Essence and Existence (Firefly) 0198854293, 9780198854296

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Essence and Existence

Essence and Existence Selected Essays

Bob Hale

edited by

Jessica Leech with an introduction by

Kit Fine

1

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 Editorial material © Jessica Leech 2020 Introduction © Kit Fine 2020 Chapters 3, 4, 7, 8, 10, 13, and 14 © the Estate of Bob Hale 2020 Chapters 1, 2, and 6 © Bob Hale 2018 Chapter 5 © the Estate of Bob Hale and Øystein Linnebo 2020 Chapter 9 © Bob Hale and Jessica Leech 2016 Chapter 11 © Springer Nature B.V. 2015 Chapter 12 © Springer Science B.V. 2011 Chapter 15 © Bob Hale and Crispin Wright 2015 The moral rights of the authors have been asserted First Edition published in 2020 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: 2019954046 ISBN 978–0–19–885429–6 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. 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, 21/5/2020, SPi

Contents Artist Bio Acknowledgements Introduction by Kit Fine 1. Essence and Definition by Abstraction

ix xi 1 9

2. Essence and Existence

24

3. The Problem of De Re Modality

46

4. Ontology Deflated

59

5. Ontological Categories and the Problem of Expressibility with Øystein Linnebo

73

6. What Makes True Universal Statements True?

104

7. Exact Truthmakers, Modality, and Essence

124

8. S5 as the Logic of Metaphysical Modality: Two Arguments for and Two Arguments against

141

9. Relative Necessity Reformulated with Jessica Leech

149

10. Definition, Abstraction, Postulation, and Magic

173

11. Second-order Logic: Properties, Semantics, and Existential Commitments

187

12. The Problem of Mathematical Objects

213

13. Properties, Predication, and Arbitrary Sets

225

14. Ordinals by Abstraction

240

15. Bolzano’s Definition of Analytic Propositions with Crispin Wright

256

Bibliography of the Writings of Bob Hale Bibliography of Works Cited Index

285 290 297

Preface Bob Hale’s last book, Necessary Beings, was published in 2013. This volume is composed almost entirely of work carried out between the publication of that book and his death in December 2017. His intention was always to collect these papers together into a final volume of his philosophy. Bob had been ill, on and off, for a long time. During one recurrence of his illness, he told me about his book plans, and did me the great honour of asking me, should he be unable to do so, if I would complete the project. That afternoon we shared his Dropbox folder containing all his book work. I hoped I wouldn’t need to use it, but unfortunately he died before the book was complete. The plan of the book closely follows the planned Table of Contents from Bob’s Dropbox, with a few changes. One notable addition is a full bibliography of Bob’s works. This volume contains a mixture of new work and papers already published or forthcoming elsewhere. Chapters 4, 5, 7, 8, 10, and 13 are new, previously unpublished material, exclusive to this volume. In some cases some editorial correction was required, especially where work wasn’t quite complete, but complete enough to warrant inclusion in the volume. Significant editorial changes are accompanied by an editor’s note. Chapter 1 first appeared as Bob Hale, ‘Essence and definition by abstraction’, Synthese (Special Issue: New Directions in the Epistemology of Modality), pp.1–17, © 2018, doi: 10.1007/s11229-018-1726-7. Reproduced under the terms of the Creative Commons Attribution 4.0 International License (CC-BY-4.0), http://creativecommons.org/licenses/by/4.0/. Chapter 2 was previously published as Bob Hale, ‘Essence and existence’, Revista de Filosofía de la Universidad de Costa Rica, volume 57, number 147 (January–April 2018), pp.137–55, © 2018. Reprinted with permission from the Publisher. Chapter 3 also appears as Bob Hale, ‘The problem of de re modality’ in Metaphysics, Meaning, and Modality: Themes from Kit Fine, edited by Mircea Dumitru, © 2020. Reproduced by permission of Oxford University Press: https://global.oup.com/ academic. Chapter 6 was originally published as Bob Hale, ‘What makes true universal statements true?’ in: The Logica Yearbook 2017, edited by Pavel Arazim and Tomáš Láviˇcka, published by College Publications, © 2018. Chapter 9 first appeared as Bob Hale and Jessica Leech, ‘Relative necessity reformulated’, Journal of Philosophical Logic, volume 46, issue 1, pp.1–26, © 2016, doi: 10.1007/s10992-015-9391-5. Reproduced under the terms of the Creative Commons Attribution 4.0 International License (CC-BY-4.0), http:// creativecommons.org/licenses/by/4.0/. Chapter 11 was previously published as Bob Hale, ‘Second-order logic: properties, semantics, and existential commitments’, Synthese, volume 196, issue 7, pp. 2643–69, © Springer Nature B.V. 2015, doi: 10.1007/s11229-015-0764-7. Reproduced by permission from Springer Nature: Springer Netherlands.

vi

preface

Chapter 12 originally appeared as Bob Hale, ‘The problem of mathematical objects’ (2011c), in: Foundational Theories of Classical and Constructive Mathematics (The Western Ontario Series in Philosophy of Science, volume 76), edited by Giovanni Sommaruga, © Springer Science+Business Media B.V. 2011, doi: 10.1007/978-94-007-0431-2. Reprinted by permission from Springer Nature: Springer Netherlands. Chapter 14 is also forthcoming in Origins and Varieties of Logicism: A Foundational Journey in the Philosophy of Mathematics, edited by Francesca Boccuni and Andrea Sereni, Oxford: Routledge. Chapter 15 was first published as Bob Hale and Crispin Wright, ‘Bolzano’s Definition of Analytic Propositions’, Grazer Philosophische Studien—International Journal for Analytic Philosophy, volume 91, issue 1, pp.323–64, © 2015, doi: 10.1163/9789004302273_014. Reprinted with permission from Koninklijke Brill NV. Warm thanks to all of those involved in permitting the reprint of these papers. Jessica Leech

Cover art: Red Bridge c. 1975, by Walter Steggles.

Artist Bio Walter James Steggles. 15/08/1908 to 05/03/1997 Wally, as he was generally known, was born in Highbury north London: he was the eldest of five children born to Annie Elizabeth and Walter Steggles. His early years were spent at various addresses in London but during WW1 the family lived for a while in Bath whilst Walter senior was serving with the Royal Flying Corps in Northern France. Around 1920 the family moved back to Ilford where Wally finished his schooling. Aged fourteen he successfully applied for a job with Furness Withy & Co and he worked for them until he retired. In May 1925 Wally joined the art classes at the Bethnal Green Men’s Institute at Wolverley Street along with his younger brother Harold, later transferring their allegiances to the Bow and Bromley Evening Institute when their tutor, John Cooper moved there from Bethnal Green in 1926. Wally exhibited in the East London Art Club’s exhibition at the Whitechapel Gallery in 1928. The East London Group, as it became known then transferred to the Alex. Reid and Lefevre gallery in November 1929 and Wally went on to exhibit at all eight shows that the Group held there up until 1936. Like several other members of the Group Wally exhibited widely in mixed shows from as early as 1929 and continued to do so up to and into World War Two. His crowning glory came in 1936 when he and fellow Group member Elwin Hawthorne were selected to represent Great Britain at the Venice Biennale that year. It was a great recognition of his talent in such a prestigious international arena but one which, in later life, he chose not to mention! Wally continued to paint almost up until his death in 1997: by his own admission, he couldn’t stop!

Acknowledgements Thank you, first and foremost, to Maggie, for her help and support throughout the book project. Thank you to Josh for finding Bob’s bibliography (bibbob.bib). Thank you to Kit Fine for his excellent introduction. Thank you to Crispin Wright, Ian Rumfitt, Alex Skiles, Andrea Sereni, Sònia Roca-Royes, Pablo Villalobos, Peter Momtchiloff, Øystein Linnebo, Mircea Dumitru, Pavel Arazim, Bahram Assadian, and an anonymous referee for the Press, who all played a part in getting this book finished and out into the public domain.

Introduction

One of the last times I saw Bob was in a hotel bar in Oslo. I was in a state of panic, having just lost a bag containing my computer and the key to all my passwords. But somehow he managed to calm me down: we had a beer, then another beer; and before I knew it, we were deep in philosophical conversation. Anyone who knew Bob will recognize this man as the man they know—imperturbable, kind, and completely engaged in the subject; and I consider our friendship to be one of the great blessings of my philosophical and personal life. When I reflect on his work, two things stand out. One is the extraordinary breadth of his interests and the other is the highly innovative character of his thought. His work touches on all of the main areas of philosophy—logic, metaphysics, philosophy of language and epistemology—with the single exception, as far as one can tell, of value theory. And his work is teeming with original ideas—not only, of course, when he is developing his own views but also when he is considering the different ways in which a problem might be framed or when he is critically examining the views of others. There are very few philosophers who wrote in such an interesting way about so many different topics or who engaged so thoroughly with points of view opposed to his own. It is, of course, impossible in a brief introduction to do justice to the full range of his work. What I would like to do instead is to discuss the two papers in the volume that are on truthmaking—chapter 6 on truthmakers for universal statements and chapter 7 on truthmakers for modal statements—which continue a line of work he began in chapter 10 of Hale (2013a). Bob’s treatment of this topic is tentative and exploratory in character yet well worthy, in my opinion, of further study; and, even though the topic is one of many that I might have profitably discussed, I hope my discussion of it will help bring out the extraordinary combination of flair and level-headedness that runs through everything he writes. Bob works within a semantic approach whose aim is to characterize the exact truthmakers for the sentences of some given language (pp.104, 114). These are truthmakers that are relevant as a whole to the sentences they make true. In regard to a universal sentence ∀xφ(x), there is what he calls a ‘standard’ truthmaker account (p.105), which has been adopted by most of the people who have dealt with the issue. This treats the universal statement as equivalent, in effect, to the conjunction φ(a1 ) ∧ φ(a2 ) ∧ . . . of its instances. In itself, this is no different from the classical model-theoretic treatment of universal quantification. The difference appears in the Kit Fine, Introduction In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © Kit Fine. DOI: 10.1093/oso/9780198854296.003.0001

2 introduction evaluation of conjunction since, under the truthmaker approach, a truthmaker for a conjunction is taken to be the fusion of truthmakers for its conjuncts. Thus a truthmaker for φ(a1 ) ∧ φ(a2 ) ∧ . . ., and hence for ∀xφ(x), will be a fusion s1  s2  . . . of truthmakers for the respective instances φ(a1 ), φ(a2 ), . . . . As Bob points out, one might also want to include a totality fact among the truthmakers (p.106). But I will follow him in ignoring this possible complication. Bob appears to be of the view that the standard account, or something like it, is correct for certain universal statements, such as all my children live in England but incorrect for others, such as every natural number has a successor or all aardvarks are insectivorous (pp.106–108); and he considers a number of good reasons as to why it might be regarded as incorrect in the latter case. One is that the domain of quantification may be ‘merely potential’ and so it may not be possible to form the completed fusion s1  s2  . . . of the instantial truthmakers; another is that taking ‘the internal structure of the sentences by which they [the universal propositions] are expressed as a guide to the internal structure of their truth- and falsity-makers’ does little ‘to encourage the standard account’ (p.112). But his main reason is that the standard account ‘pays no attention to what appears to be an obvious uniformity in the grounds for the truth of the instances’ (p.107). It is as if the truthmaker were just a jumble of particular facts with no particular organization. Bob concedes that this last consideration is not decisive, but I think we can agree with him that something significant in the semantic content appears to be lost and something, moreover, which might conceivably be recovered under some alternative form of the truthmaker approach (pp.108–109). I should add that there is another reason, arising from one of the intended applications of the truthmaker approach, for being dissatisfied with the standard instantial account. For we would like to be able to provide a plausible account of what it is for the propositional content of one statement to contain the content of another. Now any such account should take the content of a conjunction to contain the content of its conjuncts. But if we treat a universal statement as the conjunction (or conjunction plus) of its instances, then we will be led to the implausible result that the content of a universal sentence will contain the content of its instances, that Alfie is insectivorous (or insectivorous if an aardvaark), for example, is part of the content of all aardvarks are insectivorous. Bob goes on to consider how a more satisfactory non-instantial account of the truthmakers for universal statements might proceed. He remarks (p.114), ‘it is not ruled out that there should be a single state which has no proper parts (i.e., is not the fusion of simpler states) and which exactly verifies a universally quantified statement ∀xφ(x) and also exactly verifies each of its instances φ(a1 ), φ(a2 ), . . ., i.e. a state of precisely the kind [my italics] required for generic verification or falsification.’ His thought seems to be that an exact generic truthmaker for all aardvarks are insectivorous should also be an exact truthmaker for Alfie is insectivorous (or is insectivorous if an aardvark). Of course, if the generic truthmaker obtains then, necessarily, Alfie, if an aardvark, is insectivorous. But this does not mean that the generic truthmaker for the universal statement should contain, let alone be identical to, an exact truthmaker for any of its instances; and since the generic truthmaker for a universal statement would appear to contain a great deal of information that is irrelevant to the truth of its instances, it is

introduction 3 perhaps better if this requirement is dropped (which would mean, in particular, that Bob’s special clauses on p.117, whereby a disjunction or conditional might be made true via its universal generalization should also be dropped). Let us turn to Bob’s positive account (p.116ff). Suppose that φ(x) is a formula whose only free variable is x. He suggests that there are two ways in which the universal statement ∀xφ(x) might be made true. It might be made true by an instantial truthmaker, in the same way as with the standard account. But it might also be made true by a generic truthmaker. He makes various suggestions as to what these generic truthmakers might be. They might be conceptual connections or something about the nature or essence of the things under consideration or just higher-level relations between the constituent properties (pp.108–109, 113). However, from a purely formal point of view, the generic truthmakers are simply posited: the model itself associates with each formula φ(x) a set |φ(x)|+ of generic truthmakers for ∀xφ(x). This is an interesting proposal and it raises a number of interesting issues. There is, first of all, a question of its compatibility with some of Bob’s more informal remarks. As I mentioned, he says that the instantial account is incorrect for certain universal statements; and by this he seems to mean not simply that it fails to specify certain of the truthmakers but that the truthmakers that it proposes sometimes fail to be truthmakers (p.108). However, according to Bob’s more formal account, the standard account never goes wrong in this way: its instantial truthmakers always are truthmakers! It would not be difficult to render the formal semantics in conformity with his informal remarks. For we could suppose that, when |φ(x)|+ is non-empty, it is only the members of |φ(x)|+ that are truthmakers for ∀xφ(x) and that, when |φ(x)|+ is empty, it is only the instantial fusions that are truthmakers for ∀xφ(x). In this way, a universal statement would either have generic truthmakers or instantial truthmakers but never both; or perhaps we could envisage a more complicated scheme under which some, but not all, universal statements are capable of having both kinds of truthmaker. However, I very much doubt this is the way to go. For one thing, the semantics would no longer be topic-neutral; what kind of truthmakers a universal statement had would depend upon its subject matter. But also, it is not clear there is any reasonable way to demarcate the two kinds of case. Such a demarcation is perhaps relatively straightforward in the simple cases that Bob considers. But what of the universal statement ‘for all x, [(x is an aardvark ⊃ x is insectivorous) ∧ (x is a child of Bob’s ⊃ x lives in England)]? Does this have instantial or generic truthmakers? And what of other ‘mixed’ statements of this sort? My own view, for what it is worth, is that we should not be attempting to accommodate both kinds of truthmaker within a single semantics. Rather, we face a choice between two styles of semantics, one using instantial truthmakers and the other using generic truthmakers. There is another conflict—or, at least, tension—with his informal remarks. Before proposing his own view, he considers what he calls ‘the alternative logical form view’. This is the view that ‘generalizations expressed in natural language fall into two groups, corresponding roughly with this familiar contrast [between accidental and lawlike generalizations], and that the differences are such as to warrant formalizing them in different ways’ (p.109); and he takes this view to be ‘simply orthogonal to the

4 introduction concern which drives my proposal’ (p.111). He is therefore receptive to the idea that universal statements, of a given logical form, are capable of having both instantial and generic truthmakers; and he does not insist that it is only one kind of universal statement that is capable of having instantial truthmakers and only another kind of universal or quasi-universal statement, of different logical form, that is capable of having generic truthmakers. This is indeed a consistent position and it may even be the correct thing for him to say about generalizations in natural language. But his own more formal account strongly suggests that two kinds of universal statement may be distinguished. For by his own lights, a universal statement may have instantial or generic truthmakers. But then what is to prevent us from supposing that there is a kind of universal statement ∀I xφ(x) that only has instantial truthmakers and another kind of universal statement ∀G xφ(x) that only has generic truthmakers? Moreover, once these two kinds of universal statement are recognized, it seems almost irresistible to suppose that Bob’s universal statements ∀xφ(x) are to be analyzed as the disjunction ∀I xφ(x) ∨ ∀G xφ(x) of these two other kinds of universal statement. But perhaps the most pressing issue raised by Bob’s proposal concerns the noncompositional character of his semantics. The generic truthmakers for universal statements are not determined compositionally but are simply the result of stipulation. Bob is well aware of this. He writes: Whilst the idea that universal statements may be rendered true by general connections between the properties involved . . . seems straightforward enough, it is a further and much less straightforward question how to implement this idea in the framework of exact truthmaker semantics. This is largely because we must to some extent depart from the bottom-up determination of truth-values which is built into truthmaker semantics as we have it [under the standard account], and which–notwithstanding its divergence in other respects–it takes over from standard model theoretic semantics in general. (p.115)

But it is an unfortunate feature all the same and it may lead some to think that he has not properly provided a semantics at all. However, I do not believe that the non-compositional character of his account is an unavoidable feature of his general approach. For just as we may take a property under the possible worlds approach to be a function taking each individual into a set of worlds—intuitively, the set of worlds in which the proposition that the individual has the property is true, so we may take a property under the truthmaker approach to be a function taking each individual into a pair of sets of states—intuitively, the set of truthmakers and the set of falsity-makers for the proposition that the individual has the property. The truthmaker property |φ(x)| expressed by the formula φ(x) can then be compositionally determined, since it will be the property that takes an individual a into the pair consisting of the set of truthmakers of φ(x), under the assignment of a to x, and the set of falsity-makers of φ(x), again under the assignment of a to x. We may now suppose that the model assigns to each truthmaker property P a pair of sets of states P+ and P− —which, intuitively, consist of the truth- and falsity-makers for the proposition that P is a universal property. We may then suppose that ∀xφ(x)’s truthmakers are the members of |φ(x)|+ and ∀xφ(x)’s falsity-makers are the members of |φ(x)|− .

introduction 5 We might regard the present semantics as a general abstract form of semantics for variable-binding operators within the truthmaker framework. What would then distinguish the different variable-binding operators would be nothing in the form of the semantic clauses by which they were governed but something about the constraints imposed upon the sets |φ(x)|+ and |φ(x)|− that figure in those clauses. This means that we would lose something of the explanatory power that a good semantics is customarily thought to have. For the various logical principles for the different operators would no longer follow from the clauses for these operators but would have to be written directly into the constraints imposed upon |φ(x)|+ and |φ(x)|− . To be sure, Bob does not exactly have the abstract form of the semantics considered here. He requires that |φ(x)|+ should contain all instantial truthmakers for ∀xφ(x) and that any generic truthmaker in |φ(x)|+ should be simple (without proper parts); and he also requires that |φ(x)|− should consist of the falsity-makers for the instances of ∀xφ(x). These additional constraints have real substance. They embody, in particular, an interesting asymmetry between the truthmakers and the falsity-makers for a universal statement, in which the truthmakers are allowed to be generic while the falsity-makers are required to be instantial (with the falsity-makers for an instance of a universal statement being the only falsity-makers for the universal statement itself). However, there is still a great deal within his semantics that will need to be stipulated rather than explained. Consider, for example, the connection between ∀x(φ(x) ∧ ψ(x)) and (∀xφ(x) ∧ ∀xψ(x)). We might naturally suppose that these should have the same truthmakers. That they have the same truthmakers is something that follows from the standard instantial account (with or without a totality fact). For the truthmakers for ∀x(φ(x) ∧ ψ(x)) will be those for (φ(a1 ) ∧ ψ(a1 )) ∧ (φ(a2 ) ∧ ψ(a2 )) ∧ . . . and the truthmakers for (∀xφ(x) ∧∀xψ(x)) will be those for (φ(a1 ) ∧ φ(a2 ) ∧ . . .) ∧ (ψ(a1 ) ∧ ψ(a2 ) ∧ . . .); and these two sets of truthmakers will be the same. But this is not something that follows from Bob’s account. For suppose ∀xφ(x) and ∀xψ(x) have respective generic truthmakers s and t. Then s  t will be a truthmaker for ∀xφ(x) ∧∀xψ(x). But we have been given no reason to think that s  t will be a truthmaker for ∀x(φ(x) ∧ ψ(x)). Indeed, s  t will not in general be a simple truthmaker (nor, for that matter, an instantial truthmaker) and so it will follow from Bob’s account, in that case, that s  t is not a truthmaker for ∀x(φ(x) ∧ ψ(x)). To get round this difficulty, Bob should probably drop the requirement that generic truthmakers be simple. He can then allow that the truthmakers are the same. But he will still have no explanation as to how or why they are the same, either in this case or in other cases of this sort. I might also add that one of his principal motivations for going generic is to pay attention ‘to what appears to be an obvious uniformity in the grounds for the truth of the instances [of a universal statement]’; and yet nothing has so far been said as to what this uniformity might be or how it might be embodied within a truthmaker. All the same, Bob’s contribution to the topic is highly illuminating and points us in the direction in which a well-formed account along these lines should proceed. For as he makes clear, any such account should be open to an ontology of truthmakers which are not simply fusions of the truthmakers and falsity-makers for the atomic sentences

6 introduction of the language (even though they might not themselves be simple); it should pay attention to the uniformity in how a universal statement is instantiated; and, for this reason, it should treat the condition ϕ(x) by which a universal statement is given as more of an organizing principle than as a conduit to its instances. I turn now to truthmaking in the modal case. Again, Bob is concerned to provide an account of the exact truthmakers for modal statements within the framework of truthmaker semantics; and again, he is interested in providing both an instantial and a generic account of the truthmakers. However, consideration of his generic account would take us too far afield; and so I shall only consider his instantial account. Bob presupposes that there is an accessibility relation among states, just as the relational semantics for modal logic presupposes that there is an accessibility relation among worlds. He then suggests the following clauses for when a possibility or necessity statement is made true (p.129): s makes ♦A true iff some state s accessible from s makes A true, and s makes 2A true iff no state s accessible from s makes A false,

with analogous clauses for when a possibility or necessity statement is made false. Or rather these are the clauses if we allow for a necessity statement to be vacuously true. If we wish to exclude vacuous truth, then we should require that some state s accessible from s also makes A true (and similarly for the non-vacuous falsity of a possibility statement). Bob does not say much about how the notion of accessibility between states is to be understood, though he does suggest that it is akin to the notion of relative possibility between worlds (pp.128–29). We might say that a world v is possible relative to a world w if w’s obtaining entails the possibility of v’s obtaining. But this modal understanding of relative possibility is clearly not intended in the present case. For it would mean that if the state s is possible relative to s then it is also possible relative to any more inclusive state s+  s; and so if s were a truthmaker for ♦A then s+ would also be a truthmaker for ♦A, which goes against our understanding of truthmaking as exact. Although Bob does not elaborate, it seems that the only reasonable option is to understand s to be possible relative to s when s is an exact truthmaker for the proposition that s is possible. This, at the very least, would be in keeping with the clause for ♦A. For suppose that A’s truthmakers are s1 , s2 , . . . . Then presumably: s is an exact truthmaker for ♦A iff s is an exact truthmaker for ♦(s1 obtains ∨s2 obtains ∨ . . .) iff s is an exact truthmaker for ♦(s1 obtains) ∨ ♦(s2 obtains). . . iff one of the states s1 , s2 , . . ., possible relative to s, is an exact truthmaker for A.

But this still leaves open the question of what it is for s to be an exact truthmaker for the possibility that s obtains. Given the clause for necessity, it seems that we should suppose that there are certain states which are determinative of modal space. They tell us, so to speak, which states are and are not possible. A state will then be possible relative to such a state if it is one of its possibilities. Bob’s clauses for possibility and necessity statements will then be on the cards. For s can, with some plausibility, be taken to be an exact truthmaker for ♦A just in case one of its possibilities is an exact truthmaker for A. Also, if s is an exact truthmaker for

introduction 7 2A then no one of its possibilities, s1 , s2 , . . ., can be an exact falsity-maker for A. The converse is less clear. If s is indeed a modal state, one which is determinative of the modal space, then s can, with some plausibility, be taken to be an exact truthmaker for 2A if no one of its possibilities is an exact falsity-maker for A. But what if s is not a modal state but an ordinary non-modal state? We cannot sensibly take it, in this case, to be determinative of a modal space. Thus no states at all will be accessible from s. But we would not want, on that account, to say that it is a truthmaker for any necessity statement 2A. We should therefore require if s is to be a truthmaker for 2A that s be a modal state; and this may, in its turn, be defined as a state from which some state is accessible (or within the context of the modal logic T, as a state which is accessible from itself). Thus we require a non-vacuity condition, though not quite the one envisaged by Bob. We might gain a more concrete understanding of the present proposal by thinking in terms of possible worlds. Each world w determines a modal space, the set V = {v ∈ W : wRv} of worlds accessible from w. There is therefore something about the world w in virtue of which V is its modal space; and we might take this something to be a modal state—it is, so to speak, the modal content of w. A state will then be possible relative to this modal state if it obtains in one of the worlds accessible from w. (Despite Bob’s remark on p.128, ‘Obviously only possible states will be relevant— that is the verifiers and falsifiers of ♦A and 2A will be members of S♦ ’, we might also allow there to be impossible modal states, which are determinative of a modal space which cannot be realized either because some of the states in the space are not possible or because it is not possible that they be all of the possible states.) Although the present semantics makes use of an accessibility relation on states, the parallel with the standard accessibility relation on worlds only goes so far. As we have seen, even for the modal logic T we do not have an unrestricted form of reflexivity, since it is only modal states that will stand in an accessibility relation to other states; and, for a similar reason, we do not have an unrestricted form of symmetry in the case of the modal logic B. We might therefore reconsider to what extent precisely analogous conditions will hold in the present case (p.129). The present account of the exact truthmakers for modal statements is not at all discriminating. A truthmaker for a modal statement will provide us with a complete account of the possibilities and so may be thought to contain a great deal of information that is irrelevant to the truth of a particular possibility or necessity statement. As a first step towards providing a more discriminating account, we might distinguish between inclusive and exclusive modal states. An inclusive modal state says, of certain states s1 , s2 , . . ., that they are all possibilities; an exclusive modal state says, of certain states s1 , s2 , . . ., that there are no other possibilities. Thus the fusion of an inclusive modal state with the corresponding exclusive state will correspond to a modal state in the previous sense. We might then say that a state s is an exact truthmaker for ♦A if it is an inclusive modal state which includes an exact truthmaker for A and that s is an exact truthmaker for 2A if is is an exclusive modal that excludes any exact falsity-maker for A. But we might go much further still and associate with any state s two other states: a state s♦ that is the possibility of s and a state s♦¯ that is the impossibility of s. We might then take an exact truthmaker for ♦A to be a state of the form s♦ for some exact

8 introduction truthmaker s of A (we might also allow s itself to be an exact truthmaker for ♦A) and we might take an exact truthmaker for 2A to be a state of the form s1♦ ¯  s2♦ ¯  ... where s1 , s2 , . . . are the falsity-makers of A. The truthmakers will then be tailored, so to speak, to the states that are taken to be possible or necessary. A great deal more work needs to be done in developing accounts of this sort: they need to be made technically precise; the conditions on the semantic primitives within the various models need to be determined (it is clear, for example, that if s is possible relative to s then so is any part of s ); soundness and completeness results need to be established for different modal logics; the accounts need to be related to one another and to the standard possible worlds account; and, at the more philosophical level, the formal semantics need to be related to an informal understanding of what the modal truthmakers might be. In this connection, too, Bob’s work is highly illuminating and can again be seen as pointing us in the direction of how a well-formed account along these lines might proceed, with modal truthmakers treated as determinative of a modal space and with essentially the same clauses for modal statements as the ones that he gives. (The only other work I know of on this topic is an unpublished paper of Jon Litland’s on an exact semantics for intuitionistic modal logic.) I dearly wish I could have continued my conversation with Bob. I am sure he would have made many interesting suggestions and corrected some of my misunderstandings or mistakes. But whether we knew him or not, we can all continue to have a conversation with and about his work; and it is what Bob—with or without a beer in hand—would have us do.

1 Essence and Definition by Abstraction 1.1 Introductory remarks—some kinds of definition There are doubtless different ways in which the expression ‘definition by abstraction’ may be understood.1, 2 What I mean by it is definition by means of what are now widely known as abstraction principles, that is, principles of the form: (α) = (β) ↔ Eq(α, β), where Eq is to be understood as an equivalence relation on entities of the type of α and β and, if all goes well,  can be taken to be a function from entities of that type to objects. The principle is understood as equivalent to its universal closure with respect to α and β, which may be first- or higher-order variables. This fixes the order of the abstraction. Thus, to give some well-known examples: Directions: The direction of line a = the direction of line b iff lines a and b are parallel

with the universal closure: ∀x∀y(Dir(x) = Dir(y) ↔ x//y) is a first-order abstraction, while: Hume’s principle: The number of Fs = the number of Gs iff the Fs correspond 1–1 with the Gs

with the universal closure: ∀F∀G(NxF(x) = NxG(x) ↔ Eqx (F(x), G(x))) is secondorder. At least in the context of the neo-Fregean programme, definition by abstraction is viewed as a form of implicit definition which we may employ to establish the meaning of fundamental terms for certain mathematical theories. In the case on which most work has been done, and on which most critical attention has been focused, the idea has been that Hume’s principle may be used to fix the meaning of the number operator Nx. . .x. . ., in terms of which the singular term ‘0’, and the predicates ‘. . . is a cardinal number’, ‘. . . precedes__’, ‘. . . ancestrally precedes__’, and ‘. . . is a natural number’ may then be defined in well-known ways: Cardinal(x) =df ∃F(x = NxF(x)) 0 =df Nx : x  = x P(x, y) =def ∃F∃z(F(z) ∧ x = Nu(F(u) ∧ u = z) ∧ y = NuF(u)) P∗ (x, y) =df ∀F((F(x) ∧ ∀u∀v(F(u) ∧ P(u, v) → F(v))) → F(y)) Natural(x) =df (x = 0 ∨ P*(0, x))

Our emphasis has generally been on the idea that in giving the implicit definition, what we do is to stipulate that the number operator is to mean whatever it needs to 1 This chapter was originally published as Hale (2018a). 2 For an illuminating historical discussion, see Mancosu (2016), ch.1. Bob Hale, Essence and Definition by Abstraction In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0002

10 essence and definition by abstraction mean—no more, no less—for the sentence, Hume’s principle, to express a truth, when the other expressions involved are taken to have their already established meanings, and the syntax is taken at face value. Much discussion has accordingly focused, quite rightly, upon the questions: Under what conditions, if any, can such a stipulation succeed in fixing a meaning for the definiendum? And: Are those conditions met in the present case?3 This stipulative conception of how Hume’s principle might be used as a definition is sometimes contrasted with what might be called an interpretative or hermeneutical conception, according to which the aim is rather to capture the meaning of an expression as it is already used. This is of a piece with what Quine, in ‘Two dogmas. . .’, described as the kind of definition put forward by lexicographers, aimed at reporting pre-existing relations of synonymy, “implicit in general or preferred usage prior to [their] own work” (Quine, 1953b, p.24). Whether Hume’s principle could, with any plausibility, be seen as giving a definition of this sort is—in my view—not a question that is easily answered. The answer must be that it cannot, if we take Quine’s description as definitive—i.e. as requiring of an interpretative definition that it present one complete expression as synonymous with another. Such a definition is what would normally be classed as explicit, but Hume’s principle clearly cannot function as a definition in that sense. What Hume’s principle can be used to define, if it can serve as a definition at all, is the number operator; but it presents no expression as synonymous with it. If, instead, we take recording a pre-existing synonymy relation as inessential to an interpretative definition, we could consider whether Hume’s principle could be regarded as providing an interpretative definition for the expression ‘the number of . . .’ (where the . . . holds place for a count noun or noun-phrase), or perhaps of ‘the number of things which . . .’ (where the . . . holds place for a verb-phrase, i.e. a oneplace predicate). Perhaps more plausibly, we could take the question to be whether the definitions of ‘cardinal number’ and ‘natural number’ based on the definition of the number operator by means of Hume’s principle can be regarded as capturing some already established meaning borne by these expressions. My impression is that some have thought the answer to this question is obvious: that they could not. I don’t myself think it is so obvious. A case which strikes me as relevantly similar to that of number is that of length. In that case, we have the noun ‘length’, which may function as a concrete noun (as in ‘he was holding a length of rope’), but also as an abstract one, as in ‘the length of my yacht is greater than the length of yours’; and we have the cognate expressions ‘the length of . . .’ and ‘. . . (is) as long as __’. It seems to me quite plausible that of these expressions, it is the last which is basic or fundamental. Roughly, what we learn first is how to make judgements of the forms ‘x is as long as y’, ‘x is shorter than y’, etc. Only once we have got the hang of these can we, by a kind of abstraction, come to understand talk of lengths—we learn that the length of x is the same as/less than/greater than that of y just when x is as long as/shorter than/longer than y; and by a further kind of abstraction, we come to talk of length and lengths without thinking of them as lengths of anything in particular. While there are

3 For an extended discussion of such conditions, see Hale and Wright (2000), reprinted as essay 5 in Hale and Wright (2001a).

1.1 introductory remarks 11 doubtless differences between talk of length and talk of number, it does not seem to me obviously implausible that similar relations hold between the relational expression ‘. . . as many as __’, ‘the number of . . .’ and ‘number’. There is, of course, a kind of halfway house between stipulative and interpretative definition—what Carnap called explication. As Quine puts it, “In explication the purpose is not merely to paraphrase the definiendum into an outright synonym, but actually to improve upon the definiendum by refining or supplementing its meaning” (Quine, 1953b, p.25). Since both refinement and supplementation require that the preexplicative meaning of the definiendum be, to some extent, preserved, one might think of explication as a kind of constrained or guided stipulation. Since the definition of the number operator, and the further definitions based on it, are clearly meant to preserve certain aspects of ordinary talk and thought about number and counting, there is, it seems to me, good reason to view them as explications in something like Carnap’s sense. In all of these kinds of definition, the aim is to fix the meaning of a word or other expression. All are thus to be distinguished from, and contrasted with, definition in a more Aristotelian sense, sometimes called real, as opposed to verbal or nominal, definition, where what is defined is not the word or phrase but the thing—what the word stands for or applies to. The aim of this kind of definition is to capture the essence or nature of whatever it is that the corresponding word stands for. Such a definition can be assessed as correct or incorrect, not in the sense that it captures or fails to capture the meaning of the word but according as it is a true or false statement of what it is to be what the word stands for. Some of the theoretical identifications Kripke discusses in Naming and Necessity (Kripke, 1980), although not expressly presented as such, may plausibly be seen as definitions of this kind: Water is H2 O Light is electromagnetic radiation between certain wavelengths Heat is the motion of molecules Lightning is an electrical discharge.

Obviously these are only rough and ready, but they capture the essential ideas of more accurate scientific statements of the nature of water, light, etc., as contrasted with explanations of the ordinary meanings of the corresponding words. The central question I wish to discuss concerns the relations between definition by abstraction and definition of this last kind. In particular, what is the relation between Hume’s principle, taken as an implicit definition of the number operator, along with the further definitions it supports (of cardinal and natural number predicates etc.), on the one hand, and the essence or nature of the corresponding entities—the function from concepts or properties to objects, and those objects themselves, i.e. the cardinal numbers in general, and the natural numbers? Can the now familiar sequence of Fregean definitions serve both as fixing meanings and as definitions in the more Aristotelian sense, as correct statements of the essence of the numbers etc.? An obstacle to returning an affirmative answer seems to me to suggest itself right at the outset. Is there not a direct clash between treating something as a definition in accordance with the stipulative conception and treating it as a real definition? Doesn’t that require treating the definition, not as a stipulation of any sort but as a true or

12 essence and definition by abstraction false proposition aimed at articulating some independently constituted fact about the nature of the worldly correlate of the relevant word? I believe this apparent tension can be resolved, and that its resolution can shed light both on the notion of a thing’s essence or nature—at least as I wish to understand this notion—and on an important part of the epistemology of essence. To prepare the ground for the resolution I shall propose, it will be useful to begin with the question: How, in general, are verbal and real definitions related?

1.2 Aristotle and Locke Although I shall not discuss the views of Aristotle, or those of Locke, at any length, the account I shall present has affinities with both, whilst diverging from both in important respects—so I think it will be useful to comment briefly on those aspects of their views which bear most closely on my main question.

1.2.1 Aristotle Aristotle discusses both kinds of definition: Since a definition is said to be an account of what something is, it is clear that one type will be an account of what its name, or some other name-like account, means—e.g. what triangle means. When we grasp that this exists, we seek why it is. But it is difficult to take anything in this way if we do not know that it exists. (Post.An. 93b29–34)

This, as is typical of Aristotle’s logical works, is a very compressed statement. But it is clear enough that here he is talking, in part, about what is sometimes called nominal definition. On the plausible interpretation defended by David Charles (Charles, 2000, ch.2), defining a word is the first preliminary step in a three-stage process which, if successful, culminates in the kind of definition Aristotle sees as the goal of scientific enquiry—an account of the nature or essence of what the word, with that meaning, stands for. The second, intermediate stage consists in verifying that the word, with that meaning, does indeed stand for something—it is only in regard to what exists that we can discover its nature, in Aristotle’s view. As my short quotation illustrates, he seems to identify the final stage—establishing the nature of something—with explaining why it exists. This may at first appear somewhat odd, but we can see roughly what he is driving at by considering some of his examples. Thus a little further on in Post.An. B10, he writes: One definition of definition is the one we have just stated [i.e. saying what the word means]. Another is an account which shows why something exists. Hence the former type means something but does not prove it, whereas the latter will clearly be like a demonstration of what something is, differing in arrangement from a demonstration. For there is a difference between saying why it thunders and what thunder is. In the one case, you will say: Because fire is extinguished in the clouds. But: What is thunder?–A noise of fire being extinguished in the clouds. (Post.An. 93b38–94a7)

What he seems to mean here is that what the word ‘thunder’ means is ‘noise in the clouds’, whereas a definition of what thunder is will incorporate an explanation of this type of noise—it is noise made by fire being extinguished in the clouds. The explanation of thunder is repackaged, as it were, as its definition.

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1.2.2 Locke Locke discusses essence at length in the Essay BkIII, as part of his discussion of words. Central to his discussion is his distinction between ‘nominal’ and ‘real’ essence. But even before he makes that distinction, the central tenets of his position are laid down: Words become general by being made the signs of general ideas: and ideas become general by separating from them the circumstances of time and place, and any other ideas that may determine them to be this or that particular existence. By this way of abstraction they are made capable of representing more individuals than one . . . (Locke, 1924, ECHU, III.iii.6) He that thinks general natures or notions are anything else but such abstract and partial ideas of more complex ones, taken at first from particular experiences, will, I fear, be at a loss where to find them . . . This whole mystery of genera and species, which makes such a noise in the schools, and are, with justice, so little regarded out of them, is nothing else but abstract ideas, more or less comprehensive, with names annexed to them. (op. cit. III.iii.9) . . . That then which general words signify, is a sort of things; and each of them does that by being a sign of an abstract idea in the mind: to which idea, as things existing are found to agree, so they come to be ranked under that name; or, which is all one, be of that sort. Whereby it is evident that the essence of the sorts . . . are nothing else but these abstract ideas. (op. cit. III.iii.12) They [i.e. the abstract ideas, and so the essences of sorts of things] are the workmanship of the understanding, but have their foundation in the similitude of things. (op.cit. III.iii.3.13)

After drawing the inevitable conclusion that each distinct abstract idea is a distinct essence, Locke finally comes to the distinction between nominal and real essences: First, Essence may be taken for the being of anything, whereby it is what it is. And thus the real internal, but generally in substances unknown, constitution of things, whereon their discoverable qualities depend, may be called their essence . . . Secondly, . . . [the word ‘essence’] has been almost wholly applied to the artificial constitution of genus and species. It is true, there is ordinarily supposed a real constitution of the sorts of things; and it is past doubt that there must be some real constitution, on which any collection of simple ideas coexisting must depend. But it being evident that things are ranked under names only as they agree in certain abstract ideas to which we have annexed those names, the essence of each genus or sort comes to be nothing but that abstract idea. . .. These two sorts of essences, I suppose, may not unfitly be termed, the one the real, the other the nominal, essence. (op. cit. III.iii.15)

Locke mainly discusses essence in connection with general terms for substances: here, Locke claims, real and nominal essence are quite different, and it is nominal essence on which he focuses attention, arguing that it is this which does all the important work, and that talk of real essences is idle and useless—in particular, his view seems to be that it is the abstract idea (i.e. the nominal essence) which defines what it is for something to be of a certain sort, such as man or horse—we need not trouble ourselves about the generally unknown (and, Locke seems sometimes to suggest, unknowable) internal constitution of things which causally explains the features we do know about.

14 essence and definition by abstraction

1.2.3 Comments There are important differences between Aristotle’s and Locke’s positions, symptomatic of a deep opposition between them. Both focus mainly on ‘substances’—in a sense of that term which covers both kinds or sorts of thing, such as horses and meteors, and kinds of stuff, such as gold and water—and on our words for them. But whereas for Aristotle, defining the term or word for something is merely a preliminary stage in the search for essence, for Locke it is the key. The important thing for him is the nominal essence; he does not deny that there is a real essence, and indeed insists that there must be ‘some real constitution, on which any collection of simple ideas coexisting must depend’. But in general, we do not know what that is, and its importance, if known, lies only in its providing a causal explanation of those ideas which collectively constitute the nominal essence. It remains Locke’s view that it is possession of those properties corresponding to those simple ideas which make up the abstract general idea—the nominal essence—which determine whether or not something is of a certain kind—and not the ‘real internal constitution’ on which those properties depend. His view thus stands opposed not only to Aristotle’s but also to the modern essentialist view promoted by Kripke, according to which the surface properties by which we generally identify natural kinds are not what make them what they are, that being rather the essential properties underlying and explaining the presence of the surface properties in normal circumstances. On the Kripkean view, even if it is not naturally possible that things or stuff should be of a certain natural kind— be tigers, or gold, say—whilst lacking certain of those surface features, it is at least metaphysically possible; and conversely, it is metaphysically, even if not naturally, possible that there should be things possessing all the surface features, which are not of that kind (fool’s tigers or fool’s gold, say). But not on Locke’s view. Underlying this difference is a deeper divergence over essential modalities. For Kripke, as for Aristotle, the necessity with which things possess their essential properties is de re; but for Locke, it is de dicto. To be gold, say, is to conform to a certain abstract general idea which includes having a certain colour and weight, being fusible and fixed (i.e. able to be melted, but not evaporating or losing weight when heated) etc. As we might somewhat anachronistically put it, it is, for Locke, analytic that gold is yellow, heavy, fusible, etc. This is why Locke is so adamant that individuals cannot sensibly be taken to have essential properties: That essence, in the ordinary use of the word, relates to sorts, and that it is considered in particular beings no farther than as they are ranked into sorts, appears from hence; that take but away the abstract ideas by which we sort individuals, and rank them under common names, and then the thought of anything essential to any of them instantly vanishes . . . Let anyone examine his own thoughts, and he will find, that as soon as he supposes or speaks of essential, the consideration of some species, or the complex idea, signified by some general name, comes into his mind; and it is by reference to that, that this or that quality is said to be essential. . . . to talk of specific differences in nature, without reference to general ideas and names, is to talk unintelligibly. For I would ask any one, What is sufficient to make an essential difference in nature, between any two particular beings, without any regard had to some abstract idea, which is looked upon as the essence and standard of a species? All such patterns and standards

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being laid quite aside, particular beings, considered barely in themselves, will be found to have all their qualities equally essential; and everything in each individual will be essential to it. Or, which is more, nothing at all. (op. cit. III.vi.4–5)

There could not be a clearer rejection of de re necessity. Locke’s position is, in effect, an anticipation of that championed by Quine some two and a half centuries later: necessity resides, not in the things we talk about but in the way we talk about them.4 Although, in my view, Locke’s central claims about essence—driven by his animus against scholastic claims about it, cf. ‘This whole mystery of genera and species, which makes such a noise in the schools’—are mistaken, what he says about nominal and real essence in relation to non-substance terms merits separate comment. Immediately following the section in which he introduces his distinction, he remarks: Essences being thus distinguished into nominal and real, we may farther observe, that in the species of simple ideas and modes, they are always the same; but in substances, always quite different. Thus a figure including a space between three lines, is the real as well as the nominal essence of a triangle; it being not only the abstract idea to which the general name is annexed, but the very essentia, or being of the thing itself, that foundation from which all its properties flow, and to which they are all inseparably annexed. (op. cit. III.iii.18)5

The kind of coincidence of nominal with real essence to which Locke here draws attention—or, what comes to essentially the same thing, agreement between verbal and real definition—seems to me quite certainly to occur, and indeed to be well illustrated by Locke’s own example. To give another, if we ask what it is to be a circle, a good answer is that it is a collection of all and only those points in a plane equidistant from some fixed point in the plane; but this answer is every bit as good as an answer to the question: What does the term ‘circle’ mean? Similarly, to be a vixen is to be an adult female fox, and that is what ‘vixen’ means. Examples are easily multiplied. This

4 In Mackie (1974), John Mackie pointed out that Locke certainly recognizes the possibility of adopting a view like Kripke’s, according to which our intention in using a term such as ‘gold’ is to refer to the substance whose internal constitution underlies and explains the surface properties which make up our abstract general idea of gold (its nominal essence), and allows that people may use the word with that intention (cf. “. . . the mind . . . makes them [words for substances], by a secret supposition, to stand for a thing, having that real essence”, even when we do not know what the real essence is; also “there is scarce any body in the use of these words, but often supposes each of those names to stand for a thing having the real essence, on which these properties depend” (Locke, ECHU, III.x.18); but he makes it clear that he thinks this practice is misguided and misleading. 5 In Selby-Bigge’s abridged edition of Locke’s Essay, the continuation might seem to indicate that Locke is less firmly opposed to essentialism as it is usually understood than I have claimed. It runs: “But it is far otherwise concerning that parcel of matter which makes the ring on my finger, wherein the two essences are apparently different. For it is the real constitution of its insensible parts, on which depend all those properties of colour, weight, fusibility, fixedness, etc., which makes it to be gold, or gives it a right to that name, which is therefore its nominal essence.” The subject–main verb agreement virtually obliges us to read this sentence as asserting that it is the real constitution [rather than the properties of colour etc.] which makes something to be gold. In fact, this is a catastrophic editorial blunder. Locke’s original text reads: “For it is the real constitution of its insensible parts, on which depend all those properties of colour, weight, fusibility, fixedness, etc., which are to be found in it, which constitution we know not, and so having no particular idea of, have no name that is the sign of it. But yet it is its colour, weight, fusibility, fixedness, etc., which makes it to be gold, or gives it a right to that name, which is therefore its nominal essence” (see, e.g., Locke 1812, vol.1 p.451). Selby-Bigge’s compression completely changes Locke’s clear meaning.

16 essence and definition by abstraction happy coincidence, where real and verbal definitions may be given by the same words, is the starting point for my own answer to my leading questions.6

1.3 A priori knowledge of essence In many cases, then—or so I claim—a correct definition of what a word stands for may serve as a good definition of the word itself. But in many other cases, it will not. For example, a correct definition of the stuff, water, is probably something along the lines of: the liquid form of a substance composed of molecules consisting of two hydrogen atoms bonded with one oxygen atom; but this is not a good definition of the word ‘water’. You will easily think of other examples. There may also be—indeed, it seems to me that there are—examples where there may be no clear answer whether a correct definition of a thing is also a correct definition of the word, in part because word-meanings change over time, so that what starts out being a discovery about something’s nature may become part of what a word for it means. This might be the case with tigers, for example. A tiger is a large carnivorous mammal of the cat family, of the genus panthera, distinguished from other species by certain further characteristics, including its striped skin and coat. I doubt that the whole zoological definition would, even now, amount to a correct definition of the word, as used by most speakers. But it is quite plausible that it is now part of the word’s meaning that tigers are mammals and that they are large cats, even if it wasn’t when the word was first used. The coincidence of correct verbal and real definition helps to explain how it is that, in some cases, we may have a priori knowedge of something’s essence or nature. The kind of explanation I have in mind is most readily illustrated by reference to cases in which we can give an explicit definition of a word. Thus we can define the noun ‘square’ by: A square is a plane figure bounded by four straight sides of equal length meeting at right angles.

Such an explicit definition can always be rewritten as a biconditional: ∀x(x is a square ↔ x is a plane figure bounded by four straight sides of equal length meeting at right angles).

6 Locke’s general claim, that nominal and real essence are the same for all simple ideas and modes cannot, it seems to me, be entirely right. To give an admittedly debatable example, the idea of redness would rank for him as a simple idea, but being red (as a property of surfaces) might plausibly be taken to consist in the capacity to reflect light in the range 640–720nm. To give another, perhaps less debatable, example—one which Locke himself explicitly discusses—the idea of light is simple. But while he would have to say in this case nominal and real essence are the same, it is plausible that, to the contrary, light may be defined, even if the word for it, and the idea that is its meaning (for Locke), cannot be. Locke chides “those who tell us, that light is a great number of little globules, striking briskly on the bottom of the eye”—he grants that they “speak more intelligibly than the schools”, but complains: “but yet these words ever so well understood would make the idea the word light stands for no more known to a man that understands it not before, than if one should tell him, that light was nothing but a company of little tennis-balls, which fairies all day long struck with rackets against some men’s foreheads, whilst they passed by others” (op. cit. III.iv.10). Locke is of course right that the particle theory of light doesn’t explain what the word ‘light’ means, as used by ordinary speakers of English. But that, contrary to what Locke is forced to say by his identification of its nominal with its real essence, does not mean that it cannot be a good definition of what light is.

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The definition tells us that being a plane figure etc. is a necessary and sufficient condition for the noun ‘square’ to be correctly applied to something. But if the noun ‘square’ is correctly applicable to something, that thing is a square. So anyone who is in a position to give this definition of the word can recognize that being a plane figure etc. is a necessary and sufficient condition for being a square—that is, that that is all there is to being a square. But to know that much is to know what it is to be a square. On the reasonable assumption that if all one needs to know, if one is to know a that p, is the meaning of some sentence S which expresses the proposition that p, such knowledge is a priori.7 What could be easier? Of course, we cannot always give explicit definitions. But I can see no reason why this simple and straightforward model for a priori acquistition of knowledge of essence should not be extended to other cases—both those in which, although we cannot explicitly define a word, we can give an implicit definition, and others in which neither kind of definition can be given. These cases can be illustrated by reference to our understanding of logical constants. We cannot define the truth-functional connectives explicitly. But provided that the meanings of some of them can be taken as already understood, we can implicitly or contextually define others. Thus taking words for negation and disjunction as already understood, we can implicitly define the truth-functional conditional by stipulating that A → B is to be true iff ¬A ∨ B is. → stands for a certain function from propositions to propositions. Someone who understands the implicit definition is able to recognize that the function for which it stands is that function which takes an ordered pair of propositions to a true proposition iff the first of them is false or the second is true. To know this is to know the essence of the truth-functional conditional. Supposing negation and disjunction to be our basic connectives, we cannot define either of them, either explicitly or implicitly. But we can learn what they mean. Anyone who knows what truth-functional disjunction means knows that A ∨ B is true iff at least one of the pair A, B is true. This knowledge of what the word means equips them to recognize that truth-functional disjunction is that function from propositions to propositions which takes a pair of propositions to a true proposition iff at least one of the pair is true. If I am right, there are many cases in which a priori knowledge of essence can be based upon knowledge of the meanings of relevant words. Definition by abstraction can be seen as a special case. Hume’s principle, construed as an implicitly definitional stipulation, serves—if all goes well—to fix the meaning of the number operator. Once that is done, we may explicitly define x is a cardinal number ↔def ∃F(x = NxFx)

and 0 =def Nx(x  = x)

and finally x is a natural number =def x = 0 ∨ P* (0, x)

7 Of course, knowledge of the meanings of S’s ingredient expressions and its semantically significant structure will be a posteriori. But any sensible account of a priori knowledge will allow that.

18 essence and definition by abstraction where P* is ancestral predecession, where, as Frege himself proposed, predecession is defined by: Pxy = def ∃F∃z(y = NuFu ∧ x = Nu(Fu ∧ u  = z))

and its ancestral by: P* xy = ∀F((F0 ∧ ∀u∀v((Fu ∧ Puv) → Fv)) → ∀yFy)

On the basis of these definitions one can gain knowledge a priori of the nature of cardinal and natural numbers.

1.4 Definition and essence 1.4.1 An obvious lacuna If I am right, knowledge of essence in the kind of case in which my proposed explanation applies is much more easily obtained than in cases in which only a posteriori knowledge is possible. No painstaking empirical investigation or scientific breakthrough is required. Essence is, as it were, open to view. I think this explanation of how we may get a priori knowledge of essence is good as far as it goes. But it does not go far enough. There is, in particular, at least one further question which clamours for an answer: The explanation tells us how, when the definition of a word and a real definition of the thing for which it stands coincide, we can gain knowledge of the nature of the thing from our grasp of the meaning of the word. But how is this happy coincidence—between the definition of the word and the essence of the thing—itself to be explained? The meanings of words are, in some suitably broad sense, a matter of convention. We fix them, either expressly by explicit or implicit definition, or in less formal ways, by our ongoing linguistic practice. So how does it come about that in cases of the kind I have been discussing, we fix them so that they match up so well with the essence or nature of the things the words stand for?

1.4.2 Direction of fit Put like that, the question is liable to appear intractable. For putting it like that assimilates the cases with which we are concerned to those involving natural kinds, which, we assume, have independently constituted natures which we seek to capture by investigation a posteriori. If we think of our cases on this model, preserving the same direction of fit, it may well seem that achieving fit, without empirical investigation, requires some mysterious kind of pre-established harmony. But that is, fairly obviously, the wrong picture. To a first approximation, we might say that the right picture reverses the direction of fit: it is not that we somehow miraculously fix meaning so that it conforms to an independently pre-determinate essence—instead, in these cases, we fix essence by fixing meaning. I will explain why this can only be at best a first approximation shortly. If this alternative picture is approximately right, it immediately resolves the apparent tension noted previously, between endorsing the stipulative conception of how Hume’s principle may be used to define the number operator and regarding it and the further definitions based upon it as revealing essence. The tension results from a

1.4 definition and essence

19

false assimilation to cases in which we seek to capture the essence of natural kinds. Cardinal and natural numbers are not natural kinds—at least not in the same sense in which gold and tigers are. They are, as one might say, artificial kinds.

1.4.3 Some objections answered That description risks misunderstanding of a sort that relates closely to a likely cause of resistance to the proposal that we fix essence by fixing meaning. If things of a certain sort—numbers, say—are an artificial kind, doesn’t that mean that they are mere artefacts, which would not exist, were it not for our our activity? Mustn’t the view that we fix essence by fixing meaning carry with it an intolerable sacrifice of objectivity? Soberly understood, the claim that, in certain cases, we fix essence need have no such unpalatable consequences. To explain why not, I shall draw on two points—one is a point about the specific character of definition by abstraction; the other is a more general point about properties. The specific point about abstraction is one which Hale and Wright have emphasized on many occasions. Definition by abstraction creates no objects. What, if successful, it creates is only a concept. Or to put it more accurately, what it does is to fix the meaning of a term-forming operator, such as the number operator. It does this by fixing the condition for identities configuring singular terms formed by applying that operator to suitable arguments to be true. Such identities are to have the same truth-conditions as those of corresponding statements asserting those arguments to stand in a certain equivalence relation. The existence of referents for those terms—numbers, say—is required by the truth of such identities. If the definition succeeds, such identities are true iff the corresponding statements involving the relevant equivalence relation are. It follows that so long as it is a matter of objective, mind-independent fact that, say, there is a one–one correlation between the Fs and the Gs, the existence of the number of Fs and the number of Gs (and their identity) is an equally objective, mindindependent fact. That point may be thought enough, by itself, to dispel any reasonable concern about loss of objectivity. But my more general point about properties is worth making, because it illuminates the kind of existence numbers, and other pure abstract objects, enjoy. As just noted, if Hume’s principle can be used to define the number operator, standing for a certain function from properties to objects, the existence of numbers depends only upon there being properties which are one–one correlated. No more is required for the existence of numbers than that of first-level properties and a certain second-level equivalence relation. According to the abundant or deflationary conception of properties and relations, no more is required for the existence of any purely general property or relation than that there could be a predicate with the appropriate satisfaction condition.8 Provided that this very weak existence condition is met for some suitable purely general properties and the purely general relation of one–one correlation, no more is required for the existence of cardinal numbers. This brings out the sense in which numbers are ‘metaphysically lightweight’ objects.

8 See Hale and Wright (2009, 207–8), and for a somewhat fuller discussion, Hale (2015a), ch.1, esp. 1.12.

20 essence and definition by abstraction Numbers no more depend upon us for their existence than does the relation of one–one correlation or the purely general properties which stand in that relation. They all depend for their existence on the possibility of expressions of the appropriate logico-syntactic type having them as their semantic values, but the requisite facts about possibility are themselves in no way dependent upon us or any other thinkers and speakers. Provided that it is agreed that the kind of possibility involved in formulating the existence condition for general properties conforms to S5 principles, it can easily be shown that their existence, and hence that of numbers, is absolutely necessary.9 To put the point slightly paradoxically, whilst we fix what it is to be a cardinal or natural number, their being is absolutely necessary and hence quite independent of merely contingent beings, such as—on one widely but, of course, not universally accepted view—we are!

1.4.4 Direction of fit again Against the backdrop of the abundant conception of properties (and objects), it is easier to pinpoint what is wrong, and also what is right, in the suggestion that we fix essence by fixing meaning.10 What is wrong with it is that it does insufficient justice to the fact that the properties that constitute a thing’s essence or nature exist quite independently of any definitional activity on our part. We fix the meaning of the word—‘square’ say—by stipulating that to be square, a plane figure must be composed of four sides of equal length meeting at right angles. But that there exists such a property—the property of being a figure so composed—is entirely independent of any such definition being given (though not, on the abundant conception, independent of the possibility of such a definition). What is right about it—or at least very nearly right— lies in the suggestion that there is little or no room for error. But the explanation how we achieve such a happy match between our definition and the essence is not 9 Let φ be any purely general property, and let p be the statement that φ exists, and q the statement that there exists a predicate standing for φ. Then according to the deflationary account: p ↔ ♦q. Further, this statement, if true, will be so as a matter of necessity, at least on the deflationary account, so that: (a) 2(p ↔ ♦q) But (a) entails both: (b) p ↔ ♦q and (c) 2p ↔ 2♦q and, by S5, (c) entails (d) 2p ↔ ♦q But from (b) and (d), it follows by the transitivity of the biconditional that (e) p ↔ 2p Since φ may be any purely general property, it follows that each purely general property there is exists as a matter of necessity. 10 I am much indebted here to Joachim Horvath, both for pressing the objection originally, and for subsequent discussion of it, in which it emerged that the line I propose here has close affinities with one taken in his dissertation. See Horvath (2011).

appendix 21 that we fix the essence to suit our definition; it is rather that, given the abundant conception, we cannot very well miss the target. For according to that conception, provided that there could be a predicate with the appropriate satisfaction-condition, the existence of the corresponding property is guaranteed. Thus provided only that our purported definition is in good order, it cannot fail to pick out the requisite defining property.

Appendix: Note on Dummett on mathematical objects In his paper ‘What is mathematics about?’ (Dummett, 1993b, 429–45), Michael Dummett writes: In order to confer sense upon a general term applying to concrete objects—the term “star” for example . . .we consider it enough that we have a sharp criterion for whether it applies to a given object, and a sharp criterion for what is to count as one such object—one star, say—and what as two distinct ones: a criterion of application and a criterion of identity. The same indeed holds true for a term, like ‘prime number’, applying to mathematical objects, but regarded as defined over an already given domain. It is otherwise, however, for such a mathematical term as “natural number” or “real number” which determines a domain of quantification. For a term of this sort, we make a further demand: namely, that we should ‘grasp’ the domain, that is, the totality of objects to which the term applies, in the sense of being able to circumscribe it by saying what objects, in general, it comprises—what natural numbers, or what real numbers, there are. The reason for this difference is evident. For any kind of concrete object, or abstract object whose existence depends upon concrete objects, external reality will determine what objects of that kind there are; but what mathematical objects there are within a fundamental domain of quantification is supposed to be independent of how things happen to be in the world, and so, if it is to be determinate, we must determine it. (p.438)

This passage is apt to seem somewhat puzzling. We may well agree with Dummett that the composition of a fundamental mathematical domain, such as the natural or the real numbers, should be—as he puts it—independent of how things happen to be in the world, but wonder what exactly is meant by the suggestion that if it is to be determinate, we must determine it, and be reluctant to agree to it. For Dummett, the remarks I have quoted are a prelude to arguing that fundamental mathematical domains—including those of the real and even the natural numbers— are indefinitely extensible, and, on the back of that, that we may only legitimately employ constructive, non-classical principles in reasoning about them. This is hardly the place to take on those arguments. Instead, I shall try to sketch a way in which the ideas of the preceding section might be deployed to preserve a modest form of realism about the composition of such domains. I shall, however, need to take issue with one of Dummett’s early claims in making his case. He writes: One the face of it, indeed, a criterion of application and a criterion of identity do not suffice to confer determinate truth-conditions on generalizations involving some general term, even when it is a term covering concrete objects: they can only give them content construed as embodying a claim. So understood, an existential statement amounts to a claim to be able to give an instance; a universal statement is of the form “Any object to which the term is recognized as being applicable will be found to satisfy such-and-such a further condition”. (p.438)

22 essence and definition by abstraction Since Dummett offers nothing in support or even explanation, I can only conjecture that he is taking criteria of application and identity in an epistemological sense, so that they embody the means whereby judgements of application and identity may be made, in contrast with conditions for the truth of such judgements. But, at least in the case of the natural numbers, I can see no compelling reason why we should not take our Fregean definitions, based ultimately upon an implicit definition of the number operator by means of Hume’s principle, as giving necessary and sufficient conditions for the application of the predicate ‘ξ is a natural number’ and for the identity or distinctness of its instances, and so (together, of course, with any other definitions of arithmetical terms which may be needed) providing truth-conditions for numbertheoretic statements.11 If we may so take them, we may agree with Dummett that we determine what lies within the domain of natural numbers, in the sense that we fix the concept of natural number, and thereby fix what it is to be a natural number, whilst holding that what objects that domain comprises is not itself of our making in any sense that conflicts with a modest realism, but is determined by how things stand in the world, independently of us and our activities. To be sure, it is entirely independent of contingent features of the world, just as Dummett observes it should be. For the existence of the properties and the relation of one–one correspondence, and that of the operation whose values the natural numbers are for properties bearing that relation to one another, involved in the Fregean definitions, is—on the abundant or deflationary theory—a matter of necessity, entirely independent of us. It is a further—and in my view, much more difficult—question, whether the same may be said of the domain of real numbers. Of this case, Dummett writes: Cantor’s celebrated diagonal argument to show that the set of real numbers is not denumerable has precisely the form of a principle of extension for an indefinitely extensible concept: given any denumerable totality of real numbers, we can define, in terms of that totality, a real number that does not belong to it. (p.442)

To this, there is an obvious response (as Dummett himself notes): to show that no denumerable totality of real numbers contains them all is not to show that there is no determinate or definite totality of real numbers, for there may be, as Dummett’s realist opponents suppose, a non-denumerable one. Simply to assume that any definite totality must be denumerable is to beg the question against them. Dummett seeks to forestall this objection: The argument does not show that the real numbers form a non-denumerable totality unless we assume at the outset that they form a determinate totality comprising all that we shall ever recognize as a real number: the alternative is to regard the concept real number as an indefinitely extensible one. (ibid.)

But it is unclear, at best, that the realist must prove that the real numbers form a definite totality. On the face of it, he may protest, there is a set—definite totality—comprising 11 To be sure, Dummett believes the impredicativity of Hume’s principle—unavoidable if it is to serve as a foundation for arithmetic—means that it cannot be acceptable. But Wright and I have endeavoured to deal with this objection elsewhere. See Hale (1994, 6) and Wright (1998a), both reprinted in Hale and Wright (2001a).

appendix 23 all and only the finite cardinals, i.e. the natural numbers—it comprises exactly 0 together with those cardinal numbers which bear the ancestral of the immediate successor relation to it. Similarly, the set of reals has as its members precisely those numbers which are determined by Dedekind cuts in the rationals (or by Cauchy sequences of rationals), so that here too there is a prima facie case for taking them to form a definite totality. Of course, there is a parallel prima facie case for supposing there to be a set of all ordinals, but as is shown by Burali-Forti’s paradox, there can be no such definite totality, denumerable or otherwise. However, there is no parallel argument to show that the reals cannot form a definite totality. The issue is, as Dummett sees, one of where the burden of proof lies. He grants that such issues are always difficult to resolve, but thinks that in the present case it is clearly the realist who has begged the question and must shoulder the burden— his constructivist opponent “does not assume . . . that any totality of which it is possible to form a definite conception is at most denumerable; he merely has as yet no conception of any totality of higher cardinality” (p.443). But it is not clear. We can state a necessary and sufficient condition for something to be a real number. It must be granted that this does not guarantee the existence of a totality comprising all and only the real numbers—so much is the moral of the Burali-Forti. “[T]he characterization of an indefinitely extensible concept”, Dummett claims, “demands much less than the once-for-all characterization of a determinate totality” (p.442). But he gives no hint as to what more may be required, and it is unclear what further positive requirement, beyond provision of necessary and sufficient conditions for membership, might be imposed—as distinct from the absence of any defeating argument, paralleling Burali-Forti’s paradox, to show that the assumption of a definite totality leads to contradiction.

2 Essence and Existence 2.1 Essentialist theories of metaphysical necessity 2.1.1 The general form of an essentialist theory The fundamental questions in the philosophy of necessity and possibility are two, one metaphysical, the other epistemological:1, 2 1. What is the source or ground of necessities and possibilities? 2. How may we know what is necessary and what is possible? An essentialist theory answers the first, metaphysical, question: what is necessary is what is true in virtue of the nature or essence of things, and what is possible is what is not ruled out by the natures of things (i.e. what is not false in virtue of the nature of things). Using 2x p to mean ‘it is true in virtue of the nature of x that p’, and 2, ♦ to express metaphysical necessity and possibility, we may provisionally state the general theory as:3 2p =def ∃x1 . . . xn 2x1 ...xn p

♦p =def ¬∃x1 . . . xn 2x1 ...xn ¬p This statement is neutral on the answers to some further questions, different answers to which will give rise to significantly different essentialist theories. Perhaps the most important of these questions concerns the logical form of the most basic claims made by the theory—claims of the form ‘it is true in virtue of the nature of x that p’, which we are abbreviating to 2x p. The sole assumption we have thus far made about the form of these claims is that they involve two variables, x and p. The latter varies over propositions, the former over things. We’ve so far said nothing about which things are the admissible values of this variable, i.e. which things are being taken to have natures, in some interpretation of that phrase. So that is one question: What, more exactly, does x vary over?

1 This chapter was originally published as Hale (2018b) in volume LVII, number 147, of Revista de Filosofía de la Universidad de Costa Rica. 2 A succinct and compelling statement of these questions may be found in Michael Dummett’s article on Wittgenstein’s philosophy of mathematics: ‘The philosophical problem of necessity is twofold: what is its source, and how do we recognise it?’ (Dummett (1959), p.169). 3 As we shall see much later, both principles will require an important modification, if we both hold that the modal logic of metaphysical necessity and possibility is S5 and accept that what essences there are is, to some extent, a contingent matter. But I think it is best to allow the need for amendment to emerge from the more detailed discussion of these matters in what follows. See footnote 21. Bob Hale, Essence and Existence In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0003

2.1 essentialist theories of metaphysical necessity

25

However, there is another, quite separate, question about the interpretation of ‘It is true in virtue of the nature of . . . that __’, viz. Does it have further logical structure (i.e. significant syntactic structure)? In particular, should it be understood as formed by means of a binary operator ‘__ in virtue of . . .’ taking a proposition as its first argument and a term as its second, with the first argument in our case being itself formed by applying the truth-operator to a sentence and the second by applying the termforming operator ‘the nature of . . .’? This would be to interpret our basic essentialist claims as having the structure: (It is true that (p)) in virtue of (the nature of (x))

It may seem very natural to interpret our basic claims as so structured. I shall call this the structured interpretation. However, there is clearly also an alternative, austere interpretation, on which our basic claims have no such complex internal structure, and on which we should regard ‘It-is-true-in-virtue-of-the-nature-of . . . that __’ as an unbreakable binary operator, with one argument-place to be filled by a singular term (or variable) for a thing and the other to be filled by a sentence.4

2.1.2 Some questions and problems 2.1.2.1 contingently existing individuals Some of the questions a proponent of an essentialist theory faces do not depend upon interpreting ‘It is true in virtue of the nature of . . . that __’ as possessing significant internal structure, but arise equally if one takes it to be logically or semantically unstructured. Most obviously, on either interpretation we must face the question: What does x in 2x p vary over? This is tantamount to asking: What things have natures? Or at least, it is so, provided that we do not construe that question as presupposing that natures are entities over and above the things which have them—to avoid this unwanted implication, we might prefer to ask: What things x are such that 2x p is significant? English grammar requires that the x-place in 2x p be filled by a noun or nounphrase. Thus admissible replacements for x include common nouns, both singular and plural, as in: It is true in virtue of the nature of man that man is an animal It is true in virtue of the nature of whales that whales are mammals It is true in virtue of the nature of water that water is a compound

We may also have proper nouns, i.e. proper names, as in: It is true in virtue of the nature of Aristotle that Aristotle is a man

More generally, the x-place can be filled by any singular term, at least as far as grammar goes. The x-place cannot be filled by expressions of any other syntactic type—it cannot be filled, for example, by any predicate, or relational or functional expression; nor can 4 The notation 2x p to abbreviate ‘It is true in virtue of x’s nature that p’ is borrowed from Kit Fine (Fine (1994)), but it should not be assumed that my use of the notation coincides with his. In particular, while I am sympathetic to the structured interpretation, Fine may prefer the austere one (I am not sure whether he commits himself to it in published work, but I seem to recall a conversation in which he suggested this).

26 essence and existence it be filled by conjunctions such as ‘and’, ‘or’, ‘if ’, etc. If, as some of us do, we think of expressions of these types as nevertheless standing for entities, we shall need to employ nouns or other nominalizations corresponding to these non-nominal expressions. We could then have things like: It is true in virtue of the nature of being wise that anything wise is animate It is true in virtue of the nature of addition that a + b = b + a It is true in virtue of the nature of disjunction that if p is true, so is p ∨ q

If, as I have suggested, the x-place may be filled by a proper name of an individual, such as ‘Aristotle’, then, given the quite widely held view that the existence of such individuals is often a contingent matter, there is an obvious further question to be faced: Does the truth of such basic essentialist claims require the existence of the object whose name occupies the x-place? Does our statement about Aristotle’s nature entail Aristotle’s existence? Connected with this question there is a potential problem. For according to the essentialist theory as so far formulated, what is true in virtue of a thing’s nature is necessarily true, so that we may pass by the theory from our statement about Aristotle’s nature to Necessarily Aristotle is a man

But it seems that Aristotle cannot be a man if he does not exist, so that this in turn entails Necessarily Aristotle exists

which runs counter to the widely held view that his existence is a contingent matter, i.e. that Aristotle might never have existed

Thus the essentialist theory apparently leads to the conclusion, unpalatable to believers in contingently existing individuals, that Aristotle is a necessary being. Before I discuss how this problem might be solved, I want to introduce another problem which should not be confused with our first problem.

2.1.2.2 contingently existing natures If we adopt the structured interpretation of basic essentialist claims, we are—on the face of it—taking the nature of a thing x to be a further entity, distinct from x itself. But then how is a thing’s nature related to the thing itself? In particular, does the existence of x’s nature depend upon x’s existence? How one answers this question may depend upon what kind of entity one takes a thing’s nature to be. One might, for example, take x’s nature to be the conjunction of x’s essential properties, and so a complex property. Since the existence of properties is not usually thought to depend on that of their instances, and may, on a platonistic view, be independent of their being instantiated at all, one might hold that x’s nature will not in general require x’s existence. One might hold this view, even in case x is a particular individual, such as Aristotle. But there is a complication here. If the properties composing Aristotle’s nature or essence are purely general, there is no reason why they should require either his existence or that of any other particular

2.2 essence, necessity, and non-existence

27

individual. But in the case of individuals like Aristotle, we can distinguish a strong version of essentialism and a weaker one. On the weaker version, Aristotle’s nature is indeed to be specified by answering the question: What is it to be Aristotle? But the answer can be given by specifying the purely general properties essential to Aristotle, such as being a man. On the stronger version, the question: What is it to be Aristotle? is more demanding. An answer is required to say not just what kind of thing Aristotle is, but also what individuates him, or distinguishes him from every other individual of that kind. If we set aside, as more or less patently inadequate, any answer which appeals to haecceities or primitive ‘thisnesses’, we are left with answers which individuate Aristotle by his relation to other particular individuals—such as his having originated from such and such parents, etc. But properties like being the son of x and y or coming from ovum o and sperm s are not purely general; they involve certain particular objects, and so presuppose the existence of those objects. If their existence is a contingent matter, then so will be the existence of Aristotle’s nature, even if it doesn’t require Aristotle’s existence. That certain natures may exist only contingently poses a new problem for the essentialist theory. As we have seen, the theory says that it is necessary that p iff there are x1 , . . ., xn such that it is true in virtue of the natures of x1 , . . ., xn that p, and possible that p iff there are no x1 , . . ., xn such that it is true in virtue of the natures of x1 , . . ., xn that ¬p. But if some of x1 , . . ., xn might not have existed, then—on the assumptions we are currently entertaining—their natures might not have existed either, and in the absence of further natures requiring the truth of p, it might have been false that p, so that it would not have been necessary that p. But if there are possible circumstances in which p would be false, it is not absolutely necessary that p. And if we assume that there might have existed individuals other than (and perhaps in addition to) those which actually exist, so that there might have been some (additional) natures over and above those which actually exist, then there will be a matching problem about possibility. For while the truth of p may not be ruled out by the natures of any actually existing things, so that as things are, it is possible that p, there might have existed some things whose natures require p’s falsehood, so that it is not absolutely possible that p.

2.2 Essence, necessity, and non-existence In what follows, I shall try to explain how I think these problems are best solved.5 In this section, I shall focus on the first problem—the problem of contingently existing individuals, as opposed to, and in so far as it is separable from, the problem posed by the fact that the contingent existence of individuals threatens to bring with it the contingent existence of their natures or essences, and so to subvert any

5 Both problems are discussed in some detail in Hale (2013a), ch.9. My aim here is to improve upon that earlier discussion in two ways. First, I shall try to clarify some points which, with hindsight, seem to me not to have been as well explained as I should like, and are vulnerable to mis-interpretation, and to make some further points which seem to me to tell in favour of the kind of solution I propose there. Second, I have come to think that part of my proposed solution to the second problem is unsatisfactory, and so requires more radical revision.

28 essence and existence essentialist explanation of absolute necessity and possibility in terms of un-modalized quantification over the putative sources of necessity.

2.2.1 The troublesome argument As we saw, the problem arises because an essentialist theory as described allows us to pass from (a) It is true in virtue of the nature of Aristotle that Aristotle is a man

to (b) Necessarily Aristotle is a man

which, we may suppose, entails (c) Necessarily Aristotle exists

But this conflicts head on with the widely held view that many actually existing individuals, such as Aristotle, might not have existed. The troublesome argument can be more fully and explicitly formulated like this: 1 2 1,2 4 4 1,2,4

(1) (2) (3) (4) (5) (6)

2Aristotle Aristotle is a man ∀p∀x(2x p → 2p) 2 Aristotle is a man 2∀φ∀x(φx → x exists) 2(Aristotle is a man→ Aristotle exists) 2 Aristotle exists

assn assn from 1,2 assn from 4 from 3,5

(3) is inferred from (1) and (2) by two steps of ∀-elimination and modus ponens, (5) from (4) by steps of 2-elimination, ∀-elimination, and 2-introduction. The undischarged premises are (1), (2), and (4). Various doubts might be raised.

2.2.1.1 premise (4 ) This is perhaps the most obviously questionable premise. It asserts that no predication concerning a particular object can be true unless the object which is the subject of the predication exists. It may be objected that many logically complex predications do not require the existence of referents for names occurring in them—for example, it may be held that negations and disjunctions embedding an atomic predication Fa (e.g ¬Fa, Fa ∨¬∃xFx) do not require a’s existence. Thus (4) surely needs to be restricted. A restriction to atomic predications would be much more plausible (but is probably stronger than is required, since conjunctions such as Fa ∧ Ga surely require a’s existence, if at least one of F, G is atomic). But even such a severe restriction would not block our argument, since ‘Aristotle is a man’ is plausibly taken to be atomic. One might also object to (4), even for atomic predications, if one adopted a Meinongian or similar view on which some objects do not exist. Thus Meinong, or Priest, for example, may insist that ‘Sherlock Holmes was a man’ is true, even though the famous sleuth never existed. But we need not consider this further here, since even the most ardent Meinongians and fellow-travellers will not wish to claim that

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‘ordinary’ concrete objects such as Aristotle need not exist for there to be atomic truths about them.

2.2.1.2 premise (2 ) On the face of it, rejecting this is not an option for the essentialist, since it amounts to scrapping their proposed explanation of metaphysical necessity in terms of essence. However, there is a complication. Some essentialists would deny that statements of a thing’s essential properties are properly expressed by simple 2-necessitations.6 According to the essentialist, x’s nature or essence comprises what it is to be x, so that the propositions true (solely) in virtue of x’s nature must indeed be true, if x is to be at all. However, whilst we can re-express this point by saying that (1) entails: (3a)

Aristotle is necessarily a man

this last is by no means equivalent to (3)—the crucial difference being precisely that in asserting (3a), one is not committing oneself to Aristotle’s necessary existence, as one plausibly is taken to do, if one asserts (3). As far as it goes, this appears to me to be correct. But it leaves us with a problem, if we aspire to preserve, not just a viable form of essentialism (in the more familiar sense of a position according to which a thing’s properties may be divided into those which belong to it essentially, and those which are merely accidental), but an essentialist explanation of metaphysical necessities, which we are taking to expressible by means of an exterior 2, functioning as a propositional operator. If that project is to survive, we need to locate a proposition of the form 2(. . . a . . .) concerning Aristotle for which the truth of (1) is sufficient. When we see how to do that, we shall be able to accommodate misgivings about the logical form of essentialist claims whilst preserving an essentialist explanation of de re metaphysical necessities.

2.2.1.3 questionable ∀ -eliminations Our argument relies on two (implicit) steps of ∀-elimination at which we should look askance—from ∀x(2x p → 2p) to (2Aristotle p → 2p) between (2) and (3), and from ∀x(φx → x exists) to (φ(Aristotle) → Aristotle) exists between (4) and (5). These steps are, of course, classically valid. But is the assumption of classical quantification theory safe, in the present context? Well, no—it is anything but safe. For the context is one in which we are assuming that some objects exist only contingently. But in even the weakest normal quantified modal logic, if quantification is classical, we can prove: Converse Barcan: 2∀xφx → ∀x 2 φx

and in particular, we can prove that 2∀x∃y x = y → ∀x 2 ∃y x = y, so that since the antecedent is a theorem, so is the consequent. But that says, on an obvious reading, that every object necessarily exists. Cutting what could be a somewhat longer story short, the best way for a contingentist to solve this problem is to insist that the underlying quantification logic should be free—more specifically, the sensible view is that we should adopt the minimum

6 See, for example, David Wiggins’s note on the correct formulation of essentialist claims (Wiggins, 1976).

30 essence and existence departure from classical logic, i.e. a negative free logic, in which the ∀-elimination and the ∃-introduction rules require a supplementary premise which ensures, in one way or another, that the instantial term has reference. This might be accomplished by the use of a primitive existence predicate E!, or by requiring a supplementary atomic premise embedding the instantial term—the details aren’t important here. This doesn’t yet settle the matter. As it stands, our argument is now invalid, but we need to ask whether there is a suitable supplementary premise to which we can appeal to justify the questionable steps of ∀-elimination. But now there is a problem. For any such undischarged additional premise will be non-modal—it will be, say, an atomic predication ‘φ(Aristotle)’. But then, since the step of 2-introduction that gets us to (5) will rely on it, that step will be invalidated. There remains only one further possibility for saving the argument, and that is to maintain that a suitable premise can be inferred from one of the existing undischarged premises, i.e. in effect, from (1). This, at last, takes us to the heart of the matter.

2.2.1.4 premise (1 ) Whether or not (1) entails the modal proposition (3), or some other proposition of the form 2(. . . a . . .) concerning Aristotle, it appears indisputable that it entails the nonmodal proposition (3 minus) that Aristotle is a man. For surely whatever is true in virtue of something’s nature must be at least true—true simpliciter. But then why can’t (3 minus) serve as the required atomic existence entailing premise for the subsequent (suppressed) steps of ∀-elimination? Well, it should be obvious that this simply relocates our question exactly where it should be. The suppressed step of 2-introduction between (4) and (5) already relies on (1) as a premise. So there was already a question to be faced, whether (1) is a suitable premise after all. In essence, the idea behind the rule of 2-introduction is that we may necessitate the conclusion of an inference, provided that that inference depends only upon undischarged premises which hold true of necessity. In the case of the weakest modal logics, this means that the undischarged premises must themselves be necessitated—so that we may infer 2A from B1 , . . ., Bn provided that each Bi is of the form 2C. In stronger modal logics, the undischarged premises B1 , . . ., Bn of the subsidiary deduction may be of other forms, such as ♦C, ¬2C, etc., but only because in the context of the stronger logic, these are equivalent to their necessitations. But now the crucial question for our argument is whether propositions of the form 2x p are admissible. Here there is a danger of being misled by our otherwise useful notation for ‘it is true in virtue of the nature of x that p’. Our operator looks like a kind of necessity operator—but we should not assume it is one. Since our underlying modal logic of absolute necessity is—at least in my view—the strongest normal modal logic S5,7 it would suffice, to ensure the availability of (1) as a premise in the subsidiary deduction for the required 2-introduction step, that (1) be true only if its own necessitation (i.e. 22Aristotle Aristotle is a man) is so. But that is at least very plausible—for surely if something has a certain nature, it could not have

7 For supporting argument, see Hale (2013a), 5.4; for further argument of a rather different kind, see Williamson (2013), 3.3.

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had a different nature. If, for example, Aristotle is, but need not have been, by nature a man, then it is possible that he should not have been a man at all—being a man cannot have been one of his essential properties! Does this settle our issue? Well, surely not. It merely serves to direct attention to what should have been the target of our doubts all along. For if (1) really entails (3 minus), and the latter entails Aristotle’s existence, then by the transitivity of entailment, (1) entails that Aristotle exists—so that it will be true in virtue of Aristotle’s nature that he exists; his essence will entail his existence! But that is surely wrong! The fact that it is (part) of Aristotle’s essence to be a man should no more entail that Aristotle exists than the fact that it is (part) of the essence of men to be animals should entail that there are men.

2.2.2 Essence, non-existence, and transcendence If that is right, x’s existence is never—or at least, is not in general—part of x’s essence. We might put this by saying that correct statements of essence are not existenceentailing. I shall develop a solution based on this idea in the sequel. First, I want to say a little about a quite different way of implementing it.

2.2.2.1 fine’s puzzle and his unworldly solution Something very close to this claim is made by Kit Fine in his very interesting paper ‘Necessity and Non-Existence’ (Fine, 2005c). In that paper, Fine discusses a puzzling argument rather closely related to the one we have been examining: 1 2 1,2

(1) (2) (3)

It is necessary that Socrates is a man It is possible that Socrates does not exist It is possible that Socrates is a man and does not exist

The argument is an instance of the valid form 2p, ♦q  ♦(p ∧ q). Yet it appears that the premises are both true but the conclusion false. Dismissing various alternative solutions to the puzzle, Fine argues that—by analogy with the distinction that may be drawn between sempiternal and eternal truths (the former being tensed sentences which are always true, the latter tenseless sentences which are true simpliciter)— we should distinguish between worldly and unworldly truths. He introduces this distinction as follows: Just as one may distinguish between tensed and tenseless sentences according to whether they can properly be said to be true or false at a time, so one can draw a distinction between worldly and unworldly sentences according to whether they can properly be said to be true or false in a world. And just as one may draw a distinction between eternal and sempiternal truths according as to whether they are true regardless of the time or whatever the time, so one can draw a distinction between transcendental and necessary truths according as to whether they are true regardless of the circumstances or whatever the circumstances. (op. cit., p.324)

As examples of worldly and unworldly sentences respectively, Fine gives ‘Donkeys bray’ and ‘Socrates is self-identical’. The first of these, he thinks, is true, but might not have been so—it is true in our world, but there are worlds in which it is false. The second, however—or so Fine claims—cannot properly be said to be true in a

32 essence and existence world at all. Thus it is not necessary, if by that we mean ‘true in all worlds’—rather, it is a transcendental truth—true ‘regardless of the circumstances’. This is to be contrasted, he thinks, with a sentence such as ‘Either Socrates exists or Socrates doesn’t exist’, which is (merely) a necessary truth, because true in all worlds, not a transcendental truth, true regardless of the circumstances. The application of these ideas in Fine’s solution to the puzzle involves a correlated distinction he draws between worldly and unworldly predicates. Thus in his view ‘brays’ is a worldly predicate, and so—crucially for his solution—are ‘exists’ and ‘does not exist’, whereas ‘is self-identical’ is unworldly, and so—again crucially for his solution—is ‘is a man’. Thus ‘Socrates is a man’ is an unworldly sentence, but ‘Socrates does not exist’ is worldly. The crucial question for the puzzle concerns what kind of truths the more complex sentences composing the premises and conclusion of the puzzling argument should be taken to express. Fine claims that the worldly/unworldly distinction gives rise to three different grades of necessity and possibility, which he calls unextended, extended, and superextended. Unextended necessity and possibility apply only to worldly sentences. Thus: It is possible that Socrates does not exist It is necessary that Socrates exists or does not exist

express an unextended possibility and an unextended necessity. But when we apply the notions to unworldly sentences, they are extended, so that: It is possible that Socrates is self-identical It is necessary that Socrates is a man

express an extended possibility and an extended necessity. Finally, the notions are used in a superextended way when applied to compounds having both worldly and unworldly components, so that: It is possible that Socrates is a man and that he does not exist

expresses a superextended possibility. Fine has an interesting—if somewhat speculative and possibly idiosyncratic— discussion of our inclinations and disinclinations to employ the unextended, extended, and superextended notions, especially as these bear on the ways in which we are likely to interpret the sentences composing the puzzling argument. His view is that we treat ‘man’ as an unworldly predicate in (1), and so—correctly, in his view— take (1) to express an extended necessity. As for (2), we take ‘does not exist’ to be worldly, and so—again correctly, in his view—take (2) to express an unextended possibility. But when it comes to the conclusion, things are complicated. Since one component is unworldly and the other worldly, we ought to interpret it as expressing a superextended possibility. But, Fine claims, we are extremely averse to using the superextended notions. So what we do is treat ‘man’ as short for the worldly predicate ‘existent man’ and interpret (3) as trying to express an unextended possibility. But if we take (3) this way, it seems clearly false, since when ‘is a man’ is interpreted as the worldly predicate ‘is an existent man’, it cannot be true of anything which does not exist. So the upshot is that we take (1) and (2) to be both true, and (3) false. Since the argument appears valid, we have a problem.

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But, says Fine, while we are not mistaken in our judgements about the truth-values of premises and conclusion so interpreted, we are wrong to think the argument valid. For the combination of an extended necessity and an unextended possibility in the premises cannot yield an unextended possibility as a conclusion. And further, there is clearly an equivocation over ‘is a man’, which is interpreted as an unworldly predicate in its premise but as a worldly one in the conclusion.

2.2.2.2 misgivings about fine’s solution If Fine’s solution to his own puzzle is accepted, there could be a similar solution to our puzzle. We could observe that since ‘is a man’ is an unworldly predicate, the sentence ‘Necessarily Aristotle is a man’ can only be true if it is taken as expressing an extended necessity. But since ‘exists’ is a worldly predicate, the sentence ‘Necessarily Aristotle exists’ must express an unextended necessity. But clearly no unworldly truth (such as ‘Aristotle is a man’) can entail a worldly one (such as ‘Aristotle exists’), for then the unworldly truth could be at best true in all circumstances, as distinct from being true regardless of the circumstances. A Finean solution would accordingly reject premise (4) in our version of the argument. Obviously to accept the Finean solutions, we must accept his fundamental distinction between the worldly and the unworldly, and the contrast which depends upon it, between unextended and extended kinds of necessity and possibility. I do not myself find this sufficiently clear to command acceptance. Fine characterizes the contrast in several different ways—between truths which depend ‘on the circumstances’, and those which don’t; between those which depend upon how things ‘turn out’ and those which don’t; and, at the level of necessary truths, between those which hold whatever the circumstances and those true regardless of the circumstances.I find these characterizations suggestive, but elusive. I think the nearest Fine gets to a clear explanation is in the following remarks: We are accustomed to operating with an inclusive conception of what is necessary and what is true in a possible world . . . we think of any possible world as . . . settling the truth-value of every single proposition . . . [so that] the distinction between necessary and transcendental truths [disappears] . . . All the same, it seems to me we naturally operate with a more restrictive conception of what is necessary and what is true in a possible world. A possible world . . . is constituted, not by the totality of facts, or of how things might be, but by the totality of circumstances, or of how things might turn out . . . .We might think of the possible circumstances as being what is subject to variation as we go from one possible world to another; and we might think of the transcendental facts as constituting the invariable framework within which the variation takes place. (Fine, 2005c, pp.325–6)

What seems to me clearly right here is the idea that there is—as Fine puts it—an ‘invariable framework within which variation takes place’. Put another way, although things might have been different from the way they actually are in an enormous variety of ways, the scope for variation is not unconstrained. Fine’s ‘transcendental facts’— that is, the facts expressed by what he calls extended necessities, such as Socrates’s being a man, or any individual object’s being self-identical—are precisely not subject to (modal) variation. There are no possible worlds, or possible circumstances, in which Socrates exists but isn’t a man, or in which this building exists but is selfdistinct. So far, so good. The trouble is that it is then quite unclear why logical

34 essence and existence necessities, such as the necessary falsehood of any proposition of the form A ∧¬A and, perhaps, the necessary truth of any proposition of the form A ∨¬A, are not likewise ‘framework truths’—for we no more think that there are possible circmstances in which contradictions are true, or (perhaps more problematically) instances of the Law of Excluded Middle are false, than we think there are circumstances in which Socrates isn’t a man, or in which some bachelors are married, etc.8 In short, while there is a reasonably clear contrast to be drawn here, it does not divide things up in the way Fine needs—we don’t get his contrast between transcendental/unwordly necessities/truths such as ‘Socrates is a man’ and mundane/worldly necessities like ‘Socrates exists or doesn’t exist’, and we don’t get his ‘grades’ of necessity and possibility, crucial to his proposed solution.

2.2.3 Essence and non-existence without transcendence I think there is a much simpler and much less problematic way to block the troublesome argument of 3.2.1 and solve Fine’s puzzle. In a nutshell, but somewhat roughly, we should simply deny that statements of essence—including statements of individual essence—are ever existence-entailing with respect to the entities whose essences they purport to state; they are, rather, negatively existential statements of an entirely familiar kind.9 The key idea can be clearly and most straightforwardly illustrated by considering general statements of essence, as exemplified by propositions like: Whales are mammals Snails are molluscs Men are vertebrates Water is a compound of hydrogen and oxygen

The key point here is that these propositions state what, or part of what, it is to be a whale/snail/man/water. They do not state, or imply, that there are any whales/snails/ . . ., etc. They are purely general statements which we can represent as universally quantified conditionals, such as ∀x(x is a whale → x is a mammal), and thus are equivalent to corresponding negatively existential propositions, such as ¬∃x(x is a whale ∧¬x is a mammal) I suggest that we should carry this point across to statements of individual essence. Thus a proposition such as: Aristotle is a man

8 It might seem that the first of these claims is disputable, and simply begs the question against dialetheists, who think that some contradictions are true. However, dialetheists do not think that contradictions are true in any possible world—instead, they hold that besides possible worlds, there are impossible worlds— worlds at which the laws of logic break down (cf. Priest (2005), pp.15–18). Nor is any question begged against Fine. For he agrees that instances of the Law of Excluded Middle, for example, are true in all possible worlds; and there is no reason to think that he would not take the same view of instances of the Law of NonContradiction. 9 My solution is thus quite different from those which Fine discusses and rejects as inadequate, both of which rely upon a distinction between more and less demanding interpretations of the necessity operator. I shall not discuss these here. For some brief remarks, highlighting the difference between the alternatives Fine considers and my solution, see Hale (2013a), p.218 fn.42.

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taken as a perhaps partial statement of what it is to be Aristotle, is to be sharply contrasted with such propositions as: Aristotle was a philosopher Aristotle taught Alexander

These latter propositions are indeed existentially committing—neither of them can be true unless it is also true that Aristotle exists. But our statement of essence—in so far as it is understood as just that—does not say or imply that there is such a thing as Aristotle, but says only something about what it is for something (anything) to be Aristotle—what is required for something to be Aristotle. Its surface form is therefore potentially seriously misleading, for it encourages us to think of the statement as having the logical form of a simple atomic predication Fa. But really, what it does is to state a necessary condition for something (anything) to be Aristotle; it tells us that nothing is Aristotle which is not a man. Thus its real form can, just as in the case of general statements of essence, be given by a universally quantified conditional, or equally a negatively existential proposition: ∀x(x = Aristotle → x is a man) ¬∃x(x = Aristotle ∧¬x is a man)

More generally, statements of individual essence do not have the logical form, Fa, of atomic predications suggested by their surface form in natural language. Any statement of this form is logically equivalent to ∃x(x = a ∧ Fx), which is in turn logically equivalent to its expansion ∃xFx ∧ ∀x(x = a → Fx). In this, only the left conjunct requires the existence of an object to which F applies; the right conjunct does not require the existence of any such object, and in particular, it does not require a’s existence—on the contrary, if a does not exist (i.e. there exists no object identical with a), then the right conjunct is vacuously true. Properly understood, the full import of a statement of individual essence is captured by the right conjunct alone. Obviously, if our statement (a) concerning what is true in virtue of Aristotle’s nature is expressed in this way, the first few steps of our troublesome argument will run: 1 2 1,2 4 4

(1) (2) (3) (4) (5)

2Aristotle ∀x(x = Aristotle→ x is a man) ∀p∀x(2x p → 2p) 2∀x(x = Aristotle→ x is a man) 2∀φ∀x(φx → x exists) 2(Aristotle is a man→ Aristotle exists)

assn assn from 1,2 assn from 4

But now we are stuck. To proceed as before to deduce 2(Aristotle exists), we would need the necessitation of the antecedent of the necessitated conditional on line (5); but that we do not have—we have only the necessitation of the strictly weaker statement that if anything is Aristotle, it is a man. Thus the troublesome argument breaks down. Equally clearly, Fine’s puzzle is resolved. We can and should simply reject premise (1) (It is necessary that Socrates is a man)—what we can and should accept is the weaker statement (1*) It is necessary that nothing is Socrates which is not a man

36 essence and existence which we formalize: (1 ) 2∀x(x = Socrates → x is a man)

But of course, this together with the second premise of the puzzling argument yields only the conclusion: (3 ) ♦(∀x(x = Socrates → x is a man) ∧¬∃x x = Socrates)

which is entirely unproblematic. Indeed, since the right conjunct entails the left, (3 ) may be simplified to the bare statement that Socrates might not have existed, i.e. ♦¬∃x x = Socrates. But this is simply a repetition of premise (2). The argument literally makes no advance from its premises!10

2.2.4 Propositions about contingently existing individuals By a singular proposition I mean a proposition which essentially involves singular reference to at least one object.11 Singular reference to an object is essential to a proposition iff there is no equivalent proposition which does not involve singular reference to that object. Since there can be no singular reference to non-existent objects, a singular proposition depends for its existence on that of the object(s) to which it refers.12 Thus if an object to which a proposition makes singular reference exists only contingently, the existence of the proposition is likewise contingent. Suppose a proposition p involves such reference to a contingently existing object o. Then in circumstances in which o would not exist, p would likewise not exist. It is important—and will be important in our subsequent discussion—that a proposition p’s non-existence in certain circumstances C does not mean that p would not be true, were circumstances C to obtain. We can, following Adams (Adams, 1981), draw a distinction between a proposition’s being true in certain circumstances and its being true of those circumstances. p is true in C iff, were C to obtain, p would exist and 10 In the discussion of Fine’s puzzle in Hale (2013a), pp.217–8, I propose a quite different solution, observing that once we replace premise (1) with (1 ), we should, assuming we are working in a negative free logic, require an atomic premise involving ‘Socrates’, and that this will block the step of necessitation needed to get to ‘2 Socrates is a man’, which is needed if we are to reach the original conclusion, ‘♦(Socrates is a man ∧ Socrates does not exist)’. The present objection does not rely on the assumption that the underlying quantificational logic should be free. 11 This is similar to Robert Adams’s usage. According to him, “a singular proposition is, roughly, a proposition that involves or refers to an individual directly, and not by way of its qualititative properties or its relations to another individual” (Adams, 1981, p.5). 12 The requirement that any objects to which reference is made in a singular proposition must exist, if the proposition is to exist, may be understood in more and less demanding ways. The objects in question may not be eternal but transient beings like ourselves. An extremely demanding, but rather implausible, version of the requirement would allow that a singular proposition exists only at times when its objects exist. A less demanding and more plausible version would require that the relevant objects exist or existed at some earlier time. This would allow that the proposition that Aristotle taught Alexander exists now, even though Aristotle and Alexander themselves have long since ceased to be. A fuller account would need to address the somewhat delicate and controversial question whether there can be singular propositions concerning objects which will but do not yet exist, and whether we should adopt an even less demanding version which would require only that the relevant objects either exist, or existed at some past time, or will exist at some future time. As far as I can see, nothing I say here depends on how precisely these matters are resolved. When I envisage circumstances in which certain objects would not exist, I mean circumstances in which the objects in question would have existed at all, at any time past, present, or future.

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would be true, whereas for p to be true of C, it is required only that p would be true, were C to obtain. To illustrate, consider the proposition that Aristotle does not exist. This proposition exists because Aristotle exists, and of course, precisely because he does exist, it is false; but it would be true if Aristotle were not to exist, even though it would not exist in those circumstances. The proposition Aristotle does not exist cannot be true in any circumstances, but it can be true of some circumstances—precisely those in which it would not exist. Clear examples of singular propositions in this sense are provided by simple atomic predications—propositions of the forms Fa, Rab, Sabc, etc., in which F, R, S represent simple one-, two-, or three-place predicates such as ‘. . . breathes’, ‘. . .loves__’, ‘. . .is between__ and_ . _’. But not only such atomic predications are singular propositions in my sense—in particular, a complex proposition such as ∀x(x = Aristotle → x is a man) counts as singular, since it is not equivalent to any proposition which does not involve reference to Aristotle. In those possible circumstances in which Aristotle does not exist, this proposition does not exist either, and so it cannot be true in such circumstances; but that does not mean it cannot be true of them. On the contrary, the proposition does exist (i.e. as things are—since Aristotle exists), and were Aristotle not to exist, it would be true (i.e. true of those circumstances in which neither Aristotle nor it would exist).

2.3 Contingently existing essences I turn now to the problem outlined in 3.1.2.2. I attempted to resolve this problem in my book Necessary Beings (Hale, 2013a, 9.4), but have subsequently come to think that while much of what I said there is right, or at least defensible, it stands in need of some significant revision.13

2.3.1 Background and preliminary observations If, as I think we should, we take a thing’s nature or essence to be simply the conjunction of its essential properties, then essences are (typically complex) properties of a certain kind. For example, one might hold that the essence of the natural or finite cardinal numbers—what it is to be a natural number—is simply to be 0 or one of its successors, and that the essence of mammals—what it is to be a mammal—is being a warmblooded animal having a backbone and mammary glands which lays its fertilized eggs on land or retains them inside the mother. In general, a property does not depend for its existence on that of any particular instance, and on a platonistic conception, does not depend for its existence on its having any instances at all. On the conception of properties I favour—what is often called the abundant conception—all that is required for the existence of a purely general property (i.e. a property specifiable by a predicate which makes no essential use of singular terms) is that there could be a predicate associated with a suitable application or satisfaction condition. Thus it is sufficient for the existence of the geometrical 13 This change of mind has been brought about, at least in part, by two very careful and perceptive reviews of my book, one by Penelope Mackie (Mackie, 2014), and the other by Christopher Menzel (Menzel, 2015), to whom I am also grateful for an extensive correspondence.

38 essence and existence property of being a square that there is a predicate (e.g. ‘. . .is a square’) applicable to a plane figure iff that figure is composed of four straight lines of the same length meeting at right angles. If one takes the modal logic of the kind of possibility involved in this condition for property existence to be the strongest normal modal logic, S5, then one can prove that all purely general properties necessarily exist.14 It follows that if a thing X’s nature is a purely general property, it—i.e. X’s nature— does not depend for its existence on that of any individual objects whose existence may be a contingent matter. However, not all properties are purely general. In particular, there are what we might call impure or object-dependent properties—properties like being a brother of Aristotle or being a successor of 0—which cannot be specified save by means of predicates which essentially involve singular terms. In the case of some such properties—such as being a successor of 0—one may argue that the relevant objects involved are ones which exist necessarily.15 But in other cases, the existence of the relevant objects is widely held to be contingent, so that also is that of the property. Thus there exists no such property as being a brother of Aristotle unless Aristotle himself exists.

2.3.2 Individual essences 2.3.2.1 weak and strong individual essences again As we saw, individuals may be held to have essences in either a weak or a strong sense. In the weak sense, the essence of an individual such as Aristotle might just consist in his essential possession of a certain purely general property, such as being a man. An essence in this weak sense is a property which is necessary for anything to be that individual, but not sufficient. Clearly, if individuals are held to have essences only in this weak sense—or more precisely, are held only to have purely general essential properties—the contingent existence of individuals poses no threat to the essentialist theory of necessity and possibility, precisely because purely general properties are not dependent for their existence on the individuals which instantiate them. It is thus only if individuals are held to have essences in the strong sense, or to have essential properties which are not purely general but object-dependent, that the contingent existence of individuals may pose a threat to the essentialist theory. The threat it then poses is that the essence of certain individuals will consist in, or involve, their possession of some impure relational property relating them to other objects.16 Then if the further object(s) in question exist only contingently, so too will the essence.

2.3.2.2 are there strong individual essences? In Necessary Beings I took it as obvious, pretty well without argument, that individuals have essences in the strong sense. I have since come to think that this is far from

14 The argument is simple. Let φ be any purely general property, p the proposition that φ exists, and q the proposition that there exists a predicate standing for φ. Then by the abundant theory: 2(p ↔ ♦q). It follows from this by the K-principle that 2p ↔ 2♦q, and by the T-principle that p ↔ ♦q. By the S5-principle, ♦q ↔ 2♦q. So by the transitivity of ↔, p ↔ 2p. 15 For an argument for the necessary existence of 0 and the other natural numbers, see Hale (2013a), 7.4. 16 That is, the essence will be what Robert Adams labels an α-relational essence. See Adams (1981), p.5.

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obvious.17 The issue now strikes me as much more complex. To begin with, it is obviously crucially important to separate the metaphysical question here from an epistemological one. It may be held—at least plausibly, though not, of course, uncontroversially—that if we are to speak and think of a particular object, we must be able to identify it, in the sense that we have some way of distinguishing it from every other object. But this might be a matter of our being able to single it out ostensively (or more plausibly, by a combination of a demonstration coupled with a general sortal term, e.g. ‘This tortoise’), or by supplying an identifying description which relates it uniquely to some other items which we may independently identify (e.g. ‘The man from whom I bought this copy of The Times’). How we single out a particular object, in this sense, is an epistemological matter. And it seems clear that the facts which we exploit for such identificatory purposes can be perfectly contingent facts about that object. The metaphysical question concerns not how we distinguish one object from all others, but what, if anything, distinguishes the object from all other objects— what grounds the object’s distinctness from every other object. To be sure, one may be reluctant to admit an individual’s distinctness from every other object as simply a brute, inexplicable fact. But even if there has to be something more to be said about what distinguishes each object from every other, it is far from clear that this must be something which could be taken to be a strong essential property of the object. For example, it might be granted that if a = b, there will be some non-trivial property— not necessarily a purely general property, but perhaps a relational property involving some further object(s)—which one of a, b possesses and the other lacks. It may even be granted that there must be some such property. It does not follow that this has to be even a necessary property of one of a, b, let alone an essential property. Even more obviously, it fails to follow that there has to be an essential property of a which distinguishes it not just from b but from every other object. Even if there can be—as I am inclined to suspect—no general argument from less controversial principles to the conclusion that there have to be strong individual essences, it may still be true that there are strong individual essences; and if some version of the much-discussed principle of the essentiality of origin can be upheld, there will be. Although I know of no compelling argument for that principle,18 I think the principle has at least some intuitive plausibility, and do not think we can discount the possibility that it is true. So in what follows, I shall, for the sake of argument, assume that there are strong individual essences. And I shall further assume that at least some such essences are impure, object-dependent properties, where in at least some cases the relevant objects exist only contingently. My aim will be to show that even under these assumptions, one can uphold a version of the essentialist theory.

17 Here, I am especially indebted to Penelope Mackie, who plausibly conjectures, in her review of Necessary Beings (Mackie, 2014), that the general esssentialist theory of modality I defend may be independent of my more controversial and problematic claims about (strong) individual essences. 18 The most widely discussed argument is, of course, the argument Kripke gives in footnote 56 of Naming and Necessity ((Kripke, 1972, 1980). In my view, that argument fails, ultimately because it implicitly relies on the assumption of merely possible objects. Others have, of course, given wildly divergent diagnoses of what goes wrong.

40 essence and existence

2.3.2.3 more (‘new’) objects and essences Since I remain convinced of the essential correctness of the position on this which I took in Necessary Beings (Hale, 2013a, pp.224–5), I shall here simply summarize the main points. First, given the abundant conception of properties (vide supra), together with the assumption19 that the logic of the modality involved in that conception’s condition for the existence of properties is S5, it follows not only that the purely general properties there are exist necessarily, but also that they are all the purely general properties—that is, any such properties which could exist do exist. And from that it follows that any ‘new’ objects there might be—i.e. objects distinct from any actually existing objects— must be objects of general kinds which (already, as it were) exist. Thus, second, the possibility that there should exist objects other than any of those which exist (or once existed, or will exist) is a possibility which would be realized, for example, by there being horses which are distinct from any of the horses which do, have, or will ever exist. It—that is, the possibility of ‘new’ objects—is a purely general possibility. It is the possibility that there should exist objects of some kinds other than any objects of those kinds which actually exist. It is not to be confused with the view, which I reject, that there are certain objects—‘merely possible objects’— which don’t actually exist, but might have existed. The contrast here is between the claim that ♦∃x∃φ(φx ∧ ∀ψ∀y(@ψy → x = y))—‘there could be a φ-er distinct from every actual object of any kind’, which I accept, and ∃x∃φ(∀ψ∀y(@ψy → x = y) ∧ ♦φx)—‘there is something, distinct from every actual object of any kind, which could be a φ-er’, which I reject. Third, if there were to be a φ-er—a horse, say—distinct from every actually existing object, it would (on the assumption about strong individual essences we are making) have an individual essence—it would, say, be essential to it to be a horse having a certain origin. So there would be a further, ‘new’, individual essence. The question is whether such a new individual essence would impose an additional constraint, a constraint which would rule out the truth of some propositions whose truth is not already ruled out by the essences or natures there actually are. Our new individual essence would be a complex property, composed of certain general properties—those essential to being a horse, say—and those which would distinguish our new horse from all others. But now: Fourth: anything ruled out by our new horse’s being a horse is already ruled out by what it is for anything to be a horse—i.e. by the general essence of horses, which already exists. Fifth: it is (already) part of what it is to be a horse that any horse necessarily has a certain kind of origin—for simplicity, we may take this to be a matter of being engendered by a particular mare and stallion. Thus the possibility of there being a new horse is either the possibility of there being a new horse engendered by some actually existing mare and stallion, or it is the possibility of there being a new horse engendered by some mare and stallion at least one which is also new. In the first, simplest, case, what is ruled out is once again a possibility which is, in all relevant respects, purely 19 defended in Hale (2013a), 5.4.

2.3 contingently existing essences

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general: it is not possible that there should have been a new horse, originating from a certain pair of actually existing horses, which might have originated otherwise. And this candidate possibility is already ruled out by what it is to be a horse, and so by constraints which are already, independently, in place by the actually existing nature of horses. The second case, where the new horse is envisaged as originating from one or more other new horses, is more complicated, but not essentially different.

2.3.2.4 interlude: contingently existing propositions As we saw, singular propositions depend for their existence on the existence of the objects they are about. Thus when we envisage the possibility that certain actually existing objects might not have existed, we are envisaging a situation in which certain propositions which actually exist would not exist. Equally, when—as we have just been doing—we envisage the possibility of new objects, distinct from any actually existing objects, we are envisaging a situation in which there would exist singular propositions which do not actually exist. Of course, we cannot, in the nature of the case, give examples of such propositions—we can only speak of them in general terms. If, for example, there were to exist a new horse, there would also exist various singular propositions about it, including propositions asserting that it might have been a frog, or might have had a different origin. If there were to exist a new horse, its essence would ensure that no such propositions as these could be true. While we cannot, of course, refer to any particular such propositions, and so cannot say of any of them individually that its truth is already ruled out—for there are no such propositions of which we can say this—what we can correctly say is that while there are no singular propositions concerning new horses, and so no singular propositions asserting of new horses that they might have been frogs, or might have had different origins, it is already ruled out by the nature of horses that there could be any such true singular propositions. And that, I claim, is enough to dispose of the worry that the possibility of new objects would close off possibilities which would otherwise be left open by the essentialist account.

2.3.2.5 fewer (‘ old ’) objects and essences It is here that I have come to think that my defence of the essentialist theory in Necessary Beings (vide Hale 2013a, 225–6) stands in need of some revision. The key claim I there made, which now seems to me problematic, was that if, say, Aristotle’s essence had not existed—say, because it is part of his essence that he originated from Nicomachus and Phaestis, and one or both of them had not existed—then Aristotle himself would not have existed, so that there would then have been no possibilities concerning him. In consequence, I claimed, it would be a mistake to think that the non-existence of Aristotle’s essence would leave open possibilities concerning him which are, given that his essence actually exists, ruled out—such as that he might have been a frog, or born of different parents; it would be a mistake, because in the envisaged circumstances, there would be no possibilities concerning him at all. One very serious difficulty to which this gives rise concerns my claim that the logic of absolute metaphysical modality is S5. Consider the proposition that Aristotle might have been a cobbler, which we may assume to be true. Given that the logic of metaphysical modality is S5, it follows that it is necessary that Aristotle might have

42 essence and existence been a cobbler. However, my claim that had Aristotle not existed, there would have been no possibilities concerning him implies, or seems to imply, that had Aristotle not existed, it would not have been possible that he should have been a cobbler— contradicting the claim that it is necessarily possible that Aristotle might have been a cobbler. Underlying my key claim is the assumption that if Aristotle had not existed, there would have been no singular propositions concerning him, and so no singular propositions of the form ‘Aristotle might have . . .’ (i.e. of the form ♦φa). A fortiori, there would have been no true singular proposition to the effect that Aristotle might have been a cobbler. Hence it is not the case that, no matter what else were the case, it would have been possible that Aristotle should have been a cobbler. Hence it is not necessary that Aristotle might have been a cobbler. It thus appears that there is a direct clash between the quite strict form of contingentism which I adopt in Necessary Beings and my claim that the logic of absolute metaphysical modality is S5. The difficulty is brought out with admirable clarity by Christopher Menzel in his searching review of Necessary Beings (Menzel, 2015). Let us write [p] to denote the proposition that p, and E!x for ‘x exists’. Then, as Menzel observes, I appear to be committed, by my strict form of contingentism, to the following principle: P

For any proposition p, ♦p → E![p]

Consequently, the essentialist theory’s principle governing possibility cannot be simply: ♦p ↔ ¬∃X1 . . . Xn 2x1 . . . xn ¬p but should be amended to: ♦p ↔ E![p] ∧ ¬∃x1 . . . xn 2x1 . . . xn ¬p. Further, since the definition is to be understood as applying to claims about what is possible in modal contexts, it needs to be not simply a material but a necessitated (i.e. strict) biconditional: ETP*

2(♦p ↔ E![p] ∧ ¬∃x1 . . . xn 2x1 . . . Xxn ¬p)

Now, write p for the proposition that Aristotle is a cobbler, and let q be any proposition which entails that Aristotle’s essence does not exist (and so that Aristotle doesn’t exist)—say the proposition that Phaestis never existed. Recall that, according to my theory, the usual necessity operator is to be explained in terms of the generalized counterfactual, so that 2p abbreviates ∀q(q2→ p). Then we may reason as follows: (1) (2) (3) (4) (5) (6) (7) (8)

♦p q2→ ¬E![p] q2→ ¬♦p ¬(q2→ ♦p) ∃q¬(q2→ ♦p) ¬∀q(q2→ ♦p) ¬2♦p ¬(♦p → 2♦p)

assn assn from 2 by ETP* from 3 by counterfactual logic from 4 by existential generalization from 5 by quantification logic from 6 by def 2 from 1,7

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43

Thus the characteristic S5 principle fails. Part of what has gone wrong here can, I think, be put in terms of the distinction to which I adverted in 2.2.4 between a proposition’s being true of certain possible circumstances (or, in worldly terms, true of a possible world) and its being true in those possible circumstances (true in that possible world). No singular proposition concerning Aristotle can be true in circumstances in which Aristotle would not exist, because no such proposition would exist in such circumstances. But this does not mean—and it is not true—that no singular proposition concerning Aristotle can be true of circumstances in which Aristotle would not exist. For one thing, the proposition that Aristotle doesn’t exist, which is actually false, is true of circumstances in which he doesn’t exist (and indeed, it can be true only of such circumstances). But more to our present purpose, the modal proposition that Aristotle might have been a cobbler, while it would not exist and hence could not be true in such circumstances, may perfectly well be true of them. So long as what is required by the S5 principle is taken to be that if ♦p is true, it is necessarily true in the sense that there are no possible circumstances of which it would not be true, there need be no conflict with S5. This is not by itself enough to resolve the conflict, however. At least, it is not enough if, as Menzel claims, my essentialist theory forces me to hold that what is possible with respect to a world (or more generally, counterfactual situation) w is always determined entirely by the essences which happen to exist in w. It is this which obliges me, he thinks, to adopt the existence requirement in ETP* in order to avoid the original problem—the problem that, for example, Aristotle’s being a frog would have been possible, had Aristotle’s essence not existed (cf. Menzel 2015, pp.422ff and especially p.426). There is, it seems to me, no escaping the conclusion that the root of all evil lies in the key claim in my earlier discussion—viz. that had Aristotle not existed, there would have been no possibilities concerning him. In making this claim, I was implicitly identifying a possibility concerning Aristotle with a true proposition of the form ♦φ(Aristotle), and inferring that there would, had Aristotle not existed, have been no true propositions of that form because, in that situation, there would be been no propositions about Aristotle at all. But it now seems to me that this implicit reasoning was confused—or better, that it ignores a crucial ambiguity in the key claim. Given the identification of possibilities concerning Aristotle with true propositions of the form ♦φ(Aristotle), and given that a proposition of this form would exist in a certain counterfactual situation only if Aristotle existed in that situation, I should agree that if Aristotle had not existed, there would (in that situation) be no proposition of the form ♦φ(Aristotle) which is true (i.e. true in that situation). But I should not agree that there is no proposition of the form ♦φ(Aristotle) such that, had Aristotle not existed, that proposition would have been true (i.e. true of that situation). And I should insist that what is required for the necessity of a proposition p is not that this proposition be true in every situation (i.e. true in every counterfactual situation as well as in the actual situation), but rather that it should be true of every situation. Thus in particular, the proposition that necessarily Aristotle might have been a cobbler (and generally, any proposition of the form 2♦φa) will be true (i.e. true as things are, true of (and indeed in) the actual world) provided that the proposition that Aristotle might

44 essence and existence have been a cobbler (or generally, ♦φa) is true of every possible situation, including situations in which Aristotle would not exist.

2.3.2.6 the actual determinants of necessity and possibility If, as I am claiming, the necessity of a proposition consists, not in that proposition’s being true in, but in its being true of every possible situation, and the possibility of a proposition consists, not in its being true in, but in its being true of some some possible situation, then—to the extent that what essences there are is a contingent matter— which modal propositions are true of a given situation is not, in general, determined by which essences exist in that situation. So what does determine their truth-values? The answer for which I am arguing is quite simple: their truth-values are determined by those essences which exist—that is, those essences which actually exist, as distinct from any essences there merely might have been.20 If actualism is the philosophical position that everything there is exists, or is actual (Menzel, 2014, §1), then what we might term actualist essentialism is the thesis that the determinants of necessity and possibility are exactly the essences of the things which exist, and not essences of things which don’t but might have existed—for there are no such essences, any more than there are things which might have existed but don’t. What makes it necessarily true (assuming it to be true at all) that Aristotle might have been a cobbler is that the proposition that Aristotle might have been a cobbler is true of every possible situation. What makes it true of counterfactual situations in which Aristotle doesn’t exist (i.e. wouldn’t exist, were such situations not to be counterfactual) is that there is at least one possible situation (i.e. possible relative to those possible situations in which Aristotle wouldn’t exist) of which (and indeed in which) it is true that Aristotle is a cobbler. And what makes that true is that nothing in any actually existing essence—either Aristotle’s, or any other—rules out his being a cobbler. What ensures that the proposition that Aristotle might have been a frog is not true of any possible situation—including those possible situations in which neither Aristotle nor his essence would exist—is that it is part of Aristotle’s essence (his actual essence, for there is no other) to be a man, and so not a frog.21

20 Talk of essences there merely might have been is, of course, loose talk, because it encourages the bad idea that there are, in addition to the essences there are, some shadowy, merely possible essences, waiting— as it were—to attach themselves to objects, or perhaps already attached to merely possible objects. But a consistent actualist will reject merely possible essences every bit as firmly as she rejects merely possible objects. The sober truth is simply that there might have been objects distinct from any of the objects there are, and had there been, they would have had essences. The ‘they’ lies squarely within the scope of ‘there might have been’. 21 More generally, actualist essentialism requires a further modification to the essentialist theory’s explanation of metaphysical possibility—since what is possible is what is not ruled out by any of those essences which actually exist, we must replace ETP* by ETP∗@ : 2(♦p ↔ ¬@∃x1 . . . xn 2x1 ...xn ¬p). Similarly, since it is the actually existing essences which determine what is necessary, the essentialist theory’s explanation of metaphysical necessity should run: 2(2p ↔ @∃x1 . . . xn 2x1 ...xn p). Thanks, once again, to Christopher Menzel for drawing my attention to this point. It might also be noted that without the amendment, we could distribute the outer necessity operator in the necessitated formulations to obtain 22p ↔ 2∃x1 . . . xn 2x1 ...xn p and 2♦p ↔ 2¬∃x1 . . . xn 2x.1 ..xn ¬p, which in S5 simplify—disastrously—to 2p ↔ 2∃x1 . . . xn 2x1 ...xn p and ♦p ↔ 2¬∃x1 . . . xn 2x.1 ..xn ¬p.

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It may be thought that the original worry—that if, say, Aristotle’s essence had not existed, then the constraint which, as things are, renders it impossible that he should have been a frog would be removed, so that it would then be possible that he should have been a frog—could be resurrected as follows. Consider any counterfactual situation, w, in which Aristotle’s essence (and so Aristotle himself) would not exist. Since w is a situation in which Aristotle’s non-existence is a contingent matter, it would remain true (i.e. true of, though not in, w) that Aristotle might have existed. But since, by hypothesis, were w to obtain, Aristotle’s essence would not exist, there would be nothing to exclude the possibility that he should have been a frog. But this reasoning, I contend, is just confused. It is, of course, perfectly true that the singular proposition that Aristotle might have existed does not have to exist in w if it is to be true of w. But it—that very proposition—has to exist (i.e. actually exist) if it is to be true of w, and that proposition actually exists only because Aristotle actually exists.22 The possibility in question is the possibility that that very man should have existed. But that is the possibility that a particular man with certain essential properties—which include being essentially human—should have existed. It is therefore not a possibility, with respect to w, that Aristotle should have existed but been a frog (i.e. the proposition that Aristotle might have existed but been a frog is not, and cannot be, true of w). The error in the confused reasoning lies in its tacitly assuming, in effect, that Aristotle might have existed without his essence.23

22 As before, ‘exists’ here is shorthand for ‘exists or existed’. 23 I am indebted to the participants and organizers of the workshop Individuación y Permanencia de Objetos, held in July 2015 in the Universidad Autónoma Metropolitana, Mexico, for helpful discussion of an earlier incarnation of this material, to Christopher Menzel for some very helpful correspondence, and to friends and colleagues who took part in subsequent discussions at work in progess seminars in Sheffield and King’s College London.

3 The Problem of De Re Modality 3.1 De dicto and de re modalities In modern discussions, the contrast between de dicto and de re modality is typically drawn in purely syntactic terms, focusing on a first-order formal language to which modal operators have been added.1, 2 A sentence is taken to involve de re modality if it contains, within the scope of a modal operator, an individual constant or a free variable bound by a quantifier lying outside the scope of that modal operator—e.g. ♦Fa, or ∃x2(x = 172 ), as contrasted with 2∃x(x = 172 ). This does not straightforwardly translate into a precise and effective syntactic criterion applicable to natural languages. We may of course say, roughly, that a sentence of English, for example, involves de re modality if it contains a clause, governed by a modal word, which contains a proper name, or a pronoun whose grammatical antecedent lies outside that clause. But in view of the prevalence of scope ambiguities, it is to be doubted that there is any effective way of telling whether a modal word does or does not govern a whole clause. If we set aside the difficulties of framing a syntactic criterion and allow ourselves to put things in more semantic or metaphysical terms, the underlying idea seems reasonably clear—a de re modal claim is one in which something is said to be not simply necessarily (or possibly) true, but necessarily (or possibly) true of some object(s), where the object(s) in question may be definitely identified (as in ‘17 is necessarily prime’) or left unspecified (as in ‘Some number is necessarily the square of 17’).

3.2 The problem of de re modality At least, it seems reasonably clear to some of us. For the problem of de re modality— that is, the problem which has dominated modern discussion—is how, if at all, one can make sense of it. Most who have discussed this problem have assumed that modality

1 My title is intended to remind readers of one of the two papers by Kit Fine on this topic (Fine, 1989 and Fine, 1990), re-reading which led me to write the present paper. I have been been an admirer of Kit’s work since I first encountered some of it, early in our careers, in one of the early Logic & Language conferences in the 1970s. Without a trace of a reservation, I would single him out as one of the most inspired, and inspiring, thinkers of my generation—an exceptionally gifted logician and philosopher whose work has been distinguished by his erudition and careful scholarship, his technical thoroughness and ingenuity, and, above all, by his outstanding inventiveness and originality. I am glad to be able to count him one of my most treasured philosophical friends, and to contribute this paper to [Metaphysics, Meaning, and Modality: Themes from Kit Fine, edited by Mircea Dumitru] in his honour. 2 This chapter also appears as Hale (2020) in Metaphysics, Meaning, and Modality: Themes from Kit Fine, edited by Mircea Dumitru for Oxford University Press. Bob Hale, The Problem of De Re Modality In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0004

3.2 the problem of de re modality 47 de dicto is relatively unproblematic.3 It is, rather, the interpretation of sentences involving, within the scope of modal operators, singular terms or free variables (or their natural language equivalents, relative pronouns) which is thought to give rise to grave—and in the view of some, insuperable—difficulties. Why? There is no doubt that one major reason why de re modality has seemed especially problematic lies in the broadly linguistic conception of the source of necessary truth which was widely accepted by analytic philosophers throughout the middle decades of last century, in spite of Quine’s major onslaught on the notion of analytic truth or truth in virtue of meaning. Indeed, Quine himself—somewhat surprisingly, given his misgivings about analyticity—finds the essentialism to which he thinks acceptance of de re modalities commits us unpalatable precisely because it clashes with the logical empiricist orthodoxy that all necessity is rooted in meanings. For it requires that An object, of itself and by whatever name or none, must be seen as having some of its traits necessarily and others contingently, despite the fact that the latter traits follow just as analytically from some ways of specifying the object as the former do from other ways of specifying it. . . . Essentialism is abruptly at variance with the idea, favoured by Carnap, Lewis, and others, of explaining necessity by analyticity . . . . For the appeal to analyticity can pretend to distinguish essential and accidental traits of an object only relative to how the object is specified, not absolutely. (Quine, 1953a, p.155)

It is true enough that Quine does not here directly endorse ‘the idea . . . of explaining necessity by analyticity’. But only a couple of paragraphs later, he rejects essentialism as ‘as unreasonable by my lights as it is by Carnap’s or Lewis’s’, and he certainly appeared to be speaking for himself, as well as many others, when he unequivocally declared that ‘necessity resides in the way we talk about things, not in the things we talk about’. No doubt Quine viewed the idea of explaining necessity by way of analyticity with little, and rapidly decreasing, enthusiasm. Still, his thought seems to have been that if one is to make sense of necessity at all, it can only be in some such terms, since the idea that something is necessarily thus-and-so, regardless of how it is specified, is—in his view—completely unintelligible. To anyone in the grip of the broadly linguistic conception of the source of necessity, this problem should seem not just compelling but insuperable. For if necessity really is just truth in virtue of meaning, there can be nothing that is necessarily true of any objects regardless of how they are specified (and so necessities de re, or de rebus), save what is merely trivially true of them—because true of all objects whatever. In more detail: A true singular proposition, 2A(t), will be a non-trivial de re necessity only if what it says holds by necessity of its object, t, and is not, as a matter of necessity, true of all objects whatever—that is, only if it is not the case that 2∀xA(x)—and similarly for a true existentially quantified proposition, ∃x2A(x).4 According to the broadly linguistic conception, 2A(t) is true (if and) only if A(t) is analytic. What is needed, if one is to make sense of de re necessity whilst construing necessity as analyticity, is

3 This is true even of the most prominent and influential critic of modality de re, W. V. Quine. 4 There is an analogous conception of what it is for a property φ to be non-trivially essential—this is so if ∃x2φ(x) ∧ ∃x¬2φ(x). Cf. Marcus (1971, p.62).

48 the problem of de re modality to explain what it is for the open sentence A(x) to be analytically true of an object. But the only obvious way to explain this is in terms of A(x) being such as to yield an analytically true closed sentence, no matter what individual constant replaces its free variable. In this case, we might say that the open sentence A(x) is analytically satisfiable, or analytically true of every object whatever. (For example, ‘2(t is red → t is red)’ and ‘2(t is red → t is coloured)’ express de re necessities according to the linguistic conception—but merely trivial ones, because 2∀x(x is red→ x is red) and 2∀x(x is red → x is coloured)). The point obviously generalizes—according to the linguistic conception, the only propositions which are de re necessarily true are those whose universal closures are necessarily true (and so analytic); there can be no nontrivial de re necessities. If that is right, anyone who wishes to claim that there are non-trivial de re modalities must reject the broadly linguistic conception of necessity. It is, contrary to what is sometimes supposed, simply not an option—under the assumption of that conception—to attempt to rehabilitate the de re by arguing that de re necessities and possibilities are somehow reducible to acceptable de dicto ones.5 If, as I believe, we should reject the broadly linguistic conception, that disposes of one major source of the idea that de re modality is especially problematic. Does any serious problem 5 Plantinga (1974) proposes such a reduction. Plantinga explains the distinction in semantic terms. A de dicto modal assertion ‘predicates a modal property [such as being necessarily true] of another dictum or proposition’, whereas an assertion of de re necessity ‘ascribes to some object . . . the necessary or essential possession of [some property]’ (p.9). In his view (pp.29–32), one can solve the problem—at least for what is arguably the basic kind of de re modal proposition, viz. a proposition asserting of some particular object, x, that it has a certain property, P, essentially—by providing a recipe for identifying a proposition which (i) has the same truth-condition as the given de re modal proposition, but (ii) involves only de dicto modality. Plantinga’s proposal is that D2

x has P essentially iff x has P and K(x, P) is necessarily false

where K(x, P) is what he calls the ‘kernel proposition’ with respect to x and P. To a first approximation, this is defined by D1 Where x is an object and P a property, the kernel proposition with respect to x and P (K(x, P)) is the proposition expressed by the result of replacing ‘x’ and ‘P’ in ‘x has the complement of P’ by proper names of x and P. If we take as our example the de re claim that 7 has the property of being prime essentially, and we assume that ‘7’ is a proper name of 7, and ‘being prime’ of the property of being prime, then—on Plantinga’s proposal—7 is essentially prime if and only if 7 is prime and the proposition expressed by ‘7 has the complement of being prime’ is necessarily false. The purported reduction assumes that an ascription of the property of necessary falsehood to a proposition—specifically, a proposition expressed by a sentence in which all the relevant entities are directly named—is unproblematic. A Quinean sceptic should simply refuse this assumption. For him, a de dicto modal claim is relatively unproblematic only when, and then precisely because, it can be can be construed as a harmless, albeit potentially misleading, variant on the claim that some sentence is analytic. But de re modal claims—or at least, non-trivial de re claims of the sort advanced by essentialists—admit of no such saving reconstruction. There is, for example, no prospect of re-interpreting the de re proposition that Aristotle was essentially a man as a correct claim to the effect that ‘Aristotle was a man’, or some other privileged sentence which directly refers to that individual and that property, is analytic. If one explains de dicto modality as Plantinga does—so that any ascription of necessary truth (or falsehood) to a proposition counts as de dicto—then one can indeed reduce de re modal claims to de dicto ones. But such a reduction will do nothing to budge the Quinean sceptic, who will regard de dicto claims about named objects and their properties as every bit as problematic as de re ones, and for the same reason.

3.4 quine’s ‘metaphysical’ argument against quantifying in 49 remain? Is there any further, independently compelling, reason to doubt that de re modalities are intelligible?

3.3 Quine’s ‘logical’ argument against quantifying in It may plausibly be thought so. For Quine’s doubts about de re modality do not obviously or simply reduce to the blanket complaint that de re locutions defy reasonable interpretation if one accepts that necessity resides in the way we talk of things, not in the things themselves. On the contrary, he articulates a detailed case, turning essentially upon failures of substitution of co-referential terms to be truthpreserving, for the uninterpretability of quantification into modal contexts. It is at least not obvious that that case can be answered, simply by rejecting the linguistic conception of necessity. The complaint is, it seems to me, fair enough—more does need to be said to dispose of Quine’s detailed arguments against quantifying in. But, as I shall try to explain, once those arguments are deprived of support from the background linguistic conception, they cease to be compelling. Quine’s central and basic argument for the unintelligibility of quantification into modal contexts—that is, of sentences in which a quantifier lying outside the scope of a modal operator purports to bind occurrences of a variable within its scope—may be formulated like this:6 1. If ∃x2A(x) or ∀x2A(x) is to be meaningful, it must make sense to say of an object that it satisfies the open sentence 2A(x). 2. It makes sense to say of an object that it satisfies an open sentence . . . x . . . only if co-referential terms are interchangeable salva veritate in the position(s) occupied by free x. 3. Co-referential terms are not interchangeable salva veritate in 2A(x). 4. ∴ It makes no sense to say of an object that it satisfies 2A(x). 5. ∴ Neither ∃x2A(x) nor ∀x2A(x) can be meaningful. This argument is plainly valid. Premise 1 is indisputable, provided that—as Quine intends—the quantifiers are interpreted as objectual (Quine has separate objections to substitutional quantification). Premise 3 is supported by well-known examples, such as the alleged failure of the substitution of ‘the number of solar planets’ for ‘8’ in ‘2(8 > 7)’ to preserve truth, in spite of the identity of the number of solar planets with 8. Clearly the crucial question is why we should accept premise 2. We shall discuss that shortly. First I should defend my claim that this is Quine’s central and basic argument.

3.4 Quine’s ‘metaphysical’ argument against quantifying in That the ‘logical’ argument is central to Quine’s attack is surely indisputable. But my claim that it is his basic argument is, in effect, denied by Kit Fine. According to Fine, 6 This formulation closely follows Fine (1989, 2005b, p.40) and Fine (1990, p.105). But see subsequent discussion.

50 the problem of de re modality Quine gives two quite different arguments against the intelligibility of quantifying into modal contexts. Fine contrasts the ‘logical argument’ with what he terms the ‘metaphysical argument’, which can be formulated as follows: 1. If ∃x2A(x) or ∀x2A(x) is to be meaningful, it must make sense to say of an object that it satisfies the open sentence 2A(x). 2. It makes sense to say of an object that it satisfies the open sentence 2A(x) only if it makes sense to say of an object that it necessarily satisfies the non-modal open sentence A(x). 3. But this does not make sense—an object does not necessarily satisfy the open sentence A(x) in and of itself, but only relative to a description. 4. ∴ It makes no sense to say of an object that it satisfies 2A(x). 5. ∴ Neither ∃x2A(x) nor ∀x2A(x) can be meaningful. The two arguments are similar, but they diverge over premises 2 and 3. The problem that the metaphysical argument focuses upon might be called the problem of making sense of necessary but objectual satisfaction. But the notion of necessary satisfaction plays no role in the logical argument. Fine lays great stress on the differences between the two arguments, and claims that philosophers who overlook the differences do so at their peril: Philosophers are still prone to present one-sided refutations of Quine. So they cite the criticisms of Smullyan, on the one hand, or the criticisms of Kripke, on the other, without realizing that at best only one of Quine’s arguments is thereby demolished. (Fine, 1989, 2005b, p.44)

I can (and do) agree with Fine that there are some significant differences between the two arguments.7 The most important question, for my purposes, is whether he is right to claim that the two arguments have force independently of one another. But before I discuss that, it will be useful to make an observation about the metaphysical argument.

3.5 Logical and analytic satisfaction As Fine emphasizes, the metaphysical argument is operator specific—in particular, the force of the argument, and indeed, just what the argument is, depends both upon the fact that it concerns a necessity operator (rather than a belief operator, say) and upon how the necessity operator is interpreted (for example, as expressing logical necessity, or analytic necessity, or metaphysical necessity—to take Fine’s own examples). In his discussion of the argument in the earlier of his two papers on the subject, he concentrates on the cases in which the necessity operator is understood as expressing logical necessity or analytic necessity.8 In these cases, the second premise of the argument claims that one needs to make sense of the notion of an object’s logically

7 For example, one very important difference which Fine himself emphasizes is that the ‘metaphysical’ argument is ‘operator-specific’—see next section. By contrast, the ‘logical’ argument is ‘operatorindifferent’—it applies to ‘any operator which, like the necessity operator, creates opaque contexts containing terms not open to substitution’ (Fine, 1989, 2005b, p.41). 8 Fine (2005b, 1989, pp.44–54).

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(or analytically) satisfying a (non-modal) open sentence, and the third premise claims that one can’t do so, because an object can be said to logically (or analytically) satisfy an open sentence only relative to some description of the object. Fine argues that the third premise is false. Given that one understands what it is for a closed sentence to be logically (or analytically) true, one can use this to explain a notion of logical (or analytic) satisfaction that is not relative to ways of describing objects. Fine’s explanation is somewhat complicated, but the details do not matter for our purposes. It is sufficient to observe that at the very least, one can introduce a notion of logical satisfaction which is such that an object logically satisfies A(x) if and only if ∀xA(x) is a logical truth. Logical satisfaction of A(x) will be independent of how the object is described, just because every object whatever satisfies A(x)—for example, because A(x) is some such formula as F(x) ⊃ F(x), where F is a simple predicate. So if the metaphysical argument is taken to concern logical necessity, it can be blocked by denying premise 3. And it is intuitively clear that the argument can be blocked in essentially the same way, if it is understood as concerning analytic necessity. Fine also introduces an extension of the notion of logical form under which one can speak of the logical form, not only of sentences or formulae, but also of sequences of objects. If this extension is allowed, then, as he explains, one can define a more refined notion of logical satisfaction, in terms of which a sequence of objects a1 , a2 , . . . an in which a1 = a2 may be correctly said to logically satisfy ‘x = y’, even though the universal closure of this open sentence, ∀x∀y x = y, is not a logical truth. Logical satisfaction in this more refined sense is again, like the cruder notion explained above, not relative to any descriptions of the (sequences of) objects that satisfy the open sentences. The weakness in this response to the metaphysical argument should be obvious enough. If only the cruder notion of logical satisfaction is admitted, logical satisfaction is always trivial—that is, some given objects logically satisfy an open sentence only if all objects whatever do so, which is so iff the universal closure is logically true. If the more refined notion of logical satisfaction is admitted, then admittedly one gets a range of cases of non-trivial logical satisfaction, but it is a quite limited range. Much the same goes for the somewhat broader notion of analytic satisfaction. So we get an explanation of de re modalities for a limited range of cases. But it is clear that this falls a very long way short of what is needed—a notion of objectually necessary satisfaction in which the necessity operator is understood as expressing metaphysical necessity. In the later of his two papers, Fine concedes,9 albeit rather grudgingly, that ‘Quine’s misgivings do indeed have some force in regard to the de re application of metaphysical necessity’. The fact is that there is simply no obvious way to extend his treatment of logical or analytic truth and satisfaction to metaphysical necessity, and so no prospect of an effective reply, along these lines, to the metaphysical argument conceived as directed against such putative de re necessities as ‘Aristotle was necessarily a man’. Fine explicitly sets this interpretation of necessity aside. Yet this, surely, is the really problematic case, and the case in which the challenge posed by the metaphysical argument seems most difficult to answer: how to explain what it is for an object necessarily to satisfy a condition, such as being human, or being identical with

9 Fine (1990, p.112).

52 the problem of de re modality Phosphorus, or being constituted predominantly of H2 O molecules if it is water, etc. Any attempt to use the explanation of logical satisfaction as a model for explaining metaphysical satisfaction would invoke an operation that took us from the notion of metaphysically necessary truth (assumed understood) to that of metaphysically necessary satisfaction—in parallel to the move from truth in virtue of logical form (or meaning) to satisfaction in virtue of logical form (or meaning). That is, it would try to exploit the idea that we understand well enough what it is for a closed sentence to be metaphysically necessarily true in order to explain what it is for an open sentence to be metaphysically necessarily true of an object or some objects. But this, it seems, gets things exactly the wrong way round. For it seems that it is because 17 necessarily has the property of being prime that it is necessarily true that 17 is prime, and not the other way around. And it is because it is necessarily true of Aristotle that he is a man, that it is necessary that Aristotle is a man, if he exists at all. So there seems to be no prospect of meeting the metaphysical argument by an extension of Fine’s treatment of logical and analytic necessities. It is precisely here that the Quinean sceptic will press the complaint that necessity is relative to our way of describing things, so that no notion of genuine objectual satisfaction is available for the metaphysical case. If—as Fine believes—the metaphysical argument constitutes a genuinely independent objection to quantifying into modal contexts, and so to de re modality, then a more effective response of a quite different kind is needed.

3.6 Necessary satisfaction With that we may return to the most important question raised by Fine’s discussion of the two Quinean arguments: Does the metaphysical argument really raise a separate and independent objection to the intelligibility of quantifying into modal contexts, and hence to that of de re modalities—independently, that is, of the logical argument? The obvious question10 raised by the metaphysical argument is: why should it be (accepted) that an object does not necessarily fulfil a condition in and of itself, but only relative to a description? Neither part of this claim is an obvious truth. The first part of the claim is that an object does not/cannot necessarily fulfil a condition . . . x . . . in and of itself. It seems clearly crucial that this claim concerns necessary fulfilment of a condition, rather than fulfilment simplicter. There is no obvious reason to deny that, say, the number of solar planets (‘in and of itself ’) fulfils the condition x > 7. No one would want to say that whether a certain number does or does not fulfil that condition depends upon how that number is described or specified. But now why should it make such a difference, when it is necessary fulfilment of the condition that is in question? It is hard to see how to answer that question without adverting to the idea that while ‘the number of solar planets > 7’, for example, is true, it is not necessarily true, i.e. 10 One might also ask why we should accept premise 2 in the metaphysical argument. Why accept that it makes sense to speak of an object satisfying 2A(x) only if it makes sense to speak of an object necessarily satisfying A(x)? But here there is an obvious answer: so much is required by the demands of compositionality—we should expect an explanation of the satisfaction conditions for 2A(x) in terms of the satisfaction conditions for A(x). For example, in a semantics based on possibilities, we would expect a clause roughly to the effect that an object satisfies 2A(x) at a given possibility w iff it satisfies A(x) at every possibility w (or perhaps every w accessible from w).

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‘2(the number of solar planets > 7)’ is not true, even though ‘2(8 > 7)’ and ‘8 = the number of solar planets’ are both true. But this is, in effect, to appeal to the idea that co-referential terms are not interchangeable salva veritate in modal contexts, which was supposed to be the distinctive twist in the logical argument. That is, it seems that at bottom, the two arguments do depend upon the same basic idea—that one has objectual satisfaction only if different terms for the relevant object are interchangeable salva veritate. If that is right, the upshot is that the crucial issue is whether Quine is right to claim that the failure of co-referential terms to be interchangeable salva veritate in modal contexts shows that it makes no sense to say of an object that it satisfies an open sentence with its free variable lying within the scope of a modal operator—i.e. that the object itself (as opposed to the object as specified in a certain way, or relative to a certain mode of presentation) satisfies the open sentence. Why should we accept premise 2 of the logical argument, as Fine calls it? Quine’s idea is that if the function of a singular term in a certain sentential context is simply to pick out an object for the rest of the sentence to say something about, then replacing that term by any other singular term that refers to the same object should make no difference to whether the resulting closed sentence is true (or whether the open sentence is true of the object). For if it did make a difference, that could only be because something other than which object the term refers to is relevant to the determination of the sentence’s truth-value. What could that be, if not the mode of presentation—i.e. the way in which the object is referred to? But in that case, the open sentence cannot be said to be true (or false) of the object simpliciter—rather, it is true or false of the object only relative to this or that way of specifying the object. Thus if, as Quine claims, interchange of the co-referential terms ‘8’ and ‘the number of solar planets’ in the sentence: (1)

2(8 > 7)

produces a falsehood, it makes no sense to say of the object to which these terms refer that it satisfies ‘2(x > 7)’. That is, we cannot understand (1) as expressing a necessity de re—i.e. as saying, of the object 8 that it is necessarily greater than 7. At most, only a de dicto reading of the sentence is intelligible. ‘8’, as it occurs in (1) does not occur purely referentially, to use Quine’s term—serving simply to pick out its object. The position it occupies, and likewise that occupied by ‘7’, is itself nonreferential, or at least not purely referential. And because quantified variables don’t discriminate between different ways of specifying their values, it makes no sense to quantify into such positions. Put another way, ‘∃x2(x > 7)’ makes no sense—for it asserts, in effect, that there is an object which satisfies the open sentence ‘2(x > 7)’, and that is something which—if Quine is right—no object can do. If this account of Quine’s argument is right, his case turns upon the claim that substitution-failure shows that neither the term that gets replaced, nor the position it occupies, is purely referential—in the sense that the replaced term, or any that replaces it, serves simply to pick out its object, and makes no other contribution beyond that towards determining the truth-value of the sentence. But, as examples given by Fine and others show, substitution-failure does not always mean that the replaced term was not functioning purely referentially, or that the position it occupies in the sentential context is non-referential. In examples like:

54 the problem of de re modality (a) ∴

2.2=4 2=1+1 2.1+1=4

(b) ∴

Eve’s elder son was Cain Eve = the mother of Cain The mother of Cain’s elder son was Cain

the premises are true, but the most likely reading of the conclusion is false11—yet there is no question but that the supplanted terms occur purely referentially in the premises. There is nothing opaque about the contexts ‘2.x = 4’ and ‘x’s elder son was Cain’. The obvious and simple explanation is that replacing ‘2’ by the co-referential ‘1 + 1’, or ‘Eve’ by ‘the mother of Cain’, produces a structural ambiguity—in fact an ambiguity of scope—which yields the ‘bad’ reading of the conclusion as more likely than the good one. Such examples show that there can be no straightforward inference from substitution-failure to irreferentiality in the cases that matter for Quine’s argument. For they raise the possibility that the failure, or apparent failure, of interchange of co-referential terms in modal contexts to preserve truth-value may likewise be explicable without prejudice to the purely referential status of the supplanted terms. And of course, it is well known that the stock examples are susceptible of an alternative explanation very similar to the one that can be given for non-modal examples like (a) and (b). Thus when ‘8’ as it occurs in (1) is replaced by ‘the number of solar planets’, the result: (2) 2(the number of solar planets > 7)

is arguably ambiguous between a reading on which it implies ‘2(There are more than 7 solar planets)’—presumably false—and one on which it does not. The first reading results from interpreting the descriptive term with narrow scope relative to the modal operator, and the second from assigning it wider scope, so that the conclusion is read ‘The number of solar planets is such that necessarily it is greater than 7’. One may then claim that the step from the apparent failure of substitution to the conclusion that ‘8’ does not occur purely referentially in (1) depends on tacitly resolving the ambiguity in favour of the first reading of (2). But a parallel insistence upon the ‘bad reading’ of the conclusion of the inference (a) or (b) would ‘establish’ that the supplanted singular term did not occur referentially in the premise. If, as did the original proponents of this response to Quine’s alleged counterexamples to the substitutivity of identicals,12 one adopts a Russellian theory of definite

11 Under what I believe are standard arithmetical conventions, we read ‘2.1 + 1’ as denoting a sum with a product as one of its terms, rather than a product with a sum as one of its terms. It takes some effort—on my part at least—to read ‘the mother of Cain’s elder son was Cain’ as ‘(the mother of Cain)’s elder son was Cain’ (i.e. as equivalent to ‘the elder son of the mother of Cain was Cain’), rather than ‘the mother of (Cain’s elder son) was Cain’ (i.e. ‘the mother of the elder son of Cain was Cain’). 12 cf. Fitch (1949); Smullyan (1948). Quine (1953a, p.154, fn.9) claims that Smullyan requires a modified verison of Russell’s theory to allow difference in scope to affect truth-value even in cases where the definite description is uniquely satisfied, but I think no modification is needed—unique satisfaction of the description ensures that scope differences make no difference, according to Russell’s theory, only provided that the description occurs in a truth-functional context. But the necessity operator is not truth-functional. See Principia Mathematica *14.3. The point is made by Smullyan himself—see Smullyan (1948, p.37).

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descriptions, the ambiguity of sentences such as (2) is a straightforward case of scope ambiguity—the definite description may be ‘eliminated’ in two quite distinct ways, depending upon whether the replacing uniquely existential quantifier is given narrow or wide scope relative to the necessity operator.13 But one does not have to endorse a Russellian treatment of definite descriptions if one is to explain the standard examples of substitution-failure as trading on scope ambiguity. One might hold, contra Russell, that definite descriptions can and sometimes do function as genuine devices of singular reference. When, for example, an eye witness, on being invited to say which, if any, of the members of an identity parade she thinks she saw at the crime scene, replies ‘The one wearing a yellow pullover’, it is at least very plausible that she is using the description to single out, or refer to, one among the people on parade, rather than simply asserting that one and only one of them is wearing a yellow pullover and was at the crime scene. Assuming that definite descriptions do so function, it is clear that they work quite differently from simple proper names. For they are not only syntactically, but semantically complex, embodying putative information about their intended referents in a way that simple names do not. It is this information which is intended to enable the hearer or reader to identify the intended object of reference. But does the information encoded in the description form part of the content of an assertion, or other speech act, performed by uttering a sentence incorporating the description? The answer is that in some cases it does, while in others it does not, and in consequence of this, sentences involving descriptions may be ambiguous in a way that is best understood in terms of scope. Fairly clear examples are indirect speech reports, or attributions of belief, in which definite descriptions occur within the content clause, such as: (3)

The witness said that the man in a yellow pullover was at the crime scene.

It may be that the witness said, in those very words, that the man in the yellow pullover was at the crime scene—or at least, that that is what the reporter is claiming she said. But it may be that the witness did not herself pick out the suspect as the man in a yellow pullover, and that the reporter is perfectly well aware of this, but is simply using the description as a convenient way of identifying the individual of whom the witness said that he was at the scene of the crime.14 Thus what exactly the reporter is claiming depends upon whether the information encapsulated in the definite description is to be understood as lying within the scope of the indirect speech operator ‘said that’. Similarly, in modal statements such as (2), the description, and so the information encoded in it, may or may not lie within the scope of the modal operator. If it does, (2) asserts, falsely, that as a matter of necessity, some single number both numbers the solar planets and exceeds 7. If it does not, (2) asserts, concerning the number which, as a matter of presumably contingent fact, numbers the solar planets, that it—that very number—necessarily exceeds 7. 13 Of course, since on Russell’s theory, definite descriptions are a species of quantifier phrase, and so not singular terms at all, the appearance that (2) comes from (1) by substitution of a co-referential singular term is in any case entirely illusory. 14 Perhaps the individual in question was not even wearing a yellow pullover, either at the crime scene or during the identity parade, but is present and wearing one when the reporter speaks.

56 the problem of de re modality It may be questioned whether this is enough to disarm Quine’s argument. For it may be objected that while the existence of a reading of (2) on which what it says is true establishes that the inference by substitution need not fail to preserve truth, it remains the case that there is a reading of (2) on which it says something false— and that is enough, it may be claimed, to make Quine’s case that the inference is not guaranteed to preserve truth, as it should be, if the occurrence of ‘8’ in (1) is to be purely referential. The fact that the conclusion can be read as saying something true does nothing to alter the fact that it may equally well be read as saying something false, so that the inference may lead from true premises to a false conclusion. To deal with this renewed challenge, we need to be more explicit about the rôle which may be played by a singular term in determining the content of a sentence. There are essentially two cases we need to distinguish, according as the singular term is simple or complex. Since what matters is whether the term encodes information, and if it does, whether that information forms part of the content of a sentence incorporating the term, the kind of simplicity that matters is semantic, as opposed to merely syntactic. A syntactically simple singular term—a simple name, a, say—may be semantically simple; but it may not be, for it may be a definitional abbreviation for a syntactically and semantically complex term, such as a definite description the φ, or functional term f (t). Roughly, by saying that a syntactically simple term is semantically complex, I mean that there is information associated with the term which plays a part in determining its reference, and which a fully competent user of the term must grasp. Plausible examples are the name i as used in complex analysis and h as used to denote Planck’s constant—to understand i, one needs to know that it denotes -1, and to understand h, one needs to know that it denotes the ratio of the energy of a photon to the frequency of its electromagnetic wave, E/v. A semantically simple term can contribute towards determining the content of a sentence in which it occurs (and thereby, indirectly, towards determining its truthvalue) in only one way—by identifying an object. The containing sentence then expresses that propositional content which is true iff that object satisfies the open sentence in which the simple term is replaced by a free variable. We might say that the term simply contributes its object.15 But with a semantically complex term, there are two possibilities. Such a term may, just like a simple term, simply contribute its object—that is, its sole contribution towards determining the content expressed by its containing sentence may be simply to single out a certain object about which the remainder of the sentence says something. In this case, the information encoded in the term serves merely to aid identification of the object, and makes no further contribution to the propositional content expressed. But a complex term may play a further rôle, contributing not just its object, but also the information it encodes. This dual possibility is illustrated by the two ways of understanding (3). On one reading,

15 —assuming there is one. If the term is empty, there is nothing for it to contribute, and the sentence will therefore fail to express a truth-evaluable content. In that sense, it will not say anything. This is not to say that it will express no intelligible content at all. Nor is it to say that reference failure on the part of a semantically complex term must result in failure to express a truth-evaluable content. That is a further question which we may set aside here.

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the reporter is using ‘the man in a yellow pullover’ simply to identify for his audience a certain person—it simply contributes its object, and the identificatory information it encodes plays no further part in characterizing what, according to the reporter, the witness said. But on the other reading, the reporter is giving a fuller characterization of the content of the witness’s statement—the description contributes not just its object, but also its encoded information. This results in a difference in (3)’s truth-conditions. On the first reading, the report is true provided that the witness identified the man in a yellow pullover in some way or other—it doesn’t matter how, so long as it was that man she identified—and said of him that he was at the scene of the crime; but on the second reading, the report is true only if the witness identified the man in pretty much those terms. Of course, it is not necessary that she used those very words—her actual words could have been, say ‘It was that ugly one in the yellow jersey’, or ‘Es war der Mann mit dem gelb Pulli’. The capacity of complex terms to contribute in these two different ways to the content—and so, indirectly, to the truth-value—of a sentence has an obvious bearing on the conditions under which interchange of co-referential terms should be expected to preserve truth-value. Put simply, replacement of a term t by a co-referential term s in a sentential context A(t) is guaranteed to leave truth-value undisturbed only if both premise and conclusion are so understood that the terms simply contribute their objects. If t as it occurs in A(t) simply contributes its object—either because t is a simple term, or because, though complex, its encoded information plays no part beyond simply identifying its object—then we should only expect the identity t = s to ensure that A(s) does not diverge in truth-value from A(t) if A(s) is so construed that s likewise serves simply to contribute its object. If, instead, t or s contributes not just its object but the information it encodes to the content of A(t) or A(s), then we should expect replacement of t by s to preserve truth-value only if the terms share not just their reference, but also their encoded information. The application of this general point to the inference from (1) to (2) depends upon whether we take numerals to be simple terms or not. If so, then point applies quite straightforwardly—the fact that ‘8’ and ‘the number of solar planets’ co-refer licenses us in drawing (2) as a conclusion only if (2) is understood with its descriptive term having wide scope with respect to the necessity operator, since it is only then that the term simply contributes its object. If numerals are taken to be semantically complex terms—perhaps as abbreviations for functional terms involving the successor function—then the matter is slightly more complicated, but the essential point remains the same. We should only expect replacing ‘8’ by ‘the number of solar planets’ to leave truth-values undisturbed if both terms are construed as simply contributing their objects (i.e. as both having wide scope in relation to the necessity operator). An objector may complain that by appealing to a wide-scope reading of statements such as (2), this response to Quine’s argument simply takes for granted the intelligibility of the claim that an object, in and of itself, satisfies an open sentence such as ‘2(x > 7)’, thereby begging the very question at issue. But it seems to me that this objection simply mis-locates the burden of proof: I am not trying to prove that objectual satisfaction of open sentences formed with modal operators is intelligible— rather, it is the Quinean argument that is attempting to show that it is not. What my

58 the problem of de re modality reply to that argument shows is that, in the absence of independent reasons to deny that objectual satisfaction is intelligible in the modal case, the kind of substitutionfailure the argument adduces as proof that open sentences like ‘2(x > 7)’ cannot be true (or false) of objects simply, but only relative to ways of specifying them, can be explained without drawing that conclusion.16 16 I should like to thank all those (including its tireless organizer, Mircea Dumitru) who participated in the workshop on Kit Fine’s work held in Sinaia, in May 2012, for helpful discussion of an earlier version of this paper, and especially to Kit himself, not just for his stimulating and enjoyable contributions to the workshop, but, on a more personal note, for the help and support he has generously given me over many years.

4 Ontology Deflated 4.1 Introductory remarks Quine’s slogan ‘To be is to be the value of a variable’ encapsulates his view that the sole measure of our ontological commitments is what we take to be included in the range of the bound variables of any quantified statements we accept. For Quine, the bound variables were just individual variables, since the only legitimate kind of quantification was first-order quantification—second- and higher-order quantification into predicate positions are a misleading aberration of questionable intelligibility which disguise the very substantial ontological commitments of set theory, most apparent when that theory is formulated, as he thought it should be, as a first-order theory in which the individual variables range over sets. Higher-order logic, so called, is ‘set theory in sheeps’ clothing’. Bound individual variables are the sole measure of ontological commitment because the only other reasonable marks of commitment to objects of some kind— constant singular terms—are eliminable by an extension of Russell’s theory of definite descriptions which allows us to replace ‘Socrates drank hemlock’ by ‘∃!x(x socratizes ∧ x drank hemlock)’, and likewise for other examples. Ontological commitment, for Quine, is commitment to objects. Expressions other than singular terms, such as predicates and functional expressions, are not, in his view, vehicles of reference, so that their use imports no commitment to properties (or attributes, as he usually calls them), relations, or functions. My own view diverges from Quine’s on all these points. Because I believe his animus against higher-order quantification is misplaced, I see no reason to restrict the vehicles of ontological commitment to singular terms, either constant or variable. Because I believe his proposed elimination of constant singular terms is merely superficial, I see no reason why such terms should not be taken to signal commitment to objects, along with our use of first-order quantification. In my view, existentially quantified statements merely generalize a prior commitment to particular objects already carried by their instances. Because I think that bound second- and higherorder variables range over properties, relations, and functions of various levels, and that such variables import no new commitment which is not already involved in the predicate and functional expressions replaced, but simply generalize an already existing commitment, I take our use of constant predicates etc. as further vehicles of commitment along with singular terms. My departure from Quine over higher-order quantification rests on a mixture of rejection of another of his doctrines—his extensionalism—coupled with an at least quite strong inclination to accept another—his insistence on ‘no entity without Bob Hale, Ontology Deflated In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0005

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identity’. His belief that such quantification can only be quantification over sets— and so first-order quantification over objects—rests on the conviction that the only alternatives—properties and relations—are ruled out because they lack respectable identity conditions. And so they will, if we agree that such conditions must be purely extensional, as they must be, if Quine’s rejection of meaning and analyticity is correct. But in my view, that is deeply mistaken.1 However, on one, even more fundamental point, I am in agreement with Quine: the mark of our commitment to entities of a given kind is our acceptance, as strictly and literally true, of statements embedding expressions which, if they have reference at all, have entities of that kind as their referents, or semantic values. And, I would add, what there is—which (kinds of) entities exist or are real—is just the sum total of things to which reference is made in the true statements we can assert—or at least could assert, in a suitable extension of our language. This fundamental point is essentially an application of Frege’s much discussed Context Principle, and for this reason among others, I think of my approach to ontology as broadly Fregean. In a characteristically challenging and original paper, ‘The Question of Ontology’,2 Kit Fine argues that the standard quantificational account, deriving from Quine, of ontological questions—questions about what exist or is real—is mistaken, and that it should be replaced by an account of a very different kind, on which existence or reality is expressed, not by quantifiers, but by a predicate ‘exists’ or ‘is real’, itself ultimately to be explained in terms of a sentential operator R[…], which may be read as ‘It is constitutive of reality that . . .’. Although Fine’s criticisms are primarily directed against the quantificational account, it is clear that, if they are sound, they apply equally to an account such as the one I myself favour.3 My main aim in this paper is to explain why I think Fine’s criticisms are unsound. What follows falls into three parts. I begin with a brief explanation of my own view, bringing out the way in which I think it may be seen as implementing a deflationary approach to ontological questions. Then I give a reasonably detailed summary of Fine’s criticisms and his positive view. Finally, I attempt to answer those criticisms.

4.2 A broadly Fregean approach to ontological questions 4.2.1 Ontological categories The central question of ontology, at least on a broadly Quinean approach, is: What kinds of thing are there? This presupposes some way of dividing things into kinds, or ontological categories. How is this division to be explained? How are the different categories—objects, properties, relations, etc.—to be distinguished and/or defined? 1 For some defence of these claims, see Hale (2015a), 8.2–3, Wright (1986), Rayo and Yablo (2001), Wright (2007). 2 Fine (2009). Subsequent references to Fine are to this paper, unless otherwise indicated. 3 In fact, Fine claims that the view he is attacking is held, in one form or another, by all the other contributors to the volume in which his paper appears. Since that volume includes a joint paper by Crispin Wright and me (Hale and Wright, 2009), it can safely be inferred that we are among his targets.

4.2 a broadly fregean approach to ontological questions 61 On a broadly Fregean approach, the explanation draws on a prior analysis of language into expressions of various logico-syntactic types—sentences, singular terms (proper names in Frege’s inclusive sense), predicates and functional expressions of various levels and arities, including first-, second-, and higher-level quantifiers, and sentential operators. Objects are then explained as those things for which singular terms may stand, properties and relations as those which may be the semantic values of predicates of appropriate level and arity, functions as the values of functional expressions of suitable type, and so on. In view of Frege’s notorious paradox of the concept horse, and the wider problem of which it is symptomatic, this simple explanation requires a little tweaking. The paradox arises because Frege insists that only objects can be the referents of singular terms, that only suitable predicates may refer to concepts and relations, and so on. But this renders it impossible coherently to express Frege’s analysis of language and ontological categories. I think the best way to overcome this obstacle is to acknowledge that we may use singular terms (and first-order quantification) to speak of entities of any ontological category, so that we can quite properly speak—as it is very natural and seemingly perfectly intelligible to do—of the property of being a horse, the relation of being less than, the functions of negation and conjunction, and so on. But if we do so, we must modify the explanations of object, property, etc. roughly as follows: Objects are those entities for which singular terms, and only singular terms, may stand. 1st -level properties are those entities to which reference may be made by means of 1st -level 1-place predicates. 1st -level binary relations are those entities to which reference may be made by means of 1st -level 2-place predicates. ... nth -level properties are those entities to which reference may be made by means of nth -level 1-place predicates. ... 1st -level unary functions are those entities to which reference may be made by means of 1st -level 1-place functional expressions. ...

These explanations allow that reference to properties, relations, and functions may also be made by means of singular terms. However, reference to such entities by means of singular terms is secondary or derivative. This is suggested by such facts as that the most direct and straightforward way of forming such singular terms is by nominalization of expressions of other syntactic types. Thus we may make singular reference to the property expressed by the first-level predicate ‘ξ is wise’ by means of the obvious nominalizations such as ‘(the property of) being wise’, ‘the property something possesses in virtue of satisfying ‘ξ is wise’ ’, or, more remotely, by means of the abstract noun ‘wisdom’, which is to be explained in terms of them.4

4 For a fuller discussion of this issue, see Hale and Wright (2012), section VIII.

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4.2.2 Conditions for existence objects Taking Frege’s Context Principle as our guide, we may state a straighforward sufficient condition for the existence of objects of a given kind. Roughly, there are Fs if there are true statements of a suitable type embedding singular terms which, if they have reference at all, refer to Fs. Statements of a suitable type will be those in which singular terms occur in what we may call existence-entailing positions. Thus the numerical singular term ‘0’ occurs in an existence-entailing position in ‘52 − (42 + 32 ) = 0’, but arguably not in intensional contexts such as ‘Perhaps 0 does not exist’, and hyperintensional contexts such as ‘The ancient Chinese mathematicians denied that 0 exists’. On a reasonable view, singular terms in atomic statements are always existence-entailing, but there are clearly nonatomic existence-entailing contexts, such as conjunctions with atomic conjuncts etc. Although this is a sufficient condition, it is clearly not necessary. We should allow for unnamed objects. Roughly stated, a reasonable necessary and sufficient condition for the existence of Fs would be that there are or could be suitable true statements embedding singular terms which, were they to have reference, would refer to Fs. Note that this condition is essentially modal.5 Note also that a more general sufficient condition for the existence of Fs is that the bare existential quantification ∃xFx be true. This more general condition will be met if and only if my necessary and sufficient condition is.

properties and relations It is sufficient, on my broadly Fregean view, for the existence of a property being F or a relation bearing R to . . . that there be a true statement embedding the predicate F or R. But since, in my view, any statement embedding F will be existence-entailing with respect to its predicate F, an ostensibly much weaker condition suffices: it is enough for the existence of the corresponding property or relation that there be a meaningful n-place predicate F—where by a meaningful predicate I mean any predicate associated with a well-understood satisfaction- or application-condition. As with objects, this condition, while sufficient, is not necessary. Just as we must allow for unnamed objects, so we should allow for properties and relations for which our actual languages lack predicates. Thus a necessary as well as sufficient condition would require only that there could be a suitable predicate.6 An analogous condition may, with some complications, be given for the existence of functions.7

5 A more careful statement would need to deal with some further complications. In particular, although there aren’t (at least as far as we know), even-toed ungulates otherwise like horses except for having a single, spiral horn growing from their foreheads, there could be such beasts, and there could be one called Uriah, and so there could be true statements embedding that name in existence-entailing positions—but this is evidently insufficient for the actual existence of such beasts. See Hale (2015a), 1.12. 6 For an essentially similar account of the existence conditions for properties and relations (or in his terminology, universals as opposed to particulars), see Strawson’s Individuals: Strawson (1964), ch.6, especially pp.183–6. 7 For further discussion, see Hale (2015a), 7.3.

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deflation On this account, these Fregean conditions for being an object, property, relation, etc. are very inclusive, as compared with those favoured by some metaphysicians— contrast, for instance, the ‘sparse’ universals favoured by Lewis and Armstrong8—and the conditions for their existence are, in certain respects, quite undemanding. It thus seems to me reasonable to view the account as deflationary. Though relatively undemanding in some respects, the conditions for existence are far from toothless. In the case of objects, the requirement for the possibility of true statements—strictly true when taken literally and at face value—embedding suitable singular terms is a substantial requirement. And in the case of properties and relations, the requirement that there could be suitable predicates is likewise quite demanding, given that, on my view at least, a predicate should be taken to be a finitely long expression of suitable level and arity.9

necessary and contingent existence On widely accepted views, many kinds of object exist only as a matter of contingency. Such views may be readily accommodated on my broadly Fregean approach, since the statements embedding suitable singular terms which must be true for certain objects to exist may well be ones which are at best contingently true. But it is worth emphasizing that, on my view, the existence of some properties and relations, including some functional relations, and derivatively, the existence of some objects, will be a matter of necessity. Briefly, let a purely general predicate be any predicate which does not embed any singular terms. By a purely general property or relation, I mean the reference or semantic value of a suitable purely general predicate. Thus, for instance, the identity predicate ‘=’ is purely general, and the identity relation is a purely general relation, whereas the predicate ‘ξ is shorter than Aristotle’ is not purely general, and the property of being shorter than Aristotle is a mixed or object-dependent property—had Aristotle not existed, there would have been no such property (although there would still have been the distinct but co-extensive property of being 1.9m tall, assuming that that was Aristotle’s height—properties, in my view, are individuated intensionally). The necessary existence of purely general properties is a simple and direct consequence of my deflationary conception, provided that we take the modality involved in the condition for property existence to be absolute, and governed by S5 principles. Let φ be any purely general property; let p be the statement that φ exists; and let q be the statement that there exists a predicate standing for φ. Then according to the deflationary account: p ↔ ♦q

Further, this statement, if true, will be so as a matter of necessity, at least on the deflationary account, so that:

8 See Lewis (1983, 1986); Armstrong (1979). 9 This raises difficult and controversial issues, especially in the philosophy of mathematics, which I cannot go into here.

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But (a) entails both: (b) p ↔ ♦q and (c) 2p ↔ 2♦q and, by S5, (c) entails (d) 2p ↔ ♦q But from (b) and (d), it follows by the transitivity of the biconditional that (e) p ↔ 2p Since φ may be any purely general property, it follows that each purely general property there is exists as a matter of necessity.

4.3 Fine’s criticisms and opposed approach 4.3.1 Fine’s criticisms of the quantificational account the quantificational account According the standard account of ontological questions, deriving from Quine, when philosophers ask ‘Do numbers exist?’, ‘Do tables exist?’, etc., they are asking whether there are numbers, whether there are tables, etc., and so are asking whether certain existentially quantified statements are true, such as ∃x(x is a number), ∃x(x is a table). It is this quantificational interpretation of ontological questions that Fine wishes to rebut.

fine’s arguments Fine brings two direct arguments against the quantificational account: 1. The account trivializes ontological questions. For example, that ∃x(x is a number) is a trivial consequence of the arithmetic truth that there is a prime number between 9 and 12. So the answer to the question: Do numbers exist? is trvially: Yes. But the philosophers’ questions are not trivial, nor are their answers. (p.158) 2. Ontological questions are philosophical, but the account makes whether there are numbers, for example, a mathematical, non-philosophical, question. ∃x(x is a number) is a consequence of a straightforwardly arithmetical truth, as already observed, and so—Fine concludes—is settled by mathematics. (p.158) Assuming the soundness of these direct arguments, Fine then considers and rejects a number of attempts philosophers have made or might make to answer them. These generally consist in trying to put ‘some kind of distance between our ordinary [i.e. non-philosophical] commitment to objects of a certain kind and a distinctively ontological commitment’. This idea has roots in Carnap,10 who famously distinguishes

10 As Fine himself notes—see Carnap (1950), reprinted in revised form in Carnap (1956).

4.3 fine’s criticisms and opposed approach 65 between what he calls internal and external questions about existence. Internal questions are asked within a linguistic framework, and are typically straightforwardly settled within that framework. For example, that there are numbers is straightforwardly true, internally, because it is a straightforward consequence of elementary truths of arithmetic. External questions are questions about the existence of systems of entities as a whole, and, in Carnap’s view, are to be understood, not as factual or theoretical questions, but as practical questions about whether to adopt or endorse a certain linguistic framework. Carnap thinks that philosophers’ existence questions cannot be internal, since they will be settled by ordinary empirical means or by the relevant scientific discipline, and must therefore be external. Philosophers, including Fine, who do not accept Carnap’s view that such questions are practical, not theoretical, will be likely to see them as involving some special weighty philosophical sense of ‘exists’.11 Fine discusses and rejects two broadly different kinds of argument under this head: Dispensability arguments (pp.159–61) Broadly speaking, the idea here is that proper application of scientific method often shows that ordinary commitments are dispensable—serious ontological commitments are those which are not so dispensable (cf. Field’s attempt to defend his rejection of numbers by arguing that the Quine-Putnam indispensability argument fails (Field, 1980)). Arguments aimed at distancing ordinary from philosophical commitments (pp.161–5) The general drift of these is that ordinary commitments do not aim to capture the strict and literal truth, whereas philosophical commitments do. There are various forms this line can follow: it may involve claiming that ordinary commitments are a matter of make-believe, that they are somehow weaker than philosophers’ commitments, or that there is a difference in strength of some other sort, such as might be captured by distinguishing ‘thin’ and ‘thick’ senses of the quantifier. Since I am largely in agreement with Fine’s arguments against these attempts to deflect the first two direct arguments, I shan’t discuss them further. I need not do so, I think, because I shall argue that Fine over-estimates the force of those initial arguments, and that there are some straightforward moves one can make to deflect them. Before turning to his own view, Fine advances a further argument against the quantificational account, which he thinks points directly to the kind of account he himself favours. The logical form of ontological claims (pp.165–7) Consider such ontological claims as ‘Natural numbers exist’, ‘Integers exist’, and ‘Electrons exist’. According to the quantificational account, these are to be understood as existential quantifications: ∃x(x is a natural number), ∃x(x is an integer), and ∃x(x is an electron). But this, Fine 11 Fine accepts Carnap’s internal/external distinction, but not his clean break between internal and external questions and statements. The proper expression of ontological claims, on Fine’s view, requires straddling both, using ordinary ‘thin’ existential quantification binding variables lying within the scope of a reality operator. See Fine (2009), p.173, and section 4.3.2 of this chapter.

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argues, gets the logical form of such claims quite wrong. Consider the first two claims. On the most natural and plausible interpretation, the second is a significantly stronger claim than the first—for the integers comprise both the natural numbers (i.e. the nonnegative integers) together with the negative integers, while the first claim asserts only the existence of at least one non-negative integer. But on the quantificational account, it is the first claim which is the stronger—for it asserts that there is at least one integer which is non-negative, while the second asserts only that there is at least one integer, and claims of the form ∃x(Fx ∧ Gx) are in general stronger than corresponding claims of the form ∃xFx. Fine’s conclusion is that: The commitment to integers is not an existential but a universal commitment. It is a commitment to each of the integers not to some integer or other. And in expressing this commitment in the words ‘Integers exist’, we are not thereby claiming that there is an integer but that every integer exists. Thus the proper logical form of our claim is not that ∃xIx, where I is the predicate for being an integer, but ∀x(Ix ⊃ Ex), where E is the predicate for existence. (p.167)

It is clear that Fine takes this conclusion to generalize to other ontological claims: . . . the most natural reading of ‘electrons exist’ is that there are electrons while, on our view, the proper reading, for philosophical purposes, should be modeled on the reading of ‘electrons spin’, in which it is taken to mean that every electron spins.

I shall return to this argument in due course. First, it will be useful to sketch the positive view Fine takes it to suggest and support.

4.3.2 Fine’s alternative account As we have just seen, Fine concludes from his last argument that, if we are to achieve a correct statement of realist ontological claims, we need to use a predicate to express existence or reality, rather than a quantifier. Thus the claim that integers exist is not the claim that ∃x(x is an integer) but is rather the claim that ∀x(x is an integer → x exists), where ‘exists’ should be understood, not as a quantifier, but as a predicate. On this account, it is ‘Integers exist’ that is the stronger claim, as it should be—for the claim ‘Natural numbers exist’ is now the claim that ∀x(x is an integer ∧ x is non-negative → x exists ), which is clearly weaker. Fine himself prefers to avoid the use of ‘exists’ to express ontological claims, because it is so readily understood in a ‘thin’ sense, and proposes instead to express such claims using a predicate ‘real’—so that a realist position on the integers would be expressed as ∀x(x is an integer → x is real), while a strong anti-realist would claim that ∀x(x is an integer → ¬(x is real)). Obviously this prompts the question: How should this reality or existence predicate be understood? As Fine himself puts it: ‘how, if it all, is further clarification of the concept of what is real to be achieved?’ (p.171). He suggests explaining ‘real’ as a predicate in terms of a reality operator R[. . . ], where ‘. . . ’ stands in for a sentence. We may then, he proposes, ‘define an object to be real if, for some way the object might be, it is constitutive of reality that it is that way (in symbols, Rx =df ∃ϕR[ϕx])’. Thus the reality operator serves to express that the fact stated by the sentence on which it operates is ‘constitutive of reality’—as Fine puts it, ‘ontology finds its home, so to speak, in a conception of reality as given by the operator’ (p.172). He concedes (p.174)

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that this explanation is unlikely to persuade someone who is not ‘already willing to accept the metaphysical conception of reality’, but contends that there is no prospect of a deeper explanation, and that none is needed, because we seem to have a good intuitive working grasp of the concept. Thus whether we agree with Democritus or not, we understand his claim that there is nothing more to the world than atoms and the void, and we know what he needs to do to defend it (for example, he would need to argue that there being chairs, for example, ‘consists in nothing more than atoms in the void or explain in some other way how the existence of chairs is compatible with his world-view’ (p.176). Whether this is a good enough response to the seemingly quite reasonable demand for further explanation is, of course, open to debate. For even if it can be argued that there being chairs consists in nothing over and above there being atoms in the void, suitably configured, it is quite unclear that this warrants the claim that chairs do not really exist, or that they are somehow less real than atoms. I shall not press this concern further here, but will have a little more to say that bears on it in my concluding remarks.

4.4 Deflation defended—a response to Fine 4.4.1 Fine’s first two, direct arguments I believe we can make some very simple but effective responses to these arguments.

triviality As we have seen, Fine claims that a quantificational account makes trivial such claims as that numbers exist, whereas they are surely substantial claims, at least when made by philosophers. But he is wrong. ∃x(x is a number) is indeed a trivial consequence of, for example, ∃x(x is a prime number ∧ 9 ≤ x ≤ 12), and this in turn is a consequence of the Dedekind-Peano axioms, together with the relevant definitions of prime etc. But this doesn’t mean or entail that ∃x(x is a number) is itself trivial. That would follow only if it were a trivial consequence of trivial premises. And in this case, the ultimate premises—or so it may be argued (see below)—are certainly not trivial.

philosophical, not e.g. mathematical Fine claims that a quantificational account turns philosophical questions about existence into non-philosophical ones which can be settled by common observation or some appropriate non-philosophical discipline—so that whether numbers exist, for example, becomes a question straightforwardly settled by mathematics, rather than philosophy. But here too, it seems to me, his conclusion is too-swiftly drawn. Ontological questions are indeed philosophical, not mathematical. But the quantificational account does not say otherwise. As seen, the status of the question turns upon that of the ultimate premises from which the quantificational answer to it is derived, and in the present case, the ultimate premises are the Dedekind-Peano axioms. So the question concerns their status: Is their truth a straightforwardly mathematical matter? Well, obviously they are mathematical propositions. But that hardly settles

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the question, which is whether they are strictly and literally true, when taken at face value. Is it strictly and literally true that there exists an object, 0, such that (i) Nat(0) (ii) ¬∃x(Nat(x) ∧ P(0, x)) (iii) ∀x∀y((Nat(x) ∧ Nat(y) ∧ P(x, y)) → ∀z((P(x, z) → z = y) ∧ (P(z, y) → z = x))) (iv) ∀x(Nat(x) → ∃y(Nat(y) ∧ P(x, y))) (v) ∀φ((φ(0) ∧ ∀x, y(Nat(x) ∧ Nat(y) ∧ φ(x) ∧ P(x, y) → φ(y))) → ∀x(Nat(x) → φ(x))) The first four axioms jointly imply the existence of a sequence of objects with a first member and, for every member, an immediately succeeding one, and so no last member, i.e an infinite sequence. Whether there is such a sequence, I claim, is not a question that mathematics itself can answer, but a clearly philosophical one, and the answer is certainly not trivially true (or trivially false). If I am right, neither of Fine’s direct arguments against the quantificational account is effective. There is accordingly no need for the more or less desperate defences against those arguments which Fine discusses and—in my view, rightly—finds wanting.

4.4.2 What is deflation? While my replies to Fine’s direct arguments seem to me clearly available to a defender of a quantificational account, it may be felt that my contentions that answers to ontological questions are neither trivial nor non-philosophical are in some tension with deflationism. This begs the question: What exactly is deflationary about deflationary ontology? There are, I think, two main elements to deflation in the sense in which I wish to commend it. (i) The definition of ontological categories in Fregean terms, as the worldly correlates of the various logico-syntactic types of expression, is much more inclusive and undemanding than the conception of objects or particulars and properties or universals advocated by many other metaphysicians (whose favoured concepts often seem to embody some pre-conception about what ‘really’ exists, or is ‘ultimately real’). (ii) The associated conditions for the existence of entities in these categories are likewise—at least in some respects—relatively undemanding. We require only the possibility of suitable truths embedding singular terms for the existence of objects, and require only the possibility of suitable predicates for that of corresponding properties and relations. Of course, the relevant statements must be strictly and literally true, and the requirement of possible truths or possible predicates is not toothless—whether there are such possibilities may be a hard and disputed question, as it often is in philosophy of mathematics. Deflating ontological questions in this way clearly doesn’t mean that their answers must be trivially true. Nor does it mean that they must be non-philosophical. What the deflationist—or at least my sort of deflationist—opposes is the idea, going back at least to Carnap (Carnap, 1950), but surely detectable in much earlier philosophy—that

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there is some special, philosophical notion of reality or existence in play in philosophical questions about existence. Instead, we are concerned with existence in a single sense expressible using the ordinary existential quantifier ‘there are . . .’ or ‘there is at least one . . .’, such that the truth of ‘there are Fs’ is guaranteed by the truth of at least one of its instances ‘Ft’. This is entirely consistent with holding that philosophical considerations may bear on the question whether such statements are true.

4.4.3 Fine’s final argument This argument, I believe, calls for a somewhat longer, two-part, response. Let us begin with Fine’s main example, which turns on the relative strength of the ontological claims: (N) Natural numbers exist. (I) Integers exist. While (N) and (I) could be understood as claiming only that there is at least one natural number, and that there is at least one integer, I think we should agree with Fine that this is not how we are most likely to take them. But this is because we are very likely, instead, to take them as equivalent to the claims: (N*) The natural numbers exist. (I*) The integers exist. the first of which asserts not just the existence of at least one natural number, but the existence of the whole infinite sequence of natural numbers, and crucially, the second asserts the existence, not just of at least one integer, but that of the whole backwards and forwards infinite sequence of integers. And clearly, (I*) is a stronger claim than (N*), for the latter asserts only the existence of a sequence of order-type ω, while the former asserts that of a sequence of order-type ω + ω—one which itself has the former ω-sequence as a proper part. One reason why this interpretation of (N) and (I) is so much more likely than the minimally existential one which takes them to say only that there is at least one natural number (or at least one integer) is that each particular natural number (and similarly each particular integer) is essentially a member of an certain ω-sequence (or ω + ω -sequence), so that if any natural number (or integer) exists, so must all the others. Given the nature of natural numbers, it simply makes no sense to take a selective attitude towards their existence—to hold, say, that, 0, 17,234, and 5008 all exist, but no others; similarly for the integers. If (N) and (I) are understood as (N*) and (I*), then, there is a straightforward way to explain why it is (N) which is the weaker, not the stronger claim, consistently with a quantificational account. For (N) then asserts the existence of an infinite sequence of a certain kind, and (I) that of an infinite sequence of a quite different kind, and the conditions for the existence of the latter are evidently more demanding than those for the existence of the former. On this account, there is nothing intrinsically wrong with the minimally existential interpretations—the point is rather that, given the nature of natural numbers and integers, no one should be prepared to assert (N) and (I), unless willing to endorse the stronger claims (N*) and (I*). For that reason, asserting (N)

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and (I) on the minimal interpretations is, whilst not incorrect, somewhat pointless and potentially misleading. While this defence of the quantificational account is, I believe, effective in regard to examples such as (N) and (I), it relies upon what is pretty clearly a special feature of these examples—the fact that by their very nature, the individual natural numbers and integers belong to larger structures in such a way that their existence stands or falls with that of all the other elements of those structures—which is not found in other ontological claims such as: (E) Electrons exist. (T) Tables and chairs exist. While matters are perhaps less clear in regard to (E), individual tables and chairs, for example, have no such dependence upon the existence of other tables and chairs for their existence. There is no plausible reading of (T) analogous to the readings (N*) and (I*) of (N) and (I). But, Fine may claim, it remains the case that even if (T) is open to the mininally existential reading (∃x∃y(x is a table ∧ y is a chair), it should really be understood as claiming of tables and chairs quite generally that they exist. Were he to do so, would he be right? The first point to be made in response is that Fine’s case for his claim that the quantificational account gets the logical form of ontological claims wrong draws most of its strength and plausibility from the fact that (N) is naturally construed as making a weaker claim than (I), whereas on the quantificational account he considers, (N) makes the stronger claim. But nothing analogous to this holds for claims such as (E) and (T). Of course, just as the natural numbers are—or at least may be identified with—a proper subset of the integers, so electrons are a proper subset of sub-atomic particles, and tables and chairs are a proper subset of articles of furniture. But the claims: (P) Sub-atomic particles exist. (A) Articles of furniture exist. are, just as the minimal quantificational account says, weaker than the claims (E) and (T). So there is no parallel case for thinking that that account gets the logical form wrong. Can that case be made in some other way? An opponent of the (minimal) quantificational interpretation of (E) might gloss it as equivalent to: (E ) Some electrons exist. and protest that this is plainly not what we mean to claim in asserting (E). If (E ) is in turn construed as: (E ) Some, but (perhaps) not all electrons exist. he would surely be right that we intend no such claim—for the suggestion that such-and-such electrons exist while such-and-such others do not is at best simply

4.5 concluding remarks 71 confused.12 But this is simply a travesty of the quantificational account, on which (E) just says that there is at least one electron. Beyond that, it says nothing about how many electrons there are, and certainly does not involve the misbegotten idea that there are, or might be, non-existent electrons. I can see no better way to make the case that the quantificational account gets the logical form of such statements as (E) and (T) wrong, and that they are really universal statements essentially involving an existence or reality predicate. And in the absence of such a case, defenders of that account can simply stick to it that these and similar statements are to be interpreted as simple existential quantifications. Of course, since they can define an existence predicate E in terms of the existential quantifier together with identity, by Ex = df ∃y(x = y), they can indeed assert that all electrons exist, using ∀x(x is an electron → ∃y(x = y)). But this is not, of course, the universal existence predication which Fine thinks we should be making, since it simply says that if anything is an electron, it exists (i.e. is identical with something)— on whether there exist any electrons, it is silent.

4.5 Concluding remarks It will be evident that I have given no positive argument for deflating ontology, but tried only to explain the kind of deflation I favour and to defend it, along with a quantificational account of existence claims, against the particularly interesting arguments Fine sets against them. Closing remarks are scarcely the place to make a positive case. Instead, I want to comment briefly on the opposed and widespread tendency to inflation. Fine himself steadfastly resists attempts to ‘beef up’ ontological claims by introducing a ‘thick’ existential quantifier, even a thick sense defined in terms of the thin by means of his reality operator, by embedding the thin quantification within its scope: R[∃xFx].13 Nor, he argues, can a satisfactory expression of ontological claims be achieved, as some have thought, by identifying what is real with what is fundamental.14 However, he shares with both the underlying conviction that such claims cannot be expressed by means of an ordinary or ‘thin’ quantifier alone, but demand a special, philosophical notion of what is real. As we have seen, he sees no way ‘to define the concept of reality in essentially different terms’, but claims that none is needed, because we have a good intuitive, working grasp of it—Democritus’s ‘thought that there is nothing more to the world than atoms in the void’ is ‘an intelligible position, whether correct or not’ (p.175), and ‘We know in principle how to settle such claims about the constitution of reality . . . Democritus would have to argue that there being chairs consists in nothing more than atoms in the void or to explain in

12 Surely not even the most enthusiastic Meinongian would adopt a selective attitude towards the existence of electrons! 13 See especially pp.162–5, 168–74. 14 For this idea, see Chalmers (2009) and Dorr (2005), and for Fine’s criticism, pp.174–5.

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some other way how the existence of chairs is compatible with his world-view’ (p.176). Chairs, the ancient sage will likely insist, are just atoms in the void moving in a chairshaped concentration in a locally otherwise thinly populated region of the void. And, or so Fine at least will add on Democritus’s behalf, the atoms and the surrounding void are real, but the chairs are not. But why that last step? Why think that, supposing chairs to be composed of, or constituted by, just atoms in chair-shaped motion in the void, that shows or means that they are somehow unreal, or less real than their constituent atoms? Why not draw the exactly opposite conclusion, that precisely because they are entirely composed of real atoms, the chairs themselves are equally real? Of course, it would make perfectly clear sense, and not be unreasonable, to claim that chairs are not ultimate (or fundamental) constituents of reality. But that doesn’t mean they are any less real—as Fine rightly insists, we should not conflate fundamentality with reality. The same goes, it seems to me, for other relations, such as ontological or metaphysical (or even causal) dependence, which might tempt some to think that dependent entities are somehow less real than those on which they depend. Granted that Xs depend upon Ys in one of these ways for their existence, while Ys do not so depend upon Xs, just why should it be supposed that this asymmetry carries with it an asymmetry with respect to reality—so that Ys are real but Xs are not? The former asymmetry means just that there could be Ys without there being any Xs (at least in cases where the existence of Xs, and presumably that of Ys too, is a contingent matter). But given that there are Xs (as well as Ys), the mere possibility that there might not have been any seems no sort of reason to deny the reality of those there are. In short, it seems that the deflationist can acknowledge the asymmetric relations among things which may tempt others to inflate reality, and simply decline to follow suit.

5 Ontological Categories and the Problem of Expressibility with Øystein Linnebo Frege famously held that ontological categories correspond to logico-syntactic types. Something is an object just in case it can be referred to by a singular term, and likewise for all the other categories. This view faces an expressibility problem. In order to express the view, we need to generalize across categories; but by the view itself, any one variable can only range over a single category. We provide a sharp formulation of the problem, show that there is no easy way out, and then explore some of the hard ways.

5.1 The metaphysical significance of logical type distinctions One of Frege’s celebrated insights is that language, upon logical analysis, turns out to be composed of expressions of fundamentally different types. There are ‘complete’ or ‘saturated’ expressions, which comprise simple and complex singular terms (proper names, in his inclusive sense) and sentences.1 There are also ‘incomplete’ or ‘unsaturated’ expressions, which include sentential operators together with predicates, relational and (other) functional expressions of the various levels, each level and adicity corresponding to its own type. We shall refer to these types of expressions as the logico-syntactic types, where the prefix ‘logico’ indicates that the syntax has first been regimented in light of logical analysis.2 What is the metaphysical significance, if any, of the logico-syntactic type distinctions? This question will be our central concern throughout this paper. The natural starting point for any discussion of the question is Frege. As is well known, Frege held that the division of expressions into logico-syntactic types underpins a fundamental division of the entities for which expressions stand into ontological categories. The basic idea is that an entity belongs to a certain ontological category if and only if it is, or could

1 In his later work, of course, Frege takes the latter to be a special case of the former. We shall not follow him in doing so (though it would be straightforward to make the requisite adjustments). 2 We shall use ‘(logico-)syntactic type’ and ‘syntactic category’ interchangeably, as other writings on this topic often do.

Bob Hale and Øystein Linnebo, Ontological Categories and the Problem of Expressibility with Øystein Linnebo In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale and Øystein Linnebo. DOI: 10.1093/oso/9780198854296.003.0006

74 ontological categories and the problem of expressibility be, the referent of an expression belonging to a corresponding syntactic category.3 Thus, objects are those things which are or could be the referents of proper names in Frege’s inclusive sense (i.e. singular terms), monadic first-level concepts (in Frege’s sense, namely, functions from an object to a truth-values) are those things which are or could be the referents of first-level one-place predicates, and so on. This is sometimes called the Reference Principle.4 Frege does not, as far as we know, state this principle explicitly and in full generality. But it seems to us that there can be little doubt that he is committed to it. In his response to Benno Kerry’s claim that the properties of being an object and being a concept are not exclusive, and that some concepts are objects, Frege does explicity say, or make inferences which clearly presuppose, that objects are what can and can only be referred to by proper names, and that concepts are what can and can only be referred to by predicates. There is no good reason to believe that he would dissent from the obvious generalizations of these claims to other types of expression and corresponding types of non-linguistic entity.5 This approach to the ontological categories has struck many philosophers as attractive. As for instance Michael Dummett has emphasized, the approach gives us an independent handle on the categories. Without such an independent handle, ‘we should be left wholly in the dark how it is to be decided whether numbers, or entities of any other sort, are objects or not’ (Dummett, 1981, p.56). Moreover, the approach makes available a notion of object (as well as of membership in the other categories) which is neutral with respect to important philosophical questions. Were we, for instance, to define objecthood in terms of occupancy of spacetime or causal efficacy, we would beg an important ontological question in favour of nominalism.6 Although attractive, the Fregean approach to ontological categories faces serious problems. Firstly, it may be objected that the syntactic categories are too parochial to underpin the ontological categories.7 English and other natural languages contain a wealth of syntactic categories that are not found in more austere languages; and many artificial languages, such as those used in electronic databases, are even more profligate in their syntactic categories. Which languages, if any, provide a reliable guide to the real ontological categories? Frege offers at least a partial answer to 3 This use of ‘ontological category’ must be distinguished from the use of ‘category’ in other works such as Hale and Wright (2001b), as well as in works by others cited therein. In our present sense, the ontological categories are tied to the syntactic categories and thus ultimately to considerations about interchangeability without loss of grammaticality. The alternative use of ‘category’ is tied to a finer subdivision of syntactic expressions, which track one or another more demanding notion of interchangeability. One such is the tradition, associated with Ryle and others, which links categories with interchangeability without loss of felicity. But another (cf. Hale and Wright, op. cit) sees sharing the same general criterion of identity as determinative of sameness of category. On either of these approaches, the number 3 and the person Caesar are taken to belong to different categories, but to the same ontological category (namely, that of objects) in our present sense. 4 Hale and Wright (2012, p.93) formulate the principle as follows: ‘An entity of a certain kind is anything which can, and can only, be referred to by an expression of a certain correlative logico-syntactic type.’ A formalization will be provided shortly. 5 For a detailed analysis of Frege’s response to Kerry, bringing out his commitment to these principles, see Hale and Wright (2012, pp.87–91). 6 See Hale (2013a), ch.1, sections 1–6 for further discussion of this point. 7 See Quine (1969, pp.91–2) for a version of this problem. For some thoughts about the problem, see Dummett (1981, ch.s 3 and 4), Hale (1984), and Hale (2013a, sections 1.1–1.8).

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this question. His Reference Principle is meant to be applied exclusively to logically regimented languages such as his Begriffsschrift or the language of higher-order logic. The type distinctions involved in such logically regimented languages represent minimal conditions on intelligibility at all, and thus, the thought goes, there is nothing parochial about these distinctions. We shall assume for now that this answer is on the right track. Later, however, we shall have occasion to ask whether further restrictions to the Reference Principle are required. A second problem arises even under the mentioned assumption and is accordingly more fundamental. In essence, the problem is that Frege’s tight correlation of types of entity with types of expression makes it quite literally impossible to articulate Frege’s semantic theory and its accompanying ontology. Any attempt to state the theory runs afoul of the fact that, if Frege’s analysis is correct, properties (or Fregean concepts) and relations, as well as functions more generally, can only be referred to by incomplete expressions—predicates or functional expressions of appropriate type and level—and cannot be the referents of singular terms or the values of variables standing in for such terms. (Naturally, this attempt to state the problem is itself strictly nonsensical or at least ill-formed by Frege’s lights.) This is the problem of expressibility.8 Clearly, the problem is liable to afflict any semantic analysis of language according to which expressions and their semantic values—the entities for which they stand— are divided into different types in such a way that entities may only be the referents, or values, of expressions of appropriate type. How can such a theory be even so much as stated, without violating the very type restrictions it claims apply to the language? We begin by giving a clear and formally precise presentation of the problem of expressibility, which identifies its minimal assumptions. This precise and economical analysis is helpful, we believe, as it enables us to identify exactly what is needed to generate the problem and thus also exactly which options are available for getting out of trouble. We argue that there is no easy way out of the paradox. Then we explore three hard ways out, each associated with the rejection of one of the three assumptions on which the paradox is based. One option is to claim that reference is effected only by a single type of expression, namely singular terms, and that, accordingly, there is only a single category, namely that of objects. We reject this broadly Quinean view as unacceptably hostile to the idea of predicate reference. A second option is to give up on expressing the insights that Frege tried to express; or, more precisely, to deny that there are any insights there that one might even want to express. Once the restrictions inherent in the logico-syntactic types are taken seriously, we see that there is no room for any such insights. This austere option is

8 The problem is related to the famous concept horse paradox, which has received a good deal of attention in recent decades. See in particular Dummett (1981), Geach (1976), Wright (1998b), and Hale and Wright (2012), as well as Proops (2013), which provides a useful classification of problems associated with Frege’s difficulties with the concept horse and further references to discussions of the problems. Our expressibility problem is essentially the same as what Proops (2013) calls ‘the problem of the inexpressibility of logical category distinctions’; see also Hale and Wright (2012, p.92) for an informal presentation of a closely related paradox.

76 ontological categories and the problem of expressibility defended by Agustín Rayo, Timothy Williamson, and others.9 Although we suspect it enjoys considerable popularity among contemporary philosophers working in this area, this view has received little attention in discussions of the ontological categories and the problem of expressibility. We develop a new line of attack on this popular option. A third and final option seeks to achieve the full expressibility that Frege sought and which we suspect is widely believed to be unattainable. This option has received far less attention than it deserves. We explore some reasons to favour this third option over the second, and we begin a philosophical and technical investigation of how this option might be developed. As our paradox demonstrates, this will require revising— or even abandoning—the Reference Principle.10

5.2 The paradox The first assumption involved in our paradox is that there is an ontological category that is neither empty nor all-encompassing.   (OC) ∃x∃y∃c OC(x, c) ∧ ¬OC(y, c) where ‘OC(x, c)’ expresses that x belongs to ontological category c. Frege clearly makes this assumption, for instance when he writes: ‘Objects stand opposed to functions. Accordingly I count as objects everything that is not a function’ (Frege, 1964, p.2). And of course, there is little point to a theory of ontological categories unless this assumption is satisfied. The second assumption is a non-modal version of the Reference Principle:   (RP) ∀e∀x∀c Ref (e, x) → (SC(e, c) ↔ OC(x, c)) where ‘Ref (e, x)’ says that the expression e refers to x or, if e is a variable, has x as its value; and where ‘SC(e, c)’ says that e belongs to logico-syntactic category c. (The isomorphism between syntactic and ontological categories allows us to use the same variable c to represent both, as will be explained below.) The main difference between (RP) and extant formulations of the Reference Principle is that the former eliminates the modal aspects of the latter. As we shall see, this elimination of modality makes no difference in practice.11 The third and final assumption makes plain something that must be presupposed in order to ensure that (RP) and related principles really express what they are meant to express. (EXPRESSIBILITY) There is a variable (of some language or other) that has among its values all entities of all ontological categories.

9 See Williamson (2003), Rayo (2015) (although the view is also implicit in his Rayo (2006), Krämer (2014), and Jones (2016)). The view is anticipated in some respects by Furth (1968). 10 Some earlier attempts to revise the Reference Principle can be found in Wright (1998b), Hale (2010), and Hale and Wright (2012). 11 A comparison of (RP) with earlier statements of the Reference Principle is provided in Appendix A.

5.2 the paradox 77 After all, (RP) is meant to ensure not only that objects are the sort of things that are denoted by singular terms, but that monadic concepts are the sort of things denoted by one-place predicates, and so on through all the categories. (EXPRESSIBILITY) is also required to make literal sense of the ontological claims that are at the heart of Frege’s theory, such as:12 Objects are saturated, while concepts are unsaturated. No object is a concept. Every thing is either an object or some sort of concept or other type of function.13

Unfortunately, we have the following result. Fact 1 These three assumptions are inconsistent. Proof. By (OC), there is a category c to which an entity a belongs but another entity b doesn’t belong: (1)

OC(a, c) ∧ ¬OC(b, c)

By (EXPRESSIBILITY), there is a variable v such that Ref (v, a) and Ref (v, b). Hence by two applications of (RP) we get the contradiction: (2)

SC(v, c) ∧ ¬SC(v, c)

This completes our proof. It is important to notice that nothing like the full force of (EXPRESSIBILITY) is needed. It suffices that there be a variable whose values include at least one member of some category and at least one non-member. Thus, (RP) is incompatible not only with full expressibility but with any form of quantification that extends beyond a single ontological category. How should we respond to the paradox? There are of course three assumptions that we might attempt to deny. We shall discuss these three options shortly.14 12 Some commentators suggest that these ‘claims’ should be understood merely as hints or elucidations; see e.g. Weiner (1990). This interpretation results in a view akin to the ones that we discuss and criticize in Section 5.5. A different challenge is developed by Rayo (2015), who argues that our support for (EXPRESSIBILITY) is infected with a pernicious form of ‘metaphysicalism’, by which he means a languagetranscendent conception of ontology. Rayo argues that a metaphysicalist is entitled to (EXPRESSIBILITY). Whether or not he is right about that, we deny that metaphysicalism is needed in order to ensure full expressibility. Indeed, as explained in Section 1, we follow Frege (and Rayo) in favouring a logico-linguistic approach to the ontological categories. This makes us non-metaphysicalists, albeit of a different kind from the ‘compositionalist’ variety favoured by Rayo. 13 For an example of a claim like the first in this list, see e.g. Frege (1964, p.21), where Frege writes: ‘Functions of two arguments are just as fundamentally different from functions of one argument as the latter are from objects. For whereas objects are wholly saturated, functions of two arguments are saturated to a lesser degree than functions of one argument, which too are already unsaturated’, as well as several places in Frege (1892) and ‘What is a function?’. The second claim follows from the first by logic. For an example of the third claim, see the passage quoted above in connection with (OC). 14 How does our expressibility paradox relate to the more familiar ‘concept horse’ paradox (as reconstructed, e.g. in Hale (2010) or Hale (2013a, ch.1))? Both paradoxes make essential use of the Reference Principle and involve a commitment to (OC). The distinctive feature of the expressibility paradox is its last premise, (EXPRESSIBILITY), which plays no explicit role in the ‘concept horse’ paradox, although the kind of cross-categorial quantification associated with (EXPRESSIBILITY) (and sufficient for the paradox) is implicitly presupposed, e.g. by the (explicit) premise that no object is a concept. The distinctive feature of

78 ontological categories and the problem of expressibility

5.3 Some blind alleys Might the paradox result from an equivocation or some other superficial mistake? We shall begin by arguing that there is no such easy way out. (Readers prepared to take our word for this may skip to the next section.) We proceed in the order of increasing seriousness. First, it may be objected that our argument is ‘infected’ with the more general problem associated with the liar paradox and thus is old news. This is plainly incorrect. Nothing has been assumed about the reference relation other than (RP). For all that has been said, every sentence may co-refer with its own negation—and even so the paradox would follow. Second, it might be thought problematic to use one variable, c, to range over both syntactic and ontological categories. But nothing hangs on this. The values of the variables for category can be regarded as mere ‘tags’—assigned by the relation SC to expressions and by OC to all entities whatsoever—in order to divide each domain into equivalence classes: two items belong to the same logico-syntactic category iff SC assigns them the same tag; and mutatis mutandis for the ontological categories.15 Readers who prefer separate tags for the two kinds of category may introduce a one-toone mapping of the syntactic categories onto the ontological ones, say g, and modify relevant principles accordingly. (RP) would then be rewritten as:   ∀e∀x∀c Ref (e, x) → (SC(e, c) ↔ OC(x, g(c))) The paradoxical argument would still go through in precisely the same way: for the argument only ever compares two categories of the same kind; it never compares a syntactic category with an ontological one. Third, it might be objected that the paradoxical argument equivocates on the notion of reference. Arguably, the Reference Principle is concerned with reference in the narrow sense of the relation that obtains between a constant expression and its Bedeutung or semantic value. If so, the same should go for our assumption (RP), which is meant to capture the core of the Reference Principle. By contrast, (EXPRESSIBILITY) employs a broader notion of reference which encompasses the relation between a variable and each of its values. So the paradoxical argument works only if we may appeal to (RP) understood in terms of the broad notion of reference. But, the objection goes, all we are entitled to assume is (RP) understood in terms of the narrow notion of reference. We agree that there is a theoretically important distinction between reference in the narrow sense and in the broad sense. However, this can be seen to be immaterial to our the ‘concept horse’ paradox is that its premises are concerned with nominalized singular terms such as ‘the concept horse’, which play no role whatsoever in the expressibility paradox. Thus, since the expressibility paradox goes beyond the ‘concept horse’ paradox only in making explicit an assumption already implicit in the latter, whereas the latter uses assumptions not needed in the former, the expressibility paradox is the more minimal one and therefore, we believe, more fundamental. 15 This interpretation of the category variables has the added advantage of not requiring us to reify the categories. This freedom may be thought particularly valuable in the case of the ontological categories, where for instance the category of all objects is naturally conceived of as something like the class of all objects—which raises hard problems in the foundations of set theory that are orthogonal to our main concerns in this chapter.

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argument. The reason has to do with the modal force of the Reference Principle, which we noted above. By reintroducing this modal force, we can show that the narrowsense version of the principle entails the wide-sense version, although the former appears weaker than the latter.16 In light of this observation, it is in practice not very important to distinguish between the narrow and the wide sense of reference. Yet there are obvious practical reasons to favour the wide sense. By understanding reference in this sense, we avoid the need to engage in modal reasoning, which is more complicated and harder than the corresponding non-modal reasoning. A final option is to convey the desired insights in a way that eludes the scope of the Reference Principle and in this way escapes the jaws of the paradox. One may for instance try to exploit the fact that languages which have not been subjected to proper logical regimentation are exempted from the scope of the Reference Principle. Can our trouble be avoided by formulating the Fregean theory in such a non-regimented language? The hope is that such a language can make available the requisite expressibility while the paradox is avoided because this language falls outside the scope of the Reference Principle. Unfortunately, the hope is doomed to frustration. For whatever can be expressed in a non-regimented language can also be expressed in a regimented language. Once we have provided the needed regimentation, the paradox is reinstated. We shall therefore return to the three assumptions on which our paradox depends. Can one (or more) of them be given up so as to restore consistency, while still holding on to central Fregean insights?

5.4 Ontological Monism One option is to deny the first assumption, (OC), which posits an ontological category that is neither empty nor all-encompassing. This response entails that every ontological category is either empty or all-encompassing, or, in other words, that every entity belongs to one and the same category. Might this form of ontological monism provide a viable response to our paradox? To stay as close as possible to the original Fregean view, assume that we retain the other two Fregean assumptions, namely (RP) and (EXPRESSIBILITY). Then the negation of (OC) entails that all referring expressions belong to a single syntactic type. This is a conclusion that would have pleased Quine, who famously claimed that only singular terms are in the business of referring.17 Other types of expression contribute semantically in ways that do not involve any relation of reference. For instance, predicates contribute by being true or false of objects, not by standing for some form of non-linguistic entities. Thus, the contemplated response to the paradox is not only formally consistent but its similarity to views held by famous philosophers indicates that it has at least some appeal. We believe this response operates with a far too demanding conception of reference. Assume that an open formula is meaningful. Then we have defined what is required for the open formula to be true or false of any objects. This means that the open 16 See Appendix A for a defence of this claim. 17 This claim is made throughout Quine’s long career; see e.g. the discussion of second-order logic in chapter 5 of Quine (1986).

80 ontological categories and the problem of expressibility formula defines a function from objects to truth-values. According to what we shall call minimalism about reference, nothing more than such a function is required for the open formula to define a Fregean concept.18 Clearly, it would be question-begging simply to insist that only singular terms can refer. In fact, Quine had one response which did not just beg the question. He insisted that—whether or not the mentioned functions are objects—they must be provided with criteria of identity: ‘no entity without identity’! Since Quine was convinced that no such criteria could be provided, he took himself to be entitled to deny that predicates refer. A full discussion of this Quinean response would take us too far astray, so we shall be brief.19 We believe Quine was right to insist on criteria of identity for the relevant functions. But we deny that this task is as hopeless as Quine thought. Quine’s conviction relies on two assumptions: first, that entities are inadmissible unless we can state identity conditions for them, and second, that satisfactory such conditions must be formulated in purely extensional terms. Whilst we accept the first assumption, the second seems to us, as it does to many others, to be very much open to question. At a minimum, and to a first approximation, functions might be individuated by necessary co-extensiveness. More plausibly, but exploiting further intensional resources Quine rejects, they might be individuated in terms which discriminate, as necessary co-extensiveness does not, between such properties as equilaterality and equiangularity among Euclidean triangles. Summing up, the paradox can be blocked by rejecting (OC), which results in a view similar to Quine’s. But this view is ultimately untenable. In light of minimalism about predicate reference, it is not an option to let singular terms alone be the possessors of reference or semantic value.

5.5 Austere type theory and the rejection of expressibility Our overarching question, we recall, concerns the metaphysical significance, if any, of the logico-syntactic types. We have rejected two answers. Frege’s answer, taken at face value, falls prey to paradox; and the Quinean answer just considered is unduly hostile to predicate reference. We now turn to a more robust answer, which enjoys substantial support among contemporary philosophers.20 This answer is to reject the third assumption on which the paradox rests, namely (EXPRESSIBILITY), which says that there are variables that range across two or more distinct ontological categories. It is useful to begin by considering the present proposal from Frege’s point of view. There is little or no indication that Frege ever officially allowed variables to range across categories. For sure, the first-order variables of his Begriffsschrift range over all and only objects, the monadic second-order variables range over all and only monadic first-level concepts, and so on. His own eventual reaction to the paradox was to dismiss expressions like ‘the concept horse’ and the associated predicate ‘is a

18 See Hale and Wright (2009) §9 and (for a fuller discussion) Hale (2013a, ch.1.12). 19 See further Hale (2013b). 20 See the works cited in footnote 9 of this chapter.

5.5 austere type theory 81 concept’ as defective. But he never, as far as we know, drew the more radical conclusion that the paradox results from a misbegotten attempt to say what cannot be said.21 This is unsurprising. For to give up on (EXPRESSIBILITY) is to give up on verbalizing many of the key claims associated with the Fregean conception of the ontological categories.22 For example, without (EXPRESSIBILITY) we cannot properly state the Fregean theses that objects and concepts are ‘fundamentally different’ or that every entity belongs to a unique ontological category. Nor can we state the Reference Principle, which makes essential use of type-unrestricted predications such as ‘Ref (e, x)’ and ‘OC(x, c)’, where the variable ‘x’ is allowed cross-categorial range. In sum, while the rejection of (EXPRESSIBILITY) might not have troubled Frege when viewed in isolation, it would remove the ground from underneath a number of claims to which he was committed. The result would thus be a substantial departure from Frege. It might be responded that, although the Fregean insights cannot be expressed directly, they can nevertheless be conveyed in a more roundabout way. Perhaps we can formulate the desired insights in a way that admittedly uses predicates and variables in an impermissible way—and thus makes no literal sense—but which nevertheless succeeds in conveying the right idea to any reader willing to grant us a ‘pinch of salt’. This kind of response is suggested by Wittgenstein’s Tractatus and more explicitly endorsed by Geach, who thought there are insights which cannot properly be stated but which can nevertheless be shown.23 Like many other philosophers, we find this response unconvincing. Where there is an insight to be conveyed, in however devious a way, there must also be a way of articulating it clearly. There will of course be languages that are too impoverished to express some given insight; for instance, the theory of relativity cannot be stated in a language devoid of mathematical resources. But that is not the source of our problem. The claim is rather that the insights resist expression by their very nature, in any language whatsoever. It is hard to see how such alleged insights can be insights at all.24 Moreover, any type distinctions that are present in a logically regimented language— and thus fundamental enough to mirror the ontological categories—are likely to be present also in a logical regimentation of the structure of conceptual thought.25 If so, then conceptual thought does no better than language when it comes to capturing the relevant insights. What cannot be said, cannot be thought either.

21 See his posthumously published review of Schönfliess’s Der Logischen Paradoxien der Mengenlehre (Frege, 1979), pp.176–83, as well as Dummett (1981), pp.212ff. 22 Notice that while (EXPRESSIBILITY) can of course be given up, it cannot coherently be denied. For the intelligibility of the thesis presupposes its truth, and consequently, so does its denial. 23 See Geach (1976), especially p.55 where Geach writes that ‘[t]he category distinctions in question are features both of verbal expressions and also of the reality language is describing’. In fact, there is a school of interpretation which ascribes to Frege a loosely related idea, namely that there is an indispensable form of elucidation which goes beyond what can properly be said. For instance, Weiner (1990, p.251) writes that ‘the success of [Frege’s] logic as a scientific tool requires that the meaning of “concept” not be private to Frege. Frege’s elucidations must actually succeed’. 24 See Hale and Wright (2012, pp.100–4) for a more thorough discussion. 25 We need not take a stand on non-conceptual thought, if there is any, such as scalar representations of magnitudes or cartographic content. For the relevant insights would no doubt amount to conceptual thought.

82 ontological categories and the problem of expressibility Let us therefore turn to a far more austere way to abandon (EXPRESSIBILITY), which unreservedly accepts that what cannot be said cannot be whistled either. This view, which is espoused by Williamson, Rayo, and several others (cf. footnote 9), regards the type theoretic restrictions as absolute and insists that we steadfastly resist any temptation to transgress them. What are the prospects for semantic theorizing on this austere view? Clearly, we are no longer entitled to a single, type-unrestricted reference predicate. However, there is no prohibition against the introduction of one or more reference predicates provided that these strictly respect the type distinctions of our language. We can introduce a predicate ‘Ref 0 ’ which represents the relation that obtains between an expression and an object to which it refers. This enables us to state for instance that the term ‘Hesperus’ has a referent, namely ∃x0 Ref 0 (‘Hesperus’, x0 ). More interestingly, we can introduce a predicate ‘Ref 1 ’ to represent the referential relation borne by first-level predicate expressions.26 A non-austere theorist would say that the predicate represents the relation that obtains between an expression of type 1 and the first-level concept to which it refers. But the austere theorist has to reject this gloss on the grounds that it violates essential type restrictions; specifically, it uses the first-order quantifier phrase ‘a first-level concept’ where a second-order quantifier would have been appropriate. Regardless of what gloss is provided, the predicate can be used to state that some objects are referring expressions of type 1 (for instance, ‘. . . is a horse’), while others (for instance, ‘Hesperus’) are not. Finally, it is clear how to generalize to higher types and thus introduce a whole family Ref n of type-indexed reference predicates.27 Using this family of reference predicates, some semantic theorizing is possible even on the austere view. In particular, it is possible to theorize about assignments to variables of all types and about the truth of a formula relative to such an assignment.28 In short, the austere view is more robust and resourceful than one might initially suppose. The central question is whether the view nevertheless suffers from an expressibility deficit. This question places us in a difficult dialectical situation, as there appears to be no neutral way to adjudicate the debate. Considered from the point of view of non-austere theorists, their austere opponents do indeed suffer from an expressibility deficit. For instance, the austere view is unable to express the Fregean claim that ‘objects stand opposed to functions’. Considered from the point of view of the austere theorists, however, there is no expressibility deficit. For the alleged examples of inexpressible insights violate essential type restrictions and thus have to be rejected. Where a type distinction is violated, there is nothing that we should even wish to express, not even in a roundabout way.29 By stubbornly insisting on this response, the

26 To stay closer to Fregean notation, we might write ‘Refξ1 (e, Fξ )’, where ‘ξ ’ indicates an argument place of the predicate expression ‘F’. We shall use the simpler formulation ‘Ref 1 (e, F)’. 27 See Hale and Wright (2012, pp.106–7) for a discussion of a closely related proposal. 28 Two important examples of such theorizing are Williamson (2003) and Rayo (2006); see also Linnebo and Rayo (2012) for a streamlined exposition and further generalizations. See Krämer (2014) and Jones (2016) for two recent investigations. As will becomes clear shortly, we think the last two authors underestimate the expressibility deficit in the relevant typed languages. 29 See Krämer (2014) for a development of this response on behalf of the austere theorist.

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austere theorists can always hold their ground, at least to the extent of never being caught in a formal contradiction. To get past this apparent impasse, we begin with a methodological point. A striking feature of many philosophical debates is the participants’ willingness to respond to an opponent’s carefully considered statement by denying that the statement so much as makes sense. Of course, the history of philosophy abounds with claims that genuinely do not make sense. Equally obviously, the mentioned response needs to be used with great care; otherwise, open-minded philosophical exploration and eventual progress would be stifled. How, then, can we counter an interlocutor who stubbornly denies that our claims make sense? Williamson provides an apt characterization of this dialectical situation. There may be no means of persuading [an opponent that something makes sense]. But that is the usual way with expressive impoverishment. It is hard to argue with a claim not to understand. In the heyday and aftermath of logical positivism, claiming not to understand was a standard philosophical tactic, employed with varying degrees of plausibility. These days it is rather less popular. (Williamson, 2010, p.711)

A strategy for breaking the apparent impasse now naturally suggests itself. Suppose the disputed statements can be consistently formulated and shown to enable fruitful theorizing. It would still be logically coherent to reject the statements as mere nonsense. But in light of the consistency of the statements, and assuming that the theorizing that they permit is sufficiently interesting, this rejection would be dogmatic and unjustified. An attempt to carry out the strategy was made in Linnebo (2006) §4, which provided some examples of claims that austere theorists cannot properly express (at least not in any explicit way).30 Let us make a fresh start.

5.6 Unification of universes Let us begin with the task of showing how we can consistently express claims that generalize across categories. The theoretical advantages or disadvantages of permitting such claims will later have to be assessed in light of our investigation as a whole. Our approach to the consistency problem rests on a simple idea, which we shall explain by means of a comparison. Imagine a community of extreme dualists whose language involves a strict type distinction between vocabulary for concrete objects and for abstract ones. Of course, this way of describing the situation would be unacceptable to the dualists themselves, who would deny that there are predicates ‘concrete’ and ‘abstract’ capable of being true of some objects and false of others. But 30 The three main examples were: (i) Every expression of every syntactic category has a semantic value which is unique, not just within a particular category but across all categories. (ii) The principle of compositionality holds for all expressions of all syntactic categories. That is, the semantic value of any complex expression is determined as a function of the semantic values of the expression’s constituent parts. (iii) There are infinitely many categories of semantic value. The article proceeded to show that these claims can be given a consistent expression and to argue that they enable fruitful semantic theorizing when thus expressed. There are other ways to develop the strategy as well, which we cannot explore here. Let us just mention one. What is common to all the predicates Ref n? Unless we can identify and explicitly describe enough of a common core, we will be left without a reason to regard all of the predicates as reference predicates.

84 ontological categories and the problem of expressibility our description of the situation relies on ordinary English, which does not respect the mentioned type distinction. How can we English speakers convince the dualists that it is philosophically legitimate, and of at least some theoretical interest, to abandon their type distinction and allow the expression of claims such as the following? No object is both concrete and abstract Every object is either concrete or abstract

Much like the austere type theorist in the debate about the Fregean categories, the dualists can dig in their heels and claim not even to understand our alternative proposal. In this way, they will also reject our claim that their language suffers an expressibility deficit as question-begging. The only way to argue against the dualists is to convince them that they are being dogmatic and unreasonable. The dualists should be willing to proceed—even if only tentatively—on the assumption that the controversial claims do make sense and then examine whether this is logically coherent and conducive to useful theory construction. Can the austere theorists be accused of being as dogmatic and unreasonable as the extreme dualists? The austere theorists will respond that the comparison is unfair because the type distinctions that they insist on are deeper and philosophically more important than those of the extreme dualists. We agree—but only up to a point. It is indeed because the Fregean logico-syntactic type distinctions are deep and important that the associated Fregean ontological categories are more fundamental than the distinction of objects into the concrete and the abstract. Even so, we believe there is a strong push towards full expressibility. Frege’s pioneering discussion and the ensuing philosophical debate leave little doubt that the controversial ontological statements appear to make sense. And as we shall see shortly, there are logically acceptable ways of extending a typed language which allow the controversial statements to be expressed in a consistent manner. Of course, adding the Reference Principle is another matter. This is the lesson of our paradox. We therefore suspend this principle until further notice and focus for now on the challenge of regaining expressibility. We shall return to the question of whether some modified version of the principle can and should be added. We now describe two new languages into which a typed language can be translated and in which we have what appears to be full expressibility. The first approach is nicely illustrated by considering the expressibility problem that confronts our extreme dualists. Their language, we recall, has two separate logical sorts reserved for abstract objects and for concrete objects. In technical parlance, this is a two-sorted first-order language. So let us denote it L2S . There is an intuitive and straightforward way to define a related language in which we appear to secure full expressibility. First, we lift all the restrictions associated with the two sorts. For instance, we now regard ‘7 is Roman’ and ‘Caesar is prime’ as well-formed (although false). Then, we add two predicates ‘C’ and ‘A’ for being concrete and abstract, respectively. We claim that the resulting language, which we call L*2S , achieves full expressibility. To see this, we observe first that it is straightforward to state the claims that the dualists were unable to express:

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¬∃x(Cx ∧ Ax) ∀x(Cx ∨ Ax)

Next, we need to show that everything that can be expressed in L2S can also be expressed in L*2S . To do this, we describe a translation * from L2S to L*2S . Each atomic formula is translated as itself. And the translation commutes with the logical connectives; for instance, φ ∧ ψ is translated simply as φ * ∧ ψ * . Only on quantified formulas does the translation do anything non-trivial. For instance, a formula ∀x φ, where ‘x’ is a variable of the sort reserved for concrete objects, is translated as ∀x(Cx → φ * ). In effect, a special universal quantifier which is implicitly restricted to concrete objects is translated by means of a genuinely universal quantifier which is explicitly restricted to such objects. Analogously, a formula ∀α φ, where α is a variable of the sort reserved for abstract objects, is translated as ∀α(Aα → φ * ). Clearly, what φ expresses in L2S is also expressed by φ * in L*2S .31 It can be shown that the extra expressive capacity of L*2S poses no danger with regard to consistency. Consider any L2S -theory T, and let T * be the L*2S -theory whose axioms are: 1. φ * for every axiom φ of T 2. For each L2S -predicate, an axiom stating that it is true only of the ‘right’ sort of thing, for instance ∀x(Prime(x) → Ax) and ∀x(Roman(x) → Cx). 3. Axioms stating that everything is either C or A but not both. We can now establish the following pleasing fact:32 Fact 2 T is consistent iff T * is. As Quine (1956) realized, essentially the same method can be applied to higherorder languages, with analogous results. We shall use as our paradigm example the language LSTT of the simple type theory; that is, the higher-order language with variables and constants for monadic n’th level concepts for any finite n (including an identity predicate for each type).33 The type indices may thus be taken to be simply the natural numbers. Again, our method begins by simply removing all the typing and the associated syntactic restrictions. We also enrich the language by the addition of two new predicates. We add two-place predicates ‘OC’ and ‘App’. Intuitively, ‘OC(x, c)’ states that x is of ontological category c, and ‘App(x, y)’, that an entity x of a certain type applies to another entity y of lower type. Let L*STT be the resulting language. Next, we define a translation * from LSTT into L*STT . To avoid translating distinct typed expressions as one and the same non-typed expression, all type-indicating superscripts are converted to subscripts on the corresponding untyped expression. Thus, the variables ‘x1 ’ and ‘x2 ’ are translated as ‘x1 ’ and ‘x2 ’, rather than both as ‘x’, which would result in variable clashes. The translation is defined by recursion on

31 This claim can be further substantiated by considering models of the two languages and then demonstrating that φ is true in M iff φ * is true in an associated model M* . 32 Proof as for Enderton (1972/2001), Theorem 44A, p.300. 33 Readers should have no problem generalizing our account in what follows to other examples.

86 ontological categories and the problem of expressibility syntactic complexity. Its only non-trivial clauses state that a predication of the form en+1 (en ) is sent to App(en+1 , en ) ∧ OC(en+1 , n + 1) ∧ OC(en , n), and that a quantified formula ∀xn φ is sent to ∀xn (OC(xn , n) → φ * ). Things now proceed exactly as in our previous example. It is now straightforward to express the previously problematic ontological insights; for instance, that every entity belongs to a unique ontological category: (OC-UNIQUE)

∀x∃!c OC(x, c)

There is also an analogue of Fact 2, which shows that there is nothing to fear with regard to consistency.34 The problem with this approach is rather that it is heavy-handed. It is true that we gain greater expressibility in a way that provably introduces no new threat of inconsistency. However, Fregeans will object that this expressibility is gained at the cost of obliterating the logico-syntactic type distinctions whose importance they want to be able to express and investigate. Fortunately, there is a second option for gaining expressibility which does less violence to the grammar of our original typed language. Instead of trading the original language for a new untyped language, we retain it as a proper part of a new and expanded language. This second option is motivated by the phenomenon of nominalization, which is available in English and many other natural languages. Consider a predicate such as ‘. . . is wise’. This is of syntactic category 1. But nominalization provides an associated singular term, namely ‘the property of being wise’ or ‘wisdom’. Moreover, an ordinary predication (such as ‘Socrates is wise’) can be rewritten as a claim about application of properties (such as ‘The property of being wise applies to Socrates’). Again, we shall focus on our paradigm example, the language LSTT of simple type theory. The central idea is that any entity of any ontological category n can be referred to not only by expressions of type n but also by expressions of type 0. In this way, singular terms and variables take on the special role that they, and they alone, can refer to entities of any ontological category.35 In order to implement this idea, we need a way to indicate that an expression of type 0 corefers with an expression of some other type n. We do this by adding nominalization operators, which can be applied to any variable of any type to form an expression of type 0.36 The idea is that a higher-order variable, say, xn  and its (first-order) nominalization ν(xn ) have the same value

34 This may seem too good to be true. Are not type distinctions essential in order to shield us from paradoxes? In fact, as we shall see in the next section, the shield provided by the type distinctions survives their demise from the syntax. 35 Our view thus shares with those of Wright (1998b) and Liebesman (2015) a commitment to the thesis that ‘. . . is a horse’ and ‘the property of being a horse’ have the same semantic value. As we shall see, however, the views differ in other ways. 36 Strictly speaking, there is a separate nominalization operator ν n  for each type n except 0, where there is of course no need to nominalize. In what follows, however, we shall mostly suppress the type superscript on the operator ‘ν’ and sometimes even make generalizations where the operator applies to expressions of type 0, e.g. v(e0 ), with the tacit understanding that this is just longhand for e0 itself. For a similar move, see Price (2016).

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(or reference, in the extended sense of the term, which we are still using). As before, we also add predicates ‘OC’ and ‘App’, each with two argument places of type 0, for ontological category and application, respectively. (Eventually we shall also add the reference predicate ‘Ref ’.) We call the resulting language Lν . It is important to notice that the nominalization operator ‘ν’ applies only to variables, not to predicates or open formulas. This restriction is imposed in order to remain neutral on a question which will occupy us shortly, namely whether all open formulas of the extended language define Fregean concepts.37 If an open formula doesn’t define a concept, a fortiori there is no entity that can be accessed by means of a singular term, and thus no room for nominalization. The technical upshot is just as pleasing as before. We can formulate plausible principles governing the new vocabulary. Most importantly, we have a generalization of the equivalence of ‘Bucephalus is a horse’ with ‘The property of being a horse applies to Bucephalus’, namely: (Bridge-App)

App(ν(en+1 ), ν(f n )) ↔ en+1 (f n )

We can also state and prove an analogue of Fact 2, which shows that on this approach too there is no danger of inconsistency. (See Appendix B for a precise statement.) Summing up, we have shown how expressibility can consistently be regained—at least with regard to the kinds of entities over which variables of the original typed language L range over. We first considered a rather brutal option, and then a less brutal one inspired by the phenomenon of nominalization. In what follows, we focus on the less brutal alternative, which has the advantage of preserving the type distinctions that are obliterated by the brutal alternative.

5.7 A dilemma concerning the stability of our view It is one thing for full expressibility consistently to be regained. It is quite another thing for the resulting view to be stable and theoretically attractive. Let us examine the latter question. The greatest threat to the stability of our view comes from the question of whether the new predicates of our nominalization language Lν refer and which open formulas of this extended language should be allowed to define properties. Let us begin with the question of predicate reference. Consider the predicates ‘OC’ and ‘App’, as well as ‘=’ in its extended use (in which its arguments may include ν-terms referring to functions rather than objects). Do these expressions refer? If so, to what do they refer? These questions pose a tricky dilemma. On the one hand, we have been working on the Fregean assumption that not only the singular terms, but also the various types of incomplete expression have reference. We thus assumed that the predicates and functional expressions of our unextended

37 The restriction may seem needlessly severe. For we have admitted that the predicates of the unextended language refer, and most of these predicates (unlike the identity predicate) do not receive an extended meaning in the extended language. There is no problem, however, as the reference of each such predicate can also figure as the value of a variable, to which the operator ‘ν’ can then be applied.

88 ontological categories and the problem of expressibility language L refer to functions of appropriate type and level.38 There is thus substantial pressure to ascribe reference to all meaningful predicates, including the ones just mentioned from the extended language. Indeed, as explained in Section 5.4, any such predicate defines a function from objects to truth-values, namely the function that maps an object to the True if the predicate is true of the object, and to the False otherwise. (The same goes, mutatis mutandis, for polyadic predicates.) And on our minimalist conception of predicate reference, all that it takes for a predicate to refer is that it is associated with such a function from objects to truth-values. On the other hand, if we ascribe reference to the mentioned predicates, we face some troubling problems. For one thing, it becomes hard to see how anything like the Reference Principle can be re-introduced. It follows from standard Fregean assumptions that many of these predicates cannot refer to anything in the original hierarchy of Fregean categories associated with L. Assume, for instance, that the identity predicate, as it figures in Lν , refers. Then its referent would apply not only to objects but also to concepts of any finite level. So the referent would have to be located even higher up, above all of the finite levels where all the entities that are referentially accessible from L are to be found.39 What are these new entities? In particular, what is their ontological category? An even more serious threat to the stability of our view arises when we consider the question of which open formulas of our extended language Lν should be allowed to define properties. This is a question of which so-called comprehension axioms we should accept, that is, axioms such as   (Comp1 ) ∃x1 ∀y0 x1 (y0 ) ↔ ϕ(y0 ) which states that the open formula ϕ(y0 ) defines a first-level property x1 . There is substantial pressure to accept every such comprehension axiom. For if the open formula ϕ(y0 ) is so much as meaningful, it defines a function that takes an object to the True if the object satisfies the open formula, and to the False if not. And according to our minimalist conception, to say that the formula defines a property is just to say that there is such a function. Consider now the open formula ‘¬App(x, x)’, which is available in our extended language, Lν . Since the formula is meaningful, by our minimalism about properties, it should be permissible to use it in a comprehension axiom to define a first-level monadic concept F, thus ensuring (5)

∀x(Fx ↔ ¬App(x, x))

(Type indices are here suppressed for readability.) Now let r = ν(F) and ask whether Fr. By universal instantiation of (5), we obtain Fr ↔ ¬App(r, r). But by (Bridge-App),

38 This ensures that there is no special problem over the reference of ν-terms of our extended language; each such term makes singular reference to whatever function is the referent of its embedded predicate or other functional expression. 39 The argument relies on two fairly standard Fregean principles. First there is the principle (stated in Appendix B as (6)) which ensures that when an entity x applies to another entity y, then the ontological category of x must be higher than that of y. Then we need the already mentioned principle (OC-UNIQUE) to the effect that every entity belongs to a unique ontological category.

5.8 prospects for the new reference option 89 we have Fr ↔ App(ν(F), r).40 Leibniz’s Law therefore ensures Fr ↔ App(r, r), which yields App(r, r) ↔ ¬App(r, r) and hence a contradiction.41 In short, when reflecting on the predicates of our extended language, we face a dilemma with two problematic horns. One option is to ascribe reference, in normal Fregean fashion, to all the predicates of our extended language Lν and also to accept comprehension axioms for all of its (meaningful) open formulas. We then face some hard problems, including a threat of paradox. Another option is to deny reference to certain predicates of the extended language, and more generally, to deny comprehension on some of its open formulas. We then avoid all of the problems associated with the other horn. Instead we face the daunting task of explaining how the predicates in question can be exempted from the general Fregean doctrine that meaningful predicates refer. Moreover, for each comprehension axiom that we deny, we would need to explain why the open formula in question cannot be used to define a function from objects to truth-values despite being available in our language, which we presumably take to be meaningful. In slogan form, the options are thus new reference or no reference. Each option will be scrutinized in later sections, and we shall find that things are not nearly as bleak as they currently seem.

5.8 Prospects for the new reference option The most urgent threat to the new reference option is no doubt the paradox. Let us therefore begin by examining how the paradox might be avoided. One response is to qualify our claim that the unification of universes ensures full expressibility, once and for all. It is true that the first-order variables of Lν range over all entities to which the original language L can refer. But this does not mean that these variables range over everything to which the extended language Lν can refer. Yet further entities might come into view in the extended language Lν , because of its new predicates and open formulas. These further entities need not be in the range of the first-order variables of Lν —although they will be in the range of the first-order variables of the doubly extended language that would result from carrying out another unification of the universes, this time applied to Lν . This response certainly blocks the paradox. To verify this, observe that the Fregean concept F from the paradoxical argument might be one of the new entities that come into view in the extended language Lν . (After all, F is defined by an instance of comprehension involving the new predicate ‘App’, which is not guaranteed a reference in the old domain.) If so, there would be no guarantee that r = ν(F) will be in the range of the quantifier ‘∀x’ from (5), since this quantifier is only guaranteed to range over entities to which the original language L can refer. This would mean it is impermissible to instantiate the universal quantifier ‘∀x’ in (5) with respect to r, which might lie outside of the range of this quantifier. (Indeed, on pain of paradox, r must lie outside this range.) This would block the paradox. 40 To see that this follows from (Bridge-App), it is useful to recall the notational convention from footnote 36. 41 This argument is a version of Grelling’s paradox. As a referee points out, another option is a version of Russell’s paradox, based on the open formula ‘¬∃F(x = ν(F) ∧ Fx)’. See also Jones (2016, fn.22) for a closely related argument.

90 ontological categories and the problem of expressibility Although logically effective, the response just outlined has a serious philosophical drawback. Given the importance that we attach to overcoming the expressibility problem, any abrogation of full expressibility would come as a disappointment. Indeed, the partial but incomplete expressibility that would be ensured by Lν , on the approach in question, can even be matched by the austere type theorists, at least on certain ways of developing type theory.42 To be worthwhile, our attempt to ensure expressibility must achieve more than that. Let us therefore look for a better response, this time holding fixed the requirement that the first-order variables of Lν be truly universal—in the sense that they range over any entity to which reference may be made, either in this language or in any extension of it. Given this requirement, how might the paradox be blocked? The only plausible answer, we believe, is to challenge the comprehension axiom on which one of the crucial equations depends, namely: (5)

∀x(Fx ↔ ¬App(x, x))

That is, to claim that, although ‘¬App(x, x)’ is a perfectly good and meaningful open formula of the language Lν , it nevertheless fails to define a Fregean concept. For this answer to be credible, it is obviously not enough simply to deny this comprehension axioms and selected others. We need to be told why these comprehension axioms, which look to be in good standing, are nevertheless unacceptable. The hardest part of this challenge is arguably to explain how and why the problematic comprehension axioms might fail to be in good order, given the minimalist conception of higher-order reference that we endorsed in Section 5.4. According to this minimalism, any meaningful open formula defines a function from objects into truth-values, namely the function that maps some objects (in some order) to the truthvalue that this formula has when these objects (in that order) are assigned to its free variables. It is hard to see how there could fail to be such a function—and therefore also how the relevant comprehension axiom, which merely asserts the existence of such a function, could fail to be true. On a closer consideration, however, we see that the minimalist argument just rehearsed presupposes that a first-order domain has been determined and that the relevant language has been interpreted on this domain. Given such a domain and interpretation, it is indeed hard to see how the mentioned functions could fail to exist. But our situation is different. To ensure full expressibility, we are requiring that any function to which reference might later be made—including by any expression of an extended language—should already belong to the first-order domain. Can we determine this sort of domain? Until we have successfully done so, there is no reason to expect every meaningful open formula to determine a function. But simultaneously, until the existence and behaviour of the function have been determined, the domain will not have been completely determined. In short, we are trying simultaneously to

42 We have in mind the admission of cumulative limit types, say a type ω, whose expressions can meaningfully be applied to expressions of any finite type. (See Linnebo and Rayo (2012) and Williamson (2013, section 5.7).) The expressibility achieved by first-order variables, on the approach under discussion, is matched by the expressibility achieved by the austere type theorists’ variables of type ω.

5.8 prospects for the new reference option 91 determine a domain and functions on the domain, and it is not obvious how this can be done in a non-circular manner.43 A long and rich history of grappling with the logical paradoxes suggests a line of attack. In the kind of situation we have described, it is useful to try to make sense of the domain and the entities that populate it in a well-ordered series of stages. At stage 0, we assume nothing about the domain. Even so, we are able to make sense of at least some functions, whose behaviour on the domain is nevertheless determined. For example, the open formula ‘x = x’ defines a function f that sends any object in the domain—whatever exactly it might turn out to contain—to the True. At stage 1, it has thus been determined that the domain contains certain functions, including f. Having established this much, we are now able to make sense of more functions than we could previously make sense of. For example, the function g that sends f to the False and everything else to the True is now determined, irrespective of what, exactly, the domain might eventually turn out to include. More generally, assume that at some stage it has been determined that the domain includes a certain range of objects and functions. When trying to determine further functions, we are always allowed to draw on the information that has been determined about what entities populate the domain and how these entities are constituted. This enables us to determine the existence and behaviour of yet more functions, which henceforth can safely be assumed to belong to the first-order domain. We now iterate this procedure. At limit stages, we pool all the information that has been established at any of the preceding stages. Of course, the line of attack just outlined needs to be spelled out in proper technical detail. Fortunately, several ways to do so have been developed and proven to be consistent (relative to orthodox ZFC set theory or often far less). One option is a relatively traditional predicativist approach, where at any stage further functions can be determined—provided that this determination only quantifies over entities determined to be in the domain by the relevant stage. Some broadly similar, but more liberal, options have been developed by Fine (2005a) and Linnebo (2006). The last problem to be discussed concerns what ontological category should be assigned to the ‘new’ entities that are not in the original Fregean hierarchy. On the family of approaches just outlined, the traditional Fregean categories are just the tip of an enormous transfinite iceberg of further categories whose relations to one another differ in important ways from those of the traditional categories. The most striking difference is a relaxation of the Fregean requirement that each category be applicable only to entities from a single other category. Thus, in the example we sketched, at stage 1 the domain has been determined to contain a function defined by the open formula ‘x = x’. This function sends all objects and all higher-level entities of the Fregean hierarchy to the True. Another permissible mixed-level open formula is ‘x = Bucephalus ∨ x = ν(. . .is a horse)’, whose referent applies to one ordinary object and one first-level concept. This relaxation of the Fregean strictures

43 We leave it open, for present purposes, whether this talk about ‘determination’ should be understood in an epistemological or more metaphysical manner. On the former understanding, to determine is to establish or come to know. On the latter, it is a matter of objective dependencies among truths: one truth is determined by some others just in case the former is grounded, in some sense, in the latter. See Fine (2005a) and Linnebo (2006).

92 ontological categories and the problem of expressibility is the price we pay if we wish to allow new predicates to refer—and new open formulas to define functions—subject only to the paradox-blocking requirement that the resulting functions be determined in a stagewise manner. The cumulative nature of the functions to which this relaxation gives rise means that the resulting categories—if ‘category’ even remains an appropriate word—will be utterly different from the original Fregean ones. A closely related question is what to say about Frege’s Reference Principle. One thing that is clear is that the ontological category of an entity can no longer be read off from the syntactic type of any expression that denotes it. We would additionally have to take into account the sequence of nominalizations that has been undertaken in order to endow this expression with its present meaning. It is debatable whether this should be seen as refining the Reference Principle or simply as changing it beyond recognition. To sum up, we possess the philosophical and mathematical tools to carry out various versions of the new reference option. But doing so results in a system of entities that are no longer neatly organized as in the original Fregean hierarchy of ontological categories. This leaves little or no role for a Fregean Reference Principle. In short, while the resulting view may have much to recommend itself, it appears to involve a major departure from Frege.

5.9 No reference reconsidered As we saw, the option of denying reference to the new predicates of Lν —and more generally, of restricting comprehension on Lν -formulas—effectively and straightforwardly blocks the paradoxical argument. But is this an option a Fregean can take seriously? Clearly, there is no point in the introduction of the new predicates (namely, ‘Ref ’, ‘OC’, ‘App’, as well as ‘=’ in its extended use) unless it enables us to express truths about the various types of expression belonging to our unextended language and the various types of entity which are their semantic values. So it must be accepted that the new predicates are meaningful. But as we have seen, it may be claimed that any meaningful open formula defines a function from objects to truth-values, and that to ascribe reference to a predicate is just to say that it is associated with such a function. Thus it may seem that, unless this minimal conception of reference is rejected, there is simply no room for the no reference position. In this section, we shall argue that the situation is rather more complicated than has just been suggested, and that when the complications are recognized, there remains after all space for a version of the no reference view.

5.9.1 Clarifying the view A proponent of no reference must agree that the new predicates are meaningful, and that they can figure in true or false statements. But he can reply that the simple argument just rehearsed is not decisive. Properly understood, he may claim, his view is not that the new predicates do not refer at all, but that they do not refer to any entities ‘lying outside’ the original hierarchy—his view is accordingly to be understood as the strict contradictory of new reference rather than as its contrary; not as no reference but as no new reference. Whether no new reference means no reference at all, and if so,

5.9 no reference reconsidered 93 whether that conflicts head-on with a reasonable form of minimalism will be leading questions in what follows. Let us begin by reviewing the relevant languages. We are assuming L to be a strictly typed language of some finite order n. For simplicity, we are considering only monadic predicates and predicate variables. The vocabulary includes singular terms, together with predicates of first- and higher-levels up to n, and the usual sentential operators. Among the predicates of second- and higher-level are quantifers, each binding variables of the appropriate type—i.e. term-variables, first-level predicate variables, etc. We may also assume that L includes the requisite vocabulary for describing its own syntax, including names (e.g. quotation names) of its expressions and a general syntactic category predicate ‘SC’. Since L-expressions are a species of objects, as are the syntactic categories, this predicate is first-level. Lν is the extension of L obtained by the addition of nominalization operators 1 ‘ν ’, ‘ν 2 ’, . . .which apply to predicates of various level to form new singular terms; the application predicate ‘App’; the reference predicate ‘Ref ’; the ontological category predicate ‘OC’; and an extended identity predicate ‘= ’ which can be flanked by any singular terms involving nominalization operators (as well as first-order variables).44 As we have seen, this language enables us to express the kinds of claims Frege endorsed. For example, ‘no object is a concept’ and ‘first-level concepts apply to objects’ are expressed as respectively ¬∃x(OC(x, 0) ∧ OC(x, 1)) ∀x∀y(OC(x, 1) ∧ App(x, y) → OC(y, 0)) Since the new predicates available in Lν can be used to form meaningful statements, they must themselves be meaningful. On a strongly minimalist conception of predicate reference, this suffices to ensure that they have reference. But according to the no new reference view, there are no new entities—that is, entities which do not already figure as the semantic values of L-expressions, including its variables—to which they refer. This is not, so far, incompatible with minimalism—as we have seen, no new reference does not necessarily mean no reference at all. So our first question must be: (1) To which, if any, of the distinctive predicates of Lν does no new reference deny reference altogether? If the answer is ‘some’, then our next question will be: (2) How, if at all, can denying reference to . . . be reconciled with a reasonable form of minimalism about predicate reference? where ‘. . .’ is a list of Lν predicates which lack reference, according to no new reference.

5.9.2 To which predicates does no new reference deny reference altogether? As we have seen, ‘OC’ is definable in terms of ‘Ref ’ and ‘SC’ because OC(x, n) ↔ ∃e(SC(e, n) ∧ Ref (e, x))

44 Recall from Section 5.3 that we chose to understand reference in a wide sense, which includes the relation between a variable and its value. This allows us to work with a non-modal language.

94 ontological categories and the problem of expressibility And ‘SC’ can be assumed to be an L-predicate, and one which as such does have reference. Thus the question whether ‘OC’ refers reduces to the corresponding question about ‘Ref ’. So let us consider ‘Ref ’. From a syntactical point of view, this predicate is, like ‘. . . refers to—’ in ordinary English, a first-level predicate—that is its argument-places accept only singular terms (or variables of the corresponding type). But considered semantically, ‘Ref ’ cannot be assigned to any level; rather, it is a cross-categorial reference predicate—its second argument-place may be filled by a ν -term which refers to an entity in any of the ontological categories which supply the semantic values of L-expressions. It is, therefore, impossible that ‘Ref ’ should simply refer to any single ‘old’ entity. But this does not mean that, if it is not to lack reference entirely, it must refer to some ‘new’ entity, ‘outside’ the hierarchy comprising the semantic values of L-expressions (hereafter, the L-hierarchy). For there remains the possibility that it refers ambiguously to various distinct ‘old’ entities, all of them belonging to that hierarchy. To be a little more precise: while ‘Ref ’ itself does not belong to L, this language may be assumed to contain a series of type-restricted reference predicates ‘Ref 1 (ξ , ζ 0 )’, ‘Ref 2 (ξ , ζ 1 )’, . . ., Ref n+1 (ξ , ζ n ), . . ., where the superscript on ‘Ref ’ indicates that it is a first-, second-, or higher-level predicate, and that on ‘ζ ’ indicates the level of expression required by that argument-place. Thus an instance of ‘Ref 2 (ξ , ζ 1 )’ might be ‘Ref 2 (‘breathes’, breathes)’, so that what is said is (as we may put it in a nominalization language such as Lν , but not in a strictly typed language such as L) to the effect that ‘breathes’ refers to the property of breathing (or being a breather). We may understand ‘Ref (e, ν(f ))’ as schematic—so that depending upon the level of the expression ‘f ’ from which ‘ν’ forms a singular term, ‘Ref (e, ν(f ))’ is to be understood as Ref n+1 (e, f n ) for the appropriate value of ‘n’. In fact, we can take ‘Ref (e, x)’ as defined by the following disjunctive formula: Ref 0 (e, x) ∨ ∃x1 (x = ν(x1 ) ∧ Ref 1 (e, x1 )) ∨ . . . ∨ ∃xn (x = ν(xn ) ∧ Ref n (e, xn )) ∨ . . . Thus, while ‘Ref ’ is not assigned a new reference, no new reference does not deny it reference altogether. The defined predicate inherits its reference from the family of type-restricted reference predicates in terms of which it can be defined. The predicate thus refers ambiguously to a family of partial functions, each defined on arguments from a single category only. If, per impossibile, these partial functions could be ‘stitched together’, this would yield the total function on the entire Fregean hierarchy which is the new reference theorist’s desired interpretation of the new predicate ‘Ref ’. But according to the no new reference theorist, the family of partial functions suffice to interpret this predicate. So there is no need to ‘stitch’ the partial functions together to a single, new referent. Next, consider ‘= ’. Much of what has been said about ‘Ref ’ applies, mutatis mutandis, to the identity predicate in its extended use. Syntactically, ‘= ’ is, like its counterpart in ordinary English, a first-level predicate. Semantically, however, it cannot be assigned to any one definite level—its argument-places may be filled by any ν-terms, denoting entities belong to any level in the L-hierarchy. Hence ‘= ’ cannot simply refer to any single relation in that hierarchy. But as with ‘Ref ’, this does not mean that, if it is not to lack reference entirely, it must refer to some ‘new’ entity, ‘outside’ the L-hierarchy. For L may perfectly well contain a series of type-restricted

5.9 no reference reconsidered 95 identity predicates ‘ξ =1 ζ ’, ‘ξ =2 ζ ’, . . ., admitting only arguments of levels 1, 2, . . . , respectively. Thus in contrast with English as she is, ‘is obnubilated =2 is hidden by cloud’ would be a well-formed sentence, asserting (as we may say in ordinary English, but not in any strictly typed language) the identity of property for which the first predicate stands with the property for which the second stands. Simple Lν statements deploying ‘= ’ may then, with a modest complication, be understood as schematic in much the same way as simple Ref -statements—thus ν(F i ) = ν(Gj ) will be true iff F i =i+1 Gj  is. The complication concerns the case where i = j . In this case, F i =i+1 Gj  will be ill-formed, since =i+1 requires both its arguments to be of level i. We are free to stipulate that when i = j, ν(F i ) = ν(Gj ) is false—and it is reasonable to do so, given that there can be no corresponding statement F i =i+1 Gj  and hence no true such statement. Finally, we come to ‘App’. Like the extended identity predicate, ‘App(ξ , ζ )’ accepts ν-terms in both argument-places, and so—like ‘=’—carries with it the possibility of well-formed Lν -statements, ‘App(ν(e), ν(f ))’, which harbour concealed typemismatches. Thus by (Bridge-App), ‘App(ν(e), ν(f ))’ will be true iff ‘e(f )’ is true. But the latter cannot be true unless for some k, e and f are entities of levels k + 1 and k, respectively. Thus if e is of level j and f of level k for j = k + 1, there will be no well-formed L-statement ‘e(f )’ corresponding to ‘App(ν(e), ν(f ))’.45 Evidently this complicates the evaluation of formulas involving ‘App’ in much the same way as we have seen in the case of the identity predicate. Since the components of a compound statement must be well-formed if the compound as a whole is to be so, there will be no well-formed instances of (Bridge-App) when ‘e( f )’ is ill-formed (as it will be when e’s level is not greater by 1 than f ’s). Thus in these cases, (Bridge-App) cannot be used to evaluate App(ν(e)ν(f ))). Just as with the identity predicate, we are free to stipulate that when j = k + 1, App(ν(ej ), ν(f k )) is false—and it is reasonable to do so, given that there can be no corresponding statement ej (f k ) and hence no true such statement. Otherwise, App(ν(ej ), ν(f k )) may be evaluated by applying (Bridge-App). Thus far, our treatment of ‘App’ runs parallel to our treatment of ‘= ’ But the parallel breaks down at a crucial point. In contrast with ‘= ’ (and ‘Ref ’), where there are type-restricted L-predicates (‘ξ =1 ζ ’, ‘ξ =2 ζ ’,. . .) which these cross categorial Lν -predicates may be interpreted as schematically representing, there are no type45 Thus consider the perfectly well-formed Lν -statement App(ν(F12 ), ν(G21 )). By (Bridge-App), we have App(ν(F12 ), ν(G21 )) ↔ F12 (G21 ). But the right-hand component of this biconditional is simply ill-formed, since the L-predicate F12 (ξ ) accepts only expressions of level 1 in its argument-place. Concrete examples of this kind of type mismatch in English would be the non-sentences ‘everything nothing’ and ‘are rare are extinct’—‘everything’ and ‘nothing’ (and likewise ‘are rare’ and ‘are extinct’) are really second-level predicates whose argument-places must be filled by first-level predicates if well-formed sentences are to result. Matters are complicated in English by the facts (i) that English requires the argument-place in the predicate ‘. . . are rare’ to be occupied by a common noun or noun-phrase, as in ‘Golden eagles are rare’, rather than by a first-level predicate (such as ‘. . . is a golden eagle’) in the logicians’ sense, and (ii) that it does not syntactically mark the distinction between first and second levels, so that while ‘Guy is on the verge of extinction’—in contrast with ‘Guy is a gorilla’ and ‘Gorillas are on the verge of extinction’—involves a type violation, it is perfectly grammatical. The point in the text is perhaps better illustrated by App(ν(F12 ), ν(t 0 )), a concrete English example being ‘Everything Saturn’—where, to obtain a well-formed sentence from the second-level ‘Everything. . .’, we must fill its gap by a first-level predicate.

96 ontological categories and the problem of expressibility restricted counterparts of ‘App’, in terms of which simple ‘App’-statements can be understood in a way that parallels our interpretation of simple identity-statements and ‘Ref ’-statements. It is true, of course, that L could be taken to include a series of typerestricted predicates ‘App2 (ξ 1 , ζ 0 )’, ‘App3 (ξ 2 , ζ 1 )’, . . . . But simple statements formed from these predicates would have to be evaluated in accordance with (Bridge-App). Its application reduces any type-restricted ‘App’-statement to an equivalent statement not containing Appn+2  but otherwise composed of the same ingredient expressions. It thereby explains how our type-restricted ‘App’-statements are to be understood without treating Appn+2  as making reference to a distinctive relation of level n + 2. Thus, in contrast with ‘OC’, ‘Ref ’, and ‘=’, no new reference does not treat ‘App’ as having reference, but not reference to any new entity outside the L-hierarchy—rather, it denies it reference altogether. Thus in this case, we really do have a head-on conflict with minimalism about predicate reference in its uncompromising form.

5.9.3 Moderate minimalism Is there a reasonable, less uncompromising form of minimalism to which we might retreat to resolve the conflict? We shall not try to finally resolve this question, but will rehearse two arguments which might support an affirmative answer, one from deflationism about truth, the other based on truthmaker semantics. The argument from deflationism about truth On a strict deflationary account of truth, the truth-predicate ‘ξ is true’ does not refer to a special property—a property which all true propositions possess and all other propositions lack. There is no property to which reference is made in the sentence ‘The proposition that grass is green is true’ over and above the properties to which reference is made in ‘Grass is green’. But ‘ξ is true’ is a perfectly good, meaningful predicate. Hence if strong, uncompromising minimalism is correct, it does refer to something—the property, as it would be described, of being true. So there is a head-on clash between the strict deflationary account of truth and strong minimalism about predicate reference. One of them has to go. Arguably, we should reject strong minimalism in favour of a qualified minimalism which exempts the truth-predicate from its scope. But if so, then presumably that qualified minimalism should also be understood as not requiring that our application-predicate ‘App(ξ j , ζ k )’ has reference. For just as by the deflationist principle [p] is true ↔ p, so by (Bridge-App), App([ξ j ], ν k (ζ k )) ↔ ξ j (ζ k ). One might view the predicate ‘App’ as a kind of two-place counterpart of the oneplace truth-predicate, and so as no more standing for a special relation than the latter stands for a special property. The argument from truthmaking According to truthmaker semantics, statements are made true or false by states. Although much of truthmaker semantics can be developed without making any assumptions about these states s1 , s2 , . . . beyond assuming that they collectively form a space of states S ordered by a part–whole relation

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(‘s1 s2 ’ read ‘s1 is part of s2 ’), it is natural and plausible to view them as states of affairs composed of objects, properties, relations, etc.46 Thus the state which makes true the statement Socrates is wise would have as its constituents the individual Socrates and the property of being wise. If we suppose the language for which a truthmaking semantics is being given to include a truth-predicate, we would expect that it would satisfy the Equivalence Principle that for any statement A, A ↔ True[A]. And we should surely expect that whatever state makes true (or false) the statement that A would make true (or false) the statement that True[A], and that accordingly, there would be no constituent of the state that makes true the statement that True[A] which is not a constituent of the state that makes true the statement that A, and hence that there is no special property for which the predicate True stands. But we can suppose that the language contains not only a truth predicate, but a series of application predicates App, governed by parallel equivalence principles of the form Appn+2 (ξ n+1 , ζ n ) ↔ ξ n+1 (ζ n ), and related to the truth-predicate by True[Appn+2 (ξ n+1 , ζ n )] ↔ Appn+2 (ξ n+1 , ζ n ), so that whatever state makes the statement that Appn+2 (ξ n+1 , ζ n ) true (or false) would be the very state which makes the statement that ξ n+1 (ζ n ) true (or false), and so involve no constituent over and above the constituents ξ n+1 and ζ n . If we accept that a reasonable form of truthmaker semantics could be combined with minimalism, we should agree that this will be a qualified version of minimalism, which does not call for a property of truth and relations of application holding between properties and properties of lower-level or objects. In sum, while the conflict between no new reference and minimalism is considerably less extensive than might at first be supposed, there remains a clash with minimalism in its uncompromising form. That residual conflict might be resolved, if the arguments just rehearsed (of some others) may be taken to justify a retreat to a more qualified form of minimalism.

5.10 Revising the Reference Principle It remains to investigate how, on the no new reference option, the Reference Principle might be reintroduced. As our paradox shows, some revision will be required, as full expressibility is now in place. It should be clear that, once our language is extended to allow singular reference, by way of ν-terms, to entities of any of the ontological categories corresponding to incomplete expressions, no purely syntactic restriction will achieve the requisite restriction. A more discerning approach is needed. What is needed is a restriction which cuts across syntactic categories, exempting from the principle’s scope all and only those expressions belonging exclusively to the extension of our original language, whether they be new singular terms, or new predicates or functional expressions. We could, of course, simply stipulate that the Reference Principle is not to apply to nominalized expressions of the form ν(e). But in the absence of further explanation, such a stipulation is apt to appear as a merely ad hoc device for avoiding paradox. It does, however, admit of a perfectly sound motivation. 46 For a very clear account of truh-maker semantics, which emphasizes how much can be done without delving into the innards of states, see Fine (2017).

98 ontological categories and the problem of expressibility We have, thus far, simply taken for granted the idea that ν-terms, along with the various new predicates, may be introduced to facilitate singular reference to, and firstorder talk about, entities lying outside the category of objects, and paid no attention to how exactly that might be done. Nothing of importance will be lost by focusing on the first and simplest case—singular reference to first-level properties. How is the intended use of terms of the form ν(F), where F is a first-level predicate of the original language, to be explained? To suppose that we may simply introduce such terms by saying that ν(F) is to refer to the property of being F is to assume that we already possess the means of making singular reference to properties in the metalanguage, and so fails to engage with the real issue, which is how we may introduce singular reference to properties in the first place—from a standing start, as it were. To put the point another way, the question concerns how we are to explain the function for which we intend ν to stand, and in particular, what its range of values is to be. One way—indeed, perhaps the only way—to do this is by means of a form of abstraction. That is, we might introduce an operator on first-level predicates which forms a complex name for the property for which that predicate stands by means of a principle of the form: ν(F) = ν(G) ↔ Eq(F, G) where Eq(F, G) expresses a suitable second-level equivalence relation on first-level properties. Eq(F, G) will ensure at a minimum that the predicates F and G are co-extensive, but will presumably require a stronger connection. Let us not worry about what that might be just now. The obvious point is that, assuming ν-terms may be introduced in this way, they are obviously parasitic upon reference to the relevant properties by means of predicates. In other words, of our two modes of reference to first-level properties, one is clearly basic and the other derivative. It is equally clear that, whatever the details of the means of introducing singular terms for entities of other unsaturated types, the point will generalize. That is, no matter what specific syntactic category an unsaturated expression ξ may belong to, ν(ξ ) will refer only derivatively to whatever entity is the referent, in a basic sense, of ξ itself. This gives us an obvious and natural motivation for exempting ν-terms from the scope of our reference principle—for it is only basic modes of reference which may be expected to correlate reliably with ontological categories. So far, so good. But we have, as yet, no clear motivation for exempting the new predicates, ‘App’, ‘OC’, etc., from the scope of the Reference Principle. Can the kind of motivation we have just explained be somehow extended to them? In sharp contrast with ν-terms, these predicates cannot have reference to any single entity within the old hierarchy, as their referents will apply to entities at arbitrarily high levels of this hierarchy. So there can be no question of maintaining that these predicates refer in a derivative manner to entities to which basic reference is effected by means of L-expressions. This does not, however, mean that the basic-derivative distinction has no application at all here. Any temptation to conclude that the new predicates enjoy basic reference (if they refer at all) can, and should, be resisted. For there is a clear sense in which their introduction and use is dependent upon more basic uses of other expressions, even if the dependence is less direct than in the case of ν-terms. The dependence is at its simplest and clearest with the application predicate in its most fundamental uses, such as ‘App(ν(F), b)’. This might say something like: ‘The property of being wise applies to Socrates’. Our basic grasp of the truth-condition

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of this sentence relies upon its equivalence with the corresponding sentence of the form ‘F(b)’—‘Socrates is wise’, say. The point is not that the application predicate can always be eliminated in favour of ordinary predication. It can’t. It is needed for more theoretical generalizations—‘First-level properties apply to objects’, ‘Secondlevel functions take first-level functions as arguments’, and the like—which cannot be reduced to statements of the unextended language. The point, rather, is that our grasp of this predicate is anchored in ordinary predications. But there is an obvious point of wider application, viz. that while there may be no more basic way to refer to whatever, if anything, this and the other new predicates stand for, their introduction presupposes, and is clearly parasitic upon, the extension of the class of singular terms to include ν-terms for unsaturated entities. This gives us a motivation for exempting those predicates, along with ν-terms, from the scope of the Reference Principle—for even if they do refer, they do not enjoy basic reference, but refer only in a way that is dependent upon the derivative kind of reference effected by ν-terms. In sum, we can, and should, restrict the Reference Principle so that not only ν-terms, but also the new predicates whose introduction is dependent upon that of those terms, lie outside its scope. We cannot accomplish this by any purely syntactic restriction. But we can do it quite simply, by retaining (RP) as originally enunciated, but stipulating that ‘Ref ’ is to be understood to mean basic reference; equivalently, that the expression variable e is to take as values only expressions which effect basic reference.47

5.11 Conclusion The first half of the paper revealed that two of the assumptions involved in our paradox, (OC) and (EXPRESSIBILITY), are non-negotiable in any recognizably Fregean approach to the ontological categories. Since then, we have proceeded in steps. We have argued that if we are to be able coherently to express the central claims—some would say, insights—of Frege’s analysis of language and the ontology based upon it, we must revise, or at least restrict, one of the key principles of his theory, the Reference Principle, which forges a rigid link between logico-syntactic and ontological categories. Doing so makes room for the kind of extension of a strictly typed language which is needed, if we are to be able to articulate the theory. The extension consists in allowing singular reference to, along with cross-categorial variables ranging over, entities of all types. This in turn is pointless unless accompanied by the introduction of new predicates—centrally, ‘App’, ‘OC’, ‘Ref ’, and ‘=’ in an extended use—having argument-places occupiable by terms having reference to entities in any of the ontological categories.

47 As already observed, our view thus shares with those of Wright (1998b) and Liebesman (2015) a commitment to the thesis that ‘. . . is a horse’ and ‘the property of being a horse’ have the same semantic value. It is important to notice, however, the present view differs, at least from Wright’s, in another way. For Wright, all singular terms refer to objects. Since what ‘. . . is a horse’ refers to may also be the reference of ‘the property of being a horse’, it refers to an object. More generally, all functions turn out to be objects, albeit a special kind of objects. On the present proposal, by contrast, the category of objects does not overlap with any category of unsaturated entities. See Hale and Wright (2012) for further discussion.

100 ontological categories and the problem of expressibility The crucial question now arises of whether these new predicates refer. Here we seemed to confront an awkward dilemma. Suppose the new predicates refer. Then a paradox threatens, and we would need to explain to what categories, if any, the referents of the new predicates belong. Suppose instead we deny that the new predicates refer. Then these problems are avoided—but some other ones appear instead; in particular, this response seems to clash with important Fregean principles such as compositionality and a minimalist conception of predicate reference. This dilemma has been the central concern of the second half of the paper. Each horn has been explored and found to be far less problematic than it initially appears. Where, exactly, does this leave us? Our discussion in the last three sections suggest two extreme options, each in direct competition with the other. One extreme is to uphold the new reference approach and reject no reference. As we have seen, this would involve accepting a variety of new entities that lie outside of the original Fregean hierarchy, which would thus be shown to give a radically incomplete picture of reality. At the other extreme we find the option of upholding the no reference option and rejecting the uncompromising minimalism about predicate reference on which the new reference option is based. Since this approach accepts no new entities, the original Fregean hierarchy can then be taken to be complete. We wish to end by calling attention to a possible compromise between these two extreme options, which appears attractive. So far, we have asked whether reference should be ascribed to the new predicates of Lν . But in fact, there are two distinct questions. Is the mentioned ascription of reference permissible? Or is it even obligatory? We submit that our discussion of the no reference option shows that the latter question must be answered negatively. Whether or not it is permissible to ascribe reference to the new predicates, it is certainly not obligatory in order to account for their meaningfulness. We can retain this important insight from the no reference option while conceding to the new reference option that it is nevertheless permissible to ascribe reference to the new predicates (and accept comprehension axioms for suitable open formulas in which these predicates figure). The resulting compromise view—according to which ascription of reference to the new predicates is permissible but not obligatory—has several attractive features. For one thing, the view upholds a form of minimalism about predicate reference, which is plausible in its own right48 and also robust enough to support our argument against Quine in Section 5.4. For another, the view is well equipped to minimize the un-Fregean fall-out from this minimalism. Since the referents must lie outside of the original Fregean hierarchy, it is undeniable that this hierarchy is in some sense incomplete. But these counterexamples to the Fregean view of the categories can be dismissed as relatively superficial. After all, the postulation of the counterexamples is not obligatory in order to explain the workings of the language Lν that generated the counterexamples. Finally, we observe that some very important Fregean insights remain despite the counterexamples (superficial or not). The original categories of course remain, although they turn out to be supplemented with various new entities. A broadly

48 See e.g. Hale and Wright (2009).

appendix a

101

linguistic characterization of objecthood remains. And lastly, the complete/ incomplete (saturated/unsaturated) distinction remains.49

Appendix A: Some formulations of the Reference Principle We shall here compare our principle   (RP) ∀e∀x∀c Ref (e, x) → (SC(e, c) ↔ OC(x, c)) with earlier statements of the Reference Principle. An early formulation of the Reference Principle is found in Wright (1998b). Expressed in our notation, the relevant part of this formulation states: Ref (e, x) ∧ Ref (e , x) → ∃c(SC(e, c) ∧ SC(e , c)) This is easily seen to be strictly weaker than our (RP). However, the most authoritative statement of the Reference Principle is arguably the following: An entity of a certain kind is anything which can, and can only, be referred to by an expression of a certain correlative logico-syntactic type. (Hale and Wright, 2012, p. 93)

This principle is very close to ours. Assume that every entity is referred to by some expression. It is easy to see that, modulo this assumption, (RP) is equivalent with the conjunction of the universal closures of the following two principles OC(x, c) ↔ ∃e(Ref (e, x) ∧ SC(e, c)) OC(x, c) → ∀e(Ref (e, x) → SC(e, c)) which are merely the non-modal analogues of the two modal principles that the quoted passage is meant to summarize.50 Next, observe that on the wide notion of reference—on which a variable is said to refer to each of its values—the mentioned assumption is obviously true. Every entity is indeed referred to—in the wide sense—by some expression. Our next task is to defend the claim from p. 79 that the narrow-sense version of the Reference Principle entails the wide-sense version provided that the principle is given the modal formulation mentioned above. Our argument relies on two assumptions. First, any entity has its ontological category by necessity, if it exists at all. This is an extremely weak form of essentialism about kind membership, which we take to be highly plausible. Second, for any variable of any syntactic type and any assignment, it

49 Thanks to Matti Eklund, Salvatore Florio, Keith Hossack, Ansten Klev, John Litland, Agustín Rayo, Sam Roberts, Ian Rumfitt, Stewart Shapiro, Mark Textor, Tim Williamson, Crispin Wright, as well as audiences in Aberdeen, Florence, Hamburg, Leuven, Oslo, Oxford, Stirling, and Uppsala for valuable discussion and comments on this material. (This paper was finished at the beginning of December 2017, less than two weeks before Bob Hale unexpectedly passed away.) 50 When the quoted passage is considered in isolation, it might not be entirely obvious how it should be formalized. But context suggests the passage is nothing but a compact English summary of the mentioned two modal principles—which is how we shall understand it here.

102 ontological categories and the problem of expressibility is possible for there to be a constant of the relevant syntactic type which refers to the value of the variable on the relevant assignment.51 Given these two assumptions, we argue as follows. Let v be a variable of syntactic type c whose value on the relevant assignment is a. We need to demonstrate that for any c we have SC(v, c) ↔ OC(a, c). Assume SC(v, c). By the second assumption, it is possible for there to be a constant e of syntactic type c, which refers—in the narrow sense—to a. Using the modal Reference Principle and the first assumption, we derive OC(a, c). Next, assume ¬SC(v, c). Then there is c = c such that SC(v, c ). Arguing as before, we establish OC(a, c ). Assume, for contradiction, that we also have OC(a, c). Then, by the modal Reference Principle there can be a constant e which refers to a and accordingly has both types c and c . Since c = c , this is impossible. Hence it follows ¬OC(a, c), as desired.

Appendix B: A closer look at nominalization We define the nominalization language LνSTT as follows. The types are the same as in LSTT . (This contrasts with the first of the approaches described in Section 5.6, which lifts all the type restrictions of original language, thus in effect subsuming all the old types under a single new one.) Moreover, we retain all the vocabulary of LSTT but add the nominalization operator ‘ν’ and the predicates ‘OC’ and ‘App’, each with two argument places of type 0. On the intended interpretation, the nominalization operator ‘ν’ stands simply for the identity mapping. It is not obvious how this can be expressed, however. The most straightforward attempt would be ν(xn ) = xn ; but since ‘=’ takes two arguments of type 0, this is ill-formed for any n = 0. However, the intended interpretation is adequately captured by means of a series of bridging principles that connect pairs of claims that do and do not involve nominalization. First, there is a kind of abstraction principle (Bridge-=)

ν(xn ) = ν(yn ) ↔ Eq(xn , yn )

where Eq is an equivalence relation on entities of type n that intuitively stands to such entities the way identity stands to objects. Frege of course took Eq to be the relation of coextensionality, and we shall follow him in this. (The obvious concern about consistency will be addressed shortly.) Next, we lay down the following axiom schemes concerning application, ontological category, and their interrelation: (Bridge-App)

App(ν(en+1 ), ν(f n )) ↔ en+1 (f n )

51 This observation can be challenged. For instance, a non-eliminative mathematical structuralist accepts all the indiscernible points of Euclidean three-space as objects in good standing. Although each point can be the value of a variable, arguably its indiscernibility from all the other points prevents it from being the referent of a singular term. While we believe Fregeans can and should resist this challenge, there is no need to do so here. For our paradox it suffices to make the exceedingly weak assumption that the witnesses a and b to (OC) can be chosen so as to be capable of reference by constant expressions.

appendix b (Bridge-OC) (6)

OC(x, n) ↔ ∃yn (x = ν(yn ))

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for every n = 0

App(x, y) → ∃n(OC(x, n + 1) ∧ OC(y, n))

(Bridge-OC) captures the idea that an entity is of ontological category n just in case it can be referred to by an expression of syntactic type n—except when n = 0, as any entity can be referred to by a singular term. Notice that (Bridge-OC) and the uniqueness of ontological category entail that ν(xm ) = ν(yn ) for any m = n. What is the right translation from a given typed language L into the extended language Lν ? The homophonic translation is available but is easily seen to be inappropriate. The problem is that the first-order variables in L range only over objects, whereas in Lν , such variables range over all entities. Rather, the right translation from L to Lν , which we label ν, has a single non-trivial clause, namely that ∀x0 φ is translated as ∀x0 (OC(x0 , 0) → φ ν ). This translation gives rise to another pleasing result about preservation of consistency.52, 53 Fact 3 Let T be an L-theory and T ν be the Lν -theory whose axioms comprise φ ν for each axiom φ of T, as well as all of the axioms discussed above. Then T is consistent iff T ν is consistent.

52 The analogous result would be false if we translated homophonically, as can be seen by considering the L-theory whose sole axiom is ∀xPx. Since one of the axioms we add is ∀x(Px →OC(x, 0)), we can prove ∀x OC(x, 0), which contradicts another provable claim, namely ¬OC(ν(P), 0). 53 This relative consistency result may seem too good to be true. (Cf. footnote 34.) Why is there not a property-theoretic version of Russell’s paradox? The answer is that the theory T ν allows concept comprehension only on formulas with no new vocabulary (that is, formulas already in L). Whether this is in the end sufficient will depend on our answer to the crucial question of whether the new predicates refer.

6 What Makes True Universal Statements True? 6.1 Introduction What makes true universal statements true?1 In discussing this question, I shall be especially interested in how it is to be answered within the framework of what Kit Fine calls exact truthmaker semantics.2 In an extremely useful survey article (Fine, 2017), Fine locates exact truthmaking within the more general truth-conditional approach to semantics as follows. First, we distinguish between clausal approaches such as Davidson’s (Davidson, 1967), on which truth-conditions are not given as entities but by clauses through which a theory of truth specifies when a statement is true, and objectual approaches, according to which truth-conditions are not clauses but worldly entities which stand in a relation of truthmaking to the statements they make true or false. Then, within objectual approaches, we may distinguish those which take truthmakers to be possible worlds (as in what is generally known as possible world semantics) from those which take them to be states or situations—‘fact-like entities that serve to make up a world, rather than worlds themselves’ (as in the situation semantics developed by Barwise and Perry (Barwise and Perry, 1983). Finally, we may distinguish, within ‘stately’ (as opposed to ‘worldly’) semantics, between loose, inexact, and exact forms of truthmaker semantics. Loose truthmaking is a purely modal relation: a state s makes true, or verifies, a statement p just in case it is impossible that s should obtain without p being true. By contrast, both inexact and exact verification require there to be a relevant connection between the state s and the statement p—s is an inexact verifier for p if and only if s is at least partially relevant to p, and is an exact verifier if and only if it is wholly relevant.3 Any exact verifier is also an inexact verifier, and any inexact verifier is also a loose verifier, but the converses do not, of course, hold. 1 Originally published as Hale (2018c). 2 In thinking about this topic, I have benefited enormously, not only from reading Kit Fine’s recent papers (Fine (2017) and Fine (2018)), but from some very useful discussion with him. My understanding of the truthmaker approach to semantics derives almost entirely from the clear, straightforward, and searching introduction to it that his work provides. Indeed, it was reading Fine (2018) which made me aware of the interest and importance of the leading question of this paper, and first got me thinking seriously about it. I have benefited, too, from discussions with Øystein Linnebo. I find some significant agreement with points in his recent, and as yet unpublished, work on this topic, but also some disagreement—see the appendix to this chapter. 3 What, more precisely, is to be understood by this will be discussed in due course. Bob Hale, What Makes True Universal Statements True? In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0007

6.2 the standard account and its shortcomings 105 On the version of exact truthmaking which Fine presents in this and other recent papers, a true universally quantified statement, ∀xB(x), is made true by the verifiers of its instances—or a little more precisely, it is made true by a state which is composed of the states which verify its instances. Essentially the same account is endorsed by nearly all other advocates of truthmaker semantics whose work I have consulted, so I think it may fairly be labelled the standard account. Its widespead acceptance is hardly surprising, since it can appear both very natural and even inevitable, given that ∀xB(x) is true iff B(x) is true of every object which is an admissible value of its free variable. But whilst I think there is a subclass of universal statements for which the standard account is correct, I do not think it gives the right account of truthmaking for universal statements in general. My main aim here is to promote an alternative account of the truthmakers for quantified propositions. I shall also give some attention to two closely related questions: first, when, and why, we should favour an alternative to the standard account, and second, whether the alternative account I favour can be accommodated within the framework of exact truthmaker semantics, in Fine’s sense. What follows divides into two parts. In the first, I make a case for an alternative account of what makes some universal statements true, prescinding altogether from the questions how, in detail, that account might run, and whether and, if so, how it may be implemented within the framework of exact truthmaker semantics. In the second, I turn to those questions.

Part I: A case for an alternative account 6.2 The standard account and its shortcomings As Fine observes, a major difference between possible world semantics and state or situation semantics concerns completeness—worlds are standardly taken to be complete, or maximal, in the sense that they determine the truth-value—true, or false—of every statement of the language for which a semantics is being given, whereas states or situations settle the truth-values of particular statements, but typically leave those of many others unsettled.4 One consequence of this difference is that in truthmaking semantics we need to take account both of states which make statements true (their verifiers) and of states which make them false (their falsifiers). If a statement p is not true at a possible world w, it will be false at w; but a state s which does not verify p may not falsify it either, for it may—and typically will—simply leave its truth-value unsettled. Thus the clauses for conjunction, for example, will tell us that state s verifies A ∧ B iff it is the fusion s1  s2 of two states, s1 and s2 , one of which verifies A and the other B; and that s falsifies A ∧ B iff s either falsifies A or falsifies B. But a state s may neither verify A ∧ B nor falsify A or B.

4 This is equally true of alternatives to full-blooded world semantics which replace complete worlds by possibilities which are typically incomplete, such as the possibility semantics advocated in Hale (2013a), chapter 10. For somewhat different developments of the same general approach, see Humberstone (1981a), where the idea makes its first appearance, and more recently, Rumfitt (2015), chapter 6.

106 what makes true universal statements true? The usual treatment of universal quantification takes a state s to verify a universal quantification ∀xA(x) iff s is the fusion s1  s2  … of states s1 , s2 , . . . which verify A(t 1 ), A(t 2 ). . . where A(t 1 ), A(t 2 ), . . . are all the instances of ∀xA(x), i.e. s verifies the conjunction A(t 1 ) ∧ A(t 2 ) ∧ . . . of all the instances; and s is taken to falsify ∀xA(x) iff it falsifies one of A(t 1 ), A(t 2 ), . . . .5, 6 There are, certainly, universal statements for which this account, or something like it, is very plausible. Consider, for example, the statement I might make that all my children live in England. I have just three children, Thom, Charlie, and Josh. It is very plausible that my statement is true iff the conjunction: Thom lives in England ∧Charlie lives in England ∧ Josh lives in England is true, and that it is made true by the state s = s1  s2  s3 , where s1 , s2 , s3 verify its conjuncts.7 I say only that ‘something like’ this account is very plausible, rather than it is evidently correct, because, of course, it might be objected that the truth of the conjunction suffices for that of the universal generalization only given the additional facts that Thom, Charlie and Josh are my children and that I have no others, and that, accordingly, what makes the universal generalization true is not just the state s by itself, but this state together with the state which makes it true that these and only these are my children. Proponents of truthmaker semantics may diverge over whether the truthmakers for universal statements must always include such totality facts. I neither need, nor wish, to try to settle that disagreement here. I am myself inclined to think they are needed, but for simplicity, I shall for the time being stick with the simpler account which does not take totality facts to be parts of the truthmakers for universal statements (or the falsemakers for existential statements). It will be clear enough that including them would do nothing to allay the doubts about the adequacy of the standard account which I shall be raising—indeed, including them would, if anything, make matters worse for that account. It would be very easy to multiply examples of universal statements like this one, for which the standard account, plus or minus a totality fact, seems clearly right. But it seems to me equally easy to give examples for which that account seems more or less 5 Here we assume for simplicity that all the objects in the domain of the quantifier are denoted by terms of the language. We could dispense with this assumption by stipulating that, for each element d of the domain D , the fusion s1  s2  … is to have a part sd which verifies A(x) when d is taken as the value of x. 6 See Fine (2017), §7. Assuming the quantifier ranges over the individuals a1 , a2 ,. . ., denoted by the constants a1 , a2 , . . ., Fine suggests that ‘we might take the content of ∀xϕx to be the same as the content of the conjunction ϕ(a1 ) ∧ϕ(a2 ) ∧ . . . ’ and gives the clauses: a state verifies ∀xϕx if it is the fusion of the verifiers of its instances ϕ(a1 ), ϕ(a2 ), . . . a state falsifies ∀xϕx if falsifies one of its instances. More formally: s  ∀xϕ(x) if there are states s1 , s2 , . . . with s1  ϕ(a1 ), s2  ϕ(a2 ), . . . and s = s1  s2  … s ∀xϕ(x) if s ϕ(a) for some a∈A (where A is the domain of individuals). 



See also sections 6.5 and 7.2.2 of this volume. 7 This treatment of the example assumes that my general statement is to be analysed as formed with a restricted quantifier, to be treated in accordance with the modification Fine proposes at Fine (2017), §7. Without this assumption, applying his clause for unrestricted quantification given in the previous note would take the instances to be the truth-functional conditionals ai is a child of mine → ai lives in England (one for each element of the domain of individuals) and would take the verifier for the general statement to be the fusion of the states which verify them.

6.2 the standard account and its shortcomings 107 obviously wrong. For the moment, I shall give just a couple of examples, which may at first appear somewhat special cases, although I shall argue, later, that in essential respects they are representative of a large class of universal statements to which the standard account does no justice. Consider first the statement that every natural number has another natural number as its immediate successor. According to the standard account, what makes this true is the fusion s1  s2  … where s1 verifies the statement that 0 is a natural number and is immediately followed by 1, which is also a natural number, and s2 verifies the statement that 1 is a natural number and is immediately followed by 2, which is also a natural number, and so on for an additional infinity of states si each verifying the corresponding statement that i−1 is a natural number and is immediately followed by i, which is also a natural number. The implausibility of this account does not derive, primarily or mainly, from its need to postulate a completed infinitary state. At least, this is not the main reason why I find it implausible, although there are some—most obviously, perhaps, those who insist that the only legitimate conception of infinity is as incompleteable or merely potential—who would wish to object to it on this ground.8 What, primarily, seems to me implausible in the standard account is that it pays no attention to what appears to be an obvious uniformity in the grounds for the truth of the instances. As far as that account goes, the separate instances might just as well be made true by entirely separate and unrelated states concerning individual natural numbers, much as the instances of the general statement about my children are made true by separate and independent states concerning their whereabouts. Of course, the standard account does not entail that the states which make the instances of a universal generalization true are unrelated—but the point is that it pays no attention to such connections, even when they seem plainly relevant to what makes both the generalization itself and its instances true. The further difficulty to which I alluded previously comes into play if it is held that the verifier for a universal statement, on the standard account, must incorporate a totality fact. On this view, the verifier for a universal statement will comprise—be the fusion of—two states: the state s = s1  s2  …, where s1 , s2 , … are as before, together with a state t which verifies a statement to the effect that o1 , o2 , … are all the objects of the kind over which the universal statement generalizes. In our example, t will be a verifier for the statement that 0,1,. . . are all the natural numbers. The obvious worry is that that way of stating the totality fact is just a fudge—it relies on our understanding the dots not as mere ‘dots of laziness’ but as standing in for an infinite list which we cannot complete.9 I do not claim that this difficulty amounts to a decisive objection, nor that the relevant totality fact is unstatable. A resolute defender of the standard account with 8 I discuss this further in Appendix A to this chapter. 9 It perhaps bears emphasis that this difficulty is entirely distinct from any concern that may be felt over the fact that the statement of the totality fact is itself a further universal statement. If it is held that every true universal statement is verified by a complex state involving a suitable totality fact, there is an obvious threat of vicious regress. Clearly this concern does not depend upon the infinity of the domain of the universal quantifier—it arises equally if the quantifier ranges over a finite domain, whether it be closed (as in the example about my children) or open (as with, for example, the statement that all aardvarks are mammals, assuming that there will one day be no aardvarks).

108 what makes true universal statements true? totality facts might insist that the existence of infinitary totality facts is one thing, and their statability is another. Such a stance requires a robust but familiar kind of realism, of the same stripe as appears to be involved in thinking that Goldbach’s Conjecture may be insusceptible of any general, finitely articulable proof, but is nonetheless rendered true by an infinity of facts of the form: 2n is the sum of primes pi and pj . Further, it might be claimed that, while the totality fact in our example cannot be stated by completing the list in the would-be statement that 0,1,. . . are all the natural numbers, it can be stated readily enough in another way, viz. 0 and its ancestral successors are all the natural numbers. But note that this is hardly a response with which a defender of the standard account should feel comfortable. For the fact that 0 and its ancestral successors are all the natural numbers is just the fact that everything which is a natural number is either 0 or the immediate successor of a natural number—and in the presence of this totality fact, the fusion s = s1  s2  … of states which verify the instances of our universal statement is clearly redundant, since the totality fact is by itself enough to verify it. Before I venture any suggestion about what might better be taken to verify our statement about the natural numbers, let me give just one more example. Consider the statement that (all) aardvarks are insectivorous. According to the standard account, what makes it true is the fusion s = s1  s2  … of states which verify its instances— a1 is an aardvark and eats insects, a2 is an aardvark and eats insects, . . .—one for each aardvark. This seems to me pretty incredible. As with the previous example, a determined defender of the standard account could just stick to his guns. But doing so seems to me to require chewing on some pretty indigestible bullets. One important feature of the example—and in this it is typical of a great many others which might be given—is that, in contrast with our purely numerical example, no actual or potential infinity of instances need be involved; what is involved is a possibly—indeed, almost certainly—finite but open-ended domain of entities. The generalization covers not only presently existing aardvarks and the large but finite number which have since perished, but also those which are yet to be born. In consequence, the state which verifies the generalization must be the fusion of states, an indefinite number of which verify instances concerning aardvarks which do not yet exist, but will exist at some time in the future. Unless a defender of the standard account is ready to insist that the generalization is not yet true, and will not be true at any future date before Armageddon, he must accordingly maintain either that there now exist states which somehow verify statements concerning aardvarks which do not yet exist, or that there exist (now) fusions some parts (sub-states) of which do not (yet) exist. Perhaps none of these alternatives is actually self-contradictory or absurd, but the need to embrace one of them is scarcely an attractive feature of the standard account. Were there no credible alternative accounts of what makes universal generalizations such as these true, we would have no option but to swallow these rebarbative consequences. But there are credible alternatives. Thus, in the case of our first example, it might be held that what makes it true that every natural number is immediately followed by another natural number, and what also makes true each individual instance of that general truth, is a conceptual connection—a connection between the concepts of natural or finite number and of immediate succession. Or—to canvas the

6.3 accidental and lawlike generalizations

109

alternative which I myself favour—we may hold that the general statement and its instances are made true by the essence or nature of the natural numbers, i.e. by what it is to be a finite or natural number. One reason to prefer an account along these lines is that it can better handle examples like my second than a conceptual connection account. That aardvarks are insectivores is a fact of nature, not something guaranteed by some connection between concepts. The point generalizes to many other universal statements to which the standard account appears ill-suited.

6.3 Accidental and lawlike generalizations The contrast between general statements like all my children live in England, to which the standard account seems well-suited, and others like all aardvarks are insectivores, which seem to call for a different account is, at least roughly speaking, the contrast—already found in much work that pre-dates the emergence of truthmaker semantics—between what are often called ‘accidental’ and ‘lawlike’ generalizations. The former typically, and perhaps invariably, concern the elements of some fixed or closed finite totality which could, at least in principle, be specified piecemeal, whereas the latter are about an open class of things of some general sort which, even if finite, is not exhaustively specifiable.10 Perhaps the most commonly emphasized difference between the two is that lawlike generalizations imply, or in some sense support, counterfactual statements, whereas merely accidental generalizations do not do so. Thus, to vary our example, given acceptance of a lawlike generalization such as Common salt is soluble in water, we may reasonably infer the singular counterfactual Had this teaspoon of salt been immersed in water, it would have dissolved. By contrast, given the merely accidental generalization All the students in my logic class are male, it would be rash, to say the least, to infer that Had Alicia been in my logic class, she would have been male. In some relatively recent work, it has been argued that generalizations expressed in natural languages fall into at least two different groups, corresponding roughly with this familiar contrast, and that the differences are such as to warrant formalizing or regimenting them in different ways.11 I shall refer to this as the alternative logical form view. It will, I think, help to forestall some possible misunderstandings if I say a little about how I think this view is related to my project. We should note first that the class of general statements to which the standard account seems ill-suited is not happily labelled ‘lawlike’, given the very strong suggestion that label carries, that the generalization states a natural scientific law. So much is clear from the examples already discussed, since the fact that every natural number 10 It is perhaps not entirely clear that there could not be be accidental generalizations about infinite totalities. It might be suggested, for example, that God could arbitrarily select an infinite subset of the natural numbers in such a way that, as it happens, the arabic numeral for each natural number in the subset includes the numeral ‘7’. Nor is it quite obvious—even if it is tempting to think—that the number of things covered by a lawlike generalization, even if finite, will have no finite upper bound. Aardvarks are by nature insectivorous, but we can be pretty confident that there will never have been more than 10 billion of them. 11 See, for example, Drewery (2005), developing ideas in Drewery (1998). As Drewery emphasizes, the contrast between the two sorts of generalization is generally well-marked syntactically in English.

110 what makes true universal statements true? has another as its immediate successor is hardly to be regarded as a law of nature. Nor are the facts that all spinsters are unmarried, that anything red is coloured, that only people of age 18 and upwards are entitled to vote in Parliamentary Elections in the UK, and that everything is self-identical—but all, along with many others of the same sorts, are general statements for which the standard account is implausible. The more colourless ‘non-accidental’ would be a better label. This is not, however, a point of contrast with the view that different logical forms should be assigned to accidental and lawlike generalizations. Proponents of that view are likely to see the important difference between the two classes of general statement as lying in the capacity, or lack of it, to support counterfactuals, and it is clear that many general statements besides natural laws have that capacity.12 In essence, what the alternative logical form view claims is that while merely accidental generalizations such as All the students in my logic class are male may be adequately represented as having the logical form of a universally quantified material conditional ∀x(Fx → Gx), non-accidental generalizations have a different, and probably more complex, logical form which cannot adequately be represented in standard first-order logic. There are different views about what is required to capture the logical form of non-accidental generalizations. Given that their main distinguishing feature is their capacity to support counterfactuals, it is natural to think that they involve some kind of modal element. This suggests that their logical form might be better represented in a modal extension of standard first-order logic. The simplest proposal would be to represent them as necessitated universally quantified material conditions, having the form 2∀x(Fx → Gx). But this oversimplifies in at least two respects. First, it seems clear that, while some kind of necessity may be involved in all non-accidental generalizations, different kinds of necessity—natural, mathematical, logical, legal, etc.—are in play in different examples. Second, not all generalizations which might be thought to call for different treatment are exceptionless, but, as Aristotle puts it, hold normally and for the most part. This goes for what are usually called generics, typically expressed in the form Fs are Gs, without any quantifier prefix such as ‘all’, such as Kittens are born blind, Horses have four legs, etc. The general truth that aardvarks are insectivores is not upset by the occasional aardvark tucking into a Big Mac. Drewery proposes that such generics might be taken to have the logical form 2∀x((Fx ∧ Nx) → Gx) where Nx says that x is ‘non-exceptional in this case’ (cf. Drewery 2005, p.383, drawing on Drewery 1998). The additional strength of exceptionless generalizations might then, she suggests, be captured by strengthening to 2∀x(((Fx ∧ Nx) → Gx) ∧ (Fx ↔ Nx)).13 As Drewery observes, an alternative treatment of non-accidental generalizations may be given, deploying a special logic of sortal or kind terms.14 It is unnecessary for me to explore these, or any other, specific proposals about logical form here. I mention them only to contrast them with my main claim, which does not concern the logical form of either accidental or non-accidental 12 This point is emphasized by Drewery in the article previously cited—cf. Drewery (2005), pp.380–1. 13 cf. Drewery (2005), p.387. 14 cf. Drewery (2005), p.384, where she refers to work by Lowe and others. See Lowe (1989), chapters 8 and 9.

6.4 the structure of sentences and truthmakers 111 generalizations, but about what makes them true. Although I am not committed to taking all universal generalizations as having the simple logical form ∀xA(x), with many of them exemplifying the more specific form of universally quantified material conditionals ∀x(Fx → Gx), we might, as far as my claim goes, take them to exemplify that form. The alternative logical form view is simply orthogonal to the concern which drives my proposal. Even if we were to agree that certain general statements are better represented as necessitated universally quantified conditionals, or in one of the more complicated ways mentioned above, this would leave the issue with which I am concerned pretty well untouched. For on any plausible alternative representation of its logical form, a non-accidental generalization (all) Fs are Gs will at least entail ∀x(Fx → Gx), and my question would then concern what makes that entailed general statement true. Conversely, although I shall continue, for the time being, to assume that non-accidental general statements may be adequately represented as they are in standard first-order logic, my contention that what makes them true, when they are, differs from what makes merely accidental ones true does not carry any particular implications concerning their logical form.

6.4 The structure of sentences and the structure of truthmakers The standard account may seem to draw support from a plausible view about the relation between the structure of sentences and the structure of truthmaking states. It is both natural and plausible to view the sentences of a language—especially those of a regimented language—as either simple or complex, or at least as divisible into simpler and more complex sentences, with the more complex built up by means of various constructions from the simpler. Thus we have, at the bottom level as it were, simple or atomic sentences, composed out of names or other singular terms and predicates. These may then be combined by means of sentence connectives or sentential operators to make various kinds of compound, such as negations, conjunctions, disjunctions, conditions, and so on. Then there are quantified sentences, which may be viewed as in some sense composed out of their instances. When we turn to the truth- or falsity-makers (or verifiers or falsifiers) for sentences of the various kinds, it is quite natural to suppose that these too have a kind of structure which runs parallel to the structure of sentences—so that sentences of the simplest kind, devoid of sentential operators and quantifiers etc., are verified or falsified by states of a correspondingly simple kind, whereas the more complex sentences are made true by more complex states which are, in some way, built up out of simpler states. In short, the structure of the states which serve as truth- or falsity-makers matches the structure of the sentences they make true or false—bottom up in both cases, from simpler to more complex.15 15 This need not involve the idea that all states are either atomic, or are composed out of simpler, and ultimately out of atomic, states by fusion. Fine, emphasizing the abstractness and generality of his approach, explicitly distances himself from the assumption that ‘all states are constructed from atomic states which are somehow isomorphic with the atomic sentences of the language under consideration’ (Fine 2017, §4). Later in the same paper he also stresses that there is no assumption that verifiers will be minimal (where

112 what makes true universal statements true? However natural this assumption may seem, it is certainly not obviously correct—nor is it inevitable. It gains some plausibility from the equally, if not more, plausible belief that the truth- and falsity-makers for sentences should have a kind of compositional structure mirroring that of the sentences themselves. But one has only to look a little more closely at the compositional structure of sentences to see that it does not enforce the view that the states which verify universal propositions are strictly composed (say by fusion) out of the states which verify their instances, and leaves room for an alternative account. For while negative, conjunctive, and disjunctive sentences, for example, are quite literally composed of the simpler sentences of which they are the negations, conjunctions, or disjunctions, the same is not true of quantified sentences—∀xBx is not literally composed of its instances Ba1 , Ba2 , Ba3 , . . . . It is the result of applying the universal quantifier ∀x . . . x . . . to the (typically complex) predicate B(ξ ). The belief that ∀xBx is verified by a complex state composed of the simpler states which verify its instances separately—say by the fusion of the states s1  s2  . . .  . . . which make true the conjunction of each of its instances Ba1 ∧ Ba2 ∧ Ba3 ∧ . . .—is a belief about the internal structure of the state which verifies the universal proposition, and cannot properly be evaluated without a closer consideration of the internal structure of states in general, including those states which verify atomic propositions. It is certainly very plausible that among the simplest sentences of the language there will be sentences like ‘Mary is asleep’, ‘Bill is taller than Jack’, etc., and that the states which verify such sentences (or more precisely, the propositions such sentences may be used to express in particular contexts) will be in some sense made up of parts corresponding to the sub-sentential parts of these sentences—i.e. to the proper names or other singular terms involved, and to the predicates whose argument-places they occupy—and so have objects and properties or relations as constituents. Thus it is plausible that the proposition that Mary is asleep, if true, is made so by the obtaining of a state involving Mary and the property of being asleep—a state which consists, at least in part, in the instantiation of that property by that object. It is plausible, too, that the states which verify or falsify compounds such as negations, conjunctions, and disjunctions, are structured in ways that match the structure of those kinds of proposition, so that a state s verifies ¬A if it falsifies A, verifies A ∨ B if it verifies A or verifies B, and verifies A ∧ B if it is composed of two sub-states sA and sB which verify A and B respectively. When we turn to universal propositions, however, taking the internal structure of the sentences by which they are expressed as a guide to the internal structure of their truth- and falsity-makers does less to encourage the standard account. Consider, for example, the generalization that (all) cows are ruminants. If we regiment this sentence, as usual, as a universally quantified conditional, ∀x(x is a cow → x is a ruminant), its immediate constituents, besides the logical operators, are just the predicates ξ is a cow and ξ is a ruminant. There is no definite reference to the individual objects which may be involved in the states which verify or falsify its instances. Of course, these

a state minimally verifies a statement if it exactly verifies the statement but no proper part of the state does so) (Fine 2017, §6).

6.5 standard exact truthmakers 113 individuals will belong to the domain over which the bound variable is taken to range. But the statement itself says nothing about which individuals that domain comprises. It would be entirely consistent with accepting the sentence’s structure as a guide to the internal structure of its truth- and falsity-makers to take them to consist in higherlevel relations (of inclusion and non-inclusion) holding between the properties of being a cow and being a ruminant.

Part II: Alternative exact truthmakers Turning now to the questions I have been postponing, it will be useful to start with a more detailed—but still quite concise—account of exact truthmaking, and some discussion of the requirements for exactness, before introducing an alternative account.

6.5 Standard exact truthmakers In any form of truthmaker semantics, whether exact, inexact, or loose, the ‘pluriverse’ of possible worlds which forms the basis of world semantics is replaced by a space of states, as the basis for the models of the language for which we give the semantics.16 A state space is a pair S, , where S is a non-empty set (of states) and is a partial order on S (i.e. is reflexive, anti-symmetric and transitive). A model M for a first-order language L is a quadruple S, A, ,|:|, where S,  is a state space, A is a non-empty set of individuals, and |:| is a valuation function taking each n-place predicate F and any n individuals a1 , a2 , . . ., an to a pair V , F  of subsets of S. Intuitively, V is the set of states which verify F of a1 , a2 , . . ., an and F the corresponding set of falsifiers. The former is also denoted by |F, a1 , . . ., an |+ and the latter by |F, a1 , . . ., an |− . If we assume, as Fine (2017) does, that our language contains individual constants a1 , a2 , . . ., one for each of the elements a1 , a2 , . . . of A, then exact verification and falsification clauses for atomic and complex statements may be stated as follows:17 s  Fa1 . . . .an if s ∈ |F, a1 , . . ., an |+ s Fa1 . . . .an if s ∈ |F, a1 , . . ., an |− s  ¬B if s B s ¬B if s  B s  B ∧ C if for some states t and u, t  B, u  C and s = t  u s B ∧ C if s B or s C s  B ∨ C if s  B or s  C s B ∨ C if for some states t and u, t B, u C and s = t  u s  ∀xϕ(x) if there are states s1 , s2 , . . . with s1  ϕ(a1 ), s2  ϕ(a2 ), . . . and s = s1  s2  . . . 









   

(atomic)+ (atomic)− (¬)+ (¬)− (∧)+ (∧)− (∨)+ (∨)− (∀)+

16 The explanations which follow are taken from Fine (2017), §§4–7. There are some additional conditions on state spaces, but these will not be important here. To ease comparison, I have mostly retained Fine’s notation. But I have used V , F below, where Fine has plain V, F, to avoid possible confusion between the predicate letter F and the subset F ⊆ S. 17 s  A abbreviates ‘s verifies A’ and s A ‘s falsifies A’. 

114 what makes true universal statements true? 



s ∀xϕ(x) if for some a ∈ A, s ϕ(a) s  ∃xϕ(x) if for some a ∈ A, s  ϕ(a) s ∃xϕ(x) if there are states s1 , s2 , . . . with s1 and s = s1  s2  . . .

ϕ(a1 ), s2



 

(∀)− (∃)+ (∃)−

ϕ(a2 ), . . .

Our question is whether, without forsaking the framework of exact truthmaking, the clauses (∀)+ and (∃)− might be modified to allow for verification of ∀xϕ(x) and falsification of ∃xϕ(x) by generic states. We should, accordingly, first get clear what exactness requires.

6.6 Exactness As we have seen, Fine takes a state s to be an exact verifier for a statement p if and only if it is wholly relevant to p, and to be an inexact verifier if it is partially relevant.18 But how, precisely, is ‘wholly relevant’ to be understood in this context? Contrasting exact with inexact verification, Fine gives the example: The presence of rain will be an exact verifier for the statement ‘it is rainy’; the presence of wind and rain will be an inexact verifier for the statement ‘it is rainy’, though not an exact verifier. (Fine 2017, 558)

This might be taken to suggest that a state cannot be an exact verifier for (and so wholly relevant to) a statement if it has a proper part which does not verify the statement (as, presumably, the state of its being rainy and windy has the state of its being windy as a proper part which is not a verifier for ‘it is rainy’). But this cannot be what Fine intends. For, after emphasizing that ‘state’ is for him a term of art which need not stand for a state ‘in any intuitive sense of the term’, he writes: It should be noted that our approach to states is highly general and abstract. We have formed no particular conception of what they are; and nor have we assumed that there are ‘atomic’ states, from which all other states can be obtained by fusion. (Fine 2017, 561)

Not only is it not assumed that there are atomic states from which all other states are obtainable by fusion, but it is not assumed that exact verifiers will be minimal—where ‘the state s minimally verifies the formula A if s exactly verifies A and if no proper part of s exactly verifies A (i.e. if s s and s  A implies s = s)’. Fine gives the following example: Now suppose that p is the sole verifier of p and q the sole verifier of q, with q = p. Then p and p  q are both verifiers of p∨(p∧q), with p  q non-minimal since it contains the verifier p as a proper part. (Fine, 2017, 564)

This example shows that an exact verifier can have a proper part which does not verify the statement the exact verifier verifies—for p  q is an exact verifier for p∨(p∧q), but it contains as a proper part the state q, which is not a verifier for p∨(p∧q). A little later, Fine adds this clarification: The relevant sense in which an exact verifier is wholly relevant to the statement it makes true is not one which requires that no part of the verifier be redundant but is one in which each part 18 cf. Fine (2017), §2. In Fine (2018), Fine says that an exact verifier must be ‘relevant as a whole’ to the statement it verifies.

6.7 modified exact truthmakers 115 of the verifier can be seen to play an active role in verifying the statement. Thus the verifier p  q of p∨(p∧q) can be seen to play such an active role, even though the part q is redundant, because of its connection with the second disjunct (p∧q). (Fine 2017, 564)

It would be curmudgeonly to complain that ‘play an active role in verifying a statement’ lacks precision—it seems to me that Fine’s intention is clear enough. Where s = t  u and t (exactly) verifies B but not C, while u verifies neither B nor C, s has a part, u, which plays no active role in verifying B ∨ C, so that s is not wholly relevant to B ∨ C, and does not exactly verify it, although it does so inexactly. This is in clear contrast with p  q in Fine’s example, where the part q does play an active role in verifying p∨(p∧q), being an indispensable part of a verifier for the right disjunct p∧q. If being wholly relevant, and hence exactness, is understood in this way, then it is clear that there is no reason why one and the same state should not be an exact verifier of more than one statement. In particular, it is not ruled out that there should be a single state which has no proper parts (i.e. is not the fusion of simpler states) and which exactly verifies a universally quantified statement ∀xϕ(x) and also exactly verifies each of its instances ϕ(a1 ), ϕ(a2 ), . . ., i.e a state of precisely the kind required for generic verification or falsification. I conclude that there is no bar to generic verification or falsification inherent in the conception of exact truth- and falsitymaking.19

6.7 Modified exact truthmakers Whilst the idea that universal statements may be rendered true by general connections between the properties involved, rather than made true piecemeal by states which verify their instances, seems straightforward enough, it is a further—and much less straightforward—question how to implement this idea in the framework of exact truthmaker semantics. This is largely because we must to some extent depart from the bottom-up determination of truth-values which is built into truthmaker semantics as we have it, and which—notwithstanding its divergence in other respects—it takes over from standard model theoretic semantics in general. In standard first-order semantics, the determination of truth-values for all sentences is driven by an initial assignment of individual objects from the given domain, D, to whatever simple, nonlogical predicates the language may contain. That is, we have a function, v, which 19 I don’t expect Fine to disagree with this conclusion, since at the end of Fine (2017), §7, he writes: ‘it might be thought that ∀xϕ(x) is verified, in the first place, by certain general facts which, in themselves, do not involve any particular individuals. It turns out that this idea of generic verification can be developed within the framework of arbitrary objects developed in Fine [1985]. Thus the verifier of “all men are mortal” might be taken to be the generic fact that the arbitrary man is mortal.’ There is no suggestion that adopting this course would be to abandon exact verification. The proposal I shall develop certainly differs from the alternative Fine envisages here, since it makes no use of arbitrary objects or states involving them, but I can see no reason why it should be thought to sacrifice exactness. In another paper (Fine, 2018), Fine suggests that for inexact semantics, the clause for verification of universal statements might run: a state is a truthmaker for the universal quantification ∀xB(x) iff it is a truthmaker for each of B(a1 ), B(a2 ), . . . . The implication, in context, is that this clause does not specify an exact truthmaker. This does not go against the claim made in the text, as it might at first appear to do. I am not claiming that just any state which verifies each of the instances of a universal quantification will be an exact verifier—even if s exactly verifies each of B(a1 ), B(a2 ), . . . . . ., its fusion with a state t which is irrelevant to them and to ∀xB(x) will be an inexact verifier for the latter by Fine’s clause, but not an exact verifier.

116 what makes true universal statements true? assigns to each simple n-place predicate F a subset of Dn . This determines, together with an assignment of elements of D to any constant terms and perhaps with the help of an assignment to individual variables, the truth-values of closed and open atomic sentences, and thence, via the clauses for the connectives and quantifiers, those of the complex sentences of the language. In some variations on this standard semantics, predicates may be assigned both extensions and counter-extensions, exclusive of one another, but not necessarily jointly exhaustive of the individual domain—but the overall pattern of truth-value determination remains firmly bottom-up, driven by initial assignments of individuals to predicates. First-order truthmaker semantics differs in working with a domain of states as well as a domain of individuals, and its assignment function |:| assigns states to simple n-place predicates relative to n-tuples of individuals—|F, a1 , . . . , an |+ comprising the states, if any, which make F true of a1 , . . ., an , and |F, a1 , . . . , an |− comprising the states, if any, which make F false of a1 , . . . , an . But bottom-up determination remains in place. The salient point, for our present purposes, is that in each type of semantics, the initial assignments—whether they be of individuals to the extensions (and perhaps counter-extensions) of simple predicates, or of verifying and falsifying states to pairs of predicates and n-tuples of individuals—are entirely independent of one another. This may encourage, even if it does not strictly imply, a certain sort of metaphysical atomism—any relations between predicates (or better, between the properties and relations for which they stand) are pictured as consequential upon underlying independent facts about which individuals happen to lie in the extensions of those predicates taken separately. Of course, it is essential that the range of admissible models should cover all the logical possibilities. But as we shall see—if it is not already clear enough—that need not preclude extending the assignment function in ways which capture the intuitive idea that relations between properties, and thereby, the truth-values of some statements, may be determined, not at the bottom-level, but at a higher, generic level. One way—perhaps the most obvious way—to accomplish this would be to extend the function |:| so as to allow, as well as assignments of states to predicate/n-tuple pairs, further assignments to predicate/predicate pairs. Thus where F and G are predicates of the same arity, we might have the assignments |F, G|+ and |F, G|− —where the former comprises those states which verify G of any individuals a1 , . . ., an of which F is true (i.e. for which |F, a1 , . . ., an |+ = ∅), and the latter those which falsify G of any such individuals. Intuitively, we might think of a state in |F, G|+ as a kind of generic state which simultaneously verifies both the universal statement ∀x(F(x) → G(x)) and each of its instances F(a1 ) → G(a1 ), F(a2 ) → G(a2 ), . . . . How should we formulate an appropriately modified clause for quantified statements? I think that before we can answer this question, we need to see how to generalize the idea just presented to accommodate a wider range of forms of generalization. To see how we may do so, notice first that an extension of |:| which has the same effect as extending it to a predicate pair F, G in the way proposed would be to allow its application to the open statement F(x) → G(x), with |F(x) → G(x)|+ and |F(x) → G(x)|− defined to comprise, respectively, those states which verify and falsify the open statement. The obvious advantage of the shift—effectively largely notational—is that this is readily generalized. We simply allow application of |:| to any open statement. We may then modify the verification clause for universal quantification as follows:

6.7 modified exact truthmakers 117  + ∀

s  ∀xB(x) iff either s ∈ |B(x)|+ or s = s1  s2  . . . where s1  B(a1 ), s2  B(a2 ), . . .

Note that the falsification clause remains unchanged:  − s ∀xB(x) iff s B(a) for some a. ∀ 



Should there be a state s ∈ |B(x)|− , s verifies ∀x¬B(x)—i.e. the contrary of ∀xB(x), as opposed to its contradictory. There are, however, some changes to the clauses for the connectives consequential upon the possibility of generic verification. For example, if a generic state verifies B(x) ∨ C(x), where x is the sole free variable, it will also verify the specific closed formulae B(a7 ) ∨ C(a7 ), where a7 replaces free x throughout. Accordingly, we should amend (∨)+ to: (∨)+* s  B ∨ C if s  B(x) ∨ C(x) or s  B or s C A similar adjustment is required for the verification clause for the conditional.20 An extension of (exact) truthmaker semantics along these lines is, in a certain sense, the minimum required to make space for generic verification. It allows for models in which some general statements and their instances are generically verified, whilst leaving room for instance-by-instance verification for others. Of course, a simplification of the semantics would be possible, were we to exclude the possibility of instantial verification of universal statements. On the other hand, an account which leaves room for both kinds of verification is more realistic, in the sense that it recognizes that merely accidental generalizations are verified piecemeal, in contrast with non-accidental ones verified generically.21, 22

20 No change is needed in the verification clause for conjunction. For a generic state g which verifies B(x)∧C(x) no matter which element of the individual domain x takes as its value will verify each of B(x) and C(x) for any element as taken x’s value, and—since generic states have no stately parts—will verify each of them exactly. The verifier s = t  u for B ∧ C may therefore be the fusion g  g—there is no requirement in the clause for conjunction that t and u be distinct. 21 It is arguable that such an account is best developed in a framework which replaces the standard quantifiers ∀ and ∃ by generalized quantifiers of the kind proposed by Barwise and Cooper (1981). In particular, taking quantifiers to be semantically complex expressions formed by applying a determiner (‘every’, ‘some’, ‘no’, . . .) to a common noun(-phrase)—so that a generalization such as ‘Every aardvark is insectivorous’ is represented as the result of applying a restricted quantifier ‘every aardvark’ to a predicate ‘. . . is insectivorous’—allows us to see the distinction between accidental and non-accidental generalizations as grounded in different kinds of restriction implied by the common noun to which the determiner is attached to form the initial quantifier. Very roughly, while nearly all common noun(-phrase)s serve to restrict the range of quantifiers in which they are embedded, some—examples are ‘person in this room’ and ‘member of my logic class’—imply a bound on the size of that range; it is in such cases that we have an accidental generalization. The distinction between bounding and non-bounding restrictors is not usually marked syntactically, however, and often relies upon contextual factors. In consequence, it is not straightforwardly implementable for a formal language and its semantics. I must postpone further development of these ideas to another occasion. I am grateful to Ian Rumfitt for urging on me the advantages of the generalized quantifier approach. 22 Thanks to Vladimir Svoboda and Vit Puncochar for inviting me to give a talk at the 2017 Logica symposium, and to those who contributed to its discussion, and especially to Ilaria Canavotto for pursuing some issues further in correspondence. I should like also to express my gratitude to participants in workshops on modality and truthmaking and on generality held in Augsburg and Oslo, at which I presented versions of this material, and especially to Kit Fine, Wes Holliday, Jon Litand, Øystein Linnebo, and Ian Rumfitt. I regret that limitations on length have prevented me from responding here to the challenging

118 what makes true universal statements true?

Appendix 1: The availability of the standard account Although I have argued (section 6.2) that there are many universal statements to which the standard account is ill-suited, and for which we should prefer a generic account, I have not claimed that there are any cases for which it can be definitively shown that the standard account is unavailable. In this appendix I look at this question, and some of its ramifications, more closely in the light of some recent work by Øystein Linnebo (Linnebo, 2018). As well as arguing, as I have done, that there are generalizations for which an alternative, generic account of truthmakers seems preferable, Linnebo discusses the relation between the competing versions of truthmaker semantics and the choice between classical and intuitionist logic. More specifically, he claims that Where only generic explanations are available, only intuitionistic logic is warranted. This result brings out an important consequence of the potentialist conception of numbers or sets. If the conception is true, then only generic explanations of universal generalizations are available, and we are thus only entitled to intuitionistic logic. If potentialism is false, on the other hand, then there is no reason to doubt the universal availability of the instance-based explanations, and classical logic is permissible.23

There are several claims here which it is important to separate. First, however, we should note that Linnebo is not claiming—implausibly—that taking universal statements to be verified generically is in itself inconsistent with retention of classical logic. The claims which seem to me to require separate discussion are (1) . . . when only generic explanations are available, only intuitionistic logic is warranted. (2) If [potentialism] is true, then only generic explanations of universal generalizations are available. (3) If potentialism is false . . . there is no reason to doubt the universal availability of the instance-based explanations. These three claims are clearly independent of each other. By potentialism, Linnebo intends the doctrine that the only legitimate conception of infinity is that of potential infinity—an infinite domain, such as the natural numbers, is ‘always unfinished’ or incomplete. If this doctrine is correct, then—or so it would seem—there can be no such thing as the fusion of the states which individually make true the statements that 0 is immediately followed by 1, that 1 is immediately followed by 2, . . . and so on (one state for each pair of successive natural numbers), since this would require a ‘completed’ infinity. But then neither the standard account, nor any instance-based explanation of what makes a generalization about the natural numbers true, would be points and questions they raised. I hope to address them in a fuller discussion of the approach for which this paper argues. 23 Editor’s note: This passage is not in the final version of Linnebo’s published paper but was, I presume, taken from an earlier unpublished version. Nevertheless, a similar point remains (Linnebo, 2018, 183–4). I have left the quotation unchanged here, given that Hale engages quite closely with the wording. If Linnebo would no longer endorse quite this view, then the reader can take the present discussion to be directed against this view, and not specifically against Linnebo.

appendix 1 119 available—its truthmaker would have to be a generic state, which does not call for the existence of any completed infinite collection. Thus claim (2) appears to be correct.24 Claims (1) and (2) obviously entail (4) If potentialism is true, only intuitionist logic is warranted. Claim (4) may be thought independently plausible, and it is, of course, endorsed— along with its antecedent—by the classical intuitionists (if I may be allowed a somewhat paradoxical label for them). But we should note that this argument for (4) (and hence for intuitionism) requires the truth of claim (1), which we have yet to discuss. If instance-based explanations are universally available, then the antecedent of claim (1)—i.e. the consequent of claim (2)—is false. And hence, if claim (2) is true, its antecedent must be false. That is: (5) If instance-based explanations are universally available, potentialism is false. Claim (3) is, near enough, the converse of (5), and clearly independent of claims (1) and (2). (3) is equivalent to the converse of (5) if ‘there is no reason to doubt . . .’ is understood to mean that there is no adequate reason to doubt . . ., and it is also true there is no adequate reason to doubt that p if and only if p. Even granting the truth of claim (2), I do not think we have an irresistible argument for the conclusion that there are some generalizations for which there can be only generic truthmakers—for, of course, the needed minor premise—that there can only be potential infinities—may always be resisted. I turn now to claims (1) and (3). It will be clear from section 1 that I do not think that either of them is true. For as I there suggested, even in cases where the domain of quantification is finite, but open-ended, as with many ordinary non-accidental true generalizations, there is reason to doubt the availability of an instance-based explanation of their truth. If one accepts that such generalizations are true (and not merely that they will eventually—when the world ends—be true), and one further accepts that the states which make their instances true exist only if, and when, the objects involved in them exist, and that instances concerning as yet non-existent objects, even if statable, cannot be made true by states which do not (yet) exist, then there is reason to doubt that those generalizations can be made true by states which are the fusions of states verifying each of their individual instances. The reason is of the same general kind as—but entirely independent of—the reason which potentialism would, if accepted, provide for denying the availability of the standard instance-based account. As I have indicated, I think it must be conceded that this reason is not conclusive. But the possibility of resisting it—say by insisting that a statement may be made true (now) by a state which does not yet obtain, or that a state can exist or obtain now, even though it involves objects which do not yet exist—hardly justifies claims (1) and (3). And if only generic states can verify non-accidental generalizations concerning finite but open-ended domains, such as that comprising past, present and future aardvarks, that certainly does not appear to preclude the use of classical logic 24 Of course, (2)’s consequent must be understood as claiming only that only generic explanations are available for universal generalizations which quantify over infinitely many individuals—instance-based explanations might still be given for generalizations over a finite domain of individuals.

120 what makes true universal statements true? in reasoning with or about them—contrary to claim (1). There is thus at least some reason to doubt the soundness of the argument from (1) and (2) to (4), and hence, coupled with endorsement of potentialism, to the conclusion that only intuitionist logic is warranted in reasoning about statements subject only to generic verification.

Appendix 2: Truthmaker semantics, classical logic, and inextendible states The states which make statements true or false are typically incomplete, in the sense that they leave the truth-values of many, and usually most, statements unsettled. As we noted already (see fn. 4) they are in this respect similar to the possibilities favoured by some as an alternative to the full-blooded worlds of the standard possible world semantics. Proponents of possibility semantics have not seen their shift away from the complete or maximal worlds of standard semantics as in any way hostile to classical logic, or as yielding a semantics which validates only a weaker logic, such as intuitionistic logic. However, in a recent paper to which I have already made reference (Fine 2018), Kit Fine proves a surprising result which is, at first sight anyway, in tension with the claim that a semantics which deals only in incomplete possibilities can validate classical logic. Fine shows that if the truthmaking semantics he presents is to underpin classical logic, its space of states will need to include what he calls worldstates. A world-state, which ‘corresponds to a possible world’, is a state which ‘is not a proper part of any other state within the space; it is “inextendible”. The space of states, S, is a W-space iff ‘every state in the space is part of a world-state. Thus a W-space will be one that contains all the possible worlds that one might reasonably be taken to exist from within the state space itself.’ (Fine, 2018, 41)25 In this appendix, I outline the argument to Fine’s result and consider whether it really has the disturbing implications it appears to have. If I have understood Fine’s argument correctly, the main conclusion for which I have argued in the body of this paper bears directly on that question. Fine holds, surely correctly, that if a truthmaker semantics is to underpin classical logic, it must endorse the principles he calls Bivalence and Linkage. By the former, he means the combination of: Exclusivity No truthmaker for a statement is compatible with a falsity-maker; Exhaustivity Any state is compatible either with a truthmaker or with a falsitymaker for any given statement

25 Since possibility semantics is not a species of truthmaker semantics, and Fine’s result concerns the latter, it is not immediately obvious that his argument threatens the claim that a possibility semantics which eschews complete, fully-determinate possibilities may yet validate classical logic. But it would, I think, be unwise simply to assume that no argument analogous to Fine’s could be developed so as to apply directly to possibility semantics. Fine makes it very clear that he does take his result to go against the belief that a semantics which deals only in incomplete possibilities, such as the versions of possibility semantics advocated in Humberstone (1981a), in Hale (2013a), ch.10, and in Rumfitt (2015), can be adequate to validate classical logic.

appendix 2 121 and by the latter, the combination of: Exact/Inexact Link A state is an inexact truthmaker for a statement iff it contains an exact truthmaker Inexact/Loose Link A state is a loose truthmaker for a statement if any state compatible with the state is compatible with an inexact truthmaker for the statement. Immediately after enunciating these principles, Fine comments: What is somewhat surprising is that the endorsement of either type of connection (Bivalence or Linkage) would appear to incur a commitment to worlds, so that one leading motivation for a truthmaker semantics, that we need only evaluate statements with respect to partial possibilities as opposed to worlds, must be abandoned. (Fine, 2018, 41)26

A state is said to be a world-state (i.e. a state which corresponds to a possible world) if it is a state which is not a proper part of any other state within the state space S—it is inextendible, as Fine puts it. A state space S is a W-space if every state in S is part of a world state. By a commitment to worlds, Fine tells us, he means ‘a commitment to a W-space; the “working” state space involved in giving a semantics for the language under consideration must be taken to be a W-space’ (Fine, 2018, 41). The commitment to worlds, Fine argues, arises equally through the endorsement of Bivalence or of Linkage. Since the argument is in essence the same in both cases, we shall just consider Bivalence.27 In this case, Fine argues: If we are to establish that the semantics is bivalent then we would like to establish that complex statements are bivalent on the basis of the atomic sentences being bivalent. What this then means is that, given the assumptions of Exclusivity or Exhaustivity for atomic statements, we should be able to establish that they also hold for all statements whatever. This is not a problem for truth-functional compounds of atomic statements, but it is for quantificational compounds. For it would appear to require an additional assumption. (Fine, 2018, 41)

The additional assumption required is Conjunctive Possibility If a state necessitates each of the propositions P1 , P2 , . . ., then their conjunction is possible, i.e. there is a state of the form p1  p2  . . ., with p1 ∈ P1 , p2 ∈ P2 , . . . .28 In essence, Conjunctive Possibility is needed in order to show that Exhaustivity extends to universal statements ∀xB(x), on the assumption that it holds of each of their instances separately (i.e. holds of B(a) for each individual a in the domain). As Fine himself puts it 26 In conversation, Fine explained that he himself found the result surprising because he had fully expected to be able to show that his version of exact truthmaker semantics with partial (i.e. always extendible) states would suffice to validate classical logic. 27 Fine also argues that natural constraints on the consequence relation lead to reliance on Conjunctive Possibility, but once again, the argument turns on essentially the same point, which concerns the passage to the truth of a universal quantification from the truth of all its instances. 28 Propositions here are subsets of the space of states. A state s necessitates a proposition P (written s 2 P) if any state compatible with s is compatible with any state p ∈ P. Propositions P and Q are compatible if there exists a state p  q with p ∈ P and q ∈ Q.

122 what makes true universal statements true? without it, it is hard to see what might guarantee Bivalence. For we need to ensure that Bivalence (and Exhaustivity, in particular) transfers from the instances B(a1 ), B(a2 ), . . . of a universal generalization ∀xB(x) to the universal generalization itself; and the only way to ensure this is by appeal to the assumption. (Fine, 2018, 42)

The detailed argument requires more stage-setting than I have space for here, but I think the essential point can be put as follows. If a state s is to make a universal statement ∀xB(x) true, it must make true the conjunction of its instances B(t 1 ) ∧ B(t 2 )∧, . . . . But, supposing that s makes true each of the instances separately, we need to pass from this fact to s’s making true their conjunction—equivalently, we need to rely on Fine’s Lemma 4 part (i) Lemma 4 Given that S conforms to Conjunctive Possibility: (i) s 2 P1 ∧ P2 ∧ . . . iff s 2 P1 , s 2 P2 , . . . which asserts that a state necessitates a (possibly infinitary) conjunction iff it necessitates each of its conjuncts taken separately. But the proof of this lemma (right to left) requires appeal to Conjunctive Possibility. However, the assumption that the state space S conforms to Conjunctive Possibility implies that it is a W-space.29 If I have understood this argument aright, it assumes a bottom up, or instantial, conception of truthmaking (see p.111) according to which the truthmaker for a universal statement ∀xB(x) is a complex state which is the fusion of the states which verify each of its instances B(t i ) separately. Thus when the domain of quantification is infinite, there will be infinitely many ‘partial’ truthmakers for ∀xB(x), for each of the states s1 , s2 , . . . which (exactly) verifies B(t 1 ), B(t 2 ), . . . must play its part in verifying the universal statement by forming part of the infinite fusion s1  s2  . . . . It is, at bottom, for this reason that the proof of Fine’s Lemma 4 (i) from right to left requires the assumption that the state space conforms to Conjunctive Possibility.30 Lemma 4 (i) covers covers two kinds of case—those in which s necessitates each of finitely many propositions P1 , P2 , . . ., and those in which it necessitates an infinity of propositions. Conjunctive Possibility is required only for the latter kind of case. In the finite cases, the move from s 2 P1 , s 2 P2 , . . . , s 2 Pn to s 2 P1 ∧ P2 ∧ . . . ∧ Pn is unproblematic, since it requires only finitely many steps of the kind involved in the proof of Fine’s earlier Lemma 3 (iv): Lemma 3 (iv) If P necessitates Q1 and Q2 then P necessitates Q1 ∧ Q2 It is the second type of case, in which a single state s which necessitates each of an infinity of propositions is required to necessitate their conjunction, which is crucial, 29 This is the left-right half of Fine’s Theorem 24: A state space conforms to Conjunctive Possibility iff it is a W-space (Fine, 2018, 57). 30 The dependence of Fine’s argument upon taking the truthmaker for ∀xB(x) to be the fusion of the truthmakers for its instances is particularly evident in his proof of his Theorem 12, which asserts that a state s is a loose truthmaker for A (s ||∼A) iff s 2 [A]+ (i.e. any state compatible with s is compatible with a state which verifies A). The crucial case for the induction is A = ∀xB(x). Here Fine argues for the leftright direction as follows: Suppose s ||∼ ∀xB(x). Then s ||∼B(ai ) for each i. By IH, s 2 [B(ai )]+ for each individual ai . By lemma 4(i), s 2 [B(a1 )]+ ∧ [B(a2 )]+ ∧ . . . . But [∀xB(x)]+ = [B(a1 )]+ ∧[B(a2 )]+ ∧ . . .; and so s 2 [∀xB(x)]+ . Note especially the penultimate step, which identifies the universal proposition with the conjunction of its instances.

appendix 2 123 for this means that s must be an infinite fusion of states which verify each conjunct separately. Conjunctive Possibility is required to guarantee that there exists such a state. But now, if my diagnosis of the need to invoke Conjunctive Possibility is correct, it would seem that it is needed to ensure that Bivalence holds in full generality only if it is assumed that universal statements are made true by fusions of the states which individually verify their instances. Thus if, as I have argued they may be, universal statements are made true by generic states, the need to appeal to Conjunctive Possibility lapses, so that we are no longer forced to conclude that world-states (or fully determinate possible worlds) are required to validate classical logic.

7 Exact Truthmakers, Modality, and Essence 7.1 Philosophical background What follows can be viewed as a contribution to what might be termed the logic of essence—or perhaps better, the logic of a theory of essence. It relies on a number of general philosophical or metaphysical assumptions for which I give no argument here, but which I want to put in the open, partly because some of them play a large part in shaping the detailed discussion which follows and partly because they are certainly controversial, and putting them in the open now will save the potential readers the time and bother of reading much further, should they feel they have insufficient sympathy with those ideas, or with a theory which assumes them.

7.1.1 Absolute necessity and possibility I assume that some things said—propositions, as I shall usually call them—are absolutely necessary in the sense, roughly, that they are true and would be true, no matter what were the case. 1. ‘No matter what were the case’ is to be understood in a completely unrestricted way—in particular, it is not restricted to ways things might be, but includes ways thing could not be, or impossibilities as well as possibilities. 2. The explanation is not intended to be reductive—it uses a strong conditional which is itself a modal notion, as well as quantifying over ways for things to be (i.e. ways things might or might not be). 3. Absolute necessity and possibility are assumed to conform to the principles of the strongest normal modal logic, S5. When 2 and ♦ are used without explicit advice to the contrary, and when ‘necessity’ and ‘possibility’ are used without qualification, they are intended to express absolute notions.

7.1.2 Essence and the source of necessity What is the source or basis of necessities? I assume that this is a good question, and that the answer is that necessities have their source in the essences of things. This is in opposition to answers that, for example, appeal to facts about meanings or concepts (and perhaps ultimately to facts about our linguistic conventions), and those that appeal to facts about (possible) worlds.

Bob Hale, Exact Truthmakers, Modality, and Essence In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0008

7.1 philosophical background

125

1. By a thing’s essence—to which I also refer as its nature—is meant: what it is to be that thing. 2. Roughly, the essentialist theory is that it is necessary that p iff it is true in virtue of the natures of some things that p and possible that p iff it is not true in virtue of the natures of any things that ¬p. Using a handy notation from Fine: 2p iff ∃x1 , . . ., xn 2x1 ,. . . .xn p and ♦p iff ¬∃x1 , . . ., xn 2x1 ,. . . .xn ¬p. 3. The explanation is again not intended to be reductive—the notion of a thing’s nature or essence is irreducibly modal. 4. ‘Things’ is not restricted to individual objects, but ranges over things of all kinds, including properties, relations, functions, etc.

7.1.3 Metaphysical necessities Absolute metaphysical necessities are assumed to include, but not to be exhausted by, logical necessities. For concreteness, it is assumed that the non-logical ones include many Kripkean non-logical necessities, such as necessities of kind-membership and kind-inclusion (e.g. Nixon was human, tigers are animals, etc.) and what he calls theoretical identifications (e.g. Water is H 2 O, Light is a stream of photons, Heat is mean molecular energy, etc.). So a general theory of necessity will not be a purely logical theory, but a metaphysical theory.

7.1.4 Semantics For various reasons, I prefer to avoid standard possible world semantics and variants on it (e.g. to incorporate impossible or open worlds). In particular, I want to avoid the assumption of completeness or maximality usually involved—that each possible world settles the truth-values of all propositions (or at least each proposition expressible in the language in question). Instead, we shall work with what are often called possibilities (and more generally, ways for things to be—see above), conceived as generally incomplete, in the sense that they typically leave the truth-values of many (most) propositions undetermined. In particular, I am going to assume the framework of truthmaker semantics as expounded in recent work by Kit Fine. In any form of truthmaker semantics, whether exact, inexact, or loose, the ‘pluriverse’ of possible worlds which forms the basis of world semantics is replaced by a space of states, as the basis for the models of the language for which we give the semantics. A state space is a pair S, , where S is a non-empty set (of states) and  is a partial order on S (i.e.  is reflexive, anti-symmetric and transitive).1, 2 States may be thought of as roughly the same kind of thing as the ways for things to be already introduced (i.e. possibilities and impossibilities)—like them, they are typically silent about the truth-values of many propositions.

1 The explanations which follow are taken from Fine (2017), §§4–7. There are some additional conditions on state spaces, but these will not be important here. To ease comparison, I have mostly retained Fine’s notation. But I have used V , F below, where Fine has plain V, F, to avoid possible confusion between the predicate letter F and the subset F ⊆ S. I also draw on Fine (2018). 2 Editor’s note:  can be understood as an (improper) parthood relation between the states in S.

126 exact truthmakers, modality, and essence

7.1.5 Issues Essentialist explanations of necessities and possibilities in general, though non-reductive, promise to be illuminating in another way. Roughly, the idea is that they locate the source of necessities and possibilities in general in a special, basic or fundamental kind of necessities—those directly arising from the essences or natures of things. The idea is then that all remaining necessities and possibilities can be seen as grounded, more or less indirectly, in these basic necessities. If this idea is on the right lines, it ought to be possible to distinguish clearly between those necessities which are directly or immediately grounded in the natures of things, and those which are indirectly or mediately grounded in the natures of things taken collectively.3 I shall be especially interested in this question below. It is, in part, for this reason that I am focusing on developing ideas within the framework of Fine’s version of truthmaker semantics. For an attractive feature of that semantics is that it seeks to capture a certain kind of relevance between a statement’s truth (or falsehood) and what makes it true (or false). We might expect, or at least hope, that within this framework, it would be possible to capture the distinction between what is directly true in virtue of essence, and what is only indirectly true.

7.2 First-order exact truthmakers 7.2.1 Models A model M for a first-order language L is a quadruple S, A, ,|:|, where S,  is a state space, A is a non-empty set of individuals, and |:| is a valuation function taking each n-place predicate F and any n individuals a1 , a2 , . . . , an to a pair V , F  of subsets of S. Intuitively, V is the set of states which verify F of a1 , a2 , . . . , an and F the corresponding set of falsifiers. The former is also denoted by |F, a1 , . . . , an |+ and the latter by |F, a1 , . . . , an |−.

7.2.2 Truthmakers and falsemakers If we assume4 that our language contains individual constants a1 , a2 , . . . , one for each of the elements a1 , a2 , . . . of A, then exact verification and falsification clauses for atomic and complex statements may be stated as follows (where s  A abbreviates ‘s verifies A’ and s A ‘s falsifies A’, and where ‘s  t’ is the fusion of s and t:5 

s  Fa1 . . . . an if s ∈ |F, a1 , . . . , an |+ s Fa1 . . . . an if s ∈ |F, a1 , . . . , an |− s  ¬B if s B s ¬B if s  B s  B ∧ C if for some states t and u, t  B, u  C and s = t  u 





(atomic)+ (atomic)− (¬)+ (¬)− (∧)+

3 This may correspond to an interesting distinction proposed by Fine, between what he terms constitutive and consequential essence—see Fine (1995). 4 as Fine (2017) does. 5 Editor’s note: this last clarification of the meaning of ‘’ is my addition.

7.2 first-order exact truthmakers 127 



s B ∧ C if s B or s C s  B ∨ C if s  B or s  C s B ∨ C if for some states t and u, t B, u C and s = t  u s  B → C if s  ¬B or s  C (equivalently: s B or s  C) s B → C if for some t, u, t  B and u C and s = t  u6 s  ∀xϕ(x) if there are states s1 , s2 , . . . with s1  ϕ(a1 ), s2  ϕ(a2 ), . . . and s = s1  s2  . . . s ∀xϕ(x) if for some a ∈ A, s ϕ(a) s  ∃xϕ(x) if for some a ∈ A, s  ϕ(a) s ∃xϕ(x) if there are states s1 , s2 , . . . with s1 ϕ(a1 ), s2 ϕ(a2 ), . . . and s = s1  s2  . . .  



















(∀)− (∃)+ (∃)−



(∧)− (∨)+ (∨)− (→)+ (→)− (∀)+

7.2.3 Validity As Fine points out,7 it would be too strong to require the truth of A in all models for A to be valid, or A’s truth in all models in which every B in  is true for A to be a consequence of . For there may be models which include no state which verifies A, and models in which there is a state which contains a verifier for each premise but which does not contain a verifier for the conclusion but leaves its value unsettled. We may instead, he proposes, define consequence as follows: We should instead take C to be a (classical) consequence of A if every state compatible with an exact (or inexact) truthmaker for A is compatible with an exact (or inexact) truthmaker for C (or, alternatively, if no exact (or inexact) truthmaker for A is compatible with an exact (or inexact) falsity-maker for C). (Fine, 2018, 42)

Recall that states s and t are compatible if their fusion s  t exists, so that the proposed definition of classical consequence amounts to: A | C iff for any s and t, if s  A and s  t exists, then for some s , s  C and s  t exists.

Fine’s alternative is: 

A | C iff for any s and t, if s  A and t exist).8

C, then s and t are incompatible (i.e. s  t does not

6 Fine does not give clauses for the material conditional; but we shall find them useful, and these are the obvious ones. 7 Fine (2018), p.42. ‘Under the loose conception of truthmaking, we might define the statement C to be a consequence of the statement A if every loose truthmaker for A is a loose truthmaker for C. But this definition will not work, at least for the classical notion of consequence, if we substitute exact or inexact truthmakers for loose truthmakers. Thus B ∨¬B is a classical consequence of A but an exact (or inexact) truthmaker for A need not be an exact (or inexact) truthmaker for B ∨¬B.’ 8 On the alternative definition, Fine comments that it has the same form as my definition at Hale (2013a), p.241. There I define  | A to mean that no possibility in any model verifies each B in  but falsifies A. This was designed to allow for the fact that a possibility need not be downwards closed, and so might, for example, verify a disjunction without verifying either of its disjuncts, with the result that it might verify all of the premises but fail to verify the conclusion (for example, disjunctive syllogism fails under the usual definition of consequence). So in my setting the two forms of definition are not equivalent—the one I favour is less demanding than the standard definition. But in Fine’s setting, they are equivalent, since we cannot

128 exact truthmakers, modality, and essence We recall that one state necessitates another iff any state compatible with the first is compatible with the second, and that a state s necessitates the proposition s1 , s2 , . . . if any state compatible with s is compatible with one of s1 , s2 , . . . . So, letting the proposition |A| = the set of verifiers for A etc., the first definition of consequence amounts to defining C to be a consequence of A iff any state which necessitates |A| necessitates |C|. Letting s be any state and t be any member of |A|, what this requires is that if s  t exists, then there is some u in |C| such that s  u also exists. As it stands, Fine’s definition covers only single premise consequences, but we can easily adapt to it to cover multiple premises, simply by conjoining them: A1 , . . . , An | C iff for any s, if t = t 1  . . .  t n , where t1  A1 , . . . , tn  An and s  t exists, then for some u such that u  C, s  u exists.

7.3 Adding standard modal operators In a first-order model, the set of states, S, may be taken to include impossible as well as possible states. If it does, there will be a proper subset of possible states, S♦ . We may suppose that some of the states in S♦ are merely possible, some actual, so that there will be a proper subset, Sα , of actual states. Our first question now is how to state clauses for necessity and possibility operators. Obviously only possible states will be relevant—that is, the verifiers and falsifiers for 2A and ♦A will be elements of S♦ . However, since a possible state s may neither verify nor falsify A, we cannot say simply that a possible state s verifies 2A if and only if every state s which is possible relative to s verifies A, nor can we say that a possible state s falsifies ♦A if and only if every state s which is possible relative to s falsifies A. We can, of course, still say that a possible state s falsifies 2A if and only if some state s which is possible relative to s falsifies A, and that a possible state s verifies ♦A if and only if some state s which is possible relative to s verifies A. How we should formulate the positive clause for 2A and the negative clause for ♦A depends upon whether we wish to allow or exclude cases of vacuous verification.9 Allowing for vacuous verification of 2A, we may say that a possible state s verifies 2A if and only if no state s which is possible relative to s falsifies A. This condition will be satisfied non-vacuously if some state s possible relative to s verifies A and every other state t possible relative to s either verifies A or leaves A’s truth-value unsettled; it will be vacuously satisfied if every state s possible relative to s neither verifies nor falsifies A. Verification of ♦A is always non-vacuous. A possible state s verifies ♦A if and only if some state s which is possible relative to s verifies A. If we adopt these clauses, falsification—whether of 2A or of ♦A—is never vacuous. The former is falsified by s only if some state s possible relative to s falsifies A; and the latter is falsified by s only if some state s possible relative to s falsifies A and no state t possible relative to s either verifies A or leaves A’s truth-value unsettled.

have a state which exactly or inexactly verifies a disjunction, say, without so verifying one of the disjuncts. Verification and falsification are ‘downwards closed’ in exact and inexact semantics. 9 Editor’s note: Throughout this section, some instances of ’if ’ have been changed to ’if and only if ’.

7.4 generic versus instantial truthmakers 129 To exclude the possibility of vacuous verification of 2A, we would need to restrict attention to those states possible relative to s which either verify or falsify A, and require that there be at least one such state. Thus we should have to say that a possible state s verifies 2A if and only if at least one state possible relative to s either verifies or falsifies A and no state s which is possible relative to s and either verifies or falsifies A falsifies A. No complication arises for verification of ♦A, which is always nonvacuous—a possible state s verifies ♦A if and only if some state s possible relative to s verifies A. As usual, we may assume that the same individuals exist at every state, or we may allow the individual domain to vary from one state to another. In the constant domain version of the semantics, a model is a sextuple S, S♦ , A, R, ,|:|, where S,  is a state space, A is a non-empty set of individuals, and |:| is a valuation function taking each n-place predicate F and any n individuals a1 , a2 , . . . , an to a pair V , F  of subsets of S. R is a binary relation in S × S, and S♦ ⊆ S with neither empty. For the variable domain version, we add a function δ : S −→ P A − ∅ which assigns a non-empty subset of the inclusive domain A to each state s ∈ S. Intuitively, R is a relation of relative possibility—or better, accessibility—between states (better, because the relation may hold between states some of which are impossible). As usual, additional constraints on R gives semantics validating T, B, S4, and S5. If 2 and ♦ are taken to express absolute necessity and possibility, conforming to S5 laws, then reference to R may be suppressed and the semantic clauses simplified to:





♦+ ♦−

s  2A iff s A for no s ∈ S♦ s 2A iff s A for some s ∈ S♦ s  ♦A iff s  A for some s ∈ S♦ s ♦A iff s A for every s ∈ S♦ 

2−

 

2+

7.4 Generic versus instantial truthmakers for modal statements As with the more usual possible world semantics for modalities, there is a strong analogy between our clauses for the modal operators and those for quantifiers. Just as in the more usual semantics, 2 and ♦ are treated as if they were quantifiers over worlds, so in our clauses they are treated as if they were quantifiers over states. And much as a universal quantification is taken to be made true by the totality of states which separately verify each of its instances, so the necessitation of a statement A is taken to be made true by the totality of possible states which verify the statement A which is necessitated. But just as we may question whether universal quantifications are (always) best viewed as verified instantially, so we may doubt that necessitations are best seen as verified by their ‘instances’. Indeed, there is an obvious connection between the two cases. For the examples of universal statements to which instantial verification seems ill-suited, and which seem to call for some kind of generic verifiers, are typically—perhaps invariably—ones in which the truth of the universal statement is a matter of some sort of necessity.

130 exact truthmakers, modality, and essence According to an essentialist theory of modalities, necessities are to be explained in terms of the essences or natures of the things they involve. Thus it is necessary that elephants are mammals because being a mammal is part of what it is to be an elephant. It is necessary that each natural number is immediately succeeded by another natural number because to be a natural number is to be zero or one of its successors. And so for other cases. In general, it is necessary that p when and because it is true in virtue of the nature of some thing (or of some things) that p. There need be no conflict here with the idea that what is necessarily so is what holds true in all possible circumstances, or—to express the idea more figuratively, but also potentially more misleadingly—what holds in all possible worlds. For a statement’s holding true in all possible circumstances or at all possible worlds, so far from explaining its necessity, is, rather, (part of) what needs explaining. It is because being a mammal is part of what it is to be an elephant that there are no possible worlds in which there are non-mammalian elephants. Can this idea be implemented in the general framework of truthmaker semantics? As things stand, a model for a first-order language with the standard modal operators has the shape S, S♦ , A, R, ,|:|. There are two ways in which we might modify this structure to accommodate the idea that universal and necessitated statements may have non-instantial, generic truthmakers. One way would be to distinguish, within the overall state-space, a special class of states which correspond to the essences or natures of the various entities, or kinds of entity, represented by expressions of the language. The other would be to modify the assignment function |:|—perhaps by extending the range of expressions to which sets of states are assigned, or perhaps by imposing constraints on its assignment of sets of states to simple predicates and n-tuples of objects—so as to reflect necessary connections consequent upon the natures of the various kinds of entity represented. At first sight, it might seem that the first approach is in better accord with an underlying essentialist theory about the source or basis of necessities. The essence or nature of an entity is given by its definition—what is often called a real, as opposed to verbal or nominal, definition, because what is defined is the thing for which a word or other expression stands, as opposed to the word or other expression itself. A thing’s definition states what it is to be that thing. Some examples: to be a circle is to be a collection of all the points in a plane equidistant from some fixed point; to be a horse is to be an odd-toed ungulate mammal of the family equidae, genus equus; conjunction is that function from pairs of propositions to propositions whose value is a true proposition for any two true propositions as arguments and a false proposition for any pair of propositions at least one of which is false. Since (correct) definitions, in this sense, are (true) propositions of a certain sort, there is no reason why there should not be states—definitional states, as we may call them—which make them true. Thus we might modify the truthmaker semantics described in earlier sections by distinguishing, within the subclass S♦ of possible states, a further subclass SDef of definitional states, and amend the verification clause for 2 to run somewhat as follows: 2+

s  2A if s ∈ SDef ∧ s  A

Plausible as this approach may at first seem, it seems to me that it is misguided, because it is is fundamentally at odds with a leading feature of truthmaker semantics as contrasted with possible world semantics. This is that states, in sharp contrast with

7.4 generic versus instantial truthmakers 131 possible worlds, are typically incomplete, in the sense that they will, typically, verify or falsify only a quite small proper subset of the statements of the language, and leave the truth-values of many, indeed most, statements undetermined. It was precisely for this reason that we adopted the negative verification clause: s  2A iff s



2+

A for no s ∈ S♦

for 2, rather than a clause stipulating that a state s verifies 2A iff every state s (possible relative to s) verifies A, parallelling the usual clause in worldly semantics. In this context, we cannot require that every state (possible relative to s) should verify A, precisely because there will normally be many such states which neither verify nor falsify A, but leave its value undetermined. We are therefore constrained to take verification of 2A to consist, not in the verification of A by all possible states in the state-space of the model, but in the absence of possible states which falsify A. The definitional state approach consists in enlarging the state-space to include additional states which verify A and thence verify 2A, and thereby goes flat against this point. What is required is not to postulate additional states which verify A and thence verify 2A, but to restrict the class of possible states to exclude those which would falsify A (and thence falsify 2A). For this reason, I think the right approach consists in imposing constraints on the assignment of sets of states to simple predicates and n-tuples of objects—roughly, so as to reflect necessary connections consequent upon the natures of the various kinds of entity represented. It remains to discuss how this may best be accomplished.

7.4.1 Logical necessities As things stand, we are assuming that our language may contain various simple predicates F, G, . . . of first-level and finite arity. Our function |:| assigns to each such n-ary F and n-tuple of elements a1 , a2 , . . . , an from the individual domain to a pair of sets of states V , F . V , also denoted |F, a1 , . . . , an |+ , comprises the states which verify F of a1 , a2 , . . . , an , and F , also denoted |F, a1 , . . . , an |− , comprises those which falsify F of a1 , a2 , . . . , an . Our general idea is to provide for necessities involving these predicates, and perhaps more complex predicates composed from them, by imposing restrictions or constraints on |:|. Such additional restrictions will be required, of course, only if we seek to provide for various kinds of non-logical necessities. Logical necessities—or at least those expressible in the language as we have it—are already provided for. Thus consider (A ∧ B) → A and its necessitation 2((A ∧ B) → A). Let us continue to assume that 2 expresses absolute necessity, conforming to S5 laws, so that we may take all states to be mutually accessible, and suppress reference to the acccessibility relation R. Assume, for reductio, that a state s does not verify 2((A ∧ B) → A), so that some possible state s falsifies (A ∧ B) → A. Then for some t, u, t  (A ∧ B) and u A and s = t  u. Since t  (A ∧ B), there are possible states t 1 , t 2 such that t1  A, t2  B, and t = t 1  t 2 . But then s = t  u = t1  t2  u is not a possible state, because t1  A and u A. Thus no possible state s falsifies (A ∧ B) → A. Hence s verifies 2((A ∧ B) → A). Since the semantic clauses for the logical operators are held constant for all admissible models, this logical necessity will be verified in all models. Obviously other logical necessities statable in our language can be established semantically in the same way. From an essentialist standpoint, this reflects the fact 



132 exact truthmakers, modality, and essence that the semantic clauses for the logical operators reflect, or embody, the natures of the logical functions (negation, conjunction, etc.) corresponding to them—these logical necessities are true in virtue of the natures of the various logical entities involved.

7.4.2 Non-logical necessities As things stand, there are no constraints on our assignment function |:| beyond that it should assign to each simple n-ary predicate F and n-tuple of elements a1 , a2 , . . . , an from the individual domain a pair of sets of states V , F , comprising, respectively, those states which verify F of a1 , a2 , . . . , an and those which falsify F of those individuals. In particular, |:| is free to assign states to any given predicate F independently of any assignments it makes to other predicates. If we wish to provide for the verification of necessities other than purely logical ones, reflecting the natures of non-logical entities, we must impose further restrictions on the assignments to simple predicates. To illustrate how this might be accomplished, suppose we wish to ensure the verification of 2∀x(Fx → Gx), where F and G are some simple 1-place predicates—to have a plausible example of a non-logical but absolute necessity, we might suppose that F and G are ‘. . . is an elephant’ and ‘. . . is a mammal’ respectively. Obviously, 2∀x(Fx → Gx) will not be verified by just any state in any model, since there will be models in which the set of verifiers |:| assigns to F of some individual ai includes states which falsify Gai . Equally obviously, we can exclude falsifiers for ∀x(Fx → Gx) by restricting admissible models to those in which |:|’s assignment to F for given ai is + a subset of its assignment to G (i.e.|Fa|+ i ⊆ |Gai | for all i). This will ensure that no  possible state s falsifies Fx → Gx for any individual in the domain, and so no possible state s falsifies ∀x(Fx → Gx)—so that any state s verifies 2∀x(Fx → Gx). In general terms, models of this restricted kind will be structures S, S♦ , A, R, , C |:| , where C is a set of constraints on assignments to simple predicates. These constraints may take the simple form suggested just now, requiring that the states assigned to V F —the verifiers for F of n-tuples of given individuals—be included among those assigned to V G , where G is some other simple predicate. But they may be more complicated, requiring, for example, that V F be a subset of the intersection V G1 ∩ . . . ∩ V Gk of the verifiers for several predicates Gi . Constraints of both kinds may be thought of as corresponding to real definitions of the kind of entity to which the predicate F refers, and reflecting the idea that the properties which go to make up what it is to be an entity of that kind belong to its instances necessarily. Such constraints will generate what we may call necessities of kind inclusion. These should be distinguished from necessities of kind membership, which correspond to the idea that an individual’s membership of a certain kind is part of its essence or nature (e.g. Aristotle is essentially a human being, this sapling is essentially an oak, etc.). If we wish to provide for these, constraints of a somewhat different kind would be required, to the effect that if some possible state s verifies F of an individual a, no possible state falsifies F of a (i.e. if, for any individual a, s ∈ |F, a|+ then |F, a|− = ∅).10 Obviously constraints of these kinds will immediately and directly ensure the necessity of corresponding predications. But indirecly, via the semantic clauses 10 Strictly, if, for any individual a, s ∈ |F, a|+ ∩ S♦ then |F, a|− ∩ S♦ = ∅.

7.4 generic versus instantial truthmakers 133 for the logical operators, they will lead to the necessity of many more complex statements—and if the essentialist theory is correct, they will collectively generate all absolute metaphysical necessities (or at least, all those which can be expressed in the language).

7.4.3 Essence, logical consequence, and metaphysical necessity Let L be a first-order modal language as previously described, comprising the logical constants ¬, ∧, ∨, →, ∀, ∃, 2, ♦ and perhaps =, together with individual variables x, y, z, . . . and perhaps some individual constants a1 , a2 , . . . If L contains no constant predicates besides =, we shall be able to express only purely logical necessities, which will be validated by the semantics, as we have seen in the simple case of 2((A ∧ B) → A) in 7.4.1. If L is extended to include some constant non-logical predicates for which we have in mind some definite interpretation, there may be sentences of L which, on that interpretation, express non-logical, metaphysical necessities (such as ‘Elephants are mammals’ etc.). And if we augment the semantics to incorporate constraints of the assignments to such predicates, as described in section 7.4.2, we can view the interpreted language as embodying a theory whose theorems include a class of metaphysical necessities. Of course, this is not yet a formal theory, explicitly formulated in the language L, but it would obviously be straightforward to add formal axioms corresponding to the semantic constraints on |:|. Some of the theorems of this theory—in effect, the non-logical axioms—amount to real definitions, or statements of the nature or essence, of the various kinds of entity to which the non-logical predicates apply. As such they are, if true, immediately or directly necessary. In addition, there will be many more theorems which are consequences of these basic necessities.

7.4.3.1 formalizing theories of essence Basic claims about the natures of things are in general non-logical. Corresponding to them, an essentialist theory will have various non-logical axioms. We may think of them as the formal counterparts of the special, non-logical constraints discussed in 7.4.2. They can conveniently be formulated using the notation we have borrowed from Fine. Thus suppose we think it is (part of) the essence of an elephant to be a mammal, or part of the essence a mammal to breathe air. If our language contains the corresponding predicates (say, Elephant(x), Mammal(x), Air-breathing(x)). Then there may be constraints on |:| to the effect that |Elephant(x)|+ ⊆ |Mammal(x)|+ and |Mammal(x)|+ ⊆ |Air-breathing|+ . Corresponding to these, we may have the axioms: Ax(Elephant) Ax(Mammal)

2Elephant ∀x(Elephant(x) → Mammal(x)) 2Mammal ∀x(Mammal(x) →Air-breathing(x))

In general, our theory will comprise a set of axioms of the general form: 2x A. There may also be axioms of the special form ∀x(F(x) → 2x F(x)), corresponding to essentialities of kind-membership. The issue broached in 7.1.5 can now be re-stated a little more perspicuously. First, we can distinguish between things said which are true in virtue of the nature of x for some single thing x, and those which are true in virtue of the natures of two or more things x, y, . . . and so true in virtue of the natures of some things (i.e. such that ∃x1 , . . . , xn 2x1 ,... .xn p ). Further, among things said which are true in virtue of the nature

134 exact truthmakers, modality, and essence of some single thing x, we may distinguish between those which are immediately true in virtue of x’s nature, and things which are mediately true in virtue of x’s nature (i.e. without involving the natures of any other things). Let us say that p is true in virtue of x’s nature (alone) if p is immediately or mediately true in virtue of x’s nature (alone). The issue is how to pin down this idea in our theory. Let us now understand 2x p to express the idea. Clearly, since things true in virtue of x’s nature include things mediately true, we should expect 2x p to conform to some form of closure under logical consequence. But what form, exactly? It is obvious that we cannot expect closure under full classical consequence. For any logical truth is a classical consequence of the empty set of premises, and so of any premise whatever. But it is certainly not true in virtue of Socrates’s nature that p → p, etc. Little is gained by restricting to consequence in minimal logic, since this still allows for many irrelevant consequences. Clearly some much more stringently restricted form of consequence is called for. What is needed is a further restriction of minimal logic—a sub-minimal logic of what is sometimes called tautopical implication.

Appendix: Minimal and sub-minimal logics In a natural deduction setting, minimal logic results from classical by replacing the strong rule of reductio ad absurdum (  ⊥ ⇒  − ¬A  A) by a weak rule of negation introduction: ∧I ∧E

  A,   B ⇒ ,   A ∧ B   A∧B ⇒   A   A∧B ⇒   B

∨I ∨E

  A ⇒   A∨B   B ⇒   A∨B   A ∨ B,   C,   C ⇒ ,  − A,  − B  C

→I →E ¬I ¬E

  B ⇒ −A  A → B   A → B,   A ⇒ ,   B   ⊥ ⇒  − A  ¬A   ¬A,   A ⇒ ,   ⊥

Lewis’s explosive proof that anything follows from a contradiction fails in minimal logic, as does Disjunctive Syllogism (on which, in its usual form, the proof of explosion relies). The theorems of minimal logic are a proper subset of those of both classical and intuitionist logics. But where A is any theorem of minimal logic, A will be a minimal consequence of any premise(s) —just assume , conjoin A and detach it again by the conjunction rules, to obtain   A. So 2x cannot be closed under minimal consequence. This means we are looking for a consequence relation which is a proper sub-relation of minimal consequence, and, indeed, a relevant consequence relation.

Some sub-minimal consequence relations Subtension We write A ⇒ B for A tautologically implies B.

appendix 135 Define A subtends B to mean: A ⇒ B is a substitution instance of A0 ⇒ B0 where A0 is not a counter-tautology nor is B0 a tautology. Lewis’s explosive deduction fails: we have neither: A ∧¬A subtends B nor A subtends B ∨¬B. So far so good. The definition of subtension ensures that this relation will never hold just in virtue of the logical truth of the conclusion or the logical falsehood of the premise. But as Makinson—my source for this relation11—observes, it has several drawbacks: (i) The scope of the relation depends upon the choice of primitive operators. (ii) It is not transitive. (iii) The rule of conjunction in the consequent fails (i.e. A  B, A  C  A  B ∧ C), (iv) as does the rule of disjunction in the antecedent (i.e. A  C, B  C  A ∨ B  C).

Tautopical consequence A tautopically implies B iff A ⇒ B and every propositional letter occurring in B already occurs in A. This, too, obviously avoids explosion, and appears to have all the advantages of substension without some of the drawbacks. The losses Makinson notes are: (i) The axiom scheme/rule for disjunction introduction—obviously we can’t in general go from A to A ∨ B. (ii) Contraposition—since we are reversing the relation in contraposition, and the relevance constraint is asymmetric, we can’t pass from A ⇒ B to ¬B ⇒ ¬A. There is an obvious idea underlying tautopical implication. The relevance constraint is designed to ensure that the implication does no more than spell out what is in some sense already contained in its premises. Of course, precisely because the constraint is formal—in terms of syntactical as distinct from semantical features—the notion of containment is itself purely formal. We shall need to consider whether it, or indeed any other purely formal constraint, can capture what we want. It will assist us in considering this question to have some definite examples before us. Examples include both those where we might claim to have knowledge of something’s nature a priori and those in which only a posteriori knowledge is obtainable. There is some advantage in choosing examples of the first sort, at least to start with.

natural number A natural number is a finite cardinal number. I shall suppose that we have a definition of cardinal number in general—in fact, for definiteness, I shall suppose that we may define cardinal number explicitly, in terms of the number operator defined implicitly by the usual neo-Fregean route. That is, we take the number operator to be defined by Hume’s principle [#F = #G ↔ F ≈ G, where the right-hand side abbreviates a secondorder sentence expressing that there is a one-one correspondence between the Fs and 11 and for the next to be discussed—see Makinson (1973).

136 exact truthmakers, modality, and essence the Gs]; and Card(x) =def ∃F x = #F. We shall assume that the ancestral R* of a relation R is defined as in Begriffsschrift. We then explicitly define: Natural(x) =def x = 0 ∨ S* (0, x)

the natural number 17 I shall suppose that to be 17 is just to be the seventeenth successor of 0, that is: 17 =def S(S . . . S(0))

where, exploiting the fact that the successor relation is a function, S . . . S(0) abbreviates a term with sixteen iterations of S applied (successively!) to 0, S(0), S(S(0)), . . . as its arguments. Our aim is to clarify what it is for something to be true in virtue of the nature of x, for some single value of x. Our main question is: To what (kind of) closure principle does our operator 2x conform? We are setting aside, as not our present concern, what may be true in virtue of the nature of x together with the natures of some other things, y, z, etc. Some things—those which are encapsulated in the definition of x—will be immediately true in virtue of x’s nature. But we are allowing that there may also be things true in virtue of the nature of x alone which are indirectly, mediately so true, in the sense that, while no appeal to the natures of other things besides x is involved, there may be some steps of inference from the definition of x itself. Here are some examples of such things that might plausibly be held to be indirectly true in virtue of the nature of natural number: (1) Every natural number has another natural number as its immediate successor. (2) Every natural number has infinitely many successors. (3) Every natural number has only finitely many predecessors. There are some corresponding things true in virtue of the nature of 17: (4) 17 is immediately succeeded by another natural number. (5) 17 has infinitely many successors. (6) 17 has only finitely many predecessors. Our present conjecture is that 2x conforms to closure under tautopical consequence. Mechanical application of the definition to our informal examples would clearly be wrong-headed. It is quite obvious that the things said to be indirectly true in virtue of the nature of natural numbers, or of 17, involve the use of expressions which do not appear in the corresponding definitions. Some massage—or regimentation— is called for. Of course, we must anyway extend the relevance constraint so as to cover items of vocabulary other than propositional letters. Roughly, the constraint we shall want is that the consequent shall contain no non-logical expression (predicate, functional expression, etc.) which does not already figure in the antecedent. But equally importantly, we shall need to re-express our examples (1)–(6) so as to dispel the appearance that they involve new non-logical vocabulary, For example, we can re-express (1) as: (1◦ ) ∀x(Natural(x) → ∃y(x = y ∧ Natural(y) ∧ S(x, y))

appendix 137 Since (1◦ ) says of every natural number that it has a distinct natural number as its immediate successor, it might be taken to imply, even if it does not explicitly assert, that any given natural number has infinitely many (ancestral) successors. Strictly speaking, however, (1◦ ) implies this only in conjunction with other claims about the successor relation—the claims expressed by the other three ‘elementary’ Peano axioms: (DP1) Natural(0) (DP2) ¬∃x(Natural(x) ∧ S(x, 0)) (DP3) ∀x∀y((Natural(x) ∧ Natural(y) ∧ S(x, y) → ∀z((S(x, z) → z = y) ∧ (S(z, y → z = x))) But (DP1) is obviously entailed by the definition of natural number, and so too is (DP2)—since according to that definition, x is a natural number only if it is 0 or one of its successors. (DP3) can be reckoned as merely spelling out the definition of the successor relation. More interestingly, perhaps, we can re-express (2) as the conjunction of all four elementary axioms ((DP4) = (1◦ )). (3) could be re-expressed as (3◦ ) ∀y(Natural(y) → ∃n(Natural(n) ∧ ∃n z S(z, y))

Proof-theory and semantics Our definition of tautopical implication tells us little about what rules of inference are suitable (i.e. conform to its relevance constraint), and nothing about what kind of semantics would validate them. We know, so far, only that disjunction introduction and contraposition must fail. But which rules are acceptable, and which other rules, if any, must be rejected?

direct rules With the exception of the disjunction introduction rule, it is clear that there is no problem over those rules—the so-called direct rules—which do not involve appeal to subsidiary proofs or deductions, and so discharge of assumptions. That is, we may retain without change: ∧I ∧E →E ¬E

  A,   B ⇒ ,   A ∧ B   A∧B ⇒   A   A∧B ⇒   B   A → B,   A ⇒ ,   B   ¬A,   A ⇒ ,   ⊥

indirect rules The problem with indirect rules may be illustrated very simply by considering the following deduction: 2 3 3 2,3 2

(1) (2) (3) (4) (5) (6) (7)

(A ∧ B) → B ¬B A∧B B ⊥ ¬(A ∧ B) ¬B → ¬(A ∧ B)

Assn Assn 1,3 → E 2,4 ¬E 3,5 ¬I 2,6 → I

138 exact truthmakers, modality, and essence It is to be observed that since the formula on line (1) is itself a provable tautological implication, the deduction allows us to pass from a provable implication to its contrapositive (which is, of course, classically, also a provable implication). However, while the initial provable implication satisfies our relevance constraint—since any nonlogical expression occurring in its consequent B must already occur in its antecedent A ∧ B—the same may not be true of the provable implication on line (7). For there may very well occur non-logical expressions in A, and so in its consequent ¬(A ∧ B), which do not already occur in its antecedent, B. Thus the most obviously problematic step is that of ¬I at line (6), but in fact the next step, of → I, is equally problematic. The problem, in general, is that in the application of our indirect rules, assumptions are discharged, with the possible result that the resulting reduced pool of assumptions (which may of course be empty, as at our final step) fails to contain non-logical expressions which do occur in the discharged assumption, so that we end up with a consequence which fails to satisfy our relevance constraint. We may forestall this unwanted outcome by weakening the indirect rules so as to require that the assumption set—or in the limiting case where all assumptions are discharged, the antecedent of the theorem derived—contains all the necessary nonlogical expressions. This can be accomplished by adding an instance of the law of excluded middle for each discharged assumption. That is, we modify the indirect rules as follows: ¬I + → I+ ∨ E+

, A  ⊥ ⇒ , A ∨ ¬A  ¬A , A  B ⇒ , A ∨ ¬A  A → B12   A ∨ B; , A  C; , B  C ⇒ , , , A ∨ ¬A, B ∨ ¬B  C

We shall consider in due course how this modification, which may appear no more than an ad hoc fix, might be motivated, when we enquire what semantic justification might be available for it. First, let us see how this affects our problematic deduction. Displaying all assumptions explicitly, that now runs as follows: ¬B A∧B B ¬B, A ∧ B ¬B, (A ∨ B) ∨¬(A ∨ B) B ∨¬B, (A ∨ B) ∨¬(A ∨ B)

(1) (2) (3) (4) (5) (6) (7)

(A ∧ B) → B ¬B A∧B B ⊥ ¬(A ∧ B) ¬B → ¬(A ∧ B)

Assn Assn 1,3 → E 2,4 ¬E 3,5 ¬I 2,6 → I

It can be easily seen that the problem of the relevance constraint no longer arises, because at the close of the deduction there remains a pool of assumptions which 12 Note that we have also adjusted the rule to require that A shall actually be among the assumptions on which B is proved. This change does not significantly affect the power of the rule. For we can always, given   B, where A is not contained in , add A to our assumptions on which B depends, by (i) assuming A, (ii) conjoining it with B to obtain A ∧B on assumptions , A, and (iii) inferring B, so as to obtain , A  B. Parallel adjustments have been made to the other indirect rules.

appendix 139 must include all the non-logical vocabulary occurring in the final contraposed implication. The indirect rules as usually formulated are apt to appear perfectly sound and unproblematic. How, for example, if we can deduce a contradiction on the assumption that A, can we possibly go from truth to falsehood by concluding—on whatever other assumptions we have made—that ¬A? The answer, of course, is that we cannot do so, provided that A is true or false.We tacitly assume that this is so. But of course, in the context of a sub-minimal logic, this is not an assumption which we can generally make. LEM is not a theorem of minimal logic. It is precisely to avoid its assumption that we have weakened the negation rules by dropping DNE and replacing the strong classical reductio rule by the intuitionistically acceptable weak rule. The situation is comparable to that in intuitionistic reasoning—we cannot in general employ the classical rule of DNE, but that does not mean that there are never circumstances under which its use may not be justifiable intuitionistically. Provided that we are concerned only with effectively decidable propositions, it may be safely employed.

semantics From a semantic point of view, the situation as I see it is as follows. We are not assuming bivalence. More specifically, we are working with a background semantics— a form of truthmaker semantics in which propositions, when true or false, are made so by states, the elements of the underlying state-space S, —which, for any given proposition A, may verify A or falsify A or—and this is the key point here—leave A s truth-value unsettled. Under our semantic definition of validity and consequence,  | A will be true just when, for any state s which verifies each premise B ∈ , any state t compatible with s is compatible with a state u which verifies the conclusion A. An unrestricted rule—derived or primitive—of contraposition, for example, is problematic in this context because states compatible with verifiers for the conclusion of  | A may well leave the truth-values of some of the statements in  unsettled—so that applying the rule to obtain, say,  − B, ¬A| ¬B may result in a new consequence relation in which the conclusion includes non-logical expressions (e.g. proposition letters) whose truth-values are left undetermined by states compatible with a state which verifies the (new) premises. To avoid this unwanted consequence, we need additional assumptions to ensure that states which are compatible with a verifying state for the premises already ensure that all statements which form part of the conclusion have truth-values. This is what is accomplished by adding relevant instances of the LEM—adding them amounts to saying that, provided that the conclusion has a determinate truth-value at all, it must be true if the premises are.

disjunction introduction We may now return to the question left unresolved earlier, regarding the rule of disjunction introduction. We know, of course, that the unrestricted rule must be

140 exact truthmakers, modality, and essence rejected. However, in the light of what has been said in regard to the indirect rules and discharged assumptions, there is an obvious way to weaken the rule so as to render it acceptable—viz. to allow introduction of a new disjunct, with the proviso that it is truth-valued. That is, the amended rule is: ∨I +

  A ⇒ , B ∨ ¬B  A ∨ B   B ⇒ , A ∨ ¬A  A ∨ B

8 S5 as the Logic of Metaphysical Modality: Two Arguments for and Two Arguments against 8.1 Arguments for 8.1.1 The logic of absolute necessity Some kinds of modality are commonly taken to be relative. On the standard account,1 relative necessity is analysed in terms of logical necessity—where 2 is read ‘it is logically necessary that’, ‘It is physically necessary that A’, for example, is defined as 2( → A), where  is a conjunction of physical laws. Logical necessity is taken to be absolute, at least in the sense that it is not to be itself analysed as a form of relative necessity. But it may well be thought to be absolute in another, and rather stronger, sense—in which to be absolutely necessary is to be necessary without qualification or completely unconditionally. It is not clear that this notion of absoluteness can be explained without circularity. In terms of worlds, the obvious characterization is that it is absolutely necessary that A iff A is true at absolutely every possible world, without restriction or exception—absolute necessity is explained by absolutely unrestricted quantification over possible worlds.2, 3 Given this explanation, we can mount a quite simple argument for the conclusion that the modal logic of absolute necessity is S5. If it is to be absolutely necessary that A, it must be true that A at absolutely every possible world without restriction. In terms of admissible models, this means that if 2abs A is to be true at a world wj in a model M, A must be true at each world wk in M’s set of worlds W—in particular, there must be no wk ∈ W inaccessible from wj . For if there were such a world, 2A might be true at wj even though there is a possible world at which A is false. But then 2A would be at best only relatively, not absolutely, necessary. But if each world in the set W of worlds 1 q.v. Smiley (1963). There are difficulties with the standard account (q.v. Humberstone 1981b), in view of which some revision is probably required. For further discussion, see Hale and Leech (2016), reprinted as chapter 9 of this volume. These complications do not substantially affect the argument discussed here. 2 This might be thought circular because ‘possible world’ needs to be understood as ‘absolutely possible world’. I don’t myself think this is obviously correct, but can’t discuss it here. 3 Those who, like me, do not think analyses in terms of worlds can be philosophically illuminating might explain absoluteness in terms of absolutely unrestricted quantification over propositions: 2abs A ↔ ∀q(q2→ A), where q is understood to range over absolutely all propositions, whether expressible in our language or not. For discussion of this and other alternative explanations, see Hale (2015a) or Hale (2013a), ch.4. Bob Hale, S5 as the Logic of Metaphysical Modality: Two Arguments for and Two Arguments against In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0009

142 s5 as the logic of metaphysical modality in any model M is accessible from every world in W, it follows that the accessibility relation R must be an equivalence relation, i.e. reflexive, symmetric, and transitive. And that suffices for the S5 axioms. If metaphysical necessity is a kind of absolute necessity, it follows that its modal logic is likewise S5. But is the antecedent of that conditional true? One consideration in favour of taking it to be so is that Kripke, whose work has done more than any other to secure acceptance of a metaphysical notion of necessity, explicitly states that the kind of necessity he has in mind is ‘necessity in the highest degree’ (Kripke 1980, p.99. See also p.164). Another argument relies on a certain conception of the relation between logical and metaphysical modality. On one view, metaphysical lies between logical and physical necessity, weaker than the first but stronger than the second. On another, metaphysical necessity is a kind of relative necessity. On another view still, logical necessity is a species of metaphysical necessity, so that while logical necessity is stronger than metaphysical in the purely technical sense that the former implies but is not implied by the latter, in another sense metaphysical necessities are every bit as necessary as logical necessities. In particular, if logical necessity is absolute, then so is metaphysical necessity in general. This last view is especially natural and plausible if metaphysical necessity is explained as what holds true in virtue of the nature of things in general, and logical necessities are those things which are true in virtue of the natures of just the logical entities—the various logical functions such as negation, conjunction, etc. For there would then be no reason to regard logical necessities as stronger than metaphysical necessities in general, except in the purely technical sense already noted—the difference would just be that logical necessities hold in virtue of the natures of just logical entities, whereas metaphysical necessities hold in virtue of the natures of things in general.4

8.1.2 Williamson’s argument If the argument just rehearsed is sound, it shows that the logic of metaphysical modality is at least as strong as S5, the strongest normal modal logic. It does not show that S5 is exactly the right logic, since for all the argument shows, that might be some stronger, non-normal logic. An argument given by Timothy Williamson at least closes that gap, and if successful, provides an independent reason for taking S5 to be the correct logic. Williamson’s argument focuses on propositional modal logic.5 The question is: which modal logic captures all and only those truths which are metaphysically universal—those truths about metaphysical modality which are sufficiently general in the following sense. Let A be any formula of our propositional language. Then by its universal generalization is meant the result of substituting distinct propositional

4 An argument for the absoluteness of logical necessity is given in Hale (2013a), pp.105ff. The view of logical necessity as a species of metaphysical necessity is put forward by Kit Fine in Fine (1994), and further defended in Hale (2013a), ch.4.5. For useful discussion of other views about the status of metaphysical necessity as absolute or otherwise, see Nolan (2011), Rosen (2006). 5 This is not a significant limitation, given that quantified modal logics differ from propositional ones only over the underlying non-modal logic, i.e. they add no new modal principles.

8.1 arguments for 143 variables for distinct atomic sentences throughout A, and prefixing universal quantifiers on each variable. For example, the universal generalization of 2p → 22p is ∀X(2X → 22X). A is metaphysically universal iff its universal generalization is true on its intended interpretation, which treats the truth-functional operators as usual and interprets 2 as metaphysical necessity, which Williamson glosses as ‘what could not have been otherwise, what would have obtained whatever had obtained’.6 The conclusion of the argument (q.v. Williamson 2013, pp.110–15) is that ‘if possibility and necessity are non-contingent, then MU [the system which is sound and complete for metaphysical universality] is exactly the system S5’. Possibility and necessity are noncontingent if and only if whatever is possible or necessary is necessarily so, so that the antecedent amounts to assuming that the axioms 2A → 22A (S4) and ¬2A → 2¬2A (E)7 are metaphysically universal. Williamson assumes an axiomatization of S5 with modus ponens as the sole rule of inference and as axioms all formulae of the form 2A where A is a truth-functional tautology, all instances of the K, T, S4 and E schemas (hence avoiding the need for a primitive rule of Necessitation). No one doubts that MU is at least as strong as T. Thus given the antecedent assumption, MU is at least as strong as S5. The argument that it is no stronger, and is therefore exactly S5, turns on a theorem of Schiller Joe Scroggs (Scroggs 1951), which asserts that every proper extension of S5 is the logic of a single finite frame. Williamson uses this result to show that any proper extension of S5 will have a theorem which is not metaphysically universal. In essence, the argument runs as follows.8 Let S5+ be a proper extension of S5 and its single frame be the pair W, R, where W has just n worlds. Then S5+ has the theorem: Alt n

2P1 ∨ 2(P1 → P2 ) ∨ . . . ∨ 2((P1 ∧ . . . ∧ Pn ) → Pn+1 )

—there can be no countermodel, because a countermodel would require at least n + 1 worlds. For a countermodel would have to make each disjunct of Alt n false. To falsify the first disjunct, we would need a world w1 at which P1 is false. To falsify the second, we would need world w2 at which P1 → P2 is false, and hence at which P1 is true but P2 false, so that w2 = w1 . Generally, to falsify the ith disjunct we shall require a world wi distinct from each of the worlds falsifying the preceding disjuncts. But Alt n has n + 1 disjuncts, so a countermodel would require n + 1 worlds. But S5+ has only models with n worlds. So there is no S5+ countermodel for Alt n . However, Alt n is not metaphysically universal. For under an interpretation on which each Pi says that there are at least i donkeys, each disjunct of Alt n is false, since it is metaphysically possible that there should have been fewer than i donkeys, so that Alt n itself is false and so not metaphysically universal.9

6 See Williamson (2013), p.93, and p.3 for the characterization of metaphysical necessity. 7 Equivalently ♦A → 2♦A—Williamson is working in a system with 2 primitive and ♦ defined. 8 See Williamson (2013), p.111 for a fuller statement. 9 As Williamson observes, there are two complications. First, Scroggs’s result requires extensions to be quasi-normal (i.e. extends K and is closed under modus ponens and uniform substitution, but need not satisfy the rule of necessitation) and second, it does not extend to logical consequence. Williamson points out that the first gap is merely apparent (Williamson 2013, p.111, fn.37) and gives a separate argument that the usual S5 consequence relation is right for metaphysical modality (pp.112–14).

144 s5 as the logic of metaphysical modality Clearly this argument, if sound, is enough to close the gap left open by the argument of the preceding section. Does it do more than that? Williamson grants that the assumption that (metaphysical) necessity and possibility are non-contingent is controversial. It might seem that in the absence of a supplementary argument for the S4 and E principles, we have at best a limiting result: MU is no stronger than S5. Williamson provides no direct argument for these principles.10 However, he does present a kind of challenge-argument to those who would stop somewhere short of S5. We may assume that no one will propose anything weaker than T. It is unclear how anyone who proposes T or a system between it and S5 can provide a general argument to show that every metaphysically universal formula is a theorem of their favoured system. There is nothing comparable to Scroggs’s result for systems weaker than S5, and no other way apparent to secure the desired conclusion. Of course, Williamson grants, one might propose a weaker system and try to deal piecemeal with proposed counterexamples—i.e. try to show that any purported false instance of a universal generalization of a theorem of their system is in fact true, and that any purportedly true non-theorem of their system has a false instance. That is, he grants, ‘better than nothing, but a systematic, principled argument argument for the equation would be far more satisfying’ (ibid., p.115).

8.2 Two against The principal targets of those who have thought the right logic for metaphysical modality must be one weaker than S5 have been the S4 and B principles (which, in the presence of the K and T principles plus the usual rules, suffice for S5). In this final section, we comment on the main arguments levelled against them.

8.2.1 Against S4 Nathan Salmon (1989 and earlier work cited therein) argues11 that the logic of metaphysical necessity cannot be S4 (or, therefore, S5), on the ground that while a certain table could not have been composed of matter significantly different from that from which it is actually composed, it could have been made from slightly different matter, and had it been made from slightly different matter, it would have been possible that it should have been made from significantly different matter from that from which it is actually composed, even though that is not actually possible—so that something that is possibly possible is not possible simpliciter, contradicting the S4 principle that ♦♦A → ♦A (and its equivalent 2A → 22A). Salmon is obviously relying on a version of the principle of necessity of origin, which, in the form Kripke propounds it (Kripke 1980, p.114, fn.56), runs: ‘If a material object has its origin from a certain hunk of matter, it could not have had its origin in any other matter.’ Kripke himself hints that he thinks that some qualifications to the principle may be needed, and explicitly mentions the vagueness of the notion of a hunk of matter as one source of problems. Thus it may be thought that he would endorse a modification along the lines which others have thought necessary or desirable, so that the principle asserts 10 He does, however, refer to the main arguments against the S4 and B principles, indicating that he thinks them to have been convincingly answered. See Williamson (2013), p.44, fn.18. 11 In Salmon (1989), pp.4–8. See also Salmon (1981), pp.229–52. Salmon’s argument is based on one given in Chandler (1976).

8.2 two against

145

that if a material object originates from a certain portion of matter, it could have originated from a slightly different portion of matter, but could not have originated from a substantially different portion of matter. It is some such version of the principle that Salmon’s objection assumes. Since it is by no means obvious that the presupposed principle—whether in its original or some qualified form—is true, it would be open to a defender of S5, especially given an independent argument in favour of S5 as the correct modal logic for metaphysical necessity, to view Salmon’s objection as telling against the origin principle rather than against S5. However, Salmon’s argument may be resisted without jettisoning the necessity of origin. For the argument evidently relies on the complex counterfactual claim that if the table had been composed of matter slightly different from that from which it is actually composed, it would have been possible that it should have been composed of matter differing only slightly from that, but more substantially from that of which it is actually composed. And it is certainly open to a proponent of the necessity of origin to deny this conditional, by insisting that the necessity of origin principle should be understood as requiring that the table could not have been composed of matter more than slightly different from that from which it is actually composed, where the phrase ‘that from which it is actually composed’ refers rigidly. In this case, it would be fallacious to argue: ‘Suppose that the table is actually composed of matter m, but that, as might have been the case, it had been composed of slightly different matter m . Then it would be possible that it should have been composed of slightly different matter from that, i.e. from m .’12

8.2.2 Against B Perhaps the most discussed argument against the B principle, that A → 2♦A, is due to Michael Dummett, responding to Kripke’s claim that, contrary to what might be thought to be plain common sense, it is not true, given that there neither are nor ever have been any animals of the sort described and pictured in the stories and paintings which form parts of the myth of unicorns, that there might have been unicorns.13 Dummett’s argument appears to run as follows: Although there are no unicorns (that is: there are no animals conforming to the pictures and descriptions of unicorns in the tales and myths), there might have been such animals. As the pictures and descriptions do not settle whether unicorns would belong to the order Artiodactyla (even-toed ungulates, like deer) or to the order Perissodactyla (odd-toed ungulates, like horses), such animals might have been of either of these two orders. In the one case, they would have been of one species, in the other, of another quite distinct species. In world terms, as Dummett puts it: there are no unicorns in the actual world w, but there is a possible world u in which there are unicorns, which belong to the order Artiodactyla, and another possible world v in which there are also unicorns, which in that world belong to the order Perissodactyla . . . . In world u, any animal, to be a unicorn, must have the same anatomical structure as the unicorns in u, and hence, in particuar, must belong to the order Artiodactyla. It follows that the world v is not possible relatively to u, and, conversely, that u is not possible relatively to v. How about the actual world w—is that possible relatively to either u or v? . . . u is a world in which it holds good 12 This objection to Salmon’s argument is made in Hale (2013a), p.128, fn.18. Essentially the same objection is made by Sonia Roca-Royes (Roca-Royes (2006), section 3.2.1). 13 See Kripke (1980), pp.23–4, 156–7, Dummett (1993a), especially pp.346–7.

146 s5 as the logic of metaphysical modality that unicorns are necessarily of the order Artiodactyla, whereas in w it is possible for unicorns to be of the order Perissodactyla. Since a proposition necessarily true in u is possibly false in w, w cannot be possible relatively to u, although u is possible relatively to w. The relation of relative possibility (accessibility) is therefore not symmetrical. (Dummett, 1993b, p.346)

Since the B principle: A → 2♦A holds only if the accessibility relation is symmetrical, it follows that the B principle is false. Ingenious as it is, and plausible as it may seem, Dummett’s argument is certainly not irresistable. It evidently requires that there be a proposition p, expressible by the words ‘Unicorns belong to the order Artiodactyla’, which is such that both ♦2p and ♦¬p. But, starting from the point that the pictures and descriptions constituting the myth do not settle even the order to which animals conforming to them would belong, it may be argued that there is no such proposition. For it follows from this that the pictures and descriptions do not suffice for the introduction of ‘unicorn’ as a term for a certain animal species. Of course, that does not prevent us from employing ‘unicorn’ as a general descriptive term, meaning ‘horse-like creature with a single spiralling horn projecting from its forehead’. Let us use U(ξ ) for this predicate. Then it is evidently (metaphysically) possible that there should have existed creatures answering to this description, i.e. ♦∃xU(x), even if there are not in fact any such, i.e. ¬∃xU(x). Further, ‘ξ belongs to the order Artiodactyla’ (briefly A(ξ )) is a predicate in good standing, as is ‘ξ belongs to the order Perissodactyla’ (briefly P(ξ )). And it is further evidently possible both that there should have been creatures satisfying both U(ξ ) and A(ξ ) and that there should have been creatures satisfying both U(ξ ) and P(ξ ), i.e. ♦∃x(U(x) ∧ A(x)) and ♦∃x(U(x) ∧ P(x))—as we might less perspicuously (and potentially quite misleadingly) put it, there might have been unicorns of the order Artiodactyla and there might have been unicorns of the order Perisssodactyla. Further still, we may agree that had there been creatures satisfying these complex predicates, they would have necessarily satisfied A(ξ ), or P(ξ ), depending upon which possibility was realized. In world terms, the proposition that there are horse-like animals . . . which are necessarily of the order Artiodactyla is possibly true at the actual world w (because true at u which is possible relative to w). And the proposition that there are horse-like animals . . . of the order Perissodactyla (and so not of the order Artiodactyla) is likewise possibly true at w (because true at v which is also possible relative to w). But we cannot conclude from this that there is a proposition which is necessarily true at u and so possibly necessarily true at w, but also false at v, and so possibly false at w—from which it would indeed follow that w cannot be possible relative to u, so that, since u is possible relative to w, the relation of relative possibility cannot be symmetric. In purely modal terms, what is true is only that ♦∃x(U(x) ∧ 2A(x)) and ♦∃x(U(x) ∧ ¬A(x)). There is nothing here that conflicts with the B principle, or with the symmetry of relative possibility, as there would be if we had a pair of true propositions ♦2p and ♦¬p.14 14 This response seems to me to fit well with Kripke’s remark ‘Perhaps according to me the truth should not be put in terms of saying that it is necessary that there should be no unicorns, but just that we can’t say under what circumstances there would have been unicorns’ (Kripke 1980, p.24). A quite closely related, but distinct, criticism of Dummett’s argument is given by Marga Reimer (Reimer 1997). Another, significantly different, criticism, drawing upon a reformulation of Gareth Evans’s distinction between ‘deep’

appendix 147

Appendix: Note on S5 and ‘true in virtue of the nature of x’ 2x and T It is obvious that 2x validates the T axiom, 2x p → p, since if it is true in virtue of the nature of x that p, it must be true that p 2x and S4 Suppose it is true in virtue of the nature of x that p, briefly 2x p. We may ask: what makes that true? That is, what makes it true in virtue of the nature of x that p? Given that it is true in virtue of the nature of x that p, x must exist, and given that it is true that p in virtue of what it is to be x, it cannot but be that p (i.e. 2p). But further, it cannot but be that it is true in virtue of the nature of x that p. For x could not have had a different nature. So necessarily, it is true in virtue of the nature of x that p. So if the essentialist theory is correct, it must be true in virtue of something’s nature that it is true in virtue of the nature of x that p. But what could this be, if not x itself? For it is hardly going to be true in virtue of the nature of anything else. We are thus driven to the conclusion that if it is true in virtue of the nature of x that p, then it is true in virtue of the nature of x that it is true in virtue of the nature of x that p. That is: 2x p → 2x 2x p Thus 2x validates the S4 law. 2x and S5 The analogue of the S5 law for 2x is: ¬ 2x ¬ 2x p → 2x p Suppose that ¬ 2x p, i.e. it is not true in virtue of the nature of x that p. Since x could not have had a different nature, it is not possible that it should have been true in virtue of the nature of x that p, i.e. it is necessarily true that it is not true in virtue of the nature of x that p— 2 ¬ 2x p. But then on the essentialist theory, ¬ 2x p must be true in virtue of the nature of something. What could that be, if not x? Hence we should conclude that it is true in virtue of the nature of x that it is not true in virtue of the nature of x that p—that is, 2x ¬ 2x p. Thus if ¬ 2x p, then 2x ¬ 2x p. Contraposing and applying double negation elimination, ¬ 2x ¬ 2x p → 2x p. Thus 2x validates the S5 law.

Essentialist logical necessity and S4 Suppose 2p, where 2 is logical necessity. Then according to the essentialist theory, p is true in virtue of the natures of some logical entities. For simplicity, suppose p is some proposition true in virtue of the nature of a single logical entity, as—on the essentialist theory—A → A is. So we are supposing 2(A → A). Then 2→ A → A. By our reasoning above, 2→ 2→ A → A. and ‘superficial’ contingency (and necessity) (q.v. Evans 1979, pp.187–9, reprinted in Evans 1985, pp.211–13), is made by Ian Rumfitt (Rumfitt 2010, pp.60–4), who argues that Dummett’s argument requires the premise ‘Necesssarily, animals of which the term “unicorn” (meaning what it actually means) is true are unicorns’, which Kripke can and should reject. The view about species terms taken in the text is close to that defended by David Wiggins in Wiggins (1993) (reprinted in Clark and Hale 1994), according to which such terms are ‘object-involving’, i.e. lack a sense unless there are (or have been) exemplars of the relevant species. Indeed, Rumfitt tells me, and Wiggins confirms, that Wiggins once suggested (in conversation) a diagnosis of the error in Dummett’s argument very similar to mine. A related argument against B given in Stephanou (2000) is criticized in Gregory (2001)—see also Hayaki (2005).

148 s5 as the logic of metaphysical modality That is, 22(A → A). Thus the S4 principle holds in this case. The reasoning obviously generalizes to cases in which p is true in virtue of the nature of several logical entities. So the essentialist theory validates the S4 law in general. Essentialist logical necessity and S5 It is easily shown, in a similar way, that the essentialist theory validates the S5 law.

9 Relative Necessity Reformulated with Jessica Leech 9.1 Introduction Attributions of necessity and possibility are often qualified.1 We may assert, not that something is necessary or possible simpliciter, but that it is logically necessary or possible, or physically, or mathematically so, for example. It is also natural and plausible to suppose that at least some of these kinds of necessity are not absolute, but relative, in the sense, roughly, that what is said to be necessary is not being said to be necessary outright or without qualification, but only on—or relative to—certain propositions taken as assumptions, or otherwise held fixed.2 Thus on one common view, physical necessity is a matter of following from the laws of physics, and physical possibility is compatibility with them. Varying the body of laws, or other propositions, gives us other forms of relative necessity. How, in more precise terms, should the idea that a kind of necessity (possibility) is relative be explained? In broadest terms, it seems that the most promising approach will involve taking some kind of non-relative, or absolute, modality as one’s starting point, and explaining other, relative kinds as in some way relativizations of that basic kind. But how should this be done? What kind of necessity and possibility should be taken as the starting point, and what exactly is it for another kind of necessity and possibility to be a relativized form of that kind? Interest in these questions may be prompted, and answers to them shaped, by at least two quite distinct considerations. Our goal may simply be to achieve a better understanding of the contrast between absolute and merely relative forms of necessity and possibility. This may go with a view to the effect that some kinds of necessity are absolute, and others merely relative, but it need not. One might still seek to elucidate the contrast, even if one thought it empty on one side. For one might suppose that while, if one kind of necessity is to be explained as a relativized form of another, the second kind must not itself be a relativized form of the first, it does not follow that the more basic kind must itself be absolute. To be sure, it might be thought that unless at

1 This chapter was originally published as Hale and Leech (2016). 2 This contrast goes back a long way. For example, Aristotle distinguishes what is necessary outright from what is necessary only relative to certain assumptions (see, for example, Aristotle (1984), An.Pr.30b32–33, De Int.19a25–27; the distinction is implicit in De Soph.El. 166a22–30). Bob Hale and Jessica Leech, Relative Necessity Reformulated with Jessica Leech In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © Bob Hale and Jessica Leech. DOI: 10.1093/oso/9780198854296.003.0010

150 relative necessity reformulated least one kind of necessity is absolute, we shall be involved in an infinite regress. But it is at least not obvious that any such regress must be both infinite and vicious.3 We may be more ambitious. Our hope may be that, by showing that an ostensibly large and varied range of kinds of relative necessity can be exhibited as restrictions or relativizations of a single underlying kind of necessity—logical necessity, perhaps—we can achieve a conceptual reduction of ostensibly different kinds of modality to a single kind. Here we shall simply observe that this is a further, independent aim. As we shall see in due course, there is room for serious doubt whether it can be accomplished. But even if it cannot, that need not preclude an account of relative modalities that casts light on the contrast between absolute and merely relative necessities, and so answers to the first goal. Our aim in this paper is to cater to the first goal, without committing ourselves to the second. We shall proceed as follows. First, we introduce the standard account of relative necessity, and discuss some putative problems for it, most notably propounded by Lloyd Humberstone. Having resolved some of those problems, we move on to discuss Humberstone’s diagnosis and solution to the remainder, and highlight some shortcomings with his approach. We then explore some alternative remedies, ultimately unearthing a much deeper problem with the standard account and some of its amended versions. In response, we offer our own proposed account, and discuss some of its consequences, technical and philosophical. We close with a summary of our findings.

9.2 Relative necessity: the standard account The standard account defines each kind of relative necessity by means of a necessitated or strict conditional, whose antecedent is a propositional constant for the body of assumptions relative to which the consequent is asserted to be necessary. Thus in a now classic treatment, Timothy Smiley wrote: If we define OA as L(T ⊃ A) then to assert OA is to assert that T strictly implies A or that A is necessary relative to T. Since the pattern of the definition is independent of the particular interpretation that may be put on T we can say that to the extent that the standard alethic modal systems embody the idea of absolute or logical necessity, the corresponding O-systems embody the idea of relative necessity—necessity relative to an arbitrary proposition or body of propositions. They should therefore be appropriate for the formalisation of any modal notion that can be analysed in terms of relative necessity. (Smiley, 1963, p.113)4

3 In Field (1989), Hartry Field writes: ‘In this discussion, I have avoided taking a stand on whether even logical necessity should be viewed as “absolute” necessity. One view, to which I am attracted, is to reject the whole notion of “absolute” necessity as unintelligible. Another view, also with some attractions, regards the notion as intelligible but regards the only things that are absolutely necessary as logic and matters of definition’ (p.237, fn.8). Field does not here countenance the possibility of a third view, which allows that the notion is intelligible, but denies that any form of necessity is absolute. Such a position appears comparable to that to which Quine is apparently committed at the end of ‘Two Dogmas’—that the notion of analytic truth is intelligible (because one can explain it as ‘true and immune to revision on empirical grounds’), but empty. 4 Smiley uses ‘L’ as a necessity operator, more usually written ‘2’. See also Anderson (1956) and Kanger (1971).

9.3 humberstone’s problems 151 This kind of formulation of relative necessity has been quite widely endorsed in subsequent work.5 The standard account takes the kind of absolute necessity in terms of which different forms of relative necessity are to be explained to be logical necessity, so that it is relatively necessary that p just when, as a matter of logical necessity, C materially implies p, where C is a proposition of a certain kind—briefly, 2(C → p); and it is relatively possible that p just when p’s truth is not ruled out by C, i.e. when it is not the case that, as a matter of logical necessity, C implies not-p—briefly, ¬2(C → ¬p), or equivalently ♦(C ∧ p). Before considering those features which are genuinely problematic, it is worth briefly surveying some more or less obvious peculiarities of the notion of relative necessity, as formulated on the standard account. First, in the absence of any restriction upon the choice of C, each proposition p will be relatively necessary in an indefinite number of ways or senses. Any conjunction of propositions which numbers p as one of its conjuncts strictly implies p, as does p itself. Forms of relative necessity come cheap, and most of them are entirely without interest. This is not—or not obviously—a crippling drawback. A defender of the standard account can reply that it simply underlines the point that we should not expect interesting, non-trivial kinds of relative necessity to result unless the choice of C is restricted in interesting ways—to the laws of physics, say, or those of mathematics. Second, since every logical necessity is strictly implied by any proposition whatever, every logical truth is C-necessary—necessary relative to C—no matter how C is chosen. But this, again, is not problematic. In particular, the fact that what is logically necessary is automatically relatively necessary in any sense you care to specify is in no tension with the plausible claim that logical necessity is absolute. The important contrast is not with relative necessity, but with merely relative necessity, where it is merely relatively necessary that p if it is, say, C-necessary that p but absolutely possible that ¬p. Thirdly, since C always strictly implies itself, it automatically counts as C-necessary. So on the standard account, assuming that physical necessity, say, is to be analysed as a form of relative necessity, the laws of physics themselves automatically—and so, it would seem, trivially—qualify as physically necessary. But while physical necessity may not be absolute, it may be felt that there is more to the kind of necessity attaching to the laws of physics than their mere self-strict-implication.6 This may be felt to be a more serious objection and we shall have a little more to say about it below. First, we should see why the standard account appears to be in far deeper trouble, for reasons to which Lloyd Humberstone drew attention over three decades ago.7

9.3 Humberstone’s problems Humberstone raises the following problems for the standard account.8 5 Gideon Rosen, for example, writes, ‘given any proposition, φ, we can always introduce a “restricted necessity operator” by means of a formula of the form 2φ P =def 2(φ → P)’. Rosen (2006), p.33. See also Hale (1996), p.93. 6 See, for example, Fine (2002), p.266. 7 See Humberstone (1981b), also Humberstone (2004). 8 In a footnote of Humberstone (1981b, p.34), Humberstone acknowledges Kit Fine as first noticing these problems.

152 relative necessity reformulated

Modal Collapse One might suppose that there are several distinct kinds of relative necessity, and that many of them are factive, in the sense that, where 2C is our relative necessity operator, 2C p → p, for every p. In other words, the characteristic axiom of the quite weak modal logic T holds for 2C . In particular, one would expect any kind of alethic necessity operator to be factive.9 For example, it might be held that both biological and physical necessity are two distinct kinds of necessity which are both relative and factive. However, Humberstone argues that, on very modest assumptions about the logic of the absolute modality operator in terms of which, on the standard account, they are to be defined, there is at most one factive kind of relative necessity. The modest assumption is that the logic of 2 is at least as strong as K—the weakest normal modal logic. Let 2C1 A and 2C2 A be defined by 2(C1 → A) and 2(C2 → A) respectively. Then, Humberstone claims, it is readily proved that 2C1 A and 2C2 A are equivalent. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

2(C1 → p) → p 2(C2 → p) → p 2(C1 → C1 ) → C1 2(C2 → C2 ) → C2 2(C1 → C1 ) 2(C2 → C2 ) C1 C2 2(C1 ↔ C2 ) 2(C1 → q) ↔ 2(C2 → q) 2 C1 q ↔ 2 C2 q

T-axiom for 2C1 T-axiom for 2C2 (1) × C1 /p (2) × C2 /p propositional logic × Necessitation Rule propositional logic × Necessitation Rule (3), (5) × modus ponens (4), (6) × modus ponens (7), (8) × propositional logic, Necessitation Rule (9) × obvious reasoning in K (10) × Def.2C1 , Def.2C2

The key steps in this proof purport to show that when any relative modality 2C defined in the standard way satisfies the T axioms, one can prove C. By Necessitation, 2C, so that for any p, 2C p entails 2p. But 2p obviously entails 2C p.10 So relative modalities satisfying the T axioms collapse, not only into each other, but into logical modalities—relativization is a waste of time.

Unwanted interactions: imposing S4 It seems reasonable to suppose that our absolute necessity operator, 2, satisfies the S4 axiom. If 2 does obey the S4 axiom, every relative necessity operator 2C must do so as well. The proof is simple. It is a theorem of S4 that 2A → 2(B → 2A). An instance of this theorem, substituting C → p for A and C for B is 2(C→p)→2(C→2(C→p)).

9 Alethic modalities are standardly taken to be those which concern ‘modes’ of truth, in contrast with epistemic and doxastic modalities, concerned with knowledge and belief, and deontic modalities, concerned with obligatoriness and permissibility. Factivity does not define alethic modalities, since epistemic necessity is also factive. The collapsing argument is intended to apply to all factive modalities—so that it is epistemically necessary that p iff it is physically necessary that p, for example. 10 Cf. Humberstone (2004), p.50.

9.3 humberstone’s problems 153 But this is simply the S4 axiom for a relative necessity operator 2C defined by 2C p =def 2(C → p). So, if we assume that our absolute necessity operator, 2, satisfies the S4 axiom, it immediately follows that our relative necessity operator, 2C , does as well, i.e. 2C p → 2C 2C p. Yet it may seem both odd and implausible to suggest that if it is, say, physically necessary that p, then it is physically necessary that it is physically necessary that p. More generally, we should expect to be able to define kinds of relative necessity in terms of 2 which do not satisfy the characteristic S4 principles. The standard account is either overly restrictive, or defective, in only allowing one to define kinds of relative necessity that satisfy the S4 axiom. Second, assuming 2 is S4-like, one can show in a similar way that, where 2C1 and 2C2 are any two different forms of relative necessity defined in the standard way, the bizarre ‘mixed’ S4 principle 2C1 p → 2C2 2C1 p holds. Again, it is a theorem of S4 that 2A → 2(B → 2A), so in particular: 2(q → p) → 2(r → 2(q → p)). Hence for any Ci : 2(C1 → p) → 2(Ci → 2(C1 → p)). Yet it appears quite implausible that if something is, say, physically necessary, then it is biologically (or morally, and so on) necessary that it is physically necessary. To give another example, one might define drawer-necessity relative to truths about items in NN’s top desk drawer. For example, it is drawer-necessary (but not, say, physically necessary) that all the pencils in NN’s drawer are blunt. However, the consequences of the standard account include that if it is drawer-necessary that p, then it is physically necessary that it is drawer-necessary that p (and also that if it is physically necessary that p, then it is drawer-necessary that it is physically necessary that p). But surely both results are absurd. It is not a matter of the laws of physics that NN keeps only blunt pencils in his top desk drawer. Nor should it be a matter of drawer-necessity what does or doesn’t follow from the laws of physics.

Out of the frying pan? To the modal collapse proof as it stands, there is a fairly obvious objection. The problem concerns the application of Necessitation at step 9. This rule allows us to to necessitate a proposition only if it has been established as a theorem, and if 2 expresses logical necessity, the proposition to be necessitated must be a theorem of logic. But in this case, the proposition to be necessitated, viz. C1 ↔ C2 , is no theorem of logic, since it depends upon 1) and 2), which are non-logical axioms of a system for relative necessity. Consider 1): its inner antecedent, C1 , will be some non-logical proposition. For example, if we are considering an analysis of physical necessity, it will be a proposition we take to be a physical law, or perhaps a conjunction of such laws. Thus it might be, say, the proposition that force = mass × acceleration, so that 1) will assert that if the proposition that p is strictly implied by the proposition that f = m × a, then (it is true that) p. True or not, this is evidently no theorem of logic. It would seem, then, that the crucial step of Necessitation is illicit, so that the proof does not, after all, lead to modal collapse. A similar flaw occurs in other relevant proofs. In his original paper, Humberstone argues that the standard account is afflicted by two other problems. First, on the assumption of a relative necessity operator 2C satisfying the K, T and S4 axioms, one can show that the S4 axiom must also be satisfied by 2. However, this relies on the

154 relative necessity reformulated modal collapse argument to establish the crucial result, 2C, and so is also invalid.11 Second, he claims that, on the assumptions that one relative necessity operator 2C1 satisfies the K axioms, the axioms B and D, but not T, while another 2C2 satisfies the K axioms and also the T axiom, we can prove that 2C1 must satisfy the T axiom after all. We have chosen not to discuss this problem in detail not only because it is a rather less intuitive combination of modal properties, but also because the proof fails for the same reason as that of problem 1: illicit use of Necessitation. Is the standard account thus vindicated? No. Whilst some of Humberstone’s problems, in the form presented, can be dissolved, those of the unwanted imposition of S4, and ‘mixed’ S4, remain. Hence, there is still good reason to explore alternatives to the standard account. Moreover, in so doing we must take care to ensure that any alternative does not accidentally revive those problems now deemed solved. More importantly, in developing an alternative account, we will discover a far deeper problem for the standard account.

9.4 Humberstone’s solution Although, as we have observed, some of Humberstone’s problems may be set aside as relying on illicit steps of Necessitation, others remain. We wish now to consider his proposed solution to them. To understand it, it is useful to review his diagnosis of the source of the problems he takes to be fatal to the standard account. He writes: [T]he difficulties we have become entangled with result from making substitutions too generally. To see this, let us consider what happens semantically with the idea of relative necessity. We are given an arbitrary modal operator and asked to code it up as some combination of a fixed operator ‘2’ and a propositional constant. But a propositional constant, if it is to be of the same category as the usual propositional variables, is (in effect) assigned a set of worlds in the Kripke semantics whereas the operators we are trying to use these constants to express correspond instead to binary (accessibility) relations between worlds: thus there is bound to be a loss of information in the translation. (Humberstone (1981b), p.36. See also Humberstone (2004), p.51)

His thought, in other words, seems to be as follows: modal operators are quantifiers over worlds, and in particular, relative modal operators are restricted world quantifiers. Thus whereas an absolute operator, 2, quantifies unrestrictedly, so that 2p says that p is true at every world, a relative necessity operator, 2C , quantifies only restrictedly, so that 2C p says that p is true at each of a restricted range of worlds. The question is: how is this restriction to be captured? The standard account seeks

11 The purported proof was supposed to go as follows. By the argument above, 2C. But for any p, we have 2p → 2(C → p), and by hypothesis, 2(C → p) → 2(C → 2(C → p)). But from these it follows that 2p → 2(C → 2(C → p)). But since, by the K axiom for 2, 2(C→2(C→p))→(2C→22(C→p)), it follows that 2p → (2C → 22(C → p)). And since 22(C → p) → 2(2C → 2p) and 2(2C → 2p) → (22C → 22p), it follows that 2p → (2C → (22C → 22p)). But we can permute and detach the second and third antecedents, so that 2p → 22p. Thus, the assumption of a relative necessity operator 2C satisfying the K, T and S4 axioms forces the S4 axiom for 2.

9.4 humberstone’s solution

155

to capture it by relativizing to a propositional constant—a constant which expresses a proposition true at just those worlds in the restricted range intended. Instead of interpreting the relative necessity claim as that p is true at each of a restricted range of worlds, it interprets it as the claim that the conditional C → p, where C is the propositional constant, is true at all worlds. The trouble with this, Humberstone thinks, is that it loses vital information about the distinctive accessibility relation in terms of which a relative modal operator needs to be understood. What distinguishes one kind of relative necessity from another, he thinks, is that each corresponds to a different accessibility relation. Thus what is physically necessary at a given world is what holds true at every world that is physically possible relative to that world, whereas what is epistemically necessary is what holds true, rather, at every world that is epistemically possible with respect to that world. These are quite different relations, but the standard account does not do justice to their difference. In more detail, if R is a binary relation on some set W, and S is a subset of W, we define the range-restriction of R to S symbolized RS thus: xRS y iff xRy and y ∈ S. . . .translating (or defining) OA as 2(C → A) amounts to taking the accessibility relation for “O” to be the range-restriction of that for “2” to the set of worlds at which C holds. Thus instead of being able to cope with an arbitrary collection of modal operators, we are forced to deal only with collections whose accessibility relations are range-restrictions of a single relation, and it is this circumstance which underlies the difficulties . . . (Humberstone 1981b, p.36)

In terms of this diagnosis, one can, he points out, give a straightforward explanation of the difficulties. Thus defining the relative necessity operators 2C1 and 2C2 by: 2C1 A iff 2(C1 → A) and 2C2 A iff 2(C2 → A) amounts to taking their associated accessibility relations R1 and R2 to be range-restrictions of the underlying accessibility relation R associated with 2, so that they are just sub-relations of R with the same domain W but (proper) subsets S1 and S2 of R’s range W as their ranges. But then the supposition that 2C1 and 2C2 both obey the T principle is just the supposition that R1 and R2 are both reflexive—whence, since each has domain W, it must have the whole of W as its range after all. So the supposition of reflexivity undoes the range-restriction—R1 and R2 both end up with the same range as R, and hence as each other, and since they share the same domain, they are just the same relation by different names. Hence collapse. Similar explanations, he suggests, may be given for the other difficulties. See Humberstone (1981b), pp.36–7. It does not, of course, follow from the fact that some of Humberstone’s arguments rely on illicit Necessitation steps that there are no formally correct arguments to the same conclusions; so it does not straightforwardly follow that there must be something amiss with his purported diagnosis. But in fact there is a questionable assumption on which it rests. For if the addition of the T-schemes for 2C1 and 2C2 is to enforce reflexivity on the their corresponding accessibility relations R1 and R2 , Humberstone must assume that those schemes are logically valid. The assumption of validity, as opposed to mere truth at a world, is crucial. 2p → p could be true at a world w without the accessibility relation being reflexive—for we might have wRw but ¬w Rw . v(2p → p, w) = 1 requires that if v(2p, w) = 1 then v(p, w) = 1 also, and

156 relative necessity reformulated since wRw, the truth of this antecedent requires the truth of this consequent. Further, we can suppose that v(2p, w ) = 1 while v(p, w ) = 0—there is no incompatibility here, since the truth of this antecedent does not require v(p, w ) = 1, since ¬w Rw . What cannot be the case is that v(2p → p, w) = 1 is true for all w, but R is not reflexive. But the assumption that the T-schemes are logically valid is simply the model-theoretic counterpart of the equally problematic proof-theoretic assumption— that 2C1 p → p and 2C2 p → p are logical axioms—which, as we have seen, vitiates the formal proofs for problems 1 and 3. Prescinding from our reservations, Humberstone’s proposed remedy should come as no surprise given his diagnosis—since the propositional constants central to the standard treatment lose essential information about the distinctive accessibility relations which characterize different kinds of relative necessity, they must be replaced by a new type of constant which encodes the lost information. As he puts it, ‘the constants must do some of the “relational” work themselves’ (p.37). To this end, he assumes a two-dimensional framework, in which formulae, including the new type of ‘relational’ constants, are evaluated with respect not to single worlds, but pairs of worlds: [W]e want, for A a formula and “|” a truth-relation determined by a model W, R, V , to make sense of not the usual |x A but rather |xy A (where x, y ∈ W), so that when it comes to evaluating one of our special constants, which I shall now write as R instead of C, to emphasize the relationality, we can say: |xy R iff xRy. (Humberstone 1981b, p.37)

The idea is that a given such R is to be true with respect to the pair of worlds x and y just when x bears the relevant accessibility relation, R, to y. Thus If we have in mind a formalization of physical necessity, we might read “|xy R” as “the laws of x are true in y”. (Humberstone 1981b, p. 38)

Using constants of this new type—semantically interpreted as ‘dipropositions’— Humberstone’s revised definitions of relative necessity operators have the same surface form as in the standard account. That is, where O is a relative necessity operator, we have OA =def 2(R → A) But, crucially, dipropositional constants are not substitutable for normal propositional variables: The propositional variables really do range over propositions, but the sentential constants Ri cannot be substituted for them because the latter are not propositional constants. In the terminology of [Humberstone (1981)], they are semantically interpreted not as propositions but as dipropositions—sets of (ordered) pairs of worlds. (Humberstone 2004, p.53)

Clearly, since each of problems depends upon substitution of the relevant propositional constant Ci for a propositional variable, this restriction effectively blocks all of them. Crucially, this restriction blocks the problems that, on closer examination, are still valid and hence still pressing. In particular, the schema for the S4 axiom for a relative necessity operator is: (S4)

2(R → A) → 2(R → 2(R → A)), provided R does not occur in A.

9.5 shortcomings 157

9.5 Shortcomings Humberstone’s solution comes at a price. In this section we highlight two disadvantages, which seem to us sufficiently serious to warrant looking for an alternative solution. The first disadvantage concerns the modal logic of the underlying absolute necessity operator, 2. It is usual, and it seems to us overwhelmingly plausible, to take 2 to express logical necessity. And it is, further, commonly supposed, and again very plausible, that the modal logic of 2, interpreted as expressing logical necessity, should be S5.12 But Humberstone cannot both take his absolute necessity operator 2 to express logical necessity and agree that logical necessity satisfies the S5 principles. It is less than totally clear which claim he means to deny. Noting Smiley’s suggestion that we should take absolute necessity to be logical necessity, he comments that this is a suggestion which is not entirely easy to evaluate. Most people believe that logical necessity satisfies at least the T axiom and the S4 axiom; however, neither 2A → A nor 2A → 22A is valid for arbitrary A, though both are valid for R-free A. . . . it may be held that since “2” does not satisfy all instances of the familiar schemata, it cannot be regarded as expressing logical necessity. This matter cannot be settled here. (Humberstone 1981b, p.40)

Since denying that the T and S4 axioms hold unrestrictedly for logical necessity is hardly an attractive option, it may seem that Humberstone’s best course is to accept that his absolute necessity operator does not express logical necessity. But this, too, has its disadvantages. Setting aside the absence of any plausible alternative candidate, it is independently plausible that the kind of necessity in terms of which various forms of relative necessary are to be explained should be logical necessity. Further, if that rôle is assigned to some other kind of (absolute) necessity, logical necessity would, if not simply a restriction of that kind of necessity, have to be treated as a form of relative necessity.13 But it is unclear how logical necessity could be merely relative, if logical necessity is stronger than the kind of necessity of which it is supposed to be a relativization (as it would be, if it obeys unrestricted T and S4, and plausibly S5, axioms). And in any case, even if the absolute necessity is not identified with logical necessity, it may be argued that its modal logic should be S5.14 All told, if there is an alternative formulation of relative necessity that can avoid making difficult claims about logical necessity—either that it does not satisfy the S5 principles, or that it is not absolute necessity—so much the better. Our aim below is to offer such a formulation.15

12 An argument for the claim that logical necessity is the strongest kind of necessity is given in McFetridge (1990). For further defence, see Hale (1996) and Hale (2013a), ch.2. 13 One kind of necessity is a (proper) restriction of another if necessities of the first kind are a (proper) subclass of necessities of the second kind, whereas when one kind is a relativization of another, the relation is reversed, i.e. necessities of the second kind are a (proper) subclass of necessities of the first. For example, one might hold that logical necessity is a restriction of metaphysical necessity. But given its peculiarities, it would not be plausible to take Humberstone’s absolute necessity to be metaphysical necessity. 14 For argument in support of the claim that the logic of absolute necessity is S5, and that logical necessity is a species of absolute necessity, see Hale (2013a), chs.4, 5.4. Timothy Williamson argues, along quite different lines, that the logic of metaphysical necessity is S5 in Williamson (2013), ch.3.3. See also chapter 8 of this volume. 15 See Kuhn (1989) for an axiomatization of Humberstone’s proposed two-dimensionalist logic of absolute necessity and a detailed discussion of its logical properties.

158 relative necessity reformulated A second, and in our view more fundamental, disadvantage of Humberstone’s solution is that it is inextricably reliant on the assumption that modal thought and talk is to be understood as, and analysed in terms of, thought and talk about possible worlds. Our concern is not simply that the proposed solution presupposes the twodimensional framework, which in turn appears to require acceptance of a plurality of worlds. Nor is the complaint that the solution requires acceptance of some form of extreme realism about worlds, such as Lewis’s—for there is no reason to suppose that it does so. The point is rather that, while there is no denying the enormous utility of possible world semantics in model-theoretic treatments of modal systems, it is one thing to hold that the truth-conditions of modal propositions can be usefully modelled in terms of systems of worlds, and quite another to claim that such propositions are fundamentally propositions about worlds and the relations between them—that understanding such propositions requires understanding them as making claims about worlds, and relations between them. The standard account, for all its faults, made no controversial demands on the metaphysics of modality, and took no stand on disputed questions about the nature and basis of necessity. This, it seems to us, is a virtue which a better account should preserve. Humberstone’s proposal, by contrast, requires us to accept that modal propositions—or at least propositions asserting relative necessity—are really propositions about relations between worlds. This comes out most forcefully in the new dipropositional constants. Humberstone does not spell out just what, under his proposed analysis, we are saying when we assert, for example, that such-and-such is physically possible, or physically necessary—how, exactly, R is to be understood, when it is the constant for physical necessity. The closest he comes to doing so is his suggestion that |xy R might be read as ‘the laws of x are true in y’. This does not tell us how to interpret R in the context 2(R → A), but it seems that there will be no alternative but to construe it as asserting something about worlds which are physically accessible from, or physically possible relative to, a given world. What is lacking is a way to account for our understanding of claims about relative necessity which does not require us to understand them as claims about relations between worlds. There may be some world-free way to construe dipropositions, but we take the onus to be with our opponent to offer such a construal. Humberstone’s is not the only possible two dimensional treatment.16 Alternative two dimensionalist solutions may be able to avoid these disadvantages. But it is, we think, of interest to see whether a satisfactory solution can be developed which does not draw on a two dimensional framework, and this will be our course in what follows.

9.6 Remedies 9.6.1 Adding a conjunct Before we present our preferred alternative treatment of relative modalities, it will be useful briefly to discuss another remedy which has been proposed. The added

16 A rather different two dimensionalist treatment of relative necessity is proposed in van Fraassen (1977), a paper to which Humberstone does refer (Humberstone, 1981b, p.40). For a brief discussion, see the appendix to this chapter.

9.6 remedies 159 conjunct stategy, in its simplest form, consists in expanding the definiens for 2C p by adding the propositional constant C as a conjunct.17, 18 That is, for any form of relative modality for which the T axiom is to hold, we define: 2C p =def C ∧ 2(C → p)

Whilst this strategy blocks Humberstone’s modal collapse argument, that argument itself collapses anyway, so that there is no need for any further measure to block it. More importantly, this added-conjunct strategy does nothing to solve Humberstone’s S4 problem, as the reader may easily verify. If 2 satisfies the S4 axiom, so must any form of relative necessity—adding a conjunct does not block this result, because the S4 axiom for 2Ci under the revised definition, i.e. (C1 ∧ 2(C1 → p)) → (C1 ∧ (2(C1 → (C1 ∧ 2(C1 → p))))) is fairly obviously still a theorem of S4 for 2.19 The ‘mixed’ S4 problem, however, is now solved.20

9.6.2 A better diagnosis We are now in a position to draw two crucial lessons from the failings of the standard account, and attempts to remedy those failings. In brief, (1) crucial information has been lost, but (2) that information must be reintroduced in a suitably general form, if the account of relative necessity is to have any plausible application. First, then, independently of these logical shortcomings, the additional conjunct strategy suffers from another defect. Recall Humberstone’s complaint that the standard account founders because it loses vital information. We agree with Humberstone on this point—but not on what vital information is lost. Here is our alternative diagnosis. When we adopt the standard account of a relative modality—physical necessity, say—we proceed as follows: ‘Let C be a conjunction of the laws of physics. Then to say that it is physically necessary that p is just to say that p is a logical consequence of C—in symbols: 2(C → p).’ The key point here is that, in adopting this formalization, we simply leave it to be understood that C is a conjunction of physical laws—nothing in our definiens actually records that that is so. So that information is lost. And it 17 A version of this strategy is mentioned by Steven Kuhn, who writes: ‘Other remedies [i.e. other than Humberstone’s own proposal] may also be possible. Wlodzimierz Rabinowicz, in correspondence, attributes to Lars Bergstrom the idea that, when OA ⊃ A obtains, the reduction of relative necessity should be given by OA = 2(L ⊃ A) & L rather than OA = 2(L ⊃ A)’ (Kuhn 1989, note 2). 18 Clearly this proposal is appropriate only for factive kinds of relative necessity. The effect of adding C as a separate conjunct is that 2C p always implies p. However, as we discuss briefly in section 9.8, there are other problems to be considered for treating non-factive necessities. 19 If the main antecedent is true at w but the main consequent false there, then since C1 is a conjunct in the main antecedent (and so likewise true at w), it must be the right conjunct that is false at w. This requires w with C1 true and 2(C1 → p) false, and the latter requires w with C1 → p false. But this is impossible— for since we are in S4 for 2, w must be accessible from w, so that C1 → p must be true at w . So the whole conditional is valid, and thus a theorem by completeness. 20 The foregoing model-theoretic argument depends upon the identity of the left conjuncts in the main antecedent and consequent. Thus: (C1 ∧ 2(C1 → p)) → (C2 ∧ (2(C2 → (C1 ∧ 2(C1 → p))))) is not a theorem of S4. There is a simple one world counter-model with C1 and p both true, but C2 false. Thus the additional conjunct strategy does block the last part of problem 2.

160 relative necessity reformulated is, surely, vital information. Nothing explicit in the definiens distinguishes between a strict conditional purporting to express the physical necessity of its consequent, and one which purports to express some other kind of relative necessity. One moral we might draw from this is that a better definition needs explicitly to record the relevant information about the status of the antecedent C. The simple added-conjunct strategy does no better in this respect than the standard account. An obvious way to remedy this particular shortcoming would be to employ an additional operator, π say, which might, in case we are seeking to define physical necessity, be read as ‘it is a law of physics that . . .’.21 Amending the simple added-conjunct analysis to: It is physically necessary that p

iff

π(C) ∧ 2(C → p)

not only restores the lost information, but actually provides a solution to the S4 problem. This analysis preserves what is good about the added-conjunct strategy— blocking the second ‘mixed’ version of the S4 problem—and it deals with the simpler S4 problem, which eluded the simpler strategy. For now, the S4 principle for 2C1 is: (π(C1 ) ∧ 2(C1 → p)) → (π(C1 ) ∧ (2(C1 → (π(C1 ) ∧ 2(C1 → p))))) and a simple calculation reveals that this is no theorem of S4 for 2. Crucially, while antecedent and consequent here share the same left conjunct, π (C1 ), this conjunct is not necessitated—it is allowed that the laws of physics might, logically, have been otherwise. Consequently, there is a simple two world counter-model in which π (C1 ) is true along with C1 and p at w, but π (C1 ) is false at w , even though C1 is true there. The counter-model exploits the fact that being true is necessary, but not sufficient, for being a law of physics—so that there are possible circumstances in which the propositions which are the actual laws of physics would still be true, but only accidentally, and not as a matter of physical law. This brings us to our second crucial point. The additional conjunct strategy, whether in its simple or revised form, remains afflicted with a defect which was all along sufficient to justify rejecting the standard account, quite independently of the problems with which we have been occupied, but which appears to us, remarkably, to have escaped notice in previous discussion. As the use of C and π (C) makes plain, each of these accounts simply assumes that we are able actually to state, say, the laws of physics; neither provides an analysis we could advance, unless were we actually able to do so. At least, this is so, on the charitable assumption that the intention behind these accounts is that what is physically necessary should be what is required by the true laws of physics, and not just by what we take to be the laws. We do not, of course, mean to deny that it may on occasion reasonably be claimed that something is physically necessary, or physically impossible. Someone who makes such a claim may believe, on good grounds, that such-and-such propositions are laws of physics, and that they require, or rule out, the truth of this or that further proposition. Or, without being 21 Typically what is physically necessary is a consequence of several laws of physics, not just one. In order to postpone discussion of some complications best left aside pro tem, we shall assume—as seems not unreasonable—that any conjunction of laws of physics counts as a law of physics. For discussion of this and other complications, see below, section 9.7.

9.6 remedies 161 confident about any specific candidates to be laws of physics, she may believe that whatever precisely the relevant laws are, they will be such as to require, or rule out, the truth of certain propositions—such as that a human being should move by its own unaided effort at 200mph. Our point, at bottom, is that someone may understand the claim that, or the question whether, such-and-such is a physical necessity or (im)possibility without knowing, or claiming to know, what the laws of physics are, even approximately. Whatever the question is, that such a person is considering, it cannot be the question whether certain specified propositions are laws of physics which strictly imply that such-and-such. Of course, she may put her question this way: ‘Do the laws of physics require that such-and-such?’ But her question need not concern certain specific (candidate) laws—it can still be a perfectly general question: ‘Are there physical laws which require that such-and-such?’

9.6.3 Existential generalization There is an obvious way to do justice to this point—replace our propositional constant by a variable and existentially generalize through the position it occupies. That is, we should define: It is physically necessary that p

iff

∃q(π(q) ∧ 2(q → p))

where, as suggested above, π (q) may abbreviate something like ‘It is a law of physics that q’. It is not assumed that the laws are known; hence π (q) could be read as saying that q is one of the actual physical laws, whatever they are, or may be—but not ‘whatever they could have been.’22 For any kind of relative necessity, then, our proposal is to define it in line with the following schema: It is -necessary that p

iff

∃q((q) ∧ 2(q → p))

where, (q) abbreviates ‘It is a -proposition that q’. The condition captured by “” may be more or less interesting, more or less restrictive. We will continue to use the example of the laws of physics, i.e. π (q), as a representative, and philosophically interesting, example. Such a proposal captures quite naturally, or so it seems to us, the main idea of relative necessity. It introduces explicitly the information lost by the standard account. Not only does it specify the kind of proposition to which the necessity is relative, e.g. laws of physics, but also does so in a plausibly general way. If one claims, for example, that it is necessary relative to the laws of physics that p, one need not be claiming that p is necessary relative to some particular laws that one is able state, but only that there are some such laws that necessitate p. So what might at first glance appear to be an unnecessarily complex formulation, as contrasted with the standard account, is in fact simply recording explicitly the information that was all along assumed by the standard account, and in a suitably general way.

22 We are understanding the claim that it is physically necessary that p so that to make this claim is not to deny that the laws of physics could have been otherwise (in such a way as not to require that p). In other words, ‘whatever they are, or may be’ is to be given an epistemic reading, not a metaphysical one.

162 relative necessity reformulated Is this proposal subject to any of the problems Humberstone took to afflict the standard analysis? Setting aside the arguments which we saw to be flawed by faulty Necessitation steps, it is easy to see that the arguments central to the remaining problems—those involving unwanted imposition of S4-like properties—no longer go through, since they both depend upon substituting a special propositional constant or constants for propositional variables in the S4 theorems 2(q → p) → 2(q → 2(q → p)) and 2(q → p) → 2(r → 2(q → p)), both of which are derivable in S4 from the so-called paradox of strict implication 2A → 2(B → A), and we no longer have any such propositional constant(s) to substitute. Might there be some other route by which some or all of these problems could be re-instated? In the case of Humberstone’s first problem, the question is whether, when a T-axiom governing 2π , defined as now proposed, is added to an underlying modal system, say K or T, to obtain a system S, we can derive 2p from 2π p. The distinctive axiom is ∃q(π(q) ∧ 2(q → p)) → p. With this available, we could argue: 1 1

(1) (2)

∃q(π(q) ∧ 2(q → p)) p

by T-axiom for 2π

But to get 2p, we would need to apply the Rule of Necessitation or a rule of 2-introduction. We cannot use the Rule of Necessitation, as this only permits us to necessitate theorems, but p is no theorem. And we cannot use the 2-introduction rule, as this allows us to necessitate a formula only if it depends only on suitably modal assumptions, and (1) is not suitably modal. There is no other obvious strategy for a derivation, and we are confident there isn’t one to be found. Proving this—assuming we are right—is a more substantial task than we can undertake here. But we can at least sketch how we think things would go. Giving a model to show that ∃q(π(q) ∧ 2(q → p)) S 2p would require giving a semantics for the system S, and that will require, inter alia, a semantics for propositional quantification. Perhaps the most obvious method is (following Fine 1970), to interpret propositions as sets of worlds, and take propositional variables to range over subsets of the set of worlds of the model. We would need to show that S is sound with respect to the semantics, i.e. that  S A only if  S A. As well as dealing with propositional quantifiers, the semantics will include a clause governing the π -operator (and more generally, for any relative necessity system, a clause governing the -operator). We may presume that this will ensure that π (p) is true at a world w only if p is so: the laws of physics are, apart from anything else, true. Since ∃q(π(q) ∧ 2(q → p)) → p is an axiom of S, we require this formula to come out true in every S-model. To see, informally, why it will do so, let M be any S-model. M will have a set of worlds W as its principal domain, so that propositional variables range over subsets of W. If ∃q(π(q) ∧ 2(q → p)) → p is to evaluate as true at each world w, it must be that ∃q(π(q) ∧ 2(q → p)) is false at w or p is true at w. If p is true at w, so is ∃q(π(q) ∧ 2(q → p)) → p, just as required. So suppose p is false at w. Then we require that ∃q(π(q) ∧ 2(q → p)) be false at w. In effect, a propositionally quantified formula ∃pA(p) will be true in a world in a model iff A(p) is true at that world for some replacement of the variable p by some propositional constant p0 .

9.7 complications and refinements 163 So ∃q(π(q) ∧ 2(q → p)) is false at w iff (π(q0 ) ∧ 2(q0 → p)) is always false at w, no matter how q0 is chosen. Pick any q0 . We can suppose that q0 is true or false at w. If q0 is true at w, then since p is false at w, q0 → p is false at w, so that 2(q0 → p) is false and hence (π(q0 ) ∧ 2(q0 → p)) is false as required. If q0 is false at w, then given our assumption about the clause for π , π (q0 ) will likewise be false at w, so that again (π(q0 ) ∧ 2(q0 → p)) is false as required. To see that ∃q(π(q) ∧ 2(q → p)) S 2p, we shall need to show that in some S-model, for some w, ∃q(π(q) ∧ 2(q → p)) is true at w while 2p is false at w. Intuitively, this is clearly possible. Suppose 2p is false at w. For ∃q(π(q) ∧ 2(q → p)) to be true at w, we require that π(q) ∧ 2(q → p) is true at w for some choice of q. Since we are taking π to be factive, q must be true at w. And 2(q → p) must be true at w. This means that p must be true at w (we are assuming S includes the T-axiom for plain 2, so that the accessibility relation is reflexive). But this is entirely consistent with p and q being false at w for some w accessible from w, as required for truth of 2(q → p) and falsehood of 2p at w. We are reasonably confident that with further work, this informal sketch can be turned into a rigorous model-theoretic proof, and that we shall be able to show, along similar lines, that our proposed definition of relative necessity does not succumb to any of the other problems discussed in Section 9.3.

9.7 Complications and refinements The system S, lightly sketched in 9.6.3, clearly calls for more rigorous formulation and development. We cannot undertake a full-dress presentation here. Our purpose in this section is rather to draw attention to some signficant aspects of the system we intend, and to deal with some complications noted in section 9.6.2 (see fn.21). Here we have benefitted from very helpful discussion of an earlier version with David Makinson and some observations made by an anonymous referee, to whom we are much indebted. The complications, and in some cases, refinements, concern the following facts. (i) 2p → 2 p, i.e. the principle that logical necessity implies relative necessity, which Makinson calls Down, is not a theorem of S. (ii) The converse of Down, 2 p → 2p, which Makinson calls Up, is not a theorem of either S or S+Down. (iii) S appears to lack the conjunction property, i.e. that 2 A, 2 B 2 (A ∧ B). We give a brief statement of the logical facts, then comment on the issues to which they give rise.

(i) S Down Let our relative necessity operator be 2 , defined as ∃q((q) ∧ 2(q → p)). Interpret  so that v(A, w) = 0 for all w ∈ W. Then v(∃q(q), w) = 0 for all w ∈ W. Hence v(∃q(q ∧ 2(q → A)), w) = 0 for all w ∈ W, i.e. v(2 A, w) = 0 for all w ∈ W. So, vacuously, v(2 A → A, w) = 1 for all w ∈ W. That is, the T-schema holds with this interpretation of . Now let be any tautology, so that v( , w) = 1 for all w ∈ W. Hence v(2 , w) = 1 for all w ∈ W. But v(2 , w) = v(∃q(q ∧ 2(q → )), w) = 0 for all w ∈ W. Hence v(2 → 2 ), w) = 0 for all w ∈ W. That is, Down is not derivable in S.

164 relative necessity reformulated

(ii) S (S+Down)Up Let W = {w1 , w2 } and R = W × W = { w1 , w1 , w1 , w2 , w2 , w1 , w2 , w2 } so that R is an equivalence relation on W. Interpret  so that v(A, w) = 1 iff v(A, w) = 1 for all w ∈ W. It follows that v(2 A, w) = 1 iff v(A, w) = 1 for all w ∈ W. It immediately follows from this last that v(2 A → A, w) = 1 for all w ∈ W—since if v(2 A, w) = 1, then v(A, w) = 1 for all w ∈ W. So the T-schema is validated by this model. But this model falsifies Up, if we stipulate (as we may), for some p, that v(p, w1 ) = 1 but v(p, w2 ) = 0.

(iii) 2 A, 2 B  2 (A ∧ B) Under our definition, the premises are ∃q((q) ∧ 2(q → A) and ∃q((q) ∧ 2(q → B). As far as these premises go, there need be no single proposition q such that (q) which strictly implies both A and B. So we appear unable to infer ∃q((q) ∧ 2(q → (A ∧ B)), as required for the conjunction property.

Comments on (i), (ii), and (iii) For reasons which will rapidly become apparent, it makes best sense to begin with point (iii).

The conjunction property What, in essence, appears to block the derivation of 2 (A ∧ B) (i.e. ∃q((q) ∧ 2(q → (A ∧ B)))) from 2 A and 2 B (i.e. ∃q((q) ∧ 2(q → A)) and ∃q((q) ∧ 2(q → B))) is the fact that we cannot infer from the premises that there is a single proposition r such that (r) ∧ 2(r → (A ∧ B)). As our anonymous referee points out, this problem would be solved if we could generally infer (A ∧ B) from (A) and (B), since we could then conjoin the possibly distinct propositions qi , qj such that (qi ) ∧ 2(qi → A) and (qj ) ∧ 2(qj → B), whose existence is guaranteed by the premises, to obtain (qi ∧ qj ) ∧ 2((qi ∧ qj ) → (A ∧ B)), whence ∃q((q) ∧ 2(q → (A ∧ B))). As a matter of fact, we make just this assumption for our operator π above (see footnote 21). However, an objection to this remedy, also proposed by our referee, is that it re-opens and reinforces the concern, to be discussed in section 9.8, that a -operator is really nothing but a thinly disguised duplication of the relative necessity operator which we are seeking to define. In that case, rather than working via the conjunction property for , we might as well directly stipulate that (2 A ∧ 2 B) → 2 (A ∧ B). The problem raised is a special case of a wider issue, i.e. whether relative necessity operators, according to our proposed formulation, are closed under logical consequence, that is, if A1 , . . . , An B then 2 A1 , . . . , 2 An 2 B for n≥1.23 We need a solution that does not simply beg the question. And indeed, there is a simple revision of our proposal from which closure under logical consequence follows, without the complication of needing to specify and justify a conjunction property for -operators. Instead of defining 2 A by a simple existential quantification ∃q((q) ∧ 2(q → A)), we may simply use a finite string of quantifiers: 23 We should not expect closure when n = 0, for reasons to be given in our discussion of Down.

9.8 further issues

165

2 A =def ∃q1 , . . . , ∃qn ((q1 ) ∧ . . . ∧ (qn ) ∧ 2(q1 ∧ . . . ∧ qn → A))

Clearly we obtain closure under logical consequence now, without needing to insist that a conjunction of -propositions must itself be counted as a -proposition.24

Down Makinson’s own view is that the failure of Down is a defect—without it, the system is ‘rather weak’. On the contrary, we take there to be good reason why we should not have Down. Down is, obviously, closure under logical consequence for the limiting case where n = 0. It is bound to fail if, as seems reasonable, we allow that the laws of physics, say, might have been otherwise, and that it is a contingent matter whether there are any such laws at all, for then 2B may be true, but 2 B (i.e. ∃q1 , . . . , ∃qn ((q1 ) ∧ . . . ∧ (qn ) ∧ 2(q1 ∧ . . . ∧ qn → B))) false, because there are no true -propositions. Of course, under the hypothesis that there is at least one true -proposition, Down will hold; where there are, say, some laws of physics, their logical consequences will, trivially include all the logical necessities, which will harmlessly qualify as physical necessities. That Down should be validated is, no doubt, just what one would think, if one thinks about relative necesity in essentially world-terms: that is, so that what is physically necessary, for instance, is essentially just what is true throughout a restricted range of all logically possible worlds. For then, since anything logically necessary (i.e. true throughout the whole unrestricted range) must be true throughout any restriction of it, it must also be physically necessary. Our answer to this is that it is just a mistake to think of forms of relative necessity as fundamentally to be understood in world terms. If we drop that prejudice, then it can seem entirely natural and correct to characterize or define a form of relative necessity in such a way that logical necessity does not automatically ensure relative necessity.

S (S+Down)Up Evidently this is a welcome result from our point of view; Makinson’s proof confirms what we claim in 9.6.3.

9.8 Further Issues In this closing section, we anticipate and respond to an objection to our preferred account, and draw attention to a limitation on its range of application. 24 As a referee for The Journal of Philosophical Logic observed, this refinement, in allowing for variation over natural numbers serving as indices, and in using suspension dots, results in a right-hand side that is no longer a formula of a simple extension of the language of ordinary modal logic by  and propositional quantifiers. A fuller treatment of our proposal would need to settle on a formal treatment of these extra devices. As our referee points out, we might accommodate numerical indices by enriching the language to include explicit quantification over the natural numbers; or we might instead take our formulation as shorthand for an infinite set of axioms of the form: (∃q1 . . . ∃qn ((q1 ) ∧ . . . ∧ (qn ) ∧ 2(q1 ∧ . . . ∧ qn ) → A)) → 2 A. We have no aversion to enriching the language by adding quantification over the natural numbers, if that is necessary for current purposes, nor is it obvious to us what the shortcomings, if any, would be in opting for an axiom scheme in place of a fully explicit definition. We shall not attempt to adjudicate between these, and perhaps other, alternatives here.

166 relative necessity reformulated The objection focuses on the sentential operators, such as π and η, which play a key rôle in our definition scheme. These operators serve to demarcate propositions to which specific kinds of relative necessity and possibility are relative. Because we wish to avoid assuming that we are able to actually state explicitly the relevant propositions, we need to characterize them in general terms, as propositions of a certain kind. Thus, as we suggested, π (q) might be read as abbreviating something along the lines of ‘it is a law of physics that q’, where this is to be understood as making a non-specific reference to the laws of physics, whatever they are, rather than to what we happen presently to take to be laws. Put bluntly, the objection says that by making essential use of operators so understood, the account simply gives up on the reductive explanatory aspiration which informs the original Anderson-Kanger-Smiley project. For what drove that project—and what gave it its interest—was the prospect of showing that, contrary to appearances, we do not need to recognize a great variety of independent notions of necessity, because we can explain each relative form of necessity using just a single ‘absolute’ notion (probably logical necessity). This explanatory aim is completely undermined by our appeal to laws of physics, for example, because the notion of a law of physics itself involves the idea of physical necessity.25 A first point to be made in response to this objection is that, whatever force it may possess, it does not tell selectively against the explanation of relative necessity we have proposed. For essentially the same objection could be brought against Humberstone’s two-dimensionalist proposal. Of course, he makes no use of an operator comparable to our π and η operators. But for each distinct kind of relative necessity, his proposal will require a distinctive dipropositional constant, R, and this will need to be explained. The only kind of explanation he suggests is that we might understand |xy R as ‘the laws of x are true in y’. As it stands, of course, this is hopelessly vague—there are many different kinds of laws, and he presumably does not mean that they should all hold. What he intends, presumably, is that two-dimensionally understood, the diproposition will say, for example, that x’s physical laws are true in y. So his proposal, too, essentially involves an appeal to the notion of physical law, and so is no less vulnerable than ours to the envisaged objection. Thus if it should prove impossible satisfactorily to characterize physical laws without appeal to the notion of physical necessity, his account will be no better placed to subserve the reductionist aims of the standard account than our own.26 It is true enough that the objection does not apply to the standard account itself. But as we have seen, there are good reasons to explore alternatives. Moreover, it seems inescapable that the deficiencies of the standard account have their source in its suppression of vital information. It is quite unclear how the requisite information could be incorporated without making use, in the analysis of physical necessity, say, of the notion of a physical law, or something near enough equivalent to it. Second, even if it is granted that the notion of a law of physics, say, involves an implicit appeal to the notion of physical necessity or possibility, it is not clear that this need be objectionable. Whether it is so or not depends upon what the proposal 25 This objection, or something close to to it, was put to one of us in conversation by Lloyd Humberstone. 26 The same point applies equally to the alternative 2-D analysis of relative necessity proposed by van Fraassen in van Fraassen (1977), discussed briefly in the appendix to this chapter.

9.8 further issues

167

sets out to accomplish. As we observed at the outset, one might hope that an analysis of relative necessity would subserve a reductionist programme. That is, one might hope to show that, contrary to appearances, there is no need to recognize a variety of independent kinds of necessity—physical, mathematical, etc.—by showing that each of these ostensibly different kinds of modality may be fully explained using just one single kind of modality, such as logical modality. If this were one’s aim, then the objection, if sound, would indeed be fatal. However, as we saw, there is another, no less important, aim one might have in pursuing analyses of forms of relative necessity—that of achieving an improved understanding of the contrast between merely relative and absolute kinds of necessity. The achievement of that aim is in no way compromised by the irreducibility—if such it is—of the various kinds of putatively relative necessity to a single absolute necessity. Third, the claim that the notion of physical law cannot be understood without bringing in that of physical necessity may be challenged. There are various ways in which one might attempt to characterize laws of physics that make no overt appeal to the notion of physical necessity. One might characterize laws of physics without any appeal to (familiar) modal notions at all. For example, Maudlin (Maudlin, 2007) takes law-hood to be a primitive status, and indeed proposes a definition of physical modality in terms of laws.27 Or, more commonly, one might characterize laws of physics making use of some modal notion other than physical necessity. For example, Lewis’s ‘best deductive system’ account of laws of nature arguably only draws on the logical necessity built into the notion of a deductive system (see Lewis 1986; 1973; 1994). Or so-called ‘necessitarians’ take the laws of physics to be metaphysically necessary (see Carroll (2012) for a representative summary). These views would avoid the troublesome circularity of defining physical necessity in terms of physical laws, in turn defined in terms of physical necessity. The latter would of course introduce a further kind of modality—metaphysical—to be treated as relative or absolute, but the objection presently under consideration would be avoided. The success of this response to the objection depends upon the success of one of these alternative accounts of the laws of physics, but we will not be able to adjudicate on that matter here. It is certainly not obvious that these will all prove to involve a more or less thickly disguised appeal to the notion of physical necessity, and so to be unavailable as independent characterization of laws of physics.28 Our claim in this chapter is that our proposal is the preferable treatment of alethic kinds of relative necessity, such as physical necessity or mathematical necessity. There are potential problems for applying it to non-alethic relative modalities. Whether these problems can be overcome, we leave as further work to be carried out elsewhere. But we will briefly survey two key difficulties for treating non-alethic relative necessities. The first difficulty arises from the relation between relative necessity and logical necessity. On the standard account, since every logical necessity is strictly implied 27 ‘My own proposal is simple: laws of nature ought to be accepted as ontologically primitive. . . . Laws are the patterns that nature respects; to say what is physically possible is to say what the constraint of those patterns allows’ (Maudlin, 2007, p.15). 28 One of us is more sanguine about the prospects for the reductive project than the other.

168 relative necessity reformulated by any proposition whatever, every logical truth is C-necessary—necessary relative to C—no matter how C is chosen. Similarly, although, as we have seen, Down does not in general hold for our alternative account, if it is -necessary that p just when ∃q((q) ∧ 2(q → p)), then, so long as the existence condition is fulfilled (there is a proposition), every logical truth will be -necessary. This seems to be a reasonable result for alethic necessities: it would certainly be strange to claim that, although it is logically necessary that p, it is nevertheless possible, relative to existing physical laws, that ¬p. However, it has implausible consequences for non-alethic necessities for which, intuitively, even if some relevant -propositions exist, it is not always the case that if 2p, then 2 p, such as epistemic or deontic modalities. Consider kinds of necessity defined relative to a conjunction of known propositions, or a conjunction of moral precepts. The current proposal for treating these necessities would yield the result that any logical truth is thereby epistemically necessary and morally necessary. However, it seems that we should leave room for the epistemic possibility that a proposition whose truth-value is as yet undecided should turn out to be false, even if in fact it is a logical truth. It also seems wrong to take logical truth to be a matter of moral obligation—we might not think that the world would be a morally worse place if a contradiction were true in it. Indeed, one might think, to the contrary, that if ‘ought’ expresses moral obligatoriness, ‘It ought to be the case that p’ implies that it is at least metaphysically (and so logically) possible that ¬p. A different kind of problem arises from kinds of relative necessity where one might expect to find inconsistent q, although (q), such as doxastic and legal modalities. One might expect that some conjunctions of beliefs are inconsistent, or that some conjunctions of laws of a given state are inconsistent. However, for any -necessity defined in terms of an inconsistent proposition (i.e. where the existential condition is fulfilled by an inconsistent proposition), this would have the unfortunate result that everything would be -necessary, given the inference rule that everything follows from a contradiction. But again, just because the statute books are likely to contain strict inconsistencies, does not mean that everything is legally required. That would be quite bizarre.29

9.9 Conclusion Our leading question has been: How should we best formulate claims of relative necessity? We have considered several answers. First, we reviewed the standard account: 2 =df . 2(C → p). This, we argued, falls foul of the S4 and mixed S4 logical problems as presented by Humberstone. Moreover, it omits important information about the nature of the proposition (i.e. C) to which the necessity is relative. Further, and crucially, the account assumes that, in making a claim of relative necessity, one is able to state all of the relevant propositions; for example, to make a claim of physical necessity in accordance with the standard account, one would have to be able to state the laws of physics. But such an assumption is too demanding—whilst we may be unable to answer the question whether it is physically necessary that p without some knowledge of the laws of physics, no such knowledge is required merely to understand the question; nor, accordingly, should 29 This kind of problem is presented, and a solution offered, in Kratzer (1977).

appendix 169 it be presupposed by a good explanation of claims about physical necessity. These reasons led us to explore alternatives to the standard account. Second, then, we considered Humberstone’s two-dimensional alternative. We had two main concerns with this approach: first, it is, we argued, unclear how best to interpret the 2-operator of absolute necessity, without making unpalatable claims about the logical properties of logical necessity; and, second, it appeared to us implausibly to require not just that claims about relative necessity can be modelled in terms of worlds, but that they are in fact to be understood as claims about worlds. We then considered a series of amendments to the standard account. The simplest of these consists in adding the proposition C itself as an extra conjunct in the analysans. This does indeed block the mixed S4 problem, but the simple S4 problem remains. So also does the crucial problem of understanding: the account still requires that one be able to, for example, state the laws of physics in order to make a claim of physical necessity. Adding a more complex conjunct, including the information of what kind of proposition is involved (e.g. specifying that C is a law of physics), solves the remaining S4 problem, but does nothing to resolve the problem of understanding, for the proposed analysans is still one which we can offer only if we are able to state the relevant laws (e.g. the laws of physics). We then introduced our own, preferred, account. 2 A =def ∃q1 , . . . , ∃qn ((q1 ) ∧ . . . ∧ (qn ) ∧ 2(q1 ∧ . . . ∧ qn → A))

This captures the information lost by the standard account and the simple added conjunct account—that it is -propositions relative to which things are necessary— but, in contrast with the more refined additional conjunct account, it does so in a suitably general way. The proposal avoids the logical problems (imposing S4 etc.), and does so without any intrusive use of worlds semantics in the analysis. There is no longer the problem that one must (be able to) state the -propositions if one is to give the analysis—the analysans involves only the non-specific requirement that there be some propositions of that kind. Finally, we aired some technical and philosophical issues that may arise for our proposal. We have not had space here to resolve every issue, but we hope to have shown that pursuit of this account for relative necessity is at the very least a promising alternative to what, in our view, are some rather less promising accounts.30

Appendix: an alternative 2-dimensional solution? In an article to which Humberstone refers (van Fraassen 1977) Bas van Fraassen proposes a rather different two-dimensional analysis of relative (e.g. physical) necessity. The analysis replaces the strict conditional 2(A → B) by a two-dimensional 30 We are grateful to Fabrice Correia for extensive discussion of the development of the core proposal; to Lloyd Humberstone for discussion of some of the central ideas in this paper; to Bas van Fraassen for extensive and instructive correspondence about his 1977 paper; to David Makinson for detailed and thoughtful technical suggestions; to the participants in a modality workshop in Nottingham and members of a King’s College London work-in-progress group for discussion of earlier incarnations, particularly Ian Rumfitt; and to two anonymous referees for The Journal of Philosophical Logic for their comments and suggestions.

170 relative necessity reformulated conditional A ⇒ B (van Fraassen uses →, but this risks confusion with our use of the same symbol for the truth-functional conditional), defined as follows: [A ⇒ B](α) = {β : [A](β) ⊆ [B](α)}. Sentences within square brackets denote the propositions they express, which are identified with the sets of worlds at which they are true. So this says that the proposition expressed by A ⇒ B at world α is the set of worlds β such that the proposition A expresses at β is included in the proposition B expresses at α. In other words, what A ⇒ B says at α is true at β iff every world at which what A says at β is true is a world at which what B says at α is true. This contrasts with the strict conditional, which in the two-dimensional setting is true at α iff every world at which the proposition expressed by its antecedent at α is true is one at which the proposition expressed by its consequent at α is true. The conditions coincide when the world of evaluation is the same as the world of utterance, but diverge when they diverge. ‘It is physically necessary that A’ is then defined as R ⇒ A where R is a special constant which does double duty, both expressing ‘the appropriate relation of relative physical necessity’ and the corresponding proposition, defined [R](α) = {β : αRβ}, i.e. the set of worlds physically possible relative to α (op. cit., p.82). Van Fraassen tells us that R— the law sentence, as he calls it—‘may say that the laws of α hold, or that they are laws, or that they are the only laws.’ Van Fraassen’s leading idea is that the relative character of physical necessity is best understood indexically. We can approach this as follows. When we hold that it is physically necessary that p, we commit ourselves to the claim that, even if things had been different in a whole host of ways, barring some (miraculous) suspension of the laws of physics, it would still have to be the case that p. We are not just claiming that, given the fact that things are the way they actually are (right down to the last detail), it has to be that p; we are allowing that circumstances might have varied in all sorts of ways, and claiming that even so, the laws of physics require that p. We can understand van Fraassen’s new conditional as designed to capture this. It seeks to do so by allowing the context for the antecedent to vary in any ways consistent with its still saying something true, but keeping the actual context for the consequent fixed. Thus we might read R ⇒ A as something like ‘In any circumstances in which the laws of physics hold it will be that A’—we can do so, if we may read A ⇒ B as ‘Any circumstances in which A, as said in those circumstances, would evaluate as true, is one in which B, as said here and now, would also be true’. In sum, the idea is to give a way of understanding claims about physical necessity which captures the fact that they make a claim whose truth has a high degree of independence from the actual circumstances in which they are made—so long as the physical laws hold, then although things can differ in all sorts of other ways, it will still be that p. Obviously, when we envisage different circumstances here, we are restricting attention to different circumstances in which the laws of physics as they are would not be altered—that is, we are keeping the actual laws, not talking about (more radically divergent) circumstances in which there would be different physical laws. That is, we are using ‘the laws of physics’ rigidly. The same is true of our own account—we require π (q) to say that q is a statement of the laws of physics as they are—we aren’t claiming (falsely) that no matter what the laws of physics might be, they strictly imply p. The obvious questions are: Does this account avoid the kind of problems which beset the standard account? Is it otherwise satisfactory?

appendix 171 As far as we have been able to see, the answer to the first is that it does. In van Fraassen’s system, validity can plausibly be understood in one of two ways. A strong requirement would be that validity requires that ∀α∀β Tr([A](α), β), i.e. A is valid just when, for any worlds α, β, what A says at α is true at β. A weaker requirement would be that ∀β Tr([A](α), β) when α is taken to be the real world, i.e. A is valid just when what A says at the actual world is true at any world β. Whichever notion of validity one operates under, it appears that, due to the unconventional behaviour of van Fraassen’s conditional, neither A ⇒ (B ⇒ A), nor the especially relevant instance for the S4 problem, (B ⇒ A) ⇒ (B ⇒ (B ⇒ A)), is valid. Hence, van Fraassen’s system appears not to be subject to the S4 problems that afflict the standard account.31 As regards the second question, we have some doubts. A first point is that A ⇒ A is not a law of van Fraassen’s logic. For since R is indexical, there may be a world, other than the actual world, such that what R says there is true at some world at which what R says at the actual world is not true. Whence, on the proposed semantics for the conditional, R ⇒ R is not valid.32 But the law of identity, ‘If A then A’, is often taken to be fundamental, and has as strong a claim to be definitive of the conditional as the other principles which van Fraassen mentions (op. cit., p.82) as ‘earmarks’ of the concept. It is true that he views it as a ‘cluster concept’, so that some may go missing without destroying a connective’s claim to be a conditional (or implication connective—van Fraassen makes no distinction here). It remains a serious cost, and one that we, at least, are reluctant to incur.33 A further concern is that the account may suffer from the same drawback as we found in Humberstone’s, i.e. that it does not just exploit the two dimensional framework in the model theory, but builds talk of worlds into the very content of claims about, say, physical necessity. The key question here is how the special propositional constants, such as van Fraassen’s R, are to be understood. That there is unwanted worlds content is certainly suggested by his own proposed readings of R, which ‘may say that the laws of α hold, or that they are laws, or that they are the only laws.’ It is not obvious that the world variable is dispensable without unsuiting R for its purpose. That van Fraassen thinks, to the contrary, that there is no essential reference to worlds is perhaps suggested by his official agnosticism about them (‘The items in 31 The reader may amuse herself by verifying these claims. 32 There will be other casualities. To mention only the most obvious:  A ∧ A ⇒ A,  A ⇒ A ∧ A,  A ∨ A ⇒ A,  A ⇒ A ∨ A. 33 In fairness to van Fraassen, we should point out that he observes (in correspondence) that A ⇒ Awill be a theorem of the logic we may label Lsimp , which has as theorems those sentences which are true simpliciter at every world, i.e. true at α if understood as uttered in α, for every world α. But it will not be a theorem of the logic Luniv , which has as theorems those sentences which, understood as uttered in any world, are true at every world. He appears to be sceptical, not just about the claims of either of these notions and their logics to capture the traditional notion of logical necessity, but about whether there is a good notion to be captured. Here we can only record our disagreement with him on the last question. Of course, if the meanings of words, including logical words, are allowed to vary with world of utterance, then it must be doubtful that there are any sentences which, as uttered at any world, are true at every world. What is of far greater interest is whether, holding the meanings of relevant words fixed, there are sentences which express propositions which are absolutely necessary—true at absolutely all worlds. It seems to us that ‘If A then A’ can be, and often is, used to express a proposition, stronger than a merely material conditional, which is absolutely necessarily true, and is so in virtue of the nature of the conditional alone, and so has a good title to be regarded as logically necessary.

172 relative necessity reformulated the models, such as possible worlds, I regard with a suspension of disbelief, as similar to the ropes and pulleys, and little billiard balls that were introduced in nineteenthcentury physics’, p.74). But if, more generally, we ask ourselves: ‘What is the parameter with respect to which talk of physical necessity is supposed to be indexical?’ it is not clear that the answer can be anything other than ‘Possible Worlds’. Perhaps these doubts can be answered. If so, there is another, and better, two dimensional solution to Humberstone’s problems. We see nothing inimical to our own proposal in this. For it would remain the case that if we are right, those problems can be solved more simply and economically, without any recourse to the two dimensional framework.

10 Definition, Abstraction, Postulation, and Magic In recent work,1 Kit Fine proposes a highly original, ingenious, and in some ways, at least, very attractive approach to the philosophy of mathematics, which he calls procedural postulationism, and for which he claims some very significant advantages over other approaches. Although I am not, I hope, insensitive to the attractions of Fine’s proposal, there are aspects of it which I find deeply perplexing, and I am—as I am sure he will not be surprised to learn—somewhat sceptical about whether it really does enjoy the central advantages he claims for it.2

10.1 Procedural postulationism—a very brief outline Fine contrasts procedural with propositional postulationism. On the latter view, a postulate is ‘an indicative sentence, expressing a proposition that is capable of being either true or false’, and one obtains a mathematical theory by postulating a set of propositional axioms. But the procedural postulationist takes the postulates from which mathematics is derived to be imperatival, rather than indicative, in character; what is postulated are not propositions true in a given mathematical domain, but procedures for the construction of that domain—thus the postulates express or specify procedures which ‘may or may not be implemented but certainly cannot properly be said to be true or false’.3 There are, in Fine’s account, two sorts of postulates—simple and complex. Simple postulates are all of the form:

1 Fine (2005d). All page references are to this paper unless otherwise indicated. 2 An ancestor of this paper was presented at a conference celebrating Kit Fine’s work in the University of Geneva, June 2005. At that stage, Fine had not yet published the paper which the present paper criticises. My discussion then was based on an earlier version which he very kindly made available to me prior to the conference. Although most of the papers presented at the conference were subsequently published in a special issue of Dialectica, I decided—with much reluctance, as I long been a fervent admirer of his work, and very much wished to contribute to the volume in his honour—that my paper could not be included, because it relied upon passages in the earlier version of his paper which did not survive in the final published version, and I could see no way to revise it in the time available. The present paper incorporates the revisions needed, but the main ideas and arguments are those presented at that conference. It is with great affection and gratitude that I now belatedly dedicate it to him. 3 Editor’s note: these quotations are in Fine’s online draft version of the paper, but not in the published version. See http://www.nyu.edu/gsas/dept/philo/courses/rules/papers/Fine.pdf . Where appropriate, quotations in this chapter have been corrected to match the published version of the paper. Bob Hale, Definition, Abstraction, Postulation, and Magic In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0011

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Introduction: !x.C(x) which may be read: ‘Introduce an object x conforming to condition C(x)!’ Fine recognizes four sorts of complex postulates. Where β and γ , and also β(x) and β(F) with free variables, are postulates, and A is an indicative sentence, they are: Composition: β; γ which may be read: ‘Do β and then do γ !’ Conditionality: A → β read: ‘If A then do β!’—Fine explains that if A is true, A → β is executed by executing β; if A is false, by doing nothing. Universality: ∀xβ(x) read: ‘Do β(x) for each x!’—∀xβ(x) is executed by simultaneously executing each of β(x1 ), β(x2 ), β(x3 ), . . . , where x1 , x2 , x3 , . . . are the values of x (within the current domain). Similarly for the postulate ∀Fβ(F), where F is a second-order variable. Finally, we have: Iteration: β* read: ‘Iterate β!’—β* is executed by executing β then executing β again, and so on ad infinitum (pp.91–2). Fine invites us to compare postulates to computer programs—‘Just as a computer program prescribes a set of instructions that govern the state of a machine, so a postulational rule, for us, will prescribe a set of instructions that govern the composition of the mathematical domain’—and suggests that it is ‘helpful to pretend that we have a genie at our disposal who automatically attempts to execute any procedure that we might lay down. The rules are then the means by which we tell the genie what to do’ (pp.90–1). He describes the overall idea like this: All postulates may be obtained by starting with the simple postulates and then applying the various rules specified above for forming complex postulates. A simple postulate specifies a procedure for introducing a single new object into the domain, suitably related to itself and to pre-existing objects. A complex postulate therefore specifies multiple applications of such procedures; it requires that we successively, or simultaneously, apply the simple procedures to yield more and more complex extensions of the given domain. The only simple procedure the genie ever performs is to add a new object to the domain suitably related to pre-existing objects in the domain (and perhaps also to itself). Everything else the genie then does is a vast iteration, either sequential or simultaneous, of these simple procedures.4

4 Editor’s note: The published version of this is: ‘All of the postulational rules from our simple language may be obtained by starting with the simple rules and then applying the various clauses stated above for the formulation of complex rules. Each simple rule prescribes a procedure for introducing at most one new object into the domain, suitably related to itself and to pre-existing objects. A complex rule prescribes multiple applications of these simple rules, performed—either successively or simultaneously—to yield more and more complex extensions of the given domain. Thus the only simple procedure or ‘act’ that the genie ever performs is to introduce a single new object into the domain; everything else that he does is a vast iteration, in sequential or simultaneous fashion, of these simple acts of introduction’ (p.92).

10.1 procedural postulationism—a very brief outline

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Fine proposes postulates for arithmetic and cumulative type theory. Since one example will be enough for my purposes, I shall focus upon the former (pp.92–3). There is one single but complex postulate for arithmetic—it is composed out of the simple postulate: ZERO:

!x.Nx

and the iteration of the complex postulate: SUCCESSOR:

∀x(Nx →!y.(Nx ∧ Syx))

Nx is to be read as ‘x is a number’, so ZERO instructs us to introduce a number. Fine comments: ‘In application to a domain that does not contain a number, it therefore introduces a new object that is a number. We may take this new object to be 0’. Syx is to be read as ‘y is the successor of x’, so SUCCESSOR, Fine suggests, may be read: ‘for each object x in the domain that is a number, introduce a number y that is the successor of x (unless such an object already exists)’. Fine’s single complex postulate is then the composition of these two postulates: NUMBER:

ZERO; SUCCESSOR*

Taking his cue, no doubt, from the Book of Genesis, Fine invites us to read it informally as ‘Let there be numbers!’, and more literally as ‘first perform ZERO, i.e. introduce 0, and then keep on introducing the successor of numbers that do not already have a successor!’ So the genie will execute NUMBER by introducing an ω-sequence of objects starting with 0. Later on, Fine remarks that once all the finite numbers have been introduced in accordance with NUMBER, it can give rise to no further additions of objects to the domain.5 For ZERO can only be executed once in such as way as to result in the addition of a new object to the domain; and the iteration of SUCCESSOR adds no new objects once every finite number has been provided with a successor—there is no more work for it to do. On Fine’s approach, the standard axioms for a mathematical theory are to be derived from suitable procedural postulates—e.g. the Peano axioms are to be derived from NUMBER. Since postulates are non-propositional, this derivation will require a special logic of postulation. The characteristic form of inference, Fine tells us (p.94), will be: A1 , A 2 , . . . , A n ; α B i.e. from A1 , A2 , . . . , An , given α, we may infer B. Such an inference is valid if the execution of α converts a domain in which A1 , A2 , . . . , An are true to one in which B is true. He does not give details, but comments that: We obtain in this way a kind of axiom-free foundation for mathematics. The various axioms for the different branches of mathematics are derived, not from more basic axioms of the same sort, but from postulational rules. The axioms, which describe the composition of a given mathematical domain, give way to the stipulation of procedures for the construction of that domain. We therefore obtain a form of logicism, though with a procedural twist. The axioms 5 Editor’s note: This remark does not appear in the published version of Fine’s paper.

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of mathematics are derived from definitions and logic, as in the standard version of logicism, but under a very different conception of definition and of logic, since the definitions take the form of procedural rules and the logic provides the basis for reasonings with those rules. (p.95)

Fine goes on to discuss two central issues—the problem of consistency, and the problem of existence. In regard to the former, he contends that the procedural postulationist enjoys a great advantage over other approaches, because he can easily establish the consistency of his postulates. In regard to the latter, he attempts to show that the procedural postulationist can justifiably infer the existence of mathematical objects from the consistency of suitable postulates.

10.2 The nature and scope of postulation—some questions Among the questions to which we need answers, if we are to assess how far Fine’s proposal enjoys the advantages he claims for it, one of the most obvious concerns the content of postulates. On the face of it, NUMBER presupposes that the predicates for (natural) number and successor are already understood. But if so, how—in Fine’s view—are they to be explained or defined? In the last quoted passage, he compares and contrasts his form of postulationism with more familiar versions of logicism, from which it differs, in part, by employing a distinctive kind of ‘procedural’ definition. But it does not seem that the postulates themselves, such as NUMBER, could serve to define the non-logical terms involved—obviously they don’t constitute explicit definitions, and it is hard to see how they could function as any sort of implicit definition. So how exactly are we to understand his claim that the postulationist provides a distinctive kind of definition? There is also an obvious question concerning the scope of postulation. If we can secure the existence of natural numbers just by laying down NUMBER (given that it is consistent), what is to prevent us from conjuring all manner of other objects into existence, just by laying down suitable consistent postulates? Why can’t we just as well just as easily secure the existence of talking donkeys, say, just by laying down a postulate?—one which we might, in memory of David Lewis, call: LEWIS: !x.(talks(x) ∧ donkey(x)) Naturally, Fine denies that we can do so. Not just any predicates may figure in legitimate postulation. Specifically, he contends, only predicates of a certain kind may be employed in postulation. These are predicates whose meanings are completely given by certain constraints—as he puts it, ‘they are simply to be understood in terms of how postulation with respect to them is to be constrained’ (p.107). For example, the constraints governing the predicates involved in NUMBER are as follows (p.107): Uniqueness ∀x∀y∀z(Syx ∧ Szx → y = z) Numberhood ∀x∀y(Sxy → Nx ∧ Ny) Successor-Rigidity ∀x(Nx → (¬∃ySxy → 2¬∃ySxy) ∧ ∀y(Sxy → 2∀z(Sxz → z = y)))

10.3 consistency

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The idea is that our understanding of N (number) and S (successor) is entirely given by their conformity to these constraints. No independent grip on the meaning or conditions of application of either N or S is presupposed. Further, the only other vocabulary involved in formulating the constraints is logical vocabulary. So the case is quite unlike that of a constraint such as: Bachelorhood :

∀x(Bachelor(x) ↔ Unmarried(x) ∧ Man(x))

Here we can take the meaning of ‘Bachelor’ to be wholly given by the constraint, but obviously only if the meanings of ‘Unmarried’ and ‘Man’ are presupposed as understood. For if not the constraint would not differentiate between the meaning of ‘Bachelor’ and that, say, of ‘Vixen’ as given by the constraint: Vixen :

∀x(Vixen(x) ↔ Female(x) ∧ Fox(x))

This answer to our second question, about the scope of postulation, evidently carries with it an answer to our first, about how the predicates that figure in postulates are defined. Procedural definitions are given, not by the postulates themselves, but by the constraints which govern predicates which may legitimately figure in them. Even if these answers—or answers along these lines—are correct, it seems to me that they go little way towards answering one clearly central, and quite basic, question about Fine’s approach: How exactly is procedural postulation to be understood? Procedural postulates are imperatival in form, and imperatives are standardly used to issue commands. But it is not easy to see how, once we set aside Fine’s helpful (and tireless!) genie, we might be supposed to be able so to use them. Normally commands are directed at a suitable audience, which is intended to execute them. Of course, it is different with God—He just said ‘Let there be light!’, and there was light. But while Fine says ‘Let there be numbers!’, it seems a little optimistic to suppose that one of us mere mortals can just say a few words and achieve the desired effect. Fine describes his postulates as specifying procedures for the construction of mathematical domains—but how literally or seriously are we meant to take this talk of construction? Can we take it seriously without taking equally seriously talk of procedures as if they are executed in time—of introducing new objects into an already existing domain, and of first executing one postulate, then another, etc.? And if such talk is not to be taken literally and at face value, how are we to conceive of postulation? A useful place to start is with Fine’s claims about consistency.

10.3 Consistency As I already remarked, Fine contends that the procedural postulationist is in a much better position here than his ‘propositional rival’—or, indeed, a theorist of any other stripe. As I understand him, Fine takes this advantage to result from a difference in what the consistency of procedural postulates consists in, and hence in what is required to establish it. Since propositional postulates are statements—they are put forward as true—their consistency will be a matter of their being able to be (conjointly) true. But procedural postulates are imperatives—they are not put forward as true, but as things to be made true, or executed—so that their consistency will consist, Fine suggests, in their being able to be executed. In general, where α is any

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procedure and A any indicative statement, Fine uses the indexed modal claim: ♦α A to express that it is possible to execute α in such a way that A is then true. Then where  is any tautology, the consistency of α can be symbolized as: ♦α —since  cannot but be true anyway, this will hold only if it is possible to execute α. Thus the claim that arithmetic is consistent amounts, for the procedural postulationist, to the claim: ♦NUMBER  Of this, Fine claims, he can provide a ‘convincing demonstration . . ..[that is] . . . purely modal in character and [which] make[s] no appeal either to models or proofs or to any other kind of abstract object’ (p.98). This particular piece of magic is performed as follows: Say that a postulational rule is strongly consistent or, conservative, if it is necessarily consistent (2♦α ), that is, consistent no matter what the domain. Then it should be clear that the simple rule ZERO is conservative and that the simple rule !y.(Ny ∧ Syx) is conservative whatever the object x. For these rules introduce a single new object into the domain that is evidently related in a consistent manner to the pre-existing objects (or else they do nothing). It should also be clear that each of the operations for forming complex rules will preserve conservativity. For example, if β and γ are conservative then so is β; γ , for β will be executable on any given domain and whatever domain it thereby induces will be one upon which γ is executable. But it then follows that NUMBER is consistent, as is any other rule that is formed from conservative simple rules by means of the operations for forming complex rules. (p.98)6

So the idea is, first, that we can easily see that the simple postulates from which NUMBER is built up—that is: !xNx and, for any given x in the domain such that Nx, !y(Ny ∧ Syx)—are consistent. Second, Fine is claiming, we can easily see that they are, taken individually, not only consistent but conservative—in particular, since !y(Ny ∧ Syx) can be executed for any given x such that Nx, no matter what the domain, ∀x(Nx →!y(Ny ∧ Syx)) (i.e. SUCCESSOR) is not only consistent but conservative, so that—and this is clearly a further claim—its iteration must likewise be consistent and indeed conservative. But given that both ZERO and SUCCESSOR* are conservative— can be executed no matter what the domain—it follows that their composition (i.e. NUMBER) must likewise be consistent (and, indeed, conservative). We may separate two main claims here: first, that the relevant simple postulates are conservative, and that composition, conditionalization, universalization and iteration preserve conservativity, and second, that this can easily be seen to be the case. I don’t propose to raise any doubt here about the first claim—either about whether the simple postulates for number are conservative (and hence consistent), or about whether the operations on them that yield NUMBER preserve conservativity (and hence consistency). I am more interested in the question whether, and, if so, why, this can be easily seen to be the case, so that the proceduralist has a great advantage over other approaches—as Fine puts it:

6 Editor’s note: In the draft version of Fine’s paper, he uses the word ‘postulate’ rather than ‘rule’, hence in Hale’s discussion he writes of postulates rather than rules.

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The contrast with the standard postulational [propositional] approach is striking. There is nothing in the axiomatic characterization of a basic mathematical domain that enables us to determine its consistency and, in particular, the consistency of a conjunction of axioms cannot be inferred from the consistency of its separate conjuncts. But once the present postulational [procedural] approach is adopted, the consistency (and, indeed, the conservativity) of a rule can be read off ‘compositionally’ from its very formulation; [for the conservativity of the constituent simple procedures will guarantee that the postulate itself is conservative]. (pp.98–9)7

I think that if we could take Fine’s description of his postulates as specifying procedures for the construction of a mathematical domain quite literally and at face value, this claim would be—while by no means obviously true—at least somewhat plausible. How, one might wonder, could adding a new object to those already in existence— an object of a different kind from those already exemplified—lead to inconsistency? And how, given a domain containing one or more objects of a given kind, could adding a further object of that kind, perhaps related to those already in it in a certain way, cause any problem? So in particular, how could adding an initial number to a hitherto numberless domain lead to inconsistency? And how, given a domain with some numbers in it, how could adding a further number, larger than all those already present, cause a problem? It seems, in other words, that there could be no obstacle to executing ZERO and then repeatedly executing the simple postulate embedded in SUCCESSOR—i.e. !y(Ny ∧ Syx). As Fine puts it: ‘these postulates introduce a single new object into the domain that is evidently related in a consistent manner to the pre-existing objects’. But this is not the way to go. In the first place, any such constructionist or creationist view is clearly abhorrent—the suggestion that we ourselves created the natural numbers is simply preposterous; and to propose that someone, or something, else did so is to escape this absurdity only at the cost of mysticism. Secondly, Fine anyway makes it quite clear that he does not want anything to do with any sort of creationist view. He writes: it is important to guard against certain possible misconceptions. The present view is not creationist; we do not suppose that the quantifiers range over what does and did exist and that postulation works by literally creating new objects which then enter into the domain of quantification. The objects that are introduced through postulation existed prior to all acts of postulation (if indeed they exist in time at all) and would have existed even if there had been no postulation or people to postulate. By postulation, we incorporate these objects into the domain of the unrestricted quantifier, but through a re-interpretation of the quantifier rather than a re-invigoration of the ontology.8

The crucial question, clearly, is whether, once the constructionist patter is discarded as misleading, and we shift to the position that the relevant objects pre-exist any acts of postulation, or at least exist quite independently of them, the easy route to consistency remains available. I don’t myself see how it can do so. So it seems to me

7 Editor’s note: The last phrase in square brackets appears only in the draft version of Fine’s paper. 8 Editor’s note: This passage is missing from the published version of Fine’s paper.

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that, prescinding from the immediate and intrinsic implausibility of the creationist view, there is a trilemma Fine must confront here. If we take a creative view, we have to ask whether we are to think of the creation as ours (first horn), or (second) as effected by some being not subject to human limitations. If we take the first option, then, whilst we might (but only might) retain the advantage of an easy proof of consistency of the sort Fine claims, it is quite unclear how postulation could deliver even the totality of natural numbers—let alone the classical continuum or the universe of standard set theory. Like the intuitionists, we would have to think of the sequence of natural numbers as, at any given time, finite but indefinitely expandable. If, on the other hand, we suppose the relevant procedures to be executed by some being not subject to human limitations, but capable of infinitary operations, then the question whether they can be executed becomes, in effect, equivalent to the question whether there is any inconsistency in the supposition that—somehow, we know not how—they have been executed. Since that is essentially the same question as whether the relevant mathematical axioms (e.g. for number theory or analysis or set theory) are consistent, the supposed advantage of the procedural approach has again been lost. If, finally (third horn), we do not interpret postulates creatively, but as disclosing an already existing reality, then we sacrifice the alleged advantage of procedural over propositional postulation—for once we stop thinking in terms of objects being added to an already existing domain, we can no longer think of NUMBER, for example, as specifying a sequential procedure which builds up, by obviously consistency preserving steps, from a numberless initial domain by successively adding finite cardinal numbers until they are all there. We have instead to ask whether things could be as if NUMBER had been executed— i.e. whether it can be consistently supposed that there exist infinitely many objects ordered as a progression. I would not claim to have shown that Fine can’t find a way past this trilemma. At this stage, the most I would claim for it is that it helps to make clear one serious challenge he faces—to explain how his procedural postulates can be understood in a way that steers clear of a creationist position whilst retaining the one advantage which that otherwise repugnant view seems to possess, of facilitating an easy route to an assurance of consistency. Fine does in fact propose an alternative interpretation of his postulates—not in the course of discussing the problem of consistency, but later, in his discussion of the problem of existence, in the course of defending what he calls the ‘magical inference’ from the consistency of procedural postulates for a mathematical theory to the truth of the theory’s axioms. To this I now turn.

10.4 Postulates as expanding domains of quantification The leading idea, as it appears to me, is that the procedural postulationist is to be understood, not as attempting somehow to conjure objects out of nothing, but rather as enlarging or expanding the domain of discourse, specifically the domain of firstorder quantification. Fine writes: A postulate is meant to provide us with an understanding of the domain of discourse—which, for present purposes, we may identify with the domain of quantification; and it provides us

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with that understanding on the basis of a prior understanding of the domain of discourse. The postulate NUMBER, for example, take us from an understanding of a domain of discourse which does not include numbers to one that does.9

His thought here seems to be this: We start out with an understanding of the domain of discourse—the domain over which our (first-order) quantifiers range—and the (intended) effect of a procedural postulate—such as NUMBER—is to take us, on the basis of this prior understanding—to a new, enlarged domain of discourse. There are, according to Fine, two quite different ways in which we can thus advance from a prior understanding of a given domain to a new understanding of an enlarged domain: Now a procedural postulation is meant to effect an expansion in the domain of discourse and so one way of understanding how it might do this is to suppose that it relaxes a restriction on the domain that is already in force. It could do this either by lifting the current restriction—going from male professor, for example, to professor—or by loosening the current restriction—going from male professor to male or female professor—or perhaps in some other way. (pp.100–1)10

But this, Fine observes, is of no use to the procedural postulationist who is trying to justify the magical inference—the inference from 2α  to Ax, via the stipulation of α: If this is how procedural postulation is meant to work, then I see no way in which it might plausibly be taken to provide us with the kind of justification for existential claims that we are after. For there is nothing in the nature of relaxing a restriction that might warrant us in supposing that there are objects not subject to the restriction. (p.101)

But there is, he contends, another way: The other method is creative or expansive; no wider domain is presupposed and postulation works by directly effecting a genuine expansion in the given domain. (p.103)

This sounds dangerously close to the creationist position which Fine wishes to avoid, but, as we saw above, he insists that this is not how his position should be understood: ‘By postulation, we incorporate [new] objects into the domain of the unrestricted quantifier, but through a re-interpretation of the quantifier rather than a re-invigoration of the ontology.’ There are, I think, two main worries this raises: First, there is the issue of consistency again. For, granting that one can avoid creationism by understanding expansion in terms of the introduction of independently existing objects into the domain of quantification (rather than into the mind- and language-independent world, as it were), isn’t the price of doing so simply to re-instate the problem of establishing

9 Editor’s note: This passage corresponds best to the following in the published paper: ‘A procedural postulation is meant to effect an expansion in the domain of discourse’ (p.100). See also p.104. 10 Editor’s note: This passage replaces that quoted by Hale from the draft version, namely: ‘One method is familiar and unproblematic. We understand one domain of discourse as a restriction of another. If a given domain ranges over all animals, for example, we may restrict it to the sub-domain of dogs. Now a postulate, as we understand it, is meant to result in an expansion of the domain and so, if this method is to have any application to postulation, we must suppose that a postulate effects an expansion in the domain by relaxing a restriction on the domain that is already in force.’

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consistency—in its old, seemingly intractable, form? It is all very well to say that we don’t create the natural numbers—that they exist independently of postulation, and we just revise our domain of quantification to encompass them; but then doesn’t there have to be a good question what entitles us to suppose that we can do this consistently, or to suppose that there is, all along, an infinite collection of objects of the right sort for NUMBERS to introduce into the domain of quantification? Second, there is the issue of unrestricted quantification. For the enlargement of a domain of quantification to come about through genuine expansion, rather than mere de-restriction, it is necessary that the domain not be thought of as implicitly restricted—rather it must be unrestricted; and the quantifiers on such an unrestricted domain must be taken to range over everything whatever—since otherwise (i.e. if the quantifiers are taken as ranging over the Fs, say) any enlargement of the domain would be merely a de-restriction. There appears to be a tension in Fine’s view here—for on the one hand, he requires the initial domain which is to be expanded by postulation to be unrestricted, on pain of there being merely a de-restriction of an already implicitly restricted domain; but on the other, if the quantifiers over the initial domain really are unrestricted—if they range over all objects whatever—how can there be any objects left over to introduce into the domain? How can one expand a genuinely unrestricted domain? Fine’s proposed solution to this problem is to embrace a form of relativism about unrestricted quantification. On this account, prior to the postulation of numbers, numbers are not in the unrestricted domain of discourse. But as a result of the stipulation NUMBER, the unrestricted domain comes to include numbers. So the ‘pre-postulational’ domain without numbers is unrestricted, but so too is the ‘postpostulational’ domain which includes numbers but also includes the supposedly unrestricted pre-postulational domain. To resolve the apparent contradiction here, Fine holds that the status of a domain as unrestricted is relative to the postulational context. Relative to the context in which numbers have not been postulated, there is an unrestricted domain which does not include numbers, but relative to the postulational context brought about by postulating numbers, an unrestricted domain must include the numbers. A proper assessment of this intriguing, but also somewhat mysterious, proposal requires a much fuller discussion than I can undertake here. How are postulational contexts individuated, and how is relativity to such contexts to be understood? Suppose someone postulated numbers at a certain time—does this mean that any subsequently obtaining postulational context will be one relative to which a domain of quantification can be unrestricted only if it includes the numbers? Or is it rather that the postulational context is individuated by reference to speakers or thinkers, and is determined by their relevant intentions or beliefs? These are some of the most obvious questions to which we need answers. I am not sure how Fine means to answer them. However, there is one clear and controversial implication of his proposal which he quite explicitly acknowledges and defends. This is that he must reject as unintelligible the idea of absolutely unrestricted quantification. I shall conclude with a brief discussion of this.

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10.5 Expansion vs. absolutely unrestricted quantification Fine grants that there is a powerful intuition in favour of absolutely unrestricted quantification. He envisages an opponent arguing against him as follows: surely . . . we are capable of achieving an absolutely unrestricted understanding of the domain. For surely we can form the conception of an object that might be introduced through postulation. And can we not then take the domain to include all such objects?11

But there is, Fine maintains, an at least equally plausible intuition which clashes head on with this one—that ‘the domain is always susceptible of expansion’. So one of the two intuitions must be given up. Fine claims that while the intuition in favour of absolutely unrestricted quantification is very strong, the postulationist defender of domain expansion is able to accommodate it—in the sense that he can, whilst rejecting it, ‘say what is right about it and why we might have been misled into thinking it was correct’, whereas, he claims, ‘it is far from clear that the proponent of [absolutely unrestricted quantification] is in an equally good position to accommodate our intuition’ [i.e. the intuition in favour of domain expansion]. To accommodate, i.e. explain away, the appearance that it is possible to quantify over absolutely everything, Fine appeals to a distinction between what he terms ordinary or ‘actual’ quantification and a modal variant of ordinary quantification he calls ‘potential quantification.12 Thus if ‘∃x’ (‘there is an x’) is the ordinary existential quantifier, the potential quantifier is ‘♦∃x’ (‘possibly there is an x’), where the diamond expresses ‘postulational possibility’. I think the idea is that we should interpret ‘♦∃xFx’ as meaning something like ‘it could come about through postulation that there is an F in the domain’, or maybe ‘postulation could take us from an F-free domain to one containing Fs’. There is a corresponding potential universal quantifier, written ‘2∀x’. The explanation of the illusion is then as follows. The potential quantifiers, Fine says, are ‘absolute’; in contrast to the actual quantifiers, their ‘range’ is not relative to postulational context. There is also a sense in which they are unrestricted, since the inference from ∃xA to ♦∃xA is always justified. However, the potential quantifiers are not strictly quantifiers at all; they do not delimit a domain of discourse; they are incapable of providing a basis for postulation.

So the idea seems to be that we slide into thinking that absolutely unrestricted quantification is possible because we confuse potential quantification with actual, and mistake the former for a species of genuine quantification. I should like to make two comments on this, and one on Fine’s suggestion that his opponent cannot do as well at accommodating his intuition in favour of domain expansion as he can at accommodating the intuition in favour of absolutely unrestricted quantification.

11 Editor’s note: These lines do not appear in this form in the published version of Fine’s paper. 12 Editor’s note: These quantifier notions are not discussed in the published version of Fine’s paper. He discusses instead our initial quantifiers, ∀ and ∃, and our new understanding of the quantifiers once we have expanded the domain, given by ∀+ and ∃+ .

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(i) Even if I am suffering from some kind of illusion when I think I can quantify absolutely unrestrictedly over everything whatever, I am quite sure that I have not fallen victim to this illusion as a result of confusing potential with actual quantification, and failing to appreciate that the former is not, strictly, a form of quantification at all. Until very recently, I had never encountered potential quantifiers and was entirely innocent of the concept of postulational possibility. I suspect the same is true of many other believers in absolutely unrestricted quantification. This raises the question: in what sense, if any, is Fine able to accommodate the intuition in question. It appears to me that he does not accommodate it in the sense of putting forward a plausible explanation why others (mistakenly) believe in absolutely unrestricted quantification. I cannot see that he can really claim to do more than draw attention to a notion—potential quantification— which is definable in postulationist terms, which bears some resemblance to ordinary quantification, and which is, in a certain sense, both ‘absolute’ and ‘unrestricted’. (ii) Fine asserts that ‘the potential quantifiers are not strictly quantifiers at all; they do not delimit a domain of discourse’. The apposition here suggests that it is because they do not delimit a domain of discourse that the potential quantifiers are not strictly quantifiers. I am not sure what Fine is getting at here. I assume he does not mean anything that requires that for genuine quantification, there must be a definite set or class containing the possible values of the bound variables—for this plainly just begs the question against the friends of absolutely unrestricted quantification. It might—perhaps more reasonably—be claimed that we do not understand ∀x or ∃x unless we know what the bound variables range over. But why can’t we just take them as ranging over all objects whatever?—why demand further delimitation of the domain? (iii) Pace Fine, it seems to me that his opponent—or at least one kind of opponent—can do at least as well in accounting for the postulationist’s belief in domain expansion as the postulationist can do in accounting for the belief in absolutely unrestricted quantification. Indeed, in one way, it seems to me he can do better. The kind of opponent I have in mind is, naturally, the neo-Fregean. On this view, we can—subject, of course, to certain crucial constraints—use abstraction principles as a means of implicitly defining concepts. We can, to take the best-known and most discussed example, lay down Hume’s principle—that the number of Fs = the number of Gs if and only if there is a one-one correspondence between F and G— as an implicit definition of the number operator. For present purposes, two features of the neo-Fregean position are especially relevant. First, there is the use of absolutely unrestricted quantification. In the case of Hume’s principle, the first-order quantifiers involved in the definitional expansion of the right-hand side are to be understood as ranging over all objects whatever, including numbers. More generally, the abstract objects to which—if the abstraction is acceptable and instances of its right-hand side are true—its left-hand side terms make reference lie within the entirely unrestricted range of the first-order quantification explicit or implicit in its right-hand side.

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Second, the intended effect of laying down an abstraction is not to create new objects, nor—in contrast with procedural postulation—is it to expand the domain of first-order quantification to incorporate a range of independently existing objects which had hitherto lain outside it; rather, it is to disclose, by way of the introduction of a new concept, a range of objects already lying within the domain of quantification, but not previously identifiable or discernible, because the conceptual resources needed for their identification were lacking. Someone who accepts both these things can recognize a third position one can take about our understanding of the range of quantification, distinct from both de-restriction and expansion: the range of unrestricted quantification is absolutely unrestricted, but by means of abstraction, we can disclose previously unrecognized kinds of object lying within it. It is not a matter of lifting a restriction, because the domain is unrestricted. But neither is it a matter of expanding the domain to include a further range of objects, again because the domain is unrestricted. From this point of view, the domain expansionist confuses the apparent possibility of domain expansion with the real possibility that we can always introduce new concepts in terms of which previously unrecognized ranges of objects can be conceived. How is it that we can understand quantification as absolutely unrestricted? As we saw, Fine thinks its proponents might argue that we can do so because ‘we can form the conception of an object that might be introduced through postulation’. This is an argument that one might find attractive, if one was disposed to accept Fine’s claim that any domain can be expanded (i.e. by postulation, rather than creation)—for then one might think that one could still, by means of the concept object that might be introduced through postulation make room for absolutely unrestricted quantification, by getting a characterization of the domain which would include all the objects falling under that concept. But whatever the attractions this might have for a believer in domain expansion by postulation, it does not seem to me the best way to answer our question. We understand restricted quantification over objects of a certain kind by means of a concept under which all and only those objects fall. What is needed is not just any concept, but a sortal concept—that is, roughly, a concept (such as animal, human being, or cardinal number) with which we associate a criterion of identity, in contrast with merely adjectival concepts, such as red (or red thing), with which we associate only a criterion of application. If we are to understand absolutely unrestricted quantification, what we need is a concept under which all (and only) objects fall. It is sometimes supposed that we can characterize the universal domain of objects using the concept of self-identity— for surely anything that is a term of the identity relation is an object, and every object bears this relation to itself (and to nothing else)? I cannot myself endorse this suggestion, because I think we must recognize, contra Frege, that the identity relation is not restricted to objects—as philosophers, we may properly enquire after the identity conditions for properties, relations, and functions, and perhaps of entities of other kinds. However, I think we can specify the domain of absolutely unrestricted objectual quantification easily enough. For, on the broadly Fregean approach to defining ontological categories I favour, an entity is an object if and only if it is or could be the referent, or semantic value, of a singular term. With this independent characterization of objects in hand, we may simply stipulate that the range of a bound

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individual variable is to comprise all (and only) objects. Alternatively, since all (and only) objects fall under first-level sortal concepts, we may specify the intended range as containing any entity which falls under some first-level sortal concept or other— that is, an entity x such that ∃F Fx, where F ranges over first-level sortal concepts.13

13 Note that this does not constitute an independent, free-standing characterization of objects, since we would need to invoke the concept of object in explaining the restriction to first-level sortals. Note also that I am not claiming that ∃F Fx is itself a sortal concept. Such a claim would be implausible, given that a disjunction of sortal concepts—e.g. horse or number—does not in general seem to be sortal. There is no need for any such claim, since clearly anything falling under ∃F Fx will fall under some sortal concept. It must also be conceded that this alternative involves using second-order quantification to help explain unrestricted first-order quantification, albeit second-order quantification restricted to sortal concepts; it seems to me that this is unobjectionable.

11 Second-order Logic: Properties, Semantics, and Existential Commitments 11.1 Introduction Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments.1 The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with an abundant or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Kimberly Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational.

11.2 Quine’s challenge The central core of Quine’s charge that higher-order logic is really just ‘set theory in sheep’s clothing’ is that it carries massive existential commitments. Higher-order predicate calculus is, he insists, a reformulation of set theory which 1 Originally published as Hale (2015b). Reprinted by permission from Springer Nature, Synthese, “Second-order logic: properties, semantics, and existential commitments”, Hale, B., 2015. Springer Nature B.V., Second-order Logic: Properties, Semantics, and Existential Commitments In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © Springer Nature B.V. DOI: 10.1093/oso/9780198854296.003.0012

188 second-order logic gives it a deceptive resemblance to logic. One is apt to feel that no abrupt addition to the ordinary logic of quantification has been made; just some more quantifiers, governing predicate letters already present. In order to appreciate how deceptive this line can be, consider the hypothesis ‘(∃y)(x)(xy. ≡ Fx)’. It assumes a set {x : Fx} determined by an open sentence in the role of ‘Fx’. This is the central hypothesis of set theory, and the one that has to be restrained in one way or another to avoid the paradoxes. This hypothesis itself falls dangerously out of sight in the so-called higher-order predicate calculus. It becomes ‘(∃G)(x)(Gx ≡ Fx)’, and thus evidently follows from the genuinely logical triviality ‘(x)(Fx ≡ Fx)’ by an elementary logical inference. Set theory’s staggering existential assumptions are cunningly hidden in the tacit shift from schematic predicate letter to quantifiable set variable.2

Obviously Quine’s charge relies upon an assumption about how higher-order logic must be understood—since independently of any interpretation, it can have no existential commitments at all. The assumption, explicit in the final quoted sentence, is that its higher-order variables must be interpreted as set-variables, so that the comprehension scheme ∃X∀yXy ↔ φ (provided X is not free in φ) amounts, in effect, to the claim that every open sentence determines a set. Quine believes that this interpretation is forced upon us, in effect, by the requirement for a satisfactory account of the identity-conditions for the values of its variables (‘No entity without identity’), given that—as Quine further believes—no satisfactory such account can be given for the only other serious candidates to be the values of higher-order variables, viz. properties (or attributes, as he calls them). Since the assumption that we must take second-order variables to be set-variables may certainly be resisted, it is important to see that the central core of Quine’s charge does not really depend upon it. One might insist, contra Quine, that second-order variables should be interpreted as ranging over properties, not sets. But if in response to his demand that we provide clear identity-conditions for them, we propose that properties be taken to be same iff they apply to the same objects, nothing is really gained. For as long as the values of our variables are individuated purely extensionally, it makes little difference what we call them or whether they are sets or properties. So individuated, there will be just as many of them as there are sets—from a mathematical point of view, the assumption of a domain of properties is no weaker, and so no less problematic, than the assumption of a corresponding domain of sets.3 Essentially the same point applies when we turn to interpretations in the formal, model-theoretic sense. Here, an interpretation specifies some set as the domain over 2 See Quine (1970), pp.66–8. Most of his other complaints—that reading predicate letters as attribute variables is the product of a use-mention confusion, that putting predicate letters in quantfiers, as in ∃F, must involve treating predicates as ‘names of entities of some sort’, and that ‘variables eligible for quantification . . . do not belong in predicate position . . . [but] in name position’—rest on what seems to me an unjustified refusal to entertain the possibility of non-nominal quantification. It is certainly true that higher-order quantification does not read back easily into natural English, so that we are likely, faced with a sentence such as ∃F∀xFx, to resort to a nominalizing paraphrase, such as ‘There is some property which every object has’; but that hardly justifies insisting that we can only make sense of it by confusing predicates with names. 3 Charles Parsons had already argued, some years before Quine’s famous onslaught, that even when the values of second-order variables are taken to be ‘incomplete’ entities, such as Fregean concepts, secondorder logic—or at least ‘full’ second-order logic, with unrestricted impredicative comprehension—carries existential commitments comparable to those of standard set theory. See Parsons (1965).

11.3 properties 189 which the individual variables range, and interprets the second-order variables as ranging over a set of subsets of that set. This may be the complete powerset of the individual domain, as in the standard semantics, or some proper subset of the powerset, as is permitted in the more general Henkin semantics. Of course, the use of model-theoretic semantics for a second-order language does not, in and of itself, commit one to the claim that its sentences are ‘really’ about sets, any more than the use of possible world semantics for a modal language commits one to claiming that its sentences are ‘really’ about possible worlds. But even if one insists that the second-order variables range over entities one calls ‘properties’, it remains the case that, so long as one assumes the existence of a property for each of the subsets of the individual domain taken to exist in the standard semantics, or the more general Henkin semantics, one’s interpretation makes second-order logic a version of classical set theory—in effect, even if not in name—just as Quine complained.4

11.3 Properties In my view, we should interpret predicate variables as ranging over properties, suitably understood. This section sketches the conception of properties I favour. If Quine’s central charge is to be deflected by taking the second-order variables to range over properties (and relations),5 and his demand for identity-conditions met, then they must be individuated more finely, and hence non-extensionally.6, 7 How should this be done? An obvious strengthening of the purely extensional condition for property-identity canvassed above would have it that properties are the same iff they necessarily apply to the same objects. But this strengthened condition, while clearly an improvement, is insufficiently discriminating. For it seems that distinct properties might coincide in extension, not just contingently, but as a matter of necessity. 4 In his very useful exposition and defence of second-order logic, Stewart Shapiro seeks to remain officially neutral on the issue whether second-order variables range over classes, with their purely extensional identity-conditions, or properties, which ‘are often taken to be intensional’ (Shapiro 1991, p.16)—however, he makes it clear that his preference, ‘if pressed’, would be for an extensional interpretation (see pp.63–4). Shapiro’s own response to Quine’s charge is quite distinct from the one I am recommending. In essence, he counters by appealing to another Quinean doctrine—if, as Quine maintains in line with his holistic methodology, there is no useful distinction between mathematics and physics, why should he think there is one between logic and mathematics (‘especially the logic of mathematics’, op. cit. p.17, Shapiro’s emphasis)? 5 For simplicity, I discuss only properties explicitly—adapting what I say to relations is a routine matter. 6 Of course, any departure from extensionality will mean giving an account of property identity which Quine would have found unacceptable, for that very reason. It is no part of my project to meet all Quine’s demands on his terms, some of which I reject as unreasonable. For a somewhat fuller response to Quine, see Hale (2013b). 7 It might be thought that individuating properties more finely—i.e. more finely than sets—would mean that there would be more properties than sets, rather than fewer. It is true, of course, that individuating properties non-extensionally may entail recognizing distinct (co-extensive) properties where there may be but one set. But it matters much how precisely properties are individuated. The condition for property identity and existence I shall recommend later in this section will mean that properties cannot outrun the definable sets. Since these are usually (i.e. classically) taken to constitute a tiny minority of the sets, it might be thought that my proposed condition is unduly restrictive. From what might be a called a full-blooded classical standpoint, it is restrictive—but whether it is unduly so depends, of course, on the defensibility of that standpoint. In another sense, the condition I recommend is less restrictive than one might suppose. See the closing paragraphs of Section 11.4 below.

190 second-order logic Necessarily all and only equiangular triangles are equilateral, but the properties are surely distinct, even so. Necessarily all and only general recursive functions are Turing computable, but again, we have two distinct properties. Necessary co-extensiveness is surely required for predicates to stand for the same property, but not sufficient. We can see a way to remedy this shortcoming by reviewing a somewhat similar weakness in the proposal, closely related to Leibniz’s famous definition of identity, that objects are the same iff they share all their properties.8 That this gives a necessary condition for identity seems beyond reasonable dispute, alleged counterexamples to what Quine once called the Indiscernibility of Identicals involving referentially opaque contexts notwithstanding. If φ(α) is true but φ(β) false, even though α = β, because whether or not φ(−) is true of x depends upon how x is specified, that is precisely a reason for denying that φ(−) stands for a property of x. The problem lies rather with its sufficiency—in effect, with Leibniz’s principle of the Identity of Indiscernibles. Clearly the property-sharing condition would be sufficient, if properties such as the property of being identical with α are included in the range of its second-order quantifier, but only at the price of triviality and uninformativeness. If that is to be avoided, we must confine attention to purely general properties—properties which can be specified without essential identifying reference to particular objects; but then it seems perfectly possible that two objects should have all such properties in common. In this case, a more plausible necessary and sufficient condition for identity (i.e. of objects) may be got by necessitation: a = b ↔ 2∀φ(φa ↔ φb).9 Returning to the case that especially concerns us here, an obvious thought is that just as we require, for identity of objects, necessary coincidence of all first-level properties, so, for identity of first-level properties, we should require not just necessary coincidence in application (i.e. 2∀x(Fx ↔ Gx)), but necessary coinstantiation of all second-level properties—as we may put it, using λF to denote the property of being F: λF = λG ↔ 2∀X(X(F) ↔ X(G)). It is important to be clear that what is being claimed is only that a = b ↔ 2∀φ(φa ↔ φb) and λF = λG ↔ 2∀X(X(F) ↔ X(G)) state necessary and sufficient conditions for the identity of objects and of first-level properties respectively. Neither is being proposed as a definition of identity, or even as a working test or criterion. 8 Leibniz gives his definition in several places (see, for example, Gerhardt (1875–90), vol.7, pp.219–20, 225, 228, 236, Couturat (1961), p.362—these passages are all translated in Parkinson (1966), pp.34–5, 43, 52–3, 122, 131). Here is a typical formulation: ‘Those terms are “the same” of which one can be substituted in place of the other without loss of truth, such as “triangle” and “trilateral”, “quadrangle” and “quadrilateral”. ’ Although my formulation in the text is quite common, it is not, as far as I know, used by Leibniz. It is plausibly taken to be a consequence of Leibniz’s definition in terms of intersubstitutability salva veritate. Intersubstitutability guarantees that for any context φ(−), φ(α) is true iff φ(β) is, so that provided can take φ(−) to stand for a property which α and β have or lack, they must have all their properties in common. The steps are reversible. The formulation in terms of sharing all properties has the advantage of avoiding the usemention confusion with which Leibniz is sometimes—and, in my view, somewhat pedantically—charged. But it is worth noting that his own definition is more general, and not restricted to identity of objects. On the contrary, the majority of his examples involve pairs of general terms, as in the examples quoted. Thus the un-necessitated version of the definition of property-identity I go on to propose in the text has a good claim to be a further special case of Leibniz’s definition. I am grateful to Maria Rosa Antognazza for some references and advice on Leibniz’s views. 9 It is obvious that the strengthened condition is necessary. For an argument for its sufficiency, see Hale (2013a), pp.187–8.

11.3 properties 191 Viewed as definitions, they suffer from an obvious drawback—the former would presuppose an understanding of identity for first-level properties, and the latter would presuppose an understanding of identity for second-level properties. We would have, if not a vicious circle, an equally vicious-looking regress. Viewed as working tests of identity, they suffer from an equally obvious drawback. Whilst we might convince ourselves that a = b by finding that Fa ∧¬Fb for some particular F, we can scarcely hope to establish that a = b by an exhaustive inspection of all their properties; and so, mutatis mutandis, for λF = λG and λF = λG. Neither failure impugns the claim to give a necessary and sufficient condition for identity, which is quite consistent with identity being indefinable (as it probably is)10 and with there being better working tests (as there probably are—see next). The conception of properties underlying the interpretation of second-order logic I am recommending is the abundant—or as it might equally justly be termed, deflationary—theory of properties. On this view, in sharp contrast with that of some metaphysicians,11 the conditions for property existence are very undemanding— hence abundance—but, precisely for that reason, existence claims are correspondingly weak—hence deflation. Roughly, the existence of a suitable predicate is sufficient for that of a property. A suitable predicate is one which has a well-understood satisfaction condition. I say ‘well-understood’, rather than ‘well-defined’ because it is unnecessary that the predicate be definable in any sense that requires an informative statement of the conditions for its application. An obvious example is colour predicates—speakers’ grasp of the satisfaction conditions of ‘. . . is red’ (‘crimson’ etc.) is manifested in their competent application, not their ability to say what it takes for something to be red, or crimson, etc. The same examples, as well as many others, illustrate the further point that well-understood satisfaction conditions need not be, and often will not be, precise. For a more accurate statement, we need a distinction, between purely general and object-dependent properties. The latter, as the label suggests, are those which depend for their existence on the existence of particular objects to which reference is made by the corresponding predicates (e.g. the property of being a brother of Aristotle, or of being at the top of the Tour Eiffel); purely general properties are those which involve no such dependence—they are properties which can be specified by predicates whose satisfaction-conditions concern no particular objects, so that the mere existence of a suitable such predicate is sufficient for the existence of the property. In the case of object-dependent properties—properties given by predicates essentially involving singular terms—more is of course required. There is no property for which the predicate ‘ξ is Aristotle’s eldest brother’ stands unless there is an object for which the name ‘Aristotle’ stands, and this will be so—roughly—if and only if there are 10 By this, I mean only that it is probable that no definition of the word or concept can be given. I do not mean that there can be no definition of the relation of identity itself, in the sense of a specification of its essence or nature (for some general remarks on definition in the latter, broadly Aristotelian, sense, see Hale (2013a), pp.150–6). 11 I am thinking, of course, of the ‘sparse properties’ and ‘sparse universals’ advocated by David Lewis and David Armstrong. Lewis’s own usage of ‘property’ is quite different from mine, and potentially confusing. He distinguishes between what he calls sparse and abundant properties, identifying the latter with sets. See Lewis (1986), pp.55–69.

192 second-order logic true atomic statements featuring the term ‘Aristotle’ in its relevant use.12 Of course, in no case is the actual existence of a suitable predicate a necessary condition— for it is enough that there could be a suitable predicate, even if no actual language contains one. It is this—the fact that all that is required for the actual existence of a purely general property is the possible existence of a suitable predicate—that makes the conception abundant. Given the abundant conception together with a reasonable assumption about the modal logic of absolute necessity, one can prove that purely general properties exist necessarily.13

11.4 Second-order semantics On the conception I am recommending, properties are individuated intensionally. Taking predicates and the associated second-order variables to stand for or range over properties so understood does not, as might be suspected, enforce an intensional semantics, or preclude the useful deployment of an extensional, model-theoretic semantics for second-order logic—any more than taking sentence variables to range over propositions, individuated intensionally, means that we can’t do useful metalogical work using the usual truth-functional semantics. Just as with the standard extensional semantics for a first-order language, where we can assign as semantic values to constant predicates subsets of the domain of objects, consistently with retaining a conception of those predicates as standing for properties which are, perhaps, individuated intensionally, so when we come to the semantics for a secondorder language, we can take the values of the second-order variables to be subsets of the domain of objects—intuitively, possible extensions of the properties for which those predicates stand—whilst thinking of the properties themselves as individuated intensionally. But if we are not to break faith with this underlying conception of properties, there has to be a significant restriction on the kind of extensional, modeltheoretic semantics we may adopt. For according to that conception, a necessary (as well as sufficient, in the case of purely general properties) condition for the existence of a property is that there could be a predicate whose sense embodies the appropriate satisfaction-condition. By a predicate here, we mean an expression of finite length—an expression which we could, at least in principle, understand and use to speak of the property. It is this which leads to a substantial—and, of course, scarcely uncontroversial—restriction on admissible interpretations of second-order languages, when their second-order variables are taken to range over properties in our sense. In more detail, a good way to get the restriction into focus is to see why the standard semantics is unacceptable, given the abundant conception of properties. The

12 The statement of a necessary as well as sufficient condition calls for somewhat greater care than is exercised here—see Hale (2013a), pp.36–7. 13 Let P be any purely general property, p the proposition that P exists, and q the proposition that there is a predicate standing for P. Given the abundant conception of properties, 2(p ↔ ♦q). By the Law of Necessity, it follows that p ↔ ♦q, and by the K-principle, 2p ↔ 2♦q. Provided that the logic of the modalities involved is S5, we have ♦q ↔ 2♦q. Whence by the transitivity of the biconditional p ↔ 2p. For a slightly fuller statement, see Hale (2013b), p.135, Hale (2013a), pp.165–7.

11.4 second-order semantics

193

standard semantics is a special case of the more general Henkin semantics, which allows one to take one-place predicate variables to range over any subset of the power set of the domain D over which the individual variables range, two-place predicate variables to range over any subset of the power set of D2 , and so on. The standard semantics insists that we may take only the complete power set itself as the domain for the one-place predicate variables, not any proper subset of that set, and likewise, mutatis mutandis, for the two- or more-place predicate variables. According to the classical conception, which is usually assumed, if the domain of objects is infinite, the domain of the second-order variables includes, and is indeed almost entirely composed of, arbitrary infinite subsets of the domain of objects. These subsets must each be thought of as resulting from an infinite sequence of arbitrary selections from the object domain, and the second-order domain as a whole must be seen as the product of an uncountable infinity of such arbitrary infinite selections. None of these subsets is finitely specifiable—they are precisely not generated by any finitely statable rule, nor can they be given by a complete listing of their elements. This is where the standard semantics (and, a fortiori, the Henkin semantics) breaks faith with the abundant conception. For according to that conception, it is not only sufficient, but also necessary, that each property is precisely something which can be expressed or represented by a predicate, i.e. an expression of finite length whose meaning is the predicate’s satisfaction-condition. Once this point is appreciated, there is an obvious and entirely natural adjustment needed to bring second-order semantics into line with the abundant conception: the second-order domain should consist of all and only the definable (i.e. finitely specifiable) subsets of the first-order domain. Of course, proponents of the classical, standard semantics will see this as an unwelcome restriction. But it constitutes a restriction only when viewed from the classical standpoint, and this, though orthodox, is open to challenge; an advocate of the abundant conception of properties can and should insist that these are all the properties there are, and so deny that it is really any restriction at all. What are the metalogical consequences of the shift I am recommending? What impact does its adoption have on the standard metalogical results (i.e. the metalogical results obtainable on the basis of the standard second-order semantics)? As is well known, under the standard semantics, key model-theoretic metatheorems for firstorder logic (compactness, completeness and the upward and downward LöwenheimSkolem theorems) fail at second-order; on the other hand, while first-order arithmetic has non-standard models, second-order arithmetic can be proved to be categorical (as can second-order analysis). Under the more general Henkin semantics, on the other hand, these results are all turned upside down: compactness, completeness, and the Löwenheim-Skolem results are all obtainable, while the categoricity results are not.14 It might be thought, especially if one views the more modest second-order semantics just proposed as a restriction of the standard semantics, that its adoption would fail to secure what many will see as a key advantage of second-order logic plus standard semantics over its rivals, viz. the provability of the categoricity results for

14 For a very clear and detailed account of these results, see Shapiro (1991), chs.3, 4.

194 second-order logic arithmetic and analysis.15 However, whilst my modest semantics diverges significantly from the standard semantics,16 it agrees with that semantics, and diverges from the more general Henkin semantics, in one crucial respect: once the first-order domain is fixed, there is no freedom of choice over the second-order domains. It is this feature of the standard semantics which, at bottom, underpins proofs of categoricity for arithmetic and analysis, and explains why the proofs of completeness, compactness, and Löwenheim-Skolem which can be given assuming Henkin semantics fail when the semantics is standard. Of course, if the proposed restriction of the second-order domains to definable subsets is to issue in a workable model-theoretic semantics, the notion of a definable subset must be made definite. A restriction to subsets definable in the object-language—the language we are using to formulate our system of secondorder logic, and second-order theories—would be needlessly crippling. There is, as far as I can see, no good reason not take take ‘definable’ to mean ‘definable in the meta-language’. Thus if, for example, our object-language is a second-order language for arithmetic, and we are seeking to prove the categoricity of our axiomatization, a suitable meta-language might be an extension of the object-language to include a certain amount of set-theoretic vocabulary (as in Shapiro (1991), 4.2). The secondorder variables would then be interpreted as ranging over the subsets of the first-order domain definable in this meta-language. If this is done, then the standard categoricity 15 I did think this, and said as much in Hale (2013b), where I sought—needlessly, as I now see—to argue that the loss of categoricity results was not a crippling disadvantage, and made what I now see was a badly confused attempt to show that the usual first-order metatheorems hold at second-order if one adopts my proposed semantics. In essence, my mistake resulted from paying insufficient heed to a crucial point of similarity between my proposed semantics and the standard semantics, in contrast with the general Henkin semantics. The mistake was repeated in Hale (2013a), published a few months later, and went unnoticed (by me at least) until I was prompted to rethink by a helpful reminder from Ian Rumfitt that a categorical axiomatization of arithmetic can be given in an extension of first-order logic much weaker than secondorder logic (John Myhill’s ancestral logic, q.v. Myhill (1952)). Ancestral logic adds to first-order logic an operator * which applies to any dyadic relation to yield its ancestral; a categorical axiomatization is obtained by adding to the first-order Dedekind-Peano axioms a further axiom asserting that each natural number other than 0 bears the ancestral of the successor relation to 0. 16 At least, it does so from the standpoint of the set-theoretic orthodoxy which sees its second-order domains as restrictions of the standard domains. From that orthodox point of view, there are many properties on (or subsets of) the first-order domain which go unrecognized in my modest semantics. But a proponent of modest semantics who is motivated by the deflationary theory of properties should insist that the definable properties are all the properties there are, so that there is no shortfall. Although the deflationary theory of properties does not directly conflict with the idea that there are, if the first-order domain is infinite, many subsets of that domain besides the definable subsets, it is a further question whether it would do so, if combined with a similar, deflationary conception of objects, according to which it is necessary for the existence of an object that there could be a singular term having that object as its referent. If it did so, the deflationary position would be potentially revisionary, not only of the standard second-order semantics, but of the orthodox set theory which lies behind it. The question is as delicate and difficult as it is important—and too large to take on here. One source of difficulty, crucial to a proper discussion of the issue, is that whilst, for model-theoretic purposes, ‘definable’ has to be understood as definability in some fixed language, no such restriction to a fixed language is involved in the deflationary theory’s general statement of existence-conditions for properties and objects—there need not be (and indeed, surely cannot be) some single language in which all objects and properties are specifiable. A further complication is that one can prove in second-order logic both the uncountability of the real numbers, and a version of Cantor’s Theorem that any set is strictly smaller than its powerset, to the effect that the sub-properties of any property outrun its instances (see Shapiro (2000), pp.343–5 and Shapiro (1991), pp.103–4)—of course, while these results are not in question, their philosophical significance is very much open to discussion.

11.5 existential commitments 195 proofs (see again Shapiro (1991), loc. cit) still go through, i.e. when the underlying semantics is modest; and the proofs of completeness, etc., all fail for essentially the same reason as when the semantics is standard—as I explain in a little more detail in the appendix to this chapter. It is, of course, a consequence of taking the domains of the second-order variables in any given model to comprise just the properties and relations definable in the chosen meta-language that there will be just countably many properties and relations—in the model. Those last three words are crucial. On the general metaphysics and ontology of properties I am recommending, there are no properties other than those which are specifiable by suitable predicates, and so in that sense definable. But the sense of definable in which the only properties there are are definable is what we might call an absolute sense—definable in some language, actual or possible (or, what I think comes to the same thing, definable is some possible extension of some actual language). Whilst any given finitely-based language can contain at most a countable infinity of predicates, so that at most countably infinitely many properties are definable in that language, it does not follow that the totality of all properties whatever (if, indeed, there is a determinate such totality) is at most countably infinite. From the fact that we cannot specify all the properties, it does not follow that any of them must be unspecifiable.17

11.5 Existential commitments I would like now to comment further on that aspect of second-order logic which, as we have observed, has been held to tell against its acceptance as logic and to justify its assimilation to set theory—its existential commitments. We can distinguish two quite different sources of the putatively problematic existential commitments second-order logic may be held to incur.

11.5.1 ‘To be is to be the value of a variable’ First, there is Quine’s idea—encapsulated in the memorable slogan which heads this sub-section—that quantification into predicate-position brings with it a distinctive new ontological burden. The very use of second-order quantification commits us to the existence of a special range of entities to be the values of the bound second-order variables. Quine’s idea has two parts. One part is that the use of any form of quantification involves a commitment to a range of entities to serve as the possible values of its bound variables. The other is that this commitment is not just carried but introduced by quantification. Thus in Quine’s view, the step of quantifying into predicate-position is to be seen as a momentous one precisely because it brings with it a commitment to entities of a certain type which is altogether absent from statements free of that type of quantification, and so is not already carried by the predicates into whose positions higher-order variables may be inserted. I think that this second idea is utterly

17 I am grateful to an anonymous reviewer for pressing this point.

196 second-order logic implausible.18 Its implausibility stands out especially clearly when it is applied to first-order quantification. The suggestion that ‘Someone wrote Hamlet’ carries a commmitment to the existence of people that is absent from ‘Shakespeare wrote Hamlet’ makes a complete mystery of existential generalization, which is surely to be understood—as its name suggests—as generalizing a commitment, rather than introducing one. The first part of Quine’s idea—the range-of-values-conception of quantification, as it might be called—can survive rejection of the second. Obviously one could scrap the bad idea that quantification introduces new ontological commitments but still hold that its bound variables range over whatever entities—objects, properties, relations, . . .—the constant expressions they replace stand for. Although even this more modest and more plausible conception of quantification may be challenged, I shall continue to assume it here.19 Thus in my view, second-order quantification does carry a commitment to entities of a certain kind—first-level properties and relations, understood in accordance with the abundant conception—but it is not a new commitment to entities of that kind, i.e one to which we are not already committed by the use of first-level predicates. Thus if this commitment tells against second-order logic’s claim to be logic, it tells equally against the claim of first-order logic.

11.5.2 Comprehension Second, there is the more explicit existential commitment carried by the comprehension axioms—instances of a scheme such as: ∃X∀x(Xx ↔ φ) where X does not occur free in φ, or more generally ∃X n ∀x1 , ..., xn (X n x1 , ..., xn ↔ φ) where X n does not occur free in φ. In essence, what the scheme asserts is that for every formula φ of the language, there is a property or relation which holds of, or among, any objects satisfying the formula.20 If there are no further restrictions on the comprehension axioms, the formula φ may itself contain second-order quantifiers. In virtue of this, the unrestricted comprehension scheme is impredicative—it allows us, in effect, to define properties and relations in ways that involve quantifying over all

18 Nevertheless, it appears to have escaped serious challenge until relatively recently—as far as I know, until Rayo and Yablo (2001). 19 In Wright (2007), Crispin Wright argues for a ‘neutralist’ conception of quantification, according to which it is simply a device for what he terms generalization of semantic role. The idea is, roughly, that quantifying into the position occupied by an expression s of a certain syntactic type in a sentence [. . .s. . .] does not invoke a range of entities—the possible values of the variable replacing s—rather, it takes us to a content or thought whose truth-conditions requires a certain kind of distribution of truth-values over contents of the type of [. . .s. . .]. The conception is neutral in the sense that it takes no stand, either way, on whether s stands for an entity of a certain type. So neutralism is compatible with a view on which secondorder quantification carries no ontological commitment at all; but it does not entail such a view. 20 If one combines Wright’s neutralist interpretation (see footnote 19) with the view that first-level predicates don’t stand for entities of any kind, one will deny that comprehension axioms have any such existential content. As Wright puts it: ‘It goes with neutralism, as I have been outlining it, that there need be—more accurately: that for extreme neutralism, at least, there is—no rôle for comprehension axioms’ (Wright 2007, p.164).

11.5 existential commitments 197 properties and relations. If such second-order quantifiers may be embedded under other such quantifiers to any finite depth, we have fully impredicative second-order logic.21 Concern may be felt over impredicativity for any of several reasons. One is the idea that impredicative comprehension involves some kind of vicious circularity—how can we define a putative first-level property P by means of a formula which quantifies over all first-level properties including P itself ? Another is that impredicativity may give rise to inconsistency. Serious as they are, I think these concerns can be answered. At least on a standard, referential conception of quantification, understanding a quantified sentence requires knowing, in some way, what are the admissible values of the bound variables. But it does not follow—and seems clearly false—that this requires that one be somehow acquainted with, or be otherwise able to identify, each of those values individually. It is, accordingly, unclear that the fact, if it is one, that some of those values cannot be specified except via quantification over all of them must preclude understanding of the defining formula. And while some statements involving impredicative specification give rise to inconsistency, others do not—so there is no clear reason to blame inconsistency on impredicativity per se. But these are not my concern here, which is, rather, with the idea that unrestrictedly impredicative comprehension somehow engenders ‘massive existential commitments’. The idea that it does seems at least implicit in the following passage, which forms the core of Charles Parsons’s argument that even if, out of deference to Frege’s insistence on a fundamental ontological distinction between objects and functions, and hence between sets and concepts, we take the values of second-order variables to be concepts, rather than sets, higher-order logic carries existential commitments which make it ‘more comparable to set theory than to first-order logic’: Consider the full second-order predicate calculus, in which we can define concepts by quantification over all concepts. If a formula is interpreted so that the first-order variables range over a class D of objects, then in interpreting the second-order variables we must assume a welldefined domain of concepts applying to objects in D which, if it is not literally the domain of all concepts over D, is comprehensive enough to be closed under quantification. Both formally and epistemologically, this presupposition is comparable to the assumption which gives rise to both the power and the difficulty of set theory, that the class of all subclasses of a given class exists. (Parsons 1965, p.196)

As we have seen, the standard semantics for second-order logic takes the domain of the predicate-variables to be the complete power set of the individual domain, where this is assumed—in case the latter is infinite—to include not just definable subsets but arbitrary infinite subsets of the individual domain. Parsons is unquestionably right that taking the values of second-order variables to be Fregean concepts rather than sets makes no essential difference to this point. Precisely because Fregean concepts are individuated purely extensionally, they behave, modulo their unsaturatedness, just 21 Whether we have what is usually called full second-order logic depends on whether φ is allowed to contain further free first-order variables besides x or free second-order variables other than X. In what follows, I assume that φ does not contain additional free variables. I am indebted to an anonymous reviewer for stressing the need for clarity on this point. I believe what I say below can be extended to the more general case, but do not claim that here.

198 second-order logic like sets. In particular, Frege could have had no more quarrel with arbitrary infinite concepts than he would have had with arbitrary infinite subsets of a given infinite set. However, our question is whether impredicative comprehension is implicated in, or somehow responsible for, existential commitments which would justify assimilating second-order logic to set theory. On the face of it, it is not. The comprehension scheme tells us, in effect, that for each formula of the language with one free individual variable, there exists a property—the property an object has if and only if it satisfies that open sentence. If we allow the comprehension formula to include secondorder quantifiers, there will, of course, be new open sentences by means of which properties may be defined, in addition to those which would be available, were we to restrict ourselves to predicative comprehension. But there will only be countably infinitely many of them, just as there are only countably infinitely many extra open sentences available to define properties, if we admit those containing first-order quantification, on top of those formed ultimately out of just simple predicates together with individual variables and sentential operators. There is just no massive cardinality jump, such as one obtains (on the classical view of the matter) when one applies the powerset operation to an infinite set.22 The idea which seems to drive Parsons’s claim that fully impredicative comprehension requires an assumption comparable in strength to the powerset axiom is virtually explicit in a more recent paper by Øystein Linnebo in which he claims that impredicative comprehension is justified only when the second-order variables range over arbitrary subcollections of the first-order domain. He writes: A minimal requirement for definitions by impredicative comprehension to be justified is that we have a conception of a determinate range of possible values of the second-order variables. This requirement is clearly satisfied when the second-order variables are taken to range over arbitrary subcollections of the first-order domain. Moreover, since this range consists of all arbitrary subcollections of the first-order domain, it will be closed under definition by quantification over this range (or under any other mode of definition, for that matter). If, on the other hand, the range of the second-order variables does not contain all arbitrary subcollections of the first-order domain, we will have no guarantee that the range is closed under definition by quantification over this range.23

There are really two separate requirements here. One is that we should have a conception of a determinate range of possible values for the second-order variables; the other is that we should somehow be able to guarantee that that range is closed under definition by quantification—that is, that impredicative definition involving quantifcation over the range will not take us out of it. The first requirement has, it seems to me, nothing especially to do with impredicative quantification. To the extent that it is reasonable, it is best understood as

22 The argument of the last two paragraphs puts together pieces of two given in Hale (2013b), pp.137–9, 152–5. Although the present version omits much of the detail of its predecessors, I have tried to bring out more clearly what I think is right in what Parsons says and to separate it more cleanly from what I disagree with. 23 Linnebo (2004), p.169. I discuss this argument in Hale (2013b), pp.153–5, where some of the points rehearsed here were made. But revisiting the argument allows me to separate some questions I failed to distinguish there, and to explain more fully why I think it fails.

11.5 existential commitments 199 a perfectly general requirement on any quantification. It rests upon the plausible belief that fully to understand a quantified statement, one must in some sense grasp what are the intended values of its bound variables. If one lacks this knowledge, one will not know what is required for the statement to be true. Satisfaction of this requirement does not entail that the admissible values of the variables must form a set.24 At most what is required is that it should be determinate what those values are. Obviously it is not required, for understanding the quantification, that one must know, of each of the things which is in fact one of the values of variables, that it is so;25 the requirement is surely met if one is able to produce a general specification of the conditions for something to be a value. Thus it may be met, in the case of a firstorder quantification, by stipulating that the range comprises all objects of a certain kind, such as the natural numbers, or perhaps all objects whatever, or, in the case of second-order quantification, that it comprises all properties of a certain kind, such as properties of natural numbers, or all properties of objects whatever. Perhaps it can be met, in a certain context, by stipulating that the second-order variables are to range over arbitrary sub-collections of the first-order domain. But clearly, even if this is one acceptable way of meeting the requirement, it is not the only way. Turning to the second requirement, it seems that Linnebo’s thought must be that if we take the second-order variables to range over arbitrary sub-collections of the firstorder domain, definition by quantification cannot take us outside the range, because there simply is nothing outside it for definition by quantification over it to pick out; whereas if instead, the variables are taken to range over anything short of the full power set of the first-order domain—i.e. over some proper subset of the full power set—then definition by quantification over that range might take us out of it, to subsets of the first-order domain not in the range. There are two points to be made in reply to this. First, even if we set aside the fact that the argument appears simply to assume that the second-order variables will range over a domain of sets, or at least over entities which, like sets, are individuated purely extensionally, it seems to overlook a perfectly viable alternative. It is true that if we take as the range of the second-order variables just any proper subset of the power set of the first-order domain, it may well fail to be closed under definition by quantification. But there is no need, if we are to rule out that possibility, to ascend (if that is the right direction in which to travel) to the full power set; if the range is taken to be the set of all definable subsets of the first-order domain, definition by quantification over that range cannot take us out of it, since any 24 Or a collection of any other sort, such as a proper class. The assumption that ‘to quantify over certain objects is to presuppose that those objects constitute a “collection,” or a “completed collection”—some one thing of which those objects are the members’ is what Richard Cartwright calls the All-in-One Principle. See Cartwright (1994), sec.IV for some compelling arguments against it. 25 As Frege put it, in his review of Husserl’s Philosophy of Arithmetic vol. I, ‘It should be clear that someone who utters the proposition “All men are mortal” does not want to state something about a certain chief Akpanya of whom he may never have heard’, Frege (1984), p.205. He makes the same point in his 1895 elucidation of Schröder’s lectures (Frege 1984, p.227), and in his 1914 lectures on logic in mathematics (Frege 1969, p.230 or Frege 1979, p.213). I am grateful to Ian Rumfitt for assisting my failing memory of these passages by supplying precise references. It appears that Akpanya actually existed, and that Frege knew of him. He was a chief in Togo, which became a German colony in 1884—‘to Frege’s joy’ according to Wolfgang Künne (Künne 2009).

200 second-order logic subset so defined must, by its very nature, be one of the definable subsets. It makes no difference if we take ‘definable’ to mean ‘definable in the object-language’, since any subset defined by quantification in the object-language must be so definable; and if we take it to mean instead ‘definable in the meta-language’, we can secure the point by stipulating that the meta-language is to contain the object-language, or a translation of it. Linnebo concedes that ‘there are non-standard models of full impredicative second-order logic where the second-order variables do not range over all arbitrary subcollections of the first-order domain’, but objects that ‘these models are rather artificial and do not make available any alternative general conception of a range of values of the second-order variables’. This seems to me unjustified. There is nothing artificial about models in which the second-order domain is taken to be the set of all definable subsets of the first-order domain. As with the standard semantics, the choice of this as the second-order domain amounts to restricting attention to a special case of the more general Henkin semantics. But as we have seen, there is a perfectly good conception of properties on which this is exactly the restriction needed. Second, as noted, the argument simply ignores the possibility of taking the secondorder variables to range over entities which, unlike sets, are precisely not individuated extensionally. If, instead, we take them to range over first-level properties, and we understand properties in accordance with the deflationary conception I have been recommending, then, since any definition of a new predicate ensures the existence of a property, the domain cannot fail to be closed under definition by quantification over properties. We have as good a guarantee as we could possibly wish.26 Doubtless there is much more to be said, and further questions may be raised. But as things stand, there seems to be no reason why a proponent of second-order logic, interpreted as I am recommending here, should not admit fully impredicative comprehension.27 26 I am, of course, assuming that impredicative specification is not to be ruled out on other grounds, such as vicious circularity. In the present disagreement, that assumption is common ground. However, as an anonymous reviewer observes, impredicative specification of properties may be thought especially problematic when properties are conceived, as on the abundant conception, as individuated by the satisfaction-conditions of possible predicates. More specifically, it may be feared that this will lead to situations in which the satisfaction-conditions for one first-level predicate, φ, depend upon or include those of another, ψ, which in turn depend upon or include—with vicious circularity—those of φ. This issue is difficult, and I cannot discuss it properly here. For reasons indicated above, p.11.5.2, I think the circularity need not be vicious. It is not difficult to find examples where impredicative quantification over properties in the definition of further properties or relations need raise no such problem. Pertinent examples are the Fregean definitions of predecession by Pmn ↔ ∃F∃x(Fx ∧ n = NuFu ∧ m = Nu(Fu ∧ u  = x)) and its ancestral by P* mn ↔ ∀F((Fm ∧ ∀x∀y(Fx ∧ Pxy → Fy)) → Fn), which are unproblematic—at least provided that the bound property variables do not include within their range properties which can only be defined in terms of the predecession relation. A simpler example would be the definition of a property, φ, by φx ↔ ∀F(∀y(Fy → Gy) → Gx)—that is, φ holds of an object iff it possesses all those properties which suffice for being G—where the quantification over properties raises no special problem provided that G is not itself defined in terms of φ. 27 To be clear: I am not claiming that the deflationary conception of properties forces the adoption of fully impredicative comprehension, only that it does not preclude it. In Cook (2014), Roy Cook tends to give the impression that my deflationary conception of properties will not allow endorsement of fully impredicative comprehension, and recommends a weakening of what he takes to be my overly austere position, in the interests of achieving a version of second-order semantics which will support proof of some of the standard metalogical results which, he thinks, must fail on my

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11.6 Second-order logic, or schematic logic? The question towards whose answer what I have been saying is intended to contribute—the question whether second-order logic is properly taken to be part of logic—is of general interest, independently of commitment to any specific philosophical thesis or programme. But an affirmative answer to it is—or has been generally reckoned to be—essential to the success of the neo-Fregean, abstractionist programme, which claims that an epistemological foundation for arithmetic can be provided by showing that its fundamental laws, the Dedekind-Peano axioms, can be derived from Hume’s principle in a suitable formulation of second-order logic. Just what constitutes a suitable formulation—how strong a second-order logic is needed—is a question which has itself received some attention in recent work. The short answer, in broadest terms, is that predicative second-order logic is insufficient, but that no more than 11 -comprehension is needed, and that, depending upon how much impredicative comprehension is allowed, one obtains stronger versions of the induction axiom.28 My somewhat guarded formulation of the claim that acceptance of second-order logic as such is essential to the neo-Fregean programme is not driven merely by habitual cautiousness. In recently published work (Heck 2011, ch.12), Richard Kimberly Heck argues that, at least for certain purposes which have hitherto been thought to require second-order logic, we may dispense with it in favour of what they call Arché Logic, in which the only quantifiers are first-order. In this final section, I want to consider the relations between Heck’s logic and second-order logic interpreted as I have been recommending. Broadly speaking, I shall offer some considerations which seem to me to suggest that while Heck is right to contrast their logic with second-order logic as it is standardly interpreted, there is much less distance between it and secondorder logic as I think we should interpret it—so much less that such differences as there remain may lie largely in matters of presentation rather than in disagreements of substance.

approach (as I wrongly claimed—vide supra, p.193, esp. fn.13—in the earlier paper Cook is discussing). In particular, with the weakening he favours, one could prove the categoricity of arithmetic, and a weakened categoricity result for second-order analysis. In essence, the weakening consists in allowing expressions of countably infinite length, whose utterance or inscription would therefore require performance of a supertask. Obviously a footnote is not the place for the substantial response which Cook’s paper merits. Here I can only observe, without argument, first, that, for reasons given here, I do not think my position precludes acceptance of fully impredicative comprehension; second, that while Cook seems to think that the key difference between us concerns the admission, or otherwise, of the logical possibility of supertasks being completed, it seems to me that this is incorrect—at at least one crucial point, his argument depends upon the assumption of arbitrary countably infinite sequences. This assumption remains problematic, even if one grants the logical possibility of completing a supertask. For these, and perhaps some other reasons, my admiration for Cook’s paper cannot be accompanied by agreement with it. This is, perhaps, an appropriate place to record my gratitude to him, for earlier and very helpful discussion of some of the ideas in this paper. 28 That 11 -comprehension suffices is noted in Heck (2000) and Heck (2011), p.270. For detailed results and proofs, see Linnebo (2004). The insufficiency of predicative comprehension is as one might expect, given that key definitions, such as the Fregean definition of predecession by Px, y = ∃F∃z(y = NuFu ∧Fz ∧x = Nu(Fu ∧u  = z)), require impredicative comprehension.

202 second-order logic As the discussion which follows is moderately complicated, I give first a brief overview. I begin (11.6.1) with a fairly concise sketch of the grounds for Heck’s claim that second-order logic is not needed for the proof of Frege’s Theorem, because it can be proved in the strengthened form of ancestral logic they call Arché Logic. I turn next (11.6.2) to an obvious concern which may be felt—to the effect that schematic logic is really only a thinly disguised form of second-order logic proper— and explain what I take to be Heck’s response to it. Separating two questions which Heck does not distinguish, I argue first (11.6.3) that the first (whether Arché Logic is a form of second-order logic) may be largely terminological. The more important, second, question is whether understanding Arché Logic demands a philosophically problematic conception of the full power set of the first-order domain. Heck and I agree in thinking that it does not do so—but for quite different reasons, as explained in (11.6.4). There is no simple yes-or-no answer to the question whether Arché Logic has a better claim to be logic than Second-order Logic. (11.6.5) explains why I think that, modestly interpreted, it has a better claim than second-order logic as standardly interpreted, but no better claim that second-order logic as I think that should be interpreted.

11.6.1 Frege’s Theorem in Arché Logic Frege’s Theorem asserts that the Dedekind-Peano axioms for arithmetic can be derived, in second-order logic, from Hume’s Principle together with Frege’s definitions of zero, predecession, and natural number. Hume’s Principle (HP) says that the number of objects of some sort F is the same as the number of objects of some sort G if and only if the Fs and the Gs can be put into one-one correspondence; in the notation Heck uses: Nx : Fx = Nx : Gx ↔ Eqx (Fx, Gx) where the right-hand side abbreviates one of any of several second-order formulae asserting that there is a one-one relation holding between the Fs and the Gs. Frege defined zero to be Nx : x = x (i.e. the number of non-self-identical objects), the relation of predecession by Pmn =df ∃F∃x[Fx ∧ n = Nz : Fz ∧ m = Nz : (Fz ∧ z = x)] (i.e. m precedes n iff for some F, there are n Fs and m numbers all but one of the Fs), and natural number by: Nat(x) ↔ P* 0x ∨ 0 = x, where P* xy says that x bears the ancestral of P to y. Heck’s interesting claim is that whilst one can prove the axioms from (HP) in second-order logic (and indeed, in second-order logic with just 11 -comprehension),29 second-order logic is not really needed at all, because one may prove those axioms from (HP) using just Arché logic—a strengthened form of schematic ancestral logic, in which second-order quantification and impredicative uses of comprehension are avoided by means of schematic formulae involving free second-order variables. The language of ancestral logic is obtained by adding to a first-order language an operator, *xy , which forms a binary relational expression when applied to a formula

29 That is, comprehension restricted so that second-order quantification in the defining formula φ may only take the form ∀F1 ...∀Fn φ where φ has no bound second-order variables.

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φxy with two free variables. Intuitively, *xy (φxy)(a, b) says that a bears the ancestral of the relation φ to b —i.e. φab, or φax ∧ φxb for some x, or φax ∧ φxy ∧ φyb for some x and y, or . . . and so on, for some finite sequence of suitable intervening conjuncts. Minimal schematic ancestral logic (cf. Heck 2011, p.277) adds to first-order logic a pair of introduction and elimination rules for the ancestral operator, corresponding to the two directions of Frege’s famous definition:30 (*+)

∀x(φax → Fx) ∧ ∀x∀y(Fx ∧ φxy → Fy) → Fb  φ * ab

provided that F is not free in any further premise on which the displayed premise depends, and (*−)

φ * ab  ∀x(φax → Fx) ∧ ∀x∀y(Fx ∧ φxy → Fy) → Fb

Adding a rule of substitution allowing replacement of schematic predicate letters by arbitrary formulae (subject to the usual restrictions, of course) gives the power of 11 comprehension, needed to prove some theorems concerning the ancestral without second-order quantifiers (pp.278–9). Arché logic is the result of a further strengthening, which consists in generalizing the move by which the second-order Fregean definition of the ancestral is replaced by the rules (*+) and (*−). Thus if φ x (Fxy) contains only the free variables displayed (that is, F and y), 11 -comprehension allows us to define a new predicate Aφ by: Aφ (y) =def ∀Fφ x (Fxy). Generalizing the trick with the ancestral, this can be replaced by the rules: (Aφ +)

φx (Fxy)  Aφ (y)

provided that F is not free in any premise on which φ x (Fxy) depends, and: (Aφ −)

Aφ (y)  φx (Fxy)

As with the addition of the ancestral rules, so here, the addition of the definition schemata by itself produces only a relatively weak logic. Strength lies in the substitution rule, if any. Full Arché logic is obtained by adding an unrestricted substitution rule. This, as Heck shows, is enough prove a version of Frege’s Theorem.

11.6.2 Second-order logic in sheep’s clothing? What is the philosophical significance of this result? Heck’s main claim is that ‘the language of Arché arithmetic has a significantly stronger claim to be a logical language than [does] the language of second-order logic’ (Heck 2011, p.295). This claim rests on the fact that Arché logic avoids the use of second-order quantification. Heck is well aware that the significance of that fact may be contested. If the various schematic rules, together with the required substitution rules, give the power of 11 -comprehension, and if, as Heck themself emphasizes, schematic formulae are to be understood as having the closure or generality interpretation, and as true if and only if true under all assignments to their free variables, why—it may be asked—are they not simply second-order formulae in all but appearance?

30 φ * ab abbreviates *xy (φxy)(a, b)—see Heck (2011), p.275.

204 second-order logic Heck confronts what may seem to be a version this worry. It may, they think, be objected that our understanding of the introduction rule for the ancestral involves a conception of the full power set of the domain. How else, it might be asked, are we to understand ∀x(φax → Fx) ∧ ∀x∀y(Fx ∧ φxy → Fy) → Fb as it occurs in the premise of the rule (*+), except as involving a tacit second-order quantifier? Does it not say, explicit quantifier or no, that all concepts F that are thus-and-so are so-andthus? Doesn’t understanding that claim therefore require the disputed conception of the full power set? (Heck 2011, p.293)

There are, however, two quite distinct questions here. To be quite clear, if a little pedantic, they are: (1) Isn’t it necessary, if we are to understand the displayed formula as required for it to function as a premise for (*+), that we take it to involve a tacit second-order quantifier, ∀F? And: (2) Doesn’t understanding the displayed formula as required involve possessing a conception of the full power set of the (first-order) domain? It is, accordingly, disconcerting that Heck’s answer goes right past the distinction, and assumes, in effect, that our two questions are one: No, it does not. A better reading would be: A concept that is thus-and-so is so-and-thus. What understanding this claim requires is not a capacity to conceive of all concepts but simply the capacity to conceive of a concept: to conceive of an arbitrary concept, if you like. (ibid., all italicization in original)

In other words, according to Heck, we need only understand the displayed formula as saying something about a (i.e. any arbitrarily chosen) concept, and this doesn’t involve grasping a totality of all concepts—containing one for each of the subsets of the first-order domain (assuming that to be infinite, uncountably many, according to the classical conception).

11.6.3 A terminological question? In my view, it is crucial, if clarity is to be achieved, to separate the two questions which Heck appears to run together. Second-order logic is, by definition, logic in which we have quantifiers binding variables in (first-level) predicate-position as well as quantifiers binding variables in name-position. The question whether Arché logic (or, for that matter, any other schematic logic in which there is only first-order quantification plus schematic predicate letters) is or is not a form of second-order logic is, therefore, simply the question whether any of its formulae involve secondorder quantifiers. Since none of them do, the answer—strictly speaking—is: no. But of course, the interesting question is whether any of its formulae are to be understood as equivalent to formulae with second-order quantifiers, and so may be said tacitly to involve such quantifiers. This is our first question—it is not the second question, and does not concern whether the intended second-order domain is the full power set of the first-order domain, or rather something other than that. So what is the answer to our first question? That depends, obviously, on what is required for understanding a schematic formula. One might hold that to understand a formula involving schematic predicate letters (or free second-order variables), what we have to grasp is that it implies, or is a way of asserting, each and every one of its

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instances—each result of replacing its schematic predicate letters or free second-order variables by specific predicates. In the case of a fully specified language for Arché logic, there will, we may assume, be a finite stock of simple predicate constants, together with a fixed stock of constructions by means of which these can be used to form complex predicates.31 As usual, there will be just countably many such predicates. So to understand a formula of such a language involving schematic predicate letters, one would need to grasp that it implies, or ‘asserts’, each one of a well-defined class of instances, involving just those predicates. Although the formula involves no overt secondorder quantifier, understanding it does involve a conception of a definite totality of instances, and, for precisely that reason, it may be held to be equivalent to its secondorder closure, with the second-order variables taken as ranging over the properties specified by those predicates.32 The case for regarding schematic formulae as equivalent to their second-order closures is, if anything, stronger if one takes them, as Heck does, to be capable of being evaluated as true or false. For the only reasonable condition for truth is the one Heck themself gives—viz. that a schematic formula is true if and only if it is true under all assignments to its free variables (see Heck 2011, p.280). To be sure, this truth-condition becomes definite only when it is specified what the admissible assignments to the free variables are—that is, in effect, when it is specified what the domain of the bound second-order variables would be, if one allowed them. It is, for the purposes of our question, irrelevant precisely how that range of admissible assignments is specified—for however that is done, the formula will be true if and only if all of them result in true non-schematic formulae. It therefore seems to me that the answer to our first question is: yes. Whether one concludes that Arché logic is, therefore, a form of second-order logic, or merely that it is equivalent in power to a form of second-order logic, seems to me to be a largely terminological question.

11.6.4 Arché Logic and the full power set of the first-order domain What about our second question? Does understanding schematic formulae require the disputed conception of the full power set, so that Arché logic enjoys, to that extent at least, no real advantage over second-order logic as standardly interpreted? Heck thinks not, and I agree. But I fear that the agreement may not extend far beneath the surface. Let me explain why. My reason for returning a negative answer is quite straightforward. Even if I am right in my answer to the first question, and Arché logic is equivalent to some form of second-order logic (perhaps to second-order logic with a limited form of impredicative comprehension, 11 say), that does not in and of itself enforce interpreting the 31 A more precise statement would involve taking arity into account, but such details need not distract us here. 32 In this respect, the situation may differ in an important way from understanding the first-order induction schema, φ(0) ∧ ∀n(φ(n) → φ(n + 1) → ∀nφ(n), where—pending some specification of which properties the schematic φ varies over—there is no definite domain for a bound second-order variable to range over, so that the schema cannot be properly regarded as equivalent to its second-order closure. But it seems to me that an alternative and perhaps more reasonable view would be that, in the absence of a definite specification, the schema makes no clear claim, but is effectively ambiguous between a range of alternative second-order closures which differ from one another over precisely what φ’s range of variation is taken to be.

206 second-order logic second-order variables as ranging over the full classical power set of the first-order domain. For all that we have said so far, the second-order domain could be taken to be a domain of properties understood in various other ways which fall well short of assuming a property for each and every one of the subsets, including arbitrary subsets, making up the classical power set of the first-order domain. One could not, of course, take the domain to consist of just those properties specifiable by suitable open formulae of the language of Arché logic itself, since that will include only the predicatively specifiable properties. But one could take it to comprise all (and only) the properties specifiable or definable in a meta-language which allowed some degree of impredicativity—at a minimum, as much as is allowed by 11 -comprehension, but in principle, any degree up to fully impredicative comprehension. In short, there are ways to understand the language of Arché logic, or indeed a second-order language to which it is effectively equivalent, which do not require deploying a conception of the full classical power set. Heck’s position is less straightforward, and merits close inspection. As we have seen, in support of their negative answer, they insist that understanding schematic formulae does not involve a conception of any totality of concepts, and requires only a capacity to conceive of a concept—‘to conceive of an arbitrary concept, if you like’, as they put it. However, a little later, they consider the objection that if we do not have a definite conception of the full power set of the domain—if, in particular, there is nothing in our understanding of the free second-order variables that guarantees that they range over the full power set of the domain—then the meanings of the predicates we introduce by schematic definition will be radically underdetermined, at the very least. It was stipulated earlier that Aφ (a) is true if, and only if, φ x (Fx, a) is true for every assignment of a subset of the first-order domain to F. But why? If we have no conception of the full power-set, why not take the domain of the second-order variables to be smaller? Why not restrict it to the definable subsets of the domain? Surely the axioms and rules of Arché logic do not require the second-order domain to contain every subset of the first-order domain? (Heck 2011, p.294)

Before we consider Heck’s response, it will be as well to be clear what they take the objection to be. As I understand them, their thought is that our intention, in schematically defining the predicate Aφ (ξ ) to mean ∀Fφ x (Fx, ξ ), is that the definiendum is to be true of a given object a if and only if φ x (Fx, a)33 is true for every subset of the first-order domain, definable or not. But if we do not have a conception of the full power set, then nothing prevents us from taking the definiens to be true of a if and only if φ x (Fx, a) holds (only) for just the definable subsets. So the satisfaction condition for the new predicate will be indeterminate as between these (and, indeed, many other, alternatives, each exploiting a different Henkin model). In short, Heck is assuming that while the language of Arché logic is not a second-order language (because it lacks second-order quantifiers) its intended interpretation is the same as that of second-order logic understood in terms of the standard semantics. Thus for Heck, an interpretation of its schematic predicates or free second-order variables which assigns them only definable subsets of the first-order domain is an unwanted, 33 Where φ x (Fx, ξ ) is some formula with just F and ξ free.

11.6 second-order logic, or schematic logic?

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unintended interpretation. Thus their reason for answering our second question in the negative cannot be the reason I have given. They must, I think, rely entirely on their claim that understanding formulae incorporating schematic predicates or free second-order variables does not require that we can conceive of all concepts, only that we can conceive of an arbitrary concept. It is therefore important to be quite clear what is meant in this context by ‘an arbitrary concept’. Usually, part of what is meant, when we speak of (conceiving of) an arbitrary something-or-other, is that while we may think of a particular something-or-other (and perhaps cannot avoid doing so), it does not matter which something-or-other we choose—any something-or-other will do, because what we go on to say or think does not depend upon which one we choose. And clearly part of what Heck means here is that it doesn’t matter which concept we choose—any concept will do. But there is a further question here, about what exactly is meant by concept—about just what is included in the range of which our arbitrary concept is supposed to be representative. I think it is clear that Heck is taking that range to include a concept corresponding to each and every subset of the first-order domain, so that by conceiving of an arbitrary concept, they mean conceiving of a concept which could be the concept corresponding to any one of the non-definable, arbitrary subsets that make up the complete power set of the first-order domain, according to the classical conception of the matter. That this is what they mean is, I think, confirmed by what they go on to say in response to the point that one might restrict attention to the definable subsets: The existence of non-standard models is a fact of mathematics . . . . But the philosophical significance of this technical point is not so obvious. It seems to me that there is something about the axioms and rules of Arché logic that requires the second-order domain to be unrestricted, and “unrestricted” is the crucial word. The difference between the standard model and the various non-standard models is to be found not in what that standard model includes but in what the non-standard models exclude: Since the standard model includes all subsets of the first-order domain, a non-standard model must exclude certain of these subsets. (It isn’t as if it could include more.) That, however, is incompatible with the nature of the commitments we undertake when we introduce a new predicate by schematic definition. Those commitments are themselves unrestricted in the sense that we accept no restriction on what formulae may replace B(x) when we infer φ x (B(x), a) from Aφ (a).48 One might be tempted to object that, if so, we must somehow conceive of the totality of all such formuale in advance. But that is simply to repeat the same error: No such conception of the totality of all formulae is needed; what is needed is just the ability to conceive of a formula—an arbitrary formula, if you like. [48: And B(x) might be demonstrative: Those ones. Definability considerations are therefore out of place here, too.]34

34 Heck (2011), pp.294–5. I have reproduced Heck’s footnote 48 because they clearly think the point is telling. But I have to confess that it eludes me. They can scarcely mean to suggest that the putative arbitrary infinite subsets of an infinite first-order domain which we cannot define can be picked out demonstratively. But if all they mean is that, in case of small finite subsets, we might pick them out by pointing (‘Those ones’, said while gesturing at a pile of planks, say), then even if we agree that these words together with the gesture don’t amount to a definition, there is no clear reason to think that the set of objects singled out is not definable.

208 second-order logic

11.6.5 Modest and immodest Arché Logic There are several points I need to make about this. 1. The first concerns Heck’s assertion that the language of Arché logic has a better claim to be a logical language than does that of second-order logic. As far as I can see, there are just two reasons they might advance in its support: first, that the former involves only first-order quantifiers, and second, that understanding schematic formulae requires only a capacity to conceive of arbitrary concepts, in contrast with second-order formulae proper, which require a conception of the totality of concepts corresponding to subsets of the first-order domain. The first—or so I have argued—is technically correct but of no real significance. The second needs to be evaluated in the light of the point which our last extended quotation puts beyond question: that Arché logic, at least as Heck intends it, requires to be interpreted—just as, in their view, second-order logic does—in terms of a domain which comprises all the subsets of the first-order domain. The difference thus reduces to the point, or claim, that understanding an Arché formula requires only a conception of an arbitrary concept, not the totality of all concepts. 2. Whether that difference has the significance Heck attributes to it is a further, and debatable, question. Even granting, what seems to me less than clear, that there is a substantial difference between conceiving of an arbitrary concept and conceiving of the totality of them, it may be questioned whether the former is signifcantly less problematic than the latter—indeed, it seems to me that those of us who find the conception of the full classical power set of an infinite set difficult to swallow do so precisely because of the indigestibility of its putative elements. It is the combination of arbitrariness with infinitude of the crucial majority of those subsets which sticks in the gullet. 3. Clearly there will be a significant difference if, as I have suggested, we interpret the schematic predicates and free second-order variables of Arché logic as ranging over properties in the sense I am recommending, or as ranging over the definable subsets of the first-order domain. But the difference will not be between the language of Arché logic and that of second-order logic, soberly understood— it will be between both of them, on the one side, and the language of secondorder logic as interpreted by the standard semantics, on the other. 4. Any such interpretation clashes head on with Heck’s insistence that Arché logic requires the second-order domain to be ‘unrestricted’. At least, it does so, if we assume, as they clearly do, that the only genuinely unrestricted domain for the second-order variables is the set of all subsets of the first-order domain, as on the standard interpretation of second-order logic. But that assumption, it seems to me, simply begs the question at issue here. I think they are right that the rules and axioms of Arché logic require the domain to be unrestricted—at least, there is nothing in those axioms and rules to suggest that the schematic predicateletters or free variables are to be understood as varying over a restricted domain; just as there is nothing in the axioms and rules of second-order logic to suggest that its bound second-order variables are to be understood as ranging over a restricted domain. But the absence of anything to suggest a restriction no more

appendix 209 enforces the standard interpretation in the former case than it does in the latter.35 Just as we can, in the case of second-order logic proper, take the second-order variables to range unrestrictedly over all properties whatever without thereby committing ourselves to the contentious claim that those properties include properties corresponding to arbitrary subsets of the first-order domain, so we can in the case of the schematic predicates and free variables of Arché logic. If, as I have argued, a reasonable conception of properties requires models in which the second-order variables range only over definable subsets of the first-order domain, the right complaint is not that these ‘non-standard’ models exclude things which should be included; it is that the ‘standard model’ includes (or at least purports to include) things which should be excluded. Let me try to extract the main points from the admittedly somewhat tortuous discussion of this section. In Heck’s view, unless I have misunderstood them, both second-order logic proper and Arché logic should be interpreted in accordance with the standard semantics, so that their second-order vocabulary is taken to vary over the full power set of the first-order domain. But there remains, they think, a contrast between them that justifies their assertion that the language of the latter has a significantly stronger title to be counted logical than the former: there are no secondorder quantifiers, and understanding its formulae does not require a conception of the totality of concepts corresponding to subsets of the first-order domain, only a conception of an arbitrary such concept. I have sought to suggest that this contrast lacks the significance Heck attributes to it, and that Arché logic, as they understand it, is not much, if any, less problematic than second-order logic, again as they understand it. I have further argued that we can perfectly well interpret Arché logic so that its second-order expressions range unrestrictedly over properties, understood as having (only) definable subsets of the first-order domain as their extensions. So understood, it is indeed significantly less problematic than second-order logic on the standard interpretation. But it is no less—and no more—problematic than second-order logic modestly understood as involving quantification over all first-level properties. Whether it just is second-order logic, so understood, can perhaps be set aside, as a largely terminological question.36

Appendix: Categoricity etc. I claim in Section 11.4 that the usual proofs of categoricity results for second-order axiomatizations with respect to the standard semantics go through when my modest semantics is adopted, and that the proofs of completeness etc., results for second-order logic which can be given with respect to the Henkin semantics and others essentially equivalent to it (q.v. Shapiro (1991), ch.4), fail with my semantics, just as they do with the standard semantics. I shall not here seek to provide a rigorous proof of this 35 See the final paragraph of section 11.4, p.195. 36 In addition to those whose help is explicitly acknowledged in earlier footnotes, I should like to thank Stewart Shapiro for some very helpful discussion and advice, and two anonymous reviewers whose perceptive and constructive suggestions have, I hope, helped me to make this a better paper than it would otherwise have been.

210 second-order logic claim. Since the domains of the second-order variables are taken, in my proposed semantics, to comprise those properties and relations definable in the meta-language, this would, inter alia, require a much more careful specification of that language than I wish to undertake here. My aim instead is to support the claim by lightly sketching standard proofs in what I hope to be just enough detail to make clear that, and why, the categoricity proofs still go through when my modest semantics is adopted, while proofs of completeness, etc., still fail, as they do when the semantics is standard. The proofs sketched correspond to those given in detail in Shapiro (1991), ch.4.

A. Categoricity For arithmetic, we assume a second-order axiomatization with the usual axioms for successor, addition, multiplication, and induction. The categoricity theorem asserts that any two models of these axioms, M 1 = d1 , I 1 and M 2 = d2 , I 2 are isomorphic. In essence, this is proved by defining a relation f on d1 × d2 , and proving that f is a one-one function from d1 onto d2 , that f is structure preserving. A subset S ⊆ d1 ×d2 is closed under the successor relation (s-closed) iff 01 , 02 ∈ S and a, b ∈ S → s1 a, s2 b ∈ S, where si is the successor function of M i . f is defined to be the intersection of all s-closed subsets of d1 × d2 . It is easily shown that f is non-empty and s-closed. To prove that f is a function from d1 to d2 , two lemmas are needed: that every element of d1 bears f to some element of d2 , and that f is many-one. The first lemma is proved by defining P = {a ∈ d1 : ∃b ∈ d2 ( a, b ∈ f }) and proving that P = d1 and the second by defining P = {a ∈ d1 : ∃!b ∈ d2 ( a, b ∈ f )} and proving once again that P = d1 . In both cases the proofs apply the axiom of induction with respect to P. This is legitimate when the semantics is standard, since then the bound second-order variable in the axiom is interpreted as ranging over the complete power set of the first-order domain. The proofs fail under the general Henkin semantics, since then the secondorder domain may be any subset of the power set of the first-order domain, so there is no guarantee that the set P lies in the range of the bound second-order variable in the induction axiom. They fail when the axiomatization is only first-order, since then we have only an axiom scheme for induction, which can be applied only to subsets specifiable by a formula of the object-language, and clearly P is defined only in the metalanguage. However, precisely because P is so defined, the proofs still go through under my modest semantics, in which the relevant second-order domain comprises all the subsets of the first-order domain definable in the metalanguage. In a similar way, we can prove, using the fact that M 2 satisfies the successor and induction axioms that f is 1–1 and onto d2 , and, using the addition and multiplication axioms in M 1 , M 2 , that f preserves structure. The situation with respect to real analysis is similar. In this case, the stumbling block for attempts to prove categoricity when the semantics is general Henkin or the axiomatization first-order is the need to appeal to the second-order completeness axiom, asserting that every bounded property (or set) of reals has a least upper bound. In case the semantics is general Henkin, or the axiomatization first-order, there is no guarantee that the relevant property (or set) of reals lies within the relevant secondorder domain, or is covered by the first-order axiom scheme. This guaranteed when the semantics is standard, because all the second-order domain includes all properties

appendix 211 (or subsets) of elements of the first-order domain. However it is also guaranteed when the semantics is modest, for the same reason as above.

B. Completeness etc. The proof that second-order logic is complete with respect to the Henkin semantics (or the essentially equivalent ‘first-order’ semantics—see Shapiro (1991), pp.74–6) is a fairly straightforward extension of the usual completeness proof for first-order logic. That is, we show that any deductively consistent set of sentences has a denumerable model by expanding it to a maximally consistent set which includes ‘witnesses’ for each of the existentially quantified sentences of the language suitably enriched with denumerable many new constants, and then use these new constants to build a denumerable model. In the second-order case, the expansion of the language consists in adding denumerable sequences of n-place relation letters, and function symbols, as well as of individual constants. Starting with our given consistent set of sentences, we then construct an infinite sequence of sets of sentences, adding at each stage the sentences ∃x m (x) → m (ci ), ∃X n χm (X n ) → χm (Cjn ), ∃f p m (f p ) → p

p

m ( gk ), where ci , Cjn , gk are the first individual, n-ary relation and p-ary function constants not already used. It can be shown that the union of the sets in this sequence is consistent, and by Lindenbaum’s Lemma, it can be extended to a maximally consistent set. A model is then constructed, using the set of new individual constants itself as the first-order domain, and choosing the second-order domains so that one can prove that a sentence is true in this model iff it belongs to the expanded set. Since this latter includes the original set, it follows that the model is a model of that set also. The proof depends crucially on the fact that in the Henkin semantics, the choice of second-order domains is not fully determined by the choice of first-order domain—the domain over which the monadic second-order variables range, for example, can be any subset of the power set of the first-order domain. To see this, suppose instead that the second-order domains are determined by the choice of first-order domain, as in the standard semantics, so that the domain for the second-order 1-place relation variables is the set of all subsets of the firstorder domain. Let the assignments to individual constants, n-place relation letters, and function symbols be just as in the Henkin completeness proof, based on the denumerable set of new individual constants {c0 , c1 , . . . } as the first-order domain. In particular, each new constant denotes itself, and each constant c already occurring in our original consistent set of sentences denotes the first ci such that c = ci is in our maximally consistent expansion of that set; and each 1-place relation letter F is assigned the set of individual constants c such that Fc occurs in ; similarly for n-place relation-letters generally, and function-symbols. What needs to be proved, once the model M is defined, is that for any sentence of the extended language, M | iff ∈ . Proof is to be by induction on the complexity of . A crucial case is when is, say, ∀XB(X) where B(X) is a formula with only X free. Since is maximal, it will contain B(ψ) or ¬B(ψ) for each formula ψ of the language of with just one free individual variable. Suppose contains B(ψ) for every choice of ψ, so that ∀XB(X) is in . Do we have M | ∀XB(X)? Well, clearly this is not guaranteed.

212 second-order logic Our assignments defining M will ensure that B(X) comes out true whenever X is assigned a subset of the first-order domain specified by a formula ψ of the objectlanguage. But these do not exhaust the subsets of the first-order domain. Thus we have no guarantee that there are no subsets for which B(X) comes out false. In short, the completeness proof collapses when the standard semantics is employed, essentially because we no longer have the requisite freedom to restrict the choice of the secondorder domains. And clearly, for essentially the same reason, it fails under my modest second-order semantics, in which the second-order domain must be chosen as the set of all and only the subsets of the first-order domain definable in the metalanguage— crucially, the relevant subsets need not all be definable in the object-language. Given Henkin semantics, the proof of compactness is an easy consequence of completeness and soundness, as in the first-order case. Without completeness, the proof cannot be given.37 That the Löwenheim-Skolem Theorems are not obtainable with the standard semantics follows, given the categoricity results for arithmetic and analysis. Likewise for my modest semantics.

37 To be be clear on a point rightly emphasized by one reviewer, it has not been shown that secondorder logic is incomplete or non-compact with respect to my semantics; my claim is only that the usual completeness and compactness proofs fail, just as they do with respect to the standard semantics.

12 The Problem of Mathematical Objects In seeking a foundation for mathematics, one may be looking for what may be called a foundation in the logical sense: a single, unified set of principles—perhaps unified by their jointly constituting an acceptable axiomatization of some concept or concepts plausibly taken as fundamental—from which all, or at least a very large part of, mathematics can be derived.1 In this sense, some version of set theory is plausibly taken as a foundation. But one may also be interested in an epistemological foundation— roughly, an account which explains how we can know standard mathematical theories to be true, or at least justifiably believe them. A foundation in this sense could not be provided by any mathematical theory, however powerful and general, by itself. Indeed, the more general and powerful a mathematical theory is, the more problematic it must be, from an epistemological point of view. What is called for is a philosophical account of how we know, or what entitles us to accept, the mathematical theories we do accept. Since such an account cannot very well be attempted without adopting some view about the nature of the entities of which the mathematical theories treat, this is likely to involve broadly metaphysical questions as well as epistemological ones. It is not certain either that we are right to demand such a foundation, or that one can be given, but I shall proceed on the assumption that it is reasonable to seek one. If fundamental mathematical theories such as arithmetic and analysis are taken at face value, any attempt to provide such a foundation must confront the problem of mathematical objects—the problem of explaining how a belief in the existence of an infinity of natural numbers, an uncountable infinity of real numbers, etc., is to be justified. Of course, as already noted, these theories may be derived within a suitable theory of sets, but then we simply replace the problem of justifying belief in numbers of various kinds with the problem—unlikely to be easier—of justifying belief in the existence of the universe of set theory. I am going to discuss only one small, but fundamental, part of the problem—whether we can be justified in believing that there is a denumerable infinity of natural numbers—or, more generally, an infinity of objects of any kind. I shall consider two broad approaches to this problem—what I shall call object-based and property-based approaches. By a property-based approach I mean any approach which argues indirectly for an infinity of objects, in the sense that our access to an infinite sequence of objects is seen as dependent on an underlying infinity of properties. By an object-based approach I mean any attempt to argue directly that we can have access to, or knowledge of, an at least potentially infinite 1 Originally published as Hale (2011). Reprinted by permission from Springer Nature, Giovanni Sommaruga (ed.), Foundational Theories of Classical and Constructive Mathematics, volume 76 of Western Ontario Series in Philosophy of Science, “The Problem of Mathematical Objects,” Hale, B. (2011). Springer Science B.V., The Problem of Mathematical Objects In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © Springer Science B.V. DOI: 10.1093/oso/9780198854296.003.0013

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sequence of objects. These descriptions are rather vague, but I think there are some rather clear instances of the approaches I wish to contrast. The clearest—and in my view, most promising—attempt to develop the object-based approach forms part of Charles Parsons’s investigation of the popular but normally rather opaque and illexplained idea of mathematical intuition. Another example is Dedekind’s notorious argument that the realm of his own thoughts—which comprises the possible objects of his thought—is infinite. Perhaps the most obvious example of the property-based approach is to be found in Frege’s attempt to prove the infinity of the natural numbers.

12.1 Parsons on mathematical intuition 12.1.1 Intuition of and intuition that Central to Parsons’s account is a distinction between objectual and propositional intuition—between intuition of objects and intuition that p, for some proposition p.2 The former notion is to play the fundamental role—much as perceptual knowledge that p (e.g. that there is a duck on the pond) depends upon perception of the duck, so intuitive propositional mathematical knowledge is to depend, at bottom, on intuition of certain objects.

12.1.2 Pure abstract and quasi-concrete objects Numbers themselves are not—in Parsons’s view—among the objects of intuition. He distinguishes between pure abstract objects and quasi-concrete ones. Quasi-concrete objects are, roughly, objects which are closely connected with concrete objects, or ‘belong to’ such objects—e.g. shapes, edges, and, importantly for Parsons, linguistic expressions considered as types, as opposed to their concrete tokens. Pure sets and numbers are not so related to concrete objects, are usually taken to exist independently of concrete objects, and would presumably all rank as pure abstract objects for Parsons. The objects of intuition are restricted, for Parsons, to the quasi-concrete. This restriction is, I think, crucial. So long as it is in place, it is at least very plausible to claim that we may stand in a direct cognitive relation—analogous to, and perhaps often mediated by, ordinary sense perception—to the objects of intuition. But to claim that we may enjoy such a relation to abstract objects quite generally, and so to pure sets and other pure abstract objects, which have no essential connection with concrete objects, would be to cast the notion of intuition back into the depths of obscurity from which Parsons has done much to rescue it.

12.1.3 The language of stroke-strings Parsons’s detailed account of intuition works with the ‘language’ whose sole primitive is the single stroke ‘|’, and whose well-formed expressions are just the strings: |, ||, |||, ||||, . . . . , for strings of any finite length. As he observes, this sequence of stroke-strings is isomorphic to the natural numbers, if we take | as 0 and the operation of adding a stroke on the right of any given string as the successor operation. 2 Parsons (1996). This discussion of Parsons’s view draws on Hale and Wright (2002).

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In this case, we can literally perceive token stroke-strings—the objects of intuition are stroke-string types. In the simplest cases, intuiting a type-string consists in seeing a token and seeing it as a token of its type. So intuition requires possession of the concept of a type, and a grasp of some notion of type-identity. Not all intuition of types is directly parasitic upon perception of tokens, in Parsons’s account, as we shall see. Intuition of type-strings can, Parsons claims, ground propositional knowledge concerning the system of type-strings. For example, we can—he claims—know intuitively, on the basis of intuition of type-strings that ||| is the successor of ||. More ambitiously, he claims3 that intuition can give us knowledge of analogues of the elementary Dedekind-Peano axioms: DP1s DP2s DP3s DP4s

| is a stroke-string | is not the successor of any stroke-string Every stroke-string has a successor which is also a stroke-string Different stroke-strings have different successors

Parsons takes DP3s to be equivalent to the claim that each stroke-string can be extended by one more, and regards it as ‘the weakest expression of the idea that our “language” is potentially infinite’ (Parsons 1996, p.105). Obviously, if Parsons is right that these axioms can be known by intuition, then we can have intuitive knowledge that there are potentially infinitely many objects. But is he right? Part of the difficulty here is that DP2s –DP4s , unlike DP1s , are general. How can intuition of objects—which, though types, are still particular—yield knowledge of general truths? Parsons agrees that DP2s –DP4s cannot be known by intuition founded on actual perception. But he thinks we can solve the problem by extending the notion of intuition to cover also intuition of types based upon imagined tokens: But if we imagine any string of strokes, it is immediately apparent that a new stroke can be added. One might imagine the string as a Gestalt, present all at once: then, since it is a figure with a surrounding ground, there is space for an additional stroke . . . Alternatively, we can think of the string as constructed step by step, so that the essential thing is now succession in time, and what is then evident is that at any stage one can take a further step. (Parsons, 1996, p.106)

As Parsons himself emphasizes, the crucial thing—if the required generality is to be achieved—is that one has to imagine an arbitrary string of strokes. But doesn’t this run straight into an analogue of Locke’s problem of the abstract general triangle? Just as we have trouble imagining a triangle which is ‘neither oblique nor rectangle, neither equilateral, nor equicrural, nor scalenon; but all and none of these at once’ (Locke, 1924, Bk IV Ch.7, sec.9), so we shall have trouble imagining a string of strokes which is neither one stroke long, nor two, nor three, . . . , nor any other number, but which has, nevertheless, some definite length. Parsons thinks we can get around the problem, in one of two ways: [by] imagining vaguely, that is imagining a string of strokes without imagining its internal structure clearly enough so that one is imagining a string of n strokes for some particular n, 3 Parsons does not explicitly formulate the claim as stated here. I am following James Page’s exposition (Page, 1993). Parsons takes no exception to this claim in his reply to Page.

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or taking as a paradigm a string (which now might be perceived rather than imagined) of a particular number of strokes, in which case one must be able to see the irrelevance of this internal structure, so that in fact it plays the same role as the vague imagining. (ibid)

I don’t think either suggestion works. The trouble with vaguely imagining a string extended by one stroke is that it is unclear how this is supposed to give us knowledge that any string can be extended by one, rather than just knowledge that some string or other can be so extended. The difficulty with the second suggestion is to give a satisfactory explanation of what, in this kind of case, seeing the irrelevance of the internal structure of the imagined or perceived string is supposed to consist in. Consider, by way of contrast, the relatively clear case where we do a geometrical proof using a particular figure, e.g. a particular right triangle in the proof of Pythagoras’s Theorem—we can see that the other angles, and the lengths of the sides, are irrelevant, because we know that no appeal to them is involved in the reasoning. But there is no reasoning in the intuitive case—this is why it’s so obscure how the notion that internal structure is irrelevant is to be cashed in. I think there is a more general, and more fundamental, difficulty, which concerns what is taken to be required for the existence of type-strings. There are three possible positions: (i) A type-string exists (if and) only if there exists at least one token of that type. (ii) The existence of type-strings is entirely independent of their actual or possible concrete or imaginative instantiation. (iii) A type-string exists (if and) only if there could be a token of that type. On option (i), there will be infinitely many type-strings only if there are infinitely many tokens. More generally, knowing that there are infinitely many quasi-concrete objects will require knowing that there are infinitely many concrete objects. This is obviously no good. We don’t have any such knowledge. And anyway, knowledge of pure arithmetic should not depend upon it, even if we did. Adopting (ii) amounts to rampant Platonism. Quite apart from any other objections which may be raised against that, Parsons couldn’t buy into it without abandoning his claim that types are quasi-concrete, and so abandoning all hope of having intuitive knowledge of them. That leaves (iii). Defending this would require making good the claim that there is at least a potential infinity of concrete objects (token-strings). The problem here is not that this isn’t believable enough—it is rather to see how Parsons could justify the belief. His claim would be that given any perceived or imagined token-string, a singlestroke extension of it is imaginable. That is, it must be imaginable that there be such a token-string, even if none actually exists. The problem now is with the modality in imaginable. Does it involve rigidifying on our actual imaginative capacities, or not? If so, we are in trouble, since our actual capacities are limited, so it will be just false that given any perceived or imagined token-string, an extension can be imagined—this fails, beyond the limit. But if we appeal instead to possible extensions of our imaginative capacities, we have to be able to say what extensions are admissible—and now the difficulty is to see how answer that question without appealing (in the case in hand)

12.2 frege’s proof 217 to some antecedent conception of the range of forms, or types, there are for tokenstrings to instantiate! But that means that intuition is useless as a means of getting to know facts about that. I don’t think the problems I’ve described here are peculiar to Parsons’s particular approach—they seem likely, rather, to afflict any attempt to base a knowledge of infinity in apprehension of possibilities supposedly grounded in imagination. Since I cannot see how an object-based approach could get anywhere without appealing to some such grasp of possibilities, I think that that approach cannot work. Can a property-based approach do better? I shall now try to argue that it can.

12.2 Frege’s proof Frege sketched, in Grundlagen §§82–3, a proof of the proposition that the number of numbers less than or equal to n immediately succeeds n. Given other propositions he has already derived from what is now generally called Hume’s principle: HP ∀F∀G(NxFx = NxGx ↔ F ≈ G) (where NxFx and F ≈ G abbreviate, respectively, ‘the number of Fs’ and ‘the Fs and Gs correspond one-to-one’), this suffices, if acceptable, to establish that the sequence of finite numbers is infinite.4 Frege’s attempted proof exemplifies what I am calling the property-based approach. For Frege, numbers are fundamentally and essentially numbers belonging to concepts, so that a number exists only if there is a concept to which it belongs. In particular, to establish, for an arbitrary finite number n, that n has an immediate successor, one must be able to exhibit a concept to which the number n + 1 belongs. Although Frege had not, in Grundlagen, settled into using the term ‘concept’ (i.e. ‘Begriff ’) to denote the reference, as opposed to the sense, of a predicate, I think we can reasonably interpret him as holding that numbers belong to properties—in the basic case, to first-level properties, so that a statement of number such as ‘Nx(x is a moon of Jupiter = 4)’ ascribes the second-level property of having 4 instances to the first-level property of being a moon of Jupiter. I shall not go through the proof, since the issues I want to discuss don’t depend upon its detail. No one now doubts that the Dedekind-Peano axioms can be derived, in second-order logic, from Hume’s principle—i.e. that what George Boolos called Frege’s Theorem is indeed a theorem. In particular, we know that the existence of successors for all finite numbers can be established, given Hume’s principle, in pretty much the way Frege proposed.5 What remains open to question is whether Frege’s Theorem has the philosophical significance some of us claim for it. In particular, the idea that Frege (as good as) proved that there are infinitely many finite numbers has met with

4 Frege derives HP from his explicit definition of NxFx as the extension of the concept ‘equinumerous to the concept F’. 5 See Boolos and Heck (1997).

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some substantial objections. I can discuss only what I take to be the most serious of them here.6

12.3 Dummett’s objections In his Edwards Encyclopaedia article on Frege (Dummett (1967), reprinted as Dummett (1978)), Michael Dummett diagnoses what he takes to be the fundamental error in Frege’s theory of the natural numbers, including his attempted proof of their infinity, as follows: [T]he assumption that certain class terms can be formed and treated as having reference is an existential assumption and can hardly be grounded on logic alone. The minimal assumption Frege needed in Grundlagen is that there is a mapping by which each concept F is mapped onto a class of concepts containing just those concepts G such that there are just as many Fs as Gs. When classes are taken, as Frege took them, to be objects—that is, to be in the domain over which the concepts in question are defined—this is tantamount to the assumption that there are at least denumerably many objects. From this point of view it does not much matter whether the numerical operator is taken as primitive or as defined in terms of classes: the existential assumption is the same. Now, admittedly, if numbers (or classes) are taken to be objects, then it is reasonable to assert that there are infinitely many objects (far more reasonable than Russell’s “axiom of infinity” which asserts that there are infinitely many individuals—objects which are neither classes nor numbers—and is probably not even true). But then the recognition of the truth of the statement that there are infinitely many objects cannot be held to precede a grasp of the notion of number, which is required for an understanding of the domain over which the individual variables are taken as ranging. (Dummett 1967, p.236; 1978, p.99)

As Dummett observes, the notion of class and the identification of numbers with certain classes play no essential part in the objection he is presenting here. The argument loses none of whatever force it possesses if recast so as to eliminate talk of classes in favour of just numbers—that is, if recast as: The assumption that certain numerical terms can be formed and treated as having reference is an existential assumption . . . The minimal assumption Frege needed . . . is that there is a mapping by which each concept F is mapped onto an object in such a way that concepts F and G are mapped onto the same object just in case there are just as many Fs as Gs. When numbers are taken . . .

Thus if the objection is good, it must tell equally against more recent attempts to reinstate Frege’s programme by basing it on Hume’s principle, understood as an implicit definition of the number operator.7 I want to consider Dummett’s objection as directed against this position. 6 I should mention one line of objection I’m not going to discuss in any detail. This is an old objection, going back to Poincaré, and discussed subsequently by Charles Parsons and others, to the effect that any attempt to carry out anything like the logicist programme must make use of mathematical induction, and so must be viciously circular. Here I can only say that I have always had difficulty seeing the force of this objection. It is certainly true, in particular, that Frege’s proof of the successor theorem is inductive. But Frege thinks he is entitled by employ mathematical induction by his definition of the natural numbers: Nat(x) ↔ x = 0 ∨ S* (0, x) where S* is the ancestral of the successor relation. This amounts to defining the natural numbers so that induction holds, but it does not assume the existence of any objects except 0. 7 See Hale and Wright (2001a), and references therein to earlier work.

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There are two separable objections suggested by the passage I’ve quoted. The first four sentences seem to express the objection that Frege cannot proceed without an assumption that there exist infinitely many objects, and that this assumption cannot be grounded in logic alone. But the last two sentences seem rather to be claiming that Frege’s procedure involves a vicious circularity, in consequence of the fact that the numbers themselves are taken—as they must be taken, if Frege’s proof of infinity is to work—as belonging to the domain over which the individual quantifiers range. It is not clear whether Dummett sees himself as making two separable objections—in fact, I doubt it—but they should be separated anyway, because they raise quite different issues. I’ll begin with the second—circularity—objection. There is something very odd about this objection. Dummett concedes that the assertion that there are infinitely many objects is reasonable, if numbers are (taken to be) objects, but then complains that if they are so taken, there can be no recognition of that fact—by Frege’s route—without prior grasp of the concept of number, and that prior grasp of the concept of number is required for understanding the individual quantifiers, if numbers are taken as lying within their range. But Frege would surely have agreed that there can be no recognition of the existence of infinitely many objects without a prior grasp of the concept of number—at least, he would certainly agree that one cannot follow his proof, and so come in that way to see that there are infinitely many objects, unless one already has the concept of (finite) number. To think that that is an objection is, on the face of it, simply to misunderstand how Frege thought we could come to know that there are infinitely many objects. There is no reason to suppose that Frege thought that we could come to know that except by seeing that there are infinitely many numbers—i.e. that he thought we could independently establish the existence of an infinity of objects of some other kind. So what is the intended objection? There are two possibilities, between which Dummett’s objection hovers indecisively. First, he may be claiming that Frege’s definition of number (or equally, Hume’s principle put forward as replacement for it) is viciously circular because one has to possess the concept of number in order to understand the individual quantifiers involved in the definition, once their range is taken to include the numbers themselves. Second, he may be claiming that Frege’s argument that each finite number has a successor, and hence that there are infinitely many objects, is viciously circular, because it rests on the assumption that there are infinitely many objects. In this case, the alleged vicious circularity is epistemological, rather than definitional. In so far as the passage I’ve quoted offers any support for the first claim—that Frege’s definition and its neo-Fregean replacement are viciously circular—it lies in the appeal to the idea that when the embedded first-order quantifiers are understood as ranging impredicatively over a domain including the numbers themselves, one must already grasp the concept of number if one is to understand those quantifiers. However, whilst we should agree that one cannot fully understand quantifiers without knowing what their bound variables range over, this cannot plausibly be taken to require that for any specific kind of object lying in their range, we must be in possession of the concept of that kind of object. Possession of the concept aardvark, for example, is not a prerequisite for understanding quantification over all terrestrial animals. To be sure, it might be claimed, first, that one must have some concept covering all the objects

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in the range of a first-order quantifier, and second, that in the case of number there is no more general and inclusive concept standing to it as terrestrial animal stands to aardvark, so that one cannot understand quantification over numbers without having the concept number. The first part of this claim is plausible, provided it is not understood as implying that only restricted quantification is intelligible. If it is so understood, I think we should reject it, since I can see no compelling reason why one cannot quantify unrestrictedly over all objects whatever. And if one can do so, one can understand quantifiers which range over—amongst other things—the numbers, without having the concept of number. Dummett seems fairly clearly to think otherwise—that quantifiers must be understood to range over some definite domain, and that one cannot coherently quantify over a domain including all objects whatever. Earlier in the article, he writes: . . .the paradoxes of set theory reveal that it is impossible coherently to interpret bound variables as ranging simultaneously over all objects which could be comprised within a domain over which bound variables could coherently be interpreted as ranging. (Dummett 1967, p.230; Dummett 1978, p.99)

Since what the paradoxes of set theory may be taken to reveal is that there cannot be a class or totality of all objects whatever (given that any class or totality is itself an object) it seems that Dummett’s thought here must be that interpreting bound variables as ranging simultaneously over all objects involves thinking of all objects as forming or being comprised in a class, or collection, or totality. Any objects whatever may be taken to lie within a domain of quantification, but no single domain can encompass all of them, so there can be no unrestricted quantification over all objects. But this line of thought is obviously resistable—why should quantifying absolutely unrestrictedly over all objects require taking the objects quantified over to form a class or totality? If the complaint is seen rather as charging Frege’s proof with epistemological circularity, it still needs to be distinguished from the objection presented in the first part of the quoted passage, that Frege is relying upon a non-logical assumption. But I think that essentially the same reply may be made to both objections. As far as the alleged assumption that numerical terms have reference is concerned, the immediate reply should be that there is no such assumption, at least when Hume’s principle is put forward as an implicit definition. To suppose that there is is to miss the point that Hume’s principle is the second-order universal closure of the biconditional schema: NxFx = NxGx ↔ F ≈ G so that the truth of its instances need not be taken, and should not be taken, as requiring the existence of referents for their left-hand side numerical terms. This is a perfectly general point about abstraction principles—that is, principles of the shape: ∀α∀β(§(α) = §(β) ↔ Eq(α, β)) where Eq is an equivalence relation on entities of the type of α and β, and if the abstraction is good, § is a function from entities of that type into objects. The effect of laying down such a principle as an implicit definition of the abstractive operator is to ensure that any identity of the type occurring on the left-hand side of the biconditional coincides in truth-value with the corresponding right-hand component.

12.4 dummett’s objection refurbished 221 But the biconditionals instantiating the abstraction principle are to be so understood that their truth is consistent with their ingredient abstract terms lacking reference.8 The truth of the left-hand side identities—and hence the existence of the relevant abstract object—may only be inferred with the aid of the corresponding right-hand side statements as supplementary premises. Of course, in the particular case of Hume’s principle, there are instances whose truth is a matter of logic, since the identity-map ensures that F ≈ F, for any concept F—so that the existence of NxFx is guaranteed, for any suitable concept F. But the fact that something can be proved by means of a principle is not in general taken to show that one is, in asserting the principle, simply assuming its truth.

12.4 Dummett’s objection refurbished, and a related objection from Charles Parsons To the reply just given to Dummett’s objection, it may be countered that even if there is no explicit assumption to the effect that terms formed by means of the number operator have reference, or that there are infinitely many objects, there is nevertheless an assumption implicit in the use of Hume’s principle as a basis for arithmetic which is tantamount to assuming an infinity of objects. For Hume’s principle quantifies over properties, and these properties must—if the existence of numbers is to be an objective, mind-independent matter—be conceived as existing independently of the abstraction itself. To clarify the point: it may of course be maintained that we in some sense create or construct the concept of each of the requisite properties—in this case, properties in the sequence: finite number ≤ n. But the properties themselves must be conceived as existing independently of our concept-forming activity, as ‘there’ all along. However, the assumption of sufficiently many such mind-independent properties—that is, at least countably infinitely many of them—is an existential assumption every bit as problematic as the assumption of the existence of infinitely many objects. No-one, as far as I know, has pressed the existential assumption objection in quite this form. But Charles Parsons, in an essay which appeared a little before Dummett published his encyclopaedia article, makes an objection to Frege along rather closely related lines:9 As a concession to Frege, I have accepted the claim of at least some higher-order predicate calculi to be purely logical systems . . . . The justification for not assimilating higher-order logic to set theory would have to be an ontological theory like Frege’s theory of concepts as fundamentally different from objects, because “unsaturated”. But even then there are distinctions among higher-order logics which are comparable to the differences in strength of set theories. Higher-order logics have existential commitments. Consider the full second-order predicate calculus, in which we can define concepts by quantification over all concepts. If a formula is interpreted so that the first-order variables range over a class D of objects, then in interpreting the second-order variables we must 8 Obviously this presupposes that the underlying logic is free—since atomic sentences cannot be true unless their ingredient singular terms refer, some modest restrictions on the quantifier rules are needed. 9 Parsons (1965). The quotation is from §VII of the essay.

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assume a well-defined domain of concepts applying to objects in D which, if it is not literally the domain of all concepts over D, is comprehensive enough to be closed under quantification. Both formally and epistemologically, this presupposition is comparable to the assumption which gives rise to both the power and the difficulty of set theory, that the class of all subclasses of a given class exists. Thus it seems that even if Frege’s theory of concepts is accepted, higher-order logic is more comparable to set theory than to first-order logic.

Parsons’s objection, then, is that even if we take higher-order variables to range over Fregean concepts—incomplete or ‘unsaturated’ entities, fundamentally different from objects, and so from sets—it remains that higher-order logic involves existential commitments just as problematic as those of set theory. And if we think of Fregean concepts as properties—as being the referents of predicates rather than their senses— then the objection may be seen to tell against what I am calling the property-based approach. The first objection would be devastating, if it were true that the assumption of the existence of infinitely many properties (of the type: finite number ≤ n) is as problematic as the assumption of infinitely many objects. But it seems to me that whether it is problematic depends upon how one conceives of properties. Of course, the objector is right to insist that properties must be conceived as objective and so mind-independent. But so conceiving of them does not settle whether properties are to be conceived in purely extensional terms, or in some other way. The assumption would be as problematic, if properties were thought of as individuated extensionally. For then, there would only be enough properties if there were enough objects. In general, properties P and Q would be distinct only if there were an object having one of them but not the other. Of course, for any finite number of properties, only a smaller finite number objects is needed—but for infinitely many properties, we need infinitely many objects. For similar reasons, Parsons’s complaint appears entirely justified, as directed against any view on which the values of higher-order variables are, whilst not identified with sets, taken to be individuated extensionally. So in particular, it seems a fair enough complaint against Frege, since he did conceive of his concepts in a purely extensional fashion. More generally, it appears that any property-based approach is a waste of time—or at least can enjoy no ontological or epistemological advantage over an object-based approach—if it treats properties in a purely extensional way. However, it seems to me that one does not have to conceive of properties in a purely extensional way, and that a proponent of the neo-Fregean abstractionist approach should not do so. To put it very roughly, if one thinks of properties—the values of bound higher-order variables—as individuated purely extensionally, then however much one emphasizes the supposed ontological differences between properties and sets, one will not be able to get away from the fact that properties behave just like sets at least in important respects. And one will be very hard put to make out that there is really any philosophical advantage in a foundation which assumes higherorder logic—one might as well start with set theory. This was, near enough, Parsons’s point. But if one thinks instead of properties as individuated non-extensionally, there is at least some chance of philosophical advantage. In particular, the assumption that there are infinitely many properties—say, of the Fregean form: finite number ≤ n, for finite n—does not amount to the assumption that there are infinitely many objects,

12.4 dummett’s objection refurbished 223 and may be significantly weaker, and so epistemologically less problematic—perhaps weak enough for it to form part of a foundation for arithmetic. One central task, on this approach, is to make clearer just what the non-extensional conception of properties comes to. Rejecting purely extensional individuation of properties is consistent with different—more or less demanding—positive conditions for the identity and distinctness of properties, and indeed, for the existence of properties. Some of the choices which confront us here seem likely to have significant implications for how much mathematics can be recovered on an abstractionist approach. Here I can say only a little more about these questions. On what is very naturally described as an Aristotelian conception of properties, properties exist, or have being, only through their instances—there are no uninstantiated properties. The first point I need to make is that the abstractionist is, fairly obviously, committed to a non-Aristotelian conception of properties. If we focus on the argument for the infinity of the natural numbers, it is clear that it can’t get started at all unless it is accepted that there is at least one uninstantiated property—if we are to get zero as the number of Fs, for some suitable (sortal) property F, then we require a property which no objects possess. Indeed, if we are to define zero in this way, we surely need a property which is necessarily uninstantiated—so we are committed to rejecting even a weakened form of Aristotelianism, which does not require actual instantiation, only possible instantiation. The Fregean argument for the infinity of the natural numbers appeals to properties of a specific type—for each n, we require the property of being a natural number less than or equal to n. But this does not generate any additional demand for uninstantiated properties—0 is by definition a natural number, so that the existence of 0 ensures that the property of being a natural number ≤0 is instantiated. And the existence of the natural numbers from 0 up to and including n likewise ensures that the property of being a natural number ≤ n + 1 is instantiated. Indeed, the Fregean can, and in my view should, think that the relevant properties are all necessarily instantiated—but this necessity derives not from any general Aristotelian requirement on properties; it derives rather from the necessary existence of the natural numbers themselves. To explain why I think it is reasonable to adopt a non-Aristotelian conception of properties, it will help to draw attention to another distinction, between what can be called sparse and abundant conceptions of properties. Roughly speaking, on an abundant conception, given any well-formed and meaningful first-level predicate or open sentence with one free individual variable, there is a corresponding firstlevel property, or property of objects. This is a very undemanding conception of properties—according to it, there are not only such simple properties as being round and being white, but also various kinds of complex properties, including negative, conjunctive and disjunctive properties (such as not being white, being white and spherical, and being red or white), and relational properties (such as being less massive than Nelson’s Column, being less than one kilometre distant from an aardvark, etc.). In sharp contrast with this, proponents of a sparse conception wish to insist on much more stringent conditions for something to count as a genuine property—they may, for example, wish to restrict attention to properties which play some role in causal explanations of the behaviour of objects which possess them, or to properties which

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cannot change simply as a result of the change or loss of properties of objects other than objects which possess them (as I might lose the property of being less than one kilometre away from an aardvark, even though I do not move, because all the aardvarks in my vicinity head off to the nearest waterhole or ant heap). There may be good reasons to be interested in some sparse notion or notions of properties. But however good such reasons may be, they cannot, in my view, add up to reasons for denying that a less demanding abundant conception is possible or legitimate. Indeed, in my view, the best—and I would argue, only—way to understand a sparse conception is as a restriction of the more generous abundant conception. One might propose what could be called a weakly non-Aristotelian conception, according to which a property is just a way things might be—so that a first-level property is a way objects might be, a second-level property a way first-level properties might be, and so on. This stands opposed to the strongly Aristotelian conception, according to which a property is a way some things are. Although this appears to me a perfectly intelligible conception, it is, as I’ve already remarked, insufficient for the abstractionist’s purposes. The abstractionist should define 0 by reference to some property which is necessarily uninstantiated—it seems clearly unsatisfactory to define 0 by reference to some property which is merely contingently uninstantiated. And that requires a strongly non-Aristotelian conception which admits properties which are ways things couldn’t be, corresponding to predicates which are necessarily unsatisfiable. But once one embraces the abundant conception, there is no good reason to deny that there are such properties. In fact, necessarily uninstantiated properties will be unavoidable, if the class of properties is, so to speak, closed under conjunction and negation. For then where P is any property, not-P and hence P-and-not-P will also be properties, and the last will be necessarily uninstantiated. But closure under negation, conjunction, etc., is guaranteed, if any well-formed and meaningful open sentence stands for a property. A property is, to put it somewhat loosely, a way things might or might not be—more accurately, a first-level property is a condition which objects do or don’t satisfy, either contingently or as a matter of necessity. To sum up: I have argued that of the two approaches to the problem of proving the existence of an infinity of objects I contrasted, the property-based approach is best placed to succeed. Specifically, I’ve defended Frege’s attempted proof of the infinity of the natural numbers against what seems to me to be the most serious objection to it— the charge that it is in some way viciously circular. As against Dummett, I argued that a version of Frege’s argument based on Hume’s principle involves neither a vicious circle in definition, nor an epistemological circularity—contrary to what Dummett claims, there need be no assumption of the existence of a denumerable infinity of objects. I then turned to a strengthened version of Dummett’s objection, which claims that the Fregean argument avoids the assumption of an infinity of objects only by relying instead upon the equally problematic assumption of an infinity of properties. I argued that this, along with Charles Parsons’s closely related objection to Frege’s reliance on second-order logic on the ground that involves existential assumptions comparable to those of set theory, can be answered, if we understand properties in a non-extensional sense and adopt a very modest conception—the abundant conception—of what is required for their existence.

13 Properties, Predication, and Arbitrary Sets 13.1 Background and preliminaries 13.1.1 Definable properties vs. arbitrary sets and functions Shapiro defends the standard semantics for second-order logic, according to which property variables range over the full power set of the first-order domain, which, when the first-order domain is infinite, is taken to be largely composed of infinite subsets which cannot be individually defined or specified, but are, in a certain sense, arbitrary.1 I advocate what, from this point of view, must be seen as a severely restrictive interpretation, based upon what is often called an abundant or deflationary conception of properties and on a broadly Fregean approach to ontology according to which ontological categories are to be explained in terms of a prior division of expressions into logico-syntactic categories. Roughly, an object is any entity to which reference could be made by means of a singular term; a first-level property is any entity to which reference could be made by means of a first-level 1-place predicate; and similarly for properties, relations, and functions of second- or higher-level. In considering the existence conditions for objects and properties on this view, the key point for present purposes is that there is a tight connection between the notion of a property, the existence of properties, and the availability of predicates which stand for or express them. Even if there is no predicate for a given property in any actual language, it must be possible that there should be one, in some possible language or possible extension of some actual language. This leads to conflict with the standard semantics—while it allows a form of model-theoretic semantics in which the secondorder variables range over subsets of the first-order domain, their values can only be definable subsets.

13.1.2 Historical antecedents Our disagreement centres on the notion of arbitrary set or function, or in property terms, of arbitrary properties. Arbitrary sets and functions contrast with those which are given by a rule or other specification by means of a predicate. The disagreement runs parallel to the much older dispute which surrounded Ernst Zermelo’s 1904 proof 1 Editor’s Note: This chapter is formed of notes written by Bob Hale, based on discussions, mostly during 2016, with Stewart Shapiro, partly in response to Shapiro (2018), his contribution to Fred and Leech (2018). The notes were developed enough to form a coherent and interesting chapter, with only minimal editorial changes. Bob Hale, Properties, Predication, and Arbitrary Sets In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0014

226 properties, predication, and arbitrary sets of the proposition that every set can be well-ordered, using his Axiom of Choice, which asserts the existence of arbitrary functions. Let me offer two brief comments on this earlier debate: (1) Early defenders of Choice (Zermelo himself, Hadamard) accepted that Choice is not evident, but contended that we may adopt it as an hypothesis and judge it by its fruits. Thus they tend to take the key questions to be: Is Choice consistent? Does it give us results we want (but perhaps can’t get without it)? And to these questions, of course, the answers appear (to them anyway) to be obviously ‘yes’ and ‘yes’. However, early critics of Choice (the French analysts Baire, Borel, Lebesgue; some Italians including Peano, and English, especially Hobson) didn’t think that Choice is proof-theoretically inconsistent, or that it is inconsistent in some semantic sense. That would involve accepting that it is at least a perfectly intelligible hypothesis. Their doubts were (usually) about whether it really makes sense. There is, accordingly, some reason to think that early defenders and opponents of Choice are, to some extent at least, failing to engage each other’s arguments. (2) It is a well-documented historical fact that, prior to its explicit formulation, the axiom had played an unnoticed but indispensable role in proofs of results which the vast majority of mathematicians were unwilling to give up—including proofs given by its critics themselves, in their earlier mathematical work.2 Thus the adoption of definabilist restrictions would have involved sacrificing substantial parts of accepted classical mathematics, including many results in analysis. It was, arguably, the recognition of this unpalatable consequence which, more than any other consideration, silenced constructivist opposition and ensured the widespread acceptance of Choice, along with the conception of a function as an arbitrary correspondence between domains, and of a set as an arbitrary collection of objects. For these reasons, it seems to me that the philosophical issues which surfaced in that debate were largely unresolved, and remain so.

13.1.3 Cantor’s diagonal argument Cantor’s famous argument shows that certain sets—the set of real numbers, say, or the set of subsets of N—are uncountable. It might be supposed—erroneously—that this argument already shows that there must be unspecifiable or undefinable subsets of N or properties of natural numbers, and so already settles the issue I want to discuss. The argument would be: the definable sets/properties of, say, the natural numbers are countable, but Cantor’s proof shows that the sets/properties of natural numbers can’t be counted, so there must be some indefinable ones. The flaw in this argument lies in not taking seriously the point that what the proof strictly shows is that any given enumeration must leave out at least one property/set. But it does not show that there is a property/set which can’t be defined or specified at all. On the contrary, it works, roughly, by showing how to specify one. Hence there is no valid step to: There is an absolutely unspecifiable/indefinable property. The step would be an instance of the ∀∃ → ∃∀ fallacy.

2 For details, see Moore (2013), especially ch.1.7.

13.2 main points of disagreement

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13.2 Main points of disagreement 13.2.1 Set theory and modality Shapiro is happy with set-theoretic notions, and in particular with the notions of arbitrary set, relation, and function as they stand and without further explanation, but sceptical about modal ones—except, perhaps, those that are reductively explicable in set-theoretic terms. One group favours set theory and holds that set-theoretic assertions should be taken more or less literally, “at face value”. This is realism, in two senses. First, it is held that mathematical objects, sets, exist independently of the mathematician. This may be called “realism in ontology”. Second, the assertions of set theory have objective truth values, independent of the conventions, languages, and minds of the mathematicians; and the bulk of the assertions of competent set theorists are true. Call this “realism in truth value” . . . . Under Quine’s influence, some (but not all) of these philosophers are skeptical of modality or, at any rate, they don’t think modal notions can play central roles in philosophical explanations. But the logical notions of possibility, necessity, and consequence are notable exceptions to this skepticism. It may be that these are accepted because they have (presumably) been reduced to set theory, via model theory—this despite the prevailing anti-reductionist spirit. For example, a sentence or proposition, φ, is said to be logically possible if and only if there is a model that satisfies φ . . . . The second school is the opposite of the first. Its members are skeptical of set theory, at least if it is taken literally, and they accept at least some forms of modality. To be precise, these philosophers are less skeptical of modality than they are of set theory. So, they set out to reformulate mathematics, or something to play the role of mathematics, in modal terms . . . . The point of the second enterprise is to see how far we can go without asserting the existence of abstract objects like sets. The members of this school are thus anti-realists in ontology. Many of the authors, however, are realists in truth value, in that statements of mathematics, or close surrogates, have objective truth conditions that hold or fail independent of the conventions, minds, etc. of mathematicians. (Shapiro 1993, pp.455–6)

13.2.2 Properties I think any respectable notion of property must preserve a link with possible predication. Shapiro does not. He is happy to take the values of second-order variables to be properties (see Shapiro 1991, p.16), but only because he ‘glosses over the differences’ between properties, conceived as intensional, and extensionally-individuated sets, and in particular, severs any essential link between properties and predication. Properties—of the natural numbers, say—are for him effectively just subsets by another name.

13.2.3 Quantification If we understand the claims that objects are possible objects of singular reference, and that properties are possible objects of predicative reference etc., as meaning that objects, properties, etc. are entities for which there could be singular terms, predicates, etc., then it is a corollary of my account that any entities over which we quantify must be possible objects of singular reference or predicative reference, etc. We can express this concisely by saying that on my account any entities over which we quantify must be specifiable or identifiable. As against this, Shapiro claims: In maths and elsewhere, we often quantify over unspecifiable or unidentifiable entities.

228 properties, predication, and arbitrary sets

13.2.4 Further questions There is obviously a question about what notion of possibility is at issue, in the requirement that for any property, it must be at least possible that there should be a suitable predicate, i.e. a predicate associated with a satisfaction-condition with which, on the abundant or deflationary conception, the property is to be identified. But the proponent of arbitrary properties (or arbitrary sets or functions) is likewise committed to the intelligibility of some notion of possibility. For even if he denies that an infinite selection of natutral numbers belonging to one of the arbitrary infinite subsets of N which he takes to make up the vast majority of members of P (N) need be in any sense something which beings with finite powers such as ourselves could accomplish, it seems that he must hold that such selections are, in some sense, possible. Thus, just as we may ask: what is the modality involved in the putative requirement that properties be specifiable? So we should ask: what modality is involved in the claim that there exists a subset of the natural numbers corresponding to every possible selection? In what sense of ‘possible’ does this give a sufficient (and perhaps also necessary) condition for the existence of subsets?

13.3 Which modalities? 13.3.1 Possible predicates In briefest terms, I hold that what is required is an absolute possibility. Absolute possibilities are simply those possibilities which are not ruled out by any absolute necessities. It is absolutely necessary that p if it is true that p and it would be (or would have been) true that p no matter what (else) were (or had been) the case. We can express this by a generalized counterfactual conditional: ∀q(q2→ p). Here, the propositional quantifier must be understood to range unrestrictedly over absolutely all propositions—in effect, we are explaining absolute necessity in terms of absolute generality (along with the counterfactual, which is of course itself a modal construction—there is no reductive explanation here). On my view, absolute necessities have their source in the natures of things. They divide into different kinds, according to the kinds of entity whose natures give rise to them. Logical necessities are those propositions which are true in virtue of the natures of the logical entities alone (the truth-functions of negation, conjunction, etc., the quantificational functions, and so on); mathematical necessities are those propositions true in virtue of the natures of mathematical entities (and perhaps also logical entities), and so on. Not all necessities are absolute—some are merely relative, i.e. necessary relative to certain further assumptions. Given this distinction, it seems clear that our broadly Fregean ontological theory should be taken as asserting that for something to be an object, what is required is that it should be absolutely possible that there be a suitable true statement involving a name or other singular term having that thing as its referent, and that for something to be a property, it should be absolutely possible that there be a suitable predicate expressing it or standing for it.

13.3 which modalities? 229

13.3.2 Possible arbitary selections 13.3.2.1 mathematical possibility and the friends of arbitrary sets As we have seen, the friends of arbitrary sets need not reject modal notions altogether. For they may think that some kinds of modality can be reductively explained in settheoretic terms. They may accept a notion of logical possibility, explained so that it is logically possible that φ just in case there is a model of φ. But they may then define a relative notion of mathematical possibility in terms of logical possibility: ♦math p = ♦(M ∧ p) where M is a conjunction of the laws of mathematics. They can claim, in particular, that the sense in which there exists, for example, a subset of the natural numbers corresponding to every possible selection, including arbitrary infinite selections, is that such selections are mathematically possible. Indeed, since ♦math p  ♦p, they can claim that it is outright logically possible that p—and so possible in the only absolute sense they recognise.

13.3.2.2 is mathematical possibility enough? Not in my view, for two reasons: (1) In my view, logical necessity and mathematical necessity are both sub-species of absolute, metaphysical necessity. This means that logical necessity entails metaphysical necessity and that metaphysical possibility entails logical possibility. But the converses do not hold. Likewise for mathematical necessity and possibility. Thus much as—at least on a widely accepted view—the existence of water that is not H 2 O, though logically possible, is nevertheless metaphysically impossible, and so not a genuine or real possibility, so it may be that the existence of arbitrary infinite selections, though not ruled out on purely mathematical or purely logical grounds (i.e. by the natures of mathematical or logical entities), is nevertheless not a genuine possibility, because it is not metaphysically possible. (2) It is not clear that arbitrary infinite sets are mathematically possible, in my absolute sense. What is mathematically possible is what is not ruled out by the natures of mathematical entities, including the nature of sets. That may not be (fully) captured by the usual axioms of set theory.

13.3.2.3 interim conclusion These considerations will not move the friend of arbitrary sets unless he can be brought to accept the absolute notions of metaphysical necessity and possibility on which they rely. But even if he can be brought to accept them, there remains an obvious difficulty. For the case for denying that arbitrary infinite sets are absolutely possible would rest on taking sets to be, by their very nature, determined by properties in my sense, and he will simply reject that account of the nature of sets. The upshot is that reflection on the kinds of modality involved, whilst it gives rise to further disputed questions to which we need answers, is unlikely to provide a route

230 properties, predication, and arbitrary sets to resolving the main issues, because it is hard to see how those further questions can be resolved without first making progress on the main issues.

13.4 Quantification and identifying reference 13.4.1 ‘To be is to be the value of a variable’ Quine’s slogan encapsulates his view that the sole measure of our ontological commitments is what we take to be included in the range of the bound variables of any quantified statements we accept. Even setting aside my disagreement with him over the legitimacy of higher-order quantification, it is clear that my broadly Fregean conception of objects and the conditions for their existence involves a rejection of Quine’s insistence that bound individual variables are the sole vehicle of commitment to objects. But in one way—or so I have claimed3—my difference with Quine is merely superficial. At a deeper level, there is agreement: Quine, like Frege, effectively identifies being an object with being an object of singular reference. It is merely that, because he holds constant singular terms to be always eliminable by means of an extension of Russell’s theory of definite descriptions, he takes the bound variables of first-order quantification to be the sole means by which we refer to objects. Shapiro disagrees: I do not agree that this difference is superficial. My main interest is with mathematical theories, as they have developed and are presented. The language of elementary arithmetic does have singular terms to denote every object postulated by the theory: each natural number is denoted by a numeral; but arithmetic is more the exception than the rule. (Shapiro, 2018, p. 93)

We may set aside the question whether Quine would accept my assessment of our disagreement. What matters here is Shapiro’s assessment. He at least does not think the disagreement is superifical, precisely because he thinks that bound variables can and do convey (indefinite) reference to objects, not merely in the absence of any singular terms which could be used to make definite reference to them, but independently of even the possibility of singular reference to, or thought of, those objects. After some discussion of the problem allegedly posed by ‘large domains’—too large, as Quine himself urged in another connection, for us to name all their members— Shapiro comes to the crux of the matter as he sees it: [T]he real issue here [is this:] Some mathematical domains are too homogeneous for there to be any “singular thought” or “identifying reference” for the objects in them. Euclidean geometry is one such. The language has no singular terms at all, or at least no names. It can’t. There just is no sense to be made, in that theory, of thinking about or somehow referring to or picking out a particular point, or a particular line or circle for that matter. What we have are variables that range over points, lines, planes, and the like. (Shapiro, 2018, p.94)

As we have seen, my claim that the objects composing a domain of first-order quantification must be individually nameable is not an additional doctrine, over and above the central claims of my broadly Fregean approach to ontology, but a straightforward consequence of them. For if objects are simply those things to which 3 See Hale (2013a), p.11, echoing Hale (2010), p.404.

13.4 quantification and identifying reference 231 reference may be made by means of actual or possible singular terms, then any domain of first-order quantitification (i.e. domain of objects) must be composed of things which are individually specifiable or identifiable. Conversely, to insist, as Shapiro does, that some species of objects over which we may quantify—Euclidean points, say— are not individually specifiable or identifiable (a fortiori, that they need not be, for quantification to be in good order) is already to reject the broadly Fregean account of objects and their existence conditions. Likewise, at the level of properties, holding that we can quantify over arbitrary properties is already to reject any broadly Fregean treatment which ties the existence of properties to the possibility of predicates with suitable satisfaction conditions.

13.4.2 Object-sensitive truth-conditions A universal quantification is true if and only if each of its instances is true, and an existential quantification is true if and only if at least one of its instances is so. So much, one might think, is indisputable, just because it merely spells out what is understood by universal and existential quantification. I think that in some sense it is indisputable. But what, exactly, does the requirement that each, or at least one, of the instances of a quantified statement be true amount to? In some cases, we do actually have—or at least could provide—a name for each of the objects over which we quantify. In such cases, we may take the instances of a quantified statement to be the statements which result from deleting its initial universal quantifier and replacing all remaining occurrences of its bound variable by a name for an object in the domain. It is then both necessary and sufficient for the truth of ∀xB(x) that each such singular statement B(t) should be true. Shapiro may of course agree. But he must deny that, in general, the truth-conditions of quantified statements are to be given in terms of the truth-values of their instances in this sense. For a quantified statement has instances in this sense only for those objects in the domain, if any, which can be individually named or otherwise identified. The obvious question now is: How, in general, should Shapiro explain the truthconditions of quantified statements? It is insufficient simply to state the truthconditions homophonically: ∀xB(x) (respectively ∃xB(x)) is true if and only if every (respectively some) object in the domain satisfies B(x). The shortfall is perhaps most easily seen in the case of an existential quantification: when ∃xB(x) is true, it owes its truth not just to the fact that some object or other satisfies B(x) but to the fact that some particular object does so; but the same goes for universal quantification—when ∀xB(x) is true, it is so because each particular object satsifies B(x). The claim that ∀xB(x) and ∃xB(x) involve quantification over objects—albeit unnameable objects— has no substance or plausibility unless the truth-conditions of these statements can be explained in such a way that their truth-values are, as we might put it, objectsensitive—that is, depend upon or are grounded in facts about the individual objects which are held to be the values of the bound individual variables. Now it is certainly true that we can state the truth-conditions for quantified statements in a way which embodies object-sensitivity without overt appeal to instances of those statements in the problematic sense. Instead of giving the truth-condition for ∀xB(x) as requiring that it be true if and only if each of its instances B(t) is true, we can require that ∀xB(x) be true if and only if the free-variable statement B(x) is true

232 properties, predication, and arbitrary sets whichever element, d, of the domain is assigned to free x as its value. There is no need to suppose that d is in fact the denotation of some constant term t, and so no need to invoke instances of the quantified statement in the problematic sense. I don’t think this provides a satisfactory answer to our question. One objection is that taking d to be assigned as value to x amounts to treating x as a temporary name of d. Whether that is so or not, the underlying and more fundamental point is that there can be no such thing as assigning d as value to x unless d can be somehow identified or specified. So the proposed shift in stating the truth-condition for ∀xB(x) does not accomplish what Shapiro requires—viz. a way of articulating those truthconditions which both preserves object-sensitivity and avoids taking objects to be individually identifiable. The first requirement is surely essential—without objectsensitivity, it is quite unclear what ground there could be for taking a quantified statement as involving quantification over a domain of objects. But it is difficult to see how object-sensitivity can be preserved in a statement of the truth-conditions without taking the relevant objects to be somehow identifiable; I doubt that it can be done.

13.4.3 Geometrical points This leaves us with the question what to make of Shapiro’s purported examples of quantification over unidentifiable objects. His main example, as we have seen, is Euclidean geometry. The main claim I have to make here, is that Shapiro’s claim that Euclidean geometry is about unidentifiable objects presupposes his distinctive brand of structuralism, and so fails to provide independent support for his claim. One may draw an instructive contrast with Quine on geometrical objects. For Quine, an abstract—as opposed to applied—geometry is an ‘uninterpreted theoryform’, a purely structural theory which is not as such about any objects at all. Quine’s position is similar to the eliminativist structuralist’s. Quine sees no problems of identity and identifiability affecting geometrical points. He explicitly contrasts them in this regard with the ‘ideal objects of mechanics—mass points, frictionless surfaces, isolated systems, . . .’. There is a more substantial reason why mass points and the like, as objects supplementary to the full-fledged bodies, should be less welcome than geometrical objects. No sense has been made of their date and location. Evidently, to judge by what is said of them, mass points and such ideal objects are supposed to be in space-time of some sort, ours or another; but just where is each? And if we waive location, there supervenes a perplexity of identity: when do mass points (or frictionless surfaces, etc.) count as one and when as two? There is in this strong reason to define ideal objects away—say along the Weierstrassian lines of §51 [i.e. as ‘limit myths’—BH ] . . . On the other hand geometrical objects raise no such evident problems of position and identity; they are positions outright. (Quine 1960, p.252, my emphasis)

Quine contrasts physical or applied geometry with abstract geometries, including those which are ‘actually contrary to our “true” geometry of relativity physics’. Of these he holds: [that we need neither] . . . rate these as simply false [nor] seek ways of reconstructing their words that would make them true after all, whether of our same old geometrical objects or of something else . . . We need do none of this; an uninterpreted theory-form can be worthy of study

13.4 quantification and identifying reference 233 for its structure without its talking about anything. When it is brought into connection with the quantifiers of a broader scientific context in such a way as to to purport to talk unfeignedly of objects of some sort, then is time enough to wonder what the objects are. (op. cit., p.254, my emphases)

For Shapiro, an abstract geometry is a structural theory in a quite different sense: it is a theory about an abstract structure, comprising its own internal objects. To determine whether there is independent support for Shapiro’s claims about quantification, we need to look more closely at his version of structuralism, and at what justification may be given for its key tenets.

13.4.3.1 shapiro’s structuralism Shapiro distinguishes between systems and structures: a system [is] a collection of objects with certain relations . . . a structure is the abstract form of a system, highlighting the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system. (Shapiro (1997), p.74)

Shapiro endorses Resnik’s view that ‘The objects of mathematics, that is the entities which our mathematical constants and quantifiers denote, are structureless points or positions in structures. As positions in structures, they have no identity or features outside a structure’ (Resnik (1981), p.530). These purely structural objects contrast with the objects of which systems are composed. Systemic objects (normally) exist and are identifiable independently of the systems to which they belong, whereas structural objects have no existence or identity apart from the structures to which they belong—they are just positions in those structures. They may be distinguishable and identifiable within the structure to which they belong, but they may not—that depends upon the structure. For example, Shapiro’s natural number structure consists of an infinite collection of places or positions which are ordered so as to form an ω-sequence, so that there is a first position, and for each position a next position—so that each position (each natural number) is distinguished internally from every other and can be identified as the nth position in the structure. But in the plane Euclidean geometry structure, the structural relations (betweenness and equidistance) do not suffice for even internal distinction and identifcation of positions (points in Euclidean 2-space). This contrast between independently identifiable systemic objects and internal, purely structural objects lines up with Shapiro’s distinction between the places-asoffices and the places-as-objects perspectives: Sometimes the places are discussed in the context of one or more systems that exemplify the structure. . . . Call this the places-as-offices perspective . . . In contrast . . . there are contexts in which the places of a given structure are treated as objects in their own right, at least grammatically. That is, items that denote places are bona fide singular terms. . . . Call this the places-as-objects perspective. Here, the statements are about the respective structure as such, independent of any exemplification it may have. . . . The point here is that sometimes we use the ‘is’ of identity when referring to offices, or places in a structure. This is to treat the positions as objects, at least when it comes to surface grammar. . . . Places in structures are bona fide objects. (Shapiro (1997), pp.82–3; some italics mine)

234 properties, predication, and arbitrary sets

13.4.3.2 the existence of structures and purely structural objects Clearly the existence of structures in Shapiro’s sense, including structures whose places are not even intra-structurally identifiable, would settle our disagreement about quantification—there would be quantification over entities inaccessible to identifying thought or reference. Shapiro’s structuralism entails the existence of such structures. Our disagreement would therefore be settled in Shapiro’s favour, if there were compelling reason to embrace his version of structuralism. Is there? Here we need to keep in mind a further feature of Shapiro’s structuralism. Although he describes structures as the abstract forms of systems, he does not believe that structures are essentially abstracted from systems—structures do not depend for their existence upon there being systems which exemplify them. They exist independently and in their own right, regardless of whether they are exemplified by any system of (relatively) concrete, independently existing objects. Shapiro’s structuralism is ante rem, as opposed to in re structuralism, according to which structures exist only if exemplified. Because ante rem structuralism—in contrast with eliminative and modal structuralism—takes structures to exist in their own right, Shapiro’s theory of structures has at its core axioms asserting the existence of structures. First we have an axiom asserting the existence of an infinite structure: Infinity: There is at least one structure that has an infinite number of places. Others are designed to ‘assure the existence of large structures’. They include an analogue of the powerset axiom: Power structure: Let S be a structure and s its collection of places. Then there is a structure T and a binary relation R such that for each subset s ⊆ s there is a place x of T such that ∀z(z ∈ s ≡ Rxz) This asserts, in effect, that there are at least as many places in T as there are subsets of the places in S, so that there is a structure larger than S. Obviously we can iterate to obtain further, still larger, structures. The theory also includes an analogue of the set-theoretic replacement principle which gives us further, larger structures, so that every standard model of the theory is the size of an inaccessible cardinal (Shapiro 1997, pp.94–5). Structure theory bears more than a passing resemblance to set theory, and Shapiro himself describes it as ‘a reworking of second-order Zermelo-Fraenkel set theory’ (op. cit., p.95). Laying down structure-theoretic axioms is one thing, and providing reasons to believe them is quite another. We cannot provide a philosophical foundation for mathematics simply by laying down axioms for a set theory in which other mathematical theories (arithmetic, analysis, etc.) can be reformulated and derived. We need to provide good, independent reasons to believe the axioms. Clearly the same goes for structure-theory.

13.4.3.3 abstraction and structures Shapiro describes structures as the abstract forms of systems. The fact that he takes structures to exist regardless of whether there are any concrete systems which

13.4 quantification and identifying reference 235 exemplify them does not, of itself, mean that he could not take the existence of such systems, from which structures could be obtained by some form of abstraction, to provide us with reason to believe in structures in his sense. Could he do so? One obvious obstacle is that it is at least questionable whether there are—outside of mathematics—sufficiently large systems of objects from which we might abstract to even a structure with a countably infinite number of places, let alone the larger structures generated by Power structure and the analogue of Replacement. Quite apart from this, it is not clear what kind of abstraction would serve the purpose. (i) The more or less traditional conception of abstraction, according to which we abstract from the differences between otherwise similar things to what they have in common, does not give us what Shapiro needs. Suppose we abstract, in this sense, from two simple systems in Shapiro’s sense—say the system consisting of three girls differing from one another in height and the system consisting of three boys differing from one another in weight. The objects of the first system are the girls, those of the second, the boys. The relation of the first is being taller than, and of the second being heavier than. What these two systems have in common, and so what results from traditional abstraction, is a certain higher-level property, viz. that each is composed of three objects related to one another, in each case, by a relation which is irreflexive, asymmetric and transitive, and which, under our supposition, linearly orders its terms. In other words, what the two systems share or have in common is that they each comprise three objects connected by an irreflexive, asymmetric and transitive relation. We may, if you wish, call this their common structure. But whatever we choose to call it, it does not seem that abstracting from differences gives us any new, distinctively structural, objects—the concrete systems from which we abstract involve objects of perfectly ordinary sorts, but abstracting from the differences affords no ground whatever for thinking of the structure that the two systems share as composed of objects, either of an ordinary or an extraordinary sort. (ii) Fregean abstraction is thought to yield new objects, and so may seem more promising. However, the most widely discussed examples, exploiting the abstraction principles: Directions Hume

Dir(a) = Dir(b) iff a // b Card(F)= Card(G) iff ∃f (1 − 1(f ) ∧ f : F onto G)

abstract over equivalence relations, not over systems in Shapiro’s sense. We can come closer with an ordinal abstraction principle. Here, the basis of the abstraction will not be an equivalence relation simpliciter, but a relation of a certain kind together with its terms—i.e. the objects composing the field of the relation. Since Fregean abstraction principles are formulated in extensions of the language of a suitable higher-order logic, rather than in the language of (first-order) set theory, the obvious way to formulate a principle of abstraction for ordinals will be along the lines of: Ordinals

Ord(R  F) = Ord(S  G) iff R well-orders the Fs, S well-orders the Gs, and the Fs as ordered by R are isomorphic to the Gs as ordered by S

236 properties, predication, and arbitrary sets Here we abstract over things much closer to systems in Shapiro’s sense. Further, Ordinals is clearly just a particular instance of a more general form of Fregean abstraction in which we abstract over relations—not necessarily well-ordering relations— defined on some objects. This general form might be represented: Structures

Structure(R1 , . . . , Rn  F) = Structure(S1 , . . . , Sn  G) iff C(R1 , . . . , Rn ) ∧ C(S1 , . . . , Sn ) ∧ (R1 , . . . , Rn  F)  (S1 , . . . , Sn  G)

where C(. . .) is some condition on the relations R1 , . . ., Rn and S, . . ., Sn and  is the usual isomorphism relation. This comes closer to our target, but still falls short. For while our extended form of Fregean abstraction delivers new objects, the new objects are themselves whole structures and not purely structural objects internal to those structures, as places or positions in Shapiro’s structures are supposed to be. It gives us no ground to recognise places as bona fide objects in their own right.

13.4.3.4 concluding remarks on geometrical points In the absence of independent support for Shapiro’s distinctive brand of structuralism, it seems to me that his main example provides no external, independent, ground for thinking that geometrical points constitute a domain of quantification composed of unidentifiable objects. It is not part of the data to be explained, as it were, but a product of his distinctive—but controversial—brand of ante rem structuralism. Shapiro suggests some further examples to support his claim: There is nothing that distinguishes the two square roots of −1 from each other and so there is no singular thought to be had about one of them (see Shapiro 2008, 2012a). (Shapiro, 2018, p.94, note 3) Something similar seems to hold for subatomic physical particles. Consider, for example, a system that contains five electrons. There is no sense to be made of thinking about one of them, or of picking one out. At least according to contemporary theory, reference and singular thought about individual electrons is physically impossible (see French and Krause 2006). Perhaps the neo-logicist conclusion should be that electrons are not objects (even though there are bound variables ranging over them). (Shapiro, 2018, p.94)

Do these fare any better?

13.4.4 i and −i A theory is rigid if any model of the theory has only one automorphism (the identity mapping). Shapiro claims that a non-rigid structure—the structure described by a non-rigid theory—has places that are indiscernible in a clear sense. No formula in the language of the theory distinguishes them. In Shapiro’s view, the theory of the complex numbers and Euclidean geometry are non-rigid; arithmetic, real analysis, and set theory are rigid. In an earlier paper (Shapiro 2008), he writes, there is at least a potential problem with the language of complex analysis, whether or not one adopts ante rem structuralism. The term ‘i’ seems to function, grammatically, as a singular term . . . And of course, the semantic role of a singular term seems to be to denote an object—a

13.4 quantification and identifying reference 237 single object. But the linguistic and mathematical communities have done nothing to single out a unique referent for this term. They cannot, since . . . the two square roots of −1 are indistinguishable. (pp.294–5)

Introductory texts sometimes introduce i as ‘the square root of −1’. But of course, we require that −1 has two square roots, so more careful texts introduce i as one of the square roots of −1, the other being −i. Complex numbers are then defined as numbers of the form a + bi, where a is said to be the real part and bi the imaginary part. The theory of complex numbers has a non-trivial automorphism, i.e. a function f taking complex numbers onto complex numbers, other than the identity mapping, which is structure preserving—we can set f (a + bi) = a − bi. That is, we just swap −i round with i. Within the theory, they are distinct, but there is no telling them apart. Texts often introduce complex numbers in a different way, as ordered pairs of real numbers, (a, b), where a is the real part and b the imaginary part. A pair (a, 0) is a pure real number in the complex plane, while a pair (0, b) is a pure imaginary number. Pairs (a, b) and (c, d) are the same iff a = c and b = d. Addition and multiplication are defined by: (a, b) ⊕ (c, d) = (a + c, b + d) and (a, b) ⊗ (c, d) = (ac − bd, ad + bc) In this development of the theory, i is defined as the pair (0, 1) and −i as the pair (0, −1). Since these pairs differ in their second terms, there is no problem in distinguishing one from the other, and hence in distinguishing i from −i. Shapiro is, of course, well aware of this. One option is to interpret complex analysis in another, rigid, structure or, perhaps better, to replace complex analysis with a rigid structure. For example, if one thinks of the complex numbers as pairs of real numbers, then our problem is solved. One stipulates that i is the pair (0, 1), in which case −i is the pair (0, −1). Those pairs are distinguishable from one another in R2 . (p.295)

He has, however, no enthusiasm for this solution: Given how pervasive non-rigid structures are, however, I would take this as a last resort, only to be invoked if we cannot do better. In line with faithfulness, I take it that, other things being equal, it is better to take the language of mathematics at face value. (p.295)

He means, of course, taking talk of i and −i introduced from scratch as square roots of −1 at face value, rather than as defined as certain pairs of reals. We might agree that taking the language of mathematics at face-value is something we should do, other things being equal, but deny that they are equal in the present case. The presentation of complex analysis Shapiro wishes to take at face value does raise problems—if i and −i are indistinguishable, just what is the mathematician saying when he tells us that i is the positive square root of −1 and −i the negative? Further, Shapiro’s claim that the proposed solution amounts to replacing a non-rigid theory by a rigid one can be questioned—it remains the case that there is a non-trivial automorphism on

238 properties, predication, and arbitrary sets the complex numbers, using the mapping which takes (a, b) to (a, −b). We are not replacing a non-rigid theory by a rigid theory; rather, we are reconstructing a theory, preserving its non-rigidity, while using a rigid theory as our basis.

13.4.5 Electrons We may set aside Shapiro’s suggestion that we, or the neo-logicists at least, might best conclude that electrons are not objects (even though there are bound variables ranging over them)—objects or not, so long as it is held that electrons are admissible but unidentifiable values of bound variables, the problem remains. We may also set aside Shapiro’s claim that there is no sense to be made of referring to or thinking about individual electrons—the claim that it is impossible that p (and hence the claim that it is physically impossible that there be singular reference to individual electrons) itself makes sense only if the sentence governed by the impossibility operator does so. The real issue is whether—supposing that contemporary theory really does imply that such singular reference is physically impossible—this conflicts with the view that objects (and so the admissible values of individual variables) must be possible objects of singular reference. It isn’t obvious that it does, because: (i) objecthood requires (only) the absolute possibility of singular reference, but (ii) physical impossibilities may not be absolute—metaphysical—impossibilities. Further: (iii) the physical impossibility of singular reference to electrons is, presumably, not an intrinsic property of electrons, but a relational matter—what is physically impossible is, I suppose, is that electrons should be objects of singular reference for observers/thinkers such as ourselves, but (iv) the requirement that objects be identifiable—possible objects of singular reference—does not entail identifiability by us, i.e. being possible objects of singular reference for observers/thinkers such as ourselves. To elaborate a little: ad-(i),(ii): For example, it may be physically, but not metaphysically, impossible that the attractive force between two bodies should be other than inversely proportional to the square of the distance between them. In my view, a physical impossibility that p will be an absolute, metaphysical impossibility if it is a consequence of the natures of some physical entities that it cannot be the case that p. Supposing that there cannot be singular reference to electrons, it is certainly not obvious that this is a consequence of the nature of electrons alone . . . ad-(iii): . . . that is, it doesn’t seem that this could be an intrinsic property of electrons—their intrinsic properties include having a certain mass and charge, in respect of which they are indistinguishable from one another. Distinct electrons are distinguished from one another—I suppose—by their spatio-temporal position. But this does not make them distinguishable by us—roughly, because measuring position requires physical interaction with them which, given their very small mass, makes measuring their positions with sufficient accuracy to distinguish them impossible. But . . .

13.4 quantification and identifying reference 239 ad-(iv): . . . this need not conflict with the requirement that objects, including electrons, be possible objects of singular reference, because that need not be understood as requiring identifiability (and so distinguishability) by us. Here we should recall Frege’s all-important caveat in demanding a criterion of identity for numbers: If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a, even if it is not always in our power to apply this criterion. (Frege 1974, §62)4

Without the crucial qualification, Frege’s demand for a criterion would suggest a quite substantial retreat from his otherwise uncompromising realism. The epistemological overtones of the term ‘criterion’ notwithstanding, it seems clear that what Frege means to insist upon is not a means of deciding whether or not a = b is true but an informative statement of the condition for its truth, regardless of whether we can in general use it to determine whether that condition is met. Applying this to the present—and obviously closely related—case, the requirement that each object be the possible referent of a singular term is the requirement that for each object, there could be a singular term whose sense determines that object as its referent—that we could employ the term as a means of singling out the object is not part of what is required.

4 Frege’s German runs: ‘Wenn uns das Zeichen a einen Gegenstand bezeichnen soll, so müssen wir ein Kennzeichen haben, welches überall entscheidet, ob b dasselbe sei wie a, wenn es auch nicht immer in unserer Macht steht, dies Kennzeichen anzuwenden’ (Frege 1884, §62), which seems to me to carry less suggestion than does Austin’s English that a criterion is something we can typically or normally use to decide whether a is the same as b—‘welches überall entscheidet’ literally translates as ‘which decides in all cases’.

14 Ordinals by Abstraction The neo-Fregean programme in the philosophy of mathematics seeks to provide foundations for fundamental mathematical theories in abstraction principles—that is, principles of the form ∀α∀β(ƒ(α) = ƒ(β) ↔ Eq(α, β)), understood as implicit definitions of the function f in terms of the equivalence relation Eq.1 The best known and most discussed such principle is, of course, Hume’s principle: (HP) ∀F∀G(NxFx = NxGx ↔ F ≈ G) which seeks to define the cardinal number operator Nx. . .x. . . in terms of one-one correlation (≈). Abstractions for real numbers and, with less conspicuous success, for sets have also been proposed. Until quite recently, however, very little has been published on introducing ordinal numbers by abstraction. In a welcome addition to that little,2 Ian Rumfitt (Rumfitt 2018) proposes to introduce ordinal numbers by means of an abstraction principle, (ORD), which says, roughly, that ‘the ordinal number attaching to one well-ordered series is identical with that attaching to another if, and only if, the two series are isomorphic’ (Rumfitt 2018, p.125). Although I shall myself reject (ORD) in favour of an alternative form of ordinal abstraction, I shall discuss Rumfitt’s proposal in some detail, because it poses a sharp and serious challenge to those of us seeking to advance the neo-Fregean programme. As Rumfitt explains, (ORD) runs pretty directly into trouble, as one might expect, in the shape of Burali-Forti’s paradox. But, Rumfitt argues, we may and should uphold (ORD)—in his view the best abstractionist treatment of the ordinals—by blocking the derivation of Burali-Forti’s contradiction in another place, viz. its reliance upon an impredicative comprehension principle. Instead, we should adopt a significantly weaker secondorder logic, allowing no more than 11 -comprehension. However, as Rumfitt himself emphasizes, this remedy ill-suits the neo-Fregean enterprise, at least as it has been conceived by its principal proponents. For 11 -comprehension is insufficient for the derivation of arithmetic on the basis of (HP), which requires a stronger form of comprehension—either 11 or 11 . Thus if neo-Fregean foundations for elementary arithmetic are to be saved, we must explain why we do not have to block the derivation 1 This chapter is also due to appear in Origins and Varieties of Logicism: A Foundational Journey in the Philosophy of Mathematics, edited by Francesca Boccuni and Andrea Sereni for Routledge. 2 Roy Cook (Cook 2003) discusses a simple order-type abstraction principle (OAP): ∀R∀S(OT(R) = OT(S) ↔ R ∼ = S, which runs into Burali-Forti’s paradox, and proposes a Size-Restricted Abstraction Principle (SOAP): ∀R∀S[OT(R) = OT(S) ↔ (((¬WO(R) ∨ Big(R)) ∧ (¬WO(S) ∨ Big(S)) ∨ (WO(R) ∧ WO(S) ∧ R ∼ = S ∧ ¬Big(R) ∧ ¬Big(S))))], but his main focus in this paper lies elsewhere, in developing an abstractionist version of the iterative approach to set theory. Bob Hale, Ordinals by Abstraction In: Essence and Existence: Selected Essays. Edited by: Jessica Leech, Oxford University Press (2020). © the Estate of Bob Hale. DOI: 10.1093/oso/9780198854296.003.0015

14.1 rumfitt’s proposal 241 of Burali-Forti’s contradiction from an ordinal abstraction principle by restricting comprehension in the way Rumfitt proposes, but can obstruct it elsewhere. I shall be exploring the prospects for doing so.

14.1 Rumfitt’s proposal I begin with a summary of Rumfitt’s approach and his diagnosis of the Burali-Forti paradox.

14.1.1 Series, relations-in-extension, and ordinal abstraction On Rumfitt’s approach, ordinal numbers are identified with order-types of well-ordered series. An ordinal number, then, is something which isomorphic serial relations, i.e. isomorphic wellordered series, have in common. Thus one well-ordered series consists of the Mozart-da Ponte operas, arranged in order of composition: Le Nozze di Figaro, Don Giovanni, Così Fan Tutte. Another consists of the Norman Kings of England, arranged in order of succession to the throne: William the Conqueror, William Rufus, Henry I. These two series are isomorphic, so both are instances of a common order-type, viz. the ordinal number 3. (Rumfitt, 2018, pp. 190–1)

Whilst Rumfitt’s identification of ordinals with order-types is more or less standard, his conception of series is not. A series, Rumfitt insists, is not a set-theoretic entity but a certain kind of plurality: I understand a series to be some things (plural) in a particular order . . . The series is not constituted by its terms alone, but by them in tandem with a two-place relation between terms. (Rumfitt, 2018, pp. 192)

However, he argues, we do not have to think of series as composite entities, comprising both a relation and a plurality: Logicians achieve a useful economy by identifying a series with the relevant binary relation, . . . . The identification is legitimate, but only if the relation with which the series is identified is what we may call a relation-in-extension. By this, I mean a plurality of pairs of objects which stand in the universal or pure relation. Thus the series of Mozart-da Ponte operas may be identified with the relation-in-extension , , . The series, then, comprises those pairs of Mozart-da Ponte operas which stand in the pure relation of being composed before. In general, a relation-in-extension is a plurality of pairs— specifically, the pairs of objects in a domain which stand in the relevant pure (or universal) relation. (Rumfitt, 2018, pp. 192)

A relation-in-extension R is (or forms) a series if it is a total order, i.e. R is irreflexive, transitive and connected in the relevant domain,3 and is well-ordered if it is a total order and for every subdomain X of its field (i.e. plurality of objects which either bear R to some object or to which some object bears R), some object is R-minimal among the Xs (i.e. one of the Xs has no other X bearing R to it).4 3 n.b. for Rumfitt, domains are pluralities, not sets. 4 More formally (cf. Rumfitt, 2018, p.194) a binary relation-in-extension R (= x1 , y1 , x2 , y2 , . . . , xj , yj , . . .) is a series iff R is a total order, i.e.

242 ordinals by abstraction Two series R and S are isomorphic—briefly R ∼ = S—if and only if there is an order-preserving bijection f from the field of R onto the field of S (i.e. f is such that: ∀x∀y(Rxy ↔ Sf (x)f (y))). Ordinal numbers, identified with order-types, are then introduced by the abstraction: (ORD) ∀R∀S(W(R) ∧ W(S) → (ord(R) = ord(S) ↔ R ∼ = S)) On the basis of (ORD), we may define a predicate On(x) (read: ‘x is an ordinal’): (ON) On(x) =def ∃R(W(R) ∧ x = ord(R))

14.1.2 Burali-Forti In this setting, Rumfitt obtains Burali-Forti’s contradiction as follows. Let α and β be ordinals, and suppose α = ord(R) and β = ord(S). Rumfitt defines: (