Everything, more or less: A defence of generality relativism (Oxford Philosophical Monographs) 9780198719649, 0198719647

Table of contents :
Cover
Everything, More or Less: A Defence of Generality Relativism
Copyright
Dedication
Preface
Contents
Acknowledgements
1: Absolutism and Relativism
1.1 Absolutism
1.2 The argument from sortal restriction
1.3 The argument from metaphysical realism
1.4 The argument from indefinite extensibility
Collection
The russell reductio
Interpretation
The williamson–russell reductio
The naivety rejoinder
1.5 The objection from mysteriousness
1.6 The objection from ineffability
2: Russell, Zermelo, and Dummett
2.1 Self-reproductive processes and classes
A template for paradox
Burali-forti’s paradox
Russell’s paradox
Russell’s Lesson
Against a comprehensive totality
2.2 The vicious-circle principle
New grounds for relativism?
The vicious-circle principle and ramified type theory
Against ramification
Russell’s reconciliation
Interpreting second-order quantifiers
2.3 Zermelo–Fraenkel set theory
Zermelo’s axiomatic approach
Avoiding Russell’s paradox
2.4 The open-ended hierarchy
The russell reductio
The iterative conception of set
Stage 0
Stage α+1
Stage λ
Quasi-categoricity and the open-ended hierarchy
2.5 Indefinite extensibility
The Dummettian argument
The naivety rejoinder redux
3: Quantifiers
3.1 MT-semantics for the language of set theory
3.2 MT-semantics for the language of generalized quantifiers
The language of generalized quantifiers: syntax
The language of generalized quantifiers: MT-semantics
3.3 Intended MT-interpretations?
3.4 P-semantics for the language of set theory
3.5 SP-semantics for the language of generalized quantifiers
The language of generalized quantifiers: SP-semantics
Interpreting superplural quantifiers
3.6 Semantics for Quineans
The Williamson–Russell paradox
Semantic optimism
Whither Quinean absolutism?
4: Restrictionism and Expansionism
4.1 Domains and universes
4.2 Restrictionism
The indexical account
Tacit nominal variables
Pragmatic restriction
Contextual restrictionism
4.3 The objection from semantic theorizing
4.4 Expansionism
Procedural postulationism
Interpretational expansionism
The objection from creationism
The objection from crypto-restrictionism
The objection from semantic theorizing
4.5 The objection from kind-generalizations
Kind-generalizations
An intensional gap?
A hyperintensional gap?
AvoidingWilliamson’s regress
Expressive issues revisited
5: Schemas
5.1 The objection from ineffability
Absolutism stated
Is relativism self-defeating?
5.2 Open-ended schemas
5.3 Relativism schematized
5.4 Systematic ambiguity
5.5 The objection from side-conditions
6: Modal Operators
6.1 Modal generality
Beyond schematic generality
Non-circumstantial modality
Interpretational modality: a bimodal account
6.2 Modalization: first-order theories
Invariance and Mirroring
6.3 Modalization: plural theories
Adding plural resources
Indefinite extensibility
Relativism stated
6.4 Set theory for relativists
The iterative conception: a bimodal axiomatization
Linnebo’s unimodal axiomatization
6.5 Objections from unintelligibility
The objection fromKripke semantics
The objection from Kaplanian monsters
6.6 Hybrid relativism
7: Russell Reductio Redux
7.1 The schematic argument
7.2 The modal argument
7.3 Uncollectability
The Zermellian hierarchy and the Cantorian universe
Option 1: zermellian relativism
Option 2: cantorian absolutism
Russell’s and Burali-Forti’s paradox revisited
Limitation of size
The iterative conception
7.4 Instability
Option 3: third-way absolutism
Restricted quantifiers
Urelements
7.5 Non-comprehensibility
The extended modal language
The bimodal argument
Option 1a: hybrid zermellian relativism
Option 1b: thorough-going zermellian relativism
Explaining non-comprehensibility
8: How Universes Expand
8.1 The explanatory challenge
8.2 The Quantification Question
McGee’s strategy
Community Q: syntax
Semantic assumptions
Patterns of use
Community Q: use
A metasemantic assumption
An answer to the Quantification Question
8.3 Assumptions
Semantic assumptions
The metasemantic assumption
8.4 The Expansion Question
Community E: syntax
Semantic assumptions
Trans-language patterns of use
Community E: use
A metasemantic assumption
An answer to the Expansion Question
8.5 Idealization
Community E*
Accommodation
8.6 Closing summary
Appendix A: Logic
A.1 Plural logic
The first-order language
The plural language
Metalinguistic abbreviations
Interpreting the plural language
Model theory
Proof theory
A.2 Sorted plural logic
The two-sorted plural language
Model theory
Proof theory
A.3 Modal plural logic
Proof theory
Derived modal axioms
Appendix B: Modalization
B.1 Modalized invariance
B.2 Mirroring
B.3 Inextensibility
B.4 Flattening
Appendix C: Set Theory
C.1 Zermelo–Fraenkel set theory
Pure set theory
Impure set theory
C.2 Interpreting set theory, part I
Empty Set, Pairing, Union, Separation, and Powerset
Foundation and Regularity
The Rank Theorem
C.3 Kripke normal form
C.4 Interpreting set theory, part II
The Axiom of Choice
Infinity and Replacement
Bibliography
Index

Citation preview

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Everything, More or Less

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OX F O R D P H I L O S O P H I C A L M O N O G R A P H S Editorial Committee William Child, R. S. Crisp, A. W. Moore, Stephen Mulhall, Christopher G. Timpson

Other titles in this series include Vagueness and Thought Andrew Bacon

Visual Experience: A Semantic Approach Wylie Breckenridge

Discrimination and Disrespect Benjamin Eidelson

Knowing Better: Virtue, Deliberation, and Normative Ethics Daniel Star

Potentiality and Possibility: A Dispositional Account of Metaphysical Modality Barbara Vetter

Moral Reason Julia Markovits

Category Mistakes Ofra Magidor

The Critical Imagination James Grant

From Morality to Metaphysics: The Theistic Implications of our Ethical Commitments Angus Ritchie

Aquinas on Friendship Daniel Schwartz

The Brute Within: Appetitive Desire in Plato and Aristotle Hendrik Lorenz

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Everything, More or Less A Defence of Generality Relativism

J. P. Studd

1

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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 © J. P. Studd 2019 The moral rights of the author have been asserted First Edition published in 2019 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: 2018961269 ISBN 978–0–19–871964–9 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A.

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In loving memory of my parents Peter and Elizabeth

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Preface Everything, more or less? I say: more. This book defends an expansionist view of ‘everything’. The expression ‘everything’ is standardly grouped with related quantity-indicating expressions: quantifiers (e.g. ‘nothing’, ‘something’, ‘at least one thing’, ‘few donkeys’, ‘exactly four philosophers’). Quantifiers permit us to make utterances and state theories which reach out beyond the particular and generalize. But how general can we be? Is there a widest, maximally inclusive, most general sense of ‘everything’? Not according to expansionism. On this view, no matter how broad our topic of conversation or what items our theory generalizes about, what ‘everything’ and other quantifiers encompass is always open to expansion. There is always, so to speak, a more inclusive ‘everything’. More than everything? Crudely stated, the view may seem absurd. Given the choice, many instead plump for a second view: everything. It’s hard to deny the prima facie plausibility of the principal view I defend expansionism against. The absolutist about quantifiers takes the sober-seeming view that there is a widest sense of ‘everything’. The one, the absolutist might add, where ‘everything’ really means everything: everything whatsoever, everything in the entire universe, everything without restriction, everything without exception, absolutely everything. What ‘everything’ then encompasses, the absolutist claims, is not open to expansion because there is nothing— absolutely nothing—left to add. A third view is also available: less (although few of its advocates would put it this way). Restrictionism, like expansionism, opposes absolutism, but for different reasons. There is widespread agreement that what ‘everything’ encompasses—the domain quantified over—may vary according to what is under discussion, or salient in our surroundings, and so on. An occurrence of ‘everything’ posted in a shop window, for example—‘40% off everything!’—is not intended to generalize about the same items as the ‘everything’ on its competitor’s poster: ‘60% off everything!’. But the restrictionist goes further. On this view, what ‘everything’ encompasses is always subject to restriction. The two views opposing absolutism about quantifiers may be grouped together as versions of relativism about quantifiers. The immediate importance of the disputed question—do quantifiers achieve absolute generality?—comes out in the fact that absolute generality appears to be deeply embedded in our theorizing. Even if it’s comparatively uncommon in everyday conversation, how are we to engage in metaphysics, logic, or set theory without quantifying over absolutely everything? Williamson (2003) argues that relativism about quantifiers would severely diminish our capacity for systematic enquiry well beyond these disciplines: it would leave us unable to adequately capture the semantics of quantifiers and prevent our fully expressing even some kinds of limited generalization which are a mainstay of large swathes of scientific theorizing.

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On the other hand, absolutism has been contested on diverse grounds, some broadly semantic, others more metaphysical. In my view, however, the most compelling reason to doubt absolutism comes from philosophical issues surrounding the foundations of mathematics. The thought that absolute generality is somehow culpable in the set-theoretic paradoxes is an old one, almost as old as the paradoxes themselves. Some of the chapters that follow examine related lines of thought in the work of Russell, Zermelo, and Dummett. But my primary concern is not exegetical. It is to defend an expansionist version of relativism. Marshalling considerations relating to the paradoxes into a rigorous and dialectically effective argument against absolutism is an important part of this defence. And even if no such argument is ultimately to be found in the writings of these authors, the relativist has a great deal to learn from their work. Blaming absolutism for Russell’s paradox (or Burali-Forti’s) may suggest that there is a would-be decisive argument against this view in the offing. Some of Dummett’s remarks suggest this too: the failure of absolutism, he tells us, is the ‘prime lesson’ of the set-theoretic paradoxes (1981, p. 516). But I have no such argument (and nor does Dummett). The principal, broadly Dummettian, arguments I offer against absolutism rely on non-logical assumptions, specifically on liberal plenitude assumptions about, so to speak, which pluralities of zero or more items can be collected together as the elements of a single set. Doubtless, we can save absolute generality by sufficiently limiting collectability. But why prefer the former over the latter? In examining this question, I adopt a modest methodological naturalism which limits the scope of this book. I assume that set theory, as practised by set theorists, is broadly along the right track. I apply the same assumption, moreover, to widely accepted semantic theories, cast in set theory. This is not to say that I fetishize classical logic (or even that I fully accept it—see Appendix A.3). But it does mean that I set aside responses to the set-theoretic paradoxes, such as Dummett’s intuitionism, which seek to radically reform mathematical practice, and those, such as the deployment of paraconsistent logic, which are yet to make substantial progress towards recovering classical set theory. Even if set theory or model-theoretic semantics are not beyond philosophical interpretation or criticism, we must, I assume, ultimately leave things substantially as we found them. An important part of my defence of relativism is consequently to show that it is consonant with mathematical and semantic practice. If neither absolutism nor relativism is open to a knock-down objection, we are left to decide the matter on points. In my view, relativism, in its expansionist version, avoids or substantially mitigates the main objections raised against relativism in the literature (often, it seems, with restrictionism in mind). At the same time, absolutism fares worse than its advocates generally acknowledge. Overall, expansionism outperforms both its absolutist opponent and its relativist rival. That said, I have no systematic account to offer of philosophical cost–benefit aggregation. Perhaps someone who differed sufficiently on her weighting of the issues could, in principle, agree with every specific conclusion in this book and still disagree with me overall. In practice, however, I imagine that anti-expansionists will find plenty to contest in particular as well as in general. Indeed, the absolutist may be unwilling to concede that there’s a genuine tradeoff to be had between absolute generality and the unlimited collectability promised

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preface ix by his opponents. After all, he may claim, the lesson of Russell’s paradox is that the liberal attitude towards collectability is incoherent. The contradiction ensues directly from the relevant plenitude assumptions, without the need to call on absolutism. The incoherence charge is sometimes pushed further still. Not only does the Dummettian case for relativism turn on incoherent assumptions, incoherence seeps into relativism itself. Crude statements of the view reinforce the suspicion that relativism cannot be coherently maintained. In order for the relativist to successfully defend her view, then, she needs to show that there is a coherent view to be defended. To this end, the relativist can fruitfully deploy tools from philosophical logic. With the help of relativist-friendly schemas or, more ambitiously, suitably interpreted modal operators, the relativist is able to coherently frame both her position and the argument for it. In addition to philosophical questions, the use of these resources, especially the modal operators, raises some interesting technical questions. The technical developments in this book largely take place outside the main text, predominantly in the three appendices. The main chapters are not symbol-free. But, as far as possible, I’ve tried to provide plain English glosses of formal statements and theorems. My hope is that readers who are not symbol-lovers should be able to get the thrust of the argument, without the need to decipher the formalism. The small amount of set theory needed to understand the absolute generality debate is introduced in an informal way in Chapter 2. The English glosses, however, raise issues of their own. In light of the manifest differences between English and the extensions and variations of first-order languages deployed in this book, the natural language glosses should not be begrudged their granum salis. Readers of the absolute generality literature, however, may find the need to keep the salt cellar close at hand. This book is no exception. The chapters that follow make liberal use of numerous terms which cannot be accorded a literal, face-value interpretation by both sides in the debate: ‘domain’, ‘universe’, ‘extension’, ‘semantic value’, ‘absolutely comprehensive’, ‘absolutely everything’, ‘collectable’, ‘plurality’, ‘superplurality’, ‘propositional function’, ‘entity’, ‘however the lexicon is interpreted’, among others. The fact that the problematic terms include most of the key terms in the absolute generality debate highlights the utility of the formalism. What we may at best hope to metaphorically convey with a loose use of these terms may admit of a precise, metaphor-free, statement in a suitably interpreted symbolic language. It is consequently these regimentations that I take to give the official statement of the theses put forward in what follows. The accompanying English gloss may then be treated simply as a suggestive, reader-friendly shorthand for the official regimentation. Of course, when it comes to the intelligibility of the relevant notions, this paraphrase strategy only pushes the issue back into the symbolic language. But at least this way we sharply delineate exactly which terms of art we must make sense of if we are to understand the key theses disputed by absolutists and relativists. In some cases (e.g. ‘plurality’), the regimenting language admits of a reasonably faithful translation back into English. This permits us to come full circle and interpret the regimentation via its natural language translation, effectively treating the loose ‘plurality’-talk as elliptical for a plural paraphrase given in English. In other cases

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x preface (e.g. ‘superplurality’), where no such translation is available, the intelligibility issue must be settled another way. This book originated in a DPhil thesis entitled ‘Absolute and Relative Generality’, submitted late in 2011 and examined early in 2012, which in turn derived from a BPhil thesis with the same title, submitted in 2008. Both theses were completed under the primary supervision of Gabriel Uzquiano, then at the University of Oxford. Hardly any complete paragraphs survive intact from this earlier work, but many of the leading ideas do. The thrust of the argument in favour of expansionism over restrictionism derives from my BPhil thesis, and the material on modal set theory, and the modal and schematic formulations of the argument from indefinite extensibility may be found, in more or less their current form, in my DPhil thesis. The genesis and flourishing of these ideas owes a very great deal to Gabriel’s inspirational supervision. The other two people to whom I owe the greatest intellectual debt are Øystein Linnebo and Tim Williamson, who examined my DPhil. I encountered Øystein’s work on modal set theory shortly after I embarked on my BPhil thesis, in a talk he gave at a workshop in Oxford in March 2008 entitled ‘Pluralities and Sets’. While the bimodal theory presented in this book differs in some important ways from the unimodal theory Øystein developed, the influence of his approach shines through. On the other side of the absolutism–relativism debate, Tim’s anti-relativist tour de force ‘Everything’ is one of the most frequently cited works in this book. And even if I don’t expect him to renounce absolutism any time soon, seeking to formulate a version of relativism that could respond to some of his more damaging criticisms played an important role in shaping my view. Along with Geoffrey Hellman, Øystein was also one of the OUP Readers. Alex Paseau worked through the entire DPhil thesis in a single term as my secondary supervisor. And Thea Goodsell and Lorenzo Rossi painstakingly trawled through the manuscript of the current work. Their criticisms and suggestions have greatly improved this book. Thanks are also due to the many others who have generously given me their help, especially to Solveig Aasen, Tom Ainsworth, Denis Bonnay, Hannah Carnegy, Roy Cook, Vera Flocke, Salvatore Florio, Rachel Fraser, Peter Fritz, Kentaro Fujimoto, Marcus Giaquinto, Volker Halbach, Leon Horsten, Torfinn Huvenes, Dan Isaacson, Graham Leigh, Guy Longworth, Beau Mount, Carlo Nicolai, Jonathan Payne, Oliver Pooley, Agustín Rayo, Sam Roberts, Stefan Sienkiewicz, Rob Watt, Philip Welch, and Juhani Yli-Vakkuri.

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Contents Acknowledgements 1. Absolutism and Relativism 1.1 1.2 1.3 1.4 1.5 1.6

Absolutism The argument from sortal restriction The argument from metaphysical realism The argument from indefinite extensibility The objection from mysteriousness The objection from ineffability

2. Russell, Zermelo, and Dummett 2.1 2.2 2.3 2.4 2.5

Self-reproductive processes and classes The vicious-circle principle Zermelo–Fraenkel set theory The open-ended hierarchy Indefinite extensibility

3. Quantifiers 3.1 3.2 3.3 3.4 3.5 3.6

MT-semantics for the language of set theory MT-semantics for the language of generalized quantifiers Intended MT-interpretations? P-semantics for the language of set theory SP-semantics for the language of generalized quantifiers Semantics for Quineans

4. Restrictionism and Expansionism 4.1 4.2 4.3 4.4 4.5

Domains and universes Restrictionism The objection from semantic theorizing Expansionism The objection from kind-generalizations

5. Schemas 5.1 5.2 5.3 5.4 5.5

The objection from ineffability Open-ended schemas Relativism schematized Systematic ambiguity The objection from side-conditions

6. Modal Operators 6.1 6.2 6.3 6.4 6.5 6.6

Modal generality Modalization: first-order theories Modalization: plural theories Set theory for relativists Objections from unintelligibility Hybrid relativism

xiii 1 1 4 6 10 15 18 21 22 31 40 47 54 61 62 63 69 73 75 79 87 88 91 99 102 110 120 120 125 128 131 135 142 142 153 157 163 171 176

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7. Russell Reductio Redux 7.1 7.2 7.3 7.4 7.5

The schematic argument The modal argument Uncollectability Instability Non-comprehensibility

8. How Universes Expand 8.1 8.2 8.3 8.4 8.5 8.6

The explanatory challenge The Quantification Question Assumptions The Expansion Question Idealization Closing summary

178 179 184 186 195 201 214 215 217 227 231 237 240

Appendices A Logic A.1 Plural logic A.2 Sorted plural logic A.3 Modal plural logic B Modalization B.1 Modalized invariance B.2 Mirroring B.3 Inextensibility B.4 Flattening C Set Theory C.1 Zermelo–Fraenkel set theory C.2 Interpreting set theory, part I C.3 Kripke normal form C.4 Interpreting set theory, part II

Bibliography Index

245 245 249 250 254 254 255 256 257 259 259 260 263 263 265 273

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Acknowledgements The sections of this book listed below draw on, or reproduce, material that I have previously published elsewhere: • Sections 6.2 and 6.4 and Appendices A.3, B.1–B.2, and C. Reprinted by permission from Springer Nature, Journal of Philosophical Logic, Volume 42, Issue 5, J. P. Studd, ‘The iterative conception of set: (bi)-modal axiomatisation’, 697–725, Copyright © 2012, Springer Science Business Media B.V. • Sections 3.2, 3.5, 4.1, and 4.3. Reprinted by permission from Springer, J. P. Studd, ‘Absolute generality and semantic pessimism’ in: Torza A. (ed.) Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language, Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), Volume 373, Springer, Cham, 339–66, Copyright © 2015, Springer International Publishing Switzerland. • Sections 2.5, 5.1, 7.1, and 7.3–7.4. Reproduced from J. P. Studd, ‘Generality, extensibility, and paradox’, Proceedings of the Aristotelian Society, Volume 117, Issue 1; reprinted by permission of Oxford University Press and courtesy of the Editor of the Aristotelian Society: © 2017. I thank the publishers for their permission to reproduce this material here.

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1 Absolutism and Relativism Can we theorize about absolutely everything? Do we ever succeed in being maximally general, in some interestingly maximal sense of ‘maximal’? It may seem obvious that we can and do. After all, English is equipped with quantifiers such as ‘everything’ which permit us to make general claims. Absolutism about quantifiers maintains further, with considerable plausibility, that we sometimes use these quantifiers to make claims that are as general as can be: sometimes we use ‘everything’ to talk about—quantify over—absolutely everything. Nonetheless, despite the obvious appeal of absolutism about quantifiers, diverse grounds have been forthcoming for the opposing view, relativism about quantifiers. This introductory chapter aims to give an overview of the absolute generality debate and to set the scene for the defence of relativism later chapters pursue. Section 1.1 elaborates on absolutism. We then take up some of the main arguments that have been given in favour of relativism: the argument from sortal restriction (Section 1.2), the argument from metaphysical realism (Section 1.3), and the argument from indefinite extensibility (Section 1.4). Two important objections against relativism, and a number of relativist responses, follow in Section 1.5 and Section 1.6.

1.1 Absolutism To a first approximation, absolutism about quantifiers is the view that sometimes— when subject to no explicit or tacit restrictions—quantifiers such as ‘everything’ or ∀x range over an absolutely comprehensive domain.1 The key notion stands in need of explanation. What is it to quantify over an absolutely comprehensive domain? The absolutist may expand on his view as follows:2 To quantify over an absolutely comprehensive domain is simply to quantify over absolutely everything there is. ‘Absolutely everything’ means just that: absolutely everything whatsoever in the entire universe. [Thumping the table:] NO EXCEPTIONS! The absolutely comprehensive domain is simply the domain comprising absolutely everything. It contains every item we can talk about, in this context or any other, in addition to every item speakers of other languages, natural or artificial, generalize over using their quantifiers. If there are abstract objects (such as numbers or sets), these belong to the absolutely comprehensive domain; similarly, if there are theoretical objects (such as electrons or quarks) or fictional

1 Our principal templates for absolutism about quantifiers are Cartwright’s (1994) defence of speaking of everything and Williamson’s (2003) elucidation of generality-absolutism. 2 Compare Cartwright (1994, p. 1) and Williamson (2003, p. 415).

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 everything, more or less objects (such as unicorns or hippogryphs) or merely possible objects (such as Wittgenstein’s possible children), then these too belong to the absolutely comprehensive domain. The same goes, without exception, for everything else.

Three further preliminary clarifications are in order. First, the absolutist’s thesis that we sometimes quantify over absolutely everything that there is tells us nothing about what there is. For instance, the absolutist need not agree with the platonist who thinks that there are abstract objects, nor with those who posit theoretical, fictional, or merely possible objects. (Note the ‘if ’s in the absolutist’s elucidation.) The absolutist is free to adopt as austere or bloated an ontology as he pleases, so long as he claims that we can quantify without restriction over absolutely every item it comprises.3 Second, the absolutist’s thesis says only that we sometimes quantify over absolutely everything that there is. For instance, according to a perennial fiction, there are unicorns. But this does not commit the absolutist to maintaining that the absolutely comprehensive domain contains a unicorn unless he maintains, further, that there really are unicorns. Nor does this commit him to the absolutely comprehensive domain’s containing a fictional unicorn unless he maintains, further, that there are fictional unicorns. Similarly, just because there could be a golden mountain or there will one day be humans on Mars, let’s assume, doesn’t mean that a golden mountain or a human located on Mars is in the absolutely comprehensive domain; nor need the absolutist maintain that this domain contains a possible golden mountain or a future time-slice of a human located on Mars unless he maintains, further, that there are such items.4 Third, to say that we sometimes quantify over absolutely everything there is is not to say that we always do. For instance, the English quantifier ‘no donkey’ is always restricted to range over a domain comprising only donkeys. The absolutist may also maintain that even when no restriction is explicit in the syntax of the quantifier (as, for example, in ‘everything’, ‘every object’, or ‘every item’), its domain may still be subject to restrictions supplied by the context of utterance. Imagine, for example, that after much toil, having painstakingly made the final adjustments to her apparatus, a scientist makes the following utterance, her hand poised over the start button: (1) Everything is ready. On one widely-held view,5 the truth of her utterance is compatible with a great many things not being ready (the experiment pencilled in next month, for example). All the same, these non-ready items fail to be counter-instances to the general claim she makes because the context serves to restrict the occurrence of ‘everything’ in her utterance of (1) to range only over items relevant to the task at hand. The operation of any sort of quantifier domain restriction is perfectly consonant with absolutism provided it can sometimes be lifted. The absolutist need only claim that some languages contain quantifiers which in some contexts range over the absolutely comprehensive domain. It’s helpful to suppose he adds—as he typically does—that English quantifiers such as ‘everything’ are such quantifiers and the context

3 4 5

Compare Williamson (2003, p. 423). Compare Williamson (2003, pp. 421–3). The mechanism behind quantifier domain restriction is controversial. See Section 4.2.

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absolutism



in which he explains his view is such a context. (Indeed, he must add this if his elucidation is to achieve its required generality.)6 Further clarifications will be necessary in due course. But enough has already been said to outline the prima facie appeal of absolutism. Very often in science, philosophy, and everyday life it suffices to quantify over less than everything. The enterprise at hand may only call for us to generalize about, say, particles in the standard model, or agents with free will, or the contents of one’s fridge. But sometimes restricted generality doesn’t seem to be enough: some statements in logic, set theory, and metaphysics seem to cry out for an absolutely general formulation. Take, for instance, mereological nihilism. Having explained that mereologically simple things are those that have no proper parts, the nihilist attempts to state her sparse view of parthood with the following utterance: (2) Everything is mereologically simple. Well aware of the potential for such a radical claim to invite misunderstanding, the nihilist may take pains to emphasize that she does not intend her use of ‘everything’ to be restricted. If she is more charitably interpreted to quantify over a limited domain, she doesn’t want our charity. To interpret her with anything less than absolute generality seems to vitiate the statement of her view. A logician or set theorist might make similar efforts to accompany informal English renderings of the following theorems of predicate logic (with identity) and of set theory:7 (3) Everything is self-identical. (4) Everything is the sole element of its singleton set. In either case, interpreting ‘everything’ to range over a less-than-absolutelycomprehensive domain appears to deprive the theorem of its intended generality. With the initial quantifier so restricted, an utterance of (3) or (4) fails to rule out the possibility of non-self-identical things or singletonless items outside the limited domain. To capture these theorems in their intended generality seems to call, on the contrary, for quantification over an absolutely comprehensive domain.8 The prima facie case for absolutism about quantifiers is clear. All the same, absolutism has been opposed on diverse grounds. Anti-absolutist arguments draw variously on semantic, metaphysical, and mathematical considerations: (i) advocates of ‘sortal restriction’ dispute the availability of a universal sense of ‘thing’; (ii) relativists wary of ‘metaphysical realism’ contend that absolutism leads to objectionable views in metaontology; (iii) friends of ‘indefinite extensibility’ maintain that the availability of an absolutely comprehensive domain is in conflict with the open-ended nature of concepts such as set and interpretation.

6

Compare Cartwright (1994, p. 1) and Williamson (2003, n. 1). Compare the examples in Cartwright (1994, p. 1) and Williamson (2003, p. 416). Assuming domains are extensional, there is at most one absolutely comprehensive domain. But, except when expounding the views of absolutists, we generally eschew talk of ‘the absolutely comprehensive domain’ to avoid any suggestion of a presupposition that there is any such domain. 7 8

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 everything, more or less In my view, when properly developed, considerations from indefinite extensibility provide by far and away the most powerful case against absolutism. Nonetheless, a fairly brisk survey of some of the other main arguments against absolutism is helpful in order to bring this view and the principal argument against it into sharper relief. The next three sections take up each of these anti-absolutist arguments in turn.9

1.2 The argument from sortal restriction The argument against absolutism from sortal restriction trades on the distinctive determiner–nominal structure of quantifiers in natural language. In English, and many other languages, quantifiers result from combining a determiner (e.g. ‘every’, ‘some’, ‘no’, ‘most’) with a nominal (e.g. ‘thing’, ‘set’, ‘donkey’).10 The nominal serves to delimit the quantifier’s range. For instance, ‘every donkey’ generalizes about donkeys; ‘every set’ ranges only over sets; ‘everything’ quantifies over things. Absolutism consequently calls for universal nominals: in order to contend that ‘everything’ sometimes attains absolute generality, the absolutist needs to claim that the nominal ‘thing’ applies indiscriminately to any item whatsoever, regardless of its sort. The argument from sortal restriction contests this claim on the twin grounds that a quantifier’s nominal must be a sortal term and that no sortal term is universal.11 Advocates of the sortal–non-sortal distinction differ on exactly what it takes to be sortal, but terms like ‘set’, ‘cardinal number’, ‘book’, and ‘person’ are typically taken to be clear cases of sortal terms; terms such as ‘thing’ or ‘red thing’ are usually taken to be clear cases of non-sortal terms.12 On one prominent view, a sortal term is equipped with a non-trivial criterion of identity which gives identity conditions for items of the relevant sort. In the case of the term ‘set’, the Axiom of Extensionality is often given as a paradigm example of a non-trivial criterion of identity:13 Axiom of Extensionality. A set is identical to another if and only if they have the same elements. The need for nominals to be equipped with non-trivial criteria of identity is often motivated in connection with cardinality questions. P. T. Geach, for instance, maintains that there’s something problematic about attributing a cardinal number to the red things in a given room: . . . the trouble about counting the red things in a room is not that you cannot make an end of counting them, but that you cannot make a beginning; you never know whether you have counted one already, because “the same red thing” supplies no criterion of identity. (1968, pp. 38–9)

As Michael Dummett puts it, on this view, in the case of a non-sortal such as ‘red thing’: 9 Rayo and Uzquiano (2006b) and Florio (2014a) survey similar terrain, also considering attacks on absolute generality based on Skolemite scepticism. 10 See Lewis (1970, p. 40) and Barwise and Cooper (1981, pp. 161–2). 11 See Rayo and Uzquiano (2006b, sec. 1.2.5). 12 See, for instance, Wallace (1965). 13 See Geach (1968), Dummett (1981, ch. 16), Wiggins (2001), and Lowe (2009).

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the argument from sortal restriction  . . . it simply makes no sense to speak of the number of red things . . . there are some questions ‘How many?’ which can only be rejected, not answered . . . (1981, p. 547)

For sortal restriction to threaten absolutism about quantifiers, it needs to be argued further that putative universal nominals, such as ‘thing’, ‘object’, ‘item’, and so on, either fail to be universal or fail to yield meaningful quantifiers when combined with determiners such as ‘every’. To this end, the sortalist may maintain that despite functioning syntactically as a count noun, on a par with ‘set’ or ‘book’, the nominal ‘thing’ and other supposedly universal nominals fail to be equipped with a non-trivial criterion of identity. Faced with such an objection, many absolutists, I suspect, will be all too happy to simply reject the underlying metaphysics. If we lack an effective means to determine whether this red item is the same as one we already counted, we have no way to come to know how many red items occupy the room. But surely our ignorance, even if unavoidable, is no bar to there in fact being, say, exactly 88 items in the room large enough to reflect light in the red part of the visible spectrum. Nor is it a bar to the quantifier ‘Exactly 88 red items’ being a non-semantically-defective English quantifier. Contemporary sortalists often prefer to cast non-trivial criteria of identity in a less epistemic, more metaphysical role. On such views, a non-trivial criterion of identity for Fs need not give an idealized decision procedure for determining whether or not Fs are identical; instead it serves as something approaching a conceptual analysis of being the same F, an informative account of what it is that makes Fs identical or distinct.14 The difficulty of this metaphysical enquiry depends on how demanding a notion of informativeness is in play. But, once again, the link between the success of this enquiry and quantification over every member of the relevant sort is far from immediate. Not every meaningful term can be defined in terms of more basic ones. Even if we accept tight links between quantification and identity, why think the meaningfulness of the relevant quantifiers turns on there being a non-trivial criterion of identity? This is not the place for a full evaluation of a sortalist metaphysics. But it’s worth observing that, even for philosophers generally well-disposed to this programme, two sizeable gaps need to be filled if the argument from sortal restriction is to make a wellsupported case against absolutism. First, supposing we agree with the sortalist that non-trivial criteria of identity for Fs are needed in order to render contentful numerically definite quantifiers such as ‘exactly one F’, why think the same is required in the case of universal quantifiers, such as ‘every F’? The connection between identity and quantification is much less apparent in the latter case. Indeed, even if we lack the means to count the red items, as in Geach’s example, we may still be able to straightforwardly observe, for instance, that every red item in the room fails to exceed a cubic metre in volume. Second, assuming the first gap can be filled, the sortalist still needs to establish that the absolutist’s putatively universal nominal—‘thing’, let’s say—lacks a non-trivial criterion of identity. It’s not enough here simply to observe that no candidate criterion is immediately apparent. After all, the same is true for supposedly paradigm examples of sortal terms such as ‘person’ and ‘river’. Moreover, supposing the various sorts 14

See, for instance, Lowe (2009, pp. 18–19). See also Horsten (2010).

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 everything, more or less exhaust the contents of the universe, and that we are optimistic about the prospects of framing a non-trivial criterion of identity for each limited sort, the absolutist can lay down a criterion of identity for ‘thing’ which is parasitic on the others: Criterion of Identity for Thing. One thing is identical to another if and only if, for some sort F, they are both F and meet the non-trivial criterion for F-identity. This criterion of identity seems to be as good a prima facie candidate for non-triviality as any; it’s certainly no logical truth. To close the gap, the sortalist needs to frame a well-motivated sense of non-trivial, and demonstrate that the proposed criterion fails to meet it.

1.3 The argument from metaphysical realism A second anti-absolutist argument connects the absolutism–relativism debate to issues in metaontology. The absolute generality debate is usually set up in terms of universal quantifiers: ‘everything’ or ∀x. In ontology, by contrast, existential quantifiers— ‘something’ or ∃x—come to the fore. But the difference in focus is superficial. Existential quantifiers achieve absolute generality, if they do, in exactly the same way universal ones do: namely, by ranging over an absolutely comprehensive domain. Some relativists have argued against absolutism on the grounds that such generality leads to an objectionable position in metaontology. To introduce the objection, let’s rehearse a well-worn example from Putnam (1987a, 1987b). Imagine two linguistic communities whose members appear to espouse different views about mereology (while retaining the syntax of English). Members of the first community have long considered themselves staunch mereological nihilists. The second community, on the other hand, appears to be made up of devout mereological universalists. Its members uphold the Principle of Unrestricted Composition:15 ‘for any one or more things—no matter how scattered or unrelated—something is their mereological fusion’. Imagine now that the two communities meet for the first time, and a member of the nihilist community attempts to convey her mereological worldview to the universalists with the usual sort of nihilist utterance (having first done her best to remove any contextual restrictions on her quantifiers): (5) It’s not the case that something is non-simple. The universalists’ spokesperson rejoins: (6) Something is non-simple. Some philosophers claim that, contrary to appearances, there is no substantive disagreement between the two communities. We can legitimately conceptualize reality with or without non-trivial mereological structure. The members of both communities speak truly relative to their linguistic framework/conceptual scheme/language in virtue of operating with different interpretations of the existential quantifier. 15

See, for instance, Lewis (1986, pp. 212–13).

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the argument from metaphysical realism  Under the nihilist’s interpretation, even when she succeeds in removing all contextual restrictions, the existential quantifier ranges only over simples, rendering (5) true. The universalist’s utterance of (6) is likewise true with her quantifier interpreted to express existential quantification over a wider domain.16 This apparent possibility of equivocating on the (unrestricted) existential quantifier raises a metaontological question: are questions of the sort that occupy first-order ontologists—e.g. ‘is something non-simple?’—substantive questions for which the world supplies a definite framework/scheme/language independent answer? Some relativists, apparently endorsing a negative answer to this question, argue against absolutism on the grounds that it is committed to a position that sustains a positive answer. Charles Parsons writes: What seems to me a potential problem [for absolutism] is that if our quantifiers can really capture everything in some absolute sense, then some form of what Hilary Putnam calls ‘metaphysical realism’ seems to follow. As I understand it that is that there is some final answer to the question what objects there are and how they are individuated. (2006, p. 205)

Geoffrey Hellman deploys similar considerations in one of his arguments against absolutism: The absolutist must insist that . . . at most one of the frameworks is correct, the one (if any) that quantifies over only those objects in the range of the absolute quantifiers, the objects that ‘really exist’ (‘REALLY EXIST’?). (2006, p. 87, emphasis his)

In a slogan: absolutism implies metaphysical realism. Is a commitment to a species of realism of this kind a problem for absolutism? One obvious point here is that the dialectical effectiveness of this consideration is sensitive to the ambient metaphysics. Many absolutists, I suspect, will find a metaontological outlook where the absolute domain serves as the ultimate arbiter of ontological questions quite congenial. Second, and more importantly, are Parsons and Hellman right to maintain that absolutism implies metaphysical realism? To answer this question, we need a better grip on some of the key metaontological terms of art. What’s supposed to be ‘final’ about the ‘final answers’ to ontological questions sought by the metaphysical realist (according to Parsons)? What is the relevant sense of ‘really exist’ (or indeed ‘REALLY EXIST’) in Hellman’s formulation? The metaontology literature abounds with technical terms and metaphors used to describe realist attitudes of the kind Parsons and Hellman seem to be driving at. The final answer as to what ‘really exists’ is given using ‘the quantifier that God would use’, the one whose interpretation is ‘metaphysically privileged’; substantive ontology is conducted using quantification that best ‘carves nature at the joints’ or is part of the ‘fundamental structure of reality’.17 The terms of art permit us to rephrase our question in more evocative terms: why think that absolutists are committed to a metaphysically privileged/joint-carving existential-quantifier-interpretation (that God would use)? The only apparent answer

16 17

See, for instance, Carnap (1950), Putnam (1987a, 1987b) and Hirsch (2002, 2005, 2009). The first two are from Hirsch (2002, p. 61), the second two from Sider (2011, pp. 1, 3).

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 everything, more or less is because the absolutist is committed to attaching this special status to the existentialquantifier-interpretation that he takes to achieve absolute generality: Biggest is Best. If there is an absolutely general existential-quantifier-interpretation, it is the unique metaphysically privileged/maximally joint-carving existentialquantifier-interpretation. This assumption is clearly present in the passage from Hellman quoted above. And assuming that the biggest interpretation really is (metaphysically) best, the Parsons– Hellman implication from absolutism to metaphysical realism would seem to be an immediate corollary. But should we accept the Biggest is Best assumption? The assumption clearly has some intuitive pull. Granted the availability of an absolutely comprehensive domain, why make do with a less-comprehensive one? To frame metaphysical theories by quantifying over the smaller domain may seem to ‘ignore’ items that are available to be quantified over. The usefulness of pre-theoretic intuitions in such deep metaphysical territory, however, is limited. To make further progress we need to unpack the crucial terms of art. Here we confront something of an irony in the metaontology debate: in their dispute over whether the participants in first-order ontological disputes equivocate on the existential quantifier, metaontologists often seem to attach very different meanings to their preferred term for ‘metaphysically privileged’/‘joint-carving’. Significantly, however, prominent figures on both sides of the debate elucidate the crucial metaontological terms in ways that call into question the Biggest is Best assumption. Eli Hirsch (2002) is one prominent opponent of metaphysical realism. He instead advocates quantifier variance, which denies that there is ‘one metaphysically privileged sense of the quantifier’ (p. 61). On his view, the mark of metaphysical distinction is expressive power: a non-metaphysically-privileged quantifier-interpretation ‘would leave us without adequate resources to state the truth properly’ (p. 61). He elaborates on the relevant resources in glossing the ‘basic idea’ of his view: . . . the basic idea of quantifier variance can be nicely formulated by saying that the same (unstructured) facts can be expressed using different concepts of “the existence of a thing”, that statements involving different kinds of quantifiers can be equally true by virtue of the same (unstructured) facts in the world. (p. 59)

Hirsch’s focus on unstructured facts and propositions sets the expressive bar comparatively low by permitting sentences with radically different syntactic structures to express the same proposition. On the unstructured account, sentences uttered in different languages (and contexts) express the same proposition if they are true in the same possible worlds (as interpreted according to their respective languages and contexts).18 Suppose, for instance, that the universalist makes a modest start to the project of cataloguing facts by uttering: (7) Something is a table.

18

See Hirsch (2009, p. 234); compare Hirsch (2002, p. 57).

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the argument from metaphysical realism  The nihilist can plausibly capture the same (unstructured) fact by uttering a nihilisttranslation of (7), which only deploys quantification over simples but expresses the same (unstructured) proposition: (8) Some simples are arranged tablewise. This toy example is enough to show that there’s no obvious connection between how widely a quantifier ranges and whether its interpretation counts as ‘metaphysically privileged’ on Hirsch’s account. Even if we suppose that the universalists quantify over a domain that is absolutely comprehensive, why doubt that the non-absolutely-general quantification available in the nihilist’s language can match the universalist’s in terms of coarse-grained expressive power, provided we allow for enough other nihilistfriendly expressive resources?19 Understanding ‘metaphysical privilege’ in terms of Hirsch’s coarse-grained expressive criterion, then, it’s far from clear that the Biggest is Best assumption holds. Hirsch’s account of the crucial metaontological term, however, is by no means the only option. Theodore Sider, on the opposing side of the metaontology debate, offers a radically different account of what it takes for a quantifier to be ‘joint-carving’ or ‘fundamental’ (to use some of his preferred terms). Sider follows David Lewis (1983) in taking it to be a brute fact about the world that it has metaphysical structure. Lewis’s account posits that some properties are more natural, or joint-carving, than others. Inter alia, naturalness underwrites objective similarity. If we follow Lewis in conceiving of properties liberally, any two items, a and b, however similar or dissimilar, share infinitely many properties (e.g. identical-to-a-or-b, identical-to-a-orb-or-pink, and so on) and differ on infinitely many more (e.g. identical-to-a, nonidentical-to-b, and so on). But sharing natural properties makes for genuine similarity. Two gluons are, ceteris paribus, more objectively similar than a gluon and a glue stick because the property gluon shared by the first pair is more natural—better jointcarving—than the property gluon-or-glue-stick shared by the second pair.20 Sider (2009, 2011) generalizes Lewis’s account of naturalness to also apply to quantifier-interpretations. On his account, there is a wide range of existentialquantifier-interpretations available to us. These include both less comprehensive nihilist-friendly interpretations that render (5) true and more comprehensive universalist-friendly interpretations that render (6) true. Nonetheless, it is a fundamental fact about the world’s metaphysical structure that some quantifierinterpretations are better joint-carving than others.21 What becomes of the Biggest is Best assumption in this metaontological framework? Despite its apparent remoteness from ordinary enquiry, Sider (2011, sec. 2.3) contends that we can find out about the world’s fundamental metaphysical structure by following a familiar set of Quinean criteria for theory choice. We should accept the theory that does best overall in terms of theoretical virtues such as simplicity, explanatory power, integration with other good theories, and so on. Sider, 19 This style of translation is due to van Inwagen (1990); see Uzquiano (2004) and Sider (2009) for discussion of the required expressive resources. 20 See Lewis (1983, pp. 346–7). 21 See Sider (2009, esp. pp. 392, 407–8, 2011, sec. 9.2).

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 everything, more or less however, proposes one crucial addition to this methodology concerning the ideology (i.e. primitive terms) of our best theories: A good theory isn’t merely likely to be true. Its ideology is also likely to carve at the joints. For the conceptual decisions made in adopting that theory—and not just the theory’s ontology— were vindicated; those conceptual decisions also took part in a theoretical success, and also inherit a borrowed luster. So we can add to the Quinean advice: regard the ideology of your best theory as carving at the joints. We have defeasible reason to believe that the conceptual decisions of successful theories correspond to something real: reality’s structure. (p. 12)

Assuming we accept this methodology, it’s far from clear we should accept the Biggest is Best assumption. Sider (2011, ch. 13) tentatively sketches a worldview which eschews the more comprehensive interpretations of the existential quantifier that he takes to be available. Instead, this view takes a more limited nihilist-friendly interpretation to be the unique perfectly joint-carving existential-quantifier-interpretation. On this view, even if there is an absolutely general existential-quantifier-interpretation available to us, the less comprehensive interpretation is better joint-carving. In theorizing about the world’s fundamental structure, it may be better to ignore mereologically complex objects if quantifying over them requires us to carve the joints less well. Doubtless there is much to question in both Hirsch’s and Sider’s metaontology. But we’ve waded into the metametaphysics deep enough to see that the Biggest is Best assumption is far from immediate. And without it, there’s no obvious reason to sustain the Parsons–Hellman implication from absolutism to metaphysical realism.

1.4 The argument from indefinite extensibility In the previous two sections, we failed to find a compelling reason to reject absolutism about quantifiers. Of course, this brief discussion of the arguments from sortal restriction and metaphysical realism far from rules out their being developed into effective anti-absolutist arguments. But since I can see no promising way to do so, these arguments are set aside in the chapters that follow. The primary anti-absolutist argument that is developed in this book draws on considerations of a quite different kind. At least since Russell (1908) there has been a suspicion that excessive generality is somehow bound up with Russell’s paradox and the other set-theoretic antimonies. Considerations of this kind are influentially taken up by Michael Dummett (1981, chs. 14–16, 1991, ch. 24) as the basis of a popular and powerful argument against absolutism. Central to Dummett’s argument is the thesis that some concepts F are indefinitely extensible: to a first approximation, this is to say that given any domain comprising Fs, however extensive, a further F can always be specified, giving rise to a wider domain.22 This section considers two examples of putatively indefinitely extensible concepts: collection and interpretation. It’s worth emphasizing at the outset that our first pass presentation of the argument here is no more than that. We return to consider some 22 For now we set aside some of the nuances in Dummett’s presentation. We return to his view in more detail in Section 2.5.

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the argument from indefinite extensibility



of Russell’s and Dummett’s arguments in Chapter 2, and to offer my preferred version of the argument from indefinite extensibility in Chapter 7.

Collection Start with the concept collection, in something approximating its pre-theoretic use. A collection, in the relevant sense, is an arbitrary extensional collection of zero or more members. A collection is said to comprise certain items, when each of these items, and nothing else,23 is a member of the collection. To say that a collection is arbitrary is to say that there need be no non-arbitrary relation between the members it comprises. The members of a collection need not be the property of a single collector; they need not be relevantly similar or metaphysically joint-carving; and they need not be specified by a formula of a formal language or a predicate of a natural one. The collection comprising Weston-super-Mare, the electron third closest to the centre of mass of the solar system, and my favourite ordinal is no less a collection than the collection of glass flowers in the Harvard Museum of Natural History or the collection of natural numbers. To say that a collection is extensional is to say that it is individuated according to the Axiom of Extensionality: a collection is identical to another if and only if they each comprise the same members. Elucidated in this way, the concept collection is a prime candidate to be an indefinitely extensible concept.24 For suppose we initially quantify over a domain D. No matter how extensive D may be, we seem able to specify a collection that is demonstrably not a member of D. The ‘new’ collection in question is the domain’s Russell collection, the collection that comprises the collections in D which lack themselves as members (i.e. the collection of non-self-membered collections in D). Let us label this collection rD . Adapting the reasoning of Russell’s paradox, we can then show that rD is not in the domain D. Rather than leading to an outright contradiction, the Russellian argument becomes a reductio on the assumption that rD is a member of D. Indeed the argument has much in common with the argument Zermelo uses to show that every set M has a subset {x ∈ M : ¬x ∈ x} which it lacks as an element.25 To distinguish the argument from Russell’s paradox proper, let’s call it the Russell Reductio. Its demonstration is straightforward. t h e ru s s e l l r e d u c t i o Note first that rD only contains non-self-membered collections. Consequently, if rD itself is self-membered, then rD is not a member of rD —i.e. rD is non-self-membered. This suffices to show, outright, that rD is a non-self-membered collection. Now suppose for reductio that rD is in D. It follows from our intermediate conclusion that rD is a non-self-membered collection in D. But rD contains all non-selfmembered collections in D. So, rD is a member of rD , i.e. rD is self-membered. 23

We henceforth use ‘comprise’ in this exhaustive sense, usually leaving the ‘and nothing else’ clause

tacit. 24 25

Compare Dummett (1981, pp. 530–1, 1991, p. 317). See Zermelo (1908, thm. 10). For the set-theoretic notation, see Table 2.1 in Chapter 2.

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 everything, more or less This contradicts our earlier conclusion that rD is non-self-membered. Since the reductio hypothesis leads to a contradiction, we have established its negation: rD is not in D. Before we move on to our second example, four comments on the Russell Reductio are in order. First, the argument makes very few assumptions about the nature of collections. In particular (unlike Zermelo), we do not assume that the domain D is itself a collection (or any other kind of set-like object).26 The conclusion that rD is not in D follows simply from the assumption that an item is a member of rD if and only if it is a non-self-membered collection in D. Second, the argument is unusual among informal mathematical demonstrations in having a seemingly-essential performative aspect: if the argument is to succeed, its utterance needs to bring about a shift in domain. For we suppose that we initially quantify only over D. But the whole point of the argument is to identify an item outside this domain.27 Third, assuming we can always specify rD in this way, the Russell Reductio establishes that collection is indefinitely extensible. An immediate corollary is that a quantifier such as ‘every collection’ cannot range over a domain comprising absolutely every collection. For whatever domain D of collections the quantifier ranges over, we can always specify a collection—namely, rD —outside the domain. Last, however, this is not all the argument establishes. It also provides a consideration against the characteristic thesis of absolutism that our quantifiers sometimes range over an absolutely comprehensive domain. The argument immediately generalizes in this way because it relies on no assumptions about the domain D. Whatever domain D the absolutist may claim to be absolutely comprehensive, the Russell Reductio purports to show that D lacks something: namely, rD . This gives us a first pass at an instance of the argument from indefinite extensibility. The absolutist’s plausible-seeming view about quantifiers is in tension with a prima facie attractive view about collections. The absolutist, however, may be tempted to dismiss this case against his view out of hand. First, he may claim, even if Russell’s paradox was a problem, a local problem within set theory should be solved within set theory. Second, he may add, set theory has already solved this problem by dispensing with the incoherent, naive conception of collection which led to the paradoxes. I will go to some lengths to argue that this second rejoinder is mistaken: when the argument is properly developed, the case from indefinite extensibility flows from a coherent conception of set that is perfectly consonant with standard set theory.28 But before we come to that, let’s dispense with the first objection, by widening our pool of examples of apparently indefinitely extensible concepts to include examples not involving collections.

26

In other words, we do not invoke what Cartwright (1994) dubs the ‘All-in-One’ principle. See Section 2.5. 27 This point is emphasized by Glanzberg (2006). See also Fine (2006), who suggestively labels this kind of shift from one domain to another ‘the Russell jump’. We return to consider their views in more detail in Chapter 4. 28 See Section 7.3.

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the argument from indefinite extensibility



Interpretation The second apparently indefinitely extensible concept we will consider is the concept of interpretation. For present purposes, let’s focus on interpretations of a standard first-order language containing a unary predicate P. An interpretation of the language specifies which zero or more things P applies to; in other words, that is, it specifies what satisfies the formula Px. Standardly it also supplies denotations for the language’s names (if any), a domain for its quantifiers, and so on. But any such additional features of interpretations don’t matter here. Focusing on the interpretation of P, then, it seems we are free to specify which items it does and does not apply to however we choose. There’s an interpretation of P under which it applies to all and only donkeys; a second interpretation of P takes it to apply to nothing at all; a third takes P to apply to all and only sets; and so on. Elucidated in this way, the concept interpretation is another plausible candidate to be an indefinitely extensible concept. For suppose, as before, we initially quantify over a domain D. Then we seem able to specify an interpretation that is demonstrably not in D—namely, the interpretation iD which interprets P to apply to the interpretations in D which do not interpret P to apply to themselves (i.e. the interpretation under which P applies to all and only non-self-P-applying interpretations in D). The structure of the argument is just the same as the Russell Reductio; this time we apply Zermelo’s reductio strategy to an interpretation-based variant of Russell’s paradox due to Timothy Williamson:29 t h e w i l l ia m s o n – ru s s e l l r e d u c t i o Note first that iD interprets P to apply only to non-self-P-applying interpretations. Consequently, if iD is self-P-applying, then P does not apply to iD under iD : i.e. iD is non-self-P-applying. This suffices to show, outright, that iD is a non-self-P-applying interpretation. Now suppose for reductio that iD is in D. It follows from our intermediate conclusion that iD is a non-self-P-applying interpretation in D. But iD interprets P to apply to all non-self-P-applying interpretations in D. So, P applies to iD under iD , i.e. iD is self-P-applying. This contradicts our earlier conclusion that iD is non-self-P-applying. Since the reductio hypothesis leads to a contradiction, we have established its negation: iD is not in D. Note that, this time, the argument makes no assumptions about sets or collections whatsoever. It’s common practice in model theory and semantics to encode an interpretation as a set of a certain kind; in some cases, for instance, we might identify an interpretation of P with a set-extension (i.e. the set of items P is interpreted to apply to). But the Williamson–Russell Reductio relies on no such assumptions about the nature of interpretations.30 All it relies upon is the truistic-seeming assumption that there is an interpretation under which P applies to precisely the interpretations in D that fail to apply P to themselves. 29

Williamson (2003, sec. IV).

30

Compare Williamson (2003, p. 426).

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 everything, more or less As with the Russell Reductio, however, the Williamson–Russell Reductio has a similarly performative character. If it succeeds, the argument requires us to shift from quantifying over D at the start to quantifying over a domain containing an item outside D by the end. Moreover, as before, the argument impugns a domain comprising absolutely everything just as much as a domain comprising absolutely every interpretation. The absolutist cannot hope, then, to dismiss indefinite extensibility as a mere quirk of set theory. The Williamson–Russell Reductio does for the concept interpretation what the Russell Reductio does for collection. And this style of argument for indefinite extensibility extends in the obvious way to other semantic concepts, such as extension, intension, property, and so on.31 Moreover, by instead adapting the reasoning of the Burali-Forti paradox, we may argue in an analogous way that the concept ordinal is indefinitely extensible.32

The naivety rejoinder The absolutist still has his first objection to fall back on. The objection is simply stated: doesn’t the argument from the indefinite extensibility of collection trade on a naive conception of collection that Russell’s paradox shows to be incoherent? The absolutist may elaborate on his concern as follows. In the case of the Russell Reductio, the crucial step in the argument is the assumption, for a given domain D, that there is such a collection as rD (the collection comprising the non-self-membered collections in D). But in the crucial case, when D is the domain that comprises everything, this assumption is tantamount to the following: (9) There is a collection that comprises every non-self-membered collection (and nothing else). And—he continues—if anything is the prime lesson of Russell’s paradox, surely it’s that there is no such collection. Indeed the original Russellian argument (as opposed to the Russell Reductio) shows that the formalization of (9) is inconsistent in classical logic (and in weaker logics).33 In light of this, the absolutist may be tempted to offer the following rejoinder to the argument from indefinite extensibility: he replies that Russell’s paradox shows the argument to be unsound by revealing its key premiss to be inconsistent. An exactly analogous rejoinder is available against the case for the indefinite extensibility of interpretation based on the Williamson–Russell Reductio, which has the same logical structure. Once again, the naivety rejoinder goes, the argument from indefinite extensibility offers no reductio on the availability of an absolutely comprehensive domain. Instead, the key plenitude assumption employed in the argument— this time: there is an interpretation under which P applies to all and only non-selfP-applying interpretations—collapses under its own weight. Let’s call this style of response the naivety rejoinder.

31 32 33

See, for instance, Grim (1991, pp. 119–20) and Parsons (2006, p. 209). See, for instance, Shapiro and Wright (2006, pp. 256–7). See Section 2.1 for further details. Compare, for instance, Cartwright (1994, sec. VI).

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the objection from mysteriousness  Relativists are seldom moved by accusations of naivety. More significantly, the naivety rejoinder flatly ignores an important feature of the first-pass formulation of the relativist’s argument. The assumption (9) that the absolutist reads into her argument quantifies over a single domain D. It states, in effect, that there is a collection that belongs to D that contains the non-self-membered collections in D. But the relativist makes no such assumption. Instead, as we noted in Section 1.4, the first-pass presentation of the argument from indefinite extensibility has a performative aspect. It relies on the relativist coming to quantify over a new domain, D , say. And it is from this potentially more liberal perspective that she can state the key plenitude assumption driving this instance of the Russell Reductio:34 (10) There is a collection (in D ) which comprises every non-self-membered collection in D. Unlike (9) this plenitude assumption does not, on its own, engender Russell’s paradox. To obtain a contradiction we need to assume additionally that D contains every member of D . If the absolutist assumes this because he assumes that D is absolutely comprehensive, why think it is the relativist’s liberal attitude towards ‘collectability’ rather than the absolutist’s insistence on absolute generality that is to blame for the paradox? This offers the relativist the beginnings of a response to the naivety rejoinder. But in order to dispel lingering absolutist doubts about this argument for relativism, she needs to offer a much fuller account of indefinite extensibility. The most pressing task she faces is simply to establish that there’s a genuine view to defend. In particular, she needs to give a clear and coherent account of the plenitude principles she takes to drive indefinite extensibility. The trouble, as we will see in Chapter 2, comes when we try to generalize plenitude assumptions such as (10). Merely establishing the coherence of indefinite extensibility, however, is not enough. The relativist also needs a wellmotivated account of the crucial domain-shifts her argument relies on. How is it that, on her view, no matter how extensive the initial domain, running through the Russell Reductio, or similar, leads to a new, wider domain of quantification?

1.5 The objection from mysteriousness The two straightforward challenges outlined at the end of the last section run to the heart of relativism. The challenges were raised against a particular argument in favour of relativism but they are closely bound up with two of the central objections that have been raised against the view itself: (i) the objection from mysteriousness questions whether the relativist is able to adequately explain what stands between our quantifiers and absolute generality; (ii) the objection from ineffability questions whether the relativist is able to coherently express her view.35

34 Compare, for instance, the plenitude principle (R) that drives Fine’s (2006, pp. 21–2) version of the argument from indefinite extensibility. 35 The two objections are succinctly posed by Lewis (1991); the latter is taken up at greater length by McGee (2000) and Williamson (2003). See Chapter 5.

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 everything, more or less Both objections seem to me to make a clear and reasonable demand of the relativist. Her best response is to tackle both challenges head on by giving a clear, wellmotivated account of relativism. Different relativists however can be expected to go different ways. Opposition to absolutism admits of considerable variation, along several dimensions. And it will be helpful to have a sense of the lay of the land before we take up these issues in more detail later on. This section and Section 1.6 take up the two objections in turn and briefly survey some of the main relativist lines of response, contrasting the species of relativism I defend with some of its main rivals. Let’s begin with the explanatory burden facing relativism. Even if we seldom have call for absolute generality outside of metaphysics, logic, and set theory, why doubt that it is available? What’s to stop us from quantifying over an absolutely comprehensive domain simply by dropping any restrictions applied to ‘everything’? Unless she can give a compelling answer to these questions, the relativist’s ban on absolutely general quantification would seem to be unacceptably mysterious.36 Several answers have been explored. We set aside sortal restriction in Section 1.2. But even if we assume, as we henceforth do, that universal nominals such as ‘thing’ or ‘item’ are available, at least one other source of quantifier domain restriction is commonly recognized: quantifiers not subject to a restriction explicit in their syntax may still be subject to restrictions supplied by their context of utterance. What has been said, what is salient in our surroundings, and so on, often restricts the range of ‘everything’ to things that are contextually relevant. As we noted in Section 1.1, however, the widespread operation of quantifier domain restriction is perfectly compatible with absolutism provided that some contexts permit ‘everything’ to range over an absolutely comprehensive domain. Why can’t we obtain such a context simply by placing no non-vacuous restriction on the quantifier? One way to address this challenge is to deploy a sophisticated relativist-friendly account of quantifier domain restriction due to Michael Glanzberg (2006). In his view, quantifiers exhibit two sorts of context sensitivity: the context supplies both a background domain associated with the determiner and a local contextual restriction attaching to the nominal. On this view, we are free to operate in contexts where no non-vacuous local restriction comes into play. Such locally unrestricted quantifiers may then range over the entire background domain. Nonetheless, Glanzberg does not accept that such locally unrestricted quantification achieves absolute generality. Suppose we begin with a background domain D . Glanzberg develops a pragmatic account of how we may exploit Russell’s paradox to shift to a wider background domain containing D ’s Russell set. This provides one way to make sense of the performative aspect of the Russell Reductio noted in Section 1.4. Kit Fine (2005, 2006, 2007) offers a rather different account of the kind of domain-shift he takes to lie behind indefinite extensibility. In Fine’s terminology, Glanzberg’s view may be reasonably described as a version of restrictionism:37 a 36

See, for instance, Lewis (1991, p. 68). Glanzberg does not self-apply this label. On the contrary, he notes that his view seems to have an expansionist character (2006, n. 5). Although he also concedes that it’s ‘fair enough’ to describe background domain relativity as a ‘kind of contextual restriction’ (2006, n. 4). There’s a perfectly good sense of ‘restriction’ and ‘expansion’ on which these claims are not in tension. (See the discussion of 37

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the objection from mysteriousness  quantifier’s domain is always subject to restriction (specifically, on Glanzberg’s account, a quantifier invariably ranges over members of a contextually determined, non-absolutely-comprehensive background domain). Fine eschews restrictionism in favour of what he calls expansionism. This view allows for ‘unrestricted quantification’ in a more full-blooded sense than on Glanzberg’s account. On Fine’s view, quantifiers may be wholly free from restrictions, of any kind, supplied by the context of utterance. Nonetheless, according to expansionism, the domain of such a wholly unrestricted quantifier fails to be absolutely comprehensive. This is because it is open to expansion: we can come to a wider domain by introducing ‘new’ objects into the initial one. Of course, the mere introduction of expansionism does not dispel the objection from mysteriousness. Indeed the charge may seem all the more pressing: how can we expand the domain of an unrestricted quantifier? Not by a shift of context if the quantifier was already free from contextual restriction. Nor, surely, is the sort of domain-shift at issue in the Russell Reductio a question of literally bringing new items into being, by a shift in the circumstances.38 But at least two further options are available. On my preferred account, the kind of domain-shift that lies behind indefinite extensibility is due to a shift in the interpretation of quantifiers. Semantic change, of course, is commonplace. A predicate’s extension may change as its use evolves. For example, it’s not implausible to think that ‘hoover’ once applied only to a brand of domestic appliance rather than a kind of domestic appliance. If the predicate has shifted its extension in this way, this need not show that it is indexical or otherwise context-sensitive. More plausibly, such a shift is due to our coming to use the expression in a more inclusive way, and thereby liberalizing the interpretation of our lexicon to attach a wider extension to the predicate. Similarly, in the case of quantifiers, the interpretational expansionist may seek to de-mystify domain expansion by assimilating it to more familiar cases of semantic change. Fine defends a different version of expansionism, which he calls procedural postulationism. Again, domain expansion is not achieved by changing the circumstances. But nor, on his view, is it effected by changing the content of our expressions, either through a shift of context or through semantic change. Instead, he suggests that there is a third parameter, ‘the ontology’, which is ‘intermediate, as it were, between a change in content and a change in circumstance, as these are normally conceived’ (2006, p. 40). Of course, in this case, as in the others, much more needs to be said if the relativist is to dispense with the charge of mysteriousness. We return to this issue in Chapter 4, which also assesses the significance of the restrictionism–expansionism divide more generally. Sometimes the difference is unimportant, and we may continue to cluster restrictionist and expansionist accounts together under the generic heading of ‘relativism about quantifiers’. But on other occasions the difference matters. In particular, crypto-restrictionism in Section 4.4.) On the elucidation of the restrictionist–expansionist divide given in Section 4.1, however, Glanzberg’s view counts as restrictionist. (See Section 4.2.) The importance of the distinction, so drawn, is defended in Chapter 4, and it remains whatever terminology we may choose to employ. 38

See Section 2.4 for further discussion.

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 everything, more or less Williamson (2003) objects that the relativist is (i) unable to adequately capture the semantics she ascribes to quantifiers and (ii) unable to express harmless-seeming kind generalizations such as ‘No donkey talks’. Chapter 4 argues that these wouldbe general objections against relativism are ineffective against its more promising variants. We return to fill in some of the metasemantic details of my preferred account of domain expansion in Chapter 8.

1.6 The objection from ineffability While the objection from mysteriousness challenges the relativist’s ability to motivate her view, the objection from ineffability goes one step further and questions whether she has a view to motivate. The difficulty that the relativist faces in giving a satisfactory statement of relativism becomes apparent as soon as she tries. Suppose, for instance, that she attempts to capture her opposition to the absolutist’s contention that there is an absolutely comprehensive domain with the following utterance: (11) No quantifier’s domain comprises everything. On her view, the quantifier ‘everything’ in (11) ranges over a limited domain (in its context of utterance). Consequently, in uttering (11), she says, in effect, that we are unable to quantify over every member of this limited domain. But this, of course, is not her position. The relativist has no grounds to deny the availability of quantification over everything in a suitably limited domain. Still less has she grounds to deny the availability of quantification over the limited domain she just quantified over. The absolutist can offer an easy explanation of the difficulty: What the relativist would like to say, of course, is that no domain contains absolutely everything (understood my way). But to state her view that way would be to use exactly the kind of quantification she seeks to ban.

More generally, difficulties of this nature sometimes lead to the charge that relativism is self-defeating: the view cannot be coherently maintained.39 The objection from ineffability should not be dismissed lightly. Even if the relativist is confident that absolutism fails to fill up logical space, leaving room for her view to occupy the gaps, she cannot hope to give a rigorous argument for relativism unless she can frame its conclusion. Shaughan Lavine (2006) suggests a schematic characterization of relativism. Relativists, especially when motivated by indefinite extensibility, often dispense with an absolutely comprehensive domain in favour of an open-ended sequence of ever-wider ones: D0 , D1 , D2 , . . .. Given this picture, the relativist has no difficulty in coherently denying the comprehensiveness of, say, D1 : she can straightforwardly achieve this by uttering ‘D1 does not comprise everything’ with her quantifier ‘everything’ ranging over D2 . In order to capture her view, therefore, relativists of this stripe can seek to deploy a schema whose instances state that D0 does not comprise everything in D1 , that D1 does not comprise everything in D2 , and so on. Relativists tempted by this style 39

See, for instance, McGee (2000); we return to his version of the objection in Section 5.1.

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the objection from ineffability



of formulation may claim that relativism is comparable to theories such as first-order Peano Arithmetic which admit of infinite axiomatizations, using schemas, but cannot be finitely axiomatized. Lavine further claims that schemas provide a relativist-friendly proxy for absolutely general quantification. For example, what the absolutist claims to achieve with an absolutely general quantifier in an utterance of ‘Absolutely everything is self-identical’, the relativist may claim to achieve with a schema whose instances state that everything in D0 is self-identical, that everything in D1 is self-identical, and so on.40 Parsons (1977, 2006) and Glanzberg (2004) make a similar move, taking some quantifiers to be ‘systematically ambiguous’.41 The use of the intensifier ‘absolutely’ here requires brief comment. In ordinary English, the intensified quantifier ‘absolutely everything’ is often subject to the same sort of contextual restriction as its plain counterpart. But it’s useful to have a term to flag when absolute generality is intended. Were an absolutist speaking, he might make the following stipulation: The quantifier ‘absolutely everything’ is henceforth to be interpreted as expressing absolutely general quantification.

By the relativist’s lights, however, such stipulations inevitably misfire: ‘absolutely everything’ quantifies over a less-than-absolutely-comprehensive domain. She may instead treat sentences containing ‘absolutely everything’ as an invitation (where possible) to supply a suitable schema or some other relativist-friendly paraphrase. Similar remarks apply to ‘absolutely comprehensive’, and so on.42 An alternative means for relativists to simulate absolutely general quantification is proposed by Fine (2006). As a relativist about quantifiers, he opposes the absolutist’s contention that we can quantify over absolutely everything. At the same time, he claims, in effect, that—so to speak—we can generalize about absolutely everything with the help of a suitably-interpreted modal operator. On this view, even though the quantifier ∀x does not achieve absolute generality, the operator–quantifier string, or ‘modalized quantifier’, ∀x does. Related views have been suggested in the context of set-theoretic quantifiers by Parsons (1977, 1983) and Linnebo (2010, 2013).43 This use of operator–quantifier strings to generalize about items outside the current domain of ∀x is analogous to the use of ∀x, with  interpreted as metaphysical necessity, to generalize about possible items outside the domain of the actual world. But Fine (2006) is clear that the intended interpretation of the relativist’s modal operator is not a ‘circumstantial’ modality such as metaphysical or physical modality.44 His elucidation of the relevant modality is coupled to his preferred third-parameter version of relativism, but advocates of contextual and interpretational versions of relativism may instead appeal to contextual or interpretational modality. To a very first approximation, the relevant necessity operator might be schematically glossed: 40

41 See Lavine (2006, sec. 5.9). See Section 5.4 for further discussion. See Sections 5.4 and 6.2. 43 See Fine (2006, p. 41). We borrow the term ‘modalized quantifier’ from Linnebo (2010). See Section 6.1 for further discussion. 44 See Fine (2006, p. 33). 42

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 everything, more or less ‘no matter how the domain is admissibly extended’. If available, the modality provides a much fuller means for the relativist to mimic absolutely general quantification. In particular, Fine outlines how the relativist may use these resources to express her view in a single modal sentence.45 We will have much more to say about the objection from ineffability and the relativist use of schemas and modal operators respectively in Chapters 5 and 6. But for now let us confine ourselves to noting that the question of whether modalized quantifiers achieve absolute generality can be expected to lead to a schism within relativism. A relativist about quantifiers who gives a positive answer to this question might naturally be described as adopting a hybrid view: she adopts relativism about quantifiers but absolutism about modalized quantifiers. Not all relativists can be expected to follow suit. An advocate of a more thorough-going version of relativism is unprepared to accept even this much absolutism; instead she adopts relativism both about quantifiers and modalized quantifiers (although, if she uses schemas to state her view, she may yet need to concede that schemas do achieve some form of absolute generality).46 We do better, then, not to dichotomize the absolute generality debate as a straight choice between two views about quantifiers.47 Even if this has been the main flashpoint in recent discussions, it is not the only place where the absolutism–relativism line may be drawn. As we will see in Chapters 6 and 7, absolutism about modalized quantifiers differs in some substantive ways from the corresponding view about quantifiers (which helps the proponent of the hybrid view to deflect the charge that her view collapses into absolutism about quantifiers). Nonetheless, the two absolutist views also share some important similarities. And some of the most important considerations that come into play in the absolutism–relativism debate about quantifiers have analogues in the corresponding debate about modalized quantifiers.48 My view here is somewhat tentative since the latter debate has yet to play out in full. The defence of relativism offered here, however, is ultimately a defence of a thorough-going version of relativism. The hybrid view has much to recommend it, but it comes with problems of its own. Most worryingly, absolutism about modalized quantifiers falls victim to a version of the argument from indefinite extensibility. And—for the present, at least—no satisfactory means has been forthcoming to defend absolutism about modalized quantifiers from what I take to be the driving motivation for adopting relativism about quantifiers in the first place.

45

See Fine (2006, p. 30). See Section 6.3. When it’s clear what species of absolutism or relativism is in play, we continue to use the terms ‘absolutism’ and ‘relativism’ without further qualification. 47 Thanks to Øystein Linnebo for emphasizing this point to me. 48 See Section 7.5. 46

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2 Russell, Zermelo, and Dummett The thought that excessive generality is somehow to blame for Russell’s paradox is an old one. Not long after his discovery of Russell’s paradox, Russell defended a version of this thought in Principia Mathematica. The thought, in a rather different form, appears to surface in Zermelo’s attempt to resolve the paradoxes by exploiting the open-ended nature of the set-theoretic hierarchy. And it is taken up once again in Michael Dummett’s highly influential account of indefinite extensibility. When properly developed, arguments drawing on the set-theoretic paradoxes offer what, in my view, are by far and away the strongest considerations against the characteristic claim of the absolutist about quantifiers that we sometimes quantify over an absolutely comprehensive domain. But not every paradox-inspired argument is a good one, or indeed an argument for relativism about quantifiers, as this view is understood here. The aim of this chapter is to fill in the logical and philosophical background to the contemporary absolute generality debate, with an eye to disentangling my favoured indefinite-extensibility-based argument from others in its vicinity (especially those based on Russell’s vicious-circle principle or what Richard Cartwright dubs the ‘All-in-One’ principle). The first two sections assess two of Russell’s reactions to the paradoxes that have been taken to have particular relevance to the absolute generality debate: Section 2.1 takes up his diagnosis of the paradoxes based on ‘self-reproductive processes and classes’; Section 2.2 examines his solution based on the vicious-circle principle and ramified type theory. The next two sections focus on Zermelo’s contribution: Section 2.3 outlines the familiar means of avoiding Russell’s paradox in Zermelo–Fraenkel set theory; Section 2.4 offers a relativist-friendly interpretation of Zermelo’s account of the cumulative hierarchy of sets. Finally, Section 2.5 offers a preliminary assessment of one of Dummett’s formulations of the argument from indefinite extensibility. This discussion highlights some of the difficulties in framing an effective argument for relativism which inform our development of relativist-friendly expressive resources in later chapters. With these resources in hand, we return to give a refined regimentation of the argument from indefinite extensibility in Chapter 7. It bears emphasis at the outset that my aim in this chapter is not primarily exegetical (let alone does this chapter come anywhere close to a comprehensive history of the absolute generality debate). The discussion is structured around some milestone twentieth century works in logic and philosophy; but we shall be much more interested in whether effective arguments for relativism can be extracted from them than in giving any final answer as to whether our arguments offer a faithful reconstruction of the author’s views. Our regimentations, moreover, make free use of the tools of contemporary logic whenever it is helpful to do so.

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 everything, more or less

2.1 Self-reproductive processes and classes The aftermath of the discovery of Russell’s paradox was an intellectually turbulent time for Russell. He vacillated between a number of potential solutions before converging on the ramified type theory espoused in Principia Mathematica.1 This section focuses on one of his earlier responses to the paradoxes which Dummett cites as an expression of the idea of indefinite extensibility.2 In a paper read to the London Mathematical Society late in 1905,3 Russell blames Russell’s paradox, together with Burali-Forti’s paradox and Cantor’s paradox, among others, on ‘self-reproductive processes and classes’: . . . the contradictions result from the fact that, according to current logical assumptions, there are what we may call self-reproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question. Hence we can never collect all the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property. (p. 36)

The parallel between the properties associated with Russell’s ‘self-reproductive processes and classes’ and Dummett’s indefinitely extensible concepts is clear. According to our first-pass approximation of the latter offered in Section 1.4, no domain comprises absolutely every item that falls under an indefinitely extensible concept; similarly, according to Russell, no collection comprises every item that has the relevant kind of property. But do self-reproductive processes and classes, as Russell conceives them, give us grounds for relativism about quantifiers? Before we come to answer this question, let’s first examine this early diagnosis of the paradoxes more closely.

A template for paradox In his 1905 talk, Russell takes the paradoxes to conform to a common template. Each antinomy-prone property φ (e.g. ordinal, non-self-membered class) is associated with an operation f , generating ‘new’ items with the property—let’s call this a new-φ operation. The new-φ operation meets the condition that, whenever f is defined on a class u of φs, f (u) is a φ which is not a member of u. A contradiction then results from assuming both (i) there is a class w comprising every φ and (ii) the new-φ operation f is defined on every class.4 For example, we may apply Russell’s template to Burali-Forti’s and Russell’s paradox as follows: bu r a l i - f o rt i ’ s pa r a d ox Take φ to be ordinal and, when u is a class of ordinals, define f1 (u) to be the ordinal that measures the length of the sequence of ordinals less than or equal to a member of

1

Russell and Whitehead (1910). See Urquhart (1988) for a helpful survey. Dummett (1991, p. 317, n. 5). 3 Unless indicated otherwise, page references in this section are to Russell (1906), the published version of his talk. Here and throughout we modernize the notation. 4 Proof: given (ii), f (w) is a φ which is not a member of w, contradicting (i). 2

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self-reproductive processes and classes  u under their usual order (or to remain undefined if there is no such ordinal). Given some standard assumptions about ordinals, f1 meets the condition to be a new-ordinal operation.5 As per Russell’s template, a contradiction results from two assumptions: (i) there is a class w1 comprising every ordinal and (ii) the new-ordinal operation f1 is defined on each class.6 ru s s e l l ’ s pa r a d ox Take φ to be non-self-membered class and, when u is a class of non-self-membered classes, define f2 (u) to be u. Then f2 is a new-non-self-membered-class operation.7 As per Russell’s template, a contradiction results from two assumptions: (i) there is a class w2 comprising every non-self-membered class and (ii) the new-non-self-memberedclass operation f2 is defined on each class. In fact, as Russell observes, since (ii) is trivially met in this case, assumption (i), on its own, leads to a contradiction.8 The common pattern Russell finds in the paradoxes displays a tension between, as he puts it in the passage quoted above, (i) ‘collecting’ φs into a class and (ii) ‘generating’ a new φ from a given class of φs. The process metaphor apparent in Russell’s talk of ‘generating’ and ‘collecting’ risks promoting the misconception that his diagnosis relies on a heterodox, creationist interpretation of set theory where sets are brought into being by acts of collection. But as our reconstruction of Russell’s template shows, the veneer of ‘process’-talk is readily stripped away from the two paradox-inducing assumptions, (i) and (ii), which Russell identifies. In the case of Russell’s paradox, for instance, the contradiction arises simply from the assumption that there is a class of such-and-such a kind.

Russell’s Lesson The immediate lesson Russell draws from the set-theoretic paradoxes he considers is that, at least in some cases,9 we have no choice but—so to speak—to limit ‘collecting’. Russell states this lesson in terms of propositional functions: Russell’s Lesson. ‘A propositional function of one variable does not always determine a class’ (p. 37). What this lesson amounts to depends on how we interpret Russell’s ‘propositional function’-talk. Unfortunately, his use of this term poses some notorious exegetical difficulties and is understood in different ways by different commentators: some 5 Proof sketch: we assume a background theory of ordinals that proves the following: (a) the ordinals are well-ordered by their standard order (symbolized: ψ

Moreover, Stability renders valid what we shall call the Stability Axiom for ß and ∈:21 (sta-ß) (sta-∈)

∀v(◇ ßv → ßv) ∀u∀v(◇ u ∈ v → u ∈ v)

The modal logic mfo then has as theorems the axioms of the modal system s for the defined modal operator ; similarly, for each of ≥ and ≤ , we may derive the s. axioms. This completes our model-theoretic sketch of mfo. It would seem to give us a much fuller account of the modality than provided by the approximate English glosses of > and < . But it also raises an obvious question. The Kripke-style clauses seek to give truth-conditions for the modal object language in a non-modal metalanguage. Within an MT-hierarchy, the relativist’s supposedly non-quantificational means to generalize is assimilated to quantification over MT-interpretations. What, then, is the relativist to make of this account? The issue has a clear parallel in the more familiar use of Kripkean model theory to investigate languages with modal operators, such as those intended to express metaphysical necessity and possibility. Not everyone thinks that metaphysical modality is semantically reducible to quantification over possible worlds. What are advocates of irreducibly modal operators to make of the Kripkean clauses? The question is best answered by drawing a clear distinction between modeltheory and (model-theoretic) semantics. If the relevant modality is irreducibly modal, no Kripke-structure captures the language’s intended interpretation. But for many applications of model theory, this is unimportant. Indeed, we’ve already made an analogous point on the absolutist’s behalf in Section 3.6. The lack of an intended MT-interpretation for the first-order language Lsu is no 20 In the context of provability logic, Boolos observes the correlation between the löb axiom—in the equivalent form: < (< ψ → ψ) → < ψ—and the transitivity and converse well-foundedness of the accessibility relation (1995, pp. 75–6). Thanks to Gabriel Uzquiano for drawing my attention to the possibility of a backwards-looking version of this principle as a means to express well-foundedness. 21 We borrow the label ‘stability’ from Linnebo (2013, p. 211) who makes a similar assumption in his unimodal setting. Compare his axioms Stb+ -φ and Stb− -φ. The difference in formulation reflects his use of ‘forwards-looking’ modal operators and classical first-order logic. See also Parsons’ discussion of the ‘existence-dependent rigidity’ of membership (1983, pp. 299–300). The axiom sta in Studd (2013) offers a variant formulation of the Stability Axiom: replacing the mpfo Stability Axiom with this variant yields the same set of mpfo-theorems.

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modalization: first-order theories  bar to our giving the standard model-theoretic account of the logic governing this language. For that application, and indeed all the usual applications of model theory within mathematics, all we require of MT-interpretations is that they be set-structures of the relevant kind. The same goes in the modal case. If we’re simply putting Kripke-structures or MT-hierarchies to the usual sorts of application in model theory—to give a modeltheoretic account of validity, say, or to show that a given set of modal formulas is consistent in a given modal logic—it simply doesn’t matter whether any Kripke-structure captures the intended interpretation of the modal language. Calling the indices in the structure ‘possible worlds’ or ‘admissible interpretations’ may be heuristically useful but it’s model-theoretically irrelevant.22 This observation is enough to secure a minimal instrumental role for MThierarchies: the relativist is free to put them to all of the usual model-theoretic applications. And indeed, for the purposes of reasoning in her modal language, she can eschew the model theory altogether and proceed proof-theoretically. But this still leaves unanswered the question of how she is to understand her modal operators. Granted that their intended interpretation cannot be specified via the Kripke-clauses for an intended MT-hierarchy, how is it to be specified? We return to this question in Section 6.5. It turns out that there’s room for the relativist to accord Kripke-style clauses a more than purely heuristic and instrumental role. By the relativist’s lights, there’s a precise sense in which the structure of an MThierarchy, as described in this section, illustrates in miniature—so to speak—features reflected in the entire potential hierarchy. But before we come to that, we consider some applications of relativist-friendly modal operators.

6.2 Modalization: first-order theories Let us assume that we have the interpretational modal operators > and < at our disposal. The next three sections outline four applications of suitably interpreted modal operators: (i) they provide a systematic way for the relativist to interpret firstorder theories with absolute generality; (ii) they permit us to regiment indefinite extensibility; (iii) they allow both sides of the absolutism–relativism debate to frame the issue between them in a neutral way; (iv) they permit us to give a relativist-friendly motivation for standard set theory. This section discusses application (i). Applications (ii) and (iii), which call for a plural extension of the modal language, are the subject of Section 6.3. Finally, Section 6.4 takes up application (iv). For concreteness, we continue to focus on interpretational modality (subject to the logic mfo outlined in Section 6.1). But relativists who favour a more mathematical or metaphysical modality may seek to put their modal operators to similar use.

Modalization As we saw in Section 6.1, suitably interpreted modal operators promise to provide a relativist-friendly means to attain absolute generality. More generally, the 22

Compare, for instance, Lewis (1986, sec. 1.2).

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 everything, more or less modal operators offer the relativist a systematic means to ‘translate’ non-modal formulas that strive for—but, by her lights, fail to attain—absolute generality. We adapt a very natural translation scheme due to Linnebo.23 Given a non-modal Lsu -formula φ, its modalization (symbolized: [φ]◇ ) is obtained by replacing each universal quantifier ∀v in φ with the string ∀v, each existential quantifier ∃v in φ with ◇ ∃v, and each atomic subformula  in φ with ◇ . To improve readability we sometimes write φ ◇ , φ ◇ (x) ß◇ v and u ∈◇ v for [φ]◇ , [φ(x)]◇ [ßv]◇ and [u ∈ v]◇ , respectively, adapting this convention to other formulas in the obvious way.24 For example, in Zermelo–Fraenkel set theory, the Foundation Axiom serves to rule out infinite descending ∈-chains: a1 a2 a3 · · · . In the pure first-order theory, zfc,25 this axiom is standardly formulated in the theory’s non-modal language:26 Foundation. Every non-empty set has a ∈-minimal element (i.e. an element none of its elements belong to). ∀s(∃x(x ∈ s) → ∃x(x ∈ s ∧ ∀w(w ∈ s → ¬w ∈ x))) By the relativist’s lights, the axiom’s quantifiers can only be interpreted to range over a less-than-absolutely-comprehensive domain. She may however maintain that an absolutely general formulation can be obtained by modalizing the axiom: ∀s(◇ ∃x(◇ x ∈ s) → ◇ ∃x(◇ x ∈ s ∧ ∀w(◇ w ∈ s → ¬ ◇ w ∈ x))) Modalization provides the relativist with an alternative way to make sense of the term ‘absolutely’ to the schematic one outlined in Section 5.4. She may take the intensifier ‘absolutely’ as a prompt to give a modal reading: ‘absolutely everything’ may be formalized with the string ∀v. We shall refer to this string as a modalized quantifier.27 To get a sense of the effect of modalization, it’s helpful to return to the Kripkean model theory. Consider a relativist-friendly MT-hierarchy in which each attempted expansion results in a strictly wider universe (i.e. Mi ⊂ Mj whenever i < j). The absolutist may wonder why we cannot ‘flatten’ the hierarchy to obtain an MTinterpretation which is at least as liberal as any in the hierarchy. Let the flattening of an MT-hierarchy be the MT-interpretation MI , SI , EI  whose universe collects together the items from any universe in the hierarchy, and whose two extensions respectively merge together each of the relevant extensions in the hierarchy:

23 See Linnebo (2010, pp. 115–16, 2013, p. 213). On his definition for forwards-looking operators, no ◇ is added to atomic formulas. Putnam (1967) and Hellman (1989, pp. 73–9) suggest attributing modal content to set-theoretic quantifiers in a similar way. Competing translations are offered by Parsons (1983) and Fine (2006). 24 Officially, the definition of modalization for a L -formula has no separate clause for ∃v, since this su quantifier is taken to abbreviate ¬∀v¬. However, since the unofficial version is easier to read, we’ll often let it stand in for its logically equivalent official counterpart in contexts where the additional double negations it carries don’t matter. 25 For zfc, see Section 2.3 and Appendix C.1. 26 In the context of pure zfc, ‘set’ means pure set. 27 We take the terminology from Linnebo (2010, p. 162).

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modalization: first-order theories  MI =df



Mi

i∈I

SI =df



Si

i∈I

EI =df



Ei

i∈I

The would-be maximal interpretation MI , SI , EI  is not an admissible interpretation by the lights of the relativist-friendly MT-hierarchy. Nonetheless, whatever the absolutist would achieve with a non-modal Lsu -formula interpreted by MI , SI , EI , the relativist may achieve with its modalization: φ ◇ . As interpreted by any nonmaximal MT-interpretation i in the relativist-friendly MT-hierarchy, a modalized quantifier ∀v in a modalized formula has an analogous effect to (ordinary) quantification over the would-be maximal universe MI : (ki -∀v) ∀v[φ]◇ is truei,σ iff every item a in MI is such that φ ◇ is truei,σ [v/a] . Similarly, satisfying a modalized atomic formula relative to (non-maximal) i is equivalent to satisfying its non-modal counterpart with its predicates interpreted by the would-be maximal extensions SI and EI : (ki -ß◇ ) ß◇ v is truei,σ iff σ (v) ∈ SI . (ki -∈◇ ) u ∈◇ v is truei,σ iff σ (u), σ (v) ∈ EI . Consequently, a Lsu -formula φ is true under the would-be maximal MTinterpretation MI , SI , EI  if and only if its modalization φ ◇ is true relative to any non-maximal interpretation in the relativist-friendly MT-hierarchy. As before, the model-theoretic motivation needs to be taken with a pinch of salt. The absolutist will not of course take the set-universe of the MT-interpretation obtained by flattening an MT-hierarchy to be absolutely comprehensive. And the relativist has no reason, in general, to deem such an interpretation to be genuinely inadmissible. All the same, the relativist may further claim that what the relativistfriendly MT-hierarchy illustrates in miniature is reflected in the entire potential hierarchy. On this view, under each admissible interpretation, a modalized first-order formula has the same effect that its non-modal counterpart would if—per impossibile—it were to be interpreted over the absolutely comprehensive interpretation obtained by flattening the entire potential hierarchy.28 In particular, to return to our earlier example, the relativist may claim that the intended generality of the Foundation Axiom is captured by its modalization. The modalized axiom not only rules out selfmembered sets in absolutely every domain, but further rules out infinite descending ∈-chains spread across the potential hierarchy.

Invariance and Mirroring The relativist’s proposed means to achieve absolute generality in set theory may prompt an immediate objection: theories such as zfc and its impure cousin zfcsu are manifestly not formulated in a modal language.29 To read tacit modal content into the non-modal language of set theory with urelements, Lsu , would run contrary to 28 The de-miniaturized thesis is captured, without reliance on Kripke semantics, in the Flattening result stated in Section 6.3. 29 For zfc and zfcsu, see Sections 2.3–2.4 and Appendix C.1.

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 everything, more or less mathematical practice in at least two ways: (i) if set theory really concerns a potential hierarchy of interpretations, shouldn’t we expect set-theoretic investigations to begin by ascertaining which interpretation we’ve reached? (ii) if set theory is tacitly modal, shouldn’t we expect set theorists to work in an appropriate modal logic? A positive answer to either question would seem to be mathematically revisionary. The relativist, however, has a convincing response to these worries. In both cases she has good reason to give a negative answer. Against (i): modalized formulas are insensitive to which stage in the potential hierarchy the present version of our language happens to correspond with. Let us say that a Lmsu -formula is invariant if, relative to any assignment of values to its free variables, its truth-value remains the same under each admissible interpretation, always true or always false. The invariance of a formula ψ (whose free variables are v = v1 , . . . , vk ) may be regimented in Lmsu as follows:30  inv[ψ] =df ∀v(ψ ∨ ¬ψ) To respond to the first worry we may then appeal to the following result:31 Modalized Invariance. The modal logic mfo proves that each Lsu -formula’s modalization φ ◇ is invariant. mfo inv[φ ◇ ] So long as a set theorist is interested in set theories formulated in a non-modal language, which stage we have reached is unimportant. For the tacit modal content attributed to her sentences by reading them as their modalizations renders them stageinvariant.32 Against (ii): modalization preserves first-order logical relations. We may appeal to the following result:33 Mirroring FOL. A Lsu -formula φ follows from a set of zero or more Lsu -formulas in classical first-order logic if and only if its modalization follows from the set of modalized formulas in the modal logic. fol φ iff {γ ◇ : γ ∈ } mfo φ ◇ The upshot of this result is that mainstream mathematical practice is neutral between a face-value interpretation and a modalized one. So long as the set theorist is simply concerned with proving theorems in a first-order set theory, the difference Notation: for v = v1 , . . . , vk , ∀v abbreviates the string of quantifiers ∀v1 . . . ∀vk . See Appendix B.1 for a proof. Compare Linnebo (2013, lem. 5.3). 32 Compare Linnebo (2010, pp. 158–9). Note that the result does not extend in general to arbitrary Lmsu -formulas, which may fail to be invariant. Consequently in addition to the invariant formulas modalizing those in the set-theorist’s non-modal language Lsu , the modal language also contains further non-invariant formulas. In light of this, Linnebo suggests that the modal language affords a ‘finer resolution’ that helps with difficult foundational problems (2013, pp. 206, 214). In the bimodal case, however, there is also a sense in which all formulas can be treated as modalized formulas. See the Kripke Normal Form Theorem in Section 6.5. 33 See Appendix B.2 for a proof. Compare Linnebo (2013, thm. 5.4), from whom we take the label ‘mirroring’. 30 31

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modalization: plural theories  between the two interpretations fails to surface. Exactly the same proofs are in good standing on both interpretations. Reading tacit modal content into her quantifiers consequently calls for no revision to her practice.34

6.3 Modalization: plural theories We’ve so far considered modalization against the background of a first-order language. But the differences between the absolutist and relativist come out most clearly when we also allow for plural resources. This section considers the use of a modal plural language to illuminate indefinite extensibility and to give a neutral characterization of what’s at issue between absolutists and relativists about quantifiers. But first we need to extend the modal language with the relevant plural resources.

Adding plural resources The Kripke semantics from Section 6.1 is straightforwardly extended to the modal plural language of set theory with urelements—Lmpsu —which adds > and < to the plural language of set theory with urelements Lpsu . Given an MT-hierarchy {Mi , Si , Ei  : i ∈ I }, an assignment over it for the plural language assigns a member of some Mi (with i ∈ I ) to each singular variable and a subset of some Mi (with i ∈ I ) to each plural variable. Relative to such an assignment σ , the Kripkean clauses governing truthi,σ are as before, with the addition of two clauses governing the language’s plural expressions:35 (ki -≺) u ≺ vv is truei,σ iff σ (u), σ (vv) ∈ Pi . (ki -∀vv) ∀vvψ is truei,σ iff every set A with A ⊆ Mi is such that ψ is truei,σ [vv/A] . In the first clause, Pi is the intended extension for the member–plurality predicate based on Mi : Pi =df {a, B ∈ Mi × P(Mi ) : a ∈ B} The semantics sustains a plural modal logic: mpfo.36 The system mpfo enriches mfo with plural analogues of the axioms and rules for singular quantifiers (guarded with a defined plural existence predicate E!uu, whose definition we shall come to shortly). The modal plural logic also adds all instances of the Plural Comprehension Schema:37 Plural Comprehension Schema. Given a condition ψ(x), there are zero or more items that comprise every item that satisfies ψ(x). ∃xx∀x(x ≺ xx ↔ ψ(x)) Modalization is extended to the plural language by treating plural quantifiers in the same way as singular ones: the modalization of a Lpsu -formula prefixes  to each 34 35 36 37

Compare Linnebo (2010, p. 156). As usual, σ [vv/A] assigns A to vv, and otherwise assigns the same values to variables as σ does. See Appendix A.3. Side-condition: ψ(x) is a Lmpsu -formula that lacks xx free.

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 everything, more or less universal quantifier, singular or plural, and ◇ to each existential quantifier and atomic formula.38 Much as in the singular case, the relativist may take the string ∀vv to provide a relativist-friendly means to achieve absolute generality. To speak doubly loosely: the quantifier ranges not merely over every plurality of zero or more items in a given domain but over absolutely every plurality of zero or more items in absolutely any domain. In the case of plural quantifiers, however, there’s an important difference between what the absolutist takes himself to achieve with plural quantification over an absolutely comprehensive domain and the effect of modalized plural quantifiers. We can bring out the difference in miniature by returning to the relativist-friendly MT-hierarchy, with strictly increasing domains, and its absolutist-friendly flattening from Section 6.2. Interpreted by the would-be maximal flattening MI , SI , EI , ∀vv quantifies over every plurality of items in MI ; equivalently, over every plurality of zero or more items individually belonging to some universe or other in the MT-hierarchy. In contrast, the derived Kripke-clause for modalized plural quantifiers effectively takes ∀vv to have a stratified character: it ranges over every plurality of zero or more items that collectively belong to a universe in the MT-hierarchy: (ki -∀vv) ∀vv[φ]◇ is truei,σ iff every set A with A ⊆ Mi for some i ∈ I is such that φ ◇ is truei,σ [vv/A] . Consequently, the absolutist takes ∀vv to range over pluralities outside the range of ∀vv. Only the former ranges over pluralities of items that are spread out across the MT-hierarchy in such a way that no one Mi in the MT-hierarchy contains each of them. The difference in range sometimes manifests itself in a difference in truth-value. Unlike a Lsu -formula, the truth-value of a Lpsu -formula in the would-be maximal MI , SI , EI  need not coincide with the truth-value of its modalization under an interpretation in the relativist-friendly MT-hierarchy. Some instances of Plural Comprehension provide one key example of this mismatch. Each instance of Plural Comprehension is true under MI , SI , EI , and indeed under any other MT-interpretation. But the same is not true of its modalization:39 Modalized Plural Comprehension. Given a modalized condition φ ◇ (x), there are eventually zero or more items that comprise absolutely every item that satisfies φ ◇ (x). ◇ ∃xx∀x(x ≺◇ xx ↔ φ ◇ (x)) Not every instance of Modalized Plural Comprehension is true under the MT-interpretations in the relativist-friendly MT-hierarchy. Fortunately, unlike its non-modal counterpart, Modalized Plural Comprehension has no claim to be a trivial truth. To see this, it’s helpful to briefly switch back 38 Officially, the definition of modalization only has clauses for the primitive expressions of L psu . See n. 24. 39 Side-condition: φ(x) is a L psu -formula that lacks xx free.

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modalization: plural theories  to the (unintended) interpretation where  expresses metaphysical necessity. So understood, Modalized Plural Comprehension tells us that it’s metaphysically possible for there to be zero or more items that comprise every possible satisfier of φ ◇ (x). But not all such instances are obviously true. For example, is it metaphysically possible for there to be zero or more items that comprise every possible donkey? Not if, as seems quite plausible, it’s necessarily the case that, for any one or more possible donkeys inhabiting a given world, there could be a further possible donkey (the progeny of two of them say) which is not among them. Returning to the intended interpretation of , under the relativist’s noncircumstantial interpretation, an instance of Modalized Plural Comprehension may fail to express a truth for similar reasons.40 In fact, the modal language Lmpsu permits us to frame a natural necessary and sufficient condition for a modalized condition φ ◇ (x) to be ‘plurally-comprehensible’ (as per Modalized Plural Comprehension). This kind of comprehensibility is closely bound up with indefinite extensibility, in something much like Dummett’s sense. Let’s begin by giving a modal regimentation of the latter notion.

Indefinite extensibility In Section 1.4, we offered a provisional gloss of indefinite extensibility: a concept F is indefinitely extensible if given any domain comprising Fs, however extensive, a further F can always be specified, giving rise to a wider domain. Putative examples include, among others, concepts such as object, ordinal, and set. In the present setting, the relevant concepts are expressed by modalized conditions (e.g. x =◇ x, ord◇ (x), and ß◇ x).41 If indefinite extensibility is to motivate relativism, the ‘domain’-talk in our provisional gloss had better not be restricted to set-domains. Given an arbitrary domain D, whether set- or plurality-encoded, the Russell Reductio from Section 1.4 seeks to specify a set rD that lies outside D. And it is this, by the relativist’s lights, that lies behind the failure of the modalized version of the instance of Plural Comprehension for ß◇ x. For absolutely no domain comprises absolutely every set. Consequently, no plurality of items collectively available in a domain does either. And this is tantamount to the negation of the modalized instance of Plural Comprehension.42 The modal language permits us to give a precise regimentation of indefinite extensibility, using interpretational modal operators to regiment the ‘can always’ from the first-pass gloss. Once again, subject to the now-familiar caveats, the relativistfriendly MT-hierarchy provides a helpful picture. To begin with, suppose that ψ(u) is an invariant condition (with no free variables other than u). Given an interpretation i in the hierarchy, let the condition’s i-extension ψ(u)i comprise every member of Mi that satisfiesi ψ(u) (i.e. satisfies ψ(u) as this 40 Compare Linnebo (2010, pp. 157–8) and Hellman (2011, pp. 635–6), who further observes that Modalized Plural Comprehension fails if there are incompossible satisfiers of φ ◇ (x). 41 Notation: ord◇ (x) is the modalization of a L psu -formula giving a standard definition of von Neumann ordinal (i.e. a pure transitive set whose elements are well-ordered by ∈). 42 Linnebo (2010, pp. 157–8) brings similar considerations to bear in explaining failures of Modalized Plural Comprehension.

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 everything, more or less condition is interpreted by i).43 Moreover, let the condition’s super-extension ψ(u)I (relative to the whole MT-hierarchy) comprise the items in any of its i-extensions:  ψ(u)i =df {a ∈ Mi : a satisifiesi ψ(u)} ψ(u)I =df ψ(u)i i∈I

Say, further, that the concept expressed by an invariant condition ψ(u) is extensible as interpreted under i—extensiblei —if i has an admissible extension j such that the condition’s j-extension ψ(u)j contains an item that is not in its i-extension ψ(u)i .44 We may regiment extensibility in the modal language, using the defined existence predicate E!u:45 extu [ψ(u)] =df ◇> ∃u(ψ(u) ∧ < ¬E!u) In the MT-hierarchy, extu [ψ(u)] is truei if and only if ψ(u) is extensiblei , in the sense just defined. The notion of extensibility permits us in turn to define the plural existence predicate E!uu (which serves as the existential guard on the axioms and rules for plural quantifiers): E!uu =df ¬extx [x ≺◇ uu] Defined in this way, a set A encoding the plurality assigned to uu is such that E!uu is truei,σ [uu/A] if and only if every member of A is in Mi . The regimentation of indefinite extensibility modally generalizes extensibility: ind-extu [ψ(u)] =df extu [ψ(u)] In the MT-hierarchy, the formula ind-extu [ψ(u)] is truei if and only if the concept expressed by the invariant condition ψ(u) remains extensiblej no matter which admissible interpretation j in the hierarchy interprets it; equivalently, if and only if there is no interpretation j where its j-extension encompasses its whole superextension (i.e. ψ(u)j = ψ(u)I ).46 The regimentation of indefinite extensibility permits the relativist to delimit and explain failures of plural comprehensibility. Modalized Plural Comprehension says in effect—speaking loosely—that the entire super-extension of φ ◇ (x) is eventually encoded as a plurality-extension. But this is precisely what it takes for the concept expressed by φ ◇ (x) to be non-indefinitely-extensible. Setting metaphors and MT-hierarchies to one side, the point can be made in the modal language. The system mpfo proves the following theorem:47 43

In other words: an item a satisfiesi the formula ψ(u) if ψ(u) is truei,σ [u/a] for some assignment σ . The long-form definiendum retains the ‘concept’-talk Dummett deploys in his account of indefinite extensibility. But, as the formalism makes clear, such talk is eliminable. We may instead, therefore, take extensibility to attach directly to an (interpreted) condition. The same is true of indefinite extensibility. 45 See Section 6.1. 46 Similar remarks apply, relative to an assignment of values to variables other than u, when ψ(u) contains further free variables. 47 Side-condition: φ(x) is a L psu -formula that lacks free occurrences of xx. See Appendix B.3 for a proof. 44

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modalization: plural theories  Inextensibility. The φ ◇ (x) instance of Modalized Plural Comprehension holds if and only if φ ◇ (x) is not indefinitely extensible. ¬ind-extx [φ ◇ (x)] ↔ [∃xx∀x(x ≺ xx ↔ ϕ(x))]◇

Relativism stated Our penultimate application is to provide a neutral characterization of what’s at issue between absolutists and relativists about quantifiers. To begin with, let’s briefly review the dialectical issues raised in Section 5.1. The absolutist may take the following trivial thesis to capture his characteristic claim that some plurality-domain is absolutely comprehensive. Comprehensive Domain. Zero or more items comprise everything. ∃xx∀x(x ≺ xx) Assuming that his quantifier ranges over an absolutely comprehensive domain, the absolutist’s utterance of Comprehensive Domain says that some plurality is absolutely comprehensive; and, in quantifying over this plurality, his utterance shows that it is available as a domain of quantification. The obvious difficulty, from the relativist’s point of view, is that she does not accept the assumption. By her lights, she has no need to dispute the trivial truth of Comprehensive Domain, since she instead denies that it expresses anything in tension with relativism. Nor, by the same token, does she take her view to commit her to the negation of Comprehensive Domain (which is refuted by our default plural logic, pfo):48 No Comprehensive Domain. No zero or more items comprise everything. ¬∃xx∀x(x ≺ xx) The trivial falsity of the obvious attempts to state relativism gives rise to the objection from ineffability: is there any coherent way for the relativist to frame her disagreement with the absolutist? As we saw in Section 5.3, one way for the relativist to respond to this objection is by rejecting the requirement that she capture her view as a single thesis. She may instead look to give an infinite, schematic axiomatization of relativism, with each instance of the schema denying the comprehensiveness of a single domain. Provided she helps herself to suitably interpreted modal operators, however, the relativist can go one step further and capture her opposition to absolutism as a single thesis. The thesis is the modalization of No Comprehensive Domain:49 48

For pfo, see Appendix A.1. Compare Fine (2006), who proposes a statement of relativism combining procedural possibility and metalinguistic vocabulary. Setting aside niceties of use and mention (see p. 25), he writes ∃I xφ(x) to indicate that there is some x under I for which φ(x). Working through Fine’s definitions, relativism is then formulated as follows (p. 30): ∀I ◇ ∃J(∀I x∃J y(x = y) ∧ ¬∀J x∃I y(x = y)) Hellman (2006, sec. 4.2) also suggests a modal statement. 49

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 everything, more or less No Absolutely Comprehensive Domain. Absolutely no zero or more items comprise absolutely everything. ¬ ◇ ∃xx∀x(x ≺◇ xx) As before, the ‘absolutely every’ and ‘absolutely no’ in the informal statement are to be understood in the modal—∀ and ¬ ◇ ∃—way. A weak version of the absolutist’s thesis can be captured by deleting the initial negation in this formula; typically, though, the absolutist is prepared to go further and also delete the ◇. His claim is not merely that we eventually reach an absolutely comprehensive domain once we’ve sufficiently liberalized our interpretation, but rather—assuming any restrictions have been dropped—that we’ve already reached such a plurality-domain: Absolutely Comprehensive Domain. Some zero or more items comprise absolutely everything. ∃xx∀x(x ≺◇ xx)

(acd)

The two views characterized by Absolutely Comprehensive Domain and No Absolutely Comprehensive Domain lead to different attitudes towards modalization. Assuming the domain to be absolutely comprehensive effectively flattens the hierarchy into a single interpretation under which each formula is provably equivalent to its modalization:50 Flattening. Assuming Absolutely Comprehensive Domain, each modalized Lpsu formula φ ◇ is provably equivalent to φ in mpfo. acd mpfo φ ◇ ↔ φ Section 6.2 noted a semantic analogue of this result for MT-hierarchies. The present, proof-theoretic result shows that this feature of set-sized hierarchies is shared by the full, potential hierarchy. Consequently, provided he’s at least willing to play along and countenance the relevant sort of modality, the advocate of Absolutely Comprehensive Domain takes the relativist ‘translation’ mapping each non-modal formula to its modalization to preserve truth-values. Conceived in this way, modalization gives him a harmlessly baroque way to re-express what, by his lights, he could already say with his non-modal language interpreted over the absolutely comprehensive domain. On the other hand, as we saw in the case of Plural Comprehension, by the relativist’s lights, modalization may fail to preserve truth-value. No Comprehension Domain is another case in point. By the relativist’s lights, the true thesis that she uses to articulate a core feature of her view is the modalization of a trivially false formula. As before, the difference reflects the stratification of the potential hierarchy as she sees it: the modalized plural quantifier only ranges over pluralities whose items are eventually available collectively. Indeed, by her lights, there is absolutely no other kind of plurality. 50

See Appendix B.4 for a proof.

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set theory for relativists  Despite their different attitudes towards modalization, the availability of the modalized statement of relativism markedly improves the dialectical situation for both sides in the absolutism–relativism debate about quantifiers. First and foremost, the modal thesis, No Absolutely Comprehensive Domain, gives the relativist a means to capture the core of her view as a single thesis. This provides another way for her to respond to the objection from ineffability. For, unlike No Comprehensive Domain, there’s no credible charge of incoherence: in particular, the relativist’s modal thesis is consistent in the modal logic mpfo. But the availability of the modality also aids the absolutist. For it permits him to characterize his view without presupposing it to be true. Unlike Comprehensive Domain, the relativist cannot brush aside Absolutely Comprehensive Domain as a trivial truth that is perfectly consonant with relativism. The advantage of the modal statements, then, is that they permit each side to frame a coherent thesis that their opponent needs, by his or her own lights, to deny. The normal standard expected of view-stating can be met: neither side need presuppose the correctness of their view in order to state it.

6.4 Set theory for relativists Our final application of the modal operators is a relativist-friendly account of the iterative conception of set, which permits the relativist to motivate standard set theory. Our target theory is a modal analogue of the usual textbook formulation, pure firstorder Zermelo–Fraenkel set theory with Choice: zfc.51 The choice of a first-order target theory reflects the open-ended character that relativists attribute to the set-theoretic hierarchy. Zermelo distinguishes the axioms of his analogue of our default impure plural theory, zfcsup , from what he calls ‘the meta-theory of sets’ (1930, p. 429). While the former characterizes the individual models in the Zermellian hierarchy, the latter describes the open-ended Zermellian hierarchy itself. The relativist may make a similar distinction. In the former case, the (non-modal) theory zfcsup provides a faithful description of any given portion of the cumulative hierarchy Vκ (U), comprising the sets of rank less than some inaccessible cardinal κ based on some underlying domain of urelements U.52 Of course, the relativist characteristically maintains that no such Vκ (U) comprises absolutely every set. Indeed, she may well also maintain that no such Vκ (U) contains absolutely every urelement. Suppose, for instance, she takes ordinal to be an indefinitely extensible concept, but claims further that ordinals are distinct from sets. Then she can always specify a further non-set-ordinal—and thus a further urelement—outside any given Vκ (U). Metaphorically: further and further sui generis ordinals are formed in tandem with additional ranks of sets. To describe the open-ended hierarchy the relativist may instead deploy a modal theory. However, she should not take the modalization of zfcsup as the analogue of Zermelo’s ‘meta-theory of sets’. For the theory zfcsup is based on the plural logic pfo, and the relativist does not accept every instance of Modalized Plural Comprehension.

51

For zfc and zfcsup , see Sections 2.3–2.4 and Appendix C.1.

52

See Section 2.4.

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 everything, more or less In light of the Inextensibility thesis from Section 6.3, this would be tantamount to denying that any modalized condition expresses an indefinitely extensible concept, contrary to the open-ended nature she accords the potential hierarchy. When it comes to the whole potential hierarchy, then, the relativist has reason to follow the usual set-theoretic practice of deploying a first-order set theory. Moreover, to simplify the exposition in this section, we shall set aside the complications raised by the possibility of urelements being generated in tandem with sets.53 We consequently focus on the pure theory zfc, stated in the first-order language Ls , which drops the set predicate from Lsu .

The iterative conception: a bimodal axiomatization As we saw in Section 2.4, the iterative conception of set is often described in a rather metaphorical way. In the pure case, we imagine the (pure) sets in the cumulative hierarchy being formed in a succession of ordinally-indexed stages:54 we start with no sets available (and no urelements); then at each successor stage α + 1, every set of zero or more sets available at stage α will be formed; and at each limit stage λ, the sets formed at stages before λ are gathered together, available for collection at later stages. Beyond the structure of stages, then, the core of the iterative conception is captured by two driving principles: Plenitude. Any zero or more sets available at any stage in the process are such that the set comprising them will be formed at the next stage, and will then be available at all subsequent stages. Priority. Any set formed at any stage in the process is such that the items the set comprises are available at some earlier stage in the process. The iterative conception gives us a vivid picture of the set-theoretic hierarchy. But what are we to make of the process metaphor? As we observed in Section 2.4, there are compelling reasons not to take the iterative conception at face value: pure sets are not continually being formed in a transfinite series of stages. We might instead identify stages with cumulative ranks, or follow Boolos in axiomatizing the iterative conception in a stage theory, treating formal analogues of ‘stage’, ‘earlier’, and ‘formed at’ as primitive (1989, p. 91). Interpretational modality, however, provides an alternative way to dispense with the process metaphor, so as to motivate a relativist-friendly account of set theory, according the cumulative hierarchy a potential, Zermellian character. The relativist-friendly axiomatization of the iterative conception is given in the bimodal plural language of pure set theory—Lmps —which is just like the impure language Lmpsu (introduced in Section 6.1) save that it omits the set predicate. To state the theory, it’s helpful to introduce some notation. As usual we say that s comprises every item among xx (henceforth symbolized: s ≡ xx) if s has every member of xx, 53 A theory of this kind has a rather different character to Zermelo–Fraenkel set theory with urelements as standardly conceived. The dynamic theory of abstraction suggested in Studd (2016) provides one way to implement it. 54 For the remainder of this section, ‘set’ means pure set.

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set theory for relativists  and nothing else, as an element. Officially, we lay down the following definition: s ≡ xx =df ∀x(x ∈ s ↔ x ≺ xx) We write s ≡◇ xx for the modalization of this condition. When the modalized condition holds, we say that s comprises absolutely every item among xx. The bimodal language Lmps then permits us to give a very natural formalization of the Plenitude and Priority principles (accompanied by a rather more cumbersome approximate English gloss): Plenitude Axiom. However Ls is admissibly interpreted: any zero or more sets in the universe are such that, however Ls is admissibly interpreted by succeeding interpretations, the set comprising absolutely every one of them belongs to the new universe. ∀xx> ∃s(s ≡◇ xx) Priority Axiom. However Ls is admissibly interpreted: for any set in the universe, the lexicon may be admissibly interpreted by a preceding interpretation such that absolutely every item the set comprises belongs to the preceding universe. ∀s ◇< ∃xx(s ≡◇ xx) The Priority and Plenitude Axioms dispense with the process metaphor in favour of notions that can be expressed in the language Lmps (whose only non-logical predicate is ∈). The use of tense, or quantification over ‘stages’, gives way to interpretational modality generalizing about how Ls may be admissibly interpreted. Talk of formation and availability is replaced with plain quantification. Understood in this way, the ‘formation’ of sets is not a question of changing the world in order to bring new sets into being. Instead it consists in admissibly reinterpreting the lexicon of Ls to liberalize the range of its quantifiers. The non-temporal ordering of interpretations is then captured by the underlying bimodal logic. As with Boolos’s non-modal axiomatization of stage theory, the interpretationally modal account of the iterative conception recovers a fair portion of zfc. Let mstp be the modal plural theory which adds the Priority and Plenitude Axioms to mpfo, together with the modalization of the Extensionality Axiom. And let s+ be the subtheory of zfc whose axioms comprise Extensionality, Foundation, Empty Set, Pairing, Union, the Separation Schema, and Power Set, together with a zfc-theorem that gives voice to the cumulative rank structure of the hierarchy:55 Rank Theorem. Every set is a subset of some cumulative rank Vα . ∀s∃α(s ⊆ Vα ) The two theories are then related by a further mirroring result:

55 The quantifier ∃α is tacitly relativized in the usual way, as per Appendix A.1. See Appendix C.2 for details of how to formulate s ⊆ Vα in the context of s+ .

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 everything, more or less Mirroring S+ . An Ls -formula φ is a theorem of s+ if and only if its modalization is a theorem of mstp . s+ fol φ if and only if mstp mpfo φ ◇ What about the rest of zfc: the Axiom of Choice, Infinity, and Replacement? The modalization of each axiom is derivable in a natural extension of mstp . For Choice, the addition has a good claim to belong to the underlying plural logic. The language of the plural logic pfo permits us to frame an involute but truistic-seeming schema—the Plurality Choice Schema—which allows us to derive the modalization of the Axiom of Choice without the addition of further distinctively set-theoretic axioms to mstp .56 This leaves Infinity and Replacement. Whether the iterative conception sustains these axioms depends on how far the iteration continues. The use of ordinals to index its stages tells us little about its extent unless we come to the iterative conception with a prior conception of ordinals (not identified with von Neumann ordinals) to use as a yard stick. Is there an alternative? One inchoate but compelling thought is that the iteration should continue as far as possible.57 The use of ordinals to index stages may be seen, in this light, as a means of pointing to the ‘absolute infinity’ of the hierarchy. The stages, like the ordinals, are absolutely infinite in number. The modal setting provides a natural means to make this inchoate thesis about the extent of the hierarchy precise, without drawing on an antecedent theory of ordinals. The underlying thought is that the hierarchy of stages extends far enough to elude characterization by any Ls -formula. We may regiment the thought by adjoining bimodal stage theory with a suitable modal reflection principle for each Ls -formula φ(v) with at most the variables v = v1 , . . . , vk free. To indulge in the process metaphor: when the truthi,σ of φ(v) expresses that the hierarchy up to stage i has a certain feature, the truthi,σ of its modalization φ ◇ (v) expresses that the entire potential hierarchy has the corresponding feature. The reflection principle then ensures that we never reach a stage when such features of the whole hierarchy fail to be reflected in one of its later stages: it is always the case that there will be a stage when (relative to any values for v1 , . . . , vk then available) the entire hierarchy has the feature expressed by φ ◇ (v) just in case the present stage has the feature expressed by φ(v). Less metaphorically, the reflection principle may be captured as a Lmps -schema:58 Complete Modal Reflection. However an Ls -formula φ(v) with at most the variables v = v1 , . . . , vk free is admissibly interpreted, the lexicon may be admissibly interpreted by a succeeding interpretation with the following property: whatever  φ ◇ (v) holds if and only if φ(v) members of its universe are assigned to v, does.  ◇ (v) ↔ φ(v))  ◇> ∀v(φ 56

57 See Appendix C.4. See, for instance, Tait (1998, p. 475) and Linnebo (2013, p. 205). A similar reflection principle is deployed in Linnebo (2013). Thanks to Øystein Linnebo and Sam Roberts for impressing upon me the importance of complete reflection in the present setting. For the difference between complete and partial reflection, see Lévy and Vaught (1961). 58

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set theory for relativists  There is doubtless room to question whether the two additions to mstp should be seen simply as further unfolding the iterative conception, rather than embellishing it.59 Nonetheless, the two additions do, at the very least, offer us a natural elaboration of the iterative conception. Moreover, the whole of zfc may be recovered in the theory mst+ p , which extends mstp with the Plurality Choice Schema and Complete Modal Reflection:60 Mirroring ZFC. An Ls -formula φ is a theorem of zfc if and only if its modalization is a theorem of mst+ p. ◇ zfc fol φ if and only if mst+ p mpfo φ

Linnebo’s unimodal axiomatization Before we return to issues surrounding the intelligibility of the modality, it’s helpful to pause to compare mstp with what I take to be its most promising competitor. The bimodal account of the iterative conception shares some important features with a potentialist account of zf (zfc without Choice) developed in a unimodal plural language by Linnebo.61 Let’s begin with a brief run-down of Linnebo’s axiomatization. Linnebo employs a modal language with a single primitive weakly-forwards-looking modal operator (here written 2l to distinguish it from its defined counterpart in Lmps ). Working against the background of a unimodal plural logic,62 capturing analogues of the monotonicity and stability assumptions made in mpfo, Linnebo establishes the (2l -) modalizations of the axioms of zf from seven non-logical axioms (we write φ ◇l for Linnebo’s modalization of φ).63 Several of the axioms concern what Linnebo (2013, p. 212) calls extensional definiteness which, for modalized conditions, provides an analogue in his setting for the notion of inextensibility in the context of mstp . In terms of the process metaphor: a condition φ(x) is said to be extensionally definite at a stage if its extension remains unchanged at later stages. Formally, Linnebo employs the following definition: defx [ψ(x)] =df ∃xx 2l ∀x(x ≺ xx ↔ ψ(x)) 59 Boolos (1971, 1989) expresses doubt that the iterative conception motivates Extensionality, Choice, and Replacement. Likewise, Potter (2004, pp. 221–7) argues that the prospects for a motivation for reflection based on the iterative conception are poor. On the other side, Gödel’s remarks about Lévy’s (1960) reflection principles suggest that he took them to be among the axioms that ‘unfold the content of the concept of set’. (Gödel 1947, pp. 476–7 and n. 16). See Paseau (2007), who also takes a more optimistic view, for discussion. 60 See Appendices C.2–C.4 for a proof sketch. Compare Linnebo (2013, thms. 8.2, 8.3, 8.5, 8.7) and Parsons (1983, thms. 4–6). 61 Linnebo’s work in turn builds on the modal approach to set theory pioneered in Parsons (1983). See Linnebo (2013, sec. 9) for discussion of the major differences between his approach and its predecessor. 62 Linnebo’s underlying logic extends pfo with the modal axioms and rules distinctive of s. and axioms ensuring that x ≺ xx is stable and inextensible (2013, def. 4.1). For s., see Table A.4 in Appendix A.3. 63 In the unimodal setting, 2 -modalization respectively prefixes 2 and ◇ to ∀ and ∃ (but does not l l l add modal operators to atomic formulas). See n. 23.

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 everything, more or less To recover the modalized zf-axioms, Linnebo makes use of the non-modal versions of the zf-axioms Extensionality and Foundation, together with five further nonlogical axioms:64 Modalized Naive Comprehension. For absolutely any zero or more sets, there is potentially a set comprising them. (c)

2l ∀xx ◇l ∃y 2l ∀x(x ∈ y ↔ x ≺ xx)

The Extensional Definiteness of Membership. For absolutely any set, the condition of being a member of the set is extensionally definite. (ed-∈)

2l ∀y defx [x ∈ y]

The Extensional Definiteness of Subsethood. For absolutely any set, the condition of being a subset of the set is extensionally definite. (ed-⊆)

2l ∀y defx [x ⊆ y]

Partial Modal Reflection. An instance of the following reflection schema for each Ls -formula with at most the variables v = v1 , . . . , vk free: (◇ -refl)

 ◇l (v) → ◇l φ(v)) 2l ∀v(φ

Replacement for Extensional Definiteness. Given any modalized condition φ ◇l (x, y) describing a function (i.e. associating each x with a unique y), and absolutely any plurality, there will eventually be another plurality that contains every item to which the function maps a member of the first plurality.65 (ed-repl) [funcx,y [φ(x, y)]]◇l → 2l ∀xx ◇l ∃yy(∀x ≺ xx)(∃y ≺ yy)(φ ◇l (x, y)) This completes our summary of the unimodal theory. Turning now to comparison, how do the seven non-logical axioms of Linnebo’s unimodal theory compare to the four non-logical axioms of the bimodal theory mst+ p? Let’s first set aside two largely cosmetic differences. First, there’s the difference in target theory (zfc versus zf). This difference runs shallow since there’s nothing to stop the unimodal theory recovering this axiom via an analogous enrichment of the underlying modal logic to the one suggested here. Second, there’s a difference in how Infinity and Replacement are obtained. The bimodal theory recovers both via Complete Modal Reflection, whereas the unimodal theory deploys a different axiom for each, ◇-Refl and ed-repl. Once again, however, this reflects no deep difference. As Linnebo notes,66 a unimodal analogue of Complete Reflection suffices to recover both axioms without the need to employ ed-repl.

64 The axioms are stated in Linnebo (2013, secs. 6–8). We make some minor changes to the notation in accordance with our present conventions. 65 Notation: func [φ(x, y)] = ∀x∃y∀y (φ(x, y ) ↔ y = y ). Relativized quantifiers such as the x,y 1 1 1 df quantifier (∀x ≺ xx) are defined as usual, as per Appendix A.1. 66 Linnebo (2013, n. 28).

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set theory for relativists  Turn now to more substantive differences. The obvious advantage of Linnebo’s theory is that it employs a leaner ideology: the unimodal theory needs just one primitive modal operator, the bimodal theory appeals to two.67 In particular, this permits the unimodal theory to deploy a simpler modal logic. It should be noted, however, that the ideological difference has a quantitative character rather than a qualitative one. It’s doubtful, then, that the unimodal theory offers any serious advantage when it comes to the crucial issue of intelligibility (which we return to in Section 6.5). If we can come to comprehend one such modal-operator, why doubt that we can understand two? On the other hand, if we’re simply in the business of totting-up the number of primitives, we should not forget the plural quantifiers. The unimodal characterization of extensional definiteness deploys the plural quantifier; the bimodal characterization of inextensibility does not. In fact, a natural first-order subtheory of mst+ p suffices to interpret zf. The Plenitude Axiom may be replaced with a schema stating—speaking loosely—that whenever a condition becomes inextensible, the corresponding set is available at every later stage. The first-order analogue of Priority states that whenever a set is available, there was an earlier stage when the condition of being an element of the (future) set was inextensible. The recovery of zf may then proceed much as before.68 But no similar result appears to be on the cards for Linnebo’s unimodal theory. Without it, the plural unimodal theory and the first-order bimodal theory tie on the number of primitive ‘generalizing’ devices (two types of quantification plus one type of modality versus one type of quantification plus two types of modality). The obvious advantage of the bimodal theory is the additional expressive power gained from the two modal operators. The bimodal language allows us to give a simpler and more direct formalization of the theory’s non-logical axioms, permitting us to derive theorems (including one modalized zf-axiom) whose analogues are taken as basic in the unimodal theory. The principle of priority provides an immediate example. Linnebo takes ‘the principle that the elements of a set are prior to the set itself ’ to be an important part of his preferred account of the nature of sets (2013, p. 216). In the bimodal language this principle receives a straightforward regimentation as the Priority Axiom. But we cannot expect to articulate it in Linnebo’s unimodal language. The bimodal language can define an analogue of Linnebo’s operator 2l , but no formula of this language without the backwards-looking operator is equivalent to the Priority Axiom.69 As an alternative to a unimodal articulation of the priority principle, Linnebo proposes to adopt the zf-axiom Foundation as part of his unimodal theory. As Linnebo observes, this axiom flows from the priority principle and the well-foundedness of the ordering of stages. But, as he suggests, it falls short of providing an ‘ideal formalization’ of the priority principle (2013, p. 216). After all, the theory zf—Foundation 67

68 Compare Linnebo (2013, p. 226). See Studd (2013). Proof sketch: given an MT-hierarchy M = {Mi , Si , Ei  : i ∈ I } and j ∈ I , define the structure Mj to be the hierarchy {Mi , Si , Ei  : i ≥ j}. Let ψ be a formula containing only forwards-looking modal operators. For i ≥ j and σ over Mj , a routine induction shows that ψ is truei,σ in M if and only if it is truei,σ in Mj . Clearly however the Priority Axiom lacks this property (unless j is the first stage, deleting the hierarchy of sets prior to j will falsify Priority at j). Since the Kripke semantics is sound with respect to mpfo (Appendix A.3), Priority and ψ are not provably equivalent. 69

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 everything, more or less included—is available to those who vehemently deny that there is any useful, nonmetaphorical sense, in which elements are prior to their sets or that sets are formed in a well-founded series of stages. The coherence of this view is borne out by the bimodal formalization of the priority principle. Neither zf nor its modalized analogue suffice to prove the Priority Axiom.70 This use of Foundation also raises a different kind of difficulty. An important goal of the unimodal account—as in the bimodal case—is to motivate the zf-axioms.71 We might hope that such a motivation would consist in a derivation of the axiom, or its modal analogue, from more basic modal axioms that articulate the relevant potentialist conception of set. The bimodal theory mstp offers just such a motivation of Foundation. The modalization of this axiom may be derived from the Priority Axiom via a straightforward application of the löb-axiom that gives voice to the well-foundedness of the hierarchy’s stages.72 The proof serves as a mathematical explanation of why—according to the potentialist version of the iterative conception of set, articulated in the bimodal theory—every non-empty set has an ∈-minimal element. On the unimodal account, however, this kind of motivation is unavailable if we simply posit Foundation as an axiom. Might the advocate of the unimodal theory motivate Foundation in some other way? Linnebo offers what he describes as an ‘informal but compelling’ argument to show that Foundation follows from the priority principle and the well-foundedness of the ordering of stages. His argument proceeds in terms of the Kripke semantics, and may be paraphrased as follows:73 Let a be a non-empty set, as per the antecedent of Foundation. We need to show that a has an ∈-minimal element b (i.e. an element b such that there is no c ∈ a with c ∈ b). To see this, consider the (clearly non-empty) class S comprising every stage whose domain contains an element of a. Since the ordering of stages is well-founded, at least one stage in S—s0 , say—is minimal with respect to the ordering < of stages. Let b be an element of a in the domain of s0 . Then b is ∈-minimal. For otherwise we would have an element c of a such that c ∈ b. By the priority principle, c is in the domain of a stage s, with s < s0 . And this contradicts the minimality of s0 .

In effect, the argument dispenses with modality in favour of quantification over stages, as per the Kripke semantics. For instance the nonmodal priority principle it relies on—if a is in the domain of s, the elements of a belong to the domain of stage t, with t < s—dispenses with backwards-looking modality in favour of quantification over earlier stages. But what should an advocate of an irreducibly modal account of set theory make of this nonmodal argument? Linnebo emphasizes,74 as we did in Section 6.1, that the modality is not semantically reducible to quantification over stages. His potentialist account of set theory is articulated in a modal language in which ‘stage’-talk, while a useful heuristic, is ultimately eliminated in favour of modal notions. But without backwards-looking modality, the unimodal theory has no way to recast the nonmodal 70 Countermodel: the zf-axioms, and their modalizations, hold in an absolutist-friendly MT-hierarchy {Mi , Si , Ei  : i ∈ I } where each interpretation Mi , Si , Ei  is the same standard model of zf, M, S, E. The Priority Axiom fails at the least stage of this MT-hierarchy. 71 72 See Linnebo (2013, p. 206). See Appendix C.2. 73 Compare Linnebo (2013, pp. 216–17). Linnebo uses ‘world’ where we use ‘stage’. 74 See Linnebo (2013, p. 208).

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objections from unintelligibility



motivation of Foundation. If the argument is intended to supply a mathematical explanation of why Linnebo’s potentialist conception of set theory sustains Foundation, this use of Kripke semantics would seem to outstrip the minimal instrumental role that is available when we take the modality to be irreducibly modal. Of course, explanations must stop somewhere. We might simply content ourselves with zfc (or its extension with large cardinal axioms) as the final articulation of the iterative conception, taking its axioms to be beyond explanation. Linnebo’s unimodal theory shows that we can do better. Without invoking a dubious creationist metaphysics, we can explain why the cumulative hierarchy conforms to these axioms on the basis of a fairly short list of modal axioms characterizing an appealing potentialist account of the Zermellian hierarchy. My contention here is that with a bimodal theory we can do better still. The theory mstp provides a more direct axiomatization, which permits us to derive the analogues of axioms that the unimodal theory takes as basic.

6.5 Objections from unintelligibility Both sides in the absolutism–relativism debate about quantifiers, and relativists especially, stand to gain from the use of suitably interpreted modal operators. But some absolutists (and indeed some relativists) may be unprepared to countenance such exotic-seeming ideology. Those who take an austere view (as we shall call it) simply deny that the relevant kind of non-circumstantial modality is available. This section considers two ways in which austere absolutists (or austere relativists) may call the modality’s intelligibility into question: (i) the objection from Kripke semantics picks up on the worries about illicit quantification introduced in Section 6.1; (ii) the objection from Kaplanian ‘monsters’ argues that the relevant modality is unavailable in English.

The objection from Kripke semantics How is a non-austere relativist to specify the intended meaning of ? So far, approximate English glosses aside, the closest we’ve come to a specification of the interpretation of the modal language would seem to be the Kripkean semantic clauses, such as the clauses for  and ∀v:75 For any admissible interpretation i and assignment σ : ψ is truei,σ iff every admissible interpretation j in the hierarchy is such that ψ is truej,σ . (ki -∀v) For any admissible interpretation i and assignment σ : ∀vψ is truei,σ iff every item a with a ∈ Mi is such that ψ is truei,σ [v/a] . (ki -)

Within the Kripkean model theory, the clauses specify truth-conditions relative to an MT-interpretation within a set-encoded MT-hierarchy. So understood, the first clause clearly fails to capture the intended meaning of . For the metalanguage quantifier ‘every admissible interpretation’ is restricted to the set of interpretations in the MT-hierarchy. Might the relativist instead specify the intended interpretation by lifting the restriction, and taking the metalanguage quantifier to range over every admissible interpretation? 75 For simplicity we focus on the absolute modal operator  in this section; similar problems arise for > and < .

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 everything, more or less We’ve already seen two reasons to think that this approach is a non-starter. First, the clause ki - only endows  with the means to achieve the intended absolute generality if its quantifier ‘every admissible interpretation’ ranges over absolutely every admissible interpretation. But as we saw in Section 6.1, by the lights of the interpretational expansionist, quantification over absolutely every admissible interpretation faces the same objection from indefinite extensibility as quantification over absolutely every set or absolutely everything. Second, the clause ki -∀v runs into trouble of the kind we met when discussing the side-condition problem in Section 5.5. In order to succeed, the clause needs, in effect, to function as a super-semantic condition, specifying the intended semantics for ∀v under absolutely every admissible interpretation. But, as we saw, ki -∀v fails to capture the super-semantics unless the metalanguage universe is absolutely comprehensive. Taken at face value, then, the Kripkean clauses clearly fail to capture the intended truth conditions for the modal language. But why take this to be a problem for the modal operator rather than a problem with this style of semantics? The Kripkean semantics, in effect, seeks to semantically reduce the modality to quantification over admissible interpretations. The relativist may rejoin that this style of semantics fails precisely because the modal operator is not reducible in this way. One option, then, may be to fall back on a more Davidsonian, quasi-homophonic style of semantic theory which uses a modal metalanguage to frame the truthconditions for the formulas of the modal object language.76 Let’s continue to suppose (pending the second objection) that the absolute interpretational modality is expressed by English expressions such as ‘however the lexicon is admissibly interpreted’ or, more briefly, ‘whatever the admissible interpretation’. The conditions for the formulas ψ and ∀vψ to be trueσ under an assignment σ of values to their free variables are then respectively specified using interpretational modality and quantification in the metalanguage: Whatever the admissible interpretation: ψ is trueσ iff, whatever the admissible interpretation, ψ is trueσ . (t -∀v) Whatever the admissible interpretation: ∀vψ is trueσ iff every item a is such that ψ is trueσ [v/a] .

(t -)

This style of semantics side-steps the worries facing the Kripkean semantics. The clause for the modal operator does not deploy problematic quantifiers such as ‘every admissible interpretation’. And the clause for the quantifier does not need to quantify over absolutely everything in order to specify the intended truth-conditions for ∀vψ no matter what the admissible interpretation. Contextual restrictionists might look to the same style of semantics for their contextual modal operators, mutatis mutandis. In this case, there is an additional benefit. In light of the objection from semantic theorizing from Section 4.3, the contextual restrictionist is yet to give a full articulation of the semantics she attributes to quantifiers. The homophonic semantics for a language containing contextual modal operators permits her to do so. To specify the truth-conditions she attributes to 76

See, for instance, Peacocke (1978).

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objections from unintelligibility



∀vψ in an arbitrary context calls for the contextual restrictionist to generalize about absolutely every member of absolutely every contextually determined domain. But the contextual modal operators permit her to do this without illicitly quantifying over absolutely everything. All the same, there are good reasons to be dissatisfied with the Davidsonian semantics. We outlined some of them in Section 3.6. Moreover, in the present context, an outright rejection of the Kripke-style semantic clauses raises a further worry. What are we to make of the extensive use of the Kripke semantics in Sections 6.1–6.3 to motivate and explain the relevant modality and its applications? As noted in Section 6.1, we need not take the Kripkean semantic clauses to specify the intended interpretation in order to put MT-hierarchies to the usual sorts of application set-structures are put to within model theory. But our use of MThierarchies would seem to outstrip this minimal instrumental role. In particular, on several occasions our explications appealed to the thought that the relativistfriendly MT-hierarchy illustrates in miniature features reflected in the whole potential hierarchy. How can we make sense of claims of this sort if the Kripkean clause ki - fails to capture the intended interpretation of ? I propose the following answer: by modalizing the Kripkean clauses. More fully, the relativist may attribute tacit modal content to an ostensibly non-modal Kripke-style semantics in just the same way as in the case of ostensibly non-modal set theory. For instance, although, by the relativist’s lights, the non-modal clause ki - cannot hope to capture the intended generality of , no similar problem faces its modalization. The relativist-friendly set theory, outlined in Section 6.4, provides one way to implement this proposal in the case of Lmps . To begin with, let us define a Kripkean translation from the modal plural language into the first-order language Ls . In order to do so, we first need to re-index the variables of the first-order language to obtain two lots of countably many (singular) first-order variables: singular variables (x, y, z, . . .) and proxy-plural variables (xp , yp , zp , . . .). The latter variables are so called because they stand in for plural variables in the Kripkean translation, which is defined as follows:77 u = vα u ∈ vα u ≺ vvα ¬ψα ψ1 → ψ2 α ∀vψα ∀vvψα

=df u = v ∧ u ∈ Vα ∧ v ∈ Vα =df u ∈ v ∧ u ∈ Vα ∧ v ∈ Vα =df u ∈ vp ∧ u ∈ Vα ∧ vp ⊆ Vα =df ¬ψα =df ψ1 α → ψ2 α =df (∀v ∈ Vα )ψα =df (∀vp ⊆ Vα )ψα

> ψα =df (∀β > α)ψβ < ψα =df (∀β < α)ψβ

77 Relativized quantifiers (e.g. (∀β > α)) and apparent function symbols (e.g. V ) are defined away in α the usual manner; see Appendix A.1.

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 everything, more or less Call the Ls -formula ψα the Kripke normal form of ψ relative to α. Comparing the clauses of the translation with the Kripkean semantic clauses in Sections 6.1 and 6.3, it’s apparent that ψα , in effect, gives a Kripke-style truth-condition for the modal formula to be true under the α-th interpretation in the potential hierarchy. With the help of a (non-invariant) Lmps -formula Uα (M) formalizing the claim that the universe M of the present interpretation is the α-th universe in the hierarchy, modal sentences are related to their Kripke normal forms as follows:78 Kripke Normal Form Theorem. Suppose that ψ is an Lmps -sentence that contains no proxy-plural variables. Then, on the assumption that the α-th universe M is the current one, ψ is provably equivalent in mstp to the modalization of its Kripke normal form relative to α: [ψα ]◇ . Uα (M) mstp ψ ↔ [ψα ]◇ It bears emphasis that the Kripke Normal Form Theorem does not reduce the modal operators to quantifiers. Modal operators appear on both sides of the biconditional in the statement of the theorem. But the theorem does provide a way to accord a more than purely instrumental role to the truth-conditions ascribed to Lmps -sentences by a Kripke semantics. When suitably modalized, the Kripkean truth-conditions do capture the intended interpretation of the sentences of the modal language. And in this sense a relativist-friendly MT-hierarchy provides, by the relativist’s lights, a miniature representation of the potential hierarchy at large.

The objection from Kaplanian monsters In the case of contextual modal operators, Williamson (2003, p. 433) raises a further objection against the relativist’s deployment of modal operators. He argues that the obvious candidate expressions in English, such as ‘in some context it is true that’, fail to express the relevant modality. In David Kaplan’s colourful phrase, such modal operators are ‘monsters’ (1989, p. 511). To borrow Williamson’s example,79 suppose someone other than Tony Blair utters the following sentences: (6) ‘I am Tony Blair’ is true in some context. (7) In some context it is true that I am Tony Blair. As we noted, the relativist should not try to reduce contextual modality to quantification over contexts. All the same, if the English operator ‘in some context it is true that’ is to express the intended contextual modality, the truth of the metalinguistic statement (6) should at least suffice for the truth of the operator-statement (7) (against the background of a suitable indexical semantics). As Williamson observes, however, this is not the case. The metalinguistic statement (6) is true. In contexts where Tony Blair is the speaker, the indexical ‘I’ denotes Tony Blair. But the would-be operator-statement (7) is false in its context of utterance. For in this context, ‘I’ refers to whoever uttered (7). The sentence says falsely of someone who is (ex hypothesi) not Tony Blair that in some contexts he or she is. 78

See Appendices C.2–C.3.

79

Compare Kaplan (1989, pp. 510–11).

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objections from unintelligibility



The linguistic data seems perfectly clear: the locution ‘in some context it is true that’ fails to express the relevant contextual modality. On Kaplan’s view, this is no surprise. According to his semantic theory, the value of an indexical is fixed once and for all by the context of utterance, and cannot be shifted by any operators in whose scope it occurs. He consequently takes the view (i) that ‘monstrous’ operators effecting such shifts are unavailable in English and (ii) monsters could not be added to English.80 Where does this leave the intelligibility of relativist-friendly modal operators? Williamson concludes: ‘We may be unable to understand ‘in a context’ in the way that the operator strategy requires’ (2003, p. 433). In my view, however, the objection from Kaplanian monsters is far from conclusive. Note first that Kaplan’s second—much stronger—claim is needed if the objection has any hope of decisively undermining the intelligibility of the non-austere relativist’s modal operators. In the absence of (ii), the relativist may seek to reconcile herself to (i) as yet another place where English isn’t as expressive as it might be. To simply assume, without argument, that the bounds of intelligibility exactly align with the bounds of English has every appearance of linguistic chauvinism. The relativist may be quick to point out, moreover, that anti-Quinean absolutists also help themselves to expressive resources which are unavailable in English. Indeed, the non-austere relativist’s departure from English (which calls for one or two modal operators) is, prima facie, much more modest than the anti-Quinean absolutist’s (which calls for a transfinite hierarchy of higher-order or plural resources, or similar, if he is to sustain the thesis we labelled Universe-based Semantic Optimism in Section 3.5).81 Supposing, then, as Williamson suggests,82 that we might come to grasp the absolutist-friendly resources unavailable in our home language by employing the ‘direct method’—presumably by cultivating appropriate patterns of use for the expressions in question—why can’t we come to understand the relevant modality in a similar way? Loose talk deploying locutions such as ‘in some context it is true that’ is then analogous to ‘superplurality’- or ‘higher-order entity’-talk: it may be taken as elliptical for a suitable paraphrase in a language deploying the relevant expressive resource. This brings us to Kaplan’s second claim. The absolutist may seek to question the analogy between absolutist-friendly quantifiers and relativist-friendly modal operators. Do we have a special reason to doubt that English could be enriched with the latter, as per (ii)? Test cases drawn from English, such as (6)–(7), inevitably speak directly only to (i). But as noted above, Kaplan’s semantic theory takes the content of indexicals to be unshiftably fixed by the context of the utterance. If English, and its possible extensions, conform to this semantics, they do not admit of monsters. Competing semantic theories, however, make no similar prohibition. In fact, a key motivation for a monster-friendly theory is that monsters do seem to be available in natural languages, including English. As Philippe Schlenker (2003) observes, some 80

See Kaplan (1989, pp. 510–12). As we noted in Section 3.5, the availability of superplural terms in English is controversial. But even if limited superplural resources are available, it’s clear the required transfinite hierarchy of resources (plural, superplural, supersuperplural, . . .) quickly outstrips natural language. 82 See Section 2.2. 81

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 everything, more or less indexicals, such as ‘two days ago’, do seem to be shiftable under operators in examples like the following:83 (8) John has repeatedly told me over the years that he was sick two days ago. If we follow Kaplan, ‘two days ago’ can only be evaluated with respect to the context of its utterance (above) and not, as on the more natural reading, with respect to the context of John’s repeated speech acts over the years. Examples of this kind suggest that we need a semantics, such as Schlenker’s, which allows (8) to contain a ‘monstrous’ operator. And if the semantics of English allows for Kaplanian monsters after all, no reason has been given to think that it couldn’t be enriched with further monsters, such as relativist-friendly contextual modal operators.

6.6 Hybrid relativism Section 6.5 argued that, even if relativist-friendly modal operators are irreducibly modal or Kaplanian monsters, this does not show them to be unintelligible. All the same, even if this is granted, non-austere relativists may still disagree over their status. This section introduces a further schism within relativism about quantifiers which will occupy us further in Chapter 7. The dispute centres on whether modalized quantifiers achieve absolute generality. Relativists who appeal to such modal operators typically maintain that although we cannot quantify over absolutely everything, we can nonetheless generalize about absolutely everything by deploying modalized quantifiers. But to insist on talking of ‘generalizing’ may seem like the verbal evasion of a lapsed relativist in denial. If we can generalize about absolutely everything, how is this importantly different to quantifying over absolutely everything? One version of this worry is readily dispensed with. It should now be clear that, even with the addition of absolutely general modalized quantifiers, relativism about quantifiers does not simply collapse into absolutism about quantifiers. To avoid the objection from Kripke semantics,84 quantifiers and modalized quantifiers must be taken to be two, quite separate, generalizing devices. In particular, the latter is not semantically reducible to the former. There is consequently no incoherence in the non-austere relativist about quantifiers contending that one generalizing device attains absolute generality when another does not. This deals with the immediate collapse worry: the relativist about quantifiers may coherently ascribe absolute generality to modalized quantifiers. All the same, someone who adopts this combination of views may reasonably be said to endorse a species of generality-absolutism even if it is not a species of quantifier-absolutism. And not all relativists about quantifiers will be prepared to countenance even this much absolutism. When it comes to quantifiers and modalized quantifiers, three non-austere positions are available.85 For brevity, let’s speak of ∀-absolutism and ∀-relativism 83

84 We borrow the example from Schlenker (2003, p. 64). See Section 6.5. We set aside a fourth combination: ∀-absolutism combined with ∀-relativism. The coherence of this hybrid view is doubtful. For assuming modalized quantifiers are intelligible, presumably they achieve no less generality than ordinary quantifiers. 85

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hybrid relativism  as short for absolutism and relativism about quantifiers. Likewise, in the case of modalized quantifiers, ∀-absolutism takes them to achieve absolute generality, and ∀-relativism opposes this view. The three views are then as follows:86 (a) Thorough-going absolutism combines ∀-absolutism and ∀-absolutism. (b) Thorough-going relativism combines ∀-relativism and ∀-relativism. (c) Hybrid relativism (which we may also, with equal justice, call hybrid absolutism) combines ∀-relativism and ∀-absolutism. The debate between thorough-going relativists and hybrid absolutists (i.e. hybrid relativists) can be expected to follow a similar pattern to the debate between ∀-relativists and ∀-absolutists. The immediate benefit of hybrid absolutism is a form of absolute generality via modalized quantifiers. As a non-austere position, thoroughgoing relativism does not reject the intelligibility of modalized quantifiers. On this view, a modalized formula may have a perfectly intelligible intended interpretation. But it fails to achieve absolute generality (in any interestingly maximal sense). On the other hand, the obvious challenge facing the hybrid view is to show that such a mixed position is not merely coherent but genuinely tenable. The challenge is especially pressing for relativists who oppose ∀-absolutism on the basis of indefinite extensibility. On the face of it, at least, ∀-absolutism faces a closely analogous challenge. Given a would-be absolutely general modalized quantifier ∀v, why can’t we deploy a variant of the Russell Reductio from Section 1.4 to identify a set that it fails to generalize about? One obvious candidate is its Russell set—r∀ —the set comprising absolutely every non-potentially-self-membered item generalized about by ∀. Of course, the hybrid absolutist may respond by denying the availability of r∀ . But why think that this response is any more promising than the analogous ∀-absolutist move in response to the original Russell Reductio? We return to such ‘revenge’ objections against ∀-absolutism in Section 7.5. But before we come to them, we need first to revisit the argument from indefinite extensibility against ∀-absolutism.

86 Fine (2006, esp. sec. 2.7) is naturally read as defending hybrid relativism; Linnebo (2010, 2013) adopts an analogous position for modalized quantifiers in set theory.

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7 Russell Reductio Redux Michael Dummett’s influence on the absolutism–relativism debate is considerable. Even if few relativists adopt an intuitionist account of quantification in response to his arguments drawing on the paradoxes, many still take indefinite extensibility and the open-ended character of the cumulative hierarchy to be among the main driving forces behind their view.1 As we saw in Chapter 2, however, when we come to the details, the argument did not prove straightforward to regiment in a dialectically effective way. The obvious formulations of the Dummettian argument considered in Section 2.5 fell foul of the criticisms of Richard Cartwright and George Boolos: they either invoked the discredited All-in-One principle, presupposing domains to be set-like objects, or fell foul of what we labelled the naivety rejoinder, by turning on an inconsistent plenitude principle for sets. The aim of this chapter is to formulate the argument from indefinite extensibility in a way that avoids these pitfalls. As we saw in Chapters 5 and 6, the relativist can coherently capture her contention that absolutely no domain is absolutely comprehensive by using schemas or relativistfriendly modal operators. This chapter shows that these resources also permit the relativist to coherently articulate the liberal attitude towards set comprehension that drives indefinite extensibility. Given some plausible auxiliary premisses, modal and schematic plenitude principles provide the basis for a rigorous general formulation of a broadly Dummettian argument for relativism. Sections 7.1 and 7.2 present schematic and modal versions of the argument from indefinite extensiblity. The obvious absolutist response to the argument is simply to reject the relativist’s plenitude assumption. After dispensing with the naivety rejoinder, Section 7.3 assesses this option in comparison with its relativist competitor. But there’s also a less obvious way for the absolutist to respond to the argument. Following Timothy Williamson (1998a) and Gabriel Uzquiano (2015), the absolutist may seek to reconcile absolutism with a species of indefinite extensibility. After evaluating this response in Section 7.4, we return to the issue raised at the end of Chapter 6. Section 7.5 examines whether considerations relating to indefinite extensibility also give us grounds to doubt that modalized quantifiers achieve absolute generality.

1

Versions of this argument have been put forward by Parsons (1974), Fine (2006), Glanzberg (2004, 2006), Hellman (2006), Lavine (2006), Shapiro and Wright (2006). Boolos (1993), Cartwright (1994), and Williamson (2003) defend the absolutist.

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the schematic argument



7.1 The schematic argument The Russell Reductio purports to show that an arbitrary universe falls short of being absolutely comprehensive. For the sake of simplicity, let’s focus on the universe on which the interpretation of a simple formal language is based. Suppose that the plural language of set theory with urelements—Lpsu —is interpreted by M0 , S0 , E0 . As usual, M0 encodes the universe of discourse ranged over by unrestricted singular and plural quantifiers (∀v and ∀vv); and S0 and E0 are the extensions for the set and element–set predicates (ß and ∈).2 Moreover, we need not suppose that the universe is a set-universe, or that the extensions are set-extensions. Instead—speaking loosely— M0 , S0 , E0  may be encoded as a plurality-, or P-interpretation. Thus encoded, we may suppose that the absolutist takes M0 , S0 , E0  to encode the intended interpretation of Lpsu , with an absolutely comprehensive universe.3 Our first pass at the Russell Reductio in Section 1.4 went as follows: However extensive M0 may be, we seem to be able to specify a set that demonstrably lies outside it. The set in question is the universe’s Russell set rM0 , comprising the non-self-membered sets in M0 . Reasoning in the style of Russell’s paradox, the Russell Reductio refutes the assumption that rM0 belongs to M0 .4 As we noted, this style of argument is liable to provoke a curt response. The absolutist may rejoin that the argument fails because it relies on an incoherent conception of set. To briefly recapitulate the naivety rejoinder,5 arguably the most immediate lesson of the paradox is that we must reject over-strong comprehension principles. To fix terminology, let us say that a condition φ(x) defines a set s (in symbols: s ≡x φ(x)) if s comprises every item that satisfies φ(x): s ≡x φ(x) =df ∀x(x ∈ s ↔ φ(x)) Russell’s paradox then shows that we cannot accept every instance of the Naive Comprehension Schema:6 Naive Comprehension Schema. Given a condition φ(x), there is some set that is defined by φ(x). ∃s(s ≡x φ(x)) (As ever, in the impure language Lpsu , we reserve ∃s (and ∀s) to abbreviate a quantifier relativized to the set predicate; analogous conventions apply to the other languages we consider.)7 With the help of some further definitions, the inconsistent Naive Comprehension Schema can be helpfully factored into two assumptions. Say that a condition φ(x) defines a plurality xx (in symbols: xx ≡x φ(x)) if xx comprises every item that satisfies φ(x). As usual, moreover, we say that a set s comprises a plurality of zero or more items xx or that xx form s (in symbols: s ≡ xx) if s has every item among xx, and 2

3 See Section 3.1 and Appendix A.1. See Section 3.4. In Section 1.4, we formulated a version of the argument for collections. But since the premisses of the arguments regimented in Section 7.1 and 7.2 are consonant with Zermelo–Fraenkel set theory, we may suppose the relevant collections to be sets. See Section 7.3. 5 6 See Sections 1.4 and 2.4–2.5. Side-condition: φ(x) is a formula which lacks s free. 7 See Appendix A.1. 4

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 everything, more or less nothing else, as an element. Officially—dispensing with loose ‘plurality’-, ‘collection’and ‘formation’-talk—we lay down the following definitions in the plural language:8 xx ≡x φ(x) =df ∀x(x ≺ xx ↔ φ(x)) s ≡ xx =df ∀x(x ∈ s ↔ x ≺ xx) In these terms, Russell’s paradox arises from the following assumptions:9 Naive Comprehension Axiom. For any zero or more items, there is a set that comprises them. ∀xx∃s(s ≡ xx) Plural Comprehension Schema. Given a condition φ(x), there are zero or more items that are defined by φ(x). ∃xx(xx ≡x φ(x)) Speaking loosely, Plural Comprehension tells us that every condition φ(x) is, so to speak, plurally-comprehended: that is, such that φ(x) defines a plurality of zero or more items; the Naive Comprehension Axiom tells us that every plurality is collected as a set: that is, such that its members form a set. (For brevity, we usually abbreviate ‘collected as a set’ to ‘collected’.) The paradox shows that we cannot be maximally liberal in both respects. In our default plural logic—pfo—an inconsistent instance of the Naive Comprehension Schema follows from the Naive Comprehension Axiom and the Russellian instance of Plural Comprehension (taking φ(x) to be ¬x ∈ x).10 Indeed, since this plural logic includes Plural Comprehension as an axiom-schema, pfo itself refutes the Naive Comprehension Axiom.11 Against the relativist’s contention that there is a Russell set comprising the nonself-membered items in the would-be absolutely comprehensive domain M0 , then, an advocate of the naivety rejoinder may be tempted to simply refer her to the standard response to the paradox (which we earlier labelled Russell’s Lesson): not every plurality is collected.12 Assuming our default plural logic, pfo, the relativist cannot coherently accept the Naive Comprehension Axiom. But does the Russell Reductio really rely on an incoherent plenitude assumption? As we noted in Section 1.4, the naivety rejoinder ignores an important feature of the Russell Reductio. Taking the Naive Comprehension Axiom to quantify over M0 , the crucial instance says, in effect, that there is—in the universe M0 —the universe’s Russell set rM0 . But the relativist makes no such claim. Indeed the whole point of her running through the argument is to identify a set outside M0 . To regiment this apparent performative aspect to her argument, the relativist can deploy the same technique she used to capture her view in Chapter 5. Before she runs 8 9 10 11 12

As ever, such loose talk should be read as elliptical for a suitable paraphrase. See Sections 2.4 and 3.5. Side-condition: φ(x) is a Lpsu -formula which lacks xx free. For pfo, see Appendix A.1. Compare, for instance, Yablo (2006), Uzquiano (2009), and Linnebo (2010). This is a plural version of Russell’s Lesson. See Sections 2.1 and 2.5.

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the schematic argument



her argument against the absolute comprehesiveness of M0 , let us suppose that she first does whatever it is she thinks it takes in order to shift from M0 , S0 , E0  to a more inclusive interpretation of Lpsu (in conjunction with the rest of her linguistic community if need be). Label the new interpretation M1 , S1 , E1 .13 To keep track of this attempted shift, it’s helpful to add sort indices to the variables and non-logical predicates of Lpsu . Sort 0 expressions are interpreted according to the initial interpretation, M0 , S0 , E0 ; sort 1 expressions according to the potentially more liberal one, M1 , S1 , E1 . More fully, for i = 0 or 1, singular sort i variables (xi , yi , . . .) and plural sort i variables (xxi , yyi , . . .) range over Mi , and the extensions of ßi and ∈i are respectively Si and Ei . Informally, we shall refer to members of S0 as sets0 , members of M1 as things1 , and so on. The syntax remains standard for a plural firstorder language, with no ban on cross-sort predications (e.g. both ß0 x0 and ß0 x1 are well-formed).14 Equipped with the sorted language, the relativist may coherently deny the comprehensiveness of M0 using the sorted thesis from Section 5.3:15 No Comprehensive1 Domain0 . No zero or more items0 comprise every item1 . ¬∃xx0 ∀x1 (x1 ≺ xx0 ) No Comprehensive1 Domain0 tells us, in effect, that the relativist succeeds in expanding the domain: neither M0 nor any of its (plurality-encoded) subdomains comprise every member of M1 . The sorted language also provides a suitable setting in which to regiment a RussellReductio-style argument for this conclusion. The first task is to frame a suitable sorted plural logic. The logic must not, of course, prejudge whether the relativist succeeds in her attempt to shift to a wider domain. But it is technically convenient to suppose that her attempt does not backfire and shrink the domain. We shall consequently suppose—as the absolutist can happily allow—that M1 is no less inclusive than M0 (i.e. M0 ⊆ M1 ). The sorted semantics sustains a natural generalization of the plural logic pfo. Take, for example, the classical axiom of Universal Specification for singular quantifiers. This is standardly formulated as follows: (us)

∀vφ → φ(u/v)

As usual, the formula φ(u/v) results from substituting each free occurrence of v in φ with u (subject to the standard side constraints to avoid clashes of variable).16 To ensure that all instances of us are true under the intended interpretation we need to ensure that the specified variable u is assigned to a member of the domain that the quantified variable v ranges over. Given our assumption that M0 ⊆ M1 , this can be achieved by requiring the sort index of u to be no higher than the sort index of v. 13

As in Chapter 4, relativists of different stripes give different accounts of this shift. In this chapter we focus on interpretational expansionism. But contextual restrictionists and other kinds of relativist may offer analogous arguments. 14 See Appendix A.2 for a full account of the syntax. 15 In this chapter, we leave the language labels deployed in Section 5.3 tacit. 16 See Appendix A.1.

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 everything, more or less The other axioms and rules of pfo are sorted in a similar way in Appendix A.2. The axioms are rendered true and the rules are rendered truth-preserving by any pair of structures M0 , S0 , E0  and M1 , S1 , E1  that meet the condition that M0 ⊆ M1 . Such a pair also renders true a further axiom with no analogue in the unsorted system: Auxiliary Truism0,1 . Any one1 of zero or more items0 is an item0 . ∀y1 (∃xx0 (y1 ≺ xx0 ) → ∃x0 (y1 = x0 )) Call the sorted plural logic that adds this axiom to the sorted analogues of the pfoaxioms and -rules pfo0,1 . The logic pfo0,1 neither proves nor refutes No Comprehensive1 Domain0 . The sorted argument instead relies on two non-logical premisses.17 To state the argument, let’s first introduce sorted analogues of our earlier terms. We say that s1 comprises1 a plurality of zero or more items xx0 (in symbols: s1 ≡1 xx0 ) if s1 has every item1 of xx0 , and no other thing1 , as its elements1 : s1 ≡1 xx0 =df ∀x1 (x1 ∈1 s1 ↔ x1 ≺ xx0 ) We say that ss0 is collected as a set1 , or collected1 , if some set1 comprises1 ss0 . The first premiss then gives a maximally liberal answer to the question of which pluralities of sets0 are collected as a set1 : Sets0 get Collected1 . Any zero or more sets0 are collected1 . ∀ss0 ∃s1 (s1 ≡1 ss0 ) The singular and plural quantifiers in the premiss, ∃s1 and ∀ss0 , abbreviate quantifiers, ∃y1 and ∀xx0 , respectively relativized to the set1 -predicate ß1 and a defined plural condition ß0 xx0 , applying to pluralities of sets0 : ß0 xx0 =df ∀x0 (x0 ≺ xx0 → ß0 x0 ) This premiss can helpfully be thought of as placing a constraint on the relativist’s attempt at domain expansion. The truth of this premiss requires—to speak loosely— that, in shifting from M0 , S0 , E0  to M1 , S1 , E1 , the relativist succeeds in collectingtogether each plurality of sets0 available in M0 into a single item—a set1 —in the new universe M1 . The second premiss concerns urelements. As usual, urelements are objects that are not sets; similarly, urelementsi are objectsi that are not setsi . The second premiss makes a further claim about the relativist’s attempted shift of interpretation: Urelements0 remain Urelements1 . Every urelement0 is an urelement1 . ∀x0 (¬ß0 x0 → ¬ß1 x0 ) In other words, the second premiss requires—speaking loosely—that the shift from M0 , S0 , E0  to M1 , S1 , E1  never results in non-sets0 ‘becoming’ sets1 .18 For example, 17

Fine (2006, sec. 2.1) presents a somewhat similar argument that he dubs the Extendability Argument. To speak of ‘becoming’ (as with ‘collecting’-talk) is to indulge in the familiar process metaphor. As ever, this metaphor should not be taken at face value. See Section 2.4. 18

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the schematic argument



we are unable to form the Russell set1 of all non-self-membered0 sets0 by fiat, simply by stipulating that Bertrand Russell (the man himself) is in the extension S1 , and that for any item s, a pair of the form s, Russell is in the extension E1 just in case s is a non-self-membered0 set0 . Granted that Russell is an urelement0 , Urelements0 remain Urelements1 rules out his being ‘made’ a set1 in this way. This completes our statement of the argument: the relativist-friendly conclusion, No Comprehensive1 Domain0 , follows in pfo0,1 from the two premisses, Sets0 get Collected1 and Urelements0 remain Urelements1 .19 What should the absolutist make of the argument? Assuming he does not object to reasoning in pfo0,1 , he must either accept the conclusion or reject a premiss. The former option is, strictly speaking, compatible with absolutism. After all, No Comprehensive1 Domain0 only denies the comprehensiveness of subdomains of M0 . The absolutist may consequently go back on his initial claim that M0 is absolutely comprehensive without renouncing absolutism. The dialectical position this leaves him in, however, is clearly untenable. For the sorted argument made no special assumptions about the initial interpretation M0 , S0 , E0 . For the absolutist to claim, on the strength of the relativist’s argument, that M0 fails, after all, to be absolutely comprehensive but that it is instead M1 , S1 , E1  or some other interpretation that really has an absolutely comprehensive domain, simply invites the relativist to repeat her attempted domain-shift and run the argument again.20 Indeed we already know exactly the form the argument is going to take: it’s an instance of the argument schema that results from replacing the sort indices ‘0’ and ‘1’ with schematic variables ‘i’ and ‘j’. The schematic versions of the premisses give us the premiss-schemas of what we shall call the schematic argument from indefinite extensibility, or simply, the schematic argument: Setsi get Collectedj and Urelementsi remain Urelementsj . The conclusion of the schematic argument is then the schema No Comprehensivej Domaini , which we deployed to schematically axiomatize relativism in Section 5.3. Each instance of the conclusion follows from the corresponding instances of the premisses in pfoi,j . To avoid drawing this conclusion, however, the absolutist still has the second option: he can simply reject one of the relativist’s premisses, Sets0 get Collected1 or Urelements0 remain Urelements1 (or, if he takes a little while to find his resolve, some other instance of either premiss schema). We return to assess this option in Sections 7.3 and 7.4. Before that, we consider a modal formulation of the relativist’s argument.

19 Proof sketch: we give a sorted plural version of the Russell Reductio in pfo. Suppose Comprehensive 1 Domain0 , for reductio. Applying Sets0 get Collected1 (and Plural Comprehension) we obtain a Russell set1 —{x0 : ß0 x0 ∧ ¬x0 ∈1 x0 }—that has as elements1 those sets0 that lack themselves as elements1 . By Zermelo’s tweak of Russell’s reasoning, the Russell set1 is not a set0 . But nor is it an urelement0 , by Urelements0 remain Urelements1 . It follows that the Russell set1 is not a thing0 . But it is one of the zero of more things0 that Comprehensive1 Domain0 says comprise every thing1 . This contradicts Auxiliary Truism0,1 . 20 Compare Williamson (2003, pp. 434–5).

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 everything, more or less

7.2 The modal argument The schematic argument permits the relativist to indicate, schematically, the form of her argument against the comprehensiveness of M0 , and against the comprehensiveness of M1 , and M2 , and so on. But just as no instance of No Comprehensivej Domaini states relativism, no instance of her argument-schema establishes it. Relativists willing to countenance suitably interpreted modal operators can go further. Recall from Chapter 6 that hybrid relativism—which we may also, reasonably, call hybrid absolutism—combines relativism about quantifiers (∀-relativism) with absolutism about modalized quantifiers (∀-absolutism).21 On this view, no quantifier ∀v or ∀vv in a Lpsu -formula φ attains absolute generality. However, absolute generality is achieved by the modalized quantifiers that occur in the modalization of this formula—φ ◇ —which prefixes each of its universal quantifiers with a  and each of its existential quantifiers and atomic subformulas with a ◇.22 This is because, under its interpretational interpretation, the modal operator  serves to generalize about admissible interpretations of Lpsu : ψ is true if and only if ψ is true under absolutely every—∀—admissible interpretation M, S, E. The language Lmpsu , enriching Lpsu with relativist-friendly modal operators, is governed by the modal plural logic mpfo.23 The modal setting casts Russell’s Lesson in a rather different light.24 According to hybrid relativism, we can obtain an absolutely general version of the Naive Comprehension Schema, and related axioms, by modalizing them. The Modalized Naive Comprehension Schema remains inconsistent in mpfo. Moreover, an inconsistent instance of this schema follows in mpfo from the modalizations of the Naive Comprehension Axiom and the Russellian instance of the Plural Comprehension Schema:25 Modalized Naive Comprehension Axiom. For absolutely any zero or more items, there is potentially a set that comprises them. ∀xx ◇ ∃s(s ≡◇ xx) Modalized Plural Comprehension Schema. Given a modalized condition φ ◇ (x), there are potentially zero or more items that comprise absolutely every satisfier of φ ◇ (x). ◇ ∃xx(xx ≡◇ x φ(x)) In the first, the string ◇ ∃s abbreviates a modalized quantifier relativized to the modalization of the set predicate.26 To employ modal analogues of our earlier terminology, Modalized Plural Comprehension tells us that every modalized condition φ ◇ (x) is plurally-comprehsible: sooner or later, we reach an interpretation whose universe contains the plurality comprising absolutely every satisfier of φ ◇ (x). The Modalized 21

22 See Section 6.6. See Sections 6.2–6.3. Officially  is defined in terms of the two primitive modal operators in Lmpsu , > and < . See Section 6.1. 24 See Linnebo (2010, pp. 156–8). 25 Side-condition: φ(x) is a L ◇ psu -formula which lacks xx free. Notation: we write (s ≡ xx) to abbreviate (s ≡ xx)◇ , applying a similar convention to other infix symbols (defined and primitive). 26 In other words: ∀sφ ◇ (s) and ◇ ∃sφ ◇ (s) abbreviate the formulas ∀v(◇ ßv → φ ◇ (v)) and ◇ ∃v(◇ ßv ∧ φ ◇ (v)). 23

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the modal argument



Naive Comprehension Axiom tells us that absolutely every plurality is collectable: a set comprising the plurality is available in some admissible interpretation’s universe. Officially, these two notions are defined by modalizing the earlier definitions of collected and plurally-comprehended.27 In the modal setting, however, it is far from obvious that we should blame Russell’s paradox on unlimited collectability as opposed to liberal comprehensibility. Neither the axiom nor the schema is proved or refuted in mpfo. Moreover, as noted in Section 6.3, unlike its non-modal counterpart, Modalized Plural Comprehension has no claim to be a trivial truth. In particular, indefinitely extensible conditions fail to be plurallycomprehensible, in the sense captured by Modalized Plural Comprehension. The outcome of the absolutism–relativism debate is closely bound up with the status of the modalized axiom and axiom-schema. The hybrid relativist can capture the core of her view with a modal thesis denying the availability of an absolutely comprehensive (plurality-encoded) domain: No Absolutely Comprehensive Domain. Absolutely no zero or more items comprise absolutely everything. ¬ ◇ ∃xx∀x(x ≺◇ xx) On its own, the modal logic mpfo (which includes the non-modal version of Plural Comprehension) neither proves nor refutes this thesis. With the addition of the Modalized Naive Comprehension Axiom, however, mpfo proves No Absolutely Comprehensive Domain. And with the addition of Modalized Plural Comprehension, mpfo refutes it. The former derivation gives us the basis for an argument against an absolutely comprehensive domain, based on an appealing plenitude principle for sets. Our official version of what we shall call the modal argument from indefinite extensibility, however, relies on a slightly weaker non-logical premiss:28 Unlimited Collectability for Sets. Absolutely any zero or more sets are collectable. ∀xx(ßxx → ◇ ∃s(s ≡◇ xx)) The premiss weakens Modalized Naive Comprehension by restricting the plural quantifier to a defined plural condition—ßxx—that applies only to pluralities of sets: ßxx =df ∀x(x ≺ xx → ßx) The argument also relies on the mpfo-axiom capturing the stability of the setpredicate:29 27 In other words: ‘φ ◇ (x) is plurally comprehensible’ is regimented by a modalized instance of Plural Comprehension, [∃xx(xx ≡x φ(x))]◇ ; ‘xx is collectable’ is regimented as [∃s(s ≡ xx)]◇ . 28 This is not quite the modalization of the relativized version of Naive Comprehension. For it does not prefix the set predicate relativizing the leading plural quantifier with modal operators. The importance of this difference will become clear in our discussion of third-way absolutism in Section 7.4. 29 Proof sketch: the premiss, Unlimited Collectability for Sets, is equivalent in mpfo to the modalization of the following restricted version of the Naive Comprehension Axiom:

(∗)

∀ss∃s(s ≡ ss)

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 everything, more or less Stability Axiom. Absolutely anything that is a potential set is a set. ∀x(◇ ßx → ßx) Much like the second premiss of the schematic argument, the Stability Axiom rules out non-sets available in the universe of one admissible interpretation ‘becoming’ sets by the lights of another. The modal argument from indefinite extensibility takes Unlimited Collectability for Sets as its sole non-logical premiss. This suffices in mpfo, with the help of the Stability Axiom, to derive the argument’s conclusion, No Absolutely Comprehensive Domain. Henceforth, for brevity, we usually refer to the modal argument from indefinite extensibility simply as the modal argument.

7.3 Uncollectability The modal and schematic arguments share a common core. Both seek to establish that no quantifier’s domain is absolutely comprehensive on the basis of a premiss articulating a maximally liberal answer to the question of which pluralities of sets are collectable. Indeed, in the case of the modal argument, this plenitude principle is the argument’s only (non-logical) premiss. This section considers two ways to respond to the argument: accept the relativist’s conclusion and give up on an absolutely comprehensive domain (Option 1), or reject her plenitude assumption—Setsi get Collectedj or Unlimited Collectability for Sets— and limit which pluralities are collectable as a set (Option 2). Of course, there’s also a third possible response (Option 3): reject the other premiss of the schematic argument, Urelementsi remain Urelementsj (or the Stability Axiom in the modal case). But we defer discussion of this heterodox absolutist-friendly alternative until Section 7.4. This section proceeds as follows: we first establish the coherence of Option 1 and Option 2. Once we have thereby dispensed with the naivety rejoinder, we can assess the two options on their relative merits. In particular, an advocate of the absolutist-friendly Option 2 faces an important explanatory challenge: he owes us an explanation of what makes uncollectable pluralities uncollectable. The section concludes by assessing two prominent ways in which the absolutist may attempt to meet this challenge, by an appeal to ‘limitation of size’ or by an appeal to the iterative conception of set.

The Zermellian hierarchy and the Cantorian universe Each of the two responses to the modal and schematic argument is naturally twinned with a different view of Zermelo–Fraenkel set theory. Let’s begin by outlining the two competing accounts.

In light of the Flattening thesis from Section 6.3, the premiss proves the mpfo-inconsistent, unmodalized version of (∗), when taken in conjunction with the absolutist-friendly thesis, Absolutely Comprehensive Domain. Consequently, Unlimited Collectability for Sets proves the negation of Absolutely Comprehensive Domain. Since the premiss begins with a , mpfo permits us to necessitate this conclusion to obtain No Absolutely Comprehensive Domain.

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uncollectability



o p t i o n  : z e r m e l l ia n r e l at i v i s m The obvious option for the relativist is to accept the premiss and conclusion of the modal argument. The Zermellian relativist, as we call her, accepts both No Absolutely Comprehensive Domain and Unlimited Collectability for Sets, and indeed the Modalized Naive Comprehension Axiom (while also endorsing mpfo). The choice of label reflects the fact that the view lends itself to a Zermellian account of zfcsup . Zermelo’s Quasi-Categoricity Theorem shows that even when we fix an underlying domain of urelements U, zfcsup admits of standard models with greater and greater inaccessible height: M0 , S0 , E0 , M1 , S1 , E1 , and so on. As usual, each universe Mα is a cumulative rank Vκα (U) comprising the sets based on the urelements in U of rank less than the inaccessible cardinal κα —metaphorically: the sets formed in the iterative process, beginning with the urelements in U, when we pause set formation just short of stage κα . The extensions Sα and Eα then encode the intended extensions of the set and element–set predicates based on this universe. On the Zermellian account,30 the axioms of zfcsup describe well-behaved initial segments of a potential hierarchy. There is no maximally-inclusive model, whether set- or plurality-encoded. Instead, absolutely any MT- or P-interpretation encoding a standard model—no matter how extensive—may be surpassed by another, more liberal one (according to which the first is realized as a set). More inclusive models may be required for certain purposes, but none is regarded as the intended interpretation. The theory admits of an open-ended sequence of intended admissible interpretations encoded by ever more inclusive standard models. The coherence of Zermellian relativism may be demonstrated with an MThierarchy which miniaturizes the potential hierarchy as a set-structure. Let a Zermellian MT-hierarchy be an MT-hierarchy indexed by the ordinals α preceding some set-limit-ordinal λ, whose admissible interpretations are taller and taller standard models Mα , Sα , Eα  (where Mα = Vκα (U), for inaccessible κα , and κα < κβ for α < β). Any Zermellian MT-hierarchy then renders both Unlimited Collectability for Sets and No Absolutely Comprehensive Domain true relative to each of its admissible interpretations. Each admissible interpretation in a Zermellian MThierarchy also renders true the Modalized Naive Comprehension Axiom and each zfcsup -axiom.31 Similarly, in the case of the schematic argument, the Zermellian potential hierarchy sustains any admissible instance of the premisses and conclusion: Setsi get Collectedj , Urelementsi remain Urelementsj and No Comprehensivej Domaini all come out true whenever the sort index replacing ‘i’ is associated with one standard model and the sort index replacing ‘j’ is associated with a more inclusive one. o p t i o n  : c a n t o r ia n a b s o lu t i s m The obvious option for the absolutist is to reject the premiss and conclusion of the modal argument. The Cantorian absolutist, as we call him, rejects No 30

See Section 2.4. The assumption that there is a Zermellian MT-hierarchy amounts to a modest large cardinal assumption: there is an ω-sequence of inaccessibles. The mpfo-consistency of Unlimited Collectability for Sets and No Absolutely Comprehensive Domain may be demonstrated without this assumption, by replacing each universe Vκα (U) in a Zermellian hierarchy with Vα (U) (and restricting Sα and Eα to this universe). 31

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 everything, more or less Absolutely Comprehensive Domain and Unlimited Collectability for Sets (although he endorses mpfo). The view is so called because it is naturally combined with a conception of Zermelo–Fraenkel set theory closer to Cantor’s conception of a single absolutely infinite set-theoretic universe. Of course, assuming he accepts modest large cardinal axioms (e.g. the sequence of inaccessibles is cofinal with the sequence of ordinals), the absolutist is committed to there being an unbounded sequence of set-models of zfcsup of the kind Zermelo describes. Nonetheless, the Cantorian absolutist may claim, no set-model encodes an intended interpretation of this theory. Instead, on this account, the sole (non-set-encoded) intended interpretation—M∞ , S∞ , E∞ — has a universe that comprises absolutely everything, and extensions S∞ and E∞ that respectively comprise absolutely every set and absolutely every element–set pair. On this account, assuming she starts with the intended interpretation, the relativist’s iterated attempts to expand the universe are bound to fail. The best she can hope for is to leave the interpretation static, so that absolutely every admissible interpretation Mα , Sα , Eα  is the maximal one (i.e. Mα , Sα , Eα  = M∞ , S∞ , E∞ ). The relativist’s would-be Zermellian hierarchy is thus flattened to give the Cantorian universe, rendering each Lpsu -formula equivalent to its modalization.32 A flat Cantorian hierarchy is not ruled out by the underlying modal logic mpfo. But it renders false both the premiss and conclusion of the modal argument. Instead, on this account, the plurality-universe M∞ of the sole intended interpretation is absolutely comprehensive, and the familiar pluralities of sets that are not collected as a set in this universe (e.g. the non-self-membered sets, the von Neumann ordinals, etc.) are, moreover, uncollectable. As with Option 1, the coherence of this position may be established by miniaturizing the Cantorian Universe as an MT-hierarchy: a Cantorian MT-hierarchy is one whose admissible interpretations are each the very same standard set-model for zfcsup . The Cantorian universe likewise renders instances of the schematic argument unsound. For given any instance of Setsi get Collectedj and No Comprehensivej Domaini in which both sorts of expression are interpreted by the same interpretation M∞ , S∞ , E∞ , both this premiss and the conclusion come out false (although the corresponding instance of Urelementsi remain Urelementsj comes out true).

Russell’s and Burali-Forti’s paradox revisited The relativist’s response to the naivety rejoinder should now be clear. The modal argument manifestly does not rely on an incoherent conception of set. The sole premiss of the modal argument is consistent in mpfo. Moreover, assuming that there is the sequence of standard models required to construct a Zermellian MT-hierarchy, Unlimited Collectability for Sets is consistent in the modal set theory zfcsup , which adds to mpfo the result of prefixing each zfcsup -axiom with a . Analogous remarks apply in the schematic case. Let zfcsup,i be the notational variant of zfcsup that uniformly adds subscripts for the sort index i to the variables 32

See Section 6.3.

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uncollectability



and non-logical predicates in its axioms. This theory is then consistent with Setsi get Collectedj and Urelementsi remain Urelementsj in the sorted plural logic pfoi,j . The absolutist, then, cannot hope to refute the Zermellian relativist’s liberal attitude towards collectability simply by appealing to Zermelo–Fraenkel set theory as articulated in zfcsup or zfcsup,i . But nor—as the flat Cantorian MT-hierarchies illustrate—can the relativist hope to establish her modal or schematic conclusion on the same basis. The decision between Zermellian relativism and Cantorian absolutism must be settled on other grounds. One appealing feature of the relativist view is that it opens the way to the attractive Zermellian response to the set-theoretic paradoxes canvassed in Section 2.4. The attempt to run Russell’s or Burali-Forti’s paradox in the context of zfcsup quickly leads to the following zfcsup -theorems:33 (1) No set comprises every non-self-membered item. ¬∃s∀x(x ∈ s ↔ ¬x ∈ x) (2) No set comprises every (von Neumann) ordinal.34 ¬∃s∀x(x ∈ s ↔ ord(x)) Interpreted by a standard model of zfcsup , encoded as an MT- or P-interpretation, M, S, E, these theorems tell us, in effect, that neither the plurality of non-selfmembered items in M nor the plurality of ordinals in M is collected as a set in M. But Zermelo–Fraenkel set theory, as we just noted, is silent on the further question of whether these uncollected pluralities are also uncollectable. The Zermellian relativist characteristically claims that each uncollected plurality of items in M forms a set belonging to the universe of a more inclusive standard model. More generally, as per the Modalized Naive Comprehension Axiom, absolutely any zero or more items are collectable. Zermelo claims two benefits for his potential conception of the hierarchy: first it permits us to secure what he describes as ‘unlimited applicability’ for set theory (1930, p. 427); second, it permits us to resolve the paradoxes without ‘constriction and mutilation’ (1930, p. 431).35 As we saw in Chapter 3, applications of model theory to the semantics of natural language provide one nice illustration of the first benefit. A Zermellian relativist can take standard model-theoretic semantics at face value and use sets to encode arbitrary semantic values. In contrast, the absolutist cannot hope to encode arbitrary extensions as sets (or indeed, objects). Instead, if he is to endorse Universe-based Semantic Optimism— the appealing thesis that we can generalize about any possible interpretation of our expressions based on a given universe—the absolutist must help himself to an infinite hierarchy of different types of ideology, such as the generalized-plural hierarchy: singular, plural, superplural, and so on.36

33

In the first case, the theorem is also a theorem of the underlying logic. Notation: ord(x) is a Lpsu -formula giving a standard definition of von Neumann ordinal (i.e. a pure transitive set whose elements are well-ordered by ∈). 35 36 See Section 2.4. See Section 3.6. 34

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 everything, more or less The use of set-ordinals to encode order-types provides a more mathematical example of ‘unlimited applicability’. Standardly, the order-type of a well-ordered sequence is encoded by a von Neumann ordinal α—i.e {β : β < α}—whose elements make up an initial segment of the sequence of ordinals that is order-isomorphic to the sequence in question. According to the Cantorian absolutist, however, we cannot hope to encode the order-type of any one or more well-ordered items in this way. The sequence of ordinals in the universe he takes to be absolutely comprehensive provides a familiar example. On von Neumann’s approach, their order-type should be encoded by the set of von Neumann ordinals in that universe. On the absolutist’s view, however, the content of the zfcsup -theorem (2) (under its intended interpretation) is precisely that there is absolutely no such set-ordinal. But we are not forced to limit the applicability of set-ordinals in this way. As Zermelo notes,37 when we adopt his potential conception of the hierarchy, the sequence of ordinals in the universe of any admissible interpretation has its order-type encoded as a set-ordinal available in the universe of a more inclusive admissible interpretation. More generally, absolutely any zero or more well-ordered items can have their ordertype encoded as a von Neumann set-ordinal.38 The unlimited applicability of set-extensions to encode extensions and set-ordinals to encode order-types, and so on, is closely bound up with the second benefit Zermelo identifies: a non-arbitrary response to the set-theoretic paradoxes. Of course, it’s not enough simply to frame a consistent alternative to the Naive Comprehension Axiom which tells us that such and such pluralities are collectable and such and such pluralities are uncollectable. A satisfying, non ad-hoc response to the paradoxes also needs to give us a well-motivated account of what makes the collectable pluralities collectable, and what makes the uncollectable pluralities uncollectable.39 The Zermellian relativist has a neat way to side step the second, harder-looking part of this explanatory challenge: she may uphold Modalized Naive Comprehension according to which absolutely no plurality is uncollectable. What, then, does it take for a plurality to be collectable? The Zermellian relativist has available the following answer: nothing at all. This answer strikes me as both appealingly simple and clearly non-arbitrary. The relativist’s minimally demanding answer, of course, is not open to the Cantorian absolutist. The uncollectable pluralities he posits go to show that something is clearly required in order for a plurality to be collectable. What’s less easy to see is what this something might be. It’s certainly not as if a collectable plurality must have members that stand in some non-arbitrary relation. This would-be necessary condition for collectability is ruled out by zfcsup , according to which, for instance, any plurality of

37

Zermelo (1930) encodes set-ordinals in a different way. But his point also applies to von Neumann ordinals. 38 Hellman (2011) develops a similarly permissive approach to encoding order-types in the context of modal-structuralism. 39 This issue is pressed against absolutism by Hellman (2006, p. 81) and Lavine (2006, p. 145). See also Linnebo (2010).

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uncollectability



urelements is collected, no matter how gerrymandered and non-joint carving. What, then, makes the uncollectable pluralities uncollectable? To make the issue vivid, we might imagine some Cantorian deity drawing a dividing line between two kinds of plurality. On one side, the chosen superplurality comprises the pluralities that are collectable; on the other side, lie those unfavoured pluralities that are uncollectable. Modulo the dispensable imagery, which superplurality is chosen clearly constitutes an important fact about sets. Indeed the axioms of set theory, and its many proposed extensions, stand or fall according to where the dividing line is drawn. Advocates of the Cantorian conception owe us an explanation of why the line falls where it does: why is the chosen superplurality chosen? Two putative explanations immediately suggest themselves: (i) the uncollectable pluralities are uncollectable because their members are too numerous; or (ii) because there’s no stage in the iterative hierarchy when all of their members are available. The remainder of this section assesses each style of explanation in turn.

Limitation of size The thought that collectability is closely bound up with cardinality has played an important role in the development of set theory. As early as Russell (1906), or even perhaps Cantor’s description of ‘inconsistent multiplicities’ as ‘absolutely infinite’ (1899, p. 114), responses to Russell’s paradox have often been shaped by the thought that, as Russell puts it, ‘what classes have to avoid is excessive size’ (1906, p. 37). And the thought persists in more recent accounts. Fraenkel, Bar-Hillel, and Lévy describe it as their ‘guiding principle’ for set theory that the Naive Comprehension Schema be restricted to instances that ‘assert the existence of sets which are not too “big” compared to sets which we already have’ (1973, p. 32).40 But how many is too many? If the Cantorian absolutist is to use a limitation of size account to explain collectability failures, rather than merely an inchoate rule of thumb for generating new axioms, he needs to articulate and defend the upper cardinality limit he imposes on collectable pluralities. In the context of zfcsup , cardinals are standardly encoded as set-cardinals, assigned only to collected pluralities. As in the semantic case, however, the absolutist may overcome the limited applicability of set-cardinals on his view by appealing to plural resources. For instance, he may generalize some of the usual definitions from the settheoretic account of cardinals as follows. We say that zero or more items xx are at least as numerous as, or have greater-or-equal cardinality to, zero or more items yy (in symbols: |xx| ≥ |yy|) if there is a (plurality-encoded) function mapping xx onto yy (i.e. such that every member of yy has at least one member of xx mapped to it). We also say that xx are fewer than, or have lesser cardinality than, yy (in symbols: |xx| < |yy|) if yy are at least as numerous as xx, but not vice versa.41 With the rudiments of cardinality theory extended to plurality-cardinals in this way, we can give precise content to the limitation of size thought. Writing oo for the

40 41

See Hallett (1984, ch. 5) for a historical overview, and criticism. The definitions follow Linnebo (2010, p. 151).

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 everything, more or less plurality comprising the (von Neumann) ordinals, Linnebo (2010, sec. III) makes the following proposal on behalf of the Cantorian absolutist: Limitation of Size. Zero or more items are collected if and only if they are fewer than the ordinals. ∀xx(∃s(s ≡ xx) ↔ |xx| < |oo|) As he notes, whether or not the proposed cardinality-threshold admits of an explicit characterization,42 we can use the plurality of ordinals as a ‘measuring stick’, just as we might fix a unit of length as that of a designated object (p. 152). In fact, the use of this designated plurality of ordinals to encode the cardinality of its members is exactly on a par with the canonical encoding of cardinality in zfcsup , which assigns a designated set of ordinals to encode the cardinality of its elements. Limitation of Size, so articulated, flows naturally from standard set theory. For, as Linnebo demonstrates,43 Limitation of Size follows from the zfcsup -version of the Replacement Axiom in conjunction with the following assumption: Cardinal Comparability. For any pluralities xx and yy, either xx are fewer than yy or xx are at least as many as yy. ∀xx∀yy(|xx| < |yy| ∨ |xx| ≥ |yy|) The additional assumption is not provable in zfcsup ; it serves, effectively, as a strong choice assumption.44 Nonetheless, Cardinal Comparability straightforwardly generalizes a property that zfcsup ascribes to set-cardinals. And, aside from its logical strength, there seems little reason to doubt it. Setting the threshold cardinality as the cardinality of the ordinals, then, Limitation of Size draws a sharp dividing line between collectable and uncollectable pluralities. But does it also permit the Cantorian absolutist to explain why the dividing line falls where it does? The putative explanation goes beyond the universally quantified biconditional in the statement of Limitation of Size. Instead, on the proposed account, when a plurality is collectable, it is because its members are fewer than the ordinals, and when a plurality is uncollectable it is because its members are not fewer than the ordinals. Linnebo (2010, p. 152) raises the obvious doubt about this style of explanation: ‘Why should this particular cardinality mark the threshold? Why not some other cardinality?’ To bring out the issue, it’s helpful to contrast the proposed Limitation of Size account with its relativist alternative. According to Zermellian relativism, each admissible interpretation’s universe is an inaccessible rank Vκα (U). The relevant inaccessible cardinal κα encodes the cardinality of the ordinals in Vκα (U) and provides the model’s cardinality-threshold for collectedness: a plurality is collected in Vκα (U) if and only if its members are fewer than the elements of κα . 42

Some cardinalities can be thus characterized. See Shapiro (1991, sec. 5.1.2). Linnebo (2010, app. B). 44 Linnebo (2010, app. A) shows that it follows from the Global Well-Ordering principle that there is a (plurality-encoded) well-ordering of the universe. 43

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uncollectability



But the Zermellian relativist need not privilege κα above other (sufficiently large) inaccessible cardinals. Later models in the open-ended sequence set a higherthreshold. Any zero or more members of Vκα (U) reaching κα in cardinality, and thus too numerous to be collected as a set in Vκα (U), are nonetheless collected in the more inclusive universes further up the Zermellian hierarchy. As Zermelo puts it, the inaccessible cardinals are merely ‘relative way stations’ (1930, p. 431). The Cantorian absolutist sees things quite differently. For when zfcsup is interpreted by its sole intended model M∞ , S∞ , E∞ , the inaccessible size of the universe—κ∞ (plurally-encoded by the ordinals in M∞ )—has absolute significance. By the absolutist’s lights, it marks the absolute cardinality-threshold, the dividing line not merely between collected and uncollected in a given model but between collectable and uncollectable. Metaphorically: we continue forming sets through every stage before κ∞ , and then the process stops. Full stop. But the absolutist is yet to tell us why it is κ∞ , rather than any other inaccessible cardinality, that the Cantorian deity smiles upon. Moreover, as Linnebo notes, the account breaks down when we consider why, according to the Limitation of Size explanation, the plurality of ordinals is uncollectable: . . . why cannot the plurality oo form a set, which would then be an additional ordinal, larger than any member of oo? According to the view under discussion, the explanation is that oo are too many to form a set, where being too many is defined as being as many as oo. Thus, the proposed explanation moves in a tiny circle. The threshold cardinality is what it is because of the cardinality of the plurality of all ordinals, but the cardinality of this plurality is what it is because of the threshold. (2010, p. 153–4)

Can the Cantorian absolutist do better? Linnebo is doubtful: ‘it is hard to see how an explanation of why some pluralities fail to form sets can avoid appealing to some principle of limitation of size’ (2010, p. 151). His reason is that Limitation of Size follows from two widely held assumptions, namely the Replacement Axiom from zfcsup and Cardinal Comparability. But it’s not immediately clear why this derivation should commit a Cantorian absolutist who accepts these assumptions to a limitation-of-sizebased explanation of collectability failures.45 Given his assumptions, the Cantorian absolutist is committed to the line falling where Limitation of Size says it does. But it’s still open to him to search for some other account to explain why it falls where it does, one which gives him an independent reason to think, for instance, that the plurality of ordinals is uncollectable. Let us turn, then, to the second putative explanation.

The iterative conception Pace Fraenkel, et al., no limitation-of-size style cardinality-threshold is explicit in the conception of set underlying zfcsup . But the same cannot be said of the cumulative rank structure described by the iterative conception. Can the Cantorian absolutist exploit this structure in order to explain the limits of collectability as he sees them?

45

Thanks to Gabriel Uzquiano for pressing this point in discussion.

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 everything, more or less As with Limitation of Size, the proposal takes a precise form in the context of standard set theory. Recall that the rank of a set is the least ordinal α for which the set is a subset of Vα (U)—metaphorically: the first stage in the iterative process when every member of the set is available. Without extension this time, zfcsup delivers the following general theorem relating a plurality’s collectedness to the ranks of its members.46 Iterative Conception. Zero or more items are collected if and only if their ranks are bounded by some ordinal. ∀xx(∃s(s ≡ xx) ↔ ∃α∀s(s ≺ xx → rank(s) < α)) Much as before, absolutists and relativists who accept zfcsup should accept that this states a necessary and sufficient condition for collectedness. But can the Cantorian further exploit this theorem to explain where the limits of collectability lie? The putative explanation runs as follows: when a plurality is collectable, it is because some ordinal bounds the ranks of the sets it contains; when a plurality is uncollectable, it is because the ranks of the plurality’s sets are cofinal with the ordinals (i.e. there is no ordinal exceeding the rank of every set in the plurality). This chimes in with the usual sort of story about uncollectable pluralities told in the context of the iterative conception. For in the context of zfcsup , a set has rank less than α just in case it is a member of the cumulative rank Vα (U) or, in terms of the formation metaphor, it is formed no later than stage α. Thus—to indulge in metaphorical explanation—the finite ordinals, for instance, are collectable because there is some stage—namely, stage ω—by which they have all been formed, and are available to be collected into a set at the next stage; but the finite and transfinite ordinals (i.e. all of them) are uncollectable because new ordinals are formed at every successor-stage—at no stage are all of the ordinals available to be collected into a set. Does this improve on the unpromising limitation-of-size explanation? Setting aside unappealing heterodox accounts of sets that permit pure sets to be literally created,47 it’s clear that the process metaphor cannot add explanatory force. As Boolos points out, it is, after all, only a ‘narrative convention’: a vivid way to describe the cumulative rank structure (1989, pp. 90–1). The availability of the finite ordinals at some stage in the process of set formation can’t be the real reason that they are collectable because, in reality, there isn’t any such process. Instead, the metaphorical explanation is at best a colourful way of presenting a more sober explanation. Stripping away the ‘process’talk, the proposed explanation in our two test cases is that the finite ordinals are collectable because there is some ordinal that exceeds each of their ranks; the finite and transfinite ordinals are uncollectable because there is no such ordinal. In this bare form, however, the explanation based on Iterative Conception fails for much the same reasons that its counterpart based on Limitation of Size did. For once again, in crucial cases, the proposed explanation quickly comes round in a circle. Take for instance the plurality of finite ordinals. In the context of zfcsup , von Neumann ordinals measure their own rank (i.e. rank(α) = α) and are ordered by membership (i.e. α < β iff α ∈ β). We are told that the finite ordinals are collectable—on the The quantifier ∃α and apparent function symbol rank(s) (formalizing the rank of s) are defined as usual; see Appendix A.1. 47 See Section 2.3. 46

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instability



absolutist’s account: some set comprises the finite ordinals as elements—because some ordinal exceeds the rank of each finite ordinal. But the least ordinal bounding their rank is the very same ordinal that witnesses their collectability. The only reason we have to think their rank is bounded is because the finite ordinals are collected as the elements of ω. We’ve come full circle. To briefly take stock: the obvious response for the absolutist, when faced with the modal or schematic formulation of the argument from indefinite extensibility, is to deny the premiss or premiss schema articulating the relativist’s liberal attitude towards which pluralities are collectable. In conjunction with plausible auxiliary assumptions (e.g. zfcsup and Cardinal Comparability) this commits the absolutist to a sharp line between the collectable and uncollectable pluralities, as articulated by Limitation of Size and Iterative Conception. By the Cantorian absolutist’s lights, when interpreted by the sole intended interpretation of zfcsup , these two collectedness theses tell him precisely where the line falls. But neither succeeds in explaining why it falls where it does. And without even the beginnings of an independent motivation for limiting collectability, simply to reject the relativist’s plenitude premisses on the grounds that they come into conflict with absolutism seems to me an unpleasantly ad hoc response to the paradoxes.

7.4 Instability There is, however, a much less arbitrary-seeming option open to the absolutist. Rather than seeking to draw a line between collectable and uncollectable pluralities of sets, he may take all pluralities of sets to be collectable by taking set to be indefinitely extensible. Dummett sometimes seems to assume that a concept with an indefinitely extensible sub-concept is automatically itself indefinitely extensible. He explains that quantification over absolutely every object is impossible because ‘the (formal) concept object (or identical with itself ) embraces all others; since some of the others are indefinitely extensible, it must be also’ (1994a, p. 249). But this assumption is contestable. Recent work by some absolutists combines absolutism with something much like the indefinite extensibility Dummett attaches to set. Williamson suggests that attaching wider and wider extensions to the set predicate permits absolutists to accommodate the intuitions behind indefinite-extensibilitybased arguments for relativism: For given any reasonable assignment of meaning to the word ‘set’ we can assign it a more inclusive meaning while feeling that we are going on in the same way…The inconsistency is not in any one meaning . . . it is in the attempt to combine all the different meanings that we could reasonably assign it into a single super-meaning. (1998a, p. 20)

In a similar vein, Uzquiano seeks to reconcile absolute generality with what he describes as a ‘linguistic’ account of the indefinite extensibility of ‘set’. Citing Gödel’s presentation of the iterative conception of set, his aim is to ‘reframe Gödel’s procedure in terms of a cumulative process of reinterpretation of the primitive set-theoretic vocabulary’, deploying modal resources to describe the possible reinterpretations (2015, p. 150). Crucially, however, both views seek to reconcile indefinite extensibility with absolutism. In Dummett’s terms, although Williamson and Uzquiano take set to be

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 everything, more or less indefinitely extensible, they do not take object to be indefinitely extensible. The ever more liberal extensions for the set predicate are always contained within the absolutely comprehensive domain. Unlike on the Zermellian relativist’s view, therefore, new sets are not new objects: as Uzquiano puts it, ‘what is a nonset on one interpretation of the set-theoretic vocabulary can become a set on a more comprehensive interpretation of [the set predicate]’ (2015, p. 156). Williamson and Uzquiano’s take on indefinite extensibility suggests a third way to respond to the modal and schematic argument: o p t i o n  : t h i r d - way a b s o lu t i s m The third response to the modal argument is to question the underlying modal logic, mpfo. The third-way absolutist, as we shall call him, accepts the argument’s sole premiss, Unlimited Collectability for Sets. To avoid its conclusion, No Absolutely Comprehensive Domain, he rejects the Stability Axiom for the set and element–set predicate in mpfo. Third-way absolutism is naturally paired with a conception of the set-theoretic hierarchy that combines aspects of the Zermellian hierarchy and the Cantorian universe. On this view, something like iterated relativist-style attempts to liberalize the interpretation do result in a sequence of different interpretations for zfcsup : M∞ , S0 , E0 , M∞ , S1 , E1 , M∞ , S2 , E2 , . . . Only, in contrast to the relativist’s Zermellian picture, the more and more liberal extensions for the set and membership predicate lie within the absolutely comprehensive domain: S0 ⊂ S1 ⊂ · · · ⊂ M∞ and E0 ⊂ E1 ⊂ · · · ⊂ M∞ .48 As before, the coherence of the view may be established with a suitable setstructure. A third-way set-hierarchy may be obtained from a Zermellian MT-hierarchy {Mα , Sα , Eα  : α < λ} by replacing each MT-interpretation’s universe M α , with the universe M∞ comprising every item in any such universe (i.e. M∞ =df α 1). (d) Any plurality of at most two objectsi is collected by a unique seti . Each collectability thesis (a)–(d) is a theorem of the theory zfcup,i which drops the Urelementi Seti Axiom from zfcsup,i ; (b)–(d) are also theorems of its subtheory which restricts the Axiom of Replacementi to countable setsi ; each instance of (c) (for fixed finite n > 1) is a theorem of the subtheory which only contains four of its first-order axioms: Extensionalityi , Empty Seti , Pairingi , and Unioni ; finally, (d) is a theorem of the subtheory comprising just Extensionalityi , Empty Seti , and Pairingi . Nonetheless, the third-way absolutist must reject each of (a)–(d).

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 everything, more or less The trouble again concerns cardinality: the third-way absolutist package of Comprehensivej Domaini and Setsi get Collectedj refutes (d) in pfoi,j , via a plural version of Cantor’s diagonal argument.51 The same package of assumptions consequently also refutes (a), (b), and (c) in pfoi,j (assuming in the first case that there is at least one two-membered set). The price we pay for third-way absolutism, then, is to place severe limits on which pluralities are collectable. Not only must the third-way absolutist forego zfcsup , even the meagre combination of Extensionality, Empty Set, and Pairing is off limits. If the standard way to formulate set theory with urelements is unavailable to the third-way absolutist, might he instead appeal to a non-standard theory? He may object that zfcsup,i fails to respect an important distinction between ‘permanent’ urelementsi such as donkeys and spacetime points, which never enter the extension of the set predicate, and ‘temporary’ urelementsi such as the predicate’s extensions themselves, which do eventually ‘become’ setsj when we reach a sufficiently liberal structure. In much this spirit, Uzquiano (2015, p. 15) suggests interpreting set theorists’ assertions to be tacitly restricted to a domain of available objects. Restricted Pairing may then be formulated by adding a further unary predicate av (read ‘available’) to the language of zfcsup,i :52 Restricted Pairing. Any one or two available objects comprise the elements of a set. ∀xi ∀yi (avi xi ∧ avi yi → ∃si ∀zi (zi ∈ si ↔ zi = xi ∨ zi = yi )) The other axioms may be restricted to available items in a similar way. Setting aside the obvious loss of generality, the main difficulty with Restricted Pairing is that it drastically curtails the applicability of pair sets. One important application of pairing is to define ordered pairs and n-tuples as sets in the standard way due to Kuratowski. With unrestricted Pairing, the orthodox absolutist obtains an n-tuple a1 , . . . , an  for any items a1 , . . . , an . This permits him to encode arbitrary extensions as pluralities of n-tuples; for example, on his view, the intended extension of the identity predicate is the plurality-extension comprising absolutely every ordered-pair a, a whose co-ordinates are identical. But by restricting Pairing to available objects, the third-way absolutist forgoes Kuratowski n-tuples of unavailable objects, and loses this non-set-based means to encode arbitrary extensions. The thirdway absolutist consequently lacks the set-theoretic resources required to effect the plurality-encoding of the intended interpretation of the first-order language of set theory with urelements, Lsu , outlined in Section 3.4.

51 Proof sketch: suppose for reductio that (d) holds. Then assuming Comprehensive Domain , each j i setj (which is also an objecti ) is associated with a unique seti , namely its singleton. This goes to show that setsi are at least as numerous as setsj . On the other hand, Setsi get Collectedj implies that there are fewer setsi than there are setsj (by Cantor’s diagonal argument). The plural version of Cantor’s Theorem may be formulated along similar lines to the version in Florio (2014b). 52 Uzquiano’s proposal centres on the modalization of Restricted Pairing, which in his system is equivalent to prefixing each non-logical predicate with ◇. But assuming the restriction to available items is non-vacuous, the modal axiom faces the same problem as its non-modal counterpart.

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non-comprehensibility



7.5 Non-comprehensibility Sections 7.3 and 7.4 defended the premisses of the modal and schematic arguments in favour of relativism about quantifiers. In the modal case, a relativist about quantifiers who takes modalized quantifiers to achieve absolute generality has the advantage that she can articulate the conclusion of her argument (and each of its premisses) as a single modal statement. Recall that advocates of the hybrid view—which may be reasonably called either hybrid relativism or hybrid absolutism—combine ∀-relativism with ∀-absolutism. As we noted in Section 6.6, however, the hybrid relativist faces an important challenge. In brief, if considerations relating to indefinite extensibility show that quantifiers never range over absolutely everything, why don’t analogous considerations apply to modalized quantifiers? To elaborate, the immediate lesson of the set-theoretic paradoxes remains much the same in the modal setting as in the non-modal one. The following modalized formulas follow from the modalized axioms of zfcsup in mpfo just as their unmodalized counterparts, (1) and (2), are theorems of zfcsup :53 (1)◇ Absolutely no set comprises absolutely every non-self-membered item. ¬ ◇ ∃s∀x(x ∈◇ s ↔ ¬x ∈◇ x) (2)◇ Absolutely no set comprises absolutely every (von Neumann) ordinal. ¬ ◇ ∃s∀x(x ∈◇ s ↔ ord◇ (x)) In the non-modal case, we’ve seen how the paradoxes provide the beginnings of a case for ∀-relativism. To briefly rehearse the basic strategy, the ∀-relativist defends a liberal plenitude assumption ensuring that there potentially is the Russell set (or Burali-Forti ordinal) which, according to the absolutist’s would-be maximal interpretation of the non-modal theorem (1) (or (2)), does not lie in the domain of ∃s. In the modal setting, the ∀-relativist may attempt to exploit a similar strategy to make a case for her thorough-going version of relativism. Suppose, this time, that the modal language Lmpsu is interpreted by a potential hierarchy of interpretations— M0 , S0 , E0 , M1 , S1 , E1 , and so on—as per Zermellian relativism. Suppose further that the hybrid relativist claims that absolutely nothing lies outside absolutely every universe in the hierarchy, M0 , M1 , and so on, so that ∀x generalizes about absolutely everything. To argue against ∀-absolutism, then, the thorough-going relativist may seek to identify sets which demonstrably lie outside absolutely every universe in the hybrid relativist’s potential hierarchy. The obvious candidates are sets such as the hierarchy’s Russell set r∀ or the hierarchy’s Burali-Forti ordinal ∀ (where r∀ comprises absolutely every non-self-membered item in the potential hierarchy, and ∀ comprises absolutely every ordinal in the potential hierarchy). In light of (1)◇ and (2)◇ , r∀ and ∀ are unavailable in absolutely any universe in the potential hierarchy. The crux of the case against ∀-absolutism is to argue that nonetheless—from a sufficiently liberal perspective—there is such a set as r∀ or ∀ , and consequently that there 53

Like before, (1)◇ is a theorem of the underlying logic, mpfo.

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 everything, more or less is something that the modalized quantifier ∀x fails to generalize about (under the ∀-absolutist’s would-be maximal interpretation). As before, of course, the relativist’s argument is only as good as her case for whatever plenitude principle delivers r∀ or ∀ . In the present case, however, we should expect some differences between the argument for ∀-relativism and the earlier modal argument for ∀-relativism. After all, unlike the ∀-absolutist, the hybrid Zermellian relativist accepts Unlimited Collectability for Sets and mpfo. Indeed the hybrid relativist maintains further, as per the Modalized Naive Comprehension Axiom, that absolutely any plurality is collectable. On her view, then, the non-availability of r∀ and ∀ is not due to there being an uncollectable plurality. Instead, the nonavailability of r∀ and ∀ is due to the non-plural-comprehensibility of the relevant conditions ¬x ∈◇ x and ord◇ (x): absolutely no universe in the hybrid relativist’s potential hierarchy contains a plurality of items comprising absolutely every satisfier of either condition (as they are interpreted in her modal language). But this leaves a further question unsettled: are these pluralities available in a suitably liberalized potential hierarchy (with the result that the relevant sets, r∀ and ∀ , also become available in the enlarged hierarchy)? The remainder of this section assesses a thorough-going ∀-relativist attempt to articulate and defend an affirmative answer to this question. The case for ∀-relativism follows a now-familiar pattern: (i) we outline an extended modal language; (ii) we show how the ∀-relativist may articulate her disagreement with the ∀-absolutist in this language; (iii) we articulate a liberal comprehensibility principle which conflicts with ∀-absolutism; and finally, (iv) we assess the costs of rejecting this principle.

The extended modal language Before she can frame a rigorous argument for her view, the ∀-relativist needs to articulate her disagreement with the ∀-absolutist. Here the thorough-going relativist must proceed carefully. As in Section 6.2, the hybrid relativist may stipulate that talk of ‘absolutely everything’ is to be understood in terms of the modalized quantifier ∀v. Understood this way, of course, the ∀-relativist should not deny the trivial truth that the modalized quantifier ∀v sometimes generalizes about absolutely—∀— everything. This would be every bit as futile as denying that quantifiers sometimes range over everything.54 Instead, the ∀-relativist may first seek a more liberal modal language in which to express her view. The new language is equipped with a further modal operator— —which the ∀-relativist intends to express a liberalized version of the noncircumstantial modality expressed by . To distinguish the generality achieved by -modalized quantifiers (i.e. ∀v and ∀vv) and -modalized quantifiers (i.e. ∀v and ∀vv), we shall speak informally of ‘-absolutely every’ and ‘-absolutely every’. Set up this way, the dispute between the ∀-relativist and the ∀-absolutist centres on whether -modalized quantifiers generalize about a more inclusive potential hierarchy than the one associated with -modalized quantifiers. To elucidate the new modality, it’s helpful, subject to the familiar caveats,55 to start with a Kripke-style model theory. Suppose we begin with a modal language that 54

See Sections 1.6 and 5.1.

55

See Section 6.1.

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non-comprehensibility



enriches the plural language of set theory with urelements Lpsu with a single primitive modal operator .56 Recall that, given an MT-hierarchy {Mi , Si , Ei  : i ∈ I }, truthconditions for ψ are stated as follows (relative to an index k ∈ I and an assignment σ over the hierarchy): (kk -)

ψ is truek,σ iff every i ∈ I is such that ψ is truei,σ .

The extended modal language adds two further primitive modal operators, the forwards-looking > and the backwards-looking < . The new absolute modal operator  may then be defined as before: ψ =df < ψ ∧ ψ ∧ > ψ The extended modal language is interpreted by a suitable extension of the MT-hierarchy {Mi , Si , Ei  : i ∈ I }. The extended MT-hierarchy is an MT-hierarchy {Mj , Sj , Ej  : j ∈ J }, with associated order ψ, < ψ, and ψ are given by the following Kripkean clauses (relative to an index k ∈ J and an assignment σ over the extended MThierarchy): (kk -> ) > ψ is truek,σ iff every j >J k is such that ψ is truej,σ . (kk -< ) < ψ is truek,σ iff every j , < > , < > , < > , 
∃u(ψ(u) ∧ < ¬E!u) Given these modifications to pfo, the first group of axioms and rules comprises the Lmpsu instances of the following schemas: taut, mp, fus, fug, fref, sub, and pc.4 The second group comprises the axiom- and rule-schemas that capture the order structure of the potential hierarchy:5 k

L(ψ1 → ψ2 ) → (Lψ1 → Lψ2 ) is an axiom whenever L is one of the modal operators > or < .

4 The formulation of the first-order fragment of the free logic adapts the one given by Braüner and Ghilardi (2007). 5 As far as possible we adopt the labelling of Bull and Segerberg (2001).

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 appendix a nec

Lψ is derivable from  whenever ψ is a formula that is derivable outright (i.e. from Ø) and L is one of > or < .

cv

ψ → LMψ is an axiom whenever L and M are either respectively < and ◇> or respectively > and ◇< .

löb h

◇< ψ → ◇< (ψ ∧ < ¬ψ) is an axiom whenever ψ is a formula.   Mψ1 ∧ Mψ2 → M(ψ1 ∧ ψ2 ) ∨ M(ψ1 ∧ Mψ2 ) ∨ M(ψ2 ∧ Mψ1 ) is an axiom whenever M is one of ◇> or ◇< .

d

> ψ → ◇> ψ is an axiom whenever ψ is a formula.

The first three schemas ensure that > and < conform to the axioms and rules distinctive of minimal tense logic (the system whose axioms and rules are taut, k, cv, mp, and nec). The second three capture the distinctive order-structure of the potential hierarchy, in light of the Serial Well-Order constraint imposed on the hierarchy of interpretations in Section 6.1. The third group of axioms concerns the interaction of the modality with quantifiers and predicates: cbf

> ∀vψ → ∀v> ψ is an axiom whenever v is a singular or plural variable.

sta-

∀v(◇ (v) → (v)) is an axiom whenever (v) is an atomic formula, v is the string of variables it contains free and ∀v is the corresponding string of quantifiers.

e1

◇ E!v is an axiom whenever v is a singular or plural variable.

The first two axioms respectively give voice to the Monotonicity and Stability constraints on the hierarchy of interpretations stated in Section 6.1. The last requires that variables be assigned to members or subpluralities of the universe of some admissible interpretation or other. The first-order subsystem mfo simply omits all non-Lmsu -instances of axioms and rules (including all instances of Plural Comprehension). The proof theory is sound with respect to the model theory from Sections 6.1 and 6.3. Define  mpfo ψ as usual,6 and write  mpfo ψ when ψ is truei,σ under any index i of, and any assignment σ over, any MT-hierarchy which renders truei,σ each member of . Then  mpfo ψ only if  mpfo ψ. The proof theory also sustains the deduction theorem (i.e.  ∪ {ψ1 }  ψ2 only if   ψ1 → ψ2 ). The same results hold good of mfo. Derived modal axioms The absolute and weakly-forwards-looking modal operators  and ≥ are defined in Section 6.1. The bimodal logic permits us to establish that these modal operators conform to the

Table A.4. Unimodal systems Formulated in a unimodal language with  as its sole modal operator, the minimal normal modal logic has as axioms and rules all instances of taut, k, mp, and nec in its unimodal language (taking L to be ). The modal systems listed below add the following axioms to this system: s s. s. s

t: ψ → ψ, : ψ → ψ t, , g: ◇ ψ →  ◇ ψ   t, , h: ◇ ψ1 ∧ ◇ ψ2 → ◇(ψ1 ∧ ψ2 ) ∨ ◇(ψ1 ∧ ◇ ψ2 ) ∨ ◇(ψ2 ∧ ◇ ψ1 ) t, , b : ψ →  ◇ ψ or t, : ◇ ψ →  ◇ ψ 6

See Appendix A.1.

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appendix a



axioms and rules of the familiar unimodal systems listed in Table A.4. The absolute operator  conforms to the axioms and rules distinctive of s. The weakly-forwards-looking operator ≥ conforms to those distinctive of s. (replacing  with ≥ in each axiom). Linnebo (2010, 2013) stipulates that his primitive modal operator L conforms to the axioms and rules of the weaker system s..

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a ppe n d i x b

Modalization This appendix establishes the principal results about the bimodal logic mpfo appealed to in Chapter 6. Appendices B.1–B.4 respectively take up the theses Modalized Invariance, Mirroring, Inextensibility, and Flattening. Throughout this appendix, we adopt the notational conventions listed in Table A.2. Unless indicated otherwise,  is the mpfo-derivability relation.1 B.1 Modalized invariance This appendix proves a generalized version of the Modalized Invariance thesis stated in Section 6.2, extending the result to plural formulas: Modalized Invariance.  inv[φ ◇ ]. In the thesis, φ ◇ is the modalization of a Lpsu -formula, whose free variables are v = v1 , . . . , vk and inv[φ ◇ ] =df ∀v(φ ◇ ∨ ¬φ ◇ ). Proof. It suffices to show that the following are mpfo-theorems:

(a) φ ◇ ↔ φ ◇

(b)

¬φ ◇ ↔ ¬φ ◇

Modalized Invariance follows from (a) and (b) by propositional reasoning, generalization and necessitation. To establish (a) and (b), note first that mpfo proves the following s-theorems: (1) ◇ ψ ↔  ◇ ψ (2) ¬ ◇ ψ ↔ ¬ ◇ ψ (3) ψ ↔ ψ (4) ¬ψ ↔ ¬ψ (5) (◇ ψ1 → ψ2 ) → (ψ1 → ψ2 ) (6) ¬(ψ1 → ◇ ψ2 ) ↔ ¬(ψ1 → ψ2 ) The proof of (a) and (b) then goes by induction on the complexity of φ: • φ ◇ = ◇ (v) ((v) atomic): (a) and (b) are immediate by (1) and (2). • φ ◇ = ∀vφ1◇ : (a) and (b) are immediate by (3) and (4). • φ ◇ = ¬φ1◇ : (a) is immediate from (IH.b) (i.e. the induction hypothesis (b)); (b) is almost immediate from (IH.a), prefixing both occurrences of φ1◇ with a double negation. An account of the logic mpfo and the relevant bimodal plural language Lmpsu is given in Appendix A.3. For Lpsu , the plural language of set theory with urelements, see Appendix A.1. 1

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appendix b



• φ ◇ = φ1◇ → φ2◇ : (a) right-to-left is immediate from the t-axiom; left-to-right, apply (5) to the following (obtained from IH): (φ1◇ → φ2◇ ) → (◇ φ1◇ → φ2◇ ) For (b): apply (6) to ¬(φ1◇ → φ2◇ ) ↔ ¬(φ1◇ → ◇ φ2◇ ).



Note also the following corollary of Modalized Invariance: Proposition 1 (Modal Collapse). The following are provably equivalent in mpfo on the assumption that inv[ψ]: ψ, ψ, ◇ ψ, > ψ, ◇> ψ, ≥ ψ, ◇≥ ψ, ≤ ψ, ◇≤ ψ (The exceptions, < ψ and ◇< ψ, are provably equivalent to ψ assuming ◇< , where is an arbitrary tautology.) B.2 Mirroring This appendix proves the Mirroring thesis from Section 6.2. Mirroring.  fol φ iff {γ ◇ : γ ∈ } mfo φ ◇ . In the thesis, φ is a Lsu -formula, and  a set of Lsu -formulas. Proof. We prove the two directions separately:

• Left-to-right. Suppose fol (classical first-order logic) is axiomatized by adding the axiom E!v to the nonmodal subsystem of mfo. An induction on fol-proofs shows that mfo proves the modalization of each fol-axiom and is closed under the modalization of each fol-rule. We consider just the case when  fol φ1 → ∀vφ2 is obtained using fug. Consequently , E!u fol φ1 → φ2 (u/v) has a shorter fol-proof (satisfying the side-conditions on u). We then reason in mpfo as follows, after applying the induction hypothesis (IH): {γ ◇ : γ ∈ }, ◇ E!u  φ1◇ → φ2◇ (u/v) {γ : γ ∈ }, E!u  φ1◇ → φ2◇ (u/v) {γ ◇ : γ ∈ }  φ1◇ → ∀vφ2◇ {γ ◇ : γ ∈ }  φ1◇ → ∀vφ2◇ {γ ◇ : γ ∈ }  φ1◇ → ∀vφ2◇ ◇

IH mfo fug mfo prop. 1

The desired result follows since, for γ ∈ , {γ ◇ : γ ∈ }  γ ◇ . • Right-to-left. Suppose  fol φ. By the completeness of fol, some MT-interpretation

M, S, E renders each member of  true and φ false. Clearly, M, S, E is the flattening of the MT-hierarchy { Mp , Sp , Ep  : p < ω} with each Mp , Sp , Ep  = M, S, E. Thus the modalization of each member of  is true and φ ◇ false under each MT-interpretation in the MT-hierarchy.2 It follows that {γ ◇ : γ ∈ } mfo φ ◇ by the soundness of mfo. 

2

See Section 6.2.

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 appendix b Note that the left-to-right direction still holds for plural Lpsu -formulas when we add the plural versions of us and ug to fol and replace mfo with mpfo. But the result fails for pfo, with Plural Comprehension, since mpfo does not prove the modalization of this axiom.3 B.3 Inextensibility This appendix proves a generalized-version of the Inextensibility thesis from Section 6.3: Inextensibility. inv[ψ(x)]  ¬extx [ψ(x)] ↔ ◇ ∃xx∀x(x ≺◇ xx ↔ ψ(x)) In the thesis, ψ(x) is a Lmpsu -formula (lacking xx free) and extx [ψ(x)] and inv[ψ(x)] are defined as in Appendices A.3 and B.1. The former definition may be generalized to arbitrary variables (singular or plural) as follows: extv [ψ(v)] =df ◇> ∃v(ψ(v) ∧ < ¬E!v) First, note three lemmas about inextensibility (which we state without proof): Lemma 2. inv[ψ(v)], ¬extv [ψ(v)]  ψ(v) → E!v Lemma 3. Let  = {δ1 , . . . , δk } be a set of Lmpsu -formulas without v free. If   inv[ψ(v)] and   ψ(v) → E!v, then   ¬extv [ψ(v)]. Lemma 4. Whenever L is any of < , > , or : inv[ψ1 ], ¬extv [ψ1 ]  ∀v(ψ1 → Lψ2 ) → L∀v(ψ1 → ψ2 ) The Inextensibility thesis is then a near-immediate corollary of the following proposition and Modalized Invariance (Appendix B.1). Proposition 5. Assume ψ(x) lacks xx free: inv[ψ(x)]  ¬extx [ψ(x)] ↔ ∃xx∀x(ψ(x) ↔ x ≺◇ xx) Proof. We prove the two directions of the biconditional separately:

• Left-to-right. Reason in mpfo as follows under the assumptions, left tacit below, that inv[ψ(x)] and ¬extx [ψ(x)]:4

3

E!xx, ∀x(ψ(x) ↔ x ≺ xx)  ∀x(ψ(x) →  ◇ x ≺ xx)

mpfo

 ∀x(ψ(x) → ◇ x ≺ xx)

lem. 4

See Section 6.3. We henceforth adopt this convention in our displayed proofs: assumptions explicit in the statement of the result are left tacit in the displayed proofs. To further improve readability, we avoid repeating the list of assumptions if it remains unchanged from the previous line. 4

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appendix b



Similarly (the necessitated universal closure of) the converse of the conditional in the final line follows from the invariance of ◇ x ≺ xx and its assumed inextensibility in x (applying the definition of E!xx). We then reason as follows from the biconditional: E!xx, ∀x(ψ(x) ↔ x ≺ xx)  ∀x(ψ(x) ↔ ◇ x ≺ xx) ∃xx∀x(ψ(x) ↔ x ≺ xx)  ∃xx∀x(ψ(x) ↔ ◇ x ≺ xx) Deleting the assumption in the last line (an instance of pc) and applying the deduction theorem, we obtain the desired left-to-right conditional. • Right-to-left. Inextensibility is closed under necessary equivalence: ¬extx [◇ x ≺ xx], ∀x(ψ(x) ↔ ◇ x ≺ xx)  ¬extx [ψ(x)] The first-listed assumption is E!xx, so the right-to-left conditional follows by similar reasoning to before.  B.4 Flattening This appendix proves the Flattening thesis stated in Section 6.3: Flattening. acd  φ ◇ ↔ φ In the thesis, φ is a Lpsu -formula and the assumption is the Absolutely Comprehensive Domain thesis (acd): ∃xx∀x(x ≺◇ xx). First, we state without proof three lemmas. The first concerns the defined existence predicates: Lemma 6. acd  E!v ∧ E!vv The remaining two concern the sub-plurality relation:5 uu ≺ vv =df ∀x(x ≺ uu → x ≺ vv) Lemma 7.  ∀x(x ≺◇ xx) ↔ ∀yy(yy ≺◇ xx) Lemma 8. E!xx  ¬extyy [yy ≺◇ xx] The Flattening thesis may then be proved from the lemmas. Proof. We prove the following by induction on the complexity of φ, and then apply existential generalization: E!xx, ∀x(x ≺◇ xx)  φ ◇ ↔ φ

• φ ◇ = ◇ ßv. The right-to-left direction is immediate from the t-axiom. For the other direction, the sta-axiom and lem. 6 deliver ◇ ßv → ßv. Similarly for the other atomic formulas in the language. • The cases for connectives follow directly from the induction hypothesis (IH). Like E!v, the two uses of v ≺ uu may be disambiguated by the type of v, singular or plural. As usual, we write v ≺◇ uu to abbreviate [v ≺ uu]◇ , applying the same convention to other infix notation (primitive and defined). 5

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 appendix b • φ ◇ = ∀vvφ1◇ . The left-to-right direction follows from IH and the t-axiom. For the other direction, reason as follows: E!xx, ∀x(x ≺◇ xx), φ1  φ1◇ E!xx, ∀yy(yy

IH

≺◇ xx), φ1  φ1◇

lem. 7

 φ1◇ ◇

E!xx, ∀yy(yy ≺

prop. 1

xx), ∀vvφ1  ∀vv(vv ≺ xx → φ1◇ )  ∀vv(vv ≺◇ xx → φ1◇ ) ◇ ◇

 ∀vv(vv ≺

xx) → ∀vvφ1◇

 ∀vvφ1◇ • The case for the singular quantifier follows a similar argument.

mpfo lems. 4, 8 mpfo mpfo 

We may additionally prove the variant of Flattening deployed in Section 7.4. Let mpfo− be the modal plural logic that drops the Stability Axioms sta-ß and sta-∈ from mpfo (but retains sta-= and sta-≺). And write φ ◇u for the result of prefixing each occurrence of ßv and u ∈ v in φ with ◇ (but adding no modal operators to quantifiers or to other atomic formulas). We may then establish the following in the weaker logic: Stability-free Flattening. acd mpfo− φ ◇ ↔ φ ◇u The base case for ßv and u ∈ v is now rendered trivial; the rest of the proof proceeds as before (noting that the lemmas do not rely on the dropped sta-axioms).

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a ppe n d i x c

Set Theory Appendix C.1 gives a formal statement of zfc, and some of its close cousins. The remaining sections outline the interpretation of zfc in the bimodal theory mstp , stated in Section 6.4. This result was established for a first-order version of mstp in Studd (2013). Some of the omitted proofs may be found in that paper. We continue to adopt the notational conventions listed in Table A.2. Moreover, unless otherwise indicated,  henceforth stands for derivability in mpfo from the axioms of mstp . C.1 Zermelo–Fraenkel set theory Pure set theory This appendix gives a formal statement of pure and impure versions of set theory. We begin with pure first-order set theory. Pure first-order Zermelo–Fraenkel set theory with Choice (zfc) is formulated in the first-order language Ls described in Appendix A.1 with ∈ as its sole non-logical predicate. The theory is based on classical first-order logic (with identity). The non-logical axioms of zfc are the first ten axioms listed in Table C.2, from Extensionality through to the Replacement Schema.1 As usual, in the context of zfc, the variables x, y, z, s, and t (and their decorated versions) are all treated as ordinary first-order variables. The notation deployed in the statement of the axioms is defined in Table C.1. The theory zf is like zfc save that it omits the Axiom of Choice.

Impure set theory Impure Zermelo–Fraenkel set theory with Choice and the Urelement Set Axiom (zfcsup ) is formulated in the plural language Lpsu , and based on the plural logic pfo, each described in

Table C.1. Defined formulas abbreviation

short for

v1 ∈ / v2 v1 ⊆ v2 v1 = v2+ fn[φ(u, v)] dj-fm(s)

¬v1 ∈ v2 ∀u(u ∈ v1 → u ∈ v2 ) ∀u(u ∈ v1 ↔ u ∈ v2 ∨ u = v2 ) ∀u∀v1 ∀v2 (φ(u, v1 ) ∧ φ(u, v2 ) → v1 = v2 ) (∀x ∈ s)∃w(w ∈ x) ∧(∀x1 ∈ s)(∀x2 ∈ s)(x1 = x2 ∨ ¬∃w(w ∈ x1 ∧ w ∈ x2 )) (∀x ∈ s)∃!w(w ∈ x ∧ w ∈ t)

c-set(s, t)

1 Side-conditions: in the Separation and Replacement Schema, φ(x) and φ(w, x) are formulas of the first-order language that lack free occurrences of t.

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 appendix c Table C.2. Set-theoretic axioms Extensionality Empty Set Pairing Union Separation Schema Power Set Foundation Axiom of Choice Infinity Replacement Schema

∀s1 ∀s2 (s1 = s2 ↔ ∀x(x ∈ s1 ↔ x ∈ s2 )) ∃s∀x(x ∈ / s) ∀x1 ∀x2 ∃s∀x(x ∈ s ↔ x = x1 ∨ x = x2 ) ∀s∃t∀x(x ∈ t ↔ (∃w ∈ s)(x ∈ w)) ∀s∃t∀x(x ∈ t ↔ x ∈ s ∧ φ(x)) ∀s∃t∀x(x ∈ t ↔ x ⊆ s) ∀s(∃x(x ∈ s) → ∃x(x ∈ s ∧ ∀w(w ∈ s → w ∈ / x))) ∀s(dj-fm(s) → ∃t c-set(s, t)) ∃s((∃t ∈ s)∀x(x ∈ / t) ∧ (∀x ∈ s)(∃w ∈ s)(w = x+ )) fn[φ(w, x)] → ∀s∃t∀x(x ∈ t ↔ (∃w ∈ s)(φ(w, x)))

Separation Axiom Replacement Axiom

∀xx∀s∃t∀x(x ∈ t ↔ x ∈ s ∧ x ≺ xx) ∀xx(fn(xx) → ∀s∃t∀x(x ∈ t ↔ (∃w ∈ s) w, x ≺ xx))

Elements Urelement Set Axiom

∀x∀y(x ∈ y → ßy) ∃s∀x(¬ßx → x ∈ s)

Table C.3. The axioms of Modal Set Theory Extensionality◇ Plenitude Axiom Priority Axiom

[∀s1 ∀s2 (s1 = s2 ↔ ∀x(x ∈ s1 ↔ x ∈ s2 ))]◇ ∀xx> ∃s(s ≡◇ xx) ∀s ◇< ∃xx(s ≡◇ xx)

Appendix A.1. The theory’s axioms, listed in Table C.2, make four changes to the axioms of zfc. First, in the context of zfcsup , we take the quantifiers binding the variables s and t (and their decorated versions) in the axioms to be tacitly relativized to the set predicate, as per Appendix A.1. Second, we further modify Power Set by redefining x ⊆ s to apply only to sets: v1 ⊆ v2 =df ßv1 ∧ ßv2 ∧ ∀u(u ∈ v1 → u ∈ v2 ) Third, the theory replaces the Separation and Replacement Schemas with the Separation and Replacement Axioms. In the latter case, the notation used is defined as follows: fn(xx) =df ∀z∀w1 ∀w2 ( z, w1 ≺ xx ∧ z, w2 ≺ xx → w1 = w2 ) Fourth, we add two further axioms, Elements and the Urelement Set Axiom. A first-order version of this theory, zfcsu, formulated in Lsu and based on first orderlogic, replaces the Separation and Replacement Axioms with the Separation and Replacement Schemas. C.2 Interpreting set theory, part I The remaining appendices concern the interpretation of zfc in an extension of the modal theory mstp , from Section 6.4. The theory mstp is formulated in the bimodal plural language Lmps of pure set theory and governed by the modal logic mpfo described in Appendix A.3. Recall that the non-logical axioms of mstp comprise the axioms Extensionality◇ , Plenitude, and Priority (restated in Table C.3). This appendix proves modal versions of the axioms of the theory s+ . This theory has as axioms the first seven axioms listed in Table C.2, together with the Rank Theorem (stated

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appendix c  below). After showing that s+ also suffices to interpret mstp in Appendix C.3, Appendix C.4 treats Choice, Infinity, and Replacement. Empty Set, Pairing, Union, Separation, and Powerset Start with the first six axioms listed in Table C.2. The modalization of Extensionality is an axiom of mstp . This section proves the modalizations of the remaining five. Note first that a suitably restricted modal comprehension schema follows from the mstp -axioms: Lemma 9. The following is provable when ψ(x) is a Lmps -formula which lacks s free: inv[ψ(x)], ¬extx [ψ(x)]  > ∃s∀x(x ∈◇ s ↔ ψ(x)) This permits us to establish the relevant axioms of zfc via two lemmas about extensibility: Lemma 10. E!s  ◇< ¬extx [x ∈◇ s]. Lemma 11. (a) (b) (c) (d) (e)

 ¬extx [¬x =◇ x] E!s  ¬extx [x =◇ s] E!s  ¬extx [x ∈◇ s] ¬extx [ψ1 ]  ¬extx [ψ1 ∧ ψ2 ] inv[ψ1 ], inv[ψ2 ], ¬extx [ψ1 ], ¬extx [ψ2 ]  ¬extx [ψ1 ∨ ψ2 ]

Proposition 12. (a) (b) (c) (d) (e)

 Empty Set◇  Pairing◇  Union◇  Separation◇  Power Set◇

Proof. In each case, to show that there is eventually a set {x : φ ◇ (x)}, it suffices to show that the defining condition φ ◇ (x) becomes inextensible when the relevant parameters are formed:

(a) (b) (c) (d) (e)

 ¬extx [(¬x = x)◇ ] E!x1 , E!x2  ¬extx [(x = x1 ∨ x = x2 )◇ ] E!s  ¬extx [((∃w ∈ s)(x ∈ w))◇ ] E!s  ¬extx [(x ∈ s ∧ φ(x))◇ ] E!s  ¬extx [(x ⊆ s)◇ ]

Claims (a)–(d) follow from lem. 11. For (e), note that, assuming x ⊆◇ s, mstp proves ¬extz [z ∈◇ s] → ¬extz [z ∈◇ x]. We may then reason as follows:2 x ⊆◇ s, E!s  ◇< ¬extz [z ∈◇ s] ◇

 ◇< ¬extz [z ∈ 

lem. 10

x]

 ◇< > ∃x ∀z(z ∈  E!x

mpfo

◇ 

x ↔z∈

◇

x)

lem. 9 Extensionality◇

The result then follows from lem. 3.

2

We continue to employ the conventions governing displayed proofs from Appendix B. See n. 4.



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 appendix c Foundation and Regularity Regularity is the theorem (schema) of zfc, among whose instances is the axiom Foundation, that states that whenever some set satisfies a condition φ(x), there is some set minimal with respect to ∈ that satisfies φ(x): Regularity. ∃xφ(x) → ∃x(φ(x) ∧ (∀z ∈ x)¬φ(z)) The modalization of this axiom is proved from the Priority axiom via the following lemma: Lemma 13. Priority, E!x mpfo z ∈◇ x → ◇< E!z Proposition 14. Priority mpfo Regularity◇ . Proof. Reason in mpfo as follows:

< ∀z¬φ ◇ (z), E!x  z ∈◇ x → ◇< E!z ◇

lem. 13 ◇

x → ◇≤ ¬φ (z)

z∈

◇

z∈

◇

x → ¬φ (z) ◇

◇

 (∀z ∈

x)¬φ (z) ◇

 (∀z ∈

◇

x)¬φ (z)

mpfo prop. 1 mpfo lems. 4, 11

It follows that the consequent of Regularity◇ may be proved from Priority on the assumption that ◇(∃xφ ◇ (x) ∧ < ∀z¬φ ◇ (z)). The result follows by an application of the mpfo-theorem: ◇ ψ → ◇(ψ ∧ < ¬ψ).  The Rank Theorem This section sketches a proof of the modalization of the following zfc-theorem: The Rank Theorem. ∀s∃y∃α(Rα (y) ∧ s ⊆ y) In the statement of the axiom, Rα (y) abbreviates the following formula:3    ∃f dom(f ) = α + 1 ∧ f (α) = y ∧ (∀β ≤ α)∀x x ∈ f (β) ↔ (∃γ < β)x ⊆ f (γ ) Thus Rα (y) characterizes y = Vα without invoking Replacement.4 Note first that we may characterize the α-th set-universe M in the hierarchy as comprising every item currently available with the following (non-invariant) formula: Uα (M) =df ∀x(x ∈◇ M) ∧ ¬extx [x ∈◇ M] ∧ [∀β(β ∈ α ↔ β ∈ M)]◇ The modalization of the Rank Theorem is an immediate consequence of two lemmas, which we state without proof. The first states that every set is a subset of some universe; the second that universes and ranks coincide: Lemma 15.  ∀x ◇ ∃M ◇ ∃α(◇ Uα (M) ∧ x ⊆◇ M) Lemma 16.  ∀y∀α(◇ Uα (y) ↔ R◇ α (y)) Proposition 17.  The Rank Theorem◇ 3 4

Notation: dom(f ) = α + 1 is an Ls -formula formalizing that f is a function whose domain is α + 1. We follow Uzquiano’s (1999, sec. 5) formulation.

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appendix c  C.3 Kripke normal form This appendix elaborates on the interpretability results stated in Section 6.4 and the Kripke Normal Form Theorem stated in Section 6.5. In combination with the Mirroring thesis (from Appendix B.2), propositions 12, 14, and 17 establish the following: Proposition 18. s+ fol φ only if mstp mpfo φ ◇ In the proposition, φ is a Ls -formula, and s+ and mstp are the non-modal and modal theories defined in Appendix C.2. This appendix outlines a converse of this result. First note that using the Kripkean translation from Section 6.5, mstp can be interpreted in s+ : Lemma 19. mstp mpfo ψ only if s+ fol ψ α . Moreover, the non-modal theory then proves that each Ls -formula is equivalent to the Kripkean translation of its modalization: Lemma 20. s+ fol φ ◇ α ↔ φ We may then strengthen proposition 18 as follows: Proposition 21. s+ fol φ if and only if mstp mpfo φ ◇ . Proof. Right-to-left: suppose mstp mpfo φ ◇ , then s+ fol φ ◇ α (by lem. 19), and therefore s+ fol φ (by lem. 20). 

Next we deploy the Kripke Normal Form Theorem from Section 6.5. Proposition 22. Let Uα (M) be as defined in Appendix C.2. The following equivalence is then provable from the stated assumption when ψ is a sentence that lacks proxy-plural variables (and without the assumption when ψ is invariant): Uα (M) mstp ψ ↔ [ ψ α ]◇ From this, it follows that the theorems of mstp are precisely the Kripkean translations of s+ -theorems: Proposition 23. mstp mpfo ψ if and only if s+ fol ψ α . C.4 Interpreting set theory, part II This appendix completes the recovery of zfc in an extension of the modal theory mstp stated in Appendix C.2: (i) we state the Plurality Choice Schema and use it to outline an unremarkable derivation of the modalization of a standard set-theoretic formulation of the Axiom of Choice in mstp ; (ii) we recover Infinity and Replacement with the help of a modal reflection principle. The Axiom of Choice The Plurality Choice Schema may be stated as follows:5 (pcs)

   ∀xx(φ(xx) → ∃x(x ≺ xx)) ∧ ∀xx∀yy φ(xx) ∧ φ(yy)

 → (∀z(z ≺ xx ↔ z ≺ yy) ∨ ¬∃z(z ≺ xx ∧ z ≺ yy)) → ∃zz∀xx(φ(xx) → ∃!z(z ≺ xx ∧ z ≺ zz))

5

Our formulation necessitates a choice schema due to Burgess (2004, p. 213).

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 appendix c The following lemma is then an exercise in non-modal set theory: Lemma 24. s+ + pcs α fol Axiom of Choice The desired result then follows: Proposition 25. mstp + pcs mpfo Axiom of Choice◇ Proof. Apply props. 21 and 22 to lem. 24.



Infinity and Replacement We now complete the interpretation of zfc by showing that the modalizations of Infinity and Replacement may be derived in the modal theory that extends mstp with the Complete Modal Reflection Schema:6  ◇> ∀ v (φ ◇ ( v ) ↔ φ( v ))

(refl)

Note first that in mstp , the modalization of a more familiar, set-theoretic formulation of reflection may be recovered from refl:7 Proposition 26. mstp + refl  [∀α(∃λ > α)∃N(Rλ (N) ∧ (∀ v ∈ N)(φ( v ) ↔ φ( v )N ))]◇ Proof. Apply props. 21 and 22. Note first that (†) follows in s+ from refl α :

(†)

∀α(∃β > α)∃N(Rβ (N) ∧ (∀ v ∈ N)( φ ◇ ( v ) β ↔ φ( v ) β ))

v ) β ↔ φ( v )N , (†) yields the followApplying lem. 20, and the fact that Rβ (N), v ∈ N s+ φ( + ing in s , via elementary properties of ranks: (‡)

∀α(∃β > α)∃N(Rβ (N) ∧ (∀ v ∈ N)(φ( v ) ↔ φ( v )N ))

Applying a trick of Lévy’s (1960) we may replace the arbitrary ordinal β in (‡) with a limit ordinal λ. The non-modal theory s+ + refl α consequently proves the λ-version of (‡); the desired result then follows by props. 21 and 22.  The axioms of Infinity and Replacement are then recovered by applying mirroring to their familiar, nonmodal derivations from the set-theoretic reflection principle.8 Proposition 27. (a) mstp + refl  Infinity◇ (b) mstp + refl  Replacement◇ Let mst+ p be the theory that adds pcs and refl to mstp . In conjunction with Proposition 25, the results show that mst+ p proves the modalization of every zfc-axiom. Adapting the arguments of Appendix C.3, we may then establish the following theorem: ◇ Theorem 28. zfc fol φ if and only if mst+ p mpfo φ .

Side-condition: φ( v ) is an Ls -formulas whose free variables are those in the string v = v1 , . . . , vk . Notation: the relativization of a formula φ to a set N—φ N —is defined as usual, by restricting each quantifier to the elements of N; v ∈ N abbreviates v1 , . . . , vk ∈ N. 8 The theorem is due to Lévy (1960). The formulation of the non-modal reflection principle used in the text is due to Lévy and Vaught (1961), where it is labelled CR, p. 1048. 6 7

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Index  (unimodal axiom) 203, 252 b (mpfo+ -axiom) 204–5  (unimodal axiom) 203, 252 b (mpfo+ -axiom) 204–5 absolute modal operator see under modal operator Absolutely Comprehensive Domain 162–3, 186, 188, 198, 257 absolutely comprehensive domain see under domain ‘absolutely’ 19, 133, 154 see also loose talk absolutism about (classical) quantifiers vii, 1–4 about intuitionistic quantifiers 55 about modalized quantifiers 20, 177, 184, 205–6, 241 austere 171 Cantorian 187–9, 198, 241 hybrid 176–7, 184, 201, 211–12, 242–3 Quinean 79–86, 102, 122, 127 statement of 57, 122–4, 161–2 third-way 185, 195–200, 241 thorough-going 177 accommodation 239–40 Accommodation 239 admissible interpretation 130, 148–50 All-in-One principle 12, 21, 30, 56–7, 59–60, 73, 76, 83, 178, 241 applicability of sets see under sets argument from indefinite extensibility see under indefinite extensibility at (pfoi,j -axiom) see Auxiliary Truism0,1 attribute 24, 27–9 austere absolutism see under absolutism Auxiliary Truism0,1 182, 250 b (unimodal axiom) 203, 252 Bach, K. 96–7, 109, 112, 114–16 background domain see under domain backwards-looking modal operator see under modal operator Bar-Hillel, Y. 191 Barwise, J. 4, 61–5, 68–9, 84–6, 89–90, 92, 109 Barwise–Cooper semantics see semantics, Mostowski–Barwise–Cooper Biggest is Best 8–10 bimodal set theory see under set theory Bimodalized Collapse 205

Boolos, G. 27, 39, 44–6, 48–50, 56–8, 73, 79, 126, 152, 164–5, 167, 178, 194, 247 Brandom, R. 224 Bueno, O. 144 Bull, R. 251 Burali-Forti paradox viii, 14, 22, 34, 54, 188–9, 201 Burgess, J. P. 33, 44–5, 247, 263 Cantor, G. 22, 41, 82, 188, 191, 200 Cantorian absolutism see under absolutism cardinal plurality- 191–3 set- 52, 191–3 Cardinal Comparability 192 Carnap, R. 7, 68 Cartwright, R. 1, 3, 12, 14, 21, 24, 30–1, 56–7, 73, 77, 83, 122, 178 cbf (mpfo-axiom) 152, 252 charity (interpretational) 3, 228–30, 238 Choice Axiom 42, 166–7, 260, 263–4 Church, A. 32, 127 circumstantial modality see under modality Clark, P. 31 classical logic see under logic collection 10–12, 47, 56, 76, 179, 238 commitment to schemas see under schema Complete Modal Reflection 166–8, 264 compositionality 66–8, 93, 220, 226–8, 232–4, 237 comprehensibility 159–61, 185, 201–13, 243 Comprehension Schema Modalized Plural 158–61, 163, 184–5, 206–9, 243 Plural 45, 58–9, 123, 127–8, 157–63, 180, 183–5, 198, 206–10, 212, 243, 249, 252, 256 Predicative 27, 33 Ramified 34–5 Second-Order 26–9, 33–5, 45, 58 see also Naive Comprehension Comprehensive Domain 57–8, 122–4, 128, 133, 161, 163 see also Absolutely Comprehensive Domain context-sensitivity 16, 69, 91–9, 105–6, 214–15 contextual modality see under modality contextual restrictionism see under restrictionism contingentism 144–5

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 index Cooper, B. 4, 61–5, 68–9, 84–6, 89–90, 92, 109 Corcoran, J. 125 Correctness Maximization 229 creationism see under sets criterion of identity 4–6 cross-type quantification see under quantification cumulative hierarchy 47–52, 163–4 see also potential hierarchy, iterative conception of set cut (sequent rule) 223, 235, 240 cv (mpfo-axiom) 252 d (mpfo-axiom) 204, 252 Davidson, D. 79–80 Davidsonian semantics see semantics, Tarski–Davidson Degen, W. 82 determiner 4, 64 dil (sequent rule) 223, 235 direct method (for language learning) 40, 79, 86, 175 domain vii, 88–90 absolutely comprehensive 1–2, 133, 162 background 16–17, 98–101, 121–2, 132 restriction see quantifier domain restriction shift 12, 14–17, 47, 72, 98–9, 118, 130–1, 180–3, 239 see also universe Dummett, M. viii–ix, 4, 10–11, 21–2, 27, 30–1, 54–60, 159–60, 178, 195 Dummettian argument, the see under indefinite extensibility E-Compositionality 227, 234, 237 E-Expansion Theorem 237–8 E-Extensionality 227, 233–4, 237 e1 (mpfo-axiom) 204, 252 e− (mpfo+ -axiom) 204–5 1 Elbourne, P. 92 Elements (zfcsup -axiom) 260 Empty Set (zfcsup -axiom) 41, 165, 199–200, 260–1 existential quantification see under quantification Expansion Question 217–18, 227, 231–9 expansionism vii, 17, 90–1 interpretational 17, 104–10, 143–4, 241 procedural postulationism 17, 102–4, 147 extended modal language see under language extension see semantic value, extensional extensional definiteness 167–9, 209–10

Extensional Definiteness 209 Extensional Definiteness of Membership 168 Extensional Definiteness of Subsethood 168 extensional semantics see under semantics Extensionality (zfcsup -axiom) 4, 11, 41, 43–4, 100, 165, 167–8, 199–200, 259–61 Feferman, S. 36 Fine, K. 12, 16–17, 19–20, 49, 88, 91, 102–4, 109, 145–8, 154, 161, 177–8, 182 finite axiomatizability 19, 45, 125–8 finitism 211 first-order language see under language first-order logic see under logic Fitting, M. 92, 97 Flattening 155, 162, 186, 257–8 Florio, S. 4, 200 fol and other systems fol (first-order logic) 156, 255 mfo (modal first-order logic) 151–2, 252 mpfo (modal plural logic) 157, 251–3 mpfo+ (modal plural logic) 205–9, 212 pfo (plural logic) 45–6, 58–60, 254–9 pfoi,j (sorted plural logic) 181–2, 249–50 Forbes, G. 144 Forster, T. 48 forwards-looking modal operator see under modal operator Foundation (zfcsup -axiom) 42, 48, 154–6, 165, 168–71, 260, 262 Fraenkel, A. 191, 193 free logic see under logic fref (mpfo-axiom) 251 Frege, G. 27, 35, 41, 58, 63, 147, 239 fug (mpfo-rule) 251, 255 fus (mpfo-axiom) 151, 251 g (unimodal axiom) 252 Geach, P. 4–5, 24 generality modal 142–5, 202–3 quantificational see quantification schematic 131–5, 206 generalized quantifier see under quantifier generalized semantics see under semantics Ghilardi, S. 251 Glanzberg, M. 12, 16–17, 19, 78, 87, 91–2, 98–101, 116, 121, 132, 178, 214–15, 239 Gödel, K. 29, 32, 35, 46, 48, 167, 195 Grandy, R. 229 Grice, H. 96 Grim, P. 14 h (mpfo-axiom) 252 Hallett, M. 191

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index  Hazen, A. 39, 79, 86 Heck, R. 27 Hellman, G. 7–8, 10, 51–3, 146, 148, 154, 159, 161, 178, 190 Hewitt, S. 39, 146 Hirsch, E. 7–10 Hodges, W. 83 Horsten, L. 5 hybrid absolutism see under absolutism hybrid relativism see under relativism hyperintensionality 113–19, 132, 214, 244 idealization 68–9, 215–16, 237–9 ideology 10, 61, 75, 82, 85–6, 169, 171, 189 impure set theory see under set theory indefinite extensibility x, 10–15, 54, 159–61, 195–7 Dummettian argument from viii, 55–9, 178, 241 modal argument from 179–83 schematic argument from 184–6 indexical account see under quantifier domain restriction ineffability, objection from 18–20, 120–5, 142, 161, 243 inexpandable universe see under universe Inextensibility 161, 164, 211, 256–7 Infinity (zfcsup -axiom) 42, 51, 166, 168, 211, 260, 263–4 intension see semantic value, intensional intensional semantics see under semantics interpretation MT- 62–72 P- 73–5 SP- 75–7 see also MT-hierarchy, semantic value interpretational expansionism see under expansionism interpretational modality see under modality irreducibly modal modality see under modality Iterative Conception 194–5, 211 iterative conception of set 48–50, 164–7 see also set theory Jech, T. 41, 43 k (mpfo-axiom) 251 Kanamori, A. 44, 52 Kaplan, D. 92, 174–6 kind generalizations, objection from 110–16, 214, 242 Kleene, S. 127 Klement, K. 24 Knowledge Maximization 229–30 Kreisel-squeezing 83 Kreisel, G. 83, 126 Kripke Normal Form Theorem 156, 174, 263

Kripke semantics see under semantics Kripke semantics, objection from 171–4 Kripke, S. 34 language as opposed to languages 105–6, 138 extended modal 202 first-order 62, 245, 251 label 129–30, 181 modal 148, 251 plural 25, 74, 245–7, 251 second-order 26, 38–40 superplural 76–9 version 105–6 see also Ls and variants Lavine, S. 18–19, 38, 128–31, 133–5, 178, 190 Lepore, E. 80 Lévy, A. 166–7, 191, 193, 264 Lewis, D. 4, 6, 9, 15–16, 58, 64, 68, 70, 79, 87, 98–9, 103, 106, 116, 120, 138, 153, 224, 228–9, 239–40 Liberal Comprehensibility 207–9, 213, 243 limitation of size 29–30, 186, 191–5 Limitation of Size 192–5 Lindström, P. 63 Linnebo, Ø. 19–20, 27, 39, 45, 48, 51, 73, 76–8, 80, 82, 86, 134, 146–8, 152, 154, 156–7, 159, 166–71, 177, 180, 184, 190–3, 209–10, 212, 245, 253 Linsky, B. 24 löb (mpfo-axiom) 152, 170, 252 logic classical, first-order 27, 156, 252, 255 free 151, 204, 251 modal 149–53, 250–3 paraconsistent viii, 25 plural 247–9 second-order 26, 28 sorted 181–2, 249–50 see also fol and other systems, s and other unimodal systems loose talk ix–x, 77–8, 133, 175 plurality ix, 57, 61, 74, 77 process metaphor 23, 48–50, 103, 149, 164–7, 182, 194 superplurality ix–x, 76–8, 175 Lowe, E. 4–5 Ls and variants Lgq (language of generalized quantifiers) 63–4 Lmps (modal plural language) 164, 250–1 Lmpsu (modal plural language) 157, 250–1 Lmsu (modal first-order language) 148, 250–1 Lps (plural language) 246, 250–1

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 index Ls and variants (cont.) Lpsu (plural language) 45, 245–6, 250–1 Ls (first-order language) 43, 245, 250–1 Lsu (first-order language) 44, 245, 250–1 see also language McGee, V. 15, 18, 31, 50, 73, 83–4, 120, 124–8, 199, 216–18, 225–6 Menzel, C. 48 metaontology 6–10 metaphysical modality see under modality metaphysical realism, argument from 6–10 metaphysically privileged/ joint-carving quantification see under quantification metasemantics 101, 215–17, 228–30 mfo (modal first-order logic) see under fol and other systems Mirroring fol 156 Mirroring s+ 166 Mirroring zfc 167 Mitchell, J. 94 Modal Collapse 255 modal generality see under generality modal language see under language modal logic see under logic modal operator absolute 171, 197, 203 backwards-looking 250 forwards-looking 167, 169, 252 modal profile of sets see under sets modal set theory see set theory, unimodal and set theory, bimodal modal statement of relativism see under relativism modalism 144–5 modality circumstantial 146, 171, 202 contextual 147, 174–5 interpretational 146–8 irreducibly modal 152 metaphysical 144, 146, 152–3 modalization 153, 157–8 Modalized Invariance 156, 254, 256 Modalized Naive Comprehension see under Naive Comprehension Modalized Plural Comprehension see under Comprehension Schema modalized quantifier see under quantifier model theory 13, 53, 83–5, 148–53, 202–5, 247–50 model-theoretic semantics see under semantics Monotonicity 149–50, 152 monsters, objection from Kaplanian 174–6 Mostowski–Barwise–Cooper semantics see under semantics Mostowski, A. 63 mp (pfo-rule) 248

mpfo (modal plural logic) see under fol and other systems mpfo+ (modal plural logic) see under fol and other systems MT-hierarchy 47–53, 163–4, 187 MT-interpretation see under interpretation mstp (bimodal set theory) 165, 260 mstp + (extension of mstp ) 167, 264 Myhill, J. 35 mysteriousness, objection from 15–18, 87, 98–9, 103–4, 107, 214–15, 218, 242 mystical censor see mysteriousness, objection from n-tuple 65, 71, 200, 220 Naive Comprehension Axiom 26–9, 34, 53–5, 59–60, 180, 184–5, 190, 198, 206 Modalized 168, 184–5, 187, 189–90, 202, 206–9, 211, 243 Schema 24–9, 46, 81, 179–80, 184, 191, 210 naive set theory see under set theory naivety rejoinder 14–15, 47, 53–4, 59–60, 178–80, 186, 188, 241 nec (mpfo-rule) 252 negation of a schema see under schema Negri, S. 234 New Domains 206–8 New Items 206–8 new-φ operation 22–3, 30 nihilism, mereological 3, 6, 9–10, 115, 117–19, 134–5, 142–5, 242 No Absolutely Comprehensive Domain 162–3, 185–7, 196, 198 No Comprehensive Domain 58, 161, 163 No Comprehensive Totality 55 No Comprehensivej Domaini 130–1, 138, 141, 183–4, 187–8, 196, 199 No New Domains 205, 208 No New Items 205, 207–8 No Universal Set 46–8, 52, 57, 70–1, 84 no-classes theory 29, 34, 36 no-domain theory of domains 57–60, 74–5, 122, 130 nominal 4–5, 64 numerically definite quantification see under quantification Oliver, A. 45, 56 ontology 2–3, 6–10, 17, 73, 104, 147 Open-ended E-Soundness 237 Open-ended Q-Soundness 226, 230 open-ended schema see under schema ordinal 42–3, 52, 159, 190, 194–5 P-interpretation see under interpretation Pairing Axiom (zfcsup -axiom) 41 paraconsistent logic see under logic

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index  Parsons, C. 7–8, 10, 14, 19, 38, 49, 78, 111, 132, 146, 152, 154, 167, 178 Partee, B. 94 Partial Modal Reflection (unimodal axiom) 168 Paseau, A. 167 pc (pfo-axiom) see Comprehension Schema, Plural pcs (mst+ p -axiom) see Plurality Choice Schema Peacocke, C. 172 Peano Arithmetic 19, 36, 51 Pelletier, F. 64 Peters, S. 90 pfo (plural logic) see under fol and other systems pfoi,j (sorted plural logic) see under fol and other systems Platonism 2, 54 Plenitude Axiom (mstp -axiom) 165, 169, 260 Plural Comprehension Schema see under Comprehension Schema Plurality Choice Schema 166–7, 263–4 plural language see under language plural logic see under logic plurality see under loose talk Plurality Separation 58–9 plurality-cardinal see under cardinal plurality-extension see under semantic value Poincaré, H. 31, 36–7 potential hierarchy 149–50, 171–4, 187–8, 208–9 Potter, M. 44, 48, 167 Power Set Axiom (zfcsup -axiom) 41, 165, 197, 260 pragmatic account see under quantifier domain restriction Predicative Comprehension Schema see under Comprehension Schema Priest, G. 25, 222 Priority Axiom (mstp -axiom) 165, 169–70, 260, 262 procedural postulationism see under expansionism process metaphor see under loose talk property 14 propositional function 23–9, 38–40 pseudo-quantifier see under quantifier pure set theory see under set theory Putnam, H. 6–7, 105, 146, 154 Q-Compositionality 220, 226, 228, 232 Q-Extensionality 219, 221, 226, 228, 232–3 Q-Quantification Theorem 226–7, 237 quantification cross-type 37–40 existential 68

metaphysically privileged/ joint-carving 7–10 numerically definite 5 universal 63, 68 unrestricted 7, 16–17, 36–7, 90–1, 97–9, 102–5, 108–9, 214–15, 227, 242 see also domain, generality, quantifier, quantifier domain restriction Quantification Question 217–30 quantifier vii generalized 63–5, 89, 218–19 modalized 19, 154–5, 243 pseudo- 138–40 relativized 44, 197, 246, 264 quantifier domain restriction 2, 16, 91–2, 95–9, 109–10, 115–16, 215 indexical account 92–4, 97–9 pragmatic account 96–7, 109–10, 116 tacit variable account 93–8, 113–16 quantifier variance 8–9 Quasi-Categoricity Theorem 44, 48, 51, 107, 187 quasi-homophonic semantics see semantics, Tarski–Davidson Quine, W. V. O. 9–10, 24, 26–9, 127, 228–9 Quinean absolutism see under absolutism Ramified Comprehension Schema see under Comprehension Schema ramified type theory see under type theory Ramsey, F. 32, 34 Rank Theorem 50, 165, 260, 262 Rayo, A. 4, 40, 45, 48, 62, 70, 73–7, 80, 82–4, 86, 122 Recanati, F. 95–6, 98, 114–15 ref (pfo-axiom) 249–51 refl (mst+ p -axiom) see Complete Modal Reflection reflection principle 166–8, 211, 263–4 Regularity (zfcsup -theorem) 262 relativism about modalized quantifiers 20, 177, 205–6 about quantifiers vii, 1 hybrid 176–7, 184, 201, 208 modal statement of 161–3, 206 schematic axiomatization of 18–19, 128–31, 206 thorough-going 20, 177, 208–9 Zermellian 53, 70–2, 144, 148, 163–4, 171, 186–90, 192–3, 207–9, 243 see also expansionism, restrictionism relativized quantifier see under quantifier rep (sequent axiom) 35, 222 Replacement Axiom (zfcsup -axiom) 42, 46, 48, 50, 126, 192–3, 260

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 index Replacement for Extensional Definiteness (unimodal axiom) 168 Replacement Schema (zfcsu-axiom) 46, 50, 126, 166–8, 259–60, 262, 264 Resnik, M. 79 Restall, G. 138, 222–3 restricted domain see quantifier domain restriction Restricted Pairing 200 restricted quantification see quantifier domain restriction restrictionism vii, 16–17, 90–1 contextual 16–17, 97–9, 214, 241 sortal 4, 98 revenge 177, 201–13, 243 Roberts, S. 166 Russell, B. viii, 10–11, 21–40, 56–7, 134, 183, 191 Russell Postulate 102–4 Russell Reductio 11–17, 23, 47, 54–5, 60, 99, 102, 106, 159, 177–80, 183, 199, 207, 237–8 Russell’s Lesson 23–9, 60, 180, 184 Russell’s paradox viii–ix, 10–16, 21–4, 26–9, 31, 34, 40–1, 46–7, 52–3, 55, 61, 81–3, 85, 179–80, 185, 191, 198, 243 s+ (sub-theory of zfc) 165–6, 260, 263–4 s and other unimodal systems s 252 s. 157, 252–3 s. 152, 252–3 s 152, 205, 252–3 Sainsbury, R. 27, 32 Salmon, N. 117 schema 125–6 commitment to 126–8, 131, 224–5 negation of 26, 134–5, 143 open-ended 128, 131, 224–5 schematic axiomatization of relativism see under relativism schematic generality see under generality Schlenker, P. 175–6 Second-Order Comprehension Schema see under Comprehension Schema second-order language see under language second-order logic see under logic semantic change 17, 105–7, 143, 229 Semantic Optimism 80–5, 175, 189, 241 semantic pariah 101, 137 semantic theorizing, objection from 99–102, 107, 109–10, 214, 242 semantic value extensional 63, 68, 89, 112 intensional 63, 68–9, 112–13 plurality- 73–4, 110, 200 set- 13, 69–73, 90, 110, 189 super-extension 160, 195, 197–8, 207

superplurality- 76–7, 82, 110, 241 see also interpretation semantics extensional 62, 89, 110, 116, 228 generalized 62, 82 intensional 68, 111–13, 116, 228, 244 Kripke 148–53, 155, 157–8, 169–74, 176, 202–5, 210 model-theoretic viii, 61, 92, 189 Mostowski–Barwise–Cooper 63–9, 84–6, 89, 92, 95, 100, 110, 116, 123, 241 Tarski–Davidson 80, 83–4, 100–2, 136–8, 172–3, 217, 221, 227 Separation Axiom (zfcsup -axiom) 30, 44–6, 126–7, 260 Separation Schema (zfcsu-axiom) 41, 44–7, 56, 126–7, 129, 136, 165, 260 Serial Well-Order 149, 152, 196, 252 sets applicability of 51–3, 71–2, 189–91, 200 creationist account of 23, 35–6, 49, 103, 108, 146, 171 modal profile of 49, 103 see also cumulative hierarchy, iterative conception of set, limitation of size, potential hierarchy set theory bimodal x, 164–7, 169–71, 209–11, 260–4 impure 43–4, 69, 246, 259 naive 24–6 pure 43–4, 69, 164, 245–6, 251, 259–60 unimodal x, 167–71, 209–10 see also mstp , mst+ p , Zermelo–Fraenkel set theory, zf and extensions set-cardinal see under cardinal set-domain see All-in-One principle set-semantic-value see under semantic value Setsi get Collectedj 182–3, 186–9, 196, 199–200 Shapiro, S. 14, 28, 51, 178, 192, 199 side-conditions, objection from 135–41 Sider, T. 7, 9–10 simple type theory see under type theory Skolem, T. 4, 218 Soames, S. 24, 27, 117 sortal restriction, objection from 4–6 sortal restrictionism see under restrictionism sorted logic see under logic soundness 28, 140, 225–6, 230, 236–7, 255 Soysal, Z. 146 SP-interpretation see under interpretation sta (mpfo-axiom) see Stability sta+ (mpfo+ -axiom) 204–5, 207 stability 148–50, 152, 167, 185–6, 195–200, 204–5, 236–7, 252 Stability (mpfo-axiom) 151–2, 186, 196, 198, 236, 252, 257–8 Stability-free Flattening 198, 258

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index  Stalnaker, R. 68 standard model 3, 51–2, 59, 71, 107, 139, 144, 153, 170, 187–9 Stanley, J. 92–8, 109, 113, 115–16 sub (pfo-axiom) 249–51 super-extension see under semantic value superplural language see under language superplurality see under loose talk superplurality-extension see under semantic value systematic ambiguity 131–5 Szabó, Z. 92–8, 113, 115–16, 227

urelement 43–5, 71, 163–4, 182–3, 198–200 Urelement Set Axiom (zfcsup -axiom) 50–1, 125, 199, 259–60 Urelementsi remain Urelementsj 183, 186–9, 196 Urquhart, A. 22, 29, 32 us (pfo-axiom) see Universal Specification use (of language) 79, 107, 138, 175, 222–5, 235–6, 238–9 Uzquiano, G. 4, 9, 27, 39, 44, 46, 51, 57, 59, 70, 73–5, 122, 126, 152, 178, 180, 193, 195–8, 200, 262

t (unimodal axiom) 203, 252, 255, 257–8 tacit variable account see under quantifier domain restriction Tait, W. W. 166 Tarski–Davidson semantics see under semantics Tarski, A. 34, 79 taut (pfo-axiom) 248, 250, 251–2 Thingsi get Collectedj 199 third-way absolutism see under absolutism thorough-going absolutism see under absolutism thorough-going relativism see under relativism Totality Separation 55 type theory ramified 32–40 simple 37–40

van Inwagen, P. 9 vicious-circle principle 21, 31–7, 39 see also type theory, ramified Viganó, L. 234 von Fintel, K. 94, 96 von Neumann, J. 25, 127

ug (pfo-rule) see Universal Generalization unimodal set theory see under set theory unintelligibility, objections from 171–6 Union Axiom (zfcsup -axiom) 41, 165, 260–1 Universal Generalization 140, 224, 249–51, 256 universal quantification see under quantification Universal Specification 151, 181, 250 universalism, mereological 6–9 universe 88–90 expansion 17, 102–8, 215–17, 231–40 inexpandable 91, 118 see also domain Universe-based Semantic Optimism 81–2, 175, 189, 241 Unlimited Collectability for Sets 185–8, 196, 198, 202 Unlimited Collectability for Sets∞ 198 unrestricted quantification see under quantification

Wallace, J. 4 Warren, J. 105 Weir, A. 107 Westerståhl, D. 90 Whitehead, A. N. 22 Wiggins, D. 4 Williamson–Russell Reductio 13–14 Williamson, T. vii, 1–3, 13, 15, 18, 38–40, 61–2, 70, 73, 75, 77–8, 80–7, 90–1, 93, 96, 100–2, 104–5, 109–16, 118, 120, 122, 124, 128, 134–6, 138–40, 144, 174–5, 178, 183, 195–8, 214, 228–30, 242 xb (mpfo+ -axiom) 204–5 Yablo, S. 27, 180 Zermellian relativism see under relativism Zermelo–Fraenkel set theory 40–7, 259–60 see also set theory, zf and extensions Zermelo, E. viii, 11–12, 21, 30, 41–8, 51–4, 56, 59, 70, 107, 163, 183, 187–90, 193, 233 Zermelo’s Extendability Principle 52–3, 59 zf and extensions zf 167, 259 zfc 41, 259 zfcsu 50, 260 zfcsup 50, 259–60 zfcsup,i 188–9, 199–200 zfcup 46 see also s+