewed from the front, because we also experience the coin as constant in shape, namely, as round. Since we see the elliptical appearance of the tilted coin together with its roundness, the experience of the tilted coin and that of a frontal ellipse are introspectively distinguishable. The rationale for introducing “looks” or appearance properties to begin with is to explain differences in our experiences of the coin in terms of properties whose co-instantiation does not constitute a contradiction (being round, looking elliptical), rather than in terms of the paradoxical co-instantiation of incompatible properties (being round and being elliptical). Granted, experiences of the coin from an angle and of an ellipse viewed frontally may be introspectively indistinguishable, to some observers, under some circumstances. This is perfectly compatible with the model of appearance properties proposed here. After all, the experience of the coin from an angle presents the coin as looking elliptical; it is no surprise that one might

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confuse it with an ellipse. But this happens when we do not experience the tilted coin as also being round: on this scenario, we are under the illusion that it is an ellipse. Ordinarily, however, observers familiar with the phenomenon of constancy have enough cues to see the coin not as an ellipse, but as looking elliptical. The model of appearance properties enables us to understand how a primitive disposition is possible. The coin’s elliptical look is none other than a primitive disposition of the coin. Let me now articulate how the model eliminates the puzzling features of primitive dispositions. (1) Inheritance. The coin’s appearance can determine the phenomenal character of our experience in the explanatorily ambitious way required by Inheritance: we explain our experience of the coin in terms of its appearance—a property of the coin’s surface. Indeed, Inheritance—that is, explaining the phenomenology of constancy and change in terms of appearance properties of objects—is, in fact, the main thrust of the model. Similarly, the phenomenal difference between an ellipse viewed from the front and a tilted coin is explained by the differing properties of those objects, namely, ellipticalness and having an elliptical appearance, respectively. (2) Manifestation. The nature of the property is experientially transparent— we perceive the tilted coin as it is, namely, as looking elliptical, not as being elliptical. (3) Direct experience. The coin’s elliptical appearance is directly manifested to me in experience; it is not that I see it by seeing its manifestations. On this model, experience reveals the coin’s elliptical appearance to me directly, just as it directly reveals the coin’s roundness.20 The model for appearance properties is thus able to accommodate the notion of a disposition that satisfies MI, that is, the notion of a primitive disposition.

20

Thus, one might argue that I can see that a piece of fruit is such that were I to pick it up, it would be squished by my hand; or I see that it is such that pushing it will cause it to roll. Its tendency to be squished or to roll is visible (to a suitably sophisticated perceiver). This notion of a visible disposition is weaker than the one I am proposing, since it implies that I can see the coin’s tendency to look elliptical when tilted even when it is not tilted, but simply in front of me. This notion is close to Noë’s idea that we see patterns or regularities in the way things look. But on the model developed in this paper, by contrast, when the coin is tilted, I can see directly its elliptical appearance—not merely see its potentiality or regularity.

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3 . W H A T C O L O R S A R E : AP P L Y I N G T H E M O D E L I have argued that experiencing appearances—an elliptical look, a youthful look, and the like—provides a model for the experience of visual dispositions, dispositions to appear a certain way. The dispositional account identifies redness with the disposition to look red, so it is natural to apply the model suggested in the previous section to colors. In what follows, I apply the model to colors (section 3.1), and explain the metaphysical implications of doing so, thereby distinguishing the appearance-propertybased account of colors from dispositionalism and primitivism (section 3.2).

3.1 Applying the model The core contention of the dispositional account is that what it is for the apple to be red is for it to look red to normal observers under appropriate conditions for seeing color (I will omit the latter qualifying clauses in the following equations, and often, thereafter): (D1)

redness = the disposition of an object to look red

The model proposed in the last section explicates the dispositional nature of a tilted coin’s elliptical appearance. The relevance of the model is premised on the idea that “looks” or appearances are properties of physical objects— the elliptical look is a property of the coin, the youthful look, a property of my aunt, and so on. Accordingly, the dispositional account has to pick out the object’s appearance, or some feature thereof. When I point to the apple’s redness, according to the appearance-property model, I point to a distinctive feature, a distinctive look it has, just as I do when I point to the coin’s elliptical appearance when viewed from an angle: (D2)

the disposition of an object to look red = a red look

This equation states that redness is an appearance property. In being such as to look red, the apple has an appearance property, borne on its surface, namely, redness. (D3)

redness = a red look

D3, then, follows from D1+D2. As explained above, on this model, the disposition is manifested (that is, triggered) by causing experiences that reveal the disposition to us: the coin’s elliptical appearance causes experiences of its looking elliptical, not of its being elliptical. Similarly, the apple’s red appearance—its redness—causes experiences of itself, namely, of redness.

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Hagit Benbaji redness = the disposition to cause experiences of redness

On this model, the experiences referred to on the right-hand side of (D4) are experiences of the property referred to on the left-hand side of the equation. We cannot identify the experiences without specifying that they are experiences of redness, as the non-reductive account of dispositions asserts. One might object that (D4) is thus circular. The problem with circularity is that it deprives the account of “any interest” (Johnston, manuscript, 18), or renders it vacuous, inasmuch as it does not differentiate colors from shapes—after all, square things are also disposed to look square when conditions are right for seeing squares (Broackes 1997, 204). The model is indeed, circular—none of the equations purport to be an analysis or definition of redness, or color in general. They are not intended to be informative to someone who does not understand what redness is. However, the model is not vacuous, since these equations offer an informative account of redness. They do not merely affirm that redness is being disposed to look red; this is true of squareness as well: squareness is being disposed to look square. They also claim that redness is nothing over and above a red look. But as I showed in section 1.1, squareness, unlike redness, is more than a look. The equations are thus informative—they tell us what it is for an object to be red, whereas affirming that squareness is being disposed to look square does not tell us what it is for something to be square. The appearance-property model apprises us of the nature of colors, namely, that colors, unlike shapes, are looks—appearance properties—or to use the traditional characterization, dispositions to cause experiences. In order to know which property redness is, as opposed to, say, yellowness, or blueness, one has to look at a red object, under the appropriate circumstances (lighting, etc.). Accordingly, the experience of seeing redness can be characterized by pointing, under the right conditions, to something red: (D5)

a disposition to cause experiences of redness = a disposition to look like that (pointing to, say, a red apple)

The demonstrative “that” signifies the way the object looks, presents itself to us, appears—namely, red.21 While (D4) tells us what the nature of colors is, 21 Watkins (2002, 80–6) suggests the demonstrative formula as a solution to the circularity problem. I am presently working on a critique of that solution. Shoemaker provides a similar reply: “Particular ways of looking can be picked out demonstratively— ‘looking way W is looking that way’—or by descriptions like ‘the way chartreuse things look to observers with thus and such perceptual systems in thus and such conditions’ ” (2006, 466). Shoemaker, however, denies that these appearance properties are properties of external objects.

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(D5) tells what redness is. Both are circular (the demonstrative presupposes the concept of red), but not vacuous. It might be objected that the model leaves no room for illusions. The experience of a white object might, under abnormal conditions, be introspectively indistinguishable from the experience of a red apple under normal conditions. If the same experience can occur when the object does not have a red appearance, then it still seems that the experience somehow “intrudes,” as it were, between the perceiver and the color. Just as on adverbial or sensedata theories, we come to know about objects in the external world by means of knowing something internal, so too, according to this line of reasoning, the thesis that the experience reveals an alleged red “look” is no more than wordplay. Two replies to this objection suggest themselves, in accordance with the two chief conceptions of perception, the representational and the relational.22 Representationalists maintain that the phenomenal character of our experience of an object is determined by an object’s properties as represented in experience. The experience of an object’s looking red is a matter of its representing that object as having the property of redness, representing being a matter of, for example, the experience’s being reliably caused by red objects in the environment. Such representation can, when conditions are abnormal, be caused by a white object, so this account implies that there is a common factor to veridical experience and illusion. Yet the experience of an object’s looking red is a direct representation of the object’s color: it satisfies “Transparency,” the near-universal observation that in perception our attention is directed outward to the world, not to any inward sensation. On the relational view, experience is a primitive cognitive relation between an object and a perceiver, not merely a representation of it. According to the relational view, manifest properties of things themselves figure in experience, “not their mental surrogates” (Johnston, manuscript), be the latter sensations or representations. The experience does not “come between the perceiver and the color” in any sense at all; the experience is simply the presentation of the color itself. The color constitutes the experience. Hence, if the color is not instantiated, the relational experience does not hold. Since an illusion of a property is not a manifestation of an actual property of the object in question, it is not phenomenally identical to a veridical experience, which presents the color to us. We cannot introspectively distinguish an illusion of redness from a veridical experience of redness, but since only the latter 22 There is also a third conception of illusion that should be mentioned in this context. Johnston (2004) combines a relational view of experience with postulation of a common factor between veridical experience and illusory experience.

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presents to us the object’s red look, the illusory and veridical experiences are not phenomenologically identical. The disjunctive conception of experience that is committed to the possibility of introspective indistinguishability without phenomenological identity is controversial. Although I cannot defend this possibility here, let me point out that it is in line with the core motivation for primitivism, that is, the epistemic significance of direct perception, the premise that in manifesting a property to us, experience provides us with knowledge of which property is experienced. If the same experience sometimes—namely, when it is illusory—fails to provide such knowledge, it is hard to understand how perception can have this epistemic significance. In any case, I take it as a strength of the model proposed here that it leaves room for both views of experience. Neither the physicalist nor the classic dispositional account can make sense of the strong notion of perceptual presentation, as opposed to mere representation of a property. Appearance properties, by contrast, are the natural candidates, not merely for representation in experience, but for literally constituting experience. The picture that emerges is as follows: redness = a disposition to look red = a red look = a disposition to cause experiences of redness = a disposition to look like that There are, then, red looks, green looks, yellow looks, and so on. They are mind-independent dispositions of objects, and what they cause are experiences the phenomenology of which is determined by the property in question (redness, greenness, yellowness). But of course something can look red (green, yellow, etc.) to me when it isn’t, if the conditions are abnormal.

3.2 Metaphysical implications: between dispositionalism and primitivism The central import of the proposed model is that introducing appearance properties, that is, “looks,” allows us to invoke primitive dispositions. Let me elaborate on the metaphysical implications of a primitivist-dispositional account of color by distinguishing it from classic dispositional accounts, on the one hand, and from primitivist accounts, on the other hand. Recall that the underlying conditional, common to both primitivists and dispositionalists, claims that if colors are essentially connected to experiences of them, they are explanatorily redundant. The possibility of primitive dispositions proves the falsity of this conditional. It shows that the conditional is tempting only given that the dependence between colors and experiences of colors goes in one direction. The model of appearance

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properties displays a double-dependency between appearance properties and experiences. Appearance properties explain why, in gazing at the coin, we experience it as changing in some way: it is because something is, indeed, changing—the coin’s look. Nevertheless, the coin’s look is, and is experienced as, a property whose nature is to present itself, that is, as an appearance property. When the model is applied to colors, the experience of redness is dependent on and derived from redness, as Inheritance requires, and hence, not causally prior to redness. On the other hand, redness is not conceptually prior to the experience of redness, because we cannot understand what it is for an object to be red except in terms of its looking red. Since the priority and dependency of colors and experiences have different senses (namely, a causal sense, and a conceptual sense), there is no need for concern about explanatory circularity. Why, then, do dispositionalism and primitivism claim that if colors are essentially connected to experiences they cannot explain our experiences? Because this conditional is, I believe, ontologically biased. Dispositionalism accepts the consequence of the conditional, in order to downgrade the status of colors: a property that lacks explanatory power does not seem to be a genuine property. The upshot is that even a non-reductive account of dispositions does not undermine a “dualistic metaphysical scheme” according to which all properties are “either physical or mental, or some combination of the two” (McGinn 1996: 548). Primitivism rejects the consequence of the conditional, in order to upgrade the status of colors: our experiences do not constitute the nature of colors any more than they constitute the nature of shapes. The upshot is that colors are more than dispositions; redness is taken to be the categorical ground for objects’ dispositions to look red. On primitivist dispositionalism redness is manifested in the experience of redness, just as the elliptical appearance of the tilted coin is manifested in visually experiencing the tilted coin. Thus, primitivist dispositionalism denies ontological dualism. It acknowledges, with primitivism, “an extra layer for colors. . . . To the old question, ‘Are colors mental or physical, subjective or objective?’, we must answer, ‘Neither: they constitute a third category, just as real as, but distinct from, mental and physical properties’ ” (McGinn 1996, 548). Nevertheless, in contrast to primitivism, primitive dispositionalism maintains that the primitive property that is manifested on the surface of the apple is the apple’s red look, its red appearance. Although the apple’s red appearance is neither mental nor physical, or even a combination of physical and mental properties, it does not constitute a mysterious third category: all it takes for an object to be red is for it to look red.

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Acknowledging such sui generis properties might provoke the charge that primitivist dispositionalism collapses into either dispositionalism or primitivism. If colors are irreducible properties that explain our experiences of them, then there are two possible causes of our experiences: irreducible properties (primitive or dispositional), and their microphysical bases. Primitivist dispositionalism is either incompatible with physicalism, in insisting that irreducible properties are genuinely causal, or philosophically superfluous, in deeming them causally inefficacious. If primitivist dispositionalism is incompatible with physicalism, then it is no less extravagant than primitivism. If primitivist dispositionalism concedes that colors are causally inefficacious with respect to our visual experiences, then colors are explanatorily redundant, as is the case on classic dispositionalism. This dilemma is familiar from discussions of the over-determination of the mental, as are the proposed ways to resolve it. One suggested way out is to deny that the scientific explanation of color experience in terms of wavelengths excludes, overdetermines, or renders superfluous the simple explanation that the apple looks red because it is red. The scientific and the commonsense explanations have distinct explananda. Colors explain what the apple looks like, whereas invoking wavelength explains the visualneurological processes that underlie color experiences.23 A non-reductive account of colors, primitive or dispositional, is, then, committed to explanatory pluralism. The dilemma is false, then, because there is nothing mysterious in the idea of irreducible colors that explain our experiences of them. What makes primitivism mysterious is not the commitment to irreducible colors, but the claim that these irreducible colors are something over and above dispositions. If colors are more than dispositions, it is incumbent on primitivism to explain “in what a grasp of the supposed residue would consist” (Evans 1980, 273). Primitivists concede that there is no adequate non-sensory description of colors, as colors are primitive. So what is it for the apple to be red, that is, red-as-we-see-it, in the dark? In the case of shapes, it will be recalled, we can understand what it is for the apple to be round because we have a geometric account of roundness. But in what sense is an apple primitively red in the dark? “Berkeley’s ‘master argument’ ” (1713/34, }} 1.22–4) shows that if we try to imagine redness as persisting “out there,” e.g. in the dark, exactly as we see it, but without any connection to experience, 23 The two explanations belong to different “explanatory spaces” (Campbell, 1993, 182), in the same way that reasons constitute a genuine explanatory space with respect to actions, a space that is independent of the neurological explanations of complex physical movements. For alternative accounts of explanatory pluralism, see Yablo (1992), Child (1994), Bennett (2003), Benbaji (2010).

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we do not succeed in doing anything “but perceiving a color unperceived by anyone else” (Evans 1980, 272–4; for critical discussion, see Allen 2007). Primitivists may, switching on the light and pointing to the apple, insist that for an object to remain red in the dark is for it to remain like that. But as Evans notes, “switching on the light merely tests for the dispositional property; what could show whether or not objects did in fact retain these other colour properties [i.e. the primitive ones, H.B.] in the dark?” (1980, 273). The purported distinguishing feature that renders colors more than dispositions is mysterious—what could the “more” possibly be? Primitivist dispositionalism dissolves the mystery: the sui generis properties borne on objects’ surfaces are the objects’ dispositions to look colored.24

4 . D O C O L O R S LO O K N O N - D I S P O S I T I O N A L ? So far, I have sought to render intelligible the idea of primitive dispositions, utilizing the idea of an appearance property, e.g. someone’s looking old. Furthermore, I proposed that an object’s redness should be thought of as the property of presenting a certain appearance. What remains to be established is whether colors are, in fact, such appearances. For a critic might well accept my proposed analysis of seeing an appearance property, and with it, the possibility of primitive dispositions, but for phenomenological reasons, such as the objection that colors do not look like dispositions, reject the claim that colors are, in fact, such properties. This phenomenological objection is crucial, because it purports to undermine application of the appearance-property model to colors. If colors are dispositions, and look non-dispositional, experiencing them involves some error. The property revealed to us in experience cannot be redness, namely, the disposition we attribute to objects. Redness is not what it appears to be. Hence, MI is false: an experience of redness can be neither the manifestation of redness, nor derived from redness. To establish that colors are primitive dispositions, we need to refute the objection that colors look non-dispositional. The objection that we see colors as non-dispositional rests on two claims: the phenomenological observation that colors, like shapes, are experienced as simple properties of objects, and the metaphysical claim that dispositions are complex in nature, being comprised of relations between subjects and objects. Together, the two assumptions imply that we see colors as nondispositional. Let me present and respond to a few variants of this objection: 24

I develop this argument against primitivism in Why Color Primitivism (2016).

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that we see colors as monadic (section 4.1), constant (section 4.2), and categorical (section 4.3). I will argue that primitive dispositions are indeed monadic and constant, as they look, but that they neither are categorical, nor look categorical.

4.1 Monadic (i.e. non-relational) properties The first variant of this objection assumes that a disposition is a relational property, indeed, that the very idea of a “look,” and the invocation of the “to whom?” and “under what conditions?” provisos, suggest relativization to perceivers and perceptual conditions. An appearance property—the vivid redness of this tulip, say—would presumably not be the same for someone with binocular vision as it would for someone with monocular vision; the way things look to a hawk surely differs from the way they appear to us— and so on. Hence, redness is a relational property. The experience of a red object, however, is the experience of a non-relational, monadic property of the object’s surface. Colors look non-dispositional, since they look nonrelational.25 On primitivist dispositionalism, colors are not relations. “The apple is red” is not short for “The apple is red for S in C,” where “S ” refers to a species of perceiver, and “C ” specifies a set of visual conditions. Rather, it provides determinate answers to the questions “to whom?” and “under what conditions?”: to humans with normal vision, in daylight, in which case one can ascertain the colors of things by looking at them.26 Neither someone with monocular vision, nor a hawk, can see the apple as red. But human beings could disappear from the scene, and our notion of normal vision could also change; would this apple continue to be red, if we—humans with normal vision—were not around? More generally, can objects lose their color merely by changing their relations with other objects? It has been suggested that this question—“suppose that x is red; as we modify things other than x, and thereby modify the relations x bears to other things, will x (necessarily) continue to be red?” (Cohen 2009, 9)—is our pre-theoretical test for relational properties, and that dispositionalism is compelled to answer in the negative (9–12). By contrast, I argue that the apple would retain that look (pointing to the apple) even were we to disappear, because for the apple to be red is for it to look red to us, with our perceptual apparatus, under the actual conditions. The experience cannot be identified other than by actually pointing to the apple, and we 25

Johnston 1992, 142; McGinn 1996, 302. This characterization of perceivers and perceptual conditions is based on that proposed in McDowell 2004 and 2011. 26

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can only point to the color of the apple by using our actual apparatus, under the actual conditions. Hence, a primitive disposition is characterized “rigidly” (McDowell 2004, 399). Undeniably, it is only contingent that we have a visual system that provides for such uniformity and richness in our normal experience of colors. As McDowell says, “The contingent uniformity in chromatic visual sensations enters into the very constitution of the properties themselves. There is nothing to those properties . . . except being such as to look colored to subjects like us, subjects who come within the scope of a concept of normality in color vision only because there is that match in the chromatic looks that things present to them” (McDowell 2011, 10). But we are lucky enough to enjoy this uniformity in chromatic visual experiences, and given that we are so lucky, our perceptual apparatus reveals to us that the apple really is red, that is, it would continue to be red even had we never been around. I conclude that primitivist dispositionalism can be accused of anthropocentrism (Wiggins 1976, 107), but not relationalism: the apple has, over and above its physical properties, a red look that is not a relation to perceivers. However, for those who are convinced that the phenomenon of perceptual variation is widespread, at least with respect to some shades of colors, let me point to an intriguing non-relational explanation of the phenomenon, compatible with the account proposed in this paper. Kalderon develops a theory of color pluralism, according to which objects have multiple colors. Colors are not relations between object, perceiver, and perceptual circumstances; rather, the relations between these three factors determine the perceptual availability of colors. “According to the color pluralist, then, the relativist conflates the conditions for the perception of a color with the perceived color” (2007, 577). Although I cannot go into color pluralism here, it is consistent with primitivist dispositionalism (as it is with primitivism). Even though appearance properties are not relations to perceivers, it might be claimed that they are relations to the environment (Noë 2004, 149). Boghossian and Velleman argue that when one turns on the light in a dark room, “the colours of surrounding objects look as if they have been revealed, not as if they have been activated.” This implies that colors look non-dispositional, for otherwise, “they would seem to come on when illuminated, just as a lamp comes on when its switch is flipped. Turning on the light would seem, simultaneously, like turning on the colors; or perhaps it would seem like waking up the colors, just as it is seen to startle the cat” (1989, 85). The underlying assumption of this objection is the already familiar idea that we see the disposition by seeing its manifestations occurring : we see

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solubility by seeing sugar dissolving. I have already shown that this assumption is false with regard to appearance properties, since in the case of appearance properties, we do not see the manifestations—namely, the experiences—at all, so there is no reason to expect to see any occurrences. Redness, that is, the disposition to look red, manifests, that is, presents itself as a sustained property of the apple, not as an event that transpires when I “illuminate” the apple.

4.2 Constant properties Color constancy is, like shape constancy, a familiar, commonplace phenomenon. The white wall looks gray in the shade, but this is not an illusion of grayness; I can see that the wall does not change its color, as the coin does not change its shape when tilted. There seems to be a perfect parallel between constancy of shapes and constancy of colors. This parallel invites the distinction between (i) experiencing an object as red, and (ii) experiencing an object as having a red look. If there is such a distinction, as suggested by constancy, the model proffered here is actually giving an account of an object’s looking red, not its redness (just as the model is an account of the coin’s elliptical appearance, not of its shape). Redness would have to be more than its various “looks,” that is, it would have to be, not a primitive disposition, but primitive. Otherwise, if redness itself is nothing but an appearance, it cannot be the constant color that we see along with its various appearances. To explain this charge, consider a specific shade of red, red-215, say. This shade now, at dusk, seems dark and shadowy, whereas earlier in the day it was bright and bold. Yet we experience red-215 as one constant color that unites and explains its various appearances, just as the roundness of the coin is the property that unites and explains its various appearances (round from above, elliptical from an angle). The concern, presumably, is that on the dispositional account, red-215 is nothing beyond what it looks like under various circumstances, that is, it is merely a series of various more finely differentiated determinate appearances (Allen 2009; Kalderon, manuscript).27 The objection contends that the appearances are in constant flux, and the putative disposition is nothing but a set of changing appearances. However, having established that the disposition is a primitive property on the surface of objects, primitivist dispositionalism need not deny that the specific shade of color, red-215, is a constant property that unites and 27 To favor one very specific appearance, in a certain kind of lighting (say, the kind of bright daylight present here right now, at 2.45 p.m.), is completely arbitrary; red-215 has no such defining and privileged look. But see Allen (2010).

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explains its various appearances. First, it has to be shown that there is a fact of the matter as to whether a given shade is red-215. And we can indeed do so via a simple model. Following a suggestion by Broackes (2007, 59), we can group together as red-215 things that stand in a certain phenomenal equivalence relation, such as is indistinguishable in color from, to the master red-215 sample. Thus, I can check whether my bright red copy of the Tractatus is red-215 by holding it next to the sample to see if it is or is not distinguishable from it under various illumination conditions. Now if the upper half of the book cover is under illumination I1, and the lower half illumination I2, what unites these two appearances? The common factor is that both are the same shade of red, namely, the shade that is indistinguishable in color from the master red-215 sample (viewed under the same conditions). This is why I expect the following counterfactuals to be true: were the upper half of the cover viewed under I2, it would look exactly the way the lower half looks now; were the lower half viewed under I1, it would look exactly the way the upper half looks now. To experience the color of the cover as a constant color, over and above its various appearances, is to expect it to vary in appearance in “red-215 appropriate” ways, to use Noë’s expression (2004, 143), that is, in the phenomenally salient ways red-215 changes under different lighting conditions. To account for constancy, there is no need to assume a conception of color on which it is something over and above an appearance property. To sum up, I have argued that from the correct observation that colors are experienced as non-relational and constant, it does not follow that they look non-dispositional, because primitive dispositions are non-relational and constant.

4.3 Categorical properties Primitivists also argue that colors look categorical. This claim is essential to the regress argument against the non-reductive account of dispositions. According to the objection from infinite regress, the problem with the non-reductive dispositional account is that one “cannot see what colour an object has. For he cannot see that particular colour of an object except by seeing the particular way the object tends to appear; and he cannot see the way it tends to appear except by seeing the way it tends to appear as tending to appear; and so on, ad infinitum” (Boghossian and Velleman 1989, 90). This regress renders the content of the experience of redness completely vacuous (89). But, I would reply, why not stop the regress by invoking the relevant color by name—to appear red is to appear disposed to appear red— and put an end to the matter? Boghossian and Velleman concede that we do this for squareness (“objects can appear disposed to look square just by

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looking square”) but only because things look square “intrinsically and categorically” (89). The problem for the dispositional account is that when something looks red it looks “non-dispositionally red” (89). But if this is the objection, then it just restates the core objection addressed in this section, namely, the claim that colors look non-dispositional.28 The regress argument does not provide any reason for claiming that colors look categorical—it presupposes that they look non-dispositional, that is, categorical. Furthermore, “categorically” does not add anything to “intrinsically” (understood to mean non-relationally). Once we acknowledge a primitive non-relational dispositional property on the surface of objects, which we can point to, the regress can be preempted. I conclude that there is no reason to think that colors look nondispositional. This conclusion also serves another purpose: it constitutes a refutation of primitivism. I have already argued (in section 3.2) that primitivism has not articulated any coherent sense in which colors are more than dispositions. Having refuted the phenomenological charge, I can argue that there is no plausible motivation for making colors more than dispositions: there is no way to explain why more is needed. Primitive dispositions can provide for robust possibilities (e.g. the possibility that the apple will remain red even when we are no longer around); non-relational properties; and constancy.29 This covers the entire list of (metaphysical) reasons usually offered for the claim that colors are categorical properties; if there is no (phenomenological) reason to claim that colors look non-dispositional, there is no reason to take colors to be categorical.

5 . D O C O L O R S L O O K L I K E D I S P O S I T I O N S? Ironically, the response to the objection that colors look non-dispositional reinforces the phenomenological objection that they nonetheless do not look like dispositions. I have claimed that from the fact that colors, like shapes, are experienced as simple, constant, and monadic properties of objects, it does not follow that they are experienced as categorical, i.e. non-dispositional. However, if colors are experienced as simple, constant, and monadic, they seem to be on a par, phenomenologically speaking, with shapes. It does not appear to us that colors are dispositions, for they are not 28

For a different reply to Boghossian and Velleman, see Byrne and Hilbert (2011). Robust possibilities and causal explanation (Johnston1992, Campbell 1993); constancy (Allen 2009, Kalderon, manuscript); non-relational properties (McGinn 1996); direct intervention (Campbell 2006); categorical base (Johnston 1992, 147, Campbell 1993). 29

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presented in experience in a manner that differs in any way from the manner in which shapes are presented, and surely, it does not appear to us that shapes are dispositions. While the claim that colors look non-dispositional implies that the perception of colors involves an error—it deceives us as to their nature (colors are dispositions and present themselves as non-dispositional), the present objection is more modest. It merely raises the possibility that perception is neutral with regard to the question of whether colors are dispositions: we cannot tell, simply by looking at them, whether colors are dispositions. Colors do not look non-dispositional, but they do not look like dispositions either. The claim that “colors do not look like dispositions” tends to elicit a range of strong reactions, from “why would anyone think that colors look like dispositions?”30 to “how would they have to look to look like dispositions?”31 The more cautious and perhaps dominant response is to claim that the experience of perceiving color is silent about the question of whether colors are dispositions—that both dispositionality and nondispositionality are not visible properties at all.32 Granting—for the sake of argument—that colors are not generally perceived as dispositions—why should this undermine the dispositional account of colors? If a critic grants the previous point, that colors do not appear positively non-dispositional, it’s hard to see how the failure to appear dispositional should be taken as evidence for anything. Colors are presented to us in experience, as primitivism claims, but not as dispositions. Assuming that primitivist dispositionalism is not committed to Revelation, colors may well be dispositions though they do not look like dispositions, just as water can be H2O even though it doesn’t look like H2O. However, the core claim of the model for appearance properties adduced the fact that from an angle, we experience the coin as looking elliptical, not as being elliptical. The coin looks round and elliptical-looking. The model proffered for appearance properties implies, then, that an appearance property—an elliptical look, a youthful look, and the like—is a visible property. Appearance properties appear to us as they are, namely, as ways of looking. If we experience the coin as elliptical-looking, then perception is not neutral with respect to appearance properties, as it is regarding H2O. If an advocate of primitivist dispositionalism claims that colors are appearance properties, but do not look like appearance properties, she cannot simply invoke the neutrality of perception. The reply to the retort “so what if colors

30 31 32

McGinn 1983, Johnston 1992, Dancy 1986, Boghossian and Velleman 1989. Johnston 1992, 141. For a dismissive reading of the problem, see McDowell 1985, 112. Byrne 2001, Levin 2000.

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don’t look like dispositions?” is that if they do not, they might not be appearance properties! Let me close by pointing out some phenomenological data that supports the experiential disparity of colors and shapes, and thus reinforces the claim that colors are appearance properties (section 5.1). I then consider the implications of this disparity for Revelation (section 5.2).

5.1 Seeing colors as appearance properties I contend that the disparity between the phenomenology of colors and that of shapes is attested to by various experiences, and indeed falsifies the alleged parity between colors and shapes. Let me list the perceptual clues that, taken together, indicate that shapes are not presented in experience as dispositions, whereas colors are. (1) First and foremost, when I see something as square, I see it as having four equal sides and four right angles. When I see something as red, I see it as having that shade (pointing). What does it mean that the only way to capture the content of experience is by pointing to the color? It means, precisely, that redness is manifested in experience by looking red, that’s all. The fact that the geometric description of shapes is given to us in visual experience, whereas no description of color is given to us in visual experience, attests to the disparate natures of colors and shapes. The conceptual difference between colors and shapes expresses itself in the experience of these properties: colors are experienced as purely visual, shapes are not.33 (2) Consider the difference between Molyneux’s question and its counterpart vis-à-vis colors. Molyneux asks whether someone born blind, who later gains vision, would be capable of distinguishing between a square and a circle solely by sight.34 Answering in the affirmative makes sense, I would 33 It might be objected that it is just a matter of luck that when we see a square we see a figure with four equal sides, and there could, conceivably, be creatures who are able to tell squares from pentagons and circles just by looking, but cannot see that there is a foursided object out there (such a suggestion is put forward in Kulvicki 2007). I agree, of course, that a child might have a rudimentary capacity to track some visible pattern of squareness, that is, she might be able to pick out squares quite successfully from a group of circles and assorted rectangles. But this is not to say that she sees the pattern as squareness. Suppose we ask her, pointing to a rectangle, why it is not a square. She may not reply, “because it does not have four equal sides and four right angles.” But if she does not exhibit any grasp of the difference between the sides of rectangles and those of the squares, it is hard to ascribe to her anything more than sheer practical ability. What distinguishes the capacity to see something as square, as opposed to the more primitive ability to track a visible pattern of squareness, is an inkling, a modicum of comprehension that the shape has a certain geometric structure. 34 The original question is about a cube and a sphere.

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argue, precisely because there is a perspective-independent description in light of which the shape’s appearance is intelligible. Given the absence of such a description in the case of color, the parallel question—whether someone born blind who later acquires the ability to see would be able to distinguish red from blue on seeing them for the first time—cannot be answered in the affirmative. Now an affirmative answer to Molyneux’s original question means that the content of the visual experience of squareness is the same as the content of the tactile experience of squareness: the subject visually identifies the shape as that given to him in tactile perception of square things (as having four equal sides and so on). That the experience of shapes is cross-modal in this way illustrates that we experience shapes as more than “looks.” That the experience of colors is modality-specific indicates that colors are not more than “looks.” (3) A related indication of the cross-modality of shape-experience resides in the fact that to experience a shape, as well as its changing appearances, one must move oneself or the shape, and these movements are correlated to movements of tactile perception. Noë illustrates this correlation between visual and tactile experience of shapes with the following example: “if something looks square, then one would need to move one’s head in characteristic ways to look at each of the corners. One would have to move one’s hands the same way (at the appropriate level of abstraction) to feel each corner” (Noë 2004, 102). By contrast, movements of the observer and object will have no effect on the appearance of color. The correlation between movements of the perceiver or perceived object in the context of visual perception of shapes, and hand movements in the context of tactile perception, and the absence of such a correlation in the context of visual perception of colors, is a feature of the capacity to perceive shapes and colors. That is, it is inherent in our capacity to see shapes that they are accessible through touch. It is inherent in our capacity to see colors that they are inaccessible through other modalities. (4) When I see the coin from an oblique angle, I have no choice but to describe it as elliptical-looking, because it is not elliptical (see section 2.1). By contrast, I do not have to describe different experiences of the same color in terms of how the object in question looks. I can appeal to variations in illumination: the white wall is shaded here, and fully illuminated there. I can avoid the claim that the white wall looks gray in the shaded area by mentioning darkness or shadows. This isn’t to deny that it is also true that the wall does look gray in the shaded area. But in this case, we have an alternative description, which we lack in the case of shape. The fact that we can experience the variation as a change in illumination shows not merely that we take both the wall’s various appearances, and the constant color, as

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metaphysically on a par, but that we experience the constant color and its various appearances as “looks.” (5) To see something as having a certain property is a capacity entailing expectations with regard to the success and failure of our ascription of that property. The capacity to see something as having a certain shape entails different expectations than does the capacity to see something as having a certain color. For example, the capacity to see a figure as square entails the expectation that in cases of uncertainty, where we cannot tell, just by looking, whether the figure is a square or merely a rectangle, there is an effective method for resolving this uncertainty, namely, measuring the figure. By contrast, the capacity to see the shirt as green entails the expectation that in cases of uncertainty, where we cannot tell, just by looking, whether the color is green or teal, all we can do is improve the perceptual conditions: look at it in daylight, check it under non-fluorescent illumination, etc. To verify our perception of colors we only need to improve the conditions under which they are viewed; to verify our perception of shapes, we need to appeal to methods beyond how they look. One might object that these expectations set very high standards for being able to see that something has a shape or color. It seems implausible to argue that we don’t have the capacity to see squares unless we understand how to use protractors, for example, or that we can’t see that an object is red, unless we know (or at least “expect”) that fluorescent lighting can have strange effects. What about people who lived before fluorescent lighting was invented? Of course people who lived before fluorescent lighting was invented could see colors, but this is, in part, because they were “sophisticated” enough to recognize the distinction between normal and abnormal perceptual conditions, or what amounts to the same thing, the distinction between an object’s color and an illusion of that color.35 The capacity to see colors presupposes possible failures of the exercise of that capacity, which occur when there is a breakdown of normal conditions. Similarly, to have the capacity to see something as square, not merely to have a rudimentary ability to track some visible pattern of squareness, requires a grasp of what is involved in correcting misleading appearances beyond the way it looks. (1)–(5) are differences between what it is to see redness and what it is to see squareness. Although appearances are simple, constant, and monadic properties of objects, as shapes are, the experience of seeing colors is not the same as that of seeing shapes. Upon reflection, the parity thesis falsifies

35

See Sellars (1956, 48), the Myth of Jones.

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the phenomenology of experience: colors, unlike shapes, do look like dispositions, and more specifically, appearance properties.

5.2 Revelation? The conclusion that colors look like dispositions might seem to be an affirmation of Revelation. If colors are primitive dispositions, and their being primitive dispositions is given to us in experience, has not the whole nature of redness been revealed to us? Revelation raises two sorts of issues in my view. First, one might accept the model proposed here: in particular, one might accept that there are appearance properties; that we perceive them as appearance properties; and that these appearance properties are dispositions. Nevertheless, one might argue that it does not follow from these claims that we see appearance properties as dispositions. Recall, however, what it is to see an appearance property. To see the coin as looking elliptical is to experience it as a property of an object that is such as to look a certain way (elliptical) when viewed from this angle (when tilted). This exhausts the notion of seeing a property as a visual disposition: to see it as a property of objects that is such as to look a certain way under certain conditions. In other words, there is no gap between seeing something as looking F and seeing it as being disposed to look F.36 The second concern regarding Revelation is the thought that science could discover the physical basis for dispositions, and since a disposition is, by dint of its dispositional nature, constituted by a categorical physical base, part of its nature remains hidden. If having a physical base is part of the nature of primitive dispositions, then Revelation must be qualified. However, proponents of primitivist dispositionalism might concede that primitive dispositions are necessarily grounded in physical categorical bases, and yet refuse to take this fact to be part of the nature of colors.37 In this case, Revelation might indeed be true. So although it’s not obvious whether Revelation is true, I suggest a way in which it might be. I have argued for both dispositionalism and the irreducibility of colors to any physical properties of objects; and also that colors

36 For those unconvinced by this reply, let me qualify it as follows: acknowledging that there are appearance properties, that they are perceived as such and that they are dispositions, is all I need to establish the possibility of primitive dispositions and their manifestation in experience. 37 For a conception of Revelation as knowledge of things, in the spirit of Russell’s notion of acquaintance, see Campbell (2006). Gert (2008, 142) suggests a qualification of Revelation that seems to be warranted on my account as well.

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appear to be dispositions; from which it follows that Revelation can be true. Rather than simply assuming its truth, however, this paper has endeavored to argue for it. This vindication of Revelation completes my project of showing how it is possible to fulfil the promise of primitivist dispositionalism. Recall the dream of Eden, where the apple was wholly revealed in experience as “gloriously, perfectly, and primitively red ” (Chalmers 2006, 49). I have shown that colors can, indeed, be gloriously and perfectly revealed in experience—as dispositions, or as I called them, “looks” of physical objects. Hagit Benbaji Ben Gurion University

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McDowell, J. 1985. “Values and Secondary Qualities,” in T. Honderich (ed.), Morality and Objectivity (Boston: Routledge & Kegan Paul), 110–29. McDowell, J. 2004. “Reality and Colours: Comment on Stroud,” Philosophy and Phenomenological Research 68(2), 395–400. McDowell, J. 2011. “Colors as Secondary Qualities,” in J. Bridges, N. Kolodny and W. Wong (eds.), The Possibility of Philosophical Understanding: Reflections on the Thought of Barry Stroud (Oxford: Oxford University Press). McGinn, C. 1983. The Subjective View (Oxford: Oxford University Press). McGinn, C. 1996. “Another Look at Colors,” Journal of Philosophy 93(11): 537–53. Macpherson, F. 2006. “Ambiguous Figures and the Content of Experience,” Noûs 40(1): 82–117. Martin, M. G. F. 2010. “What’s in a Look?” in B. Nanay (ed.), Perceiving the World (New York: Oxford University Press), 160–225. Miscevic, N. 2007. “Is Color-Dispositionalism Nasty and Unecological?” Erkenntnis 66(1–2): 203–31. Noë, A. 2004. Action in Perception (Cambridge, MA: MIT Press). O’Shaughnessy, B. (2000). Consciousness and the World (Oxford: Oxford University Press). Peacocke, C. 1984. “Color Concepts and Color Experience,” Synthese 58(3): 365–82. Peacocke, C. 2008. “Sensational Properties: Theses to Accept and Theses to Reject.” Revue Internationale de Philosophie 62: 7–24. Pettit, P. 2003. “Looks as Powers,” Philosophical Issues 13: 221–52. Phillips, I. 2006. “Perception and Context,” paper delivered at the NPAPC, Warwick, July 2006. Russell, B. 1912. The Problems of Philosophy (London: Oxford University Press). Ryle, G. 1949. The Concept of Mind (London: Penguin). Schellenberg, S. 2008. “The Situation-Dependency of Perception,” Journal of Philosophy 105(2): 55–84. Sellars, W. 1956. “Empiricism and the Philosophy of Mind,” in H. Feigl and M. Scriven, eds., Minnesota Studies in the Philosophy of Science, volume 1, The Foundations of Science and the Concepts of Psychology and Psychoanalysis (Minneapolis: University of Minnesota Press), 253–329. Shoemaker, S. 2006. “On the Ways Things Appear,” in Gendler and Hawthorne 2006, 461–80. Siewert, C. 2006. “Is the Appearance of Shape Protean?” Psyche 12(3): 1–16. Smith, A. D. 2002. The Problem of Perception (Cambridge, MA: Harvard University Press). Watkins, M. 2002. Rediscovering Colors: A Study in Pollyanna Realism (Dordrecht: Kluwer). Watkins, M. 2010. “A Posteriori Primitivism.” Philosophical Studies 150(1): 123–37. Westphal, J. 1987. Colour: Some Philosophical Problems from Wittgenstein (Oxford: Blackwell). Westphal, J. 2005. “Conflicting Appearances, Necessity and the Irreducibility of Propositions About Colours,” Proceedings of the Aristotelian Society 105(2): 219–35.

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Wiggins, D. 1976. “Truth, Invention, and the Meaning of Life,” reprinted in his Needs, Values, Truth: Essays in the Philosophy of Value (3rd edn) (Oxford: Clarendon, 1998), 87–138. Yablo, S. 1992. ‘Mental Causation’. Philosophical Review 101(2): 245–80. Wiggins, D. 1987. “A Sensible Subjectivism?” reprinted in his Needs, Values, Truth: Essays in the Philosophy of Value (3rd edn.) (Oxford: Clarendon Press, 1998), 185–214.

7 Are There Ineffable Aspects of Reality? Thomas Hofweber 1. INTRODUCTION Should we think that some aspects of reality are simply beyond creatures like us, in the sense that we are in principle incapable of representing them in thought or language? Or should we think that beings with a mind and language like ours are able to represent every truth and every fact? In other words, should we think that some truths are ineffable for us: beyond what we can think or say? Whatever the answer is, it likely has substantial consequences. If it is no, i.e. no truth is ineffable for us, then this might shed light on what reality is like, or what our minds are like, or why the two match up so well. If the answer is yes, some truths are ineffable for us, then this might affect our attempts to understand all of reality. In particular, it might affect the project of metaphysics and its ambition to understand all of reality in its grander features. If we had reason to think that only a limited range of facts can be represented by creatures like us then this might give us reason to think that metaphysics in its ambitious form is beyond what we can hope to carry out successfully. In this paper I will argue that the question whether there are any ineffable truths or facts is an important, although somewhat neglected, question whose answer has significant consequences, and I will make a proposal about what the answer is, on what this answer depends, and what follows from it. The paper has four parts: first, I will clarify what is at issue and make the notion of the ineffable more precise in several ways. Second, I will argue that there are ineffable truths using several different arguments. These arguments will rely on a certain hidden assumption which is almost universally made implicitly and accepted by most when made explicit, but which I will critically investigate later in the paper, in part four. A third part will attempt to answer a puzzle about the ineffable connected to the relationship between effable and ineffable truths and why the ineffable seems to be more hidden from us than would be suggested merely by the fact that it is ineffable.

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The solution to this puzzle will make clear how the ineffable is significant. I will argue that it has important consequences for metaphysics in particular, which suggest modesty instead of ambition. In the fourth part I will have a closer look at a hidden assumption that was relied upon until then, but which might well be mistaken. I will argue that we have good, but not conclusive, reason to think that this assumption is indeed false, and if so then everything changes. On the natural way in which this assumption is false we get no ineffable truths, no modesty in metaphysics, but a form of idealism instead. I will try to make clear that the resulting form of idealism is coherent, significant, and quite possibly true. Whether the crucial assumption is, in the end, true I won’t be able to settle here, but we can see that there will be important consequences either way. I will pick sides at the end. But before we can get to all this we need to get clearer about what is at issue.

2 . W H A T I S T HE Q U E S T I O N ? The ineffable naturally appears as a possibility when we think about the relationship between what reality is like, on the one hand, and what we can truly say, on the other. The relationship between these two leads to one unproblematic (for present purposes) area of overlap, and to two more mysterious outlying areas (see Figure 7.1). The area of overlap is a true description of reality: we can truly say something and reality is like that. Although much can and has been said about how this is to be understood in more detail, I will leave it untouched here, since my concern is with the two more problematic cases: first whether Correct description The ineffable

The non-descriptive

What reality is like

What we can say truly

Figure 7.1 The location of the ineffable

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what we can truly say goes beyond what reality is like, and second whether what reality is like goes beyond what we can truly say, which is our main topic. The first possibility might seem incoherent. How could what we can truly say go beyond what reality is like? If we said it truly then how could reality not be like what we said? Those who think that this option is coherent generally maintain that it only seems incoherent to us because we mistake it with something else: either that reality is different from how we say it is (and thus what we say should be false) or else that we say something truly about something other than reality (which isn’t an option, since reality is all-inclusive). Instead, they hold, this option is coherent, since we can say something truly that isn’t descriptive at all, neither of reality nor anything else. Some parts of speech aim to describe, while other parts aim to do something else, for example express an attitude of the speaker. Truth applies to both, and thus we can say something truly that goes beyond what reality is like. It is true, but doesn’t aim to describe reality, and thus reality isn’t required to be as described for it to be true. Expressivism about normative discourse combined with minimalism about truth is a paradigmatic instance of this approach. The question, of course, still remains whether it indeed is coherent, but since we will not focus on this outside area in our Venn diagram, we don’t have to settle this here. This part of our diagram will only have a minor role in what is to come. My main concern is the other outside area: parts of what reality is like that outrun what we can truly say. This is the ineffable, that which we can’t say. Here there should at first be no question about its coherence, but there is a real question about whether there is anything which is ineffable. Is there anything that reality is like that goes beyond what we can say, and thus say truly? If so, how much of it is there? Is it merely a little sliver at the edge of the overlap, maybe something related to the paradoxes, or to consciousness, or is it a vast area, maybe most of what reality is like? What would follow for inquiry in general and philosophy in particular if there were a large area of the ineffable? To make progress on these questions we will first need to clarify the relevant notion of the ineffable, and how this problem is different from a number of other problems in the neighborhood. These problems are also real and interesting problems, but not the ones I am trying to make progress on here. My discussion here will focus on a notion of the ineffable that is most promising for it being metaphysically significant, in that it captures the sense in which it just might be that minds and languages like ours are not good enough, in principle, to represent some aspect of reality, and therefore are not good enough for carrying out an ambitious project of metaphysics. Whether or not we are limited in this way is what I hope to find out. And to do this we should put aside some issues that I will not try to resolve and focus in on the relevant ones instead.

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Ineffable feelings. First, there is a common use of “words are not enough” to articulate the limits of language, but these are not the limits I am concerned with here. Take, for example “I can’t tell you how happy I am to see you!” “Words are not enough to say how glad I am to see you!” It would be beside the point to answer “Are you very, very happy, or even happier than that?” It is not that the first speaker has a degree of happiness such that no words can pick out that degree of happiness. After all, “I am maximally happy” would certainly be good enough to do that. Where words give out is not in describing the degree of happiness, but rather in giving the hearer a sense of what it feels like to be that happy. Words are not enough in getting the hearer to feel the way the speaker feels, or at least give them a sense of what such a feeling is like. But they are enough for describing how happy someone is: very, very happy. That words are not enough to transmit feelings in this sense is notable, but not a limitation of language in capturing what reality is like. Words might also not be enough to get you on the last flight to Raleigh, in the sense that no matter what words you utter, you won’t get on that flight. This limitation of language is not one in its descriptive power, but in its limited effects to produce feelings or get an airline seat, a limitation I can happily accept and which isn’t my concern here. What I am concerned with here is whether there are any facts, any truths, or any true propositions, such that we cannot, in principle, state or represent these facts, truths, or true propositions in our language. Ineffable objects. The notion of the ineffable is often tied to objects, and as such it is seen as problematic and paradoxical. An ineffable object is usually understood in one of two ways. It either is one that we can’t talk about at all, or it is one about which nothing can be truly said. An example of the former is sometimes taken to be God when God replies to Moses’ question about what his name is with “I am who I am,” and leaves it more or less at that.1 One possible lesson of that is that God can’t be named, although this seems somewhat incoherent, since I just named God with “God.” It is God, after all, who is supposed to be unnameable. Another lesson might be that God shouldn’t be named, which wouldn’t make God ineffable, of course, just normatively out of the naming game. It wouldn’t be a limitation on our representational capacities, just on how we should employ them. On the other conception of ineffable objects, as ones about which nothing can be truly said, it also is generally taken to lead to a paradoxical conclusion.2 After all, can’t we at least truly say about ineffable objects that nothing can be truly said of them? In this sense ineffable objects can be tied 1

Exodus 3: 13–15. For a discussion of the ineffable in that sense, and a form of an embrace of the apparent paradoxes, see Priest (2002). 2

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to our main concern: ineffable truths or facts. If nothing can be truly said about an object o then any fact involving o should be ineffable. But the latter, ineffable facts, are not paradoxical. It is not required that nothing can be truly said about ineffable facts, only that ineffable facts can’t be effed, that is, one can’t utter a sentence such that this sentence expresses, states, or represents, that fact. I can’t state the fact in question, but I might well be able to say true things about the fact, including that I can’t state it. Ineffable facts or truths are not paradoxical, although ineffable objects, that is, objects about which nothing can be truly said, do seem to be paradoxical. An ineffable fact is simply a case where what is true outruns what we can truly say. It is not a paradox, but whether this is ever so is our concern here. A gap between language and thought? A third topic connected to the ineffable is the question whether there are certain facts or propositions that one can think, but one can’t say. That is, are there certain contents that our thoughts can have, but there is no utterance of a sentence that has that same content? Some think that there are. One candidate for this are thoughts that involve phenomenal concepts. Maybe such concepts allow us to think thoughts that we can’t put in language. Another, more traditional, example is a version of mysticism. According to it we can attain insights by various means like fasting or mediation, but we can’t communicate them to others after we achieve them. These insights are not supposed to be feelings, but instead have propositional content. They are thoughts with contents that can be true or false. However, due to the nature of these contents they cannot be put into language. Although the mystic can think a thought with that content, they can’t communicate it with language. You have to meditate/fast/etc. to gain that insight. Whether either one of these cases obtains is controversial, of course, but this controversy does not matter now. I am not concerned with whether there are some limitations of language that are not limitations of thought. Instead I am concerned with whether there are certain facts or truths that are simply beyond us in either way, be it thought or language. I want to find out whether there are truths that we cannot represent at all, be it in language or be it in thought. Thus from now on I will take the ineffable to be that which we can neither think nor say. Whether there is a gap between language and thought thus won’t matter for what is to come, interesting as the question is otherwise. Conceptual representation vs. other representation. Our issue here is not whether we can represent everything in some way or other, but rather whether for every fact or truth we can have a conceptual representation of that fact or truth. It might well be that something ineffable is going on right over there, and I could pull out my camera and take a picture of it, and thus represent it in some way. The issue is not whether I can always do that, but

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rather whether there is a conceptual representation of every fact or truth. Conceptual representations are paradigmatically the kind of representation we have in thought or language. The ineffable concerns the limits of conceptual representation, not the limit of representation more generally. Fine vs. coarse contents. If the proposition that I am hungry now is different from the proposition that TH is hungry at t then this truth likely is ineffable for everyone but me right now. You would only be able to express it if you were me and even then only at that particular time. This would make these truths ineffable, not because minds like ours can’t represent them, but because of who and when you need to be to represent it. If contents are that fine-grained then it is trivial, but insignificant, that there are many ineffable truths. To get a more interesting question we should see whether there are still ineffable truths even if we consider contents more coarse-grained where perspectival elements like who, when, and where, you are do not matter. We will thus from now on assume that contents are coarse-grained enough so that perspectival elements don’t matter, or alternatively, we consider something ineffable if it can’t be represented no matter who, where, and when, you are. To focus on coarse-grained contents in the following is not to take sides in the debate whether contents are best taken to be fine-grained or coarse-grained. It is rather to take sides on the question what the proper notion of the ineffable is for which we should find out whether there are any ineffable aspects of reality. If we use a notion of the ineffable tied to fine-grained contents the answer is clearly that there are ineffable truths, but that answer doesn’t have any interesting consequences, it is simply guaranteed by how fine-grained contents are. The interesting question remains whether there are ineffable truths when considering a notion of the ineffable tied to coarse-grain contents. That question is not trivial, and has the potential to lead to substantial consequences. We will thus consider contents to be coarse-grained in the following. De facto ineffable vs. completely ineffable. We need to be clearer on how the “can” in “can’t be thought or said” should be understood when we consider the ineffable. It is uncontroversial that there is a sense of “can” such that there are some facts or propositions that we can neither think nor say. But whether we are also limited in a more permissive sense of “can” is controversial and a harder and more interesting question. To illustrate the difference, take the complete sand-metric of planet earth: the precise distance that every grain of sand on earth presently has to every other one. Since there are about 1024 or so grains of sand on earth this is an incredibly complex fact. No human being will ever be able to say or think that content. But this is merely due to a limitation of resources, in particular time. Since we have a short lifespan we will run out of time before we will be done to think or say that truth. It is, as we can call it, de facto ineffable. We can

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represent every part of it, in that for each pair of grains of sand we can say what their distance is. The whole, ineffable fact is just a conjunction of many effable facts. This fact is beyond us in the sense that it is too long and complex, but it is not beyond us in the sense that our representational capacities are not suitable to represent it in principle. If we had more time we could do it. We can thus distinguish the de facto ineffable, which is what we can’t think or say, from the completely ineffable, which is what we can’t think or say even with unlimited resources like time and memory. Some contents we can’t think since we are limited on a certain scale. The question remains whether there are any that we can’t think in principle, even if we allow ourselves unlimited resources on this scale. How to make this precise is, of course, not completely clear, but the examples of limited time and memory are clear ones that give us a limitation on a certain scale, and there might be other similar ones. The real question for us here is whether we are also limited in other ways, in ways that overcoming limitations of time and memory won’t help. Are there contents that we are simply incapable of thinking or saying, in principle, in that a mind like ours just can’t represent them conceptually? Are there facts that are simply beyond creatures like us, even with unlimited time to say or think them? This is the question I hope to make progress on. Thus from here on, when I wonder whether there is anything ineffable I will thus ask whether anything is completely ineffable in the above sense. The completely ineffable is the notion of the ineffable of interest here. As will become clear below, this won’t settle what should count as something that we can in principle do. Here that notion can be made precise in various ways, leading to various more precise notions of the ineffable. All of them are equally good notions, but we need to focus on the one that leads to the most interesting and most significant question about whether or not there are ineffable facts. We will revisit this issue below. Ineffable for whom? The issue I hope to make progress on is not one about what can be represented in language in principle, but instead is what we human beings can represent. It is about whether the world outruns our representational capacities, the ones we can employ. We won’t be concerned here with whether there could be a language suitable for other creatures that captures everything, or whether other creatures could think everything. Our topic is whether we can capture everything. Naturally, we should be concerned with whether anything is in principle ineffable for us. And this question is not about what language in general can represent, or what we could do if we were gods, with completely different minds, but what we, the kinds of creatures we in fact are, can do. Aspects. I stated the main question as whether there are ineffable aspects of reality. This should not be taken as indicating that the issue is whether

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reality as a whole has some ineffable feature. I could have also asked whether there are certain parts of reality that are ineffable, but this might have suggested that some spatiotemporal part is full of ineffable facts. My concern is whether any truths or facts are ineffable. If the answer is “yes” then presumably it won’t just be a single instance. Any one ineffable fact will be closely tied to others, and such ineffable facts considered as groups of connected facts can be seen as giving us an ineffable aspect of reality. Nothing should hang on that terminology, though. Ineffable vs. unknowable vs. incomprehensible. Finally, we need to distinguish the ineffable from the incomprehensible or unknowable. It is unknowable, I take it, whether the number of grains of sand on planet earth on February 18, 1923, was odd or even. But it is not ineffable. I can think the thought that it was odd on that day, and the thought that it was even. Anything that is ineffable is unknowable, given standard assumptions about knowledge involving at least a representation of what is supposed to be known, but not the other way round. Similarly, some things might be beyond what we can understand or comprehend, but they are not thereby ineffable. It might be incomprehensible why there is anything at all, but it isn’t ineffable that there is anything at all. The ineffable is simply concerned with what we can represent. It is not an epistemic notion, but one about our representational capacities.3 Our question thus is this: are there any facts, truths, or true propositions, such that we cannot, in principle, represent them in either thought or language, even given arbitrary resources like time and memory, and even when we individuate facts and propositions coarsely enough to leave aside perspectival limitations? Since this is a yes–no question there are two possible answers. We need to find out which one is the right one, and what follows from it.4 If the answer is “no” then all aspects of reality are effable for us, and the following effability thesis is true: 3 Others are concerned with the ineffable in these other senses. See, for example, Moore (2003a, 2003b). Colin McGinn proposed that the reason why we make no progress in philosophy is tied to our cognitive limitations, but it is not clear whether his position is best understood as being tied to a limit of what we can represent, or instead a limit of what we can understand. See McGinn (1989) or (1993). 4 It could be that although one answer is true in letter, the other is true in spirit. Maybe there are some isolated facts tied to the paradoxes which can’t be represented in any thought or language, but we can represent the rest. One unsuccessful way to argue for this is to consider the fact that for some objecto, nothing about o is ever represented. That fact about o can’t be represented without failing to obtain. And there certainly can be some objects o about which no fact is ever represented. But this doesn’t show that this fact can’t be represented, only that when it is represented then it won’t be a fact any more. The content that nothing about o is ever represented is perfectly representable by us, even if we never do represent it. We can represent it, and in those counterfactual circumstances it is a false proposition, while in actuality it is a true proposition. The limits of what can be represented are not that easily drawn. Thanks here to A. W. Moore.

132 (1)

Thomas Hofweber The effability thesis. Everything is effable.

If the answer is “yes” then some aspects of reality are ineffable for us, and so the ineffability thesis is true: (2)

The ineffability thesis. Something is ineffable.

As we saw above, what is at issue is whether any fact, proposition, or content, is ineffable, or whether all of them are effable. And that is to ask whether it is true that for every proposition p, there is a speech act we can perform, or thought we can think, that has p as its content. A proposition is effable in speech, we can say, just in case some utterance that we can make has that proposition as its content. And for that to be true there has to be some sentence and some context in which we might utter the sentence, such that this utterance of that sentence in that context has the proposition p as its content. A proposition is effable in thought, correspondingly, if we can have some propositional attitude, a judgment or a belief, that has that proposition as its content. In either case, effability in thought or language, we would need some representation of the proposition, either mental or linguistic, that has the proposition as its content, in the particular context it is employed. Whether this is so for all propositions is what we need to find out. I will argue that we should think that the ineffability thesis is true, at least granting widely shared assumptions.

3. IN SUPPORT OF THE INEFFABLE Leaving aside those who hold certain unusual views, to which we will get later, everyone should believe that there are ineffable facts in our sense. But few live up to what follows from that, in particular for metaphysics. In this section we will see a number of good arguments for there being ineffable facts, and we will look at what we can say about what such facts or truths are like. It might be tempting to say that there can be no good argument given by us for there being an ineffable fact, since such a fact could not be specified by us, and thus no example of such a fact could ever be given. Although it is true that we cannot give an example of such a fact, in that we can never truly say that the fact that . . . is ineffable, there are nonetheless a number of powerful arguments that there are such facts. We can argue for there being certain things without giving examples of them, but instead by general arguments that make it reasonable to accept that there are such things nonetheless. The most important arguments for this seem to me to be the following.

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3.1 Built-in cognitive limitations The thoughts we can think must fit into our minds. And our minds think a certain way; they have a certain cognitive setup. So any thought we can think must have a content that a mind like ours can represent. But our minds didn’t develop with reality as a whole as their representational goal. They developed to deal with situations that creatures like us have to deal with to make it: midsize objects that are reasonably stable and have stable properties, some of which need to be eaten, some of which need to be avoided, and so on. The question is why we should think that a mind that developed to deal with problems in this limited situation and under those selection pressures should be good enough to represent everything there is to represent about all of reality. We know that not all of reality is like the world of stable midsize objects. The very small is very different than that, for example. We should thus expect that a mind that has developed like ours won’t be suitable for all of reality. Our biological setup imposes a constraint on how we must think. This constraint arose in response to selection pressures that came from a special kind of an environment. We can expect minds like ours to be good enough to deal with the situation they evolved to deal with, but not good enough to deal with any situation whatsoever. Thus we should expect that our mind has a hardwired constraint on what it can represent.5

3.2 The argument from analogy Although we can’t give an example of a fact ineffable for us, we can give examples of facts that are ineffable for other, simpler creatures. Take a honeybee, which can represent various things about its environment like where the nectar is, but it is in principle incapable of representing that there is an economic crisis in Greece. Its mind is just not suitable to represent such facts, even though it can represent other facts. That there is an economic crisis in Greece is a fact ineffable for the honeybee, but not for us. But we can imagine that there are other creatures that relate to us like we relate to the honeybee. We can imagine that there are vastly superior aliens or gods, say, who look down at us like we look down at the honeybee. And analogously, they would say that we humans can’t possibly represent that p, while they clearly can. They would be able to give examples of facts that are ineffable for us, but not for them. Whether or not there really are such 5 See Fodor (1983), 119ff; Nagel (1986), 90ff; and Chomsky (1975). Chomsky’s views on this matter are more carefully discussed in Collins (2002), which contains many references to particular passages of Chomsky’s work.

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aliens or gods doesn’t matter for this argument. The point simply is that this analogous reasoning makes clear that we should expect there to be such facts that could be mentioned by the aliens or gods as ineffable for us, but not for them. The facts are there, whether the aliens or gods are there or not.6 It might well be that the honeybee’s representations are not conceptual representations and thus might not have propositional contents at all, but merely indicate something about the world. They might carry information, but not have propositional content.7 But this doesn’t really change the situation. Just as we can point to information that the honeybee can’t carry with its representational capacities, aliens might point to propositional contents that we can’t represent conceptually, besides higher forms of representation that they have in addition. In the end we should think of what can be represented by us as being somewhere on a scale, with the honeybee on one side of us, and other creatures, real or imagined, on the other. And what those further over on the scale can represent is ineffable for us, and thus we should think that some aspects of reality are ineffable for us.

3.3 Cardinality arguments Our language is built up from finitely many basic vocabulary items with finitely many ways to combine them to give us a sentence, which has to be of finite length. Thus overall we have countably many sentences available to represent reality. Similar considerations at the level of concepts support that we can have at most countably many types of thoughts. But there are uncountably many facts or propositions to be represented. Here is a simple argument: for every real number r there is the fact that r is a real number. For different real numbers these facts are different, and since there are uncountably many real numbers there are uncountably many facts to be represented. Although such cardinality arguments are very powerful, the simple outline given above is a bit too quick. Although it is true that there are only countably many sentences in our languages, this does not guarantee that we can only represent countably many facts. We can use the same sentence to represent different facts on different uses of this sentence, as with sentences that contain indexicals or demonstratives like “I’ll have another one of those.” But while this is correct, it is not clear how this would help even with the simple argument using the real numbers. Demonstratives and context sensitivity don’t seem to help much in referring to real numbers, and that was only the most simple and straightforward cardinality argument. More generally, we can argue that whatever the cardinality of the set of effable facts might be, we can take some set S of larger cardinality, and then consider for every a which is a member of S, the fact that a is a member of 6

See Nagel (1986), 95f.

7

See Dretske (1981).

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S. There are just as many such facts as there are members of S, and thus most of these facts must be ineffable. Overall then we should thus expect that the cardinality of facts we can represent is smaller than the cardinality of all facts.

3.4 Explaining effability If nothing is ineffable then the effability thesis is true and thus what facts obtain and which facts we can represent are exactly the same. But those are two very different things which would then exactly coincide. What we can represent is one thing, what reality is like is at first a quite different thing. If they coincide then we should ask for an explanation of why these two things coincide. It is conceivable that they do coincide. Maybe our representational capacities have reached the limit of what can be represented. Maybe we just made it to the top, while other creatures, including our ancestors, were still on the way up, unlikely as all this may be. Maybe the world is simple enough so that we can represent all of it. Maybe we got lucky and are able to represent every fact. The effability thesis could be true, by accident, but does it have to be true? If the effability thesis is true, then we should ask for an explanation of why it is true. And if no such explanation is forthcoming we should expect that it isn’t true. We shouldn’t expect that two different things coincide, and without an explanation of the effability thesis we should expect it to be false. But what could explain that what reality is like and what we can represent about it coincide? The most natural way one might try to explain why what reality is like and what we can represent coincide is via a connection of what reality is like and our representational capacities. One route for such a connection is of limited use: what reality is like affects what we can represent. This route can explain why our representations are sometimes accurate, but not why they exhaust all of reality. That reality affects and forms our representational capacities makes plausible that we sometimes represent correctly, but it doesn’t explain why we can represent all of reality. The other route is more promising here: what reality is like is affected by our representational capacities, in particular, what there is to represent about reality is due to us and our minds. This is a version of idealism, and it is in a sense the natural companion of the effability thesis. Reality is guaranteed to be effable by us in its entirety, since we, in particular our representational capacities, are responsible for what there is and what it is like. No wonder our minds are good enough for all of reality, since reality somehow comes from our minds. Idealism could in principle explain why the effability thesis holds, but we have good reason to think that this form of idealism is false. That is, we have good reason to think that reality does not depend on us in the alluded to sense: what there is and what it is like does not depend on us in a natural sense of dependence. There would have been electrons and they would be

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like what they are in fact like even if we wouldn’t have existed, and so electrons and what they are like don’t depend on us. Furthermore, there was a time before there were human beings when reality was otherwise pretty much as it is now. So, in a natural sense of dependence, reality doesn’t depend on us globally. These are simple and possibly naive arguments, but if idealism should explain why the effability thesis holds it will need to be spelled out in a way that makes sense of a dependence of reality on us that can support and explain the effability thesis. There certainly are options on the table. One could try to analyze the content of statements of dependence in a way that makes them acceptable to idealism. Or one could develop the idealism in a way that places us in some sense outside of time, and connect time and the temporal aspect of reality to us as well. One version of this is well known (Kant 1781), but it is not clear whether it is coherent, what a coherent formulation would look like, and whether it is compatible with other things we take ourselves to know to be true. An idealist explanation is in principle possible, but ones along the lines outlined above seem to have little going for them. The idealist strategy to explain the effability thesis outlined above in effect connects two versions of idealism. Of those two one is reasonably taken to be false, and the other is closely tied to our main topic. To introduce some terminology, let us call ontological idealism the view that what there is depends on minds, in particular our minds, in a sense to be made more precise. Let us call conceptual idealism the view that what in principle can be truly said or thought about reality, what the range of the conceptual or propositional is that can be employed in principle to apply to reality, depends on us, in a sense to be spelled out. Conceptual idealism is in essence the view that the effability thesis holds not by mere accident, but for a reason tied to us. Conceptual idealism combines the effability thesis with a certain explanation of why it holds. Ontological idealism could support conceptual idealism. If what there is depends on us then one way this could be would tie what there is and what it is like to our representational capacities, in a way that would guarantee the truth of the effability thesis. But ontological idealism is false, or so we have good reason to think. The question remains whether conceptual idealism is nonetheless true, for other reasons. Could conceptual idealism be true even though ontological idealism is false? For that to be so the effability thesis has to be true, and it has to be true for a certain reason, not just by accident. So far we have seen no reason why that should be so; to the contrary, we have seen that there is little hope to explain why the effability thesis might hold. However, we will revisit this connection below, in section 5.8, where this possibility is seriously explored. But without such reasons being on the horizon so far, we should side with the ineffability thesis and accept that reality outruns what we can represent about it.

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3.5 The sources of ineffability We have seen the outlines of four strong arguments for there being ineffable facts. Three direct ones, and one indirect one, via there being no good explanation why the effability thesis should be true. The smart money is thus on there being ineffable facts, even though we cannot give an example. Before we can see what follows from this we should think a bit about what such facts might be like, and how they might be similar or different from effable facts in various ways. The easiest way to approach this question is to think about how our representations of facts might be limited. And the easiest way to do that is to think about the paradigmatic way in which we represent the world: with a subject–predicate sentence, representing an object having a property. How might such a way to represent the world be limited? There are three ways in which we might be limited with such representations: (1) we might not be able to represent a certain object; (2) we might not be able to represent a certain property; or (3) the structure of a subject–predicate sentence might not be suitable to represent a certain truth or fact. Let’s look at them in turn to see which ones are the likely sources of our limitation. Missing objects. We might be unable to talk or think about some objects. To illustrate, let’s consider one way this could be, namely that singular thought and singular reference require a causal connection between us and the object we think or talk about. This gives rise to two possible limitations: objects that are not causally efficacious at all, and objects that are causally efficacious in general, but that are not causally related to us. Focusing on the second case first, we can note that we are not causally connected to all objects in the universe, for example not those that are outside of our light cone. Under those conditions we would thus be unable to have singular thoughts about everything outside of our light cone. This would give us lots of ineffable truths: all those that involve the objects we can’t have singular thoughts about. Thus assuming, again only for the moment, that if causal connection is required for singular representation then all these truths are examples of ineffable truths. The notion of the ineffable on which this is true is not the notion we should be concerned with. It concerns merely the de facto ineffable, not the completely ineffable. All that is missing in this case is our causal contact to the object. This is something we could have if only we were closer to the object, close enough to have it inside of our light cone so that a causal connection could obtain. Given where we are and where the object is (and the assumed requirement on singular thought) we can’t think singular thoughts about that object. But we could think these thoughts if we were closer. So, in a sense the ineffability of truths involving such objects is due to

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our placement in the world, how we are causally isolated from them, but not due to our representational capacities not being good enough. If we were closer then our capacities would be all that is needed. The issue is what we should keep fixed and what we should allow to vary when we ask whether we can think a thought. We want to keep fixed our basic cognitive setup, but we should not fix the place in the world where we happen to be located. This is part of what we need to get clear on when we try to determine what the proper notion of the ineffable is that we should be concerned with. The ineffable is that which we can’t say or think, but “can” can be understood in many ways and on each precisification of “can” do we get a more precise and specific notion of the ineffable. On a notion where we fix our place in the world we might get ineffable truths about far away objects, but that is not the notion of the ineffable that is tied to the worries about our minds not being good enough to carry out ambitious metaphysics. The more interesting and more significant notion is thus the one where we do not fix our place in the world. One way to illustrate the relevant notion of a truth being beyond us in principle is the incommunicability test : some other creature who can represent that truth couldn’t communicate it to us in principle, no matter how hard they tried. If we were to encounter some creature who can represent that truth, and who has mastered our language and thus can communicate with us in general, then this creature would nonetheless be unable to communicate this truth to us. We are limited in this case that we can’t represent this truth no matter what help we might get. Even someone who can represent this truth couldn’t help us to do better. Consider for our case of missing objects someone who can represent a content involving an object that is beyond us. That person could communicate this content to us, and thereby allow us to think about that object. Suppose, for purposes of illustration again, a highly advanced alien creature is talking to us and it can represent everything there is to represent, and in particular it can refer to all objects. The alien could then help us to the contents that we couldn’t get without it since these contents involve objects that are otherwise beyond our referential reach. The alien could just tell us what its name for that object is, and we can then refer to the object in question with that name, just as we in general exploit the referential success of other humans when we refer to objects we learn about from them. The referential connection can be mediated via others, be they humans, aliens, or gods. That is how we manage to refer to Socrates, after all. Thus even going back to our illustration of the need for a causal connection in our unaided situation for reference to an object, this is not an in principle limitation. Gods or aliens that are not limited this way could allow us to piggyback on their success, not just with faraway objects, but even with objects that are causally

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inert. Reference to an object is thus not a limitation that applies to us in principle, unless of course some objects are in principle beyond what can be referred to, even by God, but we have no reason to think that this is the case. Even if aliens or gods could help us to access all objects, they might not be able to help us with other cases of a source of the ineffable. There might be some truths where the alien would have to tell us that although it can think and say them, it can’t communicate them to us, since creatures like us just can’t grasp them.8 If there is ineffability of this later kind it would have to go beyond merely inaccessible objects. And whether some part of reality is like that, not just in fact out of our reach, but in principle beyond creatures like us to grasp, that is the interesting question that might have significant consequences for metaphysics. The interesting notion of the ineffable is thus the one that goes beyond a limitation to think about or refer to particular objects. We could make the notion of the ineffable more precise along the lines where a limitation to refer to objects would make a fact involving that object ineffable, and we could make it precise where it wouldn’t. Both are perfectly good notions of the ineffable, but the more interesting one is the latter, which passes the incommunicability test. To make this explicit, we should take the ineffable to be the object-permitting ineffable: that which is completely ineffable even if we allow ourselves access to all objects. Our arguments above for there being ineffable facts did not on the face of it rely on just a limitation to refer to objects, and thus these arguments should still be compelling even with the notion of the ineffable as being object-permitting. What makes ineffable facts beyond us is thus not simply objects which are referentially inaccessible. The real source of ineffability lies somewhere else. Missing properties. The same issue now arises with properties. What if there are some properties that we, somehow, can’t represent? And in the case of properties we might have to distinguish two ways in which we might fail to represent them. First, it might be that the situation with properties is very much like the one with objects. Properties, one could argue, are simply things or entities, just like regular objects. We might be unable to pick out that entity with a term that denotes it. But here the aliens might again be able to help us out. They might give us a name to use that refers to the property that otherwise was beyond our referential abilities. Second, we might be able to fail to represent that property in a simple subject– predicate sentence “a is F.” Maybe we can’t have a predicate “is F” without outside help, and this might be a source of ineffability for us. Here, too, the 8 A version of this scenario is described in the novel The Ophiuchi Hotline by John Varley (1977). Thanks to David Baker for this reference.

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issue arises if the aliens could do it for us. Could they just let us use one of their predicates, say “is wallereanesfsa?” This might be enough for representing even if it isn’t enough for understanding what we thereby represent. And there is an issue whether or not the two cases are really different. After all, a is F just in case a has Fness. If the aliens give us a name for Fness then maybe this is all we need to represent that a is F, and thus the case of properties really reduces to the case of objects. All this might go too far, but it can be taken even further. Facts, truths, or propositions, too, can be seen as entities. And if we can refer to any objects or properties, why not to any fact or proposition? Any proposition p is equivalent to the proposition that p is true. And we can eff the truth predicate which is all that we need besides the name for the proposition to state something equivalent to p.9 Can we really be satisfied with a sense of the effable where we eff every truth t since aliens or gods could give us a name for t, and we can eff that t has the property of being true? This is clearly unsatisfactory in some ways, and might appear to be a too shallow victory for the effability thesis. But what precisely is unsatisfactory about it? One concern is about what counts as the same fact or proposition. The equivalence of p and it’s true that p holds in the sense that it is necessary that if one is true then so is the other. A more fine-grained notion of equivalence might separate these two propositions or contents, and effing one of them might then not be enough to eff the other. On the other hand, a too finegrained notion of equivalence trivializes the issue in favor of the ineffability thesis, as we saw above in our brief discussion of perspectival contents. It is preferable to take a theory-neutral and coarse-grained notion of contents or propositions and then to avoid trivializing the issue in other ways. And here there is a good middle ground. On the one hand we didn’t want to accept the ineffability thesis simply on the grounds that some objects are too distant from us in spacetime for us to refer to them. This is a restriction that is not one “in principle” in a sense analogous to the restrictions of time and memory not being restrictions in principle. But allowing objects to be free threatens to trivialize the issue, since properties and, in particular, propositions or contents themselves can be seen as objects, too. We can reach a middle ground by allowing us objects to be free, but explicitly excluding objects or entities that are content-like objects, for example propositions or facts. This leaves us with a substantial question to consider. We can remain neutral, as should become clear shortly, on

9 Whether we indeed have a truth predicate that could be used to apply to any proposition whatsoever is controversial (see Field (1994)), but for the moment we can leave that issue aside.

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whether we also have to exclude properties from the objects we get for free.10 The reason for that is that the real worry about ineffability comes not from there being ineffable properties, but from somewhere else. Missing structure. We represent the world paradigmatically with representations that have subject–predicate structure. But why should we think that representations that have this structure are enough to represent everything there is to represent? We can represent truths with other structures as well, but that only pushes the issue towards all those structures together. We have good reason to think that there is more to reality than objects having properties. Many of our own representations of reality are not in the subject–predicate form. Consider, for example, explanatory relationships: p because q. That is not a simple subject–predicate sentence, but a complex sentence with a sentential connective “because.” Since the negation of such a sentence is also not in subject–predicate form, some such representations are true. And thus some sentences of this kind truly describe reality, unless all non-subject– predicate sentences are systematically non-descriptive, analogous to an expressivist treatment of normative discourse. But this last qualification is too far-fetched to be a real limitation. We have lots of good reasons to think that sentences that are not in subject–predicate form are not systematically different from those that are in their attempts to describe reality. Since it isn’t clear how the same fact could be represented with a simple subject–predicate sentence it seems that some facts about reality require a representation with a different structure than that of a subject–predicate representation.11 Of course, in this case we do have the resources to represent these facts. We do have a sentential connective that allows us to represent explanatory relationships. But the worry is that if subject–predicate representations in general are not enough, and sometimes extra resources are needed, then why think that we have all these extra resources available to us? Even if we could access all objects and all properties, the worry would remain whether we can represent all the facts, since apparently some of the facts we can represent require a representation with a structure different from a subject–predicate one. And 10 A further worry about properties trivializing the debate is this. If p is the case then any object has the property of being such that p is the case. Thus if all properties are free then this connection gives us at least a truth conditional equivalent content for any proposition p. Whether it gives us the same content can be further debated, of course. Not to trivialize the issue in this way we would again need to restrict the properties that come for free, for example to ones that distinguish between objects in the same world, or some other way. 11 I am now leaving aside the issue discussed above about trivializing this by taking every proposition p to be equivalent to p being true, which is in subject–predicate form, or by taking p to be equivalent to the universe having the property of things are being such that p.

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once we recognize that we can, wonder why we should think that our minds have access to all the structures required to represent all facts. Why think that the kinds of representations that are available in human languages and to human minds are good enough to represent all facts and truths? We can say that a fact is structurally ineffable if the source of its ineffability is the required structure of a representation of it, i.e. a structure we do not have access to.12 Structural ineffability gets at the heart of the matter. Any limitation of this kind would pass the incommunicability test: if there are facts that cannot be represented by representations available to the human mind then aliens who do not have this limitation could not communicate them to us. These facts would be in principle beyond us, not just due to our location, our access to objects, but due to the nature of our minds. These facts would be ineffable for us in the sense that matters. The arguments for there being ineffable facts discussed above support there being ineffable facts in this sense: structurally ineffable facts. Just as the mind of simpler creatures doesn’t have the basic representational resources to represent all facts, so we should think by analogy and due to our built-in cognitive setup, that some facts are beyond what creatures like us can represent in principle. To represent such facts would require access to ways of representing the world which are beyond those available to our minds. This is what the arguments for the ineffable point to, and this is what has significant consequences for metaphysics. It is now time to see what follows from it.

4. THE SUB-ALGEBRA HYPOTHESIS A N D IT S C O N S E Q U E N C E S

4.1 The hiddenness of the ineffable Suppose then that we should accept that some aspects of reality are ineffable for us, in the sense that the human mind is not in principle capable of representing them in thought or language. What should we conclude from this? Even though I suspect many would accept that we are limited in what we can represent, few draw much of a conclusion from this. But it could be taken to lead to a largely negative and skeptical conclusion about inquiry in general and our attempts to find out what reality is like. I believe that there are indeed significant conclusions to be drawn, but they are neither that broad nor that skeptical. But to see what we should properly conclude from the ineffable we need to get clear on one further aspect of the ineffable that is puzzling. This is the problem why the 12

For an argument that structural ineffability is impossible, see Filcheva (2015).

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ineffable is so well hidden from us. Any fact that is ineffable would make it hidden from us in the sense that we can’t represent it. But if there are indeed ineffable facts then they are more hidden from us than simply being unrepresentable, as I hope to make clear shortly. This extra-hiddenness is puzzling, and points to an account of the relationship between effable and ineffable facts, which in turn is significant for what we should conclude from there being ineffable facts for the limits of human inquiry. To illustrate all this, let us first see how the ineffable is more hidden than it needs to be. A first thing to note is that we never seem to perceive something ineffable. This is not a triviality. We can well wonder why it is never the case that we perceive something that we simply cannot describe or represent in thought. In such a situation we would be perceptually connected to a fact which we cannot represent in thought or language, and realize that our representational capacities give out here. Why does this never happen? It is not inconceivable that one day we open a door, look inside, and recognize that what we see is simply beyond what we can describe in words. All we can say is just “wow!” We see something that we can’t represent conceptually, while at the same time realizing that what we see is simply beyond us conceptually. It is hard to imagine what that would be like in more detail, in part because it never happens, in part because we might have to rely on our concepts in imagination. The question is why this never happens. One answer, of course, would be that the effability thesis is true and nothing is ineffable, but there are also other possible explanations. One of them could come from the philosophy of perception. It might be that everything we perceive has to be conceptualized, and what we perceive has to be tied to the conceptual content of the perception. Since conceptual content, involving our concepts, can be represented by us, it is no wonder that what we perceive can be represented by us.13 But even if we could explain why we never perceive the ineffable without relying on the effability thesis, the question remains why we nonetheless never encounter the ineffable in other ways. It might be that what we perceive has to be conceptualized, and thus is effable, for reasons having to do with how perception works. But this doesn’t answer the question why we nonetheless apparently never encounter the ineffable in other ways. Even if we never perceive it directly, we might realize that there is something ineffable right here, behind this door. In such case we could realize that what we can perceive of the situation is effable, but there is more to it than what we can perceive, in the sense of represent in perception. This would be a scenario where we encounter the ineffable, and we recognize that what we encounter is ineffable. This could not just happen in unusual situations, but 13

One way this might go is, of course, in Kant (1781).

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it could conceivably be part of regular scientific theorizing. We might recognize regularly during inquiry that now we are approaching the limits of what we can represent, and that the answer to our problem lies beyond it. But why does this never seem to happen, given all the reasons we have seen for there being ineffable facts? Why is the ineffable so well hidden from us and apparently so irrelevant for our attempts to understand various parts of the world in inquiry? The ineffable seems to fall out of the picture and for all intents and purposes every fact we encounter is effable. This naturally gives rise to the impression that the effability thesis should be true, even though we have good reason to think it is false. To properly appreciate the significance of the ineffable we need to get clear on why the effable facts seem to be all the ones we ever encounter.

4.2 The sub-algebra hypothesis If we accept the arguments for there being ineffable facts then we need to understand how they might be related to the effable ones, in particular why the ineffable facts seem to be as irrelevant to ordinary inquiry as they seem to be, and why they are so well-hidden from us. What could explain that the possibly vast range of ineffable facts is systematically hidden from us in a stronger way than is suggested by their being merely ineffable? There is a way to understand this which seems to me to be the best way to make sense of it, and it is best spelled out with a mathematical analogy. Consider a simple mathematical structure, say the integers with addition, multiplication, and subtraction: . . .
3;
2;
1; 0; 1; 2; 3; . . . ðþ; ;
Þ The integers are closed under these three arithmetical operations. The sum, product, or subtraction of any two integers is always another integer. The integers with these operations thus form an algebraic structure. This structure corresponds to a language suitable to describe it: it has a name for each integer, and function symbols for each operation, and additional basic logical vocabulary.14 This we could call the language of the (particular instance of ) the structure. Now consider the world, so to speak, from the point of view of an integer, thinking about the integers with that structure. The integers, I imagine for the sake of the example, can capture the world in terms of the language of their structure. And from their point of view it will seem perfectly natural, 14 The additional vocabulary would in this case just include “=” and Boolean connectives. I don’t allow variables or quantifiers here for the sake of the example.

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even obvious, that the integers is all there is. After all, the sum of any two of them is always another integer, and so is the product or subtraction. Furthermore, any question the integers can ask about the world in their language will have an answer that can be stated in their language. What’s 7  8? It’s another integer. We can say that in their situation they enjoy question–answer completeness: if you can state the question then you can state the answer. From the point of view of the integers, they naturally take it that they can capture all of reality there is to capture. If we were integers with those representational resources it would seem compelling to us that the integers is all there is. All this would be so even if numerical reality is much richer than that, with the integers being embedded in different, larger structures, for example the rational numbers with the above operations as well as also division: 1 1 :::
3:::
2:::
1:::
:::0::: :::1:::2:::3::: ðþ; ;
; Þ 2 2 Here there are infinitely many other numbers between any two integers, getting arbitrarily close to them. The rational numbers, just like the integers, are closed under addition, subtraction and multiplication. In addition the rational numbers are closed under division (leaving out 0, as usual), whereas the integers are not. Although the integers are embedded in the rational numbers, the other rational numbers are completely hidden from them. Since the integers form a sub-structure, or sub-algebra, of the rational numbers with addition, multiplication, and subtraction, these other rational numbers can never be reached from the integers with those operations: they always lead back to the integers. But if the integers just had access to division, say, then they could get out of their sub-algebra and reach the rest of the rational numbers. And if we would add just one more rational number, say 12, to them then they could use that to reach lots of other rational numbers: 1 12, 2 12 and so on. But from the point of view of the integers, they are all there is. And this will seem clear and compelling to them, since after all, given their resources, what else could there be? Sums and products of integers are always integers. This is so even though the integers are surrounded by and thinly spread among things that are not integers. These other aspects of numerical reality are completely hidden from them. This could be our situation. The parts of reality that we can represent might form a sub-algebra of all of reality. That is to say, it might be that we can represent certain objects, events, and propositions, and certain relations or operations on them such that whenever we apply the operations to things we can represent we get something that we can represent as well. Whenever we can represent an event, say, then we can represent the cause of that event.

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Whenever we can represent a fact we can represent the explanation of that fact, and so on. Our representational system might be a closed structure analogous to an algebraic structure. And that structure might be a substructure or sub-algebra of all of reality. This hypothesis we can call the subalgebra hypothesis: the hypothesis that what we can represent forms a proper sub-structure of all there is to represent. It is based on the analogy to an algebraic structure and a sub-algebra of it. If it were correct then we would be much like the integers. We can never represent aspects of the much richer reality we are part of, but it would seem to us that the parts of reality we can represent are all there is. After all, all operations and relations that we can represent and that hold among things we can represent lead to things we can represent. So, from our point of view, what else could there be? Everything else would be systematically hidden from us. Similarly, we should expect that we have question–answer completeness, and so for any question we can ask we can state the answer (whether or not we can know that this is the correct answer). If this is our situation then it would appear to us that everything is effable, even though what we can eff is surrounded by, and possibly thinly spread among, the ineffable. We can never get there from our point of view, and it will be completely hidden from us. In considering this hypothesis we can note right away that it can’t be quite our situation. First of all, we don’t enjoy question–answer completeness in the way the integers would. If the effability thesis is true then, of course, we can state any answer to any question since we can state everything. But if the ineffability thesis is true then we can ask questions where we know we can’t state the answer, for example “What are all the ineffable truths?” Still, we might enjoy large-scale question–answer completeness, which is question– answer completeness leaving aside questions that deal with the ineffable and related questions. In general, and for almost all cases, it might be that if we can state the question, we can state the answer. And just this seems to be the case. When we ask what caused something, or what explains something, then we can in general at least state the answer, even if we don’t know that it is the right answer. This fact we can understand on the sub-algebra hypothesis even if the ineffability thesis is true: causal and explanatory relations are ones under which our structure is closed. We can represent the explanations of what we can represent, and we can represent the causes of what we can represent. And again, all this could be true even though we are surrounded by the ineffable. And just like the very same arithmetical operations of multiplication and addition can be seen as applying to the integers as well as the rational numbers, so causal and explanatory relations might hold among the facts or events we can represent as well as those that are ineffable for us. The ineffable aspects of reality might be very much like the ones we can eff, just outside of our sub-structure, or they might be completely different.

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The sub-algebra hypothesis would explain why the ineffable is systematically hidden from us even though it is possibly a vast part of reality. Since it is outside of our sub-algebra we won’t encounter it via causal or explanatory relationships. We can expect our sub-algebra to be closed under causal and explanatory connections. And we can expect our sub-algebra to be properly integrated with those things that causally effect us, for example in perception. Such a sub-algebra would be a very stable resting point for a representational system. There is little need to develop it further, even if it only captures a small part of what there is to represent. On the other hand, if a representational system does not form a closed structure, or at least something reasonably close to it, then we would expect it to develop further if it develops at all. But once it is reasonably closed it rests at a stable place. The sub-algebra hypothesis explains why the ineffable is systematically hidden from us even if the ineffability thesis is true. It accommodates what needs to be accommodated, and we have good reason to think it holds. The question remains, however, what follows from it?

4.3 Ineffability and modesty The sub-algebra hypothesis makes clear in what sense the ineffable matters, and in what sense it doesn’t matter. It doesn’t matter locally, i.e. for particular questions of fact that aren’t concerned with reality as a whole. If I ask an ordinary question like why is there a sandwich on the table, who ate my apple, or why is the sky blue, then I should expect that I can state the answer. Causal and explanatory relationships are part of my representational system, and thus are what my algebra is closed under. For ordinary local questions the ineffable will fall out of the picture.15 But not so for global questions about all of reality. Here, too, our reasonable question–answer completeness will likely allow us to state the answer if we can state the question. But the ineffable and how it is hidden from us will often mislead us into accepting the wrong answer. This is worrisome for questions about materialism and naturalism in particular. From our point of view it can seem perfectly compelling that everything is material and all there is fits into the natural world. So the questions “Is everything material?” and “Is the natural world all there is?” are questions we can state, and whose answers (yes or no) we can state as well. But our representational limitations might lead us to accept the wrong answers. If 15 Quantified claims, when relevant to local issues, should be taken to be restricted to the locally relevant domain, and thus they won’t range over all of reality, in contrast to those that are explicitly intended to be unrestricted, like the ones that are intended to make claims about all of reality.

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the sub-algebra hypothesis is correct then we might be in the materialist/ naturalist sub-algebra in a largely non-naturalist world. If this is our situation we would naturally, but in this case incorrectly, hold that materialism or naturalism is true. Just as the integers would find it compelling to think that all there is are integers, even when there are infinitely many other numbers before we get to the next integer. We might similarly find naturalism compelling even though the non-natural is infinitely close to us, and all around us, but systematically hidden from us. The ineffable is locally irrelevant, but globally central. And it is, in particular, central for the largescale metaphysical questions about all of reality. The sub-algebra hypothesis is not a skeptical hypothesis, in the sense that it is not a scenario that we can’t rule out to obtain, and which invalidates our entitlement to our ordinary beliefs. On the contrary, it is a scenario that we have reason to believe obtains, but on this scenario our ordinary beliefs are taken to be true. The parts of reality that we can represent we do represent correctly, or so we can grant here. But when we want to make claims about all of reality then we reach a limit. The sub-algebra hypothesis does not warrant a rejection of trying to answer large-scale metaphysical questions about reality, but it does warrant a form of modesty. We must recognize that our situation is one where these questions are to be approached with a sense that whatever answer seems compelling to us might simply reflect our own limitations, but not how reality actually is. The ineffability thesis combined with the sub-algebra hypothesis in particular suggests modesty about global metaphysical questions. Modesty is not agnosticism or quietism, but it is a step in that general direction.16 It does not justify the abandonment of grand metaphysical theorizing, but it does justify that such theorizing is different in its epistemic status from other parts of inquiry. How different it is will depend on how strong the reasons are that we have for the ineffability thesis and the sub-algebra hypothesis. It will be a difference in degree, and to what degree is not clear so far. Modesty for grand metaphysics follows, to what degree is left open. The argument for modesty given here is importantly different from the argument that we should be modest about judging how many people are in this room, since after all there might be lots of invisible and otherwise undetectable people all around us. We have no reason to think that there are invisible people around us; that is just a hypothesis we might not be able to rule out. But we do have reason to think that there are ineffable facts, and that these facts are systematically hidden from us. These reasons are not

16 Modesty thus contrasts with the positions taken by Gideon Rosen (1994) and Sven Rosenkranz (2007).

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conclusive, of course, but they are good reasons nonetheless. Thus reflecting on our own situation should lead us to conclude that there is a reasonable expectation that we fall short of complete effability. We have reason to believe that what we can represent is less than what is the case. We also have reason to believe that the ineffable is systematically hidden from us and that the sub-algebra hypothesis is correct. Thus we can expect to be misled in our judgments about global features of reality, and so modesty in grand metaphysics is advisable, and the ambitions of metaphysics need to be toned down. At the same time we should expect the ineffable to be locally irrelevant. The consequence of all this is modesty for metaphysics, but it is insignificant for most of the rest. This would be a natural place to end the chapter, but I am afraid it is only half of the story. Although most philosophers hold views that should make all the above arguments compelling to them, there is an assumption in the background of our discussion so far that is worth making explicit and that might well be false. I have to confess that I have argued elsewhere that it is false.17 If it is indeed false then everything changes for our discussion, or so I hope to make clear in the following part of the paper. If the assumption is false then it naturally leads to a defense of the effability thesis and its natural companion: idealism. To make this assumption explicit and to argue that everything depends on it will hopefully justify going on for a bit longer.

5 . I N T E R N A L I S M , E F F A B I L I T Y, A N D I D E A L I S M Our starting point was to wonder about the relationship between two different things: which facts obtain on the one hand, and which facts we can represent in thought or language on the other. The general relationship between these two was illustrated with the Venn diagram in section 2. Here the idea was that on the one side is reality, and on the other side are we, with our attempts to represent reality. The notion of reality can be made more precise, as usual, in two ways: the totality of what there is, or the totality of what is the case. On the latter conception it is simply the totality of all the facts that obtain. Reality, understood as all that is the case, is taken to be simply there, waiting for us to represent some of it in thought or language. The totality of facts forms an independent domain: a domain of all truths, facts, or true propositions. Reality, on this broadly propositional conception of it, just as reality on the broadly ontological conception as the totality of what there is, is simply there, independently of us, with no special place for 17

See Hofweber (2006) (2009), and, in particular (2016b).

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us in what it in general is like. But this general picture of the relationship between the totality of facts and us might be false. Although this picture is widely accepted, we in fact have good reason to think that it is mistaken. In the final parts of this paper we will look more at this standard picture of the propositional, what its alternative is, and how the issue about which one is right relates to the question about the ineffable. But first, let’s get clearer on the standard picture of the propositional and its alternative.

5.1 Talk about propositions: that-clauses When we talk about propositions or facts we do so most directly with a thatclause, for example in the ascription of content to an utterance or representation: (3)

A said that p.

The most common way to understand such sentences is to take them to involve two semantically singular terms, that is two phrases that aim to pick out, refer to, or denote some thing or entity.18 In the case of (3), one term that stands for a person: “A”; and another term that stands for a proposition: “that p”. Although the term that picks out the proposition is different in kind from the term that picks out the person both in effect do the same thing, standing for some thing or entity. And on the common way of fleshing out this picture, reality contains not just a domain of persons, it also contains a domain of propositions. When A says that p then some relation, the saying relation, holds between two things: a sayer and a proposition. On this picture it is natural to think of the domain of propositions in analogy with the domain of persons. It is something that is part of reality, something that is simply there, waiting to be picked out with our that-clauses.19 But there is an alternative way of looking at this. It is partly motivated by the fact that that-clauses are first and foremost clauses. As (complement) clauses that-clauses are of the same general category as “where I hoped it was” or “whether she ate it,” and the like. Such clauses are not naturally taken to be terms that pick out things or entities. They seem to have a different semantic function. It is tempting to hold that that-clauses in examples like (3) specify 18 Although it can be argued that we should recognize a difference between referring, denoting, and picking out, I will gloss over such differences in the following, since it won’t matter which one of them that-clauses do, only whether they do any of them. 19 A rare exception of this picture of propositions to be there, but not be language independent is Stephen Schiffer’s (2003) view. Almost all other authors who believe in a domain of propositions take them to be there independently of us. Whether Schiffer’s view makes a difference to our debate here is discussed among other issues in Hofweber (2016a), where I hold that on a natural reading it does not, although an alternative might be possible.

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what was said, but don’t refer to some object which was said. They do not refer to a content, but say what the content is. On such a non-referential picture of the function of that-clauses, “said” does not express a relation between two things, one denoted by “A” and the other denoted by “that p.” Instead, “said” predicates of one thing, A, that it said that p. Complement clauses on such a picture have a quite different function than names for objects. Which one of these two positions is correct should be seen at least as an open question in semantics, one that is widely debated in the present literature.20 Whichever side comes out on top in this debate, it will have significant consequences for our discussion of the ineffable, or so I will argue now. These two pictures of the semantic function of that-clauses correspond, on their natural development, to two different pictures of the propositional, and correspondingly two different answers to the question about what is effable. To see that we first need to say more about how the second, non-referential, picture of that-clauses is naturally developed in more detail. The non-referential view of that-clauses might intuitively be very plausible, but it faces a serious obstacle when it comes to making sense of quite obviously valid inferences like the inference from (3) to (4)

A said something.

How can a believer in that-clauses being non-referring or non-denoting expressions account for the validity of this inference? If A said something then it seems that there must be some thing or entity to which A bears the saying relation. Whatever the believer in non-referential and non-denoting clause wants to say here, it has to make sense of such inferences. But as it turns out, similar inferences are also possible in cases that have nothing to do with clauses and also involve apparently non-referential complements. For example, (5)

I need an assistant.21

implies that (6)

I need something.

even when I don’t need any particular person to be my assistant, just some assistant or other. Similarly,

20 See Schiffer (1987) and (2003), Bach (1997), Moltmann (2003b), Rosefeldt (2008), Hofweber (2007) and (2016b) and many others. There is a further issue here whether that-clauses and more generally proposition terms like “the proposition that p” have to be either all referential or all non-referential, or whether a mixed view might be true instead. This issue is discussed in more detail in Chapter 8 of Hofweber (2016b). I will leave these mixed positions aside here. 21 See Moltmann (2003a).

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My daughter wants a unicorn.

(which is true) implies that (8)

She wants something.

Even though she doesn’t want a particular unicorn, she would be happy with any one, and even though there are no unicorns at all. What all this suggests is that there is more going on with quantifiers than the simple objection to the non-referential picture of that-clauses suggests.

5.2 Talk about propositions: quantifiers Some story about quantification must make sense of all this, and my favorite one is the following.22 Quantifiers are semantically underspecified. They are polysemous expressions that have at least two different readings. These readings arise from two different functions they have in ordinary communication. One is to make a claim about the domain of objects, or entities. On this use of quantifiers we say that among all the things there are, whichever they might be, one of them has a certain feature (in the case of a use of the particular quantifier). This we will call the domain conditions, or external, reading, since it makes a claim about the domain of entities, which is something language external. On the other reading they are supposed to inferentially relate to other expressions in ones own language. This we will call the inferential, or internal, reading. On this reading the quantifier has a certain inferential role. In the case of the particular quantifier, it simply is the inferential role (9)

F(t)/F(something)

whereby “t” could be any expression of a limited range of syntactic categories, be it a singular term, or a clause, or some other complement.23 On the inferential reading of “something” the quantifier has this inferential role and thus inferentially relates some sentences to others internal to the language. To illustrate the difference, consider an example that uses the universal quantifier “everything”: (10)

Everything exists.

22 See Hofweber (2000) and (2005), and, in particular, Chapter 3 of Hofweber (2016b). 23 Which precise range of cases belong to this group is not completely clear. Quantifiers in natural language do not interact with just any syntactic category. For example, we can’t quantify into determiner position, and we will simply accept this limitation here without hoping to explain it.

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On the polysemy account outlined above this sentence has two readings. On one it should have a ring of truth, on the other it is quite clearly false. The true reading is the external one, where you are saying that all the things in the domain have in common that they all exist. And that, leaving some philosophical views to the contrary aside, seems true. But on the inferential reading it is quite clearly false. On this reading the sentence has the inferential role (11)

F(everything)/F(t)

where “t” can be any instance in my language. So (10) implies Santa exists, the Easter Bunny exists, etc. which are false. Since it implies false things it must be false itself. So, on the inferential reading, (10) is false. To say that quantifiers have an inferential reading is not to accept an inferentialist semantics in general. An inferential role can be the result of a contribution to the truth conditions. The question remains so far what contribution to the truth conditions gives quantifiers their inferential roles on the internal reading. One thing is clear: the truth conditions on the inferential reading have to be different than the truth conditions on the domain conditions reading. On the inferential reading the inference goes through no matter what the semantic function of the relevant instances is. What truth conditions would give it that inferential role? There is a simple answer to this question. The simplest truth conditions that give the particular quantifier the inferential role for which we want it is for sentences in which it occurs to be equivalent to the disjunction of all the instances that are supposed to imply it. These truth conditions are simplest in the sense that any other candidate truth conditions that give the particular quantifier that inferential role must be weaker: the simplest one would imply them. Similarly, the simplest truth condition that gives the universal quantifier its inferential role is the conjunction over all the instances that it is supposed to imply, which is for us all the instances in our language. Any other candidate truth conditions that have that inferential role would have to be stronger in that it would itself have to imply that conjunction. When it comes to the truth conditions that the quantified sentences have when the quantifiers are used in their inferential role reading we have an optimal solution, and we thus have some reason to think that quantifiers in that reading make that contribution to the truth conditions.24 As we will see shortly, this leaves one important complication aside, one we will need to address in the following section. So far we have only given the simple version of the semantics of the inferential reading of 24 The details of this approach to quantification, including how it can be extended to generalized quantifiers, are spelled out in Chapter 3 of Hofweber (2016b). How it relates to and is different from substitutional quantification is also discussed there.

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quantifiers. As we will see later, a proper version can be given as well, but let’s work with the simple one for now. Whether this is the correct understanding of quantification, in particular when related to quantification over propositions, is one question, but it should be clear that this is one natural way in which the non-referential picture of that-clauses might extend to quantification as well. We will not try to settle whether this picture is indeed correct, but instead focus on what it implies for our discussion of the ineffable.

5.3 Internalism vs. externalism To understand talk about propositions involves at least making progress on the question whether or not that-clauses are referring or denoting expressions, and how to understand quantification into that-clause position. There are two large-scale options about how such talk can be understood in a coherent way, which in effect depend on what one thinks about that-clauses and quantification into that-clause position. If that-clauses pick out things then quantification into that-clause position should be quantification over the domain of these things. If that-clauses don’t pick out entities then quantification into that-clause position should be understood along different lines. Here, I suggested, the best option is to understand it as being based on the inferential reading of quantifiers, a reading quantifiers have in general. These two large-scale views are thus: (12) (12)

a. Internalism: that-clauses are non-denoting, quantifiers into that-clause position are used in their internal, inferential, reading. b. Externalism: that-clauses are denoting, quantifiers into that-clause position are used in their external, domain conditions, reading.

Externalism is naturally connected to an ontology of propositions. If our talk about the propositional is not completely in error then the domain over which we quantify must be non-empty, and thus propositions exist. But if internalism is correct then there is no such domain of propositions, and in particular no propositions exist. I take propositions, if there are any, here simply to be whatever that-clauses refer to or pick out. They are what we talk about when we say that Fred believes that p or Sue said that p. If there are such things as propositions then they are the things that we talk about when we ascribe content. But on the internalist picture we do not talk about any things when we ascribe content since our that-clauses don’t pick out any such things. They are non-referring and non-denoting expressions. And if propositions just are whatever we talk about when we ascribe content then it follows from the internalist picture of such talk that no such things exist.25 If “that p” as well as 25

More on all this is in Chapter 4 of Hofweber (2016b).

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“the proposition that p” does not refer to anything then whatever there might be, none of it is the proposition that p. The internalist picture of talk about propositions guarantees that there are no propositions, so understood. This will be relevant later. Whether internalism or externalism is correct is a question in the philosophy of language that we can’t hope to settle here. What we can hope to see is that this question is crucial for our debate about the ineffable.

5.4 Internalism and the effability thesis Everyone should believe that there is a substantial issue about how we should understand our talk about the propositional, no matter which side one eventually ends up on. Suppose, though, that internalism turns out to be correct. Then quantification over26 propositions generalizes over the instances, in my own language. After all, the inferential role I am concerned with is the inferential behavior that quantifiers have in my language, and to have that inferential role it needs, on the simplest solution, to be equivalent to the conjunction or disjunction over all the instances. Thus (13)

I need something

on the internal reading is equivalent to the disjunction (14)

∨ (I need F )

whereby F can be any of the variety of instances of the sentence: an assistant, a unicorn, etc. The quantified sentence, on its internal reading, is thus equivalent to a disjunction over all the instances in our own language. This disjunction is infinite, but it is a disjunction of instances nonetheless. The effability thesis was the thesis that (1)

Everything is effable.

And as we noted above, the relevant way to understand it is that all true propositions, or truths, are effable, but not necessarily all feelings, objects, etc. The quantifier “everything” in the statement of the effability thesis is thus a quantifier over propositions. Assuming the truth of internalism, it is thus used in its inferential, internal reading. Internal quantifiers have the simplest truth conditions that give them the inferential role, and so the effability thesis is equivalent to the conjunction of all the instances in my own language. Thus, assuming internalism, it is equivalent to the following conjunction: 26 I take “talk about propositions” and “quantification over propositions” to have a sense in which they are neutral between the two options, and I mean it in this neutral sense here. That there is such a neutral sense is clear using a topical sense of aboutness, the sense in which you can talk about aliens all night long, whether or not there are any.

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∧ (that p is effable).

Here we have a conjunct for every instance of “p” in our own language. But for each such instance the sentence “that p is effable” is true, since we can eff all the instances in our own language. Thus on the internalist understanding of talk about propositions the effability thesis is true. When we say that everything is effable we are generalizing over the instances in our own language. And if this is so then the effability thesis is true. This answer to the question whether or not any aspect of reality is ineffable might seem very unsatisfactory and more like a cheap trick. We will discuss whether this answer could possibly be correct shortly. Before that, though, we need to see a bit more about how the internalist picture deals with the arguments against the effability thesis. And to see this we first have to formulate it properly.

5.5 The proper formulation of internalism The formulation of internalism given above was too simple, in a way that will matter for us later on for larger questions about the relationship between what there is and what is the case, and for dealing with the arguments for ineffable facts. But that it was too simple can be seen quite directly without having these issues in mind. To formulate internalism properly we need to make sure that the quantifiers properly interact with all the instances, not just the simplest ones we have considered so far. All the instances include cases like “he ate the cookies” and “it is pink.” From (16)

John said that it is pink

it follows that (17)

John said something.

But on the account given so far this inference might not be valid. So far we have not considered how to deal with an instance of a quantifier that contains context sensitive elements. Implicitly at least, we have simply ignored them, and ignoring them gives a well-defined semantics for the quantifiers: the instances are simply all sentences that have truth conditions independently of the context in which they are uttered. We could call these eternal sentences.27 But simply having disjunctions or conjunctions of eternal 27 There is a substantial further issue about such sentences, and which ones have that feature. Many sentences involve contributions from context or speakers intentions besides filling in values of demonstratives. I will largely put these aside here. For other ways in which context can affect content the question will be if there is also a way to express the

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sentences is not enough, since in “it is pink” the “it” might pick out something that is not picked out by any eternal term, and thus no eternal sentences would be equivalent to “it is pink.” How internalism is to be formulated to overcome this issue is discussed in detail in Hofweber (2006), but the main idea is simple enough. The inference from (16) to (17) has to be valid no matter what object might be referred to with “it.” Thus for any object o we need a disjunct that is true just in case o is pink. And this we can do by simply adding external quantification over objects on the outside of our disjunction, and allow new variables to be bound by this quantifier in these disjunctions. In our case this would give us the truth conditions of (17) not as (18)

∨ (John said that p)

but as (19)

∃x ∨ (John said that p½x Þ.

Here “p½x ” means that any of the new variables x may occur in the instances that replace “p.” And since we can’t in advance give an upper bound on how many such variables there might be (after all, John might have said that it is taller than that, but shorter than this, etc.) we must allow for infinitely many. Thus the truth conditions of internalist quantifiers over propositions are not merely given by infinitary conjunctions and disjunctions, but involve infinitary quantification as well. All this is at least technically unproblematic.28 In case of universal quantification the external quantifier over objects out front of the big conjunction is, of course, a universal external quantifier.29 With this new formulation we can now see that the effability thesis (1)

Everything is effable

is not simply equivalent to

same content without the effects of context. Since our instances are all instances in our language, we would avoid this issue if we could always find a context insensitive way to express a content that otherwise was expressed in a context sensitive way. This is certainly possible for most contributions of context, like disambiguation, various enrichments, etc. If it is always possible for cases other than demonstratives, it will be left as an open question for now. 28 In effect, internal quantification over propositions increases the expressive power of a language to a small fragment of what is called Lω1 ; ω1 built on top of that language. More about this is in Hofweber (2006). Infinitary extensions of first order logic are discussed, for example, in Keisler (1971). 29 How to do this for generalized quantifiers is developed in the appendices to Chapters 3 and 10 of Hofweber (2016b).

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∧ (that p is effable)

but instead to (21)

8x ∧ (that p½x  is effable).

And (21) is now not necessarily true any more with just any notion of effability. If what is effable is understood as what is effable by us in the circumstances we are actually in then this can be false, for example if there are objects outside of our light cone that we are thus not able to refer to (to use an example and assumption discussed above). But (21) is true if we use the object-permitting notion of effability, as discussed in section 3.5. On this, object-permitting, notion of effability, missing objects are never a source of ineffability. It is the notion of the ineffable that passes the incommunicability test, where reference to objects is always assumed to be possible and not a relevant source of a limitation. In our case (21), every instance of the quantifier is in effect equivalent to some sentence pða1 ; a2 ; :::; an Þ where ai is some parameter standing for an object, and p is some sentence in our language. On the object-permitting notion of effability, every one of these instances is effable. Thus the proper formulation of internalism guarantees the truth of the effability thesis using the object-permitting notion of effability. And the extra resources that are needed to give the proper formulation of internalism are the ones that explain why the ineffability thesis nonetheless seems true to us, as we will see now.

5.6 Explaining apparent ineffability If internalism is true, and with it the effability thesis, what then becomes of our arguments for ineffability? After all, we found them quite compelling above. But maybe all these arguments implicitly relied on a standard, externalist picture of propositions and facts? We considered four arguments for ineffability: built-in limitations, the argument from analogy, cardinality arguments, and there being no explanation why the effability thesis would be true. Let’s revisit them now. What could explain that a mind like ours can represent everything there is to represent? The internalist has a simple answer: “everything” here generalizes over our instances. No wonder we can represent everything, since we can represent every one of our instances: every instance in our language of “that p.” And relying on an object-permitting notion of effability we can eff every instance with parameters. It is no accident and no mystery that we can represent everything there is to represent. Internalism maintains that it is

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based on a mistake to wonder here why two separate things coincide: the propositional and our representational abilities. Instead the propositional and our representational abilities are not two unconnected things, if internalism is true. The truth of the effability thesis falls out of how talk about propositions is to be understood. It would be hard to explain why the effability thesis is true on an externalist picture, but it is quite straightforward on an internalist one. What about our built-in imitations? Maybe our mind must think a certain way, maybe we are inflexible and fixed in how we have to think, due to how our mind evolved or how our brain is structured. This naturally supports that we are limited in what we can represent on an externalist conception of facts. If facts are simply there, as part of an independent domain, then we should expect a mismatch. But on the internalist conception there is no such domain. So even our lack of flexibility does not support ineffability, since it does not support that this lack of flexibility is a limitation. We can explain why the effability thesis holds even if our minds have a fixed setup. The cardinality arguments are directly answered by the extra resources that we get from the proper formulation of internalism. Consider again the argument that there are more propositions than we can eff, since there are only countably many sentences in our language, but there are uncountably many facts about real numbers. The following is true: (22)

For every real number r there is a fact which is the fact that r is a real number.

This truth seems to be incompatible with the effability thesis as well as with internalism, since for different real numbers we get different facts, and thus overall too many facts for the effability thesis to hold. However, on the proper formulation of internalist quantification over propositions, (22) is true, and thus the truth of (22) is compatible with internalism. Here, crucially, the external quantifier over real numbers interacts with the external quantifiers that bind the new variables discussed above. Thus on the proper formulation of internalism (22) would look like this: (23)

8r∃x ∨ (that p½x  is the fact that r is a real number).

This is true when “p” is instantiated with “xi is a real number,” where “xi ” is bound from the outside with an external quantifier that ranges, amongst others, over real numbers, and thus has the relevant r as an instance. (22) appears to be in conflict with internalism, but in fact it is only in conflict with internalism on its naive formulation, but not with internalism on its proper formulation. To put it differently: the proper notion of an ineffable fact is one that is ineffable on an object-permitting notion of effability. Cardinality arguments

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like our simple argument using the real numbers argue that there is a certain cardinality of facts since there are collections of objects (the real numbers) of this cardinality. Furthermore, the argument continues, that cardinality is larger than the cardinality of effable facts and thus most are ineffable. But on the object-permitting notion of the ineffable, more objects get us more effable facts. On that notion of the ineffable it is hopeless to try to show that there more facts than effable facts, since there are lots of objects involved in all the facts. Such cardinality arguments thus won’t get off the ground.30 Finally, let’s consider the argument from analogy, which is maybe the most compelling and forceful argument for ineffability. Here the internalist answer is clear: although it appears to be coherent to imagine aliens or gods who relate to us like we relate to the honeybee, there in fact can be no such creatures. There can be no creatures who can represent more facts than we can, since we can already represent all the facts. We can represent everything there is to represent, while the honeybee can not. Since we can already represent everything, there can be no creatures who can represent more. There can be more powerful creatures with better spaceships, but they cannot represent any more facts. But whether this answer is at all satisfactory, or merely an endorsement of an absurd consequence of a view, can’t really be appreciated without looking more generally at the internalist picture of the propositional. Internalism is not merely a view about the semantics of that-clauses and quantifiers, but it incorporates a completely different picture of the propositional and of reality understood as the totality of facts. And only with that picture clearer in view can we see that this answer is not in fact absurd. Although an externalist will take it to be extreme bullet-biting, the internalist will take it to be a deep insight into the nature of the propositional or fact-like aspect of reality. We can only assess who has the upper hand once the full picture is on the table, to which we must turn now.

5.7 The internalist picture of the propositional On the externalist picture of the propositional, propositions form a domain of entities that we can refer to, and that our quantifiers range over when we quantify over propositions. Propositions on this view are most naturally understood as simply being there, as part of reality in addition to regular objects, waiting for us to refer to them.31 Facts can be seen as either being identical to or at least corresponding to true propositions, and so what holds 30 How internalism can deal with these and various other cardinality arguments is discussed in more detail in Hofweber (2006). 31 Again, an exception to this picture is Schiffer (2003).

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for propositions will hold for facts as well. Facts, propositions, states of affairs, anything we would pick out with a that-clause, we can call the propositional. On the externalist picture, the propositional is simply there, a part of reality, ready for us to discover and refer to. If facts are entities then reality as all there is determines reality as all that is the case. Under this assumption we should expect, as we saw above, that not all facts or propositions can be represented by us, and some will remain in principle ineffable. Taking propositions to form a domain that is simply there, as a part of reality, naturally leads to accepting ineffable facts as well as the sub-algebra hypothesis as the best explanation of why the ineffable is so well hidden. But the internalist account of talk about propositions is not just a semantic view of talk about propositions, it embodies a completely different picture of the propositional. On this picture the propositional is not simply there, and it is not independent of us, in a sense to be worked out. This difference is crucial when it comes to understanding why the effability thesis holds, assuming internalism, and how the arguments against effability are to be answered. In this and the next section I hope to work out more clearly what the internalist picture of the propositional is and how to understand it not being independent of us. And to do this we first should contrast the internalist picture with what it is not. The internalist does not simply establish the effability thesis with a semantic trick. Consider, as a contrast, a person who holds that they own everything. They say that the correct semantics of “everything” is that it ranges only over their things, and thus they own everything, since they own all their things. This is a bad view on several grounds, not the least of which is, of course, that this is not the correct semantics of “everything.” But the crucial difference between this view and the internalist view of talk about propositions goes beyond that. On the internalist view it is not true that when we say that we can represent everything we say that we can represent everything that is in some sense ours. We do not restrict our quantifier to some subset of the propositions which are related to us, analogous to the universal owner who restricts their quantifiers to the subset of things that they own. Internalism does not restrict the quantifier, but instead embodies a different view of what such quantifiers do, which is tied to a different view of what singular ascriptions of content do. Such quantifiers are unrestricted inferential quantifiers. As such their truth conditions give them a certain inferential role in our language, and the simplest truth conditions that do this are the ones that are equivalent to generalizations over all the instances. That these instances are instances in our language, and thus ones we can represent, is not the result of some sort of a restriction, but simply a consequence of the simplest truth conditions that give us what we need. Talk about propositions or facts, on the internalist picture, is not talk about

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some independent domain, it is not talk about any entities at all. On the internalist picture, that-clauses do not refer. They do not aim to pick out any entities. So talk about propositions is very different than talk about objects. Still, the internalist defense of the effablity thesis might seem unsatisfactory. It seems to involve too much language and not enough metaphysics and thus can’t be a defense of the view that all of reality is effable by us. Reality played little role in this defense. But that doesn’t mean that the defense was defective. The internalist’s explanation of why the effability thesis is true relied on the connection between the quantifier “everything” that occurred in the statement of the effability thesis and the instances of such quantified sentences in our own language. That this was enough to see that the effability thesis is true is surprising, but that doesn’t make it incorrect. That the explanation has to rely on reality in addition to what we do when we talk about all facts or truths is true on an externalist picture of the propositional, but not on an internalist one. For the latter, reflection on our language is enough. We can see that once we talk about the propositional at all, in the internalist way, the truth of the effability thesis follows. This is surprising, but it might just be true if internalism is indeed true. It is tempting to say that on the internalist picture the propositional is not an independent part of reality, but somehow due to us. But this on the face of it doesn’t make sense. Which facts obtain is, of course, not in general due to us. But there is something right about this, although it’s hard to put one’s finger on what precisely it is. I will make a more precise proposal about this in the next section, but first we should see some more if all this is too shallow a victory for the effability thesis. Let’s assume for the moment that internalism is true, and thus the effability thesis, as stated, is true as well. What this might be taken to show is that we need to state the question we wanted to ask differently than we did. As formulated the question has a negative answer, but maybe we need to reformulate it so that it is more substantial, more about reality, and less about language. After all, the question we intended to ask was not supposed to be settled by the semantics of quantifiers and the nonreferentiality of that-clauses. Of course, we can’t demand that the questions we ask are answered the way we expect or intend them to be answered, but still, maybe the lesson we should draw from all this is that we need to state the question we wanted to ask differently. Maybe internalism wins a shallow victory when it comes to the letter of the effability thesis, but it only pushes the real issue somewhere else. This line of thought is indeed tempting, but in the end it is mistaken. The truth and recognition of internalism does not motivate that we should ask the question differently. In fact, there is no better way of asking it. Instead

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internalism shows us the answer to the question we wanted to ask in the first place. To see this, let’s consider some attempts to ask the question differently, assuming internalism is true, and why they won’t help. Assuming again internalism, it won’t do to ask about the effability of every fact, or every proposition, or every proposition-like thing. Internal quantifiers over those make clear that the answer is in line with the effability thesis. So, maybe we should state the question in a way where we quantify over something else, something where internalism is not true: sentences, or inscriptions, or thought tokens, or something along those lines. These are simply material objects (we can assume) and so internalism doesn’t apply to talk about them (we can also assume). Should we thus ask instead whether there is some (actual or possible) concrete inscription which can’t be translated into our language, or some thought token which has a different content than any such token we can in principle have? But this won’t help. If internalism is true then we can conclude that there can be no such thought token or inscription. If such a token has a content at all then it has a content which we can think as well. Anything else has no content. Thus if there are inscriptions, sentences or tokens which we can’t translate it is not because they have a content that is beyond us, but rather because they have no content at all. And there certainly is no failure of translation if you fail to be able to translate something devoid of content. This way of trying to restate the question won’t change the issue. Another attempt could go via truth. Maybe the aliens can say things truly which we can’t say at all? Maybe true things accessible to them go beyond the truths accessible to us? But this, too, won’t help. There is a bridgeprinciple that connects things that are true to contents: (24)

x is true if and only if x has a true content.

It is hard to see, maybe inconceivable, how anything could be true, but not have any content. With this connection, moving the issue to truth doesn’t change things. Finally, one might try to throw in the towel on truth, content, and propositions, and acknowledge that we are not limited when it comes to those, but that there is a limitation nonetheless, but we can’t even properly articulate our limitations. To illustrate with the advanced aliens again, the idea is that although they are not doing better than we when it comes to truth and content, they are doing better when it comes to truth* or schmuth and content* or schmontent. When the aliens look down at us from their advanced spaceships, they will certainly take us to be limited, and maybe we can’t quite say how, but they might think of us as missing out on some important truths* or contents*. Now, this is certainly right in many ways, but is wrong in the crucial way that matters here. We are clearly limited

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when it comes to the aliens—we don’t have the spaceships, and their invasion of earth might be a walk in the park for them. But the crucial issue for us is whether we are limited in what we can represent conceptually. And here the thought experiment motivates no such thing. Truth*, whatever it is supposed to be, isn’t truth, and content* isn’t content. Whichever of our many limitations we consider they are not limitations in what we can in principle represent conceptually. If internalism is true then we can reason conclusively that every fact can in principle be represented conceptually by us, although other things can’t be conceptually represented by us, but these are not the kinds of things that conceptual representation is supposed to represent in the first place. Conceptual representation is complete in its domain, and this indicates that there is no proper separation between what is to be represented conceptually and the representations that do the representing.32 Let us now return to the argument from analogy, which was somewhat postponed in its assessment above. When the aliens look down at us like we look down at the honeybee then we are correct in thinking that they are superior to us in many ways, but incorrect in thinking that they can represent more facts than we do. When they think about us as these primitive creatures, we cannot truly report their thoughts that the primitive humans don’t get all the facts. That simply isn’t true, assuming internalism, of course. We might not be as good as the aliens in all kinds of things, but when it comes to representing the facts we are. There is no question that there is the strong sense that there is a limitation of us that is analogous to the limitation of the honeybee. It might be hard to state what that limitation is supposed to be more precisely, but there clearly is the sense of a limitation motivated by analogy. However, assuming 32 The conclusion here is similar to one drawn by Donald Davidson (1984). However, it is reached in quite different ways. Davidson held that due to his particular theory of meaning it was impossible for there to be a language that is in principle untranslatable into our language, and thus that there could not be variation like this across conceptual schemes, and thus the notion of a conceptual scheme is based on a mistake. The present view holds that there could not be content that goes beyond what we can say in principle, not because of a theory of meaning, but because of what we do when we talk about propositions. A second similarity is to Hilary Putnam’s (1981) argument that we are not brains in vats, since the question whether we are, as stated by us, is guaranteed to have a negative answer. Internalism is not tied to skepticism, but the fact that the focus on the question can be a key to its answer is similar in both Putnam’s argument as well as here. It should be noted that the internalist picture of the propositional is quite different from McDowell’s view on the matter. McDowell explicitly contrasts his view with an “arrogant anthropomorphism” (1994, 39) which holds that we can represent all facts with our present conceptual resources. In contrast he holds that what there is to represent about reality is not influenced by our conceptual resources, but an independent fact about it. See section 8 of Lecture 2 in McDowell (1994) which is devoted to this issue.

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internalism, we can reason conclusively that there is no relevant limitation. We will have to weigh the sense of a limitation against the reasons against a limitation. Since reasons make things more reasonable than merely feeling a limitation, it is thus reasonable to conclude that the sense is misguided and the argument from analogy, although powerful, is in the end mistaken. We can thus conclude, assuming internalism about our own talk about propositions, that there is nothing the aliens can truly say that we can’t say in principle as well. When it comes to conceptual representation we face no limitation, and this is a consequence of what the propositional aspect of reality is like. Internalism does not give us reason to think we are limited, but it gives us a different picture of the propositional. The propositional is not part of reality in the sense that there is no domain of propositions or facts of which we might capture more or fewer. Instead, it reflects our employment of talk about propositions, and it doesn’t and can’t go beyond that. The propositional is a reflection of our talk about propositions. This might sound like a version of idealism, and it is.

5.8 Idealism vindicated Idealism sounds bad, but it doesn’t have to be when properly stated. Idealism broadly understood holds that, in some form or other, minds, in particular minds of creatures like us, are central to reality. The most common and maybe most natural way to be an idealist is to hold that what reality is like depends on our minds. And this is naturally spelled out as ontological idealism: the view that what there is depends on our minds in some way. Many of the historically significant idealists were idealists of this sort, but it is rather problematic and rightly widely rejected. It is not clear how the notion of dependence is supposed to be understood such that idealism so formulated is compatible with other things we know to be true. We know that the universe existed before there were any humans and we know roughly what it was like before we were around. It is not clear how this is compatible with what there is depending on us, on a natural understanding of dependence. In particular, it is not clear how to even state the idealist position without it leading immediately to conflict with many other things we know. Ontological idealism is false, but idealism might still be true.33

33

A weaker form of idealism is to hold that even though what there is does not depend on us, nonetheless, there being minds like ours is no accident. Any world must contain some minds, and without minds there could be no material world at all. A version of this position was defended by Anton Friedrich Koch with an intriguing argument in (1990) and (2010). I don’t believe that his argument works, and have tried to say why not in

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An alternative way of thinking about idealism is to tie it not to ontology but rather to the conceptual or propositional aspect of reality. Although what there is is independent of us, the fact-like or propositional aspect of reality is not. That is, reality, understood as what there is, is independent of us, but reality, understood as what is the case, is not. This way of understanding idealism faces the same problems at first as ontological idealism. How can we coherently state a notion of dependence for the propositional that doesn’t immediately conflict with what we know to be the case? Here we have to distinguish two ways in which the propositional could be dependent on us. The propositional is truth-dependent on us just in case which propositions are true is dependent on us. In this sense it is, of course, false that the propositional depends on us. Which truths are true is not something that in general depends on us. We can make some propositions true or false by affecting the world, but we don’t make the true propositions true in general. Alternatively, the propositional is range-dependent on us just in case the extent of the propositional is dependent on us, which is to say, what propositions there are as candidates for being true depends on us, somehow and in some sense. The natural way to understand this is that what can be represented conceptually in principle is, somehow, dependent on us and our conceptual resources. And a natural way to understand that is that it is guaranteed that all there is to represent about reality can be represented by us, in principle. And this in turn means that the effability thesis is true for a reason connected to us. The view that this is so was called conceptual idealism in section 3.4. And this is the form of idealism that might be true. Conceptual idealism is thus the view that the propositional is rangedependent on us. It is a form of idealism, but different from and independent of ontological idealism.34 As discussed briefly above, the standard route to conceptual idealism, and with it the standard defense of the effability thesis, is via ontological idealism. But conceptual idealism might be true even if ontological idealism is false, and internalism about talk about propositions and facts supports just that option. If internalism is true then the propositional depends on us, not for its truth, but for its range. Internalism thus supports idealism, not ontological idealism, of course, but conceptual idealism. Conceptual idealism avoids the incompatibility worries that ontological idealism faces rather directly. It is a promising candidate to provide a coherent

Hofweber (2015). Further, completely different, recent versions of idealism include Adams (2007), whose paper title I borrow for this section, and Smithson (2015). 34 In Chapter VI of Nagel (1986), Thomas Nagel takes something like conceptual idealism to be the defining mark of idealism. This is slightly unusual, but I believe with Nagel that the real issue about idealism is just that. Nagel, of course, rejects idealism so understood.

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formulation of idealism. Although the idealist might be tempted to try to say more about how the world depends on us, it is not clear how we can do this coherently. It is not clear how we can on the one hand stick with our picture of a world that is there for us to discover and which has us as just a small part of it, while at the same time elaborating on how the world in the end depends on us. But we can formulate conceptual idealism coherently. And if there is at least this kind of dependence, range dependence of the propositional, then this supports a version of idealism that we can make sense of.35 If externalism is true for talk about objects, but internalism is true for talk about propositions then ontological realism and conceptual idealism are natural consequences. What there is is in general independent of us. But what can in principle, by anyone, be said about it is not independent of us. To illustrate this again one last time, consider events. They, as entities, in general occur without us having anything to do with which ones occur and which ones don’t. They are simply there. But what can be truly said about them, in the sense of what the possible range of a propositional or conceptual description of them is, is not independent of us. Internalism shows why and how we come in here, since the range of the propositional is not independent of us. That way ontological realism gets combined with conceptual idealism. Ontological realism is not compatible with the truth-dependence of propositions on us, since propositions about what there is can’t depend on us for their truth while it is independent of us what there is. But the propositional can be range-dependent on us while ontological realism is true. There is no incoherence here since we are working with an object-permitting notion of the effable, and thus the range of the propositional can accommodate all there is, whatever it might be. Even though what things there are is independent of us, both in its range and for its existence, and even though these things can figure in the facts that can in principle obtain, the range of facts that can obtain is not independent of us. We can in principle eff everyone of them, on an object-permitting notion of effability, the relevant notion of 35 Hilary Putnam, in (1981) and other places, has defended a view he calls “internal realism” that goes by the motto that there is no ready-made world. However, Putnam focuses on ontology and hopes to argue that the world does not come by itself carved into objects. What there is, for Putnam, is tied to our ways of talking about it, and thus his view is best understood as a version of ontological idealism. As Simon Blackburn (1994) has argued quite successfully, this view leads to the consequence of a conflict between our statement of the idealist position with other things we take ourselves to know to be true, and thus turns the view into an incoherent one, given what we know. Putnam’s view focuses on the world not being ready-made when it comes to the objects that inhabit it. I find it more fruitful to consider the conceptual or propositional aspect of the world to be, using the same metaphor, not ready-made. This can, I hope to make clear, be stated coherently. Even if the world of objects is ready-made, the world of facts is not, on the way to spell out the metaphor attempted here.

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effability for our debate. The object-permitting conception of effability and internalism about talk about the propositional allow for the combination of the dependence of the facts on us, on the one hand, with the independence of the things on us, on the other. 6 . C O N C LU S I O N Whether or not we should accept the ineffability thesis or the effability thesis depends on what we should think we do when we talk about propositions. If externalism is true about such talk then we should accept the ineffability thesis and the sub-algebra hypothesis. This would be insignificant for most of inquiry and ordinary life, since, on the sub-algebra hypothesis, the ineffable will be there, but it will be hidden from us in a way that makes clear why it is insignificant for these purposes. But the ineffable will be significant for metaphysics, in particular debates about what reality as a whole is like. We should expect that we will naturally be misled to believe that our sub-algebra is all there is. Here ineffability should lead to modesty in grand metaphysics. But if, on the other hand, internalism is true about talk about propositions then the effability thesis will be true, and we can explain why it seems to us that some aspects of reality should be ineffable for us. No modesty would follow for metaphysics from this, but the metaphysical picture of the propositional that is tied to the internalist view of talk about propositions is itself a substantial consequence. It combines a version of realism, in that reality as what there is is independent of us, with a version of idealism, in that what there is to say about reality can all be said by us, as we are right now, not by mere accident, but for a reason. The internalist picture of the propositional makes clear why content cannot be beyond us and thus all there is to say about reality can be said by us in principle. Internalism thus implies conceptual idealism, but is compatible with ontological realism. The question whether internalism or externalism is true about our talk about propositions is a largely empirical question about what we do when we talk about propositions. It is a question about our actual use of certain expressions in natural language, and thus something that we can’t settle on the basis of a priori reflection. The crucial question on which this issue depends is thus one about language, and a largely empirical question at that. Idealism, properly understood, and the effability thesis follow if things turn out one way; ineffability and modesty follow if they turn out another. If the former then not only would it support idealism, which might sound bad enough, but furthermore it would support idealism on empirical grounds, which might sound even worse. Nonetheless, since I have argued in other

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work that internalism about the propositional is, as best as I can tell, true, I side with idealism and the effability thesis.36 Thomas Hofweber University of North Carolina, Chapel Hill

R E F E REN C E S Adams, R. (2007). “Idealism Vindicated.” In P. van Inwagen and D. Zimmerman, eds, Persons Human and Divine, Oxford: Oxford University Press, 35–54. Bach, K. (1997). “Do Belief Reports Report Beliefs?” Pacific Philosophical Quarterly 78(3): 215–41. Blackburn, S. (1994). “Enchanting Views.” In P. Clark and B. Hale, eds, Reading Putnam, Oxford: Blackwell, 12–30. Chomsky, N. (1975). Reflections on Language, New York: Pantheon Books. Collins, J. (2002). “The Very Idea of a Science Forming Faculty.” Dialectica 56(2): 125–51. Davidson, D. (1984). “The Very Idea of a Conceptual Scheme.” In Davidson, Inquiries into Truth and Interpretation, Oxford: Clarendon Press, 183–98. Dretske, F. (1981). Knowledge and the Flow of Information Cambridge, MA: MIT Press. Field, H. (1994). “Deflationary Views of Meaning and Content.” Mind, 103(411): 249–85. Filcheva, K. (2015). “Can there be ineffable propositional structures?” Unpublished manuscript. Fodor, J. A. (1983). The Modularity of Mind. Cambridge, MA: MIT Press. Hofweber, T. (2000). Quantification and Non-Existent Objects.; In A. Everett and T. Hofweber, eds, Empty Names, Fiction, and the Puzzles of Non-Existence, Stanford: CSLI Publications, 249–73. Hofweber, T. (2005). “A Puzzle about Ontology.” Noûs, 39(2): 256–83. Hofweber, T. (2006). “Inexpressible Properties and Propositions.” In D. Zimmerman, ed., Oxford Studies in Metaphysics, volume 2, Oxford: Oxford University Press, 155–206. Hofweber, T. (2007). “Innocent Statements and their Metaphysically Loaded Counterparts.” Philosophers’ Imprint 7(1): 1–33.

36 Thanks to David Baker, Elizabeth Barnes, Mike Bertrand, Cian Dorr, Kyle Driggers, Matti Eklund, Krasimira Filcheva, Jeremy Goodman, John Hawthorne, Carrie Ichikawa Jenkins, Andreas Kemmerling, Anton Friedrich Koch, Matt Kotzen, Dan Lopez de Sa, Bill Lycan, Ofra Magidor, Anna-Sara Malmgren, A. W. Moore, Alan Nelson, Ram Neta, David Reeve, Sven Rosenkranz, Geoff Sayre-McCord, Ted Sider, Rob Smithson, Moritz Schulz, Corina Strößner, and Juhani Yli-Vakkuri for helpful discussions and/or comments on earlier drafts. Special thanks to Karen Bennett for comments on an earlier draft that led to substantial revisions.

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Hofweber, T. (2009). “Ambitious, yet Modest, Metaphysics.” In D. Chalmers, D. Manley, and R. Wasserman, eds, Metametaphysics: New Essays on the Foundations of Ontology. Oxford: Oxford University Press, 269–89. Hofweber, T. (2015). “The Place of Subjects in the Metaphysics of Material Objects.” dialectica 69(4): 473–90. Hofweber, T. (2016a). “From Remnants to Things, and Back Again.” In G. Ostertag, ed., Meanings and Other Things: Essays in Honor of Stephen Schiffer. Oxford: Oxford University Press, Chapter 2. Hofweber, T. (2016b). Ontology and the Ambitions of Metaphysics. Oxford: Oxford University Press. Kant, I. (1781). Kritik der reinen Vernunft. Riga: Johan Friedrich Hartnoch. Keisler, H. J. (1971). Model Theory for Infinitary Logic. Amsterdam: North-Holland. Koch, A. F. (1990). Subjektivität in Raum und Zeit. Frankfurt am Main: Klostermann. Koch, A. F. (2010). “Persons as Mirroring the World.” In J. O’Shea and E. Rubinstein, eds, Self, Language, and World. Atascadero: Ridgeview Publishing, 232–48. McDowell, J. (1994). Mind and World. Cambridge, MA: Harvard University Press. McGinn, C. (1989). “Can We Solve the Mind-Body Problem?” Mind 98(391): 349–66. McGinn, C. (1993). Problems in Philosophy: The Limits of Inquiry. Oxford and Malden, MA: Blackwell. Moltmann, F. (2003a). “Nominalizing Quantifiers.” Journal of Philosophical Logic 32(5): 445–81. Moltmann, F. (2003b). “Propositional Attitudes without Propositions.” Synthese 35(1): 77–118. Moore, A. (2003a). “Ineffability and Nonsense.” Proceedings of the Aristotelian Society 77: 169–93. Moore, A. (2003b). “Ineffability and Religion.” European Journal of Philosophy 11(2): 161–76. Nagel, T. (1986). The View from Nowhere. Oxford: Oxford University Press. Priest, G. (2002). Beyond the Limits of Thought. Oxford: Oxford University Press. Putnam, H. (1981). Reason, Truth, and History. Cambridge: Cambridge University Press. Rosefeldt, T. (2008). “ʽThat’-Clauses and Non-nominal Quantification.” Philosophical Studies 137(3): 301–33. Rosen, G. (1994). “Objectivity and Modern Idealism: What is the Question?” In J. O’Leary-Hawthorne and M. Michael, eds, Metaphysics in Mind. Dordrecht: Kluwer, 277–319. Rosenkranz, S. (2007). “Agnosticism as a Third Stance.” Mind, 116(461): 55–104. Schiffer, S. (1987). Remnants of Meaning. Cambridge, MA: MIT Press. Schiffer, S. (2003). The Things We Mean. Oxford: Oxford University Press. Smithson, R. (2015). “Edenic Idealism.” Unpublished manuscript. Varley, J. (1977). The Ophiuchi Hotline. New York: The Dial Press.

8 The Metaphysics of Quantities and Their Dimensions Bradford Skow 1. INTRODUCTION Quantities have dimensions. Force, mass, and acceleration are quantities— for now let this just mean that they may be faithfully measured by numbers—and the dimension of force is “ML=T 2 ,” the dimension of mass is “M ,” and the dimension of acceleration is “L=T 2 ,” where “M ” stands for mass, “L” for length, and “T ” for time, or duration. While the notion of a quantity is familiar to those with only a passing acquaintance with physics, the notion of a quantity’s dimension is probably not.1 Even though they are unfamiliar, dimensions play a variety of important roles in physics, and science generally. Perhaps the most important one is the role they play in the technique of dimensional analysis. One of the aims of physics is to discover laws, and laws of physics usually take the form of relations between quantities—Newton’s second law, F ¼ ma, is a relation between the quantities force, mass, and acceleration. Newton’s discovery of this law involved, in part, consulting a lot of observational evidence. Dimensional analysis is a different technique for discovering law-like relations between quantities, one that doesn’t involve inferring a generalization from a collection of instances. If you know only that some law or other relates force, mass, and acceleration, but you also know the dimensions of these quantities, and a few other facts, you can use dimensional analysis to conclude that the law is F ¼ ma; even if you don’t have any of the data that Newton did. How to use the technique won’t matter in what follows.2 The 1

The notion of the dimension of a quantity has nothing to do with the familiar notion of a dimension of space. 2 For the record, and simplifying a little: to apply dimensional analysis we need to know that the force on a body is some function of its mass and acceleration, so that F ¼ gðm; aÞ for some function g, and we need to know that the function g makes this

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point is that dimensions can help answer important physical questions. But should metaphysicians care about dimensions? Is the fact that the dimension of acceleration is L=T 2 in any way a metaphysically interesting fact about it? Does this fact, for example, have anything to do with the nature of acceleration? The physicist R. C. Tolman thought it did; he wrote in 1917 that a statement of the dimension of a quantity is “a shorthand restatement of its definition and hence [is] an expression of its essential physical nature” (quoted in Bridgman 1922: 26). To say that the dimension of acceleration is L=T 2 is certainly to say that there is some connection or other between acceleration and length and duration. Tolman here says that the connection has something to do with the essence, or nature, of acceleration. Could that be right? It seems right to me, at least initially, but it has had powerful opponents. Percy Bridgman, who won the Nobel Prize in physics in 1946, scorned views like Tolman’s, scorn that comes across in his description of those views: It is by many considered that a dimensional formula has some esoteric significance connected with the “ultimate nature” of an object, and that we are in some way getting at the ultimate nature of things in writing their dimensional formulas. (1922: 24)

Bridgman went on to give several arguments against the idea that dimensional formulas have “some esoteric significance,” or, as we might put it, that they have some metaphysical significance. His view was that dimension

equation “dimensionally consistent.” An equation is dimensionally consistent iff the dimension of the quantity on the left-hand side is the same as the dimension of the combination of quantities on the right. Assuming we know these things, we can reason: force has dimension ML=T 2 , so gðm; aÞ must be a combination of mass and acceleration that also has dimension ML=T 2 . Clearly if g is multiplication, so gðm;aÞ ¼ ma, we get this result, since the dimension of ma is the product of the dimension of m and the dimension of a, namely ML=T 2 . The Buckingham Π theorem tells us that, in this case, multiplication is the only function you can put in for g to make the equation dimensionally consistent. (For a more detailed explanation of dimensional analysis, including a proof of the Π theorem, see Barenblatt 1996. Technically, applying the technique requires a weaker assumption than dimensional consistency, sometimes called “dimensional homogeneity;” for more on this distinction see Lange 2009, p. 760.) Although I have contrasted dimensional analysis with the use of observational data, dimensional analysis is not an a priori technique; the things one must know to apply it, namely the dimensions of the quantities and that the quantities stand in a dimensionally homogeneous relationship, cannot be known a priori. In fact, regarding the example I have used, the usual route to coming to know that the dimension of force is ML=T 2 requires prior knowledge of the law that F ¼ ma; those of us who know the dimension of force in this way obviously could not use our knowledge to discover that F ¼ ma.

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formulas just reflect arbitrary choices we have made when setting up our scales of measurement. I’m going to examine this debate. The idea that the dimension of a quantity has something to do with its nature can be made precise in different ways; I will identify two distinct ways to make it precise, and defend one of them. This debate is not one that philosophers discuss, or have ever discussed. At least, I have not been able to find anything written on it.3 The question is, nevertheless, interesting, worthy of our attention. I think this is obvious; further evidence comes from the fact that physicists like Tolman and Bridgman saw that dimensional analysis raised questions about the metaphysical import of talk of the dimension of a quantity, and proceded to take those questions up. For this reason my primary interlocutors will be those physicists. If you have never before been exposed to talk of the dimension of quantity, either through exposure to dimensional analysis or through some other avenue, you might feel that you are in no position at all to think about this debate. It’s a little like asking someone who has had no exposure to physics to evaluate hypotheses about the essential properties of electrons. Fortunately, there is widespread agreement about part of the answer to the question of what a quantity’s dimension tells us about it. The debate is over whether a quantity’s dimension tells us any more than 3 There are certainly articles in philosophy journals that discuss dimensional analysis, for example Marc Lange’s (2009) “Dimensional Explanation” and Luce’s (1971) “Similar Systems and Dimensionally Invariant Laws.” But I haven’t found any that discuss the nature of dimensions, or what dimensions say about the nature of quantities. This is not to say that philosophers have ignored questions about the nature of quantities. In fact some questions about the nature of quantities are increasingly a focus of attention. Take the property of being 2 kg in mass. Is this a fundamental, or perfectly natural, property? If it is non-fundamental, what are the fundamental mass properties or relations, and how is being 2 kg mass defined in terms of them? Mundy (1987) can be understood as holding that the property of being 2 kg mass, and all other particular mass properties, are fundamental, while Field’s views about quantities (Field 1980) suggest that it is the “mass-less” and “mass-congruence” relations that are fundamental (intuitively, a mass-less b holds iff a is less massive than b; and a, b are mass-congruent to c, d holds iff the difference in mass between a and b is the same as the difference in mass between c and d). There are other views also (surveyed in Eddon 2013). I think that these questions about the nature of quantities are orthogonal to the question I take up, the question of what dimensions say about the nature of quantities. For these questions, the questions Mundy’s and Field’s and others’ work is relevant to, are questions “internal” to a given quantity. The question they want to answer about mass—the question of which mass properties or relations are most fundamental—is a question just about mass, not about its relation to any other quantities. But the question I’m asking, the question of what the dimension of a quantity says about its nature, is a question about the relations between that quantity and other quantities, as we will see.

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that. (Tolman’s claim that it tells us something about the quantity’s nature goes beyond the agreed-on part.) Seeing the agreed-on claims about dimensions will, I hope, help the uninitiated get a better grip on what the debate is about. So that is what I will start with, in the next section.

2 . Q U A N T I T I E S , S C AL E S, A N D D I M E N S I O N FORMULAS Dimensions are had by quantities; so I should start by saying a bit more about what a quantity is. Familiar examples are, again, force, mass, and acceleration, and also velocity, momentum, and energy. I will take a quantity like mass to be a family of properties, the “specific values” of that quantity. So the mass family includes the properties, or values, having 1 kg mass, having 2 kg mass, and so on. There is a distinction between “vector” quantities and “scalar” quantities: all quantities have “magnitude,” but vector quantities also have a “direction” associated with them. Although many quantities of interest are vector quantities (force and acceleration for example), for the sake of simplicity I will limit my attention to scalar quantities, and often pretend that some vector quantities like force are scalar quantities. An important notion for us is that of a scale of measurement of a quantity. A scale associates a number with each value of that quantity; the kilogram scale, for example, associates the number 2 with the property of having 2 kg mass. Now of course there are tons of ways to associate numbers with mass values, many of them useless—it would be futile to associate the number 3 with every mass value. What I care about are “faithful” scales, scales that adequately reflect the intrinsic structure of the quantities they are scales for. I won’t go into what that intrinsic structure is in a great deal of detail; it will be enough for our purposes to note that the property of having 4 kg mass is, in some sense, “double” the property of having 2 kg mass, and that it is in virtue of this that a scale is faithful only if the number it assigns the first property is double the number it assigns the second. In general, just as the ratio of the property of having 4 kg mass to the property of having 2 kg mass is 2, any two values of a given quantity stand in a definite numerical ratio, and a faithful scale will assign to those values numbers that stand in the same ratio. It follows that to set up a scale for measuring a quantity it is enough to choose a unit—the value v of that quantity that shall be associated with the number 1. The number the scale assigns to any other value u is then determined by the ratio of u to v: if the ratio of u to v is n, then the scale must assign n to u, if it is to be faithful.

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It follows, further, that any two faithful scales for a given quantity, like mass, differ from each other by a positive multiplicative constant: if we write SðvÞ ¼ n to mean that scale S assigns quantity value v the number n, then if S1 and S2 are faithful scales for a quantity, there is a fixed positive number K such that S2 ðvÞ ¼ KS1 ðvÞ for every value v.4 For example, the numbers the centimeter scale assigns to lengths are all 100 times larger than the numbers the meter scale assigns to lengths. Now at the beginning of this paper I said what the dimensions of some quantities (mass, acceleration) were, and quoted Tolman’s claim that the dimension of quantity has something to do with its nature or definition. This talk of the “dimension” of a quantity is, however, a little nebulous, and it is time now to introduce more precise terminology. The term we need is “the dimension formula of a quantity.” Formulas are in the same ontological category as sentences. The dimension formula of mass is “M ,” a capital “m,” and the dimension formula of acceleration is “L=T 2 ,” a capital “l” followed by a slash, etc. Just as we can ask what a sentence means, we can ask what a formula means. The most uncontroversial claim about dimension formulas says this about their meanings: (1)

The dimension formula of a quantity defines that quantity’s dimension function.

Great; so what’s a dimension function? Here is the answer: 4 I should note that this claim only applies to scales for quantities with a certain structure, which might be described as “scalar quantities with a ratio structure and a lower bound.” These are the only kinds of quantities I will be concerned with in this paper. You might worry that temperature is a counterexample to the claim I make in the text about scalar quantities. The numbers the Kelvin scale assigns to temperatures are not a fixed multiple of the numbers the Celsius scale assigns to temperatures (the temperature value to which Celsius assigns the number 0, Kelvin assigns 273.15, but there is no positive K with 0 ¼ K 273:15). This is not inconsistent with my claim, though, since the Celsius scale is not faithful. The short explanation of why it is not faithful is that it fails to adequately reflect that temperature has a lower bound—to do so it would have to assign 0 to the lower bound. The longer, and true, explanation is more complicated. In fact temperature does not have a lower bound. It is possible for a thermodynamic system to have a negative temperature on the Kelvin scale. Still, even though temperature does not have a lower bound, the Celsius scale is not faithful—and neither is the Kelvin scale. Systems with negative temperatures (on the Kelvin scale) are hotter than systems with positive temperatures (on the Kelvin scale)—the first kind of system is disposed to transfer energy to the second kind. But a faithful scale should assign higher numbers to hotter things (this is a problem for the Celsius scale, since it assigns negative numbers to the temperatures to which the Kelvin scale assigns negative numbers). Since temperature is not, in the end, a quantity with a lower bound, it technically falls outside the scope of my claim that any two scales for a quantity differ by a positive multiple.

176 (2)

Bradford Skow A quantity’s dimension function tells you how the numbers assigned to values of that quantity change, when you change from using one system of scales of measurement to another (in the same class).

This needs a lot of unpacking, of course; we will want to know what a system of scales is, what it is for two systems to be in the same class, and how the dimension function tells you the cited fact—we will want to know what the function’s inputs and outputs are. But (1) and (2) at least makes one of the dimension formula’s roles clear. If we switch from using the meter scale for length to using the centimeter scale for length, the numbers assigned to length values all go up by a factor of 100; somehow this function that the dimension formula for length defines, the dimension function for length, is going to deliver up to us this factor. To a philosopher, the job description in (1) that dimension formulas answer to may not look very interesting. The question is going to be, the interesting debate is going to be over, whether there is any other, metaphysically interesting, job that dimension formulas do. Before getting to that question I should spell out how dimension formulas define dimension functions, and how dimension functions work.5 A system of scales of measurement is a set of related scales for measuring a set of quantities. A system of scales designates some quantities as “primary” and some as “secondary.” The primary quantities are those measured by scales in which the unit is not chosen for its relation to the unit of any other scale. The secondary quantities are those measured by scales in which the unit is chosen for its relation to the units of other scales. For example, we usually think in terms of a system in which speed is a secondary quantity, and length and duration are primary. The unit for speed is chosen by reference to the units for length and duration: we say that the unit for speed is the speed at which something travels when it covers one unit of distance in one unit of time. That’s one way to choose a unit for speed, given units for other quantities. It is not the only way—we could choose the unit to be the speed at which something travels when it covers two units of distance in one unit of time. In general, there is always more than one way to choose a unit for a secondary quantity, given units for the primary quantities.6 5 Much of the material to follow comes from Chapter 1 of Barenblatt 1996, though my terminology is in some places slightly different. 6 Here is a more interesting example: suppose we want force to be a secondary quantity, and length, mass, and duration to be primary. The usual way to choose a unit for force is to say that it is the amount of force required to produce unit acceleration in something with unit mass. But we could choose a unit for force by saying that the unit

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To set up a system of scales, then, we divide the quantities we will measure into those that will be primary and those that will be secondary, select units for the primary quantities, and specify how the units for the secondary quantities are determined by the units for the primary ones. Suppose two systems of scales X1 and X2 measure all the same quantities, that the set of quantities designated as primary is the same in both systems, and that the units for secondary quantities are defined by reference to the units for primary quantities in the same way. So all differences between X1 and X2 flow from differences over which values of the primary quantities are the units. Then X1 and X2 belong to the same class of systems. Again, when we change from using a scale S1 for a given quantity Q to using a scale S2 , the numbers assigned to values of Q all get multiplied by some constant K . If S1 and S2 belong to systems of scales that are in the same class, then even if Q is a secondary quantity, the constant K is determined by the scales for the primary quantities: (3)

If S1 (the “old” scale) and S2 (the “new” scale) are scales that measure the same quantity Q, and they belong to systems of scales X1 and X2 that are in the same class, then the constant K satisfying S2 ðvÞ ¼ KS1 ðvÞ (where v is a value of Q) is a function of the ratios of the old units to the new units of the scales for the primary quantities.

This function, which takes some numbers (the ratios of the old units to the new units) and outputs K , is the dimension function of Q. To know how the dimension function is defined we look to the dimension formula for Q (recall (1)), and interpret the letters that appear in the dimension formula to denote the numbers that are the ratios of the old units to the new units of the relevant quantities (and interpret the other symbols in the dimension formula, like “/,” to denote the usual mathematical operations). To see how all this works, let us assume that we are working in a class of systems of scales in which only length, duration, and mass are primary, and in which the unit for speed, a secondary quantity, is defined in the usual way. In this case, there are three primary quantities, and so, since dimension functions take as input the ratios of old to new units of the primary quantities, all dimension functions take three arguments. The dimension formulas that define these functions contain the letters “M ,” “L,” and “T ,” and no other letters; when a dimension formula is used to define a dimension function, “M ” is interpreted to name the ratio of the old unit for mass to the new unit, “L” the ratio of the old unit for length to the

force is the strength of the gravitational force between two bodies of unit mass that are one unit of distance apart.

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new unit, and “T ” the ratio of the old unit for duration to the new unit. When Q is a quantity, ½Q is its dimension function, and the conversion factor K is given by K ¼ ½QðM;L;T Þ, the application of the function ½Q to the arguments M;L;T .7,8 This may be clearer if I work through some examples: Example 1: Mass. The dimension formula for mass, at least in the class of systems of scales we are using, is, again, “M .” So ½massðM;L;T Þ ¼ M , and M is the number that is the ratio of the old unit for mass to the new unit for mass. So suppose we are changing from a system of scales that uses grams to measure mass, to a system that is otherwise the same except that it uses kilograms to measure mass. The old unit for mass then is the gram, and the new unit is the kilogram. The ratio of the gram to the kilogram is 1/1000 (a gram is 1/1000th of a kilogram). So in this case M = 1/1000. Then if the old scale, the gram scale, assigns the number n to a mass value v, we compute that the new scale, the kilogram scale, assigns the number [mass](M, L, T)n = Mn = (1/1000)n to v. The result is familiar: the numbers assigned to masses change by a factor of 1/1000 when we change from using grams to using kilograms. Example 2: Speed. The dimension formula for speed, at least in the class of systems of scales we are using, is “L=T .” So the dimension function for speed is defined by the equation ½speedðM;L;T Þ ¼ L=T: If we change from using meters to using centimeters, and use the same scales for the other primary quantities, then L, the ratio of the old unit for length to the new unit, is equal to 100, while M ¼ T ¼ 1. So if the old scale for speed assigns the number n to a speed value v, we compute that the new scale assigns the number ðL=T Þn ¼ 100n to v. Again the result is familiar: the numbers assigned to speeds change by a factor of 100 when we change from using the “meters per second scale” to using the “centimeters per second” scale.

7 I should note that the claim that the conversion factor for a quantity in this class is a function, in the mathematical sense, of three arguments, is not the same as the claim that the conversion factor for a given quantity depends on each of those arguments. The function defined by the expression f ðx;y;zÞ ¼ x 2 is a function of three arguments, but its value only depends on the first argument. The examples below illustrate this distinction. 8 I said that everything in this section is common ground in the debate over the nature of dimensions. This may not quite be right. Bridgman says that on his view “the symbols in the dimensional formula [are] reminders of the rules of operation which we used physically in getting the numerical measure of the quantity;” they do not represent “the facts used in changing from one set of units to another” (30). But he also says that his view “cannot be distinguished” from the one I am describing “as far as any results go.” I am not entirely sure what Bridgman means when he says we should regard dimension formulas as reminders. Barenblatt clearly endorses the claim that dimension formulas just serve to define dimension functions, and that the letters “M ” and so on denote ratios of old units to new (1996: 31–2).

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One thing about dimension functions is worth emphasizing. The dimension function of a quantity gives the factor by which the scale for measuring it changes when we pass from one system of scales to another system in a given class of scales. So dimension functions, and dimension formulas too, are class relative. For example, we usually take force to be a secondary quantity, and mass a primary one; the dimension formula for force is then ML=T 2 . But we could instead use a class of systems of scales in which mass is secondary, and force primary; the dimension formula for force is then just F , while the dimension formula for mass is FT 2 =L.9 The class-relativity of dimension formulas (and dimension functions) will be important in the next section.

3 . T HE P O S I T I V I S T T H E O R Y O F D I M E N S I O N S , A N D IT S O P P O N E N T S Here are the teams. On the one side, Tolman and others, who think that the dimension of a quantity has something to do with its nature. On the other, Bridgman and his allies, who deny this. I will call the view of Bridgman and his allies The Positivist Theory of Dimensions;10 to have it set out explicitly, if a bit vaguely, the view is this: THE POSITIVIST THEORY: The dimension formula of a quantity defines the dimension function of that quantity (relative to some class of systems of scale). It does not (in addition) have anything to do with the nature or essence of that quantity. So what, exactly, do those who oppose the positivist theory believe? I will distinguish between two distinct anti-positivist theses. These theses in turn 9 To figure this out, we assume that the dimension of the right-hand side of F ¼ ma is the same as the dimension of the left-hand side. Acceleration is still secondary, and its unit still chosen in the customary way, so its dimension formula is still L=T 2 . So the dimension formula Z for mass must satisfy F ¼ Z L=T 2 , where what can go in for Z are products of powers of the primary quantities L;T; and F . The only solution is FT 2 =L. (Why products of powers? I say something about this in the appendix.) 10 The appearance of “positivist” is meant to be suggestive—Bridgman was an operationalist, the worst kind of positivist—but should not be taken too seriously. Other proponents include Max Plank: “to inquire into the ‘real’ dimension of a quantity has no more meaning than to inquire into the ‘real’ nature of an object” (1932: 46; see also page 8); Barenblatt (1996); (Langhaar 1951: 5); and (Palacios 1964: xiv). These people may not all subscribe to the positivist view exactly as I state it, but my interest is in the view I state, not the interpretive question of whether, and if so just where, its allies’ actual views deviate from it.

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share two presuppositions, so before stating the theses I want to articulate those presuppositions. The first presupposition is that there is a “natural” way to divide quantities into primary and secondary. Return to the two systems of scales I ended the last section with. One designated mass as primary and force as secondary, the other did the reverse. These systems belong to different classes (for, again, a class of systems of scales is partly determined by a choice of which quantities are to be regarded as primary and which secondary). Now there are various criteria we might use to evaluate these classes of scales. One of them might, for example, be better than another if we are intersted in making certain kinds of measurements. Anti-positivists think that we can also evaluate them with respect to whether they designated as primary only quantities that really are primary. The presupposition here is this: NON-RELATIVITY: Quantities are not just primary, or secondary, relative to this or that class of systems of scales; they are also primary, or secondary, in a non-relative way. The positivist theory rejects NON-RELATIVITY. However, despite the positivist theory’s opposition to it, I think NON-RELATIVITY is a quite plausible thing to believe. NON-RELATIVITY can appear even more tempting if we use a different word in place of “primary”: doesn’t it seem right to say that some quantities are fundamental in a non-relative way? The idea that some properties are fundamental and others are derivative has great currency in metaphysics.11 I certainly think we should accept it. Since quantities are just families of properties, we should also accept that some quantities are fundamental and others derivative.12 This gives us a way to “argue” for NON-RELATIVITY. Some quantities are fundamental, and a quantity is primary simpliciter13 iff it is fundamental; so some quantities are primary simpliciter. I don’t expect this argument to convince any skeptics— that’s why “argue” is in scare-quotes—but it at least draws a connection between being primary simpliciter and a notion, fundamentality, that is already well understood.14 11

Its popularity among contemporary metaphysicians traces back to (Lewis 1983). There are details to work out about the relationship between “fundamental” as a predicate of properties and “fundamental” as a predicate of quantities. We could say that a quantity is fundamental iff every one of it values is a fundamental property; or iff the disjunction of its values is a fundamental property; and there are other options. I will not try to work out these details here. 13 “Primary simpliciter” is just a short way to write “primary in a non-relative way.” 14 A referee objected: Gross Domestic Product (GDP) is surely not a fundamental quantity; so I must hold that it is a secondary quantity, and therefore that its dimension formula, in a class of systems of scales that designates as primary the quantities that really 12

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I do not mean to suggest that NON-RELATIVITY is a completely unproblematic thesis. If we accept that some quantities are primary simpliciter, we are going to want to know which quantities those are. How do we go about figuring this out? Can we know which quantities are primary simpliciter, or are we bound to remain ignorant? I will only say something brief about these questions. The same questions arise with respect to fundamental properties. In that case, one common suggestion is that we take our best physical theories as guides to which properties are fundamental—if a property appears in one of those theories (is expressed by a predicate in one of those theories), that is good evidence it is fundamental. To the extent that this is a good answer to the epistemic questions about fundamental properties, an analogous suggestion is a good answer to the epistemic questions about primary quantities: if physicists, when their aims involve no practical computations, but only the formulation of the laws, use a system of scales in which a quantity is primary, that is good evidence that that quantity is primary simpliciter. A lot more could be said about the epistemology of primary quantities, and how it does or does not recapitulate the epistemology of fundamental properties, but since these epistemic questions are not relevant to what follows, I will leave the topic here. Anti-positivists presuppose NON-RELATIVITY because they presuppose that each quantity has a “true” dimension formula, and a quantity cannot have a “true” dimension formula unless NON-RELATIVITY holds. For recall that a given quantity has tons of dimension formulas, one for each class of systems of scales that measures that quantity. One of those dimension formulas can only be its “true” dimension formula if nature singles out exactly one class of systems of scales as the “true” class; a quantity’s true

are primary, is some combination of “M ,” “L,” “T ,” and so on (assuming that mass, length, and duration are primary). This, however, is implausible. I am not entirely sure what to think about GDP, or other quantities that appear in economics and other special sciences, but I am tempted by this reply. Consider the quantity frequency. The unit for frequency is chosen to be the value of frequency had when the relevant system completes one “cycle” in one unit of time. But “cycles” is not a quantity: one does not choose a unit for measuring cycles, one just counts cycles. That’s why the dimension formula for frequency contains “T ” (the formula is “1=T ”) but no letters that have anything to do with cycles. The same, I suspect, goes for GDP. We (in the US) usually measure GDP in dollars per year, but (my suspicion is) we should not regard the dollar as a unit for measuring value, but as a thing we count, like cycles. (If the dollar and the euro were both units for measuring value, then one dollar should always be some fixed number of euros. But it’s not; the exchange rate fluctuates.) If this is right, then it is not after all implausible that GDP’s dimension formula contains only letters for fundamental physical quantities: its dimension formula is the same as that of frequency, “1=T .”

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dimension formula is then its dimension formula in the true class. What would nature have to do to single out exactly one class of systems of scales as the true class? It certainly must single out one set of quantities as those that are primary simpliciter, so it can say that the true class has the property of designated as primary all and only the quantities that are primary simpliciter. But it turns out—and I kind of wish this weren’t so—that accepting NON-RELATIVITY is not sufficient for defining “true class,” and so is not sufficient for defining “true dimension formula.” As I mentioned above, a class of systems of scales is determined, not just by a choice of which quantities to designate as primary, but also by a way of choosing the units of the secondary quantities, given units for the primary quantities. And there is always more than one way to choose a unit for a secondary quantity. So anti-positivists need a second assumption: RIGHT UNITS: for any quantity that is secondary simpliciter, only one way of choosing a unit for that quantity given units for the primary quantities is the “right” way. If RIGHT UNITS is correct, then the true class of systems of scales can be defined as the class that designates as primary exactly the quantities that are primary simpliciter, and chooses the units for the secondary quantities in the right way. RIGHT UNITS is, I think, harder to swallow than NON-RELATIVITY. We usually choose the unit for speed to be the value had by something that moves unit distance in unit time. But, as I said earlier, we could choose it to be the value had something that moves two times the unit distance in unit time. Which is the “right” way for defining a unit for speed? Some might find it hard to believe that this question has an answer. Different people might find it hard for different reasons; one reason might be that, if it does have an answer, it is hard to see how we could know what that answer is. This particular example, however, doesn’t lead me to doubt RIGHT UNITS; the usual choice for a unit of speed, the choice that designates as the unit the speed something has when it moves unit distance in unit time, is, at least in some sense, more natural than any alternative. But other examples are harder.15 Still, I will not take an in-depth look at the cases for and 15 It is tempting to say that the right way for choosing a unit for a secondary quantity never uses “arbitrary constants” (as the second way for choosing a unit for speed uses the number 2). But this is not the end of the story. Some quantities, like the curl of a vector field, cannot be defined in terms of more fundamental quantities in any way without using something like an arbitrary constant (in this case, an arbitrary choice of an orientation for space). One thing to say about this case is that curl is not a scalar quantity, so is of a kind of quantity that I am ignoring. A more substantive response is to say that curl is not a “real” quantity, since from a four-dimensional perspective one never needs to

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against RIGHT UNITS here. I will say, though, that one of the theses I will be interested in, which I will shortly name DEFINITIONAL CONNECTION, can get by without assuming RIGHT UNITS, even though the most natural statement of if does require that assumption. (I will explain how it can avoid RIGHT UNITS in section 6.) So DEFINITIONAL CONNECTION is still something you could believe, even if you can’t stomach RIGHT UNITS. Having flagged that the anti-positivist theses to be discussed make presuppositions which could easily be challenged, let’s press on. Tolman’s idea was that the dimension of quantity has something to do with its essential nature, that it is a “shorthand restatement of its definition.” Now Kit Fine famously linked essences to definitions in his 1994 paper “Essence and Modality.” The essence of a thing is its “real definition”; it says “what a thing is,” as a “nominal” definition says what a word means. I will read Tolman’s reference to a quantity’s “definition” as talk of its real definition, and so also as talk of its essence. For the sake of precision it is worth regimenting all such talk using the same locution, so I will use the operator “It is definitional of X that . . . .” Using this notion, just what does Tolman means when he says that the dimension of a quantity is a restatement of its definition? For now I will only give a vague answer; I will take up the task of making it more precise in section 5: DEFINITIONAL CONNECTION: Let P1, . . . , Pn be the symbols for the primary quantities that appear in quantity Q’s dimension formula in the true class of systems of scales. Then some relationship between Q and the primary quantities associated with the symbols P1, . . . , Pn is definitional of Q. That’s very abstract, so it will help to see an instance. Suppose the speed is a secondary quantity, and length and duration are primary, and that the dimension formula for speed in the true class is “L=T .” Then, if DEFINITIONAL CONNECTION is right, some relationship between speed, length, and time is definitional of speed. Just what this relationship might be, I will say more about in section 5. Before spending more time on DEFINITIONAL CONNECTION, I want to get another anti-positivist idea about dimensions on the table. The other one is,

use the curl of a vector field to do physics. (Given the theory of relativity, speed is not a real quantity either, but my argument in the paper is compatible with this; speed, even in section 5, serves only as an example.) Further problems come from very simple quantities. Here are two ways to select a unit for area: let it be the area value had by a square with unit-length sides; let it be the area value had by an equilateral triangle with unit-length sides. Which is “right”? Neither uses an arbitrary constant. If I had to take a stand, I would say that the square is right, but I don’t know how to justify this answer.

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in fact, more popular. It is suggested by another expression of anti-positivism, due to W. W. Williams: The dimensional formulae may be taken as representing the physical identities of the various quantities, as indicating, in fact, how our conceptions of their physical nature . . . are formed, just as the formula of a chemical substance indicates its composition and chemical identity. . . . The question then arises, what is the test of the identity of a physical quantity? Plainly it is the manner in which the unit of that quantity is built up (ultimately) from the fundamental units L, M , and T [ . . . ]. (Quoted in Bridgman 1922: 26; the paper quoted from was published in 1892)

Some of what Williams says is close to DEFINITIONAL CONNECTION. But Williams also draws an analogy between dimensional formulas and chemical formulas. The idea seems to be that, just as the chemical formula for glucose, “C6H12O6,” tells us that a glucose molecule is built out of six carbon atoms, twelve hydrogen atoms, and six oxygen atoms, the dimension formula for acceleration, “L=T 2 ,” tells us that, and how, acceleration values are “built up from” lengths and durations. This idea identifies a second role dimensional formulas might play, in addition to that of defining dimension functions.16 Generalizing gives us our second anti-positivist thesis: CONSTRUCTION: Let Q be any quantity, and D its dimension formula in the true class of systems of scales. If Q is non-primary, then D exhibits the way in which values of Q are constructed, or built out of, values of primary quantities. The picture here is that there are several different ways in which two quantity values may be combined to create a third. Two quantity values may be multiplied together; or one may be divided by the other; or one quantity value may be raised to some power. For example, if we assume that speed is a secondary quantity in the true class, and that length and duration are primary, then the dimension formula of speed in the true class is “L=T .” In this formula the symbols for length and duration flank the division sign. According to CONSTRUCTION, this indicates that speed values are constructed by taking length values and dividing them by duration values. The meter per second, for example, which is a speed value, is constructed by dividing the meter (the value of length assigned the number 1 when we choose the meter as our unit) by the second.

16

A formula can define a function without indicating how anything is built up from other things: the formula “x  y” defines the subtraction function, but even though 12 ¼ 10  2; the formula doesn’t indicate that 12 is built up from 10 and 2 in any sense. Conversely, chemical formulas indicate how molecules are built from atoms, but do not define any functions.

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It is worth emphasizing that CONSTRUCTION says that some quantity values—properties—are built, by operations with names like “multiplication” and “division,” out of other values. Some resistance to CONSTRUCTION is based on failing to appreciate this fact. Bridgman complains of “treat[ing] the dimensional formula as if it expressed operations actually performed on physical entities, as if we took a certain number of feet and divided them by a certain number of seconds” (29). Later on the page he asserts that “We cannot perform algebraic operations on physical lengths, just as we can never divide anything by a physical time” (29).17 If Bridgman is saying that the following instruction makes no sense, then he is absolutely right: You’ve got a meter stick (which is a “physical entity”) in your left hand. Please take it and divided it by the stretch of time during which this table exists.18

Division just is not an operation defined on meter sticks and stretches of time. But this observation does not touch CONSTRUCTION, which makes no claims or presuppositions about dividing physical things by each other. One could try to make Bridgman-like observations about quantity values. Pounding the table one could say, “Multiplication is an algebraic operation defined on numbers, not on quantity values!” But this is only kind of true. Mathematicians happily talk of multiplying functions together, and of dividing one group by another. Of course the operations going under the names “multiplication” and “division” here are different from those going under those names when we are working with numbers. But the operations are closely enough related that we use the same name for them. I see no reason why a set of quantity values could not have an algebraic structure like that had by sets of numbers, functions, and groups, so that there were operations on sets of quantity values that deserved to be called multiplication and division. We now have two anti-positivist theses on display, DEFINITIONAL CONNECTION and CONSTRUCTION (which share the presuppositions NONRELATIVITY and RIGHT UNITS). The first thing I want to emphasize about them is that they are different theses. Granted, there is probably a way of understanding talk of one property being constructed out of others on which DEFINITIONAL CONNECTION, or something quite close to it, follows from CONSTRUCTION. But DEFINITIONAL CONNECTION does not entail CONSTRUCTION; it is logically weaker. You can think that dimension formulas tell us something about the natures of quantities without thinking that quantities are constructed out of other quantities. 17

Barenblatt echoes this argument (1996: 30). Suppose substantivalism about time is true, so that there are such entities as stretches of time. 18

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My main aim in this paper is to say something in favor of DEFINITIONAL CONNECTION. Distinguishing between DEFINITIONAL CONNECTION and CONSTRUCTION is an important part of my argument, since many of the arguments proponents of the positivist view give make contact only with CONSTRUCTION, and leave DEFINITIONAL CONNECTION untouched. Before discussing DEFINITIONAL CONNECTION explicitly I want to look at the best of those arguments, partly to illustrate its irrelevance to DEFINITIONAL CONNECTION, but also because it is worth, at least briefly, coming to some conclusion about CONSTRUCTION. My conclusion will be that it is defensible, though I myself see no good motivation for defending it.

4 . B U I L D I N G S O M E Q U AN T I T I E S F R O M O T H E R S ? One argument against CONSTRUCTION is especially popular; I myself have endorsed it in the past.19 Here is a statement of it from Barenblatt’s book Scaling, Self-Similarity, and Intermediate Asymptotics:20 [I]f the relations for the derived units [that is, units for the secondary quantities] mentioned above were actually to make sense as products or quotients of the fundamental [that is, primary] units, they would have to be independent of what we mean when carrying out the multiplication or division. For example, according to the above definition [which I will not quote, but which entails something like CONSTRUCTION], kgf m is the derived unit for the moment of a force as well as for work; m2/s is the derived unit for the stream function as well as the kinematic viscosity, etc. But it is not implied that the stream function is measured in multiples of a basic amount of kinematic viscosity or that the moment of a force is measured in multiples of a basic amount of work! In contrast, using our definition [of dimension formulas, namely the positivist theory], the fact that the dimensions of two physical quantities of different nature are identical does not seem unnatural. (1996: 33)

I am not entirely sure what Barenblatt’s argument here is. But I can find an interesting argument in here. Let’s examine it and not worry about whether it is exactly what Barenblatt had in mind. Barenblatt’s central observation is that there are distinct quantities with identical dimension formulas (in some given class of system of scales). He mentions kinematic viscosity and the stream function. Roughly speaking, the kinematic viscosity of a fluid is a measure of how hard each bit of the fluid “pulls” against its neighbors when its neighbors move past it. In a 19

In my (2012), and section 7.4 of my (2015). Something like this argument also appears in (Palacios 1964: xiii) and (Duncan 1953: 123). 20

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highly viscous fluid, when a bit of fluid neighboring B moves past it, B pulls rather hard against that neighbor. (Honey has higher viscosity than water. As a practical matter, how viscous a fluid is tells you how hard it will be for you to move through it: as you move you pull the bits of the fluid that touch you along with you, and this is more difficult to the extent that those bits’ neighbors pull back.) As for the stream function: if you imagine a fluid (technically an incompressible fluid) flowing in two dimensions, the value of the stream function at a point P tells you how much fluid flows across the line from P to an arbitrarily given point A in unit time. Clearly the stream function and kinematic viscosity are distinct quantities. A blob of honey, all at the same temperature, has the same viscosity everywhere. The stream function, on the other hand, is a property, not of a thing like a blob of honey, but of an event, the motion of a fluid. It varies from one point in the flow to another. Nevertheless, in any class of systems of scales that designates length and duration as primary quantities (and designates the stream function and kinematic viscosity as secondary), the stream function and kinematic viscosity have the same dimension formula, L2 =T . Why might this be trouble for CONSTRUCTION? Here is one way to spell out a simple argument against CONSTRUCTION that uses Barenblatt’s observation as a premise: (P1)

If CONSTRUCTION is true, then there cannot be distinct quantities with the same dimension formula (in the true class of systems of scales). (P2) But there are distinct quantities with the same dimension formula. (C) Therefore, CONSTRUCTION is false.

A reason to believe (P1) is not hard to find. If CONSTRUCTION is true then values of kinematic viscosity are built out of values of length and time. Specifically, if CONSTRUCTION is true, then you get a value of kinematic viscosity by taking a value of length, multiplying it by itself, and then dividing the result by a value of duration. But if performing these operations on values of length and duration gives you a value of kinematic viscosity, then performing those operations on those same values cannot also give you a value of a distinct quantity like the stream function. In general: distinct quantities with the same dimension formula would have to be built up from the primary quantities in the same way. But it cannot happen that distinct quantities are built up from the same primary quantities in the same way. This might sound convincing, but it goes beyond what CONSTRUCTION officially says. There are two ways a proponent of CONSTRUCTION may resist (P1). First Way. CONSTRUCTION does not say that it is impossible for distinct quantity values to be built up from the same basic values in the same way.

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The argument above for (P1) smuggled in an extra premise, a “uniqueness” premise. Here is one way to put it: UNIQUENESS: For any way W of building a quantity value out of other quantity values q1;:::;qn, the application of W to q1;:::;qn is unique. If UNIQUENESS is true, and if multiplication and division are ways of building quantity values, then we do have a good argument for (P1). But UNIQUENESS may be rejected. True, multiplication and division of numbers satisfies uniqueness. But there is plenty of precedent for non-mathematical analogues of mathematical operations to fail to have all the properties of their mathematical counterparts. Take summation. Lots of metaphysicians happily embrace the thesis that a collection of material things may have more than one sum. Plenty think that you can have a bunch of clay particles that have two sums: one of the sums is a lump of clay, the other sum is a statue. The observation that distinct quantities can have the same dimension formula, then, does not refute CONSTRUCTION.21 Second Way.22 Barenblatt’s argument may refute CONSTRUCTION when it is interpreted one way, but there is a better way to interpret it that escapes the argument. Officially CONSTRUCTION just says that Q’s dimension formula “exhibits the way in which values of Q are constructed, or built out of, values of primary quantities.” I have been interpreting this to mean that, if a quantity (like viscosity) has for its dimension formula “L2 =T ” then each of its values is completely specified by saying that it is the square of a given length value, divided by a given value of duration. On this interpretation, the dimension formula contains complete information about how a quantity’s values are built up from the basic values. But there is a weaker way to interpret CONSTRUCTION: interpret it to say that a quantity’s dimension formula gives only partial information about how the quantity values are built up. This weaker interpretation becomes more plausible when you look at the precise (nominal) definitions of “kinematic viscosity” and of “the stream function” (they are rather involved, so I won’t state them here): the quantities invoked in the two definitions are quite different. On the weaker interpretation, CONSTRUCTION does not say that quantities with the same dimension formula are built up in the same way, so even if UNIQUENESS is true, (P1) is not. 21 This is not the end of the matter. Fans of uniqueness of summation say their opponents face a “grounding problem”: given that the statue and the clay are sums of the same particles, how do they manage to have different properties? Fans of UNIQUENESS might mount a similar argument. Following this thread of the dialectic would take me too far away from my main line of argument. My guess is that it would parallel the dialectic in the case of summation, in which the denial of uniqueness remains a viable position. See Bennett 2004 for more. 22 Thanks to Martin Glazier and Karen Bennett, who (separately) proposed this way to me.

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Still, even though CONSTRUCTION is defensible, I am not tempted to endorse it, and I agree with Bridgman that the main temptation to endorse it is based on a mistake. What is that temptation? Consider a law relating different quantities, for example Newton’s second law. What is Newton’s second law? Of course everyone knows what it is: F ¼ ma: But what does this string of symbols mean? How should it be translated into English? Here is a common translation: (4)

The net force on any material body is equal to the result of multiplying that body’s mass and that body’s acceleration.

But if this is right then it must make sense to multiply a mass and an acceleration. And surely mass and acceleration are not special here. So the truth of Newton’s second law supports the thesis that quantity values can be multiplied together. Again, surely multiplication is not special here; if quantity values can be multiplied together then one value can also be divided by another. Once we’re comfortable with this idea it is not a great leap to start thinking of multiplication and division as ways of constructing new quantity values from old ones, of thinking, for example, of speed values as got by dividing length values by duration values.23 But I do not think that (4) is what F ¼ ma means in English. In this denial I follow Bridgman, and I also follow him in his views about what equations like F ¼ ma do mean. He wrote that24 . . . x1 [in an equation relating some quantities] might stand for the number which is the measure of a speed, x2 the number which is the measure of a viscosity, etc. By a sort of shorthand method of statement we may abbreviate this long-winded description into saying that x1 is a speed, but of course it really is not, but is only a number which measures speed. (1922: 17–18)

What goes for speed goes for force, mass, and acceleration. Bridgman might just as well have said that while the “a” in “F ¼ ma” stands for the number which is the measure of a value of acceleration, we might as a sort of shorthand say that a is a value of the quantity acceleration, or even that a is an acceleration. But those who did not know this was shorthand would be misled about what “a” denotes.25 (4) is not true when read literally, it is only true when read as shorthand. When we read it that way, it does not support CONSTRUCTION.

23

For a recent endorsement of this argument, see van Inwagen 2014: 104–5. See also Duncan 1953: 6 and Palacios 1964: 18. 25 See also page 41 in Bridgman 1922 where he complains about people thinking that something like (4) is the reason why all true laws can be written so that the dimensions of the left side and the right side are the same. 24

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What, exactly, is the correct and fully explicit translation of F ¼ ma into English? Let Z be the class of systems of scales to which the system we currently use, the SI system, belongs; then, I think, the correct and fully explicit translation of F ¼ ma is (5)

For any material body B, and for any system of scales of measurement S in Z, the number S assigns to the net force on B is equal to the result of multiplying the number S assigns to B’s mass and the number S assigns to B’s acceleration.26

As I said, I am not tempted to endorse CONSTRUCTION. Even though I have not, in this section, defended any arguments that it is false, there is an argument against it that I like. But since my main aim in this paper is to defend DEFINITIONAL CONNECTION, not to refute CONSTRUCTION, and since I want to get on to that main aim now, I will save that argument for a later section (section 6).

5 . D E F E N D I N G A D E F IN I T I O N A L C O N N E C T I O N Now that CONSTRUCTION has had its day in the sun I want to turn to DEFINITIONAL CONNECTION, which has not received nearly as much attention. Here it is again: DEFINITIONAL CONNECTION: Let P1, . . . , Pn be the symbols for the primary quantities that appear in quantity Q’s dimension formula in the true class of systems of scales. Then some relationship between Q and

26 As I have been, I am pretending that force and acceleration are scalars. It is straightforward to write down the correct vectorial version of (5), but brings in unnecessary complications. One might concede that (5) is the translation of “F ¼ ma” into English and try to put together a more complicated argument for CONSTRUCTION. (5) makes reference to numbers. One might argue that the truth of (5) must be grounded in some facts that do not involve numbers. And perhaps those facts are facts about the result of multiplying mass values and acceleration values. But this argument does not succeed. One need not accept CONSTRUCTION to have facts available to ground the truth of (5). Instead one could say that the facts that ground (5) are facts like these: if the force on any material body were doubled, while its mass (and everything else) remained the same, then its acceleration would double; if the mass of any material body were halved, while the force on it (and everything else) remained the same, then its acceleration would double; and so on. In general: a body’s acceleration is directly proportional to the force on it, indirectly proportional to its mass, and independent of everything else. (In fact Newton first formulated his second law of motion in terms of proportions; see Newton 1999: 416.)

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the primary quantities associated with the symbols P1, . . . , Pn is definitional of Q. The first thing to do is to make DEFINITIONAL CONNECTION more specific: what relationship between Q and the primary quantities is definitional of Q? Unfortunately, I do not know how to answer this in a completely general way. I know how to do it for particular quantities, just not for an arbitrary quantity. So let’s focus on a claim about one particular quantity that is supposed to follow—given an assumption I’m about to state—from the more-specific-but-unstated version of DEFINITIONAL CONNECTION. The assumption (one I’ve made many times) is that speed is a secondary quantity, and length and duration primary. If this is true, then DEFINITIONAL CONNECTION is meant to entail the following claims about the essences of particular values of speed: • It is definitional of the property of moving at 1 m/s that anything that has this property throughout some temporal interval that is 1 second long moves along a curve in space that is 1 meter long. • It is definitional of the property of moving at 1 m/s that anything that has this property throughout some temporal interval that is 2 seconds long moves along a curve in space that is 2 meters long. • It is definitional of the property of moving at 4 m/s that anything that has this property throughout some temporal interval that is 2 seconds long moves along a curve in space that is 8 meters long.27 Using the dimension formula for speed (in the true class of systems of scales) we can wrap these and all similar claims into one generalization: (DEFINITIONAL CONNECTION—Speed): Let a;b; and c be any positive real numbers with the following property: if you interpret “L” to denote b, and “T ” to denote c, in the dimension formula for speed, and “compute the result,” you get a. (Since the dimension formula for speed is “L=T ,” the requirement is that a ¼ b=c.) Let S be any system of scales in the true class, and let r be the speed value that S assigns the number a, l the length value that S assigns the number b, and d the duration value that S assigns the number c; then it is definitional of r that anything that has r during a temporal interval that has d moves along a curve in space that has l . For every secondary quantity Q (in the true class), there is a generalization like (DEFINITIONAL CONNECTION—SPEED), in which Q’s dimension formula 27 In all of these claims, occurrences of descriptions of properties are to be read de re. That descriptive content, which contains reference to numbers, is not part of the definitions of these properties.

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in the true class appears, that advocates of DEFINITIONAL CONNECTION will accept. DEFINITIONAL CONNECTION is to be understood to be equivalent to the conjunction of these generalizations.28 As a reminder, all of this is premised on the assumption that speed is a secondary quantity. I made that assumption just so I could write down a model for the generalizations that DEFINITIONAL CONNECTION is the conjunction of. If speed is not a secondary quantity, then advocates of DEFINITIONAL CONNECTION will not accept (DEFINITIONAL CONNECTION—SPEED), and will not understand DEFINITIONAL CONNECTION to entail it. Is DEFINITIONAL CONNECTION correct? I don’t have a knock-down argument for it here. My goal is to offer it for your consideration, and try to say a few things to make it plausible. As I said earlier, just isolating it from CONSTRUCTION is some defense of it; for a lot of the controversy around CONSTRUCTION does not apply to DEFINITIONAL CONNECTION. The other main thing I have to say in defense of DEFINITIONAL CONNECTION is this: if you like the at-at theory of motion, then you should like DEFINITIONAL CONNECTION.

28 In any class, including the true class, some quantities are dimensionless: their dimension function is the constant ½Q ¼ 1. This means that every system of scales in the given class assigns the same numbers to values of Q—no matter what units those scales choose for the primary quantities. Clearly if a quantity is dimensionless in the true class, then DEFINITIONAL CONNECTION says nothing interesting about what relations between that quantity and the primary quantities are definitional of that quantity. Is the existence of dimensionless quantities any kind of problem for DEFINITIONAL CONNECTION? Well, if every secondary quantity were dimensionless (only secondary quantities can be dimensionless), DEFINITIONAL CONNECTION would be empty. But it is false that every secondary quantity is dimensionless—obviously, if there are primary quantities, which there are, then some secondary quantities are not dimensionless (for example the “squares” of the primary quantities—if length is primary, area is its square—are not dimensionless). One idea I have heard is that it may turn out, some hope that it turns out, that the quantities that appear in the laws of physics are all dimensionless. If this is right, and if the only “important” quantities are the ones that appear in the laws, then DEFINITIONAL CONNECTION concerns only unimportant quantities. This might be thought to diminish its importance. But I find it hard to imagine how the laws could be stated in purely dimensionless terms: surely acceleration will continue to appear in the laws, and surely there is no way to make acceleration dimensionless? Nor do I see why one should hope that the laws can be stated using only dimensionless quantities. The usual thought is that then the laws will be independent of our scales of measurement. But even laws involving dimensionful quantities, like F ¼ ma, can be stated in a way that is independent of our scales of measurement (see footnote 4). In passing: though DEFINITIONAL CONNECTION says nothing about what is definitional of a dimensionless quantity, it does not follow that dimensionless quantities lack real definitions. It is just that their dimension formulas play no role in stating those definitions.

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Why this is so emerges from trying to get clear on just what the at-at theory of motion says. Let’s start by looking at a classic statement of the theory, namely Bertrand Russell’s: motion consists merely in the occupation [by some given material thing] of different places at different times. (1937: 473; emphasis added)

Other presentations of the theory use similar language; Frank Arntzenius, for example, states the theory as the view that “there is nothing more to motion than the occupation of different locations at different times” (2000: 189; emphasis added). What is the “merely” doing in the Russell passage, and the “nothing more” in the Arntzenius? A natural interpretation has these words serving to mark that the theory is stronger than WEAK AT-AT: Necessarily, for any body B, B is in motion during temporal interval I iff B occupies different places at different times in I. WEAK AT-AT is consistent with the idea that motion, location, and time are “independent” things that are necessarily connected. But the language Russell and Arntzenius use suggests that this idea is not supposed to be consistent with the at-at theory. We can express a stronger version of the atat theory (stronger than WEAK AT-AT) using talk of real definitions. Talk of real definitions gives us a way to make sense of the idea of things that are independent but necessarily connected. For then we can construe “independent” not as “modally independent” but as “have independent defintitions”: two things have independent definitions iff neither’s real definition places any constraints on what is true of the other. It is part of the point of recognizing the legitimacy of talk of real definitions, or essences, or natures, that it makes at least conceptual room for necessary connections between things that do not follow from the natures of those things. This idea is the root of Kit Fine’s (1994) well-known arguments against modal theories of essence. So one way to do justice to Russell’s and Arntzenius’s talk of what motion “consists merely” in, and of what motion is “nothing more” than, is to interpret them as asserting that some connection between motion, location, and time, is definitional of motion. More specifically, a better (maybe the best) statement of the at-at theory looks like this: STRONG AT-AT: It is definitional of motion that, for any body B, B is in motion during temporal interval I iff B occupies different places at different times in I. Now the at-at theory is not supposed to be just a theory of the nonquantitative notion of motion; it is also a theory of the quantitative notions, speed and velocity. So what does the analogue of Strong At-At for speed look

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like? It must say, for every speed, that it is definitional of that speed that something cover a certain distance in a certain time. But what exactly does the theory say? Which “certain distance” and “certain time” appear in the definition of a given speed? The answer comes from having the theory assert (DEFINITIONAL CONNECTION—SPEED), which I repeat here: (DEFINITIONAL CONNECTION—SPEED): Let a;b; and c be any positive real numbers with the following property: if you interpret “L” to denote b, and “T ” to denote c, in the dimension formula for speed, and “compute the result,” you get a. (Since the dimension formula for speed is “L=T ,” the requirement is that a ¼ b=c.) Let S be any system of scales in the true class, and let r be the speed value that S assigns the number a, l the length value that S assigns the number b, and d the duration value that S assigns the number c; then it is definitional of r that anything that has r during a temporal interval that has d moves along a curve in space that has l . The thesis DEFINITIONAL CONNECTION—SPEED is part of the at-at theory of motion, properly understood. My argument has been that if you like the at-at theory, you should like DEFINITIONAL CONNECTION. That’s a bit vague so let me be more specific. One way to like the at-at theory is to think that it is true; and if you think it is true, you should definitely like DEFINITIONAL CONNECTION, for the quantitative version of the at-at theory is just part of DEFINITIONAL CONNECTION. Of course, one can consistently accept (DEFINITIONAL CONNECTION—SPEED) but reject the other parts of DEFINITIONAL CONNECTION. But presumably if you are drawn to the at-at theory of motion, you will also be drawn to theories of other secondary quantities (energy, for example, if it is a secondary quantity) that are similar in spirit. That similarity in spirit amounts to this: they also assert that it is definitional of values of the secondary quantity in question that they are related to certain values of the primary quantities. There is, however, another way to like the at-at theory, one that does not require you to believe it is true. This is important, since the at-at theory is mildly controversial even in the context of pre-relativistic and pre-quantum physics, and, I think, more controversial when relativity and quantum mechanics are taken into account.29 You can like the at-at theory even while thinking it false by thinking that it has the right form. What form is that? You could think that the at-at theory is right to presume that the correct theory of any secondary quantity will assert that it is definitional of that secondary quantity that it is related in a certain way to the primary

29

For arguments against the theory see Arntzenius 2000 and Lange 2005.

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quantities. You could think that where the at-at theory goes wrong is (merely) in holding that speed is a secondary quantity.30

6. DEFINITIONAL CONNECTION WITHOUT R IG H T U N IT S The main work of this paper is done, but there is a loose end I want to tie up. DEFINITIONAL CONNECTION, as I stated it, presupposes RIGHT UNITS, a presupposition which might seem implausible. I said that DEFINITIONAL CONNECTION could be formulated without this presupposition; in this section I will show how to do this. The fact that DEFINITIONAL CONNECTION can be formulated without presupposing RIGHT UNITS is also relevant to evaluating CONSTRUCTION. CONSTRUCTION, as far as I can tell, must presuppose RIGHT UNITS. So if RIGHT UNITS is objectionable, its being objectionable provides a reason to believe DEFINITIONAL CONNECTION rather than CONSTRUCTION. Without RIGHT UNITS we cannot say that choosing the unit for area to be the value for area had by a square with unit length sides is an “objectively better” choice than choosing the unit to be the value had by an equilateral triangle with unit length sides. It might seem that DEFINITIONAL CONNECTION requires RIGHT UNITS, because stating the generalizations that appear in DEFINITIONAL CONNECTION requires referring to the true class of systems of scales, and “true class” is defined in terms of right units. But in fact this is not so. Here are two ways of stating the definitional connection between area and length without referring to the true class: • Let m and n be any two positive reals with n ¼ m2 . Let S be a system of scales that designates length as primary and chooses the unit for area to be the area value had by a square with unit length sides. And let a be the area value S assigns n, and l the length value S assigns m. Then it is definitional of a that anything that has a has the same area as a squarepwith ffiffi sides of length l . • Let m and n be any two positive reals with n ¼ 43 m2 . Let S be a system of scales that designates length as primary and chooses the unit for area to be the area value had by an equilateral triangle with unit length sides. And let a be the area value S assigns n, and l the length value S assigns m. Then it

30 The standard alternative to the at-at theory of motion says that motion is not defined by reference to location and time. The analogous view of speed has it that speed is not defined by reference to length and duration, and so that speed is a primary quantity. This is compatible with DEFINITIONAL CONNECTION.

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is definitional of a that anything that has a has the same area as an equilateral triangle with sides of length l . We can avoid assuming RIGHT UNITS by taking either (or both!) of these to be conjuncts in DEFINITIONAL CONNECTION, and taking the other conjuncts to be stated in a similar way. DEFINITIONAL CONNECTION still presupposes NON-RELATIVITY, for the quantities that DEFINITIONAL CONNECTION concerns are meant to be all and only the quantities that are secondary simpliciter. And dimension formulas are still used to state DEFINITIONAL CONNECTION; n ¼ m2 just is (an instance of ) the dimension formula for length (in the given system), and pffiffi n ¼ 43 m2 is the dimension formula multiplied by a constant. It is harder to see how CONSTRUCTION could do without RIGHT UNITS. Is the product of the property of being 1 meter with itself identical to the area property had by things that are the same area as a square with meter-long sides, or is it identical to the area property had by things that are the same area as an equilateral triangle with meter-long sides? It cannot be identical to both; picking one goes hand-in-hand with privileging one way of choosing a unit for area over another. If RIGHT UNITS makes you a lot more nervous than NON-RELATIVITY, you should be more drawn to DEFINITIONAL CONNECTION than to CONSTRUCTION.

7 . C O N C LU S I O N My goals in this paper have been modest. I have not set out to prove that DEFINITIONAL CONNECTION is true from premises that are in any interesting sense independent of it. Instead, my aims with respect to this thesis were these. First, to get DEFINITIONAL CONNECTION on the table, and on the agenda. Something close to DEFINITIONAL CONNECTION has surfaced in discussions of the metaphysics of dimensions, but it has not until now been clearly distinguished from CONSTRUCTION. And clearly distinguishing it is a prerequisite to reaching a clear-headed conclusion about whether it is true. A second aim was to say something in defense of DEFINITIONAL CONNECTION. I myself found that appreciating the relation between DEFINITIONAL CONNECTION and the at-at theory of motion increased my confidence in it, and I suspect it may do the same for others. I will not pretend that this amounts to a comprehensive treatment of DEFINITIONAL CONNECTION—here, as elsewhere in metaphysics, it is hard to find good arguments. I hope, instead, that it is a good start on a worthy project. In the appendix that follows, I pursue a question that CONSTRUCTION raises that did not fit into the paper’s main line of argument. Bradford Skow Massachusetts Institute of Technology

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A P P E N D I X : A B I T M O R E O N CO N S T R U C T I O N CONSTRUCTION, on one reading at least, entails that some quantity values are got by “multiplying” other quantity values together, and some are got by “dividing” one quantity value by another, and some are got by “raising a quantity value to a power.” For example, if speed is a secondary quantity, the view is that the speed value 4 meters per second is the result of dividing being 4 meters long by being 1 second in duration. Here is a pointed question for fans of CONSTRUCTION: why are these the only three operations?31 Why can’t you construct a new quantity by “taking the sine of a length value,” or “subtracting the natural logarithm of a duration value from a mass value”? It might look at first like this question should embarrass proponents of CONSTRUCTION. Surely they’ll either have to say that it is just a brute fact that these are the only operations; or they’ll have to say that there is a quantity got by taking the sine of length, it is just nowhere instantiated in the actual world (or of no interest to physicists). But proponents of CONSTRUCTION do not need to say either of these things. There is a proof in most treatments of dimensional analysis that can be used to explain why the only operations are multiplication, division, and exponentiation (for Bridgman’s version see (Bridgman 1922: 21–2)). Here is the theorem proved: Let X1 and X2 be two systems of scales of measurement in the same class. Suppose also that X1 and X2 agree on the ratios of numbers assigned to quantity values. That is, if the quotient of the numbers X1 assigns to values v and u of some given quantity is r, then the quotient of the numbers X2 assigns to those values is also r. Then for any quantity Q measured by X1 and X2 , the dimension formula for Q is a “power law monomial,” that is, it has the form “KP1m P2n :::,” where symbols for the primary quantities go in for “P1 ” and so on, and symbols denoting real numbers go in for “K;” “m;” and so on. How is this relevant? Suppose some “quantity” were, say, the sine of length. Then its dimension formula (in the true class) would not be a power law monomial. So there would be systems X1 and X2 in the true class which did not agree on the ratios of numbers assigned to quantity values. Since these systems are faithful to the intrinsic structure of this “quantity” (we have always restricted ourselves to faithful scales), there must not be scale-independent facts about the ratios of values of this “quantity.” But that is definitional of quantities! This contradiction shows that no quantity can be the sine of length, and more generally, that quantities can only be constructed from other quantities using multiplication, division, and exponentiation. (I leave it open whether those who like CONSTRUCTION will want to say that there is something (which is not a quantity) that is the sine of length.)32

31 Really we only need two; division can be defined in terms of multiplication and exponentiation. 32 Thanks to Martin Glazier, an audience at the University of Rochester, and an audience at NYU. Thanks especially to Karen Bennett, without whose large investment of editorial efforts this paper would not be what it is.

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Arntzenius, Frank (2000). “Are There Really Instantaneous Velocities?” The Monist 83(2): 187–208. Barenblatt, G. I. (1987). Dimensional Analysis. Trans. Paul Makinen. New York: Gordon and Breach Science Publishers. Barenblatt, G. I. (1996). Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge: Cambridge University Press. Bennett, Karen (2004). “Spatio-Temporal Coincidence and the Grounding Problem.” Philosophical Studies 188(3): 339–71. Bridgman, Percy (1922). Dimensional Analysis. New Haven: Yale University Press. Duncan, W. J. (1953). Physical Similarity and Dimensional Analysis. London: Edward Arnold & Co. Eddon, Maya (2013). “Quantitative Properties.” Philosophy Compass 8(7): 633–45. Field, Hartry (1980). Science Without Numbers. Princeton: Princeton University Press. Fine, Kit (1994). “Essence and Modality.” Philosophical Perspectives 8: 1–16. Inwagen, Peter van (2014). Metaphysics. 4th edn. Boulder: Westview Press. Lange, Marc (2005). “How Can Instantaneous Velocity Fulfill Its Causal Role?” The Philosophical Review 114(4): 433–68. Lange, Marc (2009). “Dimensional Explanations.” Noûs 43(4): 742–75. Langhaar, Henry (1951). Dimensional Analysis and Theory of Models. New York: John Wiley & Sons. Lewis, David (1983). “New Work for a Theory of Universals.” Australasian Journal of Philosophy 61(4): 343–77. Luce, R. Duncan (1971). “Similar Systems and Dimensionally Invariant Laws.” Philosophy of Science 38(2): 157–69. Mundy, Brent (1987). “The Metaphysics of Quantity.” Philosophical Studies 51(1): 29–54. Newton, Isaac (1999). The Principia: Mathematical Principles of Natural Philosophy. Trans. I. Bernard Cohen and Anne Whitman. Berkeley: University of California Press. Palacios, Julio (1964). Dimensional Analysis. Trans. P. Lee and L. Roth. New York: St. Martin’s Press. Plank, Max (1932). General Mechanics. London: Macmillan. Russell, Bertrand (1937). Principles of Mathematics. 2nd edn. London: G. Allen & Unwin. Skow, Bradford (2012). “ ‘One Second Per Second’.” Philosophy and Phenomenological Research 85(2): 377–89. Skow, Bradford (2015). Objective Becoming. Oxford: Oxford University Press.

MODALITY AND EXISTENCE

9 Vague Existence Alessandro Torza

Debates about what there is are common and often fascinating. If there were a black hole close enough to the solar system, we would have reasons to be worried. Had the Higgs Boson turned out not to exist, that would have meant bad news for the Standard Model of particle physics. Euclid’s proof of the existence of infinitely many prime numbers was no small feat—and so on and so forth. Ontology is the study of what there is, unrestrictedly. Ontologists can argue about the existence of things which are of concern to laypersons (macroscopical objects, values, fictional characters), scientists (fundamental particles, fields) and mathematicians (numbers, sets). In other cases, ontological disagreement will turn instead on more exotic items such as substances, possible worlds, or spatiotemporally disconnected wholes. Can existence, in the unrestricted sense of ontology, be vague? One popular construal of vagueness is defined by the method of precisifications. A precisification is a way of making precise all the terms of a language. A sentence is vague when some precisification makes it true and some other makes it false. According to an influential argument due to Sider [25] [26] [28], in ontology there is no such thing as vague existence, as long as vagueness is construed precisificationally. I aim to show that existential vagueness is a coherent notion, albeit in a weaker form which I will refer to as super-vague existence. Section 1 exposes a gap in the alleged reductio of vague existence. I will wrap up the section by considering a potential objection. Section 2 develops and defends a novel framework, dubbed negative supervaluationary semantics, which models super-vague existence and its logic. Two further objections will be anticipated at the end of Section 2. 1 . AG A I N S T “ A GA I N S T V A GU E E X I S T E N C E”

1.1 Sider against vague existence The question whether existence can be vague is relevant to both ontology and metaontology. Let us start with the latter.

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One issue about ontological disputes is that it is often hard to identify the source of disagreement. This point is the main target of the recent debate in metaontology, which sees two opposing camps. Metaontological realists regard ontological disputes as genuine and substantive. When philosophers argue in the ontology room, it is claimed, their disagreement turns on different pictures of reality. This position, which took shape in Quine’s work on ontological commitment, has been developed by Peter van Inwagen, Kit Fine, and Ted Sider among others. The antirealist camp contends instead that ontological disputes are in some sense semantic or verbal. If the latter thesis is correct, the kind of disagreement taking place in the ontology room amounts to a sophisticated version of a dispute on the nature of tomatoes—whether they are fruits or vegetables. The core of this deflationist view, which originated with Carnap, was rejected by Quine and then revived by a group of philosophers including Hilary Putnam, Eli Hirsch, Amie Thomasson, and, more recently, David Chalmers.1 Some popular forms of deflationism embrace quantifier variance, the Putnam–Hirsch view “that quantifiers can mean different things, that there are multiple candidate meanings for quantifiers” (Sider [29, p. 391]). But if the quantifier used in the ontology room is semantically vague, there will be as many existence meanings as there are admissible precisifications of the quantifier. As a consequence, deflationism would likely be true of at least some ontological disputes. For instance, there may be no determinate fact of the matter as to whether, say, tables exist, provided that the meaning of “exist” is compatible with multiple and equally good ways of carving the domain of the world at the relevant joints, so that tables will exist on some but not all such existence meanings.2 Whether existence could be vague has first-order ontological implications, as well. Restricted composition is the metaphysical view that not all collections of things have a mereological fusion. According to a particular brand of restricted composition advocated by van Inwagen [31], organisms represent the only case in which a collection of objects composes a whole. On that view, tables and chairs do not exist, whereas pluralities of simples arranged table-wise and chair-wise do.3 Van Inwagen’s mereological organicism is certainly not the only flavor of restricted composition. On a different 1 A deflationary framework is also developed in Sider [27] on behalf of neo-Fregeans such as Bob Hale and Crispin Wright. 2 Truth be told, it is in principle possible that there be vague existence without ontological deflationism, namely if for every sortal P, ∃xPx is true (false) on all candidate meanings for ∃. For in this case any existence question would have an objective and determinate answer, despite the existential vagueness. Thanks to an anonymous referee for the pointer. 3 On plural quantification, see Boolos [5].

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way of constraining fusion, a collection of simples composes a whole just in case the members of the collection are topologically connected. Thus, organisms, tables, and chairs will exist, whereas table-giraffes will not. This view is prima facie more consonant with common sense intuitions about the conditions under which something can be said to exist. At the end of the spectrum we have mereological nihilism, which only accepts the existence of simples. It has been noted that restricted composition leads to vague existence.4 Assume that composition is restricted à la van Inwagen. On December 7, 43 BC, Marcus Tullius Cicero was on his deathbed. There is a time t0 at which he was definitely still alive and a time t1 at which he was definitely dead. However, there was arguably no cut-off point in the series starting with “Cicero is alive” at t0 and ending with “Cicero is dead” at t1 . For some time tk between t0 and t1 it is vague whether Cicero was dead or alive, therefore it is vague whether the simples arranged Cicero-wise lying on the bed would constitute a whole at time tk . Since there were times at which it was vague whether Cicero still existed, dying Cicero constitutes a case of vague existence. Or consider the case of viruses. Because it is vague whether a virus is an organism, it is vague whether it constitutes a whole or it is just a collection of simples arranged virus-wise. Existential vagueness also arises if we think, as most people arguably do, that objects must be connected. For, although the mathematical concept of connectedness is a sharp one, things get tricky when we apply it to physical objects. Suppose we are soldering two pieces of metal m1 and m2 . What is the minimum threshold of subatomic interaction that must occur between m1 and m2 so that the two pieces will count as connected? It does not look as though a unique non-arbitrary answer to that question could be provided. If so, there must be a time at which it is vague whether there exists a whole composed of m1 and m2 .5 Lewis [18, p. 213] objected to restricted composition on the assumption that vague existence is incoherent. This strategy was further developed by Sider [25, pp. 120–32], who appealed to unrestricted composition in his 4 Or, to be precise, that all interesting ways of constraining the relation of composition lead to vague existence. Some philosophers, however, beg to differ: Carmichael [8] and Donnelly [11] have proposed precisificational accounts of composition that do not entail existential vagueness. 5 Topology in fact distinguishes between multiple notions of connectedness. That fact seems to suggest that, if our concept of object is subject to a connectedness constraint, existential vagueness arises on multiple dimensions: relative to a notion of connectedness and relative to the application of such a notion to the physical realm. (But it might be argued, as well, that the first dimension of vagueness should instead be construed as a case of ambiguity, rather than vagueness.)

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proof of four-dimensionalism. A full-blown attempt at discharging the assumption that existence cannot be vague appears only in Sider [26, pp. 138–43]. The main claim is that (V) if vagueness is given a precisificational account and existence is expressed by the unrestricted existential quantifier, then existence cannot be vague. The first conjunct of the antecedent leaves out some theories of vagueness, most notably all degree-theoretic construals (viz. fuzzy logic) but includes the supervaluationism of Fine [13], as well as the epistemicism of Williamson [35] and the semantic nihilism of Braun and Sider [6]. Since there arguably are independent reasons to reject degree-theoretic semantics for vague language,6 Sider’s proof, if sound, would suffice to cover what happens to be the de facto standard representation of vague talk. But what does it mean to precisify a language L? Precisifications are ways of making precise all of the terms in L.7 Typically, a precisification is interpreted extensionally, in the sense that it (i) specifies a domain of quantification and (ii) assigns extensions over the domain to the non-logical constants in L. According to the precisificational framework, a statement of L is true (false) if it is true (false) in every precisification of L; it is vague if it is neither true nor false. The precisifications of L are assumed to be admissible— roughly, they must be compatible with the linguistic practice of the competent speakers of L. For instance, a precisification of English in which the extension of “hirsute” contains people with zero hairs will be inadmissible. Likewise for a precisification in which the extension of “bald” contains people with ten hairs and the extension of “hirsute” contains people with nine hairs. The task of determining the admissible precisifications of a language is far from obvious, but for present purpose it will suffice to employ a primitive notion of admissibility. The second conjunct of the antecedent in (V) serves to rule out restricted quantification, which is obviously open to vagueness. For instance, the statement “There are over 21 million people” will be vague if uttered by someone referring to the population of the Greater Mexico City, due to the unsharp nature of its urban sprawl. A quantifier is unrestricted iff it ranges over absolutely everything that exists. As a consequence, only the unrestricted 6

It has been shown that degree-theoretic interpretations of vagueness violate classical logic, misrepresent penumbral connections (i.e. logical connections among indefinite sentences) and fail to account for higher-order vagueness. See Williamson [38], Keefe [16]. 7 Varzi [32] considers a number of construals of a precisification. In the present context, however, such distinctions will not matter.

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quantifier, i.e. the one employed in the ontology room, is relevant to Sider’s argument. The strategy adopted by Sider in proving (V) consists in showing that, if an object-language quantifier ∃ is vague and absolute, a contradiction can be derived in its metalanguage. By an application of reductio ad absurdum, it is concluded that object-language quantification cannot be vague.8 The argument against vague existence can be reconstructed as follows. Assume by way of reductio that 1. ⌜ ∃xϕ⌝ is vague By the underlying semantics for vagueness, (1) is equivalent to 2. In some precisification ⌜ ∃xϕ⌝ is true and in some precisification ⌜ ∃xϕ⌝ is false hence, 3. In some precisification ⌜ ∃xϕ⌝ is true. Truth in a precisification is truth in a precise language. So, by the standard extensional truth-conditions for quantified statements in a precise language, (3) is equivalent to 4. There is something such that, in some precisification, it belongs to the domain of ∃ and satisfies ϕ and, a fortiori, 5. There is something such that, in some precisification, it satisfies ϕ. Now, ϕ is intended to be precise. For, otherwise, the main premise (1) would not amount to the assumption that the quantifier ∃ is vague. But the interpretation of a precise expression coincides across all precisifications. Therefore, (5) entails 8 In Sider’s own words: Suppose ‘ ∃’ has two precisifications, ‘ ∃1 ’ and ‘ ∃2 ’, in virtue of which ‘ ∃xϕ’ is indeterminate in truth value, despite the fact that ϕ is not vague. ‘ ∃xϕ’, suppose, comes out true when ‘ ∃’ means ‘ ∃1 ’, and false when ‘ ∃’ means ‘ ∃2 ’. How do ‘ ∃1 ’ and ‘ ∃2 ’ generate these truth values? A natural thought is: Domains ‘ ∃1 ’ and ‘ ∃2 ’ are associated with different domains; some object in the domain of one satisfies ϕ, whereas no object in the domain of the other satisfies ϕ But the natural thought is mistaken. If Domains is assertible, it must be determinately true. But Domains entails that some object satisfies ϕ (if “. . . some object in the domain of one satisfies ϕ. . .”, then some object satisfies ϕ). And so ‘ ∃xϕ’ is determinately true, not indeterminate as was supposed (Sider [28, pp. 557–8]).

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6. There is something such that, in every precisification, it satisfies ϕ. Recall that ∃ is unrestricted—it ranges over all there is. So, from (6) it can be inferred that 7. There is something such that, in every precisification, it belongs to the domain of ∃ and satisfies ϕ. By the truth-conditions for quantified statements in precise languages, (7) entails 8. In every precisification ⌜ ∃xϕ⌝ is true which, in a precisificational framework, is tantamount to 9. ⌜ ∃xϕ⌝ is true Since a statement is vague just in case it is neither true nor false, (1) and (9) are jointly inconsistent. By reductio ad absurdum, we can discharge the main premise (1) and infer its negation, namely 10. ⌜ ∃xϕ⌝ is not vague which concludes the proof.9

1.2 Super-vague existence Before we take a closer look at the merits of the argument, some preliminary remarks are due. As was pointed out, the reductio requires that, whereas ∃xϕ is vague, ϕ should be precise. Accordingly, there must exist at least one instance of ϕ meeting such conditions. What language could be such that all of its terms are sharp except at most the quantifiers? Sider [26, pp. 139–40] originally formulated the argument in the vocabulary of mereology, so that the main premise would be (E)

∃x(x is composed of the F and the G).

But if “F ,” “G,” or “compose” are vague, (E) could be vague without there being any vagueness at the quantificational level. The use of those terms is inessential, however, since (E) can be replaced with a sentence containing only logical vocabulary. Thus, instead of attempting to disprove the vagueness of (E), we could attempt to disprove, at a world containing exactly the

9 See Section 2.4 for an attempt at rephrasing Sider’s argument without appealing to reductio.

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two simples F and G, the vagueness of the sentence expressing the existence of a third object:10 (E’)

∃zðx 6¼ y ∧ y 6¼ z ∧ x 6¼ zÞ

where x and y refer to F and G, respectively. This solution faces two challenges. If the quantifiers are truly unrestricted, they will have to range over abstract as well as concrete objects. Since it is necessarily the case that there definitely exist infinitely many abstracta, (E’) would come out trivially true at the world in question. In order to obtain the desired restriction in the object language, we must introduce a concreteness predicate “C ” and replace (E’) with: (E”)

∃zðC ðxÞ ∧ C ðyÞ ∧ C ðzÞ ∧ x 6¼ y ∧ y 6¼ z ∧ x 6¼ zÞ.

But then, as Korman [17, p. 893] pointed out, the source of vagueness in (E”) could be “C ” rather than the quantifiers. If so, Sider would have failed to provide a reductio of vague existence. I find this strategy to block the argument unconvincing. Even if the concreteness predicate were vague, its vagueness would be irrelevant to the present argument. For if the vagueness in (E”) were due only to the vagueness of “C ,” there would have to be an admissible precisification in which the mereological sum of F and G is concrete and one in which it is not. But we can make the reasonable assumption that the sum (if any) of a collection of definitely concrete objects is definitely concrete. Since it was assumed that F and G are concrete simples, F and G will then have to be definitely concrete and therefore their sum (if any) must also be definitely concrete. Hence, relative to the world at which we evaluate (E”), there is no admissible precisification of the language according to which the sum of F and G—the third object in question—fails to be concrete. A related challenge concerns the semantic status of identity, for there is no point in replacing (E) with (E’), unless identity is precise. This extra assumption could be discharged by piggybacking on the well-known reductio of vague identity offered in Evans [12]. Nevertheless, the validity of Evans’ argument has been disputed.11 Be that as it may, for the time being I will concede that all logical vocabulary is precise, with the sole possible exception of quantifiers, and therefore that any reductio of the vagueness of (E’) amounts indeed to a reductio of quantifier vagueness. A separate and more crucial issue concerns the reductio step at the very end of Sider’s argument. Indeed, reductio ad absurdum is a valid form of

10 11

Sider [29, p. 390]. For a discussion, see Williams [33], Barnes [3], Heck [15], Akiba [1] [2]. Cf. Lewis [19].

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inference for bivalent languages: an inconsistent condition cannot be true, hence it must be false. When the language is not bivalent, on the other hand, reductio ad absurdum fails, since being non-true is a weaker condition than being false. In particular, reductio is invalid in the case of vague languages: if a condition is inconsistent, it must be either false or indeterminate. It follows that Sider’s final reductio step is valid insofar as the language in which the argument is formulated, viz. the metalanguage of ∃xϕ, is perfectly precise. Since the main premise (1) is equivalent to its precisificational truthcondition (2), one is vague just in case the other is. Now, (2) does two things: it quantifies over precisifications, and it says of the object-language sentence ∃xϕ that it is true in some but not all of them. By definition, for every precisification s, truth-in-s is determinate; therefore, if there is any vagueness at all in (2), it must come down to what precisifications there are. The moral is that, if quantification over precisifications is precise, so that there is exactly one candidate set of precisifications for the object-language, Sider’s reductio goes through—if not, not.12 Let me flesh out this point a bit further. It is not at all uncommon for a vague term (and, a fortiori, a vague language) to be associated with multiple sets of precisifications. Take for instance the color adjective “blue.” This could refer to the spectrum of visible light with a wavelength between 450 and 490 nanometers—each single value specifying a precisification. But there is nothing about that very interval that makes it the right one. If the set of precisifications of “blue” can be 450–490nm, then the intervals 455–495nm or 445–485nm will work just as fine, as well as many other intervals in that neighborhood. Which is to say, none of them is a better candidate set of precisifications for the meaning of “blue.” We must conclude that the term “blue” is second-order vague, insofar as the extent of its vagueness is itself vague. Could it be the case that second-order vagueness affects the language of ∃xϕ? From what we have seen so far, nothing prevents such a possibility. In a scenario of this sort, quantifying over precisifications in Sider’s argument would be vague and the reductio step, therefore, unwarranted. Let’s take stock. As it should now be clear, Sider has conclusively shown us that 1. ⌜ ∃xϕ⌝ is vague cannot be true, provided that ϕ is precise and ∃ is unrestricted. If the metalanguage of ∃ is not perfectly precise, however, there is no guarantee

12 The idea that some assumption in Sider’s argument could be indeterminate has also been explored in Barnes [4].

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that (1) is false and, therefore, that vague existence has been disproved. In particular, even though the quantifier ∃ cannot be definitely vague, there is a prima facie possibility that it might be second-order vague. In this case, there would be a set of precisifications that makes ∃xϕ vague, and another set that makes ∃xϕ precise. At this point Sider might reply by offering a second reductio, this time around of the vagueness—not the truth—of (1). The conjunction of this new argument with the original one would then amount to the desired disproof of vague existence. The second reductio goes as follows. (Note that the object language in the present argument coincides with the metalanguage in the previous one. Accordingly, what “precisification” means in this argument is not what it used to mean in the previous one. I will use the term “precisification2 ” to refer to precisifications of the meta-language of ∃xϕ, i.e. second-order precisifications of the language of ∃xϕ.) Assume 1* ⌜⌜ ∃xϕ⌝ is vague⌝ is vague Hence, 2* In some precisification2 ⌜⌜ ∃xϕ;⌝ is vague⌝ is true, and in some precisification2 ⌜⌜ ∃xϕ⌝ is vague⌝ is false Now, by virtue of Sider’s first argument, the first conjunct of (2*) entails 3* In some precisification2 ⌜⊥⌝ is true On the other hand, precisifications2 being classical, it is the case that 4* In all precisifications2 ⌜⊥⌝ is false hence a contradiction. If we could now apply reductio ad absurdum, we would be able to infer 5* ⌜⌜ ∃xϕ⌝ is vague⌝ is not vague thus disproving the second-order vagueness of ∃xϕ. But we are back to square one. For, the reductio step is licensed as long as the language in which the argument is formulated (i.e. the meta-metalanguage of ∃xϕ) is perfectly precise. In particular, quantification over precisifications2 in (2* ) will be vague if the quantifier ∃ is third-order vague. In such a scenario, there will be a set of sets of precisifications of the language of ∃xϕ according to which ⌜⌜ ∃xϕ⌝ is vague⌝ is vague, and another set of sets of precisifications of the language of ∃xϕ according to which ⌜⌜ ∃xϕ⌝ is vague⌝ is sharp. Sider’s argumentative strategy could be iterated at any order—and so the relevant rejoinder. Consequently, as long as quantifying over precisificationsn is

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vague at every order n, neither side will get the upper hand, and the possibility of vague existence will remain neither proved nor disproved. The dialectics prompts the following moral. A precisificational semantics is a framework in which truth, falsity, and vagueness 0for a language L are formulated, within some relevant metalanguage L , via the notion of truth-in-a-precisification-of-L. Since precisifications are ways of making a language precise, it cannot be vague whether an L-statement is true or false in a given precisification. This account is perfectly compatible, however, with there being a vague set of precisifications of L. When that happens, the precisificational truth-conditions for 0L-statements can be vague, insofar as they are formulated by means of L -statements of the form “there is a precisification . . . .” Crucially, Sider’s reductio of vague existence requires that the truth conditions for ∃xϕ be formulated in a perfectly precise metalanguage. (Or at least that a precise metalanguage could be found somewhere up in the hierarchy.) The existence of such a language is neither guaranteed nor required by a precisificational account of vagueness. Therefore, pace Sider, if what precisifications there are is vague at all orders, vague existence will remain an open possibility. The above discussion is complicated by the use of a hierarchy of metalanguages, each expressing whether sentences of the relevant object language are true, false, or indeterminate. Matters can be simplified as follows. Given sentence ϕ in an object language L, in lieu of the metalanguage expression ⌜ϕ⌝ is true we will use Δϕ where Δ is an object-language sentential operator with the intended meaning “it is definitely the case that.” Also, let’s define the expression I ϕ (“it is vague whether ϕ”) as ¬Δϕ ∧ ¬Δ¬ϕ. By iterating these newly introduced sentential operators, it is now possible to reduce the hierarchy of metalanguage truth/falsity/vagueness predicates to the object language L. For instance, the above condition 1* ⌜⌜ ∃xϕ⌝ is vague⌝ is vague translates into L as II ∃xϕ.

{

In general, we can express that ϕ is n-th order vague simply by iterating the I operator n times. Now, let I n be short for the concatenation I . . . I . What n the generalized reductio has shown is that, for all n, ⌜I n ∃xϕ⌝ is not true.

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Which is to say, existence cannot be said to be definitely vague, no matter what the vagueness order is (provided that ϕ is precise). I will henceforth refer to this condition as Sider-determinacy. Let us stipulate that an existence statement ⌜ ∃xϕ⌝ is anti-Sider-determinate just in case, for all n, ⌜I n ∃xϕ⌝ is not false. I have argued in the foregoing discussion that precisificational truth conditions need not be given in a precise metalanguage and, in particular, that the set of precisifications can in principle be vague at each order in an infinite hierarchy of metalanguages. Insofar as, for all n, ⌜I n ∃xϕ⌝ can be vague, it follows that Sider-determinacy is compatible with anti-Sider-determinacy. When an existence statement is both Sider-determinate and anti-Sider-determinate, I will say that it expresses an instance of super-vague existence. For example, it might be that it is vague whether there exists the sum of F and G, and it is vague whether it is vague whether there exists the sum of F and G, and it is vague whether it is vague whether there exists the sum of F and G, etc. In the next section, I will show how super-vague existence can be accommodated within a precisificational model theory. But before that, I wish to address a potential objection.

1.3 Metalanguage objection In the above reconstruction of Sider’s argument against vague existence, I pointed out that the final reductio step requires the metalanguage of ∃xϕ (viz. the language in which the argument is formulated) to be perfectly precise. The same problem cropped up, mutatis mutandis, in the higherorder generalization of the argument. Sider touches on the issue of a precise metalanguage as he describes what it means to give an account of vague statements such as “S is bald”: When confronted with vagueness, I retreated to a relatively precise background language to describe the relevant facts. In this background language I quantified over the various sets containing persons with different numbers of hairs, and said that the referent of S was in some but not all of these sets. [. . .] Moreover, in principle one could describe the sets with perfect precision by retreating to a background language employing only the vocabulary of fundamental physics. (Sider [26, p. 139], emphasis added)

It is tempting to interpret the passage as entailing that the precisificational truth-conditions of ∃xϕ can be assumed to be perfectly sharp and, therefore, that Sider’s reductio of vague existence is valid. Consequently, existence cannot be super-vague, since it is definitely not vague. The above passage, however, does not license this conclusion. Recall from the discussion in Section 1.2 that one thing is to say that (i) “true-in-s” is

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sharp, for s a given precisification; and another thing is to say that (ii) “true in some precisification” is sharp. Indeed, (i) is true by definition, insofar as each precisification is a classical interpretation described in the metalanguage with perfect precision. But (i) can obtain in the absence of (ii), namely, if it is indeterminate what precisifications there are. In the particular case of “S is bald,” every precisification for “bald” will be a sharp set of persons. Nevertheless, it could still be indeterminate what set of sets is the extension of the metalanguage term “precisification.” And since the truth conditions of a vague statement require quantification over precisifications, the language of Sider’s argument can be vague, hence the reductio is invalid. As I argued in Section 1.2, the assumption that the language be perfectly precise is not part and parcel of a precisificational account of vagueness. Moreover, the assumption that the precisificational truth-conditions of a vague statement be formulated in a perfectly precise meta-language can lead to paradox. The statement (*) ⌜Ted is bald⌝ is vague is tantamount to (**) in some precisification ⌜Ted is bald⌝ is true, and in some precisification ⌜Ted is bald⌝ is false If (**) is perfectly precise, as per hypothesis, it must be either definitely true or definitely false. It follows that ⌜Ted is bald⌝ is either definitely vague or definitely not vague. Therefore, although Ted—or anybody else, for that matter—could be a borderline case of baldness, he could not be a borderline borderline case of baldness. Sider’s hypothesis that the metalanguage is perfectly precise rules out the possibility of higher-order vagueness. However, there are independent reasons to admit the possibility of higher-order vagueness. If “bald,” for example, is only first-order vague, there will be a clear-cut border between the definitely tall and the not definitely tall, which is no less absurd than there being a clear-cut border between the tall and the not tall. Moreover, the absence of higher-order vagueness can be exploited to generate higher-order sorites paradoxes. I conclude that Sider’s assumption is unwarranted and unwelcome.

2 . SU P E R - VA G U E E X I S T E N C E A N D I T S L O GI C What is the logic of a language whose quantifiers are super-vague? This Section attempts to provide a model-theoretic answer to that problem.

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2.1 Finean supervaluationism The most popular precisificational model theory is arguably the supervaluationary semantics put forward in Fine [13]. A specification space is a model defined by (i) a set of specification points; (ii) a binary admissibility relation defined on the set of specification points; and (iii) a selected specification point called “base point.” Specification points are identified with partial models, each representing a way of making the language more precise. A partial model is defined by a domain of individuals and an interpretation function assigning to each predicate a positive and a negative extension such that the two extensions are disjoint but do not necessarily exhaust the domain. (Classical models are the degenerate case of partial models in which the positive and negative extension of any predicate are jointly exhaustive.) Since quantification is assumed here to be absolute, a quantifier must range over the whole domain of a specification point. The admissibility relation is a reflexive, antisymmetric, and transitive ordering R such that (1) if uRv and p is true (false) at u, then p is true (false) at v; (2) every specification point bears the ancestral of the admissibility relation to a complete specification point, which is a classical model (intuitively, a precisification of the language). A sentence p is said to be true in a specification space if it is true at the base point. It is said to be true at a specification point if it is true at all accessible specification points. “Δp” is said to be true at a specification point if p is true at the base point. It follows from the definitions that a sentence is true at the base point iff it is true at all complete specification points. A sentence ϕ is said to be a supervaluationary consequence of a set Γ of sentences if every specification space which makes Γ true makes ϕ true. Sentence ϕ is said to be valid if it is a supervaluationary consequence of the empty set. Fine’s framework immediately yields some desired results concerning vagueness phenomena. Bivalence fails, since it is not the case that any given p is either true or false in a specification space. This is equivalent to the object-language fact that Δp ∨ Δ¬p is invalid, as it should be. On the other hand, p ∨ ¬p is valid (and so is any classical tautology) in virtue of the classicality of complete specification points. The feature of Fine’s supervaluationary semantics relevant to the present discussion is that there is no vague existence at any order (provided that there definitely are finitely many objects). To see that, it suffices to show that (i) existence is definite, i.e. ∃xðx ¼ yÞ ! Δ ∃xðx ¼ yÞ, and that (ii) definite statements cannot be indefinitely definite, i.e. Δϕ ! ΔΔϕ.

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It follows that Fine’s framework, in its basic form, is not suitable for modeling super-vague existence.13

2.2 Variable domain frames I have just argued that the Finean supervaluationism presented here is unable to account for higher-order vagueness at the object-language level (cf. Appendix A). Since the failure of higher-order vagueness famously leads to higher-order sorites paradoxes, an alternative precisificational framework has been proposed by Timothy Williamson in which statements can be higher-order vague. In the remainder of Section 2, I aim to show that a suitable generalization of Williamson’s semantics admits of models for supervague existence, i.e. in which both of the following conditions are met: Sider-determinacy: for all n, it is not definitely the case that existence is n-th order vague. anti-Sider-determinacy: for all n, it is not definitely the case that existence is not n-th order vague The resulting model-theory will give us an idea of the logic of super-vague existence. The semantics developed in Williamson [35] [37] is designed for a sentential language with a definiteness operator. The simplest generalization to the first-order case can be defined as follows. A frame F is a structure hS;U;Ri where S is a set of points (the specifications), U a set of individuals (the universe of discourse) and R a relation over S (the admissibility relation). Let LΔ be a first-order language with a definiteness operator Δ. A model M for LΔ is a pair hF ;σi where, σ is an interpretation function such that, for every point s ∈ S, (i) σð¼; sÞ is the identity relation over U and (ii) for P an n-ary predicate, σðP;sÞ  U n . The truth condition for an atomic formula at a point given a value assignment for the variables is classical. Truth conditions for connectives and quantified formulas are as expected. Given a value assignment, Δϕ is true at s if, for every point t such that sRt, ϕ is true at t. A formula ϕ is true (false) in a model M under a value assignment if it is true (false) at every point in M. A formula ϕ is a supervaluationary consequence of a set Γ of formulas if, given a model and a value assignment, if Γ is true then ϕ is true. A formula ϕ is valid if it is a supervaluationary consequence of the empty set.

13 The proof is provided in Appendix A. (The finiteness condition can be dropped if the language is infinitary.)

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The admissibility relation R is intended to be reflexive and symmetric (so as to validate Δϕ ! ϕ and ϕ ! Δ¬Δ¬ϕ) but intransitive. This choice allows for the possibility of higher-order vagueness.14 It will be useful to stress a crucial difference between Fine’s and Williamson’s precisificational frameworks. Recall that being definitely true means being true in all precisifications. On the Finean approach, what counts as a precisification of the language is an absolute matter, since precisifications are identified with complete specification points (i.e. the points to which the base point bears the ancestral of the admissibility relation). As a consequence, facts about definiteness and vagueness are absolute, as well—there can be no instances of higher-order vagueness. In Williamson’s models, on the other hand, there is no such thing as the set of precisifications of the language. For every specification point s there is a set of admissible specification points—intuitively, those precisifications that are in the neighborhood of s. Therefore, an expression of the form “there is a precisification so-and-so” is always relativized. This important feature is meant to capture the idea, discussed in Section 1.2, that the language can have multiple sets of precisifications, and that it can be vague which of those sets is the correct one. Consequently, Williamson’s models make room for second-order vagueness. Likewise, distinct sets of sets of specifications can be admissible to distinct sets of specifications, thus making room for third-order vagueness—and so on and so forth. This kind of higher-order vagueness can be put to use in order to model super-vague existence. Nevertheless, the Williamson-style semantics just sketched does not represent a generalization over Fine’s framework in the desired direction, since all specification points in a model have constant domain and, as a result, constrain existence to be definitely not vague at all orders. I submit that, in order to be able to model super-vague existence, we should proceed as follows. A variable domain frame F* is a quadruple hS;U;R;Domi such that (i) specification points S, universe U and admissibility relation R are as before; (ii) Dom is a function mapping each point s ∈ S to a subset of U (intuitively, the objects that exist according to the precisification s); (iii) U ¼ [fDomðsÞgs ∈ S ; (iv) for every n ⩾ 1, there is some s ∈ S which is n-determinate. 14 Statements of higher-order vagueness are satisfiable if either Δϕ ! ΔΔϕ or ¬Δϕ ! Δ¬Δϕ fails. On the model-theoretic side, the failure of either condition corresponds to a non-transitive or non-euclidean admissibility relation: see Williamson [37, p. 133]. In fact, higher-order vagueness has its modal counterpart in the contingency of contingency, which also takes place only in systems weaker than S5.

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The notion of n-determinacy is defined recursively as follows: 1. s is 1-determinate iff for all s 0; s 00 : if sRs0 and sRs 00 , then Domðs 0 Þ¼ Domðs 00 Þ n þ 1. s is n þ 1-determinate iff for all s 0 ;s 00 : if sRs 0 and sRs 00 , then s 0 is n-determinate iff s00 is n-determinate. As I will discuss in Section 2.3, the property of n-determinacy is crucial, since it makes models based on a variable domain frame F* Siderdeterminate. If (iv) were dropped from the definition of a frame, the semantics would admit of unintended models, i.e. models that are not Sider-determinate. Let us now turn to the problem of finding a suitable notion of local truth, i.e. truth-at-a-point. In a model based on F* , a quantifier evaluated at s will range over DomðsÞ. The nature of the interpretation function σ will depend on the kind of semantics we choose for evaluating formulas with nonreferring terms at a specification point. If x exists at point s, what should be the truth-value of “PðxÞ” at a point t where x does not exist? Three options are on the table for dealing with local truth. Positive local semantics. We could allow an atomic formula like “PðxÞ” to be true at a point where x does not exist. On this view, for any point t, the interpretation of a predicate P is a subset of the frame domain, namely σðP; tÞ  U , and the value of a free variable an element of U . Accordingly, “PðxÞ” will be either true or false at t depending on whether x is or is not in DomðP; tÞ. However, this approach provides a misleading picture of existential vagueness. For, if the same set of objects can be referred to according to all precisifications of the language, existence will be vague only nominally. Indeed, whereas quantifiers are restricted to DomðtÞ, which varies with t, free variables will constantly range over U . Hence, there could be a precisification in which mereological nihilism is true, and yet it is also true of the sum of this mug and that table that they occupy space in my office. Positive semantics is therefore inadequate for modeling vague absolute quantification. Neutral local semantics. A more attractive approach is to (i) let a variable at a specification point pick something out of the relevant domain, or nothing at all, and (ii) restrict the interpretation of a predicate in such a way that σðP;tÞ  DomðtÞ. Neutral semantics takes an atomic formula to be indeterminate at all points where some of its terms fail to refer. The picture should then be completed by defining truth conditions for the non-atomic formulas. As it turns out, neutral semantics is not a viable candidate for local truth, either. One of the most attractive features of supervaluationism is that it preserves classical tautologies. The motivating intuition is that vagueness facts should not affect the logic of truth-functors. That nice result breaks

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down when local truth is defined via neutral semantics. If x does not refer at a specification point s, x ¼ x is indeterminate there and so is x 6¼ x (under the assumption that local semantics is truth-functional). Thus, x ¼ x∨x 6¼ x is indeterminate at s (since, in neutral semantics, a disjunction is indeterminate when both disjuncts are). Hence, some instance of p∨¬p is untrue in some model and, therefore, invalid. To restore the validity of classical tautologies without giving up neutral local semantics, we could tinker with global truth, i.e. truth-in-a-model. Let then ϕ be true in a model M* based on F* just in case it is true at all points where all of its terms are defined. Since in the evaluation of x ¼ x we rule out every world where x is non-referring, x ¼ x∨x ¼ 6 x must be valid. Out of the frying pan, into the the fire: now ∃xðx ¼ yÞ is true in a model iff it is true at all worlds where y is defined. But in neutral semantics, y is defined at point s iff it picks a value in DomðsÞ iff y exists at s. So, ∃xðx ¼ yÞ is trivially true in every model and, therefore, valid. However, Δ ∃xðx ¼ yÞ is invalid. To see that, consider a model where y is defined at a point s but not at a point t where sRt. Then Δ ∃xðx ¼ yÞ is false at s, since t is an accessible point where ∃xðx ¼ yÞ is untrue. So, Δ ∃xðx ¼ yÞ is invalid. It follows that the so-called necessitation rule (N) If ⊨ ϕ then ⊨ Δϕ fails under the proposed revision. Since it is a standard desideratum of any semantics for vague language that validity be closed under definiteness, the above theory should be rejected. One might reply that (N) can be restored by making one simple change: just let Δϕ be true at a point s iff, for every point t such that sRt and all terms in ϕ are defined at t, ϕ is true at t. This attempt at validating (N) does more harm than good, however, because now Δ ∃xðx ¼ yÞ is true—in fact valid— even when there are precisifications of the language according to which there is no y. Since we are looking for a semantics modeling existential vagueness, I take this to be a reductio of neutral local semantics. Negative local semantics. On the third approach for defining truth-at-apoint, value assignments and the interpretation of the language are exactly as in neutral local semantics: (i) a variable at s picks something out of DomðsÞ, or nothing at all, and (ii) σðP;tÞ  DomðtÞ. These two conditions codify the reasonable assumptions that reference and predication make sense only relative to what exists according to a given precisification. The essential difference is that negative local semantics takes any atomic formula to be false at a point where some of its terms are non-referring. I claim that a negative supervaluationary semantics, i.e. a supervaluationary framework based on variable domains and negative local semantics, yields the correct account of vague existence phenomena. To corroborate my claim, I will first

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provide an exact formulation of the theory, and then tease out its main semantic and logical features.

2.3 Negative supervaluationary semantics A negative supervaluationary (NS) model M* for LΔ is a pair hF* ;σ* i where F* is a variable domain frame and σ* an interpretation function such that, for every point t ∈ S, (i) σ* ð¼; tÞ is the identity relation over DomðtÞ and (ii) for P an n-ary predicate, σ* ðP;tÞ  DomðtÞn . Let VAR be the set of variables in LΔ and S the set of specification points. A value assignment for VAR over M* is a set of partial functions fξt gt ∈ S such that: 1. ξt : VAR ! DomðtÞ 2. [fξt gt ∈ S is a total function f : VAR ! U 3. if ξs ðxÞ and ξt ðxÞ are both defined, then ξs ðxÞ ¼ ξt ðxÞ The first condition allows a variable to be undefined at some precisifications, whereas the second condition forces a variable to be defined at some precisification. Consequently, negative supervaluationary semantics is not a framework for definitely non-referring terms, unlike free logic. The third condition guarantees that variable assignments are rigid across specification points. Truth-at-a-point s (local truth) for ϕ under a variable assignment fξt gt ∈ S in an NS-model M* , written ðM* ;s; fξs gs ∈ S Þ ⊨ NS ϕ, is defined recursively thus: (at) If ϕ ¼ Pðx1 ; :::;xn Þ, then ðM* ; s; fξs gs ∈ S Þ ⊨ NS ϕ iff ξs is defined for all i ∈ f1,..., ng and hξs ðx1 ),..., ξs ðxn Þi ∈ σ* ðP; sÞ ð¬Þ If ϕ ¼ ¬ψ, then ðM* ; s; fξs gs ∈ S Þ ⊨ NS ϕ iff ðM* ; s; fξs gs ∈ S Þ ⊨ NS ψ ð∧Þ If ϕ¼ðψ∧χÞ, then ðM* ; s;fξs gs∈S Þ⊨NS ϕ iff both ðM* ; s;fξs gs∈S Þ ⊨NS ψ and ðM* ; s;fξs gs∈S Þ⊨NS χ 0 ð8Þ If ϕ ¼ 8xψ,0 then ðM* ;s; fξs gs ∈ S Þ ⊨ NS ϕ iff, for every fξ s gs ∈ S such that ξ0 s is defined on x and differs from ξs at most on x, ðM* ; s; fξ s gs ∈ S Þ ⊨ NS ψ ðΔÞ If ϕ ¼ Δψ, then ðM* ;s; fξs gs ∈ S Þ ⊨ NS ϕ iff, for every t such that sRt, ðM* ; t; fξs gs ∈ S Þ ⊨ NS ψ A few more definitions are needed. A formula ϕ is true in a NS-model M* relative to a variable assignment fξt gt ∈ S (i.e. ðM*; fξt gt ∈ S Þ ⊨ NS ϕ) iff, for every s ∈ S, (M*; s; fξt gt ∈ S Þ ⊨ NS ϕ. A formula ϕ is true in a NS-model M* (i.e. M* ⊨ NS ϕ) iff, for every fξt gt ∈ S , (M*; fξt gt ∈ S Þ ⊨ NS ϕ.

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A formula ϕ is a NS-consequence of a set Γ of formulas (i.e. Γ ⊨ NS ϕ) iff, for every NS-model M* , if M* ⊨ NS Γ then M* ⊨ NS ϕ A formula ϕ is NS-valid iff it is a NS-consequence of the empty set. Let us now turn to the key properties of negative supervaluationary semantics. Standard precisificational theories of vagueness, such as Fine’s specification space semantics or Williamson-style constant domain semantics, have the virtue of being classical in a precise sense: given a purely extensional language, classical consequence ( ⊨ C ) and supervaluationary consequence ( ⊨ SV ) coincide. Namely, for ϕ and Γ formulated in a first-order language L without definiteness operator,15 (Eq1 ) Γ ⊨ C ϕ iff Γ ⊨ SV ϕ On the other hand, some classical inference rules, which have been regarded as being the source of sorites paradoxes, fail in standard supervaluationary semantics. Existential instantiation does not hold: from “some number n is the least number such that n grains of sand constitute a heap” we cannot infer the existence of any particular n0 such that “n0 is the least number such that n0 grains of sand constitute a heap.” The same applies mutatis mutandis to universal generalization.16 Notice, however, that the result of substituting NS-logical consequence for ⊨ SV in (Eq1 ) does not hold, due to the fact that local semantics for NS is nonclassical. Here is an intuitive example. (A formal countermodel is provided in Appendix B.1.) We know that “Ted is not a (mereological) simple” classically entails “Something is not a (mereological) simple.” The same inference, on the other hand, is not NS-valid. Suppose that it is vague whether mereological nihilism or universalism is true. Now, in any precisification that allows the existence of sums, Ted exists and is not a simple; and in any precisification that does not allow the existence of sums, Ted does not exist and therefore (local semantics being negative) is not a simple. Hence, Ted is not a simple. But in any precisification that does not allow the existence of sums, it is not the case that something is not a simple. Hence, “Something is not a (mereological) simple” is untrue, which shows that classical existential generalization is NS-invalid. Nevertheless, negative supervaluationary semantics can be shown to validate a weaker version of existential generalization, which typically holds in free logic: ( ∃G ) fϕðxÞ; ∃yðx ¼ yÞg ⊨ NS ∃xϕðxÞ. In fact, a general result can be proven connecting negative supervaluationary semantics to negative free logic: in a purely extensional language, the 15

See Keefe [16, pp. 174–81]. For a discussion and defense of this aspect of supervaluationism, see Keefe [16, pp. 181–8]. 16

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consequence relation of negative free logic is preserved by negative supervaluationary semantics. Negative free logic—a first-order logic for languages with non-referring terms—is sound and complete with respect to negative semantics, which is the semantics employed here for defining local truth in NS-models.17 We can think of a model of negative free logic (NF -model) as the degenerate case of a NS-model with a single specification point. A variable assignment over a NF -model is a partial function from free variables to the model domain.18 In a first-order extensional language, ϕ is a negative-free consequence of Γ (Γ ⊨ NF ϕ) if, for every NF -model and variable assignment, Γ is true only if ϕ is true. The aforementioned result connecting negative free logic and negative supervaluationary semantics is as follows. For ϕ and Γ formulated in a language L without “Δ,” (Eq2 ) if Γ ⊨ NF ϕ then Γ ⊨ NS ϕ. (For a proof, see Appendix B.2.) On the other hand, the converse of (Eq2 ) fails, since in negative free logic but not in NS it can be consistently said of something that it doesn’t exist (cf. Appendix B.2). We can tease out a few interesting facts concerning the interaction of existence and identity. First of all, notice that existence is definable via identity in both negative free logic and negative supervaluationary semantics, because each of the two frameworks validates the biconditional ∃yðx ¼ yÞ $ x ¼ x. Since ¬ ∃yðx ¼ yÞ is NF -satisfiable, so is x 6¼ x. On the other hand, in negative supervaluationary semantics nothing is nonexistent, therefore nothing is self-distinct. That is how things should be. Moreover, the indiscernibility of non-existents (IN) ¬ ∃zðx ¼ zÞ ∧ ¬ ∃zðy ¼ zÞ ! ðϕðxÞ ! ϕðyÞÞ which is valid in negative free logic19, fails in negative supervaluationary semantics (see Appendix B.3). Although the converse of (Eq2 ) fails, a weaker equivalence can be proven. Namely, negative free logic and negative supervaluationary semantics define the same class of valid formulas in a Δ-free language: (Eq3 ) ⊨ NF ϕ iff ⊨ NS ϕ. (For a proof, see Appendix B.4.)

17

For the relation between negative free logic and negative semantics see Nolt [22], Burge [7]. Note, however, that a variable assignment over a NS-model with a single specification point is a singleton fξg, where ξ is a total function. 19 Nolt [22, p. 1033]. 18

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Given a language LΔ , negative supervaluationary semantics satisfies the following conditions: (Tau) ⊨ NS ϕ, for every tautology ϕ (K) ⊨ NS Δðϕ ! ψÞ ! ðΔϕ ! ΔψÞ (MP) fϕ;ϕ ! ψg ⊨ NS ψ (N) If ⊨ NS ϕ then ⊨ NS Δϕ (T) ⊨ NS Δϕ ! ϕ (B) ⊨ NS ϕ ! Δ¬Δ¬ϕ The two conditions (Tau) and (MP) guarantee the validity of classical sentential logic. By the Kripke schema (K), the “definitely” operator distributes over the material conditional. (T), expressing the facticity of definiteness, holds because the admissibility relation R is assumed to be reflexive. The symmetry of R validates (B). The so-called necessitation rule (N) ensures that validity is closed under definiteness. In fact, a stronger result than (N) is provable in negative supervaluationary semantics (as well as in most supervaluationary frameworks), viz. the definiteness of truth, or Δ-introduction rule: (N* ) ϕ ⊨ NS Δϕ I now turn to conditions that fail in the present framework. A non-classical aspect of negative supervaluationary semantics, which it inherits from negative free logic, is that validity is not closed under uniform substitution. For P atomic, for instance, PðxÞ ! x ¼ x is valid whereas ¬PðxÞ ! x ¼ x is not. Since negative supervaluationary semantics admits of variable domain models, and in particular it could be that DomðsÞ⊂ DomðtÞ for sRt, the Barcan formula (BF) ¬Δ¬ ∃xϕ ! ∃x¬Δ¬ϕ has invalid instances, such as ¬Δ¬ ∃xðx ¼ yÞ ! ∃x¬Δ¬ðx ¼ yÞ. Think of Ted, for instance. In some sense, he exists—namely, the sense of existence of mereological universalism. But for the nihilist there is no such thing as Ted. Therefore, it is not the case that there is something which, in some sense, is Ted. The failure of the Barcan formula is, of course, not an idiosyncrasy of negative supervaluationary semantics. When “¬Δ¬” is substituted with intensional operators of other sorts, it is not hard to find counterexamples to the schema. In the modal case, the sentence If Mary could have had a daughter, somebody could have been Mary’s daughter

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is false, barring exotic semantic frameworks.20 The same occurs (unless we are eternalists) when the possibility operator is replaced with a tense operator, as in: If Mary will have a daughter, there exists somebody who will be Mary’s daughter. The Barcan formula also fails in fictional contexts. If the operator “according to fiction S” is construed as the analog of the necessity operator, we can define its dual, “according to fiction S it might be that.” Works of fiction being typically incomplete, “according to fiction S it might be that p” is true just in case the fiction does not entail the falsity of p. Thus, consider If according to Woyzeck it might be that the Captain has a mistress, then there is somebody who according to Woyzeck might be the Captain’s mistress. Since, according to Büchner’s play, the Captain does in fact have a mistress, the antecedent is obviously true. However, the consequent is absurd, unless we buy into realism about fictional characters.21 Insofar as there can be specification points s, t such that DomðtÞ⊂DomðsÞ for sRt, the Converse Barcan formula (BF)

∃x¬Δ¬ϕ ! ¬Δ¬ ∃xϕ

is also invalid. Further conditions which are invalid in negative supervaluationary semantics are the definiteness of identity (DI) x ¼ y ! Δx ¼ y and the definiteness of distinctness (DD) x 6¼ y ! Δx 6¼ y. (Proofs in Appendix B.5.) However, (DI) can never be false in a model, for if it were, that would contradict the fact that x ¼ y ⊨ NS Δx ¼ y, which is an instance of (N* ). Likewise for (DD). The invalidity of (DI) and (DD) may bring to mind the analogous case of the failure of the necessity of identity and distinctness in some versions of possible-world semantics. However, the analogy is only superficial and, at 20

One such framework, involving the use of possibilist quantification, is defended in Linsky and Zalta [20] [21], Williamson [36]. An alternative semantics which validates the Barcan schema is one in which quantifiers range over individual concepts: see Garson [14]. 21 Cf. Sainsbury [24, p. 34]. An alternative option is to go Meinongian and allow quantification over non-existents.

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bottom, misleading. For, in negative supervaluationary semantics, the reason why (DI) and (DD) are invalid is that a term may refer at some but not all precisifications. On the other hand, the run-of-the-mill counterexamples to the necessity of identity and distinctness are formulated in counterpart theory or other possible-world semantics relying on non-standard interpretations of de re truth.22 Moreover, in such modal frameworks there are true instances of contingent identity and distinctness, whereas (DI) and (DD) can at most have untrue instances in negative supervaluationary semantics. Incidentally, since variable assignments over NS-models are rigid, it follows that identity is weakly definite: 0

0

(DI ) x ¼ y ! Δð ∃z ∃z ðx ¼ z ∧ y ¼ z Þ ! x ¼ yÞ Distinctness, on the other hand, does not satisfy weak definiteness: 0

0

(DD ) x 6¼ y ! Δð ∃z ∃z ðx ¼ z ∧ y ¼ z Þ ! x 6¼ yÞ (See Appendix B.5.) The importation schema (IM)

∃xΔϕ ! Δ ∃xϕ

fails, too. (Proof in Appendix B.6.) We can finally return to the main point, quantifier vagueness. Our goal is to find some NS-model of super-vague existence. First of all, we need to check that the following condition holds: Sider-determinacy: for all M* , fξs gs ∈ S and n ⩾ 1: ðM* ; fξs gs ∈ S Þ ⊭ NS I n ∃xðx ¼ yÞ for otherwise negative supervaluationary semantics would be inconsistent with Sider’s result, or with the higher-order generalization of it. Second, we would have to show anti-Sider-determinacy: for some M* and fξs gs ∈ S and for all n ⩾ 1: ðM* ; fξs gs ∈ S Þ ⊭ ¬I n ∃xðx ¼ yÞ A proof that both conditions hold in negative supervaluationary semantics can be found in Appendix B.7. We can conclude that super-vague existence is NS-satisfiable. Let’s recap. Sider argued that (V) if vagueness is given a precisificational account and existence is expressed by the unrestricted existential quantifier, then vague existence is incoherent. 22

For example, see Lewis [18].

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According to Sider [26, p. 4], the moral of his alleged reductio is that vague existence “would be radically unlike familiar cases of vagueness. Vague quantifiers may yet be possible, but such vagueness would require an entirely different model from the usual one.” If I am correct, this moral is correct only in a qualified manner. In Section 1 I argued that, although Sider’s (generalized) argument proves the impossibility of n-th order vague existence, for all n, the result is prima facie consistent with the possibility of super-vague existence. The suggestion has been vindicated in Section 2, where I have developed a generalization of Williamson-style precisificational semantics where Sider-determinacy is satisfied and yet super-vague existence is shown to be a consistent notion. Consequently, Sider’s moral holds as long as vagueness is not super-vagueness, modulo the present choice of semantic framework.

2.4 Objection from reductio ad absurdum In Section 1.2, I remarked that Sider’s argument against vague existence is invalid unless stated in a perfectly precise language, since the argument is a reductio ad absurdum, which is a valid inference form only for bivalent languages. It might be objected that vagueness in the proof ’s language doesn’t suffice to rule out the applicability of reductio. For if that language is vague, it would be reasonable to interpret it within a supervaluationist semantics. Now, it is true that reductio is invalid in supervaluationism—for instance, we cannot infer from ϕ ∧ ¬ Δ ϕ ⊨ ⊥ to ⊨ ϕ ! Δ ϕ. However, reductio is supervaluationarily valid for Δ-free languages. To see that, let ⊨ C and ⊨ SV denote the relations of consequence for classical logic and standard supervaluationism, respectively. Now, suppose that Γ;ϕ ⊨ SV ⊥. By (Eq1 ), we know that ⊨ C and ⊨ SV are equivalent (cf. Section 2.3). So, Γ;ϕ ⊨ C ⊥ and, since reductio is classically valid, Γ ⊨ C ¬ϕ. By (Eq1 ), it follows that Γ ⊨ SV ¬ϕ. But Sider’s argument is formulated in the metalanguage of ∃xϕ, which paraphrases away Δ (as well as any other expression defined via Δ) by quantifying over precisifications which are extensional, set-theoretic objects. Thus, the language of Sider’s proof being Δ-free, reductio ad absurdum appears to be valid after all. What undermines the above objection is the tacit assumption that, if the semantics for the language of Sider’s proof is supervaluationary, then it has to be some kind of standard supervaluationism, such as Fine’s specification space semantics or Williamson-style constant domain semantics. I claim instead that, if Sider’s proof is coached in a vague language, we should model it via negative supervaluationism. The reason for this choice is quite straightforward. I argued in Section 1.2 that if Sider’s language lacks

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determinacy, this must be due to vague quantification over precisifications. We need, therefore, a precisificational semantics to deal with vague quantification which, as explained in Sections 2.2 and 2.3, should be negative supervaluationism, with the relevant consequence relation ⊨ NS . Now, a feature of negative supervaluationism, which distinguishes it from standard supervaluationism, is that it fails to validate reductio ad absurdum even for Δ-free languages. For instance, it is a fact that Ted doesn’t exist ⊨ NS ⊥ since names and free variables cannot be definitely non-referring in a NSmodel. However, it doesn’t follow that ⊨ NS Ted exists for otherwise existence would always be determinate in negative supervaluationism, which we know not to be the case due to the NS-satisfiability of super-vague existence (Appendix B.7). It is worth noting that, even though we could give a Sider-style argument against vague existence which doesn’t employ reductio ad absurdum, the new argument would still have to be formulated in a precise language, in order to be valid. For example, we could give a proof by cases: i. ⌜ ∃xϕ⌝ is either vague or not vague. ii. Suppose ⌜ ∃xϕ⌝ is vague. Therefore, it is true at some precisification, and so true at all precisifications, which means that it is true (cf. steps 3–9 in my reconstruction of Sider’s argument, Section 1.1). But if ⌜ ∃xϕ⌝ is true, then it is not vague. iii. Suppose ⌜ ∃xϕ⌝ is not vague. Therefore, it is not vague. iv. Thus, ⌜ ∃xϕ⌝ is not vague (proof by cases) The argument has the form: i. ii. iii. iv.

p ∨¬p p⊨q ¬p ⊨ q ⊨q

Now, in standard supervaluationism, proof by cases behaves just like reductio ad absurdum: although invalid in general, it is valid for arguments stated in a Δ-free language. But again, if the language in which the above proof is stated is vague due to quantification over precisification, the correct framework is negative supervaluationism, with the relevant consequence relation ⊨ NS which does not validate proof by cases even for Δ-free languages, as the following simple counterexample will show:

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a ¼ a∨a 6¼ a a ¼ a ⊨ NS ∃xðx ¼ aÞ a 6¼ a ⊨ NS ∃xðx ¼ aÞ ⊨ NS ∃xðx ¼ aÞ

Indeed, in some model the three premises are true,23 whereas the conclusion is untrue.

2.5 Inferentialist objection I have modeled super-vague existence by means of precisified quantifiers. I will now consider an objection, which is an adaptation of an argument originally formulated by Williamson [34], purporting to show that there cannot be multiple precisifications of the existential quantifier. 0 Given a language L with vague ∃, define a new language L in which ∃ is replaced with two precisifications ∃1 and ∃2 . For instance, ∃1 could be the ontologically sparse quantifier of the mereological nihilist, whereas ∃2 is the 0 promiscuous quantifier of the universalist. (Likewise, in L0 there will be “composition1 ” and “composition2 ”). Now, let ϕðxÞ be a L formula. From ∃1 xϕðxÞ we can deduce ϕðzÞ by existential1 instantiation, where z is chosen so that it does not occur free in ϕðxÞ. By existential2 generalization, ∃2 xϕðxÞ follows from ϕðzÞ. Hence, there exists a deduction of ∃2 xϕðxÞ from ∃1 xϕðxÞ. Since we can produce the same kind of argument running in the opposite direction, the two quantifiers are equivalent, which contradicts the initial assumption that ∃1 and ∃2 are distinct precisifications. To this argument I offer a two-tiered reply. For reasons that will soon become clear, I take the second part of my reply to be the more enlightening one. Firstly, the objection assumes that the precisified quantifiers are classical, in the sense that for each ∃n , the rules of generalizationn and instantiationn are the classical ones. But this assumption is unwarranted. Recall that a precisification of the language is identified with a particular specification point in a model of negative supervaluationary semantics. Local truth, i.e. truth at a specification point, is defined in terms of negative semantics. Moreover, negative free logic is sound and complete with respect to negative semantics. Therefore, the generalization and instantiation rules for a precisified quantifier are the ones of negative free logic: • from ϕðzÞ ∧ ∃n xðx ¼ zÞ infer ∃n xϕðxÞ • from Γ and ∃n xϕðxÞ infer ϕðzÞ ∧ ∃n xðx ¼ zÞ, where z does not occur free in Γ or ϕðxÞ. 23

a 6¼ a is an NS-inconsistency, hence ∃xðx ¼ aÞ follows from it trivially.

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With that being said, it is easy to see what goes wrong in the inferentialist objection. Recall that ∃1 is the less promiscuous quantifier, which does not support composition 1 , whereas ∃2 allows unrestricted composition2 . 0 Let ϕðxÞ be the L formula “x is a mereological compound2 ,” where “compound2 ” is the universalist precisification of the vague L-term “compound.” Since the underlying logic is free, from “there exists2 the fusion2 of a table and a giraffe” by existential2 instantiation we can infer “z is the fusion2 of a table and a giraffe and z exists2 .” But in order to conclude by existential1 generalization “there exists1 the fusion2 of a table and a giraffe,” we first need to be able to infer “z is the fusion2 of a table and a giraffe and z exists1 .” So, the derivation goes through if “z exists2 ” entails “z exists1 ,” which cannot be assumed without begging the question.24 I now turn to the second reply. The inferentialist objection simply 0 assumes that it is possible to define a new language L in which the vague quantifier ∃ is replaced with the sharp quantifiers ∃1 and ∃2 . As it turns out, multiple quantifiers obeying the rules of free logic cannot coexist in the same language. For if there were such ∃1 and ∃2 , a new existential quantifier ∃˄ could be defined in L0 whose range is the union of the ranges of ∃1 and ∃2 : ∃˄ xðx ¼ zÞ :¼ ∃1 xðx ¼ zÞ∨ ∃2 xðx ¼ zÞ. Now, if ∃1 is a proper restriction of ∃˄ , for some z it is true that ∃˄ xðx ¼ zÞ ∧ ¬ ∃1 xðx ¼ zÞ. By definition of ∃˄ , that is equivalent to (i) ¬ ∃1 xðx ¼ zÞ ∧ ∃2 xðx ¼ zÞ. Since ∃1 xðx ¼ zÞ $ z ¼ z is a theorem of negative free logic, from the first conjunct of (i) it follows that (ii) z 6¼ z. From (ii) and the second conjunct of (i) we can infer (iii)

∃2 xðx 6¼ xÞ.

But (iv) 82 xðx ¼ xÞ is a theorem of negative free logic. Hence, the claim (i) that ∃1 is a restriction of ∃˄ is inconsistent, provided that the precisified quantifiers obey negative free logic. But according to negative supervaluationary semantics, precisified quantifiers do obey negative free logic. We must conclude 24

Cf. Turner [30, pp. 25–6].

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that it not possible to define ∃˄ and, therefore, that we cannot use both ∃1 and ∃2 within the same language, as the inferentialist objection presupposes.25 The moral is that quantifiers behave differently from non-logical predicates in one key respect. If we speak a language where “bald” is vague, we can define a new language in which the original predicate is replaced with a multiplicity of precise predicates “bald1 ,” “bald2 ” etc. This cannot be done with quantifiers on pain of inconsistency. I hope it is now clear that the deeper reason why the inferentialist objection is unsound is that it assumes that quantifiers governed by negative free logic can coexist in a single language.

3 . C O N C LU S I O N Whether existence can be vague has consequences both in first-order ontology and in metaontology. In the former case, the possibility of vague existence makes room for vague composition. In the latter, vague existence may be a symptom of the world lacking a unique quantificational structure. Sider has famously submitted a reductio of vague existence, on the assumption that vagueness is interpreted precisificationally and existence is absolute. In Section 1, I argued that a precisificational framework per se does not allow us to disprove vague existence, i.e. to prove that it is definitely not vague. At most it can be proven that existence is not definitely vague. The same applies to a disproof of higher-order vague existence. The upshot of the discussion turned out to be that Sider’s argument is compatible with existence being neither definitely vague nor definitely precise, at every order. I named this specific phenomenon super-vague existence. In Section 2, I provided a precisificational model theory, dubbed negative supervaluationary semantics, with the aim of modeling super-vague existence and its logic. Moreover, an objection from reductio ad absurdum and an inferentialist objection have been taken care of.

25 It could be objected that a logical constant satisfying some given condition in 0 a language L need not do so in an expanded language L . For instance, identity satisfies Leibniz’s law if the language is extensional, but not if we add doxastic or epistemic satisfy the same inference rules in each operators. Likewise, the various ∃n might not 0 precisification Ln of L and in the expanded L . This observation overlooks one important bit of information, namely that the quantifiers ∃n are all precisifications of the original quantifier ∃. I assume the following principle: if c 0 is a precisification of the the logical constant c, then c 0 satisfies at least the axioms and rules of inference which c satisfies. In fact, I take that condition to partly define what it means for c 0 to be a precisification of c. It follows in particular that ∃1 and ∃2 must satisfy the axioms and rules of negative free logic.

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If existence is super-vague, we ought to accept that composition might be super-vague (i.e. vague at all orders) and that the world may lack a unique quantificational structure. Alessandro Torza National Autonomous University of Mexico

A C K N OW L E D G M E N T S I would like to thank Dean Zimmerman and one anonymous referee for the extensive and very helpful comments. I am also in debt to the audiences at the University of Turin, University of Sussex, Duke University, the XVI International Congress of the Mexican Philosophical Association, the National Autonomous University of Mexico, the 2nd Colombian Conference of Logic, Epistemology and Philosophy of Science. This work has been made possible in part by the CONACyT grant CCB 2011 166502 and the PAPIIT grant IA400412.

APPENDICES A P P E N D I X A: FI N E A N SU P E R V A L U A T I O N I S M A N D EX I S T E N C E In Fine’s supervaluationary semantics it can be proved that, as long as there definitely are finitely many objects, there is no vague existence at any order (cf. Section 2.1). For the proof, it suffices to show that (i) existence is definite, i.e. ∃xðx ¼ yÞ ! Δ ∃xðx ¼ yÞ, and that (ii) definite statements cannot be indefinitely definite, i.e. Δϕ ! ΔΔϕ. Let n be the cardinality of the domain of the largest complete specification point in a space. (If the cardinalities had no upper bound, it would not be the case that there definitely are finitely many objects.) Notice that the domain of the base point @ is a subset of the cardinality of any complete specification point, since if ∃yðx ¼ yÞ is true at @ (given an assignment for x), then it is true at every accessible point. So, there is some m ⩽ n which corresponds to the cardinality of the domain of @. Let 8yðy ¼ x1 ∨y ¼ x2 ∨:::∨y ¼ xm Þ express which things exactly exist at @. By the construction of a specification space, that sentence must be true at all complete specification points, which must therefore have constant domain. So, ∃yðx ¼ yÞ is true at a complete specification point s only if it is true at all complete specification points. It follows that ∃yðx ¼ yÞ is true at @, and so is Δ ∃xðx ¼ yÞ. Therefore, at every s it is true that ∃xðx ¼ yÞ ! Δ ∃xðx ¼ yÞ. We can conclude that existence is not vague. Moreover, Fine’s theory does not admit of cases of higher-order vagueness to be expressed in the object-language: what is true/false/ indeterminate is definitely true/false/indeterminate. Which is to say, both (a) Δϕ ! ΔΔϕ and (b) ¬Δϕ ! Δ¬Δϕ are supervaluationarily valid in Fine’s model. As to (a),

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suppose that Δϕ is true at a complete specification point s. So, ϕ is true at all complete specification points. It trivially follows that it is true at s that every complete specification point is such that ϕ is true at all complete specification points. Which is to say, ΔΔϕ is true at s. Mutatis mutandis for (b). As a consequence, since existence is precise, it must be precise at all orders. Q.E.D.

A P P E N D I X B : EL E M E N T S O F NE G A T I V E SUPERVALUATIONARY SEMANTICS

B.1 NS and existential generalization We want to show that, for ϕ and Γ formulated in a first-order language L without Δ, it is not the case that Γ ⊨ C ϕ iff Γ ⊨ NS ϕ where ⊨ C ( ⊨ NS ) indicates classical (NS-) consequence relation. To see that, let M* be a NS-model with only two specification points s and t, where DomðsÞ ¼ fag and DomðtÞ ¼ fa;bg. Also, suppose that σ*ðP;sÞ ¼ σ* ðP;tÞ ¼ fag. Consider a value assignment mapping x to b at t and leaving it undefined at s. Since PðxÞ is false at both s and t, ¬PðxÞ is true in the model. However, ∃x¬PðxÞ is true at s and false at t, therefore indeterminate in the model. Thus, existential generalization does not hold in general: ϕðxÞ ⊭ NS ∃xϕðxÞ. The same reasoning applies mutatis mutandis to universal instantiation.

B.2 NS and negative free logic I For ϕ and Γ formulated in a language L without ‘Δ’, (Eq2 ) if Γ ⊨ NF ϕ then Γ ⊨ NS ϕ. To see that, let Γ be true in the NS-model M* given a variable assignment fξt gt ∈ S . Then, for every ξt , Γ is locally true at t. Since local truth for a Δ-free language in a NSmodel is tantamount to truth in a NF -model, and since ϕ is an NF -consequence of Γ, it follows that ϕ is locally true at t under ξt . Hence, ϕ is true at M* under fξt gt ∈ S . However, the converse does not hold. Because a variable assignment over a NSmodel maps each variable to an object at some specification point, ¬ ∃yðx ¼ yÞ can only be false or indeterminate in a model. Hence, it must be that ¬ ∃yðx ¼ yÞ ⊨ NS ⊥. On the other hand, ¬ ∃yðx ¼ yÞ is NF -satisfiable.

B.3 NS and the indiscernibility of non-existents The indiscernibility of non-existents: (IN) ¬ ∃zðx ¼ zÞ ∧ ¬ ∃zðy ¼ zÞ ! ðϕðxÞ ! ϕðyÞÞ fails in negative supervaluationary semantics. In order to see that, consider a NF model with two specification points s, t such that tRs and σ* ðP;sÞ ¼ fbg. Now,

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assume that ξs ðxÞ ¼ a, ξs ðyÞ ¼ b, a 6¼ b whereas both ξt ðxÞ and ξt ðyÞ are undefined. Then, the instance of (IN) obtained by substituting Δ¬P for ϕ is false at t under fξs gs ∈ S and therefore untrue in the model.

B.4 NS and negative free logic II Negative free logic and negative supervaluationary semantics define the same class of valid formulas in a language without “Δ”: (Eq3 )

⊨ NF ϕ iff ⊨ NS ϕ.

The left-to-right direction is an immediate consequence of (Eq2 ). As to the converse, consider a NF -model and a partial function ζ mapping the free variables of ϕ to the domain. Truth in that model is tantamount to local truth at a specification point t of some NS-model M* under ξt ¼ ζ. Since ϕ is NS-valid, it is true at M* under fξs gs ∈ S and therefore locally true at t under ξt . Consequently, ϕ is true in the original NF -model under the variable assignment ζ.

B.5 NS and identity A condition which fails in negative supervaluationary semantics is the definiteness of identity (DI) x ¼ y ! Δx ¼ y To see that, just consider a NS-model and a variable assignment in which x and y co-refer to a at point s, whereas t is a point such that sRt and a 2 = DomðtÞ. Since x ¼ y is true at s and false at t, (DI) is false at s and therefore untrue in the model. A symmetrical scenario yields a counterexample to the definiteness of distinctness: (DD) x 6¼ y ! Δx 6¼ y. However, (DI) can never be false in a model, for if it were, that would contradict the fact that x ¼ y ⊨ NS Δx ¼ y, which is an instance of (N* ). Likewise for (DD). Since variable assignments over NS-models are rigid, it follows that identity is weakly definite: (DI ) x ¼ y ! Δð ∃z ∃z 0 ðx ¼ z ∧ y ¼ z 0 Þ ! x ¼ yÞ Distinctness, on the other hand, does not satisfy weak definiteness: (DD ) x 6¼ y ! Δð ∃z ∃z 0 ðx ¼ z ∧ y ¼ z 0 Þ ! x 6¼ yÞ For suppose a variable assignment maps x and y to the same object a at point t, and let sRt, where a= 2DomðsÞ. Since the assignment is rigid, x and y will fail to refer at s, and so the antecedent of (DD ) must be true at that point. The consequent, on the other hand, is false at s.

OUP CORRECTED PROOF – FINAL, 28/12/2016, SPi

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B.6 NS and the importation schema (IM) The importation schema: (IM)

∃xΔϕ ! Δ ∃xϕ

is not NS-valid. To construct a counterexample, let M* be a NS-model with specification points s and t where sRt, DomðsÞ ¼ fag and DomðtÞ ¼ fbg. Suppose that σ* ðP; sÞ ¼  and σ* ðP;tÞ ¼ fbg. Then, it is true at s that ∃xΔ¬PðxÞ, since ¬PðxÞ is true at s under ξs ¼ fhx;aig and true at t under ξt ¼ . But ∃x¬PðxÞ is false at t and therefore Δ ∃x¬PðxÞ is false at s. So, ∃xΔ¬PðxÞ ! Δ ∃x¬PðxÞ is untrue in M* .

B.7 NS and super-vague existence In order to show that negative supervaluationary semantics can model super-vague existence, it suffices to show that the following conditions hold: Sider-determinacy: for all M* , fξs gs∈S and n⩾1: ðM* ;fξs gs∈S Þ⊭NS I n ∃xðx¼yÞ anti-Sider-determinacy: for some M* and fξs gs ∈ S and for all n ⩾ 1: ðM* ; fξs gs ∈ S Þ ⊭ NS ¬I n ∃xðx ¼ yÞ The first part, Sider-determinacy, follows immediately from the fact that every variable domain frame F* of a NS-model is by definition n-determinate, for every n ⩾ 1 (cf. Section 2.2). An NS-model satisfying anti-Sider-determinacy can be constructed as follows. Let M* ¼ hS;U;R;Dom;σi be a model for a LΔ language without non-logical constants, where • • • • •

S ¼ fs0 ;s1 ;s2 ,..., sn ,...g (the specification points) U ¼ fa;bg (the individuals) R is the reflexive and symmetric closure of fhsn ;snþ1 ign ⩾ 0 Domðs0 Þ ¼ fa;bg; Domðsnþ1 Þ ¼ fag σð¼; sn Þ is the identity relation over sn

Now, let fξs gs ∈ S be an assignment over M* such that ξs0 ðyÞ ¼ b. It is easy to show that, for all n ⩾ 1, ðM* ;sn ; fξs gs ∈ S Þ ⊨ NS I n ∃xðx ¼ yÞ. Hence, ðM* ; fξs gs ∈ S Þ ⊭ NS ¬I n ∃xðx ¼ yÞ, for all n ⩾ 1. We can conclude that super-vague existence is possible in negative supervaluationary semantics.26

REFERENCES Akiba, Ken (2000). “Vagueness as a Modality,” The Philosophical Quarterly, 50(200): 359–70. 26 Notice, however, that the tautology I n ∃xðx ¼ yÞ∨¬I n ∃xðx ¼ yÞ remains NSvalid, for all n.

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Akiba, Ken (2004). “Vagueness in the World,” Noûs 38(3): 407–29. Barnes, Elizabeth (2009). “Indeterminacy, Identity and Counterparts: Evans Reconsidered,” Synthese 168(1): 81–96. Barnes, Elizabeth (2012). “Metaphysically Indeterminate Existence.” Philosophical Studies 166(3): 495–510. Boolos, George (1984). “To Be Is To Be a Value of a Variable (or to Be Some Values of Some Variables),” Journal of Philosophy 81: 430–50. Braun, David and Theodore Sider (2007). “Vague, So Untrue,” Noûs 41(2): 133–56. Burge, Tyler (1974). “Truth and Singular Terms,” Noûs 8: 309–25. Carmichael, Chad (2011). “Vague Composition without Vague Existence,” Noûs 45 (2): 315–27. Carnap, Rudolf (1950). “Empiricism, Semantics and Ontology.” Reprinted (1956) in Meaning and Necessity, 2nd edn. Chicago: University of Chicago Press: 205–21. Chalmers, David, David Manley and Ryan Wasserman, eds. (2009). Metametaphysics: New Essays on the Foundations of Ontology. Oxford: Oxford University Press. Donnelly, Maureen (2009). “Mereological Vagueness and Existential Vagueness.” Synthese 168(1): 53–79. Evans, Gareth (1978). “Can there be Vague Objects?”, Analysis 38(4): 208. Fine, Kit (1975). “Vagueness, Truth, and Logic,” Synthese 30 (3–4), 265–300. Garson, James (2001). “Quantification in Modal Logic,” in Dov Gabbay and Franz Guenthner (eds.) Handbook of Philosophical Logic, second edition, vol. 3. Dordrecht: Reidel: 267–324. Heck, Richard (1998). “That There Might Be Vague Objects (So Far As Concerns Logic),” Monist 81(2): 274–96. Keefe, Rosanna (2000). Theories of Vagueness. Cambridge: Cambridge University Press. Korman, Daniel (2010). “The Argument from Vagueness,” Philosophy Compass 5 (10): 891–901. Lewis, David (1986). On the Plurality of Worlds. Oxford: Basil Blackwell. Lewis, David (1988). “Vague Identity: Evans Misunderstood,” Analysis 48(3): 128–30. Linsky, Bernard and Edward Zalta (1994). “In Defense of the Simplest Quantified Modal Logic,” Philosophical Perspectives 8: Philosophy of Logic and Language, Atascadero, CA: Ridgeview: 431–58. Linsky, Bernard and Edward Zalta (1996). “In Defense of the Contingently Nonconcrete,” Philosophical Studies 84(2–3): 283–94. Nolt, John (2006). “Free Logics” in Dale Jacquette (ed.), Philosophy of Logic. Handbook of the Philosophy of Science. Amsterdam: North Holland: 1023–60. Quine, W. V. O. (1951). “Two Dogmas of Empiricism.” Philosophical Review 60(1): 20–43. Sainsbury, Richard (2009). Fiction and Fictionalism. London: Routledge. Sider, Theodore (2001). Four-Dimensionalism. Oxford: Oxford University Press. Sider, Theodore (2003). “Against Vague Existence,” Philosophical Studies 114(1): 135–46. Sider, Theodore (2007). “Neo-Fregeanism and Quantifier Variance,” Proceedings of the Aristotelian Society 81(1): 201–32.

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Sider, Theodore (2009a). “Against Vague and Unnatural Existence: Reply to Liebesman and Eklund,” Noûs 43(3): 557–67. Sider, Theodore (2009b). “Ontological Realism,” in Chalmers et al. [10]: 384–423. Turner, Jason (2010). “Ontological Pluralism.” Journal of Philosophy 107(1): 5–34. van Inwagen, Peter (1990). Material Beings. Ithaca: Cornell University Press. Varzi, Achille (2007). “On Supervaluationism and Its Logics,” Mind 116(463): 633–75. Williams, Robert (2008). “Multiple Actualities and Ontically Vague Identity,” Philosophical Quarterly 58(230): 134–54. Williamson, Timothy (1987). “Equivocation and Existence,” Proceedings of the Aristotelian Society 88: 109–27. Williamson, Timothy (1994). Vagueness. London: Routledge. Williamson, Timothy (1998). “Bare Possibilia,” Erkenntnis 48(2–3): 257–73. Williamson, Timothy (1999). “On the Structure of Higher-Order Vagueness,” Mind 108(429): 127–43. Williamson, Timothy (2005). “Vagueness in Reality,” in Michael Loux and Dean W. Zimmerman (eds.) The Oxford Handbook of Metaphysics, Oxford: Oxford University Press: 690–715.

10 Ersatz Counterparts Richard Woodward We find it congenial to talk not only of what goes on at possible worlds, but also of their inhabitants, the possibilia. Thus we talk not only of worlds where donkeys talk but also of talkative possible donkeys. And when we talk this way, we can think of ourselves as speaking a certain language: the possibilist language. We speak the possibilist language to talk about possibilities, by means of talking about possible things. But we also talk about possibilities in a more familiar way, by talking about what might have been and what might not, by talking about what is contingent and what is necessary. Here, we can think of ourselves as speaking a different language, one that is readily amenable to regimentation using boxes and diamonds: the modalist language. These languages share a subject matter, modal space, and the sentences of each express claims about this subject matter. It’s not so surprising, then, that we can pair sentences of the one language with sentences of the other. The modalist sentence ‘there might have been talking donkeys’ can, e.g. be paired with the possibilist sentence ‘there is a world where donkeys talk’ and the modalist sentence ‘round squares are impossible’ can be paired with the possibilist sentence ‘there is no world where squares are round’. Of course, there are questions about whether everything you can say in the one language you can also say in the other, and there are questions about whether one of these languages is more fundamental than the other. But set these issues aside. Our focus is on some delicate issues relating to de re possibilities. When we try to pair a modal claim like ‘Messi might have won’ with some sentence of the possibilist language, it’s natural to appeal to worlds where Messi wins: worlds which represent Leo Messi as winning. But how does a world represent Messi as winning? You might think it does so by containing that man. But nearly everyone thinks that this is false. Even Alvin Plantinga (1974), a card-carrying believer in transworld identity, thinks that talk of possibilia isn’t to be understood in terms of flesh and blood individuals like

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you and me, but in terms of special abstract objects: individual essences. Whatever are the constituents of other worlds, it seems, Messi is not one of them.1 And David Lewis, who does think that talk of possibilia should be understood in terms of objects like you and me, explicitly tells us that ‘nothing is in more than one world’ (1968, p. 114). So how is de re representation to be understood? A natural answer to this question begins by appealing to representatives of Messi: objects that, though numerically distinct from Messi, stand in an important relation to the man himself. The thought is then that Messi’s representative, Messi*, can go where he cannot. To see how this helps, consider how things play out for Plantinga. Plantinga has an easy time identifying Messi’s representative: it’s his individual essence, the property of being Messi. This property is distinct from Messi and stands in an important relation to the man himself. Plantinga cannot hold that Messi’s essence wins at any world: essences don’t do that kind of thing. But that just means that he needs to be creative when interpreting sentences like ‘Messi* wins’. Plantinga’s preferred option is to take ‘wins’ to express the property of being co-instantiated with the property of being a winner. In this way, Plantinga can tell a story about how it can be true at a world that Messi wins. Lewis appeals to representatives too. But his representatives are the real thing: flesh and blood concrete objects just like Messi. And whereas Plantinga only has only one representative, Lewis has many. So whereas Plantinga can hold that Messi’s actual representative is identical to his representatives elsewhere—it’s always Messi’s essence—Lewis needs a different story. Lewis’s story is familiar. Messi’s representatives are his counterparts: possible objects that are similar to him in some contextually salient way. Speaking loosely, we might say that Messi’s counterparts in other worlds are him in those worlds, but Lewis (1968, p. 114) insists that this is not identity in any literal sense, and that it would be better to say that Messi’s counterparts are the men he would have been, had things been different. But though Messi’s counterparts in other worlds are not really him, the distinctness of Messi and his representatives is not a unique feature of counterpart theory. Even someone who believes in transworld identity can accept that Messi’s representatives are not really him. Plantinga is a case in point.2

1 Phrasing the point in terms of ‘constituents’ is perhaps misleading. It’s not clear that Plantinga believes that possible worlds have constituents in any sense. What’s really at stake here is the semantic point that in a standard Kripke semantics each world is associated with a domain of possible individuals. And on Plantinga’s view, these domains are sets of essences. 2 For more on this point, see Woodward (2012).

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What is distinctively counterpart-theoretic, and what transworld identity theorists cannot accept, is Lewis’s account of when something counts as a representative of Messi. For one thing, whether one object is a counterpart of another is a context-sensitive matter: what counts as a counterpart in one context need not count as a counterpart in another. For another thing, the counterpart relation is not an equivalence relation: though reflexive, it is neither symmetric nor transitive. And as is well-known, these two features are beneficial, allowing us to account for the inconstancy of our de re modal judgements, solve puzzles generated by the conflicting modal properties of statues and lumps of clay, and avoid commitment to the idea that the logic of de re modality is S5 (see, e.g. Lewis (1968, 1971, 1986)).3 Is this beneficial package Lewis’s and Lewis’s alone? Could you jettison Lewis’s robustly realist interpretation of the possibilist language and still be a counterpart theorist? Those attracted to an ‘ersatz’ counterpart theory must answer two questions. They must tell us what counterparts are, if not the flesh and blood things that Lewis takes them to be. And once they have told us that, they must tell us in virtue of what these things are counterparts of each other. That is, they must tell us two stories: one about the relata of the counterpart relation and one about the relation itself.4

1. ERSATZ COUNTERPARTS The benefits that I just associated with counterpart theory were semantical, conceptual, and logical in character. The theory gives us a neat semantic treatment of de re modal predications; it ‘fits’ our de re modal concepts and gives us a nice logic. A further benefit is more metaphysical in character: counterpart theory allows Lewis to reduce de re representation to de dicto representation. This point is often lost, partially because there is a tendency 3 There are two ways of doing counterpart-theoretic accountancy, and I have just played fast and loose with the distinction. Going one way, we might count many relations as counterpart relations and then think that (ideally) one of these is selected in a context, with the other counterpart relations being ignored or excluded. Going this way, what we should say is that counterpart relations in general are not equivalence relations. Going the other way, we might count (in a context) a single relation as the counterpart relation, and then think that which relation counts as the counterpart relation shifts around from context to context. Going this way, what we should say is there is no semantic guarantee that the counterpart relation is an equivalence relation, even though (in some contexts) the counterpart relation will be an equivalence relation. Thanks to John Divers for discussion. 4 Question: Can’t the actualist refuse to tell a story about the counterpart relation, and take it as a primitive? Answer: The story needn’t be an analysis. Lewis never gives us a formal definition of the counterpart relation, but he does his best to explain the notion.

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in the literature to contrast Lewis’s theory with ‘representational’ theories like Plantinga’s. But don’t be misled: Lewis needs a story about representation just as much as anyone else. And he gives us two, one for the de dicto case and one for the de re. What a Lewis world represents de dicto—which purely qualitative facts it represents as holding, which purely general propositions it represents as being true—is analysed in mereological terms. So, for example, a Lewis world represents the existence of talking donkeys just in case it contains talking donkeys and has talking donkey as parts. And once we’ve settled what each Lewis world de dicto represents, we can introduce de re representation by appealing to the qualitative features of the possibilia that populate Lewis’s worlds and the similarity relations that they bear to each other. Some dislike this way of setting things up. Representation, I’m often told, is a notion that only has real application in connection with ersatz views of worlds and doesn’t do much work for someone like Lewis. He has his worlds and that is enough to get his analysis up and running. Well, what’s true is that representation isn’t part of the fundamental ideology of Lewis’s view: it’s reduced to a combination of mereology and similarity. I find it difficult to see how this reduction means that representation doesn’t do any work for Lewis, but in any case, Lewis (1983a, 1986) is clear that a possibility is not always a possible world: individuals represent possibilities too. And it bears emphasis that which possibility an individual represents depends on which counterpart relation we have in mind. Thought of as one of my counterparts, a given object may represent a possibility for me; thought of as one of yours, the very same thing may represent a possibility for you. Moral: forgo words like ‘representation’ if you want, but don’t pretend that Lewis himself didn’t make more than enough room for them, and don’t pretend that Lewis doesn’t appeal to the notion of representation to do important work. Ersatz counterpart theorists hope to reduce de re representation to its de dicto cousin too. The first thing to remember is that when the ersatzer speaks of the actual possible world, she is not speaking about concrete reality but a special abstract world: the one that is actualized. And concrete reality is the thing that does the actualizing, not the thing that is actualized.5 A similar point appeals to possible individuals. When the ersatzer speaks of actual possible individuals, she is not speaking about concrete things like you and me, but about their abstract representatives at the actual world. Real things like you and me are the things which are represented, not the things which do the representing. We are not medium but message. 5

See Divers (2002, pp. 228–9) for more on this distinction.

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At this point, the challenge to specify the relata of the ersatz counterpart relation becomes pressing. How should it be met? Well, think of all the purely qualitative predicates that Lionel Messi satisfies.6 Then think of his representative at the actual world, Messi*, as the set of those predicates: • Messi* = {x is male, x is a footballer, x is Argentinian . . . } This generalizes to give us representatives for each actual individual. We can introduce representatives for the merely possible individuals by remembering that my ersatzer aims to reduce de re representation to de dicto representation. Think of what a world represents de dicto as the purely qualitative sentences that it makes true. Each of the objects that exist at w can then be associated with the set of qualitative predicates that w says they satisfy. So if w says that there is a talking donkey called ‘Robin’, then our representative will be the set of qualitative predicates that Robin satisfies at w: • Robin* = {x is a donkey, x talks, x has a tail, . . . } In this way, our ersatzer is able to construct representatives for not only actual individuals, but merely possible ones too.7 When I asked you to think of all of the qualitative predicates that Messi satisfies, I really did mean all of them. So our set-theoretic representation doesn’t only include predicates like ‘x is male’ but also ones like ‘x is part of world containing no talking donkeys’ and ‘x is such that there are exactly n stars’. As Lewis puts it: The ersatz individual is by no means a purely intrinsic description; the description is extrinsic in a big way. By the time we are done describing an individual completely, we have en passant described the world wherein it is situated. (1986, p. 149)

Our setting thus has an important consequence: it entails that Messi* and Robin* are worldbound. This isn’t obvious, since we’ve identified them with sets of open sentences and one might think that such things exist at every world. What’s going on? Notice that to each ersatz individual there corresponds a very complex qualitative property. Let’s say that something plays the Robin-role just in case it satisfies each of the predicates in Robin*, that is, iff it is a donkey, who talks, and has a tail, and so on. Our Leo Messi, then, has the qualitative property of being such that there is no player of the Robin-role. Indeed, for each role associated with each merely possible object, Messi has the property 6

See Lewis (1986, pp. 148–50). There are famously a number of issues regarding the descriptive adequacy of this kind of ersatz setting, but I think it has become clear that they can be addressed: see, e.g. Melia (2001) and Sider (2002). 7

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of being such that there is no player of that role. Moreover, for each role associated with each actually existing object, Messi has the property of being such that there is a player of that role. The point is that Messi* is bound to the actual world in the sense that only the actual world represents the existence of a player of the Messi-role.8 If a world w represents the existence of a player of the Messi-role, then w represents that existence of a player of each qualitative role associated with each actual individual. And this is because part of what’s involved in playing the Messi-role is being worldmates with a player of the me-role, a player of the you-role, and so on. So if w contains Messi* and thereby represents the existence of a player of the Messi-role, w must also represent the existence of a player of the you-role and so must contain you*. Moreover, if w represents the existence of a player of the Messi-role, then w must also represent the non-existence of a player of each qualitative role associated with each nonactual individual. And this is because part of what’s involved in playing the Messi-role is being worldmates with no player of the Robin-role, no player of the Alice-role, and so on. So if w contains Messi* and thereby represents the existence of a player of the Messi-role, w cannot also represent that there is a player of the Robin-role and so cannot contain Robin*. To be clear: this is not to say that only the actual world represents Messi, nor that there is a bizarre necessary connection between Messi and me, nor that there is a bizarre necessary disconnection between Messi and Robin. The point is rather that ersatz worlds don’t represent the existence of Messi by representing the existence of a player of the Messi-role. Questions of de re representation—and in turn questions of de re modality itself—are answered by looking to the facts about counterparts, just as on Lewis’s account. In essence, all we have is an ersatz surrogate for Lewis’s original demand that nothing is in more than one world. So much for specifying the relata of the ersatz counterpart relation. What can the ersatzer say about the relation itself ?

2. ERSATZ COUNTERPART RELATIONS Remember that Lewis explains his counterpart relation in terms of similarity: whether two things are counterparts depends on whether they are similar to 8 Setting aside worries about whether there could be distinct worlds that were nonetheless indiscernible in terms of what they de re represent. Compare Lewis (1986, p. 224) and see Divers (1996) for discussion. A few seasons back, the Messi-role was called False Nine.

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each other in some contextually important way. When asked to explain her counterpart relation, can’t the ersatzer just say the same thing? No. Lewis’s explanation has it that two individuals are counterparts iff those very things are similar in some contextually salient way. If the ersatzer parroted this, she’d end up saying that Messi* and Robin* are counterparts iff those very things are relevantly similar. And this seems to me to get things back to front: whether Messi* and Robin* count as counterparts shouldn’t, I think, turn on whether or those two sets—for that’s what they are—are similar. Rather, whether these representations are counterparts should turn on whether the things they represent are similar.9 The trouble is that the ersatzer can’t accept this. For Robin* represents a player of the Robin-role, and the ersatzer doesn’t believe that there is any such thing. The present proposal works when two actual individuals are counterparts: the ersatzer can hold that whether Messi* is a counterpart of me* depends on whether he is similar to me in some contextually relevant way. But the proposal simply breaks down when one of the relata represents a merely possible individual. When it comes to explaining the counterpart relation, the ersatzer faces troubles that Lewis does not. But though she has fewer ontological resources than Lewis, the ersatzer has extra ideology: primitive modality. Her worlds are maximally consistent sets of sentences or maximally possible states of affairs (or whatever), where consistency and possibility are either taken as primitive or defined in primitively modal terms. So whereas Lewis uses his plentiful ontology to explain the counterpart relation, couldn’t the ersatzer deploy her extra ideological resources to do the same thing? This strategy runs into trouble too. Suppose that the ersatzer tried to explain her ersatz counterpart relation as follows: • Messi* is a counterpart of Robin* iff Necessarily, for any two things x and y, if x plays the Messi-role and y plays the Robin-role then x is relevantly similar to y The problem, as Lewis notes, is that the right-hand side of this biconditional is trivial.10 There is no possible world containing both a player of the Messirole and a player of the Robin-role. For remember that it’s built into the Messi-role that anything that plays the role is such that there is no player of

9 Sider (2002, p. 303) agrees. The point can be made a little more vivid by remembering that we could have associated each qualitative predicate with a unique number, and then taken Messi* and Robin* to be sets of numbers. But whether Messi* and Robin* are counterparts shouldn’t turn on whether {1, 2, 3, . . . } is similar to {4, 5, 6, . . . }. It should turn on whether the things that these sets represent are similar to each other. 10 Lewis (1986, p. 238). Compare Sider (2002, p. 304).

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the Robin-role, due to the fact that the ersatz descriptions are ‘extrinsic in a big way’. Our modal explanation of the counterpart relation is a failure and the ersatzer seems to be running out of options pretty quickly. (The problem with the modal analysis considered above arises because it requires an intra-world similarity relation between a player of the Messi-role and a player of the Robin-role. And so one might naturally think that the counterpart theorist should require an inter-world similarity relation instead. That is, one might think that the counterpart theorist should endorse the following modal analysis of the counterpart relation: • Messi* is a counterpart of Robin* iff Necessarily, for any thing x, if x plays the Messi-role then, necessarily, for any thing y, if y plays the Robin-role then x is relevantly similar to y Certainly this analysis fares better than its predecessor since its adequacy does not require the existence of a single world containing players of both roles. But the result comes at a price: it commits the ersatz counterpart theorist to primitive modality that is de re in character. And remember that the counterpart theorist’s project, as it is being conceived here, is similar to Lewis’s: she aims to reduce representation de re to representation de dicto. In this setting, the appeal to primitive de re modality is off-limits. To put that point otherwise, the counterpart theorist aims to reduce the de re modal facts to facts about ersatz counterparts and and our present question is whether these latter facts can themselves be further reduced. Within this context, it is clear that the ersatz counterpart theorist cannot explicate the ersatz counterpart relation in the manner suggested.) At this stage, it’s worth repeating a point that I made earlier. Even if we grant that no world contains both a player of the Messi-role and a player of the Robin-role, we are not thereby forced to deny the Humean thought that anything can co-exist with or without anything else. For one thing, the ersatz counterpart theorist, like Lewis, thinks that the question of whether a could exist without b turns on whether or not there is a world containing a counterpart of a but not b. For another thing, and as Phillip Bricker (2001) stresses, the Humean denial of necessary connections is best articulated in terms of duplication: there are no necessary connections between distinct things iff for any two distinct things, it is possible for a duplicate of the one to exist without a duplicate of the other. Crucially, however, duplication is cashed out in terms of intrinsic properties: x is a duplicate of y iff x and y have the same intrinsic properties. Even if the extrinsic information contained within the Messi-role and the Robin-role means that there is no world containing both a player of the Messi-role and a player of the Robin-role, the Humean picture is in place as there are worlds containing duplicates of Messi and Robin: things which satisfy the intrinsic information contained within the Messi-role and the Robin-role.

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Next, consider the following explanation of the counterpart relation: • Messi* is a counterpart of Robin* iff Necessarily, for any two things x and y, if x is a Messi-duplicate and y is a Robin-duplicate, then x is relevantly similar to y. This explanation is better than the original modal analysis insofar as some worlds contain both Messi-duplicates and Robin-duplicates. The problem is that it is crucial to counterpart theorists that things can count as counterparts because they are similar in highly extrinsic respects: you may count as a counterpart of mine by being twelve feet from a cat, for instance. Be that as it may, the explanation nonetheless hints at how the counterpart theorist can explain her relation. For what we have in effect done is restricted the Messirole and the Robin-role so as to ensure that there is a world containing players of both roles. This idea was a good one, but by focusing solely on the intrinsic information contained within the roles, we went too far. What we are after, then, is a way of constructing worlds that contain copies of Messi and Robin, and we know that ‘copying’ needs to be handled carefully. If we are too liberal, our resultant explanation of the counterpart relation will go trivial, but if we are too conservative, our explanation will be overly restrictive. At this point, we need to make an assumption about the extent of modal space, and grant that disjoint spacetimes are possible so that some worlds represent the existence of a plurality of disjoint ‘island’ universes. I take it that this assumption is legitimate in context: even Lewis accepts that it is a cost of his account that it cannot allow for the possibility of disjoint spacetimes. Lewis of course thinks the cost is negotiable and outweighed by other lovely features of his view but, be that as it may, the point is just that it is generally accepted that it is a benefit of ersatz proposals that they can allow for the possibility of disjoint spacetimes. Moreover, suppose that w1 and w2 each represent the existence of a single universe. Then I assume that there is a third world w3 that represents the existence of two disjoint spacetimes v1 and v2 such that v1 is a duplicate of w1 and v2 is a duplicate of w2 .11 Figure 10.1 shows what we have. 𝜔2

𝜔1

𝜐1

𝜔3

𝜐2

Figure 10.1 The copying process 11 This is the ‘hefty metaphysical assumption’ mentioned in Sider (MS). I’ll mount a defence of the assumption in due course.

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Now, if w1 contains a player of the Messi-role and w2 contains a player of the Robin-role, then w3 will contain duplicates of each. The trick is now to understand the arrows in such a way as to ensure that the extrinsic information about Messi and Robin carries over into w3 too. Theodore Sider (MS) suggests a neat way to pull off this trick. Earlier we saw that an object o plays the Messi-role only if o satisfies the open sentence ‘x is such that there are no talking donkeys’ and that an object o* plays the Robin-role only if o* satisfies the open sentence ‘x is such that there are talking donkeys’. Suppose, however, that we restrict the quantifiers of these open sentences by different predicates. In our toy example, say that an object is on the right iff it is spatio-temporally connected to a talking donkey and that an object is on the left iff it is not on the right. Then even though it is impossible for there to be two objects o and o* that satisfy the predicates ‘x is such that there are no talking donkeys’ and ‘x is such that there are talking donkeys’ respectively, it is possible for there to be two objects that satisfy the predicates ‘x is such that there are (on the left) no talking donkeys’ and ‘x is such that there are (on the right) talking donkeys’ respectively. Moreover, suppose that we restrict all of the quantifiers of all of the open sentences that are members of Messi* by the predicate ‘is on the left’ and all of the quantifiers of all of the sentences that are members of Robin* by the predicate ‘is on the right’. This allows us to delineate two complex qualitative roles: the Messi-copy role and the Robin-copy role. We can then explain the ersatz counterpart relation as follows: • Messi* is a counterpart of Robin* iff Necessarily, for any two things x and y, if x is a Messi-copy and y is a Robin-copy then x is relevantly similar to y. Given the assumption about the possibility of disjoint spacetimes, that for any two worlds w1 and w2 there is a third world w3 containing copies of w1 and w2 , it turns out that the right-hand side is non-trivial. But the copying exports into w3 not only the intrinsic information contained within Messi* and Robin* but the extrinsic information too: if w1 represents the player of the Messi-role as being twelve feet from a cat and w2 represents the player of the Robin-role as being twelve feet from a cat, then w3 will represent both the Messi-copy and the Robin-copy as being twelve feet from a cat. In these ways, our explanation of the counterpart relation avoids the problems that afflicted its predecessors. Before we move on, I want to forestall a possible misunderstanding about the role that disjoint spacetimes play within the proposed account. The worry can be illustrated if we suppose that Messi could have been lonely and thereby could have existed in isolation from any distinct thing. On the present account, representing this possibility requires that Messi* has a lonely counterpart—call this counterpart Loner*. And Messi* and Loner*

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being counterparts requires (in part) that there is a world w which represents the existence of a player of the Messi-copy role and a player of the Lonercopy role. One might now worry that this means that we have in fact not represented the possibility of Messi being lonely; what we have instead represented is the possibility of Messi and Loner co-existing. What is true on the present proposal is that Messi* and Loner* being counterparts requires (in part) that there is a world w which represents the co-existence of players of the Messi-copy role and the Loner-copy role. But even though the theory says that whether Messi* and Loner* are counterparts depends on what goes on in worlds containing copies of both, representing the possibility of Messi being lonely does not require that any player of the Loner-copy role is lonely. What is instead required is that Messi* has a counterpart who is lonely*. And given (a) that Loner* and Messi* are counterparts and (b) that Loner* is lonely* insofar as any player of the Lonerrole is lonely, we thereby represent the possibility of Messi being lonely. So note that even though the theory appeals at a crucial stage to the existence of a world w that represents the de dicto possibility of the copies co-existing, w does not itself represent the de re possibility of Messi being lonely. Given that Messi* and Loner* are counterparts, the worlds which represent that possibility are instead those which contain a player of the Loner-role. Now, as Sider (MS) points out, the explication of the ersatz counterpart relation that we are considering requires a rather hefty assumption about what possibilities there are. But as hefty as it is, the assumption seems well motivated given a Humean picture according to which there are no necessary connections between distinct things.

3. THE PRINCIPLE OF SOLITUDE Intuitively, there are necessary connections between distinct things just in case there is an x such that, necessarily, x coexists with some y that is distinct from x. But as Bricker (2001, p. 37) points out, this claim is ambiguous. If we treat ‘some y’ as having wide-scope, we get one anti-Humean principle: ∃x∃y (x is distinct from y ∧ □ (x exists
y exists)) But if ‘some y’ has narrow-scope, we get another anti-Humean principle: ∃x □(x exists
∃y (x is distinct from y) (These principles are somewhat sloppy: remember that what’s really at stake is not the necessity of x coexisting with y but the necessity of a duplicate of x coexisting with a duplicate of y. I’ll take this qualification as read.)

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Now, when Lewis (1986, p. 88), as a good Humean, says that anything can fail to coexist with anything else, he is denying the first principle above. Take any two distinct things, and it is possible for (a duplicate of) the one to exist in isolation from (a duplicate of) the other. But that does not entail that, for any x, it is possible for (a duplicate of ) x to exist all by itself. So one could deny the first anti-Humean principle whilst accepting the second. With Bricker, I think that rejecting only the first anti-Humean principle is unattractive: its denial is too weak to capture the full range of possibilities. Consider a red ball and a blue candle. Given that we are rejecting the first anti-Humean principle, we know that it is possible for the red ball to exist without the blue candle. Maybe the blue candle is replaced by a blue pencil? But, intuitively, it is also possible for the red ball to exist without anything blue existing. Maybe the blue pencil is replaced by a green one? But surely it is also possible for the red ball to exist without any other coloured thing existing, and also possible for the red ball to exist without any other extended thing existing, and also possible for the red ball to exist without . . . . We are sliding down a slippery slope, and the denial of the second anti-Humean principle lies at the end of our journey. To deny the second of the anti-Humean principles outlined above is to endorse the Principle of Solitude: 8x ⋄(x exists ∧¬∃y (y is distinct from x)) Properly understanding this principle requires us to be quite specific about the range of the quantifiers. After all, we do not want it to turn out that the Principle of Solitude rules out the necessary existence of things like numbers and sets, and to avoid this result it’s crucial that the quantifiers are taken to range only over objects that are located in space and time. Moreover, existing in insolation is not quite the same as being surrounded by empty spacetime. In the world where Messi exists all by himself, spacetime has whatever shape Messi has. Even within a Lewisian setting, endorsing the Principle of Solitude does not by itself deliver the possibility of disjoint spacetimes. For even though the quantifiers range only over objects that are located in spacetime, it is still natural to think that the quantifiers range over what John Divers (1999) calls ordinary individuals: individuals that are wholly located in a single world. But suppose that we understood the quantifiers in the Principle of Solitude as ranging over not only ordinary individuals but also over extraordinary or transworld individuals that are partially located in different worlds. Call this the Generalized Principle of Solitude, or GPS for short. Then the possibility of disjoint spacetimes follows immediately by instantiating the quantifier to any transworld individual.

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Now, Bricker’s defence of GPS is based on the thought that there should be no restriction on what can be actualized. He invites us to look at things from God’s perspective. To insist that only single worlds can be actualized ‘would be to put a limitation on God’s power to choose, one not grounded in any logical necessity’ (2001, p. 38). He continues: Suppose that, in surveying the worlds prior to actualization, God found that two or more worlds were tied for best. Why must God choose between actualizing one world, or the other? He’s all-powerful! He can simply say: ‘Actualize those!’

It’s important to see that Bricker isn’t proposing to redefine the Lewisian concept of a world so that there are worlds which contain spatiotemporally disconnected parts. Rather, what he is in effect proposing is that there is another sense in which Lewis was right to say that a possibility is not always a possible world: a possibility is sometimes some possible worlds, where the ‘some’ is plural. Two worlds taken together can represent the possibility of disjoint spacetimes even if no world taken in isolation can do the same thing. So whilst embracing Bricker’s picture might force the Lewisian to tweak her analysis of modality, the Lewisian concept of a world can remain fixed. Though Bricker mounts his defence of GPS in a Lewisian setting, his point can be developed in an ersatz setting too. But we need to tread carefully. To begin, note that there is a certain sense in which it doesn’t matter, from (Bricker’s) Lewisian perspective, which worlds are actualized. Consider two scenarios. In the first, God actualizes a single world containing talking donkeys; in the second, He actualizes a single world containing no talking donkeys. Either way, it is still true that there are, quantifiers wide open, talking donkeys: the scenarios only generate different answers to the question of whether talking donkeys are actualized. That is just to say that when we assume a possibilist metaphysic, the class of things which exist simpliciter includes but is not exhausted by the class of things which are actual simpliciter. Actualization will therefore make a metaphysical difference, not an ontological one. When we assume an actualist metaphysic, however, the class of existing things just is the class of actual things. So it makes an ontological difference whether God actualizes a world where donkeys talk as opposed to a world lacking talking donkeys. In the former scenario, God makes it the case that there are, quantifiers wide open, talking donkeys; in the second, He makes it the case that there are, quantifiers wide open, no talking donkeys. Next, consider a third scenario in which God actualizes two worlds, one which contains talking donkeys and one which doesn’t. This scenario is troubling for the actualist in a way that it is not troubling for the possibilist. For in actualizing a world where donkeys talk, God seems to make it the case that there are, quantifiers wide open, talking donkeys. But in actualizing a

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world containing no talking donkeys, God seems to make it the case that there are, quantifiers wide open, no talking donkeys. But God can’t make it the case that there are and are not talking donkeys. A second difference between the Lewisian setting and the ersatz setting is that Lewisian worlds are concrete objects whereas ersatz worlds are at best abstract representations of concrete objects. Suppose that God looks down on the worlds (be they concrete or abstract) and finds that two of them, w1 and w2 , are equally good candidates to be actualized. If the worlds God looks down upon are Lewisian, He cannot try to solve His problem by creating a third world containing copies of w1 and w2 . Lewisian worlds are spatiotemporally unified by definition, after all. Moreover, He doesn’t need to do this: He can just take w1 and w2 together and actualize both. But if the worlds He looks down on are abstract representations of worlds, God can try to solve His problem by creating a third world that represents the existence of two disjoint spacetimes. Nothing in the definition of abstract worlds requires that they represent spatiotemporally unified wholes. Moreover, there is at least a prima facie case for thinking that God has to solve His problem in this way: for, as we saw, if God were to actualize both w1 and w2 , there is a worry that He would make a contradiction the case, by making it the case both that talking donkeys exist and that talking donkeys don’t exist. Time to take stock. We have seen that there are good reasons to think that Humeans should accept the Generalized Principle of Solitude. And we have also seen that GPS suggests that it is possible for more than one Lewis world to be actualized simpliciter. Exactly how to transpose GPS into an ersatz setting is a delicate issue, however. Suppose that w1 and w2 are ersatz worlds that represent the existence of single unified spacetimes v1 and v2 . Then GPS does not require that it be possible for both w1 and w2 to be actualized: GPS will be upheld if it is possible for both v1 and v2 to coexist in isolation from each other. This would be underwritten if the ersatzer were to accept that it is possible for more than one ersatz world to be actualized. But there is a clear logical problem with this idea, and the ersatzer doesn’t need to pursue this strategy in any case. She can instead hold that GPS is satisfied because there is a third world, w3 that contains copies of v1 and v2 . Insofar as the ersatz counterpart theorist is a Humean, then, I submit that GPS is both well motivated and supports the hefty metaphysical assumption required to underpin our explanation of the ersatz counterpart relation. (Question: What if the logical problems can be solved? Won’t it then turn out that GPS is underwritten not by the hefty metaphysical assumption required for the explanation of the counterpart relation but by the possibility of more than one ersatz world being actualized? Answer: Suppose that w1 represents the existence of a player of the Messirole and w2 represents the existence of a player of the Robin-role. Even if the

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logical problems can be solved, so that w1 and w2 can be jointly actualized, it does not follow that it is possible for a player of the Messi-role to coexist with a player of the Robin-role. Indeed, the set of sentences that would have been true had w1 and w2 been jointly actualized is seemingly identical to the set of sentences that would have been true had w3 been actualized, where w3 is the world constructed by the copying process described earlier. In this sense, the possibilities generated via multiple actualization just are the possibilities generated by the copying process. Put otherwise, there is no difference, within an actualist metaphysic, between actualizing w3 and jointly actualizing w1 and w2 . Either way, we get the same range of possibilities required to underpin our explanation of the ersatz counterpart relation.)

4 . S E M AN T I C S V E R S U S M E T A P H Y S I C S Ersatz counterpart theorists hope to enjoy the benefits of counterpart theory without footing the ontological bill. But Trenton Merricks (2003) has argued that ersatz counterpart theories simply don’t work. He is owed an answer, and I shall give mine.12 Merrick’s criticisms, surprisingly, focus not on the ersatzer’s story about the counterpart relation itself, but on her story about its relata. He objects that the method by which we constructed ersatz individuals is just one of many. Instead of taking possibilia to be sets of open sentences, we could equally well have associated each sentence with a number and taken ersatz individuals to be sets of numbers. Or we could have used different sentences, written in a different language. But remember that the theory reduces the fact that Messi is possibly F to facts about Messi* and his counterparts. That Messi is possibly F might, for example, be reduced to something like: • There is some set of open sentences x such that x is a counterpart of Messi* and x is F* (Where x is F* iff it has ‘x is F’ as a member.) But why, Merricks wonders, is this any better than reducing the target modal fact to facts about some other set-theoretic candidate? The multitude of candidates is thus problematic as it ‘absurdly implies that no single analysis of a modal property is better than some incompatible analyses’ (2003, p. 533).13 12

Merricks’s main target is the ersatz counterpart theory developed by Heller (1998), and some of his criticisms focus upon quite specific details of Heller’s proposal. But others are more general, and can be levelled against any ersatz counterpart theory. 13 Worse still, there is a qualitative problem: ersatz counterparts are not cut out to represent the possibilities that they are designed to represent. For they fail to represent

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Notice that Merricks thinks of the counterpart theorist as offering us an analysis of modal properties like being possibly F. But even if we grant this, it’s far from clear exactly what kind of analytic project she’s engaged in. And to see that, just observe that philosophers use possible worlds and individuals for a variety of different purposes.14 Most obviously, some philosophers cast possible worlds and individuals in a foundational role, and this is precisely the key role that counterparts play in their original Lewisian setting: on Lewis’s view, to be possibly F just is to have a counterpart who is F. Perhaps the ersatz counterpart theorist agrees. If so, let’s say that she is deploying counterpart theory to give a metaphysical analysis of de re modality. Possible worlds and individuals aren’t only used to ground modal truths: they’re also used as semantical tools. Thus possible worlds can be used to specify the semantic values of modal operators, functions from worlds to sets of objects can be used to specify the intensions of predicates, and sets of worlds can be used to specify the conditions under which a sentence is true. When we give a semantical theory of a language which incorporates claims such as these, we need entities for our (metalinguistic) quantifiers to range over, and possible worlds, individuals, and other things constructed out of them can serve as those entities. Perhaps the ersatz counterpart theorist sees her counterparts as playing this role. If so, let’s say that she is deploying counterpart theory to give a semantical analysis of de re modality. It’s important to realize that it is one thing to hold that ersatz counterparts play a foundational role and quite another to hold that they play a semantical role. This point isn’t new: Allen Hazen (1979, p. 319) made it over thirty years ago, telling us that the question of what grounds de re modal truths is ‘irrelevant’ to the assessment of counterpart theory qua semantic theory. There has been a tendency to overlook this, however, a tendency traceable to the fact that counterparts play both roles within the context of Lewis’s modal philosophy. But the two roles should be sharply distinguished, and just as one might use possible worlds in one’s semantics without thinking that what it is for f to be possible is for f to be true at some possible world, one might use counterparts in one’s semantics without intrinsically since, being sets, they ‘just sit there’ (p. 535). Good candidates, Merricks thinks, must have their representational properties ‘in and of themselves’. I won’t say much about this worry here, since I think that it’s been adequately dealt with by Sider (MS). The basic point is that what matters is that Messi* represents Messi relative to the structure in terms of which the counterpart theorist’s semantic theory is given. And this relative notion of representation is perfectly objective, even if there is no ‘intrinsic’ sense in which Messi* represents Messi. 14 Here I agree with Agustín Rayo (2012), from whom the terminology introduced below is borrowed.

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thinking that de re modality can be metaphysically analysed in counterparttheoretic terms. Once foundational issues are divorced from semantical ones, Merricks’s complaint can be understood in two ways: either as a complaint about the adequacy of ersatz counterpart theory qua semantical analysis or as a complaint about its adequacy qua metaphysical analysis. The latter complaint can be answered quickly. For whether or not some ersatz counterpart theorists are aiming to provide a foundation from which de re modality can be constructed, that’s certainly not the role which ersatz counterparts play within the theory developed earlier. I distinguish between the counterpart* relation, which holds between ersatz individuals, and the counterpart relation, which holds between the things represented by ersatz individuals. What’s true is that the theory tells us that ‘Messi is possibly F’ is true iff Messi* has a counterpart* who is F*. But that’s not the end of the story: whether this truth condition obtains depends on whether certain de dicto modal facts obtain. And so what’s ultimately going on, foundationally speaking, is that we have a de dicto reduction of de re modality. Facts concerning Messi* and his counterparts* are, in this setting, not being appealed to in order to provide a foundation to which de re modality can be reduced. The foundational version of Merricks’s objection simply doesn’t arise.15 What if Merricks is objecting to the deployment of ersatz counterparts in a semantical analysis of de re modality? Here, it’s worth remembering that the distinction between foundational and semantical projects wasn’t lost on Lewis. He tells us early on in Plurality that when we are doing semantics: . . . we need no possible worlds. We need sets of entities which, for heuristic guidance, may be regarded as possible worlds, but which in truth may be anything you please. We are doing mathematics, not metaphysics. (1986, p. 17)

So even Lewis—arch enemy of all things ersatz—is careful not to complain that sets of sentences (or other abstract whatnots) are incapable of serving as adequate semantic values within an intensional semantics. His complaint is rather that ersatz constructions are inadequate tools when our project is metaphysical in character, and the anti-ersatzer diatribe in Plurality is meant to convince us only that ersatz constructions are not up to the foundational 15 As Jason Turner pointed out to me, there is a further aspect to this issue. When we accept a certain metaphysical analysis, there might be different routes to the analysis. Ultimately, my counterpart theorist offers a de dicto analysis of de re modality. But the route to the analysis is indirect: we first analyse de re modality in terms of counterpart* relations and then offer a de dicto analysis of counterpart* relations. That’s not the only route: I could have instead analysed de re modality in terms of counterpart** relations and then offered an analysis of those. But the end result is the same either way, and the fact that there are different routes to the ultimate analysis isn’t important.

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task. Their semantic adequacy is granted from the beginning, and Lewis (1986, pp. 143–4) is careful to point out that their foundational inadequacies need not render ersatz possibilia inadequate semantical tools.16 But hang on, how can the fact that there are different ways to construct ersatz possibilia not have a negative effect on the semantic project? Won’t we end up accepting that no single semantical analysis of a language is better than some incompatible analysis based on some other way of constructing ersatz possibilia? Isn’t that what the semantical version of Merricks’s worry amounts to? How we answer these questions will depend on how we conceive of the explanatory ambitions of semantic theorizing.17 On what we can think of as the ‘folklore’ conception of (model-theoretic) semantics, the model theory ranges over a class of interpretations of a language, amongst which we hope to find the ‘intended’ one: the one that ‘gets the interpretation right’. The role of a semantic theory emerges as that of providing a constitutive account of both logical consequence (truth-preservation under every interpretation) and truth simpliciter (true on the intended interpretation). But this conception of the explanatory ambitions of a semantic theory is not mandatory. On what we can think of as the ‘instrumentalist’ conception of semantics— a view recently defended by Hartry Field (2008)—the goal of a semantic theory is not that of providing constitutive accounts of truth or consequence but rather that of providing an extensionally correct account of logical consequence. What we want to know is what follows from what? and, for the Fieldian instrumentalist, any value that we might associate with a semantic theory lies in its provision of extensionally correct answers to this question. (Which is a relief, given that Field’s take on logical consequence is that it is best viewed as a primitive notion, meaning that semanticists trying to provide a constitutive account of it are barking up the wrong tree.) If the explanatory ambitions of semantic theorizing are understood along instrumentalist lines, the existence of many equally good semantic theories is harmless. For the goodness of a semantical analysis (of a language l ) depends, from the instrumentalist’s point of view, on how well that analysis 16 A similar point is made in a different context by Sider (1996, }3), who points out that the fact that ordered pairs can be constructed in different ways isn’t a problem unless the constructions are being used to give a metaphysical analysis of relations. But that doesn’t mean that other uses of pairs are illegitimate because on those uses no metaphysical significance is claimed: the pairs are only used to get a job done. 17 This way of putting the point is perhaps a little tendentious. Rather than think that there is this single project, semantics, and a debate about its explanatory ambitions, one might instead think that there are a bunch of different projects, each of which might be called ‘semantics’ and each of which has different explanatory ambitions. Compare the discussion of the role Plantinga’s semantics serves in Woodward (2011).

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tracks the extension of the consequence relation (of l ). And so the existence of more than one equally good semantic theory would only show that we have more than one way to provide an extensionally accurate specification of logical consequence. And the mere fact that the entities these semantic theories quantify over are not intrinsically suited to represent possibilities does not prevent them from serving this instrumental end. We are, after all, doing mathematics, not metaphysics. If we conceive of semantic theorizing as being more ambitious—if we accept something like the folklore conception—then the instrumentalist’s reply to Merricks is unavailable: now the goodness of a semantic theory amounts to more than its mere ability to offer correct answers to questions of consequence. But saying that isn’t saying very much: there is still a question of what the ambitions of semantics are. Lewis’s (1986, pp. 40–50) discussion of the role of possible worlds and individuals in the analysis of language is particularly revealing in this regard.18 Lewis’s starting point is that language is used to convey information: you know something; you want me to know it too; I take you to my source of knowledge and to speak truthfully; via this mechanism I come to possess the piece of information you wanted me to possess. But though I rely on you to speak truly, the words you use will be true on some interpretations and false on others. The right interpretation is the one that specifies those conditions under which you are indeed truthful and my trust is well-placed. So a semantics must associate each sentence of a language with a truth condition, conceived of as a set of worlds. And the assignment must be systematic, and tell us ‘which speakers at which times at which worlds are in a position to utter which sentences truly’ (p. 40). This means that we must assign, by finite means, truth conditions to an infinite number of sentences, and Lewis’s story about how this can be done is familiar: we list the vocabulary of the language, assign to each expression a syntactic category and a semantic value, list the rules for building new expressions from old ones in such a way as to ensure that the semantic values of the new are a function of the semantic values of the old, and specify truth conditions for sentences as a function of the semantic values of their subsentential constituents. Lewis then gives us a job description for semantic values, and it’s at this point where things get interesting. Semantic values are there to do two things and two things only: to generate other semantic values and to 18 Even before we get into the details, we can see that there are crucial differences between the Fieldian picture and the Lewisian one. For whereas Field thinks that semantic theories serve an instrumental role in the analysis of logical consequence, Lewis thinks that they serve a genuinely explanatory role in the analysis of linguistic communication.

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generate truth conditions for sentences. The object, Lewis (p. 41) insists, is not that we should find entities capable of deserving names ‘from the established jargon of semantics’, but just that we find entities capable of doing these two things. It’s for this reason that Lewis uses the colourless term semantic value rather than a more established piece of jargon like referent. After all, the thought that my name refers to me rather than to a set of open sentences is a Moorean fact. But ‘there is no reason not to say both that my name has me as its referent and also that it has a certain [set] as its semantic value’ (p. 42). So long as a set of open sentences carries enough information to generate other semantic values, it will be deserving of the name. Notice, then, that the mere observation that semantic values come cheap wouldn’t have bothered Lewis a great deal. For semantic theories are judged to the extent that they associate sentences with the right truth conditions and so long as the truth conditions are generated systematically, which entities are used as semantic values is unimportant. In a sense, subsentential matters are irrelevant if sentential questions are answered systematically. As Lewis put it much earlier: ‘Semantic values may be anything, so long as their job gets done’ (1980, p. 26). And again, their only job is to generate semantic values and truth conditions.19 This isn’t instrumentalism. As far as the Fieldian instrumentalist is concerned, a semantic theory can assign to sentences whatever truth conditions it likes so long as it answers questions of consequence correctly. And to see that, note that the notion of an ‘intended’ interpretation doesn’t play any role for the instrumentalist. On the Lewisian picture, by contrast, semantics does aim to specify the conditions in which speakers are in a position to utter a sentence truly and a semantic theory which assigned silly truth conditions would be a silly theory even if it delivered extensionally correct answers to questions of consequence. What is true, admittedly, is that the Lewisian picture involves an instrumentalism of sorts about semantic values: semantic theories which assign silly semantic values might still assign the right truth conditions after all. True enough, we might have pause for thought were we to think that the semantic project was to specify something like ‘what we were referring to all 19 Doesn’t naturalness constrain subsentential interpretation too? Well, Lewis (1984) does accept naturalness as a constraint on interpretation, but he prefaces that entire discussion by conceding to Putnam that semantics takes ordinary things like you and me to be semantic values. If the relevant pages of Plurality (pp. 40–50) are anything to go by, Lewis doesn’t believe this for a second. And though Lewis does mention naturalness in relation to his preferred convention-based approach to language—see, e.g. Lewis (1992)—it’s far from clear that he endorses anything like the magnetic conception of subsentential reference that he’s often associated with (cf. Schwarz (2014)) and Weatherson (2013)). Thanks to Robbie Williams for discussion of these issues.

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along’. But it’s hard to see what the problem is once the notion of a semantic value is divorced from more everyday semantic jargon. Moreover, even if we are happy with something like the picture Lewis sketches, it’s still an open question exactly how instrumental our account of semantic values needs to be. For instance, even if we are careful to distinguish the referent of a term from its semantic value, it doesn’t automatically follow that there is no connection between the two. Consider, for example, the values assigned to ordinary names in a Montagovian setting: sets of properties. Though these sets clearly aren’t what ordinary uses of proper names refer to, they can be used to specify ordinary referents: thus whereas the semantic value of ‘Leo Messi’ is a set of properties, that name refers to the object that instantiates those properties. And this point has an analogue in the context of Merricks’s complaint against ersatz counterpart theory. For though the present picture involves the thought that ‘Leo Messi’ has a set of open sentences as its semantic value, we can at least define up an intuitive referent in terms of it: the name refers in the ordinary sense to that object which in fact satisfies all of those open sentences. These issues are rich and deserving of more attention. But I hope that they can be set aside in the present context. For remember that Merricks objected to ersatz counterpart theories on the grounds that there is no good reason why ersatz individuals should be identified with sets of open sentences rather than something else. But whether the fact that ersatz individuals can be constructed in many ways is problematic depends on which roles they are playing within our account of de re modality. Perhaps the problems Merricks sees are genuine if the ersatzer were trying to deploy ersatz counterparts to provide a metaphysical reduction of de re modality. But on the version of the view I’ve defended, ersatz counterparts are not playing a foundational role. Rather, ersatz counterparts are semantical tools, to be used as semantic values. And the semantic adequacy of ersatz counterparts is neither vitiated by the fact that they can be constructed in many ways nor by the fact that they fail to be intrinsically representational. What matters is just that they get the job done.

CONCLUSION Lewis (1986, p. 4) calls his realist account of modal space ‘a paradise for philosophers’ and challenges his opponents to deliver the theoretical benefits that realism offers more cheaply. Whether she can enjoy all of the benefits of Lewisian realism is doubtful, but I contend that the ersatzer can enjoy those associated with his treatment of de re modality. For the counterpart-theoretic

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analysis of de re modality Lewis offers can be made acceptable by the lights of those of who reject Lewis’s ontology. To those of us who find counterpart theory attractive, this should come as something of a relief. Richard Woodward University of Hamburg

ACKNOWLEDGEMENTS Thanks to Elizabeth Barnes, Ross Cameron, Shamik Dasgupta, John Divers, Thomas Krödel, Ted Sider, Tatjana von Solodkoff, Jason Turner, Isabel van der Linde, Barbara Vetter, Jennifer Wang, and Robbie Williams. Earlier versions of this paper were presented at the Universities of Barcelona, Leeds, and Hamburg, and at Humboldt-Universität zu Berlin—many thanks to my audiences on those occasions for discussion. My research on this paper was conducted within the context of the DFG Emmy Noether Research Group Ontology After Quine and was supported by my involvement in the Nature of Assertion: Consequences for Relativism and Fictionalism project (FFI2010-169049); the Vagueness and Physics, Metaphysics, and MetaMetaphysics project (FFI2008-06153); and the PERSP-Philosophy of Perspectival Thoughts and Facts project (CSD2009-00056). Many thanks to the DFG, the DGI, MICINN, and the Spanish Government for supporting these projects.

REFERENCES Bricker, Phillip. 2001. ‘Island Universes and the Analysis of Modality’. In Gerhard Preyer and Frank Siebelt (eds.), Reality and Humean Supervenience: Essays on the Philosophy of David Lewis, 27–55. Lanham, MD: Rowman & Littlefield. Divers, John. 1996. ‘On the Prohibitive Cost of Indiscernible Concrete Possible Worlds’. Australasian Journal of Philosophy 72(3): 384–9. Divers, John. 1999. ‘A Genuine Realist Theory of Advanced Modalizing’. Mind 108 (430): 217–39. Divers, John. 2002. Possible Worlds. London: Routledge. Field, Hartry. 2008. Saving Truth from Paradox. Oxford: Oxford University Press. Hazen, Allen. 1979. ‘Counterpart-Theoretic Semantics for Modal Logic’. Journal of Philosophy 76(6): 319–38. Heller, Mark. 1998. ‘Property Counterparts in Ersatz Worlds’. Journal of Philosophy 95(6): 293–316. Lewis, David. 1968. ‘Counterpart Theory and Quantified Modal Logic’. Journal of Philosophy 65(5): 113–26. Lewis, David. 1971. ‘Counterparts of Persons and their Bodies’. Journal of Philosophy 68(7): 203–11. Lewis, David. 1980. ‘Index, Context, and Content’. In Stig Kanger and Sven Ohman (eds.), Philosophy and Grammar, 79–100. Dordrecht: D. Reidel.

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Lewis, David. 1983a. ‘Individuation by Acquaintance and by Stipulation’. The Philosophical Review 92(1): 3–32. Lewis, David. 1984. ‘Putnam’s Paradox’. Australasian Journal of Philosophy 62(3): 221–36. Lewis, David. 1986. On the Plurality of Worlds. Oxford: Basil Blackwell. Lewis, David. 1992. ‘Meaning Without Use: Reply to Hawthorne’. Australasian Journal of Philosophy 70(1): 106–10. Melia, Joseph. 2001. ‘Reducing Possibilities to Language’. Analysis 61(269): 19–29. Merricks, Trenton. 2003. ‘The End of Counterpart Theory’. Journal of Philosophy 100(10): 521–49. Plantinga, Alvin. 1974. The Nature of Necessity. Oxford: Oxford University Press. Rayo, Agustín. 2012. ‘An Actualist’s Guide to Quantifying-In’. Crítica 44(132): 3–34. Schwarz, Wolfgang. 2014. ‘Against Magnetism’. Australasian Journal of Philosophy 92(1): 17–36. Sider, Theodore. 1996. ‘Naturalness and Arbitrariness’. Philosophical Studies 81(2): 283–301. Sider, Theodore. 2002. ‘The Ersatz Pluriverse’. Journal of Philosophy 99(6): 279–315. Sider, Theodore. MS. ‘Beyond the Humphrey Objection’. Unpublished manuscript. Weatherson, Brian. 2013. ‘The Role of Naturalness in Lewis’s Theory of Meaning’. Journal for the History of Analytical Philosophy 1(10): 1–19. Woodward, Richard. 2011. ‘The Things That Aren’t Actually There’. Philosophical Studies 152(2): 155–66. Woodward, Richard. 2012. “Fictionalism and Incompleteness.” Noûs 46(4): 781–90.

GROUNDING AND EXPLANATION

11 The Principle of Sufficient Reason and Probability Alexander Pruss

1 . T H E P R I N C I P L E O F SU F F I C I E N T R E A S O N I will take the Principle of Sufficient Reason (PSR) to be the claim that necessarily every contingent truth has an explanation (Pruss 2006). It has been often noted that many of our ordinary epistemic practices presuppose the existence of explanations. Rescher, for instance, talks of how the investigators of an airplane crash do not conclude that there is no explanation from their inability to find an explanation (Rescher 1995, p. 2). But it is one thing to agree that the PSR holds for propositions about everyday localized matters, and another to generalize to cosmic cases, such as the origination of the universe. Of course it would be difficult to precisely define what counts as “localized.” Still, we have a rough grasp of localization: an LED lighting up is localized, while an infinite regress of events or the complete state of our large universe presumably are not. I will argue, however, that in order to make sense of our scientific epistemic practices, we need a principle like the PSR that applies to global matters. In doing so, I will draw on recent mathematical work on laws of large numbers for nonmeasurable processes (Pruss 2013). The argument begins with the local cases and generalizes to the global ones. I need to note that I understand the PSR to be compatible with indeterministic phenomena. The PSR is a claim that explanations exist, not that deterministic explanations exist. The explanations might well be probabilistic in nature, and might even involve low probabilities, as is widely accepted in the philosophy of science (Salmon 1989). After all, giving explanations is largely about making events understandable, and we understand low probability chancy events as well as high probability ones (Jeffrey 1969).

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2.1 Frequency to chance inferences I will consistently use the word “chance” to indicate probabilities that are the tendencies of stochastic processes. Consider now this simple inference. A coin has been independently flipped a thousand times, and about 750 times the coin has landed heads. There is a very natural inference: the coin is loaded in such a way as to have approximately a 3=4 chance of landing heads. This natural inference leads to further predictions of the coin’s behavior. Frequency-to-chance inferences like the above are everywhere, and inductive reasoning to a universal generalization is arguably just a limiting case. Let Cp be the hypothesis that the chance of the coin’s landing heads is p. Plausibly, the inference in the coin example was based on the fact that given Cp with p close to 3=4, it is very likely that about 750 times we will get heads, while given Cp with p far from 3=4, this is unlikely. The reason for this is a Law of Large Numbers: given a large number of independent and identically distributed chancy trials, the frequency of an outcome among the trials is likely to be close to the chance of the outcome. There are difficulties here, of course, with how we determine that the coin flips are independent and identically distributed. Independence perhaps is backed by the fact that we just cannot find any memory mechanism for the coin, and identical distribution by the fact that the coin flips appear to be identical in all relevant respects. The questions of how exactly one cashes out these considerations are difficult, but they are not the questions I want to focus on. Instead, I want to focus on why there are probabilities at all. Our intuitive Bayesian-flavored reasoning was based on comparing the probability of our observed frequencies on the different hypotheses Cp . But what about hypotheses on which there are no probabilities of frequencies? In the next subsection, I will sketch a picture of how such hypotheses might look by making use of nonmeasurable sets. A reader who wishes to avoid mathematical complication may wish to skip that subsection. The basic idea behind the technicalities is to imagine a case where we pick out a series of heads and tails results by throwing a dart at a circular target, and deeming heads to have occurred when the dart lands within some set A and tails to occur otherwise. But we will take the set to be utterly nonmeasurable (“saturated nonmeasurable”): it has no area, and doesn’t even have a range of areas. I will then argue on technical grounds that the hypothesis that coin toss results were generated in such a way can neither

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be confirmed nor disconfirmed by observation. But in order to confirm an alternate hypothesis—say that a series of heads/tails results was the result of a sequence of fair coin tosses—we need to rule out the hypothesis that it was generated in some “nonmeasurable” way like this.

2.2 Nonmeasurable sets So suppose coins are being flipped in this roundabout way. A dart with a perfectly defined tip (say, infinitely sharp or perfectly symmetrical) is uniformly randomly thrown at a circular target. There is a region A of the target marked off, and a detector generates a heads toss (maybe in a very physical way: it picks up the coin and places it heads up) whenever the dart lands in A; otherwise, it generates a tails toss. The chance of the coin landing heads now should be equal to the proportion of the area of the target lying within A, and we can elaborate Cp to the hypothesis that the area of A is pT , where T is the area of the whole target. Given the observation of approximately 750 heads, it seems reasonable to infer that probably the area of A is approximately ð3=4ÞT . But what if an area cannot be assigned to the marked region A? Famously, given the Axiom of Choice, there are sets that have no area—not in the sense that they have zero area (like the empty set, a singleton or a line-segment), but in the sense that our standard Lebesgue area measure cannot assign them any area, not even zero. In such a case, there will also be no well-defined1 chance of the dart hitting A, and hence no well-defined chance of heads. Let N be the hypothesis that A has no area, i.e. is non-measurable. Now, here is a fascinating question. If N were true, what would we be likely to observe? Of course, if we perform 1000 tries, we will get some number n of heads, and n=1000 will then be a frequency between 0 and 1. We might now think as follows. This frequency can equally well be any of the 1001 different numbers in the sequence 0:000; 0:001;:::; 0:999; 1:000. It seems unlikely, then, that it’s going to be near 0.750, and so C3=4 (and its neighbors) is still the best hypothesis given that the actual frequency is 0.750. But this reasoning is mistaken. To see this, we need to sharpen our hypothesis N a little more. There are non-measurable sets A where we can say things about the frequency with which A will be hit. For instance, it could be that although A is non-measurable, it contains a measurable set A1 of area ð0:74ÞT and is contained in a measurable set A2 of area ð0:76ÞT . 1 One wants to say “no chance,” but that suggests “zero chance,” which is not what one is saying.

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(Think of A as 74% of the target plus a nonmeasurable set localized to an area containing only 2% of the target.) But the dart will hit A1 about 74% of the time, and A2 about 76% of the time. Whenever the dart hits A1 , it hits A, and whenever it hits A, it hits A2 , so we would expect the dart to hit A approximately 74%A to 76% of the time. And so our observed frequency would be no surprise. The mere fact that A is nonmeasurable does not rule out probabilistic predictions about frequencies, because a nonmeasurable set might be “quite close” to measurable sets like A1 and A2 that bracket A from below and above. However some sets are not only nonmeasurable, but saturated nonmeasurable. A set A is saturated nonmeasurable provided that all of A’s measurable subsets have measure zero and all measurable subsets of the complement of A are also of measure zero. Given the Axiom of Choice, for Lebesgue measure on the real line, there not only are nonmeasurable sets, but there are saturated nonmeasurable sets (Halperin 1951).2 When A is saturated nonmeasurable, no method of generating predictions by bracketing a the probabilities into an interval, like the one from 74% to 76%, will work. The only measurable subsets of A will have zero area and the only measurable supersets of A will have area T . So our bracketing will only tell us the trivial fact that the frequency will be between 0 and 1, inclusive. Let M then be the hypothesis that A is saturated nonmeasurable. Can we say that given M , the frequency is unlikely to be near 0:750? The answer turns out to be negative even if we have infinitely many observations. In Pruss (2013), I gave has given a plausible mathematical model of an infinite sequence of independent identically distributed saturated nonmeasurable random variables, and it follows from Theorem 1.3 there that for any nonempty interval I which is a proper subset of ½0; 1, the event that the limiting frequency of the events is in I is itself saturated nonmeasurable. Thus, even with infinitely many independent shots, we could say nothing probabilistic about the observed frequency of hits of our saturated nonmeasurable set being near 0:750: the probability would neither be large nor small. And analogous claims can be argued for in finite cases by building on the tools in that paper (see Theorem 2 in the Appendix). So in our case above, if I is some small interval like ½0:740; 0:760 centered on 0:750, there is nothing probabilistic we can say about the If A0 is a saturated nonmeasurable subset of the interval ½0; 1, then the Cartesian product A0  ½0; 1 will be a saturated nonmeasurable subset of the square ½0; 12 . (There is also a directly two-dimensional construction in Sierpiński 1938.) Intersecting this with a disc of radius 1=2` centered on ð1=2; 1=2Þ yields a saturated nonmeasurable region for a disc-shaped area, like in our example. 2

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observed frequency being in I when the target set is saturated nonmeasurable as the event of the frequency being in I is then saturated nonmeasurable. In particular, we cannot say that the frequency is unlikely to be near 0:750 (nor that it’s likely to be near). The observation of a frequency close to 0:750 is neither surprising nor to be expected given that the target set is saturated nonmeasurable. Our probabilistic reasoning thus cannot disconfirm hypotheses of saturated nonmeasurability. Such hypotheses endanger all our local scientific inferences from observed frequencies to chancy dispositions, inferences central to our epistemic practices. Yet our local scientific inferences are, surely, good. If we cannot disconfirm saturated nonmeasurability hypotheses a posteriori, then we need to do so a priori. Kleinschmidt (2013) defends a presumption of explanation principle (EP) on which we are justified in presuming explanation, by analogy with our presumption that a student coming late to class was not struck by lightning, even though people are sometimes struck by lightning. She then writes: “In the lightning case, we can explain our reluctance by noting that we justifiably believe, partly due to induction, that lightning strikes rarely. We might hope to give a similar explanation of our endorsement of EP.” But while we can say what it would look like if lightning was more common than we think it to be—and thus we can say on empirical grounds that it is not so—the upshot of our argument is that we cannot say what it would look like if there were more unexplained events than we think, and hence cannot say on empirical grounds that it is not so.3 Another approach would be to assume low prior credences for hypotheses of saturated nonmeasurability. But it is difficult to justify this in a way that is not ad hoc. Given the Axiom of Choice, there are just as many saturated

3 Kleinschmidt (2013) also attempts to give counterexamples to the PSR on the basis of cases involving the diachronic identity and synchronic composition of persons. Why, for instance, does a cloud of atoms in group G1 compose a person while the cloud of atoms in group G2 , which differs only by a single atom, does not? (The fission case can be handled analogously.) A full response to her intricate arguments would take us too far afield, but the facts underwriting the composition claims are either necessary or contingent. Since I am only defending a PSR for contingent truths, it is only the contingent option that I need consider. But even without assuming the PSR, simply using Kleinschmidt’s own EP, it will count very strongly against a theory on which it is contingent that an arrangement of atoms composes a person if the theory can give no explanation of why it happens to do so. Consider, for instance, dualist theories on which souls attach to some groups of atoms with no explanation as to why they attach to the ones they attach to. Those theories are implausible precisely because of the explanatory failure. And the same will be true of those materialist theories on which it is a brute unexplained fact that some atoms compose a person. Kleinschmidt has not shown that there is a brute-composition theory that is still plausible after taking into account her own EP.

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nonmeasurable subsets of, say, ½0; 1 as there are measurable ones (see the Corollary in the Appendix). Granted, subjective Bayesians may not worry too much about the ad hoc in the choice of priors, but the reason they do not worry is because of the hope that evidence will swamp the priors and make them irrelevant. But evidence cannot affect the credence of the nonmeasurability hypotheses, since those hypotheses do not generate the kinds of likelihoods that are needed for confirmation or disconfirmation, and so one would simply be stuck with the ad hoc priors. Note, however, that the credences of such hypotheses would have to be very low indeed: they would have to be zero or infinitesimal. I sketch the argument in a simple case. Suppose that N is a hypothesis according to which the coin tosses are independent (in the sense of Pruss 2013) saturated nonmeasurable sets. Suppose that PðN Þ > 0 and that the only alternative to N is the hypothesis H that the coin tosses are fair and independent so that PðH Þ ¼ 1  PðN Þ. Let E be the observed evidence—a sequence of n coin toss results (say HTHHHTH if n ¼ 7). We would like to be able to use Bayes’ Theorem to say that the probability of H given the evidence is: PðH jEÞ ¼

PðE jH ÞPðH Þ PðE j H ÞPðH Þ þ PðE jN ÞPðN Þ

Unfortunately, the term PðE j N Þ in the denominator is not a number, since E is saturated nonmeasurable conditionally on N . But we can, nonetheless, use Bayes’ Theorem. We can model the probability of a nonmeasurable set by an interval of numbers. In the case of PðE j N Þ, the interval will range all the way from 0 to 1, i.e. will be ½0; 1. Fortunately, only one of the quantities on the right-hand-side corresponds to an interval: the other quantities, PðE jH Þ, PðH Þ and PðN Þ are ordinary classical probabilities. We can then use the range of the interval PðE j N Þ and Bayes’ Theorem to define an interval-valued probability for PðH j EÞ. Since PðE jN Þ is the interval ½0; 1, one endpoint for the interval corresponding to PðH jEÞ will be given by replacing PðE j N Þ in Bayes’ Theorem with zero: PðE jH ÞPðH Þ ¼ 1: PðE j H ÞPðH Þ þ ð0ÞPðN Þ This turns out to be the upper endpoint. The second endpoint—the lower one—will correspond to replacing PðE jN Þ with one, the other endpoint of its interval: PðE jH ÞPðH Þ : PðE j H ÞPðH Þ þ ð1ÞPðN Þ

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Now, PðE jH Þ ¼ ð1=2Þn . It follows that this lower endpoint of the interval given by PðE j H Þ is less than or equal to:ð1=2Þn =PðN Þ. Therefore, the interval-valued probability PðH jEÞ will contain the interval ½ð1=2Þn =PðN Þ; 1.4 As long as PðN Þ is strictly bigger than zero and not infinitesimal, it follows that as n goes to infinity—i.e. as the amount of evidence gathered increases—the hypothesis H gets to have a probability range closer and closer to ½0; 1. But in order to get confirmed, to become believable, H would have to be an interval of probability like ½a; b where a is close to one, not close to zero. So the more evidence we gather, the broader our probability interval for H . This is Bayesian divergence rather than convergence. As we observe more coin-flips, the nasty hypothesis N infects our probabilities more and more. One could try to claim that PðN Þ is infinitesimal, but that would seem to be ad hoc. Alternately, one might try to completely a priori rule out pathological hypotheses, ruling them to have probability zero. For instance, one might simply deny any version of the Axiom of Choice strong enough to imply the existence of nonmeasurable sets. Or one might deny the possibility of physical processes sensitive to whether an input falls into a pathological set.

2.3 No-explanation hypotheses But clever mathematical constructions are not the only potential source of probabilistic nonmeasurability in the physical world. Suppose a coin is tossed, but there is a hypothesis H that there will be no explanation at all of how the coin lands. Simplify by assuming that the coin must land heads or tails. Then I contend that the event of the coin landing heads has no probability conditionally on H . For suppose it has some probability. What is it? The only non-arbitrary answer seems to be the one given by symmetry or the Principle of Indifference: 1=2. And, generalizing, the only non-arbitrary probability for any particular fixed sequence of n tosses on a no-explanation hypothesis will be 2n . But if that is the answer, then the no-explanation hypothesis cannot be 4 This argument is simpler than the usual kind of reasoning with interval-valued probabilities (e.g. Whitcomb 2005), because only one of the quantities involved is an interval. We can justify the argument more rigorously by thinking of interval valued probabilities as determining a family of classical probability functions, and then applying Bayes’ Theorem to each member of the family. The only relevant variation between members of the family will be as to the value assigned to E given N , and that will range between 0 and 1, thereby generating a range for the posterior probability containing ½ð1=2Þn =PðN Þ; 1.

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empirically distinguished from the hypothesis that the coin tosses form an independent sequence of stochastic events each with probability 1=2 of heads, as both hypotheses predict the same behavior. In fact, although 1=2 is the only non-arbitrary answer, the arbitrary answers face exactly the same problem. For each possible answer, arbitrary or not, to the question of the probability that the coin lands heads conditionally on the no-explanation hypothesis, there is a possible stochastic process hypothesizing which yields the same prediction as the no-explanation hypothesis does. And then no amount of observation can rule out the no-explanation hypothesis. Whether this is a serious problem will depend on whether we can rule out the hypothesis a priori, an option we will discuss later in this section. Formal results on nonmeasurable sets like those discussed in Section 2.2 model not only those events that are saturated nonmeasurable due to their pathological set theoretic construction but also those that are nonmeasurable simply because no probabilities can be assigned to them. Within a probabilistic framework, there are now three possibilities. Either (a) conditioning on a no-explanation hypothesis yields an exact probability, or (b) it yields no probability at all, or (c) it yields a probability range or interval. If it yields an exact probability, then we cannot empirically distinguish the no-explanation hypothesis from an explanatory chance hypothesis whose chance assignment matches that exact probability. If it yields no probability at all, not even a range of probabilities, we are dealing with a saturated nonmeasurable event, and our formal results tell us that we cannot make any probabilistic predictions about frequencies, and in particular cannot say whether it is likelier or less likely that we would get the same observations from an explanatory chance hypothesis. Now consider the third option, that of a probability range. Perhaps when we condition on the no-explanation hypothesis, we get an interval ½a; b of probabilities with a 0g. For each instantaneous state of such a universe, there is an earlier state, but there is no earliest state. It has been argued by Grünbaum (1993) that our universe is like that. Let us further assume that each state is explained by an earlier state. A localized PSR will not apply to the infinite regresses of states we find in such universes. Thus, a localized PSR will be compatible with an unexplained no-beginning finite-past universe.9 But then there will be no meaningful chances for the existence of such universes, and hence no non-arbitrary reason to prefer the hypothesis we think is true—a universe that begins with a Big Bang about 13:8 billion years ago—over aberrant hypotheses such as Russell’s five-minute hypothesis on which the universe is five minutes old with the twist that there is no initial moment (compare Koons 2006, Section 2.1). This was too quick. Although there are no chances associated with the two hypotheses, maybe there are epistemic probabilities. But what reason do we have to take the precise limiting sequence of states that we think obtained after the Big Bang to be more probable than the limiting sequence of states in the five minute hypothesis? Again, we would have to assign an arbitrary epistemic probability. One might think that of all possible unexplained nobeginning finite-past universes, the five-minute universe (i.e. the one that coincides with what we take to be our universe over the past five minutes, without an initial time) has an initial segment that is too elaborate— festooned with galaxies and stars as it is, and with at least one planet of very complex life—to come into existence without an explanation, while the Big Bang universe’s initial segment is not as elaborate. But both initial sequences of states live in the same configuration space, and there does not appear to be a reasonable way to privilege the Big Bang sequence over the five-minutes-ago sequence as an option for coming into existence causelessly ex nihilo. For what it’s worth there is even an intuition preferring the five-minutes-ago sequence. For one would intuitively expect an unexplained universe would have high entropy, and the states in the five-minutes-ago sequence have higher entropy than the states in the post-Big-Bang sequence do. But suppose that we have some way of arguing that the Big Bang hypothesis is simpler than the five-minute hypothesis. We can beat the Big Bang hypothesis in respect of simplicity. Consider the hypothesis that reality consists of a short truncated backwards light cone (say, extending 9 Hume (1779, Part IX) has argued that an infinite causal regress is explained by the causes within it. However, the discussion of the beginningless cannonball flight thought experiment in Pruss (2006, Section 3.1.4) makes this implausible.

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backwards one minute), with no initial state, centered on the present state of my brain. This hypothesis supposes only a tiny fraction of the complexity of our gigantic universe (indeed even of our solar system), and there will be a lot more simplicity than in the Big Bang story. After all, it is a part of the Big Bang story that the world evolved in the kinds of complex ways that resulted in a life full of earth, many galaxies, and so on. All the details of this complexity will have free parameters, either determined at the time of the Big Bang or set by later chancy events. Thus a priori simplicity arguments do not rule out our aberrant hypotheses. We really do seem to need a PSR.

4. PSR-COMPATIBLE ABERRANT HYPOTHESES Our argument for the PSR was that it allows us to reject aberrant local and global hypotheses that we have no a posteriori way of rejecting. A natural response is that there are aberrant hypotheses that are compatible with the PSR as well, and the PSR only pushes the problem back a step. For consider the hypothesis that a necessarily existing being with the essential property of being a science-hater creates a universe that appears very much other than how it is. If the PSR is restricted to contingent states of affairs, as in Pruss (2006), then it is compatible with this aberrant hypothesis. Compare this to the hypothesis that a necessarily existing being with the essential property of being a science-lover creates a universe that allows us to make correct scientific predictions. We cannot compare the science-lover and science-hater hypotheses by means of chances. By S5, each hypothesis is either necessarily true or necessarily false, and hence has chance 1 or chance 0, but we don’t have access to these chances except by figuring out which hypothesis is true. However, in the realm of PSR-compatible hypotheses, we can make use of simplicity reasoning to rule out aberrance. For instance, the science-hater hypothesis has many free parameters: just how much does the science-hater hate science and in what respects, how powerful is it, etc. And there does not seem to be any canonical set of values for these free parameters. On the other hand, there is a canonical science-lover hypothesis in the literature, namely the theistic hypothesis that there exists a perfect being, who has maximal power, maximal knowledge, and loves all that is fine and good precisely in proportion to how fine and good it is. And of course science getting at the truth is fine and good. And while there will be many variants of the stochastic axiarchic hypothesis that universes are likely to come into existence to the extent that they are

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good, depending on just how the direct relationship between value and chance works, it could turn out that a broad range of them would produce non-aberrant stories.

5 . C O N C LU S I O N S Where there are no explanations, there are no chances. Where there are no chances, it is difficult to have non-arbitrary probabilities. In some cases, we can have non-arbitrary probabilities based on simplicity considerations. In others, perhaps by indifference considerations. But neither simplicity nor indifference considerations help with rejecting aberrant no-explanation hypotheses that undercut the scientific enterprise. We need a PSR, and not just one restricted to local situations, but a global one. Finally, while our arguments above took for granted that chances were governed by the axioms of classical probability theory, it may be that we should relax this assumption. For instance, while a rational agent tends to act in proportion to the strength of her reasons, it would be hasty to suppose that this tendency can be quantified by classical probability theory. Moreover, global explanatory hypotheses are unlikely to give rise to numerical chances, if only for cardinality reasons—likely, there is no set of all possible worlds (e.g. Pruss 2001), and classical probability theory is defined over sets of possibilities. But the main point of the paper should remain: where there are no explanations, there are no chances—whether the chances are understood as classical probabilities or are governed by some more complex calculus—and hypotheses that do not give rise to some sort of chances need a priori refutation. Alexander Pruss Baylor University AP P E N D I X : S O M E RE S U L T S A B O U T NONMEASURABLE SETS A set is saturated nonmeasurable with respect to a probability measure provided that all measurable subsets have measure zero and all measurable supersets have measure one. Theorem 1 Assuming the Axiom of Choice, the cardinality of the set of all saturated nonmeasurable subsets of ½0; 1Þ is equal to the cardinality of the set of subsets of ½0; 1Þ. For the proof, we need the following easy Lemma. Let AΔB be the symmetric difference ðA  BÞ [ ðB  AÞ of the sets A and B. Recall that every subset of a set of Lebesgue measure zero is measurable and has measure zero.

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Lemma 1 If A is a saturated nonmeasurable subset of ½0; 1Þ and AΔB is a set of Lebesgue measure zero, then B is saturated nonmeasurable. Proof of Lemma 1. For any set C , the set B \ C differs from A \ C at most by a set of measure zero since AΔB has measure zero, and any two sets that differ by a set of measure zero must either both be measurable or both nonmeasurable. Thus, if C is a measurable subset of B, it follows that A \ C differs from C by a set of at most measure zero. Thus, A \ C is a measurable subset of A, hence has measure zero, and hence C also has measure zero since it differs from it by at most a set of measure zero. This argument shows that if all measurable subsets of A have measure zero, the same is true for B. Applying this argument to the complements ½0; 1Þ  A and ½0; 1Þ  B, and using the observation that all measurable supersets of U that are subsets of ½0; 1Þ have measure 1 if and only if all measurable subsets of ½0; 1Þ  U have measure zero, we conclude that all measurable subsets of ½0; 1Þ  B have measure zero, from which we conclude that all measurable supersets of B that are subsets of ½0; 1Þ have measure 1. Thus, B is saturated nonmeasurable. Proof of Theorem 1. Famously, the Cantor middle-thirds set C  ½0; 1Þ is a set of zero measure that has the same cardinality c as all of ½0; 1Þ. Given any one saturated nonmeasurable subset S of ½0; 1Þ (these exist by Halperin 1951), for each subset A of C let SA ¼ ðS  AÞ [ A. Then SA ΔS  C , and hence SA is saturated nonmeasurable. There are as many sets SA as subsets of C , and there are as many subsets of C as of ½0; 1Þ. The fact that the number of subsets of the Cantor set equals the number of subsets of ½0; 1Þ and that every subset of a set of zero measure is measurable also yields: Corollary

Assuming the Axiom of Choice, the cardinality of the set of saturated nonmeasurable sets is equal to the cardinality of the set of measurable subsets of ½0; 1Þ.

Following Pruss (2013), I will model independent sequences of nonmeasurable events E1 ; :::; En as follows. We have a probability space ðΩ; F ; PÞ which is a product of probability spaces hΩk ; Fk ; Pk i for 1  k  n. Then for each k, we suppose that the event Ek depends only on the factor Ωk (this is how one models independence). This means that Ek can be written as

Ek ¼ fhω1 ; :::; ωn i : 8iðωi ∈ Ωi Þ& ωk ∈ Uk g; for some subset Uk of Ωk . Let N be a “nonmeasurable random variable” representing the number of events Ek that happen, i.e. N is a function from Ω to the natural numbers such that N ðωÞ equals the cardinality of the set fk : ω ∈ Ek g. Then we have: Theorem 2 Suppose E1 ; :::; En are saturated nonmeasurable. Let J be any non-empty proper subset of f0; :::; ng. Then the event NJ ¼fω∈Ω:N ðωÞ∈J g is saturated nonmeasurable.

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Hence nothing non-trivial can be probabilistically said about the number of the Ek that occur. The proof of Theorem 2 will use the terminology in Pruss (2013). In particular, for any function f , measurable or not, on a probability space there is a maximal nonmeasurable minorant f and a minimal nonmeasurable majorant f * such that * f  f  f * and such that for any measurable functions g and h such that g  f * and f  h we have g  f almost everywhere (i.e. outside of some set—perhaps * empty—of probability zero) and h  f * almost everywhere. Proof of Theorem 2. Define the function Ψk on Ωk by Ψk ðωÞ ¼ 1 if ω ∈ Uk and Ψk ðωÞ ¼ 0 otherwise. Define the function Yk on Ω by Yk ðhω1 ; :::; ωn iÞ ¼ Ψk ðωk Þ. Thus, Yk ðωÞ is 1 if ω ∈ Ek and 0 otherwise. Because Ek is saturated nonmeasurable, ðYk Þ ¼ 0 (almost everywhere—I will omit such qualifications from now on) and ðYk Þ* ¼* 1. It follows from Pruss (2013, Proposition 1.2) that ðΨk Þ ¼ 0 and ðΨk Þ* ¼ 1. It follows that Uk is a saturated nonmeasurable subset of Ω*k . Let P and P k be probability measures on Ωk extending P to a k σ-field that includes Ek and such that Pk ðUk Þ ¼ 0 and P k ðUk Þ ¼ 1. These exist by Pruss (2013, Lemma 1.5): just let f in the Lemma be Ψk . Now let j be any member of J . Let Q be the probability measure on Ω formed as the product of the probability measures P 1 ; :::; P j ; Pjþ1 ; :::; P . Then QðEk Þ ¼ 1 if n k  j and otherwise QðEk Þ ¼ 0. Thus with Q-probability 1, exactly j of the Ek occur, and since j ∈ J , we have QðNJ Þ ¼ 1. Let l be any member of f0; :::; ng  J . Let R be the product of the probability measures P 1 ;:::P l ;Pl þ1 ;:::;Pn . Then RðEk Þ ¼ 1 if k  l and otherwise RðEk Þ ¼ 1. Hence, with R-probability one exactly l of the Ek occur, and since l 2 = J , we have RðNJ Þ ¼ 0. Thus there is an extension of P on which NJ has measure zero and an extension of P on which it has measure one, from which it follows (Pruss 2013, Lemma 2.3) that NJ is saturated nonmeasurable with respect to P.

R E F E REN C E S Edwards, W., Lindman, H. and Savage, L. J. (1963). “Bayesian Statistical Inference for Psychological Research,” Psychological Review 70(3): 193–242. Grünbaum, Adolf. (1993). “Narlikar’s ‘Creation’ of the Big Bang Universe was a Mere Origination,” Philosophy of Science 60(4): 638–46. Halperin, Israel. (1951). “Non-Measurable Sets and the Equation f ðx þ yÞ ¼ f ðxÞ þ f ðyÞ.” Proceedings of the American Mathematical Society 2(2): 221–4. Hume, David. (1779). Dialogues Concerning Natural Religion. Inwagen, Peter van (1983). An Essay on Free Will. New York: Oxford University Press. Jeffrey, Richard C. (1969). “Statistical Explanation vs. Statistical Inference.” In Nicholas Rescher (ed.), Essays in Honor of Carl G. Hempel, Dordrecht: Reidel, 104–13.

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Kleinschmidt, Shieva. (2013). “Reasoning without the Principle of Sufficient Reason.” In Tyron Goldschmidt (ed.), The Puzzle of Existence: Why Is There Something Rather Than Nothing? New York: Routledge, 64–79. Koons, Robert. (2006). “Oppy and Pruss on Cosmological Arguments.” Presented at the Central Division Meeting of the American Philosophical Association, Chicago, April 2006. Leslie, John. (1979). Value and Existence. Totowa, NJ: Rowman and Littlefield. Pruss, Alexander R. (2001). “The Cardinality Objection to David Lewis’s Modal Realism.” Philosophical Studies 104(2): 167–76. Pruss, Alexander R. (2006). The Principle of Sufficient Reason: A Reassessment. New York: Cambridge University Press. Pruss, Alexander R. (2013). “On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables.” Bulletin of the Polish Academy of Sciences—Mathematics 61(2): 161–8. Rescher, Nicholas N. (1995). Satisfying Reason: Studies in the Theory of Knowledge. Dordrecht/Boston/London: Kluwer. Rescher, Nicholas N. (2010). Axiogenesis: An Essay in Metaphysical Optimalism. Lanham, MD: Lexington. Salmon, Wesley C. (1989). Four Decades of Scientific Explanation. Minneapolis: University of Minnesota Press. Sierpiński, Wacław. (1938). “Sur un problème concernant les famille d’ensembles parfaits.” Fundamenta Mathematicae 31(1): 1–3. Taylor, Richard. (1974). Metaphysics. Englewood Cliffs, NJ: Prentice-Hall. Whitcomb, Kathleen M. (2005). “Quasi-Bayesian Analysis Using Imprecise Probability Assessements and the Generalized Bayes’ Rule.” Theory and Decision 58(2): 209–38.

12 Grounding Ground Jon Erling Litland 1. INTRODUCTION If Γ’s being the case grounds ϕ’s being the case, what grounds that Γ’s being the case grounds ϕ’s being the case?1 This is the Problem of Iterated Ground. Dasgupta (2014b), Bennett (2011), and deRosset (2013) have grappled with this problem from the point of view of metaphysics. But iterated ground is a problem not just for metaphysicians: the existing logics of ground2 have had nothing to say about such iterated grounding claims. In this paper I propose a novel account of iterated ground and develop a logic of iterated ground. The account—what I will call the Zero-Grounding Account (ZGA for short)—is based on three mutually supporting ideas: (i) taking non-factive ground as a primitive notion of ground; (ii) tying nonfactive ground to explanatory arguments; and (iii) holding that true non-factive grounding claims are zero-grounded (in Fine’s sense). A notion of ground is factive if the truth of “Γ grounds ϕ” entails that each γ ∈ Γ as well as ϕ is true; the notion is non-factive otherwise. Most authors take a factive notion of ground as their primitive; I adopt a non-factive notion as primitive. Taking a non-factive notion of ground as basic allows one to solve the Problem of Iterated Ground for factive ground: if Δ factively grounds ϕ then this is grounded in Δ’s non-factively grounding ϕ together with Δ’s being the case.3 This, of course, just shifts the bump under the rug: what grounds that Δ non-factively grounds ϕ?

1 Here Γ are some (true) propositions and ϕ is a (true) proposition. For the official formulation of claims of ground, see }2 below. In the interest of readability I will not distinguish carefully between use and mention throughout. 2 Fine 2012b; Correia 2010, 2014; Schnieder 2011; Poggiolesi 2015. 3 It is not strictly speaking necessary to hold that true factive grounding claims are partially grounded in non-factive grounding claims (}10). However, assuming this allows for a smoother presentation; and, as we will see, it does no harm.

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Fine (2012a, pp. 47–8) distinguished between a truth’s being ungrounded, on the one hand, and having the empty ground or being zerogrounded on the other. Crucially, being zero-grounded is a way of being grounded. I show that if Δ’s non-factively grounding ϕ is zero-grounded we have a formally satisfactory solution to the Problem of Iterated Ground. To go beyond a merely formal solution we must answer two questions: (i) What does it mean to say that a truth is zero-grounded? (ii) Why should we believe that (true) non-factive grounding claims are zero-grounded? We answer these questions by tying non-factive ground to explanation. The basic idea is that for Δ to non-factively ground ϕ just is for there to be a special type of argument from premisses (exactly) Δ to conclusion ϕ—what we can call a metaphysically explanatory argument. If one accepts this connection between ground and metaphysically explanatory arguments, the notion of zero-grounding is unproblematic: a truth is zero-grounded if it is the conclusion of an explanatory argument from the empty collection of premisses. The seemingly mysterious distinction between being ungrounded and being zero-grounded is a special case of the familiar distinction between not being derivable and being derivable from the empty collection of premisses. In response to the second question, I do not simply postulate that nonfactive grounding claims are zero-grounded. If the claim that Δ non-factively grounds ϕ just is the claim that there exists an explanatory argument from Δ to ϕ there are compelling reasons for holding that the claim that Δ nonfactively grounds ϕ—if true—is zero-grounded. To substantiate this I show how to develop a logic of iterated ground—the Pure Logic of Iterated Strict Full Ground (PLISFG). A novel feature of PLISFG is that its deductive system distinguishes between explanatory arguments and what we may call “plain” arguments. This allows us to equip factive and non-factive grounding operators with natural introduction and elimination rules. (In fact, the rules are proof-theoretically harmonious.) Together these rules entail that true non-factive grounding claims are zero-grounded.

1.1 Overview of the paper }2 explains how the various notions of ground are to be understood. }3 rehearses a serious problem posed by claims of iterated ground. }4 formally states the ZGA. }5 sketches a graph-theoretic account of ground, discusses how the graphs are to be understood, and shows how the zero-grounding of non-factive grounding claims is a natural consequence. }6 show how we can understand ground in terms of explanatory arguments and develops a deductive system distinguishing between explanatory and merely plain

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arguments. }7 shows how to find introduction rules for the grounding operators. }8 uses an inversion principle to find matching elimination rules; these rules have the consequence that non-factive grounding claims, if true, are zero-grounded. }9 defends the ZGA against the objection that every true non-factive grounding claim has the same (empty) ground. }10 compares the ZGA with the “Straightforward Account” (SFA) offered by Bennett (2011) and deRosset (2013) and argues that even the SFA needs zero-grounding. The paper concludes with some issues for further work in }11. There are two technical appendices. Appendix A states introduction and elimination rules for the Pure Logic of Iterated Strict Full Ground (PLISFG) in an “amalgamation-friendly” form to facilitate comparison with Fine’s Pure Logic of Ground (PLG). Appendix B develops a graph-theoretic semantics and uses it to show that PLISFG is a conservative extension of a subsystem of PLG.4

2 . G R O U N D AN D EX P L AN A T I O N I take ground to be an explanatory notion. As I will understand ground, to say that ϕ0 ;ϕ1 ;::: ground ϕ just is to say that ϕ0 ;ϕ1 ;::: explain ϕ in a distinctively metaphysical way.5 The explanatory connection between the grounds and what they ground is very intimate; following Fine, I take the grounds to explain the grounded in the sense “that there is no stricter or fuller account of that in virtue of which the explanandum holds. If there is a gap between the grounds and what is grounded, then it is not an explanatory gap.” (Fine 2012a, p. 39)6 Here are some plausible cases of ground: 4 The graph-theoretic semantics can be extended to a semantics for all the grounding operators of Fine’s Pure Logic of Ground—and more besides. It is also possible to find introduction and elimination rules for these operators—and more besides. This is a task for another occasion. 5 I should flag a controversy here. That grounding is intimately connected with explanation is widely accepted; that Γ’s non-factively grounding ϕ just consists in Γ’s explaining ϕ in a distinctive way is not uncontroversial, though it is accepted by Fine (2001, 2012a), and Dasgupta (2014a, 2016). An alternative view would take grounding to be a (the?) distinctive relation of determination that “underwrites” such metaphysical explanations. On this view grounding stands to metaphysical explanation as causation stands to causal explanation. (Such a view is held by Audi (2012b, p. 688), Audi (2012a) and Schaffer (2012, 2016).) I will not attempt to refute this position here. Many of the claims made in this paper can, in any case, be appropriated by the defenders of this other view. 6 Fine (2012a, pp. 38–40) distinguishes between metaphysical, normative, and natural ground. Here we will only discuss metaphysical ground: it is only with metaphysical ground that the connection between the grounds and that which they ground is this intimate.

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Jon Erling Litland A conjunction is grounded in its conjuncts. A disjunction is grounded in its true disjuncts.7 a’s being G is grounded in a’s being F , where F is a determinate of the determinable G.

Several notions of ground have been distinguished in the literature. The one that has been the focus of the debate over iterated ground is factive, full, mediate, strict ground. Adopting the notation of (Fine 2012a, 2012b) we express claims of ground using a sentential operator “