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The Many and the One: a Philosophical Study of Plural Logic

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The Many and the One: a Philosophical Study of Plural Logic OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi The Many and the One OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi The Many and the One A Philosophical Study of Plural Logic SALVATORE FLORIO AND ØYSTEIN LINNEBO 1 OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Salvatore Florio and Øystein Linnebo 2021 The moral rights of the authors have been asserted First Edition published in 2021 Impression: 1 Some rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, for commercial purposes, 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. This is an open access publication, available online and distributed under the terms ofa Creative Commons Attribution – Non Commercial – No Derivatives 4.0 International licence (CC BY-NC-ND 4.0), a copy of which is available at http://creativecommons.org/licenses/by-nc-nd/4.0/. Enquiries concerning reproduction outside the scope of this licence should be sent to the Rights Department, Oxford University Press, at the address above 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: 2020951752 ISBN 978–0–19–879152–2 DOI: 10.1093/oso/9780198791522.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi To Aneta and Laurel OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi OUP CORRECTED PROOF – FINAL, 14/6/2021, SPi Contents Preface xi Acknowledgments xiii 1. Introduction 1 I.PRIMITIVEPLURALS 2. Taking Plurals at Face Value 9 2.1 Some prominent views of plural sentences 9 2.2 Taking plurals at face value 14 2.3 The language of plural logic 15 2.4 The traditional theory of plural logic 19 2.5 The philosophical significance of plural logic 20 2.6 Applications of plural logic 24 2.7 Our methodology 29 3. The Refutation of Singularism? 31 3.1 Regimentation and singularism 31 3.2 Substitution argument 34 3.3 Incorrect existential consequences 36 3.4 The paradox of plurality 38 3.5 Plural Cantor: its significance 41 3.6 Plural Cantor: its statement and proof 42 3.7 Conclusion 45 3.A Alternative formulations of Plural Cantor 47 II. COMPARISONS 4. Plurals and Set Theory 55 4.1 A simple two-sorted set theory 55 4.2 Plural logic and the simple set theory compared 57 4.3 Plural logic vs. set theory: classifying the options 59 4.4 Against the elimination of pluralities in favor of sets 60 4.5 Against the elimination of sets in favor of pluralities 64 4.6 The iterative conception of set 66 4.7 Zermelo-Fraenkel set theory 68 OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi viii contents 4.8 Proper classes as pluralities 70 4.9 Are two applications of plural logic compatible? 72 4.A Defining the translations 73 4.B Defining the interpretation 74 5. Plurals and Mereology 76 5.1 Mereology 76 5.2 Can mereology represent the plural? 77 5.3 One-sorted plural logic 81 5.4 Classifying some ways to talk about the many 83 5.5 Mereological singularism in linguistic semantics 85 5.6 Assessment of singularism in linguistic semantics 88 5.7 The elimination of mereology in favor of plural logic 90 5.8 Keeping both plural logic and mereology 92 5.A Partial orders and principles of decomposition 96 5.B Some notions of sum 98 5.C Atomicity 101 5.D One- and two-sorted plural logic compared 102 6. Plurals and Second-Order Logic 104 6.1 Second-order logic 104 6.2 Plural logic and second-order logic compared 107 6.3 The elimination of pluralities in favor of concepts 109 6.4 The elimination of concepts in favor of pluralities 116 6.5 Conclusion 118 III.PLURALSANDSEMANTICS 7. The Semantics of Plurals 123 7.1 Regimentation vs. semantics 124 7.2 Set-based model theory 126 7.3 Plurality-based model theory 130 7.4 Criticisms of the set-based model theory 135 7.5 The semantics of plural predication 139 7.6 The problem of choice 144 7.7 Absolute generality as a constraint 146 7.8 Parity constraints 147 7.9 Conclusion 150 8. On the Innocence and Determinacy of Plural Quantification 151 8.1 Introduction 151 8.2 A plurality-based Henkin semantics 154 8.3 The legitimacy of ascending one order 155 OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi contents ix 8.4 Does ontological innocence ensure determinacy? 157 8.5 The semantic determinacy of plural quantification 160 8.6 The metaphysical determinacy of plural quantification 162 8.7 A generalized notion of ontological commitment 163 8.8 Applications reconsidered 168 8.A Henkin semantics 170 8.B Completeness of the Henkin semantics 172 9. Superplurals 174 9.1 Introduction 174 9.2 What superplural reference would be 174 9.3 Grades of superplural involvement 178 9.4 Possible examples from natural language 179 9.5 The possible examples scrutinized 182 9.6 The multigrade analysis 185 9.7 Covers 188 9.8 Mixed-level predications 193 9.9 Mixed-level terms, order, and repetition 195 9.10 Conclusion 198 9.A The notion of upwards closure 200 IV. THE LOGIC AND METAPHYSICS OF PLURALS 10. Plurals and Modals 205 10.1 Introduction 205 10.2 Why plural rigidity matters 207 10.3 Challenges to plural rigidity 208 10.4 An argument for the rigidity of sets 210 10.5 An argument for plural rigidity 216 10.6 Towards formal arguments for plural rigidity 219 10.7 The argument from uniform adjunction 221 10.8 The argument from partial rigidification 223 10.9 The argument from uniform traversability 224 10.10 Pluralities as extensionally definite 226 10.11 The status of plural comprehension 229 10.A Traversability and quasi-combinatorial reasoning 231 10.B Proofs 232 11. Absolute Generality and Singularization 240 11.1 Absolute generality 240 11.2 A challenge to absolute generality 241 11.3 A trilemma 244 11.4 Relativism and inexpressibility 246 OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi x contents 11.5 Traditional absolutism and ascent 249 11.6 Ascent and inexpressibility 253 11.7 Lifting the veil of type distinctions 256 11.A The Ascent Theorem 262 12. Critical Plural Logic 268 12.1 Introduction 268 12.2 The extendability argument 269 12.3 Our liberal view of definitions 272 12.4 Why plural comprehension has to be restricted 276 12.5 The principles of critical plural logic 278 12.6 Extensions of critical plural logic 284 12.7 Critical plural logic and set theory 286 12.8 Generalized semantics without a universal plurality? 290 12.9 What we have learnt 293 References 297 Index 307 OUP CORRECTED PROOF – FINAL, 12/6/2021, SPi Preface Plural logic has become a well-established subject, especially in philosophical logic. We want to explore its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? Different readers may want to follow different threads running through the book. Readers interested in the application of plural logic in philosophy will find Chapters 1, 2, and 8 especially relevant. We argue that plural logic has useful applications, though not all those it is commonly thought to have. Next, questions about the correct logic of plurals are discussed in Chapters 1, 2, 4, and 9–12, where we defend an unconventional view. We reject tradi- tional plural logic in favor of a “critical” alternative. The most striking feature of this alternative is that there is no universal plurality. Chapters 1–3, 5, 7, and 9 discuss the significance of plural logic to linguistics. Advocates of plural logic often claim that linguistic semantics should avoid “singularist” prejudices and be formulated taking plurals at face value. We contest this claim. A few words about the origin of the project may be appropriate. Both authors have for a number of years been interested in questions about the logic, meaning, and metaphysics of plurals. Many of the ideas in the book were first conceived during long runs along the River Thames in the period 2010–12. A first glimpse of the book project arose in connection with the course “Plurals in Semantics and Philosophical Logic” taught at ESSLLI 2012 in Opole, Poland. There are a lot of people to thank. This book has benefited enormously from extensive comments given by Peter Fritz, Simon Hewitt, David Nicolas, Alex Oliver, Agustín Rayo, Sam Roberts, Timothy Smiley, Eric Snyder, Hans Robin Solberg, and Gabriel Uzquiano. For useful feedback and discussion, we are also indebted to Colin Caret, Aistė Čelkytė, Eyjólfur Emilsson, Vera Flocke, Olav Gjelsvik, Nicholas Jones, Jönne Kriener, Dan Marshall, Ian Rumfitt, Stewart Shapiro, Sean Walsh, Tim Williamson, the students in our course at ESSLLI, and the audiences of numerous talks where material from the book has been presented.
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